Pharmacokinetics Mathematical and Statistical Approaches to Metabolism and Distribution of Chemieals and Drugs

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Pharmacoki netics Mathematical and Statistical Approaches to Metabolism and Distribution of Chemicals and Drugs

Edited by

A. Pecile University of Milan Milan, Italy

and

A. Rescigno University of Ancona Ancona, Italy

Springer Science+Business Media, LLC

Proceedings of a NATO Advanced Study Institute on Pharmacokinetics, held June 2-13, 1987, in Erice, Italy

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NATO Advanced Study Institute on Pharmacokinetics (1987: Erice, Italy) Pharmacokinetics: mathematical and statistical approaches to metabolism

and distribution of chemicals and drugs.

(NATO ASI series. Series A, Life sciences; v. 145) "Proceedings of a NATO Advanced Study Institute on Pharmacokinetics, held

June 2-13, 1987, in Erice, Italy"-T.p. verso. "Published in cooperation with NATO Scientific Affairs Division." Includes bibliographies and indexes. 1. Pharmacokinetics-Mathematical models-Congresses. 2. Pharmacokine·

tics-Statistical methods-Congresses. 1. Pecile, A. (Antonio) II. Rescigno, Aldo. III. North Atlantic Treaty Organization. Scientific Affairs Division. IV. Title. V. Series. [DNLM: 1. Drugs-Pharmacokinetics-congresses. 2. Mathematics­congresses. 3. Tissue Distribution-congresses. QV 38 N2785p 1987] RM301.5.N37 1987 615.7 87-36043 ISBN 978-1-4684-5465-9 ISBN 978-1-4684-5463-5 (eBook) DOI 10.1007/978-1-4684-5463-5

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DIRECTORS

A, PEC I lE, HEAD OF THE DEPT, OF PHARMACOLOGY, CHEMOTHERAPY

AND MEDICAL TOXICOLOGY, UNIVERSITY OF MILAN, ITALY

A, RESCIGNO, PROFESSOR OF PHARMACOKINETICS, INSTITUTE OF

EXPER, AND CLINICAL MEDICINE, UNIVERSITY OF ANCONA, ITALY

ORGANIZING COMMITTEE

A, PECIlE, PROFESSOR OF PHARMACOLOGY, UNIVERSITY OF MILAN,

ITALY

A, RESCIGNO, PROFESSOR OF PHARMACOKINETICS, UNIVERSITY OF

ANCONA, I TAL Y

J,H, MATIS, PROFESSOR OF STATISTICS, TEXAS A&M UNIVERSITY,

COLLEGE STATION, TX, U,S,A,

A,K, THAKUR, PH,D" PRINCIPAL SCIENTIST & BIOSTATISTICIAN,

HAZLETON LABORATORIES AMERICA, INC" VIENNA, VA, U,S,A,

SECRETARY

MARIA lUISA PECIlE, INTERNATIONAL CONGRESSES & COURSES

SECRETARIAT, DEPARTMENT OF PHARMACOLOGY, CHEMOTHERAPY

AND MEDICAL TOXICOLOGY, UNIVERSITY OF MILAN, ITALY

v

PREFACE

Pharmacologists can be considered pioneers of the study of kinetics of materials introduced into biological systems. The study of drug kinetics is particularly suited to a formulation of relatively simple models which make possible an interpretation of the time-dependent nature of various important phenomena (e.g. distribution by means of diffu­sion through membranesl. The objective of the NATO ASI Course on Pharmacokinetics was that of presenting and dis­cussing the mathematical and statistical approaches current­ly available or being developed for the description, inter­pretation and prediction of the fate of drugs and tracer substances administered to living beings. Different physi­cal methods for measuring drugs and tracer substances were considered, but the emphasis was on the interpretation of the results of the measurements in terms of mathematical and statistical models. The present book contains all invit­ed lectures given in this Course by outstanding internation­al authorities and specialists from different fields. A great effort was made to keep a balance among the mathemati­cal, physical, biological and clinical aspects of the prob­lems; exchange of ideas and experiences between scientists with a physico-mathematical background and scientists with a biomedical background was encouraged and all participants were deeply involved in fruitful discussions. This unique feature of the Course is also the unique characteristic of this book which is therefore mainly directed to people interested not just in acquiring a working knowledge of the methods but in developing new methods. Thus we hope this book will be judged as achallenging presentation of new trends for future developments in Pharmacokinetics rather than a review of today's knowledge.

We take this opportunity to thank the NATO Scientific Affairs Committee for its support. The contribution of the Centro Nazionale delle Ricerche (C.N.R.l of Italy is kindly acknowledged. Warm thanks are also due to Bracco Industria Chimica S.p.A. and Istituto Oe Angeli S.p.A. for their generous financial assistance.

vii

We wish to thank each of the lecturers and ASI partici­pants for their effort in making the Course a success: the interaction of scientists from research institutions with people actually involved in the production and testing of drugs has proven particularly fruitful for both Industry and University. We thank the International School of Pharmacology for hous­ing the Course at the "Ettore Majorana Centre for Scien­t i f i c Cu 1 t ure" i n E r i ce, a d mir a b 1 y dir e c ted. T h e i m p 0 r t a n t organisational work done by Dr. Maria Luisa Pecile was highly appreciated.

A. Pecile, A. Rescigno

viii

CONTENTS

History of Pharmacokinetics .................... . E. Gladtke

Conceptual Foundations and Uses of Models in Pharmacokinetics ...................... . 11

J.S. Beck

Development of Compartmental Concepts .......... . 19 A. Rescigno and A.K. Thakur

Modeling of Pharmacokinetic Data ............... . 27 A.K. Thakur

Mathematical Foundations of Linear Kinetics ..... 61 A. Rescigno

An Introduction to Stochastic Compartmental Models in Pharmacokinetics ............... . 113

J.H. Matis

Modeling First-Pass Metabolism ................. . 129 J.G. Wagner

Saturable Drug Uptake by the Liver: Models, Experiments and Methodology .............. . 1 51

L. Bass

Physiological Models, Allometry, Neoteny, Space-Time and Pharmacokinetics .......... . 1 91

H. Boxenbaum and R. D'Souza

Equivalence of Bioavailability and Efficacy in Drug Testing .......................... . 21 5

C.M. Metzler

Modeling and Risk Assessment of Carcinogenic Dose-Response ............................ . 227

A.K.Thakur

ix

The Puzzle of Rates of Cellular Uptake of Protei n-Bound Ligands .................... . 245

L.Bass and S.M. Pond

A Pharmaeokinetie Equation Guide for Clinieians ..................... . 271

J. Mordenti

Pharmaeokinetie Studies in Man ................. . 291 J.G. Wagner

Metabolie Models in Radiation Proteetion ....... . 323 F. Breuer

Author Index ................................... . 337

Subjeet Index .................................. . 339

x

HISTORY OF PHARMACOKINETICS

INTRODUCTION

Erich Gladtke

Director of the hospital for sick children of the university of Cologne 0-5000 Cologne 41

By pharmacokinetics we mean the science of the quantitative actions between a biological organism and a pharmacological substance within it. The qualitative question of the actions exerted by a drug does not con­stitute part of the primary concern of pharmacokinetics, it belongs to pharmacology as a whole. The word "pharmacokinetics" first appeared in print on page 244 of the first comprehensi ve treatise of pharmacokinetics, F. H. Dost' s book "Der Blutspiegel" ("Blood Levels") (9); this was the point at which the term and the concept became part of the scientific vocabulary. Dost himself, when taking part in a small discussion group in 1966, asked about the or­igin and the first use of the term, and was astonished when Ekkehard Krü­ger-Thiemer promptly gave him this answer.

F.H. Dost was born 11th July 1910 in Dresden. He received his educa­tion in medicine and pharmacology at different places in Germany and be­came pediatrician at the Childrens hospital of the University of Leipzig. From 1947 to 1951 he hold the position as lecturer and head physician at Leipzig, 1951 to 1960 as professor and director of the famous childrens hospital of the Charite in Berlin, and from 1960 to 1975 he held the same position at Giessen. The "Blutspiegel" appeared in 1953, in 1949 the first paper of this topic was published by Dost (8).

There had, i t is true, been forerunners who had not only measured blood concentration curves but had also derived the same or similar for­mulae and laws to those later gathered together and summarized by Dost; but these workers had generally only applied their knowledge to the solu­tion of one or two problems, and had not come anywhere near formulating general rules.

To my mind, the classic example of the application of what was actu­ally a general law to the solution of just one single problem was Wid­mark's analysis of curves of blood ethanol and acetone concentrations. Widmark's famous work on blood alcohol is often considered to be one of the very first papers on pharmacokinetics. This monograph appeared in 1932 (26) and is important in our context, because Widmark here tries to relate blood levels of alcohol to psychic impairment. This work, however,

had one major disadventage: it arrived at a zero-order kinetics, which is required to explain the time course of alcohol levels in man, but cannot be generalized for other drugs. Nevertheless we should go back much ear­lier. In 1919 Widmark published a far less known study with the title "Narcosis and its Interdependence on the Narcotic Present in the Body" (25). In that paper you do see some basic elements of pharmacokinetics. Widmark investigated acetone, because he was able to measure it with his method and he was at least able to show quite convincingly that acetone follows an exponential decline in the late phase. He introduced terms si­milar to today's distribution volume, defines the fluctuations of blood levels following intermittent drug administration and he finally states that it is therefore evident that a knowledge of the concentration is of the greatest importance for the study of narcosis.

The curves of Widmark and even the formulation of the equations, might have been taken directly from Dost's book - but they antedate it by more than 30 years.

Widmark produced excellent formulae, but he only applied them to the problem of alcohol, wh ich was of topical interest in Sweden at the time; and in his final treatise he concentrated mainly on the forensic impor­tance of blood alcohol levels. It was a magnificent achievement, but it was not thought through to the end in the direction with which we are concerned today.

In 1933 published Walter Gehlen (12) his idea, that intravenously administered drugs follow a function of time. As he was not able to meas­ure blood concentrations of drugs at that time he approached the problem in an interesting way. Assuming that the processes of invasion and excre­tion of drugs are described by exponential equations he arrived at a sum of exponentials following the Bateman function (1) describing the time course of blood concentrations. He found out that the t is independent of dose. This is a first approach of what we would to~g~ call an effect kinetics. It might be interesting to add that the idea to describe the time course of blood concentrations by a sum of 2 exponentials was intro­duced first by Biehler in 1925 (4). Walter Gehlens paper seems to be one of the earliest papers in this regard. This paper was followed by a num­ber of articles by Dominguez in 1934 and 1935 (7). In 1937 Teorell made one of the most important contributions to this field with his famous ma­nuscript "Kinetics of distribution of substances administered to the bo­dy" (23). He derived the weIl known equations for a two-compartment mo­del. In addition the problems of intermittent and continuous drug admin­istration are extensively delt with. He had a precise idea, how to deal with the absorption compartment and he also gave a complete discussion of the problem of distribution volumes. Therefore this is a sense the first comprehensive and truly pharmacokinetic paper. It does not contain any experiments, it just tries to explain data in terms of some generalized ideas. This article was followed by a less weIl known little monograph by Beccari in 1938, with the Italian title "Distributione dei pharmaci nell organismo" (2).

In 1949 appeared Druckrey' sand Küpfmüller' s monograph "Dosis und Wirkung" (Dose and Effect), printed on brownish post-war paper which ap­peared in 1949 containing a complete theory of pharmacokinetics under the most modern aspects including effect kinetics and also some elements of system kinetics, utilizing already electrical analog circuits to visual­ize time courses (11). For these reasons this work has to be considered a very important book which did not receive adequate attention obviously due to some political reasons. Before the era of pharmacokinetics doctors have been acustomed, as it

2

were, to titrate their medication on the patient. If a dose was insuffi­cient to control pain, to lower blood pressure or to cure fever, the doctor simply.pressed on until an effect was apparent.

But this sort of approach was no longer possible where antibiotics were concerned. It was not the patient but the microbes that hat to be controlled. Minimal inhibitory concentrations were known, or could be determined on available organisms. But how to set this concentration in the body? What blood level was required? Even G. Domagk in his day had called for rules to achieve this.

Dost's reflections were not however based on antibiotic treatment, although he later produced clear guidelines and applications in this context. At the end of the war, when he returned from captivity to the Paediatric Clinic at Leipzig University, his chief Professor Peiper said to hirn, more or less: "Ah, Dr. Dost, we have something new here - they call it Clearance or something. Have a look, will you, and see whether it has anything to offer in paediatrics".

Now Dost had a grounding in mathematics. During the war, instead of drinking alcoholics and playing cards, he had studied higher mathematics with the help of a university lecturer in mathematics who had been through the retreat with hirn. Teorell, incidentally, teIls me a similar story: he too spent his military service improving his knowledge of higher mathematics.

Basic Principles Obst had the idea that the concept of clearance - a purely fictitious notion implying the complete removal of some foreign material from a gi­ven volume of blood in a given time - could in effect be regarded as a process of concentration-dependent elimination, which could be simply and accurately expressed as an exponential function. He noted that the property of eliminating foreign substances and establishing a steady­state equilibrium for native ones - the body' s tendency to maintain a constant balance of masses and forces despite the constant turnover of chemicals and energy - had been corretly described by Burton in 1940 as the "steady state" (5) and by Bertalanffy in 1942 as a "fluid equili­brium" (3).

In the literature he found numerous time-concentration curves that indicated that the substances in quest ion had been eliminated according to a first-order function, although the authors hat not drawn any gene­ral conclusions from their findings. Some had produced fanciful and com­plicated formulae to describe their curves. Dost recognised the under­lying general principles and based his conclusions on them.

Some studies carried out in Leipzig and later at the Charit§ in Berlin provided experimental confirmation of his hypotheses. His publi­cations gradually aroused more and more interest. Some groups of scien­tists, outside his own groups in Berlin and later in Giessen, took up the new study. Paediatricians, pharmacologists, pharmacists, internists and interested biomathematicians swelled the ranks of the students of pharmacokinetics.

One of Dost's great merits was that although he made use of the ra­te constant to measure elimination, he held that the elimination half­life was a concept easier for the physician to grasp than that of timed clearance.

3

The Widening Interest in Pharmacokinetics As we have said, an ever-increasing number of scientists took up this new speciality. In Europe, Ekkehard Krüger-Thiemer, while working on the sulphonamides, came upon Dost's work and was the first, in his own inimitable way, to use modern computers to describe pharmacokinetic relationships (19). He succeeded in gathering a group of collaborators around him in Borstel and converted them to this new field of study. Among them we should men­tion, first and foremost , B. Diller, a physicist, P. Bünder, a clini­cian, and J. Seydel, a chemist.

In 1962, Ekkehard Krüger-Thiemer came to public notice in the USA; this was followed by visiting professorships in pharmacology at Wiscon­sin (1964)and Boston (1967-8).

While Ekkehard Krüger-Thiemer concentrated mainly on the problems of calculating curves and deriving guidelines for dosage from them, the internist L. Dettli in Basle was concerned with how to present this so­mewhat indigestible material in teaching (6). He later also brought a third fluid, the cerebrospinal fluid, into the field of pharmacokinetics in addition to the circulatory fluid and the urine. He had a rare gift for teaching and did much to propagate pharmacokinetic ideas throughout the world.

In Milan, another paediatrician, Fabio Sereni, saw the importance of pharmacokinetics for drug therapy in children (22).

The topic gradually advanced to become one of the cornerstones of clinical pharmacology.

Meanwhile phamacists, pharmacologists and drug firms - particularly in the USA - had taken up pharmacokinetics in an intensive way. The Basle pharmaceutical firms and the great German producers also began setting up more and more large-scale departments of pharmacokinetics. Ultimately they were obliged to do, since increasing numbers of national drug licensing agencies in different countries began to require the pre­sentation of pharmacokinetic data.

One ought to mention just a few of the scientists in the USA who worked hard and very productively, especially in the early days, on va­rious aspects of pharmacokinetics. They included E. Garrett, J.G. Wag­ner, S. Riegelman, S.J. Yaffe, E. Nelson, G. Levy and W.A. Ritschel (16) .

The catalogue cannot be an exhaustive one; and by now there are countless scientists worldwide working on pharmacokinetics. Every day, all kinds of medical, pharmaceutical and pharmacological journals pub­lished scientific papers with a pharmacokinetic content, and even the popular press occasionally touches on the subject. Many universities in different countries offer their students an introduction to pharmacoki­netics, and many include the subject in their examinations.

In 1962 the first Symposium on pharmacokinetics took place at Borstel near Hamburg. It was organised by Krüger-Thiemer, and was at­tended by the small group of scientists then working on pharmacokine­tics. We all knew each other, at least through our publications. It was a friendly meeting, remembered with pleasure by all of uso In 1985 it was, as it were, commemorated by another symposium at Borstei, held to demonstrate the progress that had been made in the subject.

4

Pharmacokinetics has come a long way since its beginning as a technique for interpreting curves of changing blood concentrations with time.

Technological advances Three important technicological advances have carried the science of pharmacokinetics forward.

Isotope techniques have made it possible, using tracer methods, to detect minute quantities of a drug in biological materials. The pos si­bility of assimilation of radiolabelled drugs has also led to notable advances, and despite all one's reservations about the technique it is necessary from time to time in order to obtain data on the metabolism and pharmacokinetics of drugs.

The mathematical description of the decay of radioactive isotopes, and in particular calculations involving decay through radioactive in­termediates (Bateman functions) has demonstrated the general validity of physical and blological laws, and has also shown that formulae to de­scrlbe the decay of isotopes over BO to 100 years are also valid in pharmacokinetics. ThlS realisation has enriched the study of pharmacokl­netics to a marked degree.

Other new pathways have been opened up by work using non-radioctive isotopes. This is a very expensive and demanding but potentially very fertile technique for the study of metabolism, clinical pharmacology and pharmacokinetics. For economic reasons, unfortunately, only a few cen­tres can afford to take it up.

In the second place, I should mention modern technlques of chemlcal analysis. The progression from microlitre technology with conventional chemical methods, via newer separation techniques like liquid chromato­graphy, highpressure liquid chromatography, with increasingly sensitive methods of detection culminating in mass spectrometry, is making it pos­sible to measure lower and lower concentrations in smaller and smaller volumes.

But what could we do with all these measured quantities, without the help of the third great advance - computers? First the analogue and then the dlgital computers have anabled these figures to be used and evaluated. Their advantages include speed of calculation and the ability to match and even print curves or individual data.

Progress and Applications Thirty years ago, it was only possible to measure a few successive drug levels after a drug had been given - just until the concentration fell below the threshold of detection. The appropriate pharmacokinetic para­meters were calculated using linear or semilogarithmic graphpaper and a slide rule. Pharmacokinetlcs was still a simple subject ln those days.

As analytic techniques improved, we began to be able to measure concentrations over three or more half-lives. This led to the discovery of deep compartments, to the recognition of the alpha, beta and gamma phases of elimination curves, to microconstants, and to the demonstra­tion of cumulation effects. It also made pharmacokinetics much more complicated.

Steady-state studies showed that it was possible to subject native substances to pharmacokinetic analysis as weIl. Concentration curves following intravenous loading with iron, glucose, or bilirubin, for ex­ample, yielded im~ortant new knowledge about the metabolism, metabollc

5

rate and distribution of these substances. In particular, it was possi­ble to analyse pathological states. For instance, we could demonstrate when jaundice was due to increased production of bilirubin (as in haemo­lysis) and when it was due to delayed removal (as in liver disease).

Dost, incidenta11y, had been able to take up the work of Ther (1948) (24) on a simple method of obtaining data on water metabolism using Volhard's urine-concentration test, and to carry the work through, producing a simple method for measuring the pharmacokinetics of water.

The principle of the area under the curve states that the area un­der a concentration curve (given constant external conditions) repre­sents the mass of measured substance passing through the compartment un­der investigation. This principle was formulated by Dost and experimen­tally confirmed by the Giessen group (13). It has since been rediscover­ed afresh by quite a number of other workers. It offered a way of tes­ting the completeness of intestinal absorption; and in 1962, when Rieder in Basle (20) presented his curves of sulphonamide absorption derived from animal experiments, showing 99.6% absorption, we feIt that our work had received important support. After a11, one only believes in one's own results to a certain extent; one is always very keen on confirmation from another source.

According to a long-standing paediatric tradition, paediatricians involved in pharmacokinetics are particularly interested in drug metabo­lism in children of different ages. Oepending on the metabolic pathway involved, elimination of a drug in early life might be prolonged for weeks or even months. Immaturity of enzyme systems involved in drug me­tabolism, individual functional peculiarities of the cells of parenchy­matous organs, or differences in transport proteins might be cited as causes. The science known as developmental pharmacology has its origins in these observations. Much of the most important work in this field has been carried out by J. Rind in Giessen (21) and other m€mbers of Dost's staff (14, 15). The branch of the Giessen working group in Cologne produced a surprising finding when G. Heimann established that the intestinal absorption of native or foreign substances was also significantly slower in neonates and young infants than in older children and adults (18). Absorption still obeys a first-order function, but is markedly subject to the laws of saturation kinetics, and in this respect shows similarity to enzyme kinetics.

The more accurately measurements were made, the more difficult it was to produce a mathematical evaluation of the curves. M. von Hatting­berg has done a great deal of work on this problems (17), and with his expertise in computer technology has produced some interesting programs which are now widely used worldwide, either in their original form or as an inspiration to other programmmers.

A discovery of major interest and importance not only for clini­cians but also for pharmacologists and pharmacists was the finding that sulphonamides could have very different elimination half-lives from one another. The sulphonamides known at the time (in the mid-1950's) had half-lives ranging from three to around ten hours. This quickly led to attempts to develop so-ca11ed intermediate-acting sulphonamides with half-lives of 4-12 hours, and long-acting sulphonamides with much longer half-lives of up to 120 hours, so as to cut down the tedious need for frequent doses and achieve a once or twice daily or even a once weekly dosage regime.

6

Such very long half-lives have certain advantages. They make it pos si­ble, for instance, to give an adequately long course of antimjcrobial chemotherapy with a single dose, in areas where medical manpower is scarce; or to give effective anti-streptococcal fo11ow-up treatment using just one injection every four weeks. The corresponding disadvanta­ge is that the product remains in the body for a long period, so that if allergies or other unwanted effects occur they will persist for a long time too - just so long as the drug remains present in the body.

The study of the causes for slow excretion and long duration of ac­ting is an interesting one, and there are many theories on the subject.

There are also pharmaceutical techniques available to produce a long duration of action. Drugs can be packaged in capsules, bound to ion-exchange resins or subjected to other similar precedures in order to delay their release in the gastro-intestinal tract and prolong their ab­sorption. The timing of release and of passage through the gastro-intes­tinal tract have to be matched; control of this aspect is a pharmacoki­netic problem.

Substances administered by intramuscular or subcutaneous injection can also be modified for slow release, for instance by crystallisation, incorporation in capsules etc. Complicated calculations have been car­ried out to estimate the surface area of the incorporated particles and to relate the rate of release to the rate of elimination; these calcula­tions have spawned formulae of grotesque complexity, but even so it is possible to work with them.

We all have an interest in the pharmacokinetics of drugs given sub­lingually, percutaneously or by other routes; in the pharmacokinetics of antibodies, sera and a11 kinds of other substances; in toxicological problems such as the removal of toxins, whether by dialysis, plasma ex­change, forced diuresis or other methods. Dur interest may be mainly practical, but even so the theory is also worth some thought.

The state of the Art. The Future The concept of pharmacokinetics has by now achieved international stand­ing. No new drug can be the subject of a licensing application, nor be put on the market, unless pharmacokinetic data are available. Long-es­tablished substances are being subjected to pharmacokinetic studies and are being better characterised as a result. Pharmacokinetics has made an important contribution to developmental pharmacology. It is also invol­ved in the study of alterations in drug metabolism in sick patients and in the elderly.

Pharmacokinetics is one of the essential foundations of clinical pharmacology. It is of interest to all clinicians, particularly paedia­tricians and gerontologists - but ultimately to any doctor who is invol­ved with patients who may have some functional organ impairment. It is also of concern to the pharmacologist, the pharmacist, the specialist in nuclear medicine, the toxicologist, the biomathematician - and last but not least, the national agencies involved with drug licensing, drug ma­nufactureres themselves, and many young scientifists.

Everyone actively involved in the so-ca11ed phase I testing of a drug is concerned with obtaining pharmacokinetic data on it. In phase 11 studies all the work is then repeated worldwide, from North America to Japan and Europe.

7

Despite a11 this worldwide activity, and despite the interest of many other speciali ti es in the subj ect of pharmacokinetics, the classical pragmatic formulation propounded by F.H. Dost still holds good: "By pharmacokinetics we mean the study of the quantitative interactions between an organism and a pharmacological substance within it. That is all. "

The qualitative question of the actions exerted by a drug does not constitute part of the primary concern of pharmacokinetics; it belongs to pharmacology as a whole.

F.H. Dost founded pharmacokinetics in 1953 with his book "Der Blut­spiegel n ("Blood Levels") (9); he renewed it with a second edition in 1968 under the title of "Grundlagen der Pharmakokinetik" (Foundations of Pharmacokinetics).

Hundreds of scientists have staked out the territory, refined its study, rounded it off and gained new knowledge. The foundations es­tablished by Dost still hold good; and for most new discoveries one can still find an appropriate formulation in Dost's writings, or at least an indication that the formulation must be established and its validity tested.

This scientist, like a solitary explorer, opened up a whole new scientific territory, gave it a name and made this name part of the scientific vocabulary; he worked out and formulated the basic founda­tions of the discipline; yet today, on the international scene, he is scarcely even quoted any longer. This is the way of the world, as exemplified by a single notable case, by the fa te of one unique man of science.

References

1. Bateman, H. Proc. Cambridge Phil. Soc.15 (1910) 423

2. Beccari, E. Distributione dei farmaci nell'organismo Arch. int. Pharmacodyn.58 (1938) 437

3. v. Bertalanffy, L. Theoretische Biologie Bd. 11 Berlin-Zehlendorf 1942

4. Biehler, W. Blutkonzentration und Ausscheidung des Alkohols im Hochgebirge Arch. exp. Path. Pharmacol.107 (1925) 20

5. Burton, A.C. The Properties of the steady state compared to those Equilibrium as shown in characteristic biological behavior. J. cellul. a. comp. Physiol. 14 No.3 (1939) 327-349

6. Dettli, L. Ein hydrodynamischer Simulator für die Darstellung der Pharmakokine­tik im medizinischen Grundlagenunterricht Antibiot. et Chemotherap. Fortschr.12 (1964) 195

7. Dominguez, R., E. Promerene Studies of renal excretion of creatinine J. Biol. Chem. 104 (1934) 449

8. Dost, F.H. Die Clearance Klin. Wochenschr. 1949, 257

9. Dost, F. H .

8

Der Blutspiegel Thieme, Leipzig, 1953

10. Dost, F.H. Grundlagen der Pharmakokinetik Thieme, Stuttgart, 1968

11. Druckrey, H., K. Küpfmüller Dosis und Wirkung Die Pharmazie, 8, Beiheft, Saenger, Berlin 1949

12. Gehlen, W. Wirkungsstärke intravenös verabrelchter Arzneimittel als Zeitfunk­tion Arch. exp. Ther. pharmak. 171 (1933) 541

13. Gladtke, E. Die Bestimmung der Absorptionsrate von Pharmaka nach dem Gesetz der korrespondierenden Flächen von Dost Berichte XXIII. Internationaler Kongreß der pharmazeutlschen Wlssen­schaften 1963 Münster, S.75

14. Gladtke, E., H. Rind Der Stoffwechsel als werdende Funktion beim Kind. A. Untersuchungen mit körperfremden Stoffen Monatsschr. Kinderheilk. 113 (1965) 299

15. Gladtke, E., H.M. v. Hattlngberg Pharmacokinetics Springer, New York 1979

16. Gladtke, E. The Historical Development of Pharmacokinetics Meth. and Find. Exptl. Clin. Pharmacol. 8 (1986) 587

17. v. Hattingberg, H.M., D. Brockmeier, G.Krenter A Rotating Iterative Procedure (RIP) for estimating hybrid constants ln multi-compartiment analysis on desk computers J. Clin. Pharmacol. 6 (1972) 105

18. Heimann, G. Age dependence of gastrointestinal absorption in children In E. Gladtke, G. Heimann, Pharmacokinetics, S.211, G. Fischer Stuttgart 1980

19. Krüger-Thiemer, E. Die Anwendung programmgesteuerter Ziffernrechen-Automaten für die Lösung spezieller chemotherpeutischer Probleme Antibiot. et Chemotherapia 12 (1964) 253

20. Rieder, J. Personal communication

21. Rind, J., E. Gladtke Der Stoffwechsel als werdende Funktion beim Kind. B. Untersuchungen mit körpereigenen Metaboliten Monatsschr. Kinderheilk. 113 (1965) 302

22. Sereni, F., P. Marchisio, R.C. Moresco, N. Primeipi, D. Sher Control of Antibiotic Therapy in Prediatric Patients by a Computer System In E. Gladtke, G. Heimann, Pharmakocinetics, 201, G. Fischer Stutt­gart 1980

23. Teorell, T. Kinetics of Distribution of Substances Administered to the Body Arch. int. Pharmacodyn. 57 (1937) 205 and 226

24. Ther, L. Über einige Gesetzmäßigkeiten der Diurese Arch. exper. Path. 205 (1948) 376

25. Widmark, E.M.P. Studies in the concentration of lndifferent narcotics in blood and tissue Acta Med. Scand. 52 (1919) 87

26. Widmark, E.M.P. Die theoretischen Grundlagen und die praktische Verwendbarkeit der gerichtlich-medizinischen Alkoholbestimmung Urban u. Schwarzenberg, Berlin 1932

9

CONCEPTUAL FOUNDATIONS AND USES OF MODELS IN PHARMACOKINETICS

James S. Beck

Faculty of Medicine, University of Calgary 3330 Hospital Drive N.W. Calgary, Alberta Canada T2N 4NI

The word "model" is overused in the biomedical literature. And it is often misused in a way that blurs the distinction between the system under study and someone's concept of the system. We can get involved in wasteful disputes and fruitless efforts with such sloppy thinking. Because these failings appear so frequently and because rational uses of models can be very helpful, I will sketch here abrief picture of what - I suggest - a model should be and what it can do for uso Along the way I will point out some pitfalls I see as taking a toll on our scientific and therapeutic enterprizes.

The distinction just made between science and therapy will serve as a starting point. In practice, especially in pharmacology, science and therapy often mix and jOin, but they are different things. A pharmacologist or clinical pharmacologist who is planning a protocol for administration of a drug to a patient has the pur pose of producing a beneficial effect. Any general knowledge gained about the disease involved or the drug administered is strictly secondary. Why or how the drug produces any given effect is literally irrelevant to the immediate purpose. That is to say, the model of the disease and drug action which the clinician has in mind is simply of no interest or use except in its performance in relating the delivery of the drug to the conditions of the protocol regardless of mechanism. In contrast a scientific study of a drug action will involve a model of the conceptual framework of which is all-important while the performance in terms of imitating the system is significant only for drawing conclusions about the conceptual framework.

If we are to agree on what a model is, we must start from some common grounds. Let us agree that we are talking about science. In scientific activity we try to understand a system. Already two critical terms appear: "system" and "understand". "System" is a primitive concept for which we can assurne a common meaning. Still, we must note that a particular system requires definition before it can be

11

studied effectively. For an experimenter it is some part (or all) of the universe which is identified in an operational context. For example it may be the heart of a particular dog with its thorax open, say. and these particular catheters in place, and with this particular flow of this specific blood perfusing it. The example makes it obvious that the system may not be weIl isolated from other parts of the uni verse and that these other parts may weIl effect the system, perhaps profoundly.

Having focussed on a system we must design some observation - perhaps an experiment - appropriate to what we want to know about the system. But of course we cannot do this with only the system itself because the specification of the question and choice of observation requires anticipation of the behavior of the system. So rather that make random guesses about the behavior, we construct a model, the behavior of which we can predict and observe.

A very common model will serve as an example. Say a pharmacologist is interested in the elimination of a drug and hypothesizes that it is eliminated at a constant fractional rate and writes the equation

A(t) = A(O)e-kt , where A(t) is the amount in the body at time t, A(O) is the initial amount and k is the fraction of the amount present eliminated per unit time. Here one uses the model by changing the values of the parameter k and the initial condition A(O). The observations of the behavior of the model are the pairs (t, A) generated by choosing values of t and computing A(t). It is the model which is manipulated and observed initially to give a basis for choice of observations on the system and then to generate data for comparison with data from corresponding observations of the system.

Clearly one use of a model is to answer the question: how will (or did) the system behave under certain specific conditions with respect to the observations which might be made (or might have been made)? Responding to such arequest the model predicts (teIls us what will happen) or retrodicts (teIls us what did happen): the model functions as a simulator. Rescigno and Beck(1987) have suggested that a device (concrete or abstract) used only for retrodiction or prediction be called a nsimulator n and not a "modeln. The task of a simulator is simply to mimic in some sense the behavior of the system. It is nice if it is robust (predicts weIl even when the values of the parameters or boundary conditions within it are changed) and simple to use. But it is irrelevant whether it has any other correspondence with the system. Thus a table of numbers obtained from a physical simulator is no better or worse than an elaborate mathematical function if either of the two simulators does as weIl as the other with respect to retrodiction or prediction.

Of course, a model can be used as a simulator, but a model has a much richer, subtIer and more demanding relation to the system, associated with its purposes. In the case of the therapeutic protocol, the problem there is solved by a good simulator. But the scientist uses a model to advance understanding of the system- nUnderstanding" is not an easy term to define. Understanding a system changes its effect on

12

{81--:=::~l r--> H2--->M2~Dm:=l, Fig.l. The use of models (MI' M2) to generate data

sets (Oml, 0m2) for comparison with data from a system (Os), testing hypotheses (81, 82) and a theory (T).

us intellectually and possibly emotionally. As understanding of a system increases we are less mystified by it~ we can ask clearer questions about it. We know more about how it works. But all these reassuring things are known only in the context of a model which involves assumptions and arbitrary exclusions.

If a model is to have a scientific purpose, it must be at risk - at risk of being disconfirmed, shown to behave inconsistently with the behavior of the system it was meant to represent. In fact, we don't want a model to be robust~ we want it to be sensitive, subject to potential definitive disconfirmation. The connection between the system and the model is a set of hypotheses. The model translates a hypothesis or hypotheses about the system into data which can be compared with data taken from observations of the system. Suppose that we are interested in system S from which we can get data Os by some defined observations. We think about this system in a certain context which limits the explanations of the potential observed behavior to two possibilities, say 81 and 82 (see Fig.l). Then we derive models MI and M2 corresponding to 81 and 82, respectively. MI and M2 produce data sets Oml and 0m2' From 0ml and 0m2 we know how to look for Os. If we do a shrewdly designed experiment, then we find for example 0ml extremely far from correspondence with Os, and 0m2 very close by a test decided on in advance.

Then what we conclude? MI is an inappropriate representation and - if MI and M2 are similar except for the consequences of 81 and 82 - therefore 81 is incorrect with respect to S. 00 we know that 82 is correct? No, but we are in a position to test it further and to generate new hypotheses and design an experiment which can disconfirm one or more of the two or more new hypotheses. In this way we move closer and closer to a complete understanding of S in the context we adopted. Oon't worry about putting yourself out of work~ though it is an interesting question, probably a complete understanding, an exhaustion of all possible meaningful questions, is unreachable, if for no other reason than because system definitions and understanding can never be context-free.

It should be clear from this sketch of the scientific process that a model, to be useful, must be designed to be disproved. It exists in a context which includes assumptions and almost always simplifications and apriori has nothing to do with the da ta taken from observation of the system other

13

than having guided the scientist in deciding what to observe. At this point we should say something about the common expression "modeling the data", a rather inconsistent phrase by almost any extant meaning of "model". Deriving a mathematical expression from da ta alone is, of course, impossible. By any method one has to postulate a form of mathematical expression or physical device in advance of "modeling the data". The parameters of the expression or the device might be chosen on the basis of some relation between it and the data. But if one postulates a model from the data, then one had no real reason to make those particular measurements. There was no question to answer.

Of course, this process is not a single-pass, one-way path. In practice we double back, adjust, reinterpret. We are influenced by experience and ideas and we, hopefully, control our model (though there are probably occasions when expensive models have controlled their users more than their users have controlled them). The structure of our investigations and the structure of our arguments must be formal. We must make clear to the reader or listener what is the system of interest, what are the hypotheses to be tested, what assumptions are made. We can't state all our assumptions with every argument. The majority derive from the historical context of the communal effort of a particular discipline - and from our particular culture. But certainly we must be aware that there are many assumptions, many of which we are not consciously aware. Further, we must make clear exactly what are the models devised, what are the identity and manner of acquisition of the data (on both the system and the models) and what are the tests and criteria for comparison of sets of data. The connection between this and what we did day by day, how we came to ask the question, how many times we changed the hypotheses and/or the models need not be known by others or remembered by USo It may be interesting, inspiring or even funny and it may even instruct with respect to doing science but it has no use in drawing conclusions about the system in the context of the argument.

I find it conceptually helpful to distinguish various classes of statement. A synthetic statement relates terms which refer to objects or to relations among objects subject to comparison with experience. In contrast an analytic statement is abstract, usually symbolic and is judged valid or invalid on the grounds of logic and consistency. About y = Y + 3 we can say that it is false. On the other hand, we can say y = 4x, making an analytic statement for a set of numeric pairs (x,y). In contrast, if we say, "There are four times as many tires touching the ground as there are cars in a parking lot", we make a synthetic statement about the real word. There are subclasses of synthetic statements. The statement about tires might be a hypotheses. If it is sufficiently confirmed by sufficiently many observers and gains acceptance as a true statement about some defined class of parking lot, then it becomes a law.

In itself this law - any law - explains nothing. But if we put it in context - by defining "car" and so on - joining it to a large and weIl founded body of definitions and laws and relations, then it becomes part of this larger structure which we call a theory and thus is explained within the

14

limits of that theory. The analytic statement can be connected to the law or theory by statements - called ftcorresponcence rules ft by Carnap (1966) - which identify x with Rnumber of cars ft and y with ftnumber of tires touching the ground ft • Such statements may be hypotheses from which we define experiments. Or hypotheses might be statements expected to become, or expected to be contrary to, laws. Or they may be statements placing a system in a theory.

Thomas Kuhn (1962) argued that science progresses through periods of ftnormal science ft begun and ended by ftscientific revolutions ft • His distinction between normal science and a revolution in science is a distinction between points of challenge. All scientific investigations are essentially challenges of some conclusion or assumption or hypothesis. If we challenge statements about small parts or more superficial consequences of a theory, testing its scope of validity or elaborating it a bit, then we are doing normal science. If we challenge a broad and very successful theory and seriously constrain its range of applicability, or show it to be a special case of a grander theory, then we create a revolution in science. An example of normal science might be measuerment of the activation energy of an enzymic reaction~ of a revolution, Einstein's hypothesis that there is a finite constant maximum speed of light.

Again, hypotheses are statements about the system S. The grammar of the statements may be that of mathematics or of some scientific discipline. The terms used in the hypotheses will refer to Sand directly or indirectly to some theory T. S may be a normal adult exposed by intravenous injection to deuterated water in the blood and extravascular spaces over a period of, say, 10 days. We might make the two hypotheses Hl: the body is a system of two compartments, one blood and the other the extravascular space, connected in both directions, with elimination fron the latter~ and H2: the body is a system of three compartments connected in series in both directions - blood, extravascular 1 and extravascular 2 - with elimination from extravascular 2 only.

Because T is weIl developed in this instance we can derive the models MI and M2 corresponding respectively with Hl and H2 by drawing:

k12 k23

1~2~3 ~~{

k21 k32 ~ ke

and then writing the corresponding sets of differential equations which can be solved to yield the mathematical models which generate the data sets Dml, Dm2 for comparison with observations Ds on S. This, of course, requires assignment of values to the parameters. This can be done by regression analysis or by trial-and-error or what have you. This done, perhaps Dml is inconsistent with Ds and we reject MI and thus Hl, taking into account the variances of the measurements and so on. We do the same with Dm2. Whether H2 is confirmed or rejected, we start asking new questions and designing new experiments.

15

This example gives me an opportunity to make an important point about this type of model used so much in pharmacology. If D~l is consistent with Os' then Dm2 will be cons1stent with Os as weIl, if certain relations among the parameters hold, which relations will occur if we estimate the parameter values from the data. That is, the exponential equations corresponding to a sufficiently connected system of compartments can always fit a set of data as weIl as the equations corresponding to a smaller number of compartments. Once again, the only sure thing is disconfirmation. So if in the context of the experiment, S is not two-compartment but may be three-compartment, then S also may be four-, or five-. or six-compartment.

Philosophers generally accept the impossibility of positive and absolute proof, but scientists sometimes face this with resentment, even denial. But clearly it doesn't stop uso Nor should it. Bere is an example of use of models.

Schwarz, Fridovich and Lodish (1982) present a study of the asialoorosomucoid-binding system in a human hepatoma cell in culture. Their conceptual model of binding, internalization and recycling of receptor is:

where R, C, Ci represent free surface receptor, ligand-receptor complex and internalized complex, respectively. The constant kl incorporates the ligand concentration, assumed constant (in excess)~ k2 governs the internalization of complex. The constant k3 governs the recycling of receptor and degradation of ligand. Using the experimentally obtained values of the constants reported and the set of differential equations derived from this scheme, one gets temporal variations of system components consistent with what the authors observed experimentally.

But clearly the step Ci~R is more complex in the system than a single first-order reaction. Introducing just one additional component, internal uncomplexed receptor, to the scheme suggests two possibilities:

I II

The results of solving the corresponding sets of equations are different (Fig.2). The values of kl' k2, k3 used are the same in all cases. The values of ko and ki are chosen to produce steady-state values of internalized complex similar to the experimental results. Scheme II produces results indistinguishable from the experimental results and

16

\ "

0:: 0 .... (l. UJ U UJ 0::

...J < .... 0 40 .... -;l.

20

".

........ ".

.......... -----_._._._. ---------

10 15 20 25 TIME (MIN)

Fig. 2. Surface receptor and internalized complex for the three schemes described in the text: -----generated by the scheme of Schwartz et al.~ -------from scheme I of text~ --------from scheme II.

from the results generated by the scheme suggested by the experimenters and is a candidate for elaboration and experimental design. while scheme I is disconfirmed with the parameter values used. In the context of cell biology this is probably a useful conclusion.

These points about science and its methodology have been discussed before, of course, by philosophers and by scientists and mathematicians. From the latter group we can find discussions by Bergner (1962). Nooney (1965). Beck and Rescigno (1970) and Rescigno and Beck(1987). The frequency of erroneous claims about. and uses of, mathematical models in the current literature of medicine and its basic sciences suggests that consideration of these discussions could weIl be more wide-spread.

I am not suggesting that all pharmacologists, for example, stop what they are doing and become scholars of the philosophy of science. I arn suggesting, however, that we can avoid a lot of waste, be more thorough and effective and have more pleasure doing science if we pause once in a while and take a view from a distance of what we are doing.

Acknowledgement: My long association and occasional collaboration with Dr. Aldo Rescigno has contributed much to my interest in the conceptual foundations and uses of models.

REFERENCES

Beck, J.S.· and Rescigno, A.- 1970. Calcium kinetics: the philosophy and practice of science. Phys. Med. Biol.- 15: 566.

Bergner, P.-E.E •. 1962. The significance of certain tracer kinetical methods, especially with respect to the tracer dynamic definition of metabolic turnover, Acta Radiologica, Supplementum 210. -----

Carnap, R •• 1966. nphilosophical Foundations of Physics n, Basic Books, New York.

17

Kuhn, T.S.- 1970. "The Structure of Scientific Revolutions", 2nd edition, University of Chicago Press, Chicago.

Nooney, G.C., 1965. Mathematical models, reality and results, J. teoret. Biol., 9: 239-

Rescigno, A., and Beck, J.S.- 1987. The use and abuse of models, J. Pharmacokin. Biopharm.- 15: 327-340_

Schwartz, A.L.- Fridovich, S.E. - and Lodish, H.F.- 1982.

18

Kinetics of internalization and recycling of the Asialoglycoprotein receptor in a hepatoma cell line, J. Biol. Chem., 257: 4230.

DEVELOPMENT OF COMPARTMENTAL CONCEPTS

Aldo Rescigno l ,2 and Ajit K. Thakur 3

lSection of Neurosurgery, Yale University School of Medicine, New Haven, CT 06510

2Present address: Institute of Experimental and Clinical Medicine, University of Ancona, Ancona, Italy

3Biostatistics Department, Hazelton Laboratories America, Inc., 9200 Leesburg Turnpike, Vienna, VA 22180

HISTORICAL INTRODUCTION TO COMPARTMENTAL ANALYSIS

The first compartmenta1 models were used in Physics for the description of radioactive decay. After Becquerel (1896) discovered the radioactivity, Rutherford and Soddy (1902) found experimenta11y that Thorium X decays in time according to an exponentia1 law, i.e. that the number of radioactive atoms decaying per unit time is proportional to the number of radioactive atoms present. If X(to ) and X(t) are the quantities of radioactive substance present at time t o and t respective1y, the law of radioactive decay is

(1) dX/dt = - K.X,

whose integral is

X(t) = X(to).exP(-K(t-to »). Later Rutherford (1904) deve10ped the theory of

successive radioactive transformations. If A is transformed into B, B is transformed into C, and so forth, call Xa , Xb, Xc, ••• the amounts of A, B, C, ••• present at any given time; call also Ka , Kb, Kc the rates of such transformations. He wrote, in analogy with equation (1),

dXa/dt = - KaXa ,

dXb/dt + KaXa - KbXb, (2)

dXc!dt

and by integration,

19

Xa(t) = Xa(tO).eXP~Ka(t-to»), Xb(t) = Ka/(Kb-Ka),Xa(tO).eXPtKa(t-toY +

+ ~Kb-Ka )Xb (to )-KaXa (tO») / (Kb-Ka)'

.eXP(-Kb(t-toU,

and so forth.

Many experimental observations have shown that this compartmental model is consistent with the behavior of all known radioactive substances, thus confirming the hypothesis incorporated into equations (1) and (2), i.e •. that radioactive decay is a first order process.

The first quantitative analysis in pharmacokinetics was made by Widmark (1920). who studied both theoretically and experimentally the kinetics of distribution of several narcotics, in particular acetone. He studied the concentration curve of acetone in the blood after a single dose administration, and assumed that the fall of the curve was due principally to elimination from the lungs and chemical metabolism. The mathematical model used by widmark was

dx/dt -ax - bx

dy/dt = ax

dz/dt bx

x(O)

y(O)

z(O)

o o

where x, y, z are the amounts of acetone in the body, exhaled, and metabolized, respectively, and Xo is the amount administered initially. From the knowledge of the time behavior of the concentration c(t) of the acetone in the blood and of the so-called nreduced body volumen m, where m = x/c, Widmark computed the time behavior of x, y, z in several experimental conditions.

Later Widmark and Tandberg (1924) derived the equation of a model where there is a constant rate administration, and also when the drug is administered with rapid intravenous injections repeated at uniform intervals of time.

Another important contribution has been given by Gehlen (1933) who derived some theoretical expressions for what we would now call a two-compartment model.

Widmark (1932) studied also the elimination of ethanol and developed in this context what we would now call a zero-order compartment model.

The first systematic study of the kinetics of drugs introduced into the mammalian body in various ways was performed by Teorell (1937). As in the dynamical analysis of exchange of inert gases and of distribution of narcotics, the assumptions about the transport and the definition of the regions or compartments wherein measurements are to be made, lead to a set of linear differential equations with constant coefficients. Beyond that, however, two other interesting considerations appeared in these papers. One is the idea of a

20

chemical transformation as a route between compartments, where the latter term has a more general meaning in the sense evidenced by Widmark.

Teorell's concern was the disappearence of a drug from blood or tissue in a more general framework and the generalization of the term compartment was made to include possible inactivation of the drug via transformation to another chemical form. The other idea was the distinction between what one may call Fick kinetics and stochastic kinetics. For the resorption of a drug from a subcutaneous depot, Teorell considered that each particle has the same probability of being transported; therefore the instantaneous rate of loss is proportional tb the number of particles present at that instant. In our notation this assumption leads to the set of equations

(3 ) i=1.2 •... . n,

where Xi is the amount of substance present in compartment i, the constant kji is the fraction of the substance in compartment j transporported to compartment i per unit time, and the constant Ki is the total fractional efflux from compartment i. On the other hand. for transport between blood and tissues, Teorell assumed what we may call Fick kinetics; this may be expressed by the equation

F = A(Gj - Gi),

where F is the net flux from compartment j to compartment i, Gi and Gj are the activities in compartment i and j respectively, and A is a constant. Here the driving force for transport is activity, a thermodynamic quantity, rather than an amount of substance. Then with the assumption that the activity of a substance is adequately approximated by its concentration, and that the rate of change of concentration in a homogeneous constant volume is proportional to the net flux across its boundary, we have the equations

(4) i=1,2 •... . n,

where h" is the permeability constant for the barrier of constan~Jthickness and area between compartments i and j. These equations represent the kinetics of the system of compartments governed by Fick kinetics. Equations (3) are more general than equations (4). as equations (4) follow from physical conditions that narrow their applicability. Now define

i=1,2 •... n,

where Vi is a parameter independent of time; then equation (4) become

dYi/dt = Lj hijVi(Yj/Vj-Yi/Vi).

which are formally identical with equations (3) if we put

(5) hij Vi/Vj = kji, Zj hij = Ki.

Fick kinetics is thus formally a special case of

21

stochastic kinetics~ where definitions (5) hold. Again formally, whatever Xi is, kji is the instantaneous time rate ofoincrease of Xi due to Xj, expressed as a fraction of Xj. Given the physical interpretation of Ci and hij' one may choose to regard Vi as a volume, which:then leads to the interpretation of Xi as an amount; then kji becomes the fractional transfer rate, the frl'~ion of Xj contributed to Xi per unit time. Though equations (4) are very restrictive, the special case of Fick kinetics is an important one, having a wide use as a model for biological transport processes.

Another important step in the use of compartment equations in physiological models was made by Artom et al. (1938). To study the formation of phospholipids as affected by dietary fat, they administered inorganic phosphate containing radioactive 32p to rats and measured the radioactivity present in inorganic phosphate of blood, in the lipid of liver and in the skeleton at known times after administration. The physical correlate of compartment, then, is astate determined by the simultaneous existence of a particular location in space and a particular chemical state. For example, the variable representing the amount of 32p in inorganic form in blood is a compartment and is distinct from the variable representing inorganic 32p in the liver and distinct as weIl from that representing lipid 32p in blood.

As a basis for their analysis, Artom et ale (1938) specified four assumptions:

a) that the organism is incapable of distinguishing between 32p and 3lp;

b) that the quantity of p fixed in any form whatever (for example as lipid P) by a tissue per unit time is proportional to the amount of inorganic P in the blood; and similarly that the amount of inorganic P which, in the same interval of time, is returned to the blood from the considered form is proportional to the amount of P present in that form in that tissue;

c) that the total amount of P in the tissues remains constant during the experiment;

d} that the quantity of P administered is sufficiently small such that it does not modify the metabolism of the animal.

Then they defined the following symbols: Nb' NI' Ns represent the number of atoms of 31p of

the form of interest in blood, liver, and skeleton, respectively; nb~ nl, ns represent the analogous numbers of atoms

of 3~p; I/Nb represents the probability per unit time of fixation

in the form of interest of a given atom of inorganic P by the liver;

s/Nb represents the analogous probability of fixation by bone.

From these assumptions and definitions and the additional assumption that no other appreciable exchange of P occurs, three differential equations follow:

22

{

dnb/dt = -(l+s)nb/Nb + Inl/Nl + sns/Ns'

(6) dnl/dt = Inb/Nb - Inl/Nl'

dns/dt snb/Nb - sns/Ns'

These three._ ~uations are analogous to equations (3)­where, say, kIb = I/NI' and so forth. The solutions as functions of time are in general sums of three exponentials. The constants of the exponents are characteristic of the system, that i~t they depend upon 1, s, Nb' NI' Ns ; the coefficients on the other hand are constants dependent upon these parameters and the initial conditions of the experiment.

It is of interest to note here that the parameters 1 and s play a two-way role in this case as does the permeability parameter hij in the ca se of Fick kinetics. The reason is quite different, though. In this case 1 and s are number of atoms transported between compartments per unit time. Hence the number of atoms transported per unit time from blood inorganic P to liver lipid P, say, is 1. The probability of transport per unit time for a single atom, then, is I/Nb and the number of radioactive atoms transported per unit time is Inb/Nb. That the same parameter 1 appears in the term for transport from liver lipid P to blood inorganic P is required by assumption (c) quoted above. It should be clear that the probabilities of transport per unit time between liver lipid and blood phosphate (I/NI' I/Nb) are not necessarily equal in the two directions. Furthermore, if there were a path for transport from liver to bone not including blood inorganic P. then this steady-state assumption would not imply the single parameter 1 for both directions.

DEFINITION OF COMPARTMENT

Probably Sheppard (1948) was the first author to use the term compartment: ßThere are numerous instances in biological and chemical research where multiple compartment systems are encountered. This is undoubtedly true in other fields as weIl. In such a system, real compartments may exist whose contents are homogeneous and which are separated from one another by real boundaries. However, the concept may be generalized so that a substance, such that a chemical element, can be considered to be in a different compartment when it is in a different state of chemical combination. ß Later Sheppard and Householder (1951) made this concept more precise: ßIn isotope scudies compartments may be regions of space in which the absolute specific activity (fractional amount of the substance that is tagged) is uniform such as erythrocytes and plasma in vitro or states of uniform chemical composition such as copper ions and copper chelate compounds. ß

Other definitions, substantially equivalent to Sheppard's, can be found in Rescigno and Segre (1962). Brownell et al. (1968). Berman (1972). Jacquez (1972) and Gurpide (1975). This last author suggested the use of the term pool instead of compartment, ßto avoid the purely

23

spacial implication that may by assigned to the latter term": I prefer to use this term for a different definition, as shown in the next few lines.

A simple operational definition of compartment, proposed by Rescigno and Beck (1972). is: BA variable X(t) of a system is called a compartment if it is governed by the differential equation

(7) dX/dt = -KX + f(t)

with K constant." For a physical interpretation of equation (7), consider X as the amount of a certain substance in a particular subdivision of a system, through which the concentration is uniform at any given time: that the substance leaves that subdivision at a rate proportional to its total amount there, i.e. with a first order process, with relative rate K; f(t) measures the rate of entry of that substance in that subdivision of the system from other subdivisions or from outside the system. Thus equation (7) represents the relationship between the behavior of the precursor f(t) and the behavior of its successor X(t).

Going back to the paper by Artom et al. (1938). it is worth observing that they were aware of the necessity of defining a compartment operationally, though they did not use the term compartment explicitly (see page 257 of their paper).

From a different point of view, a compartment can be defined stochastically, as done for instance by Rescigno and Segre (1966): "A compartment can be considered as being made up of an ensemble of particles, moleeules or parts of moleeules which have the same probability of passing from their state to other possible states."

More precise stochastic definitions were given by Matis and Bartley (1971). Thakur et al. (1972). Purdue (1974) and many other authors.

In contrast to a compartment, I call a pool a set of particles defined by boundaries and chemical composition, but not uniform because of their not being weIl mixed or having different ages, or somehow having different properties as regards to the process by which they leave that set. A classical example of a pool, as shown by Shemin and Rittenberg (1946). is given by the hemin in human erythrocytes; even though the circulating erythrocytes are very weIl mixed, at least for intervals of time larger than several hours, the hemin is not eliminated with a process of order one; this is so because the hemin remains in the erythrocyte until the latter is destroyed, and the probability of destruction of an erythrocytes depends on its age.

PURPOSE OF COMPARTMENTAL ANALYSIS

According to Zierler(198l). one purpose of compartmental analysis is to calculate one or more of the parameters of the system. In his words, BIf there are compartments, the

24

investigator wants to find the quantity or concentration of material in, or the volume of, one or more compartments. n More generally, we can say that compartmental analysis is a method of computing parameters from experimental data, if the system observed behaves according to the hypotheses incorporated into the compartmental model.

Basically, a compartmental model consists in making the hypothesis that a system is composed of a finite number of identifiable components, each one of them following an equation like (7). In other words such a model implies that the system under observation can be represented. more or less closely, by a set of linear, first-order differential equations with constant coefficients. The solution of such a system is, in general, a sum of exponential terms. Compartmental analysis allows us to compute the parameters describing the properties of those compartments, if the stated hypotheses are valid.

On the other hand, a sum of n exponential terms can always be written in the form of a linear differential equation of order n, or of a set of n linear equations of order one. Thus sometimes one finds in the literature the statement that when experimental data can be fitted by a sum of exponential terms. then one has to use a multicompartmental model. But many data that do not come from compartmental systems can be fitted by sums of exponentials.

In an interesting discussion of compartmental models Bergner (1962) emphasized that experimental da ta can be used for scientific purposes, as opposed to purely descriptive purposes. only if the model postulated has correlates of meaningful physical parameters, defined apriori.

It is essential, each time one uses a compartmental (or any other) model, to ask, nIs this model appropriate? What are the physical meanings of the parameters of the equations postulated?n The statement, often found in the literature, that successful fitting of da ta with sums of exponentials implies a compartmental system, is a misconception which can block us from optimal use of data.

REFERENCERS

C.Artom, G.Sarzana and E.Segre, 1938- Arch. int. physiol. 147:245.

H.Becquerel, 1896. Comptes rendus 122:420. 501-P.-E.E.Bergner, 1962. Acta Radiologica Suppl. 210.

Stock holm. M.Berman, 1972. Iodine kinetics. In: ('Methods of

investigative and diagnostic endocrinology'/(Rall and Kopin editors). North-Holland, Amsterdam.

G.L.Brownell, M.Berman and J.S.Robertson, 1968- Int. J. Appl. Rad. Isotopes 19:249-

W.Gehlen, 1933. Arch. exp. pathol. pharmakol- 171:541. E.Gurpide, 1975. nTracer methods in hormone research~

Springer-Verlag, Berlin. J.A.Jacquez, 1972."Compartmental analysis in biology and

medicine! Elsevier, Amsterdam. J.H.Matis and H.O.Hartley, 1971. Biometrics 27:77.

25

P.Purdue, 1974. Bu11. Math. Bio1. 36:305. 577. A.Rescigno and J.S.Beck, 1972. Compartments. In:

Foundations of Mathematica1 Bio1ogy (R.Rosen editor). Vo1ume 2. Academic Press, New York.

A.Rescigno and G.Segre, 1962. IILa cinetica dei farmaci e dei traccianti radioattivi~ Boringhieri, Torino. (Eng1ish trans1ation:"Drug and Tracer Kinetics: B1aisde11, Wa1tham, Mass.- 1966.)

E.Rutherford, 1904. Royal Soc. London Phi1. Trans. A 204:169.

E.Rutherford and B.A.Soddy, 1902. Phi1. Mag. 4:370. 568. D.Shemin and D.Rittenberg, 1946. J. Bio1. Chem. 166:621. C.W.Sheppard, 1948. J. App1. Physics 19:70. C.W.Sheppard,and A.S.Househo1der, 1951. J. App1. Physics

22:510. T.Teore11, 1937. Arch. Int. Pharmacodynamie Th~rapie

57:205. A.K.Thakur, A.Rescigno and D.E.Schafer, 1972. Bu11. Math.

Bio1. 34:53. ----E.M.P.Widmark, 1920. Acta med. Scand. 52:87.

E.M.P.Widmark, 1932.'TIie wissenschaftliche Grundlagen und die praktische Verwendbarkeit der gerichtlich-medizinischen A1koho1bestimmung~ Urban & Schwarzenberg, Ber1in.

26

E.M.P.Widmark and Tandberg, 1924. Biochem. Z. 147:358-K.Zier1er, 1981. Ann. Rev. Biophys. Bioeng. 10:531.

HODELING OF PHARHACOKINETIC DATA

INTRODUCTION

Aj it K. Thakur

Biostatistics Department Hazleton Laboratories America, Inc. 9200 Leesburg Turnpike Vienna, Virginia 22180 U.S.A.

with the improvements and easy availability of digital computers,

practically all branches of biology are employing mathematical techniques

to extract every possible bit of information from experimental data.

Hodeling in the statistical sense is one of the tools which may provide

an experimenter with knowledge about some intricate parts of a system

which were otherwise inaccessible or too expensive to probe into. It

also allows one to make predictions about a system under certain condi­

tions. Finally, modeling as a dynamic tool should provide input for

better future experiments.

There are basically two types of models used to describe a system:

predictive and descriptive. The purpose of a predictive model is to

predict the local behavior of a system in response to an input variable

under given experimental conditions. This can generally be accomplished

by smoothing or empirical functions which allow one to describe the

system's behavior under some restrictions. These models are not gener­

ally based on the physical structure of a system, and as a result, it is

not advisable to be used for extrapolation beyond the observed data

range. Furthermore, the parameters of this type of models do not have

any specific physical meaning. A descriptive model, on the other hand,

is based on physical structure and function of a system. In general,

descriptive models are more complex. A good descriptive model, however,

can provide more far reaching predictions than a simpler predictive

model.

27

This deseription will be eentered around deseriptive models sueh as

the standard rate equations of ehemieal reaetions and eompartmental

systems as used in pharmaeokineties. Beeause of the eomplexities of

these models, Whieh may be deseribed in terms of algebraie, differential,

or integral equations, the deseriptions are often nonlinear both fune­

tionally and in parameters. As a result, parameter estimation from sueh

models is more diffieult than models linear in parameters. Even though

the statistieal theories of non linear estimation are well developed in

literature (1-3), the aetual modeling and estimation from experimental

data eneounter insurmountable bloeks sometimes. Unfortunately, there is

very little in standard statistieal literature to provide any effeetive

guidanee for an investigator involved in sueh modeling. Whatever is

existing in the literature is seattered around in applied statistieal,

mathematieal, engineering, and ehemieal journals, as well as in eookbooks

and manuals for some nonlinear eurve fitting programs (4-18).

The purpose of this diseussion is to systematieally -develop the

eoneept and praetiee of modeling through some examples and the problems

one eneounters in them. Examination of the raw and fitted data with

graphieal and simple statistieal evaluation will be used as the main

tools for this purpose. I would hope that onee a model is formulated,

very little mathematies beyond simple algebra and ealeulus will be

neeessary to grasp the ideas herein. The benefits derived should be the

differenee between failure and sueeess in modeling.

I have freely taken examples and teehniques of solutions from exist­

ing literature for the present exereises. All my worked examples are

taken from the HLAB manuals (15-17) with kind permission from their

ereator Dr. Gary D. Knott, my good friend and one-time eolleague. These

examples show some interesting problems assoeiated with nonlinear

modeling.

HETHOD

Unlike the sum squares (SS) surfaee of a linear model, a eomplex non­

linear model may not have any unique minimum. In general, the SS surfaee

of a nonlinear model will have several-to-many, depending on the number

of nonlinear parameters and their dependeney, loeal minima. These minima

are, to quote Fleteher and Shrager (13), ..... like eraters on the surfaee

of the moon. When one is in the eenter of a erater, one eannot tell if

there are other eraters with even deeper eenters." Also, it may not be

possible to elimb out of the wrong "erater" When the eonditions are not

28

ideal. Of course the overall minimum lies in the "crater" with the

smallest SS. Let us assume a model of the form

y (1)

where y is the dependent variable (measured), xi' i=1,2, ... ,n the inde­

pendent variables, and Pi, i=l, 2, ... ,n the parameters which need to be

estimated using a matrix M of observations of y on xi. If the associated

weights for the observations are given by a vector wand the linear

constraints on Pi as ciPi = or ;;t or ~ 0, the curve fitting problem

becomes:

(2)

In other words, find a set of numerical va lues for the parameters Pi

which minimizes the SS under the given linear constraints above. There

may not be such an assignment at all or there may be multiple such

assignments. For problems linear in Pi the solution is straightforward

and a global minimum for the SS is achieved in a single trial. For

nonlinear problems, even finding the most appropriate local minimum for

a problem requires careful selection of initial estimates for Pi. This

selection will determine the failure or success in nonlinear estimation

or what one refers to as curve fitting. One must remember that the

convergence ability, converged parameter va lues , the converged SS and

the speed of convergence - all heavily depend on how good the initial

estimates are.

Initial Estimate

Use all possible prior information which may be available - from

published or unpublished literature, previous experiments, knowledge

about the system, physical and biological considerations, steady state

or equilibrium behavior of the system under consideration, etc. Some-

times perturbing the system in both directions may provide some idea

about the possible ranges of values to search for initial estimates as

will be explained and demonstrated later. If ranges of va lues are known

for some or all of the parameters but no other information is available,

use the mid-range va lues as initial estimates and use the ranges as

constraints (to be discussed later). Remember that if there are multiple

minima on the SS surface, poor initial estimates may result in conver­

gence to a stationary point or there may be no convergence at all.

29

Parameter estimates and their statisties under these eonditions may not

have any physieal meaning at all. Let us briefly diseuss sorne of the

teehniques whieh ean be used for accomplishing this purpose.

(a) Curve Peeling: For simple one- or two-eompartment systems, it may

be possible to obtain good initial estimates or ranges of values for

the parameters by eurve peeling (18) and limiting slope-intereept

methods (6,19-21). The feasibility of these graphieal or semi­

numerieal teehniques depends on the number of eompartments, number

of data points eovering a wide range of behavior of the system, as

well as magnitudes of measurement and other errors.

(b) Simulation: Sinee it may not be praetieal or even possible to solve

all equations deseribing the model of interest analytieally, one of

course must have numerieal algebraie and differential equation

solvers whieh are well behaved. Along with this faeility, one must

also have good graphies eapability as part of the eurve fitting

paekage. One ean then try to simulate the model in question with

values obtained from the previous step, eompare against the experi­

mental data and refine the parameters further. This may be a time

eonsuming proeess partieularly for eomplex models but is worth it.

(e) Grid Seareh Hethod: A very effeetive seareh proeedure, whieh surpris­

ingly is not widely utilized, is the grid seareh teehnique (9,15).

The proeedure is very simple onee one has the right eomputing tools.

One ehooses a range of values for the parameters of a model and

eomputes and prints points on the SS surfaee using equation (2) or

some form of it. For a single-parameter nonlinear model, the seareh

ean proeeed graphieally by simply plotting the SS against different

values of the parameter. The value whieh produces presumably the

smallest value of the SS is then used as the initial estimate for

the parameter for eurve fitting. The rate and other aspeets of

eonvergenee are mueh improved by this method. For models with

multiple parameters, a visual display of the SS as a graph may be

diffieult if not impossible. The best possible means is to sort the

SS va lues on a computer and let it print out the parameter set whieh

produces the smallest SS value. This set will then beeome the

starting point for the eurve fitting.

(d) Cyclic Search: Some people use the cyclic search technique (17,22)

which under certain circumstances will provide good initial esti-

30

mates. According to this technique part of the parameter set is

fixed while the others are estimated by nonlinear least squares

method. One can then change the sequence thereby obtaining initial

estimates for curve fitting for the entire set of parameters. The

parameter set initially kept fixed should be the one consisting of

the best known parameters. If there are high dependencies among the

parameters, the method may not work well. Once aga in , this method

should be used for obtaining good initial estimates, not for the

final values of the parameters.

Model Validity Testing

Conventional hypothesis testing such as analysis of variance etc.,

may not be appropriate for nonlinear models because estimation of

variances generally is overly optimistic for these cases. As a result,

we will refrain from using the term hypothesis testing in our context.

Questions regarding the validity of a model, however, can be answered

very simply from the residuals, i.e., the difference between the observed

data points and the corresponding expected values from the fitted curve.

Additional pieces of information regarding the statistical features of a

model are obtained directly from the curve fitting procedure and most

computer programs for this purpose automatically provide them, or at

least they should. Let us briefly discuss them.

(a) Graphical Evaluation: One should always plot some graphs of the

observed data and the corresponding fitted values once the estima­

tion exercise is completed. Looking at the experimental data points

over the fitted curve often allows one to detect key punching errors,

trend, possible outliers, as well as determining visual goodness of

fit. Preferably this should be done with high quality graphics, if

possible.

(b) Uncertainty in the Parameters: The standard deviations of the para­

meters are generally approximated using the normal distribution

theory. As a result, they may be optimistic. However, if they are

unusually large, that would definitely indicate lack of fit, inappro­

priateness of the model being tried, or some problem in the design

itself. This would at least be a flag to indicate that further

evaluation of the model and/or inadequacy of the data should be

considered. Of course if the model is linear in the parameters, if

the error distribution is normal, and if the reciprocal of the

31

variance is used as the weight for each data point, then the fitted

parameter values are maximum likelihood estimates and their standard

deviations are exact. Also, exact joint confidence intervals of the

parameters can be obtained with some computational complexities (23)

for nonlinear models. Kost computer packages for this type of

analysis, however, does not provide this facility to my knowledge.

(c) RKS (Root Kean Square) Error: The RKS error is defined as:

RKS=[SS/(m-n)]% (3)

where m number of data points

n number of parameters to be estimated (m>n)

The RKS is a dimensional measure of the goodness of fit of a model.

It is not an absolute criterion because a fit with random deviations

and another with systematic deviations could both produce the same

RKS. If systematic deviations can be ruled out by some other means

such as examination of the residuals, the magnitude of the RKS can

be extremely useful as an indicator of a good fit.

(d) Oependency Value: The dependency value of a parameter is defined as:

32

o I-Var(PiI Pj fixed, i, j=1,2, ... ; j~i) (0<0<1)

(4)

Oependency is a measure of the degree of instability of a parameter.

A large value of 0, e.g., 0.98, for a parameter indicates that it is

unstable. What that implies is that even if that parameter is

constrained to a different value, one can still obtain an equally

acceptable fit of the data by changing the other parameters. Large

dependency values also imply that very small change in one or more

observations may produce large changes in the parameter estimates.

These may also give rise to situations where the parameter sets may

be significantly different from each other, yet they produce no

significant changes in the converged SS values. This may induce

large joint confidence regions for the parameters as well. In some

of these cases reparameterization of the model, transformation,

weighted regression, better experimental designs, or a combination

of any of the above may be useful. Generally, when these situations

arise, no techniques of f inding good initial estimates may produce

any acceptable results. Finally, this situation wams that the least

square solution may not be unique, the model may not be appropriate,

the range covered by the measurements may not be adequate, or there

may even be outliers among the observations. As a consequence,

dependency values provide reasons for further examination of the

data and the model as weIl as insight to design improved experiments.

(e) Examination of the Residuals: This topic has been discussed in

detail in published literature (2,3,18,24,25). We will briefly

point out some aspects of these discussions here. The residuals

provide measures for departure or lack of fit of the model from the

data. When the model is exact, the residuals are directly related

to the errors in the experimental observations. In other words, all

the information regarding the fit of a model can be statistically or

graphically extracted from the residuals, irrespective of whether the

model is linear or nonlinear in its parameters. Some rather popular

computer programs perform analysis of variance and related tests on

nonlinear models when they are not applicable yet neglect to provide

either graphical displays or simple statistical tests on residuals

which are always valid. We will mostly rely on these graphical

displays. In a model with multiple parameters the residuals are

correlated. As a resul t, some of the more sophisticated tests on

residuals rely on approximations; whereas, as Anscombe (24) pointed

out, the effect of correlated residuals is minimal on corresponding

graphical procedures. When residuals are plotted against time or

the values of the independent variables, one of the phenomena as

depicted in Figures 1-4 may take place. When the fit is good, the

residual plot as described above will be as in Figure 1 where the

residuals form an approximate horizontal band. This would indicate

no trend or time effect in data. Significance of slight deviations

from this horizontality can be tested by the exact runs (a run is a

sequence of equal signs in the residuals) test which is illustrated

in (18). An approximate runs test can be performed by counting the

numbers of negative (nI) and positive (n2) residuals. Suppose r is

the observed number of runs among the residuals. Under the assump­

tion of randomness, the expected number of runs and its variance are:

~r=2n1n2/(n1+n2)+1

a2(r)=2n1n2[2n1n2-(n1+n2)]/[(n1+n2)2(n1+n2-1)]

(5)

(6)

When both n1 and n2 are large (>5 generally will provide a conserva­

tive test):

33

z=(r-Pr+.5)/a(r) (7)

where .5 is the eontinuity eorreetion. The signifieanee of the above

z-statistie is obtained from N(O,l), the standard normal deviate.

But in most eases of praetieal importanee, a visual examination will

provide a satisfaetory test. On the other hand, the residual plots

may show bands as in Figures 2-4. In eaeh of these eases, there have

been some problems in the fit whieh may have been due to the model

ehosen as well as statistieal problems in the assumptions behind the

least square estimation. If the residual plot is as in Figure 2,

this would imply that there has been signifieant varianee nonuni­

formity whieh violates an important requirement for the least squares

analysis. Weighting of the data, or in some eases transformation of

the data (to be diseussed later), is neeessary for the curve fitting

proeedure. If the residual plot is as in Figure 3, there may have

been error in data, error in intermediate ealeulation, or there may

have been a missing term in the model .

. r-----~------~------~----~~----~------~------~----_. 1.1

1 ~-----------------------------------------------------------------------

.I

I ul------------I

-1 --------------------------------------------------------------------------

-1.1

~·O------~.~----~,~o------+.,.~----~m~-----.~----~»~----~»=-----~ TIIE Oll x

FIGURE 1: Residual plot showing a good fit.

Sometimes error in the independent variable or insuffieient da ta to

resolve a partieular segment of the model may also produee sueh

effeets on the residual plot. Similar diagnosties would be indieated

if the residual plot looks like Figure 4. Simple statistieal tests

eould be performed to verify these faets (2,24,25); however, these

plots themselves are adequate for the diagnostie purposes. If any

of the above indications is there, one must examine the experimental

detail earefully to isolate any problems. In some eases the problems

eould be eliminated statistieally. In many other eases these diag-

34

0.0

~~o------~------~,o~----~,.~----~~~----~~~------=~~----~~~----~~ 11lIE OR X

FIGURE 2: Residual plot showing variance nonuniformity (After Ref. 2)

, ........ _---,,--.. .. -- --------"--.. ------- .. -....... ... - --------

o.o~-----------------------_---_--_~~E~~._-_--------------------------~ -- "-- .. -------- --"-- .. ---- --- "'-- .. ---,1---- ----..

~~O------~------~'O~----~'5~----~~~----~~~------~~~----~~~----~~ 11IIE OR X

FIGURE 3: Residual plot showing error in data or model or similar problems (After Ref. 2)

35

Ir-----~----_,------~-----~.----~----_T------~----~

......

~·~O------~.~----~'~O------+',.~----~m~----~.~~--~.~----~»~-----.~ 1IIE Oll X

FIGURE 4: Residual plot showing problems as in Figure 3 (After Ref. 2)

nostics would provide insight for better designs for the experiments

in question.

An approximate goodness of fit test can also be performed with

the residuals as follows:

k I

i=l (8)

the i-th residual.

the variance of the i-th observation from the fitted

curve.

The significance of x2 can be obtained using the X2 distribution

with degrees of freedom given by k-n, where n is the number of

parameters.

Problems in CUrve Fitting: What to Do?

36

So far we have discussed the steps to be usec1 for moc1eling as

well as the lessons we may have learned from it. Let us now briefly

discuss some ways of remec1ying some of these problems.

Constraints: Suppose "convergence" was achieved in the estima­

tion process but one or more of the parameters showed nonsensical

values, such as negative va lues for physical parameters, or values

too large or too small based on the 1cnowledge about them. Under

these conditions one may want to impose linear constraints on those

parameters. These constraints may take the forms such as: Pi>O,

Pi<O, Pi=Pj, Pi+Pj>Pk, Pi+Pj<Pk, etc. The right handsides of the

constraints can also take any numerical values. Remember that the

constrained minimum is a conditional one and the variances obtained

und er such constraints generally do not reflect the true variances.

A second alternative under these circumstances may be weighted

regression.

Weighting: The basic principle behind weighting is very simple.

If some observations are more precise than the others, they should

have more contribution, Le., weight in the curve fitting process.

In most nonlinear models there is severe nonuniformity of variances

which pos es additional problems in the least squares estimation

process. Although the exact weights to be assigned may never be

1cnown, some guidelines can often be provided based on the experi-

mental data. Generally weights are taken to be inversely propor-

tional to the variances of the observations. For the uncorrelated

linear case, the Gauss-Karkov theorem (26) guarantees that this will

provide minimum variance parameter estimates. Even though there is

no extension of this theorem in the nonlinear case, many investi­

gators believe that any kind of weighting based on the knowledge

about the data should provide better estimates of the parameters

(3). A detail discussion of this appears elsewhere (18). Here let

us briefly mention some of the ways weights may be assigned to

observations.

If the observations are radioactive counts, then the dependent

variable Xi is approximately Poisson distributed. As a resul t,

a2(Xi)=E(Xi) and the weight should be proportional to 1/E(Xi). This

fact can be easily verified if there are replicates available at

each point. If the error source is the aliquoting process, a 2(Xi) «

[E(Xi))2 and the weight should be proportional to 1/[E(Xi))2. When

data are collected in replicates, one may fit the observed a 2(Xi) as

a function of E(Xi)' The functions generally used are polynomials,

power functions, etc. (18). Generally a polynomial of 1-2 degrees

(fractional degrees included) or apower function of power 1-2

(fractiona1 powers included) suffices for such purposes. A different

type of smoothing can also be done to accomplish this. For example,

one can draw a best fitting curve for the observed data points using

some smoothing techniques. The variance at each point then can be

37

obtained by squaring the difference between the observed point and

the corresponding value on the curve. This type of error is known

as "white noise". If there are adequate numbers of data points and

there are no wide gaps in the low frequency noise region, this is an

excellent way of estimating variances from nonreplicative data. KLAB

has a built-in operator EWT (Estimated Weights) Which provides this

facility. Its usefulness and application will be shown in soma of

the examples.

Finally, at each iteration step one may alternatively want to

readjust the weights based on the calculated theoretical curve by

using any weighting model one wishes to have.

Even though the unweighted least squares estimates are still

unbiased and the differences between the estimates from unweighted

and weighted regressions may not be spectacular, the former would

invariably be associated with more lack of precision, Le., higher

standard deviations than the latter.

Transformation: In some cases transformation of the independent

or dependent variable will produce uniformity of variances as well as

normality which are two desired properties of the estimation process.

In pharmacokinetics where one is dealing with a multi-compartment

process, it may not be a practical exercise. Even for a single­

compartment process this may have sometimes adverse effect on the

curve fitting procedure. The following example illustrates this

possibility:

dx/dt

x(t)

lnx(t)

-kx, x(O)

xoexp(-kt)

lnxo-kt

(9)

(10)

(11)

In the course of curve fitting with any model, the underlying assump­

tion is that the dependent variable has anormal error distribution

associated with it. If x(t) constitutes radioactive counts, then

the original error was probably Poisson distributed, which is asymp­

totically normal. If untransformed Equation (9) or (10) is used for

the curve fitting purpose, one maintains approximate normality. On

the other hand, Equation (11), because of logarithmic transformation,

will have a log-normal error distribution which can be long-tailed.

This may have adverse effect on the estimation process. As a result,

the obvious simplicity in linear regression rendered by Equation (11)

may be lost producing unreliable parameter estimates. A second

38

example is the case where the error is uniformly distributed. Trans­

formation of this sort will once aga in produce worse results than

using the untransformed data.

Convergence Factor: The convergence factor is a measure of the

failure to improve the SS by a prespecified fraction, i.e.

(SSold - SSnew)/SSold < convergence factor

If during a particular iteration step the above relation holds, the

process is no longer continued and convergence is assumed. It may

occasionally be helpful to decrease this factor to achieve a stable

minimum.

outliers: An outlier or erroneous observation produces an

unusually large residual. Its effect is reflected as large S.D. in

parameter estimates. This problem can easily be detected by plotting

the cumulative frequencies of the residuals on anormal probability

paper (with the assumption that the error distribution is normal).

To do this one sorts the algebraic va lues (with signs) of the

res iduals in ascending order. One then def ines Pi = (i-. 5) In, which

estimates the probability that e ~ ei (.5 is aga in the usual conti-

nuity correction). A plot of Pi vs ei, i=1,2, ... ,n, gives the

cumulative distribution of the errors. On anormal probability

paper this plot should be a straight line. If there are a few points

particularly at the low and high ends which visually deviate from

linearity, those points would correspond to outliers and should be

taken out and curve fitting should be repeated without them. When

the distinction is not clear cut, the decision is not that simple.

Some people use the rule whereby residuals which exceed by 2 or 3

S.D.s (calculated as described under Residuals) are flagged as

arising from outliers. One can alternatively plot the normalized

residuals, i.e., ei/vd2, where d2 is the residual mean squares on a

normal probability paper and accomplish the same goal. Anscombe (27)

provides a systematic approach to this problem which is left to the

statistically sophisticated individuals. For our purpose the present

approach will suffice.

Let us now examine some actual examples to see how the discus­

sions so far help us in curve fitting.

39

EXAHPLES

The examples in this presentation have been taken from the manuals

for KLAB (15-17) Which is an extremely versatile interpretive high-level

language which is itself written in SAIL (Stanford Artificial Intelli­

gence Language). At this point, consideration of some special features

and notations of KLAB is in order for clarity because this program

should serve as an example to demonstrate What computer software for

modeling should be like. KLAB' s nonlinear least squares algorithm is

based on the Harquardt-Levenberg modification. There are four differ­

ential equation solvers in this program: Adam's method, Gear-Tu method

with symbolic derivatives, the same with numerical derivatives and a

combination of Adam's and Gear-Tu, the last three for stiff differential

equations. Eigen value and eigen function computations can be performed

very elegantly with this program. KLAB allows one to impose any sets of

linear constraints and weighting mechanisms (including the operator EWT

as previously mentioned). Because of the powerful matrix arithmetic

algorithms included as operators in KLAB, it can be used as a programming

language for virtually any type of statistical or mathematical computa­

tion. On top of it all, it can provide several varieties of two and

three dimensional high resolution graphics. KLAB' s main limitation at

this time is that it can only be run on DEC (Digital Equipment Corpora­

tion) Systems 10 and 20. With the advent of new 32-bit personal

computers and the language C, it may soon be available for the general

modeling community. The terminology used by HLAB differs in some

instances from other languages such as FORTRAN. Since some actual HLAB

dialogues will be used in this section, these differences are briefly

pointed out:

A +- 5 (or A 5): The assignment command for scalars and matrices

as opposed to A = 5 in FORTRAN

A t 5 (or A A 5):

@:

Raising to apower as opposed to A ** 5 in FORTRAN

Exponentiation to the base 10, for example,

FORTRAN 1.E-3 becomes 1@-3 in HLAB.

Example 1: Let us take an artificial example first:

f(x)=h(x-c)

h(x)=[l/Coshx+6 (x2-8x+18)-x/5j ICosh(2x/5) (12)

Suppose the observations on fex) are given by the matrix as in Table 1.

Even though this example is purely artificial, it demonstrates some of

the problems encountered in curve fitting and how to get around them

40

based on our discussions so far. Let us now use MLAB dialogue to define

the data matrix Hand the functions fand h «Z> implies pressing CONTROL

and Z keys simultaneously):

*H COL 1 ~ LIST(-1:1:0.1) *H COL 2 ~ READ(TTY)

TYPE <Z> AFTER THE LAST NUHBER .69,.77,.8,.8,.83,.89,.9,1.03,1.05,1.04,1.15,1.18,1.2,1.26, 1.34,1.41,1.38,1.33,1.33,1.29,1.36<Z> *FUNCTION H(X)=(1/COSH(X)+6/(Xt2-8*X+18)-X/5)/COSH(2*X/5) *FUNCTION F(X)=H(X-C)

Let us now create the SS as a function of c and plot it as in Figure 5:

*FUNCTION G(X,C)=F(X-C) *FUNCTION SS(C)=SUH(J,1,NROWS(H),(H(J,2)-G(H(J,1),C»t2) *H1~POINTS(SS,-30:30: .5) (creating points on the SS surface between -30

to 30 by increment of .5)

TABLE 1: Table of observations for the problem in Example 1, taken from Knott (17)

x qx) -1.0 0.69 -0.9 0.77 -0.8 0.80 -0.7 0.80 -0.6 0.83 -0.5 0.89 -0.4 0.90 -0.3 1.03 -0.2 1.05 -0.1 1.04 0.0 1.15 0.1 1.18 0.2 1.20 0.3 1.26 0.4 1.34 0.4 1.41 0.5 1.38 0.6 1.33 0.7 1.33 0.8 1.33 0.9 1.29 1.0 1.36

*H2~SORT(H1,2) (sorting on the second column of the matrix which contains the SS va lues in ascending order to obtain the minimum SS and its corresponding value for c)

*TYPE H2 ROW 1 MATRIX :

.5000000 1.552710

This row-matrix tells us that given the generated SS surface, the minimum

is at 1.55271 with a va1ue of c=.5. Since our grid-search was only

incremented by .5, this indicates that minimum SS occurs for a value of

41

c between .5 and 1. Let us proceed with c=1 as our starting value for

the curve fitting:

*C+-l *QUIET FIT(C),F TO K

CONSTRAINTS NAKE= (NONE)? KAK NUKBER OF ITERATION=(3)? 10 CONVERGENCE FACTOR FOR SUK OF SQUARES=(.OOl)?

FINAL PARAMETER VALUES:

.709404

CONVERGED

NORMAL ERROR STANDARD ERRORS: .224071@-1

RKS WEIGHTED DEVIATION=.385986@-1 FINAL WEIGHTED SUK OF SQUARES=.297970@-1 #ITERATIONS USED=3

DEPENDENCY VALUES

.000000 C

Since the estimate for c was good, as can be seen from the SS vs c plot

(Figure 5), curve fitting converged rapidly to the correct region of the

SS surface. As the graph shows, for -20>c>20, the curve is f1at at both

~r-------~--------~--------~------~---------r--------,

211

18

10

-10

c

FIGURE 5: Plot of SS vs. C for Examp1e 1

ends wi th SS=26. 4 . As a resu1 t, va1ues of c in this range as initial

estimates cannot compute a direction of steepest descent.

il1ustrated in the fo11owing two runs:

*C+-500 *QUIET FIT(C), F TO K CONSTRAINTS NAKE=(NONE) ? KAK NUKBER OF ITERATIONS=(3) ? 10 CONVERGENCE FACTOR FOR SUK OF SQUARES=(.001) ?

This is

The curve fitting procedure encounters all kinds of numerical problems

42

and provides a meaningless "convergence":

FINAL PARAMETER VALUES:

500.000

CONVERGED

NORMAL ERROR STANDARD ERRORS:

.000000

RMS WEIGHTED DEVIATION=1.14820 FINAL WEIGHTED SUM OF SQUARES=26.3671 #ITERATIONS USED=1

For the second run:

*C+- -30

DEPENDENCY VALUES:

.000000 C

Without going into details, the numerical procedure fails as before and

we get the same meaningless "convergence":

FINAL PARAMETER VALUES:

-36212.6

CONVERGED

NORMAL ERROR STANDARD ERRORS: 8857.34

RMS WEIGHTED DEVIATION=1.14820 FINAL WEIGHTED SUM OF SQUARES=26.3671 HOF ITERATIONS=1

DEPENDENCY VALUES:

-.149012@-7 C

In this case, since the initial value is to the left of the minimum for

SS vs c, the method tries to overcompensate by trying negative values

for c such that Icl is large and fails to converge to the correct region.

This problem shows several other interesting aspects of curve

fitting as indicated by the next three runs:

*C+- -2

We repeat the same commands as before and obtain:

FINAL PARAMETER VALUES:

-3.08924

CONVERGED

NORMAL ERROR STANDARD ERRORS:

.289478

RMS WEIGHTED DEVIATION=.319045 FINAL WEIGHTED SUM OF SQUARES=2.03579 HOF ITERATIONS USED=7

DEPENDENCY VALUES:

.000000 C

The SS from this run corresponds to a local minimum from which the

procedure fails to jump out. Let us see what happens in the next run:

*C+-10

43

We repeat the same commands as before and obtain:

FINAL PARAMETER VALUES:

-3.08879

CONVERGED

NORMAL ERROR STANDARD ERRORS:

.289753

RHS WEIGHTED DEVIATION=.319048 FINAL WEIGHTED SUH OF SQUARES=2.03583 #ITERATIONS USED=lO

DEPENDENCY VALUES:

-.149012@-7 C

Here the search process overshoots the correct solution and converges to

the same local minimum as before. Perhaps this overshooting can be

prevented by linear constraint on c:

*CONSTRAINTS Z TYPE <Z> AFTER THE LAST CONSTRAINT !C>O !<Z> *C+-IO *QUIET FIT (C), F TO K

CONSTRAINTS NAKE=(NONE) ? Z

The rest of the commands are as before.

FINAL PARAMETER VALUES:

.709475

NORMAL ERROR STANDARD ERRORS:

.224048@-1

DEPENDENCY VALUES: .000000 C

LAGRANGE MULTIPLIERS: .000000 (each active constraint is associated with

a Lagrange multiplier)

CONVERGED RHS WEIGHTED DEVIATION=.385986@-1 FINAL WEIGHTED SUH OF SQUARES=.29790@-1 #OF ITERATIONS USED=5

In this case the search process was prevented from considering c<O by

the linear constraint c>O and as a result, it converged to the correct

minimum as in the very first run. As can be shown by the last run for

this example, we could have accomplished the same thing by weighted

curve fitting as opposed to imposing the linear constraint. In most

cases such a weighted fitting is more desirable.

*C+-IO *QUIET FIT (C), F TO K WITH WEIGHT EWT(K)

The rest of the commands are the same as before except we now do not

impose the linear constraint.

44

FINAL NORMAL ERROR PARAMETER VALUES: STANDARD ERRORS:

.709887 .217241@-1

CONVERGED RKS WEIGHTED DEVIATION ERROR=1.73435 FINAL WEIGHTED SUH OF SQUARES=60.1592 #OF ITERATIONS USED=3

DEPENDENCY VALUES:

.000000 C

The observed and the fitted values as weIl as the residuals are

displayed graphically in Figures 6-7. Both these figures indicate

excellent fit for the model under consideration.

1.1

0 0

0 0

1.311

0 0

... .711

G.DO -1.1 -1 -.I 0.0 .I 1.1

X

FIGURE 6: Observed and converged fitted va lues for Example 1

.1

0 0 0 lJIj

lJIj

0 0 JK 0 0

0

J)Z

! 0 0

G.DO 0 0 0 0 0

0 0 -.112

0 - 0

0

-.111

-.111

~1 -1.1 -1 -.I 0.0 .I 1.a

x

FIGURE 7 : Residual plot for Example 1

45

Example 2: Let us eonsider a three-eompartment model as in Equations

(13):

dA/dt=-K1A

dX/dt=K1A-K2X

dB/dt=K2X (13)

The observations on X and B as funetions of time are given in the matrix

in Table 2. The equations ean be easily solved explieitly as follows:

X(t)=AoK1[exp(-K1t )-exp(-K2t )]/(K2-K1)

B(t)=AoK1 K2 [(l-exp(-K1 t) }/K1-{1-exp(-K2t) }/K2] I (K2-K1 )

=Ao[l-exp(-K1t )]-X(t) (14)

TABLE 2: Table of observations for Example 2, taken from Knott (15)

t x~t2 B~t2 10 66.4 0.02 20 141.0 0.23 30 150.8 0.24 40 174.6 0.42 50 207.1 0.59 60 155.7 0.67 70 207.2 1.04 80 215.6 1.22 90 220.6 1.47

100 188.6 1.68

Of course one could proceed with the differential equations as weIl for

curve fitting purpose. In HLAB it does not make any numerical differ­

ences in the two methods; however, in some curve fitters, particularly

with lower order Runge-Kutta methods, there may be some significant

numerical differences. The experimenter knew that AO=200, fixed and K1

and K2 should be close to .05 and .0001 respectively. We will use those

values for K1 and K2 as initial estimates for this example. Once again,

let us proceed with the HLAB dialogue:

*DX COL 1~LIST(10:100:10) *DX COL 2~READ(TTY)

TYPE <Z> AFTER LAST NUHBER 66.4,141,150.8,174.6,207.1,155.7,207.2,215.6,220.6,188.6<Z> *DB COL l~DX COL 1 *DB COL 2~READ(TTY)

TYPE <Z> AFTER LAST NUHBER .02,.23,.24,.42,.59,.67,1.04,1.22,1.47,1.68<Z>

46

Let us abbreviate FUNCTION as FCT in this ca se and define our model:

*FCT X(T)=(AO*K1/(K2-K1»*(EXP(-K1*T)-EXP(-K2*T» *FCT B(T)=AO*(1-EXP(-K1*T»-X(T) *A0+200;K1~.05;K2~.0001

*QUIET FIT(K1,K2),X TO DX,B TO DB CONSTRAINTS NAKE=(NONE)? KAK NUKBER OF ITERATIONS=(3)? 10 CONVERGENCE FACTOR FOR SUK OF SQUARES=(.001)?

FINAL PARAMETER VALUES:

.511870@-1 -.161445@-3

CONVERGED

NORMAL ERROR STANDARD ERRORS:

.511935@-2

.335215@-3

RHS WEIGHTED DEVIATION ERROR=13.3619 FINAL WEIGHTED SUM OF SQUARES=3213.72 #OF ITERATIONS USED=2

DEPENDENCY VALUES:

.898002@-1 K1

.898002@-1 K2

We will not plot the observed and fitted values for X(t) and B(t) for

this run because there are several things wrong with it. If one plots

them, one will find that with the above estimates of K1 and K2' X(t) is

well estimated but of course K2 is negative and so are the expected

va lues of B(t). In other words, we have a completely meaningless fit as

indicated by the negative value for K2' high standard error for K2' and

the large RHS error. What happened here is that the total SS is

completely dominated by the SS associated with X(t) because of its much

larger values. As we will find out, putting a constraint on K2 to force

it to be positive does not help in this situation: *CONSTRAINTS Z

TYPE <Z> AFTER THE LAST CONSTRAINT !Kl>O !K2>0 !<Z> *Kl~.05;K2~.0001

*QUIET FIT(K1,K2),X TO DX,B TO DB CONSTRAINTS NAKE=(NONE)? Z KAK NUKBER OF ITERATIONS=(3)? 10 CONVERGENCE FACTOR FOR SUM OF SQUARES=(.001)?

FINAL NORMAL ERROR PARAMETER VALUES: STANDARD ERRORS:

.518536@-1 .526175@-2

.125056@-11 .338089@-3 LAGRANGE MULTIPLIERS:

.000000 -256864.

CONVERGED RHS WEIGHTED DEVIATION ERROR=13.4478 FINAL WEIGHTED SUM OF SQUARES=3255.19 #OF ITERATIONS USED=2

DEPENDENCY VALUES:

.802379@-1

.802379@-1 K1 K2

47

The linear constraint was active on K2 as indicated by the Lagrange

multiplier but it did not improve the situation any from before. Let us

now try a weighted regression with the same initial estimates and see

what happens:

*K1~.OS;K2~.0001

*QUIET FIT(K1,K2),X TO DX WITH WEIGHT EWT(DX),B TO DB WITH WEIGHT EWT(DB) CONSTRAINTS NAKE=(NONE)? KAX NUKBER OF ITERATIONS=(3)? 10 CONVERGENCE FACTOR FOR SUK OF SQUARES=(.OOl)?

FINAL PARAMETER VALUES:

.432368@-1

.107473@-3

CONVERGED

NORMAL ERROR STANDARD ERRORS:

.S8293S@-2

.497904@-S

RKS WEIGHTED DEVIATION ERROR=1.59324 FINAL WEIGHTED SUH OF SQUARES=4S.6914 gOF ITERATIONS USED=3

DEPENDENCY VALUES: .889717 .889717

K1 K2

The standard deviations of the parameters, the RHS error, the fitted

curves (Figure 8) and the residuals (Figure 9) all indicate now reason-

able fit to the data. This example indicates that when there are

extreme components of the total SS, no matter how good the initial

estimates are, one may end up at an incorrect minimum unless appropriate

weighting is used for the regression. For this particular example, one

may be able to do even better with a different weighting model than the

one used here. It also demonstrates that a considerable change in one of

the parameters may have very little-to-no effect on the other parameters

when there is considerably high dependency values as in this case between

the parameters.

Example 3: Finally let us examine another example as in Equations (15):

dX1/dt=-k1xl

dX2/dt=k1x1-k2x2

dX3/dt=k2x2-k3x3 (15)

The da ta for x3 as a function of t are given by the matrix in Table 3.

TABLE 3: Table of values for Example 3, taken from Knott (17)

t x3(t)

0 0.000 1 0.193 2 0.288 3 0.214 4 0.111 5 0.094 6 0.033

48

200

2211 0

0 0 0

2110

0

1711

0 >< 110 0

0

1211

100

711 0

10 0 10 20 JO 40 10 10 70 10 10 100 110

11YE

1.711

1.1

1.211

'"

o o

.I

l1lIE

FIGURE 8: Observed and converged fitted values for Example 2

49

. 0 0

0

• 0

10 0 .. l!s 0

I 0

0

0 -10

1IIE

.1

0

.1111

.l1li

ID All 0 0

l!s 0

I o.aao

0 0 -- 0

0 0 -.l1li

-.II1It-

_1 0 I I I

10 • • 40 110 10 711 10 10 1110 110

'IIIIE

FIGURE 9: Residual plots for Example 2

50

k1 was known from independent experiment. To obtain good initial esti­

mates of k2' k3 and A, all three parameters were varied from 0 to 10 by

an increment of .5 and points were generated on the SS surface as in

Example 1. After sorting, the approximate minimum SS was found to be

where k2=.5, k3=1 and A=l. These were then used as the initial estimates

for the curve fitting:

*H COL 1"'0:6 *H COL 2"'READ(TTY)

ENTER <Z> AFTER LAST NUMBER 0,.193,.288,.214,.111,.094,.033<Z>

Define the differential equations for the model:

*FCT Xl DIFF T(T)=-Kl*Xl *FCT X2 DIFF T(T)=Kl*Xl-K2*X2 *FCT X3 DIFF T(T)=K2*X2-K3*X3

The initial conditions are defined as follows:

*INITIAL Xl(O)"'A;INITIAL X2(0)"'O;INITIAL X3(0)"'0 *Kl ... l;K2 .... 5;K3 ... 1;A ... l *QUIET FIT(A,K2,K3),X3 TO M

CONSTRAINTS NAME=(NONE)? MAX NUHBER OF ITERATIONS=(3)? 10 CONVERGENCE FACTOR FOR SUH OF SQUARES=(.OOl)?

FINAL PARAMETER VALUES:

1.08716 1.06827 1.12419

CONVERGED

NORMAL ERROR STANDARD ERRORS:

3.65289 3.42906 3.75298

RMS WEIGHTED DEVIATION ERROR=.169316@-1 FINAL WEIGHTED SUH OF SQUARES=.114672@-2 #OF ITERATIONS USED=4

DEPENDENCY VALUES:

.999863

.999113

.999832

A K2 K3

Let us now integrate the differential equations with the estimated

va lues for A, k2 and k3:

*HK"'INTEGRATE(XI DIFF T,X2 DIFF T,X3 DIFF T,0:6)

The first column of matrix HK will have the values for t between 0 and 6

with increment of 1; the other six columns will contain the va lues for

the dependent variable xi (even columns of HK) and their respective

derivatives (odd columns) for each value of t. A comparison of the

estimated values of x3 (column 6 of HK) and the observed values did not

reveal any lack of fit of the model. However, the standard deviations

of the parameters were too large casting some doubt about the curve

51

fitting. Several other attempts were made to obtain better estimates of

the parameters with different combinations of initial values. They all

produced essentially similar SS values with significantly different

parameter estimates, all associated with very large S.D.s as in this

case. This example demonstrates a practical situation as described

under Dependency Values. Let us see what improvement we can make on

these parameters with weighted regression:

*K2+-.5 jK3+-1 jA+-1 *QUIET FIT(A,K2,K3),X3 TO H WITH WEIGHT EWT(H)

CONSTRAINTS NAHE=(NONE)? HAX NUHBER OF ITERATIONS=(3)? 10 CONVERGENCE FACTOR FOR SUH OF SQUARES=(.001)?

FINAL PARAMETER VALUES:

1.56256 .794497 1.59449

CONVERGED

NORMAL ERROR STANDARD ERRORS:

1.07818 .330061 1.03982

RHS WEIGHTED DEVIATION ERROR=.489240 FINAL WEIGHTED SUH OF SQUARES=.957422 gOF ITERATIONS USED=5

DEPENDENCY VALUES: .994159 .932693 .994056

A K2 K3

The integrated values of x3 as a function of time are plotted along with

the observed values in Figure 10. Even though these values do not

necessarily show any improvement over the previous fit, the residual

plot (Figure 1~) as weIl as the standard deviations indicate improvement

~r-------~------~------~------~------~------~----~

o

I

FIGURE 10: Observed and converged fitted values for Example 3

52

0 .oz

0 .DI

I a.ao 0 0

0 -.111

-.oz 0

-.G3 0 2

11IIE

FIGURE 11: Residual plot for Example 3

in the curve fitting. As a final attempt, the convergence factor was

changed to .0001 and .00001 wi thout any numerical change in any of the

computations. A closer look at the observed and fitted va lues for x3(t)

makes one believe that a better design could be suggested based on this

exercise for future experiments. For example, one probably should obtain

more information during the interval 0-2 where there is nonmonotonicity

in the curve. This should enable one to obtain more precise estimates

of the three parameters. In any case, this example once again

established the need for appropriate weighting of the observations.

DISCUSSION

We have so far discussed some methods of efficient modeling and

demonstrated their effectiveness with examples. I do not, by any means,

intend to imply that the methods will work under all circumstances. If

a model is improper, if there is degeneracy in the model or data, if it

is overparameterized, or if there are insufficient data, none of the

methods described will work. What is needed under these conditions is a

realistic understanding of the physical system under consideration and

possible redesigning of the experiments. Modeling is an iterative

dynamic process where a mathematician or a statistician and an experi­

menter actively interact to pursue the true nature of a system. Box and

Hunter (6) stated:

53

In the iterative process of model-building the experimenter, who can call upon pertinent technical knowledge that the statistician will in general lack, plays a vital role in modifying the model as the nature of its defects are revealed. Depending on the nature of the modification the calculation may go forward using the same data or further appropriate data may be required.

An improper model is generally obtained when a modeler simply picks

up a set of data from a published report without having in-depth under­

standing of the system under study and without having any appreciations

for the experiments which produced the data set in the first place. The

remedy for the problem is very simple. Interact with the experimenter

and try to und erstand the problem as well as the system. A model may be

correct but visibly "wrong" because of limitation of available data. An

example is when there is a fast absorption phase, yet since no observa­

tions have been made at very early time periods, there is severe

discrepancy between the observed and the fitted curves. This can be

easily solved by new experiments where observations are made during that

interval, if at all experimentally feasible. This also falls under the

problem of insufficient data. Alternatively, the absorption rate should

be determined by some other means.

Degeneracy is a common phenomenon in modeling. Bard (3) discusses

several examples with such problems. The consequence of degeneracy is

that some or all parameters or linear combinations of them are ill­

determined even though the residuals are small. Following Bard (3) let

us construct two such pharmacokinetic cases. The first is given by

Equation (16):

x(O)=xo (16)

where k1 and kZ are rates of elimination through two different routes.

Given only time observations on x(t), it is not possible to estimate k1

and kZ separately. This consti tutes degeneracy in the model. The

solution is to measure either kl or kZ from independent experiments or

make k1+kZ=k, a lumped parameter.

Equation (17):

The second example is given by

(17)

where xZ and x3 are two other components whose model descriptions are

not important for our discussion. In the above model measurements are

54

made on x2 and x3 to determine xl and estimate both k2 and k3 . However,

as it turned out, all va lues of x2 were approximately equal to the

values of x3 during the time interval measurements were made. Assuming

then x2=x3, the model can be rewritten as:

which has the same problem as in the case of Equation (17). In this

particular case, the degeneracy in data reduces to degeneracy in the

model. Even though in theory it is conceivable that one can take

measurements at different intervals where X2 will not be equal to X3,

from a modeling standpoint it may pose serious problems.

When a model is overparameterized the situation can be helped some­

times by removing the nonlinearity of some of the parameters by a

factorial or fractional factorial analysis of the constants estimated

from a simpler model (6). This requires expertise in statistical

methods. An alternative may be to lump some of the parameters that are

less important. Granted such a compromise is not a replica of the

system under study (after all, which model is?), it will still provide

most of the pertinent information from the model. Once again, modeler­

experimenter interaction is absolutely essential in this decision. If

the overparameterization is due to insufficient data, the solution may

be much simpler: perform additional experiments.

An approximate F-test can also be done to investigate the overpara­

meterization. This same test may be suitable to investigate whether

extra compartments should be included in a model. According to this,

one defines an F'-statistic as follows (18):

F'

where SSp

SSq

N

[(SSq-SSp)/SSp ] [(N-p)/(p-q)]

SS for the model with p parameters (more complex)

SS for the model with q parameters, q < p

Total number of observations

(19)

Under some general conditions, F' is F distributed with p-q and N-p

degrees of freedom. When there is significance at some low level, one

accepts the more complex model. The test is exact for linear models but

provides approximate guidance for nonlinear cases. Another computation­

ally rather complex but mathematically more elegant and appropriate way

of performing the same task is to make use of the eigenvalues and

55

eorresponding normalized eigenveetors. A eomplete diseussion of this

method is made by Fleteher and Shrager (13).

Additional experiments are also warranted wen the residuals are

large but otherwise aeeeptable but the parameters have large S.D.s

(ill-determined), although redueing the sampling errors may solve the

problem in some eases. Tbe rule of thumb is that the S. D. deereases

approximately as I/n%, i.e. a 10-fold improvement in parameter estimate

will require about a 100-fold inerease in the number of experiments

(3). If time and expenses are serious limitations, one may have to

deeide on how large an error one will tolerate.

Hodeling with ehemieal rate reaetions sometimes eneounters eomplex

numerieal and analytieal problems. One of them, stiffness, was mentioned

under the short diseussion for HLAB. When reaetions or exehanges involve

simultaneous fast and extremely slow processes, the integration method

is dominated by the fast proeess. Under these eonditions, standard

integration algoritluns may fail to produee solutions. As mentioned

earlier, HLAB has effieient routines to handle stiffness. Oeeasionally

the integrator may eneounter initial stiffness due to improper initial

estimates of the parameters. Under these eonditions, one should make

eareful use of the teehniques diseussed earlier to obtain good initial

estimates.

A major problem in modeling with differential equations involving

dynamie systems is instability of the system. Rumerieal instability

will often arise due to poor initial estimates. It ean be reetified by

either reparameterization or by the methods deseribed earlier.

On the other hand, the instability ean be inherent to the system,

i.e., analytieal under eertain ranges of parameter values. Depending on

these values, the system may diverge to infinity, or oseillate or may

enter limit eyeles. These phenomena will produee serious problems or

failures in modeling such systems. Bard (3) diseusses this problem in

detail. For time invariant systems, one ean, of course, easily inves~i­

gate the stability behavior analytieally (28). Problems arise wen the

system is not time invariant or eontains partial differential equations.

Fortunately, most standard pharmaeokinetie problems ean be posed under

the former framework. Let us examine a simple example deseribed by Bard

(3) which may involve instability:

56

dx

dt -kx, x(O) (20)

The solution of Equation (20) is stable for k~O and unstable for k<O. A

wrong initial estimate of k in some cases may push the curve fitting

algorithm in a wrong "crater" where k<O, which will make the system

unstable. If one knows that the system should be physically stable,

then one should constraint k~O for the curve fitting purpose. As the

model becomes more complex, finding such constraints may become more

difficult. Once again modeler-experimenter interaction becomes

essential to formulate such constraints which would make the system

stable. As Bard (3) points out, an ironic situation lIlay arise with

equations that are "too stable". Let us examine his example:

dX1/dt=-k1X1+k2X2,

dX2/dt=k1x1-k2x2'

X1(0)=X10

x2(0)-x20

The solutions of Equations (21) are:

x1=[(x10-kx20)/(1+k)]exp[-(k1+k2)t]+k(x10+x20)/(1+k)

x2=[(kx20-x10)/(1+k)]exp[-(k1+k2)t]+(x10+x20)/(1+k)

(21)

(22)

where k == k 1 /k2 . Assigning initial estimates for k1 and k2 too large

will make the exponential terms negligibly small at the smallest interval

when one may measure Xl and x2.

relations will hold:

If that is the case, the following

x1=k(x10+x20) I (l+k)

x2=(x10+x 20)/1+k) (23)

It is obvious from Equations (23) that all separate knowledge regarding

k1 and k2 is lost as a result and what is left is their relative

strength, i. e., the ratio k. A least squares algorithm will then only

estimate k, not k1 and k2 separately from Equations (21) or (22). Once

again, this is a type of degeneracy which we have already discussed.

The lesson we learned from these two examples can be best stated in

Bard's terms (3):

It seems clear then that we are most likely to avoid both instability and overstability if we start out with very small values of any unknown parameters which are rate coefficients. This gives us the best chance of obtaining solutions whose magnitude remains reasonable throughout the time intervals for which observations are available, and which are sensitive to the values of the parameters.

57

Finally, a few points should be made regarding modeling with repeated

experiments as well as experiments where observations are made on several

compartments. As has been illustrated by several investigators (3,18),

these situations should be handled by simultaneous estimation. Even

though the apparent fit may seem to be better with individual curves,

statistical criteria, in the long run , will indicate that simultaneous

estimation, whenever possible, provides better parameter estimates and

more meaningful description of a system under investigation.

Before I end this discussion I must point out two important aspects

of modeling without going into details. The first is the problem of

independent variables subject to experimental errors. Since in most

pharmacokinetic experiments the independent variable is time which, one

hopes, is correctly read, this is not a problem. Bard (3) discusses

methods to handle the situation if it arises. The solution is not simple

even for the linear case; however, my feeling is that one should be able

to handle it by appropriate weighting. The second is the question of

optimal designs for different models. Optimal designs are determined by

the purpose of the model and some prior knowledge about the system under

study. Huch of the work in the nonlinear field is due to Box and his

associates as cited earlier (5,6,12). Bard (3) has an excellent discus­

sion based on the works of the above mentioned investigators. Suffice it

to say here that the design which may be optimal for parameter estima­

tion may not be optimal for prediction purposes or model discrimination.

REFERENCES

1. Hartley, H.O. (1961) The modified Gauss-Newton method for the fitting of nonlinear regression functions by least squares, Technometrics, 3:269-280.

2. Draper, N.R. and H. Smith (1966) Applied Regression Analysis, Wiley, New York.

3. Bard, Y. (1974) Nonlinear Parameter Estimation, Academic Press, New York.

4. Berman, K. and R. Schoenfield (1956) Invariants in experimental data on linear kinetics and the formulation of models, J. Appl. Phys., 27:1361-1370.

5. Box, G.E.P. and H.L. Lucas (1959) Design of experiments in non­linear situations, Biometrika, 46:77-90.

6. Box, G.E.P. and W.G. Hunter (1962) A useful method for model­building, Technometrics, 4:301-318.

7. Berman, K., K.F. Weiss and E. Shahn (1962) Some formal approaches to the analysis of kinetic data in terms of linear compartmental systems, Biophys. J., 2:289-316.

8. Box, G.E.P. and W.G. Hunter (1965) The experimental study of physical mechanisms, Technometrics, 7:23-42.

9. Kittrel, J.R., R. Kezaki and C.C. Watson (1965) Estimation of parameters for nonlinear least squares analysis, Indust. Eng. Chem., 57:18-27.

58

10. Kittrel, J.R., W.G. Hunter and C.C. Watson (1966) Obtaining precise parameter estimates for nonlinear catalytic rate models, A.I.Ch.E. J., 12:5-10.

11. Kittrel, J .R., W.G. Hunter and R. Mezaki (1966) The use of diag­nostic parameters for kinetic model building, A.I.Ch.E. J., 12:1014-1017.

12. Box, G.E.P. and W.G. Hili (1967) Discrimination among mechanistic models, Technometries, 9:57-71.

13. Fleteher, J.E. and R.I. Shrager (1973) A User's Guide to Least Squares Model Fitting, Technical Report No. 1, Division of Computer Research and Technology, National Institutes of Health, Bethesda, Maryland, U.S.A.

14. Berman, M. and M.F. Weiss (1978) SAAH User's Manual, NIH-78-180, National Institutes of Health, Bethesda, Maryland, U.S.A.

15. Knott, G.D. (1983) MLAB- An On-line Modeling Laboratory- Beginner's Guide, National Institutes of Health, Bethesda, Maryland, U.S.A., 2nd Edition.

16. Knott, G.D. (1979) MLAB-An On-line Modeling Laboratory-Reference Manual, National Institutes of Health, Bethesda, Maryland, U.S.A., 8th Edition.

17. Knott, G.D. (1981) MLAB-An On-line Modeling Laboratory-Applications Manual, National Institutes of Health, Bethesda, Maryland, U.S.A., 3rd Edition.

18. Thakur, A. K. (1983) Some Statistical Principles in Compartmental Analysis in J. S. Robertson Ed. Compartmental Distribution of Radiotracers, CRC Press, Boca Raton, Florida, U.S.A., pp143-176.

19. Darvey, I.G. and E.J. Walker (1978) A procedure for obtaining initial estimates of parameters appearing in steady-state rate or equilibrium binding equations, Canad. J. Biochem., 56:697-701.

20. Thakur, A.K. and D. Rodbard (1979) Graphical aids to interpretation of Scatchard plots and dose-response curves, J. Theoret. Bioi., 80:383-403.

21. Thakur, A.K., M.L. Jaffe and D. Rodbard (1980) Graphical analysis of ligand binding systems: Evaluation by Monte-Carlo studies, Analyt. Biochem., 107:279-295.

22. Lapidus, L., and T.I. Peterson (1965) Analysis of Heterogeneous Catalytic Reactions by Nonlinear Estimation, A.I.Ch.E. J., 11:891-897.

23. Hartley, H.O. (1964) Exact confidence regions for the parameters in nonlinear regression laws, Biometrika, 51:347-353.

24. Anscombe, F.J. (1961) Examination of Residuals in Proceedings of the Fourth Berkely Symposium on Mathematical Statistics and Probability, 1:1-36.

25. Anscombe, F.J. and J.W. Tukey (1963) The examination and analysis of residuals, Technometries, 5:141-160.

26. Scheffe, H. (1959) The Analysis of Variance, Wiley, New York, p14. 27. Anscombe, F.J. (1960) Rejection of outliers, Technometries,

2:123-147. 28. Pontryagin, L.S. (1962) Ordinary Differential Equations, Translated

from the Russian by L. Kacinskas and W.B. Counts, Addison-Wesley, Reading, Massachusetts, U.S.A.

Acknowledgment: The author expresses his sincere gratitude to Dr. Gary

D. Knott of the Univeristy of Maryland for discussion and review of this

work, as well as kind permission for the use of examples from the MLAB

manuals. Sincere thanks are also due Terry Horner for her careful

preparation of the manuscript, and Anna Mann for the graphics.

59

MATHEMATICAL FOUNDATIONS OF LINEAR KINETICS

Aldo Rescigno

Section of Neurosurgery, Yale University School of Medicine, New Haven, CT 06510

Present address: Institute of Experimental and Clinical Medicine, University of Ancona, Ancona Italy

O. INTRODUCTION

The following is the summary of aseries of lectures I presented for a number of years to graduate students interested in Mathematical Modeling in Biology. The aim of those lectures, as weIl as the present discussion, is to give investigators an idea of the range of possibilities for more quantitative grounds behind the results of their investigations.

The analysis presented here has no pretence of being exhaustive, nor the bibliography of being complete. Interested readers are encouraged to develop their own methods and to explore new avenues for the solution of their particular problems, using these discussions as background.

1. LINEAR INVARIANT SYSTEMS

Consider a particle in a living system and suppose that it can be recognized in two different states of the system, where by state we mean a particular location or a particular chemical form, or both. If one state is aprecursor of the other (not necessarily the immediate precursor). then we can study the relationship among event ~(the particle leaving the precursor state), event ~ (transition from precursor to successor state), and event ~ (presence of the particle in the successor state).

For any t and -c: such that O~'C~t, call A(-r:)d1:: the probability ofAin the interval of time (1::, "'t:+d-r) and C(t) the probability of ~ at time t. Suppose now that ~depends only on the interval of time separating J't and te., so that we can call now B(t-~) the conditional probability that the particle is in 'e.at time t if it left c/., in the interval (1:, --c+d'l:).

61

The product A(~).B(t-~).d~therefore is the absolute probability of c.A.at a time between"t: and "'C+d"t and of ce at time t. By integration of this product we must obtain the probability of C6 at time t irrespective of"'<, Le.

O'D

(1-1) fA("l:)B(t-"t)d-1:= C(t). o

This is the weIl known conyolution integral representing the relationship among the variables of a linear, invariant system.

By linear system we mean that two different solutions of equation (1-1) can be added to give a new solution~ in fact if a solution of equation (1-1) is given by Al(t). Cl(t). and another one by A2(t). C2(t). then a third solution of equation (1-1) is Al(t)+A2(t). Cl(t)+C2(t). as can be easily verified.

By invariant system we mean that a solution does not change if the time origin is changed. In fact suppose that A(t). C(t) is a solution of equation (1-1). and consider the new function

Al (t) o for O~t<to A(t-to ) for t~to'

For this function ~

JAl (1:)B(t-"C)d1: = o

fc "~A ("(-to)B (t-"C) d "'t'

= .f.~-~(6"")B (t-to-6"')dD~ o

using now equation (1-1). t" r Al (1:)B(t-1:)d1:

o o for O~t<to

C(t-to ) for t~to'

i.e. A(t) and C(t) are shifted along the time axis by the same quantity.

These two properties, i.e. linearity and invariance, of equation (1-1). constitute the theorem of superposition.

If we think of an experiment where a very large number of identical particles is used, then the numbers of particles present )n the precursor and in the successor states are good estimators of functions A(t) and C(t) respectively. Function B (t) represents the probabili ty that a particle that left.;l(, at time zero will still be in % at time t~ therefore in a hypothetical experiment where all identical particles left the precursor near time zero, the number of particles found in the successor will be given by B(t).

2. CONVOLUTION ALGEBRA

Manipulation of the convolution integral is considerably simplified by the use of the operational calculus as developed by Mikusinski (1959).

If ~ is the class of all real-valued, defined and

62

continuous functions of the variable t~O, we shall use the symbol {f} to represent the function f of class~, while with f(t) we represent the ~ of that function for a particular value t of the independent variable.

We define the operations of addition and multiElication of functions by

(2-1 ) {f~ + fg) tt+g1, ~

(2-2) ffl fgS t (f(1:)g(t--r)dL}.

It is easy to verify that the addition is closed, i.e. the sum of two functions of class ~ is always a function of class ~~ it is commutative, i.e.

and it is associative, i.e.

There exists an additive identity, i.e.

and an additive inverse, i.e.

The multiplication of functions is also closed, commutative and associative~ in fact the convolution of two functions of class([ is a function of class «' ~ the commutativity of the multiplication is proved by writing

t:- 0 l: [f(l:)g(t-1:)d1: = J.f(t-6'")g(6")(-d6") = r g(1:)f(t-""t)d"'t. ~ t 0

To prove the

then write

Finally, distributive

it is easy to prove that multiplication is with respect to addition, i.e.

t f1.({g1+ihJ) = tf}.[g} + stfl·th}.

The functions of class ~ with the two operations (2-1) and (2-2) therefore constitute a commutative ring.

It is also useful to define the powers of a function, thus: 1. _-tl ""-i f1 = t fS.tf1, lff = [f{.[fi ' n=2,3 ....

63

3. QUOTIENT OF FUNCTIONS

According to the theorem of Titchmarsh (1926), the product {f3 .{ g\ is identically zero if and only if !Lf! or [g1, or both, are identically zero.

Now we can define the quotient of two functions

{f1/ tg~ = i h1 if i g} f (O~ and if a function {ht exists such that

{f"j = fg}.[hj.

If the quotient of two functions exists, it is unique: in fact suppose that we can find a second function f hll such that

subtracting the last two identities from each other,

and for the distributive property,

[9~.(fh\-fhl~) = tol· But from our hypothesis {g1 f fo~. therefore from the theorem of Titchmarsh it follows that th} and fhll are identical.

Not for all pairs of functions does the quotient exist. To make this operation always possible, with the only restriction that the divisor be different from zero, we define the operator, represented by two functions separated by a bar, with the fo11owing defining properties:

(3-1 )

(3-2)

(3-3)

{f}/fg1 ='i'f!/f+~ if and only if {fI·i'-/') = tg1.)'fJ,

tt~ {~\ ffLittt + {g) Al,",' fgl *Tt} ':: f g\ • t.'l't '

U\ ~t..fl ff~ • ftf1 t91· ffl ~ {g~·i+~ •

Definition (3-1) shows that the same operator can be written in different forms: for instance {2t)/h~ and lt'i.!/~t~ are the same operator because [2t\ .lt1 = Pl. {tt5 .

Definitions (3-2) and (3-3) are formally identical with the addition and multiplication of rational numbers. It is easy to verify that they are commutative, associative and distributive, as for the functions of c1ass~. Furthermore, given any two operators f fl/fgl and tc.ff/itl, with ffl HO}, we can always find their quotient: in fact

if}/ [gl [fs. [4\ t'f}/l+t :: ~9~ ·lft '

as can be proved by using definition (3-1).

64

The operators thus defined with the three operations (3-1)­(3-2). (3-3) constitute a field. We can operate on them exactly as on ordinary fractions. For example from the definition of equality,

(f3.{t{') {f\ {gl'{fl ::: 19f ,

i.e. numerator and denominator of an operator can be divided by the same function, as in the reduction of ordinary fractions.

From the definition of a quotient,

where the left hand side is a function: the right hand side also is a function, but it can also be looked at as an operator. It follows that any function can be looked at as an operator, but not the other way around: in other words the ring of functions is embedded in the field of operators.

4. NUMERICAL OPERATORS

Observe that

ta.f}/[ft = {a.gl/fg',

where a is a constant and f, gare arbitrary functions: this operator, being independent on the function {f~, can be represented simply by the letter a, i.e. we put by definition

{a. f1/tf~ = a,

and call it the numerical operator.

It is easy to verify that

a.{f1 = [a.f~:

in fact, by definition, I:

a. [ft i a .tf}.l f} = ia.f~. tft = 2 f a.f(,,()f(t-1:)d1:J;~tff _____ 0

ff~ [~t

= fCf('"t).af(t-"t)d-c1!i:ft = E'f~.\af}/l'-f} = [af1.

The numerical operator has the important property o=t01: in fact

Sum, product exactly the same real numbers. In

{a.f~ (b.g\ ---T -:::.

t fj fgJ

and quotient of numerical operators have properties as sum, product, and quotient of fact, the sum of two numerical operators is

~ ~ [a.fl ("1:)g (t-l:.)dt::f + [ b.{f ('r)g (t-'qd t~ = a+b;

~ t a. f f ('1:)g (t-1:.)d1::1 o

65

the product is E:

tao f' [b·gS \.a.b.J.f (1:)g (t-'T")dt} -- --=

f r~ ("\")g (t-1:)dt ) = a.b~

{fl fgl CI

the quotient is f;-

{a.f} {b.g} f a. {f (~)g (t-1:')d"\"1 /

{b.rf(~)g(t-~)dt} a/b~

{f} {g} 0

5. DIFFERENTIAL OPERATOR

The inverse of the operator tf}/[g!, with [f} + 0, is the operator (g}/{f\. Obviously the product of an operator by its inverse is the numerical operator 1.

Because of identity b

{ls .iff =Hf("t)d\::~ "

we can call the constant function [~, integral operator. Call s its inverse~ then

[l\.S = 1.

Now consider a function f(t) having a derivative f'(t)~ we can write

I: If ' ("'C.)d""C = f(t) - f(O) o

or {l~ • tf ,\ = t f} - ~ f (0)1 •

Divide both sides by [1\ .

(5-1 )

{f} / f 11 - t f (0 )l / {I L

s.tf! - f(O).

The derivative of a function can thus be expressed in terms of the original function, its initial value, and the operator s. For this reason s can be called differential operator.

If fft does not have a derivative, then the expression s.tf~ - f(O) is not equal to any function, but it is a weIl defined operator with a number of properties formally similar to the properties of a derivative~ we can conventionally call it nderivative of {f1 ".

If {ff has higher derivatives we can write

t f"1 = s. {f ')- - f 1 (0 )

= s2 [f! - s. f (0) - f 1 (0 ) ~

and in general

66

6. FUNCTIONAL CORRELATES

From the previous definition,

(6-1) h"} = l/s;

thence, multiplying both sides repeatedly by [r~ .

(6-2 ) s.: q = 1/s2,

tt2} = 2/s3 ,

and in general,

(6-3)

From the identities

d eoit/dt = o<.e"t, e O

we get

[ <X' • eo! t J s.{e<>lq - 1,

thence,

(6-4 ) tei t J = l/(s-o().

From the identities

d sin rt/dt r·cos~t,

d cos~t/dt -r- sin~t,

we get

t ~. cos rt~ s. tSin~t1 '

t-~. sintt) s. tcos~t1

thence.

(6-5) r sin~t1 ~ 'l. t. = /(s-r)·

(6-6 ) t cospt! = s/ (s - ) . 'L r1.

1,

sin 0 0,

cos 0 1.

- 1,

Note that all expressions thus found represent the fact that the functions listed in them are sOlutions of differential equations with given initial conditions. Thus from the differential equation

af n (t) + bf' (t) + cf (t) = 9 (t )

with initial conditions

f(O) = p, f' (0) q,

we get

67

thence

f fl = tg\ + r (as + b) + aq as + 5s + c

For instance, identity (6-5) corresponds to the differential equation

f"(t) + r~f(t) = 0

with initial conditions

f(O) = O. f'(O)=L

Similarly identity (6-1) integral of the differential condition f(O) = 1. Identity integral of the differential conditions f(O) = 0, f'(O)

simply means that {li is the equation f'(t) = 0 with initial (6-2) shows that {tl is the equation f"(t) = 0 with initial L

7. DISCONTINUOUS FUNCTIONS

So far we have considered operators defined in terms of functions of class ~. Now consider the class IK of real functions f(t) defined for any t~O, such that

l: f. f ('t:) d'r o

is of class C. Class IK includes all functions of class <C, plus those having a finite number of discontinuities in any finite interval of t such that

t-

fo If ('t:)~"l;. is finite in the whole domain of t.

Thus if f(t) is of class /Kwe can write l:

tff = iff("'-)d1:)/{ll· C>

Le. any functiQn of class IK can be written as the ratio of two functions of class ~; in other words the functions of class 11< are embedded in the field of operators defined by functions of class <f •

As an example of a function of class/K consider the jump function

HA(t) = 0 for O~t<A = 1 for t~~.

Define the translation operator

for any function f(t) we have

S.~H}J .~f1 l:

s. t ~H ~ (t-L) f ("1:)d"t.\;

68

for the definition of HA this integral vanishes for ~>t-~. therefore

and finally,

t-).

h~.ltS s.f1a f(t)dl:~ for t~).

o for t<A.

for t<~ for t~A.

This result shows that the operator hA shifts a function f(t) by a quantity A along the taxis and makes it zero for t<Ai thence its name.

The jump function can thus be written in operational form,

Many other functions can be written in operational form using the translation operator. For instance the gate function

can be written

for O~t<), or t>r for "'{,t~1

fG~,r1= (h",-hi)/s.

A single sinusoidal pulse

Sw(t) sin wt for O~t~"lT = 0 for t>TI

can be written as '2.. 2-fsw J = W. (l+hlT)/(s +w).

8. CONTINUOUS DERIVATIVE OF AN OPERATOR

Consider an operator f(A) depending upon the parameter A. If we can write

(8-1)

where p is an operator not depending upon A, and f l ('A.t) is a function such that 1 f l /~).. exists and is continuous for t~O and for a certain domain of A, then we can write

(8-2 )

called the continuous derivative of the operator f(~).

The continuous derivative so defined is unique. In fact, suppose that f(A) can be represented by (8-1) and also by

f(~) = q.tf2.(~7t)~i

in this case we have

69

Now we chose a function ~(t) such that

p = [911 Il'fl· q = \92.l/{fl·

with 9t(t) and 9'l.(t) functions of class~. It follows

r 9 a . tf} = r 9'2.1. \f1 , ~ ~ f >09, ("e)f, ().,t-~)dt) = tC9'2.(-r)f~(~,t-.. t:)d1:J,

t5~9 I ('r>-s1 f, (~, t-1:')d-t 1 = ([b9 '2. ('t~ f 2. (~, t-i:) dt l , o •

[9 11.1';) fJJr~ = {92.!.r f'1/dA};

finally dividin9 both sides by [fl we obtain

p.t';)f, /';)Al = q.fdft/'JAf, q.e.d.

Consider now the translation operator hÄ" We have seen that it can be written in the form

hj, = s.tHi,(t)\ •

but H>.(t) does not have a continuous partial derivative with respect to A, therefore we cannot use definition (8-1) in this case. Define the two functions

and observe that

for O~t<A for t~>'

~It

H)..(t)

~H~ (t)' = s.iH~(t)! = S2..fH~lf(t>?, tdH~~t),AAl = -fH~(t)1.

for O~t<>­for t>"A

The translation operator can alternatively be written in the form

and now definition (8-2) can be used; thence

dh~/dA -s? .iHr(t)} = _s2. .fH).. (t)f,

dh~/d>- = -so h)...

Other properties of the translation operator are

ho = 1, h>,. hr = h,\.,.!'-.

The only ordinary function with these three properties is the exponential function; therefore we can write

h>.. = exp(-~s)"

70

9- INTEGRAL OF AN OPERATOR

Consider an operator f(A) depending upon the parameter A~ if we can write

f(~) = p.ffl(A.t)],

where p is an operator not depending on A, and fl(A,t) is a function such that ~41(~,t)d~ exists for t~O and for a certain domain of A, then we can write

r~(A)d). = P-tf~l(A.t)d)., 01 cl.

called the integral of the operator f(A).

The integral so defined is unique. In fact, suppose that f(~) can be represented in two different ways, for instance

we can chose a function ~(t) such that

_<' 1/,n) p - ~91) Lir'

with 91(t) and 92(t) functions of class~. It follows

i911- ~f11 t: r \' 91 ("1:)f 1 ( A . t - T) d '"C

o

integrating both sides with respect to A and changing the order of integration,

H~l('"C)(f:l()..t-L)d)..)d'L = rr~2('l:')/[~2(A.t-1:)dA)d1:'}~ o cl. f3 d ~d

{91t·~rfl(A.t)dA1 = f92}·rrf 2(A.t)dA}, ~ ~

finally dividing both sides by tf5 we obtain ~ ~

p.rrfl(A,t)dA! = q.iff2(A,t)dA;, q.e.d •. l ~ «

For any numerical operator f(A) we can write

or

where

exp(-~s).f(A)

exp (- As ) _ f (,A)

91('A,t) = 0 = 9 (A)

S·fH>.1·g(A)

S.[91(A,t)~.

for O~t<A for t~A.

According to the definition of the integral of an operator,

but

therefore

)., {AI rexp(-~s).g(A)d~ = s.f 9l(A.t)dn~' o 0

>. [91 (A.t)dA o

for O?t< 'A,

for t>A • .,.,

71

l:-S.1r9(A)dA} = {gI for O~t<~1

S.1~~<'A)d>" ~ = 0 for t~).i

= 0 everywhere else. Ob ).1..

With the notation [ for limit J. we can write <3 'lI.,- 0 ~\

>- ......... 00 r exp(-'>..s).g(}.)d>' = {gl, o

formally identical with the definition of Laplace transform; but it must be clear that in the Laplace transform s is a complex variable, while here s is the differential operator.

10. ALGEBRAIC DERIVATIVE

Define the operation D on a function by

D ~f (t ß = t -t. f (t )5 ; D is not an operator because the ratio ~-t.f~/1fl does depend upon f(t). This operation has the properties

D(f.fl + t9)) = D~.f1 + D 19},

D(~f) . fgJ) = D{f1· f9} + {f1·D[9}, L.

tfl = t9VfhJ =? D{f} = D[91·t:hr - t91.D~h} /[h}.

The first property is obvious; to prove the second write I:

D (~f}. t9~) = D~ ff (1:)9 (t-"C)d'!::1 o !,-

= t -t)f(1:)9(t-T )d1:J ~ 0 ~

= ~ ~(-"t.)f(1:).9(t--c.)dL~ +~ (f("'C.). (-t+1::)g(t-"1:)d1:.)

= .[-t.f(t)1·'i9(t)~ + t f (t)j.{-t.9(t)}

= D {ß~ • ~ 9 1 + t f 1· D f 91 ' q. e . d.

To prove the third property write

D[q.i.h\ + {fJ.Dfh} = Dt9t

D{.fl Dt9}/[h} - t.f l·Dthl/{h1

= (0[91 - D(h\·t9~/~h))/{hl, q.e.d.

The operation D on an operator p = tfS/t91 is defined by

Dp = (D{f}.i91 - VJ.Dtg1)/f91L.;

it has the properties

72

O(p+q) = Op + Dq,

(10-1) O(p.q) Op.q + p.Dq, ~ O(p/q) (Op.q - p.Dq)/q ,

ana10gous to the properties of the operation 0 on a function, and easy to verify.

For a numerica1 operator,

in fact

Oa = 0:

Oa = o(fa}/t11) = (o~a1.~1\ - tal.otll)/[11l..

= (L-at). tIl - [a1·t..- t1) /t1~~ = ([-at:%../2) - {-at'''/21)/[11'l.. o. q.e.d.

For the differential operator,

Os = 1:

in fact

Os O~/{ll) = COl.{lJ - l.O[1~)/[11?. -i -t 1/ i111. = L q.e. d.

Using repeatedly property (10-1).

OS7... = 2.s, os~ = 3.S'2., ~ """ ... Os n.s

For the translation operator,

o exp(-)..s) = O(S.[H~\) = OS.[H>-~ + s.OtH>.\ = {H~1 + s.~-t.H>.~ t;

= s. HoH >.. ( ~ ) d L. - t. H }, 1 = s. t - A. H >-L o exp(-~s) = -~.exp(-As).

If R(s) is a rational expression in sand in the translation operator exp(-As). for the properties just shown the operator OR(S) can be computed by formal differentiation of R(s) with respect to s, as though s were a variable. This fact can be expressed by the formula

OR(s) = dR(s)/ds.

For instance from the identity (6-5) we get

{ 'l... ßt. '2... t.sin~q = 2~S/(S + I ) :

from (6-6) we get

[t.cos~tl = (s'2.. - pl)/(S'L + rl.)'2....

Many other functional correlates can be found by applying the operation 0 several times.

73

11. OPERATION T IX.

The operation T~ on a function is defined by the formula

TDllf (t») = t exp (-oI.t)f (t) 1; observe that T~ is not an operator because the ratio texp(-«t)f(t)1/[f(tH does depend upon f(t). This operation has the properties,

T.,l(Trtf~) = T.,(-+p{t1,

Totnf1 + (g1) To(tf1 + T.({g},

T.,(~f~ • tg1) = T!X~n • TD(fg},

[fr = tg-f/[h"J -=? Tol [fl = Tp([g}/T.(thJ.

The first two properties are obvious; to prove the third property wri te

c = T.,I[Jf('l:)g(t-t)d"C}

o /;-

= f exp (-elt )J.f ('t:)g (t-1:)dc: 1 (;-

= rfexp(-ä)f("t:).exp[-<i.(t-1:)]g(t-'1:)d"Cl o

fexp(-ctt)f(t)1·texp(-~t)g(t)1

= Td t f1 . T~tg1, q.e.d.

The fourth property is an obvious consequence of the third.

The operation T~ on an operator p ={fr/ fgl is defined by the formula

this operation has the four properties

T,,{ (T~p) T"-~f'

TD{(p+q) T,,{ p + Tot q,

(11-1) TD\(p.q) TI{ P • Tl( q,

Trt(p/q) T",P / To(q.

For a numerical operator,

in fact

T.{a Tttta1/Tc/tlf = fa.exp(-.lt)S/{exp(-o(t)l

a. Iexp(-o(t)~/iexp(-o(t)! = a, q.e.d.

For the differential operator,

74

in fact T/XS = Tol.l/Tolllt = l/iexp(-olt>},

and from (6-4) our thesis folIows.

Using property (11-1) repeatedly,

To( sn = (s + 0< ) n .

From identity (6-3) we can write

{tnexp(-~t) = n!/(s+~)n+l:

from identities (6-5) and (6-6),

texp(-~t)Sinrt~= W(s+d.)2 + f~2~,

'iexp(-olt)cosrt1= (s+P()/ (s+ol.)2 + ~2).

For the translation operator,

(s + o().exp(->.s).{exp(-Dl~)exp(-oit)!

= (s + o{).exp[-A(s+O()].fexp(-clt)l

T~exp(-As) = exp[-A(s+~)].

In general, if R(s) is a rational expression in sand in the translation operator exp(-~s), then

TI)( R (s) = R (s + O().

12. FUNCTIONS APPROXIMATING SOME OPERATORS

Define

thence ~

o

l~ A ~t~ ).,-t-~

1 >'+«t<oO

f/>" l (~)d"r = ° l/~ . (t-A)

1

If ~ is small we can write t

foF,\,t.('t)d"t ~ H),(t),

O?t<A

~~t~)...-t"e

>'~~<t < OCI

which is exactly correct for t~~ and for t~~+f: therefore

tl~ ·tF~,~(t)~-; tH>,)

exp(-As) ~fF"\,~(t)1, ~small.

75

Define now FAe. ... (t)

I ~ , o

= (t-A)/~Y A<t<~~~

l/E. Ä""'~t~A~ E.

- (t-).-~-~)/E:Y ~"'E.<t<>\H.-t""'t

o

thence. E:-r ~ ~ (1:)d"t' = 0

b A,"-I'" (t-X)'l.../(2~')

= ?/(2~)+(t-A-7)/~ 1-1/ (2 E)- (t-A- t) (t-}..-e-2,)/ (2E;~)

1

O~tö. - = >-<t<h,

A+f~tS}..-t~

~ofE,(tÜ't'C"f'''t

and if ~ and.., are small, i. e. ignoring the interval )., A"~,:?,

{li .~F).,i.,,'~ {H).. ~

exp(-).s) ~ fF~,E,'t (t)1, ~,-r small.

Furthermore,

dFA,l,.,/dt 0 O~t~>-

= 1/ (~.,) A<t<A+"

0 A+Ut~>'+~

-1/ (~t) ~ '* ~<t <>-"'E. ~1'

0 ~~h1'$t<CIO. ~

F",,~,~ (0) = 0,

thence

s.exp(-A5) ~ fdF~'1'/dt1, i,( small.

In a similar way we can approximate the operators s2. exp (-AS), s3.exp(-~s) ••.• : then if ~ is also very small, we get the functions approximating the operators l~ s, 52, •••

13. ELEMENTARY TRANSFORMATIONS

Given the operator

p(s) Pmsm + Pm_lSm- l + •.. + PIs + Po =

q(s) qnsn + qn_lsn- l + ... + qls + qo

with qn + 0, if m>n .. we can write

76

pes) p*(s) -- = am_nsm- n + am_n_lSm-n-l + ••• + ao +.~ __ q(s) q(s)

where p*(s) is a polynomial in s of degree less than the degree of q (s).

Suppose now that m<n~ the operator p(s)/q(s) can be written as a sum of operators of the forms

A Bs + C (s - o{)""

where A, B, e, 0<... r . Y are numerical operators, and f' v are integers. We first prove a Lemma and a Theorem.

Lemma. If

(13-1) an sn + an-l sn-l + .•. + als + ao=

bnsn + bn_lSn- l + .•• + bIS + bo ' then

(13-2 ) ao bo , al bl' ... , an bn ,

and vice versa.

In fact, multiplying both sides of (13-1) by [l}n+l,

an b\ + an-l {11 2 + •.. + aot11 n+l

thence = bnt11 + bn-ltlJ 2 + •.. + boh1n+l.

(13-3 ) (an + an-lt + •.. + ao t n/nl1 =

= {bn + bn-lt + ... + bo t n/nl1,

which implies the identities (13-2). On the other hand, if the identities (13-2) hold, equation (13-3) is valid for any value of t, and multiplying both sides by sn+l we obtain (13-1).

Theorem. If

amsm + •.. + als + ao

bnsn + ... + bIs + bo

then for any number x (real

bnxn + ... + blx

dqxq + •.. + dlx

it is amxm + ... + alx

(13-4 )

cpsP + ... + cIs + Co ,

dqsq + ... + dIs + do

or complex) such that

+ bo :f:

+ do f'=

+ ao =

o. O.

cpxP + •.• + clx + Co

dqxq + ••• + dlx + do

77

and vice versa.

In fact this hypothesis is true if and only if the identity

= (bnsn + •.• + bIs + bo)'(cpsP + •.. + cIs + co)

is true; for the Lemma this implies, and is implied by, the identities

aldo + aodl blco + bocl

a2do + aldl + aod2 = b2 co + blcl + boc2

and so forth; they are the necessary and sufficient conditions for (13-4) being an identity, q.e.d.

Now return to the rational operator

pes) Pmsm + PIS + Po = 1

q(s) qnsn + qls + qo

with qn:fO and m<n. If the polynomial

q(x) = qnxn + ••. + qlx + qo

contains a real zero ~ with multiplicity ~, i.e. it contains~ times the factor (x-~). we can write

where

p(x) A p* (x) ----- + ----~~------

q(x) (x-cOr (x-.t)I"-'.q*(x)·

q(x) = (x-t:Or.q*(x),

A = p(CC)/q*(o{);

furthermore the new rational function

p*(x)

contains in the denominator the factor (x-~) only r-l times, and the same decomposition can be repeated until the polynomial q(x) does not contain any more real zeros. For the theorem just proved we can find the operators A/(s-J)~ making the same computations on the operator p(s)/q(s) as though s were a variable.

If ~+ir and ~-i, are complex conjugate zeros of multiplicity v of the polynomial q(x). then we can write

78

p(x)

q(x)

Bx + C p(x) 2. ..; + ------111-----;:-2 "7'11'_'1 -----

[ (x - r) +, ~ ] [ (x - ~ ) + '6] . q * (x )

where 2. 2. V

q(x) = [(x-r) +.,] .q*(x),

B l/p.Im[p(p+iOljq*(r+io)J.

C Re (p (r+ i f) jq* (f+i?J)] -j j o· Im [p (r+ i 0) jq* (f+ib")]

and the same considerations as for the real zeros hold.

14. DEFINITION OF COMPARTMENT

For our present purposes we shall use an operational definition of compartment, as proposed by Rescigno and Beck (1972): RA variable X(t) of a system is called a compartment if it is governed by the differential equation

(14-1 ) dxjdt = -KX + f(t)

with K constant. R For a physical interpretation of equation (14-1), consider X the amount of a certain substance in a particular subdivision of a system, through which the concentration is uniform at any given time: that substance leaves the subdivision at a rate proportional to its total amount there, i.e. with a process of first order, with relative rate K: f(t) measures the rate of entry of that substance in that subdivision of the system from other subdivision or from outside the system. Thus equation (14-1) represents the relationship between the behavior of the precursor f(t) and the behavior of its successor X(t).

Using the operational notation introduced before, equation (14-1) can be written

-K~xi + [f}

where Xo is the initial value of X(t). or

(14- 2 ) [f)

{x1 = + Xo

s + K

We shall now compare this equation with equation (1-1) written in operational form,

(14- 3 )

We begin with the case when Xo=O: equation (14-2) is now

1 tX~ =--·\ßl:

s + K

the correlate of function {B~ from equation (14-3) is

1 t exp (-Kt )5 . s + K

79

The above function is the probability that a particle that ente red the compartment at time 0 will still be there at time t. On the other hand the correlates of ~Al and (Cl are the functions {f~ and {xl; except for a normalizing factor, they are equal respectively to the probability density that a given particle enters the compartment, and the probability that the same particle is present in the compartment.

We consider now the case (f)=O, Xo=O. Equation (14-2) is now

s + K

Xo is not a function, so we cannot call it the "probability density function" of the entrance of particles into the compartment; but it has a role analogous to that of "function" [f~. Following the considerations of section 12, we can say that Xo is approximated by a function equal to zero everywhere except in the small interval form 0 to E.., where it assurnes the value l/~.

In general we can say that, according to equation (14-2) ,

tf~ + Xo

is the inEut into the compartment, and

1

s + K {exp(-Kt)1

is its transfer function.

15. MATRICES

If the system under observation is formed by a finite number of compartments, i.e. if it can be decomposed into homogeneous and weIl mixed parts, then we can write the set of equations

-KIXI + k21 X2 + ••• + knlXn + fl(t)

k12Xl - K2 X2 + ••• + kn2Xn + f2(t)

where n is the number of compartments, Xi is the amount of material constituting compartment i, Ki the fractional rate of exit from compartment i, kij the fractional rate of transfer from compartment i to compartment j, and fi(t) the rate of entry into compartment i from external sourees. The above differential equations must be completed by appropriate initial conditions. We put

for i=1,2, ••• ,n.

80

The differential equation above can be written in matrix form

(15-1) d!jdt = !:! + !! where

is the n.n matrix of the transfer rates, with

kii = -Ki

the transfer rate out of compartment i, X is the row vector of the n variables of the system, d~/dt its~ime derivative, and ~ the row vector of input functions into the system. The initial conditions are

!.(O) = 10' where !o is the initial value of row vector ~

An important observation should be made at this point. I have indicated with kij the relative transfer rate from i to j, while a number of authors use that symbol for the rate to i from j~ in fact a casual survey of the literature has shown that each of the two definitions is preferred by approximately 50% of the authors. The reasons for my choice are four:

1. The physical meaning fo kij becomes more evident if the transfer of material from one compartment to another is shown by reading the subscripts from left to right~

2. If material is transfer red through a succession of compartments, the product of the transfer constants involved, written as astring with the second subscript of a constant equal to the first subscript of the following constant, has a particularly useful physical and mathematical meaning~

3. This definition is consistent with the notation used in the theory of Markov processes, as shown by Thakur et al. (1973)~

4. This notation has been recommended by the Journal of pharmacokinetics and Biopharmaceutics (Rowland and Tucker, 1980) .

On the other hand the reason why some authors prefer to use the symbol kij for the transfer to i from j is because in that ca se the constant kij appears in row i and column j when matrix [kij] is post-mult1plied by the column vector ~ I have just shown that the same result is obtained by pre-multiplying matrix [kij] by row vector ~

The

(15-2)

integral of equation (15-1) is t:

X(t) = Xo.exp(Kt) +rF(~).exP(K(t--r»)d-r, ~ ~ ~ o~ ~

where by definition

exp(_Kt) = I + Kt + K2 t 2/2! + K3 t 3/3! + .... .......... ~ """"'" ~

I being the n.n identity matrix. The properties of the exponential matrix exp (Ji.t) depend on the eigenvalues of li., therefore we shall spend the next few pages on the analysis of these eigenvalues.

81

Physical realizability of the system requires that

(15-3)

(15-4 )

kij ~ 0 for any i and j -::{:~ ,

Ki ~ ~j kij for any i and all values of j~i~

Hadamard (1903) has shown that these properties imply that all real eigenvalues of 1i. are non-positive, and all complex eigenvalues have a negative real part. For the presence of zero eigenvalues a further analysis of K is necessary. -

Matrix K is said to be decomposable if with a number of permutationfr of its rows and corresponding columns it can be put in a quasi-diagonal form, i.e. in the form

where ~l and !2 are two square matrices and ~is the zero matrix. When this is the case, if m.m is the size of Kl, then the n compartments can be partitioned in two different systems, one of m and the other of n-m compartments, completely independent of each other. Unless otherwise explicitly stated, we shall only consider non-decomposable matrices.

Matrix 11. is said to be reducible if, with appropriate permutations of its rows and corresponding columns, it can be put in the form

(15-5) K .....

where Jl is an m. (n-m) matrix, Q is (n-m) .m, Kl is m.m, !.2 is (n-m).(n-m). When this is the case, the n~compartments can be relabeled in such a way that the first m of them are independent from the remaining n-m.

Hearon (1963) has proved that, if 1t is irreducuble, then it is singular if and only if condition (15-4) is a strict equality, i.e. if

Ki = 1:j kij

for any value of i and for the sum extended to all values of j~i. When this is the case, the system is closed, i.e. material is transferred from one compartment to another, but nothing to the outside. It is also easy to prove that if .!. is irreducible, then it is of rank n or n-l, i.e. zero cannot be a multiple eigenvalue.

Suppose now that K is singular and reducible~ we can put it in the form (15-5)~ iJE either Al or !2 or both are further reducible, we can transform them im the same way, and proceed until we have

lil ~2 ~13 f] & _2.. Q.. K = 0 !3 l!,3m - --. . . . . . . . .

0 0 0 K 1M. 404 -. c.c.l m

82

where Kl' !2, .... Im are all irreducible square matrices. ~f any of these last matrices, say li, is singular, then zero is a simple eigenvalue of it, and the corresponding subsystem is closed; this implies that all matrices on the same row as Ki are zero; but if i<m, this implies that the system is deco~posable. It follows that if K is not decomposable, only matrix Km can be singular. ~

A(

Returning now to equation (15-2). consider the ca se

!..(t) = Q.;

if all eigenvalues of lL are real and distinct, then there exists a non-singular matrix l!.- such that

K = p-l. L. P, ~ ..c.- ...........

where ,4. is the diagonal matrix of the eigenvalues of !.! equation (15-2) can be written

(15-6)

where

with "1' A2.· •. ·· the eigenvalues of K, and diag(. .. 1 representing a diagonal matrix of the-elements indicated. Then equation (15-6) shows that the elements of vector X are linear combinations of the exponentials of Alt, A2t, -'" the coefficients of which depend on the elements of Xo; all the exponential terms have a negative exponent, i.e.-they decrease when t increase, except possibly one that can be constant.

If some eigenvalues of ~are multiple, then we can find two matrices Sand N such that

K = S + N, -- - ..... S. N = N. S, ..... "'" ..c-"""

where S is diagonalizable and 1i is nilpotent, Le. all its powers~above a certain one are void. In this case equation (15-6) can be written as

X = xo.exp(Nt).exp(St). """""" "'" ..... ~

Now matrix S can be diagonalized leading to a sum of exponentials; with all the eigenvalues of K, while

""'"" exp(!!.t) = I + ,!it + ~t2/2! + ••• + ~tm/m!

is a sum of a finite number of terms, since ßlis nilpotent, and m is certainly not larger that n; therefore the elements of ~ are a sum of exponential terms, each one multiplied by a polynomial in t of degree equal to the multiplicity of the corresponding eigenvalue minus one.

The case when some eigenvalues are complex will be treated in section 18.

83

16. DIRECTED GRAPHS

One important problem in the analysis of compartment systems is the study of the properties related to the structure of the model, i.e. of the properties depending upon the presence or absence of a connection between any two compartments and not upon the values of such connections.

The topological properties of a system of compartments have been studied by Rescigno and Segre (1964) using a directed graph (Ore, 1962). A directed graph was called reseau oriente by Sainte-Lague (1926) and graphe by Berge (1958). I shall use the term graph for brevity in this section when referring to a directed graph. A graph consists of a set of nodes, representing the compartments, together with a set of arms (or arrows) connecting the nodes and representing the transfers between compartments.

To each graph containing n nodes we can associate a square matrix of order n, called connectivity matrix. The element aij of row i and column j of the connectivity matrix is equal to 1 if there is an arm from node i to node j, and is equal to o if not.

The sum of two connectivity matrices A [bij] of~e same order is the matrix -

[aij] and B =

~ + ~ = [aij + bij]'

where the elements are added according to the rules of Boolean algebra, Le.

0+0=0, 0+1=1, 1+0=1, 1+1=1.

The product of ! and B is

where again addition and multiplication of elements follow the rules of Boolean algebra, i.e.

0.0=0. 0.1=0. 1.0=0. 1.1=1.

The power ~ of a connectivity matrix is defined by

r=2,3 ••••

finally the ,transpose AT of ~ is defined by

.e:.: = [aji]·

In a connectivity matrix a column of zeros means that the corresponding node is an initial node, a row of zeros means that the corresponding node is a terminal node. For convenience we shall consider only graphs with only one initial node; this kind of graph corresponds to systems of compartments where only the first compartment is fed from an outside source. Of course the linearity of the system implies that if several other compartments are fed from the outside, then that system can be considered to be the sum of several systems, each one with one initial node.

84

We shall call the initial node, node O. Node 0 does not corresponds to areal compartment of the system, but rather to the ideal point from where the system is fed (Rescigno, 1960).

A succession of arms such that the node ente red by each of them (except the last) is the node at which the next arm begings, is called a path. If the starting node of the first arm coincides with the ending node of the last arm of a path, that path is called a cycle. The length of a path is equal to the number of its arms. A path, including a cycle, is called simple if every arm of it appears only once; it is called elementary if every node of it is ente red only once.

A graph is called connected if there is at least one path from its initial node to any other node. In this section we consider only connected graphs. A graph is called strongly connected, or strong graph, if there is al least one path from every node, including node 0, to every other node, excluding node zero. In a strong graph there is at least one cycle.

A subgraph is a connected graph obtained by suppressing some nodes and their connecting arms from a given graph. The subgraph obtained by suppressing the initial node and the arms leaving it, from a given graph, is called its Go-subgraph. A subgraph in which each node occurs in exactly one cycle is called a linear subgraph. Each set of cycles in which each node of the subgraph occurs in exactly one cycle is called a strong component. A Hamiltonian cycle of a graph or subgraph is an elementary cycle that joins all the nodes of that graph or subgraph. For instance the graph of Fig.l has the Hamiltonian cycle (1-2-3-4-5-6-7-8-1). the two strong components (1-2-3-4-5-6-7-8-1) and (1-2-7-8-1;3-4-5-6-3). and the three elementary cycles (1-2-3-4-5-6-7-8-1). (1-2-7-8-1). (3-4-5-6-3).

The graph of Fig.2 is also strongly connected, but it is not the linear subgraph of any graph; it does not have Hamiltonian cycles or strong components, but it has two elementary cycles, (1-2-5-6-1) and (2-5-4-3-2).

A graph is symmetrie if for any arm connecting a non-initial node to another node, there is an arm in the opposite direction. The connectivity matrix of the Go-subgraph of a symmetrie graph is symmetrie, i.e. Ao = AT -4 0 .

1 2 3 4 1 2 3

8 7 6 5 6 5 4

Fig.l Fig.2

85

A graph is asymmetrie if there is no more than one arm between any two nodes; if an asymmetrie graph is strongly eonneeted, it admits one Hamiltonian eyele. Define the element-by-element produet A x B of two matriees A=[al."]"] and ..... .-.- ........ Jt=[bij] by

~ x!.. = [aij.bij];

then for an asymmetrie graph,

A x AT = o. - -- .-If in a graph some of the connections between nodes are symmetrie, the set of such connections is given by the non-zero elements of the element-by-element product A x AT.

- oe-

A lineal or catenary graph is a graph with all nodes ente red by no more that one arm and left by no more than one arm; it has one initial node, one terminal node, and one path; its connectivity matrix has no more than one non-zero element in each column and in each row.

A tree is a graph in which all nodes, except the initial one, are ente red by exactly one arm, and at least one node from which more than one arm starts; these last nodes are called roots. The connectivity matrix of a tree has no more than one non-zero element in each column and at least one row with more than one non-zero element; the rows with more than one non-zero element eorrespond to the roots of the tree.

A mammillary graph (Mattews, 1957) has a central node connected with all other nodes, in one or both directions, while all other nodes are not connected among thema Its connectivity matrix has all elements not on the row or column corresponding to the central compartment, equal to zero.

The successive powers of the connectivity matrix show the existence of paths in the corresponding graph; in fact the element of row i and column j of matrix Ar is equal to 1 if there is a path of length r from i to j; of course the diagonal elements of,h.r show the cycles of length r.

If a graph with n nodes does not contain any cycle, then there is a number r<n such that ~ = ~, and matrix A is said to be nilpotent. If ~r = Q. but -är - l '" Q, then r-l is the length of the longest path of the graph. Marimont (1959) has proved that a matrix is nilpotent if and only if every principal submatrix has at least one zero row or one zero column. As a practieal rule for finding out whether a matrix is nilpetent, i.e.the corresponding graph has no cycles, we can delete successively the rows (or eolumns) whose elements are all zeros, and the corresponding columns (or rows). If some non-zero elements are left, the original matrix is non nilpotent.

The sum of successive powers

shows the paths of length up to k. If k- is nilpotent and Ak+l = 0, then Rk shows the paths of any length. If A is not --

-. - -86

nilpotent, and k is the 1ength of the longest simple path in the graph, then adding higher powers of~ does not change Rk because it inc1udes all possib1e connections between nodes7 If we ca11 R or reachabi1ity matrix the limit of the sum above for k sUfficllent1y 1arge, then a non-zero element rij of R shows that there exists a path from node i to node j, 1.e. ~at compartment j can be reached from compartment i. Harary (1959) has shown that the e1ement-by-e1ement product of Rand its transpose, ]lx ~T, indicates in row i the nodes bEr10nging to the same cyc1e as node i.

For instance, from the graph of Fig.3,

1 o o 1 o

o 1 o o 1

o 1 o o o

R = A + A'1.. + A? ....... -

... , 1 1 o 1 o

o 1 o o o

therefore

1 1 1 1 1

1 1 o 1 o

1 o 1 1 o

1 o o 1 o

o 1 o 1 o

1 o o 1 o

o o 1 o 1

o 1 o o 1

o 1 o 1 o

o 1 o o o

~ 0, 1 1

~J

tJ

This last matrix shows that nodes 2 and 4 are on one cyc1e and node 3 and 5 on another one.

Another important app1ication of graphs to compartment analysis is the c1assification of the precursor-successor relationship (Rescigno and Segre, 1961). If in a graph there is a path from i to j, then i is aprecursor of j and j is a successor of i~ the 1ength of the shortest path from i to j is ca11ed the order of the precursor. For instance, in the graph of Fig.3. node 1 is the precursor of order one of node 2. of order two of nodes 3 and 4. and of order three of node 5.

1

4

2

Fig.3

5

3

87

Precursors of order one can be further classified into different types: a) Absolute precursor: the arm from i to j is the only one

leaving i and the only one entering j; b) Complete precursor: there is only one arm leaving i; no

cycle may enter j except from i; c) Complete precursor with recycling: there is only one arm

leaving i; j belongs to a cycle not including i; d) Unique precursor: there is only one arm entering j; e) Total precursor: there are no paths from i to j of length

more than one; if j belongs to a cycle, i belongs to the same cycle;

f) Total precursor with recycling: there are no paths from i to j of length more than one; there is a cycle in j not through i;

g) Partial precursor: there is a paths from i to j of length more than one; if j belongs to a cycle, no node of the cycle except i has an arm entering j;

h) Partial precursor with recycling: there is a paths from i to j of length more than one; there is a cycle in j not through i.

The following table shows how the different types of precursors of order one can be classified according to the values of the matrices A and R.

akj=l, some k=i

aij=l akj=O, any k=i rjkak~=~' rjkakj=l.

any =1 some k=1

aih=O. any h=j a b c

aih=l, aihrhj=O.

d f any h e s ome h=j

aihrhj=l, some h - 9 h

If aij = 0 but aihahj = 1 for some h, then i is a precursor of order two of j; classification of second order precursors according to different types is done as above. More details on the precursor-successor relationship will be given in section 20.

It is easy to find the relationship between the matrix K of a system, as defined in section 15. and the graph G of theiSame system. For instance if ~. is decomposable, then G is not connected; if K is reducIble and can be put in the triangular form (15-5), tlien !l and !2 correspond to two strong subgraph. and~ to the arms going from the nodes of the first subgraph to the nodes of the second.

Due to the importance of the strong components of a Go-subgraph, it is useful to describe a special notation devised by Caley (1861) to represent them. The symbol lijk ••• / represents an elementary cycle through the nodes i,j,k, ••• , and the symbol labe ••• lijk ••• Ivwx ••• / represents the set of elementary cycles labc ••• I, lijk ••• I, Ivwx.~ then the strong

88

components of a graph with n nodes can be represented by a string with apermutation of the first n natural numbers, interrupted by an appropriate number of bars. For instance the graph of Fig.l has the strong component 1123456781 and 11278/34561. In general, with four nodes the strong components have one of the two forms lili2i3i41 or lili2/i3i41; with five nodes one of the two forms lili2 i 3i 4i 5 I or lili2 i 31 i4 i 51·

A graphical rule for the construction of the strong components of a Go-subgraph was described by Rescigno and Segre (1965).

17. LINEAR GRAPHS

The differential equations of a system of compartments written at the beginning of section 15. in operational form become

thence

XIO + tfl!

X20 + ff2)

(17-1) {X..,:) = l/(S+K..;..).(X..,:. +if..:l) +fr .. kj..:/(S+K .• :>.lXjL any i.

This equation shows that function tXi\ of compartment i dependes on the numerical operator Xio of its initial condition, on the feeding function ~fi~, and on the functions iXj\ of all other compartments. The above equation can be represented graphically with anode for Xio+{fil, anode for each function tXj\, and anode for {Xi\, plus an arm from each of the former nodes to this last node. These arms are equal to the coefficients of the respective terms on the light hand side of equation (17-1). Thus node tXi~ is equal to the sum of all arms entering it times their nodes of departure. For instance to equation

(17-2) fX,1 = l/(S+K,).X IO + kl.l/(S+KI),~Xl'\ + k~I/(S+K,).{X3l

corresponds the graph

1/ (S+K, )

k2.1 / (s+K ,) k 3, !(S+K, )

89

Of course an equation like (17-1) can be written for each compartment of a system, and to each equation corresponds a graph. All those graphs can be combined together, because the arms entering anode of one graph will not change the values of the nodes determined by another equation. If equation (17-2) is holding with the two additional equations

tX2l = k12/(S+K2)·(Xl) + k32/(s+K2)·(X31,

f X33 = k13/(S+K3)·iXi},

then their two corresponding graphs

and

• can be combined with the first one thus

1/(S+K 1)

k\l!(S+KJ )

k~~/(s+Kz.)

•

This graph includes all information contained in the given differential equations and in the initial conditions. This kind of graph was introduced in 1953 by Mason (1953) who called it signal flow graph. It has been used in compartment analysis since 1.960 (Rescigno, 1960). I prefer the name linear graph as more indicative of its function, even though this term was used by Kirchhoff (1847) in a slightly different context.

Most of the definitions given for directed graphs are valid for linear graphs as weIl. In particular a path has a length, as in a directed graph, but also a value, equal to the product of its arms.

The fundamental property of a linear graph, as an immediate consequence of its definition, is: "Each node is equal to the

90

sum of the products of the arms entering it times their departing nodes. n A number of transformations can be made, including the suppression of some nodes, without changing the fundamental properties of the nodes left. The following four transformations were described by Mason (1953): a) Two tandem arms can be substituted by a single arm equal to

their product, and the intermediate node suppressed; b) Two parallel arms can be substituted by an arm equal to

their sum; c) An arm entering anode can be substituted by arms entering

all nodes immediately following it, each new arm being equal to the product of the original arm by the arm connecting the previous to the new end;

d) An arm starting and ending at the same node can be suppressed by dividing all arms entering that node by one minus the value of the suppressed arm.

Repeated application of these four rules lead to a much simpler linear graph that helps interpreting some of the properties of the system of compartments it represents. More details can be found in the literature (Robichaud et al., 1962; Chow and Cassignol, 1962; Laue, 1970); here I intend to show only some simple properties of the linear graphs.

Going back to equation (17-1), observe that the term

XIO + tfi~

represents the contribution to compartment i from outside the system, either as material put into it at time zero, or fed to it successively; we can call this term the input to compartment i. For simplicity consider only the case when only one compartment, say compartment i, has an input different from zero; then its linear graph has only one initial node, as defined in section 16. If the graph does not contain any cycle, then there is only a finite number of paths between its initial node and any other node. Repeated application of the first three rules by Mason leads to a graph containing exactly one arm between the source and any other node. For instance the graph

l/(S+K\ )

has one path from X 10 to tX.1 equal to 1/ (s+K, ), one path from X lO to {X'l.l equal to k I'l../ (s+K I) (s+K 2.) , two paths from X 10 to [X4~ equal to k12.k1.It/((S+K,) (s+K2.) (S+K 4 ») and

91

k 11 k ""./({S+K I) (s+K1) (S+K.») respectively, and SO forth. This graph therefore is equivalent to the graph

k,t. (s+K ,) (S+K,,)

X/~----------~-----------4[Xt.'

1/ (s+K, ) ----~----tx,1

kl2.k~ttk4" kl~k1.k".r (s+K,) (S+K~> (s+Kt,.) (S+Kf") + (S+K,) (S+K,) (s+K~) (s+K.", ------------------~------------------fx~

k,'Lk ... .,. kL)k,. (s+K,) (S+Kt.) (S+K.,J + (S+K,) (s+K,) (s+K+->

----------------~---------------fx41

(s+K, ) (S+K) ------------~------~~tx)l

In this new graph there is only one initial node, the source; all other nodes are terminal nodes; therefore any of them can be suppressed without altering the properties of the rest of the graph. This is important, because when the behavior of only one compartment is of interest, the graph can be reduced to only one arm, between the source and that particular node.

The problem is slightly more complicated when the original graph contains some cycles, because in this case the number of paths between some nodes is infinite. A first step in the simplification of such graph is to look for essential nodes, i.e. nodes that must be removed to interrupt all cycles; their choice is not unique, but in any ca se it should be such that the number of essential nodes is minimum. Once the essential nodes are chosen, the simplified graph contains: a) an arm from the source to the terminal node of interest, b) an arm from the source to each essential node, c) an arm from each essential node to the terminal node, d) an arm from each essential node to itself, e) an arm from each essential node to each other essential node.

The value of each of these arms is equal to the sum of the values of the elementary paths between the nodes they connect, excluding all other nodes of the simplified graph. Some of the arms listed above may be missing, and the terminal node itself may be an essential node. Fig. 4. 5. 6. 7 show some examples of simplified graphs.

From the simplified graph the closed arms can be eliminated using Mason's fourth rule; then the new graph can be further simplified as before, until only an initial and a terminal node are left.

Now write equation (15-2) in operational form,

or

92

l.! 1 = Go + l.rl )[ex p (!.t) 1 ' where t!\, !o and r~1 are row vectors of operators, and

lexp(!t)) is a square matrix of operators.

Fig.4. A simplified graph with one essential node.

Xio _. --------»-----8 Fig.5. A simplified graph where the essential

node is the terminal node.

Fig.6. A simplified graph with two essential nodes, general case.

Fig. 7. A simplified graph with two essential nodes, one of them being the terminal node.

93

Remember now that we are considering the case when only one compartment has an input different from zero, therefore the row vector Xo + 1..[1 has only one non-zero element, equal to Xio + [fi' ..... It follows that l.!J is equal to this last operator times the i-th row of {exp(Kt»). and the j-th element of lx1 is - -(17-3) f Xj \= (XiO + ffi1) {fij~ , where tfij1 is the element of row i and column j of texp~t)1.

Function ffij1 is called the ~t~r~a~n~s~f~e~r~f~u~n~c~t~i~o~n~f~r~o~m=-~th~e source of compartment i to compartment j, and it is equal to the value of the single arm left wben tne graph is simplified as described above.

The rules shown here are conveniently applied when a graph does not contain many cycles. If the simplification of the graph involves more than one or two essential nodes, it is more convenient to use the method based on the strong components of a graph (Rescigno and Segre, 1965).

18- OSCILLATIONS

We have just seen that some eigenvalues of matrix !-can be complex: we shall now examine the conditions for this to happen, and wh at the physical consequences become.

We first observe that with two compartments K cannot have complex eigenvalues: in fact the eigenvalues of--

are the roots of equation

z2 + (Kl+K2)z + KlK2-k12k2l = 0,

and the discriminant

of this equation is never negative.

With more than two compartments K can have complex eigenvalues. For instance consider a-three-compartment system: matrix

K = ......

has complex eigenvalues if and only if equation

(A+Kl)·(A+K2)·(A+K3) =

k12 k2l(A+K3) + k13 k3l(A+K2) + k23 k32(A+Kl)

+ k12 k23k3l + k13k32k2l

has complex roots. The real roots of this equation are given by

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the points where the cubic of equation

intersects the straight line of equation

i(A) = k12 k21(A+K3) + k13k31(~+K2) +

+ k23 k32(A+Kl) + k12 k23 k31 + k13 k32k21·

The cubic intercepts the abscissa at the points -Kl, -K2, -K3 and the ordinate at the point KIK2K3; the straight line intersects the ordinate at the point

r = k12 k21 K3 + k13 k31K2 + k23k32Kl +

+ k12 k 23 k31 + k13 k32k21,

with condition (15-4) requiring

the slope of the straight line is

while the slope of the tangent to the cubic at the point where it intersects the ordinate is

r = KIK2 + KIK3 + K2 K3'

with condition (15-4) requiring

Consider the extreme case ~=~, i.e.

k12 k 21 + k13 k31 + k23 k32

KIK2 + KIK3 + K2 K3;

if the system is not decomposable, this is possible only if it is a tree, i.e. if

Kl k12 + k13, K2 K3 0, or

K2 k21 + k23, Kl K3 0, or

K3 k31 + k32, Kl K2 0;

any of the three conditions above implies

~= ~ = 0,

and L has the eigenvalue 0 of multiplicity 3.

Any other values of the transfer rates make the difference ~-~ larger, with ~ and ~ both positive. The cubic and the straight line will intersect on the left of the ordinate, in general at three different points. A sufficient condition to have only one intersection, therefore two complex eigenvalues,

95

is to minimize the folding of the cubic, i.e. to make

Kl = K2 = K3,

and at the same time to maximize rand to minimize~, i.e. to make

k12k2l + k13 k3l + k23k32 = O.

k12k23k3l + k3lk32k2l = max.

These three conditions are satisfied for instance by making

Kl = k12, K2 = k23,

thence

r-Kl Kl o 1 K =

L ~l -Kl KU' ..-

0 -Kl

whose eigenvalues are

o. -1/2. (3+V-3) .Kl'

With XlO=l and fl(t);O we have

fXll = (s2+2KlS+K12)/(s3+3KlS2+3K12s)

1/3.l/S+2/3.(S+3Kl/2)/(S+3Kl/2)2+3K12/4)

[1/3 + 2/3.eXp(-3/2.Klt).Cos(43/2.Klt)}.

This function approaches the asymptotic value 1/3 through some very weak oscillations, as shown in the following table.

Klt Xl

0 1 2.42 0.32447 (minimum) 6.05 0.33337 (maximum) 9.68 0.33333 (minimum)

With four compartments and following the same line of reasoning we have the matrix

K = .......

and with the same initial conditions we get equation

or

the oscillations are now slightly more marked, as shown below:

96

Klt Xl

0 1 2.42 0.21860 (minimum) 5.53 0.25145 (maximum) 8.64 0.24994 (minimum)

but observe that any relaxation of the conditions imposed on these systems make those oscillations weaker, and then disappear altogether.

Of course strong oscillations are always possible when the system is not linear.

19. INTEGRAL EQUATIONS

We have seen that the element tfijS of row i and column j of matrix texp(Kt)l is called the transfer function from the sour ce of compartment i to compartment j. We can rewrite equation (17-3) in the form

I: (19-1) Xj(t) = Xiofio(t) +~fi("C)fij(t-'r)d't'.

This is a Volterra integral equation of the second type. If Xio and fi(t) are given and Xj(t) is measured, the transfer function can be computed.

A simple physical interpretation of the transfer function is obtained by considering equation (19-1) with fi(t)=O and Xio=l~ then

i.e. the transfer function from the source of i to j is equal to the function describing compartment j when compartment i is fed with a unitary instantaneous dose at time t=O. If instead Xio=O but fi(t)=O. then

t:

Xj(t) = ffi(~)fij(t-1::)d't', o

i.e. the behavior of compartment j is the convolution of the feeding function of compartment i with the transfer function from the sour ce of i to j when compartment i is initially empty.

The integral equation approach for the description of metabolizing systems was introduced by Branson (1946. 1947 r

1948. 1952) and reformulated in 1963~ Stephenson (1960) also used integral equations for the description of transport through linear biological systems.

Equation (17-3) or its correlate (19-1) can be made more general. Suppose that compartment i is not directly controllable, i.e. it is not possible to feed the system directly from it, but that a component of the system introduced into aprecursor of i can be measured in i and in j, and that everything that is measured in j must have first passed through i~ or alternatively suppose that, even though the system can be fed directly into i, the feeding function fi(t) cannot be measured, but Xi(t) can. In either case, from the linear

97

graph we see that

where

is called the transfer function from compartment i to compartment j. This equation is equivalent to

t:

Xj (t) = C Xi ("C)9ij (t-"r)d t ,

and a partial solution of it may be obtained with a graphical method (Beck and Rescigno, 1964). By sampling Xi and Xj at different times we can plot a number of values of Xj/(t.Xi) versus t and extrapolate those values for t=Oi if Xi(O)=O but Xi' (0)=0, using L'Hospital rule,

limit t=O

t f. Xi (1:")9i)' (t-'t:)d"'C limit _0 _______ _

Xj(t)

t.Xi(t) t=O t.Xi(t) rlc' I 1 Xi" ('l:') 9i j (t -~) d 1: + Xi (t ) 9 i j (0 )

= limit • I

t=O Xi(t) + t.Xi(t) i: 1/ I' ~Xi ('r')9ij (t- 'r)d"l: + Xi (t )9ij (0)+ Xi (t )9ij (0)

limit----------~----~-----------I Q t=O 2.Xi(t) + t.Xi(t)

1/2·9ij(0)i

if Xi(O)=X{(O)=O but X{(O)rO we apply L'Hospital rule once more and get

limit t=O

= 1/3.9ij(0)i

in general, if m is the lowest order of derivative of Xi(t) different from zero at t=O, then

limit t=O

1 =--.9ij(0).

m+l

Suppose now that it has been found that

9ij (0) = Oi

we can plot Xj/(t2Xi) versus t and extrapolate those values for t=Oi again if Xi(O)=O but X{(O).O, applying L'Hospital rule three times we get

Xj(t) limit ' = 1/6.9ij(0).

t=O t 2 ,Xi(t)

Proceeding in the same way it is easy to prove that, if

98

, C--I) (--) Xi(O) = Xi(O) = ... = Xi lO) = 0, Xi (0) + 0,

(19-2) I (~-I)

9ij (0) = 9ij (0) - ... - 9ij (0) = 0,

then

(19-3 ) (~). 1 9iJ' (0) = (m+n+l)1/ml.1imit XJ'(t)/ t n+ Xi(t) C"'O

Conditions (19-2) are not very restrictive, for if they do not hold for a specific transfer function 9ij(t). we define the function

~ , 9ij (t) = 9ij (t) - 9ij (0) - t.9ij (0) -

2 /I 1 t"'-I) - t /21.9ij(0) - ... - t n- /(n-l)1.9ij (0),

and conditions (19-2) hold for this function. Since the derivatives, up to the order n-l, of 9ij and of 9ij have the same value at t=O. from (19-3) we get

(19-4)

where

c.-) (m+n+l)1 limit

ml t=O 9ij (0) = ----

"'" c- ;It-Xj (t) = )oXi ("C)9ij (t-"!:)d"'C~

'" Xj (t)

now using the formalism of the operational calculus,

~ X j* 1 = i X i ~ r 9 ti1 tXi~ (t9ij~-9ij (0)t11-glj (0)'Ü"\2_ ... -gfj-')(oH)ln)

('~ I ~ ("'_I)!o~ ~ iXj ~-9ij (0)t)"Xi\-9ij (o)l~!xis -···-9ij (0)Ht· .. txi1·

Equation (19-4) and the above permit the determination of initial values of derivatives of all orders of the transfer function 9ij(t). The physical meaning of those initial values is explainea in the next section.

20. PRECURSOR-SUCCESSOR RELATIONSHIP

In section 16 I described qualitatively the precursor successor relationship~ now we can consider this relationship from a quantitative point of view.

Consider a linear graph without cycles. We can reduce it to a single arm simply by adding the values of all paths between the two chosen nodes~ each of those paths is a product of factors of the form kab/(s+Kb)' i.e. it is a fraction whose numerator is formed by astring of small k's, while the denominator is a polynomial in s of degree equal to the length of that path. When adding the values of the different paths, a fraction is formed whose denominator contains terms of the form s+Kb corresponding to the compartments on any path between the precursor and the successor, including the latter but excluding the former. The numerator of that fraction is a sum of strings of small k's, each string multiplied by the factors s+Kb corresponding to arms not on the path from where that particular string comes from. Of all the terms on the

99

numerator, the one with the highest degree in s will be the one coming from the shortest path; in fact the difference in degree in s between the denominator and the numerator will be exactly equal to the number of arms on the shortest path, i.e. to the order of the precursor. Furthermore, the coefficient of the highest term in s of the numerator is the string of small k's along this shortest path. This product is called the precursor's principal term (Rescigno and Segre, 1961). If cycles are present, the formation of the transfer function from the factors kab/(S+Kb) is more complicated, but the last two statements still hold. In fact each time a closed arm is eliminated, the value of some arms is divided by one minus the value of a path. This last value is an operator with the degree in s of the numerator smaller than the degree of the denominator. One minus that value generates a fraction with exactly the same degree in s in the numerator and the denominator, and with exactly the same coefficient for the highest term of the numerator and the denominator.

For instance from the graph

we get

k''L.k'Z-~ k ' "l.k2.S"kSJ k ,4 k4 ,.kS"3 ix ~} = ------ + --------- + --------~[x l J

(s+Kt.) (s+K"l) (s+K,) (s+K~) (S+K) (s+K~) (S+Ks-) (S+K;J)

k l 1. k q(S+K4 )(S+Ks-) + k\1. k 'tsKn (s+K,,) + kI4k ... S"kS"~(S+K2.) tX ~"\ = f X \ 1.

(s+K 2.) (s+K 'J) (S+K~) (S+Ko;-)

Compartment 1 is aprecursor of order two of compartment 3. and the difference of degree in s between denominator and numerator of {913} is two; the coefficient of s2 in the numerator is k12k23, the product of the transfer rates on the shortest path from \Xll tO~X3r. Now consider the graph

100

Here fx~\ is an essential node, therefore this graph ean be simplifJ.ed thus

where k 1lok 23 (s+K 4 ) (S+K, )+k '1 k'1.S"k n (S+Kt,.)+k 14 k4 S"kJ"'3 (s+K2.,)

(S+K t ) (S+K 3 ) (S+K 4 ) (S+KF )

(S+Kt,) (s+Ks--) (S+K 3 )

Using Mason's fourth rule we get

. fx,1 1 - fh21

(S+K 3 ) (S+K4 ) (s+K$"") ih ,1 . ·1x,J

(s+K 1) (S+Kt,.) (S+K)") - k}4 k4S"kS'":J

,

k I~ku (S+K 4 ) (S+K,,) + kl~k2S"kn(S+K~) + k 1t,.kt,.s-k n (S+K 2 ) r tX}l= ·LX\~

(S+K t ) (s+K 1 ) (S+K 4 ) (S+Ks-) - k., ... k4 )""k n (S+K 2 )

In this new graph the order of the preeursor is the same, and even though the denominator of the new transfer funetion is different, the differenee of degree between the denominator and the numerator, and the eoeffieient of s2 of the numerator, are the same as in the previous graph.

We have seen that in general if n+l is the preeursor order of {Xi~with respeet to rXj\, then the transfer funetion t9ij\ is a fraetion of two polynomials in s, with a denominator of degree n+l higher than the numerator; therefore the operator

has exaetly denominator

where

one as differenee between the degrees of the and the numerator. Then

~-f r~f. as + bs + •.. c. ... )l.

~ 9 i j ~ = --p,::;----.,..--_7", ---s'- + es + ...

a ~-I

1 (b-ae)s + = s + S . S r + es p-I +

I: = ia) + ~rf(1:)d-rJ,

o

101

(b-ac ) i'-l +

is a certain function and a is the precursor's principal term; but the integral above vanishes for t=O. therefore

C ..... ) 9ij (0) = a.

We have thus proved that if ( ... ·1)

9ij(O) I

9ij(O) 9ij (0) = o. ,-) 9ij (0) = a + 0,

then n+l is the order of the precursor, and a is the principal term.

21. MOMENTS

Consider an open system; if.r. == 0 7 equation (15-1) becomes

(21-1) dX/dt = X • K; - --. by successive differentiations we get

dPX/dtP - dP-1X/dtP-l K, p=1,2 .... - -thence

(21-2) dPX/dtP = X • KP, p=1,2, ••• - --From the same equation (21-1) we get

I: " [dX/dt.dt = (" Ydt • K, .... " ....... t- I: r tP/pI.dX/dt.dt = [tP/pI .Xdt • K o ....... 0 ..... ~

thence t-

X - Xo = r Ydt • K, ~ ~ D";':' ~

t tP/pI.X - f tp-l/ (p-l) I.Xdt

..- . - I: = [ tP/pI .Xdt • K; p=1,2, o -._

Remembering that the matrix lS.. of an open system is non-singular, by induction we get -(tp-l/(p-l)l.Xdt = Xo. (-K)-P.

o ~ """"'" ~ p=1.2, •••

If we put

the last equation becomes

" (21-3) (tp-l/(p-l)I.Xdt = Xo.HP, o ......... ~ ....

p=1,2, •••

We can prove that all elements of Ji and its powers are non-negative. In fact, because of inequality (15-4). matrix :!, is diagonal dominant, i.e. each diagonal element is larger

102

than, or equal to, the absolute value of the sum of the other element of the same row. The determinant of such a matrix has been called unisignant by Muir (1933). and is non-negative. In the present case ~is non-singular, therefore the determinant of -K is strictly positive • .......

The elements of ..[. have the sign of the cofactors of the corresponding elements of -~. The cofactors of the diagonal elements of -~ are its principal minors. These principal minors are also diagonal dominant, therefore they ar~ ~on-negative. The cofactor of -kij in -~, with i~j, is (-1)1+J(-!)i,j' where this last symool represents the determinant -K w1thout row i and column j. Developing this determinant accc>rding to the elements of row j, then developing the resulting non-principal minors in the same way, until only principal minors are left, this cofactor is equal to

(21-4) k., (-K).· .' ~.... - ... d", .... f-

According to inequalities negative, q.e.d.

+ ~ ka:.e,kR,~ (-!)":l,;.:~tl +

-t- 2:. k·" k" D k" . ( - K ) .' ~ .f .. j) ~ + ~'t. 1t:, :.I( .. 't~ "' .............. ~"" '/"~'t "-

(15-3) none of these terms is

We can now prove that all elements of ~ and consequantly of all its powers, are strictly positive if and only if K is irreducible. In fact if L is reducuble, there is at least a value of j and a value of i such that all terms k~ , k~~kR~, kit,kk'.ttkt~..:, . -. - for any l" Rt., ... - contained in sum (24-1), are void. On the other hand, suppose that an element of H is zero; if it is an element of the principal diagonal, say o~ row i and column i, then (-ll)i i is singular, therefore all elements of column i and rows different from i are zero, and K is reducible. If the void element of H is not on the--diagonal, then the corresponding sum (21-4) JLS zero. If any principal minor (-1i)":j.~, .. ,4~e, ... in that sum is void, the elements of its rows and the mlssing columns in K are zero; therefore the coefficient multiplying that minor in (21-4) is zero. When all coefficients in (24-1) are zero, ~is reducible, q.e.d.

We shall call kij and hij the elements of matrices !? and liP , respectively. To understand their physical meaning think of an experiment in which all compartments are initially empty except compartment i, whose initial value is 1; in this case the products

are equal to row i of KP and HP respectively. Calling Xij the element j of vecto~ X in Eluch an experiment, then ......

Cf) (r) (21-5) kij = Xij (0),

DIJ

(21-6) hI~) = (tP-l/(P-l)!.Xijdt,

for all positive integer values of p; to those identities we can add the obvious one

(21-7) ~ij = Xij(O),

where

103

J ij = 1 = 0

for i=j for i:f=j

is the Kronecker delta.

For p=l identity (21-6) becomes (I) ('0()

hij = loXijdt,

that can be written (I)

hij co

l/Kj.rKjXijdt, c)

where l/Kj is the transfer time (Rescigno, 1973) through compartment j, while

j(';'x. ·dt o J 1J

is the fraction of particles introduced into compartment i that exit from compartment j. It follows that hij is the expected total time spent in compartment j by particles introduced into compartment i. For i=j we see that hii is the permanence time in compartment i.

or

For p>l we can write (I") ,-

hij '~tP-l/(p-l)!.Xijdt

h ~l! 1J

(r') hij

---z;) = hij

<>0

.... r x· ·dt " 1J

~ tp-l/ (p-l)! .KjXijdt

(K·x· ·dt o J 1J

if we define the random variable Tij as the interval of time between the introduction of a particle into compartment i and its exit from compartment j, then

e,. ... ·) CI) p!hij /hij

is the p-th moment of Tij, and 00 Of)

(exp(ts) .Xijdt/~Xijdt is the moment generating function of Tij.

If all compartments of a system can be controlled and observed, i.e. if we can make n different experiments feeding the system from all different compartments, and in each experiment we can measure all n compartments, then all elements hij of lI... ca~ be computed, hence by inversion we obtain A, Wh1Ch descr1bes completely the compartment system. Most of the times though very few compartments are controllable and few observable, so that only a few elements of J!..r and the corresponding elements of its powers, can be computed. In that case we must reconstruct K from those known elements. -

Call

104

the m1n1mum polynomial of K: this polynomial is equal to the characteristic polynomial Cif K or to one of its divisors. By definition p( A) annihilates t;' Le.

(21-8) p(K) = Km + Cl~-l + C2Km- 2 + .•. + cmI = 0: - - - - .-multiply each term of this equation by .,!!.m, Hm+l. ...... Hm+2 •... -

I - Cl!!.. + C2 H2 + ... + ( -1 ) mCmHm = 0 - - ...-

(21-9) 1!. - CIH2 + C2 H3 + ... +(-l)mCmHm+l = 0 ...... ..- ... -li2 - CIH3 + C2!L4 + ... +(-1)mCmHm+2 = .!L ......... -

Now multiply each term of equation (21-8) by Hm-l, Hm-2,. . . --(21-10)+C2~ - c3H + ... +(-1)mCmHm-2 = -K2 -cIK

__ AoII4. N'I'I- """"""'- ,.,.-

-c3I + ... +(-1)mCmHm-3 = +K3 + CIK2 + c2~ "... ~ ~ ~ ,.--

For particular values of i and j, using definitions (21-6) and (21-7) • equations (21-9) can be written

$ij (I) ('L) (-)

-Clhij + C2 hij + ... + (-1 )mcmhij = 0 CI'

hij ('1.)

-Clhij + Cl)

C2 hij (_ ... )

+ ... +(-l)mcmhij = 0

h~~ Cl) C<.) (-"'I.) 1J -Clhij + C2 hij + ... +(-l)mcmhij = 0

If Xi' (t) has been measured for a particular set of values i,J j , then we can compute the element of row i and column j of.!!. and any of its powers: we can therefore find a unique non trivial solution for cl' c2'· ... cn from these last equations if matrix

~ .. h~? ('-) (-) hij h· . 1J 1J 1J

(I) h l,,:) h C') h'(~+') hij 1J 1J 1J

(-) hij

( .... ,) hij

( ... +1.) hij

('2. ... ) hij

has rank m. To determine the rank m we must check which of the matrices

r· Cr) (-..) r· (.JJ 1J hij hij

h' , 1J

hf1 ' hC'~ ('l.) hl~) e.r'! 1J h1j 1J ' ...

1J ('t.) h I"!) h l'Y h1j 1J 1J

is singular: then the coefficients of the minimal polynomial P(A)Of Kare given by -

105

Ci) h(~) CIk) -1 f ij -cl hij 1) hij

+C2 h(~) h f?) h~·+!> CI) 1) 1) 1) hij . . . . .

(-l)mcm l-) (111&+/) (t .. ·,) (...,-1 h· . hij Dij hij 1)

and from these equations we can compute the elements kij of ~ sequentially.

If the minimum polynomial of K has m distinct real roots, then the integral of equation (21~) is

X = xo.p.exp(tA).p-l, ..... ~..-. AIIIIIr .....

where P is an appropriate non-singular matrix and ~ is the diagonal matrix of the roots. With the same hypotheses on ~o as before, the product above gives

(21-11) Xij = ~ uRexp( Alt) 7 rc; with the conditions

5"ij = i" u.t' • C.'> ........ \ ,.

X1)(O) = ~ ".tul' or, using identities (21-5).

(21-12) =

u

From this equation we can compute the coefficients ui, if all roots ~, )..2' •.. )A ...... are real and distinct.

If tx.±r...f=i is a pair of complex roots, then the corresponding terms in the sum (21-11) must be substituted by

(/cosft + VSin~).exP(l)(t),

the corresponding columns of the square matrix in equation (21-12) are substituted by the real part and the imaginary part, respectively, of the successive powers of the complex eigenvalues, i.e.

106

1 0

0<. r r;l..t._ ~1. 2Dtr

o{) _3Dl~'L 3l~-~3 . . . . . . . .

If ~~is a root of multiplicity v, then V terms in the sum (21-11) must be substituted by

V-I '101-'1. <fIt +/''I.t + ... +h)·exP<A,l)

and in the matrix of equation (21-12) the co1umn corresponding to that eigenva1ue will be fo11owed by co1umns with its successive derivatives with respect to ~/' up to the order V-I.

22. TRANSFER FUNCTION

Equation (21-1) of the previous section can be written in operationa1 form

or

It is easy to verify that "'-1 -t. _-f·t·, _-I 't .... l.'I. .-_t 't-+-I _-e-'l.-I

(sI - K). 2..c.tK s = "2: c.t K s - '%: CIK s ....... ..... .t~~,./) - e./t-o - ~ .. 4c.O -

-:;:.-1 "",_t __ It. _-t-'L """'- 't -.-l-'l. = G- C Is + ~ c K s - z:: c K s .e .. o'P.- 1 ....... ,,1..... ~+'I.."l-

It~o 'l?"

....... M&-~ ..... -'.t. = L Cn S • I - c""I - ~ Co r-e:o ~ ~ .,...tt:I'J 7l-

........ IfII4.-( """- __ t = ~ Cn S • I - ~ c .. K &4 ~ ..-. e;iJ Je. -

where the last sum is zero according to (21-8). Therefore

1 ~' 't ..... -l'-'t-f -- --~ ( s I - K) - = &;.- CA K s /;E. c .. s - - ~~.t:." .c- t'= A::

and consequent1y -.-1 (-\.) ,.,-l-t-, __ .... _~

(22-1) [Xij 1 = ~ c.- k ., S /ZcA S €-tt:"... .",. t~~ ...

where for convenience we have put (0) r

kij = ij·

Now Xij as defined in section 21 is equa1 to the transfer function from compartrnent i to compartrnent j, therefore equation (22-1) shows the relationship between the coefficients of the minimum po1ynomia1 of ~and the coefficients of the transfer function between any two compartments.

107

23. MORE PROPERTIES OF THE MOMENTS

We suppose now that in equation (15-1) L is not zero, but Xo=O. Equation (15-1) becomes -' - (:'

X = (F ("C) • exp (K (t--r») d't". "."..... ~

Proceeding as in section 21. we get 00 CIO

LXdt = -(Fdt.K-1, v..... )0"'-'-00 t 04 ..

(t'-/(p-1)1.Xdt = -!tr/pI.Xdt.K - ftr/pI.Fdt, p=1.2, ••• " .". 0 ."....,..... ~ ~

and by induction, ... p-I '-1 r- I -(,,-1)

(23-1) [t /(p-1)1.Xdt = ~ Lt /ll.FdL(-K) • p=1.2, ••• o - ~ .. o 0 - -

With the notation o(J~

F l. = [t /il.Fdt, - 0 -

the equations above can be written co

(o~dt = !o·.!!' fo Mt • .!gt = !.0 . .[2 + LI.!!.' .-~ t 2/21 • .!dt = !o·Ie + !1·..[2 + !2·l!!

and so on. The physica1 meaning of matrix H be comes c1ear if one thinks of an experiment where on1y comiPartmeßt i is fed according to function fi(t)~ the product~~.H-c". in this case is equa1 to row i of H-c,,·eJ multiplied by -"'p

[t /€'l.fi(t)dt, o

and equation (23-1) becomes "'" "-1 00 - 1 r i ('('.-t>

[t"- /(p-1)I.XiJ"dt = ~ t /Rl.fidt.hiJ" , u ~~ 0

showing that the sequence ()Q ~ ""

{oXijdt, (0 t.Xijdt, ~ t 2/21.Xijdt.

is the convo1ution ofthe sequences tIO • D(I r fidt , (t.fidt , r t 2/2I.fidt ,

., • cl

and

Therefore if we define the random variable Ti as the time of introduction of a partic1e into compartment i, then

II(J f" pI (tp/pl .fidt/ Ilidt

o

is the p-th moment of Ti'

If the functions fi and Xij have been measured, the elements of row i and co1umn j of matrices H. H2. H3._ ••• - - --108

can be computed, and the reconstruction of matrix ~follows as in the previous case.

Equation (23-1) is a special ca se of a more general theorem. If we define the i-th moment of a generic function f(t) by

0()

Fi = !.ti/i!.f(t)dt, o i=O .1.2 ....

then equation (1-1) becomes

i. e.

and so forth.

To prove it,

invert the order of integration,

Cn = (~(1:')(l~n/n!.B(t-t)dt)d't", change the variable,

Cn = fo:(1:") (r;'t:-t~)n/n!.B(6"')d6"')d'C', and finally develop the binomial in the inner integral.

24. REFERENCES ,

C.Artom, G.Sarzana and E.Segre, 1938. Influence des grasses alimentaires sur la formation des phospholipides dans les tissues animaux. Arch. Internat. Physiol. 147:245-276.

J.Beck and A.Rescigno, 1964. Determination of precursor order and particular weighting functions from kinetic data. ~ theoret. Biol. 6:1-12.

C.Berge, 1958. nTh~orie des graphes et ses applications. n Paris. Dunod.

M.Berman, 1972. Iodine Kinetics. In: nMethods of investigative and diagnostic endocrinologyn (Rall and Kopin editors). Amsterdam, North Holland Publ. Co.

H.Branson, 1946. A mathematical description of metabolizing systems. Bull. Math. Biol. 8:159-165; 9:93-98.

H.Branson, 1947. The use of isotopes to determine the rate of a biochemical reaction. Science 106:404.

109

B.Branson, 1948. The use of isotopes in an integral equation deseription of metabi1izing systems. Cold Spring Barbor Symposium Quant. Bio1. 13:35-42.

B.Branson, 1952. Metabolie pathways from tracer experiments. Areh. Bioehem. Biophys. 36:60-70

B.Branson, 1963. The integral equation representation of reaetions in eompartment systems. Ann. N. Y. Aead. Sei. 108:4-14.

G.L.Browne11, M.Berman and J.S.Robertson, 1968. Nomenc1atur for tracer kinetics. Int. J. App1. Rad. IsotOpes 19:249-262.

A.Cay1ey, 1861. Note on the theory of determinants. Philos. Magazine 21:180-185.

Y.Chow and E.Cassigno1, 1962. "Graphs and App1ications~\ New York, Wi1ey.

E.G~rpide, 1975. "Tracer Methods in Hormone Research." Ber1in, Springer-Verlag.

F.Barary, 1959. A graph theoretic method for the comp1ete reduction of a matrix with a view toward finding its eigenva1ues. J. Math. Physics 38:104-111.

J.A.Jacquez, 1972. "Compartmenta1 Analysis in Bio10gy and Medicine." Amsterdam, Elsevier.

G.Kirchhoff, 1847. Ueber die Anf10esung der Gleichungen auf welche man bei der untersuchung der linearen Verthei1ung galvanischer Stroeme gefuehrt wird. Am. Phys. Chem. 72:497-514

R.Laue. 1970. "Elemente der Graphentheorie und ihre Anwendung in den biologischen Wissenschaften. n Leipzig, Geest & Portig.

R.B.Marimont, 1959. A new method for checking the consistency of precedence matrices. J. Assoc. Comp. Machinery 6:164-171.

S.J.Mason, 1953. Feedback theory - some properties of signal f10w graphs. Proc. Inst. Radio Eng. 41:1144-1156.

J.H.Matis and B.O.Bart1ey, 1971. Stochastic compartmenta1 analysis: Model and least square estimation from time series data. Biometrics 27:77-102.

C.M.E.Matthews, 1957. The theory of tracer experiments with 131-I-1abe11ed plasma proteins. Phys.Med.Bio1. 2:36-53.

J.Mikusinski, 1959. "Operationa1 Ca1cu1us." New York, Pergamon Press.

T.Muir, 1933. "A Treatise on the Theory of Determinants." London, Longmans, Green & Co.

O.Ore, 1962. "Theory of Graphs." Providence, Rhode IS1and, American Mathematica1 Society.

110

P.Purdue, 1974. Stochastic theory of compartments. Bu11. Math. Bio1. 36:305-309 and 577-587.

A.Rescigno, 1960. Synthesis of a mu1ticompartmented bio1ogica1 model. Biochim. Biophys. Acta 37:463-468.

A.Rescigno, 1973. On transfer times in tracer experiments. J. theoret. Bio1. 39:9-27.

A.Rescigno and J.S.Beck, 1972. Compartments. In: "Foundations of Mathematica1 Bio1ogy," Vo1. 2. (Rosen editor). New York, Academic Press.

A.Rescigno and G.Segre, 1961. The precursor-product relationship. J. theoret. Bio1. 1:498-513.

A.Rescigno and G.Segre, 1964. On some topo1ogica1 properties of the systems of compartments. Bu11. Math. Bio1. 26:31-38.

A.Rescigno and G.Segre, 1965. On some metric properties of the systems of compartments. Bu11. Math. Bio1. 27:315-323.

A.Rescigno and G.Segre, 1966. "Drug and Tracer Kinetics." Wa1tham, Mass •• B1aisde11 Pub1. Co.

L.P.A.Robichaud, M.Boisvert and J.Robert, 1962. "Signal F10w Graphs and App1ications. n Eng1ewood C1iffs, New Jersey, Prentice-Ha11.

M.Row1and and G.Tucker, 1980. Symbols in Pharmacokinetics. ~. Pharmacokin. Biopharm. 8:497-507.

M.A.Sainte-Lague, 1926. "Les Reseaux." Paris, Gauthier-Vi11ard.

C.W.Sheppard, 1948- The theory of the study of transfers within a mu1ti-compartment system using isotopic tracers. ~ App1. Physics 19:70-76.

J.L.Stephenson, 1960. Theory of transport in linear bio1ogica1 systems. Bu11. Math. Bio1. 22:1-17 and 113-138.

A.K.Thakur, A.Rescigno and D.E.Schafer, 1972. On the stochastic theory of compartments: A single compartment system. Bu11. Math. Bio1. 34:53-63.

A.K.Thakur, A.Rescigno and D.E.Schafer, 1973. On the stochastic theory of compartments: MU1ti-compartment systems. Bu11. Math. Bio1. 35:263-271.

E.C.Titchmarsh, 1926. The zeros of certain integral functions. Proc. London Math. SOCa 25:283-302.

111

AN INTRODUCTION TO STOCHASTIC COMPARTMENTAL

MODELS IN PHARMACOKINETICS

Introduc tion

James H. Matis

Departmcllt of Statistics Texas A&M Univcrsity College Station, Texas 77843, U.S.A.

Linear compartmental models are being widely used to model pharmacokinetic systems. Most of these models are deterministic and the statistical analysis of such models has been studied extensively. Many deterministic models are illustrated in other papers of this volume, and recent reviews are also given by Gibaldi and Perrier (1982), Godfrey (1983), and Jacquez (1985).

A number of papers have appeared recently in the literat ure presenting various stochastic models and deriving new methodology based on the stochastic assumptions. Many of these papers are reviewed in the previous references. In addition to their intrinsic appeal of providing a broader theoretical framework, the stochastic models have contributed two major results of interest in pharmacokinetic methodology. One is a new, rigorous approach to statistical moment theory incorporating the assumed compartmental structure of a model. The other is a generalization of the compartmental model to include nonexponential retention time distributions. These new methods have not been adopted rapidly in the literat ure, perhaps in part due to their perceived complexity.

This paper presents a simplified approach to these new, stochastic models and their related method­ology. The objective is first to present a logical development of the results as heuristic extensions of previous wellknown findings and then to illustrate the simplicity of the practical application of the methods. The paper may be considered as divided into two parts. The first part, consisting of Sections 1 to 4, reviews the single and multicompartment deterministic models and also a stochastic analog to these models. This part, though presenting some useful insights derived from the stochastic model, is intended primarily to serve as a foundation for the subsequent sections. The second part focuses on the new methodology. Section 5 develops the stochastic one-compartment model from the assumption of an exponential retention time random variable, and Section 6 extends the development to multicompart­ment models. Section 7 generalizes the stochastic one-compartment model to nonexponential retention time distributions which are common in survival analysis. Section 8 presents a generalized compart­mental analysis in which a multicompartment model is assumed to have nonexponential retention time distributions within the compartments.

For the sake of simplicity, all models will assume that the drug enters the system as a bolus (IV injection). Also, it is assumed that the observations are on the total drug, as opposed to the concentration, in the compartments at various times. A matrix approach will be used to formulate all multicompartment models in order to unify the presentation. Many extensions to continuous dosing and concentration variables, and many mathematical proofs using matrix and other solution techniques are given in subsequent references.

113

I. One-Compartment Deterministic Model

1A Derivation 0/ Model

Consider first the one-compartment deterministic model. It requires the following notation:

Notation Set 1A: Let"

1) X(t) denote the amount of drug in the compartment at time t, and

2) X(t) denote the derivative of X(t).

The basic assumption of this model is that the compartment has a constant proportionalloss rate. This may be stated symbolically as follows:

Assumption 1A: Assume [X(t + t..t) - X(t) 11 X(t) = -kt..t (1-1)

where t..t is a small increment of time and k is a constant.

The constallt k is often called the fractional (or proportional) !low rate. Dividing (1-1) by t..t, multi­plying by X(t) and then taking the limit as t..t approaches 0 yields the following model:

ModellA: X(t) = X(t)[-k]

This differential equation has the familiar solution:

Solution 1A: X(t) = X(O) exp( -kt)

where X(O) is the initial amount (i.e. the dose).

1B Parameter Estimation

(1-2)

(1-3)

Usually the parameter k, and often X(O) also, are unknown and are estimated from data on the system. Consider now the following additional notation:

Notation Set 1B: Let

1) y(tj) denote the observed amount of drug in the compartment at time t,., 2) y = [y(td, ... , y(tm )] be the (row) vector of observations at m distinct times, 3) Y(k) = [X(tIJ, ... , X(tm )] be a corresponding vector of the assumed model values with pa­

rameter k, and 4) e = [€l, ... , €m] be a vector of random errors.

The statistical model which is usually assumed for estimation is:

Assumption 1B: Let y = Y(k) +e

with expected values E(e) = 0, a vector of O's, and variance-covariance matrix E[E'e] = D, a diagonal matrix.

(1-4)

Under this assumed model, the y(t) observations may have different variances, for example one could assume that the variances are proportional to the observed values, however the assumption of a diagonal D matrix requires that all observations be uncorrelated, which for all practical purposes means that they must be independent. The errors are usually associated with the independent measurement (or sampling) process which takes place in order to ascertain a y(t) value. The parameters may be estimated by ordinary or by weighted nonlinear least squares, as discussed elsewhere in this volume and in a number of reviews.

11. Multicompartment Deterministic Model

2A Derivation 0/ Model

The subsequent generalization will hold for any n-compartment model. However, for simplicity, the concepts will be illustrated at times only for the two-compartment case. Consider now the following expanded set of notation:

114

Notation Set 2A: Let

1) X;(t) be the amount of drug in compartment i at time t, 2) X(t) = [Xr(t) , ... ,Xn(t)] be the n-vector of amounts at time t, 3) k;i' for i = 1, ... , nj j = 0,1, ... , nj i f jj denote the fractional ßow rate from i to j, where 0

represents the system exterior, 4) k;, = - :Li ki" denote the total outßow rate from i, 5) K = (k'i) be the n x n matrix of k i , coefficients, 6) .Al, ... ,.An be the eigenvalues of K, with A as the diagonal matrix of .A., and 7) Tl, ... , Tn be the corresponding right eigenvectors of K with n x n eigenvector matrix T =

(Tl' ... ' Tn ).

One could now assume a linear compartment model, namely one where each derivative, X,(t), is a linear function of the X,(t)'s. For example, in a general two-compartment model, one would assume the following:

Xl(t) = Xl(t)[-k lO - k12] + X 2 (t)[k21 ] and

X2 (t) = Xl(t)[kd + X 2 (t) [-k20 - k2l ].

Such linear compartment models may be expressed in matrix form as folIows:

Model2A:

X(t) = X(t)K (2-1)

Clearly, Model 2A is the matrix generalization of Model 1A in (1-2). It is typically presented in a different form where a matrix K premultiplies a column vector of X,(t). The present form is a bit more awkward to manipulate mathematically, but in the author's opinion, it simplifies the subsequent practical application of the theory. The above vector of differential equations has the following solution:

Solution 2A:

X(t) = X(O) exp(Kt) (2-2)

with the matrix exponential defined as exp(Kt) = I + :L~l K't' /i!

Keeping in mind that K is negative diagonal dominant by virtue of the definition of k,i, the relations hip between Solution 2A and Solution 1A in (1-3) is immediate.

Two corollaries are helpful in the application of Solution 2A. In most pharmacokinetic applications, one can assume that the system is open and at least weakly connected and further that no compartment is a sink (i.e. closed). These are necessary for the following corollary:

Gorollary 2A.l: Assuming the above regularity conditions and distinct .A., one can show 1) the .A, have negative real parts, and 2) X(t) = X(O)Texp(At)T- l

where exp(At) is a diagonal matrix with elements exp(.Ait).

All 2-compartment and most 3-compartment models have real eigenvalues. This motivates the following corollary:

Gorollary 2A . .€: Assuming that the eigenvalues in Corollary 2A.1 are distinct and real, formula (2-3) im plies that

X,(t) = 2..= A" exp(.Ait) , for i = 1, ... , n. i=1

(2-4)

8noindent Corollary 2A.2 gives the "sums of exponentials" models which generalize Solution 1A and which are common in pharmacokinetics. The A'i and .Ai are often called the "macro-parameters" and they are involved functions of the k;i "micro-parameters." The equations relating these formulations are given explicitly for the common 2- and 3- compartment models in many books, including Gibaldi and Perrier (1982), Godfrey (1983), Jacquez (1985), Rescigno and Segre (1966), and Wagner (1971).

2B Parameter Estimation

Parameter estirnation for the n-compartment model is identical in structure to that of the one­compartment model. For completeness, a few brief comments follow.

115

Notation Set J!B: Let

1) y;(tj) denote the observed amount of drug in compartment i at time tj, 2) y denote the vector of all observations arranged in the following order

[Ydtd, Ydt2),"" Yn(tm-d, Yn(tm)J, 3) k denote the vector of all parameters arranged in the following order [klO, k12, .•• , kn,n-l], 4) Y(k) denote the vector of model values corresponding to the y vector of observations, 5) E denote the vector of random errors.

It is not necessary that all mn observations be represented in y. The statistical model is:

Assumption J!B: Let (2-5)

with expected values E(E) = 0 and variance-covariance matrix E(~E) = D.

The errors are attributed to uncertainties in the measurement process and are usually considered inde­pendent, though with nonconstant variances (i.e. heteroskedastic variances).

III. One-Compartm.ent Stochastic Model Based on Transfer Probabilities.

9A. Derivation 0/ Model.

The present stochastic model is the so-called "particle" model where the substance of interest is viewed as a set of particles. The number of particles in the set may be very large. The notation is as folIows:

Notation Set 9A: Let

1) P(t) denote the probability that a particular particle introduced at time 0 is still in the compartment at time t, with p(t) as its time derivative, and

2) X(t) denote the number of particles introduced at t = 0 which are still present at time t.

The assumptions for this model are:

Assumptions 9A: Assurne

1) the conditional prob ability that any random particle present at time t leaves by t + ~t, where ~t is smalI, is k~t, and (3-1)

2) all X(O) particles are independent.

Note that the constant k in (3-1) is a probability intensity coefficient, also called a hazard rate, which de­fines a probability of particle transfer rat her than the proportional transfer rate as before in Assumption 1A.

A differential equation model may be derived for P(t) as folIows. Clearly, the necessary events for a particle to be present at time (t + ~t) are 1) that the particle be present at time t, and 2) that it remain in the compartment during the interval from t to (t + ~t). The conditional prob ability of the latter event from (3-1) is (1- k~t). Therefore, the probability of the desired joint event may be written as:

P(t + ~t) = P(t)[1 - k~tJ

Rearranging and taking the limit as ~t approaches 0, one has the model:

Model9A: p(t) = P(t)[-kJ. (3-2)

The solution to the model with initial condition P(O) = 1, is:

Solution 9A: P(t) = exp( -kt) (3-3)

P(t) may be viewed as the common "survival" prob ability of the particles. Since all of the particles are independent by assumption, the number of particles which survive to t follows a binomial distri­bution, the mean and variance of which have simple wellknown forms. These facts give the following corollary:

116

Corol/ary SA.l: The distribution, mean, and variance of the particle count X(t) are:

l)X(t) ~ binomial [n = X(O),p = P(t)] 2)E[X(t)] = X(O)P(t) = X(O) exp( -kt) 3)V[X(t)] = X(O)P(t)[l - P(t)]

(3-4)

(3-5)

(3-6)

Of course, the fundamental difference between the stochastic model for X(t) in (3-4) and the deter­ministic model in (1-3) arises from the assumed chance mechanism in (3-1) wh ich generates so-called "process" uncertainty, or process eITor. The data from this model may be fit ted to the mean value function in (3-5), which is identical to the model in (1-3). One consequence of the assumed chance mechanism is the heteroskedasticity of the process eITors, as apparent in (3-6). The variance function is 0 at time 0, i.e. V[X(O)] = 0, reaches a maximum, and then decays to 0 as t approaches 00.

An important generalization concerns the multivariate distribution of observations at different times. This leads to the following corollary.

Corollary SA.2: Let X(td and X(tz) be observed counts at times t 1 < tz. The distribution, means, variances, and covariance are

l)[X(td, X(t z)] ~ chain binomial [n = X(O), P1 = P(t1), pz = P(tz)] 2)E[X(tJ )] = X(O)P(tJ )

3)V[X(tJ )] = X(O)P(t J )[l- P(t])] 4)COV[X(t1),X(tZ)] = X(O)P(tt}[l- P(tz)]

(3-7)

(3-8)

(3-9)

(3-10)

The chain binomial distribution is discussed in Chiang (1980). Its covariance structure in (3-10) for observations at different times will be used subsequently.

SB Parameter Estimation

The parameter estimation for this model is similar to the one-compartment deterministic model, however there are significant differences which later extent also to the multicompartment stochastic mode!. This problem may be set-up as folIows:

Notation Set SB: Let

1) y(tJ ) denote the observed count of particles at time tJ ,

2) Y = [y(ttJ, ... , y(tm )] be the vector of observations, 3) Y(k) = {E[X(tt}}, ... , E[X(tm )]] be the vector of expectations, 4) 0 = [01, ... , om] be the vector of random process eITors.

One may then assume the following model:

Assumption SB.l: Let y=Y(k)+5

with E[5] = 0, and variance-covariance matrix E[0'5] = E(k), a positive matrix whose diagonal elements are given by (3-9) and off- diagonal elements by (3-10).

(3-11)

In this model, one assumes implicitly that there is no counting (or measurement) eITor but rather that all discrepancies between the observed and the expected values are due to stochastic eITors. This is a realistic assumption in a small number of applications (Matis and Hartley, 1971; Kalbßeisch et. a!., 1983). The covariance is a function of the parameter k, and consequently one could use a two­stage iterative nonlinear least squares procedure. With this method, one assumes an initial covariance matrix, such as the identity matrix, estimates the parameter k, calculates an updated covariance matrix, reestimates k, and so on until covergence of the estimate of k is attained. Other estimation procedures for the one-compartment model are the minimum XZ procedures which are often used with life tables.

In other applications, the random eITors may be a combination of measurement and process eITors. In such cases, one might assume the following mode!.

Assumption SB.I!: Let y = Y(k) + 5 + E

with E[5 + E] = 0 and E[(5 + E)'(O + E)] = D + E(k)

(3-12)

117

were D and l:(k) are given in (1-4) and (3-11).

It has been pointed out that often one observes proportional data, say r(t) = X(t)jX(O). A model for the r(t;) observations could be obtained by slight transformations of (3-11) and (3-12). The vanance-covariance matrix for the proce88 errors in the r(t.)'s would be l:(k)jX(O), which vanishes as X(O) approaches infinity. For this reason, it is dear that for many applications with very large X(O) the stochastic error is negligible in comparison with the measurement error, and it may be omitted for purposes of estimation. However it should be noted that the stochastic model and its vanance structure may be very important for other purposes, even in the case of very large X(O). üne such example is given in Rescigno and Matis (1981).

IV Multicompartment Stochastic Model Based on Transfer Probabilities

4A Derivation 0/ Model

Consider the stochastic formulation of the n-compartment model. The following definitions are necessary:

Notation Set 4A: Let

1) P.i(t) be the prob ability that a partide starting in i at time 0 will be in j at time t, 2) P(t) = [P.i(t)] be an n x n matrix of probabilities, 3) X'i(t) be the number of the Xii(O) particles starting in i at time 0 which will be in j at time

t, and 4) X.(t) = [Xil(t), ... , X.n(t)] be the random vectar denating the numbers of the X • .(O) particles

in the various compartments at time t.

For subsequent convenience, the present formulation tracks separately the distribution of particles for each compartment of origin. The assumptions defining the chance (ar prabability) mechanism for this model are:

Assumptions 4A: Assume

1) the conditional probability that, far smalll1t, a random particles in i at time t will be in j at time t + I1t is k.i l1t, and (4-1)

2) all particles are independent.

The k'i constants may again be regarded as hazard rates which define the instantaneous transfer probabilities. A set of differential equations generalizing (3-2) may be constructed for any n, but far simplicity will be illustrated only for n=2. Note first that an equation for any Pii(t + I1t) may be constructed by considering a set of events that enumerates all the possible ways in which a partide starting in i at 0 could pass through the vanous compartments at time t and end up in j at t + I1t. Far example, after considering all p088ible pathways, Pll(t + I1t) could be written as

where o(l1t) denotes all possible higher order terms of I1t. Such equations are called Chapman­Kolmogorov equations and with simple manipulation they yield differential equations such as

The complete set of these differential equations, also called the Kolmogorov equations, for the presently defined n = 2 compartment model is

Pil(t) = -(klO + k 12)Pil (t) + k21P.2(t)

Pi2(t) = k12P.2(t) - (k20 + k21)Pi2 (t)

for i = 1,2. Using the matrix definition of K given in Notation Set 2A, these equations give the following model which holds for any n:

Model4A: P(t) = P(t)K (4-2)

The matrix solution to this model has similar form to that of (2-2), however one has the initial condition prO) = I for this model. Therefore, the solution is:

Solution 4A: P(t) = exp(Kt) (4-3)

118

A number of corollaries follow which generalize the previous results. One concerns the probabilities, the other the counts.

Corollary 4A.l: Assume that the system is open and weakly connected. Then

1) if the >"s are distinct, P(t) = T exp().t)T-l, and 2) if the >"s are distinct and real,

Pii(t) = L A,;e exp(>'et ) for i, j = 1. .. , n. e

Corollary 4A.2: The distribution, means, variance and covariances for the random vector X. (t) are

1) X;(t) - multinomial [n = X,;(O), Pol(t), ... , Pm (t)], 2) E[Xii(t)] = X .. (O)P,;(t), 3) V[X,(t)] = X,,(O)P,;(t)[l- Po;(t)]. 4) Cov[Xii(t),X'k(t)] = -X .. (O)P,; (t)P'k(t) for j -I- k.

Clearly, the generalization of Corollary 3A.2 concerning the covariance matrix between random vectors observed at two distinct times, say X. (td and X. (t2), would also be of interest. The results are given in Kodell and Matis (1976) but are not given in this paper for the sake of brevity.

4B Parameter Estimation

For simplicity, it will be assumed that units are introduced only into one compartment at time o. Notation Set 4B. Let

1) y,(t) = [Ytl (t), ... , Ym(t)] denote the observed counts of particles introduced into t at time 0, 2) y = [Yil(td, ... , Ym(tl), y,I!t2), ... , Ym(tm)i denote the vector string of all observations at m

distinct times, 3) Y(k) = {E[X'l(tl)]' ... , E[Xm(t",)]} be the vector of expectations, 4) S = [Oll' ... ,on",] be the vector of random process eITors.

As before, one may then assume the model:

Assumption 4B Let y = Y(k) + S

with E[S] = 0 and variance-covariance matrix E[S'S] = 2:(k), a positive matrix.

The details of estimation were discussed in Section 3B. One could use a generalized nonlinear least squares procedure as illustrated in Kodell and Matis (1976). The conditionalleast squares procedure in Kalbfleish et al. (1983) is another convenient way to incorporate the inherent stochasticity of the observations into the estimation procedure.

Many variations of the basic approach have been derived. The generalization where initial units are introduced simultaneously into several compartments is simple but seems to have limited application. Matis and Hartley (1971) considers an example where the observations consist of the combined elimi­nation from all compartments. Matis (1970) has a further generalization where the variance-covariance matrix has three independent sources of variability.

v. One-Compartment Stochastic Model Based on Exponential Retention Time Distributions

5A. Derivation of Model

A stochastic model mayaiso be defined on the basis of its retention time (also called transit time or sojourn time) distributions. In some ways, this conceptualization of the inherent chance mechanism is more satisfactory since it re lies on a continuous time prob ability distribution rather than on a conditional flow prob ability in discretized time units of size /).t. One first needs the basic notions associated with a continuous probability distri~Jution:

Notation Set 5A: Let

1) R be the retention time of a random particle in the system, 2) F(a) = Prob[R ~ a] be the distribution function of R, i.e. the prob ability that the particle

willleave prior to attaining "age" a in the compartment, 3) f(a) = dF(a)jda be the density function, and 4) S(a) = Prob[R > a] = 1 - F(a) be the survivorship function, l.e. the prob ability that the

particle survives to age a in the compartment.

119

Suppose one assurnes a particular prob ability distribution for R, namely the exponential distribution. This gives:

Assumption 5A: Let R - exponential distribution (k), which implies

l)F(a) = 1- exp(-ka)

2)f(a) = kexp(-ka)

3)S(a) = exp( -ka)

(5-1)

(5-2) (5-3)

It is clear that the survivorship function S(a) in (5-3) is equivalent to the previous function P(t) introduced in Section 3A and solved in (3-3). The argument is changed for subsequent convenience from t to a. The former argument, t, often denotes some exterior, exogeneous time measure in a system whereas the argument a is defined as the endogenous or within-compartment measure of time since the particle introduction to the compartment. The equivalence between Assumptions 3A and 5A may also be explicitly demonstrated by deriving the conditional !low probability, or hazard rate, implied by Assumption 5A. Through basic prob ability relationships, the conditional probability that a particle present at time t leaves by t+ ll.t is equal to the joint probability that a particle is present to t and leaves by (t + ll.t), divided by the probability that the particle is present at time t (See ego Gross and Clark, 1975). In symbols, this conditional probability may be found as f(a)ll.t/S(a), from whence the hazard rate may be defined as f(a)/S(a). Therefore the assumed exponential distribution in Assumption 5A leads to the following model:

Model 5A: The conditional probability that a particle present at age aleaves by a + ll.t is

kexp(-ka)ll.t/ exp(-ka) = kll.t (5-4)

One important feature of the above model is that the hazard rate is not a function of the age a, i.e. that the particle "has no memory." One practical restriction of this model is that the transfer mechanism must not discriminate on the basis of the accrued age of a particle in the compartment. In summary it is clear that the formulations in 3A and 5A are equivalent, in 3A one assurnes an age-independent transfer rate and derives the exponential distribution whereas in 5A one assurnes an exponential distribution and derives an age-independent transfer rate. Therefore, it follows that all the results in Sections 3A and 3B, including the properties of the random vector [X(tl), X(t2)] in (3-7) to (3-10) and the model for estimation, hold for the present formulation.

5B Add~tional Results Goncerning Moments

Clearly, the formulation in Section 5A focuses on a continuous random variable, namely the reten­tion time R of a random particle, rather than on the discrete counting variable X(t), as in Section 3A. The new focus on retention times may be exploited to determine various statistical moments of R, such as its mean and variance. This leads to the following corollary to Assumption 5A:

Gorol/ary 5A: The me an and variance of R are

l)E[R] = / af(a)da = k- 1 , and

2)V[R] = / a2 f(a)da - (E[R])2 = k- 2

(5-5)

(5-6)

Of course, these moments, as weil as many others, are wellknown for the assumed exponential distribution.

VI. Multicompartment Stochastic Model Based on Exponential Retention Time Distribu­tions

6A Derivation of Model

The multicompartment model enables one to define a number of random variables to describe the transfer of a particle among the various compartments. Some of the more useful variables are as folIows:

120

Notation Set 6A: Let

1) R.] denote the reLention time during a single visit in t of a particle whose next transfer will be to i,

2) R. denote the retention time during a single visit of a random particle in i prior to its next transfer out of i (either to another compartment or to the exterior),

31' Ni; be the number of visits that a particle starting in i will make to j prior to its exit from the system,

4) Si; be the total residence time that a particle originating in i will accumulate in j during all

of its Ni; visits, i.e., S.; = l:~" R;, . 5) Si = L; Si; be the residence time in the system of a particle originating in i.

Consider now the assumption:

Assumption 6A: Let R.;, ~ exponential distribution (ki;).

The assumption implies that each hazard rate is age-independent, more specifically each rate is the constant k;;. This age-independent conditional transfer coefficient mayaiso be interpreted as a "force of transfer" to the various compartments. This gives rise to the following corollary:

Corollary 6A: The prob ability that a particle in i will transfer to j on its next departure is

p., = k.,/"Lki, = -k.,/k .. ,,,,. (6-1)

Note that this probability concerning the destination of particle transfer is also independent of particle age.

6B Additional Results Concerning Moments

Numerous expressions may be derived to describe the statistical moments of the random variables R., Si" and N". Consider first two matrix operations:

Notation Set 6B.l: Let

1) ZD be the diagonal matrix with elements (Zll,' .. , Znn), and 2) Z2 be the matrix of squared elements, !Z,~], for any n X n matrix Z.

The necessary notation for the mO::lents of R.'s is:

, Notation Set 6B.E. Let

1) 8(R) be a diagonal matrix of expectations with elements E[R.]. 2) V(R) be a diagonal matrLx cf variances with elements V[R.].

The following results are immediate generalizations of (5-5) and (5-6).

Theorem 6B.l:

1)8(R) = -Kr/ where K is the coefficient matrix, and

2)V(R) = 8(Rh (6-2)

(6-3)

The previous theorem gives the moments for the R. transit times. The moments for any conditional transit time R., may be found directly from (5-5) and (5-6), i.e. E[R.;,) = k.~l and V[R.;) = ki/.

The necessary notation for the moments of the Si, residence times is:

Notation Set 6B.9: Let

1) 8(S) be an n X n matrix of expectations with elements E[S,,), 2) V( S) be an n X n matrix of variances with elements V [S,,), 3) 8.(S) be an n-vector of expedations with elements E[S.). 4) V.(S) be an n-vector of vari3.:lces with elements V[S,).

The following results are given in Matis, Wehrly, and Metzler (1983):

Theorem 6B.E:

1)8(S) = _K- 1

2)V(S) = 28(S)8(S)D - 8(Sh 3)8.(S) = 8(S)1, where 1 is a vector of l's, and

4)V.(S) = 2[8(SW1- 8.(Sh

(6-4)

(6-5)

(6-6) (6-7)

These results are illustrated subsequently. Note that (6-4) is an inverse similar in form to (5-5) and (6-2), and that (6-5) is a difference of squared terms which is analogous to (5-6) and (6-3).

The notation for the moments of the N" numbers of visits is:

121

Notation Set 6B.4: Let

1) 9(N) be an n x n matrix of expectations with elements E[N;;], and 2) V(N) be an n x n matrix ofvariances with elements V[N;,].

The following results are given in Matis and Wehrly (1985):

Theorem 6B.9: Let K* be a normalized K matrix such that k:, = 0 anel ki; = -k;,/k;;. Tbeu

1)9(N) = (I - K*)-l, and

2)V(N) = 29(N)9(N)D - 9(Nh (6-8)

(6-9)

These moments also have the characteristic form that the expectation is an inverse and the variance is a difference of squared terms. Note that the off-diagonal elements defiiled as k:, in the theorem are age-independent transfer probabilities, i.e. the Pi, in (6-1).

6C. Application 0/ Moment Formulas

The formulas (6-2) to (6-9) involve only simple matrix inversion, multiplication and subtraction, and therefore the formulas are very easy to apply. In order to illustrate their use, consider the model schematic given in Figure 1 which was considered also in Matis and Wehrly (1985).

.......-k:n =l--l_kU =2_1 ~------------~ ~------~----~

~ k.o = 2

Figure 1. A specific two-compartment model with elimination from the peripheral compartment.

Using the previous definitions, one can obtain the following matrices involving K:

-K = [+2 -2] -1 +3

Substituting these into the theorems, oue has the following matrices of expectations:

9(R) = [1/02 0] 1/3

9(S) = [3/4 1/2] 1/4 1/2

9(N) = [3/2 3/2] 1/2 3/2

and the following matrices of variances:

V(R) = [1/04 0] 1/9

V(S) = [9/16 1/4] 5/16 1/4

V(N) = [9/4 9/4] 5/4 9/4

Also, the mean and variance for the total residence time in the system, often denoted as M RT and V RT may be found using (6-6) and (6-7) to be

[ 5/4] 9.(S) = 3/4 _ [17/16] V.(S) - 13/16

The interpretation of these moments is immediate. Suppose one introduces a particle into compartment 1 of the model in Figure 1. The expected retention times for each visit to compartments 1 and 2 are 1/2 and 1/3 respectively. Tbe expected residence times in compartments 1 and 2 are 3/4 and 1/2 respectively for a total of 5/4. The expected number of visits ia 3/2 to compartment 1 and 3/2 to 2 prior to departing the system. The variances of the residence times in 1 and 2 are 9/16 and 1/4 respectively, and the variance of the total time in the system is 17/16.

Tbe formulas in Section 6H assume that the model structure and the rate parameters are known. It is shown in Matis, Wehrly and Metzler (1983) that by substituting estimated rate parameters into tbe formulas one obtains -estimated moments. In finding the estimated moments, it is not necessary to use the covariance matrix from the stochastic model in the estimation of the k;, microparameters, any common weighting structure is sufficient though not necessarily optional for statistical precision.

122

These statistical moments for stochastic compartmental models have been very useful in practical applications. Of course, their chief limitation is that the compartmental structure is assumed known. However their advantages are many, including the following:

1) They are very easy to calculate from the k'J estimates. 2) Unlike half lives, they are easy to interpret for any compartment model and they are additive. 3) The formulas are valid for all models regardless of the paths of elimination. Non­

compartmental methods, on the other hand, are restricted to elimination only from com­partment 1 (the central compartment) and hence would not give the means nor variances of residence times for the model in Figure 1. (See e.g. Cobelli and Tofollo, 1985; DiStefano and Landaw, 1984; Matis and Wehrly 1985.)

4) The formulas can give the mean residence time of a drug at specific target compartments specified by the model. For example, in the three compartment model given in Matis, Wehrly and Metzler (1983), the model provides individual estimates for the mean time spent in each of two peripheral, target compartments.

5) The formulas also give the means for the number of visits, and variances of the variables. It is shown in Matis, Wehrly and Gerald (1985) that a treatment which might not alter the means could have a large noticeable effect on the other response variables. Apower study indicating the sensitivity of each of the variables to selected simulated treatment effec ts is given in Matis and Gerald (1986).

For these reasons, the author expects that the use of these statisLicdl moments from stochastic compart­ment models will increase substantially over the years. In light of these results, the main contribution of the stochastic model in most applications is not in its description of the process eITors, such as that given in (3-7) to (3-10). Instead, the main contribution lies in the new theoretical results and new methodology pertaining to residence time and related moments, such as those given in (6-9), which are based on the stochastic formulatioll.

VII One-Compartment Models Based on Nonexponential Retention Time Distribution

The stochastic one-compartment model with an exponential retention time was developed in Section 5, and it was observed in Model 5A, formula (5-4), that the assumption of an exponential retention time is equivalent to the assumption of an age-independent hazard rate. Most of the necessary notation for the generalized models with nonexponential retention time distributions was also given in Section 5. However, one definition will be reiterated due to its central importance in subsequent considerations:

Notation 7: Let kral denote the hazard rate function at time a, i.e. the conditional probability that a particle that has remained in the compartment for age aleaves by a + /lt is k( a)/lt.

It has also been indicated that one could solve for the hazard rate function for any retention t.i1ll'.' distribution through the relationship kral = f(a)/ S(a).

Statistical survival analysis has been used in biomedical and engineering applications to model the retention times, or "survival times," of units in a population. Ordinarily the nature of the data observed in survival studies is completely different from that observed in compartmental modeling. In survival analysis, for example, one might record the actual, censored survival times of a sam pIe of cancer patients. In compartmental analysis, the individual exit times are usually either not conveniently available or are too numerous to record individually; and instead one counts (or estimates) the number of survivors at fixed times. However, although the data and hence the estimation techniques would differ, the models in survival analysis may be used as a foundation for a generalized compartmental analysis based on nonexponential retention times.

The causal reasons for nonexponential retention times, and hence agp.-varying hazard rates, may be numerous. Two general reasons for sllch retention times in pharmacokinetk <'i'plications are non­instant mixing and comp,u·tment heterogeneiLy. Noninstant mixing, for example, is likely to occur in compartmental models with oral dosing. Nonhomogeneous compartments, on the other hand, are a natural consequence of the lumping inherent in dividing a body into only two or three compartments, for example, into a "central" and a "peripheral" compartment. In such survival modeling, the focus is not so much on the mechanism which produces the nonexponential retention time distribution, but rather on the shape of the distribution, or alternatively of the hazard rate function.

Three of the more common, nonexponential retention time distributions are considered in the subsequent models. The density and survivorship functions, the hazard rates, and the moments for the distributions are given in Johnson and Kotz (1970).

123

Model 7.1: Let R ~ Weibull (a,A > 0). This model implies the following:

l)f(a) = aAaaa - 1 exp[-(Aat]

2)S(a) = exp[-(Aa)a]

3)k(a) = aAaaa - 1

4)E[R] = r(1 + a-1)/A

(7-1) (7-2)

(7-3) (7-4)

Other moments could be written, but they become increasingly difficult. Note that for a = 1 one has the special case where R ~ exponential (A) and the above formulas reduce to those given in Section 5. The hazard rate decreases monotonically as a negative power of a for a < 1 and increases as a positive power of a for a > 1. One could fit the survival model, S(a), to observations y(tl)"'" y(tm ) from a compartmental system and thereby estimate the parameters a, A, and possibly X(O).

Model 7.t!. Let R ~ gamma (a,A > 0). This model implies the following:

l)f(a) = Aaaa - 1 exp(-Aa)/r(a)

2)E[R] = a/A 3)V[R] = a/A2

(7-5) (7-6) (7-7)

The survival and the hazard rate functions are not available in simple closed form expressions. Clearly, for a = 1, one again has the special case on an exponential. For a < 1, the hazard rate decreases monotonically, asymptoting to A; for a > 1, the hazard rate increases monotonically, also asymptoting to A. One could use numerical techniques to estimate the parameters a and A from data.

Model 7.9. Let R ~ Erlang (n, A); n = 1,2, ... ,; A > O. This model is a special case of the gamma where the shape parameter n is an integer. The model implies:

l)f(a) = Anan- 1 exp( -Aa)/(n - I)! n-l

2)S(a) = exp(-Aa) L(Aa)'/i!

n-l

3)k(a) = [Anan- 1 /(n - 1)!]f[L (Aa)' /i!]

4)E[R] = n/A 5)V[R] = n/A2

(7-8)

(7-9)

(7-10)

(7-11) (7-12)

A graph of several rate functions for various small n with A = 1 is given in Figure 2. Two qualitative features are apparent in the graph. Firstly, for n > 1, the rate function at age 0 is k(O) = 0, after which the rate increases. This provides an initial dampening of the passage probability of newly in­troduced particles. Secondly, the rate asymptotes to A as the age, a, increases. This implies that the age discrimination within the compartment diminishes, either rapidly or slowly depending on n, as the retention time increases. Both of the qualitative features are characteristic of data from nonhomoge­neous compartments and/or compartments with noninstantaneous initial mixing. The model also has a survival function which is easy to fit to data, and it has tractable moment formulas. For these many reasons, the Erlang retention time distribution has been ycry useful in practice, and it will be utilized extensively also in the subsequent generalized multicompartment modeling.

124

100 "=1

"=2

080

" 060 ~

'" ... 040 '" ~ 020

000 000 250 500 750 1000 1250 1500 17 50 2000

Time, t

Figure 2. Age-dependent hazard rate functions for some Erlang retention time distributions with small n and A = 1.

VIII Multicompartmental Models Based on Nonexponential Retention Time Distributions

Consider now the multicompartment generalization of the survival models in Section 7. The prin­cipal objective of survival analysis is to find nonexponential families of survival distributions wh ich are mathematically tractable and yet sufliciently rich in form to correlate weil with observed data. On the other hand, the multicompartment compartment models in Section 6 have a different objectivej they aim not only to describe the observed data but also to provide a rough, mechanistic description of how the data were generated. Of course, one chief limitation of the previous multicompartment models is their assumption of the exponential retention time distribution. The multicompartment models in this section are the intersection of the two previous approaches. A mechanistic system of compartments is envisaged with the drug flowing between the compartments. However, the processes determining the retention time of the drug within each compartment are viewed as being much more complex than the simple conceptualizations leading to constant hazard rates. The present multicompartment general­ization, called generalized compartmental analysis, does not attempt to describe the mechanics of the processes within each compartment but rat her to describe their net observed effect through age-varying hazard rates such as those present in SectiOll 7. The present approach therefore attempts to characterize fully the mechanistic flow pattern between compartments but to use non-mechanistic models with the smallest possible number of parameters to describe the within-compartment processes.

The most general model formulation would assume arbitrary retention time distributions for each R.) retention time, however the model is intractable. A more restricted, widely studied model is the semi-Markov formulation which was originally proposed in a compartmental context by Purdue (1975) and has more recently been investigated by Mehata and Selvam (1986). The semi-Markov model assumptions are as folIows:

Model 8Al: Let

1) R. retention time have the arbitrary distribution F. for i = i, ... , nj and

2) a.) be an age-invariant transfer probability from i to j.

The a., transfer probabilities correspond to the P.,'s given in (6-1). Therefore the mechanism deter­~ining. the sequential location of particles in the compartmental structure is the same as that given m SectlOn 6j only the retention time distributions within the compartments are different. This new model formulation is elegant, however analytical solutions for the p., (t) survival models, which would be fit ted to data, are available only for very restricted special cases.

One special case which is being used extensively by the author is the following model:

Model 8A2: Let

1) R. ~ Erlang (ni, A.), and

2) a.) = k., / Ai be the transfer prob ability from i to j, where Ai = -ku.

As noted in Section 7, the assumed Erlang distributions have hazard rate functions which are useful for a dass of problems where there is an initial dampening of the conditional transfer probability due to such phenomena as noninstantaneous mixing.

Model 8A2 may be solved by utilizing the following wellknown mathematical theorem concerning Erlang random variables:

Theorem 8: Let Ul be independent exponential (A) distributions for f. = 1,2, ... , n. Then R = Z7= 1 Ul is distributed as an Erlang (n, A) random variable.

. !he applicat~on o~ the theore~ in the present context enables one to generate the desired Erlang dl.stnbuted retentlOn tlmes by passmg through a sequence of identical exponential compartments, each wlth parameter Ai. In effect, one may define a set of n, exponential "pseudo-compartments" for each R. Erlang distributed random variable in the· model. These pseudo-compartments are not intended to have any mechanistic interpretation, rather they exist merely as a mathematical artifice to solve the model. After passing through a sequence of pseudo-compartments, a partide would transfer according to the a.) transfer probabilities. Thus Model 8A2 may be solved using the previous theory of Sections 4 and 6, although as noted subsequently it is a very special case with different types of solutions than previously outlined.

125

As an illustration, consider the model represented by the schematic in Figure 3. In this model, originally proposed in Matis and Wehrly (1984), let R1 ~ Erlang (nb Ad, R2 ~ Erlang (n2' A2), a12 = p, a10 = 1 - p, and a21 = 1. An equivalent representation with n1 + n2 pseudo-compartments is given in Figure 4. The latter model is reparameterized using the relationships Al = k lO + k 12 , A2 = k21J and a12 =.k12/ A1'

L.....------,_--J1--

P --1, __ --, R,-Eln, ,I,) R,-Eln, ",I .. ~ '-p

Figure 3. A generalized two-compartment model with Erlang retention times.

r-- .... _l .. __ ...... - .. ,r, ............... _ .............. _ ........ __ ........... l ___ .. ,: i l ........ _-_ .......... --........ 1 r .:11 I : , ~k,,~ k"~i k,. , r;-, r;-, r:--" i L.J~~~ L:...r-!~ ---..::....---rL..J-~- ~ I L ____ .... ____ ........ _ .... _____ ........ l ........ ] L ....... _ .... __ ............ _ ..... __ ...... _ .. ___ .. _ .... ~

+ k"

Figure 4. Representation of generalized (Erlang) two-compartment model using exponential pseudo-compartments.

The Kolmogorov equations may be written for the system of n = n1 +n2 exponential compartments, as before in Model 4A. Letting lItt) denote the expanded n x n matrix of probabilities, Model8A2 may be represented as follows:

Model8A9: iI(t) = iI(t)K'

The solution to Model 8A3 follows from previous theory as

Solution 8: lItt) = exp(K't)

(8-1)

(8-2)

In applying Solution 8, one musL considcr Lhe fact that the coefficient matrix K' has a special pattern form. The special structure does not affect the results given in Settion 6 for residence time mo­ments. Therefore one may obtain these moments for the pseudo-compartments and then aggregate them appropriately to obtain moments of the corresponding generalized (Erlang) compartments. However for most models of practical interest, the pattern in K' will yield equal and/or complex eigenvalues. Conse­quently the IIij(t) solutions will not be the sums of exponential models given in Corollary 4A1. Instead the solutions will tend to have other algebraic forms, thereby increasing the model flexibility without adding additional parameters. The solutions for a few simple models are presented subsequently.

Consider first small irreversible systems with Erlang retention times as specified in Model 8A2. Matis (1972) investigates the irreversible model with Rl ~ Erlang (n, A) and R2 ~ Exponential (k). The K' matrix has equal eigenvalues and the analytical solution for the survival function is

n-l

S(t) = Sn exp(-kt) + exp(-At) L (1- sn-i)(At)i li! i=O

where S = A/(A - k). The model is also discussed in France et al. (1985) and Matis (1987).

Assuming n is known, the model has only two parameters and may be easily fitted to data. The expected residence time in the system is E[Sl] = nlA + k- l . Hughes and Matis (1984) generalize this irreversible model to the case Rl ~ Erlang (nl' A d and R2 ~ Erlang (n2' A2)'

A reversible system such as that sketched in Figure 3, almost always has complex eigenvalues, which leads to solutions with damped oscillations. As an illustration, Matis and Wehrly (1984) consider the

126

model in Figure 3 with R l ~ Erlang (nl = 2, Al = 2),R2 ~ Exponential (A2 = 1), and 0<12 = 1/2. The solutions for the probabilities of parLicle loeation in the pseudo-eompartments are

IIu(t) = 0.255 exp( -0.304t) + exp( -2.35t)[0. 775 eos(1.03t) - 0.110 sin(t.03t)] (8-3)

II12 (t) = 0.266exp(-0.304t) + exp(-2.35t)[-0.266eos(1.03t) + 1.416sin(1.03t)]

IIdt) = 0.382exp(-0.304t) -I exp(-2.35t)[-0.382eos(1.03t) - 0.759sin(1.30t)]

The solutions for the generalized eompartments may be obta.ined by summing the above together, i.e.

Putt) = IIu(t) + II12 (t)

P12(t) = IIdt )

Although the analytieal solutions for the P,j(t) loeation probabilities for reversible Erlang models may be diffieult, the residenee time moments are relatively simple to ealculate from the mieroparameters. For example, the model illustrated above would have the eoeffieient matrix K' given below, from whenee using (6-4) one eould also ealculate the following matrix of mean residenee times, 8(S):

K' = [~2 ~2 ~ 1 1 0 -1 1 il

It follows that the mean residenee time (MRT) of a particle starting in eompartment 1 of the reversible generalized model may be ealculated using (6-6) to be E(Stl = 3, and it eould be shown that the eorresponding varianee (VRT) is V(Stl = 18.5. Reversible generalized models in general would have a similar striking differenee between the simplieity of the eomputations for residenee time moments and the eomplexity of the ealculations for the P" (t) loeation probabilities.

The eomplexity of the analytieal solutions of the P,j(t) for reversible Erlang models suggests a different approach to the problem of estimating the ki1 microparameters. A number of computer pro­grams, including PCNONLIN (Metzler and Weiner, 1985), solve the differential equations numerically and give the option of estimating model parameters direetly from the differential equation formulation of the model, such as from equation (8-1), rather than from the integrated solutions, such as those in (8-2) and (8-3). The differential equations formulation is immediate for the generalized eompartments with Erlang retention times, and henee the diffieult quest ions assoeiated with the nature of the eigenval­ues and the eomplexity of the analytieal solutions may be avoided using suitable numerieal proeedures and computer software. Of course, onee the parameters have been estimated, one may apply all of the results of Seetion 6B, as discussed previously, to obtain the estimated residenee time moments.

In summary, all of the advantages previously mentioned in Section 6C also hold for the present generalized eompartment models based on Erlang retention times. However the Erlang models have the obvious additional theoretieal advantage of a more generalized model formulation and often a praetieal advantage of redueed dimensionality as a result of Erlang approximations. It is the author's expectation that the use of such models will also inerease in time.

References

Chiang, C. L., 1980. "An Introduction to Stoehastie Processes and Their Applieations," Krieger, Huntington, NY.

Cobelli, C., and Toffolo, G., 1985. Compartmental and noneompartmental models as eandidate classes for kinetie modeling. TheOlY and eomputat.ional aspects, in: "Mathematies and Computers in Biomedieal Applieations," J. Eisenfeld and C. DeLisi, eds., North-Holland, Amsterdam.

DiStefano, J.J. III, and Landaw, E. M., 1984. Multiexponential, multieompartmental, and noneompart­mental modeling. 1. Methodologieallimitations and physiologieal interpretations, Am. J. Physiol. 246 (Regulatory Integrative Comp. Physiol. 15): R651.

Franee, J., Thornley, J. H. M., Dhanoa, M. S., and Siddons, R. C., 1985. On the mathematies of digesta flow kineties. J. theor. Biol. 113:743.

Gibaldi, M., and Perrier, D., 1982. "Pharmacokineties," ed. 2, Mareel Dekker, New York.

Godfrey, K., 1983. "Compartmental Models and Their Applieations," Aeademie Press, New York.

127

Gross, A. J., and Clark, V. A., 1975. "Survival Distributions: Reliability Applications in the Biomedical Sciences," Wiley, New York.

Hughes, T. H. and Matis, J. H., 1984. An irreversible two-compartment model with age-dependent turnover rates, Biometries. 40:501.

Jacques, J. A., 1985, "Compartmental Analysis in Biology and Medicine," ed. 2, Univ. of Michigan Press, Ann Arbor, MI.

Johnson, N. L., and Kotz, S., 1970. "Continuous Univariate Distributions - 1." Wiley, New York.

Kalb!leisch, J. D., Lawless, J. F., and Vollmer, V. M., 1983. Estimatioll in Markov models from aggregate data, Biometries. 39:907.

KodelI, R. L., and Matis, J. H., 1976. Estimating the rate constant in a two-compartment stochastic model, Biometries. 32:377.

Matis, J. H., 1970. Stochastic compartmental analysis: Model and least squares estimatioll from time series data. Ph.D. Dissertation. Texas A&M University, College Station, TX.

Matis, J. H., 1972. Gamma time-dependency in Blaxter's compartmental model, Biometries. 28:597.

Matis, J. H., 1987. The case for stochastic models of digesta !low. J. theor. Biol. 124:371.

Matis, J. H., and Gerald, K. B., 1986. On selecting optimal response variables for detecting treatment effects in a two-compartment model, in: "Modelling of Biomedical Systems," J. Eisenfeld and M. Whitten, eds., North-Holland, Amsterdam.

Mati~, J. H., and Hartley, H. 0., 1971. Stochastic compartmental analysis: Model and least squares estimation from time series data, Biometries. 27:77.

Matis, J. H., and Wehrly, T. E., 1984. On the use of residence time moments in the statistical analy"if; of age-dependent stochastic compartmental systems, in: "Mathematics in Biology and Medicine," S. L. Paveri-Fontana and V. Capasso, eds., Springer-Verlag, New York.

Matis, J. H., and Wehrly, T. E., 1985. Modeling pharmacokinetic variability on the molecular level with stochastic compartmental systems, in: "Variability in Drug Therapy," M. Rowland, L. B. Sheiner, and J. L. Steiner, eds., Raven, New York.

Matis, J. H., and Wehrly, T. E., and Gerald, K. B., 1985. Use of residence time moments in compart­mental analysis, Am. J. Physiol. 249 (Endocinol. Metab. 12): E409.

Matis, J. H., Wehrly, T. E., and Metzler, C. M., 1983. On some stochastic formulations and related statistical moments of pharmacokinetic models, J. Pharmacokinet. Biopharm. 11:77.

Mehata, K. M., and Selvam, D. D., 1986. A dass of general stochastic compartmental systems, Bull. Math. Biol. 48:509.

Metzler, C. M., and Weiner, D. L., 1985. "PCNONLIN User's Guide." Statistical Consultants, Lexing­ton, KY.

Purdue, P., 1975. Stochastic theory of one compartment and two compartment systems. Bull. Math. Biol. 36:577.

Rescigno, A., and Segre, G., 1966. "Drug and Tracer Kinetics," Blaisdell, Waltham, MA.

Rescigno, A., and Matis, J. H., 1981. On the relevance of stochastic compartmental models to phar­macokinetic systems, Bull. Math. Biol. 43:245.

Wagner, J. G., 1971. "Biopharmaceutics and Relevant Pharmacokinetics," Drug 1nt., Hamilton, 1L.

128

MODELING FIRST-PASS METABOLISM

John G. Wagner

John G. Searle Professor of Pharmaceutics Professor of Pharmacology and Staff Member of The Upjohn Center for Clinical Pharmacology University of Michigan, Ann Arbor, MI 48109

INTRODUC'rION

Table 1 lists 66 drugs which exhibit a first-pass effect. A reasonable definition of a first-pass drug is one that ex­hibits a significant arterial-venous concentration difference at steady-state. Also, after oral administration, and when there is complete absorption of the dose, the area under the blood concentration-time curve is less than the area under the curve when the drug is administered intravenously at the same dose. Figure 1 depicts the meaning of the area (AUC).

Figure 2 indicates the meaning of clearance. Table 2 lists examples of drugs which obey Michaelis-Menten elimination kinet­ics. Most first-pass drugs, when administered orally, or via the hepatitis artery or portal vein, exhibit Michaelis-Menten elimination ~inetics as a result of the relatively high drug concentrotions reaching the drug metabolizing enzymes in the liver.

WHAT DOES AUC r1EAN?

AUC = AREA UNDER THE CURVE

C

o T

Fig. 1. Meaning of ar~a under the curve.

WHAT IS THE CLEARANCE?

CLEARANCE = RATE OE ~LIMINATION = AMOUNT ~LIMINATED BLOOD CONCENTRATION AUC

= ~ = __ --'I~ti.t:!Pu>LJT'-"RAö..T~E~ __ _ AUC STEADy-STATE CONCENTRATION

Fig. 2. Meaning of clearance.

129

Table 1. 66 Drugs Which Have a First-Pass Effect

Aldosterone Alprenolol Amitryptyline Aspirin Beclomethasone

Dipropionate Bromocriptine

Mesylate (F~O.06) Butorphanol

Tartrate Chlorpromazine HCl Cimetidine (~O.7) Codeine Cortisone Cyclobenzamine HCl Desmethylimipramine Dihydroergotamine

Mesylate Diltiazem HCI Dobutamine HCI Dopamine HCI Epinephrine Ergoloid Mesylates Ergotamine Tartrate Estradiol Ethinylestradiol

(P::>'O.4)

Flunisolide Fluorouracil 5-Fluoro-2 -

Deoxyuridine Guanethidine

Sulfate Hydralazine HCI Imipramine HCI Iscethorine HCI

& Mesylate Isoproterenol

Sulfate Isosorbide

Dinitrate Levallorphan

Tartrate Lidocaine HCI Meperidine HCI 6-Mercaptopurine Metaproterenol

Sulfate Methoxamine HCI Methylpredniso­

lone (F::>'O. 85) Methyltestoste­

rone Metoprolol

Tartrate Morphine Sulfate Nalbuphine HCI Naloxone HCI Neostigmine

Nifedipine Nitroglycerin Norepinephrine

Bitartrate Norethindrone

(F':!O.65) Nortriptylene

HCI Oxyprenolol Oxyphenbutazone Penicillamine Pentazocine HCI

& Lactate Phenacetin Phentolamine

HCI & Mesylate Phenylephrine

HCl &

Bitartrate Prednisone

(F~O. 85) Progesterone Propoxyphene HCl

& Napsylate Propanolol HCl Ritodrine HCI Salicylamide Salbutamol Testosterone Timolol Maleate Verapamil HC 1

Source of information was Drug Information 84, American Hospital rormulary Service, American Society of Hospital Pharmacists.

Table 2. Drugs Which Obey Michaelis-Menten Kinetics in the Therapeutic Dose Range

130

Phenytoin Cinromide Zonisamide (CI-912) 2-Deoxy-5-fluorouridine

(FUDR) 5-Fluorouracil (5-Fu) 5-Bromo-2~Deoxyuridine

(BUDR) Bromouracil (BU) Theophylline Propranolol Nicardipine

Dil tiazem HCI Propoxyphene Verapamil HCI Hydralazine HCI Nitroglycerin Salicylate Salicylamide PenicilIamine Prednisolone 4-Hydroxybutyrate Ethanol Verapamil

Table 3. Symbolism

AUC - Area under the concentration-time curve after a single dose of drug (mass vol- l time).

C - Concentration of drug (mass vol- l ) * ss SS S8 ss -1 C - (Ci-Co)/ln(C/C o) or (CA - Cv )/ln CA~/CV) (mass vol )

Ci - input concentration to liver in perfused liver preparation (mass vol- l )

Co - output concentration from liver in perfused liver prepara­tion (mass vol- l )

Co - mean outflow concentration of a distribution of outflows (mass vol- l )

CÄs - steady-state arterial drug concentration in intact animal or man (mass vol- l )

C~s - steady-state venous drug concentration in intact animal or man (mass vol- l )

f CLint - intrinsic clearance of free (unbound) drug eguivalent to i -1 i CLm/fub (vol time ) ,

Clm - intrinsic metabolie clearance = ~~o [CL~sl (vol time- l ) ss

Cl m - steady-state metabolie clearance at dose rate Ro (vol time- l )

CL~s- steady-state systemic clearance (vol time- l )

cl~~- steady-state concentration in compartment #1 when drug is administered intravenously (ie. into compartment #1) at a zero order rate, Ro ' in Figure 4.

cr~~- steady-state concentration in compartment #1 when drug is administered orally (ie. into compartment #2) aT a zero order rate, Ro ' in Figure 4.

CLR - Renal clearance D - axial dispersion coefficient of drug (length 2 time- l ) DN - DA/QL (dimensionless) € - the coefficient of variation of the distribution of Vm/Q

over all sinusoids of the liver lim Ei - intrinsic extraction ratio of drug= R ~O[Essl

(dimensionless) 0

Ess - steady-state extinction ratio of drug at dose rate Ro = (Ci-Co)/Ci(dimensionless) l'

Fi - inLrinsic bioavailability = ~~O[F5S1/ (dimensionless)

F - bioavailability at dose rate R = c~s/CÄs = Co/Ci ss (dimensionless) 0

fUb fraction of drug free (unbound) in whole blood (dimensionless)

Km - l-Uchaelis constant (mass volume- l ) for venous equilibrat­ion model

K~ - Michaelis constant (mass volume- l ) for undistributed si-nusoidal perfusion model

L - Length of the liver (length) Q - flow parameter (volume time- l ) % - infusion rate (mass time- l ), equal to r RN = Vm/QKm = CL~/Q

(/2 (Vmax - V max ) 2

Vm - maximal velocity of metabolism (mass time-li v - velocity of metabolism (mass time- l ) vma ' - mean Vma of distribution Z x_ distance afong liver normalized to the length of the

liver (L)

131

~ RATE IN

SCHEMATIC DIAGRAM OF "WELL-STIRREO" MODEL

~()-D--ELIMINATION

SCHEMATIC DIAGRAM OF "PARALLEL-TuBE" MODEL

Fig. 3. Schematic of concen­tration gradients in a sinusoid.

THEORETICAL MODELS OF HEPATIC ELIMINATION

There are a number of theoretical models of hepatic elimin­ation and the steady-state equation of the four most important models are shown in Table 4. Figure 3 illustrates the principal difference between the venous equilibration (or 'well-stirred') model and the sinusoidal perfusion (or 'parallel tube') model. The two models differ in their assumptions of the concentration of drug within the hepatocyte at the site of metabolism. In the venous equilibration model it is assumed that the liver is a weIl-mixed compartment and that the concentration at the site of metabolism is the same as the concentration of drug exiting from the liver. This is the same assumption as is made in li­near compartmental analysis. In the undistributed sinusdoidal perfusion moel it is assumed that the substrate concentration declines exponentially along the sinusoid and the logarithimic

* average (C) of inflow (Ci) and outflow (Co) concentration (see eq. 4) is the concentration appearing in the Michaelis-Menten (eq. 3). Note thatequations 2 and 6 which apply to these two

C· theories, differ only in that ~ -1 in equation 2 is replaced

'-0 by In Ci/CO in equation 6. Now, Ci -1 is equal to In Ci/Co

only when Ci/Co = 1.001, which isCO a case when the drug is not even a 'first-pass one. At all other values Ci is not

Co -1

equal to In Ci/Co. Hence I believe the Km values and the in­trinsic clearances, Vm/Km, cannot be the same in the two models, although others (Roberts and Rowland, 1986) have assumed them to be equal. In the distributed sinusoidal perfusion model it is assumed that there is a distribution of Vm/Q values in the liver and that € is the coefficient of variation of this distribution and Co is the me an outflow concentration of drug. The effect of the introduction of this new variable is to change the shape of the plot of logarithm of reciprocal of bioavailability vs. infusion rate (Ro ) or rate of metabolism (v) compared with the linear relatlonship between these variables in the

132

undistributed sinusoidal perfusion model. Robinson (1979) dis­cusses these differences in detail.

The dispersion model of Roberts and Rowland (1968) involves a second order differential equation (eq. 11, Table 4) and two fundamental parameters called the axial dispersion number (ON) and the efficiency number (~).

Table 4. Steady-State Equations

Venous equilibration moael (Gillette, 1971):

v = R o v Co Q(C,-C ) = r.,

J_ 0 -l~;';';'-+'-;'C-( 1)

V !:1 K C. (2) m ( 1 _ 1)

iI1 0 Q - QC

o Unaistributed sinusoidal perfüsion mocel (Bass et al, 1976) :

v = R o

C.-C 1 0

Vm C * Ci -Co ~(3) whece C = ln[C./C 1 K +C 1 0

m

V m ( 5)

( 6)

(4)

Distributed sinusoidal perfusion ~odel (Bass, Robinson and Bracken, 1978):

* C.- C C= 1 0

In [Ci/Cl o

* where Ci> C > Co (7)

where Co is the me an outflow concentration of a

distribution of outflows so that v = (C. -C) (8) 1 0

r(1-0.5r~ 2)=ln C.- In C- (9) 1 0

where r = v max /QKm = Vm/QKm and ~= the coefficient

of variation of the distribution of V /Q over all m

sinusoids of the liver = ( 0/~)2 and 0 2 = (V =V---)2 max max max and V­max

Dispersion ~odel

a 2 c _1 'i.L. az2 DNa Z

mean Vmax (10)

(Ro~2rts and Rowland, RN

- D .C = 0 (11) N

1986) :

where C is the concentration of drug in blood within the liver normalized to the input concentration, Z is the distance along the liver normalized to the length of the liver (L), ON is the axial dispersion number and RN is the efficiency number.

133

MODELING OF STEADY-STATE DRUG CONCENTRATIONS OF FIRST-PASS DRUGS

Rowland et al (1973) and Pang and Rowland (1977) covered the venous equilibration model under first order conditions ex­tensively. Hagner et al (1985a) extended the treatment to Michaelis-Menten elimination kinetics. Figure 4 shows the two compartment open model with peripheral compartment elimination according to Michaelis-Menten elimination kinetics and central compartment elimination by renal excretion used by Wagner et al (1985a). In the model, we assumed that when you measure the blood concentration of drug you are sampling compartment #1 and that the concentration in compartment #2 is equivalent to the concentration of drug in the liver and that the instanta­neous rate of metabolism is VmC2/{~ + C2). For hepatic arter­ial, portal or oral administration, it is assumed that input is into compartment #2 as shown at the top of Figure 4, while for intravenous administration it it assumed that input is in­to compartment #1 as shown at the bot tom of Figure 4. In liver perfusion, if drug is infused into the reservoir this is analosous to intravenous administration, and if drug is pumped directly into the liver this is analogous to oral ad­ministration. Equations 12 and 13 apply under these two conditions.

134

(-L) Q

COMPARTMENT MODEL m VENOUS

EQUIIIBRATlOI~ MODEL ARE THE SAME THII~G

LIVER + OTHER TISSUES

I"PUT ~METAlllJTE<S)

HEPATIC ARTERIAL ADMINISTRATION

BOLUS OR ZERO ORDER INPUT

L1VER + OllER TISSUES

ADMINISTRATION IV eR VIA PERIPHERAL ARTERY

BOLUS OR ZERO ORDER INPUT

Fig. 4. Two· compartment open model with peripheral elimination obeying Michaelis-Menten kinetics studied by Wagner et al (1985).

( 12)

(13)

It has been stated that according to the well-stirred model, oral bioavailability is independent of hepatic blood flow (Morgan et al, 1985) and that organ blood flow does not enter into the first-pass effect (Pang, 1986), but both of these statements are in error. Since bioavailability is given by

ci~;/c~~;, equations 12 and 13 indicate that oral bioavailability

depends on Q. The later equation 20 also shows this.

Equations 14 to 19 apply to the undistributed sinusoidal perfusion model of Ba~ et al (1978) at steady-state.

* 0 Ro=V=QKmln~C +V (14) i m

Since Fss = Co/Ci then equation 14 may be written as equation 15.

* Ro = v = Q Km In F ss + Vm (15)

Equation 15 mayaIso be written as equation 16.

_ {Vm-~o} Q Km Fss = e (16)

And equation 15 may be rearranged to give equation 17, V

~l In F ss = m + Ro (17) ---*-

Q Km Q Km t 1- .,.' t

Ordinate Intercept Slope Abscissa

which is the equation of a straight line as indicated. Figure 5 shows examples of plots based on equation 17 for verapamil as reported by Wagner et al (1985c).

0.9 0.8

0.7

0.6

10.5

-ii 0.4

G

&. 0.3

= 1°.2

O. J

VERAPAMI L

SUBJECT), ," _ ...... -

," --' / _.... '"

./- .- /::~:~/

.-/'

/////

.-

/ /

/

/ .- / ..... - ,/

/ /

/ /

,/ ,,/ SUBJ ECT 4

00510g Roh rmg/hr)

511

Fig. 5. Predictions of systemic availability of verapamil according to the undistributed sinusoidal perfusion-model calculated from li­terature data (Wagner, 1985c).

135

When Ro = 0:

And:

lim R _O[Fssl

o e

* -Vm/Q Km (18)

(19)

Returning to the model shown in Figure 4, Wagner et al (1985a) showed that equations 20 through 23 apply. The shape of a plot of Fss vs Ro ' D8sed on Equation 20, is shown in Figure 6.

F ss = V R 1 + ( m - 0)

Q Km

1 (20)

Equation 20 rearranges to give equation 21. Fss - 1

R o = v = Q Km (--) + V (21) Fss m

Comparing equations 15 and 21 one can see that the un­distributed sinusoidal perfusion model differs from the venous equilibration model in that In Fss is replaced by (F sS - 1 ) Rearrangement of equation 21 gives equation 22, Fss

F ss -1 = _ Vm + [ 1 ~~- R o (22)

Fss Q Km Q Km 1 1 t t

Ordinate Intercept Slope Abscissa

which is the equation of a straight line as indicated.

136

10

09

08

07

06

~ 05

-

-~ ..

04

03

02

01

12 16 2 24

DOSE RITE (R o) IN Mg/NOUR

Fig. 6. Plot of Fss vs Ro based on equation 20.

When Ro = 0:

lim Fi = Ro~O [Fssl V

1 + m Q Km (23 )

Figure 7 illustrates how, under first order eonditions, one may diseriminate between the venous equilibration and un­distributed sinusoidal perfusion models on the basis of plots based on protein binding. This is based on assuming that in-

trinsie metabolie elearanee, CLi , may be equated to the produet m of the fraetion of drug unbound in blood, ~ or fUb, and the intrinsie elearanee of free (unbound) drug. Under these

eonditions equation 18 beeomes equal to e- fuCLif/Q and

equation 23 beeomes equal to 1/(1 + f CL~/Q). Thus, if bio­availability is plotted on a semilogalith~ie seale versus f the undistributed sinusoidal perfusion model will yield a u straight line, while the venous equilibration model will yield a bowed line as illustrated. The venous equilibration model will yield a straight line when (l-F)/F is plotted versus f u as shown inset in Figure 7.

~ -.::;

10 0.9

Oß

Q

0.5

04

0.3

02

01

0.

0.00

OOe

:Y:6

CD5

00'

OC3

J 6 08

F, = _.....!..I--;f-

~ 1 + 0

05 06 0'" :8 09 10

Fig. 7. IllustratioD of how, under first order eonditions, one may diseriminate between the venous equilibration and undistributed sinusoidal perfusion models based on plasma protein binding data.

137

1-----:=l--..,:V~M!ll.::·K:w..Ml_7 Metabohte 1 V M2 , KM2

+-~-~ Metabol~te 2 VMn • KMn -+--'=--=----7 Me t a bo 1 ~ t e n

n

_ ~ _ ~ + v M2C + ____ + VMnC '= l: ~Ml.~C dt - KMl + C KM2 + C KMn+C 1=1 MI

Where:

n = Number of parallel Mlchaells-Menten equatlons.

VM1 = MaXImum velocIty of metabol 15m for the 1 th

Mlchae lls-Menten pa thway.

KMl = MIchaelIs constant for the 1 th pathway.

C = Concentratlon of unchanged drug 1n the blood.

Fig. 8. Parallel Michaelis-Menten metabolite model of Sedman & Wagner (1974).

We must digress somewhat to consider the case when two or more metabolites are formed and both formations obey Michaelis­Menten kinetics. The situation is depicted in Figure 8 where there are N parallel paths. In such situations, Sedman and Wagner (1974) showed that often data appears as if there were only one Michaelis-Menten path and equation with a pooled V or V and a pooled K or K as depicted in Figure 9. They m p m p showed that sometimes the pooled parameter values are concen­tration or dose dependent while in other cases they are dose­independent. The meaning of the pooled parameter values in the latter case are shown in Figure 10. Examples of pooled parameter fits which appear dose-independent are shown in sub­sequent Figures 11, 12, 14, 15, 16, 19 and 20.

Considering the intact animal or man rather than the per­fused liver we may replace C. by the steady-state arterial

l

drug concentration, C~s, and Co by the steady-state hepatic

venous drug concentration, C~s, when drug is infused to steady-

state either intravenously or intraarterially. In the absence

138

Where:

V P

--t--K"---~ Metabolites p

V Pooled Michaelis-Menten maximum velocity. p

K Pooled Michaelis constant. p

C = Concentartion of unchanged drug in blood.

Fig. 9. Effect of pooling in the model of Fig. 8. is to produce the model shown he re in which there appears to be only one Micahelis-Menten path.

of urinary excretion of unchanged drug and these substitutions, equation 1 (Table 4) leads to equations 24 through 38 in Table 5.

Note that when CLR = 0, the equation for ci~; (eq. 13), based

on the model of Figure 4, becomes the same as equation 33, (Table 5) for the steady-state venous concentration and the

equation for ci~~ (eq. 12) based on the model of Figure 4 be­

comes the same as equation 36 (Table 5) for the steady-state arterial concentration. I have been most interested in applying the equations shown in Table 5 (see following pages) to the fitting of real steady-state data in the intact anima I and man.

F.LUOROURACIL

Figures 11 and 12 show the fits of hepatic aterial and he­patic venous plasma concentrations of 5-fluorouracil versus infusion rate of 5-fluorouracil in two different cancer patients (Wagner et al, 1986) to equations 36 and 33, respectively. The estimated parameter values were for patient #1 (Figure 11): Vm = 1.528~ moles/kg/min, K = 6.749)U M (plasma), plasma Q = 0.0291 m L/(kg x min) corresponding to blood Q = 0.0446 x/(kg x min) and

for patient #3 (Figure 12): Vm = 4.027JU moles/kg/min, Km = 8.688

JU M (plasma), plasma Q 0.0556 L/(kg x min) corresponding to blood Q = 0.0861 L/(kg x min).

REFERENCE: A, J, SEIfo'AN AND J, G, WAGNER

J,flHAAt.w:OKINET ,BIOPHARM,2.:149-1ffl(974)

Fig. 10. Dose-independent equations for V and K after m m Sedman and 'Wagner (1974).

160.------------------;.__--,

140

~ 80

:;; 60

10 12 14

I I I

010 30 270

Fig. 11. Bits of hepatic arte­rial and hepatic venous steady­state plasma concentrations of 5-fluorouracil to eqs. 31 & 34 (Table 5) for subject #1 from Wagner et al (1986).

139

16

"., o

Tab

le

5.

Co

mp

aris

on

o

f fir

st

ord

er

and

n

on

lin

ear

eq

uati

on

s w

hen

dru

g is

ad

min

iste

red

b

y th

e p

eri

­g

hera

l in

trav

en

ou

s ro

ute

an

d ste

ad

y-s

tate

arte

ria

l (u

suall

y h

ep

ati

c art

eri

al)

an

d h

ep

ati

c

ven

ou

s p

lasm

a co

ncen

trati

on

s are

si

mu

ltan

eo

usl

y m

easu

red

. N

ote

th

at

C

has

the

sam

e m

ean

ing

as:

Css

d

C

ss

0 an

.

has

the

sam

e m

ean

ing

as

CA

v 1

~~~E_2E_Y~E~

Ste

ad

y-s

tate

h

ep

atl

c

ven

ou

s co

ncen

trati

on

Ste

ad

y-s

tate

m

eta

bo

lic

cle

ara

nce

Ste

ad

y-s

tate

arte

ria

l co

ncen

trati

on

Ste

ad

y-s

tate

sy

stem

ic

cle

ara

nce

Bio

av

ai 1

ab

il ty

Ex

tractl

on

ratl

o

Eff

icie

ncy

n

um

ber

fir

st

ord

er

or

lntr

insic

Css

v

CL

1 m

CSS

A

CL

i S

Fi

Ei

Rn

R O

~

m

l.

c~s

Km

Ro

Vm

~

Km

1 1

R (0

-+~)

o C

L' m

Q C

Li

__

m_

1

Q+

CL

i 1

1 -

+

-.

m

Q

CL

1

( 24

)

( 25

)

Als

o:

(26

) ( 27

)

m

CSS

.J

L.,

. 1

..::L

-( 2

8)

CSS

Q

+CL

1 C

Li

A

m

1 +

.....2!

! Q

CL

i 1

-F

. m

__

___ L

(2

9)

1 ;:

~~I

1 +

-Q.,.

m

C

L1 m

V

CL

i m

m

Q

Km

0

-(3

0)

CSS

v

CL

SS

, m

R

__

0_

CL

sS

m

R

__

0

_

CSS

v

No

nli

near

K;R

O

v -

R

m

0 V

V

-R

_

_

m_

,

.J!!

.-.2

K

+CS

S

K

m

v m

CL

i -(~R

m

Km

0

CL

ss

, C

Li

_ (.

JL

..)

Km

(Css

_ C

SS)

A

v m

m

R

K R

C

SS

A ~

+

mo

Q

V

-

R

m

0

CL

ss-

1 1

( 31

)

()2

)

( 3

3)

(34

)

( 35

) S

-1 ~

1 +

:-r,

Ö +

V

-R

Ö

C

L1-(_

)R

m

0

m

Km

0

CS

S ~

1 F

..L

V

-R

(3

6)

ss

CSS

Q

+CL

SS

1 (m

0

) A

m

+

Q

Km

CL

sS

( 3

7)

1 E

1

-F

' --

!!!-

----ÖK~--

ss

SS

Q+CL~S

1+ -----

(Vm

-Ro

)

V -

R

CL

sS

RS

s,

. ...!!!

...-2

m

(38

) i'I

Q

Km

,..

60r-------------------------------------,

:: 30

~

20

- 10

~

02 04 INFUSION RAH

I , I

010 30 90 INFUSION RAH

Fig. 12. Fits hepatic venous concentrations equations 31 & #3 from Wagner

Bromouracil (BU)

molecular welght 190.98

luM = 190.98 ug/L or ng/ml

H[PAIIG AR!fRIAl

H[PA1IGVmUS

06 OB 10 12 14 16 (,M 'kg, M,n I

I I

135 180 210 Z70 I mg kg d,,1

of hepatic arterial and steady-state plasma of 5-fluorouracil to 34 (Table 5) for subject et al (1986).

-bromo-2~deoxyuridlne (BUOR)

molecular welght 307.11

luM = 307.11 ug/L or ng/ml

Fig. 13. Structures of 5-bromo-2'­deoxyuridine and bromouracil.

141

~ .:'1,0 a:: Cl ::> 111

~25

z Q !;izo a:: f-Z

'" ~ 15 0 u

'" VENOUS ~ 10

'" ..J I>.

010203040506070

INTRAVENOUS INFUSION RATE(amoles/kg/hr)

Fig. 14. Fits of arterial and hepatic venous plasma concentra­tions of 5-bromo-2'-deoxyuridine of dog LDK05 to equations 31 and 34 (Table 5).

5-BROMO-2'-DEOXYURIDINE

24

22

~20

.3 a:: 18 c ::;)

m 16 u. 0 Z 14 0

Ei 12 a:: ~ UJ 10 0 Z

8 < ::::E U) < ...J a...

8

6

4

2

o 10 20 30 40 50 60

INTRAVENOUS INFUSION RATE (,Umoles/kg/min)

Fig. 15. Fits of arterial and hepat~c venous plasma concentrations of 5-bromo-2'-deoxyuridine of rabbit P0326C to eqs. 31 and 34 (Table 5).

We have also studied 5-bromo-2-deoxyuridine (BUDR) and its major metabolite (BU) in the dog (Andrews et al, 1987) and the rabbit (Knol et al, 1987). The structures are shown in Figure 13. The fits of the steaäy-state arterial and venous BUDR con­centrations in one dog are shown in Figure 14 and in one rabbit are shown in Figure 15. For the dog the parameter values were: Vm = 0.9186)U mOles/kg/:r..Ln, Km = 2.60.,.u m (plasma) and plasma

Q = 0.0388 L/(kg x min). ?or the rabbit the parameter values were: Vm 1.591-ll mOles/kg/min, Km = 7.01-ll m (plasma) and

plasma Q = 0.0541 L/{kg x min).

PROPRANOLOL

Wagner (1985b) fitted the steady-state venous plasma con­centrations of Silber et al (1983) to equation 1 with Co = C~s. The fits for the 4 available subjects are shown in Figure 16. The V values ranged from 382 to 513 with an average of 470 mg/

m day and the K values ranged trom 33.7 to 59.7 with an average m of 44.4 ng/ml. In this case the drug was administered orally at dose rates of 40, 80, 160, 240 and 320 mg/day in divided

142

doses every 6 hours. The eSs values are average steady-state v concentrations equal to Aue O-~~where ~ is the dosage inter­val or 6 hours.

There were reports tnat sustained release formulations of propranolol had lower bioavailability than conventional medi­cation. Hence, I (Wagner, 1985) derived equation 41 in an attempt to explain this.

(AUe 0 -")'J)zero (AUe 0 -f')bOlus

where r

Q =

Ro/Vm

V'}'-D -~-----~

VKm

(39 )

( 40)

( 41)

In equation 41 (AUe 0 -.,..,) zero is the area under the blood concentration-time curve during a dosage interval at steady­state when drug is administered by constant rate (zero order) infusion and (AUe 0 -~) bolus is the corresponding area when the drug is administered by bolus injection every ~ hours. Literature values of the volume of distribution, v, and average values of Vm and Km from the fittings (Figure 16) were substi-

tuted into equations 41-43 and the value of the area ratio was calculated for various Ro values. Results are plotted in Fig.

17 and shows a minimum value of the ratio at a dose rate of about 180 mg/day which is the one usually used to test sus­tained-release formulations. Hence this approach does explain the reported lower bioavailability of the sustained-release formulations of propranolol. However, verapamil under the same conditions gives an area ratio equal to or greater than ab out 0.95 at all dose rates.

24Ü SuP JE: r A 80

60

80

20 o 80 160 240 320

o 80

80 160 240 320 R o = (mg Oayl

SJf-1_!~1 1 D

60 120 SlJ!:lJf-L T C

..::: 40

2C 40

j~O 0 80 160 240 320 Ra'" (mg da,.!

Fig. 16. Fits of steady-state venous plasma concentrations of propranolol of Silber et al (13) to eq. 13 as reported by Wagner (1985b) .

143

" v '"

l ;9 VlHAPAMll

0 U :0

'" U8

o 1,'6 Pi 1U~'RANOL UL

.< _____ '--____ -'-____ --'-____ ----J....J

100 200 30t) ..JUU

DUS!::. RATE Imgidc.lY)

Fig. 17. Plots of steady-state area ratio for zero order to bolus administrations vs. dose rate for propranolol and verapa­mil as reported by Wagner (1985b).

ADINAZOLAM MESYLATE

Wagner et al (1987) carried out a steady-state study in 8 normal volunteers in which a solution of the benzodiazepine, adinazolam mesylate, was used to provide loading doses, then aliquots of the solution were given hourly at dose rates of 1, 2 or 3 mg/hr until steady-state was achieved. Minimum steady­state venous plasma concentrations of the unchanged drug and its N-demethyl metabolite just before the next dose and hourly were measured. Four of the subjects exhibited linear kinet­ics and results with one of those subjects are shown in Figure 18. The other four subjects exhibited Michaelis-Menten elimi­nation kinetics and results with one of those subjects are shown in Figure 19.

144

125,------------------::-1

a a a

aa a

2 4 6 8 10 12 14 16

$HABf-STArE ptASn mCElTUml OF AOIUIOLAI I[sYlATE (ni)

500',----------------,

~20

~10

OL-~~~-~-~-,~O~~,~2-~,4~~,6 STEAOY-SUH PlASU mmmlln_ Of ADIU/OLU USYLm (,nil

Fig. 18. Plots indicating :inear kinetics in one of 4 sUbjects ex­hibiting such kinetics for adina­zolam mesylate at steady-state (From Wagner et al, 1987a).

NICARDIPINE

100 --;;--iii ~ 75 ~ ~

50 -- 25

50 100 150 200

SmOY-STAH PLASMA COMCEMTRATIOM OF AOIMAlOLU MESYLATE (nM)

1000 :L ;:::;::

,~ 800

m a-=~ ~-~--~ ~~ 400 ~~ ~~ ~~

=~ :;;~

~: 200

50 100 150 200

STEAOY-STATE PLASMA COMCENTRATIOM OF AOIMAlOLAM MESYLATE (nM)

Fig. 19. Plots indicating Michaelis-Menten elimination in one of 4 sUbjects exhibiting such kinetics for adinazolam mesylate at steady-state (From Wagner et al, 1987a).

Wagner et al (1987) administered oral doses of 10, 20, 30 and 40 mg of nicardipine every 8 hours for 3 days and steady­state venous plasma concentrations were measured during the 10th dosing interval. The average steady-state concentrations were calculated as Aue o-~~ and are plotted vs. dose rate in mg/hr for the six subjects who participated in Figure 20. The solid lines are the fits to the Michaelis-Menten steady-state equation 31 (Table 5). As you can see, the fits are excellent and, along with the other two drugs propranolol and adinazolam mesylate above, are excellent examples of the pooled Michaelis­Menten parameter-concept (Figures 8 to 10).

145

'[2JCBJECT' • 0

;

~ , 4 ." .-

~ SUBJECT '"

:0 'uL]

x = Css of nicardipine (ng/ml) Y = Infusion rate. Ra (mg/hr)

Fig. 20. Fits oi steady-state venous plasma concentrations of nicardipine to e~uation 31 (Table 4) as reported by Wagner et al (1987b) •

CONSEQUENCES OF MICHAELIS-MENTEN ELIMINATION KINETICS

When Michaelis-Menten Kinetics are Operative:

1. AUC increases more than proportionately with increase in dose.

2. Steady-state concentrations increase more than pro­portionately with increase in dose.

3. Clearance decreases with increase in blood concentration,

4. Steady-state clearance is less than single dose clearance.

5. The percentage of drug metabolized via the Michaelis­Menten path decreases with increase in dose.

6. The slower the rate of absorption, the smaller the AUC for a given dose - if limiting elimination half­life is less than about 12 hours.

7. The time required to reach steady-state increases with increase in the dose.

8. Bioavailability increases with increase in dose rate. 9. Rectilinear plots of blood concentration versus time

are pseudo-linear for the upper 2/3 rds of their length but this is NOT zero order kinetics.

Model Distinction Based on Bioavailability

Table 6 lists steady-state bioavailability equations under conditions of both first order kinetics and Michaelis-Menten elimination kinetics. These equations form one basis of distinguishing between the theoretical models of hepatic eli­mination.

146

Tab

le

6.

Mo

del

Ven

ou

s eq

uil

ibra

tio

n

Un

dis

trib

ute

d

sin

uso

idal

perf

usio

n

Dis

trib

ute

d

sin

uso

idal

perf

usio

n

Dis

pers

ion

~

-...I

Ste

ad

y-s

tate

B

ioav

ail

ab

ilit

y

Eq

uati

on

s w

here

F

C

IC.

css/

CA

sS

o l

v

Fir

st

Ord

er

Eli

min

ati

on

M

ich

aeli

s-M

en

ten

E

lim

inati

on

F.

1 ( 4

2)

F 1

1 (4

0)

l V

S

S

(V

-R

) V

R

1+

_m

__

1+

m

0

1+

m

(1-~)

Q K

Q

K

Q K

V

m

m

m

m

V

(Vm

-Ro

) V

R

.-

-2!L

-

m

0 (4

7)

--(1

--)

F.

e Q

Km

(4

3)

F e

Ti<

e

QK

V

l

ss

m

m

m

In

1: C

. V

R

-l

n

F In

_1

:=

--E!}

(1

--2

..)

(48

) ss

F C

Q

I<

V

ss

o m

m

C.

In

1: C

. In

..J

. r

-0

.5

E2 r

2 +

R

(44

) -l

n

F In

. ..

! C

ss

F C

r

(1-0

.5E

2r

2)-

0 ss

0

wh

ere

co

eff

icie

nt

of

vari

ati

on

R

r

2 E

-2

(1

-E2

-r-)

(49

) o

f V

IQ

&

wh

ere

r

= V

IQ

K

(5

0 )

V

m

m

m

m

e -1

F =

4

a i

(l+

a)

2ex

p [(

a-l

) I

2DN

1-

(l-a

) 2ex

p [

-(a

+l)

12D

Nl

(45

)

1/2

w

here

a

= (1

+

4DN

RN

) ,R

N

= f

. CL~

• p

i Q=

pCL

I Q

, 1

l u

b

lnt

m

DN

=

äf, D

= ax

ial

dis

pers

ion

co

eff

icie

nt

of

dru

g

wit

hin

th

e

liv

er,

A

= cro

ss-s

ecti

on

al

are

a

of

blo

od

in

th

e

liv

er,

Q=

liv

er

blo

od

fl

ow

ra

te,

L

= l~ngth

of

the

liv

er.

Til

ere

is

no

so

l uti

­ti

on

y

et

for

the

Mic

haeli

s-M

en

ten

case.

REFERENCES

Andrews, J. C., Knutsen, C., Stetson, P. L., Wagner, J. G. and Ensminger, W. O. Hepatic pharmokinetics of 5-bromo-2'-deoxyuridine (BUOR) in a canine model. Clin. Pharmacol. Ther., 41:160 (1987), Abstract 11 10-2.

Bass, L. Keiding, S. Winkler, K. and Tygstrup, N. Enzy­matic elimination of substrates flowing through the intact liver. J. Theor. Biol., 61:393-409 (1976).

Bass, L., Robinson, P. and Bracken, A. J. Hepatic elimination of flowing substrates: The distribu­model. J. Theor. BioI., 72: 161-184 (1978).

Gillette, J. R. Factors affecting drug metabolism. Ann. N.Y. Acad. Sci., 179:43-66 (1971).

Knol, J. A., Stetson, P. L., Wagner, J. and Ensminger, W. o. Hepatic pharmacokinetics of 5-bromo-2'­deoxyuridine (BUOR) in the rabbit. Clin. Pharmacol. Ther., 41:171 (1987), Abstract #PPF-3.

Morgan, o. J., Jones, o. B. and Smallwood, R. A. Modeling of substrate elimination by the liver: Has the albumin receptor model superseded the well-stirred model? Hepatology 5:1231-1235 (1985).

Pang, K. S. and Rowland, M. Hepatic clearance of drugs. I. Theoretical consideration of a "well-stirred" model and a "parallel-tube" model. Influence of hepatic blood flow, plasma and blood cell binding, and the hepatocellular enzymatic activity on hepa­tic drug clearance. J. Pharmacokin. Biopharm. 5: 625-653 (1977). 11. Experimental evidence for acceptance of the "well-stirred" modelover the "parallel-tube" model using lidocaine in the per­fused rat liver in situ preparation. ibid 5:655-680 (1977). 111. Additional experimenrar-evidence supporting the "well-stirred" model, using meta­bolite (MEGX) generated from lidocaine under varying hepatic blood flow rates and linear con­ditions in the perfused rat liver in situ preparation. ibid 5:681-699 (1977).

Pang, K. S. Metabolic first-pass effects. J. Clin. Pharmacol. 26:580-582 (1986).

Roberts, M. S. and Rowland, M. A dispersion model of hepatic elimination. 1. Formulation of the model and bolus considerations. J. Pharmacokin. Biopharm. 14:227-260 (1986). 2. Steady-state considerations - influence of hepatic blood flow, binding within blood, and hepatocellular enzyme activity. ibid 14:261-288 (1986). 3. Application to metabolite formation and elimination kinetics. ibid 14:289-308 (1986). --

Robinson, P. J. Aspects of mathematical liver kinetics. The steady state statistical mechanics of hepatic elimination, a thesis. Oepartment of Mathematics, University of Queensland, Brisbane, Australia, 1979.

Rowland, M., Benet, L. Z. and Graham, E.G. Clearance concepts in pharmacokinetics. J. Pharmacokin. Biopharm., 1:123-136 (1973).

Sedman, A. J. and Wagner, J. G. Quantitative pooling of Michaelis-Menten equations in models with parallel metabolite formation paths. J. Pharmacokin. Bio­pharm. 2:149-160 (1974).

148

Silber, B. M., Holford, N. H. G. and Riegelman, S. Dose­dependent elimination of propranolol and its major metabolites in humans. J. Pharm. Sci., 72: 725-732 (1983).

Wagner, J. G., Szpunar, G. J. and Ferry, J. J. A non­linear physiologic pharmacokinetic model: I. Steady-state. J. Pharmacokin. Biopharm., 13:73-92 (1985a).

Wagner, J. G. Propranolol: Pooled Michaelis-Menten para­meters and the effect of input rate on bioavaila­bility. Clin. Pharmacol. Ther., 37:481-487 (1985b)

Wagner, J. G. Comparison of nonllnear pharmacokinetic parameters estimated from the sinusoidal perfusion and venous eguilibration models. Biopharm. Drug Dispos., 6:23-31 (1985c). .

Wagner, J. G., Gyves, J. W., Stetson, P. L., Walker­Andrews, S. C., Wollner, I. S., Cochran, M. K. and Ensminger, W.D. Steady-state nonlinear­pharmacokinetics of 5-fluorouracil during hepa­tic arterial and intravenous infus ions in cancer patients. Cancer Res., 46:1499-1506 (1986).

Wagner, J. G., Rogge, M. C., Natale, R. B., Albert, K. S. and Szpunar, G. J. Single dose and steady-state pharmacokinetics of adinazolam after oral administration to man. Biopharm. Drug Dispos., accepted January 7, 1987a.

Wagner, J. G., Ling, T., Mrooszczak, E. J., Freeman, D., Wu, A., Huang, B., Massey, I. T. and Ridge, R. Single intravenous dose and steady-state oral pharmacokinetics of nicardipine in healthy sUbjects. Biopharm. Druq Dispos., accepted April 17, 1987b.

ACKNOWLEDGEMENTS

Figures were reproduced with permission of the journal and publishers as folIows:

Figures 5, 18, 19, 20 - Biopharmaceutics and Drug Disposition, John Wiley & Sons Ltd., England.

Figures 11 and 12 - Cancer Research, Waverly Press, Inc. rigures 16 and 17 - Clinical Pharmacology and Therapeutics,

The C. V. Mosby Company.

149

SATURABLE DRUG UPTAKE BY THE LIVER

EXPERIMENTS AND METHODOLOGY

Ludvik Bass

MODELS,

Department of Mathematics, University of Queens land,

Australia

1. INTRODUCTION: BASIC FORMULATIONS

Observations pertaining to physiological and biochemical dynamies in

vivo have to be interpreted in terms of biochemical kinetics transposed

from the familiar homogeneous (test-tube) phase into the appropriate

physiological setting. As a result of this transposition, anatomical

structures, capillary and eellular permeabilities, diffusion eoefficients,

loeal metabolie rates, flow rates, transit times etc. enter (explieitly or

implieitly) into any interpretation of data. Modelling is then threatened

from one side by losing the biology in oversimplifications, and from the

opposite side by so complicating the models for the sake of realism that a

multitude of adjustable parameters can get no grip on real data which are

limited in number and accuracy. A fruitful middle way cannot be found in

this subject without careful study of experimental data and designs.

We shall be concerned with physiological struetures ealled eapillary

beds (such as liver, museIe, brain ete.) in which blood flow entering and

leaving through large blood vessels is manifolded into a very large number

of microscopie eapillary flows, so that contact between blood and organ

cells is facilitated. There are two important classes of mathematical

models of uptake and release of substances by capillary beds (or observed

regions thereof): compartmental models and single-capillary models. For a

recent account of formulations and applications of such models see

Lambrecht and Rescigno [lJ. Except in eertain limiting circumstances,

models belonging to the two classes are incompatible: in practice their

uses are complementary rather than competitive. Compartmental models seek

to retain homogeneous-phase kinetics by representing the eapillary bed as a

151

finite set of (interacting) test-tubes: this means that an attempt is made

to replace partial differential equations by sets of ordinary differential

equations. Single-capillary models seek to represent the capillary bed as

an ensemble of functionally identical capillaries acting in parallel. In

simplifying the transposition of kinetics from the test-tube to the intact

organ, each cl ass of models is prone in its own way to the fallacy of

averages, which consists of replacing the mean of a function by the

function of the mean of its argument. The resulting errors need to be

corrected or at least bounded by quantifying such biological

heterogeneities of the organ as have been disregarded in perpetrating some

significant form of the fallacy of averages. Certain statistical sample­

splitting maneouvres will be shown to be particularly appropriate to

hypotheses-testing associated with identifying and remedying fallacies of

averages.

We illustrate the foregoing remarks in the simplest way that shall

lead into fruitful generalizations. We consider steady elimination of a

blood-borne substance by the intact liver. The elimination occurs by

irreversible enzymatic transformation of the substance (calIed substrate in

this context) by enzyme moleeules located in cells which are distributed

throughout the hepatic capillary bed.

In the homogeneous phase, the rate Vhom of the transformation of

substrate of concentration c by enzyme may be given by various kinetic laws

[2], depending on the mechanism of the enzyme-substrate interaction. Thus,

v c max~'

0.1) with positive constants Vmax ' K, describes Michaelis-Menton kinetics. It

is generalised by HilI kinetics

n > 0 (1. 2)

and by substrate-inhibition kinetics

(1. 3)

V is the greatest value of Vh attainable (at c ~ ~ in (1.1) and (1.2), max om

152

at c = JK1K2 in (1.3»; it is an extensive quantity proportional to the

number of enzyme molecules present in the system. By contrast, the

positive constants K, KH, K1, K2 are intensive quantities pertaining to

molecular enzyme-substrate interactions. In (1.1) and (1.2), Vhom increases monotonically with c to a maximum value Vmax ; these are examples

of saturation kinetics. For small values of c, Vhom in (1.1) and (1.3)

tend to proportionality to c. This limit of first-order kinetics is a

necessary feature of any realistic kinetics: (1.2) is not valid at

concentrations c « ~, where it must be replaced by a more complicated

expression [2].

We now transpose homogeneous-phase kinetics into the intact liver.

Let steady hepatic blood flow of rate F carry the substrate convectively

into the liver at the concentration ci' and out of it at the concentration

c . The steady rate of elimination is then o

v F(c.-c ) > 0 1 0

(1. 4)

For any saturation kinetics such as (1.1) or (1.2), increasing ci (and

hence co) to sufficiently high values makes V tend to Vmax ' so that Vmax/F

is the maximum possible input-output (arterial-venous) concentration

difference that can be observed across the liver. Thus V can be max determined independently of models.

It is natural to ask whether a mean value of the observed

concentrations c., c can be constructed such that if it is put ~n (1.1) in 1 0

place of c, the result will be the physiological V satisfying (1.4), in

place of Vhom . Preliminary guesses, such as ca = (ci +co)/2, or the spatial

mean c taken over the liver, yield examples of the fallacy of averages. As

a first attempt [3J, we consider a substrate rapidly equilibrated between

blood and eliminating liver cells which provide a spatially distributed

sink of the substrate. Putting the x-axis along the bood flow, with the

inlet at x = 0 at outlet at x L, the depletion of the predominantly

convective [3] substrate flux Fc by elimination in the iDterval x, x + dx

is modelIed by using (1.1) locally:

F dc c - p(x)dx c-+l[ , (1.5)

153

where now e(x) varies with x between the observed boundary values

e(O) = e. e(L) = e . l' 0'

and p(x)dx is the fraetion of the organ Vmax in the

interval (x,x+dx), SO that

L

J p(x)dx =

o

V max

Separating and integrating (1.5), we obtain

x

F(e.-e) + FK in 1 p dx , e. J 1 e

o

(1. 6)

(1. 7)

determining a spatial profile e(x) whieh ean be eompared at least semi­

quantitatively with results of autoradiography of the liver [4]. Putting

x = L and using (1.6), we obtain the input-output relation, independent of

the form of p(x),

e. 1

e o + K tn

e. 1

e o

Using (1.4) to eliminate F, and defining

we obtain

e. - e 1 0

e = tn c.je ' 1 0

(1. 8)

(1. 9)

v = V ~,( 1. 10) (1. 10) max e + K

whieh has the same form as the homogeneous-phase relation (1.1). The

fallaey of averages is eireumvented by the use of the effeetive mean value A A

e (1.9). It is easy to show [5] that e is the harmonie mean of the linear

interpolation of e. and e in the interval (O,L). Henee e < ca' and 1 0

(1.11)

It follows that the fallaey of averages beeomes unimportant for

e. »V /F (high flow rate or high input eoneentration); only then does 1 max

the liver eliminate as a homogeneous eompartment. For the kinetics (1.2),

154

it can be shown similarly [6] that

where

v

c n

1

V m~

n > 0

n ~ I.

For example, c2 = (c.c )2. We shall see below that the forms (1.9), 1 0

(1.12)

(1.13)

(1.10), (1.12) and (1.13) are important in the designs of experiments with

intact livers.

A closer study of hepatic architecture shows the blood flow manifolded

through many (107-108) microscopic conduits, called hepatic sinusoids.

These conduits are lined with the eliminating cells and are just wide

enough to pass red blood cells, which prevent the development of systematic

velocity distributions such as Poiseuille distributions found in large

vessels: the total flow rate through the sinusoid is adequate to describe

effects of flow on elimination. The convective transit time through a

sinusoid is much longer than the transverse equilibration time of the

substrate in each sinusoidal cross-section, and much shorter than the

diffusion time along the sinusoid [3]. The foregoing calculations apply

therefore to uptake by a single sinusoid if the organ values V, V and F m~

are replaced by their sinusoidal counterparts v, vm~ and f, while the

intensive constants K, ~, KI , K2 are left unchanged.

We can now reconsider elimination by the intact liver by using this

model of a single sinusoid as an element of an ensemble of N sinusoids

acting in parallel and having a common input concentration ci' The

validity for the organ of equations (1.4), (1.8) and their consequences

would be recovered if all sinusoids had the same values of v = V IN m~ m~

and f = F/N: for then vmax/f = Vm~/F, and Co would be the same for all

sinusoids according to (1.8). The resulting single-capillary

(undistributed) model of organ elimination would be merely an

interpretation of (1.4), (1.8). In the biological reality, v If is max always distributed over the ensemble of sinusoids, and the observed outflow

concentration is the flow-weighted mean of the various sinusoidal Co

155

values. In these circumstances the use of (1.4), (1.8) and of their

consequences is another form of the fallacy of averages which is the more

serious, the greater the dispersion of v If over the ensemble. We shall max

see how the quantification of this consideration permits the dispersion to

be estimated from data.

The foregoing considerations form a starting point for several

generalizations [5], such as modelling of time-dependent processes; of

consecutive enzymatic reactions; investigation of the zonal structure of

hepatic metabolism along the blood flow, and of its self-organization. We

shall illustrate the uses of some of these generalizations by applications

to selected experiments.

2. TIME-DEPENDENT ELIMINATION

We return to (1.5) in order to generalize it to time-dependence. We

note first that the flux depletion term in (1.5) is more generally d(Fc),

but that F has been assumed independent of position along the conduit.

Since the fluid is practically incompressible, this assumption holds even

if the cross-sectional area of the conduit varies with position. It is

only when the solvent is also taken up through the walls of the conduit, as

for example water from kidney tubules, that the extra term cdF = c(dF/dx)dx

becomes important in the kinetics of uptake of solutes [7]. Here we shall

not pursue this interesting set of problems, but confine ourselves to F

independent of x, though possibly varying with time t.

When concentration c is not steady, there is a time-change c(Adx) in

the amount of substrate between cross-sections of area A placed at x and

x + dx (volume Adx); and now this term plus the flux increment Fdx balance

the rate of elimination given on the right-hand side of (1.5). That

equation is thus generalized to

oc oc c A(x) äF + F(t) ox - - p(x) C'+Y ' (2.1)

provided that the previously asumed transverse equilibration of the

substrate keeps up with the time-changes to be modelied [7J. Again, (1.6)

holds for p(x). Given the functions A(x). F(t), p(x) and appropriate

boundary and initial conditions, (2.1) determines c(x,t) throughout each

sinusoid, or equivalently throughout the undistributed (single-capillary)

model of the liver.

156

Equation (2.1) is of first order because of the absence of a diffusion

2 2 term - Da c/ax. In addition to apriori reasons given in the Introduction

(comparison of convective and diffusive transit times through the

sinusoid), this essential simplification is supported by results of

experiments with single capillaries (see below), and on the intact liver.

A short input pulse c.(t) always results in a liver output transient c (t) 1 0

substantially dispersed in time. This might be attributed either to

dispersion of the pulse by longitudinal diffusion along each convective

pathway (8], or to the distribution of convective transit times amongst the

many parallel convective pathways which are reunited at the outlet

("convective spaghetti" (9]). The key result that supports the latter

interpretation, and hence the absence of a diffusion term in (2.1), comes

from the analysis of output transients of sets of substances ranging from

labelied red cells to tritiated water, all contained in the input pulse

~10]. Plausible adjustments of the different volumes of distribution of

the substances within the liver make the output transients of these

substances coincide precisely, despite orders-of-magnitude differences in

diffusion coefficients. It is therefore difficult to escape the conclusion

(9] that the convective pathways traversed by the substances vary enough to

account for all the dispersion observed.

When c « K throughout the interval (O,L), (2.1) is linearized because

its right-hand side becomes - p(x)c/K. When F is time-independent, a more

powerful method of linearization used experimentally is to superpose an

* unsteady tracer concentration c (x,t) on a steady concentration profile

c(x) of its mother substance. Putting c*+c in place of c in (2.1),

* neglecting c as compared with K, and making use of (1.5) satisfied by c(x)

we obtain

* * * A ac F ac c ar- + ax - - p C+I{" (2.2)

* As c(x) is varied experimentally by varying c., tracer pulses c (x,t) 1

explore elimination (uptake) at different levels of saturation of the

enzyme (11,12J. Linear problems of this kind, with constant A, F and p but

with a slowly equilibrating cellular layer along each sinusoid, have been

solved and applied to data [12]. By contrast, in what follows we shall

study and apply solutions of the non-linear equation (2.1), including

unsteady flows F(t) and arbitrary enzyme distributions p(x) [5J.

157

We introduce the new dependent variable

u = c + K tn c . (2.3)

As du dc(I+K/c), (2.1) becomes linear in u:

A( ) du + F(t) du ( ) x. äT dx - - p x • (2.4)

As evaries from 0 to ~, u inereases monotonieally from 4» to ~ : u and

c are in one-to-one eorrespondenee. Given a value of u, the eorresponding

unique e is obtained readily by numericalor graphical methods.

We assume that c(O,t) = c.(t) is given (observed) at all times t, and 1

we wish to calculate c(x,t) and especially the outlet concentration c(L,t)

= co(t). The corresponding boundary values u(O,t), u(L,t) follow from

(2.3). Solutions satisfying initial conditions c(x,O) are given in [7J.

The equations of characteristics of (2.1) are

dx dt YrtT = A\xT =

du prx; (2.5)

The first of these describes the motion of an element of substrate between

inlet and out let. We note that the velocity dx/dt is not, in general, the

velocity of blood [5,10J. The motion x(t') of an element which will appear

at the out let at time t is given by

xCt, ) t,

I A(Odt I F(v)dv, t - T(t) ~ t' ~ t , (2.6)

0 t-T(t)

where the transit time T(t) is given implici tly by:

L t

IA(Odt I F(v)dv (2.7)

0 t-T(t)

The family of trajectories x(t') is so parametrized by the time t of exit

that x(t'=t-T(t» = 0, x(t'=t) = L. In view of (2.6), the second of

equations (2.5) can be integrated with respect to time:

158

Putting t,

relation

t l

u[x(tl),t'] - u[O,t-T(t)] J t-T(t)

p[x(t")] dt" . A[x(fff)]

t and using (2.7) and (2.8), we obtain the input-output

(2.8)

(2.9)

which is to be taken together with (2.7) defining T(t). We now consider

three applications of these general results.

A. A liver function test in humans

The functioning mass of the liver in situ may be measured by

saturating it with a non-toxic substrate of some appropriate liver enzyme,

and estimating Vmax and other clinically relevant kinetic parameters from

the time-course of elimination of the substrate. In modelling this

elimination, we shall use the foregoing equations in the sense of the

single-capillary (undistributed) model of the organ. As the rate F of

liver blood flow is kept time-independent (as far as possible) in such

tests, we use (2.5) to substitute dx/F for dt'/A(x) in (2.9). Using (1.6),

we obtain

c (t)+ K en c (t) o 0

c.(t-T) + K en c.(t-T) - V /F, 1 1 max

(2.10)

where, from (2.7)

T L

J A(Odt/F const . (2.11)

o

We note that FT is the volume of distribution of the substrate in the liver

[5]. Differentiating (2.10) with respect to time we obtain

[e (1+K/c ) lt o 0 [e.(1+K/c·)]t T .

1 1-(2.12)

159

Thus under saturation (c »K), c (t) = c.(t-T)j in the opposite limit of o 0 1

first-order kinetics (c.«K), such parallelity holds for the logarithmic 1

derivatives.

6

4

2

15

FIG. 1.

Time course of galactose concentration in the artery (ci' upper curve) and hepatic vein (co' lower curve) in

a human subject after a single injection. Data points inserted from [13J.

Fig. 1 shows hepatic elimination of a saturating dose of galactose in

man, by irreversible phosphorylation by the enzyme galactokinase which has

K ~ 0.2 m molle. In the saturated region the parallelity of Ci and Co is

apparent, as weIl as the constancy of the slopes c. ~ c 1 0

Vd is the volume of distribution of galactose in the body.

Vmax/Vd , where

If F is

measured, V can be estimated as F[c.(t-T)-c Ct)J in the saturated max 1 0

region. Typical human values are Vd '" 10 e, F", 1 e/min, T", Imin,

V ~ 2 m mol/min ([5J and references therein). max

160

Next we consider the neighbourhood of the sharp turn of Co in Fig. 1,

which occurs at about 46 min. The corresponding value of ci (earlier by

T~l min) is about 2 m mol/t ~ lOK. We therefore neglect K/c. in (2.12) and l.

regard ci as a constant:

c o

C. l.

+ K/c ,ci o

(2.13)

We note first that c o

c./2 when c l. 0

K: the tangent to co(t) drawn with

half the constant slope ci touches the co-curve at Co = K (Fig. 1). We

thus obtain the satisfactory value K ~ 0.16 m mol/t directly from the

patient's hepatic outflow. Next we consider how sharp the turn of the

patient's co(t)-curve can be according to the model. Differentiating

(2.13) and iterating, we find

c o

c o

(c +K)3 ' o

(2.14)

. 2 which has the maximum value 4(c.) 1(27K) at c = K/2. In a time-interval

l. 0

dt the maximum relative slope change, C Ic , is 4c.dt/(9K), some 39% per o 0 l.

minute which is consistent with the appearance of a kink in the curve in

Fig. 1. In accord with the second of equations (2.13), large patients with

reduced liver function have much slower turns of the c -curve. o

Although the foregoing account of the data in Fig. 1 in terms of the

single-capillary (undistributed) model appears to be successsful, it is

amended when the dispersion of sinusoidal v If over the ensemble of max

sinusoids is introduced [5].

B. Uptake from unsteady flow through a single capillary

If a substance carried by blood through a capillary is escaping

through pores in its walls, the effect on the input-output relation between

ci(t-T) and co(t) is the same as that of irreversible elimination at the

walls, so long as none of the escaped substance returns into the capillary.

The foregoing input-output relations can therefore be used to determine the

value of capillary permeability. Crone et 81. [14] succeeded in monitoring

161

the intracapillary potassium concentration in single perfused capillaries

from the frog mesentery by two K+-sensitive microelectrodes placed at a

distance L from each other. A potassium-rich pulse was injected upstream

of both electroces, and it generated a transient excess concentrations

c.(t-T) at the upstream electrode and c (t) at the downstream electrode. 1. 0

The experimental use of a single capillary removed the problem of

heterogeneity of convective pathways, but at the cost of making the small

flow rate unsteady by the injected pulse. A fully quantitative

determination of capillary permeability requires therefore an application

of (2.9) and (2.7).

For the single capillary under consideration we take A(x)

p(x) = const = V IL. Equation (2.9) becomes therefore max

V c (t) + K In c (t) = c.(t-T(t)J + K In (c.(t-T)J - ~ T(t) o 0 1. 1. fiL

const and

(2.15)

Moreover, escape of potassium through the walls of the capillary is by

first-order kinetics (c«K), so that the first terms in both sides of

(2.15) are to be neglected. The permeability-surface area product PS of a

leaky capillary, which is to be determined, has the same quantitative

effect on input-output relations as the ratio Vmax/K in irreversible

enzymatic elimination by first-order kinetics (5]. Proceeding to the limit

of first-order kinetics in (2.15) and replacing V IK with PS, we obtain max

(2.16)

where T(t) is given by (2.7) with the left-hand side simplified to AL.

'I'his coupling of (2.16) and (2.7) was removed provisionally by Crone et a1.

[14J, who used in (2.16) the approximation T(t) = const = T(O), where t = 0

designates the first appearance of excess potassium at the downstream

electrode. Thus T(O) is the initial value of T(t), observed directly as

the time-difference between the first appearances of excess potassium at

the two electrodes. As the determination of PS was made from data spread

over aperiod t > 0 during which F(t) was falling appreciably, T(t)

actually increased montonically from T(O), so that most of the pulse of

excess potassium spent longer than T(O) in the capillary. Hence the true

PS was overestimated by using T(O) = T(t) in place of (2.7). When the data

[14] were fitted numerically to solutions of the set (2.16), (2.7), the

162

provisional estimates [14J of PS were halved. Moreover, the detailed

fitting of the accurate single-capillary data to (2.16), (2.7) subjected

the modelling to quite severe testing, free of the complications of whole­

organ studies. The quantitative success of the fitting [5J supports the

subsequent use of this modelling (without diffusion along the capillary)

for elements of the ensemble of capillaries representing an intact organ.

c. Uptake from pulsating blood flow

Blood flow through capillary beds in vivo is pulsatile; the

attenuation of pulsation in the venous outflow is due to the randomization

of the phase in the convective network, rather than to absence of pulsation

in the capillaries. We therefore consider the effect of a periodic rate of

blood flow,

F(t) F(t+T) > 0 (2.17)

on Michaelis-Menten elimination by a capillary or by a single-capillary

model of an organ. We assume again A(x) = const and p(x) = const leading

to (2.15), and assume in addition that the input concentration is steady:

ci(t) = const. Then c(x,t) is periodic throughout the capillary with the

same period c (though not with the same phase at all x) as F(t) since the

form of (2.1), and its boundary conditions, are unchanged on replacing t

with t + T. If outflow sampies over many per iods are pooled and then

analyzed, the observed concentration is equal to the flow-weighted time-

average cover one period: o

c FCiF 0 0

The uptake rate corresponding

t+r t+T

I F(v)co(v)dv/ I F(v)dv (2.18)

t t to (1.4) is then

v F(c.-c ) 1 0

(2.19)

1'0 bring out the effect of pulsation on such long-term elimination, we

compare c with the concentration c (F) which would be observed at the o 0

out let if the flow was steady at the rate F;

(1.8) with F = F.

c (F) is thus the solution of o

163

Several observable results follow from (2.17) for any form of F(t).

Without loss of generality we can write

T(t) nr + q(t)r , o S q(t) < 1 (2.20)

where n is zero or a positive integer. This is apparent by considering the

area under F(v) according to (2.7). That equation (with A=const.) thus

becomes

AL

t

nrF + J F(v)dv

t-qr

The factor T(t)/AL in (2.15) becomes, using (2.20) and (2.21):

T/AL n + q

nF + qFt-qr,t where

t

F J F(v)dv t-qr,t qT t-qr

(2.21)

(2.22)

(2.23)

is the mean flow rate between t - qT and t. The following cases are of

particular interest.

(a) When q = 0, we have T/AL = I/F from (2.20), (2.21). As ci = const,

(2.15) and (2.18) give c = const = c o 0

c (F): when an integral number of o

volumes Fr just fits into the volume of distribution AL, the pulsation has

no effect on time-averaged elimination in that volume. The same result is

obtained when n is large (T»r): as F cannot exceed the peak value t-qT,t

of F(t), T/AL from (2.22) again tends to l/F as n tends to infinitYj the

many periods r within T make the effect of elimination in the incomplete

period q insignificant.

(b) When n = 0, we have T < r from (2.10) and T/AL = I/Ft - qr ,t from

(2.22). If moreover q « 1 (T«T), Ft t tends to F(t) by (2.23): the -qT,

elimination is quasi-steady in the sense that c (t) is calculated from o

164

(2.15) by setting c. = const and T/AL = I/F(t). Then c produced by 1 0

pooling the outflux F(t)c (t) over one period T according to (2.18) is the o

same as if we mixed steady outfluxes from an ensemble of parallel conduits

all having the same ci and the same fraction of Vmax ' and steady flow rates

distributed as F(v) in t < v < t + T. We shall prove below that c (F) < c o 0

for any shape of F(v). This means that V(F) > V according to (2.19): when

there are many transits within aperiod of pulsation, mean elimination is

less than elimination from the mean flow.

These effeets of pulsation on elimination can be illustrated

explicitly by considering the pulsation

F F + F1 coS(2TIt/T) , (2.24)

with the coeffieient of variation e of the temporal distribution of flow

rate given by

t+T

e 2 } J (F-F)2dt/ F2

t

(2.25)

For small F1/F, a straightforward calculation [5] to order e 2 yields

;; /e (F) 1 + 1 2 max 1 + 0 FT [V ]2 [ c (F)]-3 [Sin(TI~L)12

2" e -- -,- AL • (2.26) o 0 FK TI--

FT

This shows explicitly how changes in elimination can come about by changes

in the pulse rate at a fixed mean flow rate.

We have e o

c (F) (no fallaey of averages) under saturation when o

c »K, or when the time AL/F of transit with the mean flow is an integral o

multiple of the period T, or when AL/F »T. When AL/F « T, the last

factor in (2.26) tends to unity; in this limit (2.26) will be met below in

steady-state elimination by a set of parallel sinusoids with a distribution

of flow rates having a small coefficient of variation e. Thus, when there

are many transits in one pulsation period, ;; /c(F) is the same no matter o

165

whether the small dispersions of F about its mean is sequential in one

sinusoid, or simultaneous in a set of steadily perfused parallel sinusoids

differing only in flow rates.

Problems of transient analysis

• When an input pulse ci(t) of substrate is transformed by the capillary

bed into an output pulse c (t), the transformation involves dispersion of o

the input pulse by the "convective spaghetti", as weIl as its reduction by

elimination along each of the convective pathways. If a sinusoidal transit

time is T, we have vmax/f = aT with a = vmax/(fT) being the maximum

elimination rate per unit volume of the sinusoid. If we assume that a is

the same constant along all sinusoids, then the distribution of v If is max the same (after normalisation) as the distribution of T. If the

distribution of T over the ensemble of sinusoids was known, the

transformation of c.(t) into c (t) could be interpreted quantitatively for 1. 0

an organ represented by an ensemble of unequal sinusoids, each of which

obeys the foregoing single-sinusoid elimination.

Unfortunately, the distribution of T is not observable. The

instruments sampling Ci and Co are inevitably placed weIl upstream and

downstream of the actual sinusoids. When labelIed inert indicators

(suitably matched to a substrate), are used to determine the distribution

of transit times T between the two sampling instruments, each T includes

Limes T' of passage through extra-sinusoidal regions, such as arterioles

and venules, which do not eliminate substrates. For any T, there may be

distributions of T' and T such that T' + T = T. Elimination by a capillary

bed with a known distribution of transit times T involves therefore also

the conditional probability density ~(TIT): given a transit time T between

sampling instruments, ~dT is the probability that T includes a time between

T and T + dT spent in eliminating sinusoids. Although classes of models of

~(TIT) have been formulated and discussed [15J, experimental support for a

choice of any particular ~ is meagre at present.

In steady elimination by a capillary bed (c.=const, F = const), the 1.

times T' are immaterial, as is apparent from the indifference of input-

output relations of any sinusoid to the form of p(x) in (1.8) (we can have

p = 0 for any part of the interval (O,L». In the steady case, to which we

now turn, modelling gets a firmer grip on existing experimental data.

166

3. STATISTICAL KINETICS OF STEADY ELIMINATION

We consider steady elimination by an organ modelied as an ensemble of

single sinusoids acting in parallel. The sinusoidal flow rates f and

maximum elimination capacities vmax are distributed over the ensemble,

while the constants K, ~, KI , K2 in (1.1)-(1.3) are the same everywhere

for any particular enzyme-substrate pair because they characterize specific

interactions at the molecular level. The sinusoids have a common constant

input concentration ci' but the sinusoidal output concentrations Co are

distributed when vmax/f is distributed. Their mixing results in a mean

concentration Co which denotes the observable output concentration in place

of Co of the undistributed (single-capillary) model.

A. Exact results

The method of averaging over the ensemble is determined by the

circumstance that output sampies are taken after the mixing of outflows

from individual sinusoids: the quantities which are additive in

determining the output concentration of the substrate are not the

concentrations co' but the oufluxes fco ' The observed output concentration

is therefore the flow-weighted mean of the sinusoidal c 's o '

where the summation is extended over the ensemble (organ). Thus

2 f = F (3.2)

is the organ rate of blood flow. Taking the flow-weighted mean on both

sides of (1.4) written for one sinusoid, we obtain from (3.1)-(3.2) the

organ uptake rate V in the distributed model:

V F(c.-c ) 1 0

(3.3)

What are now the predicted relations between the quanti~ies c., C , V and F 1 0

observed on the ensemble (i.e., on the intact organ)? This question

evidently cannot be answered without some knowledge of (or assumptions

167

about) the distributions of v and f over the ensemble. Conversely, if max we construct the relations theoretically, we can infer something about

these distributions from the data [16,17].

In this we are greatly aided by a remarkable feature of all saturation

kinetics (such as (1.1) and (1.2» when placed in the hepatic setting by

the foregoing modelling. Consider an ensemble of N sinusoids perfused by

blood at the total rate F given by (3.2), and having a total maximum

(saturated) rate of uptake V as the high-concentration limit of V given max

by (3.3). We then have the mean va lues f = F/N and v = V IN, about max max which sinusoidal values of f and v are distributed. The remarkable max

feature is that any dispersion of f or v about the ensemble means f, max

vmax must reduce the organ uptake rate V (except in saturation) or,

equivalently, must increase the output concentration c as compared with an o

organ in which all sinusoids have the mean values f = f, v max v max

(3.4)

where Ci in the arguments indicates that the comparison is made at equal

input concentrations. The equality sign refers to the limit of saturation,

in which the architecture of the organ has no effect on uptake. With that

exception, (3.4) shows the systematic direction of the fallacy of averages

inherent in the single-capillary model.

To prove (3.4) and related results, we replace (1.5) with an equation

for a single sinusoid:

f dc - p(x) g(c) dx (3.5)

where g(c) stands for Vhom/Vmax in (1.1), (1.2), (1.3) or any other

relevant test-tube kinetics. Corresponding to (1.6) for one sinusoid, the

integral over p(x)

out let we obtain

168

is v max Separating (3.5) and integrating for inlet to

dc g(cl

v max -r

which expresses the dependence of Co on ci and on vmax/f in implicit form.

For example, for Michaelis-Menten kinetics (3.6) reduces to (1.8) with

Vmax/F replaced by vmax/f. Note that c in Eq. 3.6 has become a dummy

variable, and its dependence on x is immaterial to the input-output

relation.

Because of the large number N of sinusoids in an intact liver [16], we

shall use continuous probability densities for the dispersions of vmax and

f. Because only the ratio vmax/f appears in (3.6), we prefer to work with

distributions of f and of the ratio

w v If max (3.7)

Let the probability of values in the intervals w, w + dw, and f, f + df in

the ensemble be

v(w,f)dw df ,

defining the non-negative and normalized density function v, from which all

mean values over the ensemble are calculated. Since concentrations

observed in the liver vein are flow-weighted mean values,

where

c o

fc o

v dw df

00 00

f = fofo f v dw df = F/N .

(3.8)

(3.9)

The ratio vmax/f is the maximum arterial-venous concentration difference

and accordingly its mean is also to be flow-weighted:

1 fOOfOO w = f 0 0 fw v dw df = f v dw df

Hence we have the important result

w = (v Ir) = (Nv )/(Nf) = Vmax/F max max

v max (3.10)

(3.11)

As c depends on w but not on f according to (3.6), we can simplify (3.8) o

by defining [18] the flow-weighted density function:

169

1 [co A(W) = ~ f v (w,f)df . f 0

(3.12)

Since v is normalized, A is also normalized according to (3.9).

A simplifies (3.8) to

The use of

c (c.) o 1

c (c.,w)A(w)dw , o 1

where co(ci,w) is given by (3.6). The function ci(ci ) connects two

observables and is available from experiments [19J. If there is no

(3.13)

dispersion about w, A is given by the Dirac impulse function ö(w-w) and

(3.13) is reduced to the corresponding relation for the undistributed

model: C (c.) = c (c.,w). o 1 0 1

We note that the function A(w) determines the relations between the

observables in the context of uptake; uptake experiments cannot therefore

determine the form of the density function v(w,f). Indeed, there is an

infinity of choices of v consistent with a complete set of experimental

results on uptake. For €xample, v may be the product of a function of w

and a function of f (each normalized), which describes the class of cases

in which w and f are uncorrelated in the ensemble [16J. In the context of

uptake, this non-uniqueness of the determination of v is not a deficiency,

since it is A rather than v which is the function governing the results of

steady uptake experiments.

(3.13) is a Fredholm integral equation of the first kind in which the

left-hand side is an empirically given function, A(w) is the unknown and

c (c.,w) is the kerne 1 given by (3.6) for any choice of test-tube kinetics o 1

g(c). The mathematical problem of determining A is particularly

interesting and difficult because of the implicit definition of the kerneI,

and because the solution must be non-negative and normalized for positive

values of its argument w.

For the case of Michaelis-Menten kinetics, (3.13) has been solved

analytically [18J, and numerically in relation to simulated data [20]. The

170

analytical solution proceeds through the transformation of (3.13) to

convolution form, followed by the Fourier transform using certain

convergence factors to ensure the existence of the transform, followed by

solution in the transform space and by final inversion to A(w), which is

expressed as an integral in the complex plane with a Fourier transform of

the empirical relation c (c.) involved in its integrand. Existing sets of o 1

data cannot support such a weight of mathematics. While the mathematical

structure of the problem has been greatly clarified by [18], practical

application of the analytical solution must await much more numerous and

accurate data, from a single preparation, than are available at present.

A much closer approach to practical applicability has been achieved by

direct numerical analysis of (3.13) [20], using constrained least-square

techniques in conjunction with a quadratic spline approximation to the

unknown A(w). The data were produced by simulation in terms of the model,

and included simulated random errors of the actual orders of magnitude

[19]. It was shown that if about 30 data pairs c ,Co were available from a o 1

single preparation and if they were distributed satisfactorily across the

concentration range on both sides of K, the form of A(w) could be recovered

to a good approximation. That is only three or four times more data pairs

than have been available from existing experiments [19], so that practical

use of this method is almost within range of experimental work.

It is, however, a third approach [16-18] that has proved to be so

fruitful in getting a grip on present-day data, that high levels of

statistical significance were attained [17] in testing salient features of

the distributed and undistributed uptake models. The exact as weIl as the

approximate results of this approach are obtained from expanding the kernel

c of (3.13) in aseries about w, whereby the integral is expressed in a o

series of central moments of A(w):

00

'\ I Cn) - - n L. iiT Co (ci,w) (w-w) n=O

(3.14)

where c (n) denotes the n'th derivative of c with respect to w, evaluated o 0

from (3.6) and taken at w = w. Substituting in (3.13) we get

171

00

'\ 1 (n) -L ;:-r c (c. ,w)~ n=O n. 0 1 n

(3. 15)

where the moments of Aare given by

~n n = 0,1,2,3, .... (3.16)

In particular ~O = I, ~1 = 0, and ~2 = 0 2 is the variance of A(w). The

conditions for the existence of all ~ , discussed in detail in [181, may be n

expected to hold in practice. But under what conditions are the expansions

(3.14) and (3.15) justified? We can answer this question precisely by

extending w and Co into the complex domain and using the inverse function

theorem. Differentiating w = vmax/f with respect to Co in (3.6), we obtain

dw/dc = -1/g(c ) . o 0 (3.17)

For definiteness we confine our attention to the Michaelis-Menten case in

the discussion of convergence of the expansion. Then (3.6) gives the

counterpart of (1.8) for the single sinusoid,

and

w lr=

c o

r c c. c.

i n 0 + 1 + in 1 r r r'

-1 K c

o

(3. 18)

(3.19)

The derivative of the inverse function, dw/dco ' vanishes at co/K - I,

where ~3.18) gives

w l{=

c. c. + 1+ 1 Ir in Ir in(-l)

c. C. 1 1

+ Ir + in Ir'" in(I±2m), m=O,I,2 ... (3.20)

and now w is complex, with vmax/f playing the role of the real part. This

mapping of the complex co-plane on to the complex w-plane is discussed

172

thoroughly in [18]. Here we note only that the singularities (branch

points) nearest to the expansion point w ~ v /f are those for m = 0 and max m = -1, as is seen in Fig. 2. The radius of convergence R of the expansion

(3.14) about ;/K is therefore

H = [1 -" tr=- + en tr=- - ;:x] + n2 , [ c. c. V 2 ]t

(3.20

where we have introduced organ parameters by (3.11).

We thus see how the radius of convergence res ponds to changes in the

physiological quantites. It has the minimum value n, so that if A is zero

outside the interval w - nK, w + nK, the series in (3.15) converges for a11

ci and all F. It is apparent from (3.21) that R tends to infinity as ci/Km

tends to zero or to infinity, so that the convergence in (3.15) is assured

for first-order uptake, and also close to saturation, for all shapes of A

(which go to zero sufficiently rapidly as w tends to infinity: cf. [18]).

In general, however, it may happen that the tails of A(w) will protrude out

of the interval w - nK, w + nK, in which case the series in (3.15) will

diverge, but will be asymptotic to c (c.) [18J. o 1

Independently of the problem of convergence, exact results can be

obtained by writing the series in finite terms with a remainder [17,18].

Returning to (3.6) with its general kinetics g(c), we use Taylor's theorem

with Lagrange's form of the remainder in (3.14). Dropping the argument ci

for brevity, we obtain

c (w) o

(3.22)

where 9 may depend on wand w. Substituting this in (3.13) and remembering

that Po = I and PI = 0, we get the exact result

c o

00

c (w) + t J (w-w)2c "(W+9(W-W»A(W)dw . o 0 0

(3.23)

173

174

Im(w/K)

Re(w/K)

-in

FIG. 2

The circle of convergence for the expansion of the

sinusoidal outlet concentration Co in powers of w-w =

v /f - V /F. The radius of convergence is max max

determined by the nearest singularities at Band Cj A is at 1 + c./K + en(c./K).

1. 1.

To evaluate c" from (3.6), we use (3.17) repeatedly: o

dg(co) g(co) -a""'c-­

o (3.24)

so that c" is expressed as a function o

of c rather than of w + 8(W-W). o

However, since uptake must reduce ci' all possible values of the argument

of g(co) are contained in the range 0 < Co < ci· If therefore dg/dc is

this interval, (3.24) implies that c~ is positive for

non-zero, and the last term in (3.23) is positive for

positive throughout

all w when g(c ) is o

any shape of A(w). Then c > c (w), as in (3.4), which we can now prove o 0

for the kinetics given by (1.1) or (1.2): the inequality holds for all

finite Co because g(c) rises monotonically with c, and the equality is

attained asymptotically at high Co (saturation) because g(c) tends to unity

and so dg/dco tends to zero. Indeed it is now apparent that (3.6) holds

for any saturation kinetics, if that class of kinetics is defined by a

positive dg/dc which tends to zero as c tends to infinity. [By contrast,

substrate inhibition kinetics defined by (1.3) satisfy (3.6) only so long 1

as Ci < (K1K2 )2j.

Thus c (w) is a lower bound on c· it is readily calculated from the o 0'

undistributed model, or by considering that since (3.18) holds for each

sinusoidal w, it holds also for W, which is equal to Vmax/F by (3.11). An

upper bound on c is also available from (3.23) under the same conditions o

which make c (w) a lower bound. If we replace c" by its largest value in o 0

o < Co < Ci' that is by the largest value of g(dg/dco) according to (3.24),

then the integral in (3.23) is increased to

ta2 (g dg/dc ) ? omax t Ioo

o - 2 - - -

(w-w) c" (w+8 (w-w)) (w-w)) A (w)dw o

2 where 0 = ~2 is the variance of A(w). Combining the upp~r and lower

bounds we obtain

c (w) < c ~ c (w) + fo2(g dg/dc ) o 0 0 omax

(3.25)

(3.26)

175

which holds for all Co up to ci when dg/dco > O. For example, for

Michaelis-Menten kinetics it is easily shown by differentiation that

occurs at c o

distribution

(g dg/dc ) = 4/(27K) o max

K/2. Defining the coefficient of variation of the

e. = o/w

and using (3.11), we obtain from (3.26):

c (V /F) < c < c (V /F) + 2Ke. 2 (V /FK)2/27 o max 0 - 0 max max

where c (V /F) satisfies (1.8). o max

(3.27)

(3.28)

(3.29)

In experiments with ci and Co varied over a wide range [19], detection

of a distribution is greatly facilitated when its effects are known in

advance to be insignificant in one limiting region of the concentration

range: for then V and the molecular constants implicit in g(c) can be max calibrated in that region for subsequent use at all concentrations. A

sufficient condition for finding such a calibrating concentration region is

readily obtained from (3.26) by writing

c - c (;) _0 __ 0 __ ~ !o2(g dg/dco)max/co

c o

(3.30)

If the numerator on the right-hand side is independent of concentration

(or, more generally, if it does not increase with concentration as fast as

Co), then the difference between Co and c(;) can be made relatively

insignificant by working at sufficiently high c , and effects of o

distributions on uptake disappear. For example, Michaelis-Menten kinetics

gives a maximal difference c - c(;) which is independent of concentration o

according to (3.29), so that at sufficiently high c it becomes immaterial o

whether C or c (;) is used in place of c of the undistributed model. o 0 0

This limiling situation has been called the homogeneaus regime of uptake

[16] .

176

Just how high a c is needed to attain this regime with some o

required accuracy depends also on ~, Vmax/F and K according to (3.29). In

practice this is considered in terms of the high-concentration segment of a

suitable data plot [17J.

It is now easy to show that any saturation kinetics has the advantage

of a homogeneous regime of uptake at high substrate concentrations. We

have a positive dg/dc which tends to zero at large c, while g tends to a

finite constant. Since evidently g(O) = 0 and g ~ 0 for any kinetics, the

product g dg/dc must have a maximum value at some finite concentration,

determined by the molecular constants implicit in g(c) (such as the value

4/(27K) in Michaelis-Menten kinetics). Then the numerator of the right­

hand side of (3.30) is independent of concentration, and the conclusion

follows as before.

An important example of g(c) for which no homogeneous regime exists is

first-order kinetics, g(c)

attains its greatest value in 0 < Co $ Ci at Co = Ci' so that

(g dg/dco)max = Ci· Then the upper bound (3.30) on the relative difference

in outflow concentrations is proportional to c./~ , which cannot be reduced 1 0

at will be increasing concentration. Furthermore, first-order kinetics has

the general property that nothing can be deduced about distributions from

steady-state experiments in which concentration is varied at constant organ

flow. This is because g(c) = c gives c (c.,w) = c.exp(-w) from (3.6), o 1 1

(3.7), so that the ratio c je. is independent of concentration aceording to o 1

(3.13). Experiments which do reveal distributions under first-order

kineties involve changes in the organ flow rate, as we shall see below.

B. Perturbation theory

We next wish to form an at least approximate picture of what happens

between the preeise bounds (3.26) or (3.29). The undistributed model gives

a fair approximation to data from at least some experiments ([19],

[21-24J). We therefore adopt the undistributed model as a lowest

approximation, and perturb away from it in powers of a suitable

dimensionless parameter characterizing distributions. Relation (3.29)

contains only one such parameter, namely the coefficient of variation ~ of

A(w) defined by (3.28); when ~ = 0, one recovers the undistributed model.

The perturbation theory will therefore be developed by carrying the

expansion (3.15) to at least the ~2-term on the assumption that w is

177

distributed narrowly about w, so that to successive moments of A(w) there

correspond successively smaller contributions to the values of observable

quantites. If for example A(w) is approximated by a narrow Gaussian

distribution centred on w and having the variance 0 2 = ~2' then ~3 0 by

symmetry and the next non-vanishing term in (3.15) is proportional to ~4. By using upper and lower bounds on the appropriate Taylor remainder (such

as the remainder in (3.23), but taken after more terms), the sense in which

the distribution of w should be narrow can be made precise [18]. We shall

now develop the perturbation theory to order ~2 for Michaelis-Menten

kinetics [16, 18], aiming particularly at formulations that will make

corrections to the undistributed model easily observable.

Calculating c"(w) from (3.24) for g = c/(c+K) and using the definition o

(3.28) of~, we retain from the expansion in (3.15):

(3.31)

Here the relation of (3.31) to (2.26) is apparent. As c (w) satisfies o

(3.18) for w = W, we now have two equations from which we can eliminate

co(w) in favour of the observed output concentration co' Expanding the

2 2 requisite terms in powers of ~ and working consistently to order ~ , we

thus find the correction to (1.8):

c. - c + K in(c./e ) 1 0 1 0

2 (V /F) [1 - ~2(V /FK) K 2]

max max (c +K) o

(3.32)

Evidently (3.32) exemplifies (3.4), with equality c co(w) attained only

- 2 - 2 when co/K ~ w or ~ ~ o. When co/K ~ 0, the effect of ~ in (3.32) is the

same as if the undistributed model was used with V replaced by a smaller max

V (say): max

178

V max (3.33)

2 After calibrating V and K from data at high concentrations, ~ can max

therefore be detected by the critique of the undistributed model applied to

low concentration data [16, 17].

It is particularly illuminating to perturb away from (1.9) and (1.10),

cast in the linear form

" I/V I/Vmax + (K/Vmax)(I/c) (3.34)

for the variables I/V, I/c (the Lineweaver-Burk plot [21). In applications

of the undistributed model [3, 19], data triplets (V, c., c ) yield by 1 0

(1.9) points (V,c) on the plot of (3.34), from which V and Kare readily max read off: Fig. 3A.

An interesting prediction of (3.34) is the independence of the plot of

the rate F of blood flow. From the point of view of the distributed model,

c in (1.9) must be replaced by c observable at the output: o 0

c. - C 1 0

C = (3.35) in c/co

Then (3.34) no longer holds, except at c /K ~~. As c in (3.35) is the o

mean of c. and c , high c implies high c and low I/c: (3.34), (3.35) 1 0 0

" determine the initial tangent to the plot of I/V against l/c (Fig. 3). As

c /X ~ 0 at the other extreme, I/c ~ ~; here the distributed model is o

governed by the same equations as the undistributed one if Vmax is replaced

by V from (3.33). Hence a straight line of the type (3.34), but with max

slope K/Vmax , is the asymptote of the curved plot of I/V against 1/c. The

total change AS of the slope S of the curved plot, from the initial tangent

to the asymptote, is readily obtained to order ~2 from (3.33):

2 ~ /(2F) . (3.36)

179

2 As the value of F is measured, (3.36) brings out the effect of ~ clearly

in a form suitable for statistical validation [17J. For a flow-independent

~2, a reduction in F swings the asymptote away from the flow-

180

R 1/V

Asymptote

/

/ ./

/ ./ initial tangent

--~~--~------------------------~1/c -l /K

m l /V B

/

--~~--~--------------------------~1/c

FIG. 3. Inverse organ uptake rate plot ted against inverse logarithmic mean of inlet and out let substrate concentrations. A: solid curve is predicted by distributed model; its initial tangent is the linear law predicted by the corresponding undistributed model. B: transformation of panel A when hepatic blood flow is lowered, with change indicated by arrows. Note resulting swing of asymptote to a steeper one, and unchanged initial tangent.

independent initial tangent, as in Fig. 3B. A complete classification of

the family of curved plots, to order ~2, is obtained by lengthy but

straightforward analysis [16]. The intersection of the initial tangent

with the asymptote is helpful in data analysis [173, especiallyas it is

independent of ~2:

K/c + 1 (3.37)

There is an interesting connection between functional demands on the

healthy liver, and the approximate validity of the undistributed model

resulting in the success of the foregoing perturbation theory. The liver

is so positioned in the circulation that substances from the intestines (in

particular, toxic substances) must first pass through it before reaching

other organs. If the distribution of w = v If (and hence of extractive max properties of sinusoids) was too widely dispersed about the ensemble

average w, the detoxifying function of this first-pass arrangement would be

lost (as in pathological states such as cirrhosis). The functionally

desirable limitation on the magnitude of the coefficient of variation ~ of

the distribution of w is reflected mathematically in the utility of

expansions in powers of~, outlined above to order ~2

4. TWO FUNDAMENTAL SETS OF EXPERIMENTS

As in any new application of mathematics, the ability of models to get

a statistically significant grip on real experimental data needs to be

demonstrated. The modelling developed so far suffices for the analysis of

basic experiments in hepatic elimination, sorne of which were designed [21]

to test that modelling.

A. The Keiding null-experiments

In aseries of experiments of simple yet powerful design, Keiding and

co-workers [21-24] exploited equations (1.9), (1.10) of the undistributed

model. Isolated rat livers were perfused in a recirculating system, in the

steady state of substrate elimination brought about by a steady infusion of

the substrate. As V in (1.10) is equal to the constant rate of the

infusion, and as Vmax and Kare intrinsic constants for each liver, (1.10)

predicts

181

o (3.38)

for any choice of the steady rate F of flow. While Co and ci change with F

at fixed V consistently with (1.4), c given by (1.9) is a flow-invariant.

Similarly, c given by (1.13) is a more general flow-invariant in the n

corresponding experimental design for elimination by HilI kinetics [25J.

The power of this experimental design is due to the absence of any

adjustable parameters in (3.38). When F is re-set to various chosen

va lues , all data test the undistributed model. Moreover, the null­

prediction (3.38) lends itself to testing by powerful statistical slippage

tests [21, 22, 25J. Next, with suitable choices of steady infusions of the

substrate, the two presuppositions of (3.38) can be checked [23, 24J;

aK - 0 äF" - (3.39) ,

a V max

aF o . (3.40)

The first of these must be expected to hold at a11 F from the biochemical

meaning of K, whereas the second one delimits the range of flow-changes

within which the number of actually perfused sinusoids does not change (no

recruitment of sinusoids: see B below). The complete set of experiments

relating to (3.38) - (3.39) has been performed by Keiding et 81. for

substrates including galactose, propranolol and ethanol [21-24J. For all F

for which (3.40) held, no statistically significant deviation from (3.38)

and (3.39) was found within experimental errors.

When c is interpreted by (3.35) and a distribution of v If over the max

sinusoids is admitted, a negative (a~/aF) of order ~2 is predicted [26J.

The failure of the experiments to detect this effect sets therefore an

upper bound on ~2 at each chosen level of statistical significance. Re-

2 analysis of the data [26J yielded the upper bounds ~ < 0.15 (P<O.I),

2 2 ~ < 0.18 (P<0.05), ~ < 0.27 (P<O.OI).

B. Brauer's anomaly and the failure of the undistributed model

In single-pass (once-through) experiments with isolated perfused rat

182

livers, the input concentration ci is fixed and effects of flow-changes on

the output concentration c can be observed. For first-order kinetics o

2 -(ci«K) and no heterogeneity (e =0, Vmax = Vmax ) , (3.32) is reduced to the

relation

c. I'n 2.­

c o

V max -nr ' (3.41)

predicting that the plot of en(c./c ) against I/F should be a straight line 1 0

through the origin, with the slope Vmax/K. If sinusoids collapse (de-

recruit) with decreasing flow rates (increasing I/F), V is reduced, the max

slope Vmax/K falls, and en(ci/co) falls below the straight line given by

(3.41) at high flow rates. Brauer et al. [27] discussed such plots for the

uptake of colloidal CrP04 by Kupffer cells of the rat liver, and used the

deviation of data from (3.41) to quantify the fraction of sinusoids open at

each flow rate. Following Brauer, we call this deviation the uptake

anomaly. Similar uptake anomalies have since been obtained for other

substrates, most recently for the elimination of taurocholate by rat liver

[28] shown in Fig. 4. However, the uptake anomaly is commonly found from

data at flow rates above 0.9 ml/min.g liver [27, 28], at which de­

recruitment of sinusoids does not occur [22, 24]. The high statistical

significance of the anomaly in the absence of de-recruitment, given below,

leaves little doubt that (3.41) and hence the undistributed perfusion model

itself is at fault. The remedy [30] is in admitting the heterogeneity of

the sinusoidal capillary bed by taking e 2 > 0 and replacing c with c and o 0

Vmax in (3.41) by Vmax from (3.33):

c. en 2. -

c-o

V max -nr

1 2 Ie (3.42)

where B is a remainder term, dependent on e, used to check the consistency

of the perturbation expansion [30]. As F falls (I/F increases), the e 2_

term in (3.42) accounts for the uptake anomaly (in the absence of de-

2 recruitment) in such a way that e can be quantified. This can be

demonstrated [29] in detail on the data of Pries et al. [28], as follows.

183

184

;::[[1 / ,

// x 4 o 01 02

c' .- X ~ X // /' X 0/

~ /O>/-,6 X X

e X u- / ,../ ---5; " ~~ . '" ........

2 l';~/o 0 ~ __ • . 0

.~-, ," v

O~ ________ ~ ______ ~~ ____________ L-~_

o os 10 15

1/F (min. 9 liver.ml-1)

FIG. 4.

Effect of perfusate flow rate F on extraction of taurocholate by isolated perfused rat livers. Experimental points are from Pries et a1. [28]. Only

-1 -I the 24 data points at flow rates above 1.1 ml.min. g liver (vertical arrow) are used in the analysis, split into two equal groups (open and closed circles) according to perfusate flow. The dotted line represents a linear regression through the origin and the solid circles. Open circles are used to construct

a probability density of the parameter ~2 (inset), with

a most probable value of 6 2 = 0.12. There is a 95%

probability that ~2 lies in the range (0.07, 0.17) and that the theoretical curve lies between the two broken curves.

In order to exclude effects of de-recruitment in rat liver, we

estimate ~2 and V /K from the 24 data points in Fig. 4 which belong to max -1 -1

flow-rates exceeding 1.1 ml.min. g liver [27, 28]. From (3.42) we know

that at high rates of flow (low I/F), en(c./c ) tends to (V /K)(I/F), 1 0 max

independent of ~2. This prior theoretical knowledge suggests the use of a

data-splitting procedure which reduces the two-parameter estimation problem

to two separate single-parameter estimations, and which leads naturally to

a Bayesian analysis of probabilities [17], [31].

We begin by using the 12 data points at the highest flow-rates (solid

circles in Fig. 4) to calibrate Vmax/K as the slope of the initial tangent

of the plot, constrained to pass through the origin (since co=c i when F=~).

This linear regression yields the dotted straight line in Fig. 4 with the

-1 -1 slope Vmax/K = 4.74 ml.min. g liver. The remaining 12 data points at

low flow rates (open circles in Fig. 4), which are reserved for estimation ..,

of ~~, are all located below the dotted straight line as predicted by

(3.42). -12 The probability of this occurring by chance is 2 ,a result

which bears out our choice of data--spli tting as weIl as our expectation

that information about the value of ~2 is implicit in the data. To extract

this information we proceed as folIows. We denote the 12 observed

ordinates en(c./c ) of the open cireles in Fig. 4 by Yk' k = 1,2 ... 12. 1 0

2 For any postulated value of ~ , there are 12 corresponding values of

fn(c./c ) ealeulated from (3.42) (with B=O); 1 0

2 these we denote by xk(~ ), k

= 1,2 ... 12. We then form the sum of squares of the differences,

m 2 [Yk-Xk(~2)]2, m

k=1

12.

Nl~erical computation shows that 82 is an almost symmetrie, U-shaped

function of ~2, with a distinctive minimum at ~2 = 0.12.

In a Bayesian approach (for example, [32, 33] the funetion

(3.43)

2 2 -m/2 2 [S (~ )] transforms any prior probability density pr(~ ), known before

185

considering the data Yl'Y2 ... Ym' into the (marginal) posterior probability

2 density pr(~ IY1'Y2 ... Ym):

2 2 2 2 -m/2 pr(~ IY1'Y2 ... Ym) ~ pr(~ )[S (~)J, (3.44)

where ~ denotes direct proportionality. (For the biological user abrief

summary of the steps leading to this powerful result is given by Bass [34J

in another liver-kinetic context). As to pr(~2), we know that ~2 ~ 0 and

that ~2 < ~2 ~ 0.27 in rat liver [26J: see A. above. In the absence of m~

2 other prior knowledge, it will suffice to take pr(~ ) = const. > 0 in

interval 0 5 ~2

satisfactory in

2 < ~ ,and zero elsewhere. The estimate - m~

that [S2(~2)J-m/2 is practically zero for

that the precise numerical choice of ~2 does not affect m~

2 ~ ~ 0.27 is m~

2 ~ ~ 0.20, so

pr(~2IYl'Y2 ... Ym) calculated from (3.44), and shown in the inset in

Fig. 4 in normalized form. This posterior probability density is

2 approximately symmetrie ab out a sharp peak at ~ = 0.12, with the

probability 0.95 that ~2 lies between 0.07 and 0.17 (95% Bayesian

confidence interval). Fig. 4 shows the theoretical curves given by (3.42)

for the most probable value ~2 = 0.12 (solid curve). for ~2 = 0.07 and ~2 =

0.17 (upper and lower broken curves). The odds are 19 : 1 that ~2 takes a

value such that the predicted curve lies between the two broken curves in

Fig. 4. Various iterations of the foregoing procedure converge to results

which make little quantitative difference to these conclusions. The

computed value of the remainder term B in (3.42) is satisfactorily small

[29J.

We note the similarity of pr(~2IYIY2 ... Ym) in the inset of Fig. 4 to

the coresponding probability density constructed from pig liver data,

obtained from an experiment of a different design [19J by more detailed

Bayesian methods ([31], Fig. 2a), where the most probable value of ~2 was

0.165. The foregoing data analysis illustrates again the power of data­

splitting based on a prior theoretical idea [17 J • as weIl as thc

proposition that "in the analysis of biological data involving nonlinear

models, Bayesian techniques can be used with considerable success and an

understanding of the assumptions made" [31].

186

The experimental series discussed under A and B of the pr~sent Section

show an interesting methodological difference. In flow-change experiments

with once-through perfusion [27, 28], the need for the distributed model is

revealed clearly with va lues ~2 ~ 0.12 (0.07-0.17), which however were

shown to be undetectable [26] in flow-change experiments in the re­

circulating design [21-24]. The fact that the latter design resembles more

closely the circulation in the intact body, extends considerably the

usefulness of the undistributed model.

5. METHODOLOGICAL REMARKS

The foregoing considerations interpret real data sets in terms of

physiologically based, mathematically formulated models. Confidence in

such interpretations depends on a mixture of their simplicity, rationality

as weIl as on quantitative agreement with data, expressed preferably in

statistical terms. The concepts of simplicity and rationality of models,

so familiar to practitioners of quantitative science, are difficult to

specify objectively. One difficulty arises with the question as to how

many adjustable parameters may be used to fit a given data set to a model.

lf the number of adjustable parameters approaches the oumber of data

points, the fitting evidently approaches mere interpolation of data: it

gives 00 confidence in the validity and predictive power of the model [5].

If the elusive coocepts of simplicity and rationality of models were to

become submerged in the fitting of data by adjusting parameters, using the

powerful computatiooal means now available, mathematical modelling could

lose all its scientific value.

The most powerful antidote to adjustable parameters is to cast the

predictions of a model in terms of observables alone, without any

adjustable parameters at all. This was done in the Keiding null

experiments [21-22] discussed in Section 4A. This maneouver results in an

~xtreme methodological trade-off: the experiments so designed yield no

numerical information about any parameters of the experimental system.

Consequently all data are brought to bear on the hypothesis (model) which

is thereby tested with special severity. A less drastic antidote to

arljustable parameters arises from the observation that parameters of a

system cease to be adjustable if they are determined from experiments of a

design different from that which they are used to interpret. For example,

V in equations for first-order hepatic uptake, such as (3.42), is an max

187

adjustable parameter, but ceases to be one when it had been determined by

saturating the liver with the relevant substrate (for example, [23]).

Compartmental analysis is a good testing-ground for the foregoing

dilemmas. When physiologically distinct domains are bounded unambiguously

(for example, by the erythrocyte membrane or the blood-brain barrier),

modelling by compartments leads to definite predictions that can be tested

by experiments. In contrast, for processes modelIed naturally by partial

differential equations ([5],[7],[8],[10],[12]), especially with

concentration gradients present in the steady state ([3-6J, [27-30]),

numerical analysis by discretization of the spatial variable leads to sets

of ordinary differential equations that can be interpreted in compartmental

terms [4] without any definite physiological basis. For any pre-assigned

accuracy of this approximation, the number of these "compartments" must be

changed with the steepness of the concentration gradients. Thus, a

different "compartmental model" of the same liver would be needed for each

value of the rate of the hepatic blood flow, as is apparent from (1.5) or

(2.1) .

It is interesting that compartmental interpretations can be attached

to models which have their very basis in the denial of compartments.

Consider the undistributed perfusion model of hepatic elimination (Section

1). Equations (1.9) and (1.10) assert that the steady rate of elimination

is the same as if all hepatocytes were presented with the substrate

concentration c. Should then the liver be viewed as a "well-stirred"

compartment containing substrate at the spatially uniform concentration c,

with concentration jumps from c. to c at the inlet, and from c to c at the 1 0

outlet? (Is the mass of the Earth really concentrated in its centre

because it attracts outside bodies as if it was so concentrated?) There

are two objections to such a compartmental interpretation of the perfusion

model. Firstly, it contradicts autoradiographic evidence of intrahepatic

gradients of a variety of substrates [4]. Secondly, it removes the only

rational basis for choosing c as the operative substrate concentration in

the compartment, in preference to c , c., (c.+c )/2 or any other o 1 1 0

combination of the observables c. and c [35]. The second objection 1 0

illustrates particularly weIl the elusive concept of rationality of models,

so difficult to formulate in general terms.

188

ACKNOWLEDGEMENTS

Many of the foregoing ideas and results owe much to collaboration with

A.J. Bracken, S. Keiding, P.J. Robinson, N. Tygstrup and K. Winkler. These

collaborations were support pd over the years by Australian and Danish

granting bodies (notably the Australian Research Grants Scheme), as

acknowledged in papers cited below.

REFERENCES

I. Lambrecht RM and Rescigno A: editors: Tracer Kinetics and Physiologic

Modelling. Lecture Notes in Biomathematics Vol. 48. Berlin, 1983.

Springer.

2. Dixon M and Webb EC: Enzymes. (3rd edition). London, 1983. Longman.

3. Bass L,Keiding S, Winkler K and Tygstrup, N: J. Theor. Biol. 61: 393

(1976) .

4. Weisiger RA, Mendel CA and Cavalier RH: J. Pharm. Sei. 75: 233

(1986) .

5. Bass L, Bracken AJ and Burden CJ: Tracer Kinetics and Physiologic

Modelling. RM Lambrecht and A. Rescigno, editors. Lecture Notes in

Mathematics Vol. 48, pp. 120-201. Berlin, 1983. Springer.

S. Johansen S and Keiding S: J. Theor. Biol. 89: 549 (1981).

7. Bass L and Bracken AJ: J. Theor. Biol. 67: 637 (1977).

8. Perl W and Chinard FP: Circul. Res. 72: 273 (1968).

9. Lassen NA and Perl W: Tracer Kinetic Methods in Medical Physiology,

pp. 158-160. New York, 1979. Raven Press.

10. Goresky CA: Capillary Permeability. C Crone and NA Lassen, editors.

pp.415-430. Copenhagen, 1970. Munksgaard.

11. Crone C: J. Physiol. 181: 103 (1965).

12. Goresky CA, Bach GG and Nadeau BE: J. Clin. Invest. 52: 991 (1973).

13. Tygstrup N and Winkler K: Acta Physiol. Scand. 32: 354 (1954).

14. Crone, C, Frokjaer-Jensen J, Friedman JJ and Christensen 0: J. General

Physiol. 71: 198 (1978).

15. Bass L and Robinson PJ: Clin. Exp. Pharmacol. Physiol. 9: 363 (1982).

16. Bass L, Robinson PJ and Bracken AJ: J. Theor. Biol. 72: 161 (1978).

17. Bass L and Robinson PJ: Microvasc. Res. 22: 43 (1981).

18. Bracken AJ and Bass L: Math. Biosci. 44: 97 (1979).

19. Keiding S, Johansen S, Winkler K, Tonnesen K and Tygstrup N: Amer. J.

Physiol. 230: 1302 (1976).

189

20. Holt JN and Bracken AJ: Math. Biosci. 51: 11 (1980).

21. Keiding Sand Chiarantini E: J. Pharmacol. Exp. Ther. 205: 465 (1987).

22. Keiding Sand Steiness E: J. Pharmacol. Exp. Ther. 230: 474 (1984).

23. Keiding S, Vilstrup Hand Hansen L: Scand. J. Clin. Lab. Invest. 40:

355 (1980).

24. Keiding Sand Priisholm K: Biochem. Pharmacol. 33: 3209 (1984).

25. Bass L: J. Theor. Biol. 100: 113 (1983).

26. Bass L and Robinson PJ: J. Theor. Biol. 81: 761 (1979).

27. Brauer, RW, Leong GF, McElroy RF and Holloway RJ: Amer. J. Physiol.

184: 593 (1956).

28. Pries JM, Staples AB and Hanson RF: J. Lab. Clin. Med. 97: 412 (1981).

29. Bass L, Roberts MS and Robinson PJ: J. Theor. Biol. iin press).

30. Bass L: J. Theor. Biol. 86: 365 (1980).

31. Robinson PJ, Pettitt AN, Zornig J and Bass L: Biometrics 39: 61

(1983) .

32. Jeffreys H: Theory of Probability 3rd edition, Clarendon Press, 1961.

Oxford.

33. Box, GEP, Tiao GC: Bayesian Inference in Statistical Analysis, Chapter

1, New York, 1973. Addison-Wesley.

34. Bass L: J. Theor. Biol. 89: 303 (1981).

35. Wagner JG: Pharmacol. Ther. 12: 537 (1981).

190

PHYSIOLOGICAL MODELS, ALLOMETRY, NEOTENY, SPACE-TIME AND

PHARMACOKINETICS

INTRODUCTION

Harold Boxenbaum* and Richard D'Souza**

*Merrell Dow Research Institute 2110 East Galbraith Road Cincinnati, Ohio 45215-6300 U.S.A.

**Miami Valley Laboratories Procter & Gamble Company P. o. Box 398707 Cincinnati, Ohio 45239-8707 U.S.A.

The first attempt at developing a mathematical model to describe

and predict drug distribution and elimination processes dates back only

50 years. Teorell (1937a,b) advanced a simplified scheme of the body

by grouping tissues into five compartments (connected through the

circulatory system). This pioneering work has served as the forerunner

from which all pharmacokinetic models, physiological and compartmental,

have been developed.

It has only been within the past 15-20 years, however, that

serious consideration has been given to the extrapolation of pharmaco­

kinetic da ta across species. During this period, two paradigms have

emerged. The first, the reductionist paradigm, utilizes physiological

models and concentrates on adjustments in anatomical features, blood

flows, partition characteristics, and elimination rates required for

extrapolation ac ross species. The dominant conceptual framework is to

"reduce" organisms to their constituent parts in order to explore and

characterize properties and mechanisms through which components inter­

act. The second paradigm employs allometric scaling. Allometry is the

study of size and its consequences; the term scaling, derived from the

191

chemical engineering literature, denotes the process of using one

system as a prototype for another. Allometric scaling thus utilizes

size as a variable upon which to extrapolate data. Unlike the

physiological-reductionist paradigm, the allometric approach is

predominantly empiric. Although differentiated conceptually, these two

approaches are often integrated in practice. A good example is the

work of King et al. (1986). These investigators utilized a physio­

logical model to characterize cis-dichlorodiammineplatinum (11) (DDP)

disposition in various species. It was assumed that DDP reacted with

macromolecules (mostly proteins) to form a "fixed" platinum metabolite

which could only be cleared from the body through catabolism. The rate

of loss of fixed platinum in each compartment corresponded to the rate

of protein turnover in that compartment. Turnover rates in human

plasma, liver, gut, kidney and skin were estimated from animal data

using apower law relationship, with body size as the independent

variable.

Our intent here is to explore and contrast the two paradigms.

Although different in many aspects, both approaches probe invariances,

the reductionist paradigm in terms of arrangement and mechanism and the

allometric paradigm in terms of space-time. In the latter case, the

impact of neoteny, the retardation of somatic development for selected

organs and parts, is particularly important. Neoteny is the primary

reason why humans generally metabolize drugs at a considerably slower

rate than other terrestrial mammals of the same size.

THE REDUCTIONIST PARADIGM

Pharmacokinetic systems arise from multi-leveled structures in

which each unit process is simultaneously coupled to a sub- and supra­

system. In physiological modeling, the goal usually is to demonstrate

how relevant processes at the highest level of interest (tissue uptake,

hepatic elimination, active metabolite levels, etc.) arise from lower

level unit process characteristics (partition coefficients, enzyme

levels, blood flows, etc.). Models are constructed either by grouping

organs or tissues into compartments, or by using the individual organs

of interest. The extent to which one lumps together different compo­

nents in a model is termed the degree of aggregation (Carson et al.,

1983). Relative to most other pharmacokinetic models, physiological

systems have a relatively low degree of aggregation. Typically, organs

192

that do not metabolize drug or serve as a pharmaeologie site, but that

do signifieantly uptake drug, are lumped together for eonvenienee. In

terms of the model, compartments are considered non-contiguous and are

conneeted through the circulatory system in an anatomically precise

fashion; diffusion of drug between adjacent tissues is considered small

and generally ignored (Rowland, 1984). Fig. 1 illustrates a physio­

logieal model for lidocaine disposition in rhesus monkey and man. The

VENOUS

INTRAVENOUS -----. ADMINISTRATION

LUNG

ARTERIAL

BRAIN

RET

MUSCLE

AOIPOSE

METABOllSM

IRATEe><.HBFl

Fig. 1 - Physiologieal model for lidoeaine dispo­

sition kineties in rhesus monkey and man. RET

represents "rapidly equilibrating tissues" and

ineludes heart and kidney. SET represents "slowly

equilibrating tissues" and ineludes long bone,

skull, spine, skin and ehest wall. Arrows indieate

organ blood flows. The first-order metabolism rate

eonstant is assumed proportional to hepatie blood

flow (HBF). Reprinted from Benowitz et al. (1974),

p. 91, by eourtesy of C. V. Mosby Co.

organs ineluded in the model are those deemed important for distribu­

tion (lung, musele, adipose tissue, ete.), metabolism (liver) and

193

pharmacologic (side) effects (brain). Some groups, like the rapidly

equilibrating tissues (RET) and slowly equilibrating tissues (SET), are

lumped together. As the lung is an important organ for distribution,

and different drug concentrations are expected in the venous and

arterial circulations. the blood pool was divided into separate venous

and arterial supplies.

Three classes of parameters are required for reductionist modeling:

(1) physiological parameters like organ and tissue volumes and blood

flows; (2) binding (or thermodynamic) parameters like plasma/blood free

fractions and tissue:blood partition coefficients; and (3) biochemical

parameters like metabolism rates. Hass balance differential equations

are written which describe in-flow and out-flow relationships for both

the eliminating and non-eliminating compartments. In less parametized

models, such as in Fig. 1, metabolie and tissue distribution processes

are characterized in an austere manner. Thus hepatic metabolism is

first-order, and tissue and organ uptake is blood flow limited, i.e.,

the capacity for tissue uptake greatly exceeds the ability of the

circulatory system to deliver it. This occurs when no significant

membrane permeability barriers exist (in a computational sense); lido­

caine in emergent venous blood and tissue are assumed in equilibrium,

and the tissue is said to be "well-stirred". The advantage to systems

like this is that both the model and mathematics are simplified.

Frequently, however, more complex models are required. To contend with

membrane resistance (low permeability) to actinomycin-D in dog testes,

Lutz et al. (1977) incorporated a diffusion term to account for flux

from extracellular to intracellular fluid. Also, when dealing with

metabolism or other elimination processes, first-order differential

equations may not always be satisfactory. D'Souza et al. (1987), for

example, used a second-order equation to characterize ethylene dichlo­

ride metabolism in mouse, rat and human. This compound is metabolized

in part by direct conjugation with glutathione. As glutathione levels

are significantly depleted during the course of metabolism, a second

order reaction rate was required to characterize the metabolie process

(first-order with respect to both reactants).

Both in vivo and in vitro methods have been developed for the

estimation of tissue:blood partition coefficients. The most straight­

forward in vivo method is to infuse drug to steady-state, sacrifice the

194

animal, and directly measure tissue and arterial blood/plasma concen­

trations. Benowitz et al. (1974) used this method for the estimation

of tissue to plasma partition coefficients for lidocaine in the monkey;

values ranged from 0.39 for long bone to 3.46 for spleen. ehen and

Gross (1979) derived theoretical equations for comparing partition

coefficients obtained from bolus injection (post-distribution phase)

vs. infusion experiments. They showed that these coefficients are not

equivalent for the two modes of administration. In vitro methods have

also been employed for estimating partition coefficients. Sato and

Nakajima (1979) determined tissue:air and blood:air partition coeffi­

eients for several volatile halogenated hydrocarbons using tissue

homogenates equilibrated in reaction vials; these latter ratios are

important for volatile compounds to account for the effects of inhala­

tion and exhalation. Tissue:air partition coefficients were estimated

by allowing homogenates to equilibrate in sealed reaction vials,

sampling an aliquot of the vial headspace, and analyzing it for halo­

carbon content. Tissue:blood partition coefficients were obtained by

multiplying tissue:air ratios by air:blood ratios. This method, with

some modifications, has also been used by Andersen and co-workers

(Andersen et al., 1987; D'Souza et al., 1987; Ramsey & Andersen, 1984)

for volatile hydrocarbons. Analogous in vitro methods have been

developed for non-volatile materials as weIl. Lin et al. (1982a)

determined tissue:blood partition coefficients for ethoxybenzamide

using an equilibrium dialysis approach employing tissue homogenates;

these results were in good agreement with those obtained in vivo.

Recently, Fisher et al. (1987) employed an in vitro tissue homogenate

ultrafiltration method to determine tissue:blood partition coefficients

for trichloroacetic acid (a trichloroethylene metabolite) in their

model for lactational transfer of trichloroethylene and trichloroacetic

acid in rats.

Metabolism rates may be estimated using either in vivo or in vitro

techniques, the former being more common. In linear systems with

metabolie conversion at a single site, organ clearance is conveniently

estimated from area analysis. ~hen parent compound is excreted by the

kidneys, renal elimination rate can be determined from urinary excre­

tion data. More elaborate procedures are required when elimination

involves multiple sites and in nonlinear systems. Bischoff et al.

(1971) demonstrated that the kinetic behavior of methotrexate was

highly dependent on biliary secretion and reabsorption, especially in

195

the mouse. They modeled this process in great detail, using a finite

series of discrete compartments with different intercompartmental

transfer rate constants and lag times. This chemical engineering

approach was used to simulate a very complex and poorly understood

physiologic phenomenon. Gargas et al. (1986a) used an interesting in

vivo approach to determine metabolism rates of volatile halogenated

hydrocarbons. Their apparatus was essentially a recirculating inhala­

tion chamber; metabolism rates were calculated from concentration

declines in ambient air. These investigators used this system to study

methylene bromide metabolism in order to differentiate contributions of

a saturable oxidative pathway from that of first-order glutathione

conjugation (Gargas et al., 1986b).

In vitro metabolic studies (liver homogenates, isolated liver

cells, etc.) are less commonly used, although they have enjoyed some

notable success. Dedrick et al. (1972) incorporated human liver

homogenate data in their model for 1-beta-D-arabinofuranosylcytosine

disposition. Lin et al. (1982b) performed experiments using isolated

liver cells from rats and rabbits in developing their ethoxybenzamide

model. They were the first investigators to scale their model for

another species based solelyon in vitro data. A model developed for

methylene chloride (Andersen et al., 1987) incorporated in vivo data

from gas uptake studies as weIl as in vitro literature data on relative

enzyme activities in liver and lung from several animal species.

Scaling a physiological model from one species to another requires

detailed consideration of the similarities and differences between

organisms. Two schools of thought have emerged - those of the

"splitters" and the "lumpers" [see Luria et al. (1981) for the origins

of these terms]. So me physiological modelers, frequently biologists,

approach biological systems from the point of view of species' differ­

ences (enzymatic pathways, tissue partition characteristics, etc.).

These scientists (splitters), especially on the lookout for qualitative

differences, tend to focus on distinctions. Others, oft-times chemists

and engineers, have a passion for unity (lumpers); they stress

similarities in physical design and biochemical processes. As Bertrand

Russell cogently noted many years ago: "the type of problem which a man

naturally sets to an animal depends upon his own philosophy." Despite

these apparent philosophic clashes, splitters and lumpers displaya

strong resemblance in their manipulation of symbolic generalizations.

196

Conflicts of meaning give way to shared commitments; so open-ended is

the reductionist approach, that it can account for both known differ­

ences and similarities between species.

Following adjustments for anatomical diversity, physiologie scaling

proceeds by revamping variable and parameter va lues from the old model

to that for the new species to which the model is being adapted. New

species' va lues are either assumed to be the same as with the former

species, determined experimentally, or scaled allometrically. For the

lidocaine model depicted in Fig. 1, predictions of drug concentrations

in arterial blood and other organs and tissues of man (Fig. 2) were

GI VI

o o -0

GI Ü .~ c

100

128 256 Time (min)

Fig. 2 - Simulation of the tissue distribution and

elimination of lidocaine in man (following a one

minute intravenous infusion) scaled from the model

in monkey. See Fig. 1 and text for discussion.

Reprinted from Benowitz et al. (1974), p. 93, by

courtesy of C. V. Mosby Co.

genera ted assuming partition coefficients obtained from monkey. Human

197

blood flows, tissue masses, and hepatic extraction ratio were taken

from the literature.

Physiological models have a strong appeal, since their emphasis is

on fragmentation and function - the spatial and temporal ordering of

parts and the precise way through which they interact. Although the

uni verse may very weIl be an immense cohesive continuum, the human mind

is more accustomed to looking at things as a patchwork of discrete

fragments (Yeiss, 1969). Yhat emerges in physiological scaling is a

characterization of patterns (mechanisms) which are usually applicable

to a much broader array of data. The term "global" is commonly used to

denote this characteristic; in concentrated form, the model presents a

pattern and configuration constitutive of a larger reality (Green,

1981). Through manipulation of discrete segments of the model, adjust­

ments can be made to predict concentration profiles arising from

different input functions, altered physiological states, ac ross

species, etc. Tied to the principle of causality, the reductionist

paradigm confers an emotionally gratifying sense of how things really

work (or at least how we think they ought to work). The price paid for

this understanding is considerable. Experimental methodologies are

costly and time consuming. And although the process does not embrace

curve fitting in a classical sense, "fitting" nonetheless occurs on an

even more fundamental level. Yhen a model fails (a stage frequently

encountered during early stages of model development), it is necessary

to either re-structure or re-parametize it. The most serious short­

coming of these models, however, is that as they approach the full

complexity of the modeled phenomena, the more unwieldy they become, the

less leverage they provide, and the more closely they approach a tauto­

logical reinvocation of the events they purport to model (Fethe, 1982).

THE ALLOMETRIC PARADIGM

The allometric approach is inextricably related to the dual con­

cepts of space (size) and time. Yhereas the reductionist paradigm

seeks explanation in terms of arrangement and mechanism, the allometric

paradigm concentrates on scale alterations, the goal of which is to

manipulate variables of interest so they are unified (invariant) in

their relation to one another. The organism itself frequently becomes

the coordinate system upon which the time scale is based; clock time is

transformed into a form that becomes species invariant with respect to

198

pharmacokinetic processes. Unlike physiological models, where anatomy,

physiology and biochemistry define system structure, the overall

consistency of interrelations determines network structure in the

allometric paradigm. The basic assumption is that pharmacokinetic

processes are genetically integrated through a highly coordinated

system, and this linkage places extraordinary restrictions on what can

or is likely to occur.

To better provide an understanding of the two primary variable

(size and time), each will be discussed briefly. Following this, the

two will be integrated through the'concept of space-time. Finally, the

impact of neoteny will be discussed.

Size and Its Consequences

The most obvious differences between organisms are their size. The

smallest living terrestrial mamma 1 (Barnard, 1984) is the Pygmy white­

toothed shrew (head-to-tail length from 3.5 - 4.8 cm; weight approxi­

mately 2 grams); the largest living land mamma 1 (Barnes, 1984) is the

African Savanna bull elephant (head-body length, 6 - 7.5 meters;

height, 3.3 meters; weight, up to 6,000 kg). If you " ... drop a mouse

down a thousand-yard mine shaft ..• it gets a slight shock and walks

away, provided that the ground is fairly soft. A rat is killed, a man

is broken, a horse splashes." (Haldane, 1928). A quick panoramic view

of the range of sizes of living organisms is illustrated in Fig. 3.

The largest animal that ever existed is the blue whale (a fully grown

individual may be longer than 22 meters and weigh more than 100 tons)

(McMahon and Bonner, 1983). The largest tree is the giant sequoia, and

the record land mamma 1 (Gregory, 1951; Schmidt-Nielsen, 1975) was a

herbivorous relative of the rhinoceros, the Baluchitherium (shown next

to man in Fig. 3, this animal lived during the Oligocene per iod -

approximately 25-40 million years ago).

For each animal type there is a most convenient size. Changes in

size are associated with changes in function, and in many cases,

changes in form (Haldane, 1928). Perhaps the earliest allometric

analysis was conducted by Galileo Galilei (1637). In his "Dialogues,"

Galileo noted that skeletons of large mammals, such as the elephant,

must have bones with dimensions out of proportion to their linear

scale. That is, relatively heavier bones are needed to support greater

masses (Schmidt-Nielsen, 1975).

199

Fig. 3 - The largest living things, past and

present. All organisms are drawn to scale; a

description of the different life forms is provided

elsewhere (McMahon and Bonner, 1983). From "On

Size and Life" by Thomas A. McMahon and John Tyler

Bonner, copyright 1983. Redrawn with the permis­

sion of T. A. McMahon and Scientific American

Books, Inc. The figure is an adaptation of work

first published by Yells et al. (1931).

The overall impact of size was summarized by Stahl (1963), who

tabulated many of the variables correlated to body dimensions and mass.

These included physiological variables (tidal lung volume, glomerular

filtration rate, cardiac output, etc.), body part sizes (red blood cell

volume, ehest circumference, kidney mass, etc.), biochemical variables

(metabolie rate, heat production, nitrogen excretion, etc.), physio­

logical times (breath time, pulse time, time for 50% growth, etc.) and

physical characteristics (limb movements, heat flow constants, museIe

force, etc.).

200

I I [l fll

Fig. 4 - Common nailS; the smallest is a

2 penny nail (one inch long, 847 to a

pound) , and the largest is a 60 penny

nail (6 inches long, 11 to apound) . See

McMahon and Bonner (1983) for their

discussion of nail allometry. Reprinted

from a catalog of nails published by

u. S. Steel (USX Corporation), with

permission of T. A. McMahon and USX Corp.

One need not restrict analysis to biological systems. An engaging

example of allometric variations in nails was reported by McMahon and

Bonner (1983). Fig. 4 illustrates standard sizes of common nails.

Although not readily apparent, the longer nails are relatively thinner

than the shorter ones; this is just the opposite of mammalian bone

allometry (vide supra). The nail data are plotted in Fig. 5. A

reasonable fit occurs with the allometric equations:

d = 0.07 12/3 (1)

log d 2/3 log 1 + log 0.07 (2)

where d is nail diameter and 1 in nail length. This result can be

rationalized by assuming that, under a critical load, nail buckling 4 2 depends on the ratio d 11 (McMahon and Bonner, 1983). For the ratio

of buckling force to nail diameter to be the same for all nails would

require a ratio of (d 4/1 2)/d, or d3/12 (another way of saying that d

should be proportional to 12/3 ).

201

·5 ~

~ 0.1 ;;;

:;.

• Common ~ils

o Spikes, standard si..,.

c Nonstandard Iong spikes

0.01 L-__ ....L..._...1-.....L......L.--'--'----'~L----'------'---'---'-...L....L.L-L.J.-_'

0.1 10

N.illmgth ( in .)

Fig. 5 - Double logarithmic plot of nail diameter vs.

nail length. The broken line corresponds to a slope of

1; the solid line corresponds to the allometric formula:

d = 0.07 12/ 3, where d is diameter and 1 is length. From

"On Size and Life" by Thomas A. McMahon and John Tyler

Bonner, copyright 1983. Reprinted with the permission of

T. A. McMahon and Scientific American Books, Inc. [data

supplied by U. S. Steel (USX Corp.»).

Some Aspects of Time

The concept of time is the most vexing riddle confronting man

(Ouspensky, 1920). It is therefore not surprising to find it dealt

with in a variety of ways throughout the ages. The ancient Greeks,

with their fertile imaginations, created a multiplicity of man-like

Gods (anthropomorphic polytheism) (Trivers, 1985). These Gods, who

seemed little concerned with good or evil either in themselves or in

mankind, were believed to rule the world of nature. Cronus - the

personification of Time - was a Greek God who created only to destroy

(Barthell, 1971; Guerber, 1921; Tatlock, 1917). By union with his

mother Gaea (Earth), Uranus (Heaven) produced 12 Titans, the youngest

202

of them Cronus. Mother Gaea, grieving Uranus' imprisonment of their

subsequent offspring, shaped a great adamantine sickle and gave it to

Cronus to use against his father. Laying in ambush, Cronus lopped off

his father's genitals, whereupon Uranus grimly prophesied that Cronus

would suffer similar dire retribution. Sitting on the throne and

heedful of his father's prophesy, Cronus feared he might also be

dethroned by one of his children. He therefore swallowed each of them

as they were successively born (see Fig. 6). This worked until his

(sister and) wife Rhea, after giving birth to Zeus, substituted a great

stone in swaddling cloths and presented it to Cronus (who, as usual,

promptly swallowed it). When Zeus was fully grown, he enlisted the

assistance of Metis, who convinced Cronus to sip a potion containing a

powerful emetic. As a result, the stone was disgorged, and this was

followed by Zeus' brothers and sisters (in the reverse order of their

earlier swallowing). Following over 10 years of battle (War of the

Titans), Zeus eventually overthrew his father. Atlas, who fought on

the side of Cronus, was punished by being forced to perpetually uphold

both heaven and earth.

Time was viewed quite differently by Europeans. For the early

Christian monks, time was important because they believed their souls

would be in jeopardy if they did not pray at the proper times (Landes,

1983). This provided the primary impetus for European technological

developments in clock-making. This is in contrast to the Chinese; for

them it was simply not important to tell time with any precision.

For Newton (1642-1727), time was reduced to an absolute reality,

flowing uniformly " ... without regard to anything external" (Capra,

1975). Kant (1724-1804), on the other hand, believed that time did not

inhere in objects, but rather in the subject who perceived them; for

hirn, time was not a property of the universe, but merely a reflection

of our mental apparatus for imagining the world (the mind contains the

furniture into which guests must fit) (Kimpel, 1964; Kline, 1980;

Whitrow, 1980). For Einstein, the passage of time was merely a feature

of consciousness (Barnett, 1962; Dossey, 1982; Hayward, 1984; Whitrow,

1980; Zukav, 1979). In special relativity theory, he maintained that

there is no such thing as space and time as we perceive them. Rather

the two are blended into a static, four-dimensional continuum or mani­

fold; in this manifold, events do not happen, they simply are.

203

Fig. 6 - Cronus' cannibalism of his

sons was the subject of several

paintings by old masters. None can

match this one by Goya (1746-1828)

in savagery and passion (Shickel,

1968). One of his "black paintings"

- possibly an expression of a mind

threatened by madness - Goya painted

it after becoming deaf and while

recuperating from a second near­

fatal illness. Reprinted through

the courtesy of Museo deI Prado,

Madrid, Spain.

In terms of pharmacokinetic space-time development, however, we

identify most with Jules Henri Poincarj (1854-1912): "There is not one

way of measuring time more true than another; that which is generally

adopted is only more convenient. Of two watches, we have no right to

say that the one goes true, the other wrong: we can only say that it is

advantageous to conform to the indications of the first" (Poincarj, 1929).

204

Reiser (1931) also provided a useful perspective; he divided time

relativity into 3 sub-types: (1) physical relativity; (2) biological

relativity; and (3) psychological relativity. Physical relativity does

not rest upon anything peculiar to the organism. lf you travel on a

spaceship at 86.6 % the speed of light, an on-board clock records time

at 1/2 the rate of a clock on the ground (Note: this calculation

neglects the earth's speed of 18.6 miles per second in its orbit around

the sun) (Goldsmith and Levy, 1974; Morris, 1984). Biological relativ­

ity relies upon a biological frame of reference from which properties

are deduced. Hexobarbital terminal disposition half-life is about 19

minutes in the mouse and 360 minutes in man, but it's also the duration

of 1680 gut beats in the respective species (Boxenbaum, 1982; Boxenbaum

and Ronfeld, 1983). Psychological relativity deals with the manner in

which the sensor-motor apparatus of an organism perceives objects or

events in the world. Einstein summarized it nicely: "lf you sit with a

beautiful girl, two hours seem like two minutes. lf you sit on a hot

stove, two minutes seem like two hours" (Dossey, 1982). ~hereas dila­

tion (slowing of time) in the physical realm is associated with travel

at velocities significant with respect to the speed of light (vide

supra), time dilation in the psychological realm is frequently associ­

ated with suffering [see Boxenbaum (1986) for a further discussion of

these 3 types of relativities].

Pharmacokinetic Space-Time

The allometric pharmacokinetic paradigm is founded upon the concept

of biological time. Consider the following example in which it is

assumed a drug is eliminated from the body monoexponentially. Clear­

ance (CL, ml/min) and volume of distribution (V, liters) are character­

ized by the following empiric power functions:

(3)

(4)

where B is body mass (kg), B~ is brain mass (kg), and a, x, z, band y

are empirically estimated parameters. (The reason for the apparent

dependence of clearance on brain weight will be discussed in the

205

section on neoteny). The elimination rate constant (K) is given by:

K (CLIV) (5)

and:

C = (D/V) exp [-Kt] (D/b BY) exp [-(alb) BX- y B~z t] (6)

-(alb) BX- y B~z t + In (l/b) (7)

where D is dose. A Napierian semilogarithmic plot of C/(D/BY ) versus

BX- Y B~z twill be linear with slope -(alb) and intercept In (l/b).

This is illustrated for antipyrine in Fig. 7. The values of the

002

001

0007

~OOO5

E : C) 0003 ,.....: ~ ci 0002

::z. ~

~ 0001

U I:C 00007 , e. 00005

00003

00002

B Baboon o Oog C Cynomolgus Monkey P Pig 5 Sheep R Rat Z Human

F Cow (Friesion Breed) M Rhesus Monkey E Horse (Equine) H Rabbit (Hore)

~ .. 003 - ..

~ ~ 002

~ ~ i er. c B 0 ! I~! 001rc~~'""i'~M~~-.L..JMIL-4---L u ~ 0007

e: 00050)C--~f------!;2--~3---+------! Syndesichrons

B 295 x BW -615 x time x 10-2

M

o

o

000010~-~~-~lO~-~'5--~2&O--~2~5--~~~-~35~--4~O~-~4~5--~5~O--~55~-~W

Syndesichrons B295 X BW- 615 X time x 10-2

Fig. 7 - Semilogarithmic syndesichron plot for antipyrine

disposition in 11 mammalian species. Reprinted from

Boxenbaum (1984), p. 1100, by courtesy of Marcel Dekker, Inc

parameters are x = 1.244, z = -0.615, y = 0.949, a = 0.406 ml min-1

kg-1 . 244 kgO. 615 and b = 0.831 liters kg-0 . 949 The area under the

curve is (l/a). Following Poincare (1929) (vide supra), we arbi­

trarily define the units on the abscissa of the plot illustrated in

Fig. 7 as syndesichrons. It can be shown (and observed from Fig. 7)

that antipyrine half-life is approximately 14.2 syndesichrons in all

206

species; expressed in species' minutes, this is equal to (14.2) (100)

(B-0 . 295 ) (B~0.615), where B and B~ are expressed in kg (Boxenbaum,

1984). Putting it another way, each syndesichron is equivalent to

(BY- x) (B~-z) (100), or (B-0 . 295 ) (B~0.615) (100) minutes. For a 70 kg

man with a 1.53 kg brain, the calculated half-life is 527 minutes; for

a 10 kg dog with a 0.0531 kg brain, the calculated half-life is 118

minutes. The actual half-lives (calculated directly from data from a

single member of the species, as used in Fig. 7) are 759 and 70.0

minutes, respectively. The magnitude of differences between theoreti­

cal and observed values is typical in analyses of this type. The

syndesichron is termed a space-time unit, because its component parts

are space (mass) and time (chronological). Through appropriate allo­

metric coordinate transformations, pharmacokinetic processes (volume

cleared and fraction of dose eliminated) have been converted into forms

which are species' invariant in space-time. In one syndesichron, all

relevant species have eliminated the same fraction of dose from their

bodies; also, all species have cleared the same volume of plasma per

kgy body mass. In 1054 min in the human and 236 min in the dog (corre­

sponding to 28.4 syndesichrons in both species), each organism will

have eliminated 75% of the dose (two half-lives) and cleared a volume

of 1152 ml/kgO. 949 body mass (the 0.949 exponent equals y). [See

Boxenbaum (1984) for a discussion of other space-time units.]

Broadly speaking, science has two objectives - to characterize

regularities and make predictions. The allometric paradigm offers a

great deal of leverage in satisfying the first objective; blood concen­

tration histories (and in many cases, blood component binding and

partitioning characteristics) are all that is required. Although the

literature suggests that allometric methods have had good success in

interpolating or extrapolating parameters across species, this may be

misleading. The amount of time and work involved in allometric analysis

is sufficiently minimal that investigators (and journals) may not be

inclined to publish negative results. This is in stark contrast to

reductionist models; when a failure occurs, it is only after a major

throughput of time, money and other resources. The temptation therefore

is to rationalize an inadequate model (as opposed to performing addi­

tional experiments to look for the evidence needed to transcend it).

The major shortcoming of the allometric paradigm is that although

it may tell us how size exerts its influence, it cannot tell us why; it

207

emphasizes behavior at the expense of structure. As an intellectual

tool, it perhaps reflects less the reality of nature than the ingenuity

of men (Eiduson, 1962). The dilemma of how to visualize poorly under­

stood phenomena (mechanistically) is side-stepped by imaging outputs in

unique time frames. Allometric models focus not so much on how things

work, but on resemblances between theoretical entities and subject­

matter (Kaplan, 1964). In essence, we have a cluster of conclusions in

search of apremise (Suppe, 1977). Yhereas physiological models tend

to be over-elaborated, allometric models tend to be over-simplified.

Although the relationship between the model, the modeler and the model­

ed phenomena differ markedly in the two paradigms, both approaches are

heuristically fertile. Perhaps the best methodological council is to

heed Yhitehead's advice: "Seek simplicity and distrust it." (Kaplan,

1964) .

Neoteny and Pharmacokinetics

Generally speaking, man does not clear drugs from his body at rates

(ml/min/kg) comparable to other terrestrial, mammalian species.

Boxenbaum (1980) noted, for example, that the intrinsic clearance of

unbound drug (CLuint ) for antipyrine (metabolism is predominantly phase

I) was approximately one-seventh the value ~redicted from allometric

data analysis on other mammalian species. Analogously, based solelyon

body weight, allometric analysis predicts a maximum lifespan of 27

years for a 70 kg man (Calder, 1984; Sacher, 1959), approximately 23 %

of that actually observed [reports of 140 year-old Russians notwith­

standing, we take the maximum lifespan potential (Sacher, 1959) to be

approximately 110-120 years). In both cases, allometric deviations

arise predominantly from the influence of neoteny (Yates and Kugler,

1986). Neoteny is the retention of formerly juvenile characteristics

by adult descendants (produced by the retardation of somatic develop­

ment for selected organs and parts) (Gould, 1977). Contrary to

Haeckel's Biogenetic Law (ontogeny recapitulates phylogeny), humans do

not pass through an embryological ape-like stage and then grow beyond

it (Luria et al., 1981). Rather, we develop more slowly than apes, and

as adults, retain the juvenile or embryonie characteristics of apes.

This is illustrated in Fig. 8; note the physical resemblance of the

baby chimpanzee to adul t humans (Naef, 1926). Naef re marks : "Of all

the animal pictures known to me, this is the most man-like."

208

Fig. 8 - Baby and adult chimpanzees.

Reprinted from Naef (1926), by courtesy

of Springer-Verlag Publishers. This

figure was also used by Gould (1977) in

his discussion on neoteny.

Vithin the matrix of neotenous retardation, adaptive features of

ancestral juveniles are maintained for longer periods of time, foster­

ing longer learning and greater socialization (Gould, 1977). Slower

maturation is also apparently a major factor associated with enhanced

longevity (Miller, 1978). This strategy is favorable for populations

in crowded, stable, competitively fierce environments in which low

reproductive effort, slow maturation, intense parental care, longer

lives and a high degree of socialization produce a few well-formed

individuals of extremely high competitive ability (Gould, 1977; Luria

et al., 1981). Gould (1977) views neoteny as probably the major

determinant of human evolution; retardation provided the only "escape"

away from ancestral allometry and towards favored adaptation. Overall,

mammalian neoteny is characterized by an array of biological proper­

ties: retention of juvenile characteristics of ancestors; longer

gestation period; enhanced longevity; a larger and more complex brain

(enhanced cerebralization); delayed maturation (including sexual); a

high degree of socialization; production of relatively few but well­

formed offspring; extensive parental investment; and reduced drug

metabolism rates (Gould, 1977; Luria et al., 1981; Yates and Kugler,

1986).

209

Yith their strong departures for human data, standard allometric

plots of gestation period, age at sexual maturity, lifespan, brain size

and phase I drug metabolic indices (particularly CLuint ) all express

neotenous retardation (Boxenbaum, 1986; Gould, 1977; Yates and Kugler,

1986). Calder (1984) refers to this as vertical allometry. It is the

impact of neoteny (with its resultant vertical allometry) on both both

brain mass and CLuint that is responsible for the empirical correla­

tions between the two variables. This is also the reason why drug

metabolism rates can also correlate with maximum lifespan potential

(Boxenbaum, 1984).

It has even been argued (Brown, 1959) that the prolonged infancy

and childhood made possible through neoteny is responsible for the

shaping of human desires in two contradictory directions: on the

subjective side, freed from the limitations of reality, toward omni­

potent pleasure indulgence; on the objective side, toward powerless

dependence on others. Parental discipline and restriction, religious

denunciation of bodily pleasures, and exaltation of the "life of

reason" have all left man docile; but having tasted the "fruit of the

tree of life" in infancy, man's unconscious is unconvinced and there­

fore neurotic (Brown, 1959).

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Yates, F. E., and Kugler, P. N., 1986, Similarity prineiples and

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214

EQUIVALENCE OF BIOAVAILABILITY AND EFFICACY IN DRUG TESTING

INTRODUCTION

Carl M. Metzler

Biostatistics, PR&D The Upjohn Company Kalamazoo, Michigan

In the last 15 years much attention has been given to evaluating the relative bioavailability of two formulations, and particularly to the prob­lem of deciding if a "test" formulation is equivalent to a "standard" or "reference" formulation. For much of this per iod statistics has not pro­vided good tools for helping pharmaceutical scientists in these problems, but in recent years a number of methods have been proposed for decision rules based on solid statistical foundations. I have reviewed some of these rules and reported simulations that have evaluated their performance (Metzler and Huang, 1983; Metzler, 1986; Metzler, 1987). In this paper I will briefly review that previous work, report on some extensions, and con­sider some further questions that still await answers. Iassurne that the reader is familiar with the concepts of bioavailability and with those parameters, such as AUC (area under the curve), CMAX, TMAX or total amount of drug eliminated, that are most often used to characterize the bioavaila­abilty of pharmaceutical formulations. These parameters are computed from observed concentrations of drug in blood and urine.

THE BIOEQUIVALENCE PROBLEM

It seems likely that much of the confusion and error in applying sta­tistics to the bioequivalence problem might have been avoided if communica­tion between statistician and pharmaceutical scientist had been better. One of the aids to communication is proper and precise definitions.

Definition of Bioeguivalence

The following definition states in a precise manner the generally accepted meaning of bioequivalence.

In bioequivalence testing, a Test formulation is said to be bioeguivalent to a Reference formulation when

A < THETA< B,

where A and B are relevant medical or pharmacologic limits for relative bioavailability, and THETA is a measure of the relative bioavailability of the Test formulation to the Reference.

215

For examp1e, if THETA is the ratio of the mean AUCs of the Test and Refer­ence formu1ations, then A is often taken to be 0.8 and B to be 1.2. Or if THETA is the difference of the mean CMAX of the two formu1ations, expressed as apercent of the mean CMAX of the Reference, then A is often taken to be -20% and B to be +20\. The interval (A,B), which need not be symmetric about one for ratios or zero for differences, can be ca11ed the acceptab1e interval for bioequiva1ence.

In this definition THETA is, of course, a function of the total popu­lations, not of the samp1es actua11y observed. It is the task of statisti­ca1 science to aid in making adecision about populations when all that is known is samp1e resu1ts.

The Statistica1 Problem

The too1 traditiona11y used in statistics for decision making has been the hypothesis test, but the c1assica1 hypothesis is used to "prove" that two populations are different. It can be seen from the bioequiva1ence definition that we want to prove two populations are "almost equa1". In spite of this obvious inconsistency, unti1 recent1y bioequiva1ence was usua11y decided by some app1ication of c1assica1 testing procedures. The first alternative suggested for evaluation of relative bioavai1abi1ity was the use of confidence intervals (Westlake, 1972; Metzler, 1974; Steinijans and De1etti, 1983). Later suggestions inc1uded Baysian analysis (Rodda and Davis, 1980; Se1wyn, et a1, 1981; Manda11az and Mau, 1981; Se1wyn and Hall, 1984) and the use of a proper1y stated hypothesis test (Anderson and Hauck, 1983). These proce- dures will be disucssed in more detail in the section on decision ru1es.

DECISION RULES FOR DECIDING BIOEQUIVALENCE

Since bioequiva1ence tests are usua11y carried out in order to make a decision about the relative bioavai1abi1ity of the test formu1ation, it makes sense to put it into adecision ru1e context. A1though Westlake (1976), Manda11az and Mau(1981) and F1ueh1er, et a1(1983) all ta1ked about decision making, they did not give adecision ru1e characterization of the problem. They did not discuss the errors that can be made by adecision ru1e, nor the probabi1ities of making the two kinds of errors. These and other authors have discussed the power of the statistics on which ru1es for bioequiva1ence are based, but not the operating characteristics of the decision ru1es.

The characteristics of adecision ru1es can be discussed in terms of alpha levels and power, just as is done with a statistica1 test. But in order to avoid possib1e confusion I will use the term PR, probability of reiection to study the characteristics of the proposed decision ru1es.

Characteristics of Decision Ru1es

When making decisions with incomp1ete know1edge there is a1ways some chance of making an error. In a bioequiva1ence context, the samp1e resu1ts may lead to the decision that the test formu1ation is equiva1ent to the reference formu1ation, when in truth it is not. That is, THETA may be out­side the acceptab1e interval for bioequiva1ence. Of course, the other type of error is sometimes made also; based on the samp1e it is decided that the two formu1ations are not equiva1ent, when rea11y THETA is inside the acceptab1e interval. The truth tab1e in Figure 1 i11ustrates the correct and incorrect decisions that can be made.

216

o E C I S I o N

Bioequivalent

Not Bioequivalent

TRUTH I

Bioequivalent I Not Bioequivalent

Right decision Everybody gains

Wrong decision Sponsor loses

Wrong decision Consumer loses

Right decision Consumer gains

Figure 1. Truth Table for Bioequivalence Decision Making.

Can the two types of errors be controlled? How large should the risks be? As I will show in the discussion of proposed decision rules, it is possible to formulate rules so that at least one of the risks has a set size before a study is conducted. From the consumer (or regulatory) point of view, PR, the probability of rejecting bioequivalence, should be large when THETA, the true relative bioavailability, is at the ends of the acceptable interval. In the past, use of the classical hypothesis test with an accompaning power of 0.80 has implied that it is alright to have a risk of one in five (0.20) of declaring equivalence, when the true state of affairs is that the test formulation is not equivalent to the reference.

To show the characteristics and performance of the proposed decision rules I use PR (probability of rejection) curves. These curves show the probability that the decision rule will reject bioequivalence for given true values of THETA, the true relative bioavailability. For the dis­cussions in this paper I have expressed relative bioavailability as a frac­tion and taken the acceptable interval to be (0.80,1.20). Figure 2 shows a

08 ., u c ., a; > :; 0.6 0-., 0 äi Ü ., 04 ., a:

.0 0 n: 0.2

0 07 08 09 11 12 1.3

RELATIVE BIOAVAILABILITY

Fig. 2. A Good PR (probability of rejection) Curve

217

"good" Pr curve. It is good in the sense that for values of THETA between 0.9 and 1.2 the probability of rejecting equivalence is zero, and for values of THETA outside the acceptable interval the probability of rejeet­ing equivalenee rapidly approaehes one.

SOME DECISION RULES

For the deeision rules that I have evaluated earlier(Hetzler, 1987) and reviewed here, I have taken (0.8,1.2) as the aeepetable interval. I have taken PR(0.8) = PR(1.2) = 0.90 as the aeeeptable risk for the eonsumet (regulatory ageney) to take when THETA is at the ends of the aeeeptable interval. Clearly the ideas work just as well for other ehoiees.

Previously I have diseussed seven deeision rules for bioequivalenee, but it ean be shown that some of these are exaetly the same, that is, theit PR eurves are equal for all values of THETA. There are four distinet rules, but for many situations they are so nearly equal that in praetiee one ean ehoose that one whieh is most informative or easiest to eompute.

Standard Confidenee Limits Deeision Rule

For this rule eompute the usual 80% eonfidenee interval (CL,CU) for the differenee of the means of the two formulations. This eonfidenee int­erval is based on the t-statistie and uses the appropriate error standard deviation from analysis of varianee. It is symmetrie about the sample mean differ enee. Aeeept bioequivalenee if (CL,CU) is eontained in (A,B).

A pair of one-sided tests using the same statistie has been proposed by Sehuirmann(1985). This proeedure gives exaetly the same rule.

Westlake Symmetrie Confidenee Limits Deeision Rule

Compute the 90% eonfidenee interval (WL,WU) as indieated by Westlake (1976). This eonfidenee interval is symmetrie about zero. Deeide bioe­quivalenee if (WL,WU) is ineluded in (A,B).

The Baysian analysis proposed by Rodda and Davis(1980) gives a rule that is exaetly equivalent to this. Using the eomputations outlined by Rodda and Davis eompute the posterior probability, based on the sample values, that THETA is eontained in (A,B). Deeide bioequivalenee if this probability is greater than 0.90.

Handallaz and Hau Deeision Rules

Handallaz and Hau(198l) pointed out that the methods proposed by Westlake and by Rodda and Davis were approximate in that they substituted the sample mean of the referenee bioavailabilty for the population mean. Using the eorreetions given by Handallaz and Hau gives two (equivalent) rules that are mueh like the two in the preeeeding seetion.

Hauek and Anderson Deeision Rule

The hypothesis test proposed by Anderson and Hauek(1983) is the basis for a fourth distinet rule: If the probability assoeiated with the null hypothesis is less than 0.10 deeide bioequivalenee.

For all of these rules if bioequivalenee is not eoneluded, then deeide nonbioequivalenee.

218

PERFORMANCE OF THE DECISION RULES

Like most statisical estimators and hypothesis tests, the performance of these rules will be determined by variability and sample size. It is very well known that in bioavailability studies a major source of variabil­ity is subjects. Thus I assume that most bioavailability studies will use an experimental design in which each subject is used more than once. It is then possible to identify and remove this variation source.

The four decision rules stated above all have the property, in theory, that the probability of rejecting equivalence is 0.90 if the true relative bioavailabilty, THETA, is 0.80 or 1.20. But the PR curve for other values of THETA will depend on the variance of the sample bioavailability parame­paramter and on the sample size.

Since the theoretical properties of these rules are only approximate or asymptotic, and/or can only be stated as inequalities, simulations have been used to compute their PR(THETA) curves.

Previous Simulation Results

In the simulations reported in Metzler(1987) it was assumed that the experimental design was the usual two-way crossover. The variable used was AUC, but the results hold for any other bioequivalence with similar distri­bution. Period effects were ignored; the components of the AUC were an overall mean, fixed formulation effects and random subject and error effects. In all simulations the reference mean AUC was 300, and the test mean was 300xTHETA. The coefficient of variation (CV) was the ratio of the error standard deviation to the reference mean. It is this variation which is relevant to the performance of the decision rules, since the CV of the observed AUCs is inflated by subject effects which are removed in the anal­ysis of variance.

Q) t> C Q)

';;j > '5 0-Q) 0

i.i5 ;:; Q) . .,

a::

.D 0 a:

0 .8

0 .6

OA

0 .2

",\ \\ '\ .

\', 1,

\\ >

\\. \\ ...

\(

.1/ /.i

/.,:.' '~.,,::::-,: , '~'-

// .. ~/ (/

/./

~ .. .. ;-::'~ -,;. . '

// ,-

Legend

s ta nda rd CL

~~ ~.t!9!<.t!.q

~y~ (~) .

A- Ha uck

0 4-------_+------~r-------~------,_------_+------~

0 .7 0 .8 0 .9 1,1 1.2 1. 3

TRUE RELATIVE BtOAVAILABILtTY (THETA)

Figure 3. PR Curves by Simulation. CV=20%, N=16; Metzler (1987).

219

For eomputation of the PR eurves 1000 trials were simu1ated at eaeh value of THETA. Figure 3 shows the results for the ease CV=20% and N=16. That is, the error standard deviation was 60 units and there were 8 sub­jeets in eaeh group. In this simulation standard 90% eonfidenee intervals were used rather than the 80% eonfidenee intervals proposed above. When 80% eonfidenee intervals are used the PR eurve for the standard eonfidenee limits eannot be distinguished from the PR eurves for the rules based on the Westlake eonfidenee intervals or on the Anderson-Hauek hypothesis test.

Figure 3 shows that at THETA = 0.8 or = 1.2 all of the PR eurves are very elose to, but not exaetly at 0.9. At exaet equivalenee, Theta = 1.0, the PR eurves are between .13 and .15. Thus, although the eonsumer risk has been eontrolled at the level ehosen before the trial, the risk of re­jeeting equivalenee, when in faet the two formulations are equivalenet, may be higher than the drug sponsor or manufaeturer wishes to aeeept. The only reeourse is a larger sample size.

A number or surveys, ineluding that by Fluehler, et al(1981) have shown that although a CV of 20% may be higher than average for bioavai1a­bility studies, it is by no means uneommon. These simulations show that in order to have an aeeeptable low PR between 0.85 and 1.15 sample sizes in the order on N=24 or N=32 are required.

When the approximate residual CV is known, it is not diffieult to eompute simulations to find a sample size that has an aeeeptable PR eurve. For CV - 30%, a high but eertainly not unusual amount of variation, PR eurves were eomputed at several values of N. Figure 4 shows the results for N=56, about the samllest N whieh gives aeeeptable low probability of rejeetion for true values of THETA between 0.9 and 1.1.

Extensions of Ear1ier Work

All of the work I have reported up to now (Metzler, 1987) has assumed that the observations have a symmetrie distribution. For many drugs it is quite elear that the measures of bioavailability are not symmetrie. In most eases one eannot determine whether they are skewed distributions, sueh as the lognormal, or are eontaminated with large outliers. The work on polymorphie metabolism (see, for example, Seott and Poffenbarger, 1979) would suggest that in many eases a model of eontaminated distributions might be most appropriate. Where the laek of symmetry is due to absorption or body size, a skewed distribution might be the better model.

Reeent1y we have done some simulations in whieh the subjeet effeet is a eontaminated normal, that is most of the AUC's are generated with subject effeets that have mean zero and standard deviation SO, but some pereent of the subjeet effeets have a mean of 3S0 and a standard deviation of 2S0. Figure 5 shows distributions of AUC without eontamination and with eontami­nations of 5, 10 and 15 pereent. This kind of eontamination has little effeet on the lower tail, but inereases the weight of the upper tail with a consequent lowering of the eenter of the distribution.

Figure 6 shows PR eurves for the four deeision rules applied to AUCs with a 10% eontamination. As be fore , the eurves for three of the ru1es are so nearly the same as to be barely distinguishable. The effeet of the eon­tamination is to inerease PR for values of THETA less than one and to de­erease PR for values greater than one. In a sense using these rules on eontaminated distributions is a eonservative proeedure, for it inereases the proteetion for the eonsumer when the Test formulation has bioavaila­bi1ity less than that of the Referenee formulation.

220

Q) <.J C ., m >

0.8

.~ 0.6

'" o äi Ü Q) . ., a:

J:J o a:

0.4

0 .2

o 0 .7

.... ... \ .

0 .8 0 .9 1.1

TRUE RELATIVE BIQAVAILABILITY (THETA)

1.2

Legend

standard CL

1.3

Figure 4. PR Curves by Simulation. CV=30%, N=56; l1etzler (1987).

5.5

5

4.5

>- 4 <.J C ., '" a 3.5 ., .! .,

3 .~ :0 0; 2.5 a: ~ Q)

2 ü Ci; ~

1.5

0.5

0 0 100 200 300

\ \ "~ ... .. ,:-:: .

~~~ - - -.:.., ... ... : .. : ... : . :"-"

400 500 600 700

Aue 800 900

Figure 5. Distribution of ADCs with 0, 5, 10 and 15 percent contamination.

221

It has often been suggested that bioequivalence should be evaluated by analyzing the ratios obtained from each subject in a crossover study. The distribution of ratios, however, is difficult to work with. The use of nonparametric confidence intervals has also been proposed(Steinijans and Diletti, 1983, 1985) The nonparametric confidence intervals proposed by Steinijans and Diletti, although proposed and discussed by several authors, are closely related to the work of Hodges and Lehman; thus I call them HL confidence intervals (see Hollander and Wolfe, 1973). To compute the HL

., o c .,

<0 > ·5 a ., o

co t) Q)

Q) 0:

.0 o ct

0 .8

0.6

OA

0 .2

I

!

Legend

s ta ndard CL

West lake CL

~y~ (~).

A- Ho uck

O ~--------~------~--------~-------,--------~------~ 0 .7 0 .8 0 .9 1.1 1.2 1.3

TRUE R ELATIVE BIOAVAILABILITY (THETA)

Figure 6. PR Curves by Simulation. CV=20%, N=16 ; 10% contaminated.

intervals for the difference in means I use Walsh means; to compute these intervals for ratios, I use Walsh geometric averages (Steinijans and Diletti, 1985).

Figure 7 shows PR curves for decision rules using HL intervals on ratios and on differences, and applied to the same data as in Figure 6. For comparison the PR curve for the decision rule based on Westlake sym­metric confidence intervals is also shown. For values of THETA less than 0.95 there is little difference in the curves, although the rules based on HL confidence limits are more nearly symmetric about THETA equal to one. Accordingly, these rules are closer to 0.90 at THETA equal to 1.2. But in that part of the acceptable interval from THETA = 0.95 and THETA = 1.15 the PR for the rules based on HL confidence intervals are enought higher than the PR for the Westlake rule that there does not seem to be any reason to recommend them.

FURTHER WORK AND UNANSWERED QUESTIONS

Decision Rules for Average Bioavailability

Clearly more work needs to be done on analysis of individual ratios and with nonparametric confidence intervals. Basic to that work is a better understanding of the distribution of such bioequivalence parameters as AUC and peak concentration. What causes them to be nonsymmetric? Are these factors bimodal (contaminated) distributions such as suggested by polymorphic metabolism?

222

'" () c Cl> <ii > '5 0' Cl> 0 iii Ö Cl> '0; a::

.&J 0 .t

..,.....=----,------------------- -----! . '-".

.. \

0 ,8 \

0 .6

.. . .... .. .. ..

0.4 ... .......

0 .2

...

0 0 .7 0 ,8

)..··A

.. :<';: I'

... ;: :: /

:' / .: /

/ / ,,: /

/ /

'- /

0 .9 1.1

TRUE RELATIVE BIOAVAILABILITY (THETA)

1,2

l egend

~ _L_ ~~. !~~ _r_o_i!~

We s t loke Cl

t!!o C~or ...!!,iff ~

1.3

Figure 7 . PR Curves by Simulation. CV=20%, N=16; 10% contaminated.

Some bioavailabilty trials show a very few, very large outliers; more extreme than can be obtained by the contaminated distributions we have been using. We plan more work comparing the parametric and nonparametric rules for distributions of Aues with more extreme outliers.

Another approach to analyzing ratios is to obtain their variances, and hence confidenc intervals, by bootstrapping methods. My colleague, Dr. H. J. Rostami, has been investigating that approach, with some promising results that we will be reporting soon .

As these rules are bett er understood and gain acceptance as the pre­ferred way of assessing bioequivalence, there will be a demand for software that will make their computation easy. They can all be computed with a statistical system such as SAS (Metzler, 1987). Wijnand and Timmer(1983) provide information about computer programs to compute many of these rules , along with a good survey of literature of the rules.

Variances in Bioavailability

The discussions of bioequivalence of the past 20 years have focused almost entirely on average bioavailabilities. Recently questions are being raised about the relative variability of a test formulation and the refer­ence formulation. It is suggested that a test formulation should also satisfy a requirement as to the variability of the amount bf drug it makes available or the rate at which it is available. Unfortunately, most of the questions have not been accompartied by suggestions regarding the amount of variability that should be permitted in excess of the variability present in the reference formulation.

Implied in many of the discussions of formulation variability is that the designs presently being used do not make it possible to estimate within

223

subject variability, although most people seem to think that between sub­ject variability is easily estimated. The error term in the analysis of variance of most crossover designs is an estimate of the average within subject variance. The within subject variance for each treatment can be estimated by looking at the residuals from the analysis of variance model. It is not clear at this time whether decision rules for deciding the rela­tive variability of formulations will be required in the future. All of our experience in statistics suggests that if they are they will require even larger sample sizes than are required to decide the equivalence of mean bioavailabilities.

Eguivalence in Efficacy Trials

Many clinical trials to evaluate the efficacy of a new drug are very like bioequivalence trials in that the new drug is being compared with an active drug that is already in use. The trials are not designed to show that the new drug is more efficacious, only that its efficacy is about the same. This then, is the formulation of a bioequivalence trial and the decision rules discussed here can be applied. Indeed, Anderson and Hauck (1986) have applied their interval hypothesis test to clinical trials. Blackwelder(1982) has also applied the concepts of bioequivalence assess­ment to the comparison of the efficacy of two active drugs.

Many clinical trials have a binomial response: Success or failure. In such trials the problem is easier, for one can work with the binomial dis­tribution, as shown 10 years ago by Dunnett and Gent(1977). Another factor that may make decision rules easier to apply to clinical trials than to bioequivalence studies is that often the decision can be one-sided. That is, only sample results showing the test drug poorer than the reference drug would be cause for declaring inequivalence.

224

REFERENCES

Anderson, S. and Hauck, W. W., 1983, A new procedure for testing equiva-1ence in comparative bioavai1ability and other clinical trials. Commun. Statist.-Theor. Meth. 12: 2663-2692.

Blackwe1der, W. C. 1982, "Proving the null hypothesis" in c1inica1 trials. Control1ed Clin. Trials 3: 345-353.

Dunnett, C. W., and Gent. M. 1977, Significance testing to establish equiva1ence between treatments, with special reference to data in the form of 2x2 tables. Biometrics 33: 593-602.

Fluehler, H., Grieve, A. P., Manda1laz, D., Mau, J. and Moser, H. A., 1983, Baysian approach to bioequivalence assessment: An example. Jr. Pharm.Sci. 72: 1178-1181.

Fluehler, H., Hirtz, J. and Moser, H. A. 1981, An aid to decision-making in bioequivalence assessment. J. pnarmacokin. Biopharm. 9: 235-243.

Hauck, W. W., and Anderson, S. 1986, A proposal for interpreting and re­porting negative studies. Stat. in Med. 5: 203-209.

Hollander, M., and Wolfe, D. A. 1973, "Nonparametric Statistical Methods", John Wi1ey and Sons, New York.

Mandallaz, D. and Mau, J., 1981, Comparison of different methods for decis­ion making in bioequivalence assessment. Biometrics 37: 213-222.

Metzler, C. M., 1974, Bioavai1ability - a problem in equiva1ence. Biometrics 30: 309-317.

Metzler, C. M. and Huang, D. C., 1983, Statistical methods for bioavail­ability. Clin. Res. Pract. Drug. Reg. Affairs 1: 109-132.

Metzler, C. M., 1987, Statistical methods for deciding bioequivalaence of formulations, in: "Drug Absorption from Sustained Release Formulations, " A. Yacobi and E. Halperin-Walega, eds., Pergamon Press, New York.

Rodda, B. E. and Davis, R. L., 1980, Determining the probability of an important difference in bioavai1ability. Clin. Pharmaco1. Ther. 28: 247-252.

Scott, J. and Poffenbarger, P. L., 1979, Pharmacogenetics of tolbutamide metabolism in humans. Diabetes 28: 41-51.

Selwyn, M. R., Dempster, A. P. and Hall, N. R., 1981, A Baysian approach to bioequivalence for the 2x2 changeover design. Biometrics 37: 11-21.

Se1wyn, M. R. and Hall, N. R., 1984, On Bayesian methods for Bioequiva­lence. Biometrics 40: 1103-1108.

Steinijans, V. W. and Di1etti, E., 1983, Statistica1 analysis of bioavai1-ability studies: Parametric and nonparametric confidence intervals. Eur. Jr. Clin. Pharmacol. 24: 127-136.

Steinijans, V. W. and Diletti, E., 1985, Generalization of distribution­free confidence intervals for bioavailability ratios. Eur. Jr. Clin. Pharmaco1. 28: 85-88.

Westlake, W. J., 1972, Use of confidence intervals in analysis of compara­tive bioavailability trials. Jr.Pharm.Sci. 61: 1340-1341.

Westlake, W. J., 1976, Symmetric confidence intervals for bioequivalence trials. Biometrics 32: 741-744.

Wijnand, H. P., and Timmer, C. J., 1983, Mini-computer programs for bio­equivalence testing of pharmaceutical durg formulations in two-way cross-over studies. Comput.Prog.Biomed. 17: 73-88.

225

KOOELING ANO RISK ASSESSKENT OF CARCINOGENIC OOSE-RESPONSE

INTROOUCTION

Ajit K. Thakur

Biostatistics Oepartment Hazleton Laboratories America, Inc. 9200 Leesburg Turnpike Vienna, Virginia 22180 U.S.A.

Before a pharmaceutical or agricultural compound is put in the

consumer market, it is customary to evaluate its safety for humans. In

many cases the safety is evaluated in terms of the compound' s carcino­

genic potential. The experiments for this purpose are generally per-

formed on rodents , specifically, rats and mice. Generally the human

equivalent of life-span in rodents is taken as 2 years. Since there may

be sex-specific changes taking place in humans, these studies are

performed on both sexes. The end points of such chronic toxicity-

oncogenicity studies are multiple. The dose levels at which the compound

is administered should ideally cover a "NOEL" (no-observable--effect­

level) and an "KTO" (maximum tolerable dose). At the "NOEL", the animals

in the study should not show any biologically meaningful effect, and at

"KTO" it should provide some indication of homeostatic imbalance.

Furthermore, there is a concurrent negative or vehicle control group in

the study for providing valid comparisons of treatment effect. Tradi­

tionally the design of such a study includes randomly selected equal

number of animals per group per sex in some selected strains of rats and

mice and between three to five dosed groups along with the control

(three treatment groups and a control is the common practice now-a-days

as accepted by most regulatory agencies). There are variations of this

method used in some cases with specific goals in mind. There are interim

sacrifices of randomly selected numbers of animals from generally the

control and the high-dose groups to check on disease prevalence rate or

227

TABLE 1. A Hypothetical k x 2 Table of Lesions

Level No. at Risk No. with Lesion 0 nO mo 1 n m 1 1 1

12 n2 m2 13 n3 m3

lk nk 1lItc

time of occurrence. All animals, particulary the ones in the control and

the high-dose groups, are necropsied to make sure that no unusual lesions

occurred in them. Palpable and superficial tissue masses or lesions are

also noted during the in-life phase of the study, and at the end of the

study or at death of the animals, they are examined histopathologically

for identification. This practice provides some idea about the induction

time of certain changes due to treatment. In the cases of lesions, the

data are collected as k x 2 contingency tables as in Table 1. The number

at risk is generally taken after eliminating any animals dying prema­

turely due to accidents, cannibalism, etc. The dose-response in such

tables is evaluated in terms of a positive trend by the weIl known

Cochran-Armitage method (1). A negative trend does not have any

biological meaning in these cases.

Unfortunately things do not always go smoothly in a life-time study.

There are confounding factors such as competing toxicity leading to

intercurrent mortality differences, threshold mechanisms in response

functions, and others. Some of these factors can be statistically

incorpot"aled into the appropriate dose-response evaluation techniques.

For example, if a particular lesion is "incidental", Le. was observed

in animals dying of other reasons ot" during scheduled sacrifices or at

the terminal sacrifices, and there is intercurrent mortality differences

among the groups, the logistic prevalence method or similar non-intet"val

based methods (2) are the most appropriate methods of evaluating a dose­

response. On the other hand, if the lesion was lethai or induction time

of the lesion is available, as in the case of bt"east cancer or skin

lesions, the appropriate methods are the ones as described by Thomas,

Breslowand Gart (3).

In any case, once a dose-response is evident, in many cases a partic­

ulat" compound may still be considered "safe" if the risk does not exceed

a pre-specified level. Hany people take this acceptable risk level as

less than 10-6 (one-in-a-million). A level which does not pt"oduce a

228

risk exceeding this pre-specified number is then considered virtually

safe dose (VSD). In actua1ity, these are not the numbers used for risk

evaluation purposes. The numbers used instead are the upper confidence

limit (UCL) of risk and the lower confidence limit (LCL) of VSD.

Some of the 1esions or other diseases occurring in a chronic study

may be conunon, Le., they wou1d occur norma11y during the 1ife-span of

anima1s no matter whp.ther they are treated with a chemica1 or not.

Treatment may have just acce1erated this natural process. Many of the

1iver, 1ung, and manunary 1esions in certain strains of rodents fall

under this category. On the other hand, certain types of tumors such as

hepatob1astomas, are extreme1y rare and occurrence of such 1esions even

in very small numbers may be due to treatment. This 1atter case

generally poses statistica1 nightmares because samp1e sizes used in

these studies are generally inadequate to detect any treatment effect.

The fo11owing discussion is specifica11y for dose-response analysis

and risk assessment for carcinogenicity. The mechanisms by which a

chemica1 may cause tumors in animals or human are poor1y understood, if

at all. As a resu1t, the techniques used for such evaluations are based

on empirica1 models for such dose-response curves which may or may not

be app1icab1e to certain types of data. The models sununarized here are

in their simp1est forms and there are wide variations practiced by

investigators incorporating different aspects of the life and dietary

status of the anima1s in the study.

KATHEKATICAL MODELS OF DOSE-RESPONSE MECHANISMS

All the models used in low-dose extrapolation fall into one of four

general categories: "mechanistic models" , tolerance distribution

models, time-to-tumor models, and pharmacokinetic models.

(1) "Mechanistic Models": These models are based on certain assumptions

on bio1ogica1 events taking p1ace in carcinogenesis. Unfortunate1y,

many of these assumptions cannot be verified experimenta11y because

of lack of understanding or exp1anations of these events.

(a) One-Hit Model: The one-hit model assurnes that a single critica1

mo1ecu1ar event ("hit .. ) between a target site and a proximate

carcinogen is necessary and sufficient to produce a tumor. The

probability of such an interaction is proportional to the dose

of the carcinogen. In other words:

229

230

P(d) l-exp[-(a+bd) 1 (1)

where

P(d) = Probability of a carcinogenic response as a function of

dose, d

a ~ 0

b ~ 0

background incidence rate

empirical potency of a carcinogen

At low dose levels (e.g., bd ~ 0.02), P(d) '" bd, Le., cancer

frequency at low-dose is linearly related to dose.

(b) Hulti-Stage Hodel: Originally proposed by Armitage and Doll (4)

and further extended by Crump et al. (5), the multi-stage model

assurnes that a carcinogen increases any of the events or "stages"

a cell goes through to become malignant. The model implies that

logari.thm of cancer mortality is proportional to logarithm of

dose. At low-dose, the model is linear and is virtually indis-

tinguishable from the one-hit model in most cases.

matical terms:

In mathe-

P(d)

k i

l-exp[- l: bid 1 i=O

bi ~ 0 = Empirical potency parameters

k ~ Number of dose levels = Number of stages

(2)

(c) Hulti-Hit (Gamma) Hodel: In its simplest form, the multi-hit

model (6) says that k molecular events ("hits") are needed to

induce tumor formation. The distribution of these "hits" over

time is assumed to be Poisson:

P(d)

where

b ~ 0

Q)

l: (bd)x exp(-bd)/x! x=k

Jbd[xk- 1 exp(-x)/(k-l)!ldx o

Empirical potency; k ~ 0 Number of hits

This is a generalization of the one-hit model.

(3)

(2) Tolerance Distribution Models: These models assume that each member

of a population will develop a tumor if carcinogenic exposure exceeds

a critical level or tolerance. The critical level varies from

individual to individual and is expressed by various distributions.

(a) Multi-Hit (Gamma) Model: Described in lhe previous section,

this model assumes a gamma distribution for the tolerance.

(b) Probit Model: Investigated by Mantel and Bryan (7), the Probit

model assumes that lhe tolerance distribution is log-normal.

The mathematical form of the original model is:

P(d)

with x

where

a ~ 0

b ~ 0

x [l/(2~)%] J e-u2/2 du

-co

a + b ln d

(4)

The intercept of the log-Probit plot (=background

incidence)

Slope of the log-Probit plot (=empirical potency factor)

Mantel and Bryan assumed: b=l.

(c) Logit (Logistic) Model: Doll (8) and Cornfield et al. (9)

investigated the Logi t model extensively. It assumes that the

tolerance distribution is binomial and is expressed as:

P(d) l/{l + exp[-(a+b log d)]} (5)

where

a ~ 0

b ~ 0

Intercept of the log-Logit plot (=background risk)

Slope of the log-Logit plot (=empirical potency factor)

(d) Weibull Model: Extensively investigated by Carlbor.g (10), the

Weibull model assumes that the tolerance is Weibull distri-

buted. The model takes the form:

231

P(t,d) (6)

with g(d) = a+bdm in its simplest version

where a and b have the same meaning as in the previous models

and m and kare two parameters of the model which do not have

any specific biological meaning but determine the shape of the

dose-response curve.

(3) Time-to-Tumor Models: These models assume that tumor formation is a

function of both dose of a chemical and time of exposure. The

Weibull is an example of such a model.

232

(a) Hartley Sielken Model: The Hartley-Sielken model (11), also

called a general product model, is a generalization of the

Weibull model and can be described, in its original form, as:

P(t,d)

where

k ~ 1

k 1 - exp[- r a.djh(t)]

j=O J

Number of stages of carcinogenesis

(7)

h(t) Time-to-tumor distribution, a positive non-decreasing

function of time

The underlying assumption behind the general product model is

that the living system consists of a number of linear compart-

ments. Each compartment, in its turn, comprises of a large

number of independent homogeneous units, the cells. The proba­

bility of transformation of a cell to a cancerous one is propor­

tional to the concentration of a carcinogen in a particular

compartment. The transfer of a carcinogen between compartments

is assumed to be strictly first order.

(b) Log-Linear Model: The log-linear model has been described in

detail by Kalbfleisch et al. (12) . If the time-to-tumor (T)

distribution is log-normal with mean m=m(d) and geometrie

standard deviation 0 (i. e., logT-N(log m, 0 2», then the proba­

bility of tumor formation is given by:

P(T~t) ~([log t-log m)]/a) (8)

where ~ is anormal deviate. The parameter m in equation (8) is

from the Druckery (13) equation mkd = c>O, k~l.

In this case,

log T 0<+6 log d + aw (9)

where 0< log c/k, 6= -l/k and W - N (0,1).

In its simplest form, the model does not allow for any response

at zero-dose, although it could be incorporated as an additive

background (12). A log-linear model of the above for'ffi can be

described for any error distribution; for example, with an

extreme value error distribution, one obtains a Weibull model

described previously. Similarly, one can reduce the log linear

model to the logistic or gamma (Multi-hit) models with appro­

priate error distributions.

(c) Cox's Logistic (Binary Regression) Models: This type of models

have been explored by Prentice et al. (14) in detail. A general

form of these models is:

P(t;d) 1-exp {-ItAo(u)exp[~(u)T~ldu} o

(10)

where AO(t)

6

z

unspecified baseline hazard function

(61 , 62' 6p )T a vector of regression parameters

Zp (L» T = a vector of regres-

sion variables or covariates whose components

depend on both time (t) and dose (d).

Generally, ~(t)

dence rate.

o at d o and AO is then the background inci-

(4) Pharmacokinetic Model: Cornfield's (15) pharmacokinetic model

considers a carcinogen being under simultaneous reversible activa­

tion and deactivation reactions with the probability of a carcino­

genj e response being directly pr-oportional to the concentration of

233

the activated comp1ex. If d, S, and T denote the amount of the

carcinogen, the substrate and the deactivating agent respective1y,

and Ka and ~ the ratios between the backward and forward reaction

rate constants for the activation and deactivation processes , then

the model expresses the probability of a carcinogenic response,

P(d), from steady state mass conservation 1aw, as:

P(d) A/(A + Ka ) , d > T

~d/[(S+Ka)~+KaTl , d ~ T

where A

and Q.

d-P(d)S-Q.

KaP(d)T/[P(d)Ka +{1-P(d)}Kd 1

(11)

At 10w carcinogen levels, when d < T, the dose-response curve is

almost linear. When ~=O, the model shou1d show a thresho1d at d=T,

but because of the steady state assumption, it does not actually

produce one because the model becomes independent of the time-course

of the reactions. When l<d > 0, the model shows a "hockey stick"

dose-response with a near1y f1at hitting part. In this case, once d

exceeds T, the hitting part of the "stick" rises sharp1y. The model

can be extended to incorporate a chain of such activation­

deactivation reactions without changing the above qualitative

characteristics. For some more recent deve10pment in this fie1d,

see Krewski et a1. (16).

PROBLEMS IN RISK ASSESSMENT

(1) Des; ßTI Pr'oh 1 p-ms: Chronic toxicity studies are not generally designed

for the specific purpose of risk assessment 01' va1didating any

particu1ar empirica1 model. The dose ranges covered are not wide

enough t.o revea1 the true shape of the dose-response curves. The

most important part of the dose-response curve, the low- dOH~' T'p-gion,

may not have enough information for valid extrapolation. Because of

solubi1ity problems and competing toxicity it may not be possib1e to

investigate the upper range of the curve as we11. The number of dose

levels used constilulO'f' I he number of data points which generally

cannot be replicated. As a resu1 t, maximum likelihood 01' least

squares estimation from such models may produce questionable results.

234

(2) Shape of the Oose-Response Curve: Many of the models used in the

field were designed to handle dose-response curves with specific

shapes. Most of these models cannot handle non-mono tonic curves.

If there is competing toxicity, there may be loss of monotonicity

and these models will fail to provide any realistic information from

the data.

(3) Low-dose Nonlinearity: Many people assurne that all experimental dose­

response curves must be low-dose linear (14). The apparent low-dose

nonlim~adty is taken by t.hem as noise masking linearity. They

impose linearity through upper confidence limit of estimated risk 01'

lower confidence limit of VSO (14). The U.S. Interagency Regulatory

Liason Group (17) proposed this step for low-dose extrapolation.

This assumption is debatable (10). What is needed then is a model

that should be able to handle this type of nonlinearity. One

possible means may be to extend Cornfield' s pharmacokinetic model

which would require more extensive basic experiments than most

investigators are willing to perfqrm.

(4) Presence of Threshold: Many physiological responses indicate

threshold behavior and tumor incidences are probably no exceptions.

According to such mechanisms, some chemicals may produce tumors at

high dose levels but none at lower dose levels. The Food Safety

Council (18) provides descriptions of such mechanisms with examples.

There seems to be more and more evidence of threshold in toxicity

testing with various chemicals in recent years. The argument seems

to be convincing in favor of threshold for chemicals which produce

tumors as a result of secondary effects due to metabolic overloading

at extremely high dose levels. The injury to the interna 1 control

mechanisms at 01' beyond "MTO" may be so extensive that any tumor

formation at those levels may be secondary effects. This would also

produce threshold-like dose-responses. Gehring and Blau (19) provide

vari.ous other explanations for threshold behavior in carcinogenesis.

Low-dose extrapolation on such dose-response curves with most

empirical models, particularly with the one-hit and the multi-stage

models, will provide highly undue conservative risk.

(5) Scaling from Rodents to Human: In most cases the information

necessary for appropriate scaling of information from rodents to

human is not available. As a result, empirical scaling factors are

introduced in the process. There are various chemicals which act

235

different1y in humans than in other speeies. Even the two sexes in

the same spedes, ine1uding humans, may respond differently to the

same eompound. Further, some of the hormones and metabolie enzymes

are different among different speeies. These problems have been

addressed by Dixon (20), Gi11ette (21), Krasovsky (22), Hotu1sky (23)

and others. For a eompound whieh seems to aet simi1ar1y in different

spedes, the work of Book (24) may provide some guidelines for

sealing. Boxenbaum (25) diseussp.s interspeeies pharmaeokinetie

sea1ing needed for extrapo1ating from animals to human. This is an

extreme1y diffieu1t and expensive task without Whieh risk assessment

may be eomp1ete1y meaning1ess.

Conversion from animal to human exposure dose is also a matter

of eontroversy. The method used by the U.S. Environmental Proteetion

Ageney (EPA) , for examp1e, uses a surfaee area eonversion aeeording

to whieh the human equva1ent dose (HED) is ealeulated as fo110ws:

HED (mg/kg BW/day) Animal Dose (mg/kg BW/d~ (Human BW/Animal BW)

(12)

Host people agree that this eonversion is empirieal at its best. The

Offiee of the Teehnology Assessment (OTA) states (26): "the ehoiee

of sea1ing faetor ean make a differenee up to forty-fold in

estimating human risks" from animal data.

(6) Numerieal Instability in Low-Dose Extrapolation: With the seanty

data available from standard assays, the goodness-of-fit statisties,

the residuals and the eonfidenee intervals of the estimates from the

empiriea1 equations used in eareinogenie dose-response modeling may

not have a lot of statistiea1 validity. Numeriea1 instability at

the 10w-dose region may be so great that the eonfidenee levels will

be unduly inf1ated. This is partieularly true with the one-hit and

the multi-stage models. What meanlng does an exereise in risk

assessment have if the point estimate for VSD for a risk of 10-6 is

.5 and i ts lower 95"1. eonfidenee limit is 10-6? Yet this is a very

eommon praetiee in risk assessment.

(7) Subjeetivity and Bias in Pathologieal Evaluation: The tumor inei­

denee tables used for risk assessment are eomprised of qualitative

data. Numbers of animals with tumors are not absolute eounts

independent of the observer. Different patho10gists may have

different eriteria for diagnosing eertain types of tumors. As a

236

result, it is not surprising to see a pathologist calling an adenoma

a non-neoplastic lesion or a carcinoma an adenoma when another

pathologist has diagnosed them differently. Hore serious problems

may arise from the fact that in most cases pathologists know which

treatment group the particular animal is taken from. They further

cornpare control and treated group animals back-and-forth. Until

coded or "blinded" slide reading is performed, there will always be

bias in such evaluations.

(8) Target Organ Dose: A chronic study is generally done with the

chemical in question administered through oral gavage, inhalation or

dietary intake. As a result, a target organ which may develop a

tumor may not receive the total amount administered in the whole

body. Furthermore, the chemical concerned may be metabolized by the

hormones and/or enzymes of the body and the effect one sees may be

due to byeproducts or metabolites. Generally extensive pharrnako-

kinetics and metabolism studies are not performed in association

with a chronic study to investigate these aspects. As a result what

one uses as 'dose' may be many-fold higher than what actually reaches

the particular target tissue.

EXAKI'LES

Let us consider risk assessment with three incidence tables. The

first (Table 2) is areal exarnple from the Food Safety Council (18).

The second (Table 3) and the third (Table 4) are two hypothetical

exarnples with "threshold" and "saturation" respectively. The risk

assessment is performed using the one-hit, the multi-stage, the probit,

the logit, the ganuna (multi-hit) and the Weibull models. The point

estimates and 95" LCL of the virtually safe dos es are shown in Tables

5-7. Observed incidences and the expected values frorn each of the

maximum likelihood fitted curves are presented in Figures 1-3. For the

dielddn exarnple (Fig. 1), the one-hit model shows some significant

discrepancies at all dose levels except at the zero-dose. This fact is

further confirmed by the lack of fit (p = 0.0256). The multi-stage

model was slightly better with the other four models fitted being the

best. For the "threshold" case (Fig. 2), the logit, the probit and the

Weibu11 seerned to provide the least discrepancy between the fitted and

the observed values. The one-hit, the multi-stage and the multi-hit

models show rernarkable discrepancies. For the case with apparent

"saturation" (Fig. 3), a11 the models indicated severe discrepancies

237

Dose

o 1.25 2.5 5

Dose

o 1.25 2.5 5

Dose

o 1.25 2.5 5

TABLE

TABLE 2. Dieldrin Data (Food Safety Couneil, 1980)

No. at Risk No. with Tumor Prop. w~~h Tumor

156 17 0.109 60 11 0.183 58 25 0.431 60 44 0.733

TABLE 3. Hypothetieal Example with "Threshold"

No. at Risk No. with Tumor Prop. with Tumor

75 5 0.067 75 5 0.067 75 5 0.067 75 20 0.267

TABLE 4. HypotheHeal F.xample with ··Saturation"

No. at Risk No. with Tumor Prop. with Tumor

75 5 0.067 75 10 0.133 75 20 0.267 75 20 0.267

5. Point Estimate and 95~ Lower Confidenee Limit (LCL) of Virtually Safe Dose (VSD) for the Dieldrin Example

EROB Esl 1 ma{e One-lil{ Rul{l-sfage Pro61{ [Ogl{ RUI{l-Fil{ Ae16ull ----10-2 VSD 5. 140Xl0-2 0.1722 0.6104 0.4470 0.2949 0.2943

95% LCL 3.333Xl0-2 5. 544X 10-2 0.3399 0.2154 0.1510 0.1178

10-4 VSD 5.117Xl0-4 1.940Xl0-3 0.2143 5.615Xl0-2 4. 800X 10-2 1.844X1O-2 95% LCL 3.314Xl0-4 5.517Xl0-4 8. 543Xl0-2 1. 326X 10-2 6.398Xl0-3 2.839Xl0-3

10-6 VSD 5.117Xl0-6 1.943Xl0-5 9.887Xl0-2 7.088Xl0-3 6.344Xl0-3 1. 156Xl0-3 95% LCL 3.314Xl0-6 5.516Xl0-6 7.063XlO-2 8. 165Xl0-4 2.695Xl0-4 6. 840X 10-5

10-7 VSD 5.117Xl0-7 1. 943X 1 0-6 7.086Xl0-2 2.518Xl0-3 8. 429Xl0-4 2. 895Xl0-4 95% LCL 3.314Xl0-7 5.516Xl0-7 1.971X10-2 2.024Xl0-4 5.528Xl0-5 1.061X10-5

between the fitted and observed values. For the Dieldrin ease (Table 5),

the VSD's and their 95~ LCL's are fairly eomparable for the logit, multi­

hit and the Weibull models whereas the one-hit and the multi-stage are

similar at small risk levels. The probit model stands alone on its own

with possibly the most liberal (from a eonsumer standpoint) VSD. As is

238

TABLE 6. Point Estimate and 95~ Lower Confidence Limit (LCL) of Virtually Safe Dose (VSD) for the Hypothetical

Example with 'Threshold'

~ROI! ~stlmate One-~lt RuHl-stage Prolilt [Oglt RuHl-~lt ~lliull

10-2 VSO 0.2777 1.7369 4.0648 4.4374 0.1886 4.4887 95% LCL 0.1210 0.3747 7.582X10-5 2. 725X 10-5 3.080XIO-2 4.833X10-5

10-4 VSO 2.763X10-3 0.3736 3.3540 3.7406 1. 874X10-3 3.8251 95% LCL 1.204X10-3 9.997X1O-3 1. 105X 10-5 9.452X10-6 1.339X10-5 1. 659X 10-5

10-6 VSO 2.763X10-5 8.048X10-2 2.9087 3.1545 1. 874X 10-5 3.2602 95% LCL 1.204X 10-5 3.729X10-5 7.063X10-6 5.022XIO-6 5.754X10-9 8.854X10-6

10-7 VSO 2.763X10-6 3.736X10-2 2.7358 2.8969 1.877X 10-6 3.0098 95% LCL 1. 204X 10-6 3. 729X10-6 5.967X10-6 3.892X10-6 1. 192X 10-10 6.888X10-6

TABLE 7. Point Estimate and 95~ Lower Confidence Limit (LCL) of Virtually Safe Dose (VSD) for the Hypothetical

Example with 'Saturation'

ERUI! Estlmate Une lilt lIIultl-stage Prolilt [Oglt Rultl lilt Relliull

10-2 VSO 0.1702 0.1702 0.1002 4.524X10-2 2. 472XIO-2 3. 304X 10-2 95% LCL 8.695X10-2 0.1170 3.847X10-3 6.723X10-4 1. 843X10-3 2.203X10-4

10-4 VSO 1.694X10-3 1. 694X 10-3 3.260X10-5 7.264X10-5 1.203X10-5 2. 589X 10-5 95% LCL 8.652X10-4 1. 165X10-3 5.659X10-6 2.393X10-9 8.973X10-7 2.420X10-10

10-6 VSO 1. 694X 10-5 1.694X10-5 2.576X10-4 1.184X10-7 5.857X10-9 2.045X10-8 95% LCL 8.651Xl0-6 1. 165X10-5 4.439X10-8 8.554X10-15 4.869X10-10 1.817XIO-16

10-7 VSO 1.694X10-6 1.694X 10-6 8.635X10-5 4. 778XIO-9 1.293X10-10 5.749X10-1O 95% LCL 8.651X10-7 1.165XIO-6 5.492X10-9 1.615X10-17 9.641X10-12 1.572XIO-19

OBS. P ONE-tIr wu.n-STAGE a ....•.... I

FIGURE 1. Observed and expected values for the Dieldrin Example (18)

239

OBS. P

D

p

ONE-tfI' .. .... ...

~2' .... .:,;.~~~ ... ~~ .•... . ----. .... . --

POSE

FIGURE 2. Observed and expeeted' values for the hypothetieal example with "Threshold"

OBS. P

D

ONE-tfI' ...... ...

cmtERS

•

FIGURE 3. Observed and expeeted values for the hypothetieal example with "Saturation"

obvious from this table, the VSD's and their 95~ LCL's differ by almost

a faetor of 105 . For the example with "threshold" (Table 6). the VSD' s

and their 95~ LCL's are eomparable for the logit, probit and the Weibull

models. The approximately million-fold differenee between the VSD and

its 95~ LCL in eaeh ease easts serious doubts about the meaning of the

risk assessment. On the other hand. the multi-stage model estimate of

240

the VSD does not seem to be meaningful when one examines the dose­

response curve carefully. The one-hit model, with the 95~ LCL being in

the same order of magnitudes at each risk level shows virtually a zero

VSD at smaller risk levels. The multi-hit model was even worse. When

one examines the goodness-of-fit probabilities (Table 8), the one-hit

and the multi-hit models indicated significant lack of fit., the others

did not. In fact, the logit, probit, and the Weibull models indicated

almost aperfect fit in each case. Finally, for the table with "sat.ura­

tion", as Table 7 indicat.es, the point estimates and the 95~ LCL for the

VSDjs from the one hit and the multi-stage models are virtually identi­

cal. The VSD' s at different risk levels for the probit model are

comparable to the ones from the one-hit and the multi-stage models. The

estimates from the logit, multi-hit, and the Weibull models, while

comparable within themselves, were very significantly different from the

others. Unfortunately, the goodness-of-fit probabilities for all models

(Table 8) indicated acceptable fit, confirming my previous statement that

with scanty data, these probabilities perhaps do not provide any concrete

evidence regarding the appropriateness of one modelover the others.

Hy personal view about the two hypothetical examples indicating

"threshold" and "saturation" is that none of the models tested provides

any meaningful lower confidence intervals of the VSD's at extremely small

risk regions. The same argument can be provided for EROB's as well.

DISCUSSION

Let me make a few brief comments about the different models at this

point. Both the one-hit and the multi-stage models show linearity at

small dose regions and are virtually intinguishable in their estimates

for most dose-response curves. This particular region is the most

important portion of the dose-response curve where numerical computation

is invoked. As a result, with low-dose nonlinear dose-response curves

and curves with "threshold", risk assessment with these two models may

produce unrealistic result.s. Yet that is the practice adapted by most

regulatory agencies and decision makers. Furthermore, in the multi-stage

model, the parameters of the model are constrained to be positive with

the pretense of providing physical meaning. Biologically as well as

mathematically that practice does not make any sense.

The Weibull mode 1 nlay have more general application because of the

shape parameter m except in extreme situations such as the two hypotheti­

cal examples discussed earlier. Even in those cases, the point estimates

241

TABU 8. Goodness-of-fit Probabilities for Different Hodels

Example One-hit Hulti-stage Probit Logit Hulti-hit Weibull

Dieldrin 0.0256* 0.3647 0.0670 0.1650 0.5463 0.5910

"Thresho1d" 0.0278* 0.7770 1.0000 1.0000 0.0000* 1.0000

"Saturation" 0.1021 0.2520 0.1883 0.1770 0.1669 0.1706

* Significant at 5~ level

given by the Weibull model are more realistic than many others, particu­

larly the one-hit and the multi-stage models. The logistic model comes

a close second to the Weibull in those cases. Experience with these two

models indicates that one should have more than one level at which the

response must be higher than the control; otherwise, the uncertainty in

the shape parameter m becomes too large which then is reflected particu-

larly in the confidence limits of VSD and EROB (10). Time-to-tumor

and/or t1me-to-death information ls easily superimposed in this model.

The general product model, which is a generalization of the Weibull

model, requires regular interim sacrifices, and as a result large sample

sizes are necessary to ensure enough animals at risk. Very few studies

are designed to include that kind of large sample sizes.

The gamma (multi-hit) model, when it fits data weIl, may be applic­

able in some instances. Under those cases, it provides closely similar

estimates as the Weibull or the logit model. In many other cases, the

model provides poorer estimates than the other models.

The probit model will often produce extremely liberal estimates.

Even for some proven potent carcinogens, the risk estimates given by

this model have been unrealistically too small.

Cornfield's pharmacokinetic model attempts to incorporate some

biological facts behind carcinogenesis. Unfortunately, most present-day

studies are not designed to acquire all the pieces of information

necessary to use this model. As better understanding of carcinogenesis

takes place, more cf forts should be made to extend and generalize this

model and more experiments should be designed with the purpose of risk

estimation using this model.

Two recent publications provide many of the important pieces of

information to be considered for meaningful human risk assessment

(27,28). Briefly they are: number of species, strains and sexes

affected, latency, target tissue, dose levels and duration of exposure,

proprtion of malignant and benign tumors, multiplicity of lesions,

242

chemical structure and functional analogy of a particular chemical to

known carcinogens, metabolism and pharmacokinetics of a chemical,

binding to DNA, RNA and pro teins , physiological, biochemical and

pharmacological properties of the compound under study, genotoxicity,

cytotoxicity, mutagenici ty, teratogenici ty and other such properties,

age, physiological, social and environmental states of the exposed

subjects, route of administration of the chemical, and the multiplicity

of exposure. Finally, if any epidemiological data are availahle, they

must be considered in the human risk evaluation.

In conclusion, bias toward a particular model and dogmatic approaches

should be avoided in risk assessment because all models utilized today

are basically empirical at their best. One should remember two impor­

tant facts during this type of exercise: HUMAN LIFE MAY BE AT RISK; ON

THE OTHER HAND, A LOT OF HONEY AND HUMAN ENDEAVOR WENT INTO DEVELOPING

AND TESTING A NEW CHEHICAL WHICH MAY TURN OUT TO BE BENEFICIAL FOR

HUMANS.

REFERENCES

1. Thakur, A.K., K.J. Berry and P.W. Hielke, Jr. (1985) A FORTRAN program for testing trend and homogeneity in proportions. Comp. Progr. Biomed., 19:229-233.

2. Dinse, G.E. and S.W. Lagakos (1983) Regression analysis of tumor prevalence data. J. Roy. Stat. Soc., Series C, 32:236-248.

3. Thomas, D.G., N. Breslow and J.J. Gart (1977) Trend and homogeneity analyses of proportions and life table data. Comp. Progr. Biomed., 10:373-381.

4. Armitage, P. and R. Doll (1954) The age distribution of cancer and a multistage theory of/carcinogenesis. Brit. J. Cancer, 8:1-12.

5. Crump, K.S., D.G. Hoel, C.H. Langley and R. Peto (1976) Fundamental carcinogenic processes and their implications for low dose risk assessment, Cancer Res. 36:2973-2979.

6. Rai, K. and J. Van Ryzin (1981) A generalized multi-hit dose­response model for low-dose extrapolation. Biometrics, 37:341-352.

7. Hantel, N. and W.R. Bryan (1961) "Safety" testing of carcinogenic agents. J. Nat. Cancer Inst., 27:455-470.

8. Doll, R. (1971) Age distribution of cancer. J. Roy. Stat. Soc., Series A 134:133-166.

9. Cornfield, J., F.W. Carlborg and J. Van Ryzin (1978) setting tolerance on the basis of mathematical treatment of dose­response data extrapolated to low doses, In G.L. Plaa and W.A.H. Duncan Ed., Proceedings of First International Congress on Toxicology: Toxicology as a Predictive Science, Academic Press, New York, pp. 143-164.

10. Carlborg, F.W. (1981) Dose-response functions in carcinogenesis and the Weibull model. Fd. Cosmet. Toxicol., 19:255-263.

11. Hartley, H.O. and R.L. Sielken (1977) Estimation of 'safe doses' in carcinogenic experiments. Biometrics, 33:1-20.

243

12. Kalbfleisch, J.D., D. Krewski and J. Van Ryzin (1983) Dose-response models for time to response toxicity data (with discussion). Canad. J. Stat., 11:25-49.

13. Druckery, H. (1967) QUantitative aspects of chemical carcinogenesis, In Potential Carcinogenic Hazards from Drugs (Evaluation of Risks), R. Truhaut ed., UICC Konograph Series, Vol 7, Springer­Verlag, New York, pp. 60-78.

14. Prentice, R.L., A.V. Peterson and P. Karek (l982) Dose mortality re1ationships in RFK mice following 137Cs gamma ray irradiation. Radiation Res., 90:57-76.

15. Cornfield, J. (1977) Careinogenic risk assessment. Science, 198:693-699.

16. Krewski, D., C. Brown and D. Murdoch (1984) Determining "safe" levels of exposure: Safety factors or mathematica1 models? Fund. App1. Toxico1., 4:S383-S394.

17. Interagency Regulatory Liaison Group (1979) Scientific bases for identification of potential carcinogens and estimation of risks. Fed. Register, 44:39858-39879.

18. Food safety Council (1980) QUantitative risk assessment. Fd. Cosmet. Toxicol., 18:711-734.

19. Gehring, P.J. and G.E. Blau (1977) Hechanisms of carcinogenesis: Dose-Response. J. Env. Path. Toxicol. 1:163-179.

20. Dixon, R.L. (l976) Problems in extrapolating toxicity data from laboratory animals to man. Env. Health Perspect., 12:43-50.

21. Gillette, J.R. (1976) Application of pharmacokinetic principles in the extrapolation of animal data to humans. Clin. Toxicol., 9:709-722.

22. Krasovskii, G.N. (1976) Extrapolation of experimental data from animals to man. Env. Health Perspect., 13:51-58.

23. Kotulsky, A.G. (l982) Interspecies and human genetic variation: Problems of risk assessment in chemical mutagenesis and carcinogenesis. Progr. Mut. Res., 3:75-83.

24. Book, S.A. (1982) Scaling toxicity from laboratory animals to people: An example with nitrogen dioxide. J. Toxicol. Env. Health, 9:719-725.

25. Boxenbaum, H. (1984) Interspecies pharmacokinetic scaling and the evolutionary-(:n'l1\parative paradigm. Drug Ketab. Rev. , 15: 1071-1121.

26. Office of the Technology Assessment (OTA) (1981) Assessment of Technologies for Determining Cancer \Risks from the Environment. united States Government Printing Office, Washington, D.C.

27. Park, C.N. and R.D. Snee (1983) QUantitative risk assessment: State-of-the-art for carcinogenesis. Fund. Appl. Toxicol., 3:320-333.

28. Crump, K.S., A. Silvers, P.F. Ricci and R. Wyzga (1985) Interspecies comparison for carclnogenic potency to humans, In P.F. Ricci Ed. Principles of Health Risk Assessment, Prentice Hall, Eng1ewood­Cliff, New Jersey, pp. 321-372.

Acknowledgment: The author thanks Terry Horner for her role in prepara­

tion of this manuscript.

244

THE PUZZLE OF RATES OF CELLULAR UPTAKE OF PROTEIN-BOUND LIGANDS

Ludvik Bass and Susan M. Pond

Departments of Mathematics and Medicine

University of Queensland, Brisbane, Australia

1. INTRODUCTION: UPTAKE RATES AND THEIR HYPOTHETICAL FACILITATION

The rates of hepatic and cerebral uptake of physiologically important

ligands, such as fatty acids, drugs and dyes, have long been known to be

surprisingly high when most ligand is bound to plasma proteins (Baker &

Bradley, 1966, Pardridge & Landaw, 1984). The magnitudes of these rates,

as weIl as the unexpected forms of their dependence on protein

concentration, have motivated the hypothesis of specific albumin receptors

on the hepatocyte (Weisiger, Gollan & Ockner, 1981) helping to translocate

the ligand from albumin into the cell; and the hypothesis of a catalytic

mechanism of dissociation of ligand from protein at the cellular surface

(for example, Baker & Bradley, 1966; Forker & Luxon, 1981). We shall refer

to both proposals as facilitation hyPotheses. We do not include in this

term facilitation of ce11ular uptake of unbound ligands (see for example

Stremmel, Strohmeyer & Berk, 1986).

The facilitation hypotheses have recently been brought into sharper

focus by ensuring experimenta11y that a11 hepatocytes of a liver are

presented with the same concentrations of the reactants, whereby the

currently unsettled aspects of the mode11ing of hepatic uptake by intact

livers (for example, Bass 1986) are circumvented. This was done in intact

livers by working at low extraction fractions (prazosin and antipyrine in

rat liver: Oie & Fiori, 1985); with the use of hepatocyte monolayers in

vitro (paimitate: Fleischer et al., 1986); in hepatocyte cultures (iopanic

acid: Barnhart et a1. , 1983); and in hepatocyte suspensions (BSP and

oleate: Nunes, Kiang & Berk, 1985; Mizuma et a1. , 1986). The overall

congruence of experimental results in a11 these varied situations shows

245

that the presence or absence of intact lobular architecture and of spaces

of Disse make no fundamental difference to effects of protein on uptake of

ligands.

In interpreting the resulting data in essentially compartmental terms,

a central concept has been the unbound clearance of the ligand. This is

defined as the ratio of the uptake rate V to the concentration c(oo) of

unbound ligand in equilibrium (unperturbed by uptake) with protein: c(~) is

the bulk concentration of unbound ligand, determined by separate Iaboratory

measurements. As there is no (steady) net hepatic uptake of protein, the

unbound c learance V / c( 00) may be expected to be insens i ti ve to the total

protein concentration, because i t is the uptake rate re--calculated per

unbound ligand moleeule in the bulk solution. More quantitatively,

denoting by Stot (00) the bulk concentration of binding si tes on protein

(with and without ligand), consider the relations

a (V/c(oo) ) o

Facilitation hypotheses may appear to be supported when a

derivative is found experimentally (Weisiger et al., 1981, 1984;

(1)

positive

Barnhart

et a1. , 1983; Oie & Fiori, 1985; Fleischer et a1. , 1986; Nunes et a1. ,

1985; Mizuma et al., 1986). In contrast, the vanishing derivative has been

the traditional assumption in pharmacokinetic modelling (for example,

Gillette, 1971; Pang & Rowland, 1977). In particular, that traditional

assumption underlies the group of studies aiming to discriminate between

whole-organ models of hepatic uptake by observing changes in hepatic

uptake, at intermediate and high extraction fractions, with changes in

albumin concentrations (for example, Rowland et a1. , 1984; Jones et a1. ,

1984; Byrne et al., 1985). In view of the many observations of

inequalities in relations (1), and of what follows, these studies now

appear to be inconclusive. A negative derivative in relation (1) has also

been found experimentally (Oie & Fiori, 1985) using alpha-I-acid

glycoprotein in place of albumin. Having noted this interesting exception,

we shall henceforth restriet our considerations to effects of plasma

albumins.

When much of the ligand is bound to albumin, the concept of uptake

rate per unbound ligand moleeule needs closer consideration. In response

to any local depletion of unbound ligand, possibly to zero concentration at

the hepatocyte surface (absorption boundary condition:Baker & Bradley,

1966), albumin-ligand complexes dissociate, providing thereby a source of

unbound ligand, the strength of which varies continuously with the distance

246

from the hepatocyte surface. This source is matched in the steady state by

fluxes of albumin-ligand complexes towards, and ligand-free albumin away

from, the hepatocyte surface. A non-equilibrium atmosphere of a

characteristic thickness develops at the surface of each hepatocyte; the

appearance of this new, al~umin--dependent length is central to what

fo11ows. Compartmental analysis cannot do justice to these spatially

distributed phenomena. If the aforement ioned facili tat ion hypotheses are

to be tested conclusively it is essential to use, as a reliable

quantitative standard, an exact solution of the unfacilitated uptake

problem pertaining to experimentally realizable situations.

In the present work we review and use such exact solutions for steady

uptake (Bass & Pond, 1986), and re-interpret several key experimental

results in the light of this rigorous standard. The solutions presuppose:

that only unbound ligand is taken up; that the kinetic constants of the

ligand-albumin interaction are spatially uniform up to the hepatocyte

surface (no surface catalysis); and that no albumin receptor on the

hepatocyte is invol ved in 1 igand uptake. Furthermore, we restrict our

considerations to cases when only a small fraction of albumin sites is

occupied by ligand. We consider exact solutions for two geometries: uptake

by hepatocyte plates (in vi tro monolayers, or forming hepat ic s inusoids)

separated from the bulk solution by an unstirred layer; and uptake by a

dilute suspension of spherical hepatocytes. Both solutions predict a

positive derivative in relations (1) for a11 non-zero rates of uptake.

Thus, for the monolayer experiment of Fleischer et al. (1986), the exact

solution predicts that the unbound clearance of palmitate in the presence

of 25~ albumin is between 11 and 17 times greater than in the absence of

albumin (Section 3). Fleischer et al. (1986) found a factor of about 14

experimentally. Their inference of facil itat ion illustrates the

aforementioned need for exact solutions as standards of comparison, as weIl

as the failure of compartmental concepts in the present context.

At high albumin concentrations in cases of physiological interest, as

in hepatic sinusoids, the exact solutions for the plane and spherical

geometries make identical predictions as to the effects of albumin

concentration on ligand uptake rate per unit hepatocyte area available for

uptake. The solutions predict saturation of the unbound clearance wi th

increasing albumin concentration; this we call pseudo-saturation because no

physical structures (such as an albumin receptor) are being saturated. The

resulting kinetics reproduce quantitatively the observed phenomena hitherto

motivating facilitation hypotheses, and include as a special case the

dissociation-limited uptake rate discussed by Weisiger & Ma (1985; in

247

press) . In particular, when ligand and albumin concentrations are varied

in a fixed ratio (Weisiger et a1. , 1981, 1984; Nunes et a1. , 1985), the

pseudo-saturation kinetics has the form of negatively cooperative kinetics

with the Hill constant n=t (Dixon & Webb, 1979). This yie1ds satisfactory

agreement with measurements of oleate uptake by rat livers (Weisiger et

al., 1981) and of BSP by skate liver (Weisiger et al., 1984) without the

use of any adjustable parameters in the fitting of the data (Section 5); no

facilitation hypotheses are needed.

Pseudo-saturation is seen at its simplest (in Section 4) at high rates

of uptake of unbound ligand at high albumin concentrations Stot(~) in the

bulk. Then the unbound clearance V/c(oo) , per unit accessible area of

hepatocytes, equals the diffusional permeability D/(l/A) of the

non-equilibrium atmosphere of hepatocytes (which has thickness l/A) to

unbound ligand with diffusion coefficient D1 . The thickness l/A varies as _.1

(Stot(OO» 2, so that the unbound clearance increases with Stot(~) as

In contrast, Baker & Bradley (1966) estimated the maximum

unbound clearance (per unit accessible area) as DI/eS, with c5 being the

combined thickness of endothelial cells and perisinusoidal space. Because

this eS is fixed, the maximum unbound clearance is independent of albumin

concentration and turns out to be too low at high values of Stot(~). Thjs

germinal conundrum of Baker & Bradley (1966) is now resolved quantitatively

by showing that eS is to be replaced with the albumin-dependent length l/A.

2. TWO EXACT SOLUTIONS FOR STEADY UPTAKE

We consider first steady uptake of ] igand by a plane monolayer of

hepatocytes, separated from a uniform (stirred) bulk solution by an

unstirred 1ayer of thickness eS. We put the x-axis at right angles to the

monolayer, with x=O at the exposed surface of the hepatocytes. Diffusion

takes place in the interval 0 < x < eS. We denote the steady concentration

of unbound ligand at x by c(x), that of bound ligand by ~(x), and that of

unoccupied sites on albumin by s(x). Then at any x the concentrations

Ctot(x) of all ligand, and Stot(x) of all albumin sites, are

(2)

We introduce the local deviation p(x) from the albumin-ligand equilibrium

which holds in the bulk solution:

p(x) p(oo) o (3)

248

where kl and k2 are the rate constants for the formation and decomposition

of the ligand-albumin complex (dissociation constant k2/k l ) • The

composition of the stirred bulk solution in the interval ö < x < .... is

specified by the experimentally chosen values C tot ( .... ), Stot ( .... ), which

determine c( .... ), ~( .... ) and s( .... ) by equations (2) and (3) at x = .....

We shall deal only with the case ~ (x) « s(x), so that sex) ""

Stot{x). This occurs either because of excess of albumin over ligand, or

because of a high dissociation constant (k2/k l »c(x». In either case,

albumin sites are far from saturation, and consequently Stot(x) "" const. =

Stot{ .... ) because there is no steady uptake of albumin. Then equations (3)

become

p{x) p{oo) o (4)

and in the bulk we obtain the familiar equilibrium relations from equations

(2) and (4) at x = 00'

~(oo)

C tot (00)

Denoting the diffusion coefficient of unbound ligand by Dl , and of the

ligand-albumin complex by D2 , the steady transport equations in the

interval 0 < x < ö are

p{x) (6)

with p(x), given by equations (4), being the sink of c and the source of

cb . When cb(x) is found, sex) follows from the second of equations (2). The boundary conditions at x = ö are evidently

p{(5) p(oo) 0, c(ö) = c(oo) , ~(ö) ~(oo) (7)

with c(oo) and ~( .... ) given by equations (5). There is no flux of albumin

through the hepatocyte surface:

at x",O. (8)

The highest possible rate of uptake of unbound ligand is described by the

absorption boundary condition c(O)=O (Baker & Bradley, 1966). We

generalize that condition by including a barrier of permeability P at the

249

hepatoeyte surfaee, while retaining the assumption that all ligand that has

entered a hepatoeyte is sequestered (either permanently, or on the

time-seale of the experiment). Therefore:

de -Dl ax = - Pe at x = 0 (9)

from whieh e(O)=O follows in the limit P ~~. We shall obtain the exaet

solution of equations (4)-(9) in closed form, and hence caleulate the

steady uptake rate V of ligand, whieh is equal to the flux of unbound

ligand into a plate of uptake area A:

V APc(O) (10)

Equations (6) are uncoupled in terms of the new dependent variables p(x)

given by equation (4), and u(x) given by

Linear combinations of equations (6) yield readily:

d2u = 0 dx2

d2p ,,2p 0 dxZ

with

,,2 = klstot (~)

+ k2

Dl U;

(11)

(12)

(13)

(14)

defining a characteristic distance 1/". The role of that distanee in

equation (13) is formally analogous to the Debye length defining the

thickness of the ionic atmosphere in electrolytes. The boundary eonditions

(7)-(9) are readily translated for the new variables:

(du/dx)x=O = Pc(O) (15)

p(l5) o (16)

The solution of equation (12) is a linear funetion of x, the solution

of equation (13) is a linear combination of exp(±"x). Determining the four

integration eonstants from equations (15) and (16), we find in 0 ~ x ~ 15:

u(x) (17)

250

A(26-x) Ax e -e

1 + e2AÖ (18)

from which the forms of c(x) and ~(x) are readily determined from

equations (4) and (11). The next step is the determination of c(O) (and of

~ (0» from these forms by setting x=O and solving the resulting pair of

linear equations for c(O), ~(O), simplified by the use of equation (5).

Inserting the resulting c(O) in equation (10), we find readily

tanh Aö AP 1 + (pa) 1 +0 AlS

v/doo) Dl 1 + 0 (19)

where

D2klStot(00) D2~(00) 0 Dl k2

= Dl c( 00) (20)

and A is given by equation (14) or, using equation (20), by

(14a)

Thus the unbound clearance V/c(oo) is determined (predicted) in terms of

observable parameters of the system. The solutions c(x), ~(x), giving the

exact concentration profiles in the interval 0 < x < ö, can be written

concisely in terms of the single function

E. [eA(25-X)_eAX]

1 ~+

Aö(l+e2AÖ) '(x,E.)

- ö (21)

1 +0 tanh A5 + !:!.. (l + 0) Aa pa

if we set E. = 0, then , = l-c (x)/c (00); if E.=-l, then , = l-~(x)/~(oo).

Evidently both c(x) and ~(x) are less than in the bulk.

Before discussing and applying these results, we outline the analogous

exact solution of the problem of steady uptake by a suspension of spherical

hepatocytes, so dilute that hepatocytes take up ligand independently of

each other; the requisite distance between hepatocytes will be calculated

below. We take the origin of coordinates in the cent re of a representative

hepatocyte of radius R. By symmetry, concentrations in the bathing

solution depend only on the radial coordinate r (R 5 r < 00), which is now

251

the independent variable in place of x. In equations (6), (12) and (13)

the operator d2/dx2 is replaced with (1/r2)(d/dr)(r2d/dr). The solution

u(r) is a linear function of I/r, while per) is a linear combination of

exp(:tAr)/r, with A given by equation (14) or (14a). In the boundary

conditions in equations (7), (8), (9), (15) and (16) we replace d/dx with

d/dr, and 8 with~. Determining the integration constants in u(r) and per)

from these boundary conditions, we obtain the counterparts of equations

(17) and (18) in the interval R Sr< ~ :

u(r) (22)

(23)

Using the definitions of u(r) and per) - equations (4) and (ll) with r

replacing x - we find the forms of c(r) and ~ (r) from equations (22),

(23). Setting r=R, we obtain a pair of linear equations for c(R) and

C b (R), simplified by equations (5). We thus calculate c(R) and hence V

2 from the radial counterpart of equation (10), V=APc(R), where A=4nR per

hepatocyte. We thus arrive at the counterpart of equation (19):

= I + AP [~ ]

a I + I + AR

I + a (24)

where a and Aare given by equations (20) and (14) or (14a). The profiles

c(r) and ~(r) in the interval R Sr< ~ are given by the function

I + E. e -A(r-R)

I + AR D

I + a + I (1+0) T+iJl' Plr

(25)

Analogously to equation (21), '" = I-c(r)/c(~) putting E. = a, and '" =

I-~(r)/~(~) putting E.=-I.

Equations (19) anrl (24) have an important common feature. Since A and

a both increase monotonically with Stot(~) according to equations (14) and

(20) , the functions (I +AR) -I and tanh A8/ A8 both fall monotonically as

Stot(~) increases. The right-hand sides of equations (19) and (24)

therefore both fall monotonically as Stot(~) increases. Hence the unbound

252

clearance V Ic( 00) increases monotonically wi th Stot (00) , both for planar

monolayers and for suspensions of hepatocytes: the sign of the derivative

in relations (1) is positive without the use of any facilitation

hypotheses. (The foregoing modelling therefore cannot explain the negative

derivative in relation (1), observed by Oie & Fiori (1985». By contrast,

the clearance V/ctot(oo) of total ligand, calculated from equations (19) and

(24) using the first of equations (5), can readily be shown to fall

monotonically with increasing Stot(oo) as expected, provided that Dl > D2 .

3. UPTAKE BY HEPATOCYTE MONOLAYERS IN VITRO

We exemplify the quant i tati ve power of the foregoing resul ts, and

typical orders of magnitude of the relevant parameters, by re-interpreting

the resul ts of an ingenious experiment of Fleischer et al. (1986) on the

uptake of palmitate (0.2~) by monolayers of hepatocytes covering plates of

2 geometrical area Ag=9.62 cm. The stirred solution contained either no

albumin, or 25~ of albumin; in the latter case the mean unbound fraction

of palmitate was found to be c(00)/ctot (00)=0.002 by measurements on the bulk

solution. Palmitate contents of the monolayers were determined at

intervals during the time interval between 10 sec and 50 sec after the

dipping of the plates in the solution. Treat ing the increase of the

palmitate contents with time as approximately linear (see below), the mean

unbound clearances V/c(oo) per plate (with 0.57 mg protein per plate on

average) were found to be 14.25 J.ll/sec in the absence of albumin, and

198.36 J.ll/sec in the presence of 25 ~ of albumin. Thus the albumin

enhanced the unbound clearance of palmitate about fourteen-fold. This

large enhancement factor led Fleischer et al. (1986) to infer the need for

a facilitation hypothesis.

To estimate the maximum enhancement factor that can occur in the

absence of facilitation, we take the absorption boundary condition c(O)=O,

equivalent to the limit P ~ 00 in equation (19):

1 + a 1 + 0

tanh 110 110

(26)

As the clearance of palmitate in the absence of albumin (0 = 0) is ADI/O,

equation (26) gives the maximal facilitation-free prediction of the

enhancement factor measured by Fleischer et al. (1986) . We use their

-6 2 -7 2 measured values Dl =6.5xlO cm Isec, D2=6xlO cm /sec, and the value k2=0.12

253

sec -1 at 37°C of Svenson, Holmer & Andersson (1974). With

'1, (oo)/c(oo) =1/0. 002=500, we obtain 0 = 46.15 from equation (20), and the

characteristic length l/A = 3.25xlO-4cm from equation (14a). We estimate

the thickness S of the unstirred layer from the clearance of palmitate in

the absence of albumin:

ö (27) (V/c(oo»o =0

If we take the effective uptake area to be the geometrical area of the

2 -3 plate (A=Ag=9.62 cm ), we find ö = 4.4xlO cm. If the plate was covered by

closely packed hepatocytes in the forms of hemispheres wi th bases on the

plate, then A < 2Ag and S would be almost doubled. These are typical

thicknesses of unstirred layers at planar membranes (Barry & Diamond,1984).

Hence we estimate Aö to be between 13.5 and 27, both of which give

tanh Aö = 1. From equation (26) we thus predict the enhancement factor

(V/c(oo) )/(ADl/ö) to be between 11 and 17. As the factor 14 found by

Fleischer et a1. (1986) was a mean value over the range 4.1-27.4, our

prediction is clearly consistent with their observations.

How steady was the uptake in the experiment of Fleischer et al.

(1986)? In the absence of a fu11 time-dependent solution of the uptake

problem, we consider orders of magnitude. The measurements were made 10 -

50 sec after the plate was dipped into the equilibrated bulk solution with

The formation of the non-equilibrium atmosphere

involves the diffusion times (1/A)2/2Dl < 2 -2 (l/A) /2D2<10 sec, and the

dissociation time 1/k2=8.3 sec of the ligand-albumin complex. The steady

profile c(x) is established on the time-scale ö2/2Dl , between 1.5 sec and 6

sec (ö between 4.4xlO-3 cm and 8.8xlO-3 cm). While a11 these times are

shorter than 10 sec, the time-scale ö2/2D2 for establishing the steady

profile '1,(x) is between 16 sec and 65 sec, possibly overlapping the whole

duration of the experiment. The effect of this unsteadiness on uptake is,

however, moderate. From equation (21) with ~ =-1 it follows readily

in the steady state, '1,(0)/'1,(00) is between 0.82 (S=4.4xlO-3cm) and

that,

0.67 (ö =8.8xlO-3cm). It is to these maximal depletions that '1, is falling

from '1,(00) as the steady profile is being established. Hence the source of

unbound ligand, provided by the dissociation of the ligand-albumin complex,

is stronger by (at most) 18%-33% during the transient than in the steady

254

state. This is the probable explanation of the moderate but noticeable

reduction with time of the uptake rate shown in Fig. 1 of Fleischer et al.

(1986) .

More vigorous stirring would reduce this complication by making the

unstirred layer thinner, and hence the transient state shorter. For

2 measurements taken later than the times 1/k2 and 5 /2D2 after the dipping

of the plates, the foregoing results provide a quantitative prediction of

the effect upon unbound clearance of stirring, as ref1ected in the value of

5 estimated from equation (27). That effect is given fully by equation

(19). In particular, since 5 + (O/A) tanh A5 increases monotonica11y with

5, the unbound clearance V/c(~) falls monotonically as 5 increases. Hence

the steady value of V/c(~) increases as stirring is made more vigorous.

But so does the clearance AD 1/5 in the absence of albumin, with the result

that the ratio of the two clearances, given by equation (26) under the

absorption boundary condition, falls monotonically towards unity as 5 is

reduced towards zero at fixed 0 (and hence at fixed A: see equations (20)

and (14a». In this sense, the enhancement of the unbound clearance by

albumin is an effect of the unstirred 1ayer. In practice that effect

cannot be removed by vigorous stirring. Suppose for example that the

observed enhancement factor of 14 was to be reduced to 2 by increased

stirring. Equation (26) shows that this would require approximately A5=2

or 5=7xlO-4cm, less than one third of the least thickness attained at

planar membranes by extreme stirring (Barry & Diamond, 1984).

If stirring does not affect the unbound clearance as predicted above,

the present theory of facilitation would be untenable. If the predicted

effect of stirring is present, then theories of facilitation based entirely

on thermodynamic equi1ibria between severa1 adjoining phases (Noy, Donnelly

& Zakim, 1986) become untenab1e.

4. PSEUDO-SATt~TION AND THE FAILURE OF COMPARTMENTS

We now elucidate salient features of the general results of Section

2., especially of equations (19), (24) and (26).

(a) A review of the foregoing calculations shows readily that the

albumin-free case, s tot (~) =0 (s=O, ~ =0, p~O) is a legi t imate 1 imi t of our

results. In particular, equation (19) becomes

255

AP (28) v /c( 00)

which yields equation (27) as P ... 00 (c(O)=O). The corresponding results

for suspended hepatocytes follow from equation (24): we need only to

2 replace 5 by R in equation (28) and set A=4HR per hepatocyte. For P ... 00

(c(R)=O) we then obtain V=4DRDl c(00) , which is Smoluchowski's

diffusion-limited rate, familar from reaction kinetics.

(b) An important limi ting assumption , often made in traditional

pharmacokinetic modelling (for example for diazepam: Rowland et al., 1984)

is to consider a very fast ligand-albumin interaction: formally, kl ... 00 and

k2 ... 00 such that the dissociation constant k2/k l remains finite. We see

from equations (20) and (14a) that, in this limit, 0 remains finite while

the thickness l/A of the non-equilibrium atmosphere tends to zero (A ... 00).

Since tanh A5/A5 ...0 as A ... 00, equation (19) becomes

v /c( 00) 1 + 0 AP------

P5/Dl + 1 + 0

(29)

The unbound clearance shows saturation of the Michaelis-Menten form with

increasing 1+0 (increasing Stot(oo): see equation (20». The

corresponding limit of equation (24) for suspended hepatocytes is as

equation (29) wi th 5 replaced by R. If, in addition, P ... 00, then the

unbound clearance varies proportionally to 1 + 0, and equations (29) and

(20) yield the maximal diffusion-limited uptake rate

v = (29a)

(c) The limit of very high albumin concentrations Stot(oo) is

particularly interesting, as it is approachable for ligands with any values

of kl and k2• While 0 varies as Stot(oo), A varies as (Stot(oo»+ (equations

(20), (14»; hence, in this limit,

o » A5 (0 » AR) (30)

and

A5 » 1 (AR» 1 ) (31)

wi th the relations in brackets pertaining to suspended hepatocytes. The

meaning of relation (30) is, equivalently, D2~(00)/5»Dlc(00)/(l/A): the

256

maximum possible diffusion flux of bound ligand across the unstirred layer

is much greater than that of the unbound ligand across the non-equilibrium

atmosphere. The effect of relation (31) which is important in the present

context, is to make tanh AO close to unity. For this, A6 needs to exceed

unity only moderately: for example, tanh 2=0.964. If both relations (30)

and (31) hold, then 6+a(tanh A6)/A approaches o/A in equation (19), which

becomes

V/c(OO) = AP A (32)

using also equation (14) in the limit of large Stot(oo). Here V and V/c(oo)

increase monotonically towards a maximum value as Stot(oo) increases,

obeying negatively cooperative kinetics with the Hill constant n=t, and

with kinetic parameters readily read off equations (32) (Dixon & Webb,

1979) .

(d) Next we drop relation (31) in order to consider all values of AO,

but we strengthen relation (30) slightly to 0 » AO + 1, which implies

o »A6/tanh A5 (because tanh A5 ~ .\5/0 + A5». Then equation (26) is

reduced to

V/c(oo) (34)

Now, when AO«l, we have tanhA5 = .\6 and we recover the albumin-independent

form of unbound clearance, DIA/O, of Baker & Bradley (1966). When A6»1,

we obtain the larger, albumin-dependent unbound clearance

V/c(eo) (35)

As noted in the Introduction, Dl/(l/A) is the diffusional permeability of

the non-equilibrium atmosphere to unbound ligand, a concept inaccessible to

any compartmental modelling.

A notable feature common to equations (32) and (35) is their

independence of the thickness of the unstirred layer, 5. Moreover, the

~ equations are obtained when the (bracketed) relations (30), (31) are

applied to equation (24): identical rates of ligand uptake are attained by

a hepatocyte monolayer and a dilute suspension of spherical hepatocytes

having the same total areas exposed to the perfusate.

257

In this context the concept "dilute" has the following

albumin-dependent meaning. Envisage a sphere of radius r* > R drawn about

the centre of a representative hepatocyte so that the sphere contains no

part of another hepatocyte. We say that the suspension is dilute if

l-c(r*}/c(oo) and l-~(r*)/~(oo) are as close to unity as desired for

effective kinetic independence of neighbouring hepatocytes. The radius r*

can be calculated from equation (25); it is the smaller, the higher the

albumin concentration. It suffices to calculate an upper bound on r*,

valid for all P, by going to the limit P ~ 00 in equation (25). We take

R=lO-3cm (Arias et al., 1982). Let us require, as an example, that

in the palmitate-albumin solution of Fleischer et al. (1986) discussed in

Section 3. We find r* < 1.2 R. Moreover, even at the hepatocyte surface,

implying an even smaller value at r=1.2 R. In this case a suspension is

"dilute" in the desired kinetic sense if surfaces of hepatocytes are

separated by distances of the order of their diameter.

The dependence of the uptake rate V and of the unbound clearance

V/c(oo) of ligand on albumin concentration obeys various forms of saturation

kinetics, such as was given by equations (29) and (32), without any

physical structures being saturated. We therefore speak of

pseudo-saturation, a mathematical consequence of the exact solution of the

uptake problem. It follows that when data conform to equations such as

(29),(32) or (34) (see Section 5), the presence of saturable structures,

such as albumin receptors on hepatocytes, cannot legitimately be inferred.

Such inferences belong to the domain of processes describable in

compartmental terms, and their failure signals the failure of compartmental

modelling of the uptake process in the presence of protein. The present

work provides thus another contribution to the cri tique of compartmental

models in biodynamics (Bass, 1985).

5. UPTAKE BV INTACT LIVERS.

We now apply the foregoing analysis to two data sets, pertaining to

intact perfused livers, which have been especially influential in the

formulation and subsequent discussions of facilitation hypotheses. We

258

shall show quantitatively that these data are explicable in terms of

pseudo-saturation without facilitation.

In the two studies of steady uptake, of oleate by rat liver (Weisiger

et al., 1981) and of BSP by skate liver (Weisiger et al., 1984), ligand and

albumin at the liver inlet were varied in fixed molar ratios. The molar

ratio albumin: oleate was unity, but Stot(~)/ctot(~) was larger because of

Fig. 1.

0·5

'- 0 4 QJ >

c E

I

I I I I I I I I I

-- 15

- c

I Q V1

~ 0 3 , --{---;---------- ------------------------------------ 10 ~ c I ~ QJ

~ I _ Q I ~ 02: I 8=i I CD ,_

I 0·1 I

05

OL-----L-----"-----'----'-----'O o 2 3 4 5 BSP(~M)

o 50 100 150 200 250

Albumin (11M)

w CJ

+-a E +­V1

W

Rate of uptake of BSP by skate liver, with BSP and albumin varied in a fixed molar ratio. Data points from Weisiger et al.,(1984). Solid line is predicted by theory. Broken line gives dissociation-limited uptake rate. Dotted line gives concentration of unbound BSP (right-hand scale).

259

multiple binding sites for oleate on albumin (Weisiger et al.,1981).

Oleate was varied in the range 0.02 0.5 mM. The molar ratio albumin: BSP

was 50, and BSP was varied in the range 0.5 - 5~. In both studies the

albumin concentration was kept so high that Stot(oo) »k2/kl so that,

consistent with the second of equations (5),

c(oo) k2ctot (OO)

klStot(oo) const. c(oo) (36)

with the constants c(oo) = 0.54 ~ for oleate and 1 DM for BSP: see dotted

lines in Figs. 1 and 2. We shall take Dl for oleate to be the same as for

-6 2 palmitate (6.5xlO cm /sec); for BSP we take the value of Baker & Bradley

-6 Z (1966): Dl =3xlO cm /sec. For albumin and i ts complexes wi th oleate and

BSP -7 2 we take the DZ value for albumin-palmitate complex (6xlO cm /sec).

For oleate k2 we take the value 0.142 sec-l (Weisiger & Ma, 1985). For BSP

the value of k2 has not been measured directly, so we adopt provisionally

the value 0.16 sec-l conjectured by Weisiger et al. (1984) on the basis of

multiple-site binding studies of Baker & Bradley (1966).

In plug flow through a hepatic sinusoid, the sinusoidal lumen is

unstirred, in the sense that transport of reactants transverse to flow is

by diffusion. -3 We take therefore ~10 cm, but we shall find the following

predictons of uptake insensitive to the precise choice of the value of o. We estimate the area A of hepatocyte plates per gram liver by noting that

0.8 of the parenchymal volume is taken up by hepatocytes of average volume

11000~3 (Arias et al., 1982). Taking the cube root of that volume, 22.Z~,

as the thickness of the plate, we find A=360 cm2/g liver for rat liver, and

half that value for skate liver which has plates two cells thick (Boyer,

Schwarz & Smith, 1976).

As c(oo)«ctot(oo) for all data, we shall calculate 0 from equation (20)

as D2Ctot(oo)/Dlc(OO), and hence Jo. from equation (14a). For the uptake of

BSP by skate liver, the relations o»Jo.6 and tanh Jo.&<l hold for a11 data

points reproduced in Fig. 1., because they hold at the lowest observed

concentration 0.5~ BSP:

260

~ Q)

~

CT1

c 'E --<5 E :i

QJ J:: ro

Ci.. ::J

QJ -+-ro QJ

6

03

0·2

01

/

/ /

/ /

/

/ /

/ /

/

// • 08

// 06 .... -- -- -------- ----7------- -- --- -- ----- ----------- - --------- ----------- -----

:' /

// 0·4 /

/ ~

}. 0·2

• O~------~--------~------~--------~-------- 0 o 0·1 02 OJ 0·4 OS

Fig. 2.

Oleate -Albumin (mM)

Rate of uptake of oleate by rat liver. with oleate and albumin varied in a fixed molar ratio. Data points from Weisiger et al.(1981). Solid line is predicted by theory. Broken line gives dissociation-limited uptake rate. Dotted line gives concentration of unbound oleate (right-hand scale).

L :i

QJ -+­ro QJ

6

....... V1

W

261

a = IOD, A6 = 5.2, tanh A6 = 1.00. Equation (35) pertains, therefore, to

all the skate data of Weisiger et al. (1984). In contrast, for uptake of

oleate by rat liver (Fig. 2.) we find at the lowest concentration 0.02 mM

of oleate: a 3.42, A6 = 0.51, tanh A6 = 0.45. To analyse these data of

Weisiger et al. (1981) we shall, therefore, use the more general equation

(26) (with c(~) = c(~» in place of equation (35).

Equations (26) and (35) assume that all hepatocytes are presented with

the same concentrations of the reactants. The clearance V/Ctot(~) so

calculated is commonly called the intrinsic hepatic clearance K,

K V/Ctot (00) (37)

The unbound clearance discussed above,

(38)

using equation (5), is intrinsic in the same sense. These two intrinsic

clearances are independent of ligand concentration (see equations (26) and

(35». Within an intact liver perfused at the rate F of hepatic blood

flow, ligand concentrations are in general less than the inlet

concentration c!~t(~); the total organ uptake VI of the intact liver is

less than in V=Kctot(~) as a result of ligand concentration gradients

generated by the uptake. Although extraction fractions associated with the

data in Figs. I and 2 were moderate (none exceeding 0.4), we shall take the

resul ting minor effects into consideration. We note first that here

Ctot(~) is a good approximation to the total ligand concentration

throughout any cross-section of a sinusoid, because Ctot(oo) "" ~(oo), and

~(~) is depleted only slightly even at the surfaces of hepatocytes

(Section 4). While Ctot(~) and ~(~) now vary along the hepatic blood

flow, Stot(oo) remains constant throughout hepatic blood because albumin is

not eliminated. The steady uptake rate by the intact liver is

(39)

according to the sinusoidal perfusion model for first-order uptake (for

example, Brauer et a1. , 1956). In the venous equilibration model (for

example, Gillette 1971), exp(-K/F) is replaced with l/(l+K/F), which would

262

make little numerical difference to the results of equation (39) at the low

extraction fractions we are dealing with. The flow rates F were

3.8 ml/min.g liver for rat, and 0.85 ml/min.g liver for skate (Weisiger et

al., 1981,1984). As the total ligand concentrations reported were in fact

inlet concentrations we have now assembled a11 the numerical

material needed for out ri ght predictions, involving no adjustable

parameters, of the observed uptake rates VI shown in Figs. 1 and 2. For

each data value Ci~t(~)' and its associated value Stot(~)' we calculate K

from equations (38) and (26) or (35), and then VI from equation (39).

Our outright predictions of VI (Figs. 1 and 2, solid curves) are in

good agreement with the data, considering the coarse estimates of the

values of A, and of k2 for BSP-albumin complexes. Modified ~ priori

estimates of these and other parameters may modify the outright fit, but it

will remain clear that pseudo-saturation provides a quantitative

alternative to facilitation hypotheses in interpreting the data in Figs. 1

and 2.

If VI in Fig. 2 is calculated using equation (35) in place of equation

(26), it differs little from the solid curve shown: the greatest

differences (in ~ol/min.g liver) are -0.009 at 0.02 mM, and 0.011 at 0.45

mM oleate. This illustrates the aforementioned insensitivity of the

calculations to the choice of the precise value of 6.

6. DISSOCIATION-LIMITED UPTAKE

When facilitation hypotheses are not adopted, the albumin-ligand

dissociation constant k2 (as measured in vitro) must co-determine an upper

bound on the rate of hepatic uptake of ligand. Weisiger, Gollan & Ockner

(1982) conjectured that such an upper bound has been exceeded in hepatic

uptake of fatty acids, and that facilitation therefore occurs. In

contrast, Weisiger & Ma (1985jin press) interpreted some observed rates of

hepatic uptake as a close approach to the dissociation-limited bound in the

absence of facilitation. We now deduce the dissociation-limited uptake

rate rigorously from equations (26) and (39). We show that, in the three

experiments interpreted in the present paper (Fleischer et a1. , 1986j

Weisiger et al.,1981, 1984), that rate is far from exceeded or even

attained. Nevertheless the possibility of exceeding the

dissociation-limited bound on the uptake rate conclusively in future

experiments offers the best prospects of demonstrating facilitationj for

this reason an exact formulation of that bound is important.

263

As equation (26) results from thc absorption boundary condition, it

gives an upper bound on the rate of uptake of ligand. Its right-hand side

is bounded as follows: for all (positive) a and A6,

1 + a

1 + a tanh Aö A6

1 + (40)

1n the Appendix we give a proof of this general relation by a concise

method due to R. Vyborny. Here we note only that equality in relation (40)

is approached for A6 so small that tanhAö ~ Aö - (A6)2/3 , as can be seen

readily by substi tuting this expansion and using the binomial theorem to

2 order (Aö) .

Combining equation (26) with relation (40) and using equations (14a)

and (20), we find readily

(41)

Hence the intrinsic clearance (equation (37» is bounded by

AD c(oo) ~(oo) K = V/ctot(oo) S (_1) + t A 6 k2 6 Ctot (00) Ctot (00)

AD k2 ~ (_1) 1 A 6 k2 K k2 + klstot(oo)

+ T = 6 (up) (42)

using ~(oo)S Ctot (00) and equation (5) in the second step. As VI in

equation (39) increases monotonically with K, replacement of K with K(up)

generates an upper bound Vl(up)~ VI. In Figs. 1 and 2 we plot Vl(up)

(broken lines); it is not approached by the data or by our predictions

(solid lines) except at the lowest concentrations. Had the upper bound

Vl(up) been exceeded by data, facilitation could have been inferred. Had

it been attained, VI would have varied almost linearly with concentrations,

in contrast to the concave curves actually seen.

For uptake by monolayers in vitro, relation (41) determines the

dissociation-limited upper bound on the ratio of unbound clearances of

ligand with and without albumin present, in the form

264

V/c(oo) AD/ö

(43)

~'or the experiment of Fleischer et n1. (1986) with palmitate (Sechon 3),

this upper bound is about 60, well above the observed range which does not

exceed 28.

We re- interpret the foregoing dissociat ion--limited bound in terms

currently used in the literature. The first term in relation (41) has the

form of the upper bound used by Baker & Bradley (1966)j their ö, however,

was thc thickness of a layer which they assumed to bc inaccessible to the

ligand-albumin complex. The second term is the dissociation rate of

ligand--albumin complexes eontained in the volume Aö/3 at the uniform bulk

concentration ch(oo). The blood volume of the liver would be Aö if A was

the total ares of many parallel plates distant ö from eaeh other; and it

would be Aö/4 if A was folded into many cireular eylinders of radius ö/2,

representing sinusoids. Suppose that the intermediate fraction, Aö/3, is

the blood volume of the liver. Then Aö/3=Fr, where r is the me an vaseular

transit time through thc liver. When most of thc ligand is bound (c(oo) « etot(oo)), the second ter-ms in relations (42) predominate:K(up)",r\ök2/8-Frk2 .

Tntrociudng the ht'patie f'xtraction fraction E =- V1/FC!:t(OO) , wr obtain from

equation (39) and rPlation (12):

E S 1 - (44)

The last of these expressions has been used hy Weisiger & Ma (l985j in

press) .

7. CONCLUSIONS AND DTSCUSSION

We conclude that facilitation hypotheses are not needed to explain the

results of the three experiments analyzed above. The straight forward,

quantitative account of these experimental results, in terms of a rigorous

theory free of adjustable parameters, encourages wider applieations of the

theory to uptake of protein--bound ligands by the liver and other organs,

such as the myocardium and the brain where analogous facilitation problems

occur (Pardridge & Landaw, ]984). For that purpose several experimental

and theoretical developments are desirable.

Wi th regard to experiments, we note that rigorous predict ions of

uptake rates require the knowledge of the values of hoth rate constants

265

k l ,k2 of the albumin-ligand interaction. If more of these pairs of values

were determined in vitro (at relevant temperatures) for ligands of

interest, additional data sets, some already published, could be

re-interpreted. In particular, the search for violations of the

dissociation-limited upper bound on unfacilitated uptake rates could be

extended. Next, the effective surface area available for uptake by

monolayers in vitro or by sinusoidal plates is a source of uncertainty in

the foregoing outright predictions of uptake rates. That uncertainty is

absent in calculations of uptake by hepatocyte suspensions, which seem

therefore to be the preferable experimental preparation for fully

quantitative studies of the mechanism of uptake of protein-bound ligands.

On the theoretical side, the exact solutions of the uptake problem need to

be extended to time-dependence, and to non-linear problems arising when

protein sites are partially saturated by ligand.

The foregoing work deals with problems which are essentially

inaccessible to compartmental modelling, no matter how ingenious (Weisiger,

1985) . The need for considering continuous spatial distributions of

concentrations of the reactants is seen most simply in the central role

played in the theory by the non-equilibrium atmosphere of hepatocytes,

analogous in mathematical structure to the potential of the ionic

atmosphere in electrolytes. Consequent ly, compartmental concept s can no

more grasp the present subject-matter than they could produce the Debye

length. Here, we suggest, Lumpers must yield to Distributers (Forker &

Luxon, 1985).

We are grateful to E. L. Forker and R. A. Weisiger for additional

information concerning their experiments; to A.J. Bracken, H.S. Green, P.J.

Robinson and R. Vyborny for valuable discussions; and to the Australian

Research Grants Commi ttee and the Nat ional Health and Medical Research

Council of Australia for financial support.

REFERENCES

Arias, I.M., Popper, H., Schacht er , D. & Shafritz, D.A. (1982). The Liver.

New York: Raven Press.

Baker, K.J. & Bradley, S.E.(1966). J. Clin. Invest. 43, 281.

Barnhart, J.L., Witt, B.1., Hardison, W.G. & Berk, R.N.(1983). Am. J.

Physiol. 244, G 630.

Barry, P.H. & Diamond, J.M.(1984). Physiol. Rev. 64, 763.

266

Bass, L.(1985). Cireu1ation 72 (Supp1. IV), 47.

Bass, L.(1986). J. Pharm. Sei. 75, 321.

Bass, L. & Pond, S.M.(1986). App1ied Mathematies Preprint No. 123.

University of Queens1and(ISBN 0 86776 222 5).

Brauer, R.W.,Leong, G.F., MeE1roy, R.F. & Ho11oway, R.J.(1956). Am. J.

Physio1. 184, 593.

Boyer,J.L., Schwarz,J. & Smith, N.(1976). Am. J. Physio1. 230(4), 974.

Byrne, A.J., Morgan, D.J., Harrison, P.M. & MeLean, A.J.(1985). J. Pharm.

Sei. 74, 205.

Dixon, M. & Webb, E.C.(1979). Enzymes (3rd edition). London:Longman.

Fleischer, A.B., Shurmantine, W.O., Luxon, B.A. & Forker, E.L.(1986). ~

C1in. Invest. 77, 964.

F'orker, E.L. & Luxon, B.A.(1981). J. Clin. Invest. 67, 1517.

Forker, E.L., Luxon, B.A., Snel1, M. & Shurmantine, W.O.(1982).

J. Pharmaeo1. Exp. Ther. 223, 342.

Forker, E.L. & Luxon, B.A.(1985). Hepatology~, 1236.

Gillette, J.R.(1971). Ann. N.Y. Acad. Sei. 179, 43.

Jones, D.B., Morgan, D.J., Mihaly, G.W., Webster, L.K. & Sma11wood,

R.A.(1984). J. Pharmaeol. Exp. Ther. 229, 522.

Mizuma, T., Horie, T., Hayashi, M. & Awazu, S.(1986). J. Pharmaeobio-Dyn.

~, 244.

Noy, N., Donnelly, T.M. & Zakim, D.(1986). Biochemistry 25, 2013.

Nunes, R., Kiang, L. & Berk, P.D.(1985). Hepato1ogy~, 1035 (abstract).

Oie, S. & Fiori, F.(1985). J. Pharmacol. Exp. Ther. 234, 636.

Pang, K.S. & Rowland, M.(1977). J. Pharmaeokin. Biopharm. ~, 625.

Pardridge, W.M., & Landaw, E.M.(1984). J. C1in. Invest. 74, 746.

Rowland, M., Leitch, D., Fleming, G. & Smith, B.(1984). J. Pharmacokin.

Biopharm. 12, 129.

Stremme1, W., Strohmeyer, G. & Berk, P.D.(1986). Proe. Nat1. Aead. Sei.

USA 83, 3584.

Svenson, A., Holmer, E. & Andersson, L-0.(1974). Bioehern. Biophys. Acta.

342, 54.

Weisiger, R.A., Gol1an, J. & Ockner, R.(1981). Science 211, 1048.

Weisiger, R.A., Gollan, J. & Ockner, R.(1982). Progress in Liver Disease

1, 71.

Weisiger, R.A., Zacks, C.M., Smith, N.D. & Boyer, J.L.(1984). Hepato1ogy

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Weisiger, R.A. & Ma, W-L.(in press). J. C1in. Invest.

267

APPENDIX: AN INEQUALITY FOH DISSOCIATION-LIMITED UPTAKE.

To prove the validity of relation (40), we write

All = x

a

Then relation (40) becomes

1 1 - a (1 _ tanh x)

x

O~x<oo

o ~ a < 1

(Al)

(A2)

(A3)

This is evident for a = Oj assume a > O. Re-arrangement of relation (A3)

yields the equivalent relation

W

sinh x 3(cosh x - x

s1nh x (l-a) cosh x + a --~-­x

2 x

We use the series expansions of sinh x, cosh x, valid for all lxi< 00:

sinh x 00

E n=O

2n + 1 x (2n + I)!

As cosh x ~ (sinh x)/x, we have

(l-a) cosh x sinh x + a x

Moreover,

sinh 3(cosh x x ) x ~

00 2n , cosh x E x

n=O T2iiJT

sinh x ~ x

2 00 2n + 2

+ 3 x x E (2n + 2)! n=l

Using relations (A6) and (A7) in (A4), we find

x2 1 00 2n

+ 3 E x

1 (2n + 2)! W ~ 00 n

1 + E x

1 (2n + l)!

268

(A4)

(A5)

(A6)

(A7)

(A8)

Now, for n ~ 1,

3 1 3 1 (AB) (2n + 2)! (2n + 1)! (2n + 2) (2n + 1)!

2 Hence W ~ x , so that the equivalent relation (A3) is valid.

269

A PHARMACOKINETIC EQUATION

GUIDE FOR CLINICIANS

Joyce Mordenti

SchoolofPhannacy University of Califomia San Francisco, CA, USA 94143-0446

INTRODUCTION

Pharmacokinetic monitoring plays an important role in routine patient care. With the advent of rapid and specific drug assays, the clinician can obtain results quickly and, when nesessary, modify the dosage regimen. This chapter presents guidelines for determining drug disposition parameters in adult patients based on their dosing history and their plasma drug concentrations. This information is used in conjunction with the patient's clinical response to therapy--such as a decrease in the number of seizures, cessation of arrhythmias, improvement in respiratory status, increased clotting time, or suspected toxicity--to design optimal dosage regimens. A list of the commonly monitored drugs appears in Table 1.

Four pharmacokinetic parameters are utilized for designing dosage regimens and determining the appropriate time to measure drug concentrations: clearance (CL), volume of distribution (V), elimination rate constant (k), and half-life (tl/2)' Initial estimates for clearance and volume of distribution are obtained from the literature, using references such as 1-3.

The abbreviations that appear in this chapter are defmed in Appendix I.

Clearance

Clearance (CL) is a measure of how well the body can metabolize or eliminate a drug. It is used to calculate maintenance doses or average steady state plasma concentrations.

('tCL) D=Cssave SF

271

Table 1. Drugs Commonly Monitored by the University of California, San Francisco, Clinical Pharmacokinetic Consulting Service

-=

Drug

---- -Therapeutic Plasma Conc.

- -Half·Life (hours)

Recommended Sampling Time

AJ.nmog~co~des:------------------------------------------

Amikacin peak 20-30 mg/L 2 peak: 1 h after start of trough < 10 mg/L infusion; trough: JBND

Gentamiein peak 4-8 mg/L 2 peak: 1 h after start of trough < 2 mg/L infusion; trough: JBND

Tobramycin peak 4-8 mg/L 2 peak: 1 h after start of trough < 2 mg/L infusion; trough: JBND

Carbamazepine (A) 3-12mg/L 15 just before next dose Chloramphenicol 1O-20mg/L 3 just before next dose Digoxin 0.8-2 mcg/L 48 JBND or at least 6 h after

oral and 4 h after IV dose Digitoxin (A) 1O-20mcg/L 244 JBND or at least 6 h after

oral and 4 h after IV dose Ethosuximide 40-1oomg/L 55 (adult) JBND

24 (child) Flucytosine 50-100 mg/L 5 3 h after dose

(> 125 mg/L toxic) Lidocaine (B) 2-6mg/L 2 suspected toxicity or> 8 h

after starting infusion Lithium 0.5-1.5 mEg/L 24 Just before morning dose

Methotrexate > 10-7 M for> 48 h 8 (> 12 h after last dose) By protocol

is potentially toxic; maintain leucovorin rescue until < 10-7 M

Phenobarbital 1O-40mg/L 120 just before next dose Phenytoin (A) 1O-20mg/L NA just before next dose Primidone 5-lOmg/L 8 just before next dose Procainamide 4-8 mg/L 3 just before next dose NAPA 0.25-0.66 as active 8

as procainamide Propranolol (B) 20 to > 100 mcg/L 4 just before next dose Quinidine (B) 1-4 mg/L 6 just before next dose Salicylates (A) 100-300 mg/L 2-20 just before next dose Thoephylline 1O-20mg/L 8 just before next dose

(continued on next page)

272

Tricyclic Antidepressants: Imipramine (B) 200-300 mcg/L 13

includes metabolite Desipramine (B) 150-300 mcg/L 18 Amitriptyline (B) 150-250 mcg/L 15

includes metabolite Nortriptyline (B) 50-150 mcg/L 30

Vancomycin peak< 50 mg/L 6 trough 10 ± 5 mg/L

JBND = just before next dose NA = not applicable

just before next dose

just before next dose just before next dose

just before next dose just before next dose

(A) = Therapeutic range needs to be decreased for hypoalbuminemia and renal failure. (B) = Therapeutic range needs to be adjusted for a variety of disease states.

Total body clearance is a composite term that represents elimination by all routes, such as renal, hepatic, pulmonary, and dermal. When the major pathways of drug elimination are impaired (due to disease state or concommitant medication) or changed in any way (such as enzyme induction in the liver), clearance values will require adjustment. Adjusted clearance values for select patient groups are available in many of the books listed at the end of this chapter. Journal articles are the best source of pharmacokinetic information for new drugs.

The following equation is helpful for modifying clearance values for overweight or underweight patients.

Empirie adjustment for clearance in an adult patient whose weight is significantly different from 70 kg:

CL (your patient) = CL (70 kg patient) [(wt of your patient in kg!70)O.73]

Volume

Volume of distribution (V) is the apparent volume required to ac count for all the drug in the body if it were present throughout the body in the same concentration as in the sampie obtained from the plasma. It is used to calculate a loading dose for the rapid achievement of a target plasma concentration or to calculate the expected change in plasma concentration from a single dose.

. Cssmax V Loadmg Dose = SF

SFD ~ C =Cssmax - Cssmin = V

273

The total volume of distribution can be composed of several pharmacokinetically distinct volumes, such as, VI (a rapidly equilibrating central compartment) and V 2 (a slowly equilibrating tissue compartment). The pharmacokinetic equations presented in this chapter are based on a one compartment model which assurnes all drug distributes equally to all compartments. This approach is satisfactory in the c1inical setting as long as sampies are not obtained during the initial distribution phase. Four drugs--lidocaine, quinidine, procainamide, and digoxin--are worthy of mention because they have significant two-compartment characteristics. When equations for the one compartment model are applied to these drugs, the following precautions are advised (4). Lidocaine, quinidine, and procainamide exert therapeutic and toxic effects on target organs located in the rapidly equilibrating compartment (VI). Calculation of a loading dose based on the total volume of distribution (V), when drug is only going into VI initially, will result in an initial plasma concentration larger than predicted and, possibly, toxicity. This problem can be circumvented by calculating the loading dose based on the total volume of distribution (V), but administering it at a rate slow enough to allow for drug distribution into the slowly equilibrating compartment. In contrast, digoxin exerts therapeutic and toxic effects on target organs located in the slowly equilibrating compartment (V 2).

The loading dose can be calculated based on the total volume of distribution, but one should not obtain sampies for therapeutic drug monitoring until 4 hours after an intravenous dose or 6 hours after an oral dose (e.g., the minimum time necessary to avoid the distribution phase); otherwise, the pharmacologic response will be much lower than expected based on the measured plasma concentrations.

The following two equations are helpful for modifying volume of distribution values for 'your patient when weight reflects excess adipose tissue or excess lliud.

Empiric adjustment for volume of distribution in edematous patients: V = V/kg (DBW) + (l (V/kg )(TBW - DBW)

Empiric adjustment for volume of distribution in obese patients: V = V/kg (IBW) + (l (V/kg) (TBW - IBW)

Elimination rate constant

The first order elimination rate constant (k) is the fractional rate of drug loss from the body or the fraction of the volume of distribution which is cleared of drug during a time interval. The elimination rate constant is incorporated into exponential equations to determine the fraction of drug removed, remaining, or infused during a dosing interval as follows:

274

e-kt = fraction remaining at the end of a time interval, t l-e-kt = fraction lost during decay phase (assuming no additional drug

input) l-e-kt'= fraction of steady state achieved during a constant infusion,

t' hours after starting the infusion l-e-kt = fraction lost during a dosing interval at steady state

Half-life

Half-life (t1/2) is the time required to reduce the plasma concentration to one-half of the original value. This information is important if you need an estimate of when drug will be below some critical concentration or completely removed from the body. It takes one half-life to decrese the concentration by 50%; two half-lives to decrease the concentration by 75%; three half-lives to decrease the concentration by 87.5%; four half-lives to decrease the concentration by approximately 94%, and so on. Conversely, half-life is a good indicator of when to expect the achievement of steady state during a fixed dosage regimen. Using the same mathematics as above, it will take four half-lives to achieve approximately 94% of the expected steady state concentration. In the clinical setting, three to four half-lives can be considered steady state.

Additional helpful equations are listed in Appendix ll.

PHARMACOKINETIC EVALUATION

The pharmacokinetic evaluation consists of six steps: estimating the pharmacokinetic parameters, selecting the pharmacokinetic model, selecting the proper sampling time, predicting the drug concentration, revising the pharmacokinetic parameters, and revising the dosing regimen.

Estimating the Pharmacokinetic Parameters

The pharmacokinetic drug evaluation begins by considering your patient to be an "average" patient with average values for clearance and volume of distribution. The initial estimate for these parameters can be average published values taken from literature sources such as references 1-3. These average values should be adjusted for patient specific traits such as age, weight, sex, race, fluid load, concommitant medications, concommitant illnesses, and personal habits (such as smoking) when these traits are known to impact on drug absorption, distribution, metabolism, or elimination. The other parameters (k, t1/2) are calculated as folIows:

k_ CL -Y

t1/2- 0.693 _ 0.693Y - k - CL

275

Selecting the Phannacokinetic Model

There are three phannacokinetic models: continuous infusion, intennittent infusion, and intennittent bolus dosing. The actual method of drug administration does not determine the most appropriate model for therapeutic drug monitoring; rather, the relationship between the input time, drug half-life, and the length of the dosing interval detennines the phannacokinetic model. If the actual time of drug input is short compared to the half-life of the drug, not much drug will be lost during the input period, and the intennittent bolus model can be utilized. If the time of drug input approaches one-half of the half-life of the drug, an infusion model is recommended because the amount of drug lost during the input period can not be overlooked. To detennine if the infusion is an intennittent infusion or a continuous infision, compare the length of the dosing interval to the half-life of the drug. If the dosing interval is much shorter than the half-life of the drug, the concentration of drug in the plasma will not fluctuate to a significant extent during the dosing interval, and the continuous infusion model can be utilized. If the dosing interval is longer than the half-life of the drug, the fluctuations in drug concentrations will be significant, and the intennittent infusion model is recommended. The algorithrn for phannacokinetic model selection (Table 2) offers guidelines for selecting appropriate phannacokinetic equations.

Selecting the Proper Sampling Time

The exact time that the plasma sampie is collected during the dosing interval depends on the distribution characteristics and half-life of the drug. Since one compartment models are being employed, it is imperitive that distribution and/or absorption are complete before the peak sampie (Cssmax) is obtained. Distribution can be rapid (30-minutes after the end of a 30-minute infusion for an aminoglycoside) or prolonged (six hours after an oral dose for digoxin). A large error in the estimation of the peak plasma drug concentration can occur if the plasma sampie is obtained at the wrong time; therefore, routine sampies should be drawn just before the next dose (4). If a dosing interval is greater than the expected half-life of the drug, there will be enough fluctuation in the drug concentrations during the dosing interval to obtain peak and trough sampies. If the dosing interval is less than one-half of the half-life of the drug, drug concentrations will not deviate much from the average steady state concentration (Cssave); a sampie obtained any time after the completion of absorption and/or distribution is acceptable (the trough is best).

When selecting an appropriate time to obtain a plasma sampie, it is important to consider what infonnation will be provided at the time of sampling. Volume of distribution can be detennined from the change in plasma drug concentrations following a single dose, i.e.,V = SFD/(Cmax -Cmin)' or from sampies obtained within one half-life of initiating an infusion. Clearance can be detennined from the average steady state concentration, i.e., CL = SFD/ 't Cssave·

276

Table 2. AIogorithm for pharmacokinetic model selection (5)

Is plasma concentration at steady state (D and 't constant; therapy> 3-5 tl/2)? -YES, use steady state equations

Is input continuous ('t« tl/2) or intermittent ('t > tl/2)? -CONTINUOUS, use continuous infusion model -INTERMITTENT

Is t' «tl/2? -YES, use intermittent bolus model -NO

Is t' = 't ? -YES, use continuous infusion model -NO, use intermittent infusion model

-NO, use non-steady state equations

Are D and 't constant?

-NO

-YES

Is t'« tl/2?

Is input continuous ('t« tl/2) or intermittent

( 't > tl/2)? -CONTINUOUS, use non-steady state

continuous infusion model -INTERMITIENT

Is t'« tl/2? -YES, use non-steady state

intermittent bolus model -NO

Is t' = 't? -YES, use non-steady state continuous infusion model

-NO, use non-steady state intermittent infusion model

-YES, use non-steady state, unequal dose, intermittent bolus model

-NO, use non-steady state, unequal dose, intermittent infusion model

277

Although sampies obtained less than one half-life after the initiation of therapy contain very little information on steady state drug concentrations, these sampies can indicate extremely low or high concentrations, permitting the administration of an additional bolus dose or the discontinuation of the maintenance dose. Sampies obtained at two half-lives contain some information on steady state arid, again, alert us to unusually high or low drug concentrations. Sampies obtained after three to five half-lives are the best for routine monitoring. These sampies contain the most information on steady state; however, if you wait too long to obtain a plasma sampie from your patient, drug concentrations can be in the toxic range if the c1earance is much lower than expected.

Predicting the Drug Concentration

Plasma concentration predictions are based on the patient's dosing history, the patient's expected pharmacokinetic parameters, and the selected pharmacokinetic model. Ifthe drug is known to exhibit capacity-limited metabolism, use the nonlinear (Michaelis Menten) pharmacokinetic equations (Appendix ill). If the drug is known to exhibit first-order pharmacokinetics, use the linear pharmacokinetic equations (Appendix N) and the algorithm for pharmacokinetic model selection (Table 2).

Revising the Pharmacokinetic Parameters

The pharmacokinetic parameters are revised to determine the actual values for c1earance and volume of distribution in your patient. Although the goal is to obtain values that agree with the dosing history and the observed drug concentration(s), the revised parameters must be physiologically reasonable. In general, volume of distribution is more predictable and less variable than c1earance; it rarely varies by more than ±30% from the expected value. Clearance, on the other hand, is less certain and may require revision more often; it can vary from 50 to 200% of the expected value. As a general rule, the percentage change in the pharmacokinetic parameter (revisedlexpected x 100) should be no greater than the percentage change in plasma concentration (CobservecYCexpected x 100). If a large change in a pharmacokinetic parameter is necessary to bring about a small change in the plasma concentration, the data probably do not contain enough information to accurately revise the parameter(s). The following guidelines can assist with parameter revision (6).

278

Single bolus dose Sampies obtained soon after a bolus dose reflect volume of distribution. Sampies obtained approximately 1.44 half-lives after a bolus dose reflect c1earance. Sampies obtained after 1.44 half-lives are a function of both volume of distribution and c1earance.

Constant infusion or intermittent dosing with 1'« tl12-Sampies obtained within one half-life of initiating therapy reflect volume of distribution. SampIes obtained three or more half-lives after of initiating therapy reflect c1earance.

Intennittent dosing with 't» tl!2-Since most of the drug is eliminated in a dosing interval, a large difference between the peak and trough concentration is expected. This difference is controlled by volume of distribution. The average steady state concentration is controlled by c1earance.

Revising the Dosing Regimen

Compare the dosing interval ('t) to the observed drug half-life (t1l2) in your

patient. H't > t1l2' make dosage recommendations based on the peak (Cssmax)

and trough (Cssmin) concentrations that you would like to maintain. H't < 1/2 t1l2' make dosage recommendations based on the average steady state concentration (Cssave) that you would like to maintain. Sometimes it is easier to change the dosing interval rather than the dose. If you decide to change the dosing interval, select an interval that is practical for your patient (outpatient) or the hospital staff (inpatient). Usually it is not advisable to make a dosing interval greater than 24 hours (amphotericin B and digoxin are exceptions). Recheck the new dosing regimen after three half-lives, and adjust as necessary.

279

Appendix I. Definitions

Bioavailability (F) = fraction of administered dose which reaches the systemic circulation.

Clearance (CL) = a measure of how weIl the body can metabolize or eliminate a drug.

Concentration (C or C) = dependent variable in pharmacokinetic analysis. C refers to the concentration of drug in plasma when the plasma protein binding is normal; C refers to the concentration of drug in a specific patient when plasma protein binding is abnormal.

Concentration, adjusted (C adj) = drug concentration after adjusting for changes in plasma protein binding.

Creatinine clearance (CLcr) = a measure of the kidney's ability to eliminate creatinine from the body. Total renal function is usually assumed to be proportional to creatinine clearance.

Dose (D) = amount of drug administered for desired therapeutic effect.

Dosing interval ('t') = the time interval between doses when the drug is given intermittently.

Dry body weight (DBW) = weight of patient before accumulation of excess fluid (may include obese weight).

Elimination rate constant (k) = fractional rate of drug loss from the body or the fraction of the volume of distribution which is cleared of drug during a time interval:

e-kt = fraction remaining at the end of a time interval, t l-e-kt = fraction lost during decay phase, assuming no additional

drug input l-e-kt' = fraction of steady state achieved during a constant infusion

t' hours after starting the infusion l_e-kt = fraction lost during a dosing interval at steady state

Free fraction ( a) = fraction of drug in plasma not bound to plasma proteins. Half-life (ti/2) =·time required to reduce the plasma concentration to one-half

of the original value. Ideal body weight (ffiW) = adjustment in body weight made for obese patients

when drug characterists (i.e., Cl, V) depend on lean body mass rather than obese body mass. ffiW is the amount the patient would weigh if (s)he was a weIl developed lean individual with the correct proportion of fat and muscle for their height.

Maximum concentration (Cmax) = peak drug concentration. Michaelis Menten Constant (Km) = plasma concentration at which the rate of

metabolism is occuring at one half the maximum rate. Minimum concentration (Cmin) = trough drug concentration. Number of doses (N) = parameter that is important in nonsteady state

conditions.

280

Plasma protein concentration (P or P') = P refers to the normal plasma protein concentration; P' refers to the plasma protein concentration of a specific patient.

Rate of drug administration (Ro) = average rate of drug input. Salt form (S) = fraction of administered salt or ester fonn of the drug which is

the active ingredient . Serum creatinine (Scr) = measurement of serum creatinine (mg/dl). This

parameter is needed for the estimation of creatinine c1earance. Steady State (ss) = at steady state, rate of drug input is equal to rate of drug

elimination by all routes. Css, the plasma concentration at steady state, is achieved in 3 to 5 half-lives when drug is administered on a fixed schedule.

Time (t) = independent variable in phannacokinetic analysis. For bolus dosing, t is elapsed time since drug administration. For infusion, t' is

length of infusion, t is elapsed time since end of infusion, and 't is

dosing interval ('t = t + t'). Total body weight (TBW) = the true weight of the patient. Inc1udes the

weight due to excess adipose tissue and excess fluid. Volume of distribution (V) = the apparent volume required to account for all

the drug in the body if it were present throughout the body in the same concentration as in the sampIe obtained from the plasma.

Vm = maximum rate at which metabolism can occur.

281

Appendix 11. Additional Equations

282

a f f t· unbound eoneentration = ree rae IOn = tal d . to rug eoneentratIon

Cadj= C', (l-a)(~) + a

C2 = Cl e-kt (for tenninal deeay phase)

( . )* (140-age) mW(kg) CLcr mVmm; male = 72 Ser (mg/dl)

( . )* (140-age) mw (kg) CLcr mVmm; female = 0.85 72 Ser (mg/dl)

*These CLcr equations are for adults ~ 20 years old) who not extremely museular, emaciated, or overweight; Ser is at steady state (7).

Empirie adjustment of CLcr in obese patients: Replace mw in above equations with [mW + 0.2(TBW - mW)].

CLcr(ml/min from 24-hour urine colleetion) = ~V er

where U = urine creatinine eoncentration (mg%) V = urine volume (ml) t = eolleetion time (min) Scr = serum creatinine eoneentration (mg%)

. . V (Cdesired - CinitiJ Dose to aehieve new steady state concentratIon = SF

IBW (male) = 50 kg (first 150 cm of height) plus 1 kg/cm over 150. ADD 10% to estimate for large frames; SUBTRACT 10% for small frames. If the patient's TBW is < mw, use their TBW.

mw (female) = 45 kg (first 150 cm of height) plus 1 kg/em over 150. ADD 10% to estimate for large frames; SUBTRACT 10% for small frames. If the patient's TBW is < mw, use their TBW.

In (Cl) CL C2 . .

k = V = t ' assummg no drug mput after Cl

Appendix III. Capacity-Limited (Nonlinear) Pharmacokinetic Equations

rate of elimination = (K: : C) C

SFD = ( V m \ Css 't Km + CssavJ ave

V m (desired Cssave ) 't Dose = ,-------'----------'c---

(Km + desired Cssave ) SF

time for plasma concentration to fall from Cl to C2 (assuming no drug input after Cl):

283

Appendix IV. First Order (Linear) Pharmacokinetic Equations

A. Steady State Equations

Constant infusion

Css = Ro/CL

Infusion rate = (desired Css) CL 't

Intermittent infusion

where t' = length of infusion, t = time from end of infusion, and 't = dosing interval = t + t'

Intermittent bolus dosing

where 't = dosing interval, and

SFD Cssave = 'tCL

t = time from the bolus dose

(desired Css ) 'tCL Dose = SF

B. Non-Steady State Equations

Continuous infusion

284

Intermittent infusion (same dose and interval)

where t' = length of infusion, t = time from end of infusion, and 't = dosing interval = t + t'

Intermittent infusion (unequal dose and interval)

where 01' 02' ... On = amount administered at each dose, t' = length of infusion, and tl' t2' · .. 10 = time from end of each infusion.

Intermittent bolus (same dose and interval)

where 't = dosing interval, and t = time from the bolus dose

Intermittent bolus (unequal dose and interval)

Ct = ( S~l ) e-ktl + ... + (S~n ) e-ktn

where °1, 02, ... On = amount administered for each dose,

tl' t2' ···,10 = time from each bolus dose to time of sampling.

285

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289

PHARMACOKINETIC STUDIES IN MAN

John G. Wagner

John G. Searle Professor of Pharmaceutics College of Pharmacy, Professor of Pharmacology Staff Member of the Upjohn Center for Clinical Pharmacology, Medical School, University of Michigan Ann Arbor, MI 48109

When I was invited to present this lecture, I went through my listing of over 300 articles and found that 62 of those ar­ticles were involved with original research papers involving human studies on 40 different drugs. I chose 21 of the 62 articles to present today. Before I discuss individual human studies, I wish to provide some background information.

BIOAVAILABILITY is a term used to indicate measurement of Doth the relative amount of an administered drug that reaches the general circulation and the rate at which this occurs. Several of the studies I will discuss were human bioavailabi­lity trials.

1. Generic equivalents and inequivalence - ie. comparison of drug products containing the same active ingredient(s) but made by different manufacturers.

2. Effect of food on absorption of drugs in drug products. 3. Effect of one drug on the absorption of another drug -

a specific type of drug interaction. 4. Effect of age on the absorption of drugs. 5. Effect of disease states on the absorption of drugs. 6. Assessment of so-called "fist-pass" effect to explain

differences in potency when a drug is administered ex­travascularly (particularly orally) compared with intravenously.

7. Drug-drug interaction studies.

CROSS-OVER STUDIES: Most of my human studies have been of the crössöver-type-where there is an equal number of treatments and treatment periods. For example, in a 12-subject two-way crossover study, one-half or 6 subjects are administered treat­ment A and the other 6 are administered treatment B in time per iod #1, then treatments are crossed over so the 6 who received treatment A in period #1 now receive treatment B in per iod #2 and the 6 who received treatment B in per iod #1 receive treatment A in time period #2.

291

Personally, I believe the main advantage of a crossover study is that the volume of distribution would be expected to be constant 2~_~~~_~y~~~g~ since the same subjects are involved in all treatments. Another advantage is one can test statistic­ally for time per iod effects. There are faneier crossover de­signs where one can test for treatment sequence effects also.

When discussing specific human studies in the remainder of my lecture, I will first show a slide with the name of the drug, then say what hypothesis we were testing or the objective of the study and then show some of the results of the study. You have a handout giving the literature references corresponding to the slide numbers. Also listed in the handout is the "JGW Paper Number". Should you wish a copy of the article, please supply me with this number by mail and I will send you a copy of the article.

Indoxole was an experimental non-steroidal anti­inflammatory agent studied by The Upjohn Company many years ago. The drug was about as soluble as sand and presented great pharmaceutical problems. My group prepared several different dosage forms of the drug then performed a bioavailability trial in 10 subjects of the crossover type with four different dosage forms as the treatments.

Figure 1 s~ows the average serum concentrations after the first dose on the left and those after the 6th dose on the right when aoses were administered at 8-hourly intervals.

The order of areas under the curves (the measure of bio­availability) were:

Emulsion ~ soft elastic capsule > aqueous suspension > hard-filled capsule. The drug was dissolved in the oil phase of Lipomul Oral Emulsion (Upjohn) then emulsified to form the "emulsion" and the drug was dissolved in Polysorbate 80, then encapsulated.

292

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TIME IN HOURS AFTER FIRST DOSE TIME IN HOURS AFTER SIXTH DOSE

Fig. 1. Average seru~ concentration of indoxole following the first dose (on left) and 6th dose (on right) at 8-hr intervals and for four different formulations

At the dose level of ca 400 mg g.8.h. shown here, the drug was about 7 times more bioavailable as the em~lsion or soft elastic capsule than in the hard-filled capsule. At a dose level of ca 800 mg g.8.h. (results not shown) the in­crease in bioavailability was about 14-fold.

LINCOMYCIN HYDROCHLORIDE

This is an antibiotic sold by The Upjohn Company under the trademark LINCOCIN ®. Rectal dosage forms of the antibiotic had been prepared and human trials were conducted to test the hypothesis as to whether rectal absorption was as efficient as absorption after oral administration (or not). In this case we actually performed three human studies but I will show the re­sults obtained in ten subjects (6 females and 4 males) which was common to all three studies.

Fisure 2 shows a plot of the average serum concentration of lincoymycin hydrochloride in the 10 subjects versus time after administration of a single dose of 500 mg of-Ilncomycin. The top curve with black circles were the results when the com­mercial capsule was given orally and subjects were fasted over­night and for 4 hours post during. The second highest curve with open circles were results obtained with the same dose given in aqueous solutions rectally when the dose was preceded by an enema. The third highest curve with the open diamonds was ob­tained when the solution of the drug was given rectally without an enema and when subjects were non-fasting. The lowest curve with the open squares was produced by ~ suppository given rec­tally without an enema with the subjects fasting.

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Fig. 2. Average serum concentration of linco­mycin hydrochloride in 10 subjects after 500 mg doses . • ---. capsule, orally, fasted; 0---0 aqueous solution rectally preceded-by enema; ~---~ aqueous solution rectally, no enema, non-fasting;

14

0---0 suppository, rectally, no enema, fasting.

293

We concluded that efficiency of absorption rectally was not nearly as good as orally and that the type of dosage form given rectally (ie. solution vs. suppository) was important.

CLINDAMYCIN

Clindamycin is another antibiotic sold by The Upjohn Com­pa ny and is an analogue of lincomycin.

Shown in Figure 3, we tested the hypothesis of whether food interfered with the efficiency of absorption of clindamy­cin by performing a crossover study in 12 male subjects and the results are shown here where average concentrations of clinda­mycin are plotted versus time. Curve A was obtained under fasting conditions. Curve B was obtained when the dose was given one hour after breakfast. Curve C was obtained when the dose was given one hour before breakfast. Curve 0 was obtained when the dose was given immediately following breakfast. The average areas under the individual sUbject concentration - time curves were 14.1, 15.7, 13.1 and 15.2 ~-1 x hrs for treatments

A, B, C and 0, respectively. There were significant differences among treatment means at p < 0.001 but practically the dif­ferences were small and probably not clinically important.

Figure 4 shows the range of elimination half-lives of clindamycin observed in the study. Subject #1 had the shortest while sUbject #6 had the longest half-life. The half-life of subject #9 in the middle, namely 2.40 hr was similar to the overall average half-life of 2.38 hours. Thus this Figure shows intersubject variation in half-life.

Figure S shows intrasubject variation of elimination half­life of clindamycin. Here the t~ observed in week 2 is plotted versus the t~ observed in week 1 for 9 sUbjects. The solid line is the line of identity with slope = 1.00. The dotted lines in­clude all points an~ indicate that the ratio: t~ in week 2

t;;-lil-week-I varied from 0.67 to 1.33 with an average of unity.

294

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Fiq. 3. Average serum concen­trations of c1indamycin after 250 mg doses in capsule form. A - fasting; B - dose one hr be fore breakfast; C - one hr after breakfast; 0 - dose im­mediately following breakfast.

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Illustrates the range of eli­half-lives of clindamycin 1n the study discussed in

TETRACYCLINE HYDROCHLORIDE

We wished to test the hypothesis of whether the efficiency of absorption of this antibiotic when given rectally was as good (or worse) than when the antibiotic was given orally.

From 40 to 60% of the absorbed dose of tetracycline is ex­creted ln the urine so we could use the cumulative urinary excrelion of the antiüiotic rather than serum concentrations to test the hypothosis. (See Figure 6).

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Fig. 5. Illustrates the intrasubject variation in elimination half-lives of clindamycin.

295

110

TREATMENT C

TREATMENT B

TREATMENT A

72

Fig. 6. Cumulative amounts of tetra­cycline excreted in the urine for 6 subjects after a 250 mg dose. Treat­ment A - aqueous solution rectally following breakfast; Treatment B -orally in solution immediat2ly fol­lowing breakfast; Treatment C - orally in solution after overnight fast and subjects fasted 4 hrs post dosing.

This showed average cumulative amounts oE tetracycline ex­creted in the urine versus time for 6 subjects. Treatment A was the dose (250 mg) given in aqueous solution rectally fol­lowing a specified breakfast. Treatment B were results after the same dose was given orally in solution after an overnight fast and the sUbjects were fasted for 4' hours post-dosing.

The average amount of tetracycline excreted in the urine was only 9% following treatment A relative to oral adminis­tration under fasting conditions.

SODIUM PENICILLIN G

We did a similar rectal-oral study with sodium penicillin to determine relative absorption rectally and orally.

The average cumulative amounts of penicillin excreted in the urine versus time is shown in Figure 7 for 6 subjects. Again, treatment A was the antibiotic administered in aqueous solution rectally immediately following a specified breakfast. Treatment B was the antibiotic solution given orally immediate­ly following the breakfast. Treatment C was the aqueous solution of the antibiotic administered in solution orally after an over­night fast and subjects were fasted for 4 hours post dosing. Here cumulative urinary secretion for treatment A averaged only 13.5% and that for treatment B averaged 31.2% of that for treatment C.

296

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Fig. 7. Cumulative amounts of penicillin G excret·ed in the urine in 6 subj ects af­ter a 250 mg. dose. Same treatments as in Fig. 6 above.

LINCOMYCIN HYDROCHLORIDE - SODIUM CYCLAMATE INTERACTION

During 1964, while I was working in the Medical Research Division of the Upjohn Company, I was given an intriguing problem to solve. A co-worker (Dr. Lawson) had requested that two pediatric syrups - one mint flavored and one rasp­berry flavored - be prepared to oe tested. He performed a 15-subject 3-way crossover study comparing serum concentra­tions following administrations of these syrups and the com­mercial capsule orally to adults.

In Figure 8, curve "A" was that obtained with the com­mercial ca~sule while curves "8" and "CU were obtained with the raspberry and mint syrups, respectively. The average areas under curves 0-12 hrs were 21.9, 6.1 and 5.9~~ x hr for A, B

ml and C, respectively. Hence the biovavailability of lincomycin following the syrups was only 28 and 27% for "B" and "CU, respectively, relative to "A". Obviously, something was wrong and I got the job to find out the reason for the poor results with the two syrups.

First I did a 6-subject 2-way crossover study to compare serum concentrations following the solid capsule formulation and a solution of the drug made from the capsule contents - both at a dose of 500 mg. Average serum concentrations versus time are plotted (Fig. 9) with the "A" curve from the capsule and the "B" curve from the ac;ueous solution. The average areas of the 6 individuals were 21.0 and 24.5~~ x hr for the capsule and

ml solution respectively. This study showed that results following the two pediatric syrups, shown in Figure 8, were not the re­sult of the fact that the syrups were solutions and the control a powder in a capsule.

297

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Fig. 8. Average serum concentrations of l incomycin hydrochloride in 15 adult subjects. A - Comme rcial capsulei B - raspberry flavored pediatric sirup i C - mint flavored pediatric syrup.

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Fig. 9. Average serum concentrations of lincomycin hydroch loride in 6 subjects after a 500 mgdose. A - commercial cap­sulei B - aqueous so lution prepared from the capsule.

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Fig. 10. Average serum concentrations of lincomycin hydrochloride in 15 chil­dren. A - commerical capsulei B -raspberry flavored pectiatric syrupi C - mint flavored pediatric syrup.

The results with the two syrups and capsule obtained in adults were confirmed in 15 children in a 3-way crossover study with results shown here in Fig. 10. Again, "A" re­fers to the capsule and " 8 " and "e" are the raspberry and mint syrups, r es pectiveJy. Hence the results could not be explained by the age of the subjects.

I then set up thre e simultaneous 6-subject two-way cross­over studies. In the first study we compared the capsul e and the raspberry syrup without the sweetening agent sodium cycla­mate . The difference in results here is peculiar to this study only since subsequent studies showed that the syrup without cyclamate was e gual to the capsule. (Fig. 11)

The second of the three simultaneous studies compar ed tne capsule and the raspberry syrup without sucrose - another ingredient of the syrup. The syrup however sti ll contained preservatives, sodium cyclamate and sodium saccharin. You can see from the results that obviously sucros e was not the cause of the problem. (Fig. 12)

The third of the three simultaneous studies compared the capsule with the raspberry syrup without preservatives but containing sucrose, sodium cyclamate and sodium saccharin. Obviously, the preservatives were not the cause of the problem. (Fig. 13)

These three simultaneous studies told us that sodium cy­clamate was the major cause of the problem.

299

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Fig. 11. Average serum concentrations of lincomycin hydrochloride in 6 subjects. A - commercial capsule; B - raspberry flavored pediatric syrup without sodium cyclamate.

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Fig. 12. Average serum concentrations of lincomycin hydrochloride in subjects. A - commercial capsule orally; B - raspberry flavored pediatric syrup without sucrose but containing sodium cyclamate orally.

300

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Fig. 13. Ave rage serum concentrations o f lincomycin hydrochloride in 6 sub­j ects. A - commercial capsul e orallYi B - raspberry flavored pediatric syrup without pre servative s but containing suc rose and sodium c yc lamate orally.

We then did a 16-subject two-way crossover study c omparing the raspnerry syrup without sodium c yclamte with the capsule and results sho,vn he re. (Fig. 14). Curve "A" was obtained with the caps ul e whil e c urve "B" wa s obtaine d with the s yrup without cy­clama te . The r e was no s i gnificant difference in average serum concentration a t any sampling times.

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Fig. 14. Ave rage s e rum concentrations o f lincomycin hydrochloride in 16 sub­j e cts. A - commercia l capsul e orallYi B - raspbe rry flavored pediatric syrup wi thout sod i um cyclamate orally.

301

Figure 15 is a cross-study comparison where "A" are the average serum concentrations of 16 sUbjects obtained with the raspberry syrup without cyclamate (previous Fig. 14) .and curve "B" are average serum levels of 6 subjects given a pure agueous solution (curve "B" in Fig. 9). These are statistically super­imposable and support the concept that the sodium cyclamate in­teracted with the lincomycin hydrochloride ln some way and this was the cause of the problem.

Since sodium and calcium cyclamates at that time were sold as ingredients in diet beverages I wished to know if the inter­action would occur with lincomycin if mixing occurred in the human stomach rather than in a medicine bottle. I did a 3-way crossover study in 6-subjects and results are shown here. (Fig.16) Curve "A" was the control with the lincomycin given in agueous solution. Curve "B" resulted from the administration of a so­lution containing 500 mg of lincomycin, and separately, 1 molar eguivalent (247.5 mg) of sodium cyclamate in 2 fl 02. of water. Curve "C" resulted from ingestion of 500 mg of lincomycin in 2 fl 02. of water followed by ~ bottle (8 fl 02.) of Diet-Rite Cola~ which contained 0.25% sodium cyclamate eguivalent to 2.39 molar eguivalents of sodium cyclamate per mole of lincomycin. Hence, if sodium cyclamate given in a soft drink was mixed with lincomycin in the human stomach the drug interaction occurred and absorption of lincomycin was impaired.

The mechanism of the interaction of lincomycin and cycla­mate was never elucidated.

302

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Fig. 15. This lS a cross-over study comparison of average serum concentra­tions of lincomycin hydrochlor i de. A - 16 subjects give n ra spberry flavored pediatric syrup without so­dium cyclamate orally (Fig. 14); B -6 subjects given aqueous solution of lincomycln hydrochloride orally.

T IMI IN' tiOU~S

Fig. 16. Average serum con­centrations of lincomycin hydrochloride after oral admin­istration of 500 mg A - linco­mycin HCL in aqueous solution; 8 - solution of lincomycin HCL followed by 1 molar equivalent of sodium cyclamate; C - solu­tion of lincomycin HCL followed by Y, bottle (8 fl oz) of Diet Ri te Cola ® which contained 0/25% sodium cyclamate (2.39 molar equivalents) .

TETRACYCLINE HYDROCHLORIDE

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Fig. 17. Average serum con­centrations of tetracycline hydrochloride in 6 subjects after a 250 mg dose orally. A - aqueous solution; B -aqueous solution of drug also containing 1 molar equi­valent of sodium cyclamate; C - aqueous solution of drug followed by Y, bottle (8 fl oz) of Diet Rite Cola ® containing 5.66 molar equivalents o~ sodiu~ cyclamate.

I wished to determine if an interaction of another drug with cyclamate occurred so I did a study with tertracycline hydrochloride. This was a 6-subject, 3-way crossover. Average serum concentrations of tetracycline HCL are plotted vs. time in Fig. 17. Curve "A" was obtained when 250 mg of tetracycline HCL dissolved in 2 fl oz of water taken orally. Curve "8" re­sulted from the same dose also in 2 fl oz of water but also containing 1 molar equivalent (105 mg) of sodium cyclamate. Curve "CU resulted from the ingestion of the same aqueous so­lution of the antibiotics as in "A" but followed by Y, bottle (8 fl oz) if Diet Rite Cola containing 5.66 molar equivalents of sodium cyclamate. The average areas were 15.2, 15.9 and 12.2 ug x hr for A, Band C, respectively. Analysis of variance in­ml dicated no significant differences among treatment means. Hence, unlike lincomycin, tetracycline's absorption was not inhibited by sodium cyclamate.

PHE1\)YTOIN

Dr. Gerber and I were the first to show that phenytoin eli­mination obeyed Michaelis-Menten kinetics. This shows a recti-

303

linear plot of average blood concentration of diphenyhydantoin in the rat vs. time. The lines are those based on Michaelis­Menten fitting of the data (Figure 18).

Figure 19 shows the same type of Michaelis-Menten fits of three sets of da ta in a human subject but this time on semi­logarithmic graph paper - showing this characteristic inward curvature.

DIGOXIN

Back in the 1960's and 1970's in the Uni ted States, there were many articles concerning equivalence and non-equivalence of drug products containing the same active ingredient. A study done by Lindenbaum et al in 1971 disclosed that digoxin was one such problem drug. The Lindenbaum study was faulted by the Food and Drug Administration in the United States because the tablets tested were subsequently found to not pass U.S.P. speci­fications - although they were purchased on the open market. Re: Fig. 20 - At that time, I held a contract with the Food and Drug Administration to da bioavailability studies. I performed a 2-way crossover study in 8 normal volunteers administering 0.5 mg of digoxin (as 2 - 0.25 mg tablets) . The innovator's product Lanoxin ® was one treatment and a generic tablet made by Fougera was the other treatment. Both tablets passed all U.S.P. specifications in the laboratories of the Food and Drug Administration. Subjects 1 and 2 of the 8-member panel of sub­jects also received the same dose orally in the form of an aque­ous solution and on another occasion intravenously by infusion over a 1-hour per iod. Results are shown in Figure 20.

304

T I~.E IN HOURS MEASlREO FROM FIRST SAMPL I NG TI ME

Fig. 18. Average concentration of phenytoin in whole blood of 6 rats following I.V. doses of 10, 25 and 40 mg/kg. Lines are model pre­dicted concentrtions based on fitting to the Michaelis-Menten equation.

"

,. IS. TIME IN HOURS MEASURED FRO~ ELVE HOl1'lS AFTER THE LAST DOSE

Fig. 19. Semilogarithmic plot of plasma concentrations of phenytoin in subject M. beginning at 12 hours after the last doses when doses of 2.3, 4.7 and 7.9 mg/kg were given daily for 3 days. Lines are model­predicted concentrations based on fitting to the Michaelis-Menten equation.

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Fig. 20. .----. and ~-~ are average plasma concentrations of digoxin in 8 nor­mal volunteers following oral administration of 0.5 mg doses of digoxin as Lanoxin® (B&W) and Fougera tablets, respectively. Upper two curves are average plasma c6ncentrations in 2 of the 8 subjects when digoxin was given. • IV infusion over a 1 hr period and ~-~ ocally in solution form.

305

Average plasma concentrations of digoxin measured by radio­immunoassay are plotted in Fig. 20 vs. time. The dose-corrected average area under the curves for the 8 subjects was 2.2 times greater for the Lanoxin® tablets than for the generic Fougera tablets. The lower two curves are for the Lanoxin (B&W) and Fougera tablets, while the upper two curves are for the I.V. and oral solution treatments, respectively. Suffice it to say but these results caused quite a stir in pharmaceutical circles at the time and lead to successful bioavailability testing in hu­man subjects as one criterion to market generic digoxin tablets.

4-AMINOSALICYCLIC ACID

Re Fig. 21 - We did a 4-way crossover study in 8 normal volunteers in which 1 gram doses of p-amino salicyclic acid as a solution of the sodium salt in water, a suspension of the acid form of the drug, a compressed tablet of the acid form and an enteric-coated tablet of the acid form. We did not measure the drug or its N-acetyl metabolite after the latter form. One of the subjects ingested two enteric-coated tablets as pictured at the top of the figure. He was asked to watch his feces. About 30 hours after administration we removed the tablet and fragments from his feces shown at the bot tom of the slide. The essentially intact tablet assayed 98% of label while the two fragments assayed 48% of label.

306

Fig. 21. Lower panel is an essentially in­tact (assayed 96% of label) and fragments of a second tablet (assayed 48% of label) which were excreted in the feces about 30 hr after the oral administration of the intact enteric coated tablets of 4-amino­salicyclic acid shown in the top panel.

2ig. 22. The average plasma concentrations of the bioactlve PAS (p-arnlnosalicyclic acid) in top panel and bioinactive metabolite (N-acetyl) following 1 gram oral doses of the PAS in tlle following dosage forms: A - compressed tablet of acid form; B - enteric coated tablet of acid for~; C - suspension of acid form; 0 - aqueous solution of sodium p-aminosalicytate.

The average plasma cOllcentrations measured following ad­ministration of the other three dosage forms are shown in Fi'Jure 22.

Thp top curve "0" is from the sodium salt in solution. 'l'he "C" curve is from the suspension of the acid form. The "A" curve is from the compressed tablet of the acid form. The top grapns are plasma concentrations oi the bioactive unchanged drug. The bottom graphs are the plasma concentrations of the N-acetyl metabolite. Note that the three curves of the meta­bolite are very similar whereas the drug curves are vastly different. These differences were indicative of saturable acetylation.

307

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50 )J.Ci 14_C_OPH in solution and 100 mg of pheny­toin in capsule form ~n subject #1.

PHENYTOIN + METABOLITES

Re: Fig. 23 - We hypothesized that product inhibition of the metabolism of phenytoin did not occur to any signifi­cant degree as h3d been suggested by Ashley and Levy in 1972. We developed an HPLC method to separate phenytoin, its primary metabolite, 5-(p-hydroxyphenyl)-5 phenylhydantoin ~nd its glu-

curonide. We gave 2 subjects 50~Ci of 14C_OPH in solution and a 100 mg Oilantin Kapseal. Plasma concentrations in one of the sUbjects is shown in Fig. 23. The metabolite/drug (ie.HPPH/OPH) ratio was always less than 0.06 and the major metabolite in plasma is the glucuronide of HPPH, not HPPH.

In Figure 24 are the results in the second subject and are almost identical. Thus, our hypothesis was correct. The amount of HPPH in blood is much too small to cause any significant product inhibition.

308

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Figure 25. Plasma concentration of diphenhydramine after 50 mg doses of the drug. e---e I.V. infusion over one hr; ~ aqueous solution; ----.commercial capsule. Upper panal -subject #1; Lower panel - subject #2.

DIPENHYDRAMINE

We hypothesized from scanty data in the literature that the antihistaminic drug diphenhydramine had a 'first-pass' effect.

Re: Fig. 25 - We studied 2 normal volunteers who ""ere each administered 50 mg of diphenhydramine by constant rate infu­sion over 1 hour, as an oral solution in a Coca-Cola syrup­wat e r mixture and as the commercial capsule. Plasma and urine samples were assayed by a GLC method. Both subjects went to sleep at the e nd of the infus ions but were only drowsy follow­ing the oral treatments.

Figure 25 shows the individual subject plasma concentra­tions for subject #1.

309

PUh I

Fig.26. First pass three compartment open model which explains the diphenhydra­mine da ta shown in Fig. 25.

, 0 "

0' " o. ,.

l'i 0'

z 0'

~ 0' ;:

j o.

~ 0'

0'

>

'" < 0'

! • u 0

TIJoIE HOuRS

Fig. 27. Capillary blood etha~ nol concentrations in one of 8 subjects administered doses of 15, 30, 45 and 60 ml of 95% ethanol under fasting conditions (see text).

Plasma concentrations for subject #2 are shown in Figure 25. The solid circles are from the infusion, the solid dia­monds from the aqueous solution orally and the solid squares from the capsule orally. The large differences in area under the curve for the intravenous and oral treatments are indica­tive of the 'first-pass effect'. There was about a 50% first­pass effect in those two subjects. Also, the drug following the capsule was somewhat less efficiently absorbed than from the solution.

Figure 26 shows the model which explains the diphenhy­dramine data. Elimination is from the hepato-portal system with first order rate constant k 20. When oral drug is adminis­tered, it goes directly into this system; all of the dose is exposed to the drug metabolizing enzymes. When drug is ad­ministered intravenously it enters the so-called "central compartment" and can t~en go either to the so-called "tissue compartment" or to the hepto-portal system. Hence some of the drug is protected and a greater area results.

ETHANOL

The classical pharmacology books say that ethanol is metabolized at a constant (zero order) rate independent of the concentration. But this is not so. It is metabolized accord­ing to Michaelis-Menten kinetics.

Re Figure 27: We performed an 8-subject, 4-way crossover in which doses of 15, 30, 45 and 60 ml of 95% ethanol were ad­ministered under fasting conditions. The figure shows results in one of the 8 subjects. If metabolism were zero order the down slope lines would all be parallel. But this is not so. The slope is greater the greater the dose of alcohol as seen from the numbers 0.034, 0.071, .0173 and 0.190. In fact, inset is shown a plot of the reciprocal of the slope, l/ko,

310

"

\ .. .. TIME IN HOURS

Fig. 28. Capillary blood ethanol concantrations in J.G. Wagner when he was infused I.V. for 2 hours on one occasion with 720 ml of so­lution 4% v/v EtOH and other occasion 8% v/v EtOH.

versus the reciprocal of initial concentrations, I/Co, where Co is the concentration at the start of the linear segment. This type of plotting is in conformity with Michaelis-Menten, not zero order, kinetics.

Figure 28 shows capillary whole blood concentrations of ethanol in myself when I was infused over 2 hours on two occa­sions and with a volume of 720 ml - one with 4% v/v and once with 8% v/v ethanol. Note again that the higher value of the downslope, namely 0.177, is associated with the higher concen­trat ions and larger dose of ethanol and the lower slope, namely 0.146, with the lower dose of ethanol.

Re Figure 29: We performed extensive studies of the effect of food on blood concentrations of ethanol. At low to moderate doses of ethanol there is a 'first-pass' effect and the slower the rate of absorption the greater the relative rate of meta­bolism and the less the area under the curve. Food slows ab­sorption rate of alcohol. We gave 6 normal volunteers 45 ml of 95% alconol in 150 ml of orange juice and various types of food. The figure shows mean capillary ethanol concentrations versus time. The curves from top to bottom were for the follow­ing foods. A - fasting; B - light breakfasti C - heavy break­fast and D - steak meal.

311

312

~ r------------------------------------,

• . - . •• • • •

• • -• o •

• • •

t 0 • • 0 • •• 0 •

• • 00 ••• • 0 ' 0<\ ~

o '."'E: IN ~S

Fig. 29. Average capillary blood etha­nol concentrations in 6 subjects given 45 ml of 95% ethanol in 150 ml of orange juice and various amounts of food or fasting. A - fastingi B - light break­fast i C - heavy breakfasti 0 - steak meal.

'00 '0

~ 90

~ ~ 80

;:

~ 70

~ 069

~ 60 066

ffi~ : i so

~~ 0 15 iII -0

~ ~

JO

d % 0 ~ 20 <

~

z '0

~ 0

TI", 'N HO\JRS

Fig. 30. Absorption plots for ethanol. A - fastingi B - light breakfasti C - heavy breakfasti D - steak meali E - alcohol 1 hr after a heavy breakfasti F - al­cohol 1 hr before a heavy break­fast.

~ r------------------------------.

'50

200

''''

~ rt. 100

50

o llME tHOUASI

· , , sr

• 0

• u

Fig. 31. Ave rage plasma concentra­tions of prednisolone in 12 subjects iullowing single 10 mg doses in cross­over fashion in a 4-way crossover study. Manufacturers were: R - R ond e Xi ST - StanleYi 0 - Ormont and U - Upjohn.

Kinetic analysis of the 0100d alcohol curves yielded ab­sorption plots which are shown in Figure 30. The amount of al c oho l absorbe d per ml of the volume of distribution is plot­ted versus time . Here A - fastingi F - alcohol given 1 hr be f ore a heavy breakfast (all alcohol was absorbed bef o r e the oreakfa s t) i B - light breakfasti e - heavy breakfasti 0 -ste ak meali E - alcohol 1 hr after a heavy breakfast.

PREDNISOLONE

We studie d the bioavailability of prednisolone from in­nova t ors' (Upjohn and Schering) and generic tablets. Figure 31 s hows the ave rage plasma concentrations of prednisolone following a single 10 mg oral dose in 12 subjects measure d in a 4-way cros sov e r study. The symbols and letters are for the manu f acturer s where R = Rondex, ST = Stanley, 0 = Ormont and U = Upjohn. The re wer e significant difference s in average plasma conce ntrations at 0.25 and 4 hr but not at any other time . Aue 0-12 and Aue 0-24 did not differ significantly.

We ended u p studying 9 differe nt prednisolone tabl e ts mad e by 8 diff e r e nt manufacturers. All mean plasma concentra­tion curves lay within the hatched portion shown in Figure 32 and were assumed to be bioequivalent.

313

314

T IME ~HOVR$J

Fig. 32. The hatched portion included average plasma concentra­tions of prednisolone following 10 mg oral doses as 9 different com­mercial tablets made by 8 differe nt manufacturers.

e~

~~~--,-----~----~:===~

rig. 33. TOP LEFT PANEL - Average plasma concen­trations of tolmetin following oral administration of (a) 400 mg tolmetin sodium alonei (b) 20 ml Maalox magnesium and aluminum hydroxide suspension 4 times/day for 3 days preceding the dose of tol­metin sodiuffii (c) co-administration of 20 ml Maalox and 400 mg tolmetin-sodium. OTHER PANELS are individual subject plasma con­centrations following the 3 tredtments.

TOLMETIN

Tolmetin is a non-steroidal anti-inflammatory agent which may be prescribed along with an antacid. We performed a 24-subject 3-way crossover study in which the treatments were: (a) tolmetin sodium alone (400 mg); (b) 20 ml Maalox magnesium and aluminum hydroxide suspension 4 times/day for 3 days pre­ceding the dose of tolmetin sodium; and (c) co-administration of 20 ml of Maalox and 400 mg of tolmetin sodium.

Top left in Figure 33 are the mean plasma tolmetin concen­trat ions for the 3 treatments while the other plots are indivi­dual subject plots.

The major metabolite of tolmetin is the corresponding p­carboxy compound and we measured it quantitatively also. Top left in Figure 34 are the average plasma concentrations of the metabolite and the others are the individual subject plasma con­centratrions for these treatments.

Figure 35 depicts the fall-off curves for 4 of the 72 cases. Note that the semilogarithmic plots are curved. We explained these by a nonlinear tissue binding model.

Fig. 34. TOP LEFT PANEL - Average plasma concentra­tions of the major metaboli te of tolmetin f ollowing oral administration of (a) 400 mg of tolmetin sodium alone; (h) 20 ml of Maalox magnesit:nn and aluminium hydroxide suspen­sion 4 times/day for 3 days preceding the dose of tolmetin sodium; (c) co-administation of 20 ml Maalox and 400 mg of tolmetin sodium. OTHER PANELS are individual subject plasma concentrations following the three treatments.

315

316

~

§ 10 .. O' .

~

flMf, N HOuAS

Fig. 35. Curved terminal semi­logarithmic plots of tolmetin plasma concentrations which were explained by a nonlinear tissue binding model .

•• ' , , t , t , I I I t I , t I t

. " fI I e 11 fI 1 e. 11 I!I I & 11 ~ 1 e 11

""y , CAY ~ OAV t _ ,__ D,l.y $

Fig. 36. Plots of mean [H+ l, exp.ressed as pH, versus time 0. No antacid; • Ofle dose of antacij; • Antacid for 3 days.

There is evidence of a circadian rhythm of urinary pH.

:r--------------------------,

Fig. 37. Plasma concentrations of penicillamine in anormal volunteer who ingested orally doses of 250, 500, 750 and 1000 mg of penicillamine. Note the double peaking.

Figure 36 shows a plot of me an urinary hydrogen ion con­centration, expressed as pH, versus time for the entire study period. Her e the diamonds are treatment(a) ie. no antacid; the solid dots are for treatment (b) ~here antacid given for 3 days and the solid squares are for treatment (c) concomitant tolme t in and antacid on day 4.

Note there is a circadian rhythm of urinary pR and the antacid did not change the shape of the curve but elevated t he pH.

PENICILLAMINE

Penicillamine is used in the treatment of rheumatoid arth­ritis. Fig. 37 shows pen i cillamine plasma concentrations vs. time in anormal volunteer who was administered single doses of 250, 500, 750 and 1000 mg of penicillamine. The thing of intere st here is the doubl e peaking which is not often seen.

VERAPAMIL

Figure 3 8 refers to an evaluation I made of da ta published

317

by two other groups of investigators. The diamonds are data of Freeman et al and the circles are data of Shand et al. The ordinate is the average steady-state plasma concentration - ie. the area under the steady-state concentration-time curve in a dosage interval divided by the dosage interval . The absissa is the area O-~after a single dose divided by the dosage in­terval, namely 6 hrs. If kinetics were normal this line would have a slope equal to unity but here the slope is 2.41.

In a later paper in Clinical Pharmacology and Therapeutics

I showed that the slope of this line, namely Css

CLi

CLss

Vm

Vm-R

AUC 0- 00 /,

where Cli = intrinsic clearance, CLss

steady-state clearance at dose rate, R, and R is the dose rate. Thus, the points are close to the line because the minimal ve­locity of metabolism, Vm, has low variance.

IBUPROFEN

Ibuprofen is a non-steroidal anti-inflammatory agent sold by the Upjohn Company under the trademark MOTRIN We did a so-called dose proportionality study by doing a lS-subject 3-way crossover study with 400, 800 and 1200 mg doses in the form of tablets. On the 4th week of the study all subjects received a 400 mg dose of the drug in the form of the sodium salt in aqueous solution. We measured protein binding too.

318

320 ,----------___=_.-----,

• • •

t

,A

J 20 60 100 140

Aloe Q-ex>/6 After 0 Smgle Oral Dose In Adulls (ng/ml)

Fig. 38. A plot of the average steady­state plasma concentration of Verapamil versus the area 0-00 under the single dose curve divided by the dosage interval of 6 hrs. the slope of the line is 2.41 rather than unity expected for a non-first­pass drug obeying linear kinetics.

08 • ~ a 400 • • tI'

E • ~ a 5 06 " <> i 04

~

g « d ~ • ~ ~ 02 a: a. ::>

~ !I!

~ 0 8 12 16 20 >-

DOSE (mg/kgl

16 2500

E b • E <>

" ~ 12 • • <> -' :. • c w • .1 • ü ~ • • ::> ~ 8

" ," " • " • Z ~

~ Ü

a. g 4 ::> !!l w w w !E fE

0 12 16 20

IBUPRCHN DOSE (mg Ikg I DOSE (mg/kgl

Fig. 39. TOP PANEL - Area Fig. 40. TOP PANEL - Total under the total (bound and oral clearance vs. mg/kg dose free) ibuprofen plasma con- for ibuprofen. LOWER PANEL -centration curve vs. mg/kg Free oral clearance vs. mg/kg dose. LOWER PANEL - Area dose for ibuprofen. under the free (unbound) ibu-profen plasma concentration curve vs. mg/kg dose.

At the top of Figure 39 is a plot of the area under the to­tal (bound and free) ibuprofen plasma concentration-time curve versus the mg/kg dose. The plot is curved due to nonlinear plasma protein binding.

At the bottom of Fig. 39 is a plot of area under the free (unbound) ibuprofen plasma concentration-time plot. Here the da ta are linear and pass through the origin.

Re: Figure 40 - These are the orresponding clearance versus dose plots. At the top clearance based on total drug concentra-tions changes with dose. At the bottom clearance based on free (unbound) concentrations is dose-independent.

FLURBIPROFEN

Flurbiprofen is the future successor to ibuprofen.

We did a dose proportionality study with this drug too in 15 sUbjects. The plot he re is area under the total (bound and free) plasma concentration time curve versus mq/kg dose of flurbiprofen. Here da ta are linear and area is proportional to dose (See Figure 41).

319

~

rl • e ~ • ,

~ ~ • ~ x • ~ • ~ ~ ~

8 1~ • I

0 100

u ~ ~ ~

4

Dose (mg/kg)

Fig. 41. Area under the total plasma concentration curve vs. mg/kg dose for flurbiprofen ad­ministered as tablets.

In Figure 42, the corresponding plot of clear&nce of total flurbiprofen versus mg/kg dose. Clearance is dose­independent.

I hope this survey of pharmacokinetic studies jn man has indicated many of the types of studies and how we set about to prove or disprove hypotheses.

320

~

rl • e 300 • , ~ ~

x ~

~ ~ ~

8 1~ I

0 100

u ~ ~ ~

Dose (mg/kg)

Fig. 42. Plot of plasma clearance of flurbiprofen based on total con­centrations versus dose. Clearance is dose-independent.

REFERENCES

JGW Fig. Paper !~§.! No. *

1 50

2 64

3-5 76

6,7 85

8-17 101

18,19 128

20 138

21 142

22 147

23,24 168

25,26 177

27 182

28 187

29,30191

J.G. Wagner et al, The effect of the dosage form on serum levels of indoxole, Clin. Pharmacol. Ther. 7: 610-619 (1966).

J.G. Wagner et al, Serum concentrations after rec­tal administration of lincomycin hydrochloride J. Clin. Pharmacol. 8: 154-163 (1968).

J.G. Wagner et al, Absor~tion, excretion and half­life of clinimycin in normal adult males, Am. J. Med. Sci. 256: 25-37 (1968)/

J.G. Wagner et al~-Relative absorption of both tetracycline and penicillin G administered rectally in aqueous solution, Int. J. Clin. Pharmacol. 2: 44-51 (1969).

J.G. Wagner, So~e experiences in the evaluation of dosage forms of drugs in man, Compilation of Symposia Papers Presented at the 5th National Meeting of: A.Ph.A. Academy of Pharmaceutical Sciences, American Pharmaceutical Association, Academy of Pharmaceutical Sciences, Washington, D.C., 1970, pp. 420-465.

N. Gerber and J. G. Wagner, Explanation of the dose-dependent decline of diphenylhydantoin plasma levels by fitting to the integrated form of the Michaelis-Menten equation, Res. Comm. Chem. Path. Pharmacol. 3: 455-466 (1972).

J.G. Wagner et al, Equivalence lack in digoxin plasma levels, J. Am. Med. Assoc. 224: 199-204 (1973).

J.G. Wagner et al, Failure of the USP tablet dis­integration test to predict performance in man, J. Pharm. Sc i . 62: 859 - 8 6 0 (1973).

J.G. Wagner et al, I~portance of the type of dosage form and saturable acetylation in determining the bioactivity of p-aminosalicyclic acid, Am. Rev. Resp. Dis. 108: 536-546 (1973).

K.S. Albert et al, Plasma concentrations of di­phenylhydantoin, its p-hydroxylated metabolite and corresponding glucuronide in man, Res. Comm. Chem. Path. Pharmacol. 9: 463-470 (1974).

K.S. Albert et al, Pharmacokinetics of diphenhy­dramine in man, J. Pharmacokin. Biopharm. 3: 159-170 (1975). -

J.G. Wagner et al, Elimination of alcohol from human blood, J. Pharm. Sci. 65: 152-154 (1976).

P.K. Wilkinson et al, Blood ethanöl concentrations during and following constant-rate intravenous infusion of ethanol, Clin. Pharmacol. Ther. 19: 213-223 (1976). --

Y.-J. Lin et al, Effects of solid food on blood levels of alcohol in man; Res. Comm. Chem. PATH. Pharmacol. 1~: 713-722 (1976).

*Cite this number if requesting complete article.

321

Fig. !~~!.

31,32

33 -35

36

37

38

39,40

41,42

JGW Paper No. *

214

221

222

255

260

266

A.V. Tembo et al, Bioavailability of predniso­lone tablets, J. Pharmacokin. Biopharm. 5: 207-224 (1977) -

J.W. Ayres et al, Pharmücokinetics of tolmetin with and without concomitant administration of antacid in man, Eur. J. Clin. Pharmacol.

J.W. Ayres et al, Circadian rhythm of urinary pH in man with and without chronic antacid administration, Eur. J. Clin. Pharmacol. 12: 415-420 (1977).

R.F:-Bergstrom et al, Penicillamine kinetics in normal subjects, Clin. Pharmacol. Ther. 30: 404-413 (1981)

J.G. Wagner et al, Prediction of steady-state verpamil plasma concentrations in children and adults, Clin. Pharmacol. Ther. 32: 172-181 (1982).

G.F. Lockwood et al, Pharmacokinetics of ibupro­fen in man. I. Free and total area/dose re­lationships, Clin. Pharmacol. Ther. 34: 97-103 (1983).

G.J. Szpunar et al, Pharmacokinetics of flurb­iprofen in man. I. Area/dose relationships. Biopharm. Drug Dispos. accepted October 24, 1986.

ACKNOWLEDGEMENTS

Figures were reproduced with permission of the journal and publisher as folIows:

Figures 1, 28, 37-40; Clinical Pharmacology and Therapeutics, The C.V. Moseby Company.

Figure 2: Journal 0= Clinical Pharmacology, LeJacq Publishing Inc.

Figures 3-5: American Journal of Medical Sciences, J.B. Lippincott Company.

Figures 6 & 7: International Journal of Clinical Pharmacology: Dustri-Verlag, Dr. Karl Feistle, Germany.

Figures 18, 19, 23, 24, 29, 30: Research Communications in Chemical Pathology and Pharmacology, PJD Publications Ltd.

Figure 32: Journal of Pharmokinetics and Biopharmaceutics, Plenum PUblishing Corporation.

Figures 33-36: European Journal of Clinical Pharmacology, Springer-Verlag.

322

METABOLIC MODELS IN RADIATION PROTECTION

Francesco Breuer

E.N.E.A. - D.I.S.P.

Rome, Italy

NOTES ON INTERNAL IRRADIATION DOSIMETRY

Internal irradiation happens when radioactive substances introduced into the organism, distribute themselves into the tissues, following the

normal processes of physiological metabolism. A radioactive substance, during its residence in the body,

irradiates more or less the different organs and tissues, depending on the amount of incorporated activity, its distribution, its mean residence time in the body, and other characteristics connected with the type and energy

of produced radiations, and with other parameters. The biological effects of a given irradiation is connected with the

imparted radiation dose, both for external or internal irradiation. In both cases we consider primarily the absorbed dose which is a physical parameter corresponding to the mean energy imparted by ionizing radiation

to the mass unit of the considered tissue. The unit of this parameter is the "gray" (Gy) which is defined as J!kg(2).

Absorbed dose is not sufficient Different terms of

consequences of an irradiation. different biological effects in

to predict the biological kind of radiations can give severity or probability of

deleterious effects, as the result of the same absorbed dose, and in the same irradiation conditions.

In radiation protection procedures, a further quanti ty has been

introduced, that correlates better wi th the more important deleterious effects of an irradiation, particularly with the delayed stochastic effects. This quantity, called "dose equivalent" is the absorbed dose weighted by the quality factor Q and the product N of all other modifying

323

factors:

where D Q

H = D Q N

is the absorbed dose the quality factor (depending on linear energy transfer)

N the product of modifying factors. The dose equivalent unit is the "sievert" (Sv).

Internal radiation dose cannot be measured. It must be obtained by means of appropriate calculation methods on the basis of the amount of radioactive material taken in, or other measured quantities.

The amount of a radioactive material is defined by the "activity" which is the number of spontaneous nuclear transitions in a time unit.

The unit of activity is the "becquerel" (Bq) that corresponds to 1 nuclear transition in 1 second.

The aim of internal dosimetry is to calculate the radiation dose (absorbed dose or dose equivalent) resulting from a given intake (activity), and to evaluate the intake on the basis of measured quantities (concentrations of radionuclides in air or food, human body direct measurement, concentrations of radionuclides in excreta).

Internal contamination dosimetry has two fundamental aspects: a) a physical-geometric aspect (dosimetrie aspect) where it is considered

the relationship between the activity existing in a given organ (source organ) and the dose rate to the same organ or other organs (target organs).

b) a metabolie aspect where it is considered the behaviour in time of the distribution of the radioelements in the organism, ac count the different routes of intake.

taking into

This metabolie aspect of internal dosimetry is treated following the same methods utilized in pharmacokinetics.

Phisical dosimetry

The physical aspect of internal dosimetry aims at establishing a relationship between the activi ty in a "source organ" and the dose rate (absorbed dose rate or dose equivalent rate) in a target organ. Different analytical and numerical methods have been developed to obtain the results.

The most popular is Medical Internal Radiation Nuclear Medicine Society International Commission little variations.

today the numerical method proposed by the Dose committee (MIRD) described in American

publications (4), and adopted by the on Radiological Protection (ICRP) (6) with

Following the MIRD method (4), the average absorbed dose rate R in the region r 1 due to activi ty Q2 (kBq) uniformely distributed into the

324

region r 2 is given by:

(mGy/h)

where 6. is the equilibrium dose eonstant for the radiation i and i t 1.

is

A i

where n. is the mean number of radiations i per disintegration 1.

E. is the mean energy per unit of radiation i 1.

K is a dimensional eonstant = 0.577 (2.13 for old units: rad/(h pCi)

~ ~ (11 f-r . .) is the speeifie absorbed fr~ltion of radiation i in region r 1

for the emission in region r 2 .(g ) eonsidering

~~ is the absorbed fraetion in the volume v from a souree in region r

m is the mass (g) of v. v

In MIRD publieations (4) tables of various parameters are reported. Following the method adopted in ICRP publieation 30 (6), the

average dose equivalent rate HT in a target organ due to aetivi ty QS (KBq) present in a souree organ is given by:

where SEE T,S

H = 0 577· Q' SEE T • S T,S (mSv/h)

is the_1Speeifie Effeetive Energy" (MeV g per transformation) delivered to the target organ T by the radionuelide

distributed in the souree organ S.

This method is a development of the MIRD method. It has been applied to different radionuelides, and expressed in terms of

"dose equivalent". Tables of SEE are reported in annexes of ICRP publieation 30 for

different radionuelides, target organs and souree organs.

Metabolie aspeet of dosimetry

The relationship between the aetivity in souree organ and the dose

rate in target organs represent only the first step for internal

irradiation dosimetry. We have also to know the behaviour in time of the

325

distribution of the radionuclides in the source organs of the body, to obtain a relationship between an intake of a radionuclide into the body and the consequent radiation dose to different organs.

It is important also to consider the behaviour in time of the urinary and fecal excretion rates for the interpretation of bioassays.

A mathematical description of these phenomena must have a regular terminology.

We consider two phases of internal contamination: a) Systemic Phase: describes the diffusion into the body after

absorption into body fluids. b) Non systemic phase: describes the behaviour of radionuclides in the

first deposition organs before absorption or wi thout absorption into

body fluids. The different steps of the transport into the body are:

1) Intake: is the entry of a radioactive material into the body. It is the amount entering the nose or mouth in a given event.

2) Uptake: is the transfer into the systemic phase. It is the amount absorbed into extracellular fluids.

3) Deposi tion: is the transfer to an organ

material. The set of fractional depositions

or tissue of absorbed

describes the first body distribution.

4) Retention: describes the fraction of the amount taken in or taken up

into the body, or deposited into an organ, retained after a given elapsed time.

5) Elimination: describes the removal of the substance from an organ or tissue.

6) Excretion: is the elimination from the whole body. 7) Burden: is the amount of the radioactive substance in the whole body

(body burden) or in an organ (organ burden).

Mathematical models of the metabolism of different radionuclides are set up to describe the behaviour in time of the relevant source organ burdens and of the excretion.

Mathematical models adopted in radiation protection are sometimes compartment models as those utilized in nuclear medicine and pharmacology. These models can be utilized in internal irradiation dosimetry if the metabolism of the considered element is weIl known and it can be simplified in a small number of homogeneous compartments. This is the case of the iodine and of the tritium as tritiated water. For many other elements in different chemical forms, empirical functions are used, and the part of the model described by an empirical retention function may be considered an inhomogeneous compartment.

The used model can satisfy all requirements if it gives the retention function for the whole body and for the relevant source organs,

and if the excretion through the principal routes is described. On these bases we are able to calculate the dose rates to target

organs by summing up the contribution of all source organs to

irradiation.

Using the ICRP method we have at time t

Ht(t)= 0.577 • '"\; (t).SEE. ]mSV!(h.kBq) 4[S,i T,l

\..

326

where:

rSi(t) is the fraction of the amount of the radionuclide taken in, that remains in source organ i at time t

SEE is the specific effective energy delivered in the considered T. i

target organ T for the nuclide present in source organ i

We must in any case consider also the "total dose" delivered to the target organs after a given intake. The total dose should be calculated

by integrating the dose rate over the lifetime of the single individual.

In radiation protection practice i t is used to integrate over 50 years

the dose rate for adul ts, and over 70 years for children, and this

quantity is defined "committed dose". It is not necessary to integrate for every target organ the

received dose rate. It is sufficient to consider the value of the

integral over 50 years of the fractional activity for every source organ after a unit activity of intake. This value is expressed in terms of time

and represents the mean residence time of a radionuclide in a source

organ. In annexes of ICRP publication 30 va lues of NNT are reported for

the relevant source organs for different radionuclides and for single

intakes by different routes. These parameters are the integrals over 50 years, expressed in

the in take of 1 Bq of

"number of nuclear

seconds, of the retention in source organs, and for

radioactive material they correspond to the transformation" (NNT) in the source organ.

Commi tted doses to target organs from an intake I (kBq) can be given by:

H 50,T

-4 I . 1. 6 . 10 SEE J T,i

mSv

Tables of commi tted dose equi valents related to a unitintake of

many radionuclides are reported in annexes of ICRP publication 30. The commi tted dose equivalents to target organs are important to

evaluate the risk of non-stochastic effects of interna1 irradiation.

For the evaluation of the stochastic risk, i t is important to calculate, following the recommendations of the ICRP (3), the "effective

committed dose equivalent" (effective dose) as the sum of the committed

dose equivalents to the target organs mul tiplied by the corrisponding weighting factors wT

H 50,E

This quanti ty may represent the whole body dose which resul ts in

the same risk of producing stochastic effects as the considered internal

irradiation. Weighting factors suggested by ICRP are reported in table 1.

327

Table 1.

Weighting factors recommended by the ICRP for stochastic risks

Organs or tissue

Gonads Breast Red bone marrow Lung Thyroid Bone surfaces Remainder

Operational radiation protection

0.25 0.15 0.12 0.12 0.03 0.03 0.30

Internal radiation dose is not evaluated indi vidually in routine surveillance of occupationally exposed workers. The control is mainly adressed to the possible intake, and standard intake-dose relationship are utilized.

The fundamental types of control are two: a) contamination control of the working environment to prevent that

intake of radioactive substances may exceed an estabilished limit. b) individual contamination control to confirm that the limit of intake

has not be exceeded in single individuals. Individual control is effected by means of two fundamental

monitoring methods: 1) direct measurements of radioactivity in the body by mean of radiations

detectors. 2) radiometrie or chemical analysis of excreta (urine, feces, breath).

Metabolie models are requested also in these cases to interprete the measurements data. The models can be the same utilized already for dose evaluation, but it is useful sometimes to adopt here more sofisticated models if a good picture of the early metabolie phase is required after the intake.

METABOLIC MODELS IN INTERNAL CONTAMINATION PROBLEMS

Metabolie models are utilized in radiation protection practice to solve two fundamental problems in interna 1 irradiation dosimetry. 1) They must describe the behaviour in time of the distribution of the

radioactive substances in the source organs. 2) They must describe the relationship between an intake of a radioactive

substance, the retention into the body and the rate of excretion through the different routes.

The mathematical models that are utilized to describe the body distribution of radioactive materials can be obtained by suitable compartment models, or represented by empirie functions. Mathematical models utilized in radiation dosimetry are very simplified descriptions

328

of the eomplex biologieal real i ty, and they are ehosen to deseribe

sui tably partieular biologieal phenomens related to the objeet of the

study. The first models used in radiation proteetion dosimetry were simple

l-eompartment models deseribed by a single exponential funetion. In ICRP

publieation 2, biologieal half-lives were reported for the retention of several radionuelides into the body organs, and these models were

utilized to ealeulate the internal radiation dose.

The neeessity of interpreting the results of biologieal assays led

to adopt more refined models, and in ICRP publieation 10 (11), retention

and exeretion funetions were reported for the systemie metabolie phase of

31 radionuelides.

During the last 20 years, many metabolie models were reported also

in radiation proteetion literature, and ICRP publieation 30 (6) eolleets

the most qualified and helpful both to ealeulate internal radiation dose

and to deseribe exeretion rates for a great number of radionuelides. For the systemie phase, ICRP adopts in publieation 30 the model

reported in figure 1.

It ineludes a transfer eompartment with a biologieal half-life of 6

hours generally, from whieh the radioaetive substanee is deposited in one

or more organs every one of them being eharaeterized by its own retention

funetion, deseribed by one or more exponential terms. The sum of the rates of removal of the substanee from the organs and the direet

elimination from the transfer eompartment eosti tute the total systemie exeretion.

From GI tract and respiratory system5

Transfer c •• oarhent " a ....... ,

/ \ .......

....... ....... ,

.......

Tissu p Tissue T;ssue

r-..,:).... I

--, I Tissue

co.part.ent co.parhent co.parhent I co.parh I

ent I b c d i I

L-- r - __ I

I I I I

Excretion I Fig. 1. METABOLIC MODEL: SYSTEMIC PHASE

Mathematical model usually used to describe the kinetics of radionuclides in the body exceptions to this model are for individual elements (From ICRP publ.30).

329

K 1

K2

K3

K4

K 5

330

c

~

Xl X2 X3

Kl K2 K3 K4 K5

0.000594 d -1

0.0231 d -1

-1 0.000198 d

0.00154 d -1

0.0714 d- 1

für intake of loase tritium:

o

o

and then: 1 1 = I L

far intake of bound trltlum:

f 1 - 0.869

f 2 - 0.092

f 3 - 0.039

and then: 1 1

1 2

1 3

IR f 1

- I B 1 2

J B f 3

Fig. 2. TRITIUM METABOLIC MODEL

INORGANIC IODINE IN THE BODY THYROID ORGANIC IODINE IN THE BODY

In2/?25·.3 .832 d- 1 -1

In2/80 .00866 d_1 In2/12' .9 .0520 d_1 In2/.25·.7 1.941 d_ 1 In2/12' .1 .00578 d

- 11 -

12 tu

XI'K4 - URINARY EXCRETION ~EJ ~K5

RETENTION FUNCTIONS

Whole body Rb(t) _ .700·exp(-2.77·t)-.035·exp(-.06·t)+.335·exp(-.006·t)

Thyroid Rj(t) _ -.301·exp(-2.77·t)+.015·exp(-.06·t)+.286·exp(-.006·t)

Rest of the body Rj(t) - 1.001·exp(-2.77.t)-.050·exp(-.06·t) •. 049·exp(-.006·t)

URINARY EXCRETION FUNCTION

Yu(t) - 1.940·exp(-2.77·t)-.0018·exp(-.06·t)+.0017·exp(-.006·t)

Fig. 3. SYSTEMIC IODINE

Particular systemic models are adopted for hydrogen, iodine and alkaline-earth elements. For hydrogen (tritium) a single exponential term with a half-life of 10 days is used in publication 30. A 3-compartment model is preferred to describe both retention and excretion (figure 2).

This model derived from studies on loose tritium (water) and organically bound tritium (carbohydrates, proteins , fats) , describes ei ther the turnover of water in the body and exchanges of hydrogen between water and organic substances (18).

Compartment 1 represents hydrogen of water. Compartment 2 and 3 represent organic hydrogen. For tritium taken

up as water results the following retention function:

R(t)=0.97gexp(-0.073t)+O.018exp(-O.023t)+0.003exp(-0.O015t)

Iodine metabolism is described by the 3-compartment model reported in

figure 3. Compartment 1 represents inorganic iodine in the body compartment 2 represents the iodine in the thyroid and compartment 3 the organic iodine in the body.

The transfer and elimination constants derive from nuclear medicine observations and describe in a very simple way the complex iodine metabolism in human body. It is sufficient for dosimetry purpose, and retention and excretion functions can be obtained for some abnormal

situations too. For healthy adult

excretion functions (neglecting decay).

are man, parameters reported in the

and derived figure for

retention and stable iodine

Metabolism of alkaline-earth elements was described in ICRP

publication 10, 10a, and 20 (11,12,14) by empiric power functions where retention is:

n -n R(t)=z (t+z)

being z a constant value with dimension of time, and t the variable time, n is a dimensionless number between 0 and 1.

These functions were utilized sometimes combined with an exponential function, to adjust the first part of the curve, or mul tiplied by an exponential term to obtain asymptotic va lues of their time integrals. These retention functions and related excretion functions can be expressed as sums of 5-6 exponential terms too.

The non systemic phase of internal contamination includes the model of the gastro-intestinal tract and the model of the respiratory system.

In ICRP publication 30 the gastro-intestinal tract is described by 4 compartments which represent the stornach, the small intestine, the upper large intestine and the lower large intestine. The model and its parameters are reported in figure 4.

The respiratory system (fig. 5) is described by a 10-compartment model, and i t includes a nasopharyngeal region (two compartments), a

tracheobronchial region ( two compartments), a pulmonary region (four compartments) and the pulmonary lymph nodes ( two compartments). Both

331

RegIon

N-P

T-B

Ingestion

Sectlon of GI tract Mass of walls

(g)

Stomach (ST) 150 Small Intestine (SI) 640 Upper Large Tntestlne (ULI) 210 Lower Large Intestine (LLI) 160

Mass of contents

(g)

250 400 220 135

Mean residence tIme

(day)

1/24 4/24

13/24 24/24

Fig. 4. NODEL OF GASTRO-INTESTINAL TRACT (From ICRP Pub. 30).

Class

Compartment D W Y

F T T T

F F day day day

a o 01 0.5 o 01 0.1 0.01 0.01 b o 01 0 5 o 40 o 9 o 40 o 99

0.01 0.95 0.01 0.5 0.01 0.01 d 0.2 0.05 0.2 0.5 0.2 0.99

o N·.

_._---- -------------

P

L

Fig. 5.

332

e 0.5 0.8 50 0.15 500 0.05 0.4

~: 0.4 f n.a. n.a. 1.0 0.4 1.0 g n.a. n.a. 50 0.4 500 h 0.5 0.2 50 0.05 500 0.15

0.5 1 0 50 1 0 1000 o 9 n a n a n a n a a:: 0 1

NODEL OF RESPIRATORY SYSTEM The valuer for the removal half-times, T and Compartmental fraction F are given in the tabular portion of the figure for each of the three classes of retained materials. (From ICRP Pub. 30).

models were developed for dosimetry purpose considering these body regions as source organs, but they can be used sui tably in models developed to describe urinary and fecal excretion too.

Other models can be used for particular problems. Figure 6 and 7 reports the models utilized to calculate the radiation dose to the thyroid due to iodine produced by decay of ingested or inhaled radioactive telluriums.

These were developed several years ago and in ICRP publication 30 the commi tted dose to the thyroid and effecti ve doses deri ving from intakes od radioactive telluriums are calculated using corresponding

models. These are the principal models used in internal radiation

dosimetry. One may be in doubt about the unconditioned applicability of linear compartment models to metablic kinetics. We must recognize that the transfer of substances in the body depends in many cases on the quanti ty of the substance i tself that is present in some organs. The transfer of iodine into the thyroid depends, for instance, on the amount of iodine in the thyroid. But in the case of radioactive iodine, and for many other radioactive substances, the mass of the radioelement involved in the considered phenomena is so small that its influence is absolutely negligible.

Only in some cases where the specific activity of the radioelement is low and no stable isotopes exist, the rates of transfer could depend on the mass. But also in these cases (uranium, thorium) no appreciable mass dependence has been observed, and linear compartment models meet weIl the requirements of internal irradiation dosimetry.

~

I I I I I 1 1 1 1 I I 1

1 I 1 1 1 1 I I L

Te

----t--------------,

S - S H--- BL Te

1 .-----, -W • I 1

LTcJ r- - -,

SI r---- SI -r- 1 1 L _____ J

~ ! HThyr

1 I.-1

~ 1_ BL

LI -- LI +-1 I I BL 1 1 I--1 1 I orm 1 1 ! ----f --------__ + ___ J

Fig. 6. INGESTION OF TELLURIUM

333

REFERENCES

GI

.--.., r-' I I I I

: :: I I M ~ I I' I

Te - t I

N-PI I I

Te T-I I

LrJ L[J I

t T- B : I r :r 1 l-----t Te r-: W ~ I : : l : I

LI J L[J ... , r 1 I I I I I I I I

: ~ ~ : :: : l------t L.._J l.._J

I I

P -+--: I

Te I I I

I

~

~

BL

I

BL

Te

--

---

LYMPH l-

Fig. 7. INHALATION OF TELLURIUM

1) F.H. ATTIX, W.C. ROESCH, E. TOCHILIN: Radiation

Academic Press, New York, 1968.

Dosimetry,

2) Radiation Quantities and Units. ICRU Report 33, International Commission on Radiation Units and Measurements, Washington DC, 1980.

3) Recommendations of the International Commission on Radiological

Protection. ICRP publ. 26, Annals of the ICRP, Vol.1, N.4, 1977.

4) MIRD pamphlets 1-9: Journal of Nuclear Medicine,

New York, 1968-1972.

Supplements 1-6,

MIRD pamphlets 10, 11, Society of Nuclear Medicine, New York, 1975.

5) Report of the Task Group on Reference Man. ICRP Publ. 23, Pergamon

Press, Oxford, 1975.

6) Limits of Intakes of Radionuclides by Workers.

ICRP Publ. 30 -Part 1. Annals of the ICRP, Vol. 2, N.3/4, 1979.

334

Supplement part 1. Ann. , Vol. 3, 1979. -Part 2. Annals of the ICRP, Vol. 4, N.3/4, 1980.

Supplement part 2. Ann., Vol. 5, 1980. -Part 3. Annals of the ICRP, Vol. 6, N.2/3, 1982.

Supplement A part.3. Ann. , Vol. 7, 1982. Supplement B part 3. Ann. , Vol. 8, 1982.

7) G.J. HINE, G.L. BROWNELL: Radiation Dosimetry, Academic Press, New York, 1956.

8) P. BASTAI, L. ANTOGNETTI, G.C. DOGLIOTTI, G. MONASTERIO: Diagnostica e terapia con i radioisotopi, Ed. Minerva Medica, Torino, 1962.

9) A Review of the Radiosensitivity of the Tissues in Bone. ICRP Publ. 11, Pergamon Press, New York, 1968.

10) A. RESCIGNO, G.SEGRE: La cinetica dei farmaci e dei traccianti radioattivi, Ed. Boringhieri, Torino, 1961.

11) Report of Commi ttee IV on Evaluation of Radiation Doses to Body Tissues from Internal Contamination Due to Occupational Exposure. ICRP Publ. 10, New York, 1968.

12) The Assesment of Internal Contamination Resulting from Recurrent or Prolonged Uptakes. ICRP Publ. 10 A, New York, 1971.

13) The Metabolism of Compounds of Plutonium and other Actinides. ICRP Publ. 19, Pergamon Press, Oxford, 1972.

14) Alkaline Earth Metabolism in Adult Man. ICRP Publ. 20, Pergamon Press, Oxford, 1973.

15) ICRP Task Group on Lung Dynamies: Deposition and Retention Models of the Human Respiratory Tract, Health Physics, 12, 173-207, 1966.

16) Inhalation Risks from Radioactive Contaminants. Technical Reports Series N. 142, IAEA, Vienna, 1973.

17) F. BREUER, E. STRAMBI: Considerazioni sulla sorveglianza radio­tossicologia periodica dei lavoratori esposti al rischio di contaminazione interna. Lavoro Umano, XIX, suppl. a N. 12, 53, 1967.

18) P. BELLONI, F. BREUER, G.F. CLEMENTE, S. DI PIETRO, G. INGRAO: Tritium Metabolism in the Human Body. Paper Presented: Seminar on Environmental Transfer to Man of Radionuclides Released from Nuclear Installations. IAEA, Bruxelles, 17-21 October 1983.

335

AUTHOR INDEX

Bass, L., 151, 245

Beck, J.S., 11

Boxenbaum, H., 191

Breuer, F., 323

D'Souza, R., 191

Gladtke, E.,

Matis, J.H., 113

Metzler, C.M., 215

Mordenti, J., 271

Pond, S.M., 245

Rescigno, A., 19, 61

Thakur, A.K., 19, 27, 227

Wagner, J.G., 129, 291

337

SUBJECT INDEX

Acetone in the blood, 20 Activity, 324 Adam's method, 40 Adinazolam mesylate, 144, 145 ALBERT, 144, 149 Algebraic derivative, 72 Allometric equation, 201 Allometric paradigm, 198, 205 Allometric scaling, 191 Allometry, 191 AMBRE, 287 American Nuclear Medicine Society, 324 AMES, 287 Amikacin, 272 Aminoglycosides, 272 Aminosalicyclic acid, 306 Amitriptyline, 273 Analytic statement, 14 ANDERSEN, 194-196, 210-213 ANDERSSON, 254, 267 ANDREWS, 152, 168 ANSCOMBE, 33, 35, 39, 59 Antipyrine, 206, 208, 245 ANTOGNETTI, 335 Arabinofuranosylcytosine, 196 ARCANGELO, 286 Area under the curve, 129, 215 ARIAS, 258, 260, 267 ARMITAGE, 230, 243 Arms of a graph, 84 ARTOM, 22, 24, 25, 109 Asymmetrie graph, 68 ATKINSON, 287 A.U.C., 215 AWAZU, 195, 196, 213, 245, 246, 267 Axial dispersion number, 133

BAER, 288 BAKER, 245, 246, 248, 249, 257, 260,

265 Baluchitherium, 199 BARD, 28, 33, 37, 54, 56-58 BARNARD, 199 BARNES, 199 BARNETT, 203 BARNHART, 245, 246, 267

BARRY, 254, 255, 267 BARTHELL, 202 BASS, 133, 135, 148, 153-173, 176-190,

245, 247, 258, 267 BASTAI, 335 BATEMAN, 2, 8 Bateman function, 2 Bayesian analysis, 185, 216 BECCARI, 2, 8 BECK, 12, 17, 18, 24, 26, 79, 98, 109,

111 BECQUEREL, 19, 25 Becquerel, 324 BELLMAN, 287 BELLONI, 335 BENET, 134, 148, 286-288 BENOWITZ, 195, 197, 210 Benzodiazepine, 144 BERGE, 84, 109 BERGNER, 17, 18, 25 BERK, 245, 246, 248, 267 BERMAN, 23, 25, 28, 58, 59, 109, 110 BERRY, 227, 243 BERTALANFFY, 3, 8 BIEHLER, 2, 8 Binary regression models, 233 Binding parameters, 193 Binomial distribution, 116 Bioavailability, 132, 135, 137, 146,

215, 280, 291 Biochemical parameters, 193 Bioequivalence, 215 Biological relativity, 205 BISCHOFF, 195, 211 BLACKWELDER, 224, 225 BLAU, 235, 244 BOISVERT, 91, 111 Bolus dose, 278 BONNER, 199, 201, 213 BOOK, 236, 244 Boolean algebra, 84 BORCHARDT, 287 BOURNE, 286 BOX, 28, 32, 53, 58, 59, 61, 66 BOXENBAUM, 205, 207, 208, 210, 211,

236, 244

339

BOYER. 248. 259. 260. 262. 263. 267. 268

BRACKEN. 133. 135. 148. 154-163. 168-173. 176. 178. 181. 187-190

BRADLEY. 245. 246. 248. 249. 257. 260. 265

BRANSON. 97. 109. 110 BRATER. 286 BRAUER. 183. 187. 188. 190. 262. 267 Brauer's anomaly. 182 BRESLOW. 227. 243 BREUER. 335 BROCKMEIER. 6. 9 Bromo-deoxyuridine. 142 BROWN. 219. 234. 244 BROWNELL. 23. 25. 68. 335 BRUCE. 194. 195. 210 BRYAN. 231. 243 BUNDER. 4 BURDEN. 154. 156-163. 187-189 Burden. 326 BURTON. 3. 8 BYRNE. 246. 267

CADWALLADER. 287 CALDER. 210 Capillary bed. 151

permeability. 161 CAPRA. 203 Carbamazepine. 272 Carcinogenic potential. 227 CARLBORG. 231. 235. 242. 243 CARNAP. 15. 18 CARSON. 152. 211 CASSIGNOL. 91. 110 Catenary graph. 86 CAVALIER. 154. 188. 189 CAYLEY. 88. 110 Central moments. 171 Cerebral uptake. 245 Chain binomial distribution. 117 Chapman-Kolmogorov equations. 118 Characteristic polynomial. 105 CHEN. 195. 211 CHIANG. 117. 127 CHIARANTINI. 177. 181. 182. 187. 190 CHINARD. 157. 188. 189 Chloramphenicol. 272 CHOW. 81. 110 CHRISTENSEN. 161. 162. 163. 189 CLARK. 120. 128. 288 Clearance. 3. 129. 271. 273. 278. 280 CLEMENTE. 335 CLEWELL. 195. 196. 210. 212 Clindamycin. 294 Clinical pharmacology. 7 Cmax. 215 COBELLI. 123. 127. 152. 171 COCHRAN. 139. 149 Cochran-Armitage method. 228 Committed dose. 327

340

Commutative ring. 63 Compartment. 23

definition oft 79 heterogeneity. 123

Compartmental analysis. 188. 247 Compartmental models. 19. 113. 151.

188. 329 Compartments with non-insantaneous

mixing. 124 Complete precursor. 88 Complex eigenvalues. 94. 106 Concentration. 280 Conditional flow probability. 120 Confirmation of a model. 13 Connected graph. 85 Connectivity matrix. 84 Constaints. 37 Constant infusion. 278 Continuous derivative of an operator.

69 Continuous infusion. 276 Convective spaghetti. 157. 166 Convergence factor. 39 Convolution algebra. 62

integral. 62 CORNFIELD. 231. 233. 243. 244 Cornfield's pharmacokinetic model.

235. 242 Cox's logistic models. 233 CREASEY, 288 Creatinine clearance, 280 CROCKROFT. 286 CRONE. 157. 161-163. 189 Cross-over studies, 291 CRUMP. 230. 242-244 CURRY. 287. 288 Curve fitting. 29. 36 Curve peeling. 30 Cyclamate. sodium. 297 Cycle. 85 Cyclic search. 31

DARVEY, 30, 59 Data splitting. 186 DAVIS. 216, 218, 225 Decomposable matrix, 82 DEDRICK, 192. 194-196. 211-213 Degeneracy. 54 DELETTI. 216. 222, 225 DEMPSTER. 216, 225 Dependency value, 32 Deposition. 326 Derivative. algebraic. 72

of an operator, 69 Descriptive models, 27 Desipramine, 273 Deterministic models, 113 Detoxifying function. 181 DETTLI. 4, 8 Developmental pharmacology, 6 Diagonalizable matrix. 83

DIAMOND, 254, 255, 267 Dichlorodiammineplatinum, 192 Dieldrin, 238 DIENSE, 227, 243 DIERS CAVINESS, 286, 287 Differential equations, 152 Differential operator, 66, 73, 74 Digital computers, 27 Digitoxin, 272 Digoxin, 272, 304 Dilation, 205 DILLER, 4 Dipenhydramine, 309 01 PIETRO, 335 Disconfirmation of a model, 13 Discontinuous functions, 68 Dispersion model, 133 Dissociation-limited uptake, 263 DISTEFANO, 123, 128 Distributed sinusoidal perfusion

model, 133 DITO, 288 DIXON, 153, 179, 189, 244, 245, 248,

257, 267 DOGLIOTTI, 335 DOLL, 230, 231, 243 DOMAGK, 3 DOMINGUEZ, 2, 8 DONNELLY, 255, 267 Dosage regimen, 271 Dose, 280

equivalent, 323, 325 Dosing interval, 280 DOSSEY, 203, 205 DOST, 1, 8, 9 DRAPER, 28, 33, 35, 58 DRUCKERY, 2, 9, 233, 344 Drug disposition parameters, 271 Dry body weight, 280 D'SOUZA, 194, 195, 211 DUNNETT, 224, 225

Effect kinetics, 2 Effective mean value, 154 Efficiency number, 133 EIDUSON, 208 Eigenvalues, 55, 82, 115 Eigenvectors, 56, 115 EINSTEIN, 203, 205 Elementary path, 85 Elephant, African Savanna bull, 199 Elimination, 326 Elimination half-life, 3 Elimination rate, 166, 271, 274, 280 ENDRENYI, 288 ENSMINGER, 139, 142, 148, 149 Erlang distribution, 124, 125 Essential nodes, 92 Ethanol, 182, 310 Ethosuximide, 272 Ethoxybenzamide, 195, 196

Ethylene dichloride, 194 EVANS, 286 Examination of the residuals, 33 Excretion, 326 Exponential distribution, 121 Exponential retention time, 119, 120 Exponential terms, 25

Facilitation hypotheses, 245, 247, 258, 265

Fallacy of averages, 153, 156, 168 FARRIS, 192, 212 Feeding function, 97 FERRAIOLO, 287 FERRY, 134, 136, 149 FETHE, 198 Fick kinetics, 21 Field, 65 FINKELSTEIN, 152, 211 FINN, 288 FIORI, 245, 246, 253, 267 First-pass drug, 129

effect, 129, 291 FISHER, 195, 212 Fitting of data, 187, 198 FLEISCHER, 245-247, 253-255, 258, 263,

265, 267 FLEMING, 246, 256, 267 FLETCHER, 28, 56, 59 Flow-change experiments, 187 Flow-weighted density function, 169 Flow-weighted mean of the concentra-

tion, 167 Flucytosine, 272 FLUEHLER, 216, 220, 225 Fluid equilibrium, 3 Fluorouracil, 139 Flurbiprofen, 319 Food safety council, 235, 237, 244 Force of transfer, 121 FORKER, 245-247, 253-255, 258, 263,

265-267 FORRESTER, 196, 201 FORSYTH, 195, 197, 210 Fourier transform, 170 Fractional flow rate, 113, 115 FRANCIS, 194, 195, 211 Fredholm integral equation, 170 Free fraction, 280, 282 FREEMAN, 145, 149 FRIDOVICH, 16, 18 FRIEDMAN, 161-163, 189 FROKJAER-JENSEN, 161-163, 189 F-test, 55 Functional correlates, 67, 73

Galactose, 160, 182 GALBRAITH, 194, 213 GALILEI, 199 GAMBERTOGLIO, 287 Gamma distribution, 124

341

Gamma model, 230, 231, 242 GARGAS, 195, 196, 210, 212 GARRETT, 4, 287 GART, 227, 243 Gate function, 69 GAULT, 286 Gauss-Markov theorem, 37 Gear-Tu method, 40 GEHLEN, 2, 9, 20, 26 GEHRING, 235, 244 General product model, 242 Generalized compartmental analysis,

125, 127 GENT, 224, 225 Gentamiein, 272 GERALD, 123, 128 GERBER, 303 GERSON, 288 GIBALDI, 113, 115, 128, 287, 288 GILLETTE, 133, 148, 236, 244, 246,

262, 267 GLADTKE, 4, 6, 9, 288 GODFREY, 113, 115, 128 GOLDSMITH, 205, 212 GOLLAN, 245, 246, 248, 259-263, 268 GORESKY, 157, 158, 188, 189 Go-subgraph, 85 GOULD, 196, 208-210, 213 GOYA, 204 GRAHAM, 134, 148 Graph, 84 Graphe, 84 Graphical evaluation, 31 Gray, 323 GREEN, 198 GREENBLATT, 287 GREGORY, 199 Grid search method, 30 GRIEVE, 216, 225 GROSS, 120, 128, 195, 201 GUDZINOWICZ, 287 GUERBER, 202 GURPIDE, 23, 25, 110 GYVES, 139, 149

HADAMARD, 82 Haeckel's biogenetic law, 208 HALDANE, 199 Half-life, 271, 275, 280 HALL, 216, 225 Hamiltonian cycle, 85 HANANO, 195, 196, 213 HANSEN, 177, 181, 182, 188, 190 HANSON, 183, 187, 188, 190 HARARY, 87, 110 HARDISON, 245, 246, 267 HARRISON, 246, 267 HARTLEY, 24, 25, 28, 58, 110, 117,

119, 128, 232, 243 Hartley Sielken model 232 HATTINBERG, 6, 9

342

HAUCK, 216, 218, 224, 225 HAYASHI, 245, 246, 267 HAYWARD, 203 Hazard rate, 116, 120, 123 HEARON, 82 HEIMANN, 6, 9 Hepatic elimination, 132, 160, 181

188 Hepatic sinusoids, 155 Hepatic uptake, 245 Heteroskedastic variances, 116 Heteroskedasticity, 117 HILL, 28, 58, 59 Hill kinetics, 152, 182 HILLMAN, 287 HINE, 335 HIRTZ, 220, 225, 287 HO, 196, 211 HOEL, 230, 243 HOLFORD, 142, 149 HOLLANDER, 222, 225 HOLLOWAY, 183, 188-190, 262, 267 HOLMER, 254, 267 HOLT, 170, 171, 190 Homogeneous regime of uptake, 176 Homogeneous-phase kinetics, 153 HORIE, 245, 246, 267 HOUSEHOLDER, 23, 26 HUANG, 145, 149, 215, 225 HUGHES, 127, 128 Human equivalent dose, 236 HUNTER, 28, 32, 53, 58, 59 HUXLEY, 200, 214 hypothesis testing, 31

Ibuprofen, 218 Ideal body weight, 280 Imipramine, 273 Indoxole, 292 Infusion rate, 132 INGRAO, 331, 335 Initial estimates, 29 Initial node, 84 Input into a compartment, 80 Instability, 57 Intake, 326 Integral equations, 97 Integral of an operator, 71 Integral operator, 66 Intensity coefficient, probability,

116 Intermittent bolus dosing, 276 Intermittent dosing, 278 Intermittent infusion, 276 Internal irradiation dosimetry, 323 International Commission on Radio-

logical Protection, 323-331, 334, 335

Intrinsic metabolie clearance, 137 Invariant system, 62 Iopanic acid, 245

Irreducible matrix, 103 Isotope techniques, 5

JACQUEZ, 23, 25, 110, 113, 115, 128 JAFFE, 30, 59 JEFFREYS, 185, 190 JOHANSEN, 155, 171, 176, 177, 179,

186, 188, 189 JOHNSON, 123, 128 JONES, 135, 148, 246, 267 Journal of Pharmacokinetics and

Biopharmaceutics, 81 Jump function, 68 JUSKO, 286

KALBFLEISCH, 117, 119, 128, 232, 233, 244

KANT, 203 KAPLAN, 208 KATCHER, 288 KEIDING, 133, 148, 153, 155, 171, 176,

177, 179, 181-183, 186-190 KlANG, 245, 246, 248, 267 KIMPEL, 203 KING, 192, 212 KIRCHHOFF, 90, 110 KITTREL, 28, 30, 58, 59 KLINE, 203 KNOBEN, 288 KNOL, 142, 148 KNOTT, 28, 30, 31, 40, 41, 46, 50, 59 KNUTSEN, 142, 148 KODA-KIMBLE, 288 KODELL, 119, 128 Kolmogorov equations, 118, 126 KOTZ, 123, 128 KRASOVSKII, 236, 244 KRENTER, 6, 9 KREWSKI, 232, 233, 244 KREWSKI, 234, 244 Kronecker delta, 104 KRUGER-THIEMER, 4, 9 KUGLER, 208-210, 214 KUHN, 15, 18 Kupffer cells, 183 KUPFMULLER, 2, 9

LADU, 288 LAGAKOS, 227, 243 Lagrange's remainder, 173 LAMBRECHT, 151, 189 LANDAW, 123, 128, 245, 265, 267 LANDES, 203 LANGLEY, 230, 243 LANNON, 286 LAPIDUS, 31, 59 Laplace transform, 72 LASSEN, 157, 189 LAUE, 91, 110 LAWLESS, 117, 119, 128 LEITCH, 246, 256, 267

Length of a path, 85 LEONG, 183, 187, 188, 190, 262, 267 LEVY, 4, 205, 212, 287 L'Hospital rule, 98 Lidocaine, 193, 197, 272 LIN, 195, 196, 213, 286 Lincomycin hydrochloride, 293, 297 Lineal graph, 86 Linear compartmental models, 113 Linear graph, 89, 90, 99 Linear subgraph, 85 Linear system, 62 LINEWEAVER-BURK plot, 179 LING, 145, 169 L ithi um, 272 LODISH, 16, 18 Logarithmic average of inflow and

outflow concentration, 132 Logistic model, 231, 242 Logit model, 231 Log-linear model, 232 LONGSTRETH, 195, 211 LUCAS, 28, 58 Lumpers, 196 LURIA, 196, 208, 209, 213 LUTZ, 194, 213 LUXON, 245-247, 253-255, 258, 263-267

MA, 247, 269, 263, 265, 267 Macro-parameters, 215 MAES, 287 Mammillary graph, 86 MANDALLAZ, 216, 218, 225 MANDEL, 288 MANTEL, 231, 243 MARCHISIO, 4, 9 MAREK, 233, 235, 244 MARIMONT, 86, 110 Markov process, 81 Marquardt-Levenberg modification, 40 MASON, 90, 91, 110 MASSEY, 145, 149 MASSOUD, 287 mathematical modeling, 61, 187 MATIS, 24, 25, 110, 117, 119-123, 126-

128 Matrices, 80 MATTHEWS, 86, 110 MAU, 216, 218, 225 Maximum concentration, 280 Maximum tolerable dose, 227 MCELROY, 183, 187-190, 262, 267 MCLEAN, 246, 267 MCMAHON, 199, 201, 213 Mean residence time, 123, 127 Mechanistic models, 229 Medical Internal Radiation Dose

Committee, 324 MEHATA, 125, 128 MELLETT, 194, 213 MELMON, 195, 197, 210

343

MENDEL, 154, 188, 189 Metabolie aspeets of dosimetry, 325 Metabolie models, 328 Methotrexate, 195, 272 Methylene bromide, 196 Methylene chloride, 196 METZLER, 121-123, 127, 128, 215, 216,

219, 220, 223, 225 MEZAKI, 28, 30, 58, 59 Miehaelis-Menten elimination kineties,

129, 134, 144, 146 equation, 280 kineties, 138, 152, 170, 176 path, 138 steady-state equation, 145

Miero-parameters, 115 MIELKE, 227, 243 MIHALY, 246, 267 MIKUSINSKI, 62, 110 MILLER, 209 Minimum eoneentration, 280 Minimum polynomial, 105, 107 MIZUMA, 245, 246, 267 MLAB manual, 28, 40 Model, 11, 12 . Model-building, 54 Model seleetion, 277 Modeling the data, 14 Moment generating function, 104 Moments, 102, 108

for stoehastic eompartments, 122 of the residence time, 121, 126

MONASTERIO, 335 MORDENTI, 286 MORESCO, 4, 9 MORGAN, 135, 148, 246, 267 MORRELL, 288 MORRIS, 205 MOSER, 216, 220, 225 MOTULSKY, 236, 244 MROOSZCZAC, 145, 149 MUIR, 103, 110 Multieompartment deterministic model,

113 stochastic model, 118, 120 model, 25

Multi-hit model, 230, 231, 242 Multi-stage model, 230, 241 Multivariate distribution, 117 MUNGALL, 288 MURDOCH, 234, 244

NADEAU, 157, 188, 189 NAEF, 208 NAKAJIMA, 195, 213 Napa, 272 NATALE, 144, 149 NELSON, 4 Neoteny, 192, 208 NEWTON, 203 Nicardipine, 145

344

Nilpotent matrix, 83, 86 Nodes of a graph. 84 Non-exponential retention time

distribution. 123, 125 Non-homogeneous compartments, 123, 124 Non-instant mixing. 123, 125 Non-linear model, 28 NOONEY, 17, 18 No-oxervable-effect level, 227 Nortriptyline, 273 NOTARI, 286 NOY, 255, 267 Number of doses, 280 Number of visits, 123 Numerical instability, 56 Numerical operator, 65, 73. 74 NUNES, 245, 246, 248, 267

OCKNER, 245, 246, 248, 259-263, 268 Office of the teehnology assessment,

236, 244 OIE, 245, 246, 253, 267 Once-through perfusion, 187 One-compartment deterministic model,

113 stochastic model, 116, 119, 123

One-hit model, 229, 241 Operation T, 74 Operational calculus, 62 Operator, 64 Optimal design, 58 Order of aprecursor, 87, 100 Ordinary differential equations, 152 ORE, 84, 110 Oscillations, 94 OUSPENSKY, 102 Outliers. 39 Overparameterization, 55 Overstability, 57

Palmitate, 245 PANG, 135, 148, 246, 267 Parallel tube model, 132 Parameter estimation. 117 PARDRIDGE, 245, 265, 267 PARK, 242, 244 Partial differential equations, 152 Partial precursor, 88 Particle model, 116 Partition coefficients, 194 Path, 85 PCNONLIN program, 127 PEIPER, 3 Penicillamine, 317 Penicillin g, sodium, 296 Perfusion model, 132, 133, 135-137 PERL, 157, 188, 189 Permanence time. 104 Permeability-surface area product, 162 PERRIER, 113, 115, 128, 288 Perturbation theory, 177

PETERSON, 31, 59, 233, 235, 244 PETO, 230, 243 PETTITT, 185, 186, 190 Pharmacokinetic model selection, 277 Pharmacokinetic modeling, 229, 235,

271 Pharmacokinetic parameters, 275, 278 Pharmacokinetics, 1 Phenobarbital, 272 phenytoin, 272, 303, 308 physical relativity, 205 Physiological parameters, 193 Plasma protein concentration, 281 POFFENBARGER, 220, 225 POI~CARE', 204, 206 Poiseuille distribution, 155 Poisson distribution, 37 Polymorphic metabolism, 222 POND, 247, 267 PONTRYAGIN, 56, 59 Pool, 23, 24 Pooled Michaelis-Menten parameter-

concept, 145 POPPER, 258, 260, 267 Prazosin, 245 Precursor, 24, 61, 88 Precursor order, 101 Precursor's principal term, 100, 102 Precursor-successor relationship, 99 Prediction, 12 Predictive model, 27 Prednisolone, 313 PRENTICE, 233, 235, 244 PRIBOR, 288 PRIES, 183, 187, 188, 190 PRIISHOLM, 177, 181-183, 190 PRIMElPI, 4, 9

RAMSEY, 195, 213 Rate of drug administration, 281 Rate of metabolism, 132 Rational operators, 78 Rationality of models, 187 Reachability matrix, 87 Reduced body volume, 20 Reducible matrix, 82, 103 Reductionist paradigm, 191, 192 REISER, 205 REITZ, 195, 196, 210 RESCIGNO, 12, 17, 18, 23, 24, 26, 79,

81, 84, 87, 89, 90, 94, 98, 100, 104, 109, 111, 115, 117, 128, 151, 189, 335

Residence time, 120, 123, 127 Residuals, 31 Retention, 326 Retention time, 123 Retrodiction, 12 RICCI, 242, 244 RIDGE, 145, 149 RIEDER, 6, 9 RIEGELMAN, 4, 142, 149 RIND, 6, 9 Ring, commutative, 63 Risk assessment, 234 RITSCHEL, 4, 288 RITTENBERG, 24, 26 RIZACK, 287 ROBERT, 91, 111 ROBERTS, 132, 133, 148, 183, 188, 190 ROBERTS, 287 ROBERTSON, 23, 25, 110 ROBICHAUD, 91, 111 ROBINSON, 133, 135, 148, 166, 168-171,

173, 176-178, 180, 181, 183, 185-190 Primidone, 272

Principal term of the precursor, 100, Robust model, 13 RODBARD, 30, 59 102

Probability intensity coefficient, Probability of rejection, 216 Probit model, 231, 242 Procainamide, 272 Process error, 117 Process uncertainty, 117 PROMERENE, 2, 8 Proportional flow rate, 113 Propranolol, 142, 145, 182, 272 PRYS, 287 Pseudo-compartments, 125 Pseudo-saturation, 247, 258 Psychological relativity, 205 Pulsatile blood flow, 163 PURDUE, 24, 26, 111

Quinidine, 272

Radiation dose, 323 Radioactive decay, 19 RAI, 230, 243

116 RODDA, 216, 218, 225 ROGGE, 144, 149 RONFELD, 205, 211 Root mean square error, 32 Roots of a graph, 86 ROSTAMI, 223 ROWLAND, 81, 111, 132-134, 148, 193,

195, 197, 210, 246, 256, 267, 276, 286-288

Reseau oriente, 84 Run of residuals, 33 Runs test, 33 RUSSELL, 196 RUTHERFORD, 19, 26

SADEE, 286 SACHER, 208 SADEE, 286 SAINTE-LAGUE, 84, Salicylates, 272 Salt form, 281

111

345

Sampling time, 276 SARZANA, 22, 24, 25, 109 SATO, 195, 213 Saturation kinetics, 6, 175 Scaling from rodents to human, 235 SCHACHTER, 258, 260, 267 SCHEFFE', 37, 59 SCHENTAG, 286 SCHERR, 288 SCHMDT-NIELSEN, 199, 200 SCHOENFIELD, 28, 58 SCHUIRMANN, 218 SCHWARZ, 16, 18, 260, 267 SCOTT, 220, 225 SEDMAN, 138, 148 SEGRE Giorgio, 23, 24, 26, 84, 87, 89,

94, 100, 111, 115, 128, 335 SEGRE'Emilio, 22, 24, 25, 109 SELVAM, 125, 128 SELWYN, 216, 225 Semi-Markov model, 125 Sequoia, giant, 223 SEREN I , 4, 9 Serum creatinine, 281 SEYDEL, 4 SHADER, 287 SHAFER, 24, 26, 81, 111 SHAFRITZ, 258, 260, 267 SHAHN, 28, 58 SHARGEL, 287 SHEINER, 286, 287 SHEMIN, 24, 26 SHEPPARD, 23, 26, 111 SHER, 4, 9 SHICKEL, 204 SHRAGER, 28, 56, 59, 194, 213 Shrew, pygmy white-toothed, 199 SHURMANTINE, 245-247, 253-255, 258,

263, 265, 267 SIELKEN, 232, 243 Sievert, 324 Signal flow graph, 90 SILBER, 142, 149 SILVERS, 242, 244 Simple path, 85 Simplicity of models, 187 Simulation, 30 Simulator, 12 Simultaneous estimation, 58 SINGER, 196, 208, 209, 213 Single bolus dose, 278 Single-capillary model, 151, 155, 159 Single-pass experiment, 182 Sinusoidal perfusion model, 132, 133,

135-137 Sinusoidal transit time, 166 Slow release, 7 SMALLWOOD, 135, 148, 246, 267 SMITH, 28, 33, 35, 58, 195, 196, 210,

246, 256, 259, 260, 262, 267, 268, 288

346

SNEE, 242, 244 SNELL, 267 SODDY, 19, 26 Sojourn time, 119 Specific effective energy, 325 Spline approximation, 171 Spl itters, 196 STAHL, 200 STAPLES, 183, 187, 188, 190 Statistical moments, 122 steady state, 3, 281 STEIMER, 287 STEINESS, 177, 181-183, 187, 190 STEINIJANS, 216, 222, 225 STEPHENSON, 97, 111 STETSON, 139, 142, 148, 149 Stiffness, 56 Stochastic compartment, 24 Stochastic kinetics, 21 Stochastic models, 113, 128 STRAMBI, 335 STREMMEL, 245, 267 STROHMEYER, 245, 267 Strong component, 85, 88 Strong graph, 85 Strongly connected graph, 85 Subgraph, 85 Substrate-inhibition kinetics, 152,

175 Successor, 24, 61 SUGIYAMA, 195, 196, 213 Sum of exponential terms, 25, 115 Sum squares surface of a linear model,

28 Superposition, theorem of, 62 SUPPE, 208 Survival probability, 116 Survival times, 123 Survivorship function, 120 SVENSON, 254, 267 SWIFT, 286 Symmetrie graph, 86 Syndesichrons, 206 Synthetic statement, 14 System kinetics, 2 SZPUNAR, 134, 136, 144, 149

TANDBERG, 20, 26 Target organ dose, 237 TATLOCK, 202 Taurocholate, 183 TAYLOR, 195, 212, 286-288 Taylor's theorem, 173 TEORELL, 2, 9, 20, 26, 191, 214 Terminal node, 84 Tetracycline hydrochloride, 295, 303 THAKUR, 24, 26, 28, 30, 33, 37, 45,

59, 81, 111, 227, 243 Theorem of superposition, 62 THER, 6, 9 Thermodynamic parameters, 193

Theta. 215 Thoephylline. 272 THOMAS. 227. 243 TIAO. 185. 190 Time. 281 Time-dependent processes. 156 Time-invariant systems. 56 Time-to-tumor models. 229. 232 TIMMER. 223. 225 TITCHMARSH. 64. 111 Tmax. 215 Tobramycin. 272 TOFFOLO. 123. 127 Tolerance distribution models. 229.

231 Tolmetin. 315 TONNESEN. 171. 176. 177. 179. 186. 189 Total amount of drug eliminated. 215 Total body clearance. 273 Total body weight. 281 Total dose. 327 Total outflow rate. 115 Total precursor. 88 Total residence time. 120 Toxicity-oncogenicity studies. 227 TOZER. 276. 286. 288 Tracer methods. 5 Transfer between compartments. 84

function. 80. 94. 98. 101. 107 rate. 81 time. 104

Transformation of the variable. 38 Transient analysis. 166 Transit time. 119. 158 Transition from precursor to succes-

sor. 61 Translation operator. 68. 70. 73. 75 Tree. 86 Treshold behavior. 235 Trichloroacetic acid. 195 Tricyclic antidepressants. 273 TRIVERS. 202 TTERLIKKIS. 288 TUCKER. 81. 111. 286 TUKEY. 33. 35. 59 TYGSTRUP. 133. 148. 153. 155. 160.

171. 176. 177. 179. 186. 188. 189

Unbound clearance of a ligand. 12 Uncertainty in the parameters. 31 Understanding. 12 Undistributed model. 155. 159. 175.

177.181.182.187 Undistributed sinusoidal perfusion

model. 133. 135-137 Unique precursor. 88 Unisignant matrix. 103 Uptake. 326

anomaly. 183

Validity of a model. 31 Value of a path. 90 VAN ROSSUM. 287 VAN RYZIN. 230-233. 243. 244 Vancomycin. 273 Venous equilibration model. 132-134.

136. 137 Verapamil. 135. 143. 317 VILSTRUP. 177. 181. 182. 188. 190 Vm. 281 VOLLMER. 117. 119. 128 Volterra integral equation. 97 Volume of distribution. 271. 273. 276.

278. 281 VYBORNY. 264

WAGNER. 4. 115. 128. 134-136. 138. 139. 142-145. 148. 149. 288

WALKER. 30. 59. 288 WALKER-ANDREWS. 139. 149 WARTAK. 288 WATSON. 28. 30. 58. 59 WAY. 288 WEBB. 153. 179. 189. 248. 257. 267 WEBSTER. 246. 247 WEHRLY. 121-123. 126-128 Weibull distribution. 124

model. 231. 241 Weighting. 37 WEINER. 127. 128 WEISIGER. 154. 188. 189. 245-248. 259-

263. 265. 266. 268 WEISS. 28. 58. 59. 198 WELLING. 287 WELLS. G.P .•. 200. 214 WELLS. H.G .• 200. 214 Well-stirred compartment. 188 Well-stirred model. 132 WESTLAKE. 215. 216. 218. 225 Whale. blut 199 WHELPTON. 287 White noise. 38 WHITEHEAD. 208 WHITROW. 203 WIDMARK. 1. 2. 9. 20. 26 WIJNAND. 223. 225 WILLIAMS. 286 WINKLER. 133. 148. 153. 155. 160. 171.

176. 177. 179. 186. 188. 189 WINTER. 274. 276. 277. 286 WITT. 245. 246. 267 WOLFE. 222. 225 WOLLNER. 139. 149 World Health Organization. 287 WUt 145. 149 WYZGA. 242. 244

YAFFE. 4 YATES. 208. 209. 210. 214 YOUNG. 288 YU. 287

347

ZACKS, L4, L15, L18, L19, L24 ZAHARKO, i5, i21 ZAKIM, L11, L23 ZIERLER, e6, e8 ZORNIG, h35, h36, h40 ZUKAV, ;13

348