Journal of Pharmacokinetics and Biopharmaceutics, Vol. 23, No. 1, 1995

NOTE

A General Conceptual Model for Non-Steady State Pharmacokinetic/Pharmacodynamic Data

Davide Verotta I'z'4 and Lewis B. Sheiner 1'3

Received May 19, 1994~Final June 7, 1995

INTRODUCTION

In a recent article by Dayneka et al. (1), hereafter denoted DGJ, instances of a model (IPD) based on indirect pharmacodynamic responses are proposed and compared with a pharmacodynamic model (DPD) based on the distribution of drug to a hypothetical effect compartment, first described by Segre (2).

The purpose of this note, stimulated by the incisive synthesis presented by DGJ, is to propose generalizations both of the overall structure of DGJ's IPD, and of the structure of its submodels. We arrive at a general (but not fully general) model for P K / P D that includes IPD, DPD and some other types of PD models. We also comment on some additional methodological issues raised by DGJ's work, and our generalization of it.

THE GENERAL MODEL

We suppose (as do DGJ) the existence of an endogenous substance (R) (perhaps a receptor, or an active protein) responsible for a pharmacological

This work was supported by National Institutes of Health, Education and Welfare Grants GM26691, ACTG UO1 AI 27663. ~Department of Pharmacy and Pharmaceutical Chemistry, Box 0446, University of California San Francisco, San Francisco, California 94143.

:Department of Epidemiology and Biostatistics, University of California San Francisco, San Francisco, California.

3Department of Laboratory Medicine, School of Medicine, University of California San Francisco, San Francisco, California.

4To whom correspondence should be addressed.

1 0090-466X/95/0200-0001507.50/0 �9 1995 Plenum Publishing Corporation

2 Verotta and Sheiner

effect according to

E=f (R) (1)

The function,f (R), is, in general, a positive memory-less nonlinear function. When drug is given to the system it distributes to/from an observable

site in the system following an arbitrary pharmacokinetic trajectory indicated here by C (a function of time). Drug concentration in the biophase, Ce, is given by a linear dynamic link model

Ce=C , L (2)

where " , " indicates the convolution operator and L is a nonnegative, integrable but otherwise arbitrary function (e.g., ref. 3). Most often, L has been taken to be k~o exp(-keot), whence the link model expressed in differen- tial form becomes

dCe -koo(C- Ce) (3)

dt

If L is the delta Dirac function (see ref. 4, k~o---, oo) then Ce is identical ( = ) to C.

Drug concentration in the effect compartment acts only to convey infor- mation to R, as expressed by the following differential equation:

dR - king~,(Ce) - koutgout(Ce)R (4)

dt

where the functions gi.(Ce) and gout(Ce) represent the effect of drug on the rate of formation and loss of R, respectively. These functions have the follow- ing common characteristics:

Unity for Ce (less than or) equal to zero; g~(Ce)= 1, Ce<O

Positive for all Ce; g.,(Ce)>0

Monotonic for all Ce ; either gx( Ce + e) > gx( Ce), or gx( Ce + e) < gx( Ce), e>O

where gx indicates either gi, or gout. The first condition defines ki, and kou t

as the rates of formation and (first order) loss of R in the absence of drug. The second condition guarantees positive, bounded R. The third condition reflects usual pharmacology, and aids identifiability. The exponential func- tion, exp(. ), for example, satisfies all three of these conditions.

Equations (1)-(4) generalize the IPD of DGJ by preceeding it with a linear "effect compartment," and succeeding it with a final static nonlinear transformation of R (both of these obvious generalizations are implicit in

Model for Non-Steady State PK/PD Data

Table I. Instances of the General P K / P D Model"

Mode l kl. kout gi. gou, f b

D P D <<kco keo 1 + (kr 1 e,g,, Em~x IPD l k;. kout 1 - Ce/(Cso + Ce) 1 I IPD II kl. kou, 1 1 - Ce/(C5o + Ce) I IPD III kin kou, 1 + E,.~xCe/(C5o + Ce) 1 I IPD IV kl. kou, ! I + Em~Ce/(C5o + Ce) I

"In all cases here, L is the Dirac delta function, hence Ce = - C. bl is the identity function, l ( x ) = x .

DGJ), but, more importantly, also generalize the form of the effect of Ce on R's kinetics. Table I shows how the DPD model and DGJ's IPD models are all instances of the general model of Eq. (4), with L equal to the Dirac delta function. If we drop the first constraint that gi,(0) = 1, then the DPD model more simply results from ki, = kout = k,o, gin(Ce) = Ce, and gout(Ce) = 1, thus demonstrating that the DPD is a special case of DGJ's IPD III [with the addition of Eq. (1)]. The DPD can also result when L takes its usual definition and the kinetics of R are "fast"; i.e., kout--oo. In this case, the DPD results when, for example, ki, = 1, gi,=koutCe+ 1, and gout = I, thus indicating that a priori identifiability can be a problem for models composed of sequences of submodels when measurements of intermediate entities are lacking.

Equation (1) can be further generalized by considering alternative forms for Eq. (4). For example, a model for R, involving two species, an active one, R*, and an inactive one, R, can provide a physiologically meaningful interpretation for (memoryless or otherwise) nonlinearities. For example

dR - - - - k o u t R - kout g o u t ( C e ) R dt

dR* (5) - - - - kout g o u t ( C e ) R - kout R

dt

which, for Eq. (1) changed to E=f(R*) , exhibits saturation in the equilib- rium relationship between Ce and R*, even if f = R*. [Indeed, when this is so and gout(Ce)= Ce, Eq. (5) yields the classic "Emax" model at equilib- rium.] One might also consider models like Eq. (5), but where gi,(Ce) acts on kout.

Another example, due to Ekblad and Licko (5) has

dR - - = kin - kout gout ( C e ) R dt

dR* (6) -- koutgout( C e ) R - k o u t R

dt

4 Verotta and Sheiner

This model exhibits saturation, as does Eq. (5) (again with f = R*), but also exhibits (complete) tolerance and rebound. Addition of a loss to the outside for R in the above model allows tolerance to be incomplete.

We remark that chains of linear and/or nonlinear compartments and memoryless function have been used frequently in the physiological modeling literature, e.g. ref. 6. There it has been recognized for a long time that one must expose the system to a wide range of inputs to be able to discriminate between different chains of compartments and memoryless functions. In general it is sufficient to use one input to identify a linear system, but more than one to check for linearity (over the input range). Multiple inputs (which can be deterministic, pseudorandom, or random) are necessary to identify strongly nonlinear systems. DGJ give an instance of this by showing how a DPD can fit single input data generated by an IPD model but is inadequate to fit multiple input data. Of course, one can turn the example around and show how an IPD can fit DPD generated data for single input but not for multiple inputs.

In general, some unanswered questions about all the models discussed here are the following: (i) What is the data "signature" of the various different models? That is, how might one recognize which model to try given a set of data and lacking a strong (theoretical) prior preference for one model over another? (ii) What is the minimal (or optimal) input schedule needed to identify one of these models? This question concerns a posteriori identifiability and distinguishability, and it has no easy solution. (iii) Do Eqs. (1)-(4) represent a sufficiently simple, yet sufficiently general form to represent all important classes of dose-effect models for a univariate effect? We suspect, but there can be no proof, that it does.

REFERENCES

1. N. L. Dayneka, V. Garg, and W. J. Jusko. Comparison of four basic models of indirect pharmacodynamic responses. J. Pharmacokin. Biopharm. 21:457-478 (1993).

2. G. Segre. Kinetics of interaction between drugs and biological systems. I1 Farmaco 23:906-918 (1968).

3. D. Verotta and L. B. Sheiner. Semiparametric analysis of non-steady-state pharmaco- dynamic data. J. Pharmacokin. Biopharm. 19:691-712 (1991).

4. R. F. Hoskins. Generalized Functions, Wiley, New York, 1979. 5. E. B. M. Ekblad and V. Licko. A model eliciting transient responses. Am. J. PhysioL

246:R114-R121 (1984). 6. P. Z. Marmarelis and V. Z. Marmarelis. Analysis of Physiological Systems. The White-

Noise Approach, Plenum Press, New York, 1978.