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Title: Mathematical Modelling of exogenous glucose regulation in Type 1 diabetic

subjects after multiple meals

Authors:

W. H. O. Clausen

Department of Biostatistics, University of Copenhagen

Øster Farimagsgade 5 opg. B, Postboks 2099

1014 København K, Denmark

Andrea De Gaetano, Simona Panunzi

Cnr Iasi - Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti”

Viale Manzoni, 30 - 00185 Roma

W. H. O. Clausen (corresponding author)

Department of Biostatistics

Novo Nordisk A/S, Novo Alle 9F2.23

2880 Bagsvaerd, Demark

e-mail: wanc@novonordisk.com

Tel.: +45-4442 3095

Fax.: +45-4442 1065

Word count: 262 words (Abstract); 4266 words (main text).

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Abstract

Aims/hypothesis: In most perturbation studies on glucose-insulin control based on the

intravenous or oral glucose tolerance test, single-bolus glucose administration followed by

sampling of glucose and insulin plasma concentrations is used to characterise insulin

resistance. Analyses based either on descriptive measures, such as AUC and maximum effects,

or on compartment models have been applied to these experiments. In this paper, we studied

glucose kinetics in subjects with Type 1 diabetes mellitus (T1DM) after multiple meal intakes

followed by biphasic insulin injections. The main objective is to describe the glucose profiles

using an appropriate, robust compartmental model, circumventing the difficulty in precisely

characterising the glucose contribution of each meal.

Methods: Analyses were performed on 24-h serum glucose and insulin concentrations

measured in 20 Type 1 diabetic subjects. A total of ten glucose kinetics compartmental

models were studied, while in each case insulin concentrations were estimated from a

previously identified compartmental model. All models were fitted to the glucose

concentrations measured, and the goodness of the resulting fits was compared.

Results: The analysis shows that net insulin-independent glucose elimination is

negligible, and that neither glucose absorption nor insulin action appears to be uniform across

the three after-meal periods. In particular, greater insulin actions are observed during

breakfast compared with lunch and dinner.

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Conclusion/interpretation: There is substantial between meal variation within the same

subject in terms of glucose absorption and insulin action, partially due to variation in insulin

kinetics. A robust model, taking this variation into consideration, succeeds in describing the

glucose kinetics even when no information on the composition of the meals is available.

Keywords: Mathematical models; Glucose; Insulin; Glucose kinetics; Insulin action.

Abbreviations: T1DM: Type 1 diabetes mellitus; IVGTT: intravenous glucose

tolerance test; OGTT: oral glucose tolerance test; PK: pharmacokinetics; PD:

pharmacodynamics; Cal: calorie.

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Introduction

Type 1 diabetes mellitus (T1DM), also known as insulin dependent diabetes mellitus or

juvenile-onset diabetes, is a disease caused by deficiency of circulating insulin, due to the loss

of insulin-producing β cells in the Langerhans islets of the pancreas. To this day, the standard

treatment for T1DM is the subcutaneous administration of exogenous insulin to mimic the

normal metabolic regulatory system in healthy subjects. Insulin facilitates cellular glucose

uptake, stimulates conversion of glucose into glycogen and slows down gluconeogenesis [1],

besides its actions on lipolysis/lipogenesis and the promotion of protein synthesis and growth

[2, 3]. It appears that exogenously administered insulin has essentially the same effect as

endogenous insulin produced by the β cells [4].

In healthy subjects, the amount of insulin produced varies in time, responding mainly to the

prevailing blood glucose concentrations, so that amounts of the hormone, adequate for

instance to dispose of an alimentary glucose load, are produced through stimulation of β-cell

secretion. This does not happen in subjects with type 1 diabetes: in this case, the lack of

insulin secretion after food ingestion causes hyperglycaemia. To compensate for the lack of

endogenous insulin, T1DM patients are treated with the administration of exogenous insulin,

typically by means of subcutaneous injections of fast-acting insulin preparation at meal-time,

and subcutaneous injections of slow acting insulin preparations to cover the basal needs of the

hormone. Therapy is simplified by the simultaneous administration, at meal-times, of biphasic

insulin preparations, mixtures of rapid- and intermediate-acting insulin or insulin analogues.

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Randomised long-term cohort studies of diabetic patients such as DCCT [5] have shown that

hyperglycaemia is associated with development of severe late complications, and that

intensified insulin treatment can reduce or prolong the onset of these complications. However,

excessive administration of insulin may cause unpleasant and dangerous hypoglycaemia

episodes. Therefore, individualised optimal insulin dose regimens are necessary for treating

T1DM subjects. To this end, it is of interest to investigate whether repeated subcutaneous

injections of insulin present the same pharmacodynamics or whether such effect differs

systematically. It is also interesting to assess whether the intra-individual variation of the

effect, between different doses, is of little or of great importance with respect to inter-

individual variations, since the success of an accurately titrated therapy in the individual

patient depends on the possibility of adjusting dosage in the face of an unknown but self-

repeating dynamics.

Many published studies on insulin PD are based on simple descriptive characteristics

measures derived from blood glucose measurements (such as area above measured profile

AOC, the maximum effect, Cmax, and the time to Cmax, Tmax); while these measures do

provide generic information, they are not appropriate for a structural understanding of the

involved physiological mechanisms. Clamp studies do provide more specific information (i.e.

the amounts of infused glucose needed to maintain normal constant blood glucose after

injection of insulin [6]), but they are representative of a metabolic situation severely perturbed

away from normal physiology. Time courses of plasma glucose and insulin levels after a

reasonably physiologic perturbation like an intravenous glucose tolerance test (IVGTT) or an

oral/meal glucose tolerance test (OGTT/MGTT) provide rich information for a relatively low

cost both in terms of laboratory resources, materials and labour. These data are typically

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analysed by means of compartmental models, and the problem here is that of reaching a

robust equilibrium between minimising the hypotheses underlying the model and obtaining

detailed information from the model’s fit to the available data.

In the present work, we study the PD of the premixed insulin analogue insulin aspart

(NovoMix®30) in T1DM subjects after multiple meal-time s. c. bolus injections. We

introduce several models of the appearance of glucose after food intake and of the effects that

circulating insulin has on tissue glucose disposition after meal carbohydrate delivery, and we

select one of them on the basis of goodness of fit and robustness. The qualitative and

quantitative properties of the results are evaluated, and the relevance of model predictions to

the physiological understanding of after-meal glycemia regulation is discussed.

Materials and Methods

Experimental Procedure

The analysed data are fully described in a previously published PK/PD study of NovoMix30

insulin, with 20 complete 24-h serum glucose as well as serum insulin concentrations

measured in Type 1 diabetic subjects. Briefly, all patients received three daily s. c. injections

right before they had standardised meals for breakfast, lunch and dinner, at approximately

8.00h, 13:00h and 18:00h. The selection criteria and the anthropometric characteristics are

detailed in Chen et al. [7].

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Patients were allowed to choose among three categories of standardised diet according to the

amount of calories: low (500 Cal), medium (700 Cal) and high (800 Cal) calorie food. Three

meals of the same category were served to the patient at breakfast, lunch and dinner, with the

same content in fat, protein and calories for each meal. All standardised meals consist of 49-

55% carbohydrate, 10-20% protein and 25-37% fat. Leftover food was weighed and the

amount of caloric intake was measured. Subjects were not allowed to take any snacks but

were allowed to drink water between meals. In the case that a subject encountered symptoms

of hypoglycaemia, blood glucose would be checked at the bedside. If necessary, the subject

was treated according to local practice and blood glucose was monitored after 30 and 60

minutes. If judged safe by the investigator, the subject would then continue in the trial. In the

case that glucose was administered, the remaining serum glucose measurements for that

profile day were excluded from the analysis. All patients participating in the study were asked

to avoid strenuous exercise within the 24 hours prior to the visits.

In total, 44 blood samples were collected from each subject at 15-min intervals prior to the

first injection/meal and during the first 90 min after the injection/meal, followed by 30-min

intervals for the next hour, and hourly intervals thereafter. Serum glucose concentration was

analysed at Nova Medical Medi-Lab A/S using standard enzymatic method. A total of 873

serum glucose concentrations were measured. The serum glucose profiles for all subjects are

presented in Figure 1. A representative single-subject insulin profile is shown in Figure 2.

All serum insulin profiles are shown in Clausen et al. [8].

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Modelling

The pharmacokinetics of NovoMix30 had previously been studied in the same patients, and a

compartmental model faithfully reproducing serum insulin concentrations (in the absence of

glucose-driven insulin secretion, given the fact that these are T1DM patients) had been

identified [8]. In the present work, the corresponding serum glucose data are analysed to

describe the PD of NovoMix30, i.e. the action of insulin on the glucose kinetics. All PD

models considered are based on a description of the endogenous secretion, absorption,

distribution and elimination of the glucose, according to the following hypotheses: exogenous

glucose is absorbed from the alimentary tract after digestion of carbohydrates in the meal;

endogenous glucose is supplied by the liver, through glycogenolysis or gluconeogenesis;

hepatic glucose output may be a combination of zero order (constant throughout the

experiment) and second-order (insulin-dependent) processes; blood glucose is continuously

eliminated by tissues, through insulin-dependent, second-order uptake (mainly by muscle),

possibly through insulin-independent, first-order uptake, and through essentially zero-order

(within the range of observed glucose plasma concentrations) insulin- and glucose-

independent uptake (mainly by the brain); no endogenous insulin in secreted in T1DM

subjects. The corresponding individual plasma insulin profiles, estimated from the seven-

compartment model (Model 2) of the insulin data [8] from the same trial, were used as input

function into the models.

The models are illustrated in Figure 3 and are specified by the following equations.

(1) A,K)t(tδMdt

dAjgajj --= A(0) 0=

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(2) j

ga jgh xg xgI

g

K AdG T K G K I G ,

dt V= + - - G(0)=Gb

where:

A [Kcal] is the quantity of absorbable food in the alimentary tract at time t;

t [min] is the time from delivery of the first insulin injection/meal;

Mj [Kcal] is the quantity of absorbable food eaten at meal j, j=1,2,3

(breakfast, lunch and dinner);

( )jδ t t- is the Dirac delta generalised function centered at time tj, which

makes it possible to represent a succession of impulsive additions

of calories to the alimentary tract with meals;

jgaK [c/min] is the (apparent first-order) glucose absorption rate constant from

meal j;

G [mM] is the plasma glucose concentration at time t;

Tgh [mM/min] is the net zero-order endogenous glucose production;

Vg [L/kgBW] is the apparent volume of distribution for glucose;

Kxg [min-1] is the insulin-independent elimination rate constant for glucose;

KxgIj [min-1/pM] is the insulin-dependent elimination rate constant of glucose in the

period of time after meal j;

I [pM] is the individually estimated plasma insulin concentration;

Gb [mM] is the basal glycemia.

It is to be noted that jgaK includes an unspecified conversion factor c from meal calories to

glucose millimoles; this conversion factor includes the direct conversion of meal

carbohydrates into glucose, the conversion of part of the alimentary proteins into glucose, and

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the sparing effect on glucose consumption derived from the use of alimentary lipids through

Randle’s cycle.

The differential equations (1) and (2) specify 10 possible different dynamic models,

depending on whether we consider Kga and/or KxgI to remain constant throughout the day,

i.e. gaK = 1gaK =

2gaK = 3gaK , xgIK =

1xgIK =2xgIK =

3xgIK , and/or whether we consider Kxg = 0

and/or Tgh= 0. The models considered differ therefore in the description of the four ways

through which glucose enters or leaves plasma:

1. Allowance of time-varying glucose absorption from food intake.

2. Allowance of time-varying insulin-dependent glucose elimination.

3. Inclusion of zero-order endogenous net glucose production.

4. Inclusion of first-order insulin-independent glucose elimination.

Model 1 is the basic model where both glucose absorption and insulin action are assumed to be

invariant during the three meals. Model 2 derives from Model 1 by assuming first-order insulin

independent glucose elimination to be zero (negligible).

The reason why Model 1 has been extended to Model 3 as proposed is that it is possible to

suppose that different meals might effectively provide blood glucose at different rates since the

food eaten might vary not only in terms of its proportions of carbohydrate, fat and protein, even

when the amount of calories are the same, but also that all food components may be present in

the meals in forms differing substantially in their glucose kinetics characteristics (as summarised,

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for instance, in the popular ‘Glucose Index’ of foodstuffs [9]). As an example, consider

carbohydrates in the form of orange juice (immediately available glucose) and whole-wheat

bread (starch slowly released from fiber, then converted to glucose). Additionally, the kinetics of

glucose absorption may differ, given other within-meal conditions (like the association of the

food with larger or smaller amounts of liquids) or non-meal conditions (like the level of physical

activity following the meal). Model 4 is correspondingly proposed to study the consequences of

assuming negligible insulin-independent first-order glucose elimination.

In Model 5, it is assumed that insulin-dependent glucose elimination rates could vary between

the three meals, due to variations in insulin resistance during the day. These could come about,

for instance, due to oscillations in the level of other hormones (e.g. cortisol [10]), due to varying

levels of physical activity [11]. Model 6 derives from Model 5 by assuming negligible insulin-

independent first-order glucose elimination.

Model 7 is a combination of Model 3 and Model 5 , where both variations in glucose absorption

and in insulin-dependent glucose elimination are allowed. Model 8 allows both variations as in

Model 7, but insulin-independent first-order glucose elimination is assumed to be negligible.

Model 9 derives from Model 7 by assuming net endogenous glucose production to be zero.

Finally, Model 10 derives from Model 7 by assuming both insulin-independent first-order

glucose elimination and net endogenous glucose production to be zero.

The descriptions of all the models considered are summarised in Table 1.

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Data analysis

All parameters were log-transformed to ensure positive parameter values. The results are

however expressed, for intelligibility, in the original scale. Even though log-transformation of

the parameters does in general increase the stability of iterative estimation when performing

nonlinear optimisation, it does not allow zero estimates. If such a restriction did not exist,

Model 7 (or a suitable reparametrization of it) could directly express all the above models.

All models were fitted to the measured serum glucose concentrations of each subject

individually, using nonlinear Ordinary Least Squares. The estimates for single subject fitting

from [8] were applied to get the estimated plasma insulin concentration as a known input

function. The solution for differential equations is obtained using the ordinary differential

equations solver, from the odesolve package [12], implemented in the programming language

R.

Starting values for the parameters were obtained in each case by a preliminary round of a few

(50) generations of an original genetic algorithm global optimiser routine written in R [13];

the best value found was used as a starting point for local optimisation, performed using

BFGS quasi-Newton [14] optimisation method implemented in optim procedure, using

original code written in R.

Individual parameter estimates were summarised as median, upper and lower quartile. interval.

Tests between models of individual-subject goodness-of-fit measures were performed with a

Wilcoxon sign-rank nonparametric procedure [15].

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Results

The median and inter-quartile of the parameter estimates, the negative log-likelihood (-loglik)

and the Akaike information criteria (AIC) [16] obtained from single-subject fitting for each

model are presented in Table 2. The results obtained for comparison of AIC between Model 7

and all the other models are also presented in Table 2.

In general, there is a rather large variation among the alimentary glucose absorption rates at

different meals when they are allowed to vary in the model, e.g. Model 3, 4, 7-10. The

magnitude of alimentary glucose absorption rates, regardless of other factors, appeared to be

5-10 times larger in the morning, allowing for greater plasma glucose concentrations, while

the estimated alimentary glucose absorption rates during lunch and dinner are of the same

order except in Models 8 and 9. It is also to be noticed that when both alimentary glucose

absorption rate and insulin-dependent glucose elimination rate are allowed to vary, the

estimates of alimentary glucose absorption rate and insulin-dependent elimination rate

become substantially smaller both during lunch and dinner. Some patients show near-zero

estimated alimentary glucose absorption rates, consistent with very modest elevations of their

glycemias after lunch, see Figure 4.

The apparent volume of distribution appears to be of the same order in all models, while there

is a large variation in the estimates for the insulin-independent glucose elimination rate.

The values obtained for the insulin-dependent glucose uptake rate reflect, in each model, the

choice made with respect to the glucose absorption rate from the Alimentary tract: if this last

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is allowed to vary, lower absorption is predicted at lunch, and lower insulin activity is also

estimated for the post-lunch period. If absorption is maintained constant throughout the meals,

a higher-than-normal insulin activity is needed at lunch to explain the observed lower

concentrations.

Figure 5 shows the estimated curves for subject 118 and 128 for Model 1, 3, 5, 7 and 10. It

can be seen that the estimated curves for Model 7 and 10 seem to8 actually overlap in both

cases.

Model Selection

The log-likelihood was compared between Model 7 and all the other models in the studied

subject sample, and the results of the comparison are reported in Table 2. As expected, Model

7 has the lowest negative log-likelihood, being the hierarchically most comprehensive model

tested. Greater flexibility is obviously possible by increasing the number of parameters in the

model. However, Model 7 does not perform significantly better than further appears to

provide significant improvement in the goodness of fit when compared to the other models,

except for Models 3, 4, 8, 9 or 10 when the number of free parameters in the model is also

taken into consideration by computing the AIC.

If we were to choose the model strictly according to the AIC and maximum parsimony,

Model 4, with the least number of parameters, would be chosen. However, its comparison

versus Model 7 has a p-value which is nearly significant at the 5% level, and the

simplification of structure comes at the expense of an unexpectedly (and improbably) high

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estimate of the distribution volume for glucose. For these reasons we do not feel confident in

choosing Model 4. Model 3, with essentially the same structure as Model 4 and one parameter

more, encountered from exactly the same problems. We are therefore left with the subgroup

of models (8, 9 and 10) where both glucose absorption rate and insulin action are left free to

vary throughout the day. They are largely equivalent in their AIC, and (with the possible

exception of Kga2) also in the values of the other parameters. This equivalence seems to

express the fact that if we disregard first-order glucose uptake, or net zero-order glucose

secretion, or both, we come to essentially the same results. Both for the sake of parsimony

and for the provocative conclusions which are then suggested, we would choose Model 10

over the other two, pending further experimental confirmation.

Discussion

One of the main challenges when modelling glucose concentrations after meals is the lack of

information on the composition of the food eaten. Even though in the present study each

patient was served meals with known (approximate) percentages of carbohydrate, fat and

protein, precise information was not really available. The information of the amount of

calories eaten by each subject is instead taken as a surrogate measure of the quantity of

glucose obtained from food intake. The rate of appearance of glucose in the bloodstream is

also strongly dependent on the type of carbohydrate eaten. This has been widely studied to

determine the ‘glycaemic index’ [9], ranking foodstuffs with respect to their immediate effect

on blood glucose levels. Among carbohydrate-rich foods alone, those that break down quickly

have the highest glycaemic indices (fast and high glycemia effect), while those that are slowly

digested release glucose gradually into the blood stream and have the lowest glycaemic

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indices [9]. This uncertainty in the food composition, as it reflects the rate of glucose delivery

to the bloodstream, as well as other influences on absorption may be accounted for by

represented allowing the Kga rate to freely vary between meals. It is encouraging to note that

these representations perform well and are robust to the absence of detailed information on

meal composition.

It is to be noted that serum glucose concentrations measured in both subjects shown in Figure

5 were rather high. This is mainly because the original study was designed in such a way that

dose titration was not performed prior to the day when insulin and glucose concentrations

were measured. Instead, the T1DM patients enrolled in the study were administered the same

total amount of insulin they had received on the previous day, with the ratio of 30:30:40%

during breakfast, lunch and dinner.

It is well known that the appearance of insulin inhibits hepatic glucose output. Without

glucose tracer methods, it is however not possible to isolate this effect since endogenous

glucose production cannot be separated from glucose absorption from alimentary canal. In

fact, insulin-dependent reductions of hepatic glucose output are also indistinguishable from

insulin-dependent increases of glucose tissue uptake. By the same token, relatively constant,

‘baseline’ hepatic glucose output is not distinguishable from essentially zero-order, constant

glucose consumption by the nervous system, brain and possibly the myocardium. The models

considered therefore include two terms, a zero-order net baseline liver glucose output Tgh (net

with respect to zero-order glucose consumption), and a second-order net insulin-dependent

tissue glucose uptake KxgI (net with respect to insulin-dependent hepatic glucose output

variations), which can be estimated from ‘cold’ concentration data. A tracer study would be

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needed to allow discrimination of the components which are pooled in the present study [17,

18].

Most of the modelling on glucose kinetics which has appeared in the literature so far includes

an insulin-independent, apparent first-order glucose elimination rate [19-22]. There definitely

exists a first-order process for renal elimination of glucose once the renal threshold for

glucose re-absorption is overcome [24]. Also, in the first minutes after a sudden increase in

plasma glucose concentration, distribution into interstitial space may account for an apparent

first-order kinetics. In the present situation, however, where glucose concentrations are

generally well below the renal threshold, and where absorption is gradual, the physiologic

justification of a Kxg term is doubtful, and in fact this is the one term which is not supported

by the goodness of fits comparisons. The chosen Model 10 therefore only contemplates zero-

and second-order glucose disposition terms.

It is somewhat more surprising that also the traditional zero-order net hepatic glucose output

may be disregarded without detrimental effects on model performance. This would indicate

first that, as is well known from tracer clamp studies [6], much of the hepatic glucose output

is actually insulin-dependent, and enters therefore in the second-order pooled term of the

present set of models. Secondly, these results would indicate that residual zero-order,

relatively insulin independent, glucose secretion by the liver essentially balances the zero-

order requirements of privileged tissues like the brain, and the difference between the two

zero order processes is too small to be picked up with total-body cold glucose assessment

techniques in real-life conditions.

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The modelling situation addressed in the present work is vastly simplified, with respect to the

normal volunteer or the Type 2 diabetic patient, by the fact that T1DM patients are essentially

devoid of endogenous insulin production, which means that in this case the available insulin

is entirely exogenous and insulin levels are completely independent of glucose levels. This

allows estimation of insulin kinetics to be performed prior to and independently of the

estimation of glucose kinetics. Estimates of the insulin kinetics previously obtained in the

same subjects [8] have therefore been used.

In that work, it had been reported that the mean AUC of the insulin concentration-time curves

is smaller in the morning, providing lower mean insulin concentrations. From models 5

through 10, where alimentary glucose absorption rates are allowed to vary during the different

meals, the estimates of insulin-dependent elimination rates are consistently larger after

breakfast. This result contradicts the general impression about lower insulin-sensitivity in the

morning [25, 26]. It would seem that, while tissue insulin sensitivity is even higher, insulin

absorption is reduced to such an extent that its dynamic action on glucose seems impaired.

This point merits further investigation, both as to its actual confirmation and as to the possible

causes underlying it.

One difference between the models proposed in the present work and other published models

is that an active (remote) insulin action compartment is not included in any of the models

considered here. The data used for most modelling work where the remote insulin

compartment is included come from IVGTT or clamp studies where insulin was released

rapidly or infused intravenously in a step-wise fashion, and where the remote insulin

compartment introduces a delay of insulin action. Such a delay is not expected to have any

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influence in this study where insulin is absorbed slowly from the s. c. depot, insulin

concentrations are slowly varying and a delay effect, even if appropriate, would be difficult to

estimate.

Relevant goals of the published studies of insulin PD are the determination of insulin

sensitivity in different classes of subjects and/or the characterisation of the time profiles of

insulin action after injection of different compounds [19-23]. Such studies are typically

performed analysing the results of a single intravenous or oral bolus of glucose. In this paper,

we aim to study the variations of plasma glucose concentrations after multiple normal meals

accompanied by standard sc insulin therapy. The basic question is whether there exists

significant variation of insulin dynamics across repeated administrations, and the results of

the present study indicate that this is indeed the case. The considerable variation observed in

the daily glucose kinetics is shown to be due partially due to variations in insulin dynamic

action, which are superimposed on demonstrable variations in insulin kinetics.

Although substantial variation in insulin action (insulin-dependent glucose elimination) and

variation in glucose absorption (possibly due to food composition) are observed, the

correlation between the two parameters is not so high as to indicate a simple direct

mathematical connection, possibly due to over-parameterization: the sample correlation

coefficient between the two effects is -0.14, while the asymptotic correlations of the single-

subject estimates range from -0.4 to 0.7. The across-meal variations for fast-acting insulin

absorption from the subcutaneous depot (Kpf from previously published insulin kinetics

model), glucose absorption from the alimentary tract (Kga from Model 10) and insulin action

(KxgI from Model 10) are shown in figure 6, where each bar represents the mean of the

20

estimates over the studied subjects while their sample standard errors are presented using T-

shaped segments.

Conclusions

Both the rate of absorption of glucose from food and the insulin action on tissue glucose

disposition vary significantly across the meals of a single day in T1DM patients. Glucose

absorption is faster in the morning, and is not adequately balanced by a corresponding

increase in tissue insulin sensitivity. Good glycaemic control does not merely depend on the

therapeutic regimen since treatment effects are different depending on the time of the day and

the composition of the meal. A mathematical model allowing for such variations can robustly

represent the time course of glycemia in these patients, and be the basis for further

investigations on the pharmacodynamics of injectable insulin preparations.

Acknowledgements

The authors are grateful to Jens Sandahl Christiansen and Jian-Wen Chen for carrying out the

clinical trial that allows us to perform this research. The authors are also grateful to Aage

Vølund and Susanne Ditlevsen for their advice and suggestions. This research is supported by

the Danish Ministry of Science, Technology and Innovation, and by Novo Nordisk A/S.

21

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meal tolerance tests in normal subjects: a minimal model index. 85: 4396-4402.

24. Best JD, Taborsky GJ, Halter JB, Porte D Jr (1981) Glucose disposal is not proportional to

plasma glucose level in man. Diabetes 30: 847-850.

25. Perriello G, De Feo P, Torlone E et al. (1991) The dawn phenomenon in Type 1 (insulin-

dependent) diabetes mellitus: Magnitude, frequency, variability, and dependency on

glucose counterregulation and insulin sensitivity. Diabetologia 34(1):21-28.

26. Holl RW, Heinze E (1992) Dawn or Somogyi phenomenon? High morning fasting blood

sugar levels in juvenile type 1 diabetics. Deutsche Medizinische Wochenschrift

117(40):1503-1507.

24

Figu

re 1

. Ser

um g

luco

se c

once

ntra

tion

in 2

0 T

ype

1 di

abet

ic s

ubje

cts.

050

010

0015

00

051015202530

Tim

e (m

in)

Serum concentration (mmol/l)

25

Figu

re 2

. An

exam

ple

of th

e m

easu

red

seru

m in

suli

n co

ncen

trat

ions

and

the

rela

tive

est

imat

ed c

urve

(gr

ay p

oint

s ar

e da

ta b

elow

the

low

er

limit

of q

uant

ific

atio

n at

13p

mol

/l).

020

040

060

080

010

0012

0014

00

050100150

Est

imat

ed c

urv

es fo

r S

ub

ject

104

Tim

e (m

in)

Serum concentration (pmol/l)

26

Figu

re 3

. Illu

stra

tion

of th

e pr

opos

ed c

ompa

rtm

enta

l mod

els.

A

lim

enta

ry

cana

l, A

Pl

asm

a gl

ucos

e, G

di

stri

bute

d in

vo

lum

e V

g

Kga

j

Kxg

Mj d

(t-t

j)

Kxg

I j

Plas

ma

insu

lin,

I

Tg

h

27

Figu

re 4

. Exa

mpl

e of

glu

cose

con

cent

ratio

n ex

curs

ion

for

Subj

ect 1

30 a

nd th

e re

lativ

e es

timat

ed c

urve

fro

m M

odel

10.

050

010

00

510152025G

luco

se C

on

cen

trat

ion

for

Su

bje

ct 1

30

Tim

e (m

in)

Serum concentration (mmol/l)

28

Figure 5. The serum glucose concentration and the estimated curves for subject 118 and 128 for

Model 1 (dashed-line), 4 (dotted-line), 7 (dotdash-line) and 10 (solid-line).

0 500 1000 1500

1520

25

Estimated curves for Subject 118

Time (min)

Ser

um c

once

ntra

tion

(mm

ol/l)

0 500 1000

510

1520

Estimated curves for Subject 128

Time (min)

Ser

um c

once

ntra

tion

(mm

ol/l)

29

Figu

re 6

. Bar

gra

phs

of m

ean

esti

mat

es w

ith th

e st

anda

rd e

rror

bar

s of

insu

lin

abso

rptio

n ra

te c

onst

ant (

dotte

d), g

luco

se a

bsor

ptio

n ra

te c

onst

ant

(par

alle

l lin

es)

and

insu

lin-

depe

nden

t glu

cose

elim

inat

ion

rate

(cr

osse

d li

nes)

for

Mod

el 1

0.

0

0.0

05

0.0

1

0.0

15

0.0

2

0.0

25

0.0

3

0.0

35

0.0

4

Me

al/I

nje

ctio

n 1

M

ea

l/In

ject

ion

2

Me

al/I

nje

ctio

n 3

Kp

f K

ga

Kxg

i

30

Tab

le 1

. Sum

mar

y de

scri

ptio

n of

the

ten

prop

osed

com

part

men

tal m

odel

s of

glu

cose

kin

etic

s.

Mod

el

Para

met

ers

Com

men

ts

1 ga

K, V

g, K

xg,

xgI

K, T

gh

Bas

ic M

odel

2 ga

K, V

g,

xgI

K, T

gh

Sam

e as

Mod

el 1

, exc

ept K

xg =

0

3 1

gaK

, 2

gaK

, 3

gaK

, Vg,

Kxg

, xg

IK

, Tgh

Sa

me

as M

odel

1, b

ut w

ith v

aryi

ngj

gaK

.

4 1

gaK

, 2

gaK

, 3

gaK

, Vg,

xg

IK

, Tgh

Sa

me

as M

odel

3, e

xcep

t Kxg

= 0

.

5 ga

K, V

g, K

xg,

1xg

IK

, 2

xgI

K,

3xg

IK

, Tgh

Sa

me

as in

Mod

el 1

, but

with

var

ying

jxg

IK

.

6 ga

K, V

g,

1xg

IK

, 2

xgI

K,

3xg

IK

, Tgh

Sa

me

as M

odel

5, e

xcep

t Kxg

= 0

.

7 1

gaK

, 2

gaK

, 3

gaK

, Vg,

Kxg

, 1

xgI

K,

2xg

IK

, 3

xgI

K, T

gh

Com

bina

tion

of M

odel

3 a

nd M

odel

5.

8 1

gaK

, 2

gaK

, 3

gaK

, Vg,

1

xgI

K,

2xg

IK

, 3

xgI

K, T

gh

As

Mod

el 7

, exc

ept K

xg =

0.

9 1

gaK

, 2

gaK

, 3

gaK

, Vg,

Kxg

, 1

xgI

K,

2xg

IK

, 3

xgI

K

As

Mod

el 7

, exc

ept T

gh =

0.

10

1ga

K,

2ga

K,

3ga

K, V

g,

1xg

IK

, 2

xgI

K,

3xg

IK

Sa

me

as M

odel

7, e

xcep

t Tgh

= 0

and

Kxg

= 0

.

31

Tab

le 2

. The

med

ian

of th

e es

timat

es f

or a

ll th

e m

odel

s fi

tted

are

pres

ente

d in

ori

gina

l sca

le (

in p

aren

thes

es th

e lo

wer

and

upp

er q

uart

ile).

Mod

el/

Est

imat

es

1 2

3 4

5 6

7 8

9 10

1ga

K (

c/m

in)

11.

9

(10.

3- 1

4)

3.8

(0

.2-9

.9)

9.0

(5

.4-1

4.1)

11

.5

(6.8

-16.

8)

11.1

(1

.5-1

2.8)

11

.8

(10.

2-13

.7)

8.8

(4

.5-1

1.4)

8

.2

(6.1

-11.

3)

6.2

(3

.6-8

.8)

6.4

(0.5

-8.5

)

2ga

K(c

/min

) -

- 1

.3

(0.5

-7.1

)

1.2

(0

.4-3

.9)

- -

0.6

(5

x10-5

-2.4

) 0

.1

(2x1

0-8-1

.0)

0.1

(7

x10-4

-8.8

) 0

.6

(3x1

0-5-1

.8)

3ga

K(c

/min

) -

- 1

.0

(0.5

-7.1

) 1

.9

(1.0

-7.1

) -

- 0

.9

(0.6

-4.5

) 0.

6 (0

.5-1

.9)

0.6

(0

.4-1

.7)

0.6

(0

.3-1

.1)

Vg (1

0-1)

(L/k

gBW

) 6

.8

(4.6

- 10

.3)

4.0

(2

.1-7

.4)

3.8

(2

.9-6

.0)

4.9

(3

.2-7

.2)

5.2

(3

.1-7

.7)

7.6

(4.1

-12.

2)

3.0

(2

.1-4

.8)

3.1

(2

.3-5

.2)

2.7

(0

.8-3

.5)

2.5

(1

.6-3

.6)

Kxg

(10-3

) (m

in-1

)

0.2

(1

.3 x

10-4

-2.1

) -

4.0

x10-4

(2

.7 x

10-6

-2.3

) -

3.8

x10

-2

(6.0

x10

-5-3

.4)

- 0

.2

(3.3

x10

-5-1

.6)

- 0

.2

(3.7

x10

-6-0

.9)

-

1xg

IK

(10-1

)

(min

-1/p

M)

5.1

(3

.2-

7.0)

4.

76

(2.8

-7.1

) 3.

8 (2

.8-5

.2)

4.6

(3.2

-5.1

) 3.

2 (0

.9-6

.2)

4.2

(2.7

-6.7

) 5.

2 (3

.2-6

.7)

5.4

(3.0

-7.8

) 5.

6 (3

.2-7

.2)

5.8

(3.5

-6.7

)

1xg

IK

(10-1

)

(min

-1/p

M)

- -

- -

5.2

(2.5

-8.6

) 5.

5 (3

.2-8

.5)

2.9

(1.2

-3.9

) 2.

7 (0

.9-3

.9)

1.2

(0.3

-2.7

) 2.

1 (0

.9-3

.3)

3xg

IK

(10-1

)

(min

-1/p

M)

- -

- -

4.7

(3.7

-7.5

) 4.

2 (3

.2-5

.4)

4.5

(2.0

-6.3

) 4.

0 (2

.1-5

.5)

3.7

(1.6

-6.4

) 2.

9 (1

.3-4

.6)

Tgh

(10-2

) (m

M/m

in)

4.5

(2.5

- 6.

3)

2.4

(0.7

-3.5

) 2.

2 (1

.6-3

.5)

1.7

(1.1

-2.1

) 2.

8 (1

.7-8

.1)

2.6

(1.9

-4.1

) 2.

0 (0

.01-

2.7)

0.

5 (5

x10

-4-1

.5)

- -

AIC

20

5 (1

83-2

18)

207

(194

-220

) 18

5 (1

73-1

93)

189

(181

-203

) 18

6 (1

74-2

04)

190

(176

-202

) 18

2 (1

72-1

90)

183

(172

-190

) 18

7 (1

73-1

92)

186

(173

-191

)

-log

lik

97

(86-

104)

99

(9

3-10

6)

85

(80-

90)

89

(85-

96)

86

(80-

95)

89

(82-

94)

82

(77-

86)

84

(78-

87)

85

(78-

88)

86

(79-

89)

Mod

el 7

vs.

th

e ot

her

mod

els

(AIC

)

<0.

0001

<

0.00

01

0.07

15

0.07

15

<0.

0001

<

0.00

01

- 0.

8159

0.

2853

0.

4204