A Multi-Compartmental Model Generally Applicable to Physiologically-Based Pharmacokinetics

Copyright © IFAC Modelling and Control in Biomedical Systems, Warwick, UK, 1997

A MULTI-COMPARTMENTAL MODEL GENERALLY APPLICABLE TO PHYSIOLOGICALLY-BASED PHARMACOKINETICS

Dr. Pbilip A. Arundel

Zeneca Pharmaceuticals, Lead Discovery Department, Mereside, Alderley Park, M~cc/esfield, Cheshire, SK10 4TG, UK.

Abstract: In compartmental phannacokinetics equations are fitted to data, the solutions being highly specific and incapable of extrapolation beyond the given conditions. Physiologically-based phannacokinetic (PBPK) models dealing with real tissue flows and volumes are capable of extrapolation beyond the starting regime; moreover they have the advantage that they can be used to simulate drugs which affect their own blood-flow distribution. Unfortunately they require much more data initially. This paper introduces ~ method whereby a minimal PBPK rat model can be produced without tissue sampling, which fits unrelated drugs, providing cross-species scaling of distribution volume from literature-based data. Keywords: physiological model, reduction, phannacokinetic data, simulation, lumped parameter, mathematical models.

profiles for upto 14 tissues, at a sequence of time points.

1. INTRODUCTION

In Classical Pharmacokinetics (PK) comparisons can only be made \"ithin a species regarding different dosing regimes and differing routes of drug administration. Pharmacokinetics uses 'lumped' compartments (1 to 3 usually) which bear only a weak resemblance to physiology. Thus PK offers little help in localising observations to individual organs and tissues and is of no value for organ-based control. Conceptually PK compartments are separated on the basis of speed of distribution; the perfusion rate (blood flow/ volume, [OIV] ) being used for the lumping. This leads for example to muscle and adipose tissue being lumped together in a slow compartment, which as this paper will show is often misleading.

By introducing a better discriminator (krJ, rather than simply using perfusion rate, Bernareggi & Rowland (1990) have reduced the complexity of a 14 tissue PBPK model by lumping compartments on their true kinetics. Parameter kTi , is defined in Rowland & Tozer (1989) for an individual tissue as (1)

[where Oi is the tissue blood flow rate, Vri is the tissue volume and Kp is the tissue-blood partition coefficient for a given drug in that tissue] . It can also be thought of as [perfusion rate! partition coefficient] . In the mathematical model it appears as as the rate constant of drug exit from an individual tissue when the input is teminated or alternatively as the eigenvalue for the differential equation governing the tissue kinetics.

Physiologically-based Pharmacokinetics (PBPK), in which true blood flows and tissue volumes are used. offers: estimation of individual organ parameters; extrapolation beyond the limits of the original data: and analysis of the controlling factors in individual tissues. However it initially requires more data; a full-sized PBPK model requiring drug concentration

Thus for a full PBPK model there could be a need for 14 values of kTi • However it has been shown (Bernareggi and Rowland, 1990) that for cycIosporin

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x Vda.J is effectively constant over a range of drugs. Here Vda.i is related by the following equation,

these 14 values can be successfully 'lumped' into only 6 groups as fOllows :-

1) lung; (whole blood flow) 2) heart, brain, kidney; 3) gut. stomach. spleen. pancreas; 4) liver; (metabolism) 5) muse/e. bone. skin. testes; 6) adipose;

kn kn kn kT4 kTJ kT6

highest high medium medium low lowest

(5) If

where the summation over 'n' covers all the tissues making up group 'i' . Thus a standard set of values for the left-hand side of the equation are found, which fit a range of drugs fairly well.

However even collecting data for 6 tissues is a considerable task. thus efforts to reduce the experimental work still further are desirable. The purpose of this paper is to investigate a method where, knowing the volume of distribution at steady state Vd», the values of kTi from i = 1 to 6 as defined above can be estimated for different drugs and a satisfactory PBPK model constructed without making tissue concentration measurements.

As mentioned previously adipose tissue is an exception and has to be approached in a different way. This is because the parameters 10gP (octanolwater partition coefficient) and pKa (the polarity and degree of charge on the molecule) which influence local drug distribution in fat vary widely between drugs. It is known that only free drug (that which is not bound to plasma proteins) can pass readily into fat tissue. Free drug concentration is ex-pressed as fu. Cp , where fu is the fraction of drug unbound in plasma. The fundamental parameter reflecting adipose tissue/plasma binding is now no longer Kp, but Kpu = Kp/fu, (Bernareggi and Rowland, 1990).

2. TIIEORY Initially it is necessary to estimate VdsJ . Assuming that a concentration profile of drug in plasma (Cp) vs time (t) has been obtained at a known drug dose (D) the area under this Cp:t curve (AUC) can be measured and the drug clearance rate (Cl) found as Cl = (D/AUC)

The parameters 10gP and pKa can be combin.ed into one term 10gD where D is a dispersion coeffiCIent. It has been found that for several drugs a linear correlation exists in adipose tissue between log Kpu and logD, thus by measuring 10gD (by separate physico-chemical analysis) the value of Kpu can be estimated. Parameter fu can be measured (or estimated) and thus Kpadip found. This can be inserted into the equation for adipose tissue to estimate kT6 .

(2)

Non-compamnental analysis (perrier and Mayersohn, 1982; Benet and Galeazzi, 1979) shows that fonning the 'first moment' plot [c;,i x ti] vs t then taking the area under this curve (AUMC) the mean-residence time (MRT) can be found as

:MRT = AUMC / AUC

(3)

3. RESULTS

[This has its parallel in bioreactor control systems]. Combining these equations the volume of distribution at steady state can be estimated as Vds.s = ( D / AUC ) . ( AUMC / AUC )

For each tissue group a standard value of (kT. VdsJ)i can be found by taking the mean oyer a number (10) of PBPK drug models. This is ShO"'ll in Table 1.

(~)

Table 1 Literature data on tissue volumes. flow and the ' standard ' values of (kTi x V!!a,j) for general use.

2.1 lvlodel reduction Several examples have been examined from the literature where full-sized ( 8-14) tissue PBPK models have been assembled. For each model the values of kTi have been found for all tissues and then lumped according to the 6 groups discussed previously. This has been successful for groups 1 - 5 but not for adipose tissue which \vill be tackled in an alternative way, shown later.

tissue group

1, lung 1.2 2, ht,... 4.8 3, gu,.. 13.0 ~, liver 10.5 5, mu, .. 193 6, adip 10.0

Where Vd» is large all the values of kTi (i = 1 to 5) are small and vice versa. This leads to a useful generalisation; for each tissue group the product (kTi

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standard (kTiX Vc!ss.i)

VT ml ~3 . 0

1~ . 5

9.8 11.8 16.3 0.4

35.8 3.0 0.75 1.12 0.085 0.040

6200 450 170 100 25

The values of Vr . and Qi are obtained from the literature (Bernareggi and Rowland, 1990: Bjorkman et al., 1993). To obtain a prediction of kr for an individual drug T the following equation is applied,

The predicted value of kr6 for adipose tissue has been found using the correlation shown in Fig. I which is applicable to some drugs in rat tissue. log (Kpu)

= . 0.6 +

0.8 • logO

(6) where Vdsa.j is the volume of distribution of drug 'j' found by the methods outlined above. An example is shown below in Table 2 for the drug Tolbutamide (Sugita et al., 1982), for which Vdsa.j = 20.0 rol, logD = 0.37 and fu = 0.24.

" "

~\

I, lung 2, ht, ... 3, gu, .. 4, liver 5, mu, .. 6, adip

standard (krix V=.J

" .os·

310 22.5 8.5 5.0 1.25 <0.45>

J~

2.S

1.S

/.l \

3.S

4.S

tolbutamide

.1

predicted actual kTi (min-I) kTi

6200 450 170 100 25

.

Imrpl1lrrune

"

Table 2 Actual and predicted values of kri for the drug Tolbutamide in the rat using the 'standard' lkri x V~) values found for a range of drugs.

tissue group

..

... 0 ...... 11 . .

Fig. 1. Plot of 10gKpu for rat adipose tissue against 10gD for 9 barbiturates and 3 other drugs:

236 39 13.0 11.7 1.0 0..+8

An alternative way to express the main finding of this paper is that the ratio between any pair of kr values (knikn and so on) is approximately constant between drugs. This is sho\\'1l in Fig. 2 below.

2

.£

E tolbutMnM::je

t16ml) 136ml)

1179mll

(336ml) (389".,1) (612m1)

(748mll

compartments

Fig. 2. Plot of log (krJ against Tissue group for a range of drugs

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dnJg (Vdss)

Given the values ofkTi obtained for Tolbutamide the concentration: time profile predicted using a 6 tissue compartment PBPK model is shown in Fig. 3 below. For this simulation the blood compartment has been assumed to total 20.0mI and be divided in the ratio 2:1 (venous:anerial). Tolbu\e. j ~e .

~I

I

0

I

"6 1

5

co.per\.~ol

~ode l

ven~~s oj~s~e ~on~~n t rot i on i n RA T Sue ; t .... , 00 1 (1 982) . JP8 . 10. 297-316

.-

I

I

I ,

~

0

1

0

I

~

f

-

,

"<;"

I

z

·01

1

UJ

I

I

~.-;-I--+-+---+---+-~ I I I

I

I

~-o~I· ----~-----r----~----~----~ eo

160

1 tTl i "

2'0

I

320

'00

Fig. 3. Plot of Tolbutamide concentration against time for a dose of 80mglkg in the rat (280gm), predicted using kTi values derived as above, v,ith an hepatic clearance rate of 0.3 mllmin. The 6 tissue compartment model used for the simulations is given in Fig. 4. It shows elimination only from the liver, with the veins and arteries acting as reservoirs with Kp=l.O. (It is found that Tolbutamide binds preferentially to red cells with a factor of 1.33 , this has been included). An i.v. dose is shown being applied to the veins, this would be in the form of an impulse function, while if an oral dose was to be modelled then it would be considered as a mono-exponential input to the gut [group 3] . This group drains into the liver whereas the others pass into the veins. [First-pass effects can thus be modelled for oral doses] . To check the basic validity of the model equations a mass balance is performed continuously during a run to see that no drug is 'lost'. The lungs take the full blood flow.

Fig. 4. Block diagram of a 6 tissue companment model with elimination from the liver.

4. DISCUSSION The column of predicted kT values in Table 2 has been obtained without any e:\1ensive tissue analyses and as can be seen is tolerably close to the actual e:\-perimental values. The liver is less reliable than the other four tissue groups. a possible cause being that it is the organ of elimination here and free drug levels are being affected both by distribution and elimination simultaneously. This can also be seen in Fig. 2. The general dependence of 10g(kTJ on the tissue group and V is clear. In this 3-D graph Vdss is the volume of distribution at steady stare other than the volume of the blood. The ,vide range of values of Vdss covering more than two orders of magnitude is clear, this would cover about 70% of the clinically available compounds at present (Rowland and Tozer, 1989, p22). Given the data on the rate constants for Tolbutamide, then applying the rule that after 6 halflives equilibrium has been achieved. it is clear that

Each compartment is modelled as a first order differential equation, 8 in all. These are solved using the software package ACSL I which provides a number of algorithms, notably Runge-Kutta (fixed step size, fourth order) and Gear's stiff algorithm (variable step size, variable order). The latter is particularly useful when including very rapid chemical changes. e.g. enzyme reactions, within the simulation.

I Edition 1U. Copyright 1995 by Mitchell and Gautier Software. ACSL is a registered tr:ldemark of MGA Software, Concord MA 01742 USA

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only groups 5 (and 6) will contribute to the later part of a disposition profile the other groups ha\ing equilibrated within 2 minutes, [the period of time generally assumed for adequate mixing in a pharmacokinetic model]. It is useful to consider that 80% of the total tissue volume is contained in groups 5 and 6, thus accurate assessment or estimation of the partition coefficients in these regions (especially muscle) is important. If experimentation was to be perfonned with Tolbutamide, the most useful tissue to examine would be muscle in group 5; in fact the average of several muscle samples would be of more value regarding the kinetics than data from a range of other tissues, say heart or kidney.

'standard' curve of values applicable over a range of drugs in the rat. This curve is the mean of several sets of PBPK data and provides estimates \\ithin a factor of two compared with real data. Normally the workload needed to produce one PBPK model would make regular analysis of scores of compounds by this technique quite prohibitive. However \\ith the method described the necessity of taking tissue samples is obviated. If any tissue data is available it can be included in the model and used to improve its correctness. Scaling of volume of distribution across species by this method is possible. What cannot be avoided is the need for an accurate in vitro estimate of the drug clearance mechanism and its rate.

In order to simulate the behaviour of one tissue it is possible to graft on the parameters found from an individual tissue study (e.g. in the brain) while the underlying 6 compartment model remains unaffected. In contrast, using PK directly the only approach to individual compartment simulation would be to make a one-tissue model and 'drive' it with the known plasma profile for the drug. Effects on dispostion to other organs could not then be tackled simultaneously. For a PBPK model, drug effects on disposition can be modelled simultaneously, for example where drug targeted at the brain also affects blood supply to the liver, thus altering metabolic elimination.

ACKNOWLEDGEMENT'S The author would like to thank several members of Zeneca staff for helpful discussions on aspects of this work, computing facilities and data interpretation.

REFERENCES Benet, L.Z. and R.L. Galeazzi (1979). Noncompartmental detennination of the ste;idy-state volume of distribution. J. Ph arm. Sci., 68, 10711074. Bernareggi, A., and M. Rowland (1990). Physiologic Modeling of Cyclosporin Kinetics in Rat and Man. J. Phamlacokinet. Biophaml. , 19, 21-30. Bjorkman, S., D.R. Sranski, H. Hideyoshi, R. Dowrie, S.R. Harapat, D.R. Wada, W.F. Ebling (1993). Tissue distribution of Fentanyl and Alfentanil in the rat cannot be described by a blood flow limited model. J. Pharmacokinet. Biopharm ., 21, 255-278. Igari , Y , Y Sugiyama, Y Sawada, T Iga, M. Prediction of Diazepam Hanano (1983). disposition in the rat and man by a physiologically based phannacokinetic model. J. Pharmacokinet. Biopharm., 11, 577Lin, J.H., Y Sugiyama. S. Awazu, M. Hanano (1982). In vitro and in vivo evaluation of the tissue-to-blood partition coefficient for physiological phannacokinetic models. J. Pharmacokinet. Biopharm., 10,637-647. Perrier, D. and M. Mayersohn (1982). Noncompartmental determination of the steady-state volume of distribution for any mode of administration. J. Pharm. Sci., 71, 372-373. Rowland, M. and TN. Tozer (1989). Clinical Pharmacokinetics. Lea and Febiger, Philadelphia. 2nd Ed, Chap. 10, pp. 131-147. Sugita, 0., Y Sawada, Y Sugiyama, T Iga, M. Hanano (1981). Prediction of drug-drug interaction from in vitro plasma protein binding and metabolism. Biochem. Pharmacol., 30. 33473354.

Earlier it was assened that the unbound blood-tissue partition coefficient was the fundamental parameter governing distribution if protein binding is involved. 111is term Kpu is transferable across species (e.g. Lin et al., 1982; Igari et aI., 1983), thus if the value of fu is available in both species the values of Kp can be estimated for the new species, especially Man. for the same tissue groups. It is this aspect of the work which makes the anempt to estimate group Kp in the rat so important, even if the initial accuracy is equivocal. The emphasis on acquiring earlier information on likely drug levels in Man, \\ith the additional need to pro,;de estimates of dose levels in Man are further incentives. Without a confident prediction of Vdss in man all other estimates of human kinetics \\ill be less reliable. Direct scale-up on a weight basis is insufficient as the proportion taken by adipose tissue in the final steady state value is highly variable, for several anaesthetics it is 50% or more (e.g. Bjorkman et aI., 1993) while for Tolbutamide it is only around 10% (Sugita et al .. 1981).

5. CONCLUSION A PBPK model containing 6 tissue compartments and 2 blood compartments has been presented in which compartment size has been estimated from the overall volume of distribution at steady state and a

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