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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Suggested Citation:"II. Mathematical Modeling." National Research Council. 1987. Drinking Water and Health, Volume 8: Pharmacokinetics in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/1015.
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Below is the uncorrected machine-read text of this chapter, intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text of each book. Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

PART 11 Mathematical Modeling

Modeling: An Introduction Ellen ]. O'Flaherty Fick's First Law states that the rate of diffusion of a solute down a concentration gradient is proportional to the magnitude of the gradient: dM dC d = —DA dx' (1) where M is mass, C is concentration, D is the diffusion constant with dimensions distance2/time, A is the cross-sectional area of the diffusion volume, and dx is the distance over which the infinitesimally small con- centration difference dC is measured. When Fick's First Law is restated for diffusion across a membrane barrier of thickness dx, the concentration gradient dC is approximated by the concentration difference across the membrane, Car —C2, and DAIRY is the first-order transfer constant kit for diffusion across the membrane, with dimensions distance3/time, or vol- ume/time: dt MIX ~ ~ 2) = —kit (C~ —C2) (2) Fick's Law states that transfer of freely diffusible molecules across a membrane should be first order. A large body of experimental observations supports this interpretation of the kinetic nature of diffusion. There are, of course, exceptions: Excretion from liver or kidney may not be first order, and gastrointestinal absorption may take place by active processes 27

28 ELLEN J. O'F~HERh C FIGURE 1 Lit - for a few chemicals. But, in general, it is reasonable to assume as a working hypothesis that absorption and distribution of exogenous chem- icals are first order. The rate of diffusion is dependent on the partition coefficient and molecular size and configuration of the chemical, and on its degree of ionization. Thus, diffusion out of a single compartment is a first-order process whose rate constant, k', depends both on the chemical and on the tissue and has the dimensions volume/time. If both sides of Equation 2 are divided by V so that it expresses the rate of change of concentration, not of mass since concentration is what is measured in viva then the rate constant becomes the elimination rate constant ke' with dimensions time- (Figure 11. dC = —keC. dt (3) Equation 3 is integrated to obtain the familiar first-order expression for C as a function of t, C C - ket (4) Equation 4 has a single exponential term so that if the natural logarithm of the concentration is plotted against time, the graph takes the form of a straight line whose slope is —ke and whose ordinate intercept is the logarithm of CO. The half-life is estimated from the value of ke' and the volume of distribution from the dose and the value of CO. The model is, of course, the one-compartment body model with first-order elimination. The body is not a single physiological compartment, however, and rarely behaves as if it were a single kinetic compartment. More sophis- ticated models of the body are created by the addition of peripheral com- partments. The essence of different approaches to modeling lies in how

Modeling: An Introduction 29 these compartments are defined and in what kinds of variables mea- surements of concentrations or amounts, or values of physiological pa- rameters are used to drive the development of a quantitative model. In the 1940s and 1950s, it was recognized that concentration behavior in the central compartment of a multicompartment model could be rep- resented by a sum of exponential terms like the single term describing the one-compartment model, one for each compartment in the model. Thus, for any model with more than one compartment, there will be more than one term, and the dependence of the logarithm of the concentration in the central compartment on time cannot be linear. Instead, it takes a curvilinear shape with a terminal straight-line portion. By a process variously known as feathering, peeling, or the method of residuals, curvilinear plots of in C versus t were resolved into their component exponential terms. As many terms were included in this feathering process as were required to account fully for the curvature of the data. Such fits are now, of course, carried out by nonlinear regression computer programs, but there was a time when they were not. In early modeling applications, model compartments were taken to have exact physiological correlates. Because of the correspondence between the number of compartments in a model and the number of terms required in the equation describing the model, the number of exponential terms necessary to account for the curvature in the data was taken to represent the number of distinguishable exchanges between the central compartment and peripheral tissues or organs, plus one term roughly equated with whole-body loss. An example of this approach appears in a paper on the kinetics of the rapid phases of plasma free cholesterol turnover (Porte and Havel, 19611. Free cholesterol labeled with i4C was incorporated into plasma lipoproteins in vitro and administered to dogs by intravenous injection. Resolution of the entire free cholesterol curve by successive subtraction of each com- ponent that is, by feathering gave five exponential terms with half- times of 4 min. 30 min. 65 min. 7 h, and 96 h. Porte and Havel compared these half-lives with turnover times reported for different pools of cho- lesterol, and concluded that the slowest, 96-h component represented metabolic turnover plus equilibration with very slowly exchanging com- partments; the 7-h component represented formation of plasma ester cho- lesterol; the 65-min component represented exchange of free cholesterol between plasma and red blood cells; and the 30-min component represented exchange of free cholesterol between plasma and liver. The most rapid component, with a half-life of 4 min. could not be related to any known physiological compartment. It quickly became apparent that forcing such a rigid correspondence between exponential terms and physiological compartments generated a number of problems, two of which are illustrated by the cholesterol ex-

30 ELLEN J. O'FLAHERTY cl i ~ k21 k12 FIGURE 2 He ample. It was not always possible to identify physiological correlates of exponential half-lives, particularly the shorter ones. And often more than one process for example, metabolism and slow exchange presented themselves as candidates for the source of an exponential term. With general dissemination of explicit mathematical solutions of mul- ticompartment models and recognition of the implications of these solu- tions, in the mid- 1960s a reaction set in. If there is only one compartment, the half-life is the half-life of elimination, and dose/CO is the physiological volume of distribution. When there is more than one kinetically distin- guishable compartment, the slopes and intercepts of the successive linear segments that are revealed by the curve-peeling process are expressed in appropriate units for calculation of half-lives and volumes of distribution. But these are not half-lives that are descriptive of a single process, nor are they physiological volumes of distribution. The reason is apparent on consideration of the two-compartment model shown in Figure 2 and below.

Modeling: An Introduction 31 dt = k2~C2—(kit + ketch, and dC2 = kick —ketch. (5) Simultaneous integration of these two equations gives the explicit so- lution of the two-compartment model, in which the intercepts AD and Be and the kinetic rate constants cx and ~ are expressed in terms of the rate constants kit and kin for transfer between compartment 1 and compartment 2, the elimination rate constant ke, and the volumes of the compartments: Car = AOe-at + BOe-~t D(or—ken) ° Vat (a - Fj B = D(k2~ — ~) Vat (a - P) = 112~(kI2 + k21 + ke) + t~kI2 + k21 + ke)2 - 4k2Ike] }/2} ,13 = 112~(k~2 + k2~ + key [(kl2 + k21 + ke)2 - 4k21ke] 1/2}. (6) Alpha and ~ are functions of all of the rate constants, and Ao and Be are functions of the volumes as well as of the rate constants. Being hybrid constants, they need have no direct physiological significance, although of course they reflect the biochemical and physiological basis of the chem- ical's disposition. Consequently, the volume of distribution (calculated as dose/B0) and half-life (calculated as in 2/~) need have no physiological correlates. Other apparent volumes of distribution can also be calculated. All are constants that relate a concentration to an amount under a particular set of conditions. But because kinetically determined volumes of distribution usually do not correspond to real volumes of distribution, it became commonplace in the 1970s to consider them simply as proportionality constants. The apparent volume of distribution is a useful pharmacokinetic parameter that relates the plasma or serum concentration of a drug to the total amount of drug in the body. Despite its name, this parameter usually has no direct physiologic meaning and does not refer to a real volume (Gibaldi and Perrier, 19751. This philosophy of modeling was, of course, in some respects a reaction to what was correctly perceived as unproductive and in some cases mis- leading data interpretations as a result of insistence on too exact a cor-

32 ELLEN ]. O'F~HERU respondence between the terms of the equation describing loss of the chemical from the blood and the physiological nature of compartments- organs and tissues exchanging with the blood. During this period also, it became commonplace to minimize the significance of half-lives derived from any but the terminal slope of the plasma or blood concentration curve. The terminal slope is used in the calculation of the biological half- life, which is generally accepted as an index of the persistence of the chemical in the body. The impetus for physiological modeling arose independently of classical pharmacokinetics, and physiological modeling coexisted with classical pharmacokinetic approaches during the 1960s and 1970s. In the 1980s, it is beginning to emerge as the preeminent approach to pharmacokinetic modeling. Because physiologically based pharmacokinetic modeling has received so much recent attention, it is important to make the point that physiological pharmacokinetics and classical pharmacokinetics are not fundamentally incompatible. Although the philosophy behind the two ap- proaches is different and dictates their application for different purposes, there is a direct link between the two approaches. Let us return for a moment to Fick's First Law. Fick's First Law describes the change in amount of a chemical with time. Thus, in the closed two-compartment model, the rates of transfer across the membrane separating the two compartments are expressed in terms of mass and kinetic rate constants, or of concentrations and transfer constants or clearances: = —ki2M~ + k2iM2 dt = —kl2V1 C1 + k2l V2C2 (7) At steady state, when dMlldt = 0, kl2VlCl = k2lV2C2; and since Cl = C2, kl2Vl = k2lV2 = kt. The equality applies, of course, not only at equilibrium but at disequilibrium as well. Thus, (h —k,fC1 —C2) (8) It is not coincidental that the transfer constant of classical pharmaco- kinetics has the dimensions of a flow rate. In the referent fluid volume for that flow rate lies the link between classical and physiological pharma- cokinetics. If transfers are perfusion limited that is, flow limited or first order then the transfer constant is the rate of blood flow to the tissue In the two-compartment closed model, let compartment 1 be the blood subcompartment and compartment 2 the fluid subcompartment of a tissue. Then, k~2V~ is the rate of blood flow to the tissue, which in this model is equal to total blood flow since there is only one peripheral compartment.

Modeling: An Introduction 33 Substituting blood flow rate for knave and knave, we find that the rates of transfer out of the blood and into the tissue fluid are expressed as blood flow rate times the concentration difference: dMi V dCI _n (r _ dt -~B\~1 — ~2J (9) With a single refinement, this is the fundamental equation of physio- logical pharmacokinetics. The refinement takes into account the fact that chemicals do not simply equilibrate between body fluids but, depending on their physicochemical properties, may be bound to tissue macromol- ecules or incorporated into tissue lipids. Thus, what is measured experi- mentally when a tissue is sampled is not C2 but Cal, the concentration in the tissue including fluid subcompartment V2 and bound or sequestered chemical. Because transfer is assumed to be flow limited, the concentration of the chemical in efferent blood from the tissue should be equal to its concentration in the fluid subcompartment of the tissue. Equilibration of the chemical between the fluid subcompartment of the tissue and its bound or sequestered forms is assumed to be very rapid, so that the partition coefficient R = CIT/C2 = C/concentration in efferent blood) can be measured and used to obtain C2 at any time from a measurement of tissue concentration: C2 = CHAIR. Often, the partition coefficient is determined in a separate in vitro vial equilibration experiment. Substituting CHAIR for C2 in Equation 9, we obtain an expression for the rate of change in the amount of the chemical in blood or tissue as a function of blood flow rate, partition coefficient, and momentary blood and tissue concentrations: dMi _ \! dC dt ~ 1 dt QB (C 1 — CII/R) . The form of this equation suggests that it should be possible to substitute physiological values of flow rates and volumes, and values of partition coefficients, in order to obtain predictive, physiologically based phar- macokinetic models. In fact, this is the fundamental relationship on which such models are based. The expressions for all peripheral, nonelimination tissues will be of this form, with QB replaced by blood flow to the tissue in question. The equation for the blood will be more complex but is directly derivable from the same kinds of considerations of flow and partitioning. It will include contributions from major tissue groups characterized by different perfusion rates, and it may include input rate or elimination rate terms. The equation for the liver may also include terms for metabolism or for input by absorption from the gastrointestinal tract. Some of these terms, particularly those describing metabolism, may not be first order.

34 ELLEN J. O'F~HERTY The difference between classical and physiological pharmacokinetic models, then, lies not in how they are constructed but in how they are driven. In classical pharmacokinetics, no effort is made to assign physi- ological correlates to model parameters. A compartment is simply defined as a volume (strictly speaking, as a fluid volume) that is kinetically ho- mogeneous. It is generally recognized that only in a very few instances are more than three exponential terms required to describe satisfactorily, within the precision of the data, the behavior of a declining concentration curve; most often, two suffice. This understanding has given rise to a group of models in which the body is represented by a central compartment and one or two peripheral compartments which may be "shallow" or "deep"; i.e., they may exchange relatively rapidly or relatively slowly with blood plasma. Such important concepts as volume of distribution, biological half-life, clearance, integrated total exposure following a single dose, and achievement of steady state during chronic exposure arise nat- urally from these classical models. Their utility for characterization of the behavior of a chemical, and for comparison of its behavior with that of other chemicals, is firmly established. Classical pharmacokinetic models support certain kinds of extrapola- tions in particular, extrapolation to different exposure conditions, with reasonable assurance. Capacity-limited or other nonlinear kinetic behavior can be incorporated into classical pharmacokinetic models. A specific advantage of the models is that because the kinetic characteristics of the compartments of which they are composed are not constrained, a best possible fit to a data set can be arrived at by varying the values of the parameters. Best estimates of parameter values can be compared across experimental conditions, treatments, or chemicals to establish whether apparent differences (effects) are statistically significant. This strength of classical pharmacokinetic models is also their greatest weakness. Lacking a physiological or biochemical basis, the models can- not take into account intraspecies changes such as growth, sexual matur- ation, or aging, and cannot reliably be used in interspecies conversion of pharmacokinetic data. The need for interspecies conversion of laboratory animal data, in particular, has led to the development of physiological pharmacokinetic models, in which the unspecified compartments of the classical pharmacokinetic models are replaced by actual organs and tissues with their known blood flows. Because tissue volumes, blood flow rates, and enzyme activities can be varied only within physiological limits in these models, the models are not fit to experimental data in the classical sense. Instead, gross discrepancy between the predictions of a physiolog- ical model and experimental observation requires reformulation of the model in such a way as to account for the observed behavior.

Modeling: An Introduction 35 In a sense, then, we have come full circle, from early insistence on correspondence between exponential terms and identifiable plasma-tissue interchanges to recognition that those interchanges do indeed give form to the plasma concentration curve, although not in the sense in which they were originally believed to do so. Physiological pharrnacokinetic models have tremendous potential, particularly for species-to-species conversion of dose-effect data. But classical pharmacokinetic models still have their place. Specifically, they are amenable to statistical treatment and, thus, to hypothesis testing, whereas the purely physiological pharmacokinetic models are not as readily treated statistically. - Both physiological and classical pharmacokinetic models have valid applications today. Both are capable of predicting the dose delivered to a target organ, within somewhat different limits. The assumptions on which the physiological pharmacokinetic models are based make them uniquely suited to cross-species applications. On the other hand, the de- pendence of classical pharmacokinetic models on experimental measure- ment of concentration or amount makes them especially well suited to examination of questions about mechanisms of effects that involve changes in pharrnacokinetic behavior. REFERENCES Gibaldi, M., and D. Perrier. 1975. Pharmacokinetics. P. 175. New York: Marcel Dekker. Porte, D., Jr. and R. J. Havel. 1961. The use of cholesterol-4-C '4-labeled lipoproteins as a tracer for plasma cholesterol in the dog. J. Lipid Res. 2:357-362.

Physiologically Based Pharmacokinetic Modeling Kenneth B. Bischof~f INTRODUCTION Pharmacokinetic models are used to permit the rational prediction, as much as possible, of the events occurring during the processes of drug disposition throughout the body, thus yielding tissue levels. As will be seen, it appears that it is feasible to do this with certain parts of the overall problem, although other aspects are more elusive. There are several specific reasons for pursuing this approach. One is the scientific intellectual satisfaction of having quantitative predictive models based on underlying knowledge, rather than the more empirical, curve- fitting approaches often used. The latter are always needed to some extent, of course, but should be minimized if possible. Another important purpose is to aid in the constant problems of interpreting animal experiments in drug screening, dosage regimen formulation, and similar matters. In quan- titative terms this can be called scaling the results from one species to another, and ultimately to man (Dedrick, 1973a). It seems clear that it would be of benefit to have pharmacokinetic models that specifically incorporated known animal physiology and pharmacology in these en- deavors. Yet another use of quantitative predictive models is in the de- velopment of optimal dosage regimens for clinical applications. This chapter is primarily intended to document this approach which, as usual, is in many scattered publications. It is hoped that other investigators will find it useful to have these ideas, methods, formulations, and basic data gathered in one place. 36

Physiologically Based Pharmacokinetic Modeling 37 The history and bases of physiological pharmacokinetics will be briefly reviewed, and some misconceptions will be pointed out, e.g., that mem- brane transport cannot be incorporated into these models and that only the flow-limited case can be handled. Several recent literature reviews will be given for those readers wanting further details on the modeling and/or specific drugs. This will be followed by a brief description of a few examples, and the chapter will conclude with my views of the most useful future research in the area. The basic idea of physiological pharmacokinetics was to extend phar- macokinetic modeling so that quantitative aspects of other biological areas could be incorporated. For example, this includes what is known about physiological differences and similarities among species, membrane bio- physics, biochemical kinetics, and others to be illustrated later. The ap- proach will be to focus the models on anatomically real local tissue regions, including their blood flow, binding, and transport characteristics. Certain aspects are similar to the compartmental modeling methods of mathe- matical biology (see, e.g., Rescigno and Segre, 1966, or Riggs, 1970) or of what will be termed classical pharmacokinetics, which is primarily concerned with the prediction of blood levels of various dosage regimens (see Gibaldi and Perrier, 1982, for a comprehensive treatment). Often, however, these compartments were rather abstract mathematical constructs, whose number and properties were only able to be ascertained by curve-fitting of experimental blood sample data. Useful insights into the quantitative operation of the body were obtained, although specific organ levels were usually not considered. Physiological pharmacokinetics, however, also attempts to predict the various organ and tissue levels, even extra- versus intracellular concentrations. This concept of utilization of known anatomical and physiological func- tions as a basis for pharmacokinetic models was proposed early on by Teorell (19371. This remarkable work was not able to be fully utilized, however, because of the lack of reasonable computing capabilities. When computing capabilities became feasible, the number of differential equa- tions that needed to be solved in comprehensive models was not of crucial importance, and multicompartment models based on known physiology were formulated by Bischoff and Brown (19661. The basis was to use a compartment as an actual local tissue region, as proposed by Bellman et al. ( 1960), although the term physiological pharmacokinetics was not used until about 1973 by Dedrick (1973b). The philosophical basis of the present approach resides in chemical engineering modeling and design, in which several of the problems, such as combined flow, diffusion, and chemical reactions, are similar to the present problem (see Himmelblau and Bischoff, 1968~.

38 KENNETH B. BISCHOFF BIOLOGICAL BASIS OF PHYSIOLOGICAL PHARMACOKINETICS There are many similarities in the anatomy and physiology of mam- malian species, and a general belief in this similarity has been the cor- nerstone of most biomedical research. Mammalian species share a remarkable geometric similarity. The same blood flow diagram could be used for all mammals, and most organs and tissues are similar fractions of the body weight. Major qualitative differences, such as the absence of a gallbladder . . . . In some species, are the exception. In a classic article, Adolph (1949) summarized an orderly variation of numerous anatomic and physiologic properties with body weight. Many physiologic processes vary as the 0.7-0.8 power of body weight, and the anatomic variables show a more nearly first-degree dependence on body weight. The result of this is that the physiologic process per unit of body weight or per unit of organ weight tends to decrease as body size increases, although blood perfusion (in milliliters per minute gram of tissue) may only vary by a factor of 2 across a variety of species. It is well known, for example, that the mouse has a heart rate and a relative cardiac output about an order of magnitude higher than those of a human. This does not pose any theoretical limitation to the use of the mouse as a model for cardiovascular dynamics; however, it does require that appropriate time scaling be done if the results are to be generalizable. Purely physiochemical interactions of exogenous chemicals with bio- logical tissues and fluids might be expected not to show a great variation among species. We reported a pharmacokinetic model for thiopental (Bis- choff and Dedrick, 1968) in which data from experiments as diverse as peanut oil-water distribution ratios and binding to bovine serum albumin and rabbit tissue homogenates were used to predict tissue and blood levels in the dog and the human. The most significant species differences that can confound pharmaco- kinetic predictability are in the qualitative pathways and kinetic charac- teristics of metabolism. As discussed by Williams (1974), foreign organic compounds tend to be metabolized in two phases. Phase I reactions lead to oxidation, reduction, and hydrolysis products. Phase II reactions lead to synthetic or conjugation products that are relatively polar and thus more easily excreted by the kidney in the urine and, in some cases, the liver in the bile. Within this general framework, however, there are large species variations. Williams points out that species variations in phase I reactions are very common and often appear to be unpredictable. If a species dif- ference is found for a particular compound, similar compounds may show similar variations. Phase II reactions are much more limited in number than phase I reactions, and it may be possible to identify patterns of these.

Physiologically Based Pharmacokinetic Modeling 39 Krasovskii (1976) reviewed the hepatic microsomal enzyme activities of 15 enzymes in several mammalian species. He observed that all except phosphatase tended to decrease (in units of activity per kilogram of body weight) as body weight is increased. Despite these observations, it does not appear safe to make generalizations concerning rates of unknown metabolic reactions. Many exceptions could be found to the apparent tendency of the intrinsic rate to decrease with increasing body size; quite different qualitative pathways can dominate in different species, and tox- icity sometimes can correlate with the concentration of an active inter- mediate that represents only a minor elimination pathway. Further aspects of these species similarities have been given by Dedrick and Bischoff (1980~. DEVELOPMENT OF MODELS A comprehensive discussion of the details of constructing these phar- macokinetic models has been provided by Bischoff (19751. There are two parts to this: choice of body regions and compartments, and the formulation of the proper mathematical relations to represent the drug transport, clear- ance, etc., in these compartments. The second of these is the more straight- forward, being based on the laws of physical chemistry and biology, for example; but, of course, we by no means have a satisfactory, complete quantitative description in hand. In fact, because of individual genetic differences, especially in the human population, we may never be able to precisely define all of the required parameters, as discussed by Gillette (1985~. The first aspect of constructing these models, however, requires even more judgment as to what are the important features that must be considered for a given problem in pharmacology or toxicology. CHOICE OF COMPARTMENTS The natural basis for the choice of compartments is the anatomy and physiology of the body, from the cellular level to the whole body. It is clear that the main question is how much detail needs to be used to provide an adequate description of the events. Even though consideration of these events in individual cells throughout the whole body is not feasible, special collections of cells (e.g., tumors) may have to be modeled in this much detail. The main features of drug distribution, however, can often be described with models that have surprisingly little detail. Thus, certain parts of the body can be lumped together (e.g., an organ) and described by a single concentration level. We will often term these parts body regions, with a.definite anatomic and physiologic basis, to avoid confusion

40 KENNETH B. BISCHOFF Right heart L Lung f Upper body } Left heart 1 ''in 1 WK< ,, ,~ \\,// Small Intestine | - l l Trunk t r Large Intestine Lower extremity FIGURE 1 Flow diagram for mammals.

Physiologically Based Pharmacokinetic Modeling 4} with the term compartment, which is often used in a more abstract sense. A general flow diagram for mammals is shown in Figure 1. This still does not resolve the question of exactly how many body regions, or compartments, are needed. In fact, there is no simple way to decide this, because judgment is required as to the important aspects of the drug distribution events. We base the initial choicely) on whatever is known about the physiochemical (binding, lipid solubility, ionization) and pharmacologic (mechanisms of transport, siteLs] of action) properties of the drug. For example, if the drug is not lipid soluble, the details of the adipose tissues of the body are not particularly important. One of the most difficult aspects concerns the importance of considering membrane resistances for adequate pharmacokinetic results (see Lutz et al., 19801. The main problem is the lack of knowledge of membrane transport for almost all drugs. The processes are usually agent, species, and tissue specific and often include saturable and/or active transport steps; and not much is known about generally useful models. Experimentation is difficult and tedious, leading to slow progress. Once again, however, there are some commonalities that can be used, and some workable models will be described later in this chapter. Because of these difficulties, it is often expedient to assume that the membrane resistances are not rate controlling, and the dominant variable is how much drug is presented to the tissue by the blood flow the so-called flow-limited or perfusion models. It is most important, however, to state that this approximation is not fundamental to the physiological pharmacokinetic approach, and is commonly used merely for the lack of sufficient information. Another basis for the initial determination of the number of required body regions is the speed or the time scale of events. Perhaps the simplest example of this is to consider the necessity of including the finite time of passage around the circulatory system (in man this is about 1 min). Thus, if one is attempting to describe tracer concentrations in an indicator- dilution experiment, the observations of interest are obviously occurring with a time scale the same as that of the mean circulation time, and the details of the transit times in the arteries and veins must be accounted for. (It should be mentioned here that if only an overall property Ouch as the total cardiac output] of the tracer concentration curves is of interest, then this detail may not be required.) Pharmacokinetic models containing this amount of anatomic geography are rather complicated, as is illustrated in Figure 2; the numbers in the compartments are estimates of the various volumes (in cubic centimeters) of the capillary, interstitial, and intracel- lular regions (from Bischoff, 19671. The time scale of interesting events for most drugs is usually many minutes, hours, days, or even longer. After an hour, about 60 circulations have elapsed, and this is more than adequate to have "mixed" the drug such that blood in the circulatory system has an essentially overall uniform concentration of drug, even though there could be local arterial-venous

42 KENNETH B. BISCHOFF I. ES3 EN . Liver L 54 1 32 Lit Am, L: 1 _ 300 Lung So 1 So 288 550 Head 33 1 23 430 815 Upper 23 1 15 ~ . 1755 3370 1 443 850 120 _ L H rN1 _ 46 1 28 L . . 575 1 G.l. ~ ~ . ~ _m 1100 Kidney 22 1 14 84 160 Lower 130 80 8710 16700 FIGURE 2 Flow diagram for disposition of rapidly acting substances anesthetic agents or tracers (numbers are volume estimates, in cubic centimeters).

Physiologically Based Pharmacokinetic Modeling 43 differences across certain organs. For a bolus injection, in fact, this is a good approximation after about three transient times. Thus, most phar- macokinetic models for drugs ignore the very early time events and lump the entire circulatory system into one blood pool. If the local (physiolog- ical) behavior is of interest, this approximation is probably not suitable. Also, to model very fast acting drugs, such as anesthetic agents, more detailed pharmacokinetic models than those normally used may be re- quired. For most of the discussion here, however, only relatively long time scales will be considered. If one carries this lumping to the extreme, the entire body can be assumed to have uniform (water) concentrations of the drug, with no detailed distinction between body regions. What is actually being assumed here will be discussed later, but pharmacologists know from experience that this approach often leads to reasonably accurate descriptions of the drug disposition. From the above discussion, one can reason that lumping is likely to be most useful for a drug that, for example, is rather slow acting, is primarily water soluble, and has no complicated transport mech- anisms. Empirical evidence of this is that a plot of log concentration versus time gives a single straight line for abolus injection. The mathematical equations resulting from this simple model are easy to manipulate and are the basis for most of the current detailed calculations of optimal dosage regimens by, for example, Kruger-Thiemer (1968, 1969~. However, it is probably more common to find that a semilog plot of blood (or plasma) concentration versus time has two rather pronounced phases with different slopes. The first, or earliest, time phase (sometimes denoted by a) is dominated by the distribution of drug between the central and peripheral parts of the body, and the second phase (or Q) is dominated by the ultimate drug removal processLes). Riegelman and Rowland (1968a,b) have explored extensively the use of this two-compartment model for many specific drugs, and they have discussed the proper evaluation of the various parameters. A detailed analysis of this model was given by Bis- choff and Dedrick ( 1970), and Wagner ( 1971 ) has provided a compendium of the detailed mathematical features of a collection of these and similar models. Even though the biphasic model can often give a reasonable fit to pharmacokinetic data, the specific interpretation of the model parameters, such as that of the central compartment, is often ambiguous. Thus, one of the original goals of having a predictive model incorporating physio- logical and pharmacological data is not entirely met by this approach. It should be emphasized, though, that for semiempirical projections of clin- ical data, these models are often most convenient. For our purposes, however, we still wish to construct the pharmacokinetic models from (close to) first principles.

44 L I m~ I) no ) ~ 1 '1 1 1 ,W, ~ , ( I, Cat (I in 5 ._ - Ct ._ . _ Cat so ._ Ct so Ct - - c) so ._ 04 ._ ~3 - C) r: ._ o ._ ._ o Ct Ct ._ C) ._ Ct Cat V)

Physiologically Based Pharmacokinetic Modeling 45 The above matters are the most intellectually difficult on the theoretical side, because no one set of specific rules can enable us to automatically choose the appropriate compartments. The writing of the mathematical equations, and their use with experimental data, is more straightforward, although actually doing this may be equally difficult and certainly more time-consuming. I will now outline the modeling details to illustrate the concepts in a concrete manner. I will often refer to, or imply, the above general discussion in constructing the models, and thereby clarify the philosophy of the approach. First, consider a body region consisting of cells randomly distributed in the interstitial fluid and supplied by a typical capillary, as shown in Figure 3. Even this may not be truly applicable to organs with a specific structure, such as the liver, but more appropriate models are only now being developed (see, e.g., Roberts and Rowland, 19861. Mass balances for the drug in each of the three components could be written, including three-dimensional diffusion, any flows of the fluids, membrane transport, etc., but we normally have nowhere near the amount of information to justify this. Therefore, some approximate form of these balances is used for a capillary, and these are then combined according to the architecture of the capillary bed, e.g., the Krogh tissue cylinder (see Leonard and Jorgensen, 1974, for a review of many of these approaches and a rather complete citation of the literature). For drug pharmacokinetic models, this model of a body region is even further lumped, as shown in Figure 4, based on what is normally done in an experimental biopsy. Examples of the use of these conceptual diagrams will be given below. BASIC MASS BALANCES All of the above discussions and developments will now be incorporated into mass balance equations of various degrees of completeness that can be used in actual pharmacokinetic models. Only the fully lumped models for body regions will be considered, but the physiochemical complications will be retained. A complete set of anatomically and physiochemically complicated mass balances could be formulated, but these are probably much too involved for any practical use at this time. The balances will explicitly account for free un-ionized, ionized, and bound forms of the drug, assuming that the ionized and bound forms are in instantaneous dynamic equilibrium with the free concentration. This seems to be a reasonable approximation right from the start, since the ionization and binding processes are normally very rapid.

46 KENN ETH B. BISCHOFF 1 0~ ~ Symbols: Blood Compartment 1 I 1 Tissue Compartment c2 :=> Flow ~ Mass Transfer FIGURE 4 Typical lumped tissue region. Mass Balance: Blood Pool dCB WB VB d t = ~ 01 V dCB* (unbound drug) (ionized drug) QB(WBCB + WBCB + PBXB) (flow out to organs) + si Qi(WBiCBi + WBiCBi + PBiXBi) (flow in from organs) + (injections). + V dXB (bound drug) (1) Each of the terms in Equation 1 should be clear. To use the balance, the relationships between ionized and free drug concentrations, Cal (C, pH, . . .), must be known from the ionization equilibrium constant, and like- wise the bound drug concentration, x (C, . . .), from Equation 2. SiKA CA 1 + KAi CA (2) where Si is the number of binding sites of type i. Equation 2 is valid for whole blood for a solute that instantaneously equilibrates with erythrocytes

Physiologically Based Pharmacokinetic Modeling 47 or for plasma when the solute is excluded from the erythrocytes. Cases with an intermediate degree of cell uptake are usually handled semiem- pirically. The tissue regions will be described as above in the section "Choice of Compartments." Mass Balance: Tissue Region i For equilibrium blood: VBi dt = Qi (qB qBi) ~ jA)B~,i (3) where qB is total concentration of drug and is equal to WBCB + WBCB~ + PBXB, (iA)B~i iS the membrane flux-area product and is equal to PA(CBi—CTi) for simple passive transport of free drug. For interstitial fluid: VIi dt = ~ jA)sl i— ~ jA)IC i (4) For intracellular fluid: VCi d! i = ~ jA)/c' + reaction, secretion, etc. (5) In the above equations, the symbols are defined as follows: C, free (unbound) un-ionized concentration; C*, free ionized concentration; x, bound concentration; V, volume; Q. flow rate; w, fraction of water; p, protein concentration; P. membrane permeability. The combination of Equation 1 and Equations 3 to 5 for each of the appropriate body regions, together with Equation 2 and the ionization equilibria, would permit a solution for all of the various concentration levels. To actually do this, of course, numerical values of all the parameters must also be known. Rather good estimates are possible for the various organ volumes and flows, and plasma protein binding can be measured in in vitro experiments, as can ionization. The most difficult to obtain are tissue binding and the specific membrane permeabilities. SIMPLIFICATIONS OF MASS BALANCES Since the membrane permeabilities are not commonly available, Equa- tions 3 to 5 given above are usually simplified into blood and tissue regions,

48 KENNETH B. BISCHOFF also corresponding to usual biopsy methods. Then, Equations 4 and 5 are combined to give the following for tissue mass balance: dqT VT; {] ' = ( jA)B, i + reaction, etc. where . , (6) VTi qTi = VIi qIi + VCi qci - (6a) This, of course, only gives an approximation to the true intracellular concentration, unless the cell membrane permeability is very large (the mass transfer resistance is very low). To completely get around this problem of membrane permeabilities, the concept offlow-limited conditions is most often introduced. Physically, this means that the membrane permeability is so large that any molecules that flow into the region have an easy time moving throughout the space. Thus, the part of the process that determines the drug concentration level Is now much drug is able to flow in from the blood pool. A corollary is that the free concentration is essentially identical throughout the region, since this is the driving force for mass transfer across the membrane, for simple passive diffusion. Mathematically, a very large permeability in Equations 3 and 6 is PA x, and CBi— CTi—Ci (equal free concentrations). Addition of Equa- tions 3 and 6, then, eliminates the mass transfer term, and the result is: (WBi VBi + WTi VTi) ,dt- + PBi VBi d' + PTi VTi dt = Qi (WB CB + PB XB WBi Ci PBiXBi) + ri(Ci)- (7) (The ionized moiety terms have been dropped for simplicity.) Thus, one equation results for each tissue region, but more importantly, the exact value of PA need not be known, just that it is very large. Dedrick and Bischoff (1968) have shown that the criterion for flow limiting conditions is (PA)ilQi >> 1. This relation is about what one would expect; that is, if the rate of mass transfer is much greater than the regional perfusion, one can assume that the former is relatively very rapid and that the only important resistance to transport is the flow limitation. Some knowledge of the permeability is still required to use the criterion, and such data are scarce. Dedrick and Bischoff (1968) provided some information on how these can be estimated, and Lutz et al. (1980) have given specific values. Data from biological handbooks could also be recast into an appropriate form for the use of this criterion. Despite this there definitely is still a paucity of information and more studies are needed.

Physiologically Based Pharmacokinetic Modeling 49 Flow-limited conditions will usually be assumed, but the true validity of this often cannot be assessed. The opposite extreme of having the membrane permeability be the dominant mass transfer resistance has the implication that the blood con- centration in the region is constant; there is essentially no concentration gradient across the region because more than enough drug is supplied by the blood flow. Mathematically, Equation 3 basically gives CBi = CB, and then Equation 6 reduces to: dCT dXT Ti Ti At + PTi VTi At (PA)i (CB CTi) + ri (CTi) - (8) For no binding (or linear binding, see below), Equations 7 and 8 have essentially the same form as a function of CB, and so the forms of the mathematical solutions would also be the same. With strong (nonlinear) binding, there could be some differences between the equations repre- senting the two extreme limiting cases, but it is not clear how easily these differences might be observed. Thus, it is of interest that both limits have the same form of mathematical equation, except that the parameters have quite different physical meanings. For flow-limited models, the most important property is the perfusion of a tissue. This then leads to four main compartments: the blood pool, highly perfused viscera, low-perfusion lean tissue, and low-perfusion adi- pose tissue. The last two would be kept separate for lipid-soluble drugs because of the greatly different physicochemical nature of the two types of regions. For primarily water-soluble drugs, the two low-perfusion re- gions could be combined. Given the parameter values, sets of mass balances like the various combinations given above can be readily solved for the important inter- connected body regions. Naturally, this question of parameter values is the crucial one, but it is also a key advantage of the physiological phar- macokinetics approach. Most of the parameters in the above equations are either known as average values for many animal species (e.g., volumes and flows) or can be measured in separate experiments (e.g., protein binding, although characterization of tissue binding is still difficult; see Jusko and Gretch, 1976, and Shen and Gibaldi, 1974~. For a compre- hensive discussion of scaling see Dedrick (1973), and for hemodynamic considerations see Wilkinson (19751. Further predictions are possible for special cases. If the definition of tissue total concentration (Equation 6a) is combined with the equilibrium blood total concentration, the combined tissue level, as used in the flow- limited models, is obtained: qi = ~ (WBi VBi + WTi VTi ~ Ci + PBi VB XBi + PTiVTiXTi)~/( VBi + VTi) If this is substituted into the complete mass

50 KENNETH B. BISCHOFF balance for this situation, Equation 7, the following simple form is ob- tained (see Bischoff, 1975, for details): VB {I = —QBqB + ~ Qi R + injection, and (9) (VBi + VTi) = Qi qB + ri(Ci), dt ( Ri) ~ (10) where Ri = tissue/blood distribution ratio at equilibrium. In these equations, Ri is equal to a constant only for linear binding, x = KC. It should be emphasized that Equations 9 and 10 involve no further assumptions beyond that of flow-limited conditions, which is really the only situation in which the total tissue concentration has a unique meaning. Equations 9 and 10 are valid for either a variable or constant Ri, although they are much more useful only in the latter case. The arrow under 2.200 2.520 6.200 , 0.524 39.200 0.160 1 12.200 FIGURE 5 Model for thiopental pharmacokinetics. 5.615 QB Blood Pool not In Equlilbrlum with Tissues B QV 4.080 QL 1.275 Q .A 0.260 Viscera V Lean Tissue L Adipose Tissue A

Physiologically Based Pharmacokinetic Modeling 51 the metabolic rate term indicates the question that the enzyme reactions presumably depend on the free concentration, and for nonlinear binding this has no simple relationship to the total concentrations in the rest of the mass balance. The practical importance of ignoring this and just re- rlefinin~ the rates in terms of qi is not oresentlY known. _ __AA~ EXAMPLES Several sources of reviews and literature references now exist, and so only a brief set will be presented here. The second edition of the excellent book by Gibaldi and Perrier (1982) has a chapter, and other recent review articles have been presented by Himmelstein and Lutz (1979) and by Chen and Gross (19791; a comprehensive collection of drugs modeled by this approach is given by Gerlowski and Jain (19831. These provide a wealth of examples and sources of parameter values and other data, as do the usual physiological and pharmacological handbooks. One of the first drugs simulated by us was thiopental (Bischoff and Dedrick, 19681. Figure 5 is the lumped, longer time scale model used, with an important compartment being the adipose tissue because the drug is highly lipid soluble. This drug is also strongly and nonlinearly bound, and so the detailed mass balances (Equations 1 and 7) were used, along with some approximate relations for the tissue binding. Figure 6 is one illustration of the success of the physiological model in simulation of the thiopental levels in the blood of humans and dogs. Similar comparisons for other tissues have been shown by Bischoff and Dedrick (19681. Bis- choff et al. (1971) considered in some detail the pharmacokinetics of methotrexate. Because this drug appears to be essentially linearly bound for higher concentrations (an additive saturable binding term is needed for low concentrations), the simplified Equations 9 and 10 could be used in conjunction with the flow diagram shown in Figure 7. For low con- centrations, the concentration-dependent term Ri(Ci) was used. The mass balance equations are given below. For plasma: r V dCp P dt For muscle: = (injection) + QL — + QK — + QM — RL RK RM (QL + QK + QM) CP. VM dt = QM ( CP R ) (1 la) (lib)

52 KENNETH B. BISCHOFF BLOOD 100 y E - o he Cal He o c,, 10 On - 2 Dog Human _ ~ .g ·: - , 0 1/2 1 2 TIME, hours FIGURE 6 Comparison of data with model predictions for thiopental. For kidney: VK ddCtK = QK(CP CK ~ CK —J — kK — (TIC)

Physiologically Based Pharmacokinetic Modeling 53 , rPLASMA QL QG LIVFR G l TRACT QG r~ Blilarysecretlon ~ ~ ~ ~ Gut Absorp1 lon —6~—Feces Gut Lumen KIDNEY 1 . I Urlne MUSCLE L ~ FIGURE 7 Model for methotrexate pharrnacokinetics. For liver: 1~ -1~-- QK QM VL d = (QL — QG) (CP — R ) + QG (R R ) rO, (l ld) where r0 = [KL(CLIRL)]I[KL + (CLIRL)] For bile ducts: \ / ~ d—t = ri-1 —ri, (i = 1, 2, 3) For gut tissue: (lie) dCG C CG 1 kC C, dt G( P RG) [Z 4 (KG + C + bCi) (llf) \ For gut lumen: dCGL 1 dCi = -Y dt 4 t-~i dt (1 la) 4 dt = r3—kFVGLCI—4(K C+IC + bCI),and (llh)

54 KENNETH B. BISCHOFF —4 d— = —kFVGL(Ci-~ — Ci) 4 (KG + Ci + bCi), (i = 2, 3, 41. (lli) Most of the terms and their origins should now be clear. The new ones are as follow. In Equation tic, the term kKCK/RK represents the renal excretion of methotrexate, which is close to the glomerular filtration rate; in Equation 1 ld-e, ri is the bile flow of the drug, with the liver secretion being saturable at high doses (the three compartments in series represent the actual tubular or distributed nature of the real system); in Equation 1 if-i, the motion down the gut lumen is modeled by four compartments, and the absorption term has both a saturable and a nonsaturable component that is important only for very high doses, presumably passive diffusion. The various parameters in Equation 1 la-i were either estimated from the values given earlier or were independently measured in the case of the complex secretion steps. Then, the pharmacokinetic behavior was able to be predicted in mice, rats, dogs, monkeys, and man over a dose range of 3,000, all with the same model and with a consistent set of parameters. Two examples are shown in Figures 8 and 9. Thus, the details of this model must be a reasonably faithful representation of the actual physio- logical and pharmacological events, and should be of aid in interpreting results of experiments. Also, valid predictions of local drug concentrations for various dosage regimens are possible. The flow diagram in Figure 7 has a rather complicated configuration because of the importance of the enterohepatic cycle in the methotrexate pharmacokinetics. There was little direct metabolism of the drug, however, so metabolism is not a major route of elimination; the ultimate excretion was by the renal or the fecal route. Another example considers the opposite extreme, in which a straightforward anatomic flow diagram is the basis but the metabolism is dominant. Dedrick et al. (1972) have considered the drug cytosine arabinoside on the basis of the compartments seen in Figure 10. The sizes of the boxes in Figure 10 signify the relative im- portance of the various regions. Again, the same types of balances were used, but with metabolic terms in each based on known enzyme kinetics and levels. Figure 11 shows one prediction of the concentrations of cy- tosine arabinoside and its metabolite, uracil arabinoside. A final reduction in the complexity of the models is possible when the excretion/metabolism processes are relatively slow compared with the intercompartment blood flows. In this case, the entire body has an essen-

Physiologically Based Pharmacokinetic Modeling 55 ~ 10 of o F z 1.0 z o ~ 0.1 o I t~. ~ tN 1 ~ . \ ~ - - - - - - ~ GL o K 0~01 _ 0 60 120 180 240 MINUTES FIGURE 8 Comparisons of data with model predictions for methotrexate. Mice, 3 mg/kg. Abbreviations: GL, small intestine; L, liver; K, kidney; P. plasma; M, muscle. tially identical time response, and a one-compartment whole-body model is useful. In terms of Equations 9 and 10 this implies that: r1 <<Ql~°°, qB——' R1 (12) and then the addition of all the tissue mass balances to that for blood gives: EVB + ~ (VBi + VTi) Ri] 4! = ~ (injections) + ~ ri(Ci). (13) The term in brackets would be the theoretical basis of the volume of distribution. An alternate form can be derived from Equations 1 and 7 in terms of free concentrations. These types of equations are very familiar

56 KENNETH B. BISCHOFF £ 10 o ~ 1.0 CD Ad o C' 0.1 a: O ~ 0.01 _ \ - -- M 1 1 1 1 0 90 180 MINUTES 270 360 FIGURE 9 Comparison of data with model predictions for methotrexate. Man, 1 mg/kg. in pharmacology, although they are usually used with empirical param- eters, and therefore, no specific examples will be given. DISCUSSION Various special situations can require the use of combinations of all of the types of mass balance equations given above. The examples showed the type of reasoning used in several instances, although the flow-limited case was used in all of them. It appears that this quite often gives a good estimate of the overall drug concentrations throughout the body, even though the drug may be membrane limited in certain specific organs. Thus, a combination of the analyses illustrated in the examples, plus the use of Equation 6 for the specific region, might be a reasonable scheme for both the overall drug distribution and the details of, for example, tumor uptake. The work of Dedrick et al. (1975) is in some ways an illustration of this.

Physiologically Based Pharmacokinetic Modeling 57 FIGURE 10 Model for cytosine arabinoside pharmacokinetics. E3 |Heartl Am_ :3 G.l. Tract , _ L l E , , ~ Kidneys ~ Urine _ Lean 4.04 I/min n2d 0.35 1.10 0.18 1.24 0 93 . _ _ _ Finally, it should also be mentioned that the precise definition of the various anatomic regions is often somewhat flexible and can depend on the exact situation. For example, the major barrier to the transport of many drugs is not the capillary membrane but the cellular membrane; in this case, tissue can be defined as intracellular space, and blood can be defined as vascular plus interstitial space. The important point is that the physi-

5~3 KENNETH B. BISCHOFF 10 at o - E z Cal z o CD AS g 0.1 a o _) \^ _ - ~ ARA - C + ARA - U - ~A—C 1 1 1 1 1 1 - - - - - 1 0 20 40 60 80 100 120 TIME (minutes) 140 FIGURE 11 Comparison of data with model predictions for cytosine arabinoside and adenine arabinoside. ological and pharmacological information should be used in formulating the model, so that the several goals mentioned in the introduction to this paper might be achieved. The reviews quoted above provide many ex- amples of this; all of these are based on flow diagrams similar to those shown in Figure 7 or 10, with appropriate modifications for the specific drug.

Physiologically Based Pharmacokinetic Modeling 59 FUTU RE RESEARCH N EEDS I would like to end with a brief discussion of my view of some of the important future research needs. The first is the use of nonlumped tissue region models, with spatial variation of concentrations, for certain critical tissues. This could be important in tubular-type regions or thick, quasi- homogeneous regions where diffusion must be accounted for other than across a thin membrane. Of course, this also includes more detailed de- scr~ptions of intracellular fluid, which are probably necessary to quantitate truly the biochemical drug effects. The above types of reaction-diffusion models lead to partial differential equations, which are more difficult mathematically, but with modern computing technology they should not cause severe problems in calculations. A related issue is the use of more realistic descriptions of the local blood flows in the microcirculation. This is not a totally new area, of course, because the Krogh tissue cylinder has been used for many years in physiology to model oxygen transport, and two papers in this volume are concerned with a very detailed distributed model of the lung to model transport and reaction of ozone (see J. H Overton, lit. C. Graham, and F. J. Miller and F. J. Miller, J. H. Overton, R. C. Graham, E. D. Smolko, and D. B. Menzel in this volume). A sort of middle ground model combining lumped compartments with distributed regions where needed was used by Flessner et al. (1984, 1985) to study peritoneal-plasma transport, and by Morrison and Bedeck (1986) to study the transport of cisplatin in the brain. The papers by Roberts and Rowland (1986) mentioned above also report comprehensive studies and refer to previous papers, one of which was an early approach by Pang and Rowland (1977) for hepatic metabolism. All this work is quite recent and has not yet been incorporated into very many pharrnacokinetic studies. A second broad area is the addition of more realistic biochemical rate expressions that involve known pathways and the like. This will virtually always lead to nonlinear terms, and so simple mathematical solutions will no longer be feasible. Finally, we must move forward with the modeling of drug effects with the same type of fundamental philosophy and combine these improved pharmacodynamic models with the pharrnacokinetics to provide simulation of the actual problem: improved knowledge of tissue levels at the site of action for improved risk assessment. A few examples of this are discussed by Bischoff (19731. REFER ENCES Adolph, E. F. 1949. Quantitative relations in the physiological constitutions of mammals. Science 109:579-585. Bellman, R., R. Kalaba, and J. A. Jacquez. 1960. Some mathematical aspects of che- motherapy. Bull. Math. Biophys. 22:181-190.

60 KENNETH B. BISCHOFF Bischoff, K. B. 1967. Applications of a mathematical model for drug distribution in mam- mals. In Chemical Engineering in Medicine and Biology, D. Hershey, ed. New York: Plenum. Bischoff, K. B. 1973. Pharmacokinetics and cancer chemotherapy. J. Pharmacokinet. Bio- pharm. 1:465-480. Bischoff, K. B. 1975. Some fundamental considerations in the application of pharmaco- kinetics to cancer. Cancer Chemother. Rep. Part 1 59:777-793. Bischoff, K. B., and R. G. Brown. 1966. Drug distribution in mammals. Chem. Eng. Prog. Symp. Ser. 66:33-45. Bischoff, K. B., and R. L. Dedrick. 1968. Thiopental pharmacokinetics. J. Pharm. Sci. 57:1346-1351. Bischoff, K. B., and R. L. Dedrick. 1970. Generalized solution to linear, two-compart- ment, open model for drug distribution. J. Theor. Biol. 29:63-83. Bischoff, K. B., R. L. Dedrick, D. S. Zaharko, and J. A. Longstreth. 1971. Methotrexate pharmacokinetics. J. Pharm. Sci. 60:1128-1133. Chen, H.-S. G., and J. F. Gross. 1979. Physiologically based pharmacokinetic models for anticancer drugs. Cancer Chemother. Pharm. 2:85-94. Dedrick, R. L. 1973a. Animal scale-up. J. Pharmacokinet. Biopharm. 1:435-461. Dedrick, R. L. 1973b. Physiological pharmacokinetics. J. Dyn. Syst. Meas. Cont. Trans. ASME Sept.: 255-257. Dedrick, R. L., and K. B. Bischoff. 1968. Pharmacokinetics in applications of the artificial kidney. Chem. Eng. Prog. Symp. Ser. No. 84, 64:32-44. Dedrick, R. L., and K. B. Bischoff. 1980. Species similarities on pharmacokinetics. Fed. Proc. 39:54-49. Dedrick, R. L., D. D. Forrester, and D. H. W. Ho. 1972. In vitro-in vivo correlation of drug metabolism deamination of 1-~-D-arabinofuranosylcytosin. Biochem. Pharmacol. 21: 1-16. Dedrick, R. L., R. L. Zaharko, R. A. Bender, W. A. Bleyer, and R. J. Lutz. 1975. Pharmacokinetic considerations on resistance to anticancer drugs. Cancer Chemother. Rep. 59:795-804. Flessner, M. F., R. L. Dedrick, and J. S. Schultz. 1984. A distributed model of peritoneal- plasma transport: Theoretical considerations. Am. J. Physiol. 246:R597-R607. Flessner, M. F., R. L. Dedrick, and J. S. Schultz. 1985. A distributed model of peritoneal- plasma transport; analysis of experimental data in the rat. Am. J. Physiol. 248:F413- F424. Gerlowski, L. E., and R. K. Jain. 1983. Physiologically based pharmacokinetic modeling: Principles and applications. J. Pharm. Sci. 72:1103-1127. Gibaldi, M., and D. Perrier. 1982. Pharmacokinetics, 2nd ed. New York: Marcel Dekker. Gillette, J. R. 1985. Biological variation: The unsolvable problem in quantitation extrap- olations from laboratory animals and other surrogate systems to human populations. Banbury Report 19: Risk Quantitation and Regulatory Policy. Cold Spring Harbor, N.Y.: Cold Spring Harbor Laboratory. Himmelblau, D. M., and K. B. Bischoff. 1968. Process Analysis and Simulation. New York: John Wiley & Sons. Himmelstein, K. J., and R. J. Lutz. 1979. A review of the applications of physiologically based pharmacokinetic modeling. J. Pharmacokinet. Biopharm 7:127-145. Jusko, W. J., and M. Gretch. 1976. Plasma and tissue protein binding of drugs in phar- macokinetics. Drug Metab. Rev. 5:43-140. Krasovskii, G. N. 1976. Extrapolation of experimental data from animals to man. Environ. Health Perspect. 13:51 -58.

Physiologically Based Pharmacokinetic Modeling 61 Kruger-Thiemer, E. 1968. Pharmacokinetics and dose-concentration relationships. Pro- ceedings of the 3rd International Pharmacological Meeting, Sao Paulo, Brazil. Pp. 63- 113 in Physico-Chemical Aspects of Drug Actions, Vol. 7. New York: Pergamon Press. Kruger-Thiemer, E. 1969. Formal theory of drug dosage regimens. II. The exact plateau effect. J. Theor. Biol. 23:169-170. Leonard, E. F., and S. B. Jorgensen. 1971. The analysis of convection and diffusion in capillary beds. Annul Rev. Biophys. Bioeng. 3:293-339. Lutz, R. J., R. L. Dedrick, and D. S. Zaharko. 1980. Physiological pharmacokinetics: An in vivo approach to membrane transport. Pharmacol. Ther. 11:559-592. Morrison, P. F., and R. L. Dedrick. 1986. Transport of cisplatin in rat brain following microinfusion: An analysis. J. Pharm. Sci. 75:120-128. Pang, K. S., and M. Rowland. 1977. Hepatic clearance of drugs. I. Theoretical consid- erations of a "well-stirred" model and a "parallel tube" model. J. Pharmacokinet. Biopharm. 5:625-653. Rescigno, A., and G. Segre. 1966. Drug and Tracer Kinetics. Waltham, Mass.: Blaisdell. Riegelman, S., and M. Rowland. 1968a. Shortcomings in pharmacokinetic analysis by conceiving the body to exhibit properties of a single compartment. J. Pharm. Sci. 57:117- 123. Riegelman, S., and M. Rowland. 1968b. Concept of a volume of distribution and possible errors in evaluation of this parameter. J. Pharm. Sci. 57:117-123. Riggs, D. S. 1970. The Mathematical Approach to Physiological Problems. Cambridge, Mass.: MIT Press. Roberts, M. S., and M. Rowland. 1986. A dispersion model of hepatic elimination, 1,2,3. J. Pharmacokinet. Biopharm. 14:227-260, 261-288, 289-308. Shen, D., and M. Gibaldi. 1974. Critical evaluation of use of effective protein fractions in developing pharmacokinetic models for drug distribution. J. Pharm. Sci. 63:1698- 1703. Teorell, T. 1937. Kinetics distribution of substances administered to the body. Arch. Int. Pharmacodyn. Ther. 57:205-240. Wagner, J. G. 1971. Biopharmaceutics and Relevant Pharmacokinetics. 1971. Hamilton, Ill.: Drug Intelligence Publications. Wilkinson, G. R. 1975. Pharmacokinetics of drug disposition: Hemodynamic considera- tions. Annul Rev. Pharm. 15:11-27. Williams, R. T. 1974. Inter-species variations in the metabolism of xenobiotics. Biochem. Soc. Trans. 2:359-377.

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Pharmacokinetics, the study of the movement of chemicals within the body, is a vital tool in assessing the risk of exposure to environmental chemicals. This book—a collection of papers authored by experts in academia, industry, and government—reviews the progress of the risk-assessment process and discusses the role of pharmacokinetic principles in evaluating risk. In addition, the authors discuss software packages used to analyze data and to build models simulating biological phenomena. A summary chapter provides a view of trends in pharmacokinetic modeling and notes some prospective fields of study.

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