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PART 11 Mathematical Modeling

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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

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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

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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-

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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.

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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(orken) 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-

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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.

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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.

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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.

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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.

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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)

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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: \ / ~ dt = 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 = r3kFVGLCI4(K C+IC + bCI),and (llh)

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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-

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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 < OCR for page 25
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.

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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-

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5~3 KENNETH B. BISCHOFF 10 at o - E z Cal z o CD AS g 0.1 a o _) \^ _ - ~ ARA - C + ARA - U - ~AC 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.

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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.

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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.

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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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