S2 ch07 toxicokinetics

CHAPTER 7

TOXICOKINETICS

Michele A. Medinsky and John L. Valentine

CompartmentsParameters

AnatomicPhysiologicThermodynamicTransport

Perfusion-Limited CompartmentsDiffusion-Limited CompartmentsSpecialized Compartments

LungLiverBlood

CONCLUSION

INTRODUCTION

CLASSIC TOXICOKINETICS

One-Compartment ModelTwo-Compartment ModelEliminationApparent Volume of DistributionClearanceHalf-LifeSaturation ToxicokineticsBioavailabilityComputer Software

PHYSIOLOGIC TOXICOKINETICS

Basic Model Structure

INTRODUCTION

The study of the kinetics of chemicals was originally initiated fordrugs and consequently was termed pharmacokinetics. However,toxicology is not limited to the study of adverse drug effects butentails an investigation of the deleterious effects of all chemicals.Therefore, the study of the kinetics of xenobiotics is more prop-erly called toxicokinetics. Toxicokinetics refers to the modeling andmathematical description of the time course of disposition (ab-sorption, distribution, biotransformation, and excretion) of xeno-biotics in the whole organism. The classic way to describe the ki-netics of drugs is to represent the body as consisting of one or twocompartments even if those compartments have no apparent phys-iologic or anatomic reality. An alternate approach, physiologicallybased toxicokinetics, represents the body as a series of mass bal-ance equations that describe each organ or tissue on the basis ofphysiologic considerations. It should be emphasized that there isno inherent contradiction between the classic and physiologicallybased approaches. Classic pharmacokinetics, as will be shown, re-quires certain assumptions that the physiologically based modelsdo not require. Under ideal conditions, physiologic models can pre-dict tissue concentrations, whereas classic models cannot. How-ever, the values of the appropriate parameters are often unknownor inexact, hampering meaningful physiologically based toxicoki-netic modeling.

CLASSIC TOXICOKINETICS

Often, it is difficult to obtain relevant biological tissues in order toascertain a chemical or chemical concentration in the body andthen relate that chemical concentration to a toxicologic response.The least invasive and simplest method to gather information onabsorption, distribution, metabolism, and elimination of a com-pound is by sampling blood or plasma over time. If one assumesthat the concentration of a compound in blood or plasma is in equi-

librium with concentrations in tissues, then changes in plasmachemical concentrations reflect changes in tissue chemical con-centrations, and relatively simple pharmacokinetic models can adequately describe the behavior of that chemical in the body. Compartmental pharmacokinetic models consist of a central compartment representing plasma and tissues that rapidly equili-brate with chemical, connected to one or more peripheral com-partments that represent tissues that more slowly equilibrate withchemical (Fig. 7-1). Chemical is administered into the central com-partment and distributes between central and peripheral compart-ments. Chemical elimination occurs from the central compartment,which is assumed to contain rapidly perfused tissues capable ofeliminating chemical (e.g., kidneys and liver). Advantages of com-partmental pharmacokinetic models are that they require no infor-mation on tissue physiology or anatomic structure. These modelsare valuable in predicting the plasma chemical concentrations atdifferent doses, in establishing the time course of chemical inplasma and tissues and the extent of chemical accumulation withmultiple doses, and in determining effective dose and dose regi-mens in toxicity studies (Gibaldi and Perrier, 1982).

One-Compartment Model

The simplest toxicokinetic analysis entails measurement of theplasma concentrations of a xenobiotic at several time points afterthe administration of a bolus intravenous injection. If the data ob-tained yield a straight line when they are plotted as the logarithmsof plasma concentrations versus time, the kinetics of the xenobi-otic can be described with a one-compartment model (Fig. 7-2).Compounds whose toxicokinetics can be described with a one-compartment model rapidly equilibrate, or mix uniformly, betweenblood and the various tissues relative to the rate of elimination. Theone-compartment model depicts the body as a homogeneous unit.This does not mean that the concentration of a compound is thesame throughout the body, but it does assume that the changes that

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Copyright © 2001 by The McGraw-Hill Companies Retrieved from: www.knovel.com

226 UNIT 2 DISPOSITION OF TOXICANTS

occur in the plasma concentration reflect proportional changes intissue chemical concentrations (Rowland and Tozer, 1980).

In the simplest case, a curve of this type can be described bythe expression

C � C0 � e�kel�t

where C is the blood or plasma chemical concentration over timet, C0 is the initial blood concentration at time t � 0, and kel is thefirst-order elimination rate constant with dimensions of reciprocaltime (e.g., t�1).

Two-Compartment Model

After the rapid intravenous administration of some chemicals, thesemilogarithmic plot of plasma concentration versus time does notyield a straight line but a curve that implies more than one dispo-sitional phase. In these instances, the chemical requires a longertime for its concentration in tissues to reach equilibrium with theconcentration in plasma, and a multicompartmental analysis of theresults is necessary (Fig. 7-2). A multiexponential mathematicalequation then best characterizes the elimination of the xenobioticfrom the plasma.

Generally, a curve of this type can be resolved into two monoexponential terms (a two-compartment model) and is de-scribed by

C � A � e��t � � � e��t

where A and B are proportionality constants and � and � are thefirst-order distribution and elimination rate constants, respectively(Fig. 7-2). During the distribution (�) phase, concentrations of thechemical in the plasma decrease more rapidly than they do in thepostdistributional elimination (�) phase. The distribution phasemay last for only a few minutes or for hours or days. Whether thedistribution phase becomes apparent depends on the time when thefirst plasma samples are obtained, and on the relative difference inthe rates of distribution and elimination. If the rate of distributionis considerably rapid relative to elimination, the timing of bloodsampling becomes critical in the ability to distinguish a distribu-tion phase. The equivalent of kel in a one-compartment model is �in a two-compartment model.

Occasionally, the plasma concentration profile of many com-pounds cannot be described satisfactorily by an equation with twoexponential terms—for example, if the chemical has an excep-tionally slow distribution into and redistribution out of a deep pe-ripheral compartment or tissue. Sometimes three or four exponen-tial terms are needed to fit a curve to the plot of log C versus time.Such compounds are viewed as displaying characteristics of three-or four-compartment open models. The principles for dealing withsuch models are the same as those used for the two-compartmentopen model, but the mathematics are more complex and beyondthe scope of this discussion.

Elimination

Elimination includes biotransformation, exhalation, and excretion.The elimination of a chemical from the body whose disposition isdescribed by a one-compartment model usually occurs through afirst-order process; that is, the rate of elimination at any time isproportional to the amount of the chemical in the body at that time.First-order reactions occur at chemical concentrations that are notsufficiently high to saturate elimination processes.

The equation for a monoexponential model, C � C0 �e�kel�t, can be transformed to a logarithmic equation that has thegeneral form of a straight line, y � mx � b:

log C � �(kel�2.303) � t � log C0

where log C0 represents the y-intercept or initial concentration, and�(kel�2.303) represents the slope of the line. The first-order elim-ination rate constant (kel) can be determined from the slope of thelog C versus time plot (i.e., kel � �2.303 � slope). The first-orderelimination rate constants kel and � have units of reciprocal time(e.g., min�1 and h�1) and are independent of dose.

Mathematically, the fraction of dose remaining in the bodyover time (C�C0) is calculated using the elimination rate constantby rearranging the equation for the monoexponential function andtaking the antilog to yield

C�C0 � Anti log [(�kel�2.303) � t]

Thus, if the elimination rate constant is, for example, 0.3 h�1,the percentage of the dose remaining in the body (C�C0 � 100)and the percentage of the dose eliminated from the body after 1 h,

Figure 7-1. Compartmental pharmacokinetic models where ka is the first-order extravascular absorption rate constant into the central compart-ment (1), kel is the first-order elimination rate constant from the centralcompartment (1), and k12 and k21 are the first-order rate constants fordistribution of chemical into and out of the peripheral compartment (2)in a two-compartment model.

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CHAPTER 7 TOXICOKINETICS 227

i.e., 1 � (C�C0 � 100), are 74 and 26 percent, respectively, re-gardless of the dose administered (Table 7-1). The percentage ofthe total dose eliminated at one hour is said to be independent ofdose.

Apparent Volume of Distribution

In a one-compartment model, all chemical is assumed to distrib-ute into plasma and tissues instantaneously. The apparent volumeof distribution (Vd) is a proportionality constant that relates the to-

tal amount of chemical in the body to the concentration of a xeno-biotic in plasma, and is typically described in units of liters or litersper kilogram of body weight (Kato et al., 1987). Vd is the appar-ent space into which an amount of chemical is distributed in thebody to result in a given plasma concentration. For example, en-vision the body as a tank containing an unknown volume (L) ofwell mixed water. If a known amount (mg) of dye is placed intothe water, the volume of that water can be calculated indirectly bydetermining the dye concentration (mg/L) that resulted after thedye has equilibrated in the tank simply by dividing the amount ofdye added to the tank by the resultant concentration of the dye inwater. Synonymously, the apparent volume of distribution of achemical in the body is determined after intravenous bolus ad-ministration, and is mathematically defined as the quotient of theamount of chemical in the body and its plasma concentration. Vd

is calculated as

Vd � Doseiv�(� � AUC0�)

where Doseiv is the intravenous dose or known amount of chemi-cal in body at time zero; � is the elimination rate constant; andAUC0

� is the area under the chemical concentration versus timecurve from time zero to infinity. The product, � � AUC0

�, is theconcentration of xenobiotic in plasma.

Figure 7-2. Concentration versus time curves of chemicals exhibiting behavior of a one-compartment phar-macokinetic model (top) and a two-compartment pharmacokinetic model (bottom) on a linear scale (left) anda semilogarithmic scale (right).

Elimination rate constants, kel and � are determined from the slope of the log-linear concentration versus timecurve. Half-life (T1�2) is the time required for blood or plasma chemical concentration to decrease by one-half.C0 is the concentration of the chemical at t � 0 determined by extrapolating the log-linear concentration timecurve to the Y-axis (t � 0).

Table 7-1Elimination of Four Different Doses of a Chemical at 1 HourAfter Administration as Described by a One-CompartmentOpen Model and First-Order Toxicokinetics with a kel of 0.3 h�1

CHEMICAL CHEMICAL CHEMICAL

DOSE, REMAINING, ELIMINATED, ELIMINATED,mg mg mg % of dose

10 7.4 2.6 2630 22 8 2690 67 23 26

250 185 65 26

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For a one-compartment model, Vd can be simplified by theequation Vd � Doseiv�C0, where C0 is the concentration of chem-ical in plasma at time zero. C0 is determined by extrapolating theplasma disappearance curve after intravenous injection to the zerotime point (Fig. 7-2). Vd is correctly called the apparent volume ofdistribution because it has no direct physiologic meaning and usu-ally does not refer to a real biological volume. The magnitude ofthe Vd term is chemical-specific and represents the extent of dis-tribution of chemical out of plasma and into other body tissues(Table 7-2). Thus, for chemicals that readily distribute into ex-travascular tissues, Vd often exceeds actual body spaces. A chem-ical with high affinity for tissues will also have a large volume ofdistribution. In fact, binding to tissues may be so avid that the Vd

of a chemical is much larger than the actual body volume. Alter-natively, a chemical that predominantly remains in the plasma willhave a low Vd that approximates the volume of plasma (Wang etal., 1994). Once the Vd for a chemical is known, it can be used toestimate the amount of chemical remaining in the body at any timeif the plasma concentration at that time is also known by the rela-tionship Xc � Vd � Cp, where Xc is the amount of chemical in thebody and Cp is the plasma chemical concentration.

Clearance

Chemicals are cleared from the body by various routes, for exam-ple, via excretion by the kidneys or intestines, biotransformationby the liver, or exhalation by the lungs. Clearance is an importanttoxicokinetic concept that describes the rate of chemical elimina-tion from the body (Shargel and Yu, 1993). Clearance is describedin terms of volume of fluid containing chemical that is cleared perunit of time. Thus, clearance has the units of flow (milliliters perminute). A clearance of 100 mL/min means that 100 mL of bloodor plasma containing xenobiotic is completely cleared in eachminute that passes. The overall efficiency of the removal of a chem-ical from the body can be characterized by clearance. High valuesof clearance indicate efficient and generally rapid removal, whereaslow clearance values indicate slow and less efficient removal of axenobiotic from the body. Total body clearance is defined as thesum of clearances by individual eliminating organs:

Cl � Clr � Clh � Cli � . . .

where Clr depicts renal, Clh hepatic, and Cli intestinal clearance.Clearance of xenobiotics from the blood by a particular organ can-

not be higher than blood flow to that organ. For example, for axenobiotic that is eliminated by hepatic biotransformation, hepaticclearance cannot exceed the hepatic blood flow rate even if themaximum rate of metabolism in the liver is more rapid than therate of hepatic blood flow, because the rate of overall hepatic clear-ance is limited by the delivery of the xenobiotic to the metabolicenzymes in the liver via the blood. After intravenous, bolus ad-ministration, total body clearance is defined as

Cl � Doseiv�AUC0�

Clearance can also be calculated if the volume of distribution andelimination rate constants are known, and can be defined as Cl �Vd � kel for a one-compartment model and Cl � Vd � � for a two-compartment model. Clearance is an exceedingly importantconcept.

Half-Life

Another important and frequently used parameter that character-izes the time course of xenobiotics in an organism is the half-lifeof elimination (T1�2). Half-life is the time required for the bloodor plasma chemical concentration to decrease by one-half, and isdependent upon both volume of distribution and clearance. T1�2

can be calculated if Vd and Cl are known:

T1�2 � (0.693 � Vd)�Cl

The above relationship among T1�2, Vd and Cl demonstratesthat care should be taken in analyses of data when relying uponT1�2 as the sole determinant parameter in toxicokinetic studies,since T1�2 is influenced by both the volume of distribution for achemical and the rate by which the chemical is cleared from theblood. For a fixed Vd, T1�2 decreases as Cl increases, because chem-ical is being removed from this fixed volume faster as clearanceincreases (Fig. 7-3). Conversely, as the Vd increases, T1�2 increasesfor a fixed Cl since the volume of fluid that must be cleared ofchemical increases but the rate of clearance does not.

Because of the relationship T1�2 � 0.693 � kel, the half-lifeof a compound can be calculated after kel (or �) has been deter-mined from the slope of the line that designates the eliminationphase on the log C versus time plot. The T1�2 can also be deter-mined by means of visual inspection of the log C versus time plot,as shown in Fig. 7-2. For compounds eliminated by first-order ki-

Table 7-2Volume of Distribution (Vd) for Several Chemicals Compared with Volumes of Body Fluid Compartments

Vd

CHEMICAL (L/kg) BODY COMPARTMENT

Chloroquine 200Desmethylimipramine 40Tetracycline 1.3

0.6 Total body waterDigitoxin 0.5

0.27 Extracellular body waterSalicylic acid 0.15

0.045 Plasma

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netics, the time required for the plasma concentration to decreaseby one-half is constant. Therefore, xenobiotics eliminated from thebody by first-order processes are theoretically never completelyeliminated. However, during seven half-lives, 99.2 percent of achemical is eliminated, and for practical purposes this can beviewed as complete elimination. The half-life of a chemical obey-ing first-order elimination kinetics is independent of the dose, anddoes not change with increasing dose.

Saturation Toxicokinetics

As already mentioned, the distribution and elimination of mostchemicals occurs by first-order processes. However, as the dose ofa compound increases, its volume of distribution or its rate of elim-ination may change, as shown in Fig. 7-4. This is usually referredto as saturation kinetics. Biotransformation, active transportprocesses, and protein binding have finite capacities and can besaturated. When the concentration of a chemical in the body ishigher than the KM (chemical concentration at one-half Vmax, themaximum metabolic capacity), the rate of elimination is no longerproportional to the dose. The transition from first-order to satura-tion kinetics is important in toxicology because it can lead to pro-longed residency time of a compound in the body or increased con-centration at the target site of action, which can result in increasedtoxicity.

Some of the criteria that indicate nonlinear toxicokinetics in-clude the following: (1) The decline in the levels of the chemicalin the body is not exponential, (2) AUC0

� is not proportional to thedose, (3) Vd, Cl, kel (or �) or T1�2 change with increasing dose, (4)the composition of excretory products changes quantitatively orqualitatively with the dose, (5) competitive inhibition by otherchemicals that are biotransformed or actively transported by thesame enzyme system occurs, and (6) dose-response curves show anonproportional change in response with an increasing dose, start-ing at the dose level at which saturation effects become evident.

The elimination of some chemicals from the body is readilysaturated. These compounds follow zero-order kinetics. Ethanol is

an example of a chemical whose elimination follows zero-order ki-netics, with its biotransformation being the rate-limiting step in itselimination (York, 1982). The elimination of ethanol is dependentupon the amount of dose remaining to be eliminated rather thanthe fraction of dose to be eliminated. As shown in Table 7-3, a con-stant amount, rather than a constant proportion of ethanol is bio-transformed per unit of time regardless of the amount of ethanolpresent in the body. Important characteristics of zero-orderprocesses are as follows: (1) An arithmetic plot of plasma con-centration versus time yields a straight line, (2) the rate or amountof chemical eliminated at any time is constant and is independentof the amount of chemical in the body, and (3) a true T1�2 or kel

does not exist, but differs depending upon ethanol dose.By comparison, under first-order elimination kinetics, the

elimination rate constant, apparent volume of distribution, clear-ance and half-life are expected not to change with increasing dose.The important characteristics of first-order elimination are as fol-lows: (1) The rate at which a chemical is eliminated at any time isdirectly proportional to the amount of that chemical in the body atthat time. (2) A semilogarithmic plot of plasma concentration ver-sus time yields a single straight line. (3) The elimination rate con-stant (kel or �), apparent volume of distribution (Vd), clearance (Cl)and half-life (T1�2) are independent of dose. (4) The concentrationof the chemical in plasma and other tissues decreases similarly by

Figure 7-3. The dependence of T1�2 on Vd and Cl.

Renal Cl values of 60, 130, and 650 mL/min represent partial reabsorption,glomerular filtration, and tubular secretion, respectively. Values for Vd of3, 18, and 40 L represent approximate volumes of plasma water, extracel-lular fluid and total body water, respectively, for an average-sized person.

Figure 7-4. Vd, Cl and T1�2 following first-order toxicokinetics (left panels) and changes following saturable toxicokinetics (right panels).

Vertical dashed lines represent point of departure from first-order to satu-ration toxicokinetics. Pharmacokinetic parameters for chemicals that fol-low first-order toxicokinetics are independent of dose. When protein bind-ing or elimination mechanisms are saturated with increasing dose,pharmacokinetic parameter estimates become dose-dependent. Vd may in-crease, for example, when plasma protein binding is saturated, allowingmore free chemical to distribute into peripheral tissue spaces. Conversely,Vd may decrease with increasing dose if tissue protein binding saturates.Then chemical may redistribute more freely back into plasma. When chem-ical concentrations exceed the capacity for biotransformation by metabolicenzymes, overall clearance of the chemical decreases. These changes mayor may not have effects on T1�2 depending upon the magnitude and direc-tion of changes in both Vd and Cl.

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some constant fraction per unit of time, the elimination rate con-stant (kel or �).

Bioavailability

For most chemicals in toxicology, exposure occurs by extravascu-lar routes (e.g., inhalation, dermal or oral), and absorption into thesystemic circulation is incomplete. The extent of absorption of axenobiotic can be experimentally determined by comparing theplasma AUC0

� after intravenous and extravascular dosing. The re-sulting index quantitates the fraction of dosed absorbed systemi-cally and is called bioavailability (F). Bioavailability can be de-termined by using different doses, provided that the compound doesnot display dose-dependent or saturable kinetics. Pharmacokineticdata following intravenous administration is used as the referencefrom which to compare extravascular absorption because all chem-ical is delivered (or 100 percent bioavailable) to the systemic cir-culation. For example, bioavailability following an oral exposurecan be determined as follows:

F � (AUCpo�Dosepo) � (Doseiv�AUCiv)

where AUCpo, AUCiv, Dosepo, and Doseiv are the respective areaunder the plasma concentration versus time curves and doses fororal and intravenous administration. Bioavailabilities for variouschemicals range in values between 0 and 1. Complete absorptionof chemical is demonstrated when F � 1. When F 1, incompleteabsorption of chemical is indicated. Bioavailability is an importantconcept in toxicokinetics. As was discussed earlier, the most crit-ical factor influencing toxicity is not necessarily the dose but ratherthe concentration of a xenobiotic at the site of action. Xenobioticsare delivered to most organs by the systemic circulation. There-fore, the fraction of a chemical that reaches the systemic circula-tion is of critical importance in determining toxicity. Several fac-tors can greatly alter this systemic availability, including (1) limitedabsorption after oral dosing, (2) intestinal first-pass effect, (3) he-patic first-pass effect, and (4) mode of formulation, which affects,for example, dissolution rate or incorporation into micelles (forlipid-soluble compounds).

In summary, for many chemicals, blood or plasma chemicalconcentration versus time data can be adequately described by aone- or two-compartment, classical pharmacokinetic model whenbasic assumptions are made (e.g., instantaneous mixing of com-partments and first-order kinetics). In some instances, more so-phisticated models with increased numbers of compartments willbe needed to describe blood or plasma toxicokinetic data; for ex-

ample if the chemical preferentially distributes into deep periph-eral tissues. Modeling and knowledge of toxicokinetic data can beused in deciding on what dose or doses of chemical to use in theplanning of toxicology studies (e.g., if specific blood concentra-tions are desired), in evaluating dose regimens (e.g., intravascularversus extravascular, bolus injection versus infusion or single dos-ing versus repeated doses), in choosing appropriate sampling times,and in aiding in the evaluation of toxicology data (e.g., what bloodor plasma concentrations were achieved to produce a specific re-sponse, effects of repeated dosing on accumulation of chemical inthe body, etc.).

Computer Software

Several computer software programs are available for com-partmental modeling of pharmacokinetic data (e.g., WinNonlin,PKAnalyst, Summit and SAS, among others). Many are menu-driven and operate on Microsoft Windows or Macintosh systems.In general, concentration and time data are entered into a spread-sheet format. The operator then chooses a user-defined model or aspecified model from a built-in library to fit curves to concentra-tion versus time data. Convergence of the model occurs throughan iterative process to satisfy specific statistical requirements; forexample minimizing the sum of square residuals between simu-lated and observed data. Program outputs include pharmacokineticparameter estimations and descriptive statistical estimations. Anumber of software programs also offer graphic output of both testdata and model simulations. In summary, a wide variety of optionsand costs are available that can fit the user’s needs.

PHYSIOLOGIC TOXICOKINETICS

The primary difference between physiologic compartmental mod-els and classic compartmental models lies in the basis underlyingthe rate constants that describe the transport of chemicals into andout of the compartments (Andersen, 1991). In classic kinetics, therate constants are defined by the data; thus, these models are of-ten referred to as data-based. In physiologic models, the rateconstants represent known or hypothesized biological processes,and these models are commonly referred to as physiologicallybased. The concept of incorporating biological realism into theanalysis of drug or xenobiotic distribution and elimination is notnew. For example, one of the first physiologic models was pro-posed by Teorell (1937). This model contained all the importantdeterminants in chemical disposition that are considered valid to-day. Unfortunately, the computational tools required to solve the

Table 7-3Elimination Over Time of a Chemical That Follows Zero-Order Toxicokinetics

ETHANOL

TIME, REMAINING, ETHANOL ELIMINATED, ETHANOL ELIMINATED,h mL mL % of that remaining

0 50 0 01 40 10 202 30 10 253 20 10 334 10 10 505 0 10 100

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underlying equations were not available at that time. With advancesin computer science, the software and hardware needed to imple-ment physiological models are now well within the reach oftoxicologists.

The advantages of physiologically based models comparedwith classic pharmacokinetics are that (1) these models can pro-vide the time course of distribution of xenobiotics to any organ ortissue, (2) they allow estimation of the effects of changing physi-ologic parameters on tissue concentrations, (3) the same model canpredict the toxicokinetics of chemicals across species by allomet-ric scaling, and (4) complex dosing regimes and saturable processessuch as metabolism and binding are easily accommodated (Gargasand Andersen, 1988). The disadvantages are that (1) more infor-mation is needed to implement these models compared with clas-sic models, (2) the mathematics can be difficult for many toxicol-ogists to handle, and (3) values for parameters are often ill definedin various species, strains, and disease states. Nevertheless, phys-iologically based toxicokinetic models are conceptually sound andare potentially useful tools for gaining insight into the kinetics ofxenobiotics beyond what classic toxicokinetics can provide.

Basic Model Structure

Physiologic models often look like a number of classic one-compartment models that are linked together. The actual model

structure, or how the compartments are linked together, dependson both the chemical and the organism being studied. For exam-ple, a physiologic model describing the disposition of a chemicalin fish would require a description of the gills (Nichols et al., 1994),whereas a model for the same chemical in mammals would requirea lung (Ramsey and Andersen, 1984). Model structures can alsovary with the chemicals being studied. For example a model for anonvolatile, water-soluble chemical, which might be administeredby intravenous injection (Fig. 7-5), has a structure different fromthat of a model for a volatile organic chemical for which inhala-tion is the likely route of exposure (Fig. 7-6). The route of ad-ministration is not the only difference between these two models:For example, the first model has a compartment for the intestinesbecause biliary excretion, fecal elimination, and enterohepatic cir-culation are presumed important in the disposition of this chemi-cal. The second model has a compartment for fat since fat is animportant storage organ for organics. However, the models are notcompletely different. Both contain a liver compartment because thehepatic metabolism of each chemical is an important element ofits disposition. It is important to realize that there is no genericphysiological model. Models are simplifications of reality and ideally should contain elements believed to be important in de-scribing a chemical’s disposition.

In view of the fact that physiologic modeling requires moreeffort than does classic compartmental modeling, what accountsfor the increase in the popularity of the kinetic approach amongtoxicologists? The answer lies in the potential predictive power ofphysiologic models. Toxicologists are constantly faced with the is-sue of extrapolation—from laboratory animals to humans, fromhigh to low doses, from intermittent to continuous exposure, and

Figure 7-5. Physiologic model for a hypothetical xenobiotic that is sol-uble in water, has a low vapor pressure (not volatile), and has a relativelylarge molecular weight (MW 100).

This hypothetical chemical is eliminated through metabolism in the liver(Km), biliary excretion (Kb), renal excretion (Kr) into the urine, and fecalexcretion (Kf). The chemical can also undergo enterohepatic circulation(Kec). Perfusion-limited compartments are noted in white and diffusion-limited compartments are noted in blue.

Figure 7-6. Physiological model for a typical volatile organic chemical.

Chemicals for which this model would be appropriate have low molecularweights (MW 100), are soluble in organic solvents, and have significantvapor pressures (volatile). Transport of chemical throughout the body byblood is depicted by the black arrows. Elimination of chemical as depictedby the model includes metabolism (dashed arrow) and exhalation (blackarrow). All compartments are perfusion limited.

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from single chemicals to mixtures. Because the kinetic constantsin physiologic models represent measurable biological or chemi-cal processes, the resultant physiologic models have the potentialfor extrapolation from observed data to predicted situations.

One of the best illustrations of the predictive power of phys-iologic models is their ability to extrapolate kinetic behavior fromlaboratory animals to humans. For example, physiologic modelsdeveloped for styrene and benzene correctly simulate the concen-tration of each chemical in the blood of rodents and humans(Ramsey and Andersen, 1984; Travis et al., 1990). Simulations arethe outcomes or results (such as a chemical’s concentration in bloodor tissue) of numerically integrating model equations over a sim-ulated time period, using a set of initial conditions (such as intra-venous dose) and parameter values (such as organ weights). Bothstyrene and benzene are volatile organic chemicals; thus, the modelstructures for the kinetics of both chemicals in rodents and humansis identical to that shown in Fig. 7-6. However, the parameter val-ues for rodents and humans are different. Humans have larger bodyweights than rodents, and thus weights of organs such as the liverare larger. Because humans are larger, they also breathe more airper unit of time than do rodents, and a human heart pumps a largervolume of blood per unit of time than does that of a rodent, al-though the rodent’s heart beats more times in the same period. Theparameters that describe the chemical behavior of styrene and ben-zene, such as solubility in tissues, are similar in the rodents andhuman models. This is often the case because the composition oftissues in different species is similar.

For both styrene and benzene there are experimental data forboth humans and rodents and the model simulations can be com-pared with the actual data to see how well the model has performed(Ramsey and Andersen, 1984; Andersen et al., 1984; Travis et al.,1990). The conclusion is that the same model structure is capableof describing the chemicals’ kinetics in two different species. Be-cause the parameters underlying the model structure representmeasurable biological and chemical determinants, the appropriatevalues for those parameters can be chosen for each species, form-ing the basis for successful interspecies extrapolation.

For both styrene and benzene, even though the same modelstructure is used for both rodents and humans, both the simulatedand the observed kinetics of both chemicals differs between ratsand humans. The terminal half-life of both organics is longer inthe human compared with the rat. This longer half-life for humansis due to the fact that clearance rates for smaller species are fasterthan those for larger ones. Even though the larger species breathesmore air or pumps more blood per unit of time than does the smallerspecies, blood flows and ventilation rates per unit of body massare greater for the smaller species. The smaller species has morebreaths per minute or heartbeats per minute than does the largerone, even though each breath or stroke volume is smaller. Thesefaster flows per unit mass bring more xenobiotic to organs re-sponsible for elimination. Thus, a smaller species can eliminate axenobiotic faster than a larger one can. Because the parameters inphysiologic models represent real, measurable values such as bloodflows and ventilation rates, the same model structure can resolvesuch disparate kinetic behaviors among species.

Compartments

The basic unit of the physiologic model is the lumped compart-ment, which is often depicted as a box (Fig. 7-7). A compartmentis a single region of the body with a uniform xenobiotic concen-tration (Rowland, 1984; Rowland, 1985). A compartment may be

a particular functional or anatomic portion of an organ, a singleblood vessel with surrounding tissue, an entire discrete organ suchas the liver or kidney, or a widely distributed tissue type such asfat or skin. Compartments consist of three individual well-mixedphases, or subcompartments, that correspond to specific physio-logic portions of the organ or tissue. These subcompartments are(1) the vascular space through which the compartment is perfusedwith blood, (2) the interstitial space that forms the matrix for thecells, and (3) the intracellular space consisting of the cells in thetissue (Gerlowski and Jain, 1983).

As shown in Fig. 7-7, the xenobiotic enters the vascular sub-compartment at a certain rate in mass per unit of time (e.g., mil-ligrams per hour). The rate of entry is a product of the blood flowrate to the tissue (Qt, in liters per hour) and the concentration ofthe xenobiotic in the blood entering the tissue (Cin, in milligramsper liter). Within the compartment, the xenobiotic moves from thevascular space to the interstitial space at a certain net rate (Flux1)and moves from the interstitial space to the intracellular space atdifferent net rate (Flux2). Some xenobiotics can bind to cell com-ponents; thus, within a compartment there may be both free andbound xenobiotics. The xenobiotic leaves the vascular space at acertain venous concentration (Cout). Cout is equal to the concen-tration of the xenobiotic in the vascular space.

Parameters

The most common types of parameters, or information required,in physiologic models are anatomic, physiologic, thermodynamic,and transport.

Anatomic Anatomic parameters are used to physically describethe various compartments. The size of each of the compartmentsin the physiologic model must be known. The size is generallyspecified as a volume (milliliters or liters) because a unit densityis assumed even though weights are most frequently obtained ex-perimentally. If a compartment contains subcompartments such asthose in Fig. 7-7, those volumes also must be known. Volumes ofcompartments often can be obtained from the literature or fromspecific toxicokinetic experiments. For example, kidney, liver,brain, and lung can be weighed. Obtaining precise data for vol-umes of compartments representing widely distributed tissues suchas fat or muscle is more difficult. If necessary, these tissues can be

Figure 7-7. Schematic representation of a lumped compartment in aphysiologic model.

The blood capillary and cell membranes separating the vascular, intersti-tial, and intracellular subcompartments are depicted in black. The vascularand interstitial subcompartments are often combined into a single extra-cellular subcompartment. Qt is blood flow, Cin is chemical concentrationinto the compartment and Cout is chemical concentration out of the com-partment.

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removed by dissection and weighed. Among the numerous sourcesof general information on organ and tissue volumes, Brown et al.(1997) is a good starting point.

Physiologic Physiologic parameters encompass a wide variety ofprocesses in biological systems. The most commonly used physi-ologic parameters are blood flow, ventilation, and elimination. Theblood flow rate (Qt, in volume per unit time, such as mL/min orL/h) to individual compartments must be known. Additionally, in-formation on the total blood flow rate or cardiac output (Qc) isnecessary. If inhalation is the route for exposure to the xenobioticor is a route of elimination, the alveolar ventilation rate (Qp) alsomust be known. Blood flow rates and ventilation rates can be takenfrom the literature or can be obtained experimentally. Renalclearance rates and parameters to describe rates of biotransforma-tion, or metabolism, are a subset of physiologic parameters and arerequired if these processes are important in describing the elimi-nation of a xenobiotic. For example, if a xenobiotic is known tobe metabolized via a saturable process both Vmax (the maximumrate of metabolism) and KM (the concentration of xenobiotic atone-half Vmax) must be obtained so that elimination of the xeno-biotic by metabolism can be described in the model.

Thermodynamic Thermodynamic parameters relate the totalconcentration of a xenobiotic in a tissue (C ) to the concentrationof free xenobiotic in that tissue (Cf). Two important assumptionsare that (1) total and free concentrations are in equilibrium witheach other and (2) only free xenobiotic can enter and leave the tis-sue (Lutz et al., 1980). Most often, total concentration is measuredexperimentally; however, it is the free concentration that is avail-able for binding, metabolism, or removal from the tissue by blood.Various mathematical expressions describe the relationship be-tween these two entities. In the simplest situation, the xenobioticis a freely diffusible water-soluble chemical that does not bind toany molecules. In this case, the free concentration of the xenobi-otic is exactly equal to the total concentration of the xenobiotic:total � free, or C � Cf. The affinity of many xenobiotics for tis-sues of different composition varies. The extent to which a xeno-biotic partitions into a tissue is directly dependent on the compo-sition of the tissue and independent of the concentration of thexenobiotic. Thus, the relationship between free and total concen-tration becomes one of proportionality: total � free � partition co-efficient, or C � Cf � P. In this case, P is called a partition or dis-tribution coefficient. Knowledge of the value of P permits anindirect calculation of the free concentration of xenobiotic orCf.Cf � C�P.

Table 7-4 compares the partition coefficients for a number oftoxic volatile organic chemicals. The larger values for the fat/bloodpartition coefficients compared with those for other tissues sug-gests that these chemicals distribute into fat to a greater extent thanthey distribute into other tissues. This has been observed experi-

mentally. Fat and fatty tissue such as bone marrow contain higherconcentrations of benzene than do tissues such as liver and blood.Similarly, styrene concentrations in fatty tissue are higher thanstyrene concentrations in other tissues.

A more complex relationship between the free concentrationand the total concentration of a chemical in tissues is also possi-ble. For example, the chemical may bind to saturable binding siteson tissue components. In these cases, nonlinear functions relatingthe free concentration in the tissue to the total concentration arenecessary. Examples in which more complex binding has been usedare physiologic models for dioxin and tertiary-amyl butyl ether(Andersen et al., 1993; Collins et al., 1999).

Transport The passage of a xenobiotic across a biological mem-brane is complex and may occur by passive diffusion, carrier-mediated transport, facilitated transport, or a combination ofprocesses (Himmelstein and Lutz, 1979). The simplest of theseprocesses—passive diffusion—is a first-order process describedby Fick’s law of diffusion. Diffusion of xenobiotics can occuracross the blood capillary membrane (Flux1 in Fig. 7-7) or acrossthe cell membrane (Flux2 in Fig. 7-7). Flux refers to the rate oftransfer of a xenobiotic across a boundary. For simple diffusion,the net flux (milligrams per hour) from one side of a membrane tothe other is described as Flux � permeability coefficient � drivingforce, or

Flux � [PA] � (C1 � C2) � [PA] � C1 � [PA] � C2

The permeability coefficient [PA] is often called the perme-ability-area cross-product for the membrane (in units of liters perhour) and is a product of the cell membrane permeability constant(P, in micrometers per hour) for the xenobiotic and the total mem-brane area (A, in square micrometers). The cell membrane perme-ability constant takes into account the rate of diffusion of the spe-cific xenobiotic and the thickness of the cell membrane. C1 and C2

are the free concentrations of xenobiotic on each side of the mem-brane. For any given xenobiotic, thin membranes, large surface ar-eas, and large concentration differences enhance diffusion.

There are two limiting conditions for the transport of a xeno-biotic across membranes: perfusion-limited and diffusion-limited.An understanding of the assumptions underlying the limiting con-ditions is critical because the assumptions change the way in whichthe differential equations are written to describe the compartment.

Perfusion-Limited Compartments

A perfusion-limited compartment is also referred to as bloodflow–limited, or simply flow-limited. A flow-limited compartmentcan be developed if the cell membrane permeability coefficient[PA] for a particular xenobiotic is much greater than the blood flowrate to the tissue (Qt) or [PA] Qt. In this case, the rate of xeno-

Table 7-4Partition Coefficients for Four Volatile Organic Chemicals

CHEMICAL BLOOD/AIR MUSCLE/BLOOD FAT/BLOOD

Isoprene 3 0.67 24Benzene 18 0.61 28Styrene 40 1 50Methanol 1350 3 11

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biotic uptake by tissue subcompartments is limited by the rate atwhich the blood containing a xenobiotic arrives at the tissue, notby the rate at which the xenobiotic crosses the cell membranes. Inmost tissues, the rate of entry of a xenobiotic into the interstitialspace from the vascular space is not limited by the rate ofxenobiotic transport across vascular cell membranes and is there-fore perfusion rate limited. In the generalized tissue compartmentin Fig. 7-7, this means that transport of the xenobiotic through theloosely knit blood capillary walls of most tissues is rapid comparedwith delivery of the xenobiotic to the tissue by the blood. As aresult, the vascular blood is in equilibrium with the interstitial sub-compartment and the two subcompartments are usually lumped to-gether as a single compartment that is often called the extracellu-lar space. An important exception to this vascular-interstitialequilibrium relationship is the brain, where the tightly knit bloodcapillary walls form a barrier between the vascular space and theinterstitial space.

As indicated in Fig. 7-7, the cell membrane separates the ex-tracellular compartment from the intracellular compartment. Thecell membrane is the most important diffusional barrier in a tissue.Nonetheless, for molecules that are very small (molecular weight100) or lipophilic, cellular permeability generally does not limitthe rate at which a molecule moves across cell membranes. Forthese molecules, flux across the cell membrane is fast comparedwith the tissue perfusion rate ([PA] Qt), and the moleculesrapidly distribute through the subcompartments. In this case, theintracellular compartment is in equilibrium with the extracellularcompartment, and these tissue subcompartments are usuallylumped as a single compartment. This flow-limited tissue com-partment is shown in Fig. 7-8. Movement into and out of the en-tire tissue compartment can be described by a single equation:

Vt � dC�dt � Qt � (Cin � Cout )

where Vt is the volume of the tissue compartment, C is the con-centration of free xenobiotic in the compartment (Vt � C equalsthe amount of xenobiotic in the compartment), Vt � dC�dt is thechange in the amount of xenobiotic in the compartment with timeexpressed as mass per unit of time, Qt is blood flow to the tissue,Cin is xenobiotic concentration entering the compartment, and Cout

is xenobiotic concentration leaving the compartment. Equations ofthis type are called mass balance differential equations. Differen-tial refers to the term dx/dt. Mass balance refers to the requirementthat input into one equation must be balanced by outflow from an-other equation in the physiologic model.

In the perfusion-limited case, Cout, or the venous concentra-tion of xenobiotic leaving the tissue, is equal to the free concen-

tration of xenobiotic in the tissue, Cf. As was noted above, Cf (orCout) can be related to the total concentration of xenobiotic in thetissue through a simple linear partition coefficient, Cout � Cf �C�P. In this case, the differential equation describing the rate ofchange in the amount of a xenobiotic in a tissue becomes

Vt � dC�dt � Qt � (Cin � C�P)

The physiologic model shown in Fig. 7-6, which was devel-oped for volatile organic chemicals such as styrene and benzene,is a good example of a model in which all the compartments aredescribed as flow-limited. Distribution of xenobiotic in all the com-partments is described by using equations of the type noted above.In a flow-limited compartment, the assumption is that the concen-trations of a xenobiotic in all parts of the tissue are in equilibrium.For this reason, the compartments are generally drawn as simpleboxes (Fig. 7-6) or boxes with dashed lines that symbolize the equi-librium between the intracellular and extracellular subcompart-ments (Fig. 7-8). Additionally, with a flow-limited model, estimatesof flux are not required to develop the mass balance differentialequation for the compartment. Given the information required toestimate flux, this is a simplifying assumption that significantly re-duces the number of parameters required in the physiologic model.

Diffusion-Limited Compartments

When uptake into a compartment is governed by cell membranepermeability and total membrane area, the model is said to be dif-fusion-limited, or membrane-limited. Diffusion-limited transportoccurs when the flux, or the transport of a xenobiotic across cellmembranes, is slow compared with blood flow to the tissue. In thiscase, the permeability-area cross-product [PA] is small comparedwith blood flow, Qt, or PA Qt. The distribution of large polarmolecules into tissue cells is likely to be limited by the rate atwhich the molecules pass through cell membranes. In contrast, en-try into the interstitial space of the tissue through the leaky capil-laries of the vascular space is usually flow-limited even for largemolecules. Figure 7-9 shows the structure of such a compartment.The xenobiotic concentrations in the interstitial and vascular spacesare in equilibrium and make up the extracellular subcompartmentwhere uptake from the incoming blood is flow-limited. The rate ofxenobiotic uptake across the cell membrane (into the intracellularspace from the extracellular space) is limited by cell membranepermeability and is thus diffusion-limited. Two mass balance dif-ferential equations are necessary to describe this compartment:

Figure 7-8. Schematic representation of a compartment that is blood-flow–limited.

Rapid exchange between the extracellular space (blue) and intracellularspace (light blue) maintains the equilibrium between them as symbolizedby the dashed line. Qt is blood flow, Cin is chemical concentration into thecompartment and Cout is chemical concentration out of the compartment.

Figure 7-9. Schematic representation of a compartment that is mem-brane-limited.

Perfusion of blood into and out of the extracellular compartment is depictedby thick arrows. Transmembrane transport (flux) from the extracellular tothe intracellular subcompartment is depicted by thin double arrows. Qt isblood flow, Cin is chemical concentration into the compartment and Cout ischemical concentration out of the compartment.

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Extracellular space: Vt1 � dC1�dt � Qt � (Cin � Cout) � [PA]� C1 � [PA] � C2

Intracellular space: Vt2 � dC2�dt � [PA] � C1 � [PA] � C2

Qt is blood flow, and C is free xenobiotic concentration in enter-ing blood (in), exiting blood (out), extracellular space (1), or in-tracellular space (2). Both equations contain terms for flux, or trans-fer across the cell membrane, [PA] � (C1 � C2). The physiologicmodel in Fig. 7-5 is composed of two diffusion-limited compart-ments each of which contain two subcompartments—extracellularand intracellular space—and several perfusion-limited compart-ments.

Specialized Compartments

Lung The inclusion of a lung compartment in a physiologicmodel is an important consideration because inhalation is a com-mon route of exposure to many toxic chemicals. Additionally, thelung compartment serves as an instructive example of the as-sumptions and simplifications that can be incorporated into phys-iologic models while maintaining the overall objective of describ-ing processes and compartments in biologically relevant terms. Forexample, although lung physiology and anatomy are complex, Hag-gard (1924) developed a simple approximation that sufficiently de-scribes the uptake of many volatile xenobiotics by the lungs. A di-agram of this simplified lung compartment is shown in Fig. 7-10.The assumptions inherent in this compartment description are asfollows: (1) ventilation is continuous, not cyclic; (2) conductingairways (nasal passages, larynx, trachea, bronchi, and bronchioles)function as inert tubes, carrying the vapor to the pulmonary or gasexchange region; (3) diffusion of vapor across the lung cell andcapillary walls is rapid compared with blood flow through the lung;(4) all xenobiotic disappearing from the inspired air appears in thearterial blood (i.e., there is no storage of xenobiotic in the lung tis-sue and insignificant lung mass); and (5) vapor in the alveolar airand arterial blood within the lung compartment are in rapid equi-librium and are related by Pb, the blood/air partition coefficient(e.g., Calv � Cart�Pb). Pb is a thermodynamic parameter that quan-tifies the distribution or partitioning of a xenobiotic into blood com-pared with air.

In the lung compartment depicted in Fig. 7-10, the rate of in-halation of xenobiotic is controlled by the ventilation rate (Qp) and

the inhaled concentration (Cinh). The rate of exhalation of a xeno-biotic is a product of the ventilation rate and the xenobiotic con-centration in the alveoli (Calv). Xenobiotic also can enter the lungcompartment via venous blood returning from the heart, repre-sented by the product of cardiac output (Qc) and the concentrationof xenobiotic in venous blood (Cven). Xenobiotic leaving the lungsvia the blood is a function of both cardiac output and the concen-tration of xenobiotic in arterial blood (Cart). Putting these fourprocesses together, a mass balance differential equation can be writ-ten for the rate of change in the amount of xenobiotic in the lungcompartment (L):

dL�dt � Qp � (Cinh � Calv) � Qc � (Cven � Cart)

Because of some of these assumptions, the rate of change inthe amount of xenobiotic in the lung compartment becomes equalto zero (dL�dt � 0). Calv can be replaced by Cart�Pb, and the dif-ferential equation can be solved for the arterial blood concentra-tion:

Cart � (Qp � Cinh � Qc � Cven)�(Qc � Qp�Pb)

This algebraic equation is incorporated into physiologic mod-els for many volatile organics. Because the lung is viewed here asa portal of entry and not as a target organ, the concentration of axenobiotic delivered to other organs by the blood, or the arterialconcentration of that xenobiotic, is of primary interest. The as-sumptions of continuous ventilation, dead space, rapid equilibra-tion with arterial blood, and no storage of vapor in the lung tissueshave worked extremely well with many volatile organics, espe-cially relatively lipophilic chemicals. Indeed, the use of these as-sumptions simplifies and speeds model calculations and may beentirely adequate for describing the chemical behavior of relativelyinert vapors with low water-solubility.

Inspection of the equation for calculating the arterial concen-tration of the inhaled organic vapor indicates that the term Pb, thexenobiotic-specific blood/air partition coefficient, becomes an im-portant term for simulating the uptake of various volatile organicxenobiotics. As the value for Pb increases, the maximum concen-tration of the xenobiotic in the blood increases. Additionally, thetime to reach the steady-state concentration and the time to clearthe xenobiotic also increase with increasing Pb. Fortunately, Pb isreadily measured by using in vitro techniques in which a volatilechemical in air is equilibrated with blood in a closed system, suchas a sealed vial (Gargas and Andersen, 1988).

Liver The liver is often represented as a compartment in physi-ologic models because hepatic biotransformation is an importantaspect of the toxicokinetics of many xenobiotics. The effects ofmultiple factors such as concentration, dose rate, and species onthe metabolism of xenobiotics are important in assessing risk. Be-cause the liver is often the major organ for the biotransformationof xenobiotics, the task of metabolism is generally assigned to theliver compartment in physiologic models. A simple compartmen-tal structure for the liver is depicted in Fig. 7-11, where the livercompartment is assumed to be flow-limited. This liver compart-ment is similar to the general tissue compartment in Fig. 7-8, ex-cept that the liver compartment contains an additional process formetabolic elimination. One of the simplest expressions for thisprocess is first-order elimination, which is written

Figure 7-10. Simple model of gas exchange in the alveolar region of therespiratory tract.

Rapid exchange in the lumped lung compartment between the alveolar gas(blue) and the pulmonary blood (light blue) maintains the equilibrium be-tween them as symbolized by the dashed line. Qp is alveolar ventilation(L/h); Qc is cardiac output (L/h); Cinh is inhaled vapor concentration (mg/L);Cart is concentration of vapor in the arterial blood; Cven is concentration ofvapor in the mixed venous blood. The equilibrium relationship between thechemical in the alveolar air (Calv) and the chemical in the arterial blood(Cart) is determined by the blood/air partition coefficient Pb, e.g., Calv �

Cart�Pb.

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R � Cf � Vl � Kf

R is the rate of metabolism (milligrams per hour), Cf is the freeconcentration of xenobiotic in the liver (milligrams per liter), Vl isthe liver volume (liters), and Kf is the first-order rate constant formetabolism in units of h�1. A widely used expression for metab-olism in physiologic models is the Michaelis-Menten expressionfor saturable metabolism (Andersen, 1981), which employs twoparameters, Vmax and KM and is written as follows:

R � (Vmax � Cf )�(KM � Cf)

where Vmax is the maximum rate of metabolism (in milligrams perhour) and KM is the Michaelis constant, or xenobiotic concentra-tion at one-half the maximum rate of metabolism (in milligramsper liter). Because many xenobiotics are metabolized by enzymesthat display saturable metabolism, the above equation is a key factorin the success of physiologic models for simulation of chemicaldisposition across a range of doses.

Other, more complex expressions for metabolism also can beincorporated into physiologic models. Bisubstrate second-order re-actions, reactions involving the destruction of enzymes, the inhi-bition of enzymes, or the depletion of cofactors, have been simu-lated using physiologic models. Metabolism can be also includedin other compartments in much the same way as described for theliver.

The usefulness of physiologic models for describing the com-plex toxicokinetic profiles resulting from saturable metabolism isresponsible to a large extent for the popularity of these models.The ability of physiologic models to describe experiments in whichmetabolism is altered using enzyme inhibitors or genetically mod-ified animals is illustrated in Fig. 7-12. The model used to producethe simulations in this figure accounted for inhalation of xenobi-otic, distribution to and uptake by tissues, and various states of me-tabolism (Jackson et al., 1999). The curves in Fig. 7-12 show thatwhen the maximum metabolic capacity, or Vmax, is diminished, theuptake of the chemical from the inhalation chamber into the bodyis reduced.

Blood In a physiologic model, as in a living organism, the tissuecompartments are linked together by the blood. Figures 7-5 and 7-6 represent different approaches toward describing the blood inphysiologic models. In general, a tissue receives a xenobiotic inthe systemic arterial blood. Exceptions are the liver, which receives

arterial and portal blood, and the lungs, which receive mixed ve-nous blood. In the body, the venous blood supplies draining fromtissue compartments eventually merge in the large blood vesselsand heart chambers to form mixed venous blood. In Fig. 7-5, ablood compartment is created in which the input is the sum of thexenobiotic efflux from each compartment (Qt � Cvt). Efflux fromthe blood compartment is a product of the blood concentration inthe compartment and the total cardiac output (Qc � Cbl). The dif-ferential equation for the blood compartment in Fig. 7-5 looks likethis:

dVbl � Cbl�dt � Qbr � Cvbr � Qtb � Cvtb � Qk � Cvk � Ql

� Cvl � Qc � Cbl

where Vbl is the volume of the blood compartment; C is concen-tration; Q is blood flow; bl, br, tb, k, and l represent the blood,brain, total body, kidney and liver compartments, respectively; andvbr, vtb, vk, and vl represent the venous blood leaving the organs.The venous blood is assumed to contain unbound chemical. Qc isthe total blood flow equal to the sum of the blood flows exitingeach organ.

In contrast, the physiologic model in Fig. 7-6 does not havea blood compartment. For simplicity, the blood volumes of the heartand the major blood vessels that are not within organs are assumedto be negligible. The venous concentration of xenobiotic returningto the lungs is simply the weighted average of the xenobiotic con-centrations in the venous blood emerging from the tissues:

Cv � (Ql � Cvl � Qrp � Cvrp � Qpp � Cvpp � Qf � Cvf)�Qc

where C is concentration; Q is blood flow; v, l, rp, pp, and f rep-resent the venous blood, liver, richly perfused, poorly perfused, and

Figure 7-11. Schematic representation of a flow-limited liver compart-ment in which metabolic elimination occurs.

R, in milligrams per hour, is the rate of metabolism. Ql is hepatic bloodflow, Cin is chemical concentration into the liver compartment, and Cout ischemical concentration out of the liver compartment.

Figure 7-12. Disappearance of butadiene vapor after injection (at t � 0)into a closed chamber containing mice.

The experimental data are the results of measurements of air samples takenfrom the chamber. The lines are physiologic model simulations of the ex-periments. After an initial period of equilibration, further decline in thechamber concentration is due to metabolism of butadiene by the mice. Thegreater the rate of metabolism the faster the decline in the butadiene con-centration in the chamber. Mice that do not express the gene for the en-zyme CYP2E1 (black circles) metabolize less butadiene than mice that doexpress the gene for CYP2E1 (blue squares). [From Jackson et al., 1999,p. 4. Reproduced with permission from CIIT Centers for Health Research.

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fat tissue compartments, respectively; and vl, vrp, vpp, and vfrepresent the venous blood leaving the organs. Again the ven-ous blood is assumed to contain unbound chemical. Qc is the to-tal blood flow equal to the sum of the blood flows exiting each organ.

In the physiologic model in Fig. 7-6, the blood concentrationgoing to the tissue compartments is the arterial concentration (Cart)that was calculated above for the lung compartment. The decisionto use one formulation as opposed to another to describe blood ina physiologic model depends on the role the blood plays in dispo-sition. If the toxicokinetics after intravenous injection are to besimulated or if binding to or metabolism by blood components issuspected, a separate compartment for the blood that incorporatesthese additional processes is the best solution. If, as in the case of the volatile organics shown in Fig. 7-6, the blood is simply aconduit to the other compartments, the algebraic solution is acceptable.

CONCLUSION

This chapter provides a basic overview of the simpler elements ofphysiologic models and the important and often neglected as-sumptions that underlie model structures. Detailed descriptions ofindividual models of a wide variety of xenobiotics have been pub-

lished. Several review articles describing how to construct a modelstep by step are included in the references. Computer software ap-plications are available for numerically integrating the differentialequations that form the models. Investigators have successfullyused Advanced Continuous Simulation Language (Pharsight Corp.,Palo Alto, CA), Simulation Control Program (Simulation Re-sources, Inc., Berrien Springs, MI), MATLAB (The MathWorks,Inc., Natick, MA), Microsoft Excel, and SAS software applicationsto name a few. Choice of software depends on prior experience,familiarity with the computer language used, and cost of the ap-plication.

The field of physiologic modeling is rapidly expanding andevolving as toxicologists and pharmacologists develop increasinglymore sophisticated applications. Three-dimensional visualizationsof xenobiotic transport in fish and vapor transport in the rodentnose, physiologic models of a parent chemical linked in series withone or more active metabolites, models describing biochemical in-teractions among xenobiotics, and more biologically realistic de-scriptions of tissues previously viewed as simple lumped com-partments are just a few of the most recent applications. Finally,physiologically based toxicokinetic models are beginning to belinked to biologically based toxicodynamic models to simulate theentire exposure � dose � response paradigm that is basic to thescience of toxicology.

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