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m\l( <)l <X,V \\l> VITI II OPIIAKMACOI (X A 102, 300-3 I 5 ( 1990) Precision and Sensitivity of Pharmacokinetic Models for Cancer Risk Assessment: Tetrachloroethylene in Mice, Rats, and Humans Frederic Y. Bois,* Lauren Zeise^ and Thomas N. Tozer* *Department ofPharmacy, University ofCalifornia. San Francisco, California 94143-0446, and ^California Department olHealth Scrviees/RCHAS. Annex 2, 2ISI Berkeley Way, Berkeley, California 94704 Received February 24, 1989: accepted October 3, 1989 Precision and Sensitivity of Pharmacokinetic Models for Cancer Risk Assessment: Tetrachlorocthvlene in Mice, Rats, and Humans. Bots, F. Y,, Zeise, L., and Tozer, T. N. (1990), Toxi col Appl. Pharmacol. 102, 300-315. Pharmacokinetic analyses have recently been incorpo rated in risk assessments, with resultant risks sometimes lower and associated "allowable" expo sures higher, than would have been otherwise calculated. Predictions of coupled pharmacokinetic and multistage models, as used for regulatory purposes, are evaluated here for tetrachloroethylene carcinogenicity in mice, rats, and humans. Precision is studied by treating parameters as random variables and determining the range of risk estimates once parameter uncertainties are considered via Monte Carlo simulations. The methods developed in this study are of interest for any similar application. The resultant median risk estimate for humans ex posed continuously to I ng/liter of tetrachloroethylene in the air is 1.6 per million and 5, 25, 75, and 95 percentiles are 0, 0.04, 2.8, and 6.8 per million. Sensitivity of the pharmacokinetic model predictions to its parameters is assessed by analyzing the results ofthe Monte Carlo simu lations. The kinetic parameters defining the metabolic rate are the most important for the case Studied. 1990 Academic Press, Inc. 1 A f The coupling of physiologically based phar macokinetic models (PBPK) to multistage models of carcinogenesis has been recently used as a means to improve the assessment of risk from exposure to xenobiotics. Physio logic models describe the distribution and the biotransformation of chemicals and poten tially provide estimates of the target tissue ex posure to active carcinogens (e.g., rate of pre sentation of the active metabolites to the affected tissue). These models have been pro posed to scale exposures among different spe cies, different experimental conditions, or different doses (Andersen and Ramsey, 1983; Bischoff et al,, 1971; Clewell and Andersen, 1985; Gerlowski and Jain, 1983; Lutz et al,, 1984; Ramsey and Andersen, 1984; Ramsey andGehring, 1980; Travis, 1987; Ward et al,, 1988). To estimate the cancer risk in humans a dose-response model for carcinogenesis, for example the Crump multistage polynomial model (Anderson et al,, 1983; Crump, 1984) used for regulatory purposes, is needed to ob tain a relationship for cancer risk as a func tion of target tissue exposure. Following the U.S. Environmental Protection Agency (EPA, 1986), the dose-response relationship obtained in animals is assumed to be the same for humans when exposure is expressed in terms ofthe carcinogenic metabolite(s) for mation rate, scaled to the two-thirds power of the body weight. Typically, the uncertainties associated with model misspecification or pa rameter variability are not explicitly taken into account and the model output is as sumed to provide exact estimates of risk. Quantitative knowledge of these uncertain ties should aid the user of this approach to risk assessment in determining the confi dence to be placed on the results (Bogen and McKone, 1988; Bogen and Spear, 1987; Bois 0041-008X/90 $3.00 Copyright 1990 by Academic Press, Inc. All rights of reproduction in any form reserved. 300 d oL L In thisp evaluate based oi nr these we desci rather t samplir 1964; P tions is risk, Rv PBPK t present lore g dose ar Fori perforr used vc by the (EPA, and by on Cat carcin' were f (NTP, ried o F344/ PBPK The dez et and A tissues cles ai and Vi model ancec taboli satuic ship. P450 subse urabl than er Risk ans tCalifornia 94704 chloToxi>rpoxpoi pled e for 'ting leter ludy i ex. 25, netic imucase polynomial rump, 1984) eeded to ob* >k as a func. Following tion Agency relationship d to be the is expressed bolite(s) forrds power of mcertainties ration or palicitly taken utput is asites of risk. ; uncertainapproach to the confi(Bogen and ", 1987; Bois PRECISION 01- PHARMACOKINETIC models 301 ci al., 1988, 1989; Hakanoglu el ai, 1985). In this paper, a statistical approach is used to evaluate the precision of a risk assessment based on the coupling of PBPK and multi stage models. Because the parameter values of these models are not known with certainty, we describe them by probability distributions rather than by a fixed value. Monte Carlo sampling (Hammersley and Handscomb, 1964; Press et ai, 1986) over these distribu tions is used to generate a distribution of the risk. Results of a sensitivity analysis of the PBPK component to its parameters are also presented. Model bias is discussed in terms of more general assumptions pertaining to low dose and interspecies extrapolations. For illustrative purposes, risk assessment is performed on tetrachloroethylene, a widely used volatile solvent which has been classified by the U.S. Environmental Protection Agency (EPA, 1985) as a probable human carcinogen and by the International Agency for Research on Cancer (IARC, 1979) as a known animal carcinogen. The cancer bioassay data used were from the National Toxicology Program (NTP, 1986) in which experiments were car ried out in both sexes of B6C3F1 mice and F344/N rats. METHODS The Models Km Fig. 1. Schematic representation of the physiological model used to simulate the distribution and metabolism of tetrachloroethylene. The symbols used are VR, alveo lar ventilation rate; P,, blood over air partition coeffi cient; <2,, total blood flow; KTM*, maximum rate of me tabolism; Km, Michaelis constant for the kinetic model of metabolism. Subscripts representing the tissues are used to identify the blood flows to the tissues, Q, the vol umes. V, and the tissue over air partition coefficients, P. The units of the parameters are given in Table 1. PBPK Model The model used (Fig. 1) is the same as that of Fernan dez et al- (1977), Droz and Guillemin (1983), Ramsey and Andersen (1984), and Andersen et al (1987). The tissues are grouped as follows', fat, slowly perfused (mus cles and skin), richly perfused (central nervous system and viscera, except liver), and metabolizing (liver). The model is mathematically described by a set of mass-bal ance differential equations. For tetrachloroethylene, me tabolism is assumed to occur only in the liver and to be saturable, according to the Michaelis-Menten relation ship. The formation of an epoxide by the cytochrome P450 system is a common step to the formation of the subsequent major metabolites and is assumed to be a sat urable mechanism (Dekant, 1986). The epoxide, rather than the parent compound, is considered to be the "effective" or carcinogenically active form (EPA, 1985). As such, its daily rate of formation at steady state, or equivalently the rate of formation of metabolites, is taken as the measure of the exposure to the carcinogen. The same approach has been used by EPA (1985) and Bogen and McKone (1988). Multistage Model The so-called linearized multistage, or Crump polyno mial, model (Crump, 1984) is used to relate the lifetime probability or "risk" of cancer, R(E), to the effective ex posure, E, to the carcinogen, namely, the rate of forma tion of the epoxide or its surrogate measure, the rate of metabolism SL 035496 WMMHM 302 BOIS, ZEISE, AND TOZER (/:)= I -exrt-Xtf,-/;') o 0) with a, si 0 for all i. The coefficients a,, through u,, are estimated by maxi mum-likelihood analysis, using the program "Mstagc" (Crouch, 1985). The NTP data for tetrachloroethylene allow the fitting of at most three parameters, so that n was fixed to 2 in our analysis. The coefficient a,, or some statistical upper bound, is a measure of the carcinogenic activity of the chemical studied. It is defined as the "unit risk" by the EPA (Anderson et at.. 1983) and is similar to the "potency" of Crouch and Wilson (1981). "Extra risk," R, from exposure E is taken as the difference be tween the lifetime nsk R(E) and the background risk in the absence of exposure R(0), relative to the fraction of the population expected to be cancer-free in the absence of exposure to the chemical (Anderson et at., 1983; Crump, 1984), and is given by R(E)~R(0) R~ i-*(0) (2) At low doses, extra risk then becomes R=*a,-E. (3) herty (1985), Dekant cl at. (1986). Mnoma et at. (1985) and Schumann el at (1980). All exposures were by ga. vage, except for those of Schumann cl at in which an inhalation exposure was also assessed. The conditions and animal characteristics described in the studies were simulated. For simulating the data of Buben and O'Fla herty it was assumed that urinary metabolites represent 65% of the total metabolites and that this value is inde pendent of the administered dose (EPA, 1986; Ward et at., 1988). For rats, the experiments of Dekant et at. (1986), Frantz and Watanabc (1983), Mitoma et at. (1985), and Pegg et at. (1979) were used. In contrast to the other investigators, Dekant et at. (1986) used female animals. For humans, the data used were from occupational surveys of Japanese female and male workers (Ikeda et at., 1972;Ohtsukie/o/.. 1983). Selected parameter values were adjusted, according to the anatomical model of Kerr et at. (1976), for differences in body weight and composition between average Japanese and American adults (Bogen and McKone, 1988). On the basis ofexper imental data in animals (Dekant et at.. 1986; Frantz and Watanabe, 1983; Mitoma et at . 1985; Pegg el a!.. 1979; Schumann et at.. 1980), the fraction of total metabolites recovered in the urine, in man, was assumed to be 65%. PBPK Model Parameters and Their Probability Distributions Parameter Estimates The majority of physiologic and kinetic parameter val ues correlate with measures of body sizefe.g., body weight and body surface area). Normalization of the pa rameters gives a scaling coefficient which, when multi plied by the measure, yields the parameter value. Table I lists such scaling coefficients and the appropriate multi plier (measure) for the parameters used. Median values for organ volumes, blood flows, and pulmonary ventila tion rate are independent of the chemical in question. Partition coefficients were estimated from data obtained mv/Vro (EPA, 1986). The values for and K,, scaling coefficients were derived by fitting the PBPK. model to in vivo data on rate of metabolite formation in response to tetrachloroethyl ene exposure. Statistical likelihood procedures were used: the likelihood function was obtained by tabulation under the assumption that the differences between pre dicted and experimentally observed values are normally distributed (Edwards, l972),The Km,, and K,, scaling co efficients corresponding to the maximum tabulated like lihood were taken as "maximum-likelihood estimates." During this fitting the other parameters were fixed to their median values. For mice, and K,, scaling coefficients values were estimated from the experiments of Buben and O'Fla Parameter Distributions All the model parameters are subject to uncertainty. Parameter correlations must be taken into account, so that only physiologically realistic values are chosen. To do so we assigned probability distributions to the scaling coefficients (see Table 1), rather than to the parameters themselves. This ensured that the covariance structure imposed by the scaling is incorporated into the subse quent Monte Carlo simulations. For example, in the Monte Carlo simulations, liver volume and body weight are not allowed to vary independently, (i.e., with inde pendent probability distributions); instead, the ratio of liver volume to body weight is allowed to vary within physiologically realistic values. The strong covariance of the estimates of the fitted kinetic parameters, fmt< and Km, was also taken into account. All the distributions were put in tabulated form for computation. The lower and upper limits of the scaling coefficient distributions are given in Table 1, except for Vm,, and K,,, for which reduction to one dimension was not appropriate (see Re sults). We now describe how these distributions were ob tained. Volumes. The probability distributions for the volume coefficients (percentages ofthe body weight) were derived from experimental data(S. Oie, University ofCalifornia, personal communication). Individual measurements of liver volumes and body weight were obtained in 130 rats. The observed frequency distribution of their ratio was used to describe the shape of the percentage volumes for all organs and species. u < n at Cu- Qwtfi o tr a; < C Z ecc 7 C SL 035497  y ii a 3 *>' g i SSS i, t 3 p 3 3- 5 S. I Oan "n/HI- ^e. oo Ico. sSg 5<f-oS57? j3 5 3 |o ; o rit S-- n3* u$$i 54 k ^ Ji $ (V pf y^ a*1 o S' g. - g 3. ill s=s * 5 2S a 3 R3- S 22, Iff a a e. * *1;' ~ ioi 5 Q | ? S g- it N <53 :1 ik- _a. 32 iai''3's35 .=.. ;3! Ono;-'' III 3 S-oH-^-S 9 2 4> 2 I 3 5 jj*.- ,^?- J* Vc-> 'V -W PRECISION O f PHARMACOKINETIC MODELS TABLEi Parameter Scaling Coefficients" and Their Variation Bounds Used for Tetrachloroethylene in Mice, Rats, and Humans Mice Rats Humans Scaled parameter Multiplier Median Bounds Median Bounds Kerr" EPA` Bounds*' Body weight (Bw) Alveolar ventilation rate (0.) Total blood (low (Q,) Blood flows Metabolizing tissue group (Q,) Fat tissue group (0f) Richly perfused tissue group(0r)` Poorly perfused tissue group (0P) Volumes Metabolizing tissue group (kj) Fat tissue group(vf) Richly perfused tissue group (1',) Poorly perfused tissue group (k',,)' Blood/air partition coefficient (P,) Tissue/air partition coefficients Metabolizing tissue group (P|) Fat tissue group (Pr) Richly perfused tissue group (Pr) Poorly perfused tissue group (Pp) Maximum rate of metabolism (kmi>)* Affinity constant {K,,,Y Intestinal absorption constant (K, J* 1 Bw01* Bw"'4 a a a a Bw Bw Bw Bw 1 1 1 1 1 Bw"' 1 l 0.035 0.19 0.24 0.37 0.09 0.42 0.12 0.04 0.09 0.05 0.72 16.9 70.3 2060 70.3 20 0.06 12.8 0.01 [0.03,0.04] [0.12,0.27] [0.19,0.30] [0.30,0.45] [0.07,0.11] -- [0.09,0.15] [0.03,0.05] (0.06,0.1 1] [0.03,0.06] -- [13,21] [56,84] [1600,2500] [56,84] [16,24] ~ -- -- 0.35 0.18 0.23 0.37 0.09 0.42 0.12 0.04 0.09 0.04 0.73 18.9 70.3 2060 70.3 20 0.017 6.7 0.01 [0.3,0.4} [0.11,0.25] [0.18,0.27] [0.30,0.45] [0.07,0.11] -- [0.09,0.15] [0.03,0.05] [0.06,0.11] [0.03,0 05] -- [15,23] [56,84] [1600,2500] [56,84] [16, 24] -- -- -- 55 0.26 0.27 0.25 0.04 0.52 0.19 0.02 0.15 0.10 0.63 10.3 38.3 1123 38.3 10.9 0.007 1.5 0.01 70 0.15 0.19 0.37 0.09 0.42 0.12 0.02 0.20 0.05 0.63 10.3 38.3 1123 38.3 10.9 0.007 1.5 0.01 [30, 110] [0.07,0.22] [0.13.0 24] [0.26,0 48] [0.06,0 12] -- [0.08,0.16] [0 01,0 03] [0.12.0.28] (0.03.0.07) -- (6. 15] [23.54] [670. 1570] [23. 54] |6 5. 15) -- -- -- Note. Scaled parameter = multiplier x scaling coefficient. Units: body weights in kg, flows {Q) in liter/m in, volumes (V) in liters, Fm,, in mg/min, and K,,, in mg/liler. * Median values are from EPA (1986). In addition human values from Kerr et al. (1976) are listed. The Vm,, and K,,, values are our own estimates (see text). * These values were used to fit Pm,, and K,, from data obtained in Japan (see text). Volumes and flows are taken from the anatomical model of Kerr et a! (1976) c These values were used in the Monte Carlo simulations to compute the rate of metabolism and the risk ofcancer in the American adult (see text). `'These bounds apply to the human parameter values of the EPA (1986). ' Values for this parameter were computed at each simulation so that the sum of the blood flows was equal to 100% of the total Bow. /Values for this parameter were computed at each simulation so that the sum of the volumes was equal to 90% ofthe body volume. * Median values are the maximum-likelihood estimates obtained in this study. The ranges ofvariation are defined by joint probability functions (see Results). * This parameter was not varied in the Monte Carlo simulations as inhalation exposure was the only one simulated. I 304 HOIS. ZMISi:. AND lOZl.K FIG. 2. Approximation of a truncated normal density function (solid line) by a cosine density function (dotted line), defined by the derivative of Eq. (4) (in the text). The histogram was constructed by random sampling of 500 values using the cosine function. I m,i, and A.,,, The dismbulion function for the scaling coefficients of these parameters is obtained from their likelihood function. The likelihood function, under tab ulated form, was obtained for about 500 regularly spaced pairs of I and A,, scaling coefficient values during the search for the maximum-likelihood estimates. The fitting procedure, the noise in the data, and the structure of the kinetic model result in both a correlation between the estimates of F'm,, and A,,,and an error distri bution (or uncertainty) for each of them. It is important to take these "statistical artifacts" into account. In fact, the covariance and error distribution of these estimates is exhaustively described by their joint, or bivariate, dis tribution function. This distribution function is propor tional to the likelihood function of FTM, and K,,, given the data observed (Edwards, 1972; Kalbfleisch, 1985). The proportionality constant must be such that the inte gral of the distribution function equals 1. Multiplication of the tabulated values by the appropriate constant gives the joint distribution function of the estimates. Determination ofUncertainty in Risk Estimates Body weight,flows, andpartition coefficients. For these scaling coefficients, for which frequency distribution in formation is not readily available, we assumed the cumu lative distribution function F{x) (1 -- COSfT-x)) Fix) = (0=gjr 1) (4) for which b- a' <5: The parameter P is defined only over the closed inter val [a, 6], to avoid negative and other unrealistic values. This function approximates a truncated normal distribu tion function (Fig. 2). To define the bounds a and b, the body weights of mice and rats in the carcinogenicity study ofthe National Tox icology Program (NTP, 1986) were assumed to deviate by at most 15% from the average weight, at a given age. The average weight was approximately 35 g for mice and 350 g for rats during the NTP experiment. For humans the body weight had a median of 70 kg and bounds of 30 and 110 kg. In the absence ofgood data on interindividual variabil ity, the limits of flows and partition coefficients were best guesses from inquiries of animal experimentalists. For rats and mice, the scaling coefficient bounds for blood flows, ventilation rate, and partition coefficients were, 20, 40, and 20% of the median value, respectively. For humans these bounds were 30, 50, and 40% of the median value, respectively. The probability distribution for risk estimates in hu mans is derived from probability distributions of the rate of metabolite formation in animals. the carcinogenic potency, expressed in terms of me tabolite formation. the rate of metabolite formation in humans. The distributions for metabolite formation were gen erated using the PBPK model, by Monte Carlo tech niques, with runs involving 500 simulations. The uni form random number generator, provided with the Sili con Valley Software FORTRAN version 2.5, was used. Distribution ofthe Rate ofMetabolite Formation in Ani mals The parameter values of the physiologic model were picked at random, according to their tabulated distribu tions. For Fm,, and K,, the tabulated joint distribution function was used. A linear interpolation between the tabulated points of the distribution functions was made for both the univariate and the joint distribution cases. For both mice and rats, 500 sets of randomly selected parameter values were used to run the PBPK model and determine the average daily rate offormation oftotal me tabolites for the two inhalation exposures in the NTP (1986) experiment (100 and 200 ppm in air for mice, and 200 and 400 ppm for rats, for 6 hr/day, 5 days/week). Each run represented 5 days of exposure. Following the EPA (1986), the rate of metabolism on the last day, mul tiplied by 5/7, was taken as an estimate of the average stcady-sta innate doIlie averai emulated were div. power, as Msure ti oumed estimatci tabohtes the twoi used to i the rate sure Icvi Potency Each was use mullist. occurrc mice, a male r, general tions w the po1 ered. 3 nved i tabolii dence accun convo cumu Distri As conti in aii form laine as it' ulati extr. of fc pote obU met con SL 035499 (ion for ihc scaling lamed from their lunclion, under lab500 regularly spaced mi values during the estimates. m the data, and the in both a correlation ,, and an error distrihem. It is important no account. In fact, n of these estimates nt, or bivariate, dis function is propor(/m. and Km, given Kalbfleisch, 1985). esuch that the inteils 1. Multiplication iriate constant gives estimates. ertainty :'S k estimates in huibutions of n animals. ed in terms ofme- ffation were genlonte Carlo techilations. The uniided with the Sili'n 2.5, was used. ''ormation in Ani- logic model were ibulated distribuloint distribution non between the ictions was made tribution cases, indomly selected jBPK model and ation oftotal me-tres in the NTP air for mice, and y, 5 days/week), e. Following the he last day, mule of the average PRECISION OF PHARMACOKINETIC MODELS 305 steady-state value. Preliminary runs showed that this es timate does not deviate by more than a few percent from the average steady-state value obtained alter 3 weeks of simulated exposure. The rates of metabolite formation were divided by the body weights to the two-thirds power, as done by EPA (1986) to normalize effective ex posure to surface area. The same normalized rale was assumed for both sexes of mice or rats. In this way, 500 estimated pairs of effective exposure (formation of me tabolites at steady state) were obtained, corresponding to the two inhalation exposures levels. These 500 pairs were used to construct a cumulative distribution function of the rate of formation of total metabolites for each expo sure level. Potency Distribution Each one of the 500 pairs of exposure estimates, E, was used to generate a potency distribution by fitting the multistage model (Eq. (I)) to the cancer data (lifetime occurrence of hepatocellular tumors in male and female mice, and mononuclear cell leukemia for male and fe male rats). In this way 500 potency distributions were generated for each species and sex. These 500 distribu tions were convolved by Monte Carlo sampling to obtain the potency distribution for the species and sex consid ered. Thus, carcinogenic potency distributions were de rived in terms of the daily rate of formation of total me tabolites for rats and mice of each sex. As there is no evi dence that any one of these potency distributions more accurately represents the picture in humans, they were convolved using Monte Carlo sampling to produce the cumulative distribution of potency for humans. Distribution ofRiskfor Humans As an example case the cancer risk to humans from a continuous exposure to I ng/liter of tetrachloroethylene in air was assessed. The distribution of the daily rate of formation of total metabolites, at steady state, was ob tained by Monte Carlo simulations in the same fashion as it was for animals. Steady state was achieved in all sim ulations after 14 days ofexposure. Following Eq. (3), the extra risk for humans is equal to the product of the rate of formation of total metabolites E by the carcinogenic potency a,. The probability distribution of the risk was obtained by convolving the distributions of the rate of metabolism in humans and carcinogenic potency. This convolution was achieved by Monte Carlo sampling. Sensitivity Analysis The sensitivity of the pharmacokinetic model was studied by examining the correlation between the rate of metabolism, normalized for body surface area, and, for each physiologic parameter, the values generated during the Monte Carlo simulations. This analysis was per formed for condition of low human exposure by continu ous inhalation RESULTS Adjustment ofVmay and Km Scaling Coefficients Maximum-likelihood estimates of Um,, and Km scaling coefficients obtained by fitting the full model to data for mice, rats, and hu mans are provided in Table 1. The contour plots of the joint distribution functions of these two parameters, for mice, rats, and hu mans, are presented in Fig. 3. In this figure the statistical correlation between V__and Km is evident: the major axes of the concen tric shapes are not parallel to either the x- or the y-axis. Model predictions of the rate of metabolite formation, made by taking the maximum-likelihood estimates for Fm,, and K,,,, and median values for the other parame ters are compared to observed rates in Fig, 4. On average, model predictions do not deviate by more than 22% from the observations. As an example, the maximum-likelihood fit of the model to the data of Buben and O'Fla herty (1985) is presented in Fig. 5. The pre dictions fit the data well. Frequency Distributions Figure 6 presents the results of the Monte Carlo simulations for the rate of metabolite formation in mice and rats after exposures to tetrachloroethylene. In mice, the median rate of metabolite for mation is 7.3 mg/day/kg2/3 (with 5 and 95 percentiles of 5.7 and 9.3 mg/day/kg2/3) at 100 ppm. On doubling the exposure to 200 ppm the median rate increases to a value of 12.7 mg/day/kg2/3 (with 5 and 95 percentiles of 10.4 and 15.3 mg/day/kg2/3). 306 100 hois. zi:isl. and iozir .01 .1 Vmax Scaling Coefficient (mg/min/kg^ ^) Fig. 3. Contour plots of the joint distribution functions obtained for and K,, in mice (A), rats (B), and humans (C). The concentric curves define the conditional confidence intervals for the two parameters and are labeled according to their p values. The two innermost curves define the 20% (p = 0.2) and 50% (p = 0.5) confidence intervals. The shapes were smoothed using spline interpolation. In rats, the median rate of metabolite for mation is 9.8 mg/day/kg2'3 (with 5 and 95 percentiles of 6.9 and 12.4 mg/day/kg2/1) at 200 ppm. At 400 ppm exposure, the median rate of formation increases to a value of 13.9 mg/day/kg2/3 (with 5 and 95 percentiles of SL 035501 PRECISION Ol PHARMACOKINETIC MODELS 307 l. rats(B), jrameters id50%(p the median ilue of 13.9 rcentiles of 11.8 and 16.8 mg/day/kg2/3). In both mice and rats the medians of metabolite formation rates increase with the rate of exposure to ward a limiting value (saturable metabolism). The frequency distributions ofthe carcino genic potency, for mice and rats ofboth sexes, and their convolved distribution are pre sented in Fig. 7. After convolution the me dian potency value is 0.032 (mg/day/kg2/3)_1 with 5 and 95 percentiles of 0 and 0.11 (mg/ day/kg2/3)~'. Figure 8 shows the distribution obtained for the rate of formation of total metabolites in humans, for a continuous exposure to 1 ng/liter of tetrachloroethylene. The rate of metabolism at steady state (after surface area correction) has a median of 58 ng/day/kg2/3 and 5 and 95 percentiles of 34 and 104 ng/ day/kg2/3. Due to the covariance of Vmax and Km estimates, the variability of the rate of for mation of metabolites is much lower than might be expected from the variability of these parameters (the 5 and 95 percentiles of Vmax and Km marginal distributions were 0.07 and 0.18 mg/min and 0.4 and 2.1 mg/liter. respectively). Figure 9 presents the cumula tive distribution of the predicted cancer risk for humans. The corresponding frequency distribution is skewed and shows a median of 1.6 per million with 5,25, 75, and 95 percen tiles of 0, 0.04, 2.8, and 6.8 per million, re spectively. These results take into account variation in both the rate of formation of the total metabolites and the carcinogenic po tency. Model Sensitivity The results of the sensitivity analysis are presented in Table 2. It appears that the model predictions of steady-state rates of me tabolite formation are particularly sensitive to changes in Km, Vm,,, and the partition co efficients. Examples of representative scatterplots are shown in Fig. 10. DISCUSSION Precision The analyses presented here address the de gree to which the precision of risk estimates Si 3SSo3 308 HOIS. /.MSI.. AND lOZIiR 11! 10; Jf o O mice 0) X/ mice A mice <2> (3) / 'A O mice rote <4> (2) rate (4) 3 o rate (5) x rate M> V humans <7,> .01 .01 .1 1 10 100 Observed Amount of Metabolites Formed (mg) Fig. 4. Model predictions based on maximum-likelihood estimates of Fm,, and K,, are compared to in vivo observations for rats, mice, and humans. The numbers between parentheses in the legend correspond to the following references: (I) Schumann et al (1980); (2) Mitoma et al. (1985); (3) Buben and O'Flaherty (1985); (4) Dekant et al. (1986); (5) Pegg et al (1979); (6) Frantz and Watanabe (1983); (7) Ikeda et al. (1972); (8) Ohtsuki et al (1983). is influenced by uncertainty in parameter val ues. This precision can be described by the frequency distribution of the risk estimates and may be somewhat underestimated here, due to several simplifications that were made in specifying the distribution functions of the physiologic model parameters. First, we took into account the interdepen dencies among parameters (e.g., organ vol umes and blood flow rates) by scaling to body size. The relationships used to scale parame ters were assumed to be known with cer tainty. This assumption leads to an underesti mate of the variance associated with the pa rameters of the model. Yet, sensitivity analysis, as discussed below, shows that un certainty in the parameters, other than Fmax and K,,, has a limited effect on the results. Second, in the absence ofsuitable data, cer tain parameter distributions were based on the judgement of expert researchers. For blood flows, ventilation rates, and partition coefficients, an idealized symmetric distribu tion was used, which may not be realistic. Third, although, for Vm,, and Km, a joint distribution function was obtained, which de scribes adequately the major part of their variability and interaction, these parameters were assumed not to covary with the partition coefficients or the rate of intestinal absorp tion. No data were available to assess such covariances. Measures of exhalation rate after oral gavage would allow a statistical adjust ment of the oral absorption rate and a more extensive study of the covariance structure of the model. Furthermore, only limited data too O' -* 80 60 4C 035503] Ih reared to m orrespond ' VFlaheny da el al. ble data, cerre based on rchers. For nd partition ric distriburealistic. Am, a joint d, which de art of their parameters he partition nal absorp;ess such con rate after :ical adjustand a more structure of mited data I'iiU'ISION ()t I'llMtMACOkINFTIC MODFl.S Sensitivity 309 Fig. 5. Relationship between the oral rate of adminis tration of tetrachloroethylene and the rate of formation of metabolites as assessed by the rate of their urinary ex cretion in mice (Bubcn and O'Flaherty, 1985). The model predictions (solid line) fit the data well. The means (SE) at each rate of administration assayed are shown. sets with few dose groups or individuals were available for fitting Kmax, Km, and the param eters of the multistage model. The distribu tions of estimates for these parameters may not correspond to their distribution in a large population. For example, FmiX and Km values for humans were derived from data in a ra cially homogeneous population of healthy adult male workers (Ikeda et al., 1972; Ohtsuki et al., 1983). Although other data in hu mans are available (Monster et al., 1979; Monster, 1979; Monster and Houtkooper, 1979), they were not used, because only the urinary excretion of trichloroacetic acid was monitored, when the data of Ikeda and Ohtsuji (1972) show that other metabolites may also be excreted. We decided to use the total metabolites in that trichloroacetic acid alone would have underestimated the rate of me tabolism of tetrachloroethylene in humans. Finally, there is an error associated with the use ofany finite number of Monte Carlo runs (in particular in the estimates of the tails of the distributions). With 500 runs, the errors in the percentiles reported here should be rel atively small. The relationship between the rate of me tabolism and the estimates of Fmax shows par adoxical behavior: the rate decreases when Vmax increases (Fig. 10c), when just the oppo site is expected. This is due to the fact that Fmax and K,,, estimates are statistically corre lated. The form of the correlation, described by the joint distribution function shown on Fig. 3c, implies that the ratio, Kmax/Am (to which the rate of metabolism is proportional at the low dose studied), increases more rap idly than Kmax. Choosing at random a high Fmax in the Monte Carlo simulations implies selecting a Km such that, on average, the ratio Vmm/Km and the corresponding rate of me tabolism are lower than when a low Fmax is picked. For example, for the rat data shown in Fig. 3b: when Fmax is sampled around 0.1, Km is most likely, given the data, to be in the neighborhood of 2 (Kmax/^fm = 0.05), but when Fmax is around 0.2, K,, is at about 10, Fig. 6. Monte Carlo-generated cumulative distribu tion of the rate of metabolite formation (mg/day/kg3'3) in mice (dotted lines, 100 and 200 ppm exposures) and rats (solid lines, 200 and 400 ppm exposures). Exposure conditions of the NTP (1986) inhalation bioassays for tetrachloroethylene were simulated. To obtain the rate in units of mg/day these results may be multiplied by the body weight to the two-thirds power. 310 BOIS. 7.FISF. AND lOZl'K COKRI'I \ Mi l MM" >' 1)1 H" I'l PUKING' I'll lllOKOm Fig. 7. Monte Carlo-generated cumulative distribu tions of the carcinogenic potency of tetrachloroethylene in male and female mice (dotted lines) and rats (dashed lines) and the potency distribution (solid line) obtained by convolution ofthe preceding ones. NTP (1986) cancer data were used to fit the multistage model. The variability in potency incorporates the expected variability in the rates of metabolite formation (cf. Fig. 6). on average, so that VmiJKm = 0.02. The ratio drives the behavior of the model at low expo sure, this is why Vm3X/K,, and the rate of me- Rote of Metabolism (ng/day per kg2'3) Fig. 8. Monte Carlo-generated cumulative distribu tion of the rate (mg/day/kgJ/3) of metabolite formation in humans. A continuous inhalation exposure to I ng/ liter oftetrachloroethylene was simulated. Fig. 9. Cumulative distribution of the human cancer risk estimate associated with a continuous inhalation ex posure to I ng/liter of tetrachloroethylene. tabolism tend to decrease when in creases. The sensitivity of the model results to the choice of parameter values was assessed by examining their one-to-one correlations (Ta ble 2, Fig. 10). Low correlations indicate model insensitivity to the parameter selection (for example, at steady state, following low inhalation exposure, the model is insensitive to the values of blood flows, alveolar ventila tion rate, and organ volumes). On the other hand, high correlations may be due to covari ance between parameters and not necessarily to a high sensitivity of the model. Only part of the covariance structure of the parameters was implemented, namely the dependence on body weight and the covariance between I'niiix and K,, estimates. The lack of relevant data prevented us from obtaining a better de scription of the correlations among parame ters. Due to these limitations, the high figures presented in Table 2 may be somewhat mis leading. However, the major features exhib ited by this approximation are likely to be valid in this case. The sensitivity analysis indicates that the kinetic model may be simplified without loosing any appreciable accuracy in the f Body wcigl Aleolaf vei Total bloo) Blood flov> Mctabol Fat tissu Poorly | Volumes Metabo Fat tissi Richly: Blood/au Tissue/aii Metatx Fat list Richly Poorly Maximu Affinity i "The: paramet this para model partm with Mich; analy check SL 035505 15 p#r million) tuman cancer inhalation ex- 3 Knax in- SUltS to th tssessed b ^jtitiions (Te indicat selectioi lowing lov insensitive iar ventila n the othei e to covarinecessarily Only part parameters ependence ze between of relevant 1 better deig paramerigh figures -what mis-tres exhibkely to be es that the d without cy in the PRECISION Ol PHARMACOKINETIC MODELS 311 TABLE 2 Corriiation Corn k ii;n!.s of ihe Raii oi Metabolite Formation wiiii iiii: Pakameiers oi the Pharmacokinetic Model as Observed ujring l he Mon i e Carlo Simulations of Tetra- i IILOROETH YLENE EXPOSURE IN HUMANS Parameter Correlation coefficient" Body weight Aleolar ventilation rate Total blood flow Blood flows Metabolizing tissue group Fat tissue group Poorly perfused tissue group Volumes Metabolizing tissue group Fat tissue group Richly perfused tissue group Blood/air partition coefficient Tissue/air partition coefficients Metabolizing tissue group Fat tisue group Richly perfused tissue group Poorly perfused tissue group Maximum rate of metabolism (Fm,,) Affinity constant (Km) 0.042 0.184 0.040 0.046 0.017 0.061 0.045 -0.016 0.006 0.391 -0.237 -0.329 -0.287 -0.225 -0.466 -0.727 The absolute value of the correlation coefficient for a parameter increases with the sensitivity of the model to this parameter. model predictions. The multiple body com partments can be replaced by a single one with reversible pulmonary exchange and Michaelis-Menten metabolism. Sensitivity analysis of the complete model is a means of checking the validity of such simplifications. Bias Errors due to misspecification ofthe PBPK or cancer models are likely to be many times larger than those due to parameter selection, particularly at low dose. Because low-dose data on metabolism and cancer risk are not available in both humans and animals, the models cannot be fully validated and their bi ases directly assessed. We chose to investigate the behavior of the multistage polynomial model ofcarcinogene sis, the one most commonly used by regula tory agencies. Bias in the predictions of this model has been discussed by a number of au thors (e.g., Freedman and Zeisel, 1988; Zeise et ai,, 1987). Major sources of bias occur in both the relationship between exposure and response and the measure of exposure. There may be multiple mechanisms ofcarcinogene sis for tetrachloroethylene, some of which are unlikely to be adequately described by the multistage polynomial. Furthermore, large interspecies differences, even for a given can cer mechanism, may be present. For exam ple, peroxisome proliferation may have a role in the induction of liver tumors by tetrachlo roethylene in mice, whereas this may not be true for rats (Dekant, 1986; Odum et al., 1988). The use of the rate of formation ofto tal metabolites (assumed to be proportional to the rate of formation of an epoxide) as a measure of effective exposure can be ques tioned. Other possible measures would be, for example, the cumulative exposure to the car cinogens, as suggested by Dedrick (1981), or the concentration ofactive metabolites in tar get tissues. It was assumed that both the hepatocellular tumors in mice and the mononuclear cell leu kemias in rats were equally indicative ofcan cer risk in humans. Under this assumption, the convolution of the potencies obtained from the four bioassay data sets gives a mea sure of potency in humans less biased than the value derived from selecting the most sen sitive species. Another issue here is whether the use of the PBPK model results in better estimates of effective exposure. A host of simplifications are made and several potentially nonlinear phenomena were not considered. For exam ple, in the physiologic model presented, it is assumed that metabolism occurs in the liver and that the active metabolite is too unstable to be transported far, and yet leukemia is ob served in the rat. Furthermore, detoxification reactions are not taken into account; the pos- SL 035506 312 ' 200-1 150- HOIS, Z.I.ISL. AND IOZIIR 200-i 150- B 100- 100- 50-1 50- 4 30 ` 200-, 150 100- 50 Human Body Weight (kg) 110 51i0 i "1 I 12 14 Blood/Air Partition Coefficient ` 200i D . * .* u 150100- 50- 50- *.. 0.0 0.2 0.3 Vmax (mg/L per min) 46 Km (mg/L) Fig. 10. Plot ofthe rate of metabolism of tetrachloroethylene (ng/day/kg2/,> in humans, observed during the Monte Carlo simulations, versus randomly generated values of (A) body weight; (B) blood over air partition coefficient; (C) maximum rate of metabolism (Fm,,); (D) affinity constant (Km). The correlation coefficients are 0.042,0.39,0.47, and 0.73, respectively. Large absolute values ofthe correlation coefficient indicate a high sensitivity. I sexes, so mals wen data do n der ditfei for extra arcinogt other ha\ the valid the effect tion for question procedu of our n tetrachlc Little these sir introduc mation) All thesi ranee ol to descr data. T proach' corpora analysis sibility of autoinduction of metabolism is not examined; important cellular (and pharma cokinetic) processes, such as DNA repair or DNA methylation, are ignored; in vitro data on partition coefficients for homogenized tis sues are assumed to be representative of the situation in vivo; and only a single enzymatic reaction following Michaelis-Menten kinet ics is assumed. It should be noted that unlike Ward et al. (1988), we included no parallel linear pathway for tetrachloroethylene me tabolism. For two reasons, there is no con vincing evidence of its contribution to the to tal elimination of tetrachloroethylene, and a good fit of the metabolic data, even for mice, can be obtained without such a component (Fig. 5). Sex specificity in tetrachloroethylene metabolism was not considered explicitly in the model: the fit to the animal data (Fig. 4) did not point to large differences between the 3 3550^1 B feJlV * --------- 1-- 2 14 fficient 16 D I--------- 1 id during I over air >rrelation ^efficient iylene meis no conn to the tolene, and a n for mice, omponent roethylene xplicitly in ita (Fig. 4) etween the i'ki.osion or imiarmACOKiNr/nc models 313 sexes, so that data on female and male ani mals were pooled. For humans, the available data do not allow for the investigation ofgen der differences. Finally, general procedures for extrapolating metabolic parameters or carcinogenic potency from one species to an other have not been validated. In this respect the validity of surface area normalization for the effective exposure, in addition to a correc tion for the metabolic parameters, can be questioned. We adopted this normalization procedure in order to enable the comparison of our results with EPA risk assessment for tetrachloroethylene. Little is known about the importance of these simplifications or the biases that they introduce. The direction (under- or overesti mation) of the bias is frequently not certain. All these simplifications are due to our igno rance of the processes involved, our inability to describe them adequately, and a lack of data. To account for these biases, the ap proach we have taken may be extended by in corporating some procedures of decision analysis. ylenc in air. Taking the uncertainty in the model parameters into account yields, for the same exposure level, values for the 5, 25, 75, and 95 percentiles ofO, 0.04, 2.8, and 6.8 per million. Because the experimental data, in particular in humans, are still inadequate, these percentiles may not be accurate. Never theless, they point to the precision that might be expected from regulatory risk assessment procedures, if the models used are assumed to be correct. Unfortunately, as discussed, the biases introduced by an improper choice of models could be many times larger than the errors due to the variability ofthe parameters. Considerable time may elapse before scien tific understanding of the processes involved allows for the confident use of any particular model of carcinogenesis. The use of pharma cokinetics is an area in which considerable improvement over the current approach might be achieved. Our results emphasize that elucidation of both the metabolic pro cesses and the mechanism of carcinogenesis is of primary importance. CONCLUSION The approach developed here has many potential applications. Monte Carlo simula tions are a powerful tool for studying the be havior of complex biomathematical models. Likelihood techniques allow for a rigorous description of parameter covariance, such a description being essential during the Monte Carlo sampling. Sensitivity analysis gives fur ther information on the importance of each parameter and offers a convenient way of checking for possible model simplifications. Finally, this approach can also be used to identify those parameters which may require additional experimentation. As a result ofthe use of coupled pharmaco kinetic and multistage models of carcinogen esis, a median cancer risk estimate of 1.6 per million was determined for humans continu ously exposed to 1 ng/liter of tetrachloroeth- ACKNOWLEDGMENTS This work was funded by the California Department of Health Services, IMA 85-87088 and, in part, by Grant ES 04705 from the National Institute of Environmental Health Sciences. The authors express their gratitude for the helpful comments of Dr. Stuart Beal and of the re viewers. REFERENCES Andersen, M. E., Clewell, H. 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