Frederic Yves Bois
email: frederic.bois@ineris.fr

Statistical Analysis of Fisher et al. PBPK Model of Trichloroethylene Kinetics

Keywords: Bayesian, human, Markov chain Monte Carlo, mouse, PBPK model, TCE. toxicokinetics, tricholoroethylene, uncertainty analysis, variability

Two physiologically based pharmacokinetic modeis for trichloroethylene (TCE) in mice and humans were calibrated with new toxicokinetic data sets. Calibration is an important step in model development, essential to a legitimate use of modeis for research or regulatory purposes. A Bayesian Statistical framework was used to combine prior information about the model Parameters with the data likelihood to yield posterior parameter distributions. For mice, these distributions represent uncertainty. For humans, the use of a population Statistical model yielded estimates of both variability and uncertainty in human toxicokinetics of TCE. After adjustment of the modeis by Markov chain Monte Carlo sampling, the mouse model agreed with a large part of the data. Yet, some data on secondary metabolites were not fit well. The posterior parameter distributions obtained for mice were quite narrow (coefficient of Variation (CV1 of about 10 or 20%), but these CVs might be underestimated because of the incomplete fit of the model. The data fit, for humans, was better than for mice. Yet, some improvement of the model is needed to correctiy describe tnchloroethanol concentrations over long time periods. Posterior uncertainties about the population means corresponded to 10-20% CV. In terms of human population variability, volumes and flows varied across subject by approximately 20% CV. The variability was somewhat higher for partition coefficients (between 30 and 40%) and much higher for the metabolic Parameters (Standard deviations representing about a factor of 2). Finally, the analysis points to differences between human males and females in the toxicokinetics of TCE. The significance of these differences in terms of risk remains to be investigated.

New physiologically based pharmacokinetic (PBPK) modeis of trichloroethylene (TCE) in mice and humans have recently been published (1,2), rogether wich new toxicokinetic data secs. These modeis, prior to their use in research or regularion, should be carefully calibrated by confroncation with the data. It is mosr efficient in terms of information use and from a Statistical point ofview to combine two forms of information: prior knowledge and data. Neither source of information is compleie. If prior knowledge were sufficient, experiments would not need to be performed, but data alone are insufficient to pin down all parameters ofa complex PBPK model to reasonable values. Prior knowledge can be summarized äs prior parameter distributions. These can be obtained from the scientific literature, from specific in vitro experiments, and, äs is the case here, from fitting previous data. The data [in this case the new data provided to us by Fisher (2)] are incroduced through the use of a likelihood function. A Bayesian Statistical framework gives rigorous ruies co combine ehe prior distributions with the data likelihood, to yield posterior parameter distributions. In the case of complex modeis, like those described here, the posrerior parameter distributions can be obtained by numerical Markov chain Monte Carlo (MCMC) techniques (3,4}. This staris-' tical methodology is gaining interest and is establishing itself for the calibration and validation ofPBPK models [START_REF] Gelman | Physiological pharmacokinetic analysis using population modeling and informative prior distributions[END_REF]. Beyond improving the fit, this method also provides distributions of prediction estimates direcdy usable äs inputs for uncenainry analysis of cancer dose-response relarionships.

An interesring aspect of the human data analyzed here is that they have been collecred individually on a number of male and female volunteers. A hierarchical population model (4.5) deconvolves the various levels of variabiliry present in the data. Such a model, of which the PBPK model is just a componenc, is easily calibrated with Bayesian numerical methods. This öfters a unique oppormniry to examine separately the important issues of variability and uncertainty in human toxicokinetics of trichloroethylene.

Methods

Data

Mice. Groups of male B6C3F] mice (body weight [bw] 25-30 g) were exposed to TCE or its mecabolites by intravenous (i.v.) injection [START_REF] Abbas | Pharmacokinetic analysis of chloral hydrate and its metabolism in B6C3Fi mice[END_REF] or by oral gavage (l).

All i.v. doses were equal to 100 mg ofthe compound administered per kilogram of body weighr. After chloral hydrate (CH) administration, venous concentrations of CH, free trichloroethanol (TCOH), trichloroacetic acid (TCA), and dichloroaceric acid (DCA), äs well äs the amount of urine excreted metabolites, were recorded at various rimes after dosing. Similarly, after TCA i.v. dosing, venous concentrations of TCA and DCA, äs well äs the amount of DCA excreted in urine, were recorded. Following TCOH dosing, venous concentrarions ofCH, TCOH, TCA, and the urine-excreted amounts of TCOH and glucuronidated trichloroethanol (TCOG), were measured. Finally, after DCA dosing, venous concentrations and urineexcreted amounts of DCA were measured.

When trichloroethylene was administered by gavage in corn oil, doses of 300, 600, 1,200, and 2,000 mg/kg bw were used. TCE concentrations were measured in venous blood, liver, fat, lung, and kidney. CH and TCOH concentrations were recorded in venous blood, liver, lung, and kidney. TCA, DCA, and TCOG concentrations were obtained in venous blood, liver, and kidney. The quanrities of TCA and TCOG excreted in urine were also recorded.

Additional data, published by Templin et al. [START_REF] Templin | Relative formation of dichioroacetate and trichloroacetate from trichloroethylene in male B6C3Fi mice[END_REF], were partly used. Groups offour male B6C3F i mice (bw 27 g) were gavaged with 3.8 mmol/kg (500 mg/kg) TCE. The venous blood concentrations of TCE, TCA, and DCA were measured ar various rimes.

Humans. A group of21 volunteers (10 females and 11 males) were exposed by Inhalation to 50, 60, or 100 ppm TCE for 4 hr (2). Body weight and adiposity were recorded for each subject. The venous blood concentrations ofTCE, TCOH, and TCA were measured during and after exposure. For some subjects, the exhaled air concentration of TCE was also recorded at various rimes after exposure. Finally, the cumulaced quanrities ofTCA and TCOH glucuronide excreted in urine were recorded, The dara for each individual were available to us.

Toxicokinedc and Statistical Models

Mice. The description of the physiological model used for mice can be found in Abbas and Fisher (/). The model equarions were transcribed to a format suitable for MCSim [START_REF] Maszie | Program MCSim -User Manual[END_REF]. Three modificarions were made to the model, afrer consukarion with the original auchors. a) Metabolism of TCE in the lung was turned off. b) For several parameters, the scaling power of body weight was changed from 0.74 to 0.75 for standardization with the model ofClewell et al. {12}. c) The volumes ofthe "body" and poorly perfüsed compartments, and the blood ßow to the body compartment, were computed by difFerence at each Iteration so that the sum of the organ volumes equaled 82% ofthe body weight and the sum of organ flows equaled cardiac output. Given Ais reparameterizarion, the model had a total of 57 independent parameters.

The starisrical model used for mice lumps uncertainry and variability, since the animal data were aggregated (only the mean and Standard deviation [SD] for several animals were provided). For a schemaric ofthe model, see for example, Bois (13}. Various concentrations or quantities (y) were measured. The expected values of these measuremenis are a fünction (f) of exposure level (E), time (t), a set of physiological parameters of unknown values (6), and a set of measured, covariate parameters ((p), such äs body weight. E, t, 6, and <p are experiment specific. All animals in an experiment were supposed to have behaved similarly, from a toxicokineric point of view. The funcrion / is the pharmacokineric model described above. The concentrations or quantities actually observed are also affected by measurement error and interindividual variability. Only aggregated data were available, and the corresponding errors were assumed to be independent and log-nor-' mally distributed, with mean zero and variance <T 2 (on the log scale). The variance vecror O 2 had 27 components for mice, most of them known by repearing the experiments (Table l). The unknown experimental variances were sampled with a Standard noninformative prior distribution P(c^) -Of 2 [START_REF] Bernardo | Bayesian Theory[END_REF]. It was assumed that each 9 paramerer was a priori distributed log-normally (except for PCCH, the fraction TCE converred to CH, distributed normally), with averages M and variances S (in log scale).

A major advantage of physiological modeling is to provide a priori information on several of the mean parameter values for mice. Values for the hyperparameters M were set on the basis of the parameter values used by Fisher (2), or when applicable, on the basis of the posterior parameter distributions obtained from the analysis of the Clewell et al. model [START_REF] Clewell | Development of a physiologically based pharmacokinetic model of trichloroethylene and its metabolites for use in risk assessment[END_REF][START_REF] Bois | Statistical analysis of Clewell et al. PBPK model of trichlofoethylene kinetics[END_REF]. To set S, a distinction .was made between the physiological parameters or partition coefficients (which are quite well known) and the other metabolic or pharmacokincric parameters (which are model specific and little known a priori). For the first group of parameters uncertainries of the order of 20-50% were assigned [START_REF] Gelman | Physiological pharmacokinetic analysis using population modeling and informative prior distributions[END_REF][START_REF] Bois | Population toxicokinetics of tetrachloroethylene[END_REF]6). For the second group of parameters, a "vague" distribution was assumed and S was set to correspond to a factor between 2 ("quite uncertain") or 5 ("very uncertain"). All priors on p were truncated to ±2 X S or ±3 X S to avoid reaching unrealistic values. Table 2 gives for mice the values used for exp(M) (i.e., the geometric means) and exp(S) (i.e., the geometric Standard deviation [GSD]), both of which lie on the natural scale.

Humans. The PBPK model for humans is a simplification of the mouse model. Basically, in addition to TCE, only the distribution ofTCOH and TCA in blood, liver, kidney, and rest of the body is described by physiological compartments. CH is not modeled explicidy: a fraction ofTCE metabolized forms TCOH while the rest forms TCA. Formation of DCA is not modeled. A fraction ofTCOH is glucuronidated to give TCOG, which is supposed to be immediately excreted in urine. Four modificarions to the original model were introduced. For several parameters, the scaling power of body weight was changed from 0.74 to 0.75 for standardization with the model ofClewell et al. [START_REF] Clewell | Development of a physiologically based pharmacokinetic model of trichloroethylene and its metabolites for use in risk assessment[END_REF]. The volumes ofthe body and poorly perfüsed compartments and the blood flow to the body compartment were computed by dirferences at each Iteration so that the sum of the organ volumes equaled 82% of the body weight and the sum of organ flows equaled cardiac Output; the quantity of TCOG excreted is in TCOH equivalents, rather than in TCOG equivalents. The Michaelis-Memen coefficient K^ (labeled KMTCOH) for the formarion of TCOG was reexpressed äs the ratio VmaJ^-m ^ot easier Monte Carlo sampling (preliminary runs showed that Vnux wi1 ^ Kn, for that reaction are highiy correlated). The model had 33 independent parameters.

The starisrical model describing uncertainries and variabilities in human data was censtructed using a hierarchical popularion approach [START_REF] Gelman | Physiological pharmacokinetic analysis using population modeling and informative prior distributions[END_REF][START_REF] Wakefield | The Bayesian analysis of population pharmacokinetic models[END_REF]. The model has two major components: the individual level and the popularion level. At the individual level, various concentrarions or quantities (y) were measured. The expected values of these measurements are a fünction (/) of exposure level (E), time (t), a set of physiological parameters of unknown values (Q), and a set of measured, covariate parameters ((p), such äs body weight. E, t, 9, and (p are experiment specific. The fünction f is the human pharmacokinetic model. The concentrations or quantities actually observed are also affected by measurement errors. These errors were assumed to be independent and log-normally distributed, with mean zero and variance (T 2 (on the log scale). The variance vecror O 2 had six components for humans.

At the popularion level, each component of the 9 parameter set was assumed to be distributed log-normally, with averages p and variances 2? (in log scale). A priori knowledge of p and S 2 is available under the form of Standard values for some parameters. Since uncenainty in these average values and variances has to be acknowledged, a priori lognormal distributions were assumed for the popularion means p (with hyperparameters M and S) and a Standard inverse gamma distribution [START_REF] Gelman | Bayesian Data Analysis[END_REF] for the popularion variances 2?.

Values for the hyperparameters M were set on the basis of the parameter values used by Fisher (2), or when applicable, on the basis of the posterior parameter distributions obtained from the analysis of the Clewell et al. model [START_REF] Clewell | Development of a physiologically based pharmacokinetic model of trichloroethylene and its metabolites for use in risk assessment[END_REF][START_REF] Bois | Statistical analysis of Clewell et al. PBPK model of trichlofoethylene kinetics[END_REF]. To set S, a distinction was made between the physiological parameters or partition coefficients (which are quite well known) and the other metabolic or pharmacokineric parameters (which are model specific and little known a priori). For first group of parameters uncertainries of the order of 20-50% were assigned [START_REF] Gelman | Physiological pharmacokinetic analysis using population modeling and informative prior distributions[END_REF][START_REF] Bois | Population toxicokinetics of tetrachloroethylene[END_REF]6). For the second group of parameters, a vague distribution was assumed and S was set to correspond to a factor of 5. All priors on p were truncated to ±2 X S.

The ß coefficient of the inverse-gamma prior on £ 2 was set to the posterior values found in the previous analysis [START_REF] Bois | Statistical analysis of Clewell et al. PBPK model of trichlofoethylene kinetics[END_REF] when applicable and to 0.22 otherwise (this corresponds to a CV of 50%). The a coefficients of the inverse-gamma priors were set to l or 6 [the latter when a posterior distribution had already been obcained from analysis of the Clewell et al. data [START_REF] Bois | Statistical analysis of Clewell et al. PBPK model of trichlofoethylene kinetics[END_REF], which had several groups ofsubjects]. Table 3 gives for humans the values used for exp(M)-i.e., the geometric means-exp(S) -the GSDand the a and ß coefficienrs. For O 2 , a Standard noninformative prior distribution .PtOi 2 ,...^,, 2 ) -or 2 x ... x 0"~2 was used.

Statistical Computation ofPosterior Parameter Distributions

From Bayes' theorem, the joint posterior distribution of the parameters to estimate, P(Q, O 2 , p, 2?ly, (p, E, t. M, S, 2fl), in the case of humans, is proportional to the likelihood of the data multiplied by the parameters' priors:

P(6, CT 2 , p, 2?ly, (p, E, t, M, S, S 2 ) - /'(yie.O^E.t) • fl(91p,Z 2 ) • P(0 2 ) • P(plM,S) • P(2?IS 2 ) [l]
A similar, but simpler, expression can be obtained in the case of the mouse model. For naice and humans, the likelihood term is given by the normal measurement model:

log(y) -Mlog /(9,<P,E,t,)0 2 ) [2]
As mentioned above, the prior distribution for O 2 is: P((5^,...,<S^) -or 2 X ... X O," 2 , with evenrually some fixed components.

For humans, the prior distribution of each component of6 is an independent normal distribution in log space: Table 2. Prior and posterior distributions for the scaling coefficients of the mouse PBPK model (1). All distributions are log-normal with truncations at ±2 SDs, except where indicated. [START_REF] Gelman | Physiological pharmacokinetic analysis using population modeling and informative prior distributions[END_REF] with truncation constraints. Each component of p, or S 2 is assigned an independent hyperprior distribution, p -N(M,S 2 ) and 2? -inverse-gamma(a,ß), äs described in the previous section. For mice, the p and £ 2 levcls are not defined and prior distribution of 9 depends directiy on M and S.

Current practice in Bayesian starisrics is to summarize a complicated high-dimensional posterior distribution by random draws ofthe vector of parameters. This is currently the most effective way to perform highdimensional numerical integrarion. Because there are many parameters to estimate, Metropolis-Hasting sampling was used to perform a random walk through the posterior distribution. This iterative sampling procedure is particularly convenient in the case of hierarchical modeis. It belongs to a class of MCMC techniques that has recendy received much interesc (. [START_REF] Gelman | Physiological pharmacokinetic analysis using population modeling and informative prior distributions[END_REF][START_REF] Gelfand | Sampling-based approaches to calculating marginal densities[END_REF][START_REF] Gelfand | Illustration of Bayesian inference in normal data models using Gibbs sampling[END_REF][START_REF] Gelfand | Bayesian analysis of constrained Parameter and tmncated data problems using Gibbs sampling[END_REF][START_REF] Gelman | Iterative and non-iterative Simulation algorithms[END_REF][START_REF] Tanner | Tools for Statistical Inference -Observed Data and Data Augmentation Methods[END_REF][START_REF] Wakefield | Bayesian analysis of linear and non-linear population models using the Gibbs sampler[END_REF]. Three independent MCMC runs were performcd for each species. Convergence was monirored using the method ofGelman and Rubin [START_REF] Gelman | Inference from iterative Simulation using multiple sequences (with discussion)[END_REF]

Results

Mice

For mice, 20,000 iterarions were necessary to reach convergence of the sampler. One of every 5 of the last 5,000 simularions of three independent Markov chains were recorded, yielding 3,000 sets of parameter values from which the inferences and predictions presented in the following were made. Quality o f data adjustment. Figure l presents the data values observed compared with their predicted counterparts (data values are concentrations or excreted quantities). Predictions were made with the parameter set of highest posterior density. For a perfect fit, all points would fall on the diagonal (equality of predicted and observed values). Such an adjustment is not expected given the analytical measurement errors in the data.

The graph is presented on log-log scale, since the errors are assumed to be lognormally distributed and the data span a wide ränge. Most of the residuais are contained within a factor of 3 along the diagonal, but quite a few reach a factor of 10 or even a factor of 100. Figures 234567show all the gavage data and corresponding predictions in the liver äs a fünction of time. Basically the model reproduces correcdy the profiles for TCE and TCA. TCOH and TCOG concentrations are less well predicted. CH and DCA concentrations and TCOG excreted are rather poorly fitted. The fact that each data point represents a different group of animals may explain the noise present in die data. However, some differences between model and data appear to be systemaric (Figure 3, for example).

Posterior parameter distributions. The joint distribution of all parameters is obtained in Output of die MCMC simularions. This allows considerarion of marginal distributions (distributions of the parameters considered individually) and also ofcorrelations ofany order. Table l summarizes die posterior distributions of the mouse parameter values obtained in the last 1,000 iterarions ofthe three runs performed (results of the three runs are pooled, and the distributions are established with 3,000 values). The geometric means can be interpreied äs represenring the. values for an "average" mouse. Note that the columns of geometric Standard deviations represent the sum of group variability among mice and experimental error. Overall, the parameters retain physiologically plausible values.

Yet the posterior means for several parameters are quite far from the prior means. This is observed for metabolic parameters or for some partition coefficients. In particular, the value of KM for TCE is 20 times that found with the previous model and experiments. SDs about these geometric means are quite low and hover around 1.1 (corresponding to a 10% CV, which includes uncenainry and variability). The parameter values are therefore quite well identified by the data.

Very high correlations are observed between parameter estimates for mice (up to 0.95 for KFTCC and KFDCC). Any parameterization neglecting to estimate these covariances will produce incorrect predictions, sincc these parameters cannot be sampled independently without producing highiy improbable combinations and hence highiy improbable predictions.

Humans

More than 100,000 iterarions were necessary ro reach convergence of the sampler in the case of human data. One of every 10 ofthe last 50,000 simularions of three independent Markov chains were recorded, yielding 15,000 sets of parameter values from which ehe inferences and predictions presented in the following were made.

Qwdity ofdata adjwtment. Figure 8 gives the observed data values versus their predicted counterparts (data values are concentrations or excreted quantities). Predictions were made with the parameter set of highest posterior densiry. The graph is presented on log-log scale, since the errors are assumed to be lognormally distributed and the data span a wide ränge. Most of the residuais are contained within a factor of2 along the diagonal. The fit here is substanrially better than for mice. Figures 91011show all the data and corresponding predictions äs a function of time. The model was adjusted to each individual's data, using a population toxicokinetic approach. The data are reasonably well fitted, overall, and particularly the TCA data. However, the concentration ofTCE in alveolar air and venous blood is not adjusted satisfactorily, given the quality of adjustment expected from a PBPK model: The curves do not "bend" enough (Figure 9). Another area ofsignificant roisfit is the prediction of TCOH concentration in venous blood. The predicced terminal half-life is clearly too short' (data not shown). Yet, a good adjustment of TCOH can be obtained for the volunteers observed up to 22 hr (Figure 10). Most ofthe variability observed on the figures is due to differences in Inhalation exposure levels (concentrations of 50, 60, or 100 ppm TCE were used). However, a large part ofthe variability in TCOG elimination to urine seems due to factors other than exposure levels (data not shown). Somewhat troublesome is the disagreement between measured (albeit, approximarely) fractions of body weight äs fat (VFC) and the corresponding toxicokinetic estimates. The estimates take into account the measured values äs well äs what can be inferred about the size of the tat compartment from the TCE and other concentration data. The measured values of VFC reach much more extreme values (han the estimates (Figure 12).

Posterior parameter distributions. Table 3 summarizes the posterior distributions of the population means and SDs for all human model parameters (the distributions were established with 15,000 values). The population means represent the values for an average person. They are affected by uncenainty and were each assigned a geometric mean and a GSD (note that an approximate CV can quickly be computed, since CV " GSD -l). The population SDs measure between-subject variability. They are also affected by uncertainty (i.e., the population SD cannoi be exacdy computed from a finite sample).

The posterior means for several parameters are quite far from their prior estimates. The location of the blood over air partition coefficient, PB, is shiftcd from 13.7 (±20%) to 18 (±10%). The location of other partition coefricients (slowly perfused muscie over blood and richly perfused liver over blood) is also changed. The largest shifts are observed for metabolic parameters. The estimate of the scaling coefficient of TCE Vmax i" l 1 '^1" (VMAXC) is 4.22 (±20%) instead of43.8. A large uncerrainry exisred on that prior estimate, but 4.22 is still outside its prior 95% confidence inrerval (lower bound at 12.6). Therefore, there appears to be a conflict between the data studied here and the data studied previously. Note also that the posterior mean of the fracrion TCE converted to TCOH (POH) is quite lower than a priori estimared. The posterior means of the scaling coefficient of Vmax ^ot TCOH metabolism (VMTCOC) and of the scaling coefficient of the rate constant from TCOH to TCA (KOCHC) are much increased but not incompatible with the vague priors used. In terms of uncenainty, SDs about the posterior means are quite low, approximately 1.1 or 1.2 (corresponding to a 10-20% CV). The parameter values are overall quite well idenrified by the daia.

Estimates of population variability are given by the posteriors ofthe population SDs, E. Volumes and flows appear to vary across subject by approximately 20% (CV). The variability is somewhat higher for partition coefficients (berween 30 and 40% CVs). It is much higher for the metabolic parameters (SDs representing approximately a factor of 2 difference). For example, die lowest rate constant scaling coefiicient for TCA metabolism (KTMETC) is 0.12 S 1.6 hr-1 (subject HOF) and the highest is 1.1 2 1.25 hr"' (subject 101F, the subject with lowest TCA levels on Figure 11). For the scaling coefficient ofthe rate constant ofTCA loss to urine (KRTCC), the values ränge from 0.13 $ 1.2 hr'" 1 (Figure 13). Subject 111F excreted the highest amount of TCA in urine (data not shown).

Note that the estimaies of variability given are themselves affected by uncertainty (up to 50% CV for the variability estimaies of the metabolic parameters). For example, the 95% posterior confidence imerval of S for KRTCC is [1.77, 2.85].

Examination of the subjects' paramerer values points to peculiar pharmacoldneric behaviors for some ofthem:

• Subject 106M has the highest fraction tat (VFC), particularly for a male (some female subjects approach bis VF, but no males of the group studied approach it). He is also an "outlier" with a TCE fat over blood partition coefficient 3 times that of the others. His maximal rate scaling coefficient for TCE metabolism (VMAXC) and his rate constant of TCA loss to urine (KRTCC) are the lowest. Note that this subject shows the lowest amount of TCA excreted (data not shown), and his TCOH blood concentrations are not well modeled. • Subject 201 M is also outstanding for his highest alveolar Ventilation rate scaling coefficient (OPC), highest fractional blood flow to the kidney (QKC), and high TCOH body over blood partition coefficient (POHB), particularly for a male (females tend to have higher POHBs, see below). This subject has the second lowest TCA excretion in urine and one ofthe lowest TCOG elimination. It seems that this subject can be characterized äs "fast" in intake and elimination, at least from a physiological (ifnot enzymadc) point ofview. • Subject 103M has a Michaelis-Menten constant (KM) about 3 times äs high äs everyone eise. He has one of the highest excretions ofTCA in urine. Splitting the subjects by sex reveals some differences berween males and females. The visual examination of the marginal dismbutions of a given parameter for each subject reveals little pattern. For example, in Figure 14, it appears that female subjects exhibit higher values of KOCHC, the rate constant scaling coefficient for TCOH to TCA. However, a large part of the variability observed is due to uncenainty about the populacion mean. When the effecc of the Overall mean is removed, the difference is highiy significant: P(KOCHCfi^l, > KOCHC^) = 0.9998 (i.e., die female average was superior to the male average 99.98% of the times in the sample of 15,000 parameter veccors obtained by MCMC sampling). Other significant sex differences were found: • The scaling coefficient of the alveolar Ventilation rate is higher for males P(QPCn,aic > QPCfi^k) = 0.9997. This implies that the data are sufficiently informative to show such a difference and also that the surface area correcrion is not sufficient to explain sex difFerences in alveolar Ventilation. The level of physical training might be another covariate to consider for alveolar Ventilation in human studies. • The posterior estimates of the VFC are higher on average for females than for males: P(VFCf^> VFC^ > 0.9999. That was expected and is influenced by the covariate measurements of adiposity and by the toxicokineric data. • Female subjects also seem to have higher TCOH body over blood partition coefficients: P{POHB^i, > POHB^) > 0.9999; rhey also have a higher V^mx wa K^, ratio (i.e., rate constant at low concentrarion) for TCOH glucuronidarion:

P(KMTCOH^ > KMTCOH^ >

0.9999, and a higher rate constant scaling coefficient for TCA urinary excrerion: P(KRTCCf^> KRTCC^ > 0.9943. • Males were found to have higher TCA body over blood partition coefficients: P(PTCB^> PTCB^ = 0.992. All other male versus female parameter averages do differ slighdy, but the P values are less than 0.95, most ofthem being around 0.5.

Discussion

The mice data show a rather large interindividual or interlot variability. There is litde to do about that, unless the experimental design were to allow the observation of individual animals. The data still permit an extensive calibration ofthe model. The human data have the advantage of being available for each individual. There appear to be, a posteriori, a few oudying points (for example, Figure 10), which could be checked and eventually removed. These few apparent outliers have little weight and should not affect sensibly the results presented here. There could be a problem with the measuremenc of fat content; this will be discussed below, in light ofthe model fit.

The mouse model is quite complicated, and yet some aspecrs ofthe data are not so well described by k (CH and DCA concentrarions, and TCOG excreted). This is somewhat disappointing, given the complexity of the model, and one wonders what will have ro be done ro fit such an extensive data set. It is possible that some metabolic reactions do not obey simple Michaelis-Menten reactions or that prior opinions about the model parameters were for some of them overly confidcnt. Note also that even in large modeis, model uncenainty can be large. The model, for example, does not include heparic recycling or all possibilides of extrahepatic metabolism. Yet, overall, the model fits well a large part of the data, in particular TCE and TCA distribuuon. It would be interesting to see die performance of other modeis [e.g., those presented in {12,23)} wirh this data set, which would allow a formal comparison of the competing modeis on the basis of a common measure of goodness of fit, such äs likelihood ratios.

The posterior parameter distributions obtained for mice are quite narrow (with CVs of about 10 or 20%), indicadng that the data are strongly informative for most parts of the model. Indeed, äs indicated above, some parameters or processes might need to be added for a betler fit and the model is somewhat minimal with respect to the data. However, it should be kept in mind that the fit is not excellent, and that may overconstrain the posterior distributions. Part of the high covariance between parameters may also be due to overconstraining. It is possible ro model staristically the lack of fit by including an autocorrelation between data points (24). This has not been attempted here and could be a usefui improvement. Note that the posterior uncenainty for the metabolic parameters would have been further underesrimated if all physiological parameters had been set to predefined values.

The human model, even though complex, also has difficulties in fitring all the data. This is true in parricular for TCOH concentrarions over a long period oftime, and some improvement of the model in that respect may be needed. Similarly, the adiposity ofthe subjects (a measured covariate) does not fit well with the estimated fracrion ofbody weight äs fat. Ir is possible that the pharmacokineric companment "fat" is not well estimated by external adiposity measurements, in parricular for extreme values. It also appears that the model may not be able to describe correcdy oudying subjects like subject 106M. This could be due to the PBPK model, which lacks some component important for such a subject. The misfit for that subject could also be due to a lack of flexibility of the starisrical model adopted here (log-normal distributions of the parameters in the popularion). A possibilky for checking would be to fit the data of only that subject to determine ifa good fit could be obtained.

The human posterior parameter distributions agree in general quite well with the corresponding priors, with reduced uncertainty (since information from the data has been gained). SDs about the posterior means are quite low and correspond to a 10-20% CV: The parameter values are overall quite well idenrified by the data. There is, however, a conßict for the values ofVMAXC between the values previously found (3) and the ones obtained here. This is an imponant parameter and a good characterizarion is important. The difference could be due to conflicts between the data analyzed here and the previous data. For example, extensive TCOH data are available here. It is also possible that the conflict is due to differences in human model structures. A solurion to this dilemma would be to take the model of Clewell et al. [START_REF] Clewell | Development of a physiologically based pharmacokinetic model of trichloroethylene and its metabolites for use in risk assessment[END_REF] and fit it ro the data of Fisher (2) to obtain an esrimate of VMAXC with the same data set. In any case, it is not obvious that the differences in parameter values would result in notable differences when predicting toxicologically relevant end points, such äs internal TCA concentrarions. This remains to be checked.

Vblumes and flows appear ro vary across subject by 20% (CV), approximately. The variability is somewhat higher for parririon coefficients (between 30 and 40% CVs) and much higher for the metabolic parameters (SDs represenring about a facror of 2 difference). This is similar ro what was found for a small group of human volunteers exposed to tetrachloroethylene [START_REF] Bois | Population toxicokinetics of tetrachloroethylene[END_REF]. Differences appear to exist between sexes in the toxicokinerics of TCE. There are differences between males and females in alveolar Ventilation and adiposity for the popularion sample scudied. This should be true for compounds other than TCE and shows that the model scaling could be improved. More important for TCE Idnerics are the findings that females have higher TCOH body over blood panirion coefficients, lower TCA body over blood partition coefficients, higher T^n over K^ ratios for TCOH glucuronidarion, higher rate constant scaling coefficients for TCOH ro TCA, and higher rate constant scaling coefficients for TCA urinary excrerion. Note that the staristically highiy significant differences found for these parameters should be interpreted with some caurion. They are condirional on the model strucmre being correct. At least, at this point, it can be said that there are most certainly differences in the kinetic behavior ofTCE berween the males and females of the sample studied. It would have been hard to reach that conclusion without the starisrical adjustment of a model, given the multiple exposure levels, differences in body weight, nonlinear kinetics, etc. Still, these differences may not be significant in terms ofTCE toxicity (i.e., biologically significant), It would be interesting to assess by simularions whether internal metabolke concentrarions are much different for males and females for ehe same TCE exposure.

Time(hr)Figure 2 .

 2 Figure 2. Predicted (solid line) and observed (points) time course of TCE concentration in liver of mice dosed with various quantities of TCE in com oil. Error bars correspond to ±2 SD.

Figure 3 .

 3 Figure 3. Predicted (solid line) and obsen/ed (points) time course of CH concentration in liver of mice dosed with various quantities of TCE in corn oil. Error bars correspond to ±2 SD.

Figure 4 .

 4 Figure 4. Predicted (solid line) and obsen/ed (points) time course of TCOH concentration in liver of mice dosed with various quantities of TCE in corn oil. Error bars correspond to ±2 SD.

Figure 5 .

 5 Figure 5. Predicted (solid line) and observed (points) time course of TCA concentration in liver of mice dosed with various quantities of TCE in corn oil. Error bars correspond to ±2 SD.

Figure 7 .

 7 Figure 7. Predicted (solid line) and observed (points) time course of TCOG concentration in liver of mice dosed with various quantities of TCE in corn oil. Error bars correspond to ±2 SD.

Figure 8 .

 8 Figure 8. Observed versus predicted human data values (all concentrations or quantities) for the Monte Carlo iteration of highest posterior probability.

Figure 9 .

 9 Figure 9. Predicted [solid lines) and observed (points) time eourse of TCE concentration in venous blood of human volunteers exposed by inhalation to various concentrations of TCE. The error bars presented on a data point correspond to ±2 estimated measurement SD (the size of the error is the same for all points).

Figure 10 .Figure 11 .Figure 12 .

 101112 Figure10. Predicted (solid lines) and observed (points) time eourse of TCOH concentration in venous blood of human volunteers exposed by inhalation to various concentrations of TCE. These volunteers were followed for up to 22 hr. The error bars presented on a data point correspond to ±2 estimated measurement SD (the size of the error is the same for all points).

Figure 13 .

 13 Figure[START_REF] Bois | Statistical analysis of Clewell et al. PBPK model of trichlofoethylene kinetics[END_REF]. Individual estirnates of the rate constant scaling coefficient for TCA metabolism (KTMETC) and of the rate constant of TCA loss to urine (KRTCC). The enw bars span the 95% confidence intervals of the estimates. Large individual differences are observed, with some degree of correlation.

Figure 14 .

 14 Figure 14. Individual estimates of the rate constant scaling coefficient for TCOH to TCA, grouped by sex. The error bars span the 95% confidence intervals of the estimates.

Table 1 .

 1 Values (2} or distributions used for the assay SDs, for each of the measured end points of the mouse

	data set.
	Parameter	SD
	CV	1.25
	CL	1.25
	CLU	LogUniform distribution
	GF	1.39
	CK	1.33
	CCHL	1.11
	CCHLU	1.27
	CCHV	1.25
	CCHK	1.13
	CTCL	1.16
	CTCV	1.19
	CTCK	1.21
	CDGL	1.24
	CDCV	1.24
	CDCK	1.40
	COHL	1.17
	COHLU	1.16
	COHV	• 1.21
	COHK	1.19
	COGL	1.41
	COGV	1.27
	COGK	1.34
	ACHKR	LogUniform distribution
	ATCKR	LogUniform distribution
	ADCKR	LogUniform distribution
	AOHKR	LogUniform distribution
	AOGKR	LogUniform distribution

  . Prior set by ourselves. 'Prior based on the posterior distribution obtained previously (3|. "Prior based on Fisher's values |2). 'truncation at ±3 SDs. • For this parameter a normal distribution was used, with truncation at 0.95 and 1.

		rior di	stribution		Posterior	distribution
		Geometric	Geometric	Geometric	Geometric	2.5%	97.5%
	Parameter	mean	SO	mean	SD	percentile	percentile
	BW 3	0.03	1.11	0.0354	1,02	0.0334	0.0365
	QCC"	20.1	1.08	14.4	1.05	13.1	16.1
	QPC"	21.8	1.08	19.3	1.07	16.8	22.0
	QLC 6	0.2	1.08	0.259	1.05	0.235	0.284
	QKC 6	0.09	1.22	0.0944	1.21	0.0646	0.129
	QFC"	0.054	1.14	0.0455	1.11	0.0373	0.0561
	VLUC'	0.007	1.22	0.00654	1.18	0.00480	0.00900
	VLC 1 '	0.05	1.07	0.0478	1.06	0.0424	0.0529
	VKV	0.018	1.22	0.0221	1.12	0.0180	0.0265
	FC 1 '	0.063	1.12	0.0773	1.10	0.0646	0.0944
	WC"	0.05	1.07	0.0503	1.06	0.0442	0.0567
	Kl"	1	2.72	4.18	1.14	3.22	5.47
	K2"	17.3	1.24	3.46	1.10	2.86	4.14
	K3''	1.14	1.36	0.0916	1.01	0.0898	0.0944
	PB"	16.4	1.13	22.9	1.08	19.7	26.6
	PLU'	2.61	1.6	1.81	1.10	1.50	2.19
	p|ft	1.74	1.2	3.10	1.04	2.83	3.35
	pKC.d	2.07	1.6	4.81	1.06	4.31	5.42
	PF 6	30.6	1.15	15.6	1.05	14.3	17.5
	PS" .	0.756	1.21	1.41	1.13	1.08	1.75
	PR"	1.75	1.21	1.97	1.22	1.31	2.89
	PCHLU'	1.65	1.6	1.96	1.07	1.71	2.23
	PCHL"	1.42	1.6	1.61	1.05	1.45	1.78
	PCHKP	0.98	1.6	0.361	1.06	0.323	0.399
	CHB'-"	1.35	1.6	1.51	1.08	1.29	1.74
	POHLU' pom-P	0.78 1.3	1.6 1.6	1.71 3.97	1.04 . 1.04	1.58 3.71	1.85 4.31
	OHK 1 '-"	1.02	1.6	3.90	1.04	3.63	4.14
	POHB C	1.11	1.6	0.595	1.08	0.516	0.683
	POGLU'	1.06	2	1.01	1.73	0.361	3.00
	POGLP	0.56	2	0.897	1.08	0.771	1.04
	OGK"	1.44	2	2.65	1.13	2.12	3.42
	POGB 1 --"	1.11	2	0.449	1.25	0.275	0.668
	PTCUJ'	0.54	1.6	0.518	1.46	0.249	1.07
	PTCL'	1.18	1.6	0.780	1.03	0.733	0.830
	PTW	0.74	1.6	0.758	1.04	0.706	0.812
	PTCB''	0.88	1.6	0.857	1.05	0.773	0.945
	PDCLLF	1.23	1.6	1.14	1.43	0.572	2.33
	FDCI.P	1.08	1-6	0.636	1.05	0.576	0.700
	OCK'	0.74	1.6	0.843	1.08	0.730	0.973
	PDCB'	0.37	1.6	0.548	1.06	0.487	0.619
	VMAXC 1 '	38.1	1.15	45.2	1.03	42.5	47.5
	KM"	0.47	1.57	10.1 '	1.05	9.21	11.1
	PCW	0.99	0.01	0.995	0.00380	0.986	1.00
	KRCHC'	0.06	5	0.0493	1.17	0.0365	0.0679
	PCTCOP	309	5	403	1.09	347	478
	PCTCACV	115	5	119	1.08	103	141
	MTCOC 6	16.5	5	74.4	1.11	60.9	90.9
	KMTCOH'	15.7	5	49.9	1.14	39.3	64.1
	KOCHtK	1.32	5	5.58	1.13	4.39	7.03
	ROHC'	1.14	5	0.133	1.18	0.0983	0.183
	KRW	0.35	5	0.546	1.08	0.472	0.639
	KRTW	1.55	5	0.596	1.12	0.489	0.749
	KFDCC'	20.5	5	41.7	1.09	35.5	48.4
	KRDCC 3	1	2.72	2.62	1.14	2.13	3.49
	KROGK	32.8	5	19.9	1.16	15.2	26.8
	GBLt^P 4.61	5	16.4	1.08	14.2	19.1
	•

Table 3 .

 3 Prior and posterior distributions for the population averages (|i), and population SDs (£), of the scaling coefficients for the human PBPK model.

		Prior an p	Prior on £	Posterior on |i	Posterior on £
	Parameter	Geometrie mean (GSD) Geometrie mean |GSD) Geometrie mean (GSD) Geometrie mean JGSD)
	QCC 3 QPC 8 QLC 3	15.2(1.16) 16.1(1.16) 0.232(1.17)	1.41(1.12) 1.39(1.10) 1.42(1.11)	16.3(1.08) 16.9(1.08) 0.244(1.07)	1.30(1.15) 1.36(1.17) 1.20(1.12)
	QKC" QFC 3 VLUC 6	0.198(1.22) 0.0523(1.16) 0.014(1.22)	-1.47(1.12) -	0.194(1.08) 0.0639(1.10) 0.0141(1.11)	1.26(1.14) 1.19(1.12) 1.25(1.14)
	VLC"	0.026(1.13)	1.37(1.10)	0.0257(1.10)	1.14(1.08)
	VKC" VFC 3 VRC 3 PB 3	0.00401(1.22) 0.198(1.13) 0.0483(1.13) 13.7(1.19)	-1.37(1.10) 1.37(1.10) 1.52(1.14)	0.00401(1.13) 0.196(1.05) 0.0437(1.10) 18.0(1.09)	1.26(1.14) 1.21(1.11) 1.13(1.08) 1.21(1.12)
	PLU" p|a	0.391(1.6) 6.69(1.32)	-1.52(1.16)	0.386(1.18) 5.81(1.15)	1.37(1,24) 1.22(1.15)
	PK" PF 3 PS 3 PR 3	1.08(1.6) 53(1.22) 2.7(1.23) 5.05(1.28)	-1.50(1.15) 1.49(1,15) 1.51(1.16)	1.07(1.18) 50.9(1.15) 1.50(1.14) 3.67(1.11)	1.37(1.24) 1.32(1.20) 1.22(1.17) 1.17(1.10)
	POHIDr	0.67(1.6)	-	0.685(1.18)	1.37(1.24)
	POHL" POHK 6 POHB" PTCLU"	0.589(1.6) 2.15(1.6) 0.91(1.6) 0.47(1.6)	----	0.616(1.18) 2.16(1.18) 1.26(1.10) 0.472(1.18)	1.37(1.25) 1.37(1.24) 1.47(1.24) 1.37(1.24)
	PTCL" PICK" PTCB 6 VMAXC' 1 KM 8 POHc VMTCOC"	0.66(1.6) 0.66(1.6) 0.519(1.6) 43.8(1.97) 0.542 (2.35) 0.9(0.1) 1(5)	---1.70(1.28) 1.76(1.35) --	0.708(1.21) 0.668(1.18) 0.601(1.10) 4.22(1.19) 0.801(1.40) 0.730(0.041) 5.26(1.56)	1.37(1.24) 1.37(1.24) 1.41(1.23) 1.48(1.33) 1.96(1.65) 0.187(0.10) 1.76(1.72)
	KMTCOH" KOCHC" KTMETC"	1.57(5) 1(5) 0.1 (5)	---	2.72(1.12) 8.58(1.14) 0.301(1.22)	1.70(1.35) 1.51 (1.29) 1.98(1.57)
	KRTCC''	1(5)	-	1.15(1,19)	2.18(1.56)
		10-' 10° io'	10'
		Predicted Data Value
	Figure 1. Observed versus predicted mice data values
	(all concentrations or quantities) for the Monte Carlo
	iteration of highest posterior probability.

"Prior distribution based on the posterior distribution obtained previously

[START_REF] Gelman | Physiological pharmacokinetic analysis using population modeling and informative prior distributions[END_REF]

. 'Prior distribution on |i based on Fisher's values (2). A »ague inverse-gamma (1, 0.22) was used. Tor this Parameter a normal distribution was used, with truncation at 0.6 and 1. so the Parameters are mean and SD in natural space.

This work has been supported by the D.S. Environmental Protection Agency.