* Toxicology and Risk Assessment, National Institute for Working Life, S-112 79 Stockholm, Sweden;
Institut National de l'Environnement Industriel et des Risques, Verneuil en Halatte, France; and
Department of Medical Sciences, Occupational and Environmental Medicine, Uppsala University Hospital, Sweden
Received June 20, 2000; accepted October 23, 2000
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ABSTRACT |
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Key Words: dichloromethane; inhalation; pharmacokinetics; physiologically based modeling; Markov chain Monte Carlo simulations; population kinetics.
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INTRODUCTION |
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Andersen and coworkers developed a PBPK model for DCM in humans in order to estimate the target dose, defined as the amount metabolized by the GST pathway (Andersen et al., 1987). In that model, the toxicokinetics in humans was predicted from the in vitro metabolism and the in vivo toxicokinetics of DCM in three other species. The model was then validated against kinetic data from human subjects exposed to DCM at 100 and 350 ppm. The kinetic data were pooled, and the means of the DCM concentrations in the six subjects were used in the validation. Reitz and coworkers refined the model by introducing enzymatic activities determined in vitro using human and animal subcellular fractions (Reitz et al., 1988
). The individual metabolic capacities for DCM in humans were estimated using four human liver samples (Reitz et al., 1989
). The Andersen et al. model (1987), as subsequently modified by Reitz et al., will be referred to in this paper as the Andersen/Reitz model. Later on, variability was introduced in the metabolic parameters in the Andersen/Reitz model in order to improve the reliability of the target dose estimates (Dankovic and Bailer, 1994
), but without validation against human data. The recent risk assessment for DCM by the U.S. Occupational Safety and Health Administration (OSHA, 1997
) included a new PBPK analysis published by Andersen et al. (1991). However, the data used were again the aggregates for six subjects and did not allow for an assessment of interindividual variability. To our knowledge, there are no estimates of human DCM metabolism based on individual in vivo data, and thus, no reliable estimates of the population variability of DCM metabolism. This is unfortunate, as knowledge of population variability is essential in risk assessment.
Population variability in physiological pharmacokinetic parameters can be estimated from experimentally derived individual concentrationtime profiles for human subjects. By combining the information in the experimental toxicokinetic data with the information from the in vitro estimates of DCM metabolism, new knowledge on population variability in target dose can be gained. One way of doing this is by Bayesian statistics. The Bayesian approach yields statistical distributions (so-called posterior distributions) of parameter values rather than single point estimates. These posterior distributions are consistent with both the experimental data and the prior assumptions (specified as so-called prior distributions for the parameters). The distributions can be approximated by random draws using so-called Markov chain Monte Carlo (MCMC) simulations (Wakefield and Smith, 1994). MCMC simulations have previously been linked successfully to PBPK models embedded in population frameworks (Bois et al., 1996a
,b
). The parameter vectors thus generated can then be used for further simulations to compute posterior distributions of quantities of interest, including target doses. The risk assessment for DCM by OSHA (1997) used techniques similar to those presented here. Although MCMC is a Monte Carlobased technique, it should not be confused with the Monte Carlo techniques commonly employed for predictions in conjunction with PBPK models (Thomas et al., 1996
). Regular Monte Carlo techniques can be used to estimate variability in model output (such as in target dose), but not to estimate model parameters. In contrast, MCMC is a parameter estimation technique.
Åstrand et al. (1975) exposed male volunteers to DCM vapors at rest and various levels of exercise on a bicycle ergometer. Extensive data collection was made. PBPK modeling has previously been performed on the Åstrand data set (Johanson and Näslund, 1988). This modeling successfully described the changes in DCM uptake and perfusion with increasing workload. However, that model did not account for interindividual variability in parameters other than intrinsic liver clearance and did not use the extensive data on DCM levels in exhaled air postexposure.
The aim of the present study was to develop a population model (Sheiner, 1984) for DCM in order to gain more knowledge on the variability in DCM inhalation toxicokinetics in humans, with main emphasis on the MFO metabolic pathway. This was done by merging published in vitro data on DCM metabolism and partitioning with the Åstrand toxicokinetic data (Åstrand et al., 1975
) from five human volunteers using the MCMC technique within a population PBPK model.
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MATERIALS AND METHODS |
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PBPK model.
The basic PBPK model used in this analysis was first published by Andersen et al. (1987) and subsequently modified by Reitz et al. (1988). A compartment for working muscle was added to the Andersen/Reitz model. This was done in order to reflect the increased perfusion of leg muscle during bicycle exercise (Johanson and Näslund, 1988). Thus, the present model incorporates compartments for resting muscle, working muscle, well-perfused tissue, fat, lung, and liver. The revised model is illustrated in Figure 1
. Metabolism by the MFO pathway is believed to be saturable and is described by a maximum rate of metabolism (Vmax) and by the concentration at which the rate of metabolism is half-maximal (km). This pathway produces a rise in blood HbCO levels. The GSTT1 pathway is assumed to be nonsaturable in the present concentration range and is described in the model by a first-order rate constant (Kf). Symbols are also explained in the Appendix.
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Statistical computations: the Bayesian approach.
One fundament of Bayesian inference, as opposed to purely likelihood-based inference, is to incorporate the uncertainty about assumptions into the statistical analysis. That means, in this case, quantifying the prior belief about the model parameters before taking the observed data into account. In a Bayesian analysis, two forms of information are combined logically: prior knowledge about parameter values (often drawn from the scientific literature), which may be known with various degrees of confidence, and data from experimental studies. In our case, the analysis is made within the context of a PBPK model. Neither source of information alone is sufficient to fully parameterize the PBPK model. If the prior knowledge of the parameters was sufficient, the experimental data from Åstrand et al. (1975) would not have been needed at all in the present study. However, the available experimental data alone are not sufficient to estimate all the parameters of the model independently, which is why they cannot be estimated simultaneously using the maximum likelihood-based techniques often employed in empirical pharmacokinetic modeling. Fitting only two or three parameters, while using reference literature values for the others, would produce estimates that would be conditional on the assumed values of the fixed parameters. As the values of the physiological and physicochemical parameters in vivo are not known with precision, such a procedure may result in erroneous estimates of the fitted parameters. In addition, the variability in the fixed parameters is ignored, whereas the variability in the fitted ones tends to be overestimated. An additional disadvantage is that any correlations between parameters tend to be ignored or underestimated (Woodruff and Bois, 1993).
The Bayesian approach produces posterior estimates (in the form of distributions) of all the parameters of the PBPK model, along with information on the full joint covariance structure. The posterior distributions of the parameter values (posterior estimates) are consistent with both the experimental data and the prior information. It should be noted that if the experimental data lack any information on a certain parameter, the posterior estimate is equal to the prior estimate. Additionally, if the prior distributions are assigned to be uniform in shape (i.e., complete ignorance about plausible values), the posterior will be proportional to the likelihood of the data, and asymptotically equivalent to the standard likelihood techniques referred to above. When compared with standard analytical tools for population modeling, the Bayesian population method has been shown to be more reliable from a statistical point of view (Bennett and Wakefield, 1996).
The complicated high-dimensional joint posterior distribution was summarized by random draws using Metropolis-Hastings Monte Carlo sampling (Smith, 1991). Random draws from the prior were made in order to obtain initial parameter vectors. The individual parameter vectors were used to simultaneously simulate the experimental data for each individual, and the value of the likelihood function was calculated (see following section). The sampled values were either kept or rejected on the basis of the current value of the posterior distribution, computed by multiplying the prior by the value of the likelihood function. This procedure was iterated several thousand times, with the posterior parameter vector of one step used as prior for the next, until all parameters had reached equilibrium.
Five independent Markov chains, using different seeds for the random number generator, were run to approximate equilibrium. Convergence was monitored using the method of Gelman and Rubin (Gelman and Rubin, 1992), with an
value lower than 1.1 as a criterion for convergence. At perfect convergence, all
values should be equal to 1. An
of 1.3, for example, indicates that the use of longer chains would have reduced the variance estimates by 30%. If convergence tests resulted in
values greater than 1.1 during model development, additional Markov iterations were performed until acceptable convergence was obtained. Posterior distributions were obtained by running the MCMC simulations further. Each independent equilibrated Markov chain was run for 10,000 iterations while printing one iteration in ten. The MCSim software (Bois and Maszle, 1997
) was used throughout the study.
A priori parameter distributions and data likelihood.
Alveolar ventilation was measured in the original study, and the measured values (Table 1) were used. Body weight and height were also measured in the original study (Table 1
), and thus the compartment volumes could be estimated with reasonable accuracy from the measured quantities via scaling functions (Table 2
). Interindividual variability in all tissue volumes and some blood flows was assumed to be accounted for adequately by the scaling functions (Table 2
), based on the estimated affinity of DCM for these tissue groups. Prior distributions were assigned to all other parameters, thereby quantifying the uncertainty of these (Table 3
).
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The prior uncertainties of the parameters were summarized numerically by assigning a priori truncated lognormal distributions (from now on called prior estimates) to the population parameters on the basis of the literature (Bülow and Madsen, 1978; Gargas et al., 1989
; Williams and Leggett, 1989
; Åstrand, 1983
). The choices of these values and bounds for truncation are given in Table 3
. In the prior estimates, each parameter is characterized by its expected population mean, an uncertainty around the population mean, as well as a population standard deviation (SD), expressing the expected variability in the population.
For all tissue:air partition coefficients except blood:air, prior means were set at the value for rat (Gargas et al., 1989). The prior uncertainties for these coefficients were set to 1.1 on the log scale, corresponding to an approximate coefficient of variation of 10%.
Population SDs were set at higher values for the fat blood flow than for other parameters, based on published observations on variability in fat tissue perfusion (Bülow and Madsen, 1976; Linde et al., 1989
; Samra et al., 1995
).
Truncation bounds were generally set at wide intervals in order to give the model a considerable amount of freedom to explore the parameter space.
We had no information on the values of the parameters at the individual level. Thus, starting values for individual parameters were generated by sampling from the priors for the population parameters. The individual parameters were updated on the basis of their conditional distribution with respect to the population parameters and the data likelihood. The data likelihood was calculated assuming that the observations in both blood and alveolar air were associated with lognormally distributed residual errors () due to intraindividual variability and measurement error. As the two types of observations have different experimental protocols, the vector
has two elements, reflecting differing measurement errors in blood and alveolar air, respectively.
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RESULTS |
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Figure 2 shows the predicted versus observed values of the dependent variable for all individuals. Predictions were made using one randomly chosen parameter vector among the 5,000 vectors recorded. As the chains had reached approximate equilibrium, the fit illustrated in Figure 2
can therefore to be considered as representative of the set.
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The posterior estimates of the individual parameters are summarized in Table 5. The individual estimates of fat perfusion are notably different from the corresponding prior estimates, although all values are physiologically plausible. There is also considerable interindividual variability among the estimates of VmaxC (range of posterior estimates 2.04.0 µmol/min).
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DISCUSSION |
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As our estimated individual values for Vmax are based on PBPK modeling of individual concentrationtime profiles in humans, we consider the estimates quite reliable. Our individual Vmax values indicate that the metabolic capacity for the MFO pathway in vivo is higher, and associated with lower degree of interindividual variability, than estimated previously from in vitro samples. However, there is still considerable uncertainty as to the representativeness of these five young, healthy, Swedish males to the general population. In contrast, the Bayesian estimate of population variability in Vmax illustrated in Figure 4 is based on the toxicokinetic data, plus the in vitro metabolic data from four human livers of unknown gender, age, and ethnicity, and is thus more useful for extrapolations to the general population. An even more reliable estimate of population variability in Vmax would be obtained by studying a larger and more heterogeneous population. There may also be interindividual variabilities in Vmax (or other parameters) associated with covariates such as age, ethnicity, body build, sex, or smoker status. Thus, although differences in body build may be easily incorporated into the model, extrapolations to larger, more heterogenous populations should be done with caution.
The individual posterior estimates of GST metabolism are very close to the prior estimates (Table 5). This indicates, as expected, that the experimental data contain very little information about the parameter governing GST metabolism (see "Statistical calculations" in the Materials and Methods section). Thus, most of the retained DCM is apparently metabolized by the oxidative pathway, even at the high exposure levels in the data set of Åstrand et al. (1975). DCM metabolized by the GST pathway would result in the appearance of metabolites such as glutathione conjugates in blood and mercapturic acid in the urine. In order to estimate GST metabolism in vivo, we would need information on the levels of these metabolites. Unfortunately, these were not monitored by Åstrand and coworkers, and we cannot derive an estimate of GST metabolism in vivo in the present study.
The individual posterior estimates of the increase in fat perfusion with exercise predict a considerably smaller increase than the prior estimate. The prior estimate was based on experiments performed at prolonged exercise (Bülow and Madsen, 1976). Change in fat tissue perfusion during exercise is a slow process. When Bülow and Madsen measured fat tissue perfusion during a constant workload of 120 W, fat perfusion did not reach a constant level until approximately 2 h had passed. Estimating the effect of different levels of exercise on fat perfusion with any precision using the present data set seems unfeasible and would most likely demand prolonged exposures at each level of exercise. The lack of information in the data set is also reflected in large uncertainties of the posterior estimates of the parameters governing the increase in fat perfusion during exercise, both at the population and at the individual level (Tables 4 and 5
).
All posterior estimates of the individual fat:air partition coefficients are lower than the prior estimate, which was based on an in vitro estimate for rat (Gargas et al., 1989). The shift might be an indication that the size of the fat compartment was predicted with insufficient accuracy by the algorithms being used (Table 2
). In reality, the fat compartment would then be smaller than predicted by the algorithm. On the other hand, fat tissue is heterogeneously perfused. Individual ranges in the perfusion of perirenal and subcutaneous fat have been reported to be 0.18 and 0.32.3 ml/min/100 g fat, respectively (Bülow and Madsen, 1978
). The fat compartment in our model might correspond to the well-perfused portions of fat tissue. The less well-perfused portions of fat tissue would receive only a minute amount of DCM, and the washout from the less well-perfused fat tissue would not be visible until later points in time for which we have no data.
The use of MCMC simulations and of a population hierarchical model in a Bayesian context allowed us to reach deeper conclusions about toxicokinetic data and modeling practice than before. In addition to disentangling variability (inter- or intraindividual, etc.) from uncertainty and providing improved uncertainty estimates for model predictions (Bois, 1999, 2000a
,b
), PBPK/MCMC coupling also helped us with model checking.
The present study provides a Bayesian estimate of the population distribution of metabolic capacity for the MFO pathway in humans. We also detect a need to incorporate the highly variable perfusion of various kinds of fat tissue when applying the PBPK model. Our approach illustrates the importance of implementing existing knowledge on variability in these parameters into the PBPK modeling process.
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APPENDIX |
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A1 Lung:liver ratio of rate of MFO metabolism
A2 Lung:liver ratio of rate of GST metabolism
BHt Body height (cm)
blo Venous blood
BV Lean body volume (l)
BWt Body weight (kg)
C Coefficient for scaling to physiological quantity
CV Coefficient of variation
DCM Dichloromethane
exh Exhaled
FFM Fat-free mass (kg)
GSTT1 Glutathione-S-transferase T1
h Hepatic
Kf First-order rate constant for GST metabolism (min1)
km Michaelis constant for oxidative pathway (µmol/l)
m Muscle
MCMC Markov chain Monte Carlo
MFO Mixed-function oxidases
PBPK Physiologically based pharmacokinetic
PC Partition coefficient
pul Pulmonary
Q Flow (l/min)
SD Standard deviation
TBW Total body water (l)
wm Working muscle
tot Total
V Compartment volume (l)
Vmax Maximum rate of oxidative metabolism (µmol/min)
wp Well perfused
VPR Ventilation/perfusion ratio
VO2 Excess oxygen uptake above rest (l/min)
Residual error encompassing intra-individual variability and measurement error
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ACKNOWLEDGMENTS |
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NOTES |
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