Computational Approaches for Developing Informative Prior

Chapter 18

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Computational Approaches for Developing Informative Prior Distributions for Bayesian Calibration of PBPK Models Jimena L. Davis,1 Rogelio Tornero-Velez,2 and R. Woodrow Setzer*,1 1National Center for Computational Toxicology, Office of Research and Development, U.S. Environmental Protection Agency, 109 T.W. Alexander Drive, Research Triangle Park, North Carolina 27711 2National Exposure Research Laboratory, Office of Research and Development, U.S. Environmental Protection Agency, 109 T.W. Alexander Drive, Research Triangle Park, North Carolina 27711 *E-mail: [email protected]

Using Bayesian statistical methods to quantify uncertainty and variability in human physiologically-based pharmacokinetic (PBPK) model predictions for use in risk assessments requires prior distributions (priors), which characterize what is known or believed about parameters’ values before observing in vivo data. Experimental in vivo data can then be used in Bayesian calibration of PBPK models to refine priors when it exist. However, when little or no in vivo data are available for calibration efforts, parameter estimates and uncertainties can be obtained from priors. In this chapter, we present approaches for specifying informative priors for chemical-specific PBPK model parameters based on information obtained from chemical structures and in vitro assays. Means and standard deviations (or coefficients of variation) for priors are derived from comparisons of predicted values from computational (e.g., QSAR) methods or in vitro assays and experimentally-determined chemical-specific values for a data set of chemicals.

Not subject to U.S. Copyright. Published 2011 by American Chemical Society In Parameters for Pesticide QSAR and PBPK/PD Models for Human Risk Assessment; Knaak, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2012.


Introduction Calibrating physiologically-based pharmacokinetic (PBPK) models can be quite a challenge due to the often limited amount of time course data available to determine parameter estimates for the large number of model parameters that are typical of these models. Since parameters of PBPK models reflect actual physiological and biological characteristics, additional sources of information found in the literature can be used directly to compute parameter estimates albeit with uncertainty. These sources include: direct measurements for body weight and other organ weights; “typical” values for physiological characteristics like organ weights, cardiac output, and fraction of cardiac output for each organ; in vitro determined rates of enzymatically catalyzed clearance; and partition coefficients, determined in vitro through equilibration experiments or using one of the computational approaches (e.g., Poulin and Theil (1), Rodgers et al. (2), Schmitt (3)). Bayesian methods allow one to combine this type of information, which represents prior knowledge about the unknown PBPK model parameters, with in vivo data (e.g., blood and tissue concentrations) in one analysis so that the resulting uncertainty is carried through to parameter estimates, and thus to model predictions (4–10). Although Bayesian statistical methods are used often to calibrate PBPK models because of the ability to easily include additional prior information about model parameters into the estimation process, data limitations as well as the complexity of these types of models can sometimes result in parameters that cannot be uniquely identified from the given data set. Since uncertainty in model predictions depends on uncertainties associated with the model parameters, non-identifiable parameters with large uncertainties can lead to large uncertainties in the derived predictions. Long computational runtimes can also be an issue when determining PBPK model parameter estimates and uncertainties due to the computational algorithms used in these efforts. While these two issues can pose challenges to Bayesian calibration, the use of prior information from the literature reflecting what is known about model parameters can help in addressing these challenges. Therefore, it is advantageous to pursue testable scientific approaches for using prior information from valid literature sources that can then be refined through Bayesian methods. In this chapter, we provide examples to illustrate such approaches. A Bayesian analysis of a model characterizes the values of the parameters for that model as probability distributions instead of point estimates that are sometimes determined for unknown model parameters. These distributions, termed posterior distributions (posteriors for short), give a fuller characterization of the uncertainty associated with parameter values and, indirectly, model predictions. Bayesian analysis is based on Bayes’ rule, which allows one to determine posterior densities p(θ|y) (and distributions P(θ|y)) for the unknown model parameter θ

292 In Parameters for Pesticide QSAR and PBPK/PD Models for Human Risk Assessment; Knaak, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2012.


by combining the likelihood function (or the data distribution) p(y|θ) that captures the information available in the data y with the prior densities (distributions), or priors, p(θ), which characterize the information we have about model parameters outside the data being immediately considered (11). The likelihood function, p(y|θ), is driven by an error model for the data and a model for the population distribution of some of the model parameters, all conditional on unknown model parameters. The ability to combine prior knowledge about the unknown model parameters with data to determine posteriors is one of the major advantages to using Bayesian analysis to calibrate PBPK models for model parameters.

Hierarchical Bayesian Analysis Quantifying the variability in estimated PBPK model parameters along with uncertainty is important when using these models in human health risk assessment applications. PBPK model parameters, such as blood flows, tissue volumes, absorption rates, and metabolic clearance rates, can vary from individual to individual across a population. This type of variability, known as interindividual variability, can account for the differences observed in blood and tissue concentrations for various individuals that are exposed to a chemical or drug under the same conditions as a result of varying PBPK model parameters among individuals. Another source of variability that is considered in PBPK modeling efforts is intraindividual variability, and this type of variability can be used to describe the differences in observed concentrations within the same individual that can result from variations within that individual as well as measurement error. Both intraindividual and interindividual variability can be characterized along with uncertainty in a hierarchical Bayesian analysis by using different levels or stages in the statistical model (7, 11). At the first level, or stage, of the hierarchical model, a measurement error model is used to describe the observed intraindividual variability. This model describes how predicted concentrations deviate from measured concentrations in each individual, and gaussian or log-gaussian distributions are often used to model the behavior of the deviations (7). The population model describing the interindividual variability, or how the PBPK model parameters vary across the population, is specified at the second stage, or level, of the hierarchical model. Since most PBPK model parameters reflect biological characteristics and are often skewed across a population, a log-gaussian distribution is typically used for the population model (7). Prior distributions are then assigned to the parameters that characterize the population model, e.g. means and variances, for the PBPK model parameters that are being estimated. In the following section, we describe how priors, which reflect what is known or believed about parameters before evaluating in vivo data, are derived for calibrating PBPK models and provide examples that illustrate how information from the literature can be used to develop more informative priors for chemical-specific parameters. 293 In Parameters for Pesticide QSAR and PBPK/PD Models for Human Risk Assessment; Knaak, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2012.


Priors Priors for PBPK model parameters are conveniently characterized by a probability distribution with a small number of parameters, such as mean and standard deviation or variance, which are easy to compute or derive from information found in the literature. The level of information in priors that can be assigned to parameters for Bayesian calibration efforts ranges from point priors, implying perfect knowledge, to informative priors to weakly informative (diffuse) priors. Due to the large number of parameters typical of PBPK models, parameters that are thought to be well-known (e.g., physiological parameters) are often treated as perfectly known in calibration exercises. This is equivalent to assigning point priors, like the example shown in Figure 1, to these parameters in a Bayesian analysis. . Assigning point priors to parameters understates the overall uncertainty in the estimates when the true values have not been directly measured, and may distort some estimated values, which have to compensate for deviations between true parameter values and those specified in the model.

Figure 1. Point prior example. Diffuse or weakly informative priors can be broad uniform distributions with every value in some range being equally likely. Diffuse priors, such as the example shown in Figure 2, can be constructed by specifying broad uncertainty distributions (for example, gaussian or log-gaussian distributions with large 294 In Parameters for Pesticide QSAR and PBPK/PD Models for Human Risk Assessment; Knaak, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2012.


variances or coefficients of variation); then, following Bayesian calibration, an assessment can be made as to whether the distributional properties of the posterior have changed with respect to the prior. If the data used in the Bayesian calibration process contain no information on parameters that are assigned diffuse priors (i.e., the parameters are non-identifiable with respect to the data set), then the marginal posteriors will be equal to the priors (12), which lead to much larger uncertainty in model predictions. Computations, which can already be lengthy for these types of models due to the high dimensionality and complex nature of the parameter space, can be even more inefficient due to algorithms exploring areas of the parameter space that are highly unlikely as a result of very broad distributions. Optimally, priors for PBPK model parameters should reflect what can be derived from the literature (e.g., mean values, standard deviations, and plausible ranges) for the parameters of interest.

Figure 2. Diffuse (weakly informative) prior example.

Parameters for PBPK models can be divided into two categories: physiological and chemical-specific. Physiological parameters in PBPK models include body weight, cardiac output, tissue volumes and blood flows. Values for some of these parameters, such as body weight, are usually measured during experiments and can be treated essentially as perfectly known. For the remaining physiological parameters not observed experimentally, mean values, standard 295 In Parameters for Pesticide QSAR and PBPK/PD Models for Human Risk Assessment; Knaak, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2012.


deviations (or coefficients of variation CVs), and ranges are typically found in the literature (13). Reference values found in the literature for physiological parameters provide a good starting place for informative priors for these parameters (14). Reported mean values in Brown et al. (13) for physiological parameters, such as blood flows and tissues volumes, are often used as the prior means for these parameters. However, the accompanying standard deviations reported by Brown et al. are more reflective of the variability across the multiple sources from which the mean values were taken and less of a measurement of the underlying uncertainty in these parameter values (13). Instead of using the standard deviations reported in Brown et al. (13), several authors have used relatively small CVs (in the range of 20 – 50%) to characterize the uncertainty for physiological parameters (4, 6, 7, 15). We followed this same practice and found that the range of values for these parameters based on small CVs were similar to the reported ranges for these same parameters in (13). Therefore, this approach seemed to provide reasonable, informative priors for physiological parameters. For this reason, we chose to focus on developing approaches that would lead to more informative priors for chemical-specific parameters.

Chemical-Specific Parameters Chemical-specific parameters, such as partition coefficients, oral absorption rates, diffusion-limited tissue permeability rates, and metabolic parameters, are oftentimes assigned weakly informative priors as a result of more uncertainty in the values for these parameters. While values obtained from the literature are used in many PBPK calibration efforts for physiological parameters with less uncertainty, there seems to be less consensus in the literature on the uncertainty one should use when specifying priors for chemical-specific parameters (4–9, 14–16). In many cases, priors for these parameters are based on values determined from in silico or in vitro methods with large CVs ranging from 50 – 200% being used due to the limited amount of information available in the literature for these parameters (4, 5, 7, 9, 15). There is readily available information in the literature that can be used in developing informative priors for chemical-specific parameters. Computational predictors, such as Quantitative Structure Activity Relationship (QSAR), Quantitative Structure Property Relationship (QSPR), and mechanistic models, can be used to predict chemical-specific parameters (e.g., oral absorption rates, permeability coefficients, partition coefficients) a priori based on their inherent physicochemical properties (1–3, 17–24). There also exist in vitro assays that measure some chemical-specific parameters, such as intrinsic in vitro clearance rates, and these can also be used to inform the priors for in vivo predictions of these parameters. We have developed approaches that can make use of these sources of information in specifying more informative priors for chemical-specific parameters. We discuss and illustrate our two different approaches for developing more informative priors for chemical-specific parameters in the following examples. 296 In Parameters for Pesticide QSAR and PBPK/PD Models for Human Risk Assessment; Knaak, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2012.


Partition Coefficients Partition coefficients are important in PBPK models because they describe the distribution of a chemical into various tissues within the body. They also represent a large set of chemical-specific parameters with more associated uncertainty; hence, weakly informative or vague priors are often used to characterize the uncertainty in these parameters in Bayesian analysis. In an effort to develop more informative priors for tissue-plasma partition coefficients, we surveyed the literature for computational predictors for these parameters. A number of mechanistic models using a small number of physicochemical properties, such as the octanol-water partition coefficient and the unbound fraction of a chemical in plasma, have been developed to predict tissue-plasma partition coefficients (1–3, 21–24). These models take into account lipophilicity and binding properties of the compound as well as the composition of the tissues. The underlying assumption of these models is that the chemical tissue solubility is an additive function of the chemical solubility in the various components that comprise the tissue (e.g., water, neutral lipids, and phospholipids). One widely used approach developed by Poulin and Theil (1) predicted tissue-plasma partition coefficients by considering neutral lipids and phospholipids only. Rodgers et al. (2) extended the idea of Poulin and Theil (1) by including the interaction of the compound with acidic phospholipids in the tissue in addition to neutral lipids and phospholipids. Both of these models were used to predict tissue specific partition coefficients for a diverse set of compounds and were shown to reasonably predict partition coefficients when compared to in vivo data (1, 2). However, issues with universal applicability to all types of chemicals were cited in (3) for the models of both Poulin and Theil (1) and Rodgers et al. (2). In an effort to address this issue, Schmitt’s 2008 model (3) combined principles from the models of Poulin and Theil (1) and Rodgers et al. (2) to formulate a model that improves the predictions for acidic, basic, neutral, and multiply charged compounds. Schmitt’s model extends the methods of Poulin and Theil (1) and Rodgers et al. (2) by dividing the tissues into several components: blood plasma, blood cells, interstitial space, and cellular space, and characterizing the binding to each component. Tissue-plasma partition coefficients in Schmitt’s model are determined via the following equation:

where Fint represents the volume fraction of interstitial space and fuint, the unbound fraction in interstitium, is given by



Fwint, Fwpl, Fpint, and Fppl are the water and protein fractions in interstitium and plasma, respectively. Fcell represents the volume fraction in cellular space, while fucell, the unbound fraction in cellular space, is given by

where F is the volume fraction and K is the corresponding partition coefficient for water (W), neutral lipids (nL), neutral phospholipids (nPL), acidic phospholipids (aPL), and protein (P). The unbound fraction of compound in plasma is denotd by fup, while κcell:plasma is the ratio of the unbound concentrate in the cellular space and plasma. This model was validated with a data set of 59 compounds for 12 different tissues (adipose, bone, brain, gut, heart, kidneys, liver, lung, muscle, skin, spleen, and testes) as well as red blood cells; 73% of the predicted values deviated less than 3-fold from the experimentally measured values (3). After selecting a computational model from the literature that provides predictions for partition coefficients, we then compiled an extensive data set, including the data from Schmitt (3) as well as additional in vivo measured values for tissue-plasma partition coefficients for a number of tissues (1, 2, 21–26). In our data collection, we tried to be as thorough as possible so as to avoid biased results by exploring only a small subset of chemical space. We also documented the conditions under which the experimentally measured partition coefficients were collected to make sure that our model comparisons were based on valid data. In some cases, observations for partition coefficients can be based on the free concentration, while in other experimental designs the total concentration is used to determine the value of partition coefficients. Incorrect conclusions can result from comparing predictions from Schmitt’s model, which considers total concentration of a chemical, to data that are produced by measuring free concentration as opposed to total concentration. For example, one might incorrectly conclude that Schmitt’s model does not provide accurate predictions for partition coefficients based on comparisons with in vivo partition coefficient data for which free concentration, and not total concentration, was measured. While surveying the literature for experimental tissue-plasma partition coefficients for various chemicals, we also collected the associated physicochemical property information when reported in the same source for those chemicals. When this information was not reported in the same source, we performed additional literature searches for values for these properties. We determined the relationship between the experimental and predicted values by regressing the experimental values for the partition coefficients on Schmitt’s predicted values on the log scale. Regression analysis is used to adjust for any bias that may exist. The regression corrected value that we derived from the computational predictor is then used as the mean for the prior for the chemical-specific parameter of interest. The variance from the experimental versus the predicted regression lines (or the root mean square error) is used as the variance (or standard deviation) in the prior for the chemical-specific parameter. 298 In Parameters for Pesticide QSAR and PBPK/PD Models for Human Risk Assessment; Knaak, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2012.


Figure 3 and Figure 4 show the results for the partition coefficients for adipose tissue and brain, respectively. In each plot, the predicted tissue-plasma partition coefficients are plotted on the x-axis while the experimentally measured tissue-plasma partition coefficients are plotted on the y-axis. Each data point represents one chemical from the data set we collected and used in the regression analysis (chemicals may be included in the data set more than once if there were multiple reported values in the literature from different laboratories). If the model predictions and experimental in vivo tissue-plasma partition coefficients were exactly the same, all of the data points shown in Figure 3 and Figure 4 would be along the identity line (dashed line in the figures). The (solid) regression line depicts how the model predictions deviate from the measured partition coefficients. Based on the two figures shown here, Schmitt’s model appears to do a better job of predicting the tissue-plasma partition coefficient for brain in comparison to the results for adipose tissue. The model overpredicts the adipose-plasma partition coefficient for most of the compounds included in the data set used here.

Figure 3. Regression analysis results for adipose-plasma partition coefficient. 299 In Parameters for Pesticide QSAR and PBPK/PD Models for Human Risk Assessment; Knaak, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2012.


Figure 4. Regression analysis results for brain-plasma partition coefficient. The log standard deviations that we computed from the regression analysis are given in Table I for the various tissues considered in this analysis. These values correspond to coefficients of variation (CVs) in the range of 50 – 70% for most of the tissue-plasma partition coefficients considered here. This is an improvement over CVs of 150% that have been used in vague, or weakly informative, priors for partition coefficients in some PBPK modeling efforts cited in the literature (4, 7). While in some of the Bayesian modeling efforts found in the literature, the means for the priors were based on published experimentally determined partition coefficients values (for example, in Bois et al. Bayesian analysis of tetrachloroethylene (4)), predicted partition coefficients from the Poulin and Theil approach (1) were used in cases where measured partition coefficients were not available (8, 16). As noted from Figure 3 and Figure 4, predictions from Schmitt’s model were better for brain than for adipose tissue, so using the bias corrected predicted value from our approach for the prior mean might be more plausible for certain tissues for which the accuracy of the predictive models is lower. In Figure 5, comparisons between a diffuse or weakly informative prior with a log standard deviation of 1.5 and a more informative prior with a log standard deviation of 300 In Parameters for Pesticide QSAR and PBPK/PD Models for Human Risk Assessment; Knaak, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2012.


0.6093 are shown for brain. The weakly informative prior (dashed line) is much broader and includes values that are more unlikely in comparison to the tighter informative prior (solid line) based on the approach outlined above.

Figure 5. Comparison of diffuse (weakly informative) and informative prior for brain-plasma partition coefficient.

Table I. log Standard Deviations (log SD) for Partition Coefficient Priors Tissue

log SD

Tissue

log SD

Adipose

0.5423

Lung

0.9153

Bone

0.597

Muscle

0.5751

Brain

0.6093

Skin

0.4474

Gut

0.7169

Spleen

0.724

Heart

0.5766

Testes

0.414

Kidney

0.614

Thymus

0.2685

Liver

0.6926



We note that this same approach can also be applied when developing more informative priors for additional chemical-specific parameters, such as oral absorption rates and dermal permeability, based on computational predictors that exist provided that substantial data sets can be compiled via thorough surveys of the literature. While this general approach allows one to use readily available data from the literature to construct more informative priors for chemical-specific parameters, one must note that the priors developed via this approach are very much dependent on how well the selected computational method predicts the parameter of interest and how well the chemicals included in the data set for the regression analysis cover chemical space. Clearance In our second example, we propose the use of a more empirical approach to develop informative priors for metabolic clearance parameters due to the availability of in vitro measure rate constants. We demonstrate this approach through a specific example involving the kinetics of the pyrethroid permethrin, which consists of two isomers (cis- and trans-). These two isomers behave differently metabolically with cis-permethrin metabolized slower by hepatic microsomes than trans-permethrin. Instead of using computational predictors and in vivo measurements of clearance rates, we use Scollon et al. (27) in vitro measures of intrinsic hepatic clearance rates for both cis- and trans-permethrin (e.g., mean and standard deviations for intrinsic clearance measured in rat hepatic microsomes). These in vitro measurements collected in hepatic microsomes, which include both oxidative and hydrolytic processes, are initially scaled through some function f to get an in vitro clearance value as described by the expression below

where CLin vitro represents the intrinsic clearance measurements in rat hepatic microsomes in (27). For this example, determining in vitro clearance measurements for cis- and trans-permethrin from the in vitro measurements made in rat hepatic microsomes involved scaling first order metabolic rate constants by several appropriate factors (e.g., mg microsomes/g liver and g liver/kg body weight) (27). We then apply in vitro-in vivo scaling factors to the scaled in vitro measurements of clearance to describe the in vivo clearance of permethrin in its PBPK model

Hence, the in vivo clearance is given by the product of the in vitro-in vivo scaling factors (represented by γ in the expression above) and the scaled in vitro measurements of intrinsic clearance from Scollon et al. (27). This process represents one step in the parallelogram approach (28), which can be used not only to describe extrapolation from rodent to human models but can also be used to predict in vivo characteristics based on in vitro (or in silico) 302 In Parameters for Pesticide QSAR and PBPK/PD Models for Human Risk Assessment; Knaak, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2012.


models (see Figure 6). The scaling factors are adjusted through the Bayesian estimation process to account for deviations from the observed in vivo data when compared to predictions obtained with the in vitro clearance data. These deviations may be the result of actual uncertainty in the clearance parameters, misspecified biology in the PBPK model, or a combination of both; all of which is captured through the use of the in vitro-in vivo scaling factors. The uncertainty in the in vivo clearance is quantified through the uncertainty characterized for the in vitro-in vivo scaling factor, which is assigned a weakly informative prior (due to no information being available in the literature on this parameter) that is refined via the Bayesian calibration efforts.

Figure 6. Parallelogram approach for species extrapolation.

Summary and Conclusions Uncertainty in PBPK model parameters and predictions can be assessed via the use of Bayesian statistical methods by combining prior distributions characterizing what is known about the parameters of interest with in vivo data. However, challenges can arise when attempting to quantify parameter (and hence model) uncertainty in PBPK applications with limited or no in vivo data. As stated earlier, these uncertainties can still be assigned in the absence of pharmacokinetic data, but the issue then becomes developing informative prior distributions that truly reflect what is known about a parameter as opposed to either broad weakly informative priors that can overestimate the uncertainty in both the parameters and model predictions or unrealistically precise point priors which erroneously imply certainty. We have demonstrated in this chapter how more informative priors can be developed for chemical-specific PBPK model parameters (e.g., partition coefficients and clearance rates) through the use of appropriate computational predictors (e.g., QSAR and mechanistic models), in vitro methods, and readily available data in the literature. These approaches can be expanded upon and used for other chemical-specific parameters in PBPK models as well as model parameters in other dynamical systems. The priors derived in these approaches are currently being used in Bayesian calibration efforts for PBPK models describing the kinetics of cis- and trans-permethrin with limited in vivo data and large parameter sets. In addition to determining parameter estimates and their uncertainties, we are also using the Bayesian framework to test some of the underlying biological assumptions that describe partition coefficients in Schmitt’s 303 In Parameters for Pesticide QSAR and PBPK/PD Models for Human Risk Assessment; Knaak, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2012.


model (3). Comparisons of model results from the permethrin PBPK model with unconstrained partition coefficients that are allowed to vary independently with results from the model with partition coefficients computed directly from Schmitt’s model (3) and constrained to vary only as a result of differences in the tissue compositions will give us insights into how well Schmitt’s model (3) describes the partitioning of permethrin. Ultimately, our efforts will provide statistical approaches for a) quantifying prior uncertainty in model parameters that can be used to more accurately quantify uncertainty in PBPK model predictions and b) testing and evaluating biological hypotheses made for PBPK models.

Acknowledgments The views expressed in this chapter are those of the authors and do not necessarily reflect the views or policies of the U.S. Environmental Protection Agency. Mention of trade names or commercial products does not constitute endorsement or recommendation for use.

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Computational Approaches for Developing Informative Prior

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