Correlations Between Parameters in Risk Models: Estimation and Propagation of Uncertainty by Markov Chain Monte Carlo

Monte Carlo simulation has become the accepted method for propagating parameter uncertainty through risk models. It is widely appreciated, however, that correlations between input variables must be taken into account if models are to deliver correct assessments of uncertainty in risk. Various two-stage methods have been proposed that first estimate a correlation structure and then generate Monte Carlo simulations, which incorporate this structure while leaving marginal distributions of parameters unchanged. Here we propose a one-stage alternative, in which the correlation structure is estimated from the data directly by Bayesian Markov Chain Monte Carlo methods. Samples from the posterior distribution of the outputs then correctly reflect the correlation between parameters, given the data and the model. Besides its computational simplicity, this approach utilizes the available evidence from a wide variety of structures, including incomplete data and correlated and uncorrelated repeat observations. The major advantage of a Bayesian approach is that, rather than assuming the correlation structure is fixed and known, it captures the joint uncertainty induced by the data in all parameters, including variances and covariances, and correctly propagates this through the decision or risk model. These features are illustrated with examples on emissions of dioxin congeners from solid waste incinerators.     

A Trichloroethylene Risk Assessment Using a Monte Carlo Analysis of Parameter Uncertainty in Conjunction with Physiologically-Based Pharmacokinetic Modeling

A Monte Carlo simulation is incorporated into a risk assessment for trichloroethylene (TCE) using physiologically-based pharmacokinetic (PBPK) modeling coupled with the linearized multistage model to derive human carcinogenic risk extrapolations. The Monte Carlo technique incorporates physiological parameter variability to produce a statistically derived range of risk estimates which quantifies specific uncertainties associated with PBPK risk assessment approaches. Both inhalation and ingestion exposure routes are addressed. Simulated exposure scenarios were consistent with those used by the Environmental Protection Agency (EPA) in their TCE risk assessment. Mean values of physiological parameters were gathered from the literature for both mice (carcinogenic bioassay subjects) and for humans. Realistic physiological value distributions were assumed using existing data on variability. Mouse cancer bioassay data were correlated to total TCE metabolized and area-under-the-curve (blood concentration) trichloroacetic acid (TCA) as determined by a mouse PBPK model. These internal dose metrics were used in a linearized multistage model analysis to determine dose metric values corresponding to 10^-6 lifetime excess cancer risk. Using a human PBPK model, these metabolized doses were then extrapolated to equivalent human exposures (inhalation and ingestion). The Monte Carlo iterations with varying mouse and human physiological parameters produced a range of human exposure concentrations producing a 10^-6 risk.     

Effects of Uncertainties on Exposure Estimates to Methylmercury: A Monte Carlo Analysis of Exposure Biomarkers versus Dietary Recall Estimation

This article presents a general model for estimating population heterogeneity and "lack of knowledge" uncertainty in methylmercury (MeHg) exposure assessments using two-dimensional Monte Carlo analysis. Using data from fish-consuming populations in Bangladesh, Brazil, Sweden, and the United Kingdom, predictive model estimates of dietary MeHg exposures were compared against those derived from biomarkers (i.e., [Hg]hair and [Hg]blood). By disaggregating parameter uncertainty into components (i.e., population heterogeneity, measurement error, recall error, and sampling error) estimates were obtained of the contribution of each component to the overall uncertainty. Steady-state diet:hair and diet:blood MeHg exposure ratios were estimated for each population and were used to develop distributions useful for conducting biomarker-based probabilistic assessments of MeHg exposure. The 5th and 95th percentile modeled MeHg exposure estimates around mean population exposure from each of the four study populations are presented to demonstrate lack of knowledge uncertainty about a best estimate for a true mean. Results from a U.K. study population showed that a predictive dietary model resulted in a 74% lower lack of knowledge uncertainty around a central mean estimate relative to a hair biomarker model, and also in a 31% lower lack of knowledge uncertainty around central mean estimate relative to a blood biomarker model. Similar results were obtained for the Brazil and Bangladesh populations. Such analyses, used here to evaluate alternative models of dietary MeHg exposure, can be used to refine exposure instruments, improve information used in site management and remediation decision making, and identify sources of uncertainty in risk estimates.     

Bayesian Estimation of Pharmacokinetic and Pharmacodynamic Parameters in a Mode-of-Action-Based Cancer Risk Assessment for Chloroform

Chloroform is a carcinogen in rodents and its carcinogenicity is secondary to events associated with cytotoxicity and regenerative cell proliferation. In this study, a physiologically based pharmacokinetic/pharmacodynamic (PBPK/PD) model that links the processes of chloroform metabolism, reparable cell damage, cell death, and regenerative cellular proliferation was developed to support a new cancer dose-response assessment for chloroform. Model parameters were estimated using Markov Chain Monte Carlo (MCMC) analysis in a two-step approach: (1) metabolism parameters for male and female mice and rats were estimated against available closed chamber gas uptake data; and (2) PD parameters for each of the four rodent groups were estimated from hepatic and renal labeling index data following inhalation exposures. Subsequently, the resulting rodent PD parameters together with literature values for human age-dependent physiological and metabolism parameters were used to scale up the rodent model to a human model. The human model was used to predict exposure conditions under which chloroform-mediated cytolethality is expected to occur in liver and kidney of adults and children. Using the human model, inhalation Reference Concentrations (RfCs) and oral Reference Doses (RfDs) were derived using an uncertainty factor of 10. Based on liver and kidney dose metrics, the respective RfCs were 0.9 and 0.09 ppm; and the respective RfDs were 0.4 and 3 mg/kg/day.     

A Comparative Pharmacokinetic Estimate of Mercury in U.S. Infants Following Yearly Exposures to Inactivated Influenza Vaccines Containing Thimerosal

The use of thimerosal preservative in childhood vaccines has been largely eliminated over the past decade in the United States because vaccines have been reformulated in single‐dose vials that do not require preservative. An exception is the inactivated influenza vaccines, which are formulated in both multidose vials requiring preservative and preservative‐free single‐dose vials. As part of an ongoing evaluation by USFDA of the safety of biologics throughout their lifecycle, the infant body burden of mercury following scheduled exposures to thimerosal preservative in inactivated influenza vaccines in the United States was estimated and compared to the infant body burden of mercury following daily exposures to dietary methylmercury at the reference dose established by the USEPA. Body burdens were estimated using kinetic parameters derived from experiments conducted in infant monkeys that were exposed episodically to thimerosal or MeHg at identical doses. We found that the body burden of mercury (AUC) in infants (including low birth weight) over the first 4.5 years of life following yearly exposures to thimerosal was two orders of magnitude lower than that estimated for exposures to the lowest regulatory threshold for MeHg over the same time period. In addition, peak body burdens of mercury following episodic exposures to thimerosal in this worst‐case analysis did not exceed the corresponding safe body burden of mercury from methylmercury at any time, even for low‐birth‐weight infants. Our pharmacokinetic analysis supports the acknowledged safety of thimerosal when used as a preservative at current levels in certain multidose infant vaccines in the United States.     

Development of a Standard Soil-to-Skin Adherence Probability Density Function for Use in Monte Carlo Analyses of Dermal Exposure

It has recently been suggested that "standard" data distributions for key exposure variables should be developed wherever appropriate for use in probabilistic or "Monte Carlo" exposure analyses. Soil-on-skin adherence estimates represent an ideal candidate for development of a standard data distribution: There are several readily available studies which offer a consistent pattern of reported results, and more importantly, soil adherence to skin is likely to vary little from site-to-site. In this paper, we thoroughly review each of the published soil adherence studies with respect to study design, sampling, and analytical methods, and level of confidence in the reported results. Based on these studies, probability density functions (PDF) of soil adherence values were examined for different age groups and different sampling techniques. The soil adherence PDF developed from adult data was found to resemble closely the soil adherence PDF based on child data in terms of both central tendency (mean = 0.49 and 0.63 mg-soil/cm²-skin, respectively) and 95th percentile values (1.6 and 2.4 mg-soil/cm²-skin, respectively). Accordingly, a single, "standard" PDF is presented based on all data collected for all age groups. This standard PDF is lognormally distributed; the arithmetic mean and standard deviation are 0.52 ± 0.9 mg-soil/cm²-skin. Since our review of the literature indicates that soil adherence under environmental conditions will be minimally influenced by age, sex, soil type, or particle size, this PDF should be considered applicable to all settings. The 50th and 95th percentile values of the standard PDF (0.25 and 1.7 mg-soil/cm²-skin, respectively) are very similar to recent U.S. EPA estimates of "average" and "upper-bound" soil adherence (0.2 and 1.0 mg-soil/cm²-skin, respectively).     

Estimation of Error and Bias in Bayesian Monte Carlo Decision Analysis Using the Bootstrap

Bayesian Monte Carlo (BMC) decision analysis adopts a sampling procedure to estimate likelihoods and distributions of outcomes, and then uses that information to calculate the expected performance of alternative strategies, the value of information, and the value of including uncertainty. These decision analysis outputs are therefore subject to sample error. The standard error of each estimate and its bias, if any, can be estimated by the bootstrap procedure. The bootstrap operates by resampling (with replacement) from the original BMC sample, and redoing the decision analysis. Repeating this procedure yields a distribution of decision analysis outputs. The bootstrap approach to estimating the effect of sample error upon BMC analysis is illustrated with a simple value-of-information calculation along with an analysis of a proposed control structure for Lake Erie. The examples show that the outputs of BMC decision analysis can have high levels of sample error and bias.     

Uncertainties in Pharmacokinetic Modeling for Perchloroethylene: II. Comparison of Model Predictions with Data for a Variety of Different Parameters

In this paper we compare expectations derived from 10 different human physiologically based pharmacokinetic models for perchloroethylene with data on absorption via inhalation, and concentrations in alveolar air and venous blood. Our most interesting finding is that essentially all of the models show a time pattern of departures of predictions of air and blood levels relative to experimental data that might be corrected by more sophisticated model structures incorporating either (a) heterogeneity of the fat compartment (with respect to either perfusion or partition coefficients or both) or (b) intertissue diffusion of perchloroethylene between the fat and muscle/VRG groups. Similar types of corrections have recently been proposed to reduce analogous anomalies in the fits of pharmacokinetic models to the data for several volatile anesthetics.^(17-20) A second finding is that models incorporating resting values for alveolar ventilation in the region of 5.4 L/min seemed to be most compatible with the most reliable set of perchloroethylene uptake data.     

Inferring an Augmented Bayesian Network to Confront a Complex Quantitative Microbial Risk Assessment Model with Durability Studies: Application to Bacillus Cereus on a Courgette Purée Production Chain

The Monte Carlo (MC) simulation approach is traditionally used in food safety risk assessment to study quantitative microbial risk assessment (QMRA) models. When experimental data are available, performing Bayesian inference is a good alternative approach that allows backward calculation in a stochastic QMRA model to update the experts’ knowledge about the microbial dynamics of a given food‐borne pathogen. In this article, we propose a complex example where Bayesian inference is applied to a high‐dimensional second‐order QMRA model. The case study is a farm‐to‐fork QMRA model considering genetic diversity of Bacillus cereus in a cooked, pasteurized, and chilled courgette purée. Experimental data are Bacillus cereus concentrations measured in packages of courgette purées stored at different time‐temperature profiles after pasteurization. To perform a Bayesian inference, we first built an augmented Bayesian network by linking a second‐order QMRA model to the available contamination data. We then ran a Markov chain Monte Carlo (MCMC) algorithm to update all the unknown concentrations and unknown quantities of the augmented model. About 25% of the prior beliefs are strongly updated, leading to a reduction in uncertainty. Some updates interestingly question the QMRA model.