Bayesian Analysis of A Rat Formaldehyde DNA–Protein Cross-Link Model

Abstract

As the initial effort in a multi-step uncertainty analysis of a biologically based cancer model for formaldehyde, a Markov chain Monte Carlo (MCMC) analysis was performed for a compartmental model that predicts DNA–protein cross-links (DPX) produced by formaldehyde exposure. The Bayesian approach represented by the MCMC analysis integrates existing knowledge of the model parameters with observed, formaldehyde-DPX-specific data, providing a statistically sound basis for estimating model output uncertainty. Uncertainty and variability were evaluated through a hierarchical structure, where interindividual variability was considered for all model parameters and that variability was assumed to be uncertain on population levels. The uncertainty of the population mean and that of the population variance were significantly reduced through the MCMC analysis. Our investigation highlights several issues that must be dealt with in many real-world analyses (e.g., issues of parameters' nonidentifiability due to limited data) while demonstrating the feasibility of conducting a comprehensive quantitative uncertainty evaluation. The current analysis can be viewed as a case study, for a relatively simple model, illustrating some of the constraints that analysts will face when applying Bayesian approaches to biologically or physiologically based models of increasing complexity.

The authors thank Drs. Rory Conolly, Julia Kimbell, Andy Nong, Jerry Campbell, and Miyoung Yoon and Betsy Gross Bermudez for their valuable reviews and comments. This work was supported by the Formaldehyde Council, Inc.

Present address for Yu-Mei Tan is U.S. Environmental Protection Agency, National Exposure Research Laboratory, Research Triangle Park, NC 27711, USA. Present address for Kai H. Liao is Drug Safety and Metabolism, Wyeth Research, 401 N. Middletown Rd, Pearl River, NY 10965, USA.

Notes

¹For the analysis presented here, the CFD model has been fixed and was considered to be without uncertainty. Although the CFD model is undoubtedly subject to uncertainty with respect to its form and its parameter estimates, and the fluxes predicted by it undoubtedly vary from one animal to another, the analysis of those uncertainties and variabilities has been considered outside the scope of the present analysis. In essence, we have considered the CFD model as the fixed starting point for the current work, including the definition of 20 “flux bins” that partition the nasal passages in such a way that cells within each bin are assumed to have a constant (per unit of formaldehyde concentration) flux of formaldehyde into them. Each bin was assumed to have a known surface area for the whole nose and for the high- and low-tumor regions as defined by Casanova et al. (1994).

² Conolly et al. (2000) and Hubal et al. (1997) did not encounter any problems with parameter identification in their work because they did not consider a formal parameter optimization as in MCMC analysis.

³The Bayesian Output Analysis (BOA) program (Smith, 2007) is an R or S-plus package. It includes four convergence diagnosis methods: Brooks; Gelman, and Rubin; Geweke, Heidelberger, and Welch; and Raftery and Lewis. For more information on BOA program and its applications, please see its website: http://www.public-health.uiowa.edu/boa.

⁴ Chiu and Bois (2006) found that the 1.2 value for CSRF may not be sufficient in some cases; they recommend a value of 1.05.

⁵A reviewer suggested that this might also indicate a shift toward an area of greater density or greater autocorrelation than indicated by the initial samples.

⁶We did not apply the method proposed by Chiu and Bois (2007) in this analysis so we have no direct comparison to report.

⁷One possibility that we could have attempted, but did not, is to use a subset of the DPX data to get priors for the parameters in question. For example, with three or four data points per formaldehyde concentration and region, we could have used one point per concentration and region to suggest priors, and then updated with the remaining two or three data points per concentration and region, plus the whole nose data. A reviewer suggested that we could have just used completely uninformative priors, but even in that case, one would want to “verify” that those priors were biologically reasonable, and that would have entailed some examination of the only data available to do so.