# Steady-State Solutions to PBPK Models and Their Applications to Risk Assessment I: Route-to-Route Extrapolation of Volatile Chemicals Authors' Response to Letter By Dr. Kenneth Bogen

Dear Editor:
We are disappointed that Dr. Bogen felt our paper(1) “adds little new” to previously published work utilizing steady state solutions to PBPK models. Moreover, it was not our intention to be either “dismissive” or “misleading” in our admittedly brief citation of these previously published studies, and we apologize if we gave that impression. Nor were we attempting to claim that the mathematical content of the analysis, which required no more than basic algebra and calculus, was particularly complex. Rather, as alluded to in the same paragraph to which Dr. Bogen refers, our goals were (1) to provide a more conceptual understanding of the steady state solutions and (2) to derive and discuss conceptually the implications for oral-to-inhalation route extrapolation.
With respect to our first goal, we thought it pedagogically useful to re-derive the steady state solutions in a notation that we found to be intuitive. Even though our “generic” PBPK model, as Dr. Bogen notes, is nearly identical to those he used, we believe that our derivations provide some important additional insights and utility. For instance, we hoped that our Figure 1, with its related discussion, would help a broad range of readers to more easily understand the correspondence between the PBPK model and the steady state system. In addition, while our expression for the rate of metabolism in Appendix B (equation B.7) is of course mathematically equivalent to the solutions published by others (e.g., Reference 2, equations 4 and 11a–11d), we thought it conceptually useful to rearrange the expression the solution in terms of three terms: the rate of metabolism in the low dose limit Rmet,0, , the maximum rate of metabolism Vmax, and a dimensionless factor 0 corresponding to the fraction of clearance that is metabolism rather than exhalation (discussed in section 2.1). Finally, similar to Pelekis et al. (1997),(3) we thought it would be useful for the general risk analyst for us to present common PBPK model dose metrics given our new notation (Table II). However, we do regret that we neglected to explicitly point out the equivalence of our equations to those previously published. With respect to our second goal, it is our belief that the discussion in section 3.4 of the route-to-route equivalences under a range of dose metrics is a new and useful contribution to the field of risk assessment. For instance, as we write in our abstract, we show that “[e]ven if the appropriate dose metric is unknown, bounds can be placed on the route-to-route equivalence with very limited data.” This could be particularly useful given increasing emphasis of characterizing uncertainty over the range of plausible hypotheses.(4,5)
The underlying motivation for our paper in Risk Analysis was to bring more prominence to the utility of steady state solutions so as to facilitate their use beyond the handful of existing published applications, a goal we presume is shared by Dr. Bogen. Perhaps Dr. Bogen’s letter, which reminds readers of several long-standing applications of this method, also serves that end, and for that we are grateful.
Weihsueh A. Chiu and Paul White
Disclaimer: The views expressed in this article are those of the authors, and do not necessarily reflect the views or policies of the U.S. Environmental Protection Agency.
References
1. Chiu, W.A., & White, P. (2006). Steady-State Solutions to PBPK Models and Their Applications to Risk Assessment I: Route-to-Route Extrapolation of Volatile Chemicals. Risk Analysis, 26, 769-780.
2. Bogen K.T. (1988). Pharmacokinetics for regulatory risk analysis: The case of trichloroethylene. Regulatory Toxicology and Pharmacology, 8, 447-466.
3. Pelekis, M. Krewski, D., & Krishnan, K. (1997). Physiologically based algebraic expressions for predicting steady state toxicokinetics of inhaled vapors. Toxicology Methods, 7, 205-225.
4. OMB (Office of Management and Budget) (2006). Proposed Risk Assessment Bulletin. Released January 9, 2006. Washington, DC: Office of Management and Budget, Executive Office of the President.
5. U.S. EPA (Environmental Protection Agency) (2005). Guidelines for Carcinogen Risk Assessment. EPA/630/P-03/001F. Washington, DC: U.S. Environmental Protection Agency.

## Citation

CHIU, W. AND P. WHITE. Steady-State Solutions to PBPK Models and Their Applications to Risk Assessment I: Route-to-Route Extrapolation of Volatile Chemicals Authors' Response to Letter By Dr. Kenneth Bogen. RISK ANALYSIS. Blackwell Publishing, Malden, MA, 26(3):769-780, (2006).