Symmetry, Identifiability, and Prediction Uncertainties in Multistage Clonal Expansion (MSCE) Models of Carcinogenesis

Abstract

Many models of exposure-related carcinogenesis, including traditional linearized multistage models and more recent two-stage clonal expansion (TSCE) models, belong to a family of models in which cells progress between successive stages-possibly undergoing proliferation at some stages-at rates that may depend (usually linearly) on biologically effective doses. Biologically effective doses, in turn, may depend nonlinearly on administered doses, due to PBPK nonlinearities. This article provides an exact mathematical analysis of the expected number of cells in the last ("malignant") stage of such a "multistage clonal expansion" (MSCE) model as a function of dose rate and age. The solution displays symmetries such that several distinct sets of parameter values provide identical fits to all epidemiological data, make identical predictions about the effects on risk of changes in exposure levels or timing, and yet make significantly different predictions about the effects on risk of changes in the composition of exposure that affect the pharmacodynamic dose-response relation. Several different predictions for the effects of such an intervention (such as reducing carcinogenic constituents of an exposure) that acts on only one or a few stages of the carcinogenic process may be equally consistent with all preintervention epidemiological data. This is an example of nonunique identifiability of model parameters and predictions from data. The new results on nonunique model identifiability presented here show that the effects of an intervention on changing age-specific cancer risks in an MSCE model can be either large or small, but that which is the case cannot be predicted from preintervention epidemiological data and knowledge of biological effects of the intervention alone. Rather, biological data that identify which rate parameters hold for which specific stages are required to obtain unambiguous predictions. From epidemiological data alone, only a set of equally likely alternative predictions can be made for the effects on risk of such interventions.

Could Removing Arsenic from Tobacco Smoke Significantly Reduce Smoker Risks of Lung Cancer?

“…The following discussion defines the fraction of smoker lung cancers "caused by" arsenic in cigarette smoke as the fraction of smoker lung cancers that would be prevented if arsenic were removed from cigarette smoke. (1) This fraction will be estimated with the help of the following MSCE conceptual framework (61) for lung carcinogenesis, diagrammed in Fig. 1.…”

“…This conceptual framework can be developed into a full quantitative model for the dose-timeresponse relationship between chemical exposure and lung cancer risk (Cox & Huber, 2007). The flow balance equations corresponding to Fig.…”

“…where N(t), P(t), and F(t) are the expected sizes of the normal, patch, and field states at time t, respectively. This system of equations can be solved for the expected number of tumor cells formed by age t (Cox & Huber, 2007). For a constant average daily smoking exposure starting at age 20 with approximately 100% of cells and compartments in state N at age t = 20 years, the solution for the expected cumulative number of tumor cells by age 20 + t years, or dose-time-response model, is:…”

“…The symmetries in this solution imply that if [a, b, c, d] is any set of values for [μ NP , μ PF , μ FM − μ F , μ FM ] that fits available data, then so is [b, a, c, d]. Cox and Huber (2007) show that six different sets of parameter values, related to each other as members of a symmetry group, provide identical predictions of M(t) for any smoking history s(t), but four of them can be eliminated when μ FM − μ F < 0, as implying unrealistic negative values for some transition rates.) Following Schollnberger et al, dose dependence of transition rates is incorporated into the model via the linear formulas:…”

Between Epidemiology and Basic Genetic Research – Systems Epidemiology

“…The situation today is not different from the early papers, namely that there is a lack of observational data of the stages of carcinogenesis. Due to this lack of observed data the parameters in the mathematical model can not be solved uniquely (16). At the same time the importance of fixing one parameter in the mathematical model has been stressed, this could be the duration or changes related to the last stage.…”