Universality of J-Shaped and U-Shaped Dose-Response Relations as Emergent Properties of Stochastic Transition Systems

Abstract

Dose-response data for many chemical carcinogens exhibit multiple apparent concentration thresholds. A relatively small increase in exposure concentration near such a threshold disproportionately increases incidence of a specific tumor type. Yet, many common mathematical models of carcinogenesis do not predict such threshold-like behavior when model parameters (e.g., describing cell transition rates) increase smoothly with dose, as often seems biologically plausible. For example, commonly used forms of both the traditional Armitage-Doll and multistage (MS) models of carcinogenesis and the Moolgavkar-Venzon-Knudson (MVK) two-stage stochastic model of carcinogenesis typically yield smooth dose-response curves without sudden jumps or thresholds when exposure is assumed to increase cell transition rates in proportion to exposure concentration. This paper introduces a general mathematical modeling framework that includes the MVK and MS model families as special cases, but that shows how abrupt transitions in cancer hazard rates, considered as functions of exposure concentrations and durations, can emerge naturally in large cell populations even when the rates of cell-level events increase smoothly (e.g., proportionally) with concentration. In this framework, stochastic transitions of stem cells among successive events represent exposure-related damage. Cell proliferation, cell killing and apoptosis can occur at different stages. Key components include: An effective number of stem cells undergoing active cycling and hence vulnerable to stochastic transitions representing somatically heritable transformations. (These need not occur in any special linear order, as in the MS model.) A random time until the first malignant stem cell is formed. This is the first order-statistic, T = min{T1, T2, …, Tn} of n random variables, interpreted as the random times at which each of n initial stem cells or their progeny first become malignant. A random time for a normal stem cell to complete a full set of transformations converting it to a malignant one. This is interpreted very generally as the first passage time through a network of stochastic transitions, possibly with very many possible paths and unknown topology. In this very general family of models, threshold-like (J-shaped or multi-threshold) dose-response nonlinearities naturally emerge even without cytotoxicity, as consequences of stochastic phase transition laws for traversals of random transition networks. With cytotoxicity present, U-shaped as well as J-shaped dose-response curves can emerge. These results are universal, i.e., independent of specific biological details represented by the stochastic transition networks.