Extending the Stochastic Two‐Stage Model of Carcinogenesis to Include Self‐Regulation of the Nonmalignant Cell Population

One of the challenges of introducing greater biological realism into stochastic models of cancer induction is to find a way to represent the homeostatic control of the normal cell population over its own size without complicating the analysis too much to obtain useful results. Current two‐stage models of carcinogenesis typically ignore homeostatic control. Instead, a deterministic growth path is specified for the population of “normal” cells, while the population of “initiated” cells is assumed to grow randomly according to a birth‐death process with random immigrations from the normal population. This paper introduces a simple model of homeostatically controlled cell division for mature tissues, in which the size of the nonmalignant population remains essentially constant over time. Growth of the nonmalignant cell population (normal and initiated cells) is restricted by allowing cells to divide only to fill the “openings” left by cells that die or differentiate, thus maintaining the constant size of the nonmalignant cell population. The fundamental technical insight from this model is that random walks, rather than birth‐and‐death processes, are the appropriate stochastic processes for describing the kinetics of the initiated cell population. Qualitative and analytic results are presented, drawn from the mathematical theories of random walks and diffusion processes, that describe the probability of spontaneous extinction and the size distribution of surviving initiated populations when the death/differentiation rates of normal and initiated cells are known. The constraint that the nonmalignant population size must remain approximately constant leads to much simpler analytic formulas and approximations, flowing directly from random walk theory, than in previous birth‐death models. As in other stochastic models, our analysis implies that individual susceptibility to carcinogens affecting the second (malignant) transformation rate should follow an age‐dependent frequency distribution in a population of initially identical individuals; hence, individual risks are heterogeneous even in the absence of individual differences in exposure patterns or pharmacokinetics. A theoretical formula is developed for the approximate frequency distribution of individual susceptibilities in a population exposed to a simple initiator/ delay/completor regimen.