An exact representation for the generating function for the Moolgavkar-Venzon-Knudson two-stage model of carcinogenesis with stochastic stem cell growth

The two-stage clonal expansion model of carcinogenesis provides a convenient biologically based framework for the quantitative description of carcinogenesis data. Under this stochastic model, a cancer cell arises following the occurrence of two critical mutations in a normal stem cell. Both normal cells and initiated cells that have sustained the first mutation undergo birth-and-death processes responsible for tissue growth. In this article, a new expression for the probability generating function (pgf) for the two-stage model of carcinogenesis is derived. This characterization is obtained by solving a partial differential equation (pde) satisfied by the pgf derived from the corresponding Kolmogorov forward equation. This pde can be reduced to the hypergeometric differential equation of Gauss, which leads to a closed-form expression for the pgf requiring only the evaluation of hypergeometric functions. This result facilitates computation of the exact hazard function for the two-stage model. Several approximations that are simpler to compute are also given. Numerical examples are provided to illustrate the accuracy of these approximations.