Denes J, Krewski D
Health Canada, Ottawa, Ontario, Canada.
Math Biosci. 1996 Jan 15;131(2):185-204. doi: 10.1016/0025-5564(95)00046-1.
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.
致癌作用的两阶段克隆扩增模型为定量描述致癌作用数据提供了一个基于生物学的便捷框架。在这个随机模型中，癌细胞是在正常干细胞发生两个关键突变之后产生的。正常细胞和经历了第一次突变的起始细胞都会经历负责组织生长的生死过程。在本文中，推导了致癌作用两阶段模型的概率生成函数（pgf）的新表达式。这种表征是通过求解由相应的柯尔莫哥洛夫前向方程导出的pgf所满足的偏微分方程（pde）得到的。这个pde可以简化为高斯超几何微分方程，从而得到一个仅需计算超几何函数的pgf的封闭形式表达式。这一结果有助于计算两阶段模型的精确风险函数。还给出了几个计算更简单的近似值。提供了数值示例来说明这些近似值的准确性。
