Multistage Models of Carcinogenesis: An Approximation for the Size and Number Distribution of Late-Stage Clones

Multistage models have become the basic paradigm for modeling carcinogenesis. One model, the two-stage model of carcinogenesis, is now routinely used in the analysis of cancer risks from exposure to environmental chemicals. In its most general form, this model has two states, an initiated state and a neoplastic state, which allow for growth of cells via a simple linear birth-death process. In all analyses done with this model, researchers have assumed that tumor incidence is equivalent to the formation of a single neoplastic cell and the growth kinetics in the neoplastic state have been ignored. Some researchers have discussed the impact of this assumption on their analyses, but no formal methods were available for a more rigorous application of the birth-death process. In this paper, an approximation is introduced which allows for the application of growth kinetics in the neoplastic state. The adequacy of the approximation against simulated data is evaluated and methods are developed for implementing the approximation using data on the number and size of neoplastic clones.