Calculation of the cumulative distribution function of the time to a small observable tumor

Multistage mathematical models of carcinogenesis (when applied to tumor incidence data) have historically assumed that the growth kinetics of cells in the malignant state are disregarded and the formation of a single malignant cell is equated with the emergence of a detectable tumor. The justification of this simplification is, from a mathematical point of view, to make the estimation of tumor incidence rates tractable. However, analytical forms are not mandatory in the estimation of tumor incidence rates. Portier et al. (1996b, Math. Biosci.135, 129–146) have demonstrated the utility of the Kolmogorov backward equations in numerically calculating tumor incidence. By extending their results, the cumulative distribution function of the time to a small observable tumor may be numerically obtained.