A stochastic model of tumor growth

Abstract

A stochastic model for tumor growth is derived as a diffusion approximation of a continuous-time, density dependent branching process with a Gompertz growth law as the deterministic part. For the diffusion process, the conditional probabilities of extinction, reaching a size c, and doubling are computed along with the expected time of these events. The results are given in terms of integrals, which are evaluated by numerical methods that account for the logarithmic singularity introduced by the Gompertz growth law. When the variance of the branching process is small compared to the deterministic term, simplified asymptotic expressions are given using methods that are modified for the Gompertzian logarithm. The results are used to find the probability of implant take in limiting-dilution assay experiments and the probability of a tumor becoming detectable in carcinogenesis.