Branching Process Models of Cancer

In this chapter, we use continuous time multi-type branching processes to model various aspects of cancer growth, progression, and the development of resistance to therapy.. We develop all the theory we need, beginning with basic results for the single type Markovian branching process in continuous time. Most of the results should be accessible to someone who has had a first course in stochastic process, although along the way martingales and stable laws will sometimes appear in the discussion. Throughout these notes, we consider an exponentially growing cell population in which type i cells are those that have accumulated i ≥ 0 mutations compared to the type 0 cells, and we let Zi(t) be the number of type i cells at time t. Type i cells give birth at rate ai and die at rate bi, where the growth rate λi = ai − bi> 0. Thinking of cancer, we will usually restrict our attention to the case in which i → λi is increasing, but we will see an application in Section 15 with λ1 < λ0. To take care of mutations we suppose that individuals of type i in addition give birth at rate ui+1 to individuals of type i + 1. This is slightly different than the approach of having mutations with probability ui+1 at birth, which translates into a mutation rate of aiui+1 and thi