Branching Process Models of Cancer

In this chapter, we will use multitype branching processes with mutation to model cancer. With cancer progression, resistance to therapy, and metastastis in mind, we will investigate τk, the time of the first type k mutation, and σk, the time of the first type k mutation that founds a family line that does not die out, as well as the growth of the number of type k cells. The last three sections apply these results to metastasis, ovarian cancer, and tumor heterogeneity. Even though martingales and stable laws are mentioned, these notes should be accessible to a student who is familiar with Poisson processes and continuous time Markov chains. 1 Multistage theory of cancer Throughout these notes, we model cancer as an exponentially growing cell population in which type i cells are those that have accumulated i ≥ 0 mutations compared to the type 0 cells. To motivate the study of these models we begin with a brief and incomplete history lesson and a description of some situations in which our model may be applied. The idea that carcinogenesis is a multistage process goes back to the 1950s. Fisher and Holloman [52] pointed out that when the logarithm of cancer mortality from stomach cancer in women was plotted versus log of age, the result was line with slope 6. Nordling [65] and Armitage and Doll [43] suggested that the observed relationship would be explained if a cancer cell was the end result of seven mutations. There was no model. The conclusion was based on the fact that if Xi are independent and exponential with rate ui then P (X1 + · · ·+Xk ≤ t) ∼ k∏ i=1 ui tk−