The waiting time for a second mutation: an alternative to the Moran model

The idea of successive mutations to trigger the appearance of a cancerous cell goes back to at least Muller (1951). The first mathematical model proposed for this phenomenon goes back to Armitage and Doll (1954). There has been a great deal of discussion on the number of successive mutations necessary to get a cancerous cell. It seems that this number depends on the organ, see Knudson (1971) and Moolgavkar and Luebeck (1992) . However, some authors have argued that two mutations models are flexible enough to model most cancers, see Armitage and Doll (1957) and Moolgavkar and Knudson (1981). This is the point of view we adopt. We now describe our model. We are interested in the time it takes for a given organ to have a first cancerous cell. We assume that all cells are in one of three stages: healthy, pre-cancerous (i.e. type 1) and cancerous (i.e. type 2). We start the process with all cells healthy. As the cells divide pre-cancerous cells may appear due to a type 1 mutation on a healthy cell. A type 2 mutation on a pre-cancerous cell makes the cell cancerous. The number of type 1 mutations is modeled by a Poisson process with rate μ1N . We think of μ1 as a mutation rate and N as a division rate. Every time a type 1 mutation appears there is a chance that a type 2 mutation appears. We model the appearance of a type 2 mutation by using exponential random variables with rate μ2. More precisely, let N1(t) be the number of type 1 mutations that have occurred up to time t. Let T1 < T2 < . . . be the arrival times of this Poisson process. Given that N1(t) = k and that T1 = t1, T2 = t2, . . . , Tk = tk let S1, S2, . . . , Sk be random variables with density