Waiting Times For Cancerous Mutations In Two Mathematical Models

Abstract

This dissertation explores the distribution of [tau]k , the first time for some cell to accumulate k mutations, under two different multistage models for cancer growth. The first model considers a freely mixing, exponentially growing cell population, while the second model considers a spatially fixed cell population of constant size. The first model is inspired by previous work of Iwasa, Nowak, and Michor (2006), and Haeno, Iwasa, and Michor (2007). We consider an exponentially growing population of cancerous cells that will evolve resistance to treatment after one mutation, or display a disease phenotype after two or more mutations. We use multi-type branching processes to prove results about [tau]k and about the growth of the number of type k cells, and apply our results to re-derive proofs in Iwasa, Nowak, and Michor (2006) and Haeno, Iwasa, and Michor (2007) concerning the likelihood of a type k mutant by the time the tumor reaches size M. The second model is inspired by Komarova (2006). We consider a multi-type Moran model in which cells inhabit the d-dimensional integer lattice. Starting with all wild-type cells, we prove results about the distribution of [tau]2 in dimensions d = 1, 2, and 3, and use results from neutral and biased voter models to consider the effects neutral and advantageous mutations, respectively.