UC San Diego

Cancer is currently viewed as an evolutionary process. In an organ there is a population of cells that give birth, die and mutate according to population dynamics that are determined by the types of cells under consideration. If certain cell mutations are acquired then the cells can become cancerous. In this manuscript we consider two evolutionary models that may each be viewed as a model of cancer. The first model we consider is a Moran-type model. Individuals each have an integer valued fitness. Individuals with a higher fitness value are more likely to give birth and individuals with a lower fitness value are more likely to die. We fix the mutation rate and consider the limiting rate of adaptation as the population size tends to infinity. Similar models have been used to model cancers in liquids such as leukemia. The second model we consider is a hierarchical model which differentiates between stem cells and progenitor cells. It has been proposed as a model of colorectal cancer. We find the limiting distribution for the time it takes for a cell to acquire two mutations as the population size tends to infinity and the mutation rates tend to 0. There are several different results depending how the mutation rates tend to 0 as a function of the population size. The two mutations represent the loss of two tumor suppressor genes. We also determine whether or not the mutations occur on stem cells or progenitor cells