We consider four models of cancer growth:

1. Mathematical models of stem cell renewal and dedifferentiation. We study the effects of dedifferentiation on the time to cancer onset and found that the effect of dedifferentiation depends critically on how stem cell numbers are controlled by the body. We consider both space-free and spatial versions of this process to look at effect that tissue architecture can play in this process.  Our results suggest that dedifferentiation may be a very important factor in cancer and that more study of dedifferentiation and stem cell control is necessary to understand and prevent cancer onset.

2. New findings on cancer via stochastic modeling and statistical analysis. Very recent developments in our understanding of this dynamical process due to high throughput sequencing methodologies have allowed us to formulate a more detailed mathematical model of cancer initiation and evolution. In this talk the model will be presented with some of its mathematical results and validated predictions.

3. A multitype infinite-alleles Galton Watson process and applications to cancer evolution. The infinite-allele Galton Watson process proposed by Griffiths and Pakes is extended to a multitype process allowing for different types and labels. Limit behavior of the process has been determined, particularly the asymptotic growth of the number of labels for each type and the limit of the frequency spectrum for each type. The process has applications to cancer evolution, with different types representing different sets of driver mutations, or mutations that affect the rate of growth of cells. Our asymptotic results show the number of labels grow exponentially along with the total number of individuals, so the number of labels can be considered as a surrogate for age of a particular subclone of cells. This can be used to help determine the order of events in the clonal evolution of a population of cancer cells.

4. Cancer evolution model based on the Moran model with selection and co-localization. We model the evolution of the cancer using discrete and continuous versions of the Moran model with selection (Durret, 2008). In this model, the population of granulocyte precursors is constant and consists of N biological cells, including mutant cells, number of which is variable in time (starting from i mutants at the beginning). The mutant has selective advantage expressed by the relative fitness r = 1+s > 1, equal to the ratio of average progeny count of the mutant to that of the wildtype. We also incorporate the colocalization factor to the model - we assume that it is more probable, by a factor of 1+α >1, that mutant (wildtype) cell will be replaced by a cell of the same type. It is recognized that about 70% of Severe congenital neutropenia patients who developed secondary myelodysplastic syndrome will express a truncation mutant GCSFR. Using the model we estimate the probabilities and expected times to the fixation of the mutant cells as well as the number of initial mutant cells leading to fixation. We also study the dynamics of the model and the relationship between fitness and colocalization factors.