Cancer progression is somatic evolution. Stem cells in skin or intestinal epithelia keep dividing. After many years, some cells accumulate multiple mutations of key genes, finally giving rise to cells that proliferate without being checked by immune system, causing cancer. The risk of cancer depends on the reproductive rate of cells of intermediate mutants (somatic selection), mutation rate, and the population size.

[1] Chromosomal instability (CIN) is a defining characteristic of most human cancers. Mutation of CIN genes increases the probability that whole chromosomes or large fractions of chromosomes are gained or lost during cell division. The consequence of CIN is an imbalance in the number of chromosomes per cell (aneuploidy) and an enhanced rate of loss of heterozygosity (LOH). We develop a mathematical framework for studying the effect of CIN on the somatic evolution of cancer. Specifically, we calculate the conditions for CIN to initiate the process of colorectal tumorigenesis prior to the inactivation of tumor suppressor genes.

[2] Many epithelial tissues are separated into many compartments. Also there is a clear separation of stem cells that keep dividing and differentiated cells that will be discarded after a finite number of cell division. We examine the role of tissue architecture in reducing the risk of cancer initiation. We also discuss their effect to the age-specificity of cancer incidence.

To formulate these problems, we derived new formulas for the fixation of the second or third mutation without fixation of the intermediate mutants (stochastic tunnels).