Propagation of weakly advantageous mutations in cancer cell population

Studies on somatic mutations in cancer cells DNA, their roles in tumour growth and progression between successive stages are of high importance for improving understanding of cancer evolution. Important insights into scenarios of cancer growth, roles of somatic mutations and types/strengths of evolutionary forces they introduce are gained by using mathematical and computer modelling. Previous studies developed mathematical models of cancer cell evolution with driver and passenger somatic mutations. Driver mutations were assumed to have a strong advantageous effect on cancer cell population growth, while passenger mutations were considered as fully neutral or mildly deleterious. In this paper, following several experimental results, we develop models of cancer evolution with somatic mutations introducing weakly advantageous force to the evolution of cancer cells. Our models belong to two classes, deterministic and stochastic. Deterministic models are systems of differential master equations of balances of average numbers of cells and mutations in evolution. We, additionally introduce a modification in equations for balances aimed at accounting for effects on the finite size of the cancer cell population. The novelty of our deterministic modelling is deriving parameters of travelling waves of advantageous mutations and quantification of the pattern of the population size growth. A stochastic model based on the Gillespie algorithm is used to verify the results of deterministic modelling. We confront predictions of modelling with some observational data on cancer evolution. We propose the scenario of evolution driven by a large number of weakly advantageous mutations in transition from the early stage to the invasive form of breast cancer. We also fit our models to observed patterns of rates of growth of tumours.