  Research into somatic mutations in cancer cell DNA and their role in tumour
growth and progression between successive stages is crucial for improving our
understanding of cancer evolution. Mathematical and computer modelling can
provide valuable insights into the scenarios of cancer growth, the roles of
somatic mutations, and the types and strengths of evolutionary forces they
introduce. Previous studies have developed mathematical models of cancer
evolution, incorporating driver and passenger somatic mutations. Driver
mutations were assumed to have a strong advantageous effect on the growth of
the cancer cell population, while passenger mutations were considered fully
neutral or mildly deleterious.
  However, according to several studies, passenger mutations may have a weakly
advantageous effect on tumour growth. In this paper, we develop models of
cancer evolution with somatic mutations that introduce a weakly advantageous
force to the evolution of cancer cells. The models used in this study can be
classified into two categories: deterministic and stochastic. Deterministic
models are based on systems of differential equations that balance the average
number of cells and mutations during evolution. To verify the results of our
deterministic modelling, we use a stochastic model based on the Gillespie
algorithm. We compare the predictions of our modelling with some observational
data on cancer evolution.
