Oncogenes, anti-oncogenes and the immune response to cancer: a mathematical model

We develop a mathematical model for the initial growth of a tumour after a mutation in which either an oncogene is expressed or an anti-oncogene (i.e. tumour suppressor gene) is lost. Our model incorporates mitotic control by several biochemicals, with quite different regulatory characteristics, and we consider mutations affecting the cellular response to these control mechanisms. Our mathematical representation of these mutations reflects the current understanding of the roles of oncogenes and anti-oncogenes in controlling cell proliferation. Numerical solutions of our model, for biologically relevant parameter values, show that the different types of mutations have quite different effects. Mutations affecting the cell response to chemical regulators, or resulting in autonomy from such regulators, cause an advancing wave of tumour cells and a receding wave of normal cells. By contrast, mutations affecting the production of a mitotic regulator cause a slow localized increase in the numbers of both normal and mutant cells. We extend our model to investigate the possible effects of an immune response to cancer by including a first order removal of mutant cells. When this removal rate exceeds a critical value, the immune system can suppress tumour growth; we derive an expression for this critical value as a function of the parameters characterizing the mutation. Our results suggest that the effectiveness of the immune response after an oncogenic mutation depends crucially on the way in which the mutation affects the biochemical control of cell division.