Cellular Growth Control and Travelling Waves of Cancer

Source abstract

One of the major differences between cancer cells and normal cells is an increase in cell proliferation, caused by the escape of cancer cells from the normal biochemical regulation of mitosis. This increased proliferation results from genetic mutations causing the expression of oncogenes, or the loss of anti-oncogenes. A reaction diffusion model is developed for the initial growth of a tumour following an oncogenic mutation of a single cell. This model incorporates the possibility of an immune response to the cancer cells. Numerical solutions of the model rapidly evolve into an advancing wave of tumour cells and a receding wave of normal cells. The author analyses the ordinary differential equation system governing these travelling wave solutions and obtains a lower bound on the wave speed. Under biologically relevant approximations, a necessary and sufficient condition is derived for the existence of a travelling wave solution, and it is shown how the qualitative form of the wave fronts of normal and mutant cells depends on the parameter values. Finally, an analytic approximation for the wave fronts in the case of small mutations is derived. Biologically, these results suggest that, for certain types of mutations, which are quantified, growing tumours can initially contain a significant proportion of normal cells. Moreover, this model predicts that there is a critical level of immune response,which again is quantified, above which the immune system prevents the initial growth of the tumour.