Nonlinear analysis of a fractional reaction diffusion model for tumour invasion

Abstract

Mathematical models in general and Reaction Diffusion Models in particular have been rigorously studied and applied in different forms to explain situation in biomedical and allied sciences including the complex tumour microenvironment. They have been proven to be really significant in cancer research. The not so extensively known fractional reaction-diffusion model is presented in this research as a plausible model for the invasion of tumour and its consequences in the human organ in which it resides. In this model we make use of fractional derivatives as a replacement for the normal derivatives to express the diffusion of the tumour cells, normal cells and hydrogen ion concentration. The nonlinear analysis of this model predicts that a death situation always arise if some of the parameters are of a certain magnitude. In this situation, the tumour cells population increases (with an eventual bound after death) by eating up all the normal cells of their host organ. Effectively, the model predicts that the relative interaction between the tumor and normal cells population gives rise to a bifurcation parameter for which a Hopf bifurcation occurs between the period of attack and death. Also the death situation using this model comes faster than the predictions of other models for tumour invasion. This research may therefore give a useful indication of the possible timing of the drugs and dosages required for the treatment or control of tumour progression in human. Key words: Tumour, fractional reaction diffusion equation, trotter product formula, hopf bifurcation, vasculature, oxygenation, Hypoxia.