Travelling waves arising in mathematical models of tumour angiogenesis and tumour invasion

In order to accomplish the transition from avascular to vascular growth, solid tumours secrete a number of diffusible substances, generically known as tumour angiogenesis factors (TAF), into the surrounding tissue. Neighbouring endothelial cells respond to this chemotactic stimulus in a well-ordered sequence of events leading eventually to the formation of a network of capillary sprouts which migrates towards and eventually links up with and penetrates the tumour. The subsequent vascular growth of the tumour is rapid and leads to invasion of the surrounding tissue with all its insidious consequences. In this paper we present some relatively simple mathematical models describing the processes of angiogenesis and invasion. Numerical simulations are presented which capture the qualitative features of both processes. Subsequent analysis in terms of travelling wave solutions yield biologically relevant information regarding the contributions of endothelial cell migration and proliferation and also an estimate for the rate of invasion.