A mathematical model of the first steps of tumour-related angiogenesis: capillary sprout formation and secondary branching.

The growth of a solid tumour is dependent on an adequate supply of nutrients. A tumour can establish a blood supply by inducing neighbouring blood vessels to sprout and grow towards it, a process known as angiogenesis. The tumour cells may secrete a number of diffusible chemicals which stimulate endothelial cell to migrate, to rearrange themselves into capillary tubes or sprouts, and to proliferate. In this paper we focus firstly upon the early stage of angiogenesis wherein the endothelial cells group together in the parent vessel to form the initial capillary-sprout buds. A mathematical model for the formation of the capillary buds is presented which focuses on the potential role that haptotaxis may play. In Section 2 we turn attention to the endothelial cells within the growing and developing capillary sprouts as they migrate towards the tumour cells. Once again the potential role of haptotaxis is focused upon.