MATHEMATICAL-MODELS FOR TUMOR ANGIOGENESIS - NUMERICAL SIMULATIONS AND NONLINEAR-WAVE SOLUTIONS

To ensure its sustained growth, a tumour may secrete chemical compound s which cause neighbouring capillaries to form sprouts which then migr ate towards it, furnishing the tumour with an increased supply of nutr ients. In this paper a mathematical model is presented which describes the migration of capillary sprouts in response to a chemoattractant h eld set up by a tumour-released angiogenic factor, sometimes termed a tumour angiogenesis factor (TAF). The resulting model admits travellin g wave solutions which correspond either to successful neovascularizat ion of the tumour or failure of the tumour to secure a vascular networ k, and which exhibit many of the characteristic features of angiogenes is. For example, the increasing speed of the vascular front, and the e volution of an increasingly developed vascular network behind the lead ing capillary tip front (the brush-border effect) are both discernible from the numerical simulations. Through the development and analysis of a simplified caricature model, valuable insight is gained into how the balance between chemotaxis, tip proliferation and tip death affect s the tumour's ability to induce a vascular response from neighbouring blood vessels. In particular, it is possible to define the success of angiogenesis in terms of known parameters, thereby providing a potent ial framework for assessing the viability of tumour neovascularization in terms of measurable quantities.