Mathematical Modelling of Tumour Angiogenesis and the Action of Angiostatin as a Protease Inhibitor*

Tumour angiogenesis is the process whereby a capillary network is formed from a pre-existing vasculature in response to tumour secreted growth factors (TAF). The capillary network is largely composed of migrating endothelial cells which organise themselves into dendritic structures. In this paper we model angiogenesis via the theory of reinforced random walks, whereby the chemotactic response of the endothelial cells to TAF and their haptotactic response to the matrix macromolecule fibronectin is accomplished through transition probability rate functions. Tumour secreted growth and inhibitory factors are modelled on the basis of Michaelis–Menten kinetics. Particular attention is focussed on the action of anti-angiogenic agents (i.e. angiostatins). That is as a mechanism whereby angiostatin acts as a protease inhibitor. Numerical simulations yield results, which are in good agreement with the experimental observations obtained by Folkman and his co-workers in their classical rabbit eye cornea experiments. The model offers a theoretical understanding of how some angiostatins work to inhibit tumour growth.