The mathematical modelling, simulation and prediction of tumour-induced angiogenesis

Angiogenesis, the formation of blood vessels from a pre-existing vasculature, is a process whereby capillary sprouts are formed in response to externally supplied chemical stimuli. The sprouts then grow and develop, driven by endothelial cell migration and proliferation, and organise themselves into a dendritic structure. Angiogenesis occurs during embryogenesis, wound healing, arthritis and during the growth of solid tumours. In this paper we present a novel mathematical model which describes the formation of the capillary sprout network in response to chemical stimuli (tumour angiogenesis factors, TAF) supplied by a solid tumour. The model also takes into account endothelial cell-extracellular matrix interactions via the inclusion of fibronectin in the model. The model consists of a system of nonlinear partial differential equations describing the response in space and time of endothelial cells to the TAF and the fibronectin (migration, proliferation, anastomosis, branching). Using the discretized system of partial differential equations, we use a deterministic cellular automata (DCA) model, which enables us to track individual endothelial cells and incorporate branching explicity into the model. Numerical simulations are presented which are in very good qualitative agreement with experimental observations. Certain experiments are suggested which could be used to test the hypotheses of the model and various extensions and developments of the model with particular applications to anti-angiogenesis strategies are discussed.