2-D mathematical models of Tumor-induced angiogenesis.

Angiogenesis, the formation of blood vessels from a pre-existing vasculature, is a process in which capillary sprouts are formed in response to externally supplied chemical stimuli. Tumor-induced angiogenesis plays a significant role in tumor's growth, metastasis, in which endothelia-cell is the most active member engaging in various stages, including the secretion and diffusion of various tumor-induced angiogenesis factors (TAF); the nascent endothelial cell migrates towards a particular direction and position under the interaction of a series of internal and external factors; eventually the new formed capillaries reach and penetrate the tumor. At present there is no technique of medical image to accurately investigate the micro-vessel net of tumor, it makes some hard in clinical treatment of tumor. In this paper we present a 2-D mathematical model, which takes into account essential endothelial cell-extra cellular matrix interactions, and the response to chemical stimuli (TAF) mentioned above, to describes the formation of the capillary sprout network in response to physiological conditions around tumor especially investigate the process inside tumor in terms of the complex features of tumor. This theoretical capillary networks generated by computer simulations provide a premise for the further research.