Angiogenesis, the growth of new blood vessel from existing ones, is a pivotal stage in cancer development,and is an important target for cancer therapy. We develop a hybrid mathematical model to understand the mechanisms behind tumor-induced angiogenesis. This model describes uptake of Tumor Angiogenic Factor(TAF)at extracellular level, uses partial differential equation to describe the evolution of endothelial cell density including TAF induced proliferation, chemotaxis to TAF, and haptotaxis to extracellular matrix. In addition we also consider the phenomenon of blood perfusion in the micro-vessels. The model produces sprout formation with realistic morphological and dynamical features, including the so-called brush border effect, the dendritic branching and fusing of the capillary sprouts forming a vessel network. The model also demonstrates the effects of individual mechanisms in tumor angiogenesis: Chemotaxis to TAF is the key driving mechanisms for the extension of sprout cell; endothelial proliferation is not absolutely necessary for sprout extension; haptotaxis to Extra Cellular Matrix(ECM) gradient provides additional guidance to sprout extension, suggesting potential targets for anti-angiogenic therapies.
