Abstract

Angiogenesis and mathematical modeling

Angiogenesis (The formation of new blood vessels) has been studied because it is related to cancer cure, wound healing etc. When tumors need extra channel for food and deposit of waste, they initiate angiogenesis by sending out signal to surounding capillary. Then, Endothelial cells (EC's) near capillary move to tumor by differentiation and proliferation and finally penetrate the tumor so that cancerous cell now can go to any place in body (metastasis). Study of cell movement with chemotaxis can explain some features of angiognesis. People has been observing the behavior of EC's for angiogenesis in vitro which resemble the angiogensis in vivo. There are several approach to explain angiogenesis:

continuum model which consists of system of PDEs describing chemotaxis, haptotaxis, diffusion, etc.
discrete model which describe the behavior of one cell movement
mechano-chemical continuum model which add the mechanical effects to continuum model.

Some of them were successful in explaining and predicting the some aspects of angiogenesis. Even though it's quite complicated phenomenon with thousands of unknown parameters, one can do something meaningful by constructing the rather simple models.

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