Two dimensional models of tumor angiogenesis
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Angiogenesis, the formation of new capillaries from pre-existing vessels, is essential for tumor progression. It is critical for the growth of primary cancers. In this thesis we present a new approach to angiogenesis, based on the theory of reinforced random walks, coupled with a Michaelis-Menten type mechanism. This views the endothelial cell receptors as the catalyst for transforming angiogenic factor into proteolytic enzyme in order to model the first stage. In our model we use a curvature-induced proliferation term for the endothelial cell equation. Our numerical results indicate that the proliferation of endothelial cells is high at the tip. Also, we observe that the tip movement speeds up as it gets close to the tumor;A coupled system of ordinary and partial differential equations is derived which, in the presence of an angiogenic agent, predicts the aggregation of the endothelial cells and the collapse of the vascular lamina, opening a passage into the extracellular matrix. (ECM). We have dynamical equations not only in a two-dimensional region, the ECM, but also in a one-dimensional region, the capillary. We also consider the effect of the angiostatin on the endothelial cell proliferation and fibronectin;Our computations are compared with the results of Judah Folkman's classical rabbit eye experiments in which he demonstrated that tumors can produce angiogenic growth factors. Using only classical enzyme kinetics and reinforced random walk cell transport equations, we are able to "predict" how long it should take for a new capillary to grow from the limbus of the rabbit eye to an implanted malignancy. The "predictions" agree very well with the experiments.