This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely

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... t reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overiaps. ABSTRACT 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. .A,lso, 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 pres ence of an angiogenic agent, predicts the aggregation of the endothelial cells and the collapse of the vascular lamina, opening a passage into the e.xtracellular matrix (ECM). We have dynam ical 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. 1 1 foBp, q = li, p = ll-Scaling for the equation Af : A = -, /a = ^, -, (j = 1,2). « fjj J Ij 3