It is well accepted that neo-vascular formation can be divided into three main stages (which may be overlapping): (1) changes within the existing vessel, (2) formation of a new channel, (3) maturation of the new vessel.

The purpose of this paper is to present a simple model, based on the theory of reinforced random walks, coupled with a Michaelis-Menten type mechanism which views the endothelial cell receptors as the catalyst for transforming angiogenic factor into proteolytic enzyme in order to model the first stage. In this model, a single layer of endothelial cells is separated by a vascular wall from an extracellular tissue matrix. 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. We refer to this as the onset ofvascular sprouting. The model admits extension to several angiogenic agents as well as to angiostatic agents (angiostatins). Indeed, we give a mathematical definition of what it means to be angiogenic or angiostatic.

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1998-1999 Mathematics in Biology