Pii: S0025-5564(00)00034-1

In this paper, a simple mathematical model developed in H.A. Levine, B.D. Sleeman, M. NilsenHamilton [J. Math. Biol., in press] to describe the initiation of capillary formation in tumor angiogenesis is extended to include the roles of pericytes and macrophages in regulating angiogenesis. The model also allows for the presence of anti-angiogenic (angiostatic) factors. The model is based on the observation that angiostatin can prevent the degradation of ®bronectin in the basal lamina by inhibiting the catalytic action of active proteolytic enzyme. That is, it is proposed that the inhibitor `deactivates' the protease but that it does not reduce the over all concentration of the protease. It consequently explores the possibility of preventing neovascular capillaries from migrating through the extra-cellular matrix toward the tumor by inhibiting protease action. The model is based on the theory of reinforced random walks coupled with Michaelis±Menten mechanisms which view endothelial cell receptors as the catalysts for transforming both tumor and macrophage derived angiogenic factors into proteolytic enzyme which in turn degrade the basal lamina. A simple catalytic reaction is proposed for the degradation of the basal lamina by the active proteases. A mechanism, in which the angiostatin acts as a protease inhibitor is discussed which has been substantiated experimentally. A second mechanism for the production of protease inhibitor from angiostatin by endothelial cells is proposed to be of Michaelis±Menten type. Mathematically, this mechanism includes the former as a subcase. Ó 2000 Elsevier Science Inc. All rights reserved.