A Mathematical Model for Capillary Formation and Tumor Vascularization Based on the Principles of Reinforced Random Walks

The purpose of this talk is to present a mathematical model for the tumor vascularization theory of tumor growth proposed by Judah Folkman in the early 70's and subsequently established experimentally by him and his coworkers. In the simplest version of this model, an avascular tumor secretes a tumor growth factor, (TGF) which is transported across an extracellular matrix (ECM) to a neighboring vasculature where it stimulates endothelial cells to produce a protease that acts as a catalyst to degrade the fibronectin of the capillary wall and the ECM. The endothelial cells then move up the TGF gradient back to the tumor, proliferating and forming a new capillary network.

In the model presented here, we include two mechanisms for the action of angiostatin. In the first mechanism, substantiated experimentally, the angiostatin acts as a protease inhibitor. 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 sub case. A third mechanism, receptor inhibition, can be incorporated into this model although we do not discuss this mechanism in this talk.

Our model is different from other attempts to model the process of tumor angiogenesis in that it focuses (1) on the biochemistry of the process at the level of the cell; (2) the movement of the cells is based on the theory of reinforced random walks; (3) standard transport equations for the diffusion of the various molecular species in porous media.

One consequence of our numerical simulations is that we obtain very good computational agreement with the time of the onset of vascularization and the rate of capillary tip growth observed in rabbit cornea experiments. Furthermore, our numerical experiments agree with the observation that the tip of a growing capillary accelerates as it approaches the tumor.