In this presentation we introduce several mathematical models for the angiogenesis of endothelial cells. We first introduce a model at
the level of the partial differential equation, describing the spatiotemporal evolution of the cell population, the extracellular matrix macromolecule, the proteases, the tumor angiogenic factors and the possible presence of inhibitors. We mainly focus, however, on a complementary, more physiologically realistic
approach in which the cells are treated as individual particles. We examine the model numerically in 1-D and 2-D settings, discussing
its comparison with experimental results. In our second method, we begin with a particle model for the motion of the endothelial cells. The
motion of the tumor angiogenic factors are described by a partial differential equation. This is analyzed numerically in a 2-D
setting. We will also discuss briefly the beginnings of some laboratory experiments that we have commenced.
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