Mathematical modeling of signal transduction and cell motility in tumor angiogenesis

In recent years, tumor-induced angiogenesis has become an important field of research because it represents a crucial step in the development of malignant tumors. The existing mathematical models have been able to reproduce some characteristics of angiogenesis, such as capillary loop formation and the dynamics of blood vessels, but much remains to be done until a full understanding of angiogenesis is achieved. There are a number of basic questions about cell movement that are unresolved, including the microscopic issues of how a cell decides when and how long to move. We have developed a discrete cell model for capillary tube formation of endothelial cells (ECs) that incorporates a realistic model for signal transduction, vascular endothelial growth factor (VEGF) production, and VEGF release. This allows us to explore the effect on macroscopic networks of changes in the microscopic rules by which cells determine their direction and duration of movement. In our hybrid discrete/continuum model the cells are treated as individual units and the extracellular VEGF evolves according to a continuum reaction-diffusion equation. A detailed description of signal transduction is possible in such a model and movement rules based on intracellular dynamics can be explored. The signal transduction part incorporates known biology more completely than any other model and involves 18 variables and 14 biochemical reactions. Using the law of mass action, these lead to a system of ten differential equations and auxiliary algebraic equations for the time evolution of the intracellular species. The evolution of extracellular VEGF is incorporated and described by a reaction-diffusion equation. The algorithm used to solve these equations follows an earlier algorithm developed by Dallon and Othmer in studying the aggregation of Dictyostelium discoideum, The hybrid mathematical model reproduces a number of experimental observations and gives further insight into the capillary network formation process. We investigate the effect of different model parameters on the final aggregation patterns. Our results provide strong evidence that the endothelial cell density, their activity, and the range of activity of intracellular and extracellular VEGF regulate the vascular network formation and its size.