A one-dimensional mathematical model was built to describe the initial process of angiogenesis, where tumors grow blood vessels in order to sustain further development. The experiment used two partial differential equations: one modeling the concentration of chemicals that promote angiogenesis and one modeling the behavior of endothelial cells during the forming of the blood vessels. The equations for both concentrations of chemicals and endothelial cells involve a diffusion term. While having a diffusion term, the equation for the endothelial cells also includes a chemotaxis term, which contributes to the movement of the cells toward higher concentrations of chemicals. The computational approach uses a combination of backward and forward finite difference methods for time-stepping and a centered difference method in mesh spacing. Our results show several intriguing properties when chemicals and endothelial cells react during diffusion. Through experimentation, we found a set of parameters that give rise to angiogenesis. By solely increasing the chemotaxis term in this parameter set, our results show no angiogenesis. Another property is that the population of endothelial cells will continue to decay until the concentration of chemicals approaches equilibrium. Although many of the coefficients are arbitrary, we have successfully created a computational approach to solving the partial differential equations in this model. Further research for the coefficients of this model will aid cancer research since tumors are usually benign prior to angiogenesis.

Mason Chow, ’16
Washington, DC
Lainey Drevlow, 16

Mathematics & Statistics

Sponsor: Tyler Skorczewski