Coupling of discrete random walks and continuous modeling for three-dimensional tumor-induced angiogenesis

The growth of new vascular networks from pre-existing capillaries (angiogenesis) plays a pivotal role in tumor development. Mathematical modeling of tumor-induced angiogenesis may help understand the underlying biology of the process and provide new hypotheses for experimentation. Here, we couple an existing deterministic continuum theory with a discrete random walk, proposing a new model that accounts for chemotactic and haptotactic cellular migration. We propose an efficient numerical method to approximate the solution of the model. The accuracy, stability and effectiveness of our algorithms permitted us to perform large-scale three-dimensional simulations which, in contrast to two-dimensional calculations, show a topological complexity similar to that found in experiments. Finally, we use our model and simulations to investigate the role of haptotaxis and chemotaxis in the mobility of tip endothelial cells and its influence in the final vascular patterns.