Hybrid modeling of tumor-induced angiogenesis

When modeling of tumor-driven angiogenesis, a major source of analytical and computational complexity is the strong coupling between the kinetic parameters of the relevant stochastic branching-and-growth of the capillary network, and the family of interacting underlying fields. To reduce this complexity, we take advantage of the system intrinsic multiscale structure: we describe the stochastic dynamics of the cells at the vessel tip at their natural mesoscale, whereas we describe the deterministic dynamics of the underlying fields at a larger macroscale. Here, we set up a conceptual stochastic model including branching, elongation, and anastomosis of vessels and derive a mean field approximation for their densities. This leads to a deterministic integro-partial differential system that describes the formation of the stochastic vessel network. We discuss the proper capillary injecting boundary conditions and include the results of relevant numerical simulations.