Stochastic Models of Blood Vessel Growth

Angiogenesis is a complex multiscale process by which diffusing vessel endothelial growth factors induce sprouting of blood vessels that carry oxygen and nutrients to hypoxic tissue. There is strong coupling between the kinetic parameters of the relevant branching - growth - anastomosis stochastic processes of the capillary network, at the microscale, and the family of interacting underlying biochemical fields, at the macroscale. A hybrid mesoscale tip cell model involves stochastic branching, fusion (anastomosis) and extension of active vessel tip cells with reaction-diffusion growth factor fields. Anastomosis prevents indefinite proliferation of active vessel tips, precludes a self-averaging stochastic process and ensures that a deterministic description of the density of active tips holds only for ensemble averages over replicas of the stochastic process. Evolution of active tips from a primary vessel to a hypoxic region adopts the form of an advancing soliton that can be characterized by ordinary differential equations for its position, velocity and a size parameter. A short review of other angiogenesis models and possible implications of our work is also given.