Mostly Biomathematics Lunchtime Seminar

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Stochastic modeling of blood vessel growth

Speaker: Luis L. Bonilla, Universidad Carlos III de Madrid, Department of Materials Science and Engineering and G. Millan Institute for Fluid Dynamics, Nanoscience and Industrial Mathematics

Location: Warren Weaver Hall 1314

Date: Tuesday, November 6, 2018, 12:30 p.m.

Synopsis:

Angiogenesis is the multiscale process of blood vessel formation and growth that brings oxygen and nutrients to hypoxic cells in tissues. While angiogenesis is fundamental in healthy organ growth and repair, its imbalance is behind many diseases, including cancer or forms of age-related macular degeneration. We consider simple two-dimensional stochastic models of angiogenesis representing cells at the tips of new blood vessels as active particles whose trajectories are the blood vessels themselves. Vessel tips are subject to chemotaxis and haptotaxis forces in Langevin equations and branch stochastically producing new tips. When one active tip meets a preexisting vessel (trajectory of another tip) joins it and ceases to be active, a process called anastomosis. The same occurs when it arrives at the hypoxic region. Thus anastomosis is a killing point process that depends on the past history of the given realization. We have derived a deterministic equation for the density of active vessel tips containing source terms with memory characterizing anastomosis and branching. For simple geometries, the density of active tips evolves to a soliton-like wave whose shape and velocity follow simple differential equations. Numerical simulations of the stochastic process confirm our findings, which could be helpful for angiogenesis control.