A hybrid stochastic model of retinal angiogenesis

Understanding the biological principles that govern blood vessel growth in the retina has important clinical implications, for the prevention of possible retinopathies, which may eventually lead to blindness. The availability of a realistic mathematical model of the relevant phenomenon may support the medical community in both issues, diagnosis being related to inverse problems, and therapy to optimal control strategies. The mathematical modeling of retinal angiogenesis leads to an highly complex problem, because of dimension, nonlinearity, and intrinsic randomness. In this paper, we propose a reduced model which leads to numerical simulations that somehow reproduce normal vascularization and predict possible pathologies. We call our model hybrid because it includes the coupling of a fully stochastic model for the construction of a vessel network in the retina, with continuum underlying fields describing relevant factors, such as growth factors and oxygen. We perform numerical simulations of a stochastic particle system coupled with partial differential equations (PDEs)' so to obtain images of vessel structure resembling real retina vasculatures. We then derive a possible mean field approximation of the stochastic vessel network, so to obtain a fully deterministic PDE system for the evolution of the underlying fields. Actually, in order to reproduce the geometric structure of the retina vessel network, we have to keep a stochastic model (though simplified) for its construction. Future investigations may concern the use of such a mean field approximation in the numerical simulations of the retina vasculature. Inverse and optimal control problems being the final goal of our research plan.