The role of stochasticity in a model of retinal angiogenesis

Angiogenesis is an important natural process occurring in the body, both in health and in disease. It is an example of a complex system: the endothelial cells are the building blocks for the vessels and they interact by regulation signals, forming a network of capillaries in order to reach every part of the body. Here, we investigate the case of embryonic retinal vascularization. Such a system is extremely complex, due to its intrinsic multiscale structure; a major source of complexity in the mathematical modelling derives from the strong coupling of the kinetic parameters of the relevant stochastic branching-and-growth of the capillary network at a microscale, with a family of underlying fields interacting at a macroscale. For such a complex system, a mathematical and even a computational analysis seems quite unaffordable so that other methods of model reduction are required. Thus, our main goal is not in providing additional models for the angiogenic phenomenon, but in addressing the mathematical problem of reducing the complexity of such systems, by taking advantage of its intrinsic multiscale structure. A mesoscale may be introduced, at which we may approximate, by suitable laws of large numbers, the contribution due to the vascularization process by local mean values, in the equations for the underlying fields, thus providing a family of deterministic underlying fields. We may then use these approximate mean fields to drive the evolution of the relevant stochastic processes of the cells at the microscale. In this way it is possible to generate a nontrivial and realistic geometric pattern of the capillary network. These kinds of models are known as hybrid models. For the time being, we have produced only a heuristic derivation of the mean fields. Numerical simulations confirm that substituting mean geometric densities of tips, or of full vessels, to the corresponding stochastic quantities leads to an acceptable model behaviour only when a law of large numbers can be applied.