A multiscale approach leading to hybrid mathematical models for angiogenesis: the role of randomness

In biology and medicine we may observe a wide spectrum of formation of patterns, usually due to self-organization phenomena. This may happen at any scale; from the cellular scale of embryonic tissue formation, wound healing or tumor growth, and angiogenesis to the much larger scale of animal grouping. Patterns are usually explained in terms of a collective behavior driven by \u201cforces,\u201d either external and/or internal, acting upon individuals (cells or organisms). In most of these organization phenomena, randomness plays a major role; here we wish to address the issue of the relevance of randomness as a key feature for producing nontrivial geometric patterns in biological structures. As working examples we offer a review of two important case studies involving angiogenesis, i.e., tumor-driven angiogenesis and retina angiogenesis. In both cases the reactants responsible for pattern formation are the cells organizing as a capillary network of vessels, and a family of underlying fields driving the organization, such as nutrients, growth factors, and alike