“The Role of Geometric Randomness in the Mathematical Modeling of Angiogenesis”
In biology and medicine we may observe a wide spectrum of formation of patterns, usually due
to self-organization phenomena. Patterns are usually explained in terms of a collective behavior
driven by “forces”, 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 a couple of important case studies
involving angiogenesis, i.e. tumor-driven angiogenesis [1], and retina angiogenesis [2]. 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.
The strong coupling of the kinetic parameters of the relevant stochastic branching-and-growth of the
capillary network, with the family of interacting underlying fields is a major source of complexity
from a mathematical and computational point of view.
Thus our main goal is to address the mathematical problem of reduction of the complexity of such
systems by taking advantage of its intrinsic multiscale structure; the (stochastic) dynamics of cells
will be described at their natural scale (the microscale), while the (deterministic) dynamics of the
underlying fields will be described at a larger scale (the macroscale).
References
[1] Capasso, V., Morale, D.: Stochastic Modelling of Tumour-induced Angiogenesis. J. Math. Biol., 58, 219{33 (2009)
[2] Capasso, V., Morale D., Facchetti, G.: The Role of Stochasticity for a Model of Retinal Angiogenesis, IMA J. Appl. Math. (2012) ; 19 pages; doi:10.1093/imamat/hxs050