Mathematical modeling of structure formation in angiogenesis

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Angiogenesis, the formation of blood vessels from a pre-existing vasculature, is a process whereby capillary sprouts are formed in response to chemical stimuli that can be either supplied externally or produced locally. Understanding of the fundamental mechanisms that govern angiogenesis suggests a powerful therapeutic approach that will allow to combat a variety of severe pathological conditions. The development of in-vitro angiogenesis provides researches with controllable tool that helps study blood vessel formation. We model endothelial cell pattern formation in-vitro as the first step in understanding angiogenesis and multiple factors that influence it. We formulate a mathematical model that governs endothelial cell pattern formation on a biogel surface using a five-species reaction mechanism and justify its reduction to a two-species problem. We study this simplified problem with two different forms of cell diffusion coefficient both numerically and analytically to determine whether spatially nonuniform steady patterns can appear in the system when its basic states become unstable. We perform linear stability analysis and weakly nonlinear stability analysis near the instability threshold to describe formation of certain spatial structures observed in experiments and to determine the parameter ranges where these structures can occur. We derive amplitude equations that govern the interaction of hexagonal and stripe patterns. We also derive the Sivashinsky and Cahn-Hilliard equations. Next we formulate a discrete-continuous mathematical model of angiogenesis in-vivo and perform numerical simulations of the model. The model accounts for both a continuous chemoattractant field and a discrete set of growing sprouts, propagation of which is governed by prescribed laws that involve both deterministic and random ingredients. We extend the previous model by introducing the action of repulsive factors and we show that their activity results in a larger degree of reorganization of cellular matter and in a more robust control over the size of the growing vascular network. The numerical results demonstrate new vessel growth towards the source of the growth factor and provide an insight into capillary network formation.

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  • 08/14/2018
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