Workshop on Pattern Formation and Functional Morphology, Tue, 08 Jan, 2008
Speaker: Vincenzo Capasso
Abstract
Tumour driven angiogenesis is an extremely complex process, possessing multiple, integrated modulators and feedback loops; all these phenomena are subject to random fluctuations, together with
suitable underlying fields. A major difficulty derives from the strong coupling of the kinetic parameters of the stochastic branching-and-growth of the capillary network with the interacting underlying fields.
Methods for reducing complexity include homogenization at larger scales, e. g. by (locally) averaging the stochastic cell, or vessel densities in the evolution equations of the underlying fields, while keeping stochasticity at lower scales, possibly at the level of individual cells or vessels. In this way only the simple tochasticity of the geometric processes of birth (branching) and growth is kept, having reduced the local dependence of the relevant kinetic arameters upon (now deterministic) mean underlying fields. This kind of models are known as hybrid models.
For modelling the stochastic morphology of such systems, an original approach will be presented which leads to a rigorous definition of the mean length density of a fibre process, and the mean density of branching points, based on a representation of mean geometric densities of random closed sets, as expectations of suitably defined random distributions à la Dirac-Schwartz. This is of great importance also for defining appropriate statistical methods for the estimation of the relevant mean densities.
In this lecture, as a matter of example, we present a simplified stochastic geometric model for a spatially distributed angiogenic process, strongly coupled with a set of relevant chemotactic fields. The branching mechanism of blood vessels is modelled as a stochastic marked counting process describing the branching of new tips, while the network of vessels is modelled as the union of the trajectories developed by tips; consequently capillary extensions are modelled by a system of a random number of Langevin type stochastic differential equations, coupled with the random PDE's describing the evolution of the underlying fields involved in the process. We perform a heuristic law of large numbers as the number of tips increases, showing that, when the number of tips, and then of trajectories, is large enough, the stochastic branching and growth of vessels can be described, at the macroscale, by the system of evolution equations for the relevant mean densities, coupled with the evolution equations for the underlying fields.
URL: www.ricam.oeaw.ac.at/specsem/ssqbm/participants/abstracts/index.php
This page was made with 100% valid HTML & CSS - Send comments to Webmaster
Today's date and time is 04/18/24 - 16:31 CEST and this file (/specsem/ssqbm/participants/abstracts/index.php) was last modified on 12/18/12 - 11:01 CEST