A Multiscale Approach Leading to Hybrid Mathematical Models for Angiogenesis: Role of Randomness - Mathematical Methods and Models in Biomedicine - page 82

A Multiscale Approach Leading to Hybrid
Mathematical Models for Angiogenesis:
The Role of Randomness

Vincenzo Capasso and Daniela Morale

1 Introduction

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 “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 two important
case studies involving angiogenesis, i.e., tumor-driven angiogenesis [ 7 ] and retina
angiogenesis [ 8 ]. 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 [ 18 , 19 ].
A fruitful approach to the mathematical description of such phenomena, sug-
gested since long by various authors [ 16 , 22 , 26 , 27 , 30 , 31 ], is based on the so-called
individual based models , according to which the “movement” of each individual is
described, embedded in the total population. This is also known as Lagrangian ap-
proach . Possible randomness is usually included in the motion, so that the variation
in time of the (random) locations X N (

d

t ) ∈ R , k = 1 ,..., N ( t ) , of individuals in a

group of size N 0 is described by a system of stochastic differential

equations driven by gradients of suitable underlying fields. On the other hand the

( t ) at time t ≥

V. Capasso ( )￿D.Morale
Department of Mathematics, University of Milan, 20133 Milan, Italy
CIMAB (Interuniversity Centre for Mathematics Applied to Biology, Medicine and
Environmental Sciences), University of Milan, 20133 Milan, Italy
e-mail: [e-mail] ; [e-mail]