A Mathematical Model of an Angiogenic Process

Abstract

Tubular growth of blood vessels in 2-dimensional space is described in the present study by using a phase field model. In contrast with previous studies, we propose a biomechanical model based on Canham-Helfrich energy, coupled to an angiogenic agent through a spontaneous curvature term. The concentration of this angiogenic agent is static and non uniform, generating a wellde fined gradient through time. The model is very compact consisting of only one partial differential equation, and has the clear advantage of a reduced number of parameters. Following a phase-field methodology, this model allows us to relate sprout growth with the spontaneous curvature term from the Canham-Helfrich model. The importance of the capillary shape at the initial conditions has also been addressed. Additionally, capillaries grown on other growing capillaries have been obtained by combining multiple distributions of growth factor