Ensemble Averages, Soliton Dynamics and Influence of Haptotaxis in a Model of Tumor-Induced Angiogenesis

Abstract

Abstract:In this work, we present a numerical study of the influence of matrix degrading enzyme (MDE) dynamics and haptotaxis on the development of vessel networks in tumor-induced angiogenesis. Avascular tumors produce growth factors that induce nearby blood vessels to emit sprouts formed by endothelial cells. These capillary sprouts advance toward the tumor by chemotaxis (gradients of growth factor) and haptotaxis (adhesion to the tissue matrix outside blood vessels). The motion of the capillaries in this constrained space is modelled by stochastic processes (Langevin equations, branching and merging of sprouts) coupled to continuum equations for concentrations of involved substances. There is a complementary deterministic description in terms of the density of actively moving tips of vessel sprouts. The latter forms a stable soliton-like wave whose motion is influenced by the different taxis mechanisms. We show the delaying effect of haptotaxis on the advance of the angiogenic vessel network by direct numerical simulations of the stochastic process and by a study of the soliton motion.

A convergent numerical scheme for integrodifferential kinetic models of angiogenesis

“…Branching. At time T i , the velocity of the newly created tip i is selected out of a normal distribution (6) and, for a given realization ω of the stochastic process, the number of active tips at time t is N (t, ω). The probability that a tip branches from one of the existing ones during an infinitesimal time interval (t, t + dt] is taken proportional to…”

“…Previous authors did not incorporate them to equations for probability densities (Fokker-Planck equations) that did not take into account generation of new tips and their annihilation due to anastomosis; see the review papers [27,35]. The generation and annihilation source terms are crucial to explain soliton-like behavior of solutions to the system of equations (1)- (3) and of ensemble averages of the associated stochastic process [4,5,6]. These references consider fluxes of the form |j| where…”

“…The chemotactic force F(C) depends on the average tip speed through j because consumption of tumor angiogenic factor is mostly performed by the additional endothelial cells that promote vessel extensions [3,8,42]. Other mechanisms of cellular motion such as haptotaxis can be included in F(C) as indicated in [6,8].…”

“…These boundary conditions differ from those employed in slab geometries in [4,5,6] because we must take into account the presence of velocity cut-offs in the flux (5). Conditions with a similar structure were used in [14] for theoretical studies in annular domains.…”

“…Sections 3 and 4 check the performance of the scheme by comparing with explicit soliton-like solutions constructed by asymptotic methods and with direct solutions of stochastic equations. Unlike [4,5,6], we consider here fluxes (5) in which the euclidean norm affects the velocity |v| instead of a vector flux |j| as in (7), and, additionally, a velocity cut-off is present. This avoids uniqueness issues for some classes of initial data due to lack of lipschitzianity of the flux and ensures existence of global stable solutions for the kinetic model.…”