Continuous and Discrete Mathematical Models of Tumor-induced Angiogenesis

Abstract

Angiogenesis, the formation of blood vessels from a pre-existing vasculature, is a process whereby capillary sprouts are formed in response to externally supplied chemical stimuli. The sprouts then grow and develop, driven initially by endothelial-cell migration, and organize themselves into a dendritic structure. Subsequent cell proliferation near the sprout tip permits further extension of the capillary and ultimately completes the process. Angiogenesis occurs during embryogenesis, wound healing, arthritis and during the growth of solid tumors. In this paper we present both continuous and discrete mathematical models which describe the formation of the capillary sprout network in response to chemical stimuli (tumor angiogenic factors, TAF) supplied by a solid tumor. The models also take into account essential endothelial cell-extracellular matrix interactions via the inclusion of the matrix macromolecule fibronectin. The continuous model consists of a system of nonlinear partial differential equations describing the initial migratory response of endothelial cells to the TAF and the fibronectin. Numerical simulations of the system, using parameter values based on experimental data, are presented and compared qualitatively with in vivo experiments. We then use a discretized form of the partial differential equations to develop a biased random-walk model which enables us to track individual endothelial cells at the sprout tips and incorporate anastomosis, mitosis and branching explicitly into the model. The theoretical capillary networks generated by computer simulations of the discrete model are compared with the morphology of capillary networks observed in in vivo experiments.

Vascular remodelling of an arterio-venous blood vessel network during solid tumour growth

“…In particular the geometry and characteristics of the initial vasculature has to be taken into account, which is missing in almost all existing tumour growth models. Not only the morphology of the emerging tumour vasculature is completely different in vessel-in-growth models (Anderson and Chaplain, 1998;Zheng et al, 2004;Friboes et al, 2007;Wise et al, 2008;Milde et al, 2008) when compared with our results, also their blood flow characteristics and their determinants for blood-borne drug transport are completely different (McDougall et al, 2002;McDougall et al, 2006).…”

“…Depending on the local oxygen concentration tumour cells represent the sources 4 A c c e p t e d m a n u s c r i p t of the diffusion determined VEGF concentration field, which triggers either the generation of tip cells for angiogenic sprouting from existing vessels outside or at the periphery of the tumour, or circumferential growth inside the tumour. In contrast to vessel in-growth models as in (Anderson and Chaplain, 1998) tip-cells are not dominantly generated by branching of existing tip-cells but by sprouting from vessels of the original network. This tip-cell generation from existing vessels is initiated by activation of VEGF receptors with subsequent up-regulation of the delta-like-4 (Dll4) ligand in one EC (Liu et al, 2003), and the Dll4-mediated activation of Notch1 receptors in the neighbouring EC (Sainson et al, 2005;Williams et al, 2006).…”

“…These often contain a stochastic element and model at the level of the individual cell, see for example (Stokes and Lauffenburger, 1991) and (Turner and Sherratt, 2002). (Anderson and Chaplain, 1998) derived a discrete model for EC movement in angiogenesis via a discretisation of a continuous model. Here tip cells located at discrete lattice points are assigned probabilities to migrate to adjacent lattice points within the next time-step.…”

“…Such a rule is necessary when the initial vasculature is ubiquitous as in our model since otherwise the presence of VEGF generats tip-cells in an uncontrolled fashion. 2) Instead of modelling angiogenic sprout migration and branching involving matrix metalloproteasies and collagen fibers (Bauer et al, 2007;Milde et al, 2008) we used commonly accpeted stochastic rules (Anderson and Chaplain, 1998). The ubiquity of original vessels in our model diminishes the importance of the migration and banching process per se, since the intercapillary distances are rather short, which is different from models without initial vasculature.…”

A simple mechanistic model of sprout spacing in tumour-associated angiogenesis

“…Anderson and Chaplain (1998) have developed a model for the generation of new sprouts in the brush border region, relating this to the age of the current sprout, the existence of sufficient space locally for a sprout to form, and a requirement that the endothelial sprout density is greater than a threshold level that is inversely proportional to the local attractant concentration. We note that such a brush border effect could be predicted with the present model naturally if the quantity of inhibitor produced by formation of new sprouts in this area were less than the quantity of inhibitor produced on formation of sprouts from the limbal vessels.…”