A deterministic model of growth factor-induced angiogenesis

Understanding the formation and structure of a capillary network is critical for any reparative strategy since the capillary network dictates tissue survival, hemodynamics, and mass transport. Vascular assembly and patterning has largely been studied using a reductionist approach where a particular endothelial cell molecular pathway or cellular mechanism is investigated as a relatively closed system. This trend of research has yielded a staggering wealth of genes, proteins, and cells that play critical roles in angiogenesis and some have resulted in successful targeted angiogenic therapies. However, these genes, proteins, and cells do not exist in discrete closed systems, rather they are intimately coupled across spatial and temporal dimensions. Designing experiments to study a single or group of perturbations is fraught with confounding complications. An integrative tool is required that incorporates gene, protein, and cell information and appropriately describes the complex systems behavior of vascular assembly and patterning. In this paper, we propose a new deterministic mathematical formulation to model growth factor-induced angiogenesis. Conductivity, of the extracellular matrix for the movement/extension of capillary sprouts is a new concept introduced to account for the heterogeneity and anisotropy of the extracellular matrix. The replacement of traditional endothelial cell density by the capillary indicator function enhances the capabilities of capturing the capillary network sharply in a fine scale (i.e., tracking the dynamics of the tip in uni-cellular scale). Major mechanisms including cell proliferation, sprout branching, and anastomosis are incorporated directly into this continuous mathematical model and model parameters are perturbed to determine the strength of their effect on angiogenesis. The model is fully deterministic and generates the overall dendritic structure of the capillary network morphologically similar to those observed in vivo. The simulations capture significant vascular patterning, such as vascular loops and backward growth. Moreover, the simulations provide a deeper understanding of the influence of extracellular matrix on angiogenesis and vascular patterning. An advantage of this model is that the complex physical, chemical and biological processes in angiogenesis can be described and consequently analyzed by a mathematical system with self-contained information. (C) 2004 Society for Mathematical Biology. Published by Elsevier Ltd. All rights reserved.