A mathematical model for the capillary endothelial cell-extracellular matrix interactions in wound-healing angiogenesis

Abstract

Angiogenesis, the process by which new blood capillaries grow into a tissue from surrounding parent vessels, is a key event in dermal wound healing, malignant-tumour growth, and other pathologic conditions. In wound healing, new capillaries deliver vital metabolites such as amino acids and oxygen to the cells in the wound which are involved in a complex sequence of repair processes. The key cellular constituents of these new capillaries are endothelial cells: their interactions with soluble biochemical and insoluble extracellular matrix (ECM) proteins have been well documented recently, although the biological mechanisms underlying wound-healing angiogenesis are incompletely understood. Considerable recent research, including some continuum mathematical models, have focused on the interactions between endothelial cells and soluble regulators (such as growth factors). In this work, a similar modelling framework is used to investigate the roles of the insoluble ECM substrate, of which collagen is the predominant macromolecular protein. Our model consists of a partial differential equation for the endothelial-cell density (as a function of position and time) coupled to an ordinary differential equation for the ECM density. The ECM is assumed to regulate cell movement (both random and directed) and proliferation, whereas the cells synthesize and degrade the ECM. Analysis and numerical solutions of these equations highlights the roles of these processes in wound-healing angiogenesis. A nonstandard approximation analysis yields insight into the travel ling-wave structure of the system. The model is extended to two spatial dimensions (parallel and perpendicular to the plane of the skin), for which numerical simulations are presented. The model predicts that ECM-mediated random motility and cell proliferation are key processes which drive angiogenesis and that the details of the functional dependence of these processes on the ECM density, together with the rate of ECM remodelling, determine the qualitative nature of the angiogenic response. These predictions are experimentally testable, and they may lead towards a greater understanding of the biological mechanisms involved in wound-healing angiogenesis.

Simulation of tumor-induced angiogenesis and its response to anti-angiogenic drug treatment: mode of drug delivery and clearance rate dependencies

“…To specify these terms, the authors assumed classic Fickian diffusion, with no net production/death of endothelial cells; linear uptake, production/loss rates for each substance; and a specific nonlinear response of endothelial cells to chemotaxis of TAF. For diffusion of endothelial cells E, D E (E;T;F)= " VJ E , where J E , the flux of endothelial cells, has three components: flux J rand = -a e " VE, where a e is a positive random motility coefficient; J chemo = a c (T)E " VT, where a c T ð Þ ¼ ack kþT (Lauffenberger and Kennedy 1984;Olsen et al 1997;Anderson and Chaplain 1998), a c is the saturable chemotaxis coefficient and k is a positive constant; and J hapto = a h E " VF, where a h is the haptotaxis constant. Therefore, endothelial cell flux is: J E ¼ Àa e rE þ a c T ð ÞErT þ a h ErF .…”

A hybrid bioregulatory model of angiogenesis during bone fracture healing

“…The random motion of mesenchymal stem cells was modeled as a haptokinetic process, based on Olsen et al (1997). The resulting functional form is:…”

“…The haptotactic coefficient was taken from Olsen et al (1997), based on a kinetic analysis of a model mechanism for the cell-surface-receptor-extracellular ligand binding dynamics (Sherrat 1994). The resulting functional form is:…”

“…Proliferation of cells was modeled by a logistic growth function (Olsen et al 1997;Weinberg and Bell 1985;Yoshizato et al 1985). The resulting functional forms are:…”

Analysis of Collagen and Glucose Modulated Cell Growth within Tissue Engineered Scaffolds

“…On the theoretical side, mathematical modeling, motivated by the experimental findings, has proceeded to describe the duel role of collagen, demonstrating that cells cannot proliferate in the absence of collagen but may be restricted in growth when ECM becomes sufficiently dense. 24 This work aims to further the tissue engineering modeling by incorporating the interplay between ECM collagen and cell growth, considering cell growth modulated by both nutrient and ECM collagen. The simulation results were compared to the experimental data valid in literature.…”