Olsen L, Sherratt J A, Maini P K, Arnold F
Centre for Mathematical Biology, Mathematical Institute, Oxford, UK.
IMA J Math Appl Med Biol. 1997 Dec;14(4):261-81.
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 travelling-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.
血管生成是指新的毛细血管从周围的母血管长入组织的过程，是皮肤伤口愈合、恶性肿瘤生长及其他病理状况中的关键事件。在伤口愈合过程中，新的毛细血管将重要的代谢物（如氨基酸和氧气）输送到伤口处参与复杂修复过程序列的细胞。这些新毛细血管的关键细胞成分是内皮细胞：尽管伤口愈合血管生成的生物学机制尚未完全明了，但最近已充分证明了它们与可溶性生化蛋白和不溶性细胞外基质（ECM）蛋白的相互作用。近期大量研究，包括一些连续介质数学模型，都聚焦于内皮细胞与可溶性调节因子（如生长因子）之间的相互作用。在本研究中，我们使用类似的建模框架来研究不溶性ECM底物的作用，其中胶原蛋白是主要的大分子蛋白。我们的模型由一个关于内皮细胞密度（作为位置和时间的函数）的偏微分方程与一个关于ECM密度的常微分方程组成。假定ECM调节细胞运动（包括随机运动和定向运动）及增殖，而细胞合成并降解ECM。对这些方程的分析和数值解突出了这些过程在伤口愈合血管生成中的作用。一种非标准近似分析揭示了系统的行波结构。该模型扩展到二维空间（平行和垂直于皮肤平面），并给出了数值模拟结果。该模型预测，ECM介导的随机运动性和细胞增殖是驱动血管生成的关键过程，这些过程对ECM密度的功能依赖细节以及ECM重塑速率决定了血管生成反应的定性特征。这些预测可通过实验验证，并且可能有助于更深入地理解伤口愈合血管生成所涉及的生物学机制。
