We propose a new model for the emergence of blood capillary networks. We assimilate the tissue and extra cellular matrix as a porous medium, using Darcy's law for describing both blood and interstitial fluid flows. Oxygen obeys a convection-diffusion-reaction equation describing advection by the blood, diffusion and consumption by the tissue. Discrete agents named capillary elements and modelling groups of endothelial cells are created or deleted according to different rules involving the oxygen concentration gradient, the blood velocity, the sheer stress or the capillary element density. Once created, a capillary element locally enhances the hydraulic conductivity matrix, contributing to a local increase of the blood velocity and oxygen flow. No connectivity between the capillary elements is imposed. The coupling between blood, oxygen flow and capillary elements provides a positive feedback mechanism which triggers the emergence of a network of channels of high hydraulic conductivity which we identify as new blood capillaries. We provide two different, biologically relevant geometrical settings and numerically analyze the influence of each of the capillary creation mechanism in detail. All mechanisms seem to concur towards a harmonious network but the most important ones are those involving oxygen gradient and sheer stress. A detailed discussion of this model with respect to the literature and its potential future developments concludes the paper.

中文翻译：

---

毛细血管网出现的新模型

我们为毛细血管网络的出现提出了一种新的模型。我们使用达西定律（Darcy's law）描述血液和间质液的流动，将组织和细胞外基质吸收为多孔介质。氧气遵循对流扩散反应方程式，该方程式描述了血液对流，组织的扩散和消耗。根据涉及氧浓度梯度，血流速度，剪切应力或毛细管元件密度的不同规则，创建或删除名为毛细管元件和内皮细胞模型组的离散剂。一旦创建，毛细管元件将局部增强水力传导性矩阵，从而有助于局部增加血流速度和氧气流量。毛细管元件之间没有连接。血液之间的耦合 氧气流和毛细管元件提供了一种积极的反馈机制，该机制触发了高水力传导性通道网络的出现，我们认为这是新的毛细血管。我们提供两种不同的，生物学上相关的几何设置，并在数值上详细分析每种毛细管产生机制的影响。所有机制似乎都同意建立一个和谐的网络，但最重要的机制是那些涉及氧梯度和纯粹应力的机制。本文总结了有关该模型的详细讨论，并参考了其潜在的未来发展。生物学相关的几何设置，并在数值上详细分析每种毛细管产生机理的影响。所有机制似乎都同意建立一个和谐的网络，但最重要的机制是那些涉及氧梯度和纯粹应力的机制。本文总结了有关该模型的详细讨论，并结合了文献及其潜在的未来发展。生物学相关的几何设置，并在数值上详细分析每种毛细管产生机理的影响。所有机制似乎都同意建立一个和谐的网络，但最重要的机制是那些涉及氧梯度和纯粹应力的机制。本文总结了有关该模型的详细讨论，并参考了其潜在的未来发展。