Department of Applied Mathematics, University of A Coruña, Campus de Elviña, 15071, A Coruña, Spain.
Int J Numer Method Biomed Eng. 2013 Oct;29(10):1015-37. doi: 10.1002/cnm.2552. Epub 2013 May 7.
Tumor angiogenesis, the growth of new capillaries from preexisting ones promoted by the starvation and hypoxia of cancerous cell, creates complex biological patterns. These patterns are captured by a hybrid model that involves high-order partial differential equations coupled with mobile, agent-based components. The continuous equations of the model rely on the phase-field method to describe the intricate interfaces between the vasculature and the host tissue. The discrete equations are posed on a cellular scale and treat tip endothelial cells as mobile agents. Here, we put the model into a coherent mathematical and algorithmic framework and introduce a numerical method based on isogeometric analysis that couples the discrete and continuous descriptions of the theory. Using our algorithms, we perform numerical simulations that show the development of the vasculature around a tumor. The new method permitted us to perform a parametric study of the model. Furthermore, we investigate different initial configurations to study the growth of the new capillaries. The simulations illustrate the accuracy and efficiency of our numerical method and provide insight into the dynamics of the governing equations as well as into the underlying physical phenomenon.
肿瘤血管生成，即新毛细血管从先前存在的毛细血管中生长，由癌细胞的饥饿和缺氧所促进，会产生复杂的生物模式。这些模式通过一个混合模型来捕捉，该模型涉及高阶偏微分方程和移动的基于代理的组件。模型的连续方程依赖于相场方法来描述脉管系统和宿主组织之间的复杂界面。离散方程在细胞尺度上提出，并将尖端内皮细胞视为移动代理。在这里，我们将模型置于一个连贯的数学和算法框架中，并引入一种基于等几何分析的数值方法，该方法结合了理论的离散和连续描述。使用我们的算法，我们进行了数值模拟，展示了肿瘤周围血管的发育。新方法使我们能够对模型进行参数研究。此外，我们研究了不同的初始配置，以研究新毛细血管的生长。模拟说明了我们数值方法的准确性和效率，并提供了对控制方程动力学以及潜在物理现象的深入了解。
