Capillary networks in tumor angiogenesis: From discrete endothelial cells to phase-field averaged descriptions via isogeometric analysis

SUMMARYTumor 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. Copyright © 2013 John Wiley & Sons, Ltd. In this paper, we develop a numerical method for a continuum‐discrete, tumor‐induced angiogenesis model. The formulation of the continuous description of the model involves high‐order partial differential equations, which are solved by means of isogeometric analysis. The method couples the discrete mobile cells with the continuous description of the capillary network and allows us to perform several simulations that provide insight into the governing equations and the biological phenomenon.