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

Abstract

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.

An energy-stable time-integrator for phase-field models

“…The diffuse interface is introduced through an energetic variational procedure that results in a thermodynamically consistent coupling. This consistency gives the method solid mathematical and physical footings, and explains why applications range from spinodal decomposition of immiscible binary mixtures [7,21,61], tumor angiogenesis [58,59], wetting [14] and elasto-capillarity [5,53], image processing [40] to water infiltration in porous media [22]. Another advantage of the phase-field method over sharp interface descriptions comes from the fact that under appropriate assumptions [6], a diffuse-interface description can asymptotically converge to its sharp-interface counterpart by decreasing the interfacial thickness.…”

Isogeometric analysis of the Cahn–Hilliard equation – a convergence study

“…Typical applications include the modelling and simulation of solidification processes, spinodal decomposition, coarsening of precipitate phases, shape memory effects, re-crystallisation, and dislocation dynamics [1,2,3,4]. Phase-field models have been successfully applied to predict microstructural changes under external fields [5,6], model tumor growth [7,8,9], and image impainting [10]. The phase-field approach has also been used to model crack propagation [11,12,13,14,15].…”

Computational Phase‐Field Modeling

“…Among these problems, we include the interaction of a large number of cracks in three-dimensional solids of complicated geometry (Borden et al, 2014), fully three-dimensional air-water flows with surface tension (Ceniceros et al, 2010), liquid-vapor phase transitions and cavitation (Liu et al, 2013), nucleate and film boiling (Liu et al, 2015), phase-change-driven implosion of thin structures (Bueno et al, 2014), cellular migration (Shao et al, 2012) and tumor growth (Hawkins-Daarud et al, 2012;Vilanova et al, 2013Vilanova et al, , 2014Vilanova et al, , 2012Xu et al, 2015;Lorenzo et al, 2015) and others Juanes, 2012, 2014;Gomez et al, 2013). However, even though the last few years witnessed very significant advances in the area, there are still many challenges ahead.…”