A recent physical approach to angiogenesis modeling: mixture models

Angiogenesis modeling has generally tended towards (1) discrete models of individual sprout tip cells (cellular automata, Plank-Sleeman free-swimming models, etc.), or (2) continuum models of the evolving blood vessel density. At the Mathways into Cancer II workshop last week, I learned of a new approach using mixture (or phase field) models: computational voxels are treated as a conserved mixture of water, matrix, and endothelial cells. Applying conservation laws to each of these phases, plus an energy formulation of how the phases mix, leads to fourth-order Cahn-Hilliard equations for the mixture components. These require some fairly tricky numerics to solve correctly, but the results can be beautiful. And notably, it allows the inclusion of sophisticated cell and tissue mechanics.

Without further ado, I present two papers (by two different groups) on this new approach to angiogenesis modeling, combined as hybrid models with "traditional" sprout tip agents:

Capillary networks in tumor angiogenesis: From discrete endothelial cells to phase-field averaged descriptions via isogeometric analysis
Guillermo Vilanova, Ignasi Colominas, Hector Gomez

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.

Preprint / advance copy: http://dx.doi.org/10.1002/cnm.2552

(AFAIK) The first work using this approach was by Rui Travasso and colleagues in 2011, with a focus on the role of mechanistic modeling in driving biological hypotheses:

Tumor Angiogenesis and Vascular Patterning: A Mathematical Model
Rui D. M. Travasso et al.

Abstract: Understanding tumor induced angiogenesis is a challenging problem with important consequences for diagnosis and treatment of cancer. Recently, strong evidences suggest the dual role of endothelial cells on the migrating tips and on the proliferating body of blood vessels, in consonance with further events behind lumen formation and vascular patterning. In this paper we present a multi-scale phase-field model that combines the benefits of continuum physics description and the capability of tracking individual cells. The model allows us to discuss the role of the endothelial cells' chemotactic response and proliferation rate as key factors that tailor the neovascular network. Importantly, we also test the predictions of our theoretical model against relevant experimental approaches in mice that displayed distinctive vascular patterns. The model reproduces the in vivo patterns of newly formed vascular networks, providing quantitative and qualitative results for branch density and vessel diameter on the order of the ones measured experimentally in mouse retinas. Our results highlight the ability of mathematical models to suggest relevant hypotheses with respect to the role of different parameters in this process, hence underlining the necessary collaboration between mathematical modeling, in vivo imaging and molecular biology techniques to improve current diagnostic and therapeutic tools.

Open access: http://dx.doi.org/10.1371/journal.pone.0019989