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Capillary networks in tumor angiogenesis: From discrete endothelial cells to phase‐field averaged descriptions via isogeometric analysis
Author(s) -
Vilanova Guillermo,
Colominas Ignasi,
Gomez Hector
Publication year - 2013
Publication title -
international journal for numerical methods in biomedical engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.741
H-Index - 63
eISSN - 2040-7947
pISSN - 2040-7939
DOI - 10.1002/cnm.2552
Subject(s) - isogeometric analysis , angiogenesis , parametric statistics , partial differential equation , computer science , numerical analysis , mathematics , biological system , mathematical analysis , finite element method , physics , biology , statistics , cancer research , thermodynamics
SUMMARY 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. Copyright © 2013 John Wiley & Sons, Ltd.