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Review
. 2018 Jan;109(1):15-23.
doi: 10.1111/cas.13395. Epub 2017 Dec 4.

Study on the tumor-induced angiogenesis using mathematical models

Takashi Suzuki  1 , Dhisa Minerva  1 , Koichi Nishiyama  2 , Naohiko Koshikawa  3 , Mark Andrew Joseph Chaplain  4
Affiliations
Review

Study on the tumor-induced angiogenesis using mathematical models

Takashi Suzuki et al. Cancer Sci. 2018 Jan.

Abstract

We studied angiogenesis using mathematical models describing the dynamics of tip cells. We reviewed the basic ideas of angiogenesis models and its numerical simulation technique to produce realistic computer graphics images of sprouting angiogenesis. We examined the classical model of Anderson-Chaplain using fundamental concepts of mass transport and chemical reaction with ECM degradation included. We then constructed two types of numerical schemes, model-faithful and model-driven ones, where new techniques of numerical simulation are introduced, such as transient probability, particle velocity, and Boolean variables.

Keywords: angiogenesis; extracellular matrix degradation; hybrid simulation; mathematical model; tumor microenvironment.

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Figures

Figure 1
Figure 1
Illustration of angiogenesis. Vascular endothelial growth factor (VEGF) is released by hypoxia cells, diffusing through the ECM, and finally reaching the nearby pre‐existing blood vessel. Tip cells are formed by endothelial cell activation, which is caused by VEGF induction to the pre‐existing vessel. Then, tip cells migrate in response to VEGF gradient chemotactically. Tip cells degrade ECM (orange lines) during this migration. Branching and anastomosis occurs during the angiogenesis process
Figure 2
Figure 2
Density of tip cell, stalk cell, and vascular endothelial growth factor (VEGF) on 1D model are defined as the average density on the green dash‐line for each position
Figure 3
Figure 3
Initial condition setting of (A) five tip cells position, B, ECM distribution, and C, vascular endothelial growth factor (VEGF) distribution
Figure 4
Figure 4
A, Types of anastomosis. (Left) Tip cell meets other vessel or what we call tip to sprout anastomosis. (Right) One tip cell meets another tip cell or what we call tip to tip anastomosis. B, Branching form possibilities
Figure 5
Figure 5
Snapshots of vessel growth on (A) vascular endothelial growth factor (VEGF) distribution and B, ECM distribution in time
Figure 6
Figure 6
Snapshots of ECM distribution in time
See this image and copyright information in PMC

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