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Pattern formation of vascular network in a mathematical model of angiogenesis

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Abstract

We discuss the characteristics of the patterns of the vascular networks in a mathematical model for angiogenesis. Based on recent in vitro experiments, this mathematical model assumes that the elongation and bifurcation of blood vessels during angiogenesis are determined by the density of endothelial cells at the tip of the vascular network, and describes the dynamical changes in vascular network formation using a system of simultaneous ordinary differential equations. The pattern of formation strongly depends on the supply rate of endothelial cells by cell division, the branching angle, and also on the connectivity of vessels. By introducing reconnection of blood vessels, the statistical distribution of the size of islands in the network is discussed with respect to bifurcation angles and elongation factor distributions. The characteristics of the obtained patterns are analysed using multifractal dimension and other techniques.

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Acknowledgements

The authors would like to thank Prof. Hiroki Kurihara and Dr. Kazuo Tonami for helpful discussions about angiogenesis. They also would like to thank Dr. Tatsuya Hayashi and Mr. Kazuma Sakai for useful discussions on mathematical modelling, and Dr. Naoko Takubo for providing the photo image of Fig.1. TT is grateful for financial support to Arithmer Inc.

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Authors and Affiliations

  1. College of Industrial Technology, Nihon University, 2-11-1 Shin-ei, Narashino, Chiba, 275-8576, Japan

    Jun Mada

  2. Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan

    Tetsuji Tokihiro

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  1. Jun Mada
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  2. Tetsuji Tokihiro
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Correspondence to Jun Mada.

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Mada, J., Tokihiro, T. Pattern formation of vascular network in a mathematical model of angiogenesis. Japan J. Indust. Appl. Math. 39, 351–384 (2022). https://doi.org/10.1007/s13160-021-00493-9

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