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.
Similar content being viewed by others
References
Risau, W.: Mechanisms of angiogenesis. Nature 386(6626), 671–674 (1997)
Chung, A.S., Lee, J., Ferrara, N.: Targeting the tumour vasculature: insights from physiological angiogenesis. Nat. Rev. Cancer 10(7), 505–514 (2010)
Kerbel, R.S.: Tumor angiogenesis. N. Engl. J. Med. 358(19), 2039–2049 (2008)
Zhang, L., Bhaloo, S.I., Ting, C., Zhou, B., Xu, Q.: Role of resident stem cells in vessel formation and arteriosclerosis. Circ. Res. 122(11), 1608–1624 (2018)
Anderson, A.R.A., Chaplain, M.J.: Continuous and discrete mathematical models of tumorinduced angiogenesis. Bull. Math. Biol. 60, 857–900 (1998)
Tong, S., Yuan, F.: Numerical simulations of angiogenesis in the cornea. Microvasc. Res. 61, 14–27 (2001)
Gamba, A., Ambrosi, D., Coniglio, A., de Candia, A., DiTalia, S., Giraudo, E., Serini, G., Preziosi, L., Bussolino, F.: Percolation, morphogenesis, and burgers dynamics in blood vessels formation. Phys. Rev. Lett. 90(11), 118–101 (2003)
Bauer, A.L., Jackson, T.L., Jiang, Y.: A cell-based model exhibiting branching and anastomosis during tumor-induced angiogenesis. Biophys. J . 92, 3105–3121 (2007)
Scianna, M., Preziosi, L.: Multiscale developments of the cellular potts model. Multiscale Model. Simul. 10, 342–382 (2012)
Secomb, T.W., Alberding, J.P., Deshirst, M.W., Pries, A.R.: Angiogenesis: an adaptive dynamic biological patterning problem. PLoS Comput. Biol. 9, e1002,983 (2013)
Daub, J.T., Merks, M.H.: A cell-based model of extracellular-matrix-guided endothelial cell migration during angiogenesis. Bull. Math. Biol. 75, 1377–1399 (2013)
Sugihara, K., Nishiyama, K., Fukuhara, S., Uemura, A., Arima, S., Kobayashi, R., Köhn- Luque, A., Mochizuki, N., Suda, T., Ogawa, H., Kurihara, H.: Autonomy and non-autonomy of angiogenic cell movements revealed by experiment-driven mathematical modeling. Cell Rep. 13, 1814–1827 (2015)
Arima, S., Nishiyama, K., Ko, T., Arima, Y., Hakozaki, Y., Sugihara, K., Koseki, H., Uchijima, Y., Kurihara, Y., Kurihara, H.: Angeogenetic morphogenesis driven by dynamic and heterogeneous collective endothelial cell movement. Development 138, 4763–4776 (2011)
Matsuya, K., Yura, F., Mada, J., Kurihara, H., Tokihiro, T.: A discrete mathematical model for angiogenesis. SIAM J. Appl. Math. 76, 2243–2259 (2016)
Mada, J., Matsuya, K., Yura, F., Kurihara, H., Tokihiro, T.: A mathematical modeling of angiogenesis [in japanese]. JSIAM 26, 105–123 (2016)
Takubo, N., Yura, F., Naemura, K., Yoshida, R., Tokunaga, T., Tokihiro, T., Kurihara, H.: Cohesive and anisotropic vascular endothelial cell motility driving angiogenic morphogenesis. Sci. Rep. 9. Article number 9304 (2019)
Family, F., Masters, B.R., Platt, D.E.: Fractal patterns formation in human retinal vessels. Physica D 38, 98–103 (1989)
Masters, B.R.: Fractal analysis of the vascular tree in the human retina. Ann. Rev. Biol. Eng. 6, 427–452 (2004)
Mandelbrot, B.B.: The Fractal Geometry of Nature, W. H. Freeman and Co. (1982)
Mandelbrot, B.B.: Fractals: Form, Chance and Dimension, W. H. Freeman and Co. (1977)
Falconer, K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, Amsterdam (2014)
Halsey, T.C., Jensen, M.H., Kadanoff, L.P., Procaccia, I., Shraiman, B.I.: Fractal measures and their singularities: the characterization of strange sets. Phys. Rev. A 33, 1141–1151 (1986)
Lopes, R., Betrouni, N.: Fractal and multifractal analysis: a review. Med. Image Anal. 13, 634–649 (2009)
Stošić, T., Stošić, B.D.: Multifractal analysis of human retinal vessles. IEEE Trans. Med. Imag. 25, 1101–1107 (2006)
Salat, H., Murcio, R., Arcaute, E.: Multifractal methodology. Phys. A 473, 467–487 (2017)
Murray, C.D.: The physiological principle of minimum work. I. The vascular system and the cost of blood volume. Proc. Natl. Acad. Sci. USA 12(3), 207–214 (1926)
Iizuka, O., Kawamura, S., Tero, A., Uemura, A., Miura, T.: Remodeling mechanisms determine size distributions in develo** retinal vasculature. PLoS One (2020). https://doi.org/10.1371/journal.pone.0235373
Okabe, K., Kobayashi, S., Yamad, T., Kurihara, T., Tai-Nagara, I., Miyamoto, T., Mukouyama, Y.S., Sato, T.N., Suda, T., Ema, M., Kubota, Y.: Neurons limit angiogenesis by titrating VEGF in retina. Cell 159, 584–596 (2014)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13160-021-00493-9