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References
Anderson, A.R.A., Chaplain, M.A.J.: Continuous and discrete mathematical model of tumour-induced angiogenesis. Bull. Math. Biol. 60, 857–899 (1998)
Bentley, K., Gerhardt, H., Bates, P.A.: Agent-based simulation of notch-mediated tip cell selection in angiogenic sprout initialisation. J. Theor. Biol. 250, 25–36 (2008)
Cristini, V., Lowengrub, J., Nie, Q.: Nonlinear simulation of tumor growth. J. Math. Biol. 46, 191–224 (2003)
Figg, W.D., Folkman, J.: Angiogenesis: An Integrative Approach from Science to Medicine. Springer, New York (2008)
Gerhardt, H., Golding, M., Fruttiger, M.: Vegf guides angiogenic sprouting utilizing endothelial tip cell filopodia. J. Cell Biol. 161, 1163–1177 (2003)
Godde, v., Kurz, H.: Structural and biophysical simulation of angiogenesis and vascular remodeling. Dev. Dyn. 220, 387–401 (2001)
Levine, H.A., Sleeman, B.D., Nilsen-Hamilton, M.: Mathematical modeling of the onset of capillary formation initiating angiogenesis. Math. Biol. 42, 195–238 (2001)
Lowengrub, J.S., Frieboes, H.B, Jin, F., Chuang, Y.-L., Li, X., Macklin, P., Wise, S.M., Cristini, V.: Nonlinear modeling of cancer: bridging the gap between cells and tumors. Nonlinearity 23, R1–R91 (2010)
Macklin, P., McDougall, S., Anderson, A.R.A., Chaplain, M.A.J., Cristini, V., Lowengrub, J.: Multiscale modelling and nonlinear simulation of vascular tumour growth. J. Math. Biol. 58, 765–798 (2009)
McDougall, S.R., Anderson, A.R.A., Chaplain, M.A.J.: Mathematical modelling of dynamic adaptive tumour-induced angiogenesis: clinical implications and therapeutic targeting strategies. J. Theor. Biol. 241, 564–589 (2006)
Owen, M.R., Alarcón, T., Maini, P.K., Byrne, H.M.: Angiogenesis and vascular remodelling in normal and cancerous tissues. J. Math. Biol. 58, 689–721 (2009)
Pries, A.R., Secomb, T.W., Gaehtgens, P.: Structural adaptation and stability of microvascular networks: theory and simulations. Am. J. Physiol. Heart Circ. Physiol. 275, H349–H360 (1998)
Risau, W.: Mechanisms of angiogenesis. Nature 386 (1997)
Sinek, J.P., Sanga, S., Zheng, X. Frieboes, H.B., Ferrari, M., Cristini, V.: Predicting drug pharmacokinetics and effect in vascularized tumors using computer simulation. Math. Biol. 58, 485–510 (2009)
Welter, M., Barthab, K., Riegera, H.: Vascular remodelling of an arterio-venous blood vessel network during solid tumour growth. J. Theor. Biol. 259, 405–422 (2009)
Zheng, X., Wise, S.M., Cristini, V.: Nonlinear simulation of tumor necrosis, neo-vascularization and tissue invasion via an adaptive finite-element/level-set method. Bull. Math. Biol. 67, 211–259 (2005)
Acknowledgements
The authors thank Hermann Frieboes for the valuable discussions. The authors are grateful for the partial funding from the National Science Foundation, Division of Mathematical Sciences, and the National Institutes of Health through grants NIH-1RC2CA148493-01 and NIH-P50GM76516 for a National Center of Excellence in Systems Biology at UCI.
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Wu, M., Lowengrub, J.S. (2015). Angiogenesis, Mathematical Modeling Perspective. In: Engquist, B. (eds) Encyclopedia of Applied and Computational Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70529-1_62
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