# American Institute of Mathematical Sciences

2013, 10(1): 19-35. doi: 10.3934/mbe.2013.10.19

## Model of tumour angiogenesis -- analysis of stability with respect to delays

 1 Institute of Applied Mathematics and Mechanics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland, Poland, Poland, Poland

Received  March 2012 Revised  September 2012 Published  December 2012

In the paper we consider the model of tumour angiogenesis process proposed by Bodnar&Foryś (2009). The model combines ideas of Hahnfeldt et al. (1999) and Agur et al. (2004) describing the dynamics of tumour, angiogenic proteins and effective vessels density. Presented analysis is focused on the dependance of the model dynamics on delays introduced to the system. These delays reflect time lags in the proliferation/death term and the vessel formation/regression response to stimuli. It occurs that the dynamics strongly depends on the model parameters and the behaviour independent of the delays magnitude as well as multiple stability switches with increasing delay can be obtained.
Citation: Marek Bodnar, Monika Joanna Piotrowska, Urszula Foryś, Ewa Nizińska. Model of tumour angiogenesis -- analysis of stability with respect to delays. Mathematical Biosciences & Engineering, 2013, 10 (1) : 19-35. doi: 10.3934/mbe.2013.10.19
##### References:
 [1] Z. Agur, L. Arakelyan, P. Daugulis and Y. Ginosar, Hopf point analysis for angiogenesis models, Discrete Contin. Dyn. Syst. B, 4 (2004), 29-38. [2] L. Arakelyan, Y. Merbl and Z. Agur, Vessel maturation effects on tumour growth: validation of a computer model in implanted human ovarian carcinoma spheroids, European J. Cancer, 41 (2005), 159-167. [3] L. Arakelyan, V. Vainstein and Z. Agur, A computer algorithm describing the process of vessel formation and maturation, and its use for predicting the effects of anti-angiogenic and anti-maturation therapy on vascular tumor growth, Angiogenesis, 5 (2002), 203-214. [4] M. Bodnar and U. Foryś, Angiogenesis model with carrying capacity depending on vessel density, J. Biol. Sys., 17 (2009), 1-25. doi: 10.1142/S0218339009002739. [5] K. L. Cooke and P. van den Driessche, On zeroes of some transcendental equations, Funkcj. Ekvacioj, 29 (1986), 77-90. [6] A. d'Onofrio and A. Gandolfi, Tumour eradication by antiangiogenic therapy: analysis and extensions of the model by Hahnfeldt et al. (1999), Math. Biosci., 191 (2004), 159-184. doi: 10.1016/j.mbs.2004.06.003. [7] _______, The response to antiangiogenic anticancer drugs that inhibit endothelial cell proliferation, Applied Mathematics and Computation, 181 (2006), 1155-1162. [8] _______, A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy, Math. Med. Biol., 26 (2009), 63-95. [9] A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On optimal delivery of combination therapy for tumors, Math. Biosci. Eng., 222 (2009), 13-26. doi: 10.1016/j.mbs.2009.08.004. [10] J. M. L. Ebos and R. S. Kerbel, Antiangiogenic therapy: Impact on invasion, disease progression, and metastasis,, Nat. Rev. Clin. Oncol., 8 (): 1. [11] U. Foryś, Biological delay systems and the {Mikhailov criterion of stability}, J. Biol. Sys., 12 (2004), 45-60. doi: 10.1142/S0218339004001014. [12] U. Foryś, Y. Kheifetz and Y. Kogan, Critical point analysis for three-variable cancer angiogenesis model, Math. Biosci. Eng., 2 (2005), 511-525. [13] M. Gałach, Dynamics of the tumor-immune system competition - the effect of time delay, Int J Appl Math Comput Sci, 3 (2003), 395-406. [14] A. Gilead and M. Neeman, Dynamic remodeling of the vascular bed precedes tumor growth: MLS ovarian carcinoma spheroids implanted in nude mice, Neoplasia, 1 (1999), 226-230. doi: 10.1038/sj.neo.7900032. [15] P. Hahnfeldt, D. Panigrahy, J. Folkman and L. Hlatky, Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Res., 59 (1999), 4770-4775 (eng). [16] J. K. Hale, "Theory of Functional Differential Equations," Springer, New York, 1977. [17] J. K. Hale and S. M. V. Lunel, "Introduction to Functional Differential Equations," Springer, New York, 1993. [18] S. J. Holash, G. D. Wiegandand and G. D. Yancopoulos, New model of tumour angiogenesis: Dynamic balance between vessel regression andgrowth mediated by angiopoietins and VEGF, Oncogene, 18 (1999), 5356-5362. [19] R. K. Jain, Normalization of tumor vasculature: an emerging concept in antiangiogenic therapy, Science, 307 (2005), 58-62 (eng). [20] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press Inc., 1993. [21] V. A. Kuznetzov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunologenic tumors: Parameters estimation and global bifurcation analysis, Bull Math Biol, 56 (1994), 295-321. [22] U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem, SIAM J. Control Optim., 46 (2007), 1052-1079. [23] _______, Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis, J. Theor. Biol., 252 (2008), 295-312. [24] M. J. Piotrowska and U. Foryś, Analysis of the Hopf bifurcation for the family of angiogenesis models, J. Math. Anal. Appl., 382 (2011), 180-203. doi: 10.1016/j.jmaa.2011.04.046. [25] _______, The nature of Hopf bifurcation for the Gompertz model with delays, Math. and Comp. Modelling, 54 (2011), 2183-2198. doi: 10.1016/j.mcm.2011.05.027. [26] A. Świerniak, Comparison of six models of antiangiogenic therapy, Appl. Math., 36 (2009), 333-348. [27] A. Świerniak, A. Gala, A. Gandolfi and A. d'Onofrio, Optimalization of anti-angiogenic therapy as optimal control problem, in "Proc: IASTED Biomechanics 2006" Actapress, (2006). [28] H. Ch. Wu, Ch. T. Huang and D. K. Chang, Anti-angiogenic therapeutic drugs for treatment of human cancer, J. Cancer Mol. 4 (2008), 37-45.

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##### References:
 [1] Z. Agur, L. Arakelyan, P. Daugulis and Y. Ginosar, Hopf point analysis for angiogenesis models, Discrete Contin. Dyn. Syst. B, 4 (2004), 29-38. [2] L. Arakelyan, Y. Merbl and Z. Agur, Vessel maturation effects on tumour growth: validation of a computer model in implanted human ovarian carcinoma spheroids, European J. Cancer, 41 (2005), 159-167. [3] L. Arakelyan, V. Vainstein and Z. Agur, A computer algorithm describing the process of vessel formation and maturation, and its use for predicting the effects of anti-angiogenic and anti-maturation therapy on vascular tumor growth, Angiogenesis, 5 (2002), 203-214. [4] M. Bodnar and U. Foryś, Angiogenesis model with carrying capacity depending on vessel density, J. Biol. Sys., 17 (2009), 1-25. doi: 10.1142/S0218339009002739. [5] K. L. Cooke and P. van den Driessche, On zeroes of some transcendental equations, Funkcj. Ekvacioj, 29 (1986), 77-90. [6] A. d'Onofrio and A. Gandolfi, Tumour eradication by antiangiogenic therapy: analysis and extensions of the model by Hahnfeldt et al. (1999), Math. Biosci., 191 (2004), 159-184. doi: 10.1016/j.mbs.2004.06.003. [7] _______, The response to antiangiogenic anticancer drugs that inhibit endothelial cell proliferation, Applied Mathematics and Computation, 181 (2006), 1155-1162. [8] _______, A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy, Math. Med. Biol., 26 (2009), 63-95. [9] A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On optimal delivery of combination therapy for tumors, Math. Biosci. Eng., 222 (2009), 13-26. doi: 10.1016/j.mbs.2009.08.004. [10] J. M. L. Ebos and R. S. Kerbel, Antiangiogenic therapy: Impact on invasion, disease progression, and metastasis,, Nat. Rev. Clin. Oncol., 8 (): 1. [11] U. Foryś, Biological delay systems and the {Mikhailov criterion of stability}, J. Biol. Sys., 12 (2004), 45-60. doi: 10.1142/S0218339004001014. [12] U. Foryś, Y. Kheifetz and Y. Kogan, Critical point analysis for three-variable cancer angiogenesis model, Math. Biosci. Eng., 2 (2005), 511-525. [13] M. Gałach, Dynamics of the tumor-immune system competition - the effect of time delay, Int J Appl Math Comput Sci, 3 (2003), 395-406. [14] A. Gilead and M. Neeman, Dynamic remodeling of the vascular bed precedes tumor growth: MLS ovarian carcinoma spheroids implanted in nude mice, Neoplasia, 1 (1999), 226-230. doi: 10.1038/sj.neo.7900032. [15] P. Hahnfeldt, D. Panigrahy, J. Folkman and L. Hlatky, Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy, Cancer Res., 59 (1999), 4770-4775 (eng). [16] J. K. Hale, "Theory of Functional Differential Equations," Springer, New York, 1977. [17] J. K. Hale and S. M. V. Lunel, "Introduction to Functional Differential Equations," Springer, New York, 1993. [18] S. J. Holash, G. D. Wiegandand and G. D. Yancopoulos, New model of tumour angiogenesis: Dynamic balance between vessel regression andgrowth mediated by angiopoietins and VEGF, Oncogene, 18 (1999), 5356-5362. [19] R. K. Jain, Normalization of tumor vasculature: an emerging concept in antiangiogenic therapy, Science, 307 (2005), 58-62 (eng). [20] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Academic Press Inc., 1993. [21] V. A. Kuznetzov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunologenic tumors: Parameters estimation and global bifurcation analysis, Bull Math Biol, 56 (1994), 295-321. [22] U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem, SIAM J. Control Optim., 46 (2007), 1052-1079. [23] _______, Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis, J. Theor. Biol., 252 (2008), 295-312. [24] M. J. Piotrowska and U. Foryś, Analysis of the Hopf bifurcation for the family of angiogenesis models, J. Math. Anal. Appl., 382 (2011), 180-203. doi: 10.1016/j.jmaa.2011.04.046. [25] _______, The nature of Hopf bifurcation for the Gompertz model with delays, Math. and Comp. Modelling, 54 (2011), 2183-2198. doi: 10.1016/j.mcm.2011.05.027. [26] A. Świerniak, Comparison of six models of antiangiogenic therapy, Appl. Math., 36 (2009), 333-348. [27] A. Świerniak, A. Gala, A. Gandolfi and A. d'Onofrio, Optimalization of anti-angiogenic therapy as optimal control problem, in "Proc: IASTED Biomechanics 2006" Actapress, (2006). [28] H. Ch. Wu, Ch. T. Huang and D. K. Chang, Anti-angiogenic therapeutic drugs for treatment of human cancer, J. Cancer Mol. 4 (2008), 37-45.
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