American Institute of Mathematical Sciences

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

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

Marek Bodnar 1, , Monika Joanna Piotrowska 1, , Urszula Foryś 1, and  Ewa Nizińska 1,

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.
Keywords: angiogenesis, stability analysis, tumour growth., Hopf bifurcation, Delay differential equations.
Mathematics Subject Classification: Primary: 34K11, 34K13, 34K18, 34K20, 49K25; Secondary: 92B05, 93C2.
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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[4]

M. Bodnar and U. Foryś, Angiogenesis model with carrying capacity depending on vessel density,, J. Biol. Sys., 17 (2009), 1.  doi: 10.1142/S0218339009002739.  Google Scholar

[5]

K. L. Cooke and P. van den Driessche, On zeroes of some transcendental equations,, Funkcj. Ekvacioj, 29 (1986), 77.   Google Scholar

[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.  doi: 10.1016/j.mbs.2004.06.003.  Google Scholar

[7]

_______, The response to antiangiogenic anticancer drugs that inhibit endothelial cell proliferation,, Applied Mathematics and Computation, 181 (2006), 1155.   Google Scholar

[8]

_______, A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy,, Math. Med. Biol., 26 (2009), 63.   Google Scholar

[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.  doi: 10.1016/j.mbs.2009.08.004.  Google Scholar

[10]

J. M. L. Ebos and R. S. Kerbel, Antiangiogenic therapy: Impact on invasion, disease progression, and metastasis,, Nat. Rev. Clin. Oncol., 8 (): 1.   Google Scholar

[11]

U. Foryś, Biological delay systems and the {Mikhailov criterion of stability},, J. Biol. Sys., 12 (2004), 45.  doi: 10.1142/S0218339004001014.  Google Scholar

[12]

U. Foryś, Y. Kheifetz and Y. Kogan, Critical point analysis for three-variable cancer angiogenesis model,, Math. Biosci. Eng., 2 (2005), 511.   Google Scholar

[13]

M. Gałach, Dynamics of the tumor-immune system competition - the effect of time delay,, Int J Appl Math Comput Sci, 3 (2003), 395.   Google Scholar

[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.  doi: 10.1038/sj.neo.7900032.  Google Scholar

[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.   Google Scholar

[16]

J. K. Hale, "Theory of Functional Differential Equations,", Springer, (1977).   Google Scholar

[17]

J. K. Hale and S. M. V. Lunel, "Introduction to Functional Differential Equations,", Springer, (1993).   Google Scholar

[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.   Google Scholar

[19]

R. K. Jain, Normalization of tumor vasculature: an emerging concept in antiangiogenic therapy,, Science, 307 (2005), 58.   Google Scholar

[20]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Academic Press Inc., (1993).   Google Scholar

[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.   Google Scholar

[22]

U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem,, SIAM J. Control Optim., 46 (2007), 1052.   Google Scholar

[23]

_______, Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis,, J. Theor. Biol., 252 (2008), 295.   Google Scholar

[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.  doi: 10.1016/j.jmaa.2011.04.046.  Google Scholar

[25]

_______, The nature of Hopf bifurcation for the Gompertz model with delays,, Math. and Comp. Modelling, 54 (2011), 2183.  doi: 10.1016/j.mcm.2011.05.027.  Google Scholar

[26]

A. Świerniak, Comparison of six models of antiangiogenic therapy,, Appl. Math., 36 (2009), 333.   Google Scholar

[27]

A. Świerniak, A. Gala, A. Gandolfi and A. d'Onofrio, Optimalization of anti-angiogenic therapy as optimal control problem,, in, (2006).   Google Scholar

[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), 4 (2008), 37.   Google Scholar

show all references

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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[4]

M. Bodnar and U. Foryś, Angiogenesis model with carrying capacity depending on vessel density,, J. Biol. Sys., 17 (2009), 1.  doi: 10.1142/S0218339009002739.  Google Scholar

[5]

K. L. Cooke and P. van den Driessche, On zeroes of some transcendental equations,, Funkcj. Ekvacioj, 29 (1986), 77.   Google Scholar

[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.  doi: 10.1016/j.mbs.2004.06.003.  Google Scholar

[7]

_______, The response to antiangiogenic anticancer drugs that inhibit endothelial cell proliferation,, Applied Mathematics and Computation, 181 (2006), 1155.   Google Scholar

[8]

_______, A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy,, Math. Med. Biol., 26 (2009), 63.   Google Scholar

[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.  doi: 10.1016/j.mbs.2009.08.004.  Google Scholar

[10]

J. M. L. Ebos and R. S. Kerbel, Antiangiogenic therapy: Impact on invasion, disease progression, and metastasis,, Nat. Rev. Clin. Oncol., 8 (): 1.   Google Scholar

[11]

U. Foryś, Biological delay systems and the {Mikhailov criterion of stability},, J. Biol. Sys., 12 (2004), 45.  doi: 10.1142/S0218339004001014.  Google Scholar

[12]

U. Foryś, Y. Kheifetz and Y. Kogan, Critical point analysis for three-variable cancer angiogenesis model,, Math. Biosci. Eng., 2 (2005), 511.   Google Scholar

[13]

M. Gałach, Dynamics of the tumor-immune system competition - the effect of time delay,, Int J Appl Math Comput Sci, 3 (2003), 395.   Google Scholar

[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.  doi: 10.1038/sj.neo.7900032.  Google Scholar

[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.   Google Scholar

[16]

J. K. Hale, "Theory of Functional Differential Equations,", Springer, (1977).   Google Scholar

[17]

J. K. Hale and S. M. V. Lunel, "Introduction to Functional Differential Equations,", Springer, (1993).   Google Scholar

[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.   Google Scholar

[19]

R. K. Jain, Normalization of tumor vasculature: an emerging concept in antiangiogenic therapy,, Science, 307 (2005), 58.   Google Scholar

[20]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Academic Press Inc., (1993).   Google Scholar

[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.   Google Scholar

[22]

U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem,, SIAM J. Control Optim., 46 (2007), 1052.   Google Scholar

[23]

_______, Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis,, J. Theor. Biol., 252 (2008), 295.   Google Scholar

[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.  doi: 10.1016/j.jmaa.2011.04.046.  Google Scholar

[25]

_______, The nature of Hopf bifurcation for the Gompertz model with delays,, Math. and Comp. Modelling, 54 (2011), 2183.  doi: 10.1016/j.mcm.2011.05.027.  Google Scholar

[26]

A. Świerniak, Comparison of six models of antiangiogenic therapy,, Appl. Math., 36 (2009), 333.   Google Scholar

[27]

A. Świerniak, A. Gala, A. Gandolfi and A. d'Onofrio, Optimalization of anti-angiogenic therapy as optimal control problem,, in, (2006).   Google Scholar

[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), 4 (2008), 37.   Google Scholar

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