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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 |
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.
|
[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. |
[5] |
K. L. Cooke and P. van den Driessche, On zeroes of some transcendental equations,, Funkcj. Ekvacioj, 29 (1986), 77.
|
[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. |
[7] |
_______, The response to antiangiogenic anticancer drugs that inhibit endothelial cell proliferation,, Applied Mathematics and Computation, 181 (2006), 1155.
|
[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. |
[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. |
[12] |
U. Foryś, Y. Kheifetz and Y. Kogan, Critical point analysis for three-variable cancer angiogenesis model,, Math. Biosci. Eng., 2 (2005), 511.
|
[13] |
M. Gałach, Dynamics of the tumor-immune system competition - the effect of time delay,, Int J Appl Math Comput Sci, 3 (2003), 395.
|
[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. |
[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).
|
[17] |
J. K. Hale and S. M. V. Lunel, "Introduction to Functional Differential Equations,", Springer, (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. 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).
|
[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.
|
[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. |
[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. |
[26] |
A. Świerniak, Comparison of six models of antiangiogenic therapy,, Appl. Math., 36 (2009), 333.
|
[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.
|
[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. |
[5] |
K. L. Cooke and P. van den Driessche, On zeroes of some transcendental equations,, Funkcj. Ekvacioj, 29 (1986), 77.
|
[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. |
[7] |
_______, The response to antiangiogenic anticancer drugs that inhibit endothelial cell proliferation,, Applied Mathematics and Computation, 181 (2006), 1155.
|
[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. |
[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. |
[12] |
U. Foryś, Y. Kheifetz and Y. Kogan, Critical point analysis for three-variable cancer angiogenesis model,, Math. Biosci. Eng., 2 (2005), 511.
|
[13] |
M. Gałach, Dynamics of the tumor-immune system competition - the effect of time delay,, Int J Appl Math Comput Sci, 3 (2003), 395.
|
[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. |
[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).
|
[17] |
J. K. Hale and S. M. V. Lunel, "Introduction to Functional Differential Equations,", Springer, (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. 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).
|
[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.
|
[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. |
[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. |
[26] |
A. Świerniak, Comparison of six models of antiangiogenic therapy,, Appl. Math., 36 (2009), 333.
|
[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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