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Distributed delays in a hybrid model of tumorImmune system interplay
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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, 02097 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), 2938. 
[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), 159167. 
[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 antiangiogenic and antimaturation therapy on vascular tumor growth, Angiogenesis, 5 (2002), 203214. 
[4] 
M. Bodnar and U. Foryś, Angiogenesis model with carrying capacity depending on vessel density, J. Biol. Sys., 17 (2009), 125. doi: 10.1142/S0218339009002739. 
[5] 
K. L. Cooke and P. van den Driessche, On zeroes of some transcendental equations, Funkcj. Ekvacioj, 29 (1986), 7790. 
[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), 159184. 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), 11551162. 
[8] 
_______, A family of models of angiogenesis and antiangiogenesis anticancer therapy, Math. Med. Biol., 26 (2009), 6395. 
[9] 
A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On optimal delivery of combination therapy for tumors, Math. Biosci. Eng., 222 (2009), 1326. 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, 112. 
[11] 
U. Foryś, Biological delay systems and the {Mikhailov criterion of stability}, J. Biol. Sys., 12 (2004), 4560. doi: 10.1142/S0218339004001014. 
[12] 
U. Foryś, Y. Kheifetz and Y. Kogan, Critical point analysis for threevariable cancer angiogenesis model, Math. Biosci. Eng., 2 (2005), 511525. 
[13] 
M. Gałach, Dynamics of the tumorimmune system competition  the effect of time delay, Int J Appl Math Comput Sci, 3 (2003), 395406. 
[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), 226230. 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), 47704775 (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), 53565362. 
[19] 
R. K. Jain, Normalization of tumor vasculature: an emerging concept in antiangiogenic therapy, Science, 307 (2005), 5862 (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), 295321. 
[22] 
U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem, SIAM J. Control Optim., 46 (2007), 10521079. 
[23] 
_______, Optimal and suboptimal protocols for a class of mathematical models of tumor antiangiogenesis, J. Theor. Biol., 252 (2008), 295312. 
[24] 
M. J. Piotrowska and U. Foryś, Analysis of the Hopf bifurcation for the family of angiogenesis models, J. Math. Anal. Appl., 382 (2011), 180203. 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), 21832198. doi: 10.1016/j.mcm.2011.05.027. 
[26] 
A. Świerniak, Comparison of six models of antiangiogenic therapy, Appl. Math., 36 (2009), 333348. 
[27] 
A. Świerniak, A. Gala, A. Gandolfi and A. d'Onofrio, Optimalization of antiangiogenic therapy as optimal control problem, in "Proc: IASTED Biomechanics 2006" Actapress, (2006). 
[28] 
H. Ch. Wu, Ch. T. Huang and D. K. Chang, Antiangiogenic therapeutic drugs for treatment of human cancer, J. Cancer Mol. 4 (2008), 3745. 
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), 2938. 
[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), 159167. 
[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 antiangiogenic and antimaturation therapy on vascular tumor growth, Angiogenesis, 5 (2002), 203214. 
[4] 
M. Bodnar and U. Foryś, Angiogenesis model with carrying capacity depending on vessel density, J. Biol. Sys., 17 (2009), 125. doi: 10.1142/S0218339009002739. 
[5] 
K. L. Cooke and P. van den Driessche, On zeroes of some transcendental equations, Funkcj. Ekvacioj, 29 (1986), 7790. 
[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), 159184. 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), 11551162. 
[8] 
_______, A family of models of angiogenesis and antiangiogenesis anticancer therapy, Math. Med. Biol., 26 (2009), 6395. 
[9] 
A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On optimal delivery of combination therapy for tumors, Math. Biosci. Eng., 222 (2009), 1326. 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, 112. 
[11] 
U. Foryś, Biological delay systems and the {Mikhailov criterion of stability}, J. Biol. Sys., 12 (2004), 4560. doi: 10.1142/S0218339004001014. 
[12] 
U. Foryś, Y. Kheifetz and Y. Kogan, Critical point analysis for threevariable cancer angiogenesis model, Math. Biosci. Eng., 2 (2005), 511525. 
[13] 
M. Gałach, Dynamics of the tumorimmune system competition  the effect of time delay, Int J Appl Math Comput Sci, 3 (2003), 395406. 
[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), 226230. 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), 47704775 (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), 53565362. 
[19] 
R. K. Jain, Normalization of tumor vasculature: an emerging concept in antiangiogenic therapy, Science, 307 (2005), 5862 (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), 295321. 
[22] 
U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem, SIAM J. Control Optim., 46 (2007), 10521079. 
[23] 
_______, Optimal and suboptimal protocols for a class of mathematical models of tumor antiangiogenesis, J. Theor. Biol., 252 (2008), 295312. 
[24] 
M. J. Piotrowska and U. Foryś, Analysis of the Hopf bifurcation for the family of angiogenesis models, J. Math. Anal. Appl., 382 (2011), 180203. 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), 21832198. doi: 10.1016/j.mcm.2011.05.027. 
[26] 
A. Świerniak, Comparison of six models of antiangiogenic therapy, Appl. Math., 36 (2009), 333348. 
[27] 
A. Świerniak, A. Gala, A. Gandolfi and A. d'Onofrio, Optimalization of antiangiogenic therapy as optimal control problem, in "Proc: IASTED Biomechanics 2006" Actapress, (2006). 
[28] 
H. Ch. Wu, Ch. T. Huang and D. K. Chang, Antiangiogenic therapeutic drugs for treatment of human cancer, J. Cancer Mol. 4 (2008), 3745. 
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