2013, 10(3): 551-563. doi: 10.3934/mbe.2013.10.551

Gompertz model with delays and treatment: Mathematical analysis

1. 

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

Received  June 2012 Revised  August 2012 Published  April 2013

In this paper we study the delayed Gompertz model, as a typical model of tumor growth, with a term describing external interference that can reflect a treatment, e.g. chemotherapy. We mainly consider two types of delayed models, the one with the delay introduced in the per capita growth rate (we call it the single delayed model) and the other with the delay introduced in the net growth rate (the double delayed model). We focus on stability and possible stability switches with increasing delay for the positive steady state. Moreover, we study a Hopf bifurcation, including stability of arising periodic solutions for a constant treatment. The analytical results are extended by numerical simulations for a pharmacokinetic treatment function.
Citation: Marek Bodnar, Monika Joanna Piotrowska, Urszula Foryś. Gompertz model with delays and treatment: Mathematical analysis. Mathematical Biosciences & Engineering, 2013, 10 (3) : 551-563. doi: 10.3934/mbe.2013.10.551
References:
[1]

M. Bodnar, The nonnegativity of the solutions of delay differential equations,, Appl. Math. Lett., 13 (2000), 91. doi: 10.1016/S0893-9659(00)00061-6. Google Scholar

[2]

M. Bodnar and U. Foryś, Three types of simple DDE's describing tumor growth,, J. Biol. Sys., 15 (2007), 1. doi: 10.1142/S0218339007002313. Google Scholar

[3]

O. Diekmann, S. van Giles and S.M.V. Lunel, "Delay Equations,", Springer-Verlag, (1995). doi: 10.1007/978-1-4612-4206-2. Google Scholar

[4]

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

[5]

A. d'Onofrio and A. Gandolfi, A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy,, Math. Med. Biol., 26 (2009), 63. doi: 10.1093/imammb/dqn024. Google Scholar

[6]

A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On optimal delivery of combination therapy for tumors,, Math. Biosci., 222 (2009), 13. doi: 10.1016/j.mbs.2009.08.004. Google Scholar

[7]

U. Foryś, J. Poleszczuk and T. Liu, Logistic tumor growth with delay and impulsive treatment,, Accepted for Math. Pop. Studies., (). Google Scholar

[8]

G. Gompertz, On the nature of the function expressive of the law of human mortality, and on the new mode of determining the value of life contingencies,, Philos. Trans. R. Soc. London, 115 (1825), 513. Google Scholar

[9]

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

[10]

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

[11]

G. E. Hutchinson, Circular casual systems in ecology,, Ann. N. Y. Acad. Sci., 50 (1948), 221. Google Scholar

[12]

U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem,, SIAM J. Control Optim., 46 (2007), 1052. doi: 10.1137/060665294. Google Scholar

[13]

U. Ledzewicz and H. Schättler, Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis,, J. Theor. Biol., 252 (2008), 295. doi: 10.1016/j.jtbi.2008.02.014. Google Scholar

[14]

J. D. Murray, "Mathematical Biology: {I.} An Introduction,", Springer, (2007). Google Scholar

[15]

M. J. Piotrowska, M. Bodnar and U. Foryś, Logistic equation with treatment function and discrete delays,, (submitted)., (). Google Scholar

[16]

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

[17]

M. J. Piotrowska and U. Foryś, 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

[18]

J. Poleszczuk, M. Bodnar and U. Foryś, New approach to modeling of antiangiogenic treatment on the basis of Hahnfeldt et al. model,, Math. Biosci. Eng., 8 (2011), 591. doi: 10.3934/mbe.2011.8.591. Google Scholar

[19]

R. Schuster and H. Schuster, Reconstruction models for the Ehrlich Ascites tumor for the mouse,, in, (1995), 335. Google Scholar

show all references

References:
[1]

M. Bodnar, The nonnegativity of the solutions of delay differential equations,, Appl. Math. Lett., 13 (2000), 91. doi: 10.1016/S0893-9659(00)00061-6. Google Scholar

[2]

M. Bodnar and U. Foryś, Three types of simple DDE's describing tumor growth,, J. Biol. Sys., 15 (2007), 1. doi: 10.1142/S0218339007002313. Google Scholar

[3]

O. Diekmann, S. van Giles and S.M.V. Lunel, "Delay Equations,", Springer-Verlag, (1995). doi: 10.1007/978-1-4612-4206-2. Google Scholar

[4]

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

[5]

A. d'Onofrio and A. Gandolfi, A family of models of angiogenesis and anti-angiogenesis anti-cancer therapy,, Math. Med. Biol., 26 (2009), 63. doi: 10.1093/imammb/dqn024. Google Scholar

[6]

A. d'Onofrio, U. Ledzewicz, H. Maurer and H. Schättler, On optimal delivery of combination therapy for tumors,, Math. Biosci., 222 (2009), 13. doi: 10.1016/j.mbs.2009.08.004. Google Scholar

[7]

U. Foryś, J. Poleszczuk and T. Liu, Logistic tumor growth with delay and impulsive treatment,, Accepted for Math. Pop. Studies., (). Google Scholar

[8]

G. Gompertz, On the nature of the function expressive of the law of human mortality, and on the new mode of determining the value of life contingencies,, Philos. Trans. R. Soc. London, 115 (1825), 513. Google Scholar

[9]

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

[10]

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

[11]

G. E. Hutchinson, Circular casual systems in ecology,, Ann. N. Y. Acad. Sci., 50 (1948), 221. Google Scholar

[12]

U. Ledzewicz and H. Schättler, Antiangiogenic therapy in cancer treatment as an optimal control problem,, SIAM J. Control Optim., 46 (2007), 1052. doi: 10.1137/060665294. Google Scholar

[13]

U. Ledzewicz and H. Schättler, Optimal and suboptimal protocols for a class of mathematical models of tumor anti-angiogenesis,, J. Theor. Biol., 252 (2008), 295. doi: 10.1016/j.jtbi.2008.02.014. Google Scholar

[14]

J. D. Murray, "Mathematical Biology: {I.} An Introduction,", Springer, (2007). Google Scholar

[15]

M. J. Piotrowska, M. Bodnar and U. Foryś, Logistic equation with treatment function and discrete delays,, (submitted)., (). Google Scholar

[16]

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

[17]

M. J. Piotrowska and U. Foryś, 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

[18]

J. Poleszczuk, M. Bodnar and U. Foryś, New approach to modeling of antiangiogenic treatment on the basis of Hahnfeldt et al. model,, Math. Biosci. Eng., 8 (2011), 591. doi: 10.3934/mbe.2011.8.591. Google Scholar

[19]

R. Schuster and H. Schuster, Reconstruction models for the Ehrlich Ascites tumor for the mouse,, in, (1995), 335. Google Scholar

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