October  2018, 15(5): 1243-1254. doi: 10.3934/mbe.2018057

The four-dimensional Kirschner-Panetta type cancer model: How to obtain tumor eradication?

1. 

Bauman Moscow State Technical University, 2-aya Baumanskaya ul., 5, Moscow 105005, Russia

2. 

Instituto Politecnico Nacional, CITEDI, Avenida IPN N 1310, Nueva Tijuana, Tijuana 22510, B.C., Mexico

* Corresponding author: kstarkov@ipn.mx; konstarkov@hotmail.com

Received  January 03, 2017 Revised  March 06, 2018 Published  May 2018

In this paper we examine ultimate dynamics of the four-dimensional model describing interactions between tumor cells, effector immune cells, interleukin -2 and transforming growth factor-beta. This model was elaborated by Arciero et al. and is obtained from the Kirschner-Panetta type model by introducing two various treatments. We provide ultimate upper bounds for all variables of this model and two lower bounds and, besides, study when dynamics of this model possesses a global attracting set. The nonexistence conditions of compact invariant sets are derived. We obtain bounds for treatment parameters $s_{1, 2}$ under which all trajectories in the positive orthant tend to the tumor-free equilibrium point. Conditions imposed on $s_{1, 2}$ under which the tumor population persists are presented as well. Finally, we compare tumor eradication/ persistence bounds and discuss our results.

Citation: Alexander P. Krishchenko, Konstantin E. Starkov. The four-dimensional Kirschner-Panetta type cancer model: How to obtain tumor eradication?. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1243-1254. doi: 10.3934/mbe.2018057
References:
[1]

J. C. ArcieroT. L. Jackson and D. E. Kirschner, A mathematical model of tumor-immune evasion and siRNA treatment, Discrete Contin. Dynamic. Syst. Ser. B, 4 (2004), 39-58.   Google Scholar

[2]

D. Kirschner and J. Panetta, Modelling immunotherapy of the tumor-immune interaction, J. Math. Biol., 37 (1998), 235-252.   Google Scholar

[3]

D. Kirschner and A. V. Tsygvintsev, On the global dynamics of a model for tumor immunotherapy, Math. Biosci. Engin., 6 (2009), 573-583.  doi: 10.3934/mbe.2009.6.573.  Google Scholar

[4]

A. P. Krishchenko, Estimations of domains with cycles, Comput. & Math. Appl., 34 (1997), 325-332.  doi: 10.1016/S0898-1221(97)00130-2.  Google Scholar

[5]

A. P. Krishchenko, Localization of invariant compact sets of dynamical systems, Differential Equations, 41 (2005), 1669-1676.  doi: 10.1007/s10625-006-0003-6.  Google Scholar

[6]

A. P. Krishchenko and K. E. Starkov, Localization of compact invariant sets of the Lorenz system, Phys. Lett. A, 353 (2006), 383-388.  doi: 10.1016/j.physleta.2005.12.104.  Google Scholar

[7]

A. P. Krishchenko and K. E. Starkov, On the global dynamics of a chronic myelogenous leukemia model, Commun. Nonlin. Sci. Numer. Simul., 33 (2016), 174-183.  doi: 10.1016/j.cnsns.2015.10.001.  Google Scholar

[8]

F. Salazar-Onfray, Interleukin-10: A cytokine used by tumors to escape immunosurveillance, Medical Oncology, 16 (1999), 86-94.  doi: 10.1007/BF02785841.  Google Scholar

[9]

K. E. Starkov, On dynamic tumor eradication conditions under combined chemical/anti-angiogenic therapies, Phys. Lett. A, 382 (2018), 387-393.  doi: 10.1016/j.physleta.2017.12.025.  Google Scholar

[10]

K. E. Starkov and S. Bunimovich-Mendrazitsky, Dynamical properties and tumor clearance conditions for a nine-dimensional model of bladder cancer immunotherapy, Math. Biosci. Eng., 13 (2016), 1059-1075.  doi: 10.3934/mbe.2016030.  Google Scholar

[11]

K. E. Starkov and D. Gamboa, Localization of compact invariant sets and global stability in analysis of one tumor growth model, Mathematical Methods in the Applied Sciences, 37 (2014), 2854-2863.  doi: 10.1002/mma.3023.  Google Scholar

[12]

K. E. Starkov and L. Jimenez Beristain, Dynamic analysis of the melanoma model: from cancer persistence to its eradication Internat. J. Bifur. Chaos Appl. Sci. Engrg. , 27 (2017), 1750151, 11pp. doi: 10.1142/S0218127417501516.  Google Scholar

[13]

K. E. Starkov and A. P. Krishchenko, On the global dynamics of one cancer tumor growth model, Commun. Nonlin. Sci. Numer. Simul., 19 (2014), 1486-1495.  doi: 10.1016/j.cnsns.2013.09.023.  Google Scholar

[14]

K. E. Starkov and A. P. Krishchenko, Ultimate dynamics of the Kirschner-Panetta model: Tumor eradication and related problems, Phys. Lett. A, 381 (2017), 3409-3416.  doi: 10.1016/j.physleta.2017.08.048.  Google Scholar

[15]

K. E. Starkov and A. Pogromsky, Global dynamics of the Owen-Sherratt model describing the tumor-macrophage interactions, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350020, 9pp.  doi: 10.1142/S021812741350020X.  Google Scholar

[16]

K. E. de Visser and W. M. Kast, Effects of TGF-ß on the immune system: Implications for cancer immunotherapy, Journal of Immunotherapy, 20 (1997), 165-177.   Google Scholar

[17]

S. Wojtowicz-Praga, Reversal of tumor-induced immunosuppression: A new approach to cancer therapy, Journal of Immunotherapy, 20 (1997), 165-177.  doi: 10.1097/00002371-199705000-00001.  Google Scholar

show all references

References:
[1]

J. C. ArcieroT. L. Jackson and D. E. Kirschner, A mathematical model of tumor-immune evasion and siRNA treatment, Discrete Contin. Dynamic. Syst. Ser. B, 4 (2004), 39-58.   Google Scholar

[2]

D. Kirschner and J. Panetta, Modelling immunotherapy of the tumor-immune interaction, J. Math. Biol., 37 (1998), 235-252.   Google Scholar

[3]

D. Kirschner and A. V. Tsygvintsev, On the global dynamics of a model for tumor immunotherapy, Math. Biosci. Engin., 6 (2009), 573-583.  doi: 10.3934/mbe.2009.6.573.  Google Scholar

[4]

A. P. Krishchenko, Estimations of domains with cycles, Comput. & Math. Appl., 34 (1997), 325-332.  doi: 10.1016/S0898-1221(97)00130-2.  Google Scholar

[5]

A. P. Krishchenko, Localization of invariant compact sets of dynamical systems, Differential Equations, 41 (2005), 1669-1676.  doi: 10.1007/s10625-006-0003-6.  Google Scholar

[6]

A. P. Krishchenko and K. E. Starkov, Localization of compact invariant sets of the Lorenz system, Phys. Lett. A, 353 (2006), 383-388.  doi: 10.1016/j.physleta.2005.12.104.  Google Scholar

[7]

A. P. Krishchenko and K. E. Starkov, On the global dynamics of a chronic myelogenous leukemia model, Commun. Nonlin. Sci. Numer. Simul., 33 (2016), 174-183.  doi: 10.1016/j.cnsns.2015.10.001.  Google Scholar

[8]

F. Salazar-Onfray, Interleukin-10: A cytokine used by tumors to escape immunosurveillance, Medical Oncology, 16 (1999), 86-94.  doi: 10.1007/BF02785841.  Google Scholar

[9]

K. E. Starkov, On dynamic tumor eradication conditions under combined chemical/anti-angiogenic therapies, Phys. Lett. A, 382 (2018), 387-393.  doi: 10.1016/j.physleta.2017.12.025.  Google Scholar

[10]

K. E. Starkov and S. Bunimovich-Mendrazitsky, Dynamical properties and tumor clearance conditions for a nine-dimensional model of bladder cancer immunotherapy, Math. Biosci. Eng., 13 (2016), 1059-1075.  doi: 10.3934/mbe.2016030.  Google Scholar

[11]

K. E. Starkov and D. Gamboa, Localization of compact invariant sets and global stability in analysis of one tumor growth model, Mathematical Methods in the Applied Sciences, 37 (2014), 2854-2863.  doi: 10.1002/mma.3023.  Google Scholar

[12]

K. E. Starkov and L. Jimenez Beristain, Dynamic analysis of the melanoma model: from cancer persistence to its eradication Internat. J. Bifur. Chaos Appl. Sci. Engrg. , 27 (2017), 1750151, 11pp. doi: 10.1142/S0218127417501516.  Google Scholar

[13]

K. E. Starkov and A. P. Krishchenko, On the global dynamics of one cancer tumor growth model, Commun. Nonlin. Sci. Numer. Simul., 19 (2014), 1486-1495.  doi: 10.1016/j.cnsns.2013.09.023.  Google Scholar

[14]

K. E. Starkov and A. P. Krishchenko, Ultimate dynamics of the Kirschner-Panetta model: Tumor eradication and related problems, Phys. Lett. A, 381 (2017), 3409-3416.  doi: 10.1016/j.physleta.2017.08.048.  Google Scholar

[15]

K. E. Starkov and A. Pogromsky, Global dynamics of the Owen-Sherratt model describing the tumor-macrophage interactions, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350020, 9pp.  doi: 10.1142/S021812741350020X.  Google Scholar

[16]

K. E. de Visser and W. M. Kast, Effects of TGF-ß on the immune system: Implications for cancer immunotherapy, Journal of Immunotherapy, 20 (1997), 165-177.   Google Scholar

[17]

S. Wojtowicz-Praga, Reversal of tumor-induced immunosuppression: A new approach to cancer therapy, Journal of Immunotherapy, 20 (1997), 165-177.  doi: 10.1097/00002371-199705000-00001.  Google Scholar

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