Dynamical properties and tumor clearance conditions for a nine-dimensional model of bladder cancer immunotherapy

K. E.  Starkov; Svetlana Bunimovich-Mendrazitsky

doi:10.3934/mbe.2016030

Article Contents

2016, Volume 13, Issue 5: 1059-1075. Doi: 10.3934/mbe.2016030

This issue Previous Article Development of a computational model of glucose toxicity in the progression of diabetes mellitus Next Article A two-sex matrix population model to represent harem structure

Dynamical properties and tumor clearance conditions for a nine-dimensional model of bladder cancer immunotherapy

K. E. Starkov^1, and
Svetlana Bunimovich-Mendrazitsky^2,

1.
Instituto Politecnico Nacional, CITEDI, Avenida IPN N 1310, Nueva Tijuana, Tijuana, BC 22435
2.
Department of Computer Science and Mathematics, Ariel University Center of Samaria, Ariel, 40700

Received: October 2015

Revised: March 2016

Published: October 2016

Abstract / Introduction Related Papers Cited by

Abstract

Understanding the global interaction dynamics between tumor and the immune system plays a key role in the advancement of cancer therapy. Bunimovich-Mendrazitsky et al. (2015) developed a mathematical model for the study of the immune system response to combined therapy for bladder cancer with Bacillus Calmette-Guérin (BCG) and interleukin-2 (IL-2) . We utilized a mathematical approach for bladder cancer treatment model for derivation of ultimate upper and lower bounds and proving dissipativity property in the sense of Levinson. Furthermore, tumor clearance conditions for BCG treatment of bladder cancer are presented. Our method is based on localization of compact invariant sets and may be exploited for a prediction of the cells populations dynamics involved into the model.

Keywords:

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

Related Papers

Cited by

References

[1]	S. Bunimovich-Mendrazitsky, E. Shochat and L. Stone, Mathematical model of BCG immunotherapy in superficial bladder cancer, Bull. Math. Biol., 69 (2007), 1847-1870.doi: 10.1007/s11538-007-9195-z.
[2]	S. Bunimovich-Mendrazitsky, H. M. Byrne and L. Stone, Mathematical model of pulsed immunotherapy for superficial bladder cancer, Bull. Math. Biol., 70 (2008), 2055-2076.doi: 10.1007/s11538-008-9344-z.
[3]	S. Bunimovich-Mendrazitsky, S. Halachmi and N. Kronik, Improving Bacillus Calmette Guerin (BCG) immunotherapy for bladder cancer by adding Interleukin-2 (IL-2): A mathematical model, Math. Med. Biol., 33 (2016), 159-188.doi: 10.1093/imammb/dqv007.
[4]	S. Bunimovich-Mendrazitsky and Y. Goltser, Use of quasi-normal form to examine stability of tumor-free equilibrium in a mathematical model of BCG treatment of bladder cancer, Math. Biosci. Eng., 8 (2011), 529-547.doi: 10.3934/mbe.2011.8.529.
[5]	V. A. Boichenko, G. A. Leonov and V. Reitmann, Dimension Theory for Ordinary Differential Equations, Teubner Wiesbaden, 2005.doi: 10.1007/978-3-322-80055-8.
[6]	D. Kirschner and J. Panetta, Modelling immunotherapy of the tumor-immune interaction, J. Math. Biol., 37 (1998), 235-252.
[7]	A. P. Krishchenko, Localization of invariant compact sets of dynamical systems, Differ. Equ., 41 (2005), 1669-1676.doi: 10.1007/s10625-006-0003-6.
[8]	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.
[9]	A. P. Krishchenko and K. E. Starkov, Localization analysis of compact invariant sets of multi-dimensional nonlinear systems and symmetrical prolongations, Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 1159-1165.doi: 10.1016/j.cnsns.2009.05.068.
[10]	A. Morales, D. Eidinger and A. W. Bruce, Intracavity Bacillus Calmette-Guérin in the treatment of superficial bladder tumors, J. Urol., 116 (1976), 180-183.
[11]	M. R. Owen and J. A. Sherratt, Modelling the macrophage invasion of tumors: Effects on growth and composition, IMA J. Appl. Math., 15 (1998), 165-185.
[12]	K. E. Starkov, Compact invariant sets of the Bianchi VIII and Bianchi IX Hamiltonian systems, Phys. Lett. A, 375 (2011), 3184-3187.doi: 10.1016/j.physleta.2011.06.064.
[13]	K. E. Starkov, Bounding a domain that contains all compact invariant sets of the Bloch system, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), 1037-1042.doi: 10.1142/S0218127409023457.
[14]	K. E. Starkov, Bounds for compact invariant sets of the system describing dynamics of the nuclear spin generator, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 2565-2570.doi: 10.1016/j.cnsns.2008.08.005.
[15]	K. E. Starkov and L. N. Coria, Global dynamics of the Kirschner-Panetta model for the tumor immunotherapy, Nonlinear Anal. Real World Appl., 14 (2013), 1425-1433.doi: 10.1016/j.nonrwa.2012.10.006.
[16]	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.
[17]	K. E. Starkov and D. Gamboa, Localization of compact invariant sets and global stability in analysis of one tumor growth model, Math. Methods Appl. Sci, 37 (2014), 2854-2863.doi: 10.1002/mma.3023.
[18]	J. T. Wu, H. M. Byrne, D. H. Kirn and L. M. Wein, Modeling and analysis of a virus that replicates selectively in tumor cells, Bull. Math. Biol., 63 (2001), 731-768.doi: 10.1006/bulm.2001.0245.