American Institute of Mathematical Sciences

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2016, 13(5): 1059-1075. doi: 10.3934/mbe.2016030

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, Mexico
2. 

Department of Computer Science and Mathematics, Ariel University Center of Samaria, Ariel, 40700

Received  October 2015 Revised  March 2016 Published  October 2016

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: IL-2, BCG, localization, ultimate dynamics, compact invariant set., Dynamic modelling, combined therapy.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: K. E. Starkov, Svetlana Bunimovich-Mendrazitsky. Dynamical properties and tumor clearance conditions for a nine-dimensional model of bladder cancer immunotherapy. Mathematical Biosciences & Engineering, 2016, 13 (5) : 1059-1075. doi: 10.3934/mbe.2016030
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.

show all references

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.

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