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