American Institute of Mathematical Sciences

Advanced
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]

Bull. Math. Biol., 69 (2007), 1847-1870. doi: 10.1007/s11538-007-9195-z.  Google Scholar

[2]

Bull. Math. Biol., 70 (2008), 2055-2076. doi: 10.1007/s11538-008-9344-z.  Google Scholar

[3]

Math. Med. Biol., 33 (2016), 159-188. doi: 10.1093/imammb/dqv007.  Google Scholar

[4]

Math. Biosci. Eng., 8 (2011), 529-547. doi: 10.3934/mbe.2011.8.529.  Google Scholar

[5]

Teubner Wiesbaden, 2005. doi: 10.1007/978-3-322-80055-8.  Google Scholar

[6]

J. Math. Biol., 37 (1998), 235-252. Google Scholar

[7]

Differ. Equ., 41 (2005), 1669-1676. doi: 10.1007/s10625-006-0003-6.  Google Scholar

[8]

Phys. Lett. A, 353 (2006), 383-388. doi: 10.1016/j.physleta.2005.12.104.  Google Scholar

[9]

Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 1159-1165. doi: 10.1016/j.cnsns.2009.05.068.  Google Scholar

[10]

J. Urol., 116 (1976), 180-183. Google Scholar

[11]

IMA J. Appl. Math., 15 (1998), 165-185. Google Scholar

[12]

Phys. Lett. A, 375 (2011), 3184-3187. doi: 10.1016/j.physleta.2011.06.064.  Google Scholar

[13]

Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), 1037-1042. doi: 10.1142/S0218127409023457.  Google Scholar

[14]

Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 2565-2570. doi: 10.1016/j.cnsns.2008.08.005.  Google Scholar

[15]

Nonlinear Anal. Real World Appl., 14 (2013), 1425-1433. doi: 10.1016/j.nonrwa.2012.10.006.  Google Scholar

[16]

Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350020, 9pp. doi: 10.1142/S021812741350020X.  Google Scholar

[17]

Math. Methods Appl. Sci, 37 (2014), 2854-2863. doi: 10.1002/mma.3023.  Google Scholar

[18]

Bull. Math. Biol., 63 (2001), 731-768. doi: 10.1006/bulm.2001.0245.  Google Scholar

show all references

References:
[1]

Bull. Math. Biol., 69 (2007), 1847-1870. doi: 10.1007/s11538-007-9195-z.  Google Scholar

[2]

Bull. Math. Biol., 70 (2008), 2055-2076. doi: 10.1007/s11538-008-9344-z.  Google Scholar

[3]

Math. Med. Biol., 33 (2016), 159-188. doi: 10.1093/imammb/dqv007.  Google Scholar

[4]

Math. Biosci. Eng., 8 (2011), 529-547. doi: 10.3934/mbe.2011.8.529.  Google Scholar

[5]

Teubner Wiesbaden, 2005. doi: 10.1007/978-3-322-80055-8.  Google Scholar

[6]

J. Math. Biol., 37 (1998), 235-252. Google Scholar

[7]

Differ. Equ., 41 (2005), 1669-1676. doi: 10.1007/s10625-006-0003-6.  Google Scholar

[8]

Phys. Lett. A, 353 (2006), 383-388. doi: 10.1016/j.physleta.2005.12.104.  Google Scholar

[9]

Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 1159-1165. doi: 10.1016/j.cnsns.2009.05.068.  Google Scholar

[10]

J. Urol., 116 (1976), 180-183. Google Scholar

[11]

IMA J. Appl. Math., 15 (1998), 165-185. Google Scholar

[12]

Phys. Lett. A, 375 (2011), 3184-3187. doi: 10.1016/j.physleta.2011.06.064.  Google Scholar

[13]

Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), 1037-1042. doi: 10.1142/S0218127409023457.  Google Scholar

[14]

Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 2565-2570. doi: 10.1016/j.cnsns.2008.08.005.  Google Scholar

[15]

Nonlinear Anal. Real World Appl., 14 (2013), 1425-1433. doi: 10.1016/j.nonrwa.2012.10.006.  Google Scholar

[16]

Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013), 1350020, 9pp. doi: 10.1142/S021812741350020X.  Google Scholar

[17]

Math. Methods Appl. Sci, 37 (2014), 2854-2863. doi: 10.1002/mma.3023.  Google Scholar

[18]

Bull. Math. Biol., 63 (2001), 731-768. doi: 10.1006/bulm.2001.0245.  Google Scholar

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