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

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.
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.  doi: 10.1007/s11538-007-9195-z.  Google Scholar

[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.  doi: 10.1007/s11538-008-9344-z.  Google Scholar

[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.  doi: 10.1093/imammb/dqv007.  Google Scholar

[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.  doi: 10.3934/mbe.2011.8.529.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[8]

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

[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.  doi: 10.1016/j.cnsns.2009.05.068.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[12]

K. E. Starkov, Compact invariant sets of the Bianchi VIII and Bianchi IX Hamiltonian systems,, Phys. Lett. A, 375 (2011), 3184.  doi: 10.1016/j.physleta.2011.06.064.  Google Scholar

[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.  doi: 10.1142/S0218127409023457.  Google Scholar

[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.  doi: 10.1016/j.cnsns.2008.08.005.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2012.10.006.  Google Scholar

[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).  doi: 10.1142/S021812741350020X.  Google Scholar

[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.  doi: 10.1002/mma.3023.  Google Scholar

[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.  doi: 10.1006/bulm.2001.0245.  Google Scholar

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.  doi: 10.1007/s11538-007-9195-z.  Google Scholar

[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.  doi: 10.1007/s11538-008-9344-z.  Google Scholar

[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.  doi: 10.1093/imammb/dqv007.  Google Scholar

[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.  doi: 10.3934/mbe.2011.8.529.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[8]

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

[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.  doi: 10.1016/j.cnsns.2009.05.068.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[12]

K. E. Starkov, Compact invariant sets of the Bianchi VIII and Bianchi IX Hamiltonian systems,, Phys. Lett. A, 375 (2011), 3184.  doi: 10.1016/j.physleta.2011.06.064.  Google Scholar

[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.  doi: 10.1142/S0218127409023457.  Google Scholar

[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.  doi: 10.1016/j.cnsns.2008.08.005.  Google Scholar

[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.  doi: 10.1016/j.nonrwa.2012.10.006.  Google Scholar

[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).  doi: 10.1142/S021812741350020X.  Google Scholar

[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.  doi: 10.1002/mma.3023.  Google Scholar

[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.  doi: 10.1006/bulm.2001.0245.  Google Scholar

[1]

Adam Glick, Antonio Mastroberardino. Combined therapy for treating solid tumors with chemotherapy and angiogenic inhibitors. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020343

[2]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378

[3]

Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336

[4]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[5]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002

[6]

Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443

[7]

Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169

[8]

Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266

[9]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301

[10]

Haixiang Yao, Ping Chen, Miao Zhang, Xun Li. Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020166

[11]

Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, 2021, 20 (1) : 427-448. doi: 10.3934/cpaa.2020275

[12]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031

[13]

Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 117-131. doi: 10.3934/jimo.2019102

[14]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

[15]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

[16]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

[17]

Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116

[18]

Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020426

[19]

Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164

[20]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (81)
  • HTML views (0)
  • Cited by (8)

[Back to Top]