Mathematical analysis of tumor-free equilibrium in BCG treatment with effective IL-2 infusion for bladder cancer model

Irina Volinsky; Svetlana Bunimovich-Mendrazitsky; Irina Volinsky; Svetlana Bunimovich-Mendrazitsky

doi:10.3934/math.2022896

AIMS Mathematics

2022, Volume 7, Issue 9: 16388-16406. doi: 10.3934/math.2022896

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Mathematical analysis of tumor-free equilibrium in BCG treatment with effective IL-2 infusion for bladder cancer model

Department of Mathematics, Ariel University, Ariel 4076414, Israel

Received: 12 April 2022 Revised: 03 June 2022 Accepted: 19 June 2022 Published: 06 July 2022
MSC : 34K20, 34D20

We present a theoretical study of bladder cancer treatment with Bacillus Calmette-Guerin (BCG) and interleukin 2 (IL-2) using a system biology approach to translate the treatment process into a mathematical model. We investigated the influence of IL-2 on effector cell proliferation, presented as a distributed feedback control in integral form. The variables in the system of Ordinary Differential Equations (ODE) are the main participants in the immune response after BCG instillations: BCG, immune cells, tumor cells infected with BCG, and non-infected with BCG. IL-2 was involved in the tumor-immune response without adding a new equation. We use the idea of reducing the system of integro-differential equations (IDE) to a system of ODE and examine the local stability analysis of the tumor-free equilibrium state of the model. A significant result of the model analysis is the requirements for the IL-2 dose and duration, depending on the treatment regimen and tumor growth. We proved that the BCG+IL-2 treatment protocol is more effective in this model, using the spectral radius method. Moreover, we introduced a parameter for individual control of IL-2 in each injection using the Cauchy matrix for the IDE system, and we obtained conditions under which this system would be exponentially stable in a tumor-free equilibrium.
- combined BCG + IL-2 treatment,
- system of integro-differential equations,
- exponential stability,
- feedback control, Cauchy matrix
Citation: Irina Volinsky, Svetlana Bunimovich-Mendrazitsky. Mathematical analysis of tumor-free equilibrium in BCG treatment with effective IL-2 infusion for bladder cancer model[J]. AIMS Mathematics, 2022, 7(9): 16388-16406. doi: 10.3934/math.2022896

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Abstract

We present a theoretical study of bladder cancer treatment with Bacillus Calmette-Guerin (BCG) and interleukin 2 (IL-2) using a system biology approach to translate the treatment process into a mathematical model. We investigated the influence of IL-2 on effector cell proliferation, presented as a distributed feedback control in integral form. The variables in the system of Ordinary Differential Equations (ODE) are the main participants in the immune response after BCG instillations: BCG, immune cells, tumor cells infected with BCG, and non-infected with BCG. IL-2 was involved in the tumor-immune response without adding a new equation. We use the idea of reducing the system of integro-differential equations (IDE) to a system of ODE and examine the local stability analysis of the tumor-free equilibrium state of the model. A significant result of the model analysis is the requirements for the IL-2 dose and duration, depending on the treatment regimen and tumor growth. We proved that the BCG+IL-2 treatment protocol is more effective in this model, using the spectral radius method. Moreover, we introduced a parameter for individual control of IL-2 in each injection using the Cauchy matrix for the IDE system, and we obtained conditions under which this system would be exponentially stable in a tumor-free equilibrium.

References

[1]	V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor, A. S. Perelson, Nonlinear dynamics of immunogenic tumours: parameter estimation and global analysis, B. Math. Biol., 56 (1994), 295–321. https://doi.org/10.1016/S0092-8240(05)80260-5 doi: 10.1016/S0092-8240(05)80260-5
[2]	L. G. De Pillis, A. E. Radunskaya, C. L. Wiseman, A validated mathematical model of cell-mediated immune response to tumor growth, Cancer Res., 65 (2005), 7950–7958. https://doi.org/10.1158/0008-5472.CAN-05-0564 doi: 10.1158/0008-5472.CAN-05-0564
[3]	L. G. De Pillis, W. Gu, A. E. Radunskaya, Mixed immnotherapy and chemotherapy of tumors: modeling, applications and biological interpretations, J. Theor. Biol., 238 (2006), 841–862. https://doi.org/10.1016/j.jtbi.2005.06.037 doi: 10.1016/j.jtbi.2005.06.037
[4]	L. G. De Pillis, A. Eladdadi, A. Radunskaya, Modeling cancer-immune responses to therapy, J. Pharmacokinet Phar., 41 (2014), 461–478. https://doi.org/10.1007/s10928-014-9386-9 doi: 10.1007/s10928-014-9386-9
[5]	D. Kirschner, J. C. Panetta, Modeling immunotherapy of the tumor–immune interaction, J. Math. Biol., 37 (1998), 235–252. https://doi.org/10.1007/s002850050127 doi: 10.1007/s002850050127
[6]	F. Castiglione, B. Piccoli, Cancer immunotherapy, mathematical modeling and optimal control, J. Theor. Biol., 247 (2007), 723–732. https://doi.org/10.1016/j.jtbi.2007.04.003 doi: 10.1016/j.jtbi.2007.04.003
[7]	H. M. Byrne, Dissecting cancer through mathematics: from the cell to the animal model, Nat. Rev. Cancer, 10 (2010), 221–230. https://doi.org/10.1038/nrc2808 doi: 10.1038/nrc2808
[8]	N. Kronik, Y. Kogan, V. Vainstein, Z. Agur, Improving alloreactive CTL immunotherapy for malignant gliomas using a simulation model of their interactive dynamics, Cancer Immunol. Immun., 57 (2008), 425–439. https://doi.org/10.1007/s00262-007-0387-z doi: 10.1007/s00262-007-0387-z
[9]	A. B. Weiner, A. S. Desai, J. J. Meeks, Tumor Location May Predict Adverse Pathology and Survival Following Definitive Treatment for Bladder Cancer: A National Cohort Study, Eur. Urol. Oncol., 2 (2019), 304–310. https://doi.org/10.1016/j.euo.2018.08.018 doi: 10.1016/j.euo.2018.08.018
[10]	F. Bray, J. Ferlay, I. Soerjomataram, R. L. Siegel, L. Torre, A. Jemal, Global cancer statistics 2018: Globocan estimates of incidence and mortality worldwide for 36 cancers in 185 countries, CA: a cancer journal for clinicians, 68 (2018), 394–424. https://doi.org/10.3322/caac.21492 doi: 10.3322/caac.21492
[11]	M. P. Zeegers, F. E. Tan, E. Dorant, P. A. van Den Brandt, The impact of characteristics of cigarette smoking on urinary tract cancer risk: a meta-analysis of epidemiological studies, Cancer, 89 (2000), 630–639. https://doi.org/10.1002/1097-0142(20000801)89:3<630::AID-CNCR19>3.0.CO;2-Q doi: 10.1002/1097-0142(20000801)89:3<630::AID-CNCR19>3.0.CO;2-Q
[12]	A. Morales, D. Eidinger, A. W. Bruce, Intracavity Bacillus Calmette-Guerin in the treatment of superficial bladder tumors, J. Urol., 116 (1976), 180–182. https://doi.org/10.1016/S0022-5347(17)58737-6 doi: 10.1016/S0022-5347(17)58737-6
[13]	C. Pettenati, M. A. Ingersoll, Mechanisms of BCG immunotherapy and its outlook for bladder cancer, Nat. Rev. Urol., 15 (2018), 615–625. https://doi.org/10.1038/s41585-018-0055-4 doi: 10.1038/s41585-018-0055-4
[14]	C. F. Lee, S. Y. Chang, D. S. Hsieh, D. S. Yu, Immunotherapy for bladder cancer using recombinant bacillus Calmette-Guerin DNA vaccines and interleukin-12 DNA vaccine, J. Urol., 171 (2004), 1343–1347. https://doi.org/10.1097/01.ju.0000103924.93206.93 doi: 10.1097/01.ju.0000103924.93206.93
[15]	R. L. Steinberg, L. J. Thomas, S. L. Mott, M. A. O'Donnell, Multi-perspective tolerance evaluation of bacillus Calmette-Guerin with interferon in the treatment of non-muscle invasive bladder cancer, Bladder Cancer, 5 (2019), 39–49. https://doi.org/10.3233/BLC-180203 doi: 10.3233/BLC-180203
[16]	A. Shapiro, O. Gofrit, D. Pode, The treatment of superficial bladder tumor with IL-2 and BCG, J. Urol., 177 (2007), 81–82. https://doi.org/10.1016/S0022-5347(18)30509-3 doi: 10.1016/S0022-5347(18)30509-3
[17]	S. Bunimovich-Mendrazitsky, E. Shochat, L. Stone, Mathematical Model of BCG Immunotherapy in Superficial Bladder Cancer, B. Math. Biol., 69 (2007), 1847–1870. https://doi.org/10.1007/s11538-007-9195-z doi: 10.1007/s11538-007-9195-z
[18]	O. Nave, S. Hareli, M. Elbaz, I. H. Iluz, S. Bunimovich-Mendrazitsky, BCG and IL-2 model for bladder cancer treatment with fast and slow dynamics based on SPVF method—stability analysis, Math. Biosci. Eng., 16 (2019), 5346–5379. https://doi.org/10.3934/mbe.2019267 doi: 10.3934/mbe.2019267
[19]	T. Lazebnik, N. Aaroni, S. Bunimovich-Mendrazitsky, PDE based geometry model for BCG immunotherapy of bladder cancer, Biosystems, 200 (2021), 104319. https://doi.org/10.1016/j.biosystems.2020.104319 doi: 10.1016/j.biosystems.2020.104319
[20]	E. Guzev, S. Halachmi, S. Bunimovich-Mendrazitsky, Additional extension of the mathematical model for BCG immunotherapy of bladder cancer and its validation by auxiliary tool, Int. J. Nonlin. Sci. Num., 20 (2019), 675–689. https://doi.org/10.1515/ijnsns-2018-0181 doi: 10.1515/ijnsns-2018-0181
[21]	S. Bunimovich-Mendrazitsky, I. Chaskalovic, J. C. Gluckman, A mathematical model of combined bacillus Calmette-Guerin (BCG) and interleukin (IL)-2 immunotherapy of superficial bladder cancer, J. Theor. Biol., 277 (2011), 27–40. https://doi.org/10.1016/j.jtbi.2011.02.008 doi: 10.1016/j.jtbi.2011.02.008
[22]	S. Bunimovich-Mendrazitsky, S. Halachmi, N. Kronik, Improving Bacillus Calmette Guérin (BCG) immunotherapy for bladder cancer by adding interleukin-2 (IL-2): a mathematical model, Math. Med. Biol., 33 (2016), 159–188. https://doi.org/10.1093/imammb/dqv007 doi: 10.1093/imammb/dqv007
[23]	L. Shaikhet, S. Bunimovich-Mendrazitsky, Stability analysis of delayed immune response BCG infection in bladder cancer treatment model by stochastic perturbations, Comput. Math. Method. M., 2018 (2018), 9653873. https://doi.org/10.1155/2018/9653873 doi: 10.1155/2018/9653873
[24]	E. Fridman, L. Shaikhet, Simple LMIs for stability of stochastic systems with delay term given by Stieltjes integral or with stabilizing delay, Syst. Control Lett., 124 (2019), 83–91. https://doi.org/10.1016/j.sysconle.2018.12.007 doi: 10.1016/j.sysconle.2018.12.007
[25]	I. Volinsky, S. D. Lombardo, P. Cheredman, Stability Analysis and Cauchy Matrix of a Mathematical Model of Hepatitis B Virus with Control on Immune System near Neighborhood of Equilibrium Free Point, Symmetry, 13 (2021), 166. https://doi.org/10.3390/sym13020166 doi: 10.3390/sym13020166
[26]	R. P. Agarwal, L. Berezansky, E. Braverman, A. Domoshnitsky, Nonoscillation Theory of Functional Differential Equations with Applications, Springer: New York, NY, USA, 2012.
[27]	I. Volinsky, Stability Analysis of a Mathematical Model of Hepatitis B Virus with Unbounded Memory Control on the Immune System in the Neighborhood of the Equilibrium Free Point, Symmetry, 13 (2021), 1437. https://doi.org/10.3390/sym13081437 doi: 10.3390/sym13081437
[28]	R. F. M. Bevers, K. H. Kurth, D. H. J. Schamhart, Role of urothelial cells in BCG immunotherapy for superficial bladder cancer, Brit. J. Cancer, 91 (2004), 607–612. https://doi.org/10.1038/sj.bjc.6602026 doi: 10.1038/sj.bjc.6602026
[29]	L. M. Wein, J. T. Wu, D. H. Kirn, Validation and analysis of a mathematical model of a replication-competent oncolytic virus for cancer treatment: Implications for virus design and delivery, Cancer Res., 63 (2003), 1317–1324.
[30]	J. Wigginton, D. Kirschner, A model to predict cell-mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis, J. Immunol., 166 (2001), 1951–1967. https://doi.org/10.4049/jimmunol.166.3.1951 doi: 10.4049/jimmunol.166.3.1951
[31]	F. Biemar, M. Foti, Global progress against cancer-challenges and opportunities, Cancer Biol. Med., 10 (2013), 183–186.

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