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Ultimate tumor dynamics and eradication using oncolytic virotherapy. (English) Zbl 07274877
Summary: In this paper we study ultimate dynamics of one three-dimensional model for tumor growth under oncolytic virotherapy which describes interactions between cytotoxic T-cells, uninfected tumor cells and infected tumor cells. Using the localization theorem of compact invariant sets we derive ultimate upper bounds for all cell populations and establish the property of the existence of the attracting set. Next, we find several conditions under which our system demonstrates convergence dynamics to equilibrium points located in invariant planes corresponding cases of the absence of uninfected or infected tumor cells. These assertions mean global eradication of uninfected or infected tumor cell populations and are presented as algebraic inequalities respecting virus replication rate \(\theta\). In particular, we find in Theorems 4 and 5 the following curious phenomenon. Namely, when we vary \(\theta\) from the instability range of the infected tumor free equilibrium point to the stability range we obtain in the latter range convergence dynamics to one of tumor free equilibrium points; this means that the local eradication of infected tumor cells implies their global eradication. Besides, we give conditions under which the infected tumor cell population persists. Our theoretical studies are supplied by results of numerical simulation.
34C60 Qualitative investigation and simulation of ordinary differential equation models
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
92C37 Cell biology
Full Text: DOI
[1] Sinkovics, J.; Horvath, J., New developments in the virus therapy of cancer: a historical review, Intervirology, 36, 193-214 (1993)
[2] Lawler, S. E.; Speranza, M. C.; Cho, C. F.; Chiocca, E. A., Oncolytic viruses in cancer treatment: a review, JAMA Oncol, 3, 841-849 (2017)
[3] Varghese, S.; Rabkin, S. D., Oncolytic herpes simplex virus vectors for cancer virotherapy, Cancer Gene Ther, 9, 967-978 (2002)
[4] Vähä-Koskela, M. J.; Heikkilä, J. E.; Hinkkanen, A. E., Oncolytic viruses in cancer therapy, Cancer Lett, 254, 178-216 (2007)
[5] Karev, G. P.; Novozhilov, A. S.; Koonin, E. V., Mathematical modeling of tumor therapy with oncolytic viruses: effects of parametric heterogeneity on cell dynamics, Biol Direct, 1, 30 (2006)
[6] Kasuya, H.; Takeda, S.; Nomoto, S.; Nakao, A., The potential of oncolytic virus therapy for pancreatic cancer, Cancer Gene Ther, 12, 725-736 (2005)
[7] Bajzer, Ž.; Carr, T.; Josić, K.; Russell, S. J.; Dingli, D., Modeling of cancer virotherapy with recombinant measles viruses, J Theor Biol, 252, 109-122 (2008) · Zbl 1398.92108
[8] Bridle, B. W.; Stephenson, K. B.; Boudreau, J. E.; Koshy, S.; Kazdhan, N.; Pullenayegum, E., Potentiating cancer immunotherapy using an oncolytic virus, Mol Ther, 18, 1430-1439 (2010)
[9] Crivelli, J.; Földes, J.; Kim, P.; Wares, J., A mathematical model for cell cycle-specific cancer virotherapy, J Biol Dynam, 6, 104-120 (2012) · Zbl 1447.92193
[10] Eftimie, R.; Dushoff, J.; Bridle, B. M.; Bramson, J. L.; Earn, D. J.D., Multi-stability and multi-instability phenomena in a mathematical model of tumor-immune-virus interactions, Bull Mathem Biol, 73, 2932-2961 (2011) · Zbl 1319.92026
[11] Eftimie, R.; Macnamara, C. K.; Dushoff, J.; Bramson, J. L.; Earn, D. J., Bifurcations and chaotic dynamics in a tumour-immune-virus system, Math Model Nat Phenom, 11, 65-85 (2016) · Zbl 1384.92038
[12] Mukhopadhyay, B.; Bhattacharyya, R., A nonlinear mathematical model of virus-tumor-immune system interaction: deterministic and stochastic analysis, Stochast Anal Applic, 27, 409-429 (2009) · Zbl 1173.34032
[13] Wang, Y.; Tian, J. P.; Wei, J., Lytic cycle: a defining process in oncolytic virotherapy, Appl Mathem Modelling, 37, 5962-5978 (2013) · Zbl 1274.92036
[14] Starkov, K. E.; Andres, G., Dynamics of the tumor-immune-virus interactions: convergence conditions to tumor-only or tumor-free equilibrium points, Math Biosci Eng, 16, 421-437 (2018)
[15] Tian, J. P., The replicability of oncolytic virus: defining conditions in tumor virotherapy, Math Biosci Eng, 8, 841-860 (2011) · Zbl 1259.34029
[16] Wang, Z.; Guo, Z.; Peng, H., A mathematical model verifying potent oncolytic efficacy of M1 virus, Math Biosci, 276, 19-27 (2016) · Zbl 1341.92035
[17] Malinzi, J.; Sibanda, P.; Mambili-Mamboundou, H., Analysis of virotherapy in solid tumor invasion, Math Biosci, 263, 102-110 (2015) · Zbl 1328.35252
[18] Mambili-Mamboundou, H.; Sibanda, P.; Malinzi, J., Effect of immunotherapy on the response of ticls to solid tumour invasion, Math Biosci, 249, 52-59 (2014) · Zbl 1308.92051
[19] Kirschner, D.; Panetta, J., Modelling immunotherapy of the tumor-immune interaction, J Mathem Biol, 37, 235-252 (1998) · Zbl 0902.92012
[20] Krishchenko, A. P., Localization of invariant compact sets of dynamical systems, Differential Equations, 41, 1669-1676 (2005) · Zbl 1133.34342
[21] Krishchenko, A. P.; Starkov, K. E., Localization of compact invariant sets of the Lorenz system, Phys Lett A, 353, 383-388 (2006) · Zbl 1181.37044
[22] Giacomini, H.; Neukirch, S., Integrals of motion and the shape of the attractor for the Lorenz model, Phys Lett A, 227, 5-6, 309-318 (1997) · Zbl 0962.37501
[23] Starkov, K. E.; Pogromsky, A. Y., On the global dynamics of the Owen-Sherratt model describing the tumor-macrophage interactions, Intern J Bifurcation and Chaos, 23, 2, 1350020 (2013) · Zbl 1270.34137
[24] Starkov, K. E.; Gamboa, D., Localization of compact invariant sets and global stability in analysis of one tumor growth model, Math Meth Appl Sci, 37, 2854-2863 (2014) · Zbl 1309.34085
[25] Chen, G.; Kuznetsov, N. V.; Leonov, G. A.; Mokaev, T. N., Hidden attractors on one path: Glukhovsky-Dolzhansky, Lorenz, and Rabinovich systems, Int J of Bifurcation and Chaos, 27, 8, 1750115 (2017) · Zbl 1377.34021
[26] Kuznetsov, N. V.; Leonov, G. A.; Vagaitsev, V. I., Analytical-numerical method for attractor localization of generalized Chua’s system, IFAC Proceedings Volumes, 43, 11, 29-33 (2010)
[27] Leonov, G. A.; Kuznetsov, N. V., Hidden attractors in dynamical systems. from hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractors in chua circuits, Int J Bifurcation and Chaos, 23, 1, 1330002 (2013) · Zbl 1270.34003
[28] Krishchenko, A. P.; Starkov, K. E., On the global dynamics of a chronic myelogenous leukemia model, Commun Nonlin Sci Numer Simul, 33, 174-183 (2016) · Zbl 07246598
[29] Krishchenko, A. P.; Starkov, K. E., The four-dimensional Kirschner-Panetta type cancer model: how to obtain tumor eradication?, Math Biosci Eng, 15, 1243-1254 (2018) · Zbl 1406.92311
[30] Starkov, K. E., On dynamic tumor eradication conditions under combined chemical/anti-angiogenic therapies, Phys Lett A, 382, 387-393 (2018) · Zbl 1383.92037
[31] Starkov, K. E.; Jimenez Beristain, L., Dynamic analysis of the melanoma model: from cancer persistence to its eradication, Intern J Bifurcation and Chaos, 27, 10, 1750151 (2017) · Zbl 1379.92022
[32] Starkov, K. E.; Krishchenko, A. P., Ultimate dynamics of the Kirschner-Panetta model: Tumor eradication and related problems, Phys Lett A, 381, 3016-3409 (2017) · Zbl 1374.92072
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