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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

A Cancer Model Study to See the Dynamical Effect of a Therapeutic Approach through Virus Injecting

Journal of Applied Nonlinear Dynamics 12(4) (2023) 739--756 | DOI:10.5890/JAND.2023.12.008

Aktar Saikh$^1$, Kalyan Das$^2$, Nurul Huda Gazi$^1$

$^1$ Department of Mathematics and Statistics, Aliah University, IIA/27, New Town, Kolkata-700160, India

$^2$ Department of Mathematics, National Institute of Food Technology Entrepreneurship and Management,

HSIIDC Industrial Estate, Kundli, Haryana 131028, India

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Abstract

A basic simple cancer model is studied to see the effectiveness of a therapeutic approach to control or reduce the cancer cells through a specific virus injecting into the patient's tumour site. Existence, uniqueness, positivity, persistency, boundedness of the system's solutions are established. The system's local and global behaviour about all possible equilibria are studied. Bifurcation analysis of the system is discussed as well. The basic reproduction number $\mathcal{R}_0$ is calculated using Next Generation Matrix method, which plays a significant role in the disease endemicity. We have analyzed the centre manifold near the tumour-free equilibrium point $E_0$, $\mathcal{R}_0=1$. For the proposed cancer model system, it is shown that there exists an optimal control of the virus replication rate. The characterization of the optimal control is explained as well. The optimality of the viral cytotoxicity is also studied. Numerical simulations are presented to validate the analytical findings. Finally, we conclude some epidemiological remarks made through analytical and numerical observations.

References

  1. [1]  Aghi, M. and Martuza, R.L. (2005), Oncolytic viral therapies--the clinical experience, Oncogene, 24, 7802-7816.
  2. [2]  Jebar, A.H., Errington-Mais, F., Vile, R.G., Selby, P.J., Melcher, A.A., and Griffin, S. (2015), Progress in clinical oncolytic virus-based therapy for hepatocellular carcinoma, Journal of General Virology, 96, 1533-1550.
  3. [3]  Lawler, S.E., Speranza, M.-C., Cho, C.-F., and Chiocca, E.A. (2017), Oncolytic viruses in cancer treatment: a review, JAMA Oncology, 3, 841-849.
  4. [4]  Prestwich, R.J., Harrington, K.J., Pandha, H.S., Vile, R.G., Melcher, A.A., and Errington, F. (2008), Oncolytic viruses: a novel form of immunotherapy, Expert Review of Anticancer Therapy, 8, 1581-1588.
  5. [5]  Russell, S.J., Peng, K.-W., and Bell, J.C. (2012), Oncolytic virotherapy, Nature Biotechnology, 30, 658-670.
  6. [6]  Wang, Y.-G., Huang, P.-P., Zhang, R., Ma, B.-Y., Zhou, X.-M., and Sun, Y.-F. (2016), Targeting adeno-associated virus and adenoviral gene therapy for hepatocellular carcinoma, World Journal of Gastroenterology, 22, 326.
  7. [7]  Khuri, F.R., et~al. (2000), A controlled trial of intratumoral onyx-015, a selectively-replicating adenovirus, in combination with cisplatin and 5-fluorouracil in patients with recurrent head and neck cancer, Nature Medicine, 6, 879-885.
  8. [8]  Kirn, D.H. and Thorne, S.H. (2009), Targeted and armed oncolytic poxviruses: a novel multi-mechanistic therapeutic class for cancer., Nature Reviews Cancer, 9, 64-71.
  9. [9]  Liu, T.-C., Hwang, T.-H., Bell, J.C., and Kirn, D.H. (2008), Translation of targeted oncolytic virotherapeutics from the lab into the clinic, and back again: a high-value iterative loop., Molecular Therapy, 16, 1006-1008.
  10. [10]  Andtbacka, R., et~al (2015), Talimogene laherparepvec improves durable response rate in patients with advanced melanoma, Journal of Clinical Oncology, 33(25), 2780-2788.
  11. [11]  Bridle, B.W., Stephenson, K.B., Boudreau, J.E., Koshy, S., Kazdhan, N., Pullenayegum, E., Brunelli{\`e}re, J., Bramson, J.L., Lichty, B.D., and Wan, Y. (2010) Potentiating cancer immunotherapy using an oncolytic virus, Molecular Therapy, 18, 1430-1439.
  12. [12]  Lichty, B.D., Stojdl, D.F., Taylor, R.A., Miller, L., Frenkel, I., Atkins, H., and Bell, J.C. (2004), Vesicular stomatitis virus: a potential therapeutic virus for the treatment of hematologic malignancy, Human Gene Therapy, 15, 821-831.
  13. [13]  Downs-Canner, S., et~al (2016), Phase 1 study of intravenous oncolytic poxvirus (vvdd) in patients with advanced solid cancers, Molecular Therapy, 24, 1492-1501.
  14. [14]  Msaouel, P., D~Iankov, I., Dispenzieri, A., and Galanis, E. (2012), Attenuated oncolytic measles virus strains as cancer therapeutics, Current Pharmaceutical Biotechnology, 13, 1732-1741.
  15. [15]  Jha, B.K., Dong, B., Nguyen, C.T., Polyakova, I., and Silverman, R.H. (2013), Suppression of antiviral innate immunity by sunitinib enhances oncolytic virotherapy, Molecular Therapy, 21, 1749-1757.
  16. [16]  Wollmann, G., Rogulin, V., Simon, I., Rose, J.K., and van~den Pol, A.N. (2010), Some attenuated variants of vesicular stomatitis virus show enhanced oncolytic activity against human glioblastoma cells relative to normal brain cells, Journal of virology, 84, 1563-1573.
  17. [17]  Workenhe, S.T., Verschoor, M.L., and Mossman, K.L. (2015), The role of oncolytic virus immunotherapies to subvert cancer immune evasion, Future Oncology, 11, 675-689.
  18. [18]  Woller, N., G{\"u}rlevik, E., Ureche, C.-I., Schumacher, A., and K{\"u}hnel, F. (2014), Oncolytic viruses as anticancer vaccines, Frontiers in Oncology, 4, 188.
  19. [19]  Alvarez-Breckenridge, C.A., Choi, B.D., Suryadevara, C.M., and Chiocca, E.A. (2015), Potentiating oncolytic viral therapy through an understanding of the initial immune responses to oncolytic viral infection, Current Opinion in Virology, 13, 25-32.
  20. [20]  Prestwich, R.J., Errington, F., Diaz, R.M., Pandha, H.S., Harrington, K.J., Melcher, A.A., and Vile, R.G. (2009), The case of oncolytic viruses versus the immune system: waiting on the judgment of solomon., Human Gene Therapy, 20, 1119-1132.
  21. [21]  De~Boer, R., Hogeweg, P., Dullens, H., De~Weger, R.A., and Den~Otter, W. (1985), Macrophage t lymphocyte interactions in the anti-tumor immune response: a mathematical model., The Journal of Immunology, 134, 2748-2758.
  22. [22]  de~Pillis, L.G., Radunskaya, A.E., and Wiseman, C.L. (2005), A validated mathematical model of cell-mediated immune response to tumor growth, Cancer Research, 65, 7950-7958.
  23. [23]  Goldstein, B., Faeder, J.R., and Hlavacek, W.S. (2004), Mathematical and computational models of immune-receptor signalling, Nature Reviews Immunology, 4, 445-456.
  24. [24]  Kronik, N., Kogan, Y., Vainstein, V., and Agur, Z. (2008), Improving alloreactive ctl immunotherapy for malignant gliomas using a simulation model of their interactive dynamics, Cancer Immunology, Immunotherapy, 57, 425-439.
  25. [25]  Wodarz, D. (2003), Gene therapy for killing p53-negative cancer cells: use of replicating versus nonreplicating agents, Human Gene Therapy, 14, 153-159.
  26. [26]  Komarova, N.L. and Wodarz, D. (2014), Targeted cancer treatment in silico, Modeling and Simulation in Science, Engineering and Technology, Springer.
  27. [27]  Berezovskaya, F.S., Novozhilov, A.S., and Karev, G.P. (2007), Population models with singular equilibrium, Mathematical Biosciences, 208, 270-299.
  28. [28]  Karev, G.P., Novozhilov, A.S., and Koonin, E.V. (2006), Mathematical modeling of tumor therapy with oncolytic viruses: effects of parametric heterogeneity on cell dynamics, Biology Direct, 1, 1-19.
  29. [29]  Komarova, N.L. and Wodarz, D. (2010), Ode models for oncolytic virus dynamics, Journal of Theoretical Biology, 263, 530-543.
  30. [30]  Novozhilov, A.S., Berezovskaya, F.S., Koonin, E.V., and Karev, G.P. (2006), Mathematical modeling of tumor therapy with oncolytic viruses: regimes with complete tumor elimination within the framework of deterministic models, Biology Direct, 1, 1-18.
  31. [31]  Titze, M.I., Frank, J., Ehrhardt, M., Smola, S., Graf, N., and Lehr, T. (2017), A generic viral dynamic model to systematically characterize the interaction between oncolytic virus kinetics and tumor growth, European Journal of Pharmaceutical Sciences, 97, 38-46.
  32. [32]  Bajzer, {\v{Z}}., Carr, T., Josi{c}, K., Russell, S.J., and Dingli, D. (2008), Modeling of cancer virotherapy with recombinant measles viruses, Journal of Theoretical Biology, 252, 109-122.
  33. [33]  Dingli, D., Cascino, M.D., Josi{c}, K., Russell, S.J., and Bajzer, {\v{Z}}. (2006), Mathematical modeling of cancer radiovirotherapy, Mathematical biosciences, 199, 55-78.
  34. [34]  Dingli, D., Offord, C., Myers, R., Peng, K.-W., Carr, T., Josic, K., Russell, S.J., and Bajzer, Z. (2009), Dynamics of multiple myeloma tumor therapy with a recombinant measles virus, Cancer Gene Therapy, 16, 873-882.
  35. [35]  Jenner, A.L., Yun, C.-O., Kim, P.S., and Coster, A.C. (2018), Mathematical modelling of the interaction between cancer cells and an oncolytic virus: insights into the effects of treatment protocols, Bulletin of Mathematical Biology, 80, 1615-1629.
  36. [36]  De~Boer, R.J. and Perelson, A.S. (1995), Towards a general function describing t cell proliferation., Journal of Theoretical Biology, 175, 567-576.
  37. [37]  Wodarz, D. and Thomsen, A.R. (2005), Effect of the ctl proliferation program on virus dynamics, International Immunology, 17, 1269-1276.
  38. [38]  Wodarz, D., Nowak, M.A., and Bangham, C.R. (1999), The dynamics of htlv-i and the ctl response, Immunology Today, 20, 220-227.
  39. [39]  Wang, Y., Zhou, Y., Brauer, F., and Heffernan, J.M. (2013), Viral dynamics model with ctl immune response incorporating antiretroviral therapy, Journal of Mathematical Biology, 67, 901-934.
  40. [40]  Wodarz, D. (2001), Viruses as antitumor weapons: defining conditions for tumor remission, Cancer Research, 61, 3501-3507.
  41. [41]  Birkhoff, G. and Rota, G. (1982), Ordinary Differential Equation, Ginn. and Co.
  42. [42]  Gard, T.C. and Hallam, T.G. (1979), Persistence in food webs—i lotka-volterra food chains., Bulletin of Mathematical Biology, 41, 877-891.
  43. [43]  Van~den Driessche, P. and Watmough, J. (2002), Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission., Mathematical Biosciences, 180, 29-48.
  44. [44]  Diekmann, O., Heesterbeek, J., and Roberts, M.G. (2010), The construction of next-generation matrices for compartmental epidemic models, Journal of the Royal Society Interface, 7, 873-885.
  45. [45]  May, R.M. (2001), Stability and Complexity in Model Ecosystems, Princeton university press.
  46. [46]  Murray, J.D. (2002), Mathematical Biology, Springer-Verlag.
  47. [47]  Saikh, A. and Gazi, N. (2017), Mathematical analysis of a predator-prey eco-epidemiological system under the reproduction of infected prey, Journal of Applied Mathematics and Computing, 58, 621-646.
  48. [48]  Saikh, A. and Gazi, N. (2020), A mathematical study of a two species eco-epidemiological model with different predation principles, Discontinuity, Nonlinearity, and Complexity, 9, 309-325.
  49. [49]  La~Salle, J. and Lefschetz, S. (1961), Stability by Liapunov's Direct Method with Applications, Academic Press.
  50. [50]  Poincar{e}, H. (1885), Sur l'{e}quilibre d'une masse fluide anim{e}e d'un mouvement de rotation, Acta Mathematica, 7, 259-380.
  51. [51]  Wiggins, S. (2003), Introduction to Applied Nonlinear Dynamical Systems and Chaos, vol.2. Springer Science \& Business Media.
  52. [52]  Perko, L. (2001), Differential Equations and Dynamical Systems, Springer-Verlag, 3rd edn.
  53. [53]  Lenhart, S. and Workman, J.T. (2007), Optimal Control Applied to Biological Models, CRC press.
  54. [54]  Pontryagin, L.S., Boltyanskii, V., Gamkrelidze, R., and Mishchenko, E. (1962), The mathematical theory of optimal processes.

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