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Mathematical modeling and dynamic analysis of anti-tumor immune response

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Abstract

The competitive interaction of tumor-immune system is very complex. We aim to establish a simple and realistic mathematical model to understand the key factors that impact the outcome of an antitumor response. Based on the principle that lymphocytes undergo two stages of development (namely immature and mature), we develop a new anti-tumor-immune response model and investigate its property and bifurcation. The corresponding sufficient criteria for the asymptotic stabilities of equilibria and the existence of stable periodic oscillations of tumor levels are obtained. Theoretical results indicate that the system orderly undergoes different states with the flow rate of mature immune cells increasing, from the unlimited expansion of tumor, to the stable large tumor-present equilibrium, to the periodic oscillation, to the stable small tumor-present equilibrium, and finally to the stable tumor-free equilibrium, which exhibits a variety of dynamic behaviors. In addition, these dynamic behaviors are in accordance with some phenomena observed clinically, such as tumor dormant, tumor periodic oscillation, immune escape of tumor and so on. Numerical simulations are carried out to verify the results of our theoretical analysis.

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References

  1. World Health Organization: Media centre, Cancer http://www.who.int/mediacentre/factsheets/fs297/en/

  2. Chen, W., Zheng, R., Baade, P., et al.: Cancer statistics in China, 2015. CA Cancer J. Clin. 66, 115–132 (2016)

    Article  Google Scholar 

  3. Parish, C.: Cancer immunotherapy: the past, the present and the future. Immunol. Cell Biol. 81, 106–113 (2003)

    Article  Google Scholar 

  4. Smyth, M., Godfrey, D.: A fresh look at tumor immunosurveillance and immunotherapy. Nat. Immunol. 2, 293–299 (2001)

    Article  Google Scholar 

  5. Galach, M.: Dynamics of the tumor-immune system competition-the effect of time delay. Int. J. Appl. Math. Comput. Sci. 13, 395–406 (2003)

    MathSciNet  MATH  Google Scholar 

  6. Raluca, E., Bramson, J., Earn, D.: Interactions between the immune system and cancer: a brief review of non-spatial mathematical models. Bull. Math. Biol. 73, 2–23 (2011)

    Article  MathSciNet  Google Scholar 

  7. Rosenberg, S., Yang, J., Restifo, N.: Cancer immunotherapy: moving beyond current vaccines. Nat. Med. 10, 909–915 (2004)

    Article  Google Scholar 

  8. Riddell, S.: Progress in cancer vaccines by enhanced self-presentation. Proc. Natl. Acad. Sci. USA 98, 8933–8935 (2001)

    Article  Google Scholar 

  9. Hirayama, M., Nishimur, Y.: The present status and future prospects of peptide-based cancer vaccines. Int. Immunol. 28, 319–328 (2016)

    Article  Google Scholar 

  10. Scott, A., Wolchok, J.: Antibody therapy of cancer. Nat. Rev. 12, 278–287 (2012)

    Article  Google Scholar 

  11. Pincetic, A., Bournazos, S., DiLillo, D., et al.: Type I and type II Fc receptors regulate innate and adaptive immunity. Nat. Immunol. 15, 707–16 (2014)

    Article  Google Scholar 

  12. Weiner, L., Surana, R., Wang, S.: Monoclonal antibodies: versatile platforms for cancer immunotherapy. Nat. Rev. 10, 317–27 (2010)

    Google Scholar 

  13. Adam, J., Bellomo, T.: Survey of models for tumor-immune system dynamics. Birkhauser, Boston (1997)

    Book  Google Scholar 

  14. Chaplain, M., Matzavions, A.: Mathematical modeling of spation-temporal phenomena in tumor immunology. Tutor. Math. Biosci. 3, 131–183 (2006)

    Article  Google Scholar 

  15. Kirschner, D., Panetta, J.: Modeling immunotherapy of the tumor-immune interaction. J. Math. Biol. 37, 235–252 (1998)

    Article  Google Scholar 

  16. Mallet, D., Pillis, L.: A cellular automata model of tumor-immune system interactions. J. Theor. Biol. 239, 334–350 (2006)

    Article  MathSciNet  Google Scholar 

  17. d’Onofrio, A.: A general framework for modeling tumor-immune system competition and immunotherapy: mathematical analysis and biomedical inferences. Physica D 208, 220–235 (2005)

    Article  MathSciNet  Google Scholar 

  18. Kirschner, D., Tsygvintsev, A.: On the global dynamics of a model for tumor immunotherapy. Math. Biosci. Eng. 6, 573–583 (2009)

    Article  MathSciNet  Google Scholar 

  19. Pang, L., Zhao, Z., Hong, S.: Dynamic analysis of an antitumor model and investigation of the therapeutic effects for different treatment regimens. Comput. Appl. Math. 36, 537–560 (2017)

    Article  MathSciNet  Google Scholar 

  20. Lejeune, O., Chaplain, M., Akili, I.: Oscillations and bistability in the dynamics of cytotoxic reactions medicated by the response of immune cells to solid tumours. Math. Comput. Model. 47, 649–662 (2008)

    Article  Google Scholar 

  21. Pang, L., Zhao, Z., Song, X.: Cost-effectiveness analysis of optimal strategy for tumor treatment. Chaos Solitions Fractals 87, 293–301 (2016)

    Article  MathSciNet  Google Scholar 

  22. Pang, L., Shen, L., Zhao, Z.: Mathematical modeling and analysis of the tumor treatment regimens with pulsed immunotherapy and chemotherapy. Comput. Math. Methods Med. 2016, 1–12 (2016)

    Article  Google Scholar 

  23. Kuznetsov, V.A., Zhivoglyadov, V.P., Stepanova, L.A.: Kinetic approach and estimation of parameters of cellular interaction between immunity system and a tumor. Arch. Immunol. Ther. Exp. 2, 465–476 (1992)

    Google Scholar 

  24. Bell, G.I.: Predator–prey equations simulating an immune response. Math. Biosci. 16, 291–314 (1973)

    Article  Google Scholar 

  25. Kuznetsov, V.A., Makalkin, L.A., Talor, M.A., perelson, A.S.: Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. Bull. Math. Biol. 56, 295–321 (1994)

    Article  Google Scholar 

  26. de Pillis, L.G., Radunskaya, A.E., Wiseman, C.L.: A validated mathematical model of cell-mediated immune response to tumor growth. Cancer Res. 65, 7950–7958 (2005)

    Article  Google Scholar 

  27. de Pillis, L.G., Radunskaya, A.: A mathematical tumor model with immune resistance and drug therapy : an optimal control approach. J. Theor. Med. 3, 79–100 (2000)

    Article  Google Scholar 

  28. de Pillis, L.G., Fister, K.Renee, et al.: Mathematical model creation for cancer chemo-immuntherapy. Comput. Math. Methods Med. 10, 165–184 (2009)

    Article  MathSciNet  Google Scholar 

  29. Liu, D., Ruan, S., Zhu, D.: Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions. Math. Biosci. Eng. 9, 347–368 (2012)

    Article  MathSciNet  Google Scholar 

  30. DeLisi, C., Rescigno, A.: Immune surveillance and neoplasia-I: a minimal mathematical model. Bull. Math. Biol. 39, 201–221 (1977)

    MathSciNet  MATH  Google Scholar 

  31. Skipper, H., Schabel, F.: Quantitative and cytokinetic studies in experimental tumor systems. Cancer Med. 2, 636–648 (1982)

    Google Scholar 

  32. Roitt, I., Brostoff, J., Male, D.: Immunology. Mosby, St. Louis (1993)

    Google Scholar 

  33. Shilnikov, L., Shilnikov, A., Turaev, D., Chua, L.: Methods of Qualitative Theory in Nonlinear Dynamics, Part 1. World Scientific Publishing Co. Pte. Ltd., Singapore (1998)

    Book  Google Scholar 

  34. Zhang, X., Chen, L.: The periodic solution of a class of epidemic models. Comput. Math. Appl. 38, 61–71 (1999)

    Article  MathSciNet  Google Scholar 

  35. Hassard, B., Kazarinoff, N., Wan, Y.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981)

    MATH  Google Scholar 

  36. Kumar, S., Srivastava, S., Chingakham, P.: Hopf bifurcation and stability analysis in a harvested one-predator–two-prey model. Appl. Math. Comput. 129, 107–118 (2002)

    MathSciNet  MATH  Google Scholar 

  37. Allison, E., Coltobetal, A.: A mathematical model of the effector cell response to cancer. Math. Comput. Model. 39, 1313–1327 (2004)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This research is partially supported by the National Natural Science Foundation of China (Nos. 11871238, 11871060 and 61973177), the Natural Science Foundation of Henan Province of China under Grants 182102410021 and 182102410067, the young backbone teacher of Henan Province (2018GGJS148), Henan International Joint Laboratory of Behavior Optimization Control for Smart Robots, file No.[2018]19, the programme of Henan Innovative Research Team of Cooperative Control in Swarm-based Robotics, and the self-determined research funds of CCNU from the colleges basic research and operation of MOE (Grant No. CCNU16JCZX10).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huanghuai University, Zhumadian, 463000, People’s Republic of China

    Liuyong Pang

  2. School of Mathematics and Statistics, Hubei University of Science and Technology, Xianning, 437100, People’s Republic of China

    Sanhong Liu

  3. School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, People’s Republic of China

    Xinan Zhang

  4. School of Mathematical Sciences, Monash University, Melbourne, VIC, 3800, Australia

    Tianhai Tian

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  1. Liuyong Pang
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Correspondence to Liuyong Pang or Xinan Zhang.

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Pang, L., Liu, S., Zhang, X. et al. Mathematical modeling and dynamic analysis of anti-tumor immune response. J. Appl. Math. Comput. 62, 473–488 (2020). https://doi.org/10.1007/s12190-019-01292-9

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  • DOI: https://doi.org/10.1007/s12190-019-01292-9

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