# American Institute of Mathematical Sciences

January  2014, 19(1): 55-72. doi: 10.3934/dcdsb.2014.19.55

## Mathematical modeling on helper T cells in a tumor immune system

 1 Graduate School of Science and Technology, Shizuoka University, 3-5-1 Johoku, Naka-ku, Hamamatsu-shi 432-8561, Japan 2 Department of Mathematical and Systems Engineering, Shizuoka University, 3-5-1 Johoku, Naka-ku, Hamamatsu-shi 432-8561, Japan 3 Department of Physics and Mathematics, Aoyama Gakuin University, 5-10-1 Fuchinobe, Chuo-ku, Sagamihara-shi 252-5258, Japan

Received  December 2012 Revised  October 2013 Published  December 2013

Activation of CD$8^+$ cytotoxic T lymphocytes (CTLs) is naturally regarded as a major antitumor mechanism of the immune system. In contrast, CD$4^+$ T cells are commonly classified as helper T cells (HTCs) on the basis of their roles in providing help to the generation and maintenance of effective CD$8^+$ cytotoxic and memory T cells. In order to get a better insight on the role of HTCs in a tumor immune system, we incorporate the third population of HTCs into a previous two dimensional ordinary differential equations (ODEs) model. Further we introduce the adoptive cellular immunotherapy (ACI) as the treatment to boost the immune system to fight against tumors. Compared tumor cells (TCs) and effector cells (ECs), the recruitment of HTCs changes the dynamics of the system substantially, by the effects through particular parameters, i.e., the activation rate of ECs by HTCs, $p$ (scaled as $\rho$), and the HTCs stimulation rate by the presence of identified tumor antigens, $k_2$ (scaled as $\omega_2$). We describe the stability regions of the interior equilibria $E^*$ (no treatment case) and $E^+$ (treatment case) in the scaled $(\rho,\omega_2)$ parameter space respectively. Both $\rho$ and $\omega_2$ can destabilize $E^*$ and $E^+$ and cause Hopf bifurcations. Our results show that HTCs might play a crucial role in the long term periodic oscillation behaviors of tumor immune system interactions. They also show that TCs may be eradicated from the patient's body under the ACI treatment.
Citation: Yueping Dong, Rinko Miyazaki, Yasuhiro Takeuchi. Mathematical modeling on helper T cells in a tumor immune system. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 55-72. doi: 10.3934/dcdsb.2014.19.55
##### References:
 [1] J. A. Adam and N. Bellomo, A Survey of Models for Tumor-immune System Dynamics,, Birkhauser, (1997). doi: 10.1007/978-0-8176-8119-7. [2] J. C. Arciero, T. L. Jackson and D. E. Kirschner, A mathematical model of tumor-immune evasion and siRNA treatment,, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 39. [3] T. Boon and P. Van der Bruggen, Human tumor antigens recognized by T lymphocytes,, J. Exp. Med., 183 (1996), 725. doi: 10.1084/jem.183.3.725. [4] C. Bourgeois, B. Rocha and C. Tanchot, A role for CD40 expression on CD8$^+$ T cells in the generation of CD8$^+$ T cell memory,, Science, 297 (2002), 2060. doi: 10.1126/science.1072615. [5] L. G. de Pillis, A. E. Radunskaya and C. L. Wiseman, A validated mathematical model of cell-mediated immune response to tumor growth,, Cancer Res., 65 (2005), 7950. [6] L. G. de Pillis, W. Gu and A. E. Radunskaya, Mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretations,, J. Theor. Biol., 238 (2006), 841. doi: 10.1016/j.jtbi.2005.06.037. [7] K. E. de Visser, A. Eichten and L. M. Coussens, Paradoxical roles of the immune system during cancer development,, Nat. Rev. Cancer, 6 (2006), 24. [8] A. d'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: mathematical analysis and biomedical inferences,, Phys. D, 208 (2005), 220. doi: 10.1016/j.physd.2005.06.032. [9] R. Eftimie, J. L. Bramson and D. J. D. Earn, Modeling anti-tumor Th1 and Th2 immunity in the rejection of melanoma,, J. Theor. Biol., 265 (2010), 467. doi: 10.1016/j.jtbi.2010.04.030. [10] R. Eftimie, J. L. Bramson and D. J. D. Earn, Interactions between the immune system and cancer: A brief review of non-spatial mathematical models,, Bull. Math. Biol., 73 (2011), 2. doi: 10.1007/s11538-010-9526-3. [11] B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students,, SIAM, (2002). doi: 10.1137/1.9780898718195. [12] P. Fortin and M. C. Mackey, Periodic chronic myelogenous leukaemia: spectral analysis of blood cell counts and aetiological implications,, Brit. J. Haematol., 104 (1999), 336. doi: 10.1046/j.1365-2141.1999.01168.x. [13] M. Galach, Dynamics of the tumor-immune system competition-the effect of time delay,, Int. J. Appl. Math. Comput. Sci., 13 (2003), 395. [14] K. Hung, R. Hayashi, A. Lafond-Walker, C. Lowenstein, D. Pardoll and H. Levitsky, The central role of CD4+ T cells in the antitumor immune response,, J. Exp. Med., 188 (1998), 2357. doi: 10.1084/jem.188.12.2357. [15] D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction,, J. Math. Biol., 37 (1998), 235. doi: 10.1007/s002850050127. [16] D. Kirschner and A. Tsygvintsev, On the global dynamics of a model for tumor immunotherapy,, Math. Biosci. Eng., 6 (2009), 573. doi: 10.3934/mbe.2009.6.573. [17] V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis,, Bull. Math. Biol., 2 (1994), 295. [18] Y. P. Lai, C. J. Jeng and S. C. Chen, The roles of CD4$^+$ T cells in tumor immunity,, ISRN Immunology, 2011 (2011), 1. doi: 10.5402/2011/497397. [19] C. Letellier, F. Denis and L. A. Aguirre, What can be learned from a chaotic cancer model?, J. Theor. Biol., 322 (2013), 7. doi: 10.1016/j.jtbi.2013.01.003. [20] O. Lejeune, M. A. J. Chaplain and I. EI Akili, Oscillations and bistability in the dynamics of cytotoxic reactions mediated by the response of immune cells to solid tumours,, Math. Comput. Modelling, 47 (2008), 649. doi: 10.1016/j.mcm.2007.02.026. [21] D. Liu, S. Ruan and D. Zhu, Bifurcation analysis in models of tumor and immune system interactions,, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 151. doi: 10.3934/dcdsb.2009.12.151. [22] D. Liu, S. Ruan and D. Zhu, Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions,, Math. Biosci. Eng., 9 (2012), 347. doi: 10.3934/mbe.2012.9.347. [23] M. F. Mackey, R. J. Jr Barth and R. J. Noelle, The role of CD40/CD154 interactions in the priming, differentiation, and effector function of helper and cytotoxic T cells,, J. Leukoc. Biol., 63 (1998), 418. [24] J. E. Marsden and M. McKracken, The Hopf Bifurcation and Its Applications,, Springer-Verlag, (1976). [25] D. Mucida, M. M. Husain, S. Muroi et. al, Transcriptional reprogramming of mature CD4$^+$ helper T cells generates distinct MHC class II-restricted cytotoxic T lymphocytes,, Nat. Immunol., 14 (2013), 281. doi: 10.1038/ni.2523. [26] A. Perez-Diez, N. T. Joncker, K. Choi, W. F. N. Chan, C. C. Anderson, O. Lantz and P. Matzinger, CD4 cells can be more efficient at tumor rejection than CD8 cells,, Blood, 109 (2007), 5346. [27] S. A. Quezada, T. R. Simpson, K. S. Peggs, T. Merghoub, J. Vider, X. Z. Fan, R. Blasberg, H. Yagita, P. Muranski, P. A. Antony, N. P. Restifo and J. P. Allison, Tumor-reactive CD4(+) T cells develop cytotoxic activity and eradicate large established melanoma after transfer into lymphopenic hosts,, J. Exp. Med., 207 (2010), 637. doi: 10.1084/jem.20091918. [28] S. A. Rosenberg, Progress in human tumour immunology and immunotherapy,, Nature, 411 (2001), 380. [29] S. Ruan and G. S. K. Wolkowicz, Bifurcation analysis of a chemostat model with a distributed delay,, J. Math. Anal. Appl., 204 (1996), 786. doi: 10.1006/jmaa.1996.0468. [30] 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. [31] E. Stockert, E. Jager, Y. T. Chen, M. J. Scanlan, I. Gout, J. Karbach, M. Arand, A. Knuth and L. J. Old, A survey of the humoral immune response of cancer patients to a panel of human tumor antigens,, J. Exp. Med., 187 (1998), 1349. doi: 10.1084/jem.187.8.1349. [32] Z. Szymanska, Analysis of immunotheray models in the context of cancer dynamics,, Int. J. Appl. Math. Comput. Sci., 13 (2003), 407. [33] A. Tsygvintsev, S. Marino and D. E. Kirschner, A mathematical model of gene therapy for the treatment of cancer,, in Mathematical Methods and Models in Biomedicine (eds. U. Ledzewicz, (2013), 367. doi: 10.1007/978-1-4614-4178-6_13. [34] M. Villasana and A. Radunskaya, A delay differential equation model for tumor growth,, J. Math. Biol., 47 (2003), 270. doi: 10.1007/s00285-003-0211-0. [35] L. Zamai, C. Ponti, P. Mirandola, G. Gobbi, S. Papa, L. Galeotti, L. Cocco and M. Vitale, NK cells and cancer,, J. Immunol., 178 (2007), 4011.

show all references

##### References:
 [1] J. A. Adam and N. Bellomo, A Survey of Models for Tumor-immune System Dynamics,, Birkhauser, (1997). doi: 10.1007/978-0-8176-8119-7. [2] J. C. Arciero, T. L. Jackson and D. E. Kirschner, A mathematical model of tumor-immune evasion and siRNA treatment,, Discrete Contin. Dyn. Syst. Ser. B, 4 (2004), 39. [3] T. Boon and P. Van der Bruggen, Human tumor antigens recognized by T lymphocytes,, J. Exp. Med., 183 (1996), 725. doi: 10.1084/jem.183.3.725. [4] C. Bourgeois, B. Rocha and C. Tanchot, A role for CD40 expression on CD8$^+$ T cells in the generation of CD8$^+$ T cell memory,, Science, 297 (2002), 2060. doi: 10.1126/science.1072615. [5] L. G. de Pillis, A. E. Radunskaya and C. L. Wiseman, A validated mathematical model of cell-mediated immune response to tumor growth,, Cancer Res., 65 (2005), 7950. [6] L. G. de Pillis, W. Gu and A. E. Radunskaya, Mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretations,, J. Theor. Biol., 238 (2006), 841. doi: 10.1016/j.jtbi.2005.06.037. [7] K. E. de Visser, A. Eichten and L. M. Coussens, Paradoxical roles of the immune system during cancer development,, Nat. Rev. Cancer, 6 (2006), 24. [8] A. d'Onofrio, A general framework for modeling tumor-immune system competition and immunotherapy: mathematical analysis and biomedical inferences,, Phys. D, 208 (2005), 220. doi: 10.1016/j.physd.2005.06.032. [9] R. Eftimie, J. L. Bramson and D. J. D. Earn, Modeling anti-tumor Th1 and Th2 immunity in the rejection of melanoma,, J. Theor. Biol., 265 (2010), 467. doi: 10.1016/j.jtbi.2010.04.030. [10] R. Eftimie, J. L. Bramson and D. J. D. Earn, Interactions between the immune system and cancer: A brief review of non-spatial mathematical models,, Bull. Math. Biol., 73 (2011), 2. doi: 10.1007/s11538-010-9526-3. [11] B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students,, SIAM, (2002). doi: 10.1137/1.9780898718195. [12] P. Fortin and M. C. Mackey, Periodic chronic myelogenous leukaemia: spectral analysis of blood cell counts and aetiological implications,, Brit. J. Haematol., 104 (1999), 336. doi: 10.1046/j.1365-2141.1999.01168.x. [13] M. Galach, Dynamics of the tumor-immune system competition-the effect of time delay,, Int. J. Appl. Math. Comput. Sci., 13 (2003), 395. [14] K. Hung, R. Hayashi, A. Lafond-Walker, C. Lowenstein, D. Pardoll and H. Levitsky, The central role of CD4+ T cells in the antitumor immune response,, J. Exp. Med., 188 (1998), 2357. doi: 10.1084/jem.188.12.2357. [15] D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction,, J. Math. Biol., 37 (1998), 235. doi: 10.1007/s002850050127. [16] D. Kirschner and A. Tsygvintsev, On the global dynamics of a model for tumor immunotherapy,, Math. Biosci. Eng., 6 (2009), 573. doi: 10.3934/mbe.2009.6.573. [17] V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis,, Bull. Math. Biol., 2 (1994), 295. [18] Y. P. Lai, C. J. Jeng and S. C. Chen, The roles of CD4$^+$ T cells in tumor immunity,, ISRN Immunology, 2011 (2011), 1. doi: 10.5402/2011/497397. [19] C. Letellier, F. Denis and L. A. Aguirre, What can be learned from a chaotic cancer model?, J. Theor. Biol., 322 (2013), 7. doi: 10.1016/j.jtbi.2013.01.003. [20] O. Lejeune, M. A. J. Chaplain and I. EI Akili, Oscillations and bistability in the dynamics of cytotoxic reactions mediated by the response of immune cells to solid tumours,, Math. Comput. Modelling, 47 (2008), 649. doi: 10.1016/j.mcm.2007.02.026. [21] D. Liu, S. Ruan and D. Zhu, Bifurcation analysis in models of tumor and immune system interactions,, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 151. doi: 10.3934/dcdsb.2009.12.151. [22] D. Liu, S. Ruan and D. Zhu, Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions,, Math. Biosci. Eng., 9 (2012), 347. doi: 10.3934/mbe.2012.9.347. [23] M. F. Mackey, R. J. Jr Barth and R. J. Noelle, The role of CD40/CD154 interactions in the priming, differentiation, and effector function of helper and cytotoxic T cells,, J. Leukoc. Biol., 63 (1998), 418. [24] J. E. Marsden and M. McKracken, The Hopf Bifurcation and Its Applications,, Springer-Verlag, (1976). [25] D. Mucida, M. M. Husain, S. Muroi et. al, Transcriptional reprogramming of mature CD4$^+$ helper T cells generates distinct MHC class II-restricted cytotoxic T lymphocytes,, Nat. Immunol., 14 (2013), 281. doi: 10.1038/ni.2523. [26] A. Perez-Diez, N. T. Joncker, K. Choi, W. F. N. Chan, C. C. Anderson, O. Lantz and P. Matzinger, CD4 cells can be more efficient at tumor rejection than CD8 cells,, Blood, 109 (2007), 5346. [27] S. A. Quezada, T. R. Simpson, K. S. Peggs, T. Merghoub, J. Vider, X. Z. Fan, R. Blasberg, H. Yagita, P. Muranski, P. A. Antony, N. P. Restifo and J. P. Allison, Tumor-reactive CD4(+) T cells develop cytotoxic activity and eradicate large established melanoma after transfer into lymphopenic hosts,, J. Exp. Med., 207 (2010), 637. doi: 10.1084/jem.20091918. [28] S. A. Rosenberg, Progress in human tumour immunology and immunotherapy,, Nature, 411 (2001), 380. [29] S. Ruan and G. S. K. Wolkowicz, Bifurcation analysis of a chemostat model with a distributed delay,, J. Math. Anal. Appl., 204 (1996), 786. doi: 10.1006/jmaa.1996.0468. [30] 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. [31] E. Stockert, E. Jager, Y. T. Chen, M. J. Scanlan, I. Gout, J. Karbach, M. Arand, A. Knuth and L. J. Old, A survey of the humoral immune response of cancer patients to a panel of human tumor antigens,, J. Exp. Med., 187 (1998), 1349. doi: 10.1084/jem.187.8.1349. [32] Z. Szymanska, Analysis of immunotheray models in the context of cancer dynamics,, Int. J. Appl. Math. Comput. Sci., 13 (2003), 407. [33] A. Tsygvintsev, S. Marino and D. E. Kirschner, A mathematical model of gene therapy for the treatment of cancer,, in Mathematical Methods and Models in Biomedicine (eds. U. Ledzewicz, (2013), 367. doi: 10.1007/978-1-4614-4178-6_13. [34] M. Villasana and A. Radunskaya, A delay differential equation model for tumor growth,, J. Math. Biol., 47 (2003), 270. doi: 10.1007/s00285-003-0211-0. [35] L. Zamai, C. Ponti, P. Mirandola, G. Gobbi, S. Papa, L. Galeotti, L. Cocco and M. Vitale, NK cells and cancer,, J. Immunol., 178 (2007), 4011.
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