A mathematical model of cell-mediated immune response to tumor

Ge Song; Tianhai Tian; Xinan Zhang; Ge Song; Tianhai Tian; Xinan Zhang

doi:10.3934/mbe.2021020

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 1: 373-385. doi: 10.3934/mbe.2021020

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A mathematical model of cell-mediated immune response to tumor

Ge Song ¹,
Tianhai Tian ^{2
,
,},
Xinan Zhang ^{1
,
,}

1.
School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China
2.
School of Mathematics, Monash University, Melbourne, VIC 3800, Australia

Received: 10 September 2020 Accepted: 08 November 2020 Published: 04 December 2020

Mathematical models of tumor-immune interactions provide an analytic framework for studying tumor-immune dynamics. In this paper, we present a mathematical model to describe tumor-immune cell interactions, focusing on the role of the natural killer (NK) cells and CD8+ cytotoxic T lymphocytes (CTLs) in immune surveillance. According to the experimental and clinical results, we determine part of the model parameters to reduce the model parameter space. Then we analyze the local geometric properties of the equilibria of model and carry out numerical simulations to verify the conditions for the stability properties of equilibrium points. Numerical results suggest that the host immune system alone is not fully effective against progression of tumor cells, and CTLs play a crucial role in immune surveillance.
- mathematical model,
- tumor cells,
- natural killer cells,
- cytotoxic T lymphocytes
Citation: Ge Song, Tianhai Tian, Xinan Zhang. A mathematical model of cell-mediated immune response to tumor[J]. Mathematical Biosciences and Engineering, 2021, 18(1): 373-385. doi: 10.3934/mbe.2021020

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Abstract

Mathematical models of tumor-immune interactions provide an analytic framework for studying tumor-immune dynamics. In this paper, we present a mathematical model to describe tumor-immune cell interactions, focusing on the role of the natural killer (NK) cells and CD8+ cytotoxic T lymphocytes (CTLs) in immune surveillance. According to the experimental and clinical results, we determine part of the model parameters to reduce the model parameter space. Then we analyze the local geometric properties of the equilibria of model and carry out numerical simulations to verify the conditions for the stability properties of equilibrium points. Numerical results suggest that the host immune system alone is not fully effective against progression of tumor cells, and CTLs play a crucial role in immune surveillance.

References

[1]	F. Bray, A. Jemal, N. Grey, J. Ferlay, D. Forman, Global cancer transitions according to the Human Development Index (2008-2030): a population-based study, Lancet Oncol., 13 (2012), 790-801. doi: 10.1016/S1470-2045(12)70211-5
[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. doi: 10.1158/0008-5472.CAN-05-0564
[3]	A. G. López, J. M. Seoane, M. A. F. Sanjuán, A validated mathematical model of tumor growth including tumor-host interaction, cell-mediated immune response and chemotherapy, Bull. Math. Biol., 76 (2014), 2884-906.
[4]	A. G. López, J. M. Seoane, M. A. F. Sanjuán, Destruction of solid tumors by immune cells, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 390-403. doi: 10.1016/j.cnsns.2016.08.020
[5]	L. Chiossone, P. Y. Dumas, M. Vienne, E. Vivier, Natural killer cells and other innate lymphoid cells in cancer, Nat. Rev. Immunol., 18 (2018), 671-688. doi: 10.1038/s41577-018-0061-z
[6]	H. Fujisaki, H. Kakuda, N. Shimasaki, C. Imai, J. Ma, T. Lockey, et al., Expansion of highly cytotoxic human natural killer cells for cancer cell therapy, Cancer Res., 69 (2009), 4010-4017. doi: 10.1158/0008-5472.CAN-08-3712
[7]	L. V. Hooper, D. R. Littman, A. J. Macpherson, Interactions between the microbiota and the immune system, Science, 336 (2012), 1268-1273. doi: 10.1126/science.1223490
[8]	G. Gasteiger, X. Fan, S. Dikiy, S. Lee, A. Rudensky, Tissue residency of innate lymphoid cells in lymphoid and nonlymphoid organs, Science, 350 (2015), 981-985. doi: 10.1126/science.aac9593
[9]	K. D. Moynihan, D. J. Irvine, Roles for innate immunity in combination immunotherapies, Cancer Res., 77 (2017), 5215-5221. doi: 10.1158/0008-5472.CAN-17-1340
[10]	E. Vivier, D. H. Raulet, A. Moretta, M. Caligiuri, L. Zitvogel, L. L. Lanier, et al., Innate or adaptive immunity? The example of natural killer cells, Science, 331 (2011), 44-49. doi: 10.1126/science.1198687
[11]	H. Arase, E. S. Mocarski, A. E. Campbell, A. B. Hill, L. L. Lanier, Direct recognition of cytomegalovirus by activating and inhibitory NK cell receptors, Science, 296 (2002), 1323-1326. doi: 10.1126/science.1070884
[12]	R. B. Herberman, M. E. Nunn, D.H. Lavrin, Natural cytotoxic reactivity of mouse lymphoid cells against syngeneic acid allogeneic tumors. I. Distribution of reactivity and specificity, Int. J. Cancer, 16 (1975), 216-229. doi: 10.1002/ijc.2910160204
[13]	J. M. Roda, R. Parihar, C. Magro, G. J. Nuovo, S. Tridandapani, W. E. Carson, Natural killer cells produce T cell-recruiting chemokines in response to antibody-coated tumor cells, Cancer Res., 66 (2006), 517-526. doi: 10.1158/0008-5472.CAN-05-2429
[14]	J. P. Bottcher, E. Bonavita, P. Chakravarty, H. Blees, M. Cabeza-Cabrerizo, S. Sammicheli, et al., NK cells stimulate recruitment of cDC1 into the tumor microenvironment promoting cancer immune control, Cell, 172 (2018), 1022-1037. doi: 10.1016/j.cell.2018.01.004
[15]	F. Hoffman, D. Gavaghan, J. Osborne, I. P. Barrettet, T. You, H. Ghadially, et al., A mathematical model of antibody-dependent cellular cytoxicity (ADCC), J. Theor. Biol., 436 (2018), 39-50. doi: 10.1016/j.jtbi.2017.09.031
[16]	H. Dianat-Moghadam, M. Rokni, F. Marofi, Y. Panahi, M. Yousefi, Natural killer cell-based immunotherapy: From transplantation toward targeting cancer stem cells, J. Cell Physiol., 234 (2019), 259-273. doi: 10.1002/jcp.26878
[17]	C. Hong, H. Lee, Y. K. Park, J. Shin, S. Jung, H. Kim, et, al., Regulation of secondary antigen-specific CD8+ T-cell responses by natural killer T cells, Cancer Res., 69 (2009), 4301-4308. doi: 10.1158/0008-5472.CAN-08-1721
[18]	L. G. De Pillis, A. E. Radunskaya, A mathematical model of immune response to tumor invasion, Comput. Fluid Solid Mech., (2003), 1661-1668.
[19]	C. C. Ku, M. Murakami, A. Sakamoto, J. Kappler, P. Marrack, Control of homeostasis of CD8+ memory T cells by opposing cytokines, Science, 288 (2000), 675-678. doi: 10.1126/science.288.5466.675
[20]	M. Barry, R. C. Bleackley, Cytotoxic T lymphocytes: All roads lead to death, Nat. Rev. Immunol., 2 (2002), 401-409. doi: 10.1038/nri819
[21]	J. Brummelman, E.M. Mazza, G. Alvisi, F. Colombo, A. Grilli, J. Mikulak, et al., High-dimensional single cell analysis identifies stem-like cytotoxic CD8+ T cells infiltrating human tumors, J. Exp. Med., 215 (2018), 2520-2535. doi: 10.1084/jem.20180684
[22]	M. Robertson-Tessi, A. Elkareh, A. Goriely, A mathematical model of tumor-immune interactions, J. Theor. Biol., 294 (2012), 56-73. doi: 10.1016/j.jtbi.2011.10.027
[23]	K. J. Mahasa, R. Ouifki, A. Eladdadi, L. G. Pillis, Mathematical Model of Tumor-Immune Surveillance, J. Theor. Biol., 404 (2016), 312-330. doi: 10.1016/j.jtbi.2016.06.012
[24]	L. G. Pillis, W. Gu, A. E. Radunskaya, Mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretations, J. Theor. Biol., 238 (2006), 841-862. doi: 10.1016/j.jtbi.2005.06.037
[25]	A. Lanzavecchia, F. Sallusto, Dynamics of T lymphocyte responses: intermediates, effectors, and memory cells, Science, 290 (2000), 92-97.
[26]	E. Sercarz, A. H. Coons, The exhaustion of specific antibody producing capacity during a secondary response, Mech. Immunol. Tolerance Conf., Czechoslovak Academy of Sciences Press Prague, New York, 1962, 78-83.
[27]	C. Delisi, A. Rescigno, Immune surveillance and neoplasia. I. A minimal mathematical model, Bull. Math. Biol., 39 (1977), 201-221.
[28]	V. Kuznetsov, I. Makalkin, M. Taylor, A. Perelson, Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis, Bull. Math. Biol., 56 (1994), 295-321. doi: 10.1016/S0092-8240(05)80260-5
[29]	A. Diefenbach, E. Jensen, A. Jamieson, D. Raulet, Rae1 and H60 ligands of the NKG2D receptor stimulate tumor immunity, Nature, 413 (2001), 165-171. doi: 10.1038/35093109
[30]	T. Berris, M. Mazonakis, J. Stratakis, A. Tzedakis, A. Fasoulaki, J. Damilakis, Calculation of organ doses from breast cancer radiotherapy: a Monte Carlo study, J. Appl. Clin. Med. Phys., 14 (2013), 133-146.
[31]	D. Kirschner, J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction, J. Math. Biol., 37 (1998), 235-252. doi: 10.1007/s002850050127
[32]	A. Yates, R. Callard, Cell death and the maintenance of immunological memory, Discrete Contin. Dyn. Syst.-B, 1 (2001), 43-59.
[33]	M. E. Dudley, J. R. Wunderlich, P. F. Robbins, J. Yang, P. Hwu, D. Schwartzentruber, et al., Cancer regression and autoimmunity in patients after clonal repopulation with antitumor lymphocytes, Science, 298 (2002), 850-854. doi: 10.1126/science.1076514
[34]	L. Chen, Mathematical Models and Methods in Ecology (in Chinese), Science Press, Beijing, 1988,174-198.
[35]	L. G. De Pillis, A. Eladdadi, A. E. Radunskaya, Modeling cancer-immune responses to therapy, J. Pharmacokinet. Pharmacodyn., 41 (2014), 461-478. doi: 10.1007/s10928-014-9386-9
[36]	L. G. De Pillis, W. Gu, K. R. Fister, T. Head, K. Maples, A. Murugan, et al., Chemotherapy for tumors: An analysis of the dynamics and a study of quadratic and linear optimal controls, Math. Biosci., 209 (2007), 292-315. doi: 10.1016/j.mbs.2006.05.003

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