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June  2016, 21(4): 1279-1295. doi: 10.3934/dcdsb.2016.21.1279

Mathematical and numerical analysis of a mathematical model of mixed immunotherapy and chemotherapy of cancer

1. 

Department of Applied Mathematics, Feng Chia University, Seatwen, Taichung 40724, Taiwan

Received  January 2015 Revised  November 2015 Published  March 2016

In this study, a previously published mathematical model of mixed immunotherapy and chemotherapy of tumors is considered. The stability analysis of the tumor-free equilibrium obtained in the previous study of this model is flawed. In this paper, a suitable analysis is performed to correct this error, and the parameter conditions for the stability of the tumor-free equilibrium are obtained. The stability condition gives an indicator of the host's ability to fight a cancer. The parameter conditions are examined using experimental data from clinical studies to show that the immune system is able to control a small tumor, and the host's ability to fight a cancer depends on individual variation. A numerical method based on the continuation technique is employed for one-parameter bifurcation analysis of the mathematical model with periodically pulsed therapies. The unstable fixed point curve provides a good approximation of the maximum tumor burden as a function of the dosage. Chemotherapy-induced lymphocyte damage, which may cause treatment failure, is observed in the numerical simulation. The numerical method also produces a set of combined chemotherapy and immunotherapy dosages from which an efficient and safe combination of dosages can be determined.
Citation: Hsiu-Chuan Wei. Mathematical and numerical analysis of a mathematical model of mixed immunotherapy and chemotherapy of cancer. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1279-1295. doi: 10.3934/dcdsb.2016.21.1279
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C. Bourquin, S. Schreiber, S. Beck, G. Hartmann and S. Endres, Immunotherapy with dendritic cells and CpG oligonucleotides can be combined with chemotherapy without loss of efficacy in a mouse model of colon cancer,, Int. J. Cancer., 118 (2006), 2790. doi: 10.1002/ijc.21681. Google Scholar

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S. Bunimovich-Mendrazitsky, H. Byrne and L. Stone, Mathematical model of pulsed immunotherapy for superficial bladder cancer,, Bull. Math. Biol., 70 (2008), 2055. doi: 10.1007/s11538-008-9344-z. Google Scholar

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S. Bunimovich-Mendrazitsky, E. Shochat and L. Stone, Mathematical model of BCG immunotherapy in superficial bladder cancer,, Bull. Math. Biol., 69 (2007), 1847. doi: 10.1007/s11538-007-9195-z. Google Scholar

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D. Catovsky, S. Richards, E. Matutes, D. Oscier, M. J. S. Dyer, R. F. Bezares, A. R. Pettitt, T. Hamblin, D. W. Milligan, J. A. Child, M. S. Hamilton, C. E. Dearden, A. G. Smith, A. G. Bosanquet, Z. Davis, V. Brito-Babapulle, M, Else, R. Wade and P. Hillmen, Assessment of fludarabine plus cyclophosphamide for patients with chronic lymphocytic leukaemia (the LRF CLL4 Trial): a randomised controlled trial,, Lancet, 370 (2007), 230. doi: 10.1016/S0140-6736(07)61125-8. Google Scholar

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M. E. Dudley, J. C. Yang, R. Sherry, M. S. Hughes, R. Royal, U. Kammula, P. F. Robbins, J. Huang, D. E. Citrin, S. F. Leitman, J. Wunderlich, N. P. Restifo, A. Thomasian, S. G. Downey, F. O. Smith, J. Klapper, K. Morton, C. Laurencot, D. E. White and S. A. Rosenberg, Adoptive cell therapy for patients with metastatic melanoma: Evaluation of intensive myeloablative chemoradiation preparative regimens,, J. Clin. Oncol., 26 (2008), 5233. doi: 10.1200/JCO.2008.16.5449. Google Scholar

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T. Fehm, V. Mueller, R. Marches, G. Klein, B. Gueckel, H. Neubauer, E. Solomayer and S. Becker, Tumor cell dormancy: Implications for the biology and treatment of breast cancer,, APMIS, 116 (2008), 742. doi: 10.1111/j.1600-0463.2008.01047.x. Google Scholar

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J. Folkman and R. Kalluri, Cancer without disease,, Nature, 427 (2004). doi: 10.1038/427787a. Google Scholar

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D. I. Gabrilovich, Combination of chemotherapy and immunotherapy for cancer: A paradigm revisited,, Lancet Oncol., 8 (2007), 2. doi: 10.1016/S1470-2045(06)70985-8. Google Scholar

[19]

M. Ghielmini, Multimodality therapies and optimal schedule of antibodies: rituximab in lymphoma as an example,, Hematology, 2005 (2005), 321. doi: 10.1182/asheducation-2005.1.321. Google Scholar

[20]

H. S. Hochster, M. M. Oken, J. N. Winter, L. I. Gordon, B. G. Raphael, J. M. Bennett and P. A. Cassileth, Phase I study of fludarabine plus cyclophosphamide in patients with previously untreated low-grade lymphoma: results and and long-term follow-up-a report from the eastern cooperative oncology group,, J. Clin. Oncol., 18 (2000), 987. Google Scholar

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M. Itik and S. P. Banks, Chaos in a three-dimensional cancer model,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 71. doi: 10.1142/S0218127410025417. Google Scholar

[22]

D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction,, J. Math. Biol., 37 (1998), 235. doi: 10.1007/s002850050127. Google Scholar

[23]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory,, $2^{nd}$ edition, (1998). Google Scholar

[24]

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., 56 (1994), 295. Google Scholar

[25]

H. Li, C. Wang, J. Yu, S. Cao, F. Wei, W. Zhang, Y. Han and X. Ren, Dendritic cell-activated cytokine-induced killer cells enhance the anti-tumor effect of chemotherapy on non-small cell lung cancer in patients after surgery,, Cytotherapy, 11 (2009), 1076. Google Scholar

[26]

P. Lissoni, M. Chilelli, S. Villa, L. Cerizza and G. Tancini, Five years survival in metastatic non-small cell lung cancer patients treated with chemotherapy alone or chemotherapy and melatonin: a randomized trial,, J. Pineal Res., 35 (2003), 12. doi: 10.1034/j.1600-079X.2003.00032.x. Google Scholar

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F. K. Nani and M. N. Oguztoreli, Modelling and simulation of Rosenberg-type adoptive cellular immunotherapy,, IMA J. Math. Med. Biol., 11 (1994), 107. doi: 10.1093/imammb/11.2.107. Google Scholar

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A. K. Nowak, B. W. S. Robinson and R. A. Lake, Synergy between chemotherapy and immunotherapy in the treatment of established murine solid tumors,, Cancer Res., 63 (2003), 4490. Google Scholar

[30]

J. C. Panetta, A mathematical model of periodically pulsed chemotherapy: Tumor recurrence and metastasis in a competitive environment,, Bull. Math. Biol., 58 (1996), 425. doi: 10.1007/BF02460591. Google Scholar

[31]

A. S. Perelson and G. Weisbuch, Immunology for physicists,, Rev. Mod. Phys., 69 (1997), 1219. doi: 10.1103/RevModPhys.69.1219. Google Scholar

[32]

B. A. Pockaj, R. M. Sherry, J. P. Wei, J. R. Yannelli, C. S. Carter, S. F. Leitman, J. A. Carasquillo, S. M. Steinberg, S. A. Rosenberg and J. C. Yang, Localization $of ^{111}$Indium-labeled tumor infiltrating lymphocytes to tumor in patients receiving adoptive immunotherapy. Augmentation with cyclophosphamide and correlation with response,, Cancer, 73 (1994), 1731. Google Scholar

[33]

R. Ramakrishnan, D. Assudani, S. Nagaraj, T. Hunter, H. I. Cho, S. Antonia, S. Altiok, E. Celis and D. I. Gabrilovich, Chemotherapy enhances tumor cell susceptibility to CTL-mediated killing during cancer immunotherapy in mice,, J. Clin Invest., 120 (2010), 1111. doi: 10.1172/JCI40269. Google Scholar

[34]

S. A. Rosenberg, Development of effective immunotherapy for the treatment of patients with cancer,, J. Am. Coll. Surg., 198 (2004), 685. doi: 10.1016/j.jamcollsurg.2004.01.025. Google Scholar

[35]

S. A. Rosenberg and M. E. Dudley, Adoptive cell therapy for the treatment of patients with metastatic melanoma,, Curr. Opin. Immunol., 21 (2009), 233. doi: 10.1016/j.coi.2009.03.002. Google Scholar

[36]

S. Suki, H. Kantarjian, V. Gandhi, E. Estey, S. O'Brien, M. Beran, M. B. Rios, W. Plunkett and M. Keating, Fludarabine and cytosine arabinoside in the treatment of refractory or relapsed acute lymphocytic leukemia,, Cancer, 72 (1993), 2155. doi: 10.1002/1097-0142(19931001)72:7<2155::AID-CNCR2820720715>3.0.CO;2-V. Google Scholar

[37]

T. Trisilowati, S. McCue and D. Mallet, Numerical solution of an optimal control model of dendritic cell treatment of a growing tumour,, ANZIAM J., 54 (2013). Google Scholar

[38]

H. C. Wei, A numerical study of a mathematical model of pulsed immunotherapy for superficial bladder cancer,, Jpn. J. Ind. Appl. Math., 30 (2013), 441. doi: 10.1007/s13160-013-0107-3. Google Scholar

[39]

H. C. Wei, S. F. Hwang, J. T. Lin and T. J. Chen, The role of initial tumor biomass size in a mathematical model of periodically pulsed chemotherapy,, Comput. Math. Appl., 61 (2011), 3117. doi: 10.1016/j.camwa.2011.03.102. Google Scholar

[40]

H. C. Wei and J. T. Lin, Periodically pulsed immunotherapy in a mathematical model of tumor-immune interaction,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013). doi: 10.1142/S0218127413500685. Google Scholar

show all references

References:
[1]

N. Almog, Molecular mechanisms underlying tumor dormancy,, Cancer Lett., 294 (2010), 139. doi: 10.1016/j.canlet.2010.03.004. Google Scholar

[2]

A. J. Barrett and B. N. Savani, Does chemotherapy modify the immune surveillance of hematological malignancies?, Leukemia, 23 (2009), 53. doi: 10.1038/leu.2008.273. Google Scholar

[3]

M. J. Besser, R. Shapira-Frommer, A. J. Treves, D. Zippel, Orit Itzhaki, L. Hershkovitz, D. Levy, A. Kubi, E. Hovav, N. Chermoshniuk, B. Shalmon, I. Hardan, R. Catane, G. Markel, S. Apter, A. Ben-Nun, I. Kuchuk, A. Shimoni, A. Nagler and J. Schachter, Clinical responses in a phase II study using adoptive transfer of short-term cultured tumor infiltration lymphocytes in metastatic melanoma patients,, Clin. Cancer Res., 16 (2010), 2646. doi: 10.1158/1078-0432.CCR-10-0041. Google Scholar

[4]

C. Bourquin, S. Schreiber, S. Beck, G. Hartmann and S. Endres, Immunotherapy with dendritic cells and CpG oligonucleotides can be combined with chemotherapy without loss of efficacy in a mouse model of colon cancer,, Int. J. Cancer., 118 (2006), 2790. doi: 10.1002/ijc.21681. Google Scholar

[5]

S. Bunimovich-Mendrazitsky, H. Byrne and L. Stone, Mathematical model of pulsed immunotherapy for superficial bladder cancer,, Bull. Math. Biol., 70 (2008), 2055. doi: 10.1007/s11538-008-9344-z. Google Scholar

[6]

S. Bunimovich-Mendrazitsky, E. Shochat and L. Stone, Mathematical model of BCG immunotherapy in superficial bladder cancer,, Bull. Math. Biol., 69 (2007), 1847. doi: 10.1007/s11538-007-9195-z. Google Scholar

[7]

F. Castiglione and B. Piccoli, Cancer immunotherapy, mathematical modeling and optimal control,, J. Theor. Biol., 247 (2007), 723. doi: 10.1016/j.jtbi.2007.04.003. Google Scholar

[8]

D. Catovsky, S. Richards, E. Matutes, D. Oscier, M. J. S. Dyer, R. F. Bezares, A. R. Pettitt, T. Hamblin, D. W. Milligan, J. A. Child, M. S. Hamilton, C. E. Dearden, A. G. Smith, A. G. Bosanquet, Z. Davis, V. Brito-Babapulle, M, Else, R. Wade and P. Hillmen, Assessment of fludarabine plus cyclophosphamide for patients with chronic lymphocytic leukaemia (the LRF CLL4 Trial): a randomised controlled trial,, Lancet, 370 (2007), 230. doi: 10.1016/S0140-6736(07)61125-8. Google Scholar

[9]

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. Google Scholar

[10]

L. G. de Pillis and A. E. Radunskaya, A mathematical model of immune response to tumor invasion,, in Computational Fluid and Solid Mechanics (ed. K.J. Bathe), (2003), 1661. doi: 10.1016/B978-008044046-0.50404-8. Google Scholar

[11]

L. G. de Pillis and A. E. Radunskaya, The dynamics of an optimally controlled tumor model: A case study,, Math. Comput. Model., 37 (2003), 1221. doi: 10.1016/S0895-7177(03)00133-X. Google Scholar

[12]

A. Diefenbach, E. R. Jensen, A. M. Jamieson and D. H. Raulet, Rae1 and H60 ligands of the NKG2D receptor stimulate tumour immunity,, Nature, 413 (2001), 165. Google Scholar

[13]

M. E. Dudley, J. R. Wunderlich, P. F. Robbins, J. C. Yang, P. Hwu, D. J. Schwartzentruber, S. L. Topalian, R. Sherry, N. P. Restifo, A. M. Hubicki, M. R. Robinson, M. Raffeld, P. Duray, C. A. Seipp, L. Rogers-Freezer, K. E. Morton, S. A. Mavroukakis, D. E. White and S. A. Rosenberg, Cancer regression and autoimmunity in patients after clonal repopulation with antitumor lymphocytes,, Science, 298 (2002), 850. doi: 10.1126/science.1076514. Google Scholar

[14]

M. E. Dudley, J. R. Wunderlich, J. C. Yang, P. Hwu, D. J. Schwartzentruber, S. L. Topalian, R. M. Sherry, F. M. Marincola, S. F. Leitman, C. A. Seipp, L. Rogers-Freezer, K. E. Morton, A. Nahvi, S. A. Mavroukakis, D. E. White and S. A. Rosenberg, A phase I study of nonmyeloablative chemotherapy and adoptive transfer of autologous tumor antigen-specific T lymphocytes in patients with metastatic melanoma,, Immunother., 25 (2008), 243. doi: 10.1097/00002371-200205000-00007. Google Scholar

[15]

M. E. Dudley, J. C. Yang, R. Sherry, M. S. Hughes, R. Royal, U. Kammula, P. F. Robbins, J. Huang, D. E. Citrin, S. F. Leitman, J. Wunderlich, N. P. Restifo, A. Thomasian, S. G. Downey, F. O. Smith, J. Klapper, K. Morton, C. Laurencot, D. E. White and S. A. Rosenberg, Adoptive cell therapy for patients with metastatic melanoma: Evaluation of intensive myeloablative chemoradiation preparative regimens,, J. Clin. Oncol., 26 (2008), 5233. doi: 10.1200/JCO.2008.16.5449. Google Scholar

[16]

T. Fehm, V. Mueller, R. Marches, G. Klein, B. Gueckel, H. Neubauer, E. Solomayer and S. Becker, Tumor cell dormancy: Implications for the biology and treatment of breast cancer,, APMIS, 116 (2008), 742. doi: 10.1111/j.1600-0463.2008.01047.x. Google Scholar

[17]

J. Folkman and R. Kalluri, Cancer without disease,, Nature, 427 (2004). doi: 10.1038/427787a. Google Scholar

[18]

D. I. Gabrilovich, Combination of chemotherapy and immunotherapy for cancer: A paradigm revisited,, Lancet Oncol., 8 (2007), 2. doi: 10.1016/S1470-2045(06)70985-8. Google Scholar

[19]

M. Ghielmini, Multimodality therapies and optimal schedule of antibodies: rituximab in lymphoma as an example,, Hematology, 2005 (2005), 321. doi: 10.1182/asheducation-2005.1.321. Google Scholar

[20]

H. S. Hochster, M. M. Oken, J. N. Winter, L. I. Gordon, B. G. Raphael, J. M. Bennett and P. A. Cassileth, Phase I study of fludarabine plus cyclophosphamide in patients with previously untreated low-grade lymphoma: results and and long-term follow-up-a report from the eastern cooperative oncology group,, J. Clin. Oncol., 18 (2000), 987. Google Scholar

[21]

M. Itik and S. P. Banks, Chaos in a three-dimensional cancer model,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 71. doi: 10.1142/S0218127410025417. Google Scholar

[22]

D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor-immune interaction,, J. Math. Biol., 37 (1998), 235. doi: 10.1007/s002850050127. Google Scholar

[23]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory,, $2^{nd}$ edition, (1998). Google Scholar

[24]

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., 56 (1994), 295. Google Scholar

[25]

H. Li, C. Wang, J. Yu, S. Cao, F. Wei, W. Zhang, Y. Han and X. Ren, Dendritic cell-activated cytokine-induced killer cells enhance the anti-tumor effect of chemotherapy on non-small cell lung cancer in patients after surgery,, Cytotherapy, 11 (2009), 1076. Google Scholar

[26]

P. Lissoni, M. Chilelli, S. Villa, L. Cerizza and G. Tancini, Five years survival in metastatic non-small cell lung cancer patients treated with chemotherapy alone or chemotherapy and melatonin: a randomized trial,, J. Pineal Res., 35 (2003), 12. doi: 10.1034/j.1600-079X.2003.00032.x. Google Scholar

[27]

J. H. Machiels, R. T. Reilly, L. A. Emens, A. M. Ercolini, R. Y. Lei, D. Weintraub, F. I. Okoye and E. M. Jaffee, Cyclophosphamide, doxorubicin, and paclitaxel enhance the antitumor immune response of granulocyte/macrophage-colony stimulating factor-secreting whole-cell vaccines in HER-2/neu tolerized mice,, Cancer Res., 61 (2001), 3689. Google Scholar

[28]

F. K. Nani and M. N. Oguztoreli, Modelling and simulation of Rosenberg-type adoptive cellular immunotherapy,, IMA J. Math. Med. Biol., 11 (1994), 107. doi: 10.1093/imammb/11.2.107. Google Scholar

[29]

A. K. Nowak, B. W. S. Robinson and R. A. Lake, Synergy between chemotherapy and immunotherapy in the treatment of established murine solid tumors,, Cancer Res., 63 (2003), 4490. Google Scholar

[30]

J. C. Panetta, A mathematical model of periodically pulsed chemotherapy: Tumor recurrence and metastasis in a competitive environment,, Bull. Math. Biol., 58 (1996), 425. doi: 10.1007/BF02460591. Google Scholar

[31]

A. S. Perelson and G. Weisbuch, Immunology for physicists,, Rev. Mod. Phys., 69 (1997), 1219. doi: 10.1103/RevModPhys.69.1219. Google Scholar

[32]

B. A. Pockaj, R. M. Sherry, J. P. Wei, J. R. Yannelli, C. S. Carter, S. F. Leitman, J. A. Carasquillo, S. M. Steinberg, S. A. Rosenberg and J. C. Yang, Localization $of ^{111}$Indium-labeled tumor infiltrating lymphocytes to tumor in patients receiving adoptive immunotherapy. Augmentation with cyclophosphamide and correlation with response,, Cancer, 73 (1994), 1731. Google Scholar

[33]

R. Ramakrishnan, D. Assudani, S. Nagaraj, T. Hunter, H. I. Cho, S. Antonia, S. Altiok, E. Celis and D. I. Gabrilovich, Chemotherapy enhances tumor cell susceptibility to CTL-mediated killing during cancer immunotherapy in mice,, J. Clin Invest., 120 (2010), 1111. doi: 10.1172/JCI40269. Google Scholar

[34]

S. A. Rosenberg, Development of effective immunotherapy for the treatment of patients with cancer,, J. Am. Coll. Surg., 198 (2004), 685. doi: 10.1016/j.jamcollsurg.2004.01.025. Google Scholar

[35]

S. A. Rosenberg and M. E. Dudley, Adoptive cell therapy for the treatment of patients with metastatic melanoma,, Curr. Opin. Immunol., 21 (2009), 233. doi: 10.1016/j.coi.2009.03.002. Google Scholar

[36]

S. Suki, H. Kantarjian, V. Gandhi, E. Estey, S. O'Brien, M. Beran, M. B. Rios, W. Plunkett and M. Keating, Fludarabine and cytosine arabinoside in the treatment of refractory or relapsed acute lymphocytic leukemia,, Cancer, 72 (1993), 2155. doi: 10.1002/1097-0142(19931001)72:7<2155::AID-CNCR2820720715>3.0.CO;2-V. Google Scholar

[37]

T. Trisilowati, S. McCue and D. Mallet, Numerical solution of an optimal control model of dendritic cell treatment of a growing tumour,, ANZIAM J., 54 (2013). Google Scholar

[38]

H. C. Wei, A numerical study of a mathematical model of pulsed immunotherapy for superficial bladder cancer,, Jpn. J. Ind. Appl. Math., 30 (2013), 441. doi: 10.1007/s13160-013-0107-3. Google Scholar

[39]

H. C. Wei, S. F. Hwang, J. T. Lin and T. J. Chen, The role of initial tumor biomass size in a mathematical model of periodically pulsed chemotherapy,, Comput. Math. Appl., 61 (2011), 3117. doi: 10.1016/j.camwa.2011.03.102. Google Scholar

[40]

H. C. Wei and J. T. Lin, Periodically pulsed immunotherapy in a mathematical model of tumor-immune interaction,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 23 (2013). doi: 10.1142/S0218127413500685. Google Scholar

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