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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

Hsiu-Chuan Wei 1,

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.
Keywords: Cancer therapy, mathematical analysis, impulsive differential equation, bifurcation analysis, numerical simulation..
Mathematics Subject Classification: Primary: 37N25, 34H20, 92B05; Secondary: 34D2.
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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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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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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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

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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

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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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[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

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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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