Abstract
Cancer therapies that harness the actions of the immune response, such as targeted monoclonal antibody treatments and therapeutic vaccines, are relatively new and promising in the landscape of cancer treatment options. Mathematical modeling and simulation of immune-modifying therapies can help to offset the costs of drug discovery and development, and encourage progress toward new immunotherapies. Despite advances in cancer immunology research, questions such as how the immune system interacts with a growing tumor, and which components of the immune system play significant roles in responding to immunotherapy are still not well understood. Mathematical modeling and simulation are powerful tools that provide an analytical framework in which to address such questions. A quantitative understanding of the kinetics of the immune response to treatment is crucial in designing treatment strategies, such as dosing, timing, and predicting the response to a specific treatment. These models can be used both descriptively and predictively. In this chapter, various mathematical models that address different cancer treatments, including cytotoxic chemotherapy, immunotherapy, and combinations of both treatments, are presented. The aim of this chapter is to highlight the importance of mathematical modeling and simulation in the design of immunotherapy protocols for cancer treatment. The results demonstrate the power of these approaches in explaining determinants that are fundamental to cancer-immune dynamics, therapeutic success, and the development of efficient therapies.
Similar content being viewed by others
References
Bray F, Jemal A, Grey N, Ferlay J, Forman D (2012) Global cancer transitions according to the Human Development Index (20082030): a population-based study. Lancet Oncol 13:790–801
Biemar F, Foti M (2013) Global progress against cancer-challenges and opportunities. Cancer Biol Med 10:183–6
Topalian SL, Weiner GJ, Pardoll DM (2011) Cancer Immunotherapy Comes of Age. J Clin Oncol 29:4828–4836
Gorelik B, Ziv I, Shohat R, Wick M, Hankins WD, Sidransky D et al (2008) Efficacy of weekly docetaxel and bevacizumab in mesenchymal chondrosarcoma: a new theranostic method combining xenografted biopsies with a mathematical model. Cancer Res 68(21):9033–9040
Besse IM, Madsen MT, Bushnell DL, Juweid ME (2009) Modeling combined radiopharmaceutical therapy: a linear optimization framework. Technol Cancer Res Treat 8(1):51–60
Marqa MF, Mordon S, Betrouni N (2012) Laser interstitial thermotherapy of small breast fibroadenomas: numerical simulations. Lasers Surg Med 44(10):832–839
Berris T, Mazonakis M, Stratakis J, Tzedakis A, Fasoulaki A, Damilakis J (2013) Calculation of organ doses from breast cancer radiotherapy: a Monte Carlo study. J Appl Clin Med Phys 14(1):133–146
Satti J (2009) The emerging low-dose therapy for advanced cancers. Dose-Response 7(3):208–220
Traina TA, Theodoulou M, Feigin K, Patil S, Tan KL, Edwards C et al (2008) Phase I study of a novel capecitabine schedule based on the Norton-Simon mathematical model in patients with metastatic breast cancer. J Clin Oncol 26(11):1797–1802
Comen E, Morris PG, Norton L (2012) Translating mathematical modeling of tumor growth patterns into novel therapeutic approaches for breast cancer. J Mammary Gland Biol Neoplasia 17(3—-4, SI):241–249
Newton PK, Mason J, Bethel K, Bazhenova L, Nieva J, Norton L et al (2013) Spreaders and sponges define metastasis in lung cancer: a Markov Chain Monte Carlo mathematical model. Cancer Res 73(9):2760–2769
The annual 2013 Pharmaceutical Industry Profile. http://www.phrma.org/industryprofile2013; 2013. Accessed March 23, 2014
Agur Z, Vuk-Pavlovic S (2012) Mathematical modeling in immunotherapy of cancer: personalizing clinical trials. Mol Ther 20(1):1
Eftimie R, Bramson JL, Earn DJ (2011) Interactions between the immune system and cancer: a brief review of non-spatial mathematical models. Bull Math Biol 73:2–32
Ledzewicz U, Faraji M, Schaettler H (2012) On Optimal Protocols for Combinations of Chemo- and Immunotherapy. Proceedings of 51st IEEE Conference on Decision and Control, Maui 2012;pp. 7492–7497
d’Onofrio A, Ledzewicz U, Schaettler H (2012) New challenges for cancer systems biology. Springer Verlag, New York
d’Onofrio A (2008) Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy. Math Comput Model 47(5):614–637
d’Onofrio A (2005) A general framework for modeling tumor-immune system competition and immunotherapy: mathematical analysis and biomedical inferences. Physica D 208(3):220–235
Adam JA, Bellomo N (1997) A survey of models for tumor immune systems dynamics. Springer, New York
Agur Z, Arakelyan L, Merbl Y, Daugulis P, Ginosar Y, Vainstein V, et al. (2003) Cancer Modeling and Simulation. CRC Press/Chapman & Hall, ed: Luigi Preziosi; 2003. p. 185–219
Bar-Or R (2000) Feedback mechanisms between T helper cells and macrophages in the determination of the immune response. Math Biosci 163(1):35–58
Castiglione FF, Piccoli B (2006) Optimal control in a model of dendritic cell transfection cancer immunotherapy. Bull Math Biol 68(2):255–274
De Boer RJ, Hogeweg P, Dullens HF, De Weger RA, Den Otter W (1985) Macrophage T lymphocyte interactions in the anti-tumor immune response: a mathematical model. J Immunol 134(4):2748–2758
Rihan F, Abdel Rahman D, Lakshmanan S, Alkhajeh A (2014) A time delay model of tumourimmune system interactions:Global dynamics, parameter estimation, sensitivity analysis. Appl Math Comput 232:606–623
Kronik N, Kogan Y, Vainstein V, Agur Z (2008) Improving alloreactive CTL immunotherapy for malignant gliomas using a simulation model of their interactive dynamics. Cancer Immunol Immunother 57(3):425–439
Kuznetsov V (1997) Basic models of tumor-immune system interactions - identification, analysis and predictions. In: Adam J, Bellomo N (eds) A survey of models for tumor-immune system dynamics. Springer, New York, pp 237–294
Lin A (2004) A model of tumor and lymphocyte interactions. Discrete Contin Dyn Syst Ser B 4(1):241–266
Mallet DG, de Pillis LG (2006) A cellular automata model of tumor-immune system interactions. J Theor Biol 239(3):334–350
Nazari S, Basirzadeh H (2014) Natural Killer or T-lymphocyte cells: which is the best immune therapeutic agent for cancer? an optimal control approach. Int J Control Autom Syst 12:84–92
Takayanagi T, Ohuchi A (2001) A mathematical analysis of the interactions between immunogenic tumor cells and cytotoxic T lymphocytes. Microbiol Immunol 45(10):709–715
de Vladar HP, Gonzalez JA (2004) Dynamic response of cancer under the influence of immunological activity and therapy. J Theor Biol 227(3):335–348
Wein LM, Wu JT, Kirn DH (2003) Validation and analysis of a mathematical model of a replication-competent oncolytic virus for cancer treatment: implications for virus design and delivery. Cancer Res 63(6):1317–1324
Yueping Dong Rinko Miyazaki YT (2014) Mathematical modeling on helper T cells in a tumor immune system. Discret Contin Dyn Syst Ser B 19:55–72
Kuznetsov V, Makalkin I, Taylor M, Perelson A (1994) Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis. Bull Math Biol 56(2):295–321
de Pillis L, Radunskaya AE (2014) Modeling of tumor-immune dynamics. In: Eladdadi A, Kim P (eds) Mathematical modeling of tumor-immune dynamics. Springer, New York, pp 67–115
Roesch K, Hasenclever D, Scholz M (2014) Modelling Lymphoma Therapy and Outcome. Bull Math Biol 76(2):401–430
Pfreundschuh M, Trmper L, Kloess M, Schmits R, Feller AC, Rudolph C et al (2004) Two-weekly or 3-weekly CHOP chemotherapy with or without etoposide for the treatment of young patients with good-prognosis (normal LDH) aggressive lymphomas: results of the NHL-B1 trial of the DSHNHL. Blood 104(3):626–633
Pfreundschuh M, Trmper L, Kloess M, Schmits R, Feller AC, Rbe C et al (2004) Two-weekly or 3-weekly CHOP chemotherapy with or without etoposide for the treatment of elderly patients with aggressive lymphomas: results of the NHL-B2 trial of the DSHNHL. Blood 104(3):634–641
Pfreundschuh M, Schubert J, Ziepert M, Schmits R, Mohren M, Lengfelder E et al (2008) Six versus eight cycles of bi-weekly CHOP-14 with or without rituximab in elderly patients with aggressive \(\text{ CD20 }^+\) B-cell lymphomas: a randomised controlled trial (RICOVER-60). Lancet Oncol 9:105–116
Thomlinson R (1982) Measurement and management of carcinoma of the breast. Clin Radiol 33(5):481–493
de Pillis L, Radunskaya A (2001) A mathematical tumor model with immune resistance and drug therapy: an optimal control approach. J Theor Med 3:79–100
de Pillis L, Radunskaya A (2003) The dynamics of an optimally controlled tumor model: a case study. Math Comput Model (Special Issue) 37:1221–1244
de Pillis L, Gu W, Fister K, Head T, Maples K, Murugan A et al (2007) Chemotherapy for tumors: an analysis of the dynamics and a study of quadratic and linear optimal controls. Math Biosci 209:292–315
de Pillis L, Radunskaya A (2012) Best practices in mathematical modeling. In: Mayeno A, Reisfeld B (eds) Computational toxicology, methods in molecular biology, part 2. Springer, New York, pp 51–74
dePillis LG, Radunskaya AE, Wiseman CL (2005) A validated mathematical model of cell-mediated immune response to tumor growth. Cancer Res 65(1):7950–7958
Lai R, Jackson T (2004) A mathematical model of receptor-mediated apoptosis: dying to know why FasL is a trimer. Math Biosci Eng 1(2):325–338
Diefenbach A, Jensen E, Jamieson A, Raulet D (2001) Rae1 and H60 ligands of the NKG2D receptor stimulate tumor immunity. Nature 413:165–171
Dudley ME, Wunderlich JR, Robbins PF, Yang JC, Hwu P, Schwartzentruber DJ et al (2002) Cancer regression and autoimmunity in patients after Clonal repopulation with antitumor lymphocytes. Science 298(5594):850–854
Castiglione FF, Castiglione V, Agur Z (2003) Cancer Modelling and Simulation. Chapman & Hall/CRC Mathematical and Computational Biology, Luigi Preziosi, ed.; 2003. p. 333–366
Roeder I, Horn M, Glauche I, Hochhaus A, Mueller M, Loeffler M (2006) Dynamic modeling of imatinib-treated chronic myeloid leukemia: functional insights and clinical implications. Nat Med 12:11811184
Michor F, Hughes T, Iwasa Y, Branford S, Shah N, Sawyers C et al (2005) Dynamics of chronic myeloid leukemia. Nature 435:1267–1270
Kim PS, Lee PP, Levy D (2008) Modeling imatinib-treated chronic myelogenous leukemia: reducing the complexity of agent-based models. Bull Math Biol 70(3):728–744
Kim PS, Lee PP, Levy D (2008) A PDE model for imatinib-treated chronic myelogenous leukemia. Bull Math Biol 70(7):1994–2016
Paquin D, Kim PS, Lee PP, Levy D (2011) Strategic treatment interruptions during imatinib treatment of chronic myelogenous leukemia. Bull Math Biol 73(5):1082–1100
Rosenberg SA (2004) Development of effective immunotherapy for the treatment of patients with cancer. J Am Coll Surg 198(5):685
Wainwright DA, Nigam P, Thaci B, Dey M, Lesniak MS (2012) Recent developments on immunotherapy for brain cancer. Expert Opin Emerg Drugs 17:181–201
Kirschner DD, Panetta JC (1998) Modeling immunotherapy of the tumor - immune interaction. J Math Biol 37(3):235–252
Goldsby RA, Kindt TJ, Osborne BA, Kuby J (2003) Immunology, 5th edn. W. H. Freeman, New York
Rosenberg S, Yang J, Schwartzentruber D, Hwu P, Marincola F, Topalian S et al (1999) Prospective randomized trial of the treatment of patients with metastatic melanoma using chemotherapy with cisplatin, dacarbazine, and tamoxifen alone or in combination with interleukin-2 and interferon alfa-2b. J Clin Oncol 17(3):968–975
Cappuccio A, Elishmereni M, Agur Z (2006) Cancer immunotherapy by interleukin-21: potential treatment strategies evaluated in a mathematical model. Cancer Res 66(14):7293–7300
Bellomo N, Bellouquid A, Delitala M (2004) Mathematical topics on the modelling complex multicellular systems and tumor immune cells competition. Math Models Methods Appl Sci 14(11):1683–1733
Bellomo N, Preziosi L (2000) Modelling and mathematical problems related to tumor evolution and its interaction with the immune system. Math Comput Model 32(3):413–452
Bellomo N, Bellouquid A, DeAngelis E (2003) The modelling of the immune competition by generalized Kinetic (Boltzmann) MModel: review and research Perspectives. Math Comput Model 37:65–86
Bellomo N, Bertotti ML, Motta S (2003) Cancer Modelling and Simulation. Chapman & Hall/CRC Mathematical and Computational Biology, Luigi Preziosi, ed.; 2003. p. 299–332
Bellomo N, Forni G (2006) Looking for new paradigms towards a biological-mathematical theory of complex multicellular systems. Math Model Methods Appl Sci 16:1001–1029
Antony P, Restifo N (2005) CD4+CD25+ T regulatory cells, immunotherapy of cancer, and interleukin-2. J Immunother 28(2):120–128
Radunskaya A, Hook S (2012) Modeling the Kinetics of the immune response. In: d’Onofrio A, Cerrai P, Gandolfi A (eds) New challenges for cancer systems biomedicine. Springer-Verlag, New York, pp 267–282
Cheever MA (2011) PROVENGE (Sipuleucel-T) in Prostate Cancer: The first FDA-Approved Therapeutic Cancer Vaccine. Clin Cancer Res 17(11):3520–3526
de Pillis L, Gallegos A, Radunskaya A (2013) A model of dendritic cell therapy for melanoma. Front Oncol 3(56):1–14
Ludewig BB, Krebs P, Junt T, Metters H, Ford NJ, Anderson RM et al (2004) Determining control parameters for dendritic cell-cytotoxic T lymphocyte interaction. Eur J Immunol 34(9):2407–2418. doi:10.1002/eji.200425085
Lee TH, Cho YH, Lee MG (2007) Larger numbers of immature dendritic cells augment an anti-tumor effect against established murine melanoma cells. Biotechnol Lett 29(3):351–357
Preynat-Seauve O, Contassot E, Schuler P, French LE, Huard B (2007) Melanoma-infiltrating dendritic cells induce protective antitumor responses mediated by T cells. Melanoma Res 17:169–176
de Pillis L, Caldwell T, Sarapata E, Williams H (2013) Mathematical modeling of the regulatory T cell effects on renal cell carcinoma treatment. Discret Contin Dyn Syst Ser B 18(4):915–943
Meropol N, Barresi G, Fehniger T, Hitt J, Franklin M, Caligiuri M (1998) Evaluation of natural killer cell expansion and activation in vivo with daily subcutaneous low-dose interleukin-2 plus periodic intermediate-dose pulsing. Cancer Immunol Immunother 46:318326
Jea Ko (2009) Sunitinib mediates reversal of myeloid-derived suppressor cell accumulation in renal cell carcinoma patients. Clin Cancer Res 6:2148–2157
Nanda S, dePillis LG, Radunskaya AE (2013) B cell chronic lymphocytic leukemia — a model with immune response. Discret Contin Dyn Syst Ser B 18(4):1053–1076
Messmer BT, Messmer D, Allen SL, Kolitz JE, Kudalkar P, Cesar D et al (2005) In vivo measurements document the dynamic cellular kinetics of chronic lymphocytic leukemia B cells. J Clin Invest 115:755–764
Ramos RA, Zapata J, Condat CA, Deisboeck TS (2013) Modeling cancer immunotherapy: assessing the effects of lymphocytes on cancer cell growth and motility. Physica A 392:2415–2425
Kang-Ling Liao Xue-Feng Bai AF (2014) Mathematical modeling of Interleukin-27 induction of anti-tumor T cells response. PLoS ONE 9(3):e91844
Dong Y, Miyazaki R, Takeuchi Y (2014) Mathematical modeling on helper T cells in a tumor immune system. Discret Contin Dyn Syst Ser B 19:55–72
Galach M (2003) Dynamics of the tumor-immune system competition - the effect of time delay. Int J Appl Math Comput Sci 13:395–406
de Pillis L, Gu W, Radunskaya A (2006) Mixed immunotherapy and chemotherapy of tumors: modeling applications and biological interpretations. J Theor Biol 238(4):841–862
Yee C, Thompson JA, Byrd D, Riddell SR, Roche P, Celis E et al (2002) Adoptive T cell therapy using antigen-specific CD8+ T cell clones for the treatment of patients with metastatic melanoma: In vivo persistence, migration, and antitumor effect of transferred T cells. Proc Natl Acad Sci USA 99(25):16168–16173
Dillman RO, DePriest C, McClure SE (2014) High-Dose IL2 in metastatic melanoma: better survival in patients immunized with antigens from autologous tumor cell lines. Cancer Biother Radiopharm 29(2):53–57
Machiels J, Reilly R, Emens L, Ercolini A, Lei R, Weintraub D et al (2001) 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(9):3689–3697
de Pillis L, Fister KR, Gu W, Collins C, Daub M, Gross D et al (2009) Mathematical model creation for cancer chemo-immunotherapy. Comput Mat Methods Med 10(3):165–184
de Pillis LG, Fister KR, Gu W, Collins C, Daub M, Gross D et al (2007) Seeking bang-bang solutions of mixed immuno-chemotherapy of tumors. Electron J Diff Eqns 171:1–24
de Pillis LG, Fister KR, Gu W, Head T, Maples K, Neal T et al (2008) Optimal control of mixed immunotherapy and chemotherapy of tumors. J Biol Syst 16(1):51–80
de Pillis LG, Radunskaya AE, Savage H (2014) Mathematical model of colorectal cancer with monoclonal antibody treatments. Br J Med Med Res 4(16):3101–3131
Lenz HJ (2007) Cetuximab in the management of colorectal cancer. Biol: Targ Ther 2:77–91
Grothey AM (2006) Defining the role of panitumumab in colorectal cancer. Commun Oncol 3:10–16
Gravalos C, Cassinello J, Garcia-Alfonso P, Jimeno A (2010) Integration of panitumumab into the treatment of colorectal cancer. Crit Rev Oncol/Hematol 74(1):16–26
De Vita VJ, Hellman S, Rosenberg S (2000) Cancer: principles and practice of oncology, 7th edn. Lippincott Wiliams & Wilkins, Philadelphia
Kim PS, Lee PP, Levy D (2008) Dynamics and potential impact of the immune response to chronic myelogenous leukemia. PLoS Comput Biol 4(6):e1000095
Peet MM, Kim PS, Niculescu SI, Levy D (2009) New computational tools for modeling chronic myelogenous leukemia. Math Model Nat Phenom 4:119–139
Radunskaya A, de Pillis L, Gallegos A (2013) A model of dendritic cell therapy for melanoma. Front Oncol 3:56
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
dePillis, L.G., Eladdadi, A. & Radunskaya, A.E. Modeling cancer-immune responses to therapy. J Pharmacokinet Pharmacodyn 41, 461–478 (2014). https://doi.org/10.1007/s10928-014-9386-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10928-014-9386-9