Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Mathematical modeling of tumor growth: the MCF-7 breast cancer cell line

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

Special Issues: Differential Equations in Mathematical Biology

Breast cancer is the second most commonly diagnosed cancer in women worldwide. MCF-7 cell line is an extensively studied human breast cancer cell line. This cell line expresses estrogen receptors, and the growth of MCF-7 cells is hormone dependent. In this study, a mathematical model, which governs MCF-7 cell growth with interaction among tumor cells, estradiol, natural killer (NK) cells, cytotoxic T lymphocytes (CTLs) or CD8+ T cells, and white blood cells (WBCs), is proposed. Experimental data are used to determine functional forms and parameter values. Breast tumor growth is then studied using the mathematical model. The results obtained from numerical simulation are compared with those from clinical and experimental studies. The system has three coexisting stable equilibria representing the tumor free state, a microscopic tumor, and a large tumor. Numerical simulation shows that an immune system is able to eliminate or control a tumor with a restricted initial size. A healthy immune system is able to effectively eliminate a small tumor or produces long-term dormancy. An immune system with WBC count at the low parts of the normal ranges or with temporary low NK cell count is able to eliminate a smaller tumor. The cytotoxicity of CTLs plays an important role in immune surveillance. The association between the circulating estradiol level and cancer risk is not significant.
  Article Metrics

Keywords breast cancer; MCF-7 cell line; long-term dormancy; mathematical modeling; numerical simulation

Citation: Hsiu-Chuan Wei. Mathematical modeling of tumor growth: the MCF-7 breast cancer cell line. Mathematical Biosciences and Engineering, 2019, 16(6): 6512-6535. doi: 10.3934/mbe.2019325


  • 1. D. R. Jutagir, B. B. Blomberg, C. S. Carver, et al., Social well-being is associated with less pro-inflammatory and pro-metastatic leukocyte gene expression in women after surgery for breast cancer, Breast Cancer Res. Treat., 165 (2017), 169–180.
  • 2. S. Katkuri and M. Gorantla, Awareness about breast cancer among women aged 15 years and above in urban slums: a cross sectional study, Int. J. Community Med. Public Health, 5 (2018), 929–932.
  • 3. A. Pawlik, M. Slomi´ nska-Wojewódzka and A. Herman-Antosiewicz, Sensitization of estrogen receptor-positive breast cancer cell lines to 4-hydroxytamoxifen by isothiocyanates present in cruciferous plants, Eur. J. Nutr., 55 (2016), 1165–1180.
  • 4. D. L. Holliday and V. Speirs, Choosing the right cell line for breast cancer research, Breast Cancer Res., 13 (2011), 215–215.
  • 5. R. L. Sutherland, R. E. Hall and I. W. Taylor, Cell proliferation kinetics of MCF-7 human mammary carcinoma cells in culture and effects of tamoxifen on exponentially growing and plateau-phase cells, Cancer Res., 43 (1983), 3998–4006.
  • 6. B. S. Katzenellenbogen, K. L. Kendra, M. J. Norman, et al., Proliferation, hormonal responsiveness, and estrogen receptor content of MCF-7 human breast cancer cells grown in the short-term and long-term absence of estrogens, Cancer Res., 47(1987), 4355–4360.
  • 7. A. Maton, Human biology and health, 1st edition, Prentice Hall, New Jersey, 1997.
  • 8. L. V. Rao, B. A. Ekberg, D. Connor, et al., Evaluation of a new point of care automated complete blood count (CBC) analyzer in various clinical settings, Clin. Chim. Acta., 389 (2008), 120–125.
  • 9. S. Bernard, E. Abdelsamad, P. Johnson, et al., Pediatric leukemia: diagnosis to treatment a review, J. Cancer Clin. Trials, 2(2017), 131.
  • 10. A. Shankar, J. J. Wang, E. Rochtchina, et al., Association between circulating white blood cell count and cancer mortality: a population-based cohort study, Arch. Intern. Med., 166 (2006), 188–194.
  • 11. K. Kim, J. Lee, N. J. Heo, et al., Differential white blood cell count and all-cause mortality in the Korean elderly, Exp. Gerontol., 48 (2013), 103–108.
  • 12. C. Ruggiero, E. J. Metter, A. Cherubini, et al., White blood cell count and mortality in the Baltimore Longitudinal Study of Aging, J. Am. Coll. Cardiol., 49 (2007), 1841–1850.
  • 13. D. S. Bell and J. H. O'Keefe, White cell count, mortality, and metabolic syndrome in the Baltimore longitudinal study of aging, J. Am. Coll. Cardiol., 50(2007), 1810.
  • 14. G. D. Friedman and B. H. Fireman, The leukocyte count and cancer mortality, Am. J. Epidemiol., 133 (1991), 376–380.
  • 15. M. H. Andersen, D. Schrama, P. thor Straten, et al., Cytotoxic T cells, J. Invest. Dermatol., 126 (2006), 32–41.
  • 16. J. Folkman and R. Kalluri, Cancer without disease, Nature, 427 (2004), 787.
  • 17. T. Fehm, V. Mueller, R. Marches, et al., Tumor cell dormancy: implications for the biology and treatment of breast cancer, Apmis, 116 (2008), 742–753.
  • 18. N. Almog, Molecular mechanisms underlying tumor dormancy, Cancer Lett., 294 (2010), 139–146.
  • 19. A. Friedman, Cancer as multifaceted disease, Math. Model. Nat. Pheno., 7(2012), 3–28.
  • 20. R. Eftimie, J. L. Bramson and D. J. Earn, Interactions between the immune system and cancer: a brief review of non-spatial mathematical models, Bull. Math. Biol., 73 (2011), 2–32.
  • 21. S. Banerjee and R. R. Sarkar, Delay-induced model for tumor–immune interaction and control of malignant tumor growth, Biosystems, 91 (2008), 268–288.
  • 22. H. Moore and N. K. Li, A mathematical model for chronic myelogenous leukemia (CML) and T cell interaction, J. Theor. Biol., 227 (2004), 513–523.
  • 23. L. Anderson, S. Jang and J. Yu, Qualitative behavior of systems of tumor-CD4+-cytokine interactions with treatments, Math. Method. Appl. Sci., 38 (2015), 4330–4344.
  • 24. A. d'Onofrio, Metamodeling tumor–immune system interaction, tumor evasion and immunotherapy, Math. Comput. Model., 47 (2008), 614–637.
  • 25. A. Khar, Mechanisms involved in natural killer cell mediated target cell death leading to spontaneous tumour regression, J. Biosci., 22 (1997), 23–31.
  • 26. T. Boon and P. van der Bruggen, Human tumor antigens recognized by T lymphocytes, J. Exp. Med., 183 (1996), 725–729.
  • 27. D. Kirschner and J. C. Panetta, Modeling immunotherapy of the tumor–immune interaction, J. Math. Biol., 37 (1998), 235–252.
  • 28. 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), 841–862.
  • 29. H. P. de Vladar and J. A. González, Dynamic response of cancer under the influence of immunological activity and therapy, J. Theor. Biol., 227 (2004), 335–348.
  • 30. U. Fory´ s, J. Waniewski and P. Zhivkov, Anti-tumor immunity and tumor anti-immunity in a mathematical model of tumor immunotherapy, J. Biol. Syst., 14 (2006), 13–30.
  • 31. R. W. De Boer, J. M. Karemaker and J. Strackee, Relationships between short-term blood-pressure fluctuations and heart-rate variability in resting subjects I: a spectral analysis approach, Med. Biol. Eng. Comput., 23 (1985), 352–358.
  • 32. A. Cappuccio, M. Elishmereni and Z. Agur, Cancer immunotherapy by interleukin-21: potential treatment strategies evaluated in a mathematical model, Cancer Res., 66 (2006), 7293–7300.
  • 33. N. Kronik, Y. Kogan, V. Vainstein, et al., Improving alloreactive CTL immunotherapy for malignant gliomas using a simulation model of their interactive dynamics, Cancer Immunol. Immunother., 57 (2008), 425–439.
  • 34. A. M. Jarrett, J. M. Bloom, W. Godfrey, et al., Mathematical modelling of trastuzumab-induced immune response in an in vivo murine model of HER2+ breast cancer, Math. Med. Biol., (2018), dqy014.
  • 35. K. Annan, M. Nagel and H. A. Brock, A mathematical model of breast cancer and mediated immune system interactions, J. Math. Syst. Sci., 2(2012), 430–446.
  • 36. R. Roe-Dale, D. Isaacson and M. Kupferschmid, A mathematical model of breast cancer treatment with CMF and doxorubicin, Bull. Math. Biol., 73 (2011), 585–608.
  • 37. B. Ribba, N. H. Holford, P. Magni, et al., A review of mixed-effects models of tumor growth and effects of anticancer drug treatment used in population analysis, CPT Pharmacometrics Syst. Pharmacol., 3(2014), 1–10.
  • 38. R. Bhat and C. Watzl, Serial killing of tumor cells by human natural killer cells–enhancement by therapeutic antibodies, PloS One, 2 (2007), e326.
  • 39. T. Sutlu and E. Alici, Natural killer cell-based immunotherapy in cancer: current insights and future prospects, J. Intern. Med., 266 (2009), 154–181.
  • 40. T. R. Stravitz, T. Lisman, V. A. Luketic, et al., Minimal effects of acute liver injury/acute liver failure on hemostasis as assessed by thromboelastography, J. Hepatol., 56 (2012), 129–136.
  • 41. Y. Zhang, D. L. Wallace, C. M. De Lara, et al., In vivo kinetics of human natural killer cells: the effects of ageing and acute and chronic viral infection, Immunotherapy, 121(2007), 258–265.
  • 42. P. Wilding, L. J. Kricka, J. Cheng, et al., Integrated cell isolation and polymerase chain reaction analysis using silicon microfilter chambers, Anal. Biochem., 257(1998), 95–100.
  • 43. V. Pascal, N. Schleinitz, C. Brunet, et al., Comparative analysis of NK cell subset distribution in normal and lymphoproliferative disease of granular lymphocyte conditions, Eur. J. Immunol., 34(2004), 2930–2940.
  • 44. L. de Pillis, T. Caldwell, E. Sarapata, et al., Mathematical modeling of the regulatory T cell effects on renal cell carcinoma treatment, Discrete Continuous Dyn. Syst. Ser. B, 18(2013), 915–943.
  • 45. T. D. To, A. T. T. Truong, A. T. Nguyen, et al., Filtration of circulating tumour cells MCF-7 in whole blood using non-modified and modified silicon nitride microsieves, Int. J. Nanotechnol., 15(2018), 39–52.
  • 46. C. Chen, Y. Chen, D. Yao, et al., Centrifugalfilter device for detection of rare cells with immuno-binding, IEEE T. Nanobiosci., 14(2015), 864–869.
  • 47. P. Dua, V. Dua and E. N. Pistikopoulos, Optimal delivery of chemotherapeutic agents in cancer, Comput. Chem. Eng., 32(2008), 99–107.
  • 48. A. G. López, J. M. Seoane and M. A. Sanjuán, A validated mathematical model of tumor growth including tumor–host interaction, cell-mediated immune response and chemotherap, Bull. Math. Biol., 76(2014), 2884–2906.
  • 49. K. Liao, X. Bai and A. Friedman, The role of CD200–CD200R in tumor immune evasion, J. Theor. Biol., 328(2013), 65–76.
  • 50. M. C. Martins, A. M. A. Rocha, M. F. P. Costa, et al., Comparing immune-tumor growth models with drug therapy using optimal control, AIP Conf. Proc., 1738(2016), 300005.
  • 51. M. Fernandez, M. Zhou and L. Soto-Ortiz, A computational assessment of the robustness of cancer treatments with respect to immune response strength, tumor size and resistance, Int. J. Tumor Ther., 7(2018), 1–26.
  • 52. D. F. Tough and J. Sprent, Life span of naive and memory T cells, Stem Cells, 13(1995), 242–249.
  • 53. C. M. Rollings, L. V. Sinclair, H. J. M. Brady, et al., Interleukin-2 shapes the cytotoxic T cell proteome and immune environment–sensing programs, Sci. Signal., 11(2018), eaap8112.
  • 54. E. M. Janssen, E. E. Lemmens, T. Wolfe, et al., CD4+ T cells are required for secondary expansion and memory in CD8+ T lymphocytes, Nature, 421(2003), 852.
  • 55. I. Gruber, N. Landenberger, A. Staebler, et al., Relationship between circulating tumor cells and peripheral T-cells in patients with primary breast cancer, Anticancer Res., 33(2013), 2233–2238.
  • 56. D. Homann, L. Teyton and M. B. Oldstone, Differential regulation of antiviral T-cell immunity results in stable CD8+ but declining CD4+ T-cell memory, Nat. Med., 7(2001), 913–919.
  • 57. R. J. De Boer, D. Homann and A. S. Perelson, Different dynamics of CD4+ and CD8+ T cell responses during and after acute lymphocytic choriomeningitis virus infection, J. Immunol., 171(2003), 3928–3935.
  • 58. G. T. Skalski and J. F. Gilliam, Functional responses with predator interference: viable alternatives to the Holling type II model, Ecology, 82(2001), 3083–3092.
  • 59. H. Nawata, M. T. Chong, D. Bronzert, et al., Estradiol-independent growth of a subline of MCF-7 human breast cancer cells in culture, J. Biol. Chem., 256(1981), 6895–6902.
  • 60. R. Clarke, N. Brünner, B. S. Katzenellenbogen, et al., Progression of human breast cancer cells from hormone-dependent to hormone-independent growth both in vitro and in vivo, Proc. Natl. Acad. Sci. USA, 86(1989), 3649–3653.
  • 61. N. T. Telang, G. Li, M. Katdare, et al., The nutritional herb Epimedium grandiflorum inhibits the growth in a model for the Luminal A molecular subtype of breast cancer, Oncol. Lett., 13 (2017), 2477–2482.
  • 62. T. A. Caragine, M. Imai, A. B. Frey, et al., Expression of rat complement control protein Crry on tumor cells inhibits rat natural killer cell–mediated cytotoxicity, Blood, 100 (2002), 3304–3310.
  • 63. M. R. Müller, F. Grünebach, A. Nencioni, et al., Transfection of dendritic cells with RNA induces CD4-and CD8-mediated T cell immunity against breast carcinomas and reveals the immunodominance of presented T cell epitopes, J. Immunol., 170 (2003), 5892–5896.
  • 64. J. Chen, E. Hui, T. Ip, et al., Dietary flaxseed enhances the inhibitory effect of tamoxifen on the growth of estrogen-dependent human breast cancer (mcf-7) in nude mice, Clin. Cancer Res., 10(2004), 7703–7711.
  • 65. P. V. Sivakumar, R. Garcia, K. S. Waggie, et al., Comparison of vascular leak syndrome in mice treated with IL21 or IL2, Comparative Med., 63 (2013), 13–21.
  • 66. C.Wu, T.Motosha, H.A.Abdel-Rahman, etal., Freeandprotein-boundplasmaestradiol-17during the menstrual cycle, J. Clin. Endocrinol. Metab., 43(1976), 436–445.
  • 67. E. Nikolopoulou, L. R. Johnson, D. Harris, et al., Tumour-immune dynamics with an immune checkpoint inhibitor, Lett. Biomath., 5(2018), S137–S159.
  • 68. P. Vacca, E. Munari, N. Tumino, et al., Human natural killer cells and other innate lymphoid cells in cancer: friends or foes? Immunol. Lett., 201(2018), 14–19.
  • 69. A. Cerwenka, J. Kopitz, P. Schirmacher, et al., HMGB1: the metabolic weapon in the arsenal of NK cells, Mol. Cell. Oncol., 3(2016), e1175538.
  • 70. I. J. Fidler, Metastasis: quantitative analysis of distribution and fate of tumor emboli labeled with 125i-5-iodo-2'-deoxyuridine, J. Natl. Cancer Inst., 4(1970), 773–782.
  • 71. G. G. Page and S. Ben-Eliyahu, A role for NK cells in greater susceptibility of young rats to metastatic formation, Dev. Comp. Immunol., 23(1999), 87–96.
  • 72. O. E. Franco, A. K. Shaw, D. W. Strand, et al., Cancer associated fibroblasts in cancer pathogenesis, Semin. Cell Dev. Bio., 21(2010), 33–39.
  • 73. G. P. Dunn, A. T. Bruce, H. Ikeda, et al., Cancer immunoediting: from immunosurveillance to tumor escape, Nat. Immunol., 3(2002), 991.
  • 74. D. Mittal, M. M. Gubin, R. D. Schreiber, et al., New insights into cancer immunoediting and its three component phases-elimination, equilibrium and escape, Curr. Opin. Immunol., 27(2014), 16–25.
  • 75. J. Nowak, P. Juszczynski and K. Warzocha, The role of major histocompatibility complex polymorphisms in the incidence and outcome of non-Hodgkin lymphoma, Curr. Immunol. Rev., 5(2009), 300–310.
  • 76. J. G. B. Alvarez, M. González-Cao, N. Karachaliou, et al., Advances in immunotherapy for treatment of lung cancer, Cancer Biol. Med., 12(2015), 209–222.
  • 77. M. E. Dudley and S. A. Rosenberg, Adoptive-cell-transfer therapy for the treatment of patients with cancer, Nat. Rev. Cancer, 3(2003), 666–675.
  • 78. M. Su, C. Huang and A. Dai, Immune checkpoint inhibitors: therapeutic tools for breast cancer, Asian Pac. J. Cancer Prev., 17 (2016), 905–910.
  • 79. M. Ebbo, L. Gérard, S. Carpentier, et al., Low circulating natural killer cell counts are associated with severe disease in patients with common variable immunodeficiency, EBioMedicine., 6(2016), 222–230.
  • 80. S. H. Jee, J. Y. Park, H. Kim, et al., White blood cell count and risk for all-cause, cardiovascular, and cancer mortality in a cohort of Koreans, Am. J. Epidemiol., 162 (2005), 1062–1069.
  • 81. W. B. Kannel, K. Anderson and T. W. Wilson, White blood cell count and cardiovascular disease: insights from the Framingham Study, Jama, 267 (1992), 1253–1256.
  • 82. K. L. Margolis, J. E. Manson, P. Greenland, et al., Leukocyte count as a predictor of cardiovascular events and mortality in postmenopausal women: the Women's Health Initiative Observational Study, Arch. Intern. Med., 165(2005), 500–508.
  • 83. B. K. Duffy, H. S. Gurm, V. Rajagopal, et al., Usefulness of an elevated neutrophil to lymphocyte ratio in predicting long-term mortality after percutaneous coronary intervention, Am. J. Cardiol., 97 (2006), 993–996.
  • 84. B. D. Horne, J. L. Anderson, J. M. John, et al., Which white blood cell subtypes predict increased cardiovascular risk? J. Am. Coll. Cardiol., 45(2005), 1638–1643.
  • 85. R. N. O. Cobucci, H. Saconato, P. H. Lima, et al., Comparative incidence of cancer in HIV-AIDS patients and transplant recipients, Cancer Epidemiol., 36(2012), e69–e73.
  • 86. C. Bodelon, M. M. Madeleine, L. F. Voigt, et al., Is the incidence of invasive vulvar cancer increasing in the United States? Cancer Causes Control, 20(2009), 1779–1782.
  • 87. M. R. Shurin, Cancer as an immune-mediated disease, Immunotargets Ther., 1 (2012), 1–6.
  • 88. J. S. Lawrence, Leukopenia: its mechanism and therapy, J. Chronic. Dis., 6(1957), 351–364.
  • 89. P. Venigalla, B. Motwani, A. Nallari, et al., A patient on hydroxyurea for sickle cell disease who developed an opportunistic infection, Blood, 100(2002), 363–364.
  • 90. M. Iwamuro, S. Tanaka, A. Bessho, et al., Two cases of primary small cell carcinoma of the stomach, Acta. Med. Okayama, 63(2009), 293–298.
  • 91. A. O. O. Chan, I. O. L. Ng, C. M. Lam, et al., Cholestatic jaundice caused by sequential carbimazole and propylthiouracil treatment for thyrotoxicosis, Hong Kong Med. J., 9(2003), 377–380.
  • 92. J. H. Goodchild and M. Glick, A different approach to medical risk assessment, Endod. Topics, 4(2003), 1–8.
  • 93. S. E. Hankinson, Endogenous hormones and risk of breast cancer in postmenopausal women, Breast Dis., 24(2006), 3–15.
  • 94. R. Kaaks, S. Rinaldi, T. J. Key, et al., Postmenopausal serum androgens, oestrogens and breast cancer risk: the European prospective investigation into cancer and nutrition, Endocr. Relat. Cancer, 12(2005), 1071–1082.
  • 95. S. A. Missmer, A. H. Eliassen, R. L. Barbieri, et al., Endogenous estrogen, androgen, and progesterone concentrations and breast cancer risk among postmenopausal women, J. Natl. Cancer Inst., 96(2004), 1856–1865.
  • 96. A. A. Arslan, R. E. Shore, Y. Afanasyeva, et al., Circulating estrogen metabolites and risk for breast cancer in premenopausal women, Cancer Epidemiol. Biomarkers Prev., 18(2009), 2273–2279.
  • 97. S. B. Brown and S. E. Hankinson, Endogenous estrogens and the risk of breast, endometrial, and ovarian cancers, Steroids, 99(2015), 8–10.
  • 98. L. C. Houghton, D. Ganmaa, P. S. Rosenberg, et al., Associations of breast cancer risk factors with premenopausal sex hormones in women with very low breast cancer risk, Int. J. Environ. Res. Public Health, 13(2016), 1066.
  • 99. R. Kaaks, K. Tikk, D. Sookthai, et al., Premenopausal serum sex hormone levels in relation to breast cancer risk, overall and by hormone receptor status-results from the EPIC cohortk, Int. J. cancer, 134(2014), 1947–1957.
  • 100. A. Diefenbach, E. R. Jensen, A. M. Jamieson, et al., Rae1 and H60 ligands of the NKG2D receptor stimulate tumour immunity, Nature, 413(2001), 165.
  • 101. A. Iannello and D. H. Raulet, Cold Spring Harbor symposia on quantitative biology, 1st edition, Cold Spring Harbor Laboratory Press, New York, 2013.
  • 102. M. B. Pampena and E. M. Levy, Natural killer cells as helper cells in dendritic cell cancer vaccines, Front. Immunol., 6 (2015), 1–8.
  • 103. E. Vivier, S. Ugolini, D. Blaise, et al., Targeting natural killer cells and natural killer T cells in cancer, Nat. Rev. Immunol., 12(2012), 239–252.
  • 104. G. Liu, X. Fan, Y. Cai, et al., Efficacy of dendritic cell-based immunotherapy produced from cord blood in vitro and in a humanized NSG mouse cancer model, Immunotherapy, 11(2019), 599–616.
  • 105. M. Schnekenburger, M. Dicato and M. F. Diederich, Anticancer potential of naturally occurring immunoepigenetic modulators: A promising avenue? Cancer, 125(2019), 1612–1628.
  • 106. X. Feng, L. Lu, K. Wang, et al., Low expression of CD80 predicts for poor prognosis in patients with gastric adenocarcinoma, Future Oncol., 15 (2019), 473–483.
  • 107. X. Lai and A. Friedman, Combination therapy of cancer with cancer vaccine and immune checkpoint inhibitors: a mathematical model, PloS One, 12 (2017), e0178479.


This article has been cited by

  • 1. Regina Padmanabhan, Hadeel Shafeeq Kheraldine, Nader Meskin, Semir Vranic, Ala-Eddin Al Moustafa, Crosstalk between HER2 and PD-1/PD-L1 in Breast Cancer: From Clinical Applications to Mathematical Models, Cancers, 2020, 12, 3, 636, 10.3390/cancers12030636
  • 2. Clara Burgos-Simón, Juan-Carlos Cortés, David Martínez-Rodríguez, Rafael J. Villanueva, Modeling breast tumor growth by a randomized logistic model: A computational approach to treat uncertainties via probability densities, The European Physical Journal Plus, 2020, 135, 10, 10.1140/epjp/s13360-020-00853-3

Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved