Involve: A Journal of Mathematics

  • Involve
  • Volume 11, Number 3 (2018), 361-382.

A mathematical model of treatment of cancer stem cells with immunotherapy

Zachary J. Abernathy and Gabrielle Epelle

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Using the work of Shelby Wilson and Doron Levy (2012), we develop a mathematical model to study the growth and responsiveness of cancerous tumors to various immunotherapy treatments. We use numerical simulations and stability analysis to predict long-term behavior of passive and aggressive tumors with a range of antigenicities. For high antigenicity aggressive tumors, we show that remission is only achieved after combination treatment with TGF-β inhibitors and a peptide vaccine. Additionally, we show that combination treatment has limited effectiveness on low antigenicity aggressive tumors and that using TGF-β inhibition or vaccine treatment alone proves generally ineffective for all tumor types considered. A key feature of our model is the identification of separate cancer stem cell and tumor cell populations. Our model predicts that even with combination treatment, failure to completely eliminate the cancer stem cell population leads to cancer recurrence.

Article information

Involve, Volume 11, Number 3 (2018), 361-382.

Received: 2 September 2014
Revised: 21 April 2016
Accepted: 27 June 2017
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34D05: Asymptotic properties 34D20: Stability 92B05: General biology and biomathematics 92C37: Cell biology

cancer stem cells immunotherapy recurrence ordinary differential equations stability


Abernathy, Zachary J.; Epelle, Gabrielle. A mathematical model of treatment of cancer stem cells with immunotherapy. Involve 11 (2018), no. 3, 361--382. doi:10.2140/involve.2018.11.361.

Export citation


  • R. J. Akhurst and R. Derynck, “TGF-$\beta$ signaling in cancer: a double-edged sword”, Trends Cell Bio. 11:11 (2001), S44–S51.
  • J. C. Arciero, T. L. Jackson, and D. E. Kirschner, “A mathematical model of tumor-immune evasion and siRNA treatment”, Discrete Contin. Dyn. Syst. Ser. B 4:1 (2004), 39–58.
  • T. P. Cripe, P.-Y. Wang, P. Marcato, Y. Y. Mahller, and P. W. K. Lee, “Targeting cancer-initiating cells with oncolytic viruses”, Molecular Therapy 17:10 (2009), 1677–1682.
  • M. P. Deonarain, C. A. Kousparou, and A. A. Epenetos, “Antibodies targeting cancer stem cells: a new paradigm in immunotherapy?”, Mabs 1:1 (2009), 12–25.
  • O. Dreesen and A. H. Brivanlou, “Signaling pathways in cancer and embryonic stem cells”, Stem Cell Rev. 3:1 (2007), 7–17.
  • B. J. P. Huntly and D. G. Gilliland, “Leukaemia stem cells and the evolution of cancer-stem-cell research”, Nat. Rev. Cancer 5:4 (2005), 311–321.
  • C. T. Jordan, M. L. Guzman, and M. Noble, “Cancer stem cells”, New Eng. J. Med. 355:12 (2006), 1253–1261.
  • B. Joshi, X. Wang, S. Banerjee, H. Tian, A. Matzavinos, and M. A. J. Chaplain, “On immunotherapies and cancer vaccination protocols: a mathematical modelling approach”, J. Theor. Biol. 259:4 (2009), 820–827.
  • D. Kirschner and J. C. Panetta, “Modeling immunotherapy of the tumor-immune interaction”, J. Math. Biol. 37:3 (1998), 235–252.
  • L. Li and W. B. Neaves, “Normal stem cells and cancer stem cells: the niche matters”, Cancer Res. 66:9 (2006), 4553–4557.
  • K.-J. Malmberg, “Effective immunotherapy against cancer”, Cancer Immunology, Immunotherapy 53:10 (2004), 879–892.
  • L. Mishra, K. Shetty, Y. Tang, A. Stuart, and S. W. Byers, “The role of TGF-$\beta$ and Wnt signaling in gastrointestinal stem cells and cancer”, Oncogene 24:37 (2005), 5775–5789.
  • F. Nani and H. I. Freedman, “A mathematical model of cancer treatment by immunotherapy”, Math. Biosci. 163:2 (2000), 159–199.
  • A. Soltysova, V. Altanerova, and C. Altaner, “Cancer stem cells”, Neoplasma 52:6 (2005), 435–440.
  • T. J. Stewart and M. J. Smyth, “Improving cancer immunotherapy by targeting tumor-induced immune suppression”, Cancer Metastasis Rev. 30:1 (2011), 125–140.
  • Y. Tang, K. Kitisin, W. Jogunoori, C. Li, C.-X. Deng, S. C. Mueller, H. W. Ressom, A. Rashid, A. R. He, J. S. Mendelson, J. M. Jessup, K. Shetty, M. Zasloff, B. Mishra, E. P. Reddy, L. Johnson, and L. Mishra, “Progenitor/stem cells give rise to liver cancer due to aberrant TGF-$\beta$ and IL-6 signaling”, Proc. Nat. Acad. Sci. 105:7 (2008), 2445–2450.
  • M. Terabe, E. Ambrosino, S. Takaku, J. J. O'Konek, D. Venzon, S. Lonning, J. M. McPherson, and J. A. Berzofsky, “Synergistic enhancement of CD8+ T cell-mediated tumor vaccine efficacy by an anti-transforming growth factor-$\beta$ monoclonal antibody”, Clin. Cancer Res. 15:21 (2009), 6560–6569.
  • S. Wilson and D. Levy, “A mathematical model of the enhancement of tumor vaccine efficacy by immunotherapy”, Bull. Math. Biol. 74:7 (2012), 1485–1500.