Bulletin of Mathematical Biology

, Volume 75, Issue 1, pp 161–184 | Cite as

The Tumor Growth Paradox and Immune System-Mediated Selection for Cancer Stem Cells

  • Thomas Hillen
  • Heiko Enderling
  • Philip Hahnfeldt
Original Article


Cancer stem cells (CSCs) drive tumor progression, metastases, treatment resistance, and recurrence. Understanding CSC kinetics and interaction with their nonstem counterparts (called tumor cells, TCs) is still sparse, and theoretical models may help elucidate their role in cancer progression. Here, we develop a mathematical model of a heterogeneous population of CSCs and TCs to investigate the proposed “tumor growth paradox”—accelerated tumor growth with increased cell death as, for example, can result from the immune response or from cytotoxic treatments. We show that if TCs compete with CSCs for space and resources they can prevent CSC division and drive tumors into dormancy. Conversely, if this competition is reduced by death of TCs, the result is a liberation of CSCs and their renewed proliferation, which ultimately results in larger tumor growth. Here, we present an analytical proof for this tumor growth paradox. We show how numerical results from the model also further our understanding of how the fraction of cancer stem cells in a solid tumor evolves. Using the immune system as an example, we show that induction of cell death can lead to selection of cancer stem cells from a minor subpopulation to become the dominant and asymptotically the entire cell type in tumors.


Cancer stem cells Cell death Immune system Integro-differential equation model Geometric singular perturbation analysis 



The authors wish to thank Gerda de Vries and Jeff Bachman for fruitful discussions and remarks. The work of TH was supported by the Canadian NSERC. The work of HE was supported by the American Association for Cancer Research award number 08-40-02-ENDE (to HE) and the work of HE and PH was supported by the Office of Science (BER), US Department of Energy, under Award Number DE-SC0001434 (to PH).


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Copyright information

© Society for Mathematical Biology 2012

Authors and Affiliations

  • Thomas Hillen
    • 1
  • Heiko Enderling
    • 2
  • Philip Hahnfeldt
    • 2
  1. 1.Centre for Mathematical Biology, Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada
  2. 2.Center of Cancer Systems Biology, Steward Research & Specialty Projects Corp., St. Elizabeth’s Medical CenterTufts University School of MedicineBostonUSA

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