The Tumor Growth Paradox and Immune SystemMediate

Authors: Thomas Hillen Heiko Enderling Philip Hahnfeldt

Publish Date: 2012/11/30

Volume: 75, Issue: 1, Pages: 161-184

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Abstract

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 tumorsThe 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 084002ENDE to HE and the work of HE and PH was supported by the Office of Science BER US Department of Energy under Award Number DESC0001434 to PHHere we show that the three models for cancer stem cells that are illustrated in Fig 2 are equivalent in the situation where the stem cell population is not declining Let Ut and Vt denote the CSC and TC density at time t and k the rate of CSC division For the purpose of demonstrating this equivalence we ignore TC divisions We first describe a hypothetical “complete model” i that has all three features then demonstrate that dropping feature ii maintains model generality while dropping feature iii also maintains generality provided parameters are chosen in the complete model such that the CSC compartment never decreases in timeWe introduce the complete model that includes all three division fates described above Let α 1 denote the fraction of symmetric division α 2 the fraction of asymmetric division and α 3 the fraction of symmetric commitment events with α 1+α 2+α 3=1 A schematic is shown on the left in Fig 2This model assumes that CSC is a robust state that cannot be lost during mitosis Enderling et al 2009 Therefore the dividing CSC always remains CSC and the second daughter cell is either a CSC or a TC Fig 2 This model is the Complete Model with the additional condition of no chance of commitment ie α 3=0 From a simple inspection of Equation System 27 with α 3=0 though we see this model remains just as general as the Complete Model since the leading coefficients on the right sides of the equations for U and V can range from 0 to 1 as beforeThe model most often used in the literature ignores asymmetric CSC division Ganguli and Puri 2006 MarciniakCzochra et al 2009 Wise et al 2008 A mitotic CSC event either yields two CSC or two TC Fig 2 This model is the Complete Model with the additional condition of no chance of asymmetric division ie α 2=0 From a simple inspection of Equation System 27 with α 2=0 though we see this model remains just as general as the Complete Model since the leading coefficients on the right sides of the equations for U and V can range from 0 to 1 as beforeIn summary we have shown that the “No Symmetric Commitment” and “No Asymmetric Division” models are individually equivalent to the “Complete Model” and so to each other We therefore discuss a mathematical model that essentially exploits the “No Symmetric Commitment” model above with the appreciation that it will not only provide analytic confirmation of the tumor growth paradox revealed by our agentbased studies Enderling et al 2009 but will simultaneously confirm the applicability of various sets of cell division rules we could alternatively have employed to build the model

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