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

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 tumors.

Appendix: Equivalence of Basic Stem Cell Models

Here, 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 U(t) and V(t) 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 time.

Complete Model

We 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. 2.

The change in cell populations due to CSC division events can then be described by:

Invoking the identity α ₁=1−α ₂−α ₃, we obtain the system

$$ \begin{array}{rcl} \dot{U} &=& (1-\alpha_2-2\alpha_3) k U,\\[4pt] \dot{V} &=& (\alpha_2+2\alpha_3) k U, \end{array} $$

(27)

where α ₂+2α ₃≠(0,1) and α ₂+2α ₃<1 (or equivalently, α ₁>α ₃). The last condition arises from the assumption that the number of CSCs does not decrease in time.

No Symmetric Commitment Model

This 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, i.e., 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 before.

No Asymmetric Division Model

The model most often used in the literature ignores asymmetric CSC division (Ganguli and Puri 2006; Marciniak-Czochra 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, i.e., 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 before.

In 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 agent-based 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.