^{1}

^{2}

^{1}

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Conceived and designed the experiments: KL EH FM. Performed the experiments: KL. Analyzed the data: KL FM. Contributed reagents/materials/analysis tools: KL FM. Wrote the paper: KL EH FM.

The authors have declared that no competing interests exist.

The cancer stem cell hypothesis suggests that tumors contain a small population of cancer cells that have the ability to undergo symmetric self-renewing cell division. In tumors that follow this model, cancer stem cells produce various kinds of specified precursors that divide a limited number of times before terminally differentiating or undergoing apoptosis. As cells within the tumor mature, they become progressively more restricted in the cell types to which they can give rise. However, in some tumor types, the presence of certain extra- or intracellular signals can induce committed cancer progenitors to revert to a multipotential cancer stem cell state. In this paper, we design a novel mathematical model to investigate the dynamics of tumor progression in such situations, and study the implications of a reversible cancer stem cell phenotype for therapeutic interventions. We find that higher levels of dedifferentiation substantially reduce the effectiveness of therapy directed at cancer stem cells by leading to higher rates of resistance. We conclude that plasticity of the cancer stem cell phenotype is an important determinant of the prognosis of tumors. This model represents the first mathematical investigation of this tumor trait and contributes to a quantitative understanding of cancer.

Traditionally, many different cell types within a tumor have been considered to have tumorigenic potential and possess the ability to cause cancers in secondary recipients. By contrast, the cancer stem cell hypothesis suggests that only a small subpopulation of tumor cells has that potential

These findings have led to discussions about the applicability of the cancer stem cell hypothesis to all tumor types, and also the ability of xenotransplantation assays to reliably identify cancer stem cells

The phenotypic plasticity of stem cells has been a topic attracting great interest. Studies of cells in the central nervous system, for instance, have shown that certain extracellular signals can induce oligodendrocyte precursor cells to dedifferentiate into multipotential neural stem cells

Observations parallel to those observing a dedifferentiation potential of normal cells have also been made with regard to cancer cells. A recent study identified signaling within the perivascular niche as a driving force for tumor cells to acquire stem cell characteristics. In this study, nitric oxide was shown to activate Notch signaling via cGMP and PKG in a subset of glioma cells resulting in acquisition of the side population phenotype and increased neurosphere and tumor formation

The ability of committed cancer progenitors to dedifferentiate to a stem-like state has important implications for the dynamics of tumor progression and the response to therapy. In this paper, we design a novel mathematical model to quantify the effects of the dedifferentiation rate on disease outcome. As all mathematical modeling approaches, our framework represents an abstraction of the biological system and as such should be considered as a toy model to investigate several characteristics of the system. This work is part of a growing literature describing mathematical investigations of cancer stem cells

vWe designed a simple mathematical model to investigate the dynamics of different cell populations during tumor progression and treatment. The model considers three differentiation stages for both the healthy and the cancer cell differentiation hierarchies. Stem cells reside at the top of the hierarchy and give rise to progenitor cells, which in turn give rise to differentiated cells (_{0}, _{1}, and _{2}, respectively, and the abundances of the corresponding cancer cell types by _{0}, _{1}, and _{2}. Healthy stem cells proliferate at rate _{x}_{0}, and give rise to healthy progenitors at rate _{x}_{x}_{1} and give rise to healthy differentiated cells at rate _{x}_{2}_{y}_{y}_{y}_{y}_{0}, _{1}, and _{2}. In the simplest form of our model, we consider these parameters to be constant unless external factors – such as the administration of treatment – are applied to the system. However, the model can easily be extended to include more complex scenarios such as variability in the microenvironment, involvement of the immune system, and interactions between cancer and stromal cells. Such situations may be described by considering a distribution of parameters from which the values are selected. In the absence of estimates for the parameters and their distributions, however, we chose to analyze the model in its simpler form of constant parameter values.

The mathematical model considers three levels of the differentiation hierarchy of cells: stem cells, progenitors and differentiated cells. These cell types are present in the system as healthy cells (left), drug-sensitive cancer cells (middle) and drug-resistant cancer cells (right). Stem cells give rise to progenitors which in turn give rise to differentiated cells. Additionally, cancer progenitors may have the ability to dedifferentiate to stem cells. The rate of dedifferentiation is denoted by

In addition to their ability to produce differentiated cancer cells, cancer progenitors may regress to a stem-like state via genetic, epigenetic, or other mechanisms _{x}_{y}_{x}_{y}

Then the basic mathematical model is given by

For shorthand we will write

The model outlined above considers the dynamics of treatment response without the possibility of acquired resistance. Even drugs that elicit a dramatic initial response often fail later on due to the emergence of resistance mutations which render the drug ineffective. Two prominent examples are the point mutations in BCR-ABL and EGFR that confer resistance against the small molecule inhibitors imatinib/dasatinib and erlotinib/gefitinib

Note that a resistant cell can arise during a division of a sensitive cancer stem cell or during a dedifferentiation event of a sensitive cancer progenitor cell. Then the probability that at least one resistant cell that will persist in the population has arisen by time

Furthermore, the basic mathematical model as given by equation (1) can be expended to include a differentiation hierarchy of drug-resistant cancer cells. Denote the abundance of resistant stem, progenitor and differentiated cancer cells by _{0}, _{1} and _{2}, respectively. Then the dynamics of the system including resistant cells is given by

Here the growth, death and differentiation rates of resistant cancer cells are denoted by the parameters _{y}_{z}, b_{z},_{0}, c_{1},_{2}

Let us first discuss the effects of the dedifferentiation parameter on the dynamics of treatment response.

In panel _{x}_{y}_{0}_{1}_{2}_{x}_{1}_{x}_{2}_{y}_{x}_{y}_{x}_{x}^{6}, _{y}^{7}, ^{−9}, and ω = 0.1. The initial condition for the panels is found by simulating system (1) using the pretreatment parameter values and the initial condition _{0}(0) = 10^{6}, _{1}(0) = 10^{8}, _{2}(0) = 10^{10}, _{0}(0) = 1, and _{1}(0) = _{2}(0) = 0. We simulate this system until detection time _{2}(^{12}, and then simulate the treatment phase by running system (1) with the initial conditions _{0}(_{1}(_{2}(

Let us now consider specific examples for the treatment response of a tumor for a fixed level of the dedifferentiation parameter,

Stem Cells | Progenitor Cells | Differentiated Cells | |

Treatment 1 | − | + | + |

Treatment 2 | + | + | + |

Treatment 3 | + | + | + |

Treatment 4 | + | − | − |

First, let us investigate a hypothetical drug that reduces the production rate of both progenitor and differentiated cells.

The figure shows the abundance of differentiated cancer cells, _{2}

In contrast to the scenario above, a drug may inhibit the production of all cancer cell types but still fail to completely eradicate the cancer cell population. To illustrate this point, consider a drug that inhibits all three cancer cell types. A drug that elicits this response is shown as

Let us now consider a drug that reduces the growth rate of all cancer cell types to a larger extent. An example of this type of therapy is shown as

Lastly, let us consider a drug that inhibits cancer stem cells only. In this setting, the rate of depletion of the total cancer cell population may be too slow to be considered effective; an example is shown as

These four treatment strategies represent idealized therapies; however, their study leads to insights into how heterogeneous tumor cell populations respond to treatments that affect particular types of cells, and suggests the most desirable subpopulation to target.

Instead of considering the proportion of patients that develop resistance, it is also useful to investigate the expected number of resistant cells present within a patient for a given mutation rate,

The figure shows the time until the disease burden increases despite continuous therapy versus the birth rate (panels ^{−9}. The parameter ^{−4}. The parameter ^{−7}, ^{−4},

When comparing drugs that affect the birth and death rates of cancer stem cells, drugs that target the production of cancer stem cells lead to a longer time during which treatment is effective and before resistance emerges. This effect can be seen by comparing panels a and b with panels c and d of

Let us now compare the efficacy of different treatment protocols while also taking into consideration the possibility of resistance (

Panels ^{−9} and we set

A reduction of the dedifferentiation rate has a beneficial effect regardless of the cell type that the drug targets (_{y}

Panel ^{12} and then evaluate the probability of resistance at that time. Panel

In many cases of treatment failure due to the evolution of acquired resistance, resistant cells are present at the time of diagnosis. Let us thus discuss such pre-existent resistance and its effects on the prognosis of cancer patients (

Anti-cancer therapy is often administered in treatment pulses to limit the toxicity of these agents. The advantage of treatment pulses is that higher drug concentrations can be reached using such a strategy as compared to the continuous dosing regimen. The disadvantage to pulsatile therapy, however, is that during treatment breaks, the cancer cell population expands exponentially and leads to rebounds as well as an increased risk of acquired resistance. So far, the effects of dedifferentiation of progenitor cells to a stem cell-like state have not been investigated with respect to pulsed treatment strategies. Our mathematical model is useful for evaluating the impact of pulsed therapy with regard to recovery of the cancer stem cell population by dedifferentiation of progenitors.

The figure shows the dynamics of differentiated cancer cells in response to a treatment strategy in which the drug is administered for 30 days, followed by a treatment holiday of 30 days during each pulse. Panel

During normal development, differentiation from stem cells to final products is unidirectional. Some data suggest that oncogenic mutations lead to loss of the ability for cells to maintain their differentiated state. In the case of tumor suppressors, their normal function may therefore be to maintain the unidirectional flow of differentiation during development. Their inactivation or alteration of certain signaling pathways may result in the loss of unidirectionality of this process. Dedifferentiation does not necessarily refer to a scenario in which every cell reverses its differentiation phenotype. Instead, a small fraction of committed progenitor cells may acquire stem cell-like characteristics in response to genetic or environmental changes, and this small cell number may lead to qualitatively different dynamics. We refer to the fraction of cells that dedifferentiate per time unit as gamma. In this paper, we determine the effects of various values of gamma on the response of tumors to therapy, which specifically targets either stem cells or non-stem cells. We have studied four hypothetical treatment strategies (see

We chose to formulate a simple mathematical framework that only incorporates the essential considerations of cancer stem, progenitor and differentiated cells. While our mathematical model can easily be extended to describe more complicated scenarios such as interactions of cancer cells with the stroma and immune system, and the generation of tumor cell heterogeneity through other avenues such as clonal diversification, we have concentrated on the analysis of the basic model during different treatment options. This model will be extended in future work to consider more complex situations in cancer.

The results obtained from this modeling study indicate that the response of tumors capable of dedifferentiating is qualitatively different from a scenario in which treatment cannot completely eradicate the bulk of tumor cells and the remaining cells lead to a rebound post-therapy. In the latter case, the remaining cells likely are sensitive to being re-challenged with treatment and therefore, pulsed therapy has the potential to eradicate the disease. In a scenario including the potential of dedifferentiation of cells, pulsed therapy targeting stem cells is incapable of curing the disease, since the cancer stem cell pool is continuously repopulated by progenitor cells during treatment breaks. Therefore, the consideration of a dedifferentiation potential of cancer cells is important for an accurate understanding of anti-cancer therapy.

There has been significant discussion of the effects of tumor stem cells that are insensitive to anti-cancer therapy. Standard therapy inhibits proliferating cells of the tumor bulk but the tumors recur from drug-insensitive stem cells. Because of this, it has been suggested that a strategy targeting cancer stem cells is required for curative therapy. However, if non-stem cells can acquire stem cell properties with a sufficiently high probability but still much lower than the differentiation rate, then a stem cell-specific treatment strategy will be futile. The results of our mathematical modeling studies described in this paper suggest that higher levels of dedifferentiation substantially reduce the effectiveness of therapy directed at cancer stem cells. During pulsed treatment strategies, the possibility of dedifferentiation leads to higher rebounds of the cancer cell population during treatment breaks as well as lower levels of cancer cell reduction during treatment pulses. In addition, we see that increasing the level of dedifferentiation significantly increases the number of stem cell replications and therefore increases the probability of acquiring a resistance mutation in a stem cell. In summary, plasticity of the cancer stem cell phenotype is an important determinant of the effectiveness of therapy, and its possibility cannot be neglected to gain an accurate understanding of the treatment response of human tumors.

We would like to thank the Michor lab and three anonymous referees for comments and advice.