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**|**PLoS One**|**v.5(12); 2010**|**PMC3003707

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PLoS One. 2010; 5(12): e14366.

Published online 2010 December 17. doi: 10.1371/journal.pone.0014366

PMCID: PMC3003707

Sui Huang, Editor^{}

University of Calgary, Canada

* E-mail: ude.dravrah.ymmij@rohcim

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.

Received 2010 August 23; Accepted 2010 November 24.

Copyright Leder et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

This article has been cited by other articles in PMC.

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 [1]. This hypothesis has been shown consistent with data from such diverse cancer types as chronic and acute myeloid leukemias [2], [3], breast cancer [4], colorectal cancer [5], mesenchymal neoplasms [6], head and neck squamous cell carcinoma [7], and pancreatic cancer [8]. The investigation of cancer stem cells in melanoma, however, has led to controversial findings. Some studies suggested that melanoma cells that are capable of transplanting the disease are exceedingly rare [9] while others, using more severely immunocompromised mice, found that cells with those capabilities are very common within the tumor [10]. Similarly, the frequency of tumor cells positive for stem cell-like markers in breast cancer varies according to the stage and subtype of the tumor [11].

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 [12], [13]. The differential ability of mouse models to detect cancer stem cells may be explained by a context-dependent phenotype of those cells, as supported by evidence from co-injection experiments of stromal and cancer cells [10]. In these studies, the efficiency of transplantation of putative cancer stem cells was higher when stromal cells were co-injected as compared to injection of cancer stem cells alone. This data suggests that the ability of cells to initiate neoplastic growth may not only depend on the severity of immunodeficiency of assay mice, but also on the microenvironmental context of these cells [14].

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 [15]. These extracellular signals are provided through exposure to fetal calf serum and certain cytokines, including some bone morphogenic proteins, as well as basic fibroblast growth factor (bFGF), and cause many purified oligodendrocyte precursors to revert to a state that resembles that of multipotential neural stem cells [15]. Similarly, a study in which mature astrocytes were exposed to transforming growth factor α (TGFα) demonstrated that a single extracellular factor is sufficient to induce differentiated cells of the central nervous system to regress into a stem-like cell stage [16]. This observed plasticity of normal tissue stem cells has implications for tissue organization in general, and the view of rigid differentiation hierarchies of cells must be revised in light of these findings.

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 [17]. These alterations occurred within as little as two hours of treatment and had long-term effects on the phenotype generally associated with stem cell character. This plasticity of tumor stem cells may also apply to liquid tumors, as it was recently shown that leukemia-initiating cells in AML patients harboring mutations in nucleophosmin (NPM) can reside in the CD34+ as well as CD34- fraction [18].

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 [19], [20], [21], [22], [23], [24].

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 (Fig. 1). Denote the abundances of healthy stem, progenitor, and differentiated cells by *x*_{0}, *x*_{1}, and *x*_{2}, respectively, and the abundances of the corresponding cancer cell types by *y*_{0}, *y*_{1}, and *y*_{2}. Healthy stem cells proliferate at rate *r _{x}* and die at rate

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 [17]. The rate at which cancer progenitors dedifferentiate per day is denoted by *γ*. This rate may be a function of the microenvironmental conditions of these cells and can also vary over time as tumors become more aggressive. There may also be a similar propensity of healthy progenitor cells to regress to a stem cell like state [15], [16]; a rate of dedifferentiation of healthy progenitors can be included in the model but is neglected for purposes of clarity. We consider the production of stem cells to be limited by the density of both healthy and cancer stem cells; this modeling assumption is made because stem cell numbers are limited by the availability of extracellular resources as well as spatial constraints within the tissue, and therefore the production of stem cells is constrained by those resources and spatial considerations regardless of whether they are produced by symmetric division or dedifferentiation. The density dependence is achieved by the functions and for these two cell types. The terms *k _{x}* and

Then the basic mathematical model is given by

(1)

For shorthand we will write and instead of equation (1). In order to study the dynamics of these cells in response to therapy, we denote the rates of production of cancer cells during treatment by , , and . We may also investigate an additional or alternative effect of therapy on the death rates of cancer cells, leading to increased rates during treatment: , , and . Note that we do not allow the terminally differentiated cells to dedifferentiate to give rise to progenitors; this modeling assumption is made since in most tumor types, terminally differentiated cancer cells are post-mitotic and therefore incapable of dedifferentiating.

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 [25], [26]. In the context of our model, the first cell carrying a resistance mutation can only initiate a long-lived clone if it is a stem cell, or alternatively a progenitor that dedifferentiates to a stem-like state. Denote the rate at which a resistance mutation arises per cell division by *u*. The probability of resistance depends on the total number of stem cell divisions and a fraction of progenitor cell divisions, given by

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 *t* is given by .

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 *z*_{0}, *z*_{1} and *z*_{2}, respectively. Then the dynamics of the system including resistant cells is given by

(2)

Here the growth, death and differentiation rates of resistant cancer cells are denoted by the parameters *r _{y}*,

Let us first discuss the effects of the dedifferentiation parameter on the dynamics of treatment response. Figure 2a shows the number of differentiated cancer cells as a function of time after the initiation of therapy. Note that the dedifferentiation parameter *γ* has a strong affect on the tumor's response to treatment; in particular, a larger value of *γ* corresponds to a poorer response to treatment. In Figure 2b, we investigate the effect of the dedifferentiation rate, *γ*, on the probability of resistance, *R*(*t*), finding that larger values of *γ* lead to a substantially higher risk of developing resistance.

Let us now consider specific examples for the treatment response of a tumor for a fixed level of the dedifferentiation parameter, *γ*. In the following we investigate the effects of a variety of hypothetical drugs that target different cells within the differentiation hierarchy. Note that these scenarios describe idealized treatments which exert the specified effects onto cancer cells; these scenarios serve as examples of the dynamics of the system during drug treatment. Table 1 provides a summary of the four treatment strategies considered.

First, let us investigate a hypothetical drug that reduces the production rate of both progenitor and differentiated cells. *Treatment 1* in Figure 3 provides a numerical example for such a situation. The panels of the figure show the abundance of differentiated cancer cells over time in response to treatment (Fig. 3a) as well as the probability of resistance emerging during treatment (Fig. 3b). Note that this type of treatment leads to a reduction in the number of cancer progenitors and differentiated cells, but is incapable of depleting cancer stem cells. Such interventions might reduce symptoms for a limited time by debulking the tumor. However, the persistence of cancer stem cells in this scenario prevents tumor eradication and permits the evolution of acquired resistance, which renders the response to treatment short-lived.

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 *Treatment 2* in Figure 3. The performance of a drug with these properties is a slight improvement over the example considered previously; however, the probability that acquired resistance will evolve is still substantial since the drug cannot deplete the cancer stem cell population. This type of drug allows a stable population of cancer stem cells to remain and continuously repopulate the progenitor and differentiated cells.

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 *Treatment 3* in Figure 3. A drug eliciting these effects is capable of eradicating the disease. Furthermore, the probability of resistance is small since the cancer stem cell population is driven to extinction and thus, fewer chances for the emergence of a resistance mutation to arise. Once treatment has eradicated cancer stem cells, cancer progenitor and differentiated cells equally go extinct since the latter cell types do not have (sufficient) self-renewing abilities.

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 *Treatment 4* in Figure 3. The advantage of such a drug is that the chance of drug-resistant tumors is diminished since there are so few stem cells. However, the tumor burden remains at a relatively high level for a prolonged period of time because differentiated cancer cells are unaffected by the drug.

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, *u*. In the context of our mathematical model, drug-resistant cancer stem cells initiate their own cellular differentiation hierarchy (Fig. 1 and equation (2)). With this extension of the mathematical model, we can investigate an additional aspect of treatment: the propensity of a drug to delay progression due to resistance, i.e. the onset of a resistance-driven rebound of the cancer cell population. Figure 4 shows how drugs that target various parameters of the cancer cell differentiation hierarchy can have vastly different effects on the duration of successful treatment before the onset of resistance.

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 Figure 4, and results from the fact that a reduction in the number of cancer stem cell divisions leads to fewer opportunities per unit time for resistant cells to arise. Figure 4 also shows that dedifferentiation can have a very strong effect on the time until disease progression. In particular, a change in the order of magnitude of the dedifferentiation rate has approximately twice the effect as compared to a change in magnitude of the mutation rate. Lastly, note that an increased net growth rate of stem cells delays the rebound of the tumor population. This fact is due to the maintenance of the stem cell population near its carrying capacity, which prevents resistant cell populations from arising since we consider density-dependent growth dynamics.

Let us now compare the efficacy of different treatment protocols while also taking into consideration the possibility of resistance (Fig. 5). We investigate two types of treatment: a drug that causes a decline in the growth rates of all cancer cell types (Fig. 5 a and c), and a drug that only inhibits progenitor and differentiated cells (Fig. 5b and d). When comparing the effectiveness of these two drugs over short time spans (panels a and b), we find that a drug that inhibits proliferation for all cell types is preferable as compared to the other type of treatment – i.e., the total cell number is significantly lower in the former case. However, over longer time spans (panels c and d) the drug that only inhibits progenitor and differentiated cells is more effective at preventing the emergence of resistance. This effect results from the density dependent growth of cancer stem cells; since this drug does not inhibit the stem cell population, resistant stem cells never become established due to the density constraint mechanism. Any resistant stem cells that arise during administration of this treatment will have suppressed growth since the stem cell population has already reached its maximum population size. Note that the short time span (panels a and b) refers to 500 days after the initiation of therapy, while the long time span (panels c and d) refers to 5000 days since the start of treatment. The drug shown in panels a and b does significantly decrease the population of stem cells, and therefore any resistant stem cell that arises will not be inhibited by the density constraint mechanism and be able to establish a resistant clone. Hence in the short term, it is preferable to inhibit cancer stem cells (panels a and b), while during longer periods of time (panels c and d), this strategy may backfire because it allows the resistant tumor stem cells to grow.

A reduction of the dedifferentiation rate has a beneficial effect regardless of the cell type that the drug targets (Fig. 5). However, note in Figure 6d that the sensitivity of the system to the dedifferentiation parameter is decreased with increasing progenitor birth rate, *a _{y}*. An increase in the production of progenitor cells leads to a larger number of those cells, and thus a decrease in the dedifferentiation rate will need to be enhanced in order to have a substantial effect on the stem cell population.

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 (Fig. 6). First we study the probability of developing resistance prior to detection. Fig. 6a shows that as the dedifferentiation rate increases, the probability of pre-existing resistance also increases. This effect arises because there are more opportunities for resistant cancer stem cells to arise if an increasing fraction of progenitor cells dedifferentiate to become cancer stem cells, since we consider the contribution of dedifferentiating progenitor cells to the total risk of resistance in the stem cell pool. Fig. 6b displays the number of tumor cells as a function of time after the initiation of therapy under the assumption that a small number of resistant cells is present at the time of diagnosis. Note that the time until the resistant cells cause a disease rebound is strongly dependent on the dedifferentiation rate – the larger this rate becomes, the more rapidly the cancer cell population rebounds due to the expansion of resistant cells. This finding is consistent with the results shown in Figure 4.

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.

Figure 7 shows that higher levels of dedifferentiation, i.e. a larger rate *γ*, lead to larger rebounds of the cancer cell population during treatment breaks as well as lower levels of cancer cell depletion during treatment. Hence the possibility of dedifferentiation of cancer progenitors renders otherwise successful therapy less effective, to the point of being unsuccessful if the capacity of progenitors for dedifferentiation is sufficiently large. Fig. 7a demonstrates the effects of treatment with a drug that only inhibits the production of cancer stem and progenitor cells. This scenario leads to a drastic difference between situations with different levels of the dedifferentiation parameter *γ*. Figure 7b shows the treatment response to a drug that additionally inhibits the production of differentiated cancer cells. In this situation, the difference between varying values of *γ* is not as pronounced since the drug has a substantial effect on differentiated cancer cells.

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 Table 1 and the results section) and have also evaluated the effects of dedifferentiation on the risk of pre-existing therapy and the success of pulsatile treatment.

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.

**Competing Interests: **The authors have declared that no competing interests exist.

**Funding: **This work was supported by Award Number U54CA143798 from the National Cancer Institute. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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