# A Mathematical Model of Average Dynamics in a Stem Cell Hierarchy Suggests the Combinatorial Targeting of Cancer Stem Cells and Progenitor Cells as a Potential Strategy against Tumor Growth

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## Abstract

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## Simple Summary

## Abstract

## 1. Introduction

_{j}that represents the rate at which each cellular event takes place.

## 2. Methods

#### 2.1. Hypothesis

- Residual CSCs;
- More differentiated cells regaining a CSC ability;
- Different CSC populations within the same tumor;
- Intermediate progenitor (P) cells that possess enough potency to generate tumors.

^{high}CD24

^{low}phenotypes with CSC characteristics [15]. The authors hypothesized that EMT could explain macroscopic metastasis. In the same line, Tsai et al. found that reversible EMT was required for metastasis to occur in a squamous cell carcinoma model [19]. However, further on, it was demonstrated that EMT is dispensable for the metastasis of lung and pancreatic cancers [20,21]. More recently, Fumagalli et al. investigated the role of Lgr5(+)-CSCs and their negative counterpart in colorectal cancer metastasis [29]. They showed that more differentiated Lgr5(−)-non-CSCs were able to metastasize and form secondary tumors by dedifferentiation to Lgr5(+)-CSCs. Lgr5(−) cells were more invasive and circulated in the blood with a greater frequency than Lgr5(+)-CSCs. This suggests that EMT was not necessary for the initial steps of metastasis, but plasticity of the non-stem cell fraction was fundamental for reconstitution of the secondary tumor.

_{initial}× 2

^{10}. Then, if we suppose P

_{initial}= 3.2 × 10

^{5}cells, and the tumor density is constant and equal to 3.2 × 10

^{5}cells/mm

^{3}(see the section below), the new volume will be 1 mm

^{3}× 2

^{10}= 1024 mm

^{3}~ 1 mL, which becomes a palpable tumor. Therefore, P cells might be an important subpopulation to consider when designing CSC-directed strategies.

#### 2.2. Mathematical Model

_{j}(Figure 1). By doing so, each cellular event can be followed during tumor development.

_{j}, for example, k

_{1}is the intrinsic reaction rate constant that indicates the natural tendency of CSCs to divide symmetrically to produce two CSCs (R1). The rates of production and consumption of each cellular event depend on both the number of precursor cells for that event and the constant k

_{j}that will multiply that number. For example, the rate of production of CSCs in R1 will be two-fold (2k

_{1}CSC) its rate of consumption due to cell division (k

_{1}CSC). By implementing this strategy, the calculations of production and consumption rates for each event where CSCs are involved are as shown in Table 1. The sum of all rates in each column represents the total rate of production (left) and the total rates of consumption due to cell division (center) and cell death (right) for this cell type.

^{3}. The average cellular density was estimated from the literature to be in the order of magnitude of 10

^{8}cells/cm

^{3}when considering 100–1000 mm

^{3}tumors [43,44,45]. This order of magnitude is more appropriate for tumors of epithelial origin [46].

^{3}.

_{i/j}= [Φ

_{2/1}, Φ

_{3/1}, Φ

_{4/1}, Φ

_{5/4}, Φ

_{6/1}, Φ

_{7/1}, Φ

_{8/1}]. This parameter indicates the proportions of the defined relations between specific rate constants k

_{j}. For example, Φ

_{2/1}= k

_{2}/k

_{1}is the ratio between asymmetrical and symmetrical CSC specific renewal rate constants, corresponding to cellular reactions denoted by R2 and R1, respectively. Similarly, Φ

_{5/4}= k

_{5}/k

_{4}indicates the relative magnitude between the specific rate constants associated with the symmetrical differentiation (R5) and renewal (R4) of progenitor cells.

_{i/j}) was maintained for all experimental datasets, i.e., Φ

_{2/1}= 1, Φ

_{3/1}= 0.01, Φ

_{4/1}= 5.35, Φ

_{5/4}= 0.8, Φ

_{6/1}= 0.01, Φ

_{7/1}= 0.1, and Φ

_{8/1}= 1. This solution was obtained by varying the value of Φ

_{i/j}so that the model fitted the experimental tumor volume points and, at the same time, satisfied the following constraints:

_{1}, then, k

_{2}= 1, k

_{3}= 0.01, k

_{4}= 5.35, k

_{5}= 4.28, k

_{6}= 0.01, k

_{7}= 0.1, and k

_{8}= 1. The higher values of k corresponding to P cell divisions compared to the k values related to CSC divisions mean that the frequency of CSC divisions is slower than for P cells, which is in accordance with experimental data [22,27]. Similarly, the rate of cell death of terminally differentiated (D) cells is 10-fold higher than for P cells and 100-fold higher than for CSCs [11]. For further details, please refer to Molina and Alvarez [38].

_{j}values corresponding to each scenario that we will present in the next section. The model output is presented in figures where the CSC and P fractions are also plotted as means of following the evolution of both cell subpopulations during the treatment.

## 3. Results and Discussion

#### 3.1. Tumor Relapse after the Selective Targeting of CSCs

_{i/j}relationships previously defined were maintained) and, hence, that the model was applicable to the new experimental scenarios. This was shown to be the case, as the model curves fitted the control experiment data points reported by Zielske et al. [14] (Figure 2A) and Gupta et al. [13] (Figure 2B), with coefficients of determination (R

^{2}) of 0.93 and 0.95, respectively.

^{2}= 0.95) the experimental data points well (Figure 3A). Furthermore, the model can accurately describe the experimental observations of the increase in the fraction of CSCs, which reached a new equilibrium (Figure 3A, blue dashed line).

^{2}= 0.52) and tumor cell composition with the gradual eradication of CSCs (Figure 3B, blue dashed line). This result supports our hypothesis, suggesting that even if CSCs were eliminated, more differentiated cells might be able to continue growing and, hence, allow the tumor to relapse, albeit with a different composition of cell subpopulations.

#### 3.2. Comparing Strategies to Combat Cancer

#### 3.2.1. Nonselective Treatment of Tumor Cells

#### 3.2.2. Selective Targeting of Less Differentiated Tumor Cells

_{6}, representing the CSC-specific death rate in the cellular reaction R6 (see the Mathematical Model section). Although the fraction of CSCs is diminished, the tumor continues to grow at the same rate as in the absence of treatment (Figure 4B, yellow line). On the other hand, when 99% of all cells except CSCs are killed at day 120, the tumor relapses at day 140 (Figure 4B, green line). However, when this last strategy is combined with a 100× enhancement of k

_{6}, there is a better response than when using these approaches separately (Figure 4B, dark red line).

_{7}15-fold (P cell death rate in R7 in the cellular model). We observed that this therapeutic approach was as effective as a 99% eradication of total cells in terms of the tumor volume, with relapse occurring at day 140 (Figure 5A, yellow and green lines, respectively). Moreover, when we combined both treatments, the tumor relapse was delayed to day 180 (Figure 5A, dark red line), indicating a 3-fold improvement compared to the use of either individual treatment.

_{5/4}. Interestingly, a 37.5% increase in Φ

_{5/4}(from 0.8 to 1.1), while keeping the remaining Φ

_{i/j}values fixed, retards the relapse of the tumor by approximately 20 days (Figure 5B, yellow line), similar to the nonselective elimination of 99% of total tumor cells (Figure 5B, green line) and the selective 15-fold increase in the P-cell specific death rate (Figure 5A, yellow line). Moreover, the combination of both treatments, involving the selective differentiation of P cells and nonselective 99% elimination of bulk tumor cells, slowed down tumor relapse by approximately 60 days (Figure 5B, dark red line).

#### 3.2.3. Combinatorial Targeting of Less Differentiated Tumor Cells

- (A)
- Selective killing of CSCs and P cells by a 100- and 15-fold augmentation of k
_{6}and k_{7}, and their corresponding death rates in cellular reactions R6 and R7, respectively (Figure 6A); - (B)
- Enhancement of P cell differentiation into D cells by increasing k
_{5}by 20% in cellular reaction R5, and selectively eliminating CSCs by augmenting k_{6}100-fold in R6 (Figure 6B); - (C)
- Inhibition of CSC symmetrical renewal by a 1000-fold reduction in k
_{1}(R1) and selectively killing P cells by a 100-fold increase in k_{7}(R7) (Figure 6C).

_{6}, which is the rate constant of CSC death (R6). In this case, the model output indicated no tumor relapse, even when the therapy was stopped at day 200, due to the elimination of all CSCs and P cells before that point, as can be observed in the fraction curves (Figure 7B). As our model suggests, it is important not only to simultaneously target CSCs and P cells, but also to do so for a sufficiently long enough period of time, in order to ensure their effective and complete elimination.

#### 3.3. Further Methods and Models to Verify the Hypothesis

#### 3.3.1. Selective Targeting of CSCs in Combination with Standard-of-Care Chemotherapy in Ovarian Cancer

#### 3.3.2. Sketching an In Vivo Experiment

#### 3.4. Model Limitations

_{1}= 0 during that time; but if they resume proliferation later at a rate of k

_{1}= 1, then the average k

_{1}would be 0.5. Notwithstanding this important remark, the rates of cellular divisions where CSCs are involved are of a relative lower value compared to P cells, i.e., Φ

_{4/1}= 5.35 and Φ

_{5/4}= 0.8, which is consistent with the slower average proliferation rate of CSCs.

## 4. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Compartmentalized representation of the proposed model. Each cell event is mediated by a rate constant k

_{j}. Cancer stem cells (CSCs) can self-renew (k

_{1}) or give rise to intermediate progenitor (P) cells through either asymmetric (k

_{2}) or symmetric (k

_{3}) division. P cells, in turn, can proliferate (k

_{4}) and give rise to terminally differentiated (D) cells (k

_{5}). Each cell type has a death rate represented by k

_{6}for CSCs, k

_{7}for P cells, and k

_{8}for D cells.

**Figure 2.**Model fitted to control experiment scenarios. (

**A**) Human-derived breast cancer cells (3 × 10

^{4}) were implanted in the mammary fat pads of mice [14]. (

**B**) Human breast cancer cells (1 × 10

^{6}) were implanted in the inguinal mammary glands of mice [13]. Experimental data are represented by black points, model fitting by the yellow line, the CSC fraction by the dashed blue line, and the P fraction by the dashed red line.

**Figure 3.**Model fit of experimental treatment scenarios targeting CSCs. (

**A**) Radiation-treated cancer cells (3 × 10

^{4}) were implanted in the mammary fat pads of mice [14]. This treatment reduced the percentage of CSCs in the inoculation culture. Tumor growth was delayed compared to the control, but the tumor relapsed and the fraction of CSCs increased over time to reach a new equilibrium. (

**B**) Breast cancer cells (1 × 10

^{6}) were implanted in the mammary glands of mice, which, after 24 h, were then administered daily with salinomycin for four weeks [13]. Selective elimination of CSCs considerably delayed the tumor burden. However, the tumor regrew, despite the elimination of CSCs. Experimental data are represented by black points, model fitting by a yellow line, the control treatment curve by a dashed black line, the CSC fraction by a dashed blue line, and the P fraction by a dashed red line.

**Figure 4.**Nonselective treatment and directed therapy against CSCs. (

**A**) Progression of a tumor without intervention is depicted from day 0 to 120 (black points, experimental data; dotted black curve, model fitting). At day 120 (purple arrow), the bulk of the tumor, CSCs, and P and D cells, is nonselectively eradicated: 95% (yellow line), 99% (green line), 99.9% (dark red line), and 99.99% (dark blue line). The CSC fraction (dashed blue line) and P fraction (dashed red line) remain unaltered for all nonselective treatments. (

**B**) Selective targeting of CSCs at day 120: 100-fold enhancement of k

_{6}does not affect tumor progression (yellow line), although the fraction of CSCs diminishes (dashed blue line). When 99% of all cells except CSCs are eradicated at day 120 (green line), in addition to a 100-fold enhancement of k

_{6}, there is a better response (dark red line) than using either treatment alone.

**Figure 5.**Targeting of intermediate P cells. (

**A**) Selective elimination of P cells by increasing k

_{7}15-fold at day 120 (yellow line). This effect is comparable to the nonselective eradication of 99% of total tumor cells (green line). However, when both treatments are performed simultaneously, there is an increment of time before tumor relapse (dark red line). All treatments were started at day 120 (purple arrow). (

**B**) Increasing P cell differentiation (Φ

_{5/4}is increased from 0.8 to 1.1, which is equivalent to increasing 37.5% k

_{5}, at day 120; yellow line), eliminating 99% of total cells (green line), and the combination of both treatments (dark red line) have similar effects, as depicted in (

**A**). The CSC fraction (dashed blue lines) and P cell fraction (dashed red lines) are shown for combination treatments in (

**A**,

**B**).

**Figure 6.**Combination therapy against CSCs and P cells. Yellow curves represent simulations of the simultaneous targeting of CSCs and P cells at day 120 (purple arrow). Scenario (

**A**): Selective killing of CSCs and P cells by a 100- and 15-fold augmentation of their corresponding death rate kinetic parameters, k

_{6}and k

_{7}, respectively. Scenario (

**B**): Enhancement of P cell differentiation into D cells by increasing k

_{5}by 20% and the selective elimination of CSCs by a 100-fold augmentation of k

_{6}. Scenario (

**C**): Inhibition of symmetrical CSC renewal by a 1000-fold reduction in k

_{1}and selectively killing P cells by increasing k

_{7}100-fold. For all cases: Experimental control data are represented by black points; model curve fit to experimental control data by the dotted black line; model curve for treatment simulation by solid yellow lines; and P and CSC tumor fractions by dashed red and blue lines, respectively.

**Figure 7.**Time lapse for combination treatment. (

**A**) When the treatment in Figure 6C (inhibition of symmetrical CSC renewal by a 1000-fold reduction in k

_{1}and selectively killing P cells by increasing k

_{7}100-fold) was selected and stopped at day 250, there was a tumor relapse at day 280 due to residual tumorigenic cells that were not eliminated when the treatment was stopped (see CSC and P cell fractions). (

**B**) Selectively killing CSCs by increasing k

_{6}200-fold in addition to the treatment in (

**A**) eradicated the tumor, which did not relapse, even when therapy was stopped at day 200. P and CSC tumor fractions are represented by dashed red and blue lines, respectively.

**Figure 8.**Combination treatment of ovarian cancer. (

**A**) Control treatment with a vehicle. (

**B**) Selective targeting of CSCs by treatment with CPI613. (

**C**) Chemotherapy based on carboplatin and paclitaxel. (

**D**) Combination therapy (chemotherapy + selective targeting of CSCs). (

**E**) Augmentation of the chemotherapy dose retarded tumor relapse, but did not eliminate the disease. (

**F**) Treatment in (

**E**), combined with an increased dose of CPI613, produced a better response than either treatment alone. Furthermore, tumor relapse was not observed. For all cases: Experimental data are represented by black points; model simulation by solid yellow lines; control treatment by a dotted black line; and P and CSC tumor fractions by dashed red and blue lines, respectively. The treatment time is represented with purple arrows.

Reaction | ${\mathit{r}}_{\mathit{p}\mathit{r}\mathit{o}\mathit{d}\mathit{u}\mathit{c}\mathit{t}\mathit{i}\mathit{o}\mathit{n},\mathit{C}\mathit{S}\mathit{C}\mathit{s}}$ | ${\mathit{r}}_{\mathit{c}\mathit{e}\mathit{l}\mathit{l}\text{}\mathit{d}\mathit{i}\mathit{v}\mathit{i}\mathit{s}\mathit{i}\mathit{o}\mathit{n},\mathit{C}\mathit{S}\mathit{C}\mathit{s}}$ | ${\mathit{r}}_{\mathit{c}\mathit{e}\mathit{l}\mathit{l}\text{}\mathit{d}\mathit{e}\mathit{a}\mathit{t}\mathit{h},\mathit{C}\mathit{S}\mathit{C}\mathit{s}}$ |
---|---|---|---|

R1 | $2{k}_{1}CSC$ | ${k}_{1}CSC$ | $0$ |

R2 | ${k}_{2}CSC$ | ${k}_{2}CSC$ | $0$ |

R3 | $0$ | ${k}_{3}CSC$ | $0$ |

R6 | $0$ | $0$ | ${k}_{6}CSC$ |

Total | $2{k}_{1}CSC+{k}_{2}CSC$ | ${k}_{1}CSC+{k}_{2}CSC+{k}_{3}CSC$ | ${k}_{6}CSC$ |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Molina-Peña, R.; Tudon-Martinez, J.C.; Aquines-Gutiérrez, O. A Mathematical Model of Average Dynamics in a Stem Cell Hierarchy Suggests the Combinatorial Targeting of Cancer Stem Cells and Progenitor Cells as a Potential Strategy against Tumor Growth. *Cancers* **2020**, *12*, 2590.
https://doi.org/10.3390/cancers12092590

**AMA Style**

Molina-Peña R, Tudon-Martinez JC, Aquines-Gutiérrez O. A Mathematical Model of Average Dynamics in a Stem Cell Hierarchy Suggests the Combinatorial Targeting of Cancer Stem Cells and Progenitor Cells as a Potential Strategy against Tumor Growth. *Cancers*. 2020; 12(9):2590.
https://doi.org/10.3390/cancers12092590

**Chicago/Turabian Style**

Molina-Peña, Rodolfo, Juan Carlos Tudon-Martinez, and Osvaldo Aquines-Gutiérrez. 2020. "A Mathematical Model of Average Dynamics in a Stem Cell Hierarchy Suggests the Combinatorial Targeting of Cancer Stem Cells and Progenitor Cells as a Potential Strategy against Tumor Growth" *Cancers* 12, no. 9: 2590.
https://doi.org/10.3390/cancers12092590