# Epithelial-Mesenchymal Transition in Metastatic Cancer Cell Populations Affects Tumor Dormancy in a Simple Mathematical Model

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## Abstract

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## 1. Introduction

## 2. Results

#### 2.1. Deterministic Population Dynamics of Dormant Metastatic Tumors

**Figure 1.**Tumor population dynamics in models of cancers derived from pioneering stem cells. (

**A**) Diagrammatic description of the basic model with M (mesenchymal cells), E (epithelial cells) and their rate parameters; (

**B**) trajectories for M (grey lines) and E (black lines), for the two model variations and two different values of ${\beta}_{3}$ (senescence of E).

**Table 1.**The parameters that describe the linear and saturated feedback model. The values given are those used for the simulation of the saturated feedback model. The units of all of the α and β are $[1/div]$, where $div$ is the time for one cell division in days. The unit of k is $[1/Ndiv]$, and the unit of m is $[1/N]$, where N is the number of cells of species E at time t. EMT, epithelial-mesenchymal transition.

Parameter | Value | Definition |
---|---|---|

${\alpha}_{1}$ | 0.32 | Proliferative capacity of M |

${\alpha}_{2}$ | 0.32 | Differentiation of $M\to E$ |

${\alpha}_{3}$ | 0.02 | Cell death of M |

${\beta}_{1}$ | 0.87 | Proliferative capacity of E |

${\beta}_{2}$ | 0.25 | Senescence of E |

${\beta}_{3}$ | 0.4 | Cell death of E |

${\beta}_{4}$ | 0.0025 | Rate of EMT |

k | 0.0026 | Defines carrying capacity of E |

m | 0.0036 | Defines maximum per-capita growth rate |

#### 2.2. EMT Dramatically Alters Population Dynamics in the Deterministic Model

**Figure 2.**Effect of EMT on the size of metastatic tumors. (

**A**) Modified model diagram, including EMT; (

**B**) each plot shows the trajectories of M and E for different values of ${\beta}_{4}$ (rate of EMT). As ${\beta}_{4}$ is increased, the dynamics change from close-to-equilibrium to exponential growth, where eventually, the size of M will overtake E.

**Table 2.**Sensitivity in model output (steady-state level of M) to changes in parameters by ±5%. We see that the parameters with the greatest effect on output are ${\beta}_{4}$ (the rate of EMT) and ${\alpha}_{1}$ (the proliferative capacity of M).

Parameter (p) | Sensitivity Coefficient ($\frac{\partial M}{\partial p}$) |
---|---|

${\alpha}_{1}$ | 10,200 |

${\alpha}_{2}$ | 1,840 |

${\alpha}_{3}$ | 3,260 |

${\beta}_{1}$ | 449 |

${\beta}_{2}$ | 355 |

${\beta}_{3}$ | 358 |

${\beta}_{4}$ | 29,700 |

#### 2.3. Modeling Chemotherapeutic Treatment of Dormant Tumors

**Figure 3.**Population dynamics in response to chemotherapy. Modeling chemotherapeutic treatment and recurrence; tumor population dynamics before, during and after a treatment regime (denoted by the dashed vertical lines). (

**A**) Treatment consisting of an agent that increased cell death (${\beta}_{3}$) in the E cell population. At the end of treatment, ${\beta}_{3}$ returns to its pre-treatment level; (

**B**) The E cell population acquires resistance mutations following treatment (${\beta}_{3}$ is lower than before treatment); (

**C**) Treatment consisting of an agent that prevents EMT; (

**D**) Treatment consisting of both ${\beta}_{3}$ reduction and EMT prevention in combined therapy. In (C) and (D), recovery was to original pre-treatment parameter values; (

**E**) The phase plane plot shows the behavior of the system around the fixed Point A (pre-treatment parameter values) and nullclines. Point B marks the system state after treatment; (

**F**) Phase plane plot around the fixed Point C; parameter values after E cells have acquired mutations. In both (E) and (F), the fixed points shown are the only stable fixed points for the system in this state.

**Figure 4.**Model sensitivity to changes in cell death. We study the effect of increasing the cell death rate of M (${\alpha}_{3}$) and E (${\beta}_{3}$) simultaneously. The output is colored by the extent to which the steady state for E is shifted ($\Delta E$). Increasing the rate of M cell death has a much more dramatic effect than increasing the rate of E cell death alone.

**Figure 5.**Transition from tumor to tumor-free state by induction of cell death. (

**A**) Theoretical dose response curve of stable solutions of E and M as ${\beta}_{3}$ is varied; (

**B**) solution curves of E at two different rates of cell death (by chemotherapy) with different initial numbers of cells. Red curves represent the solution of E for ${\beta}_{3}<{\beta}^{*}$; blue curves represent the solution of E for ${\beta}_{3}>{\beta}^{*}$.

#### 2.4. Defining a Stochastic Model of Dormant Metastatic Tumors

**Figure 6.**Generational capacity affects dormant tumor steady-state population sizes in a stochastic model. (

**A**) The diagram represents the basic model, where mesenchymal stem cells M divide asymmetrically to generate differentiated epithelial cells E with a generational capacity (g) that measures the number of possible cell divisions that these cells can undergo prior to senescence; (

**B**) steady-state population of tumors derived from a single pioneering M cell with varying g; (

**C**) steady-state population sizes over time for tumors derived from a single M cell.

#### 2.5. A Stochastic Model of EMT for Small Differentiated Cell Populations

**Figure 7.**EMT in a stochastic model of dormant tumor recurrence. (

**A**) The fraction of tumors in which EMT events occur with a fixed generational capacity of 18 and for different probabilities of EMT; (

**B**) the fraction of tumors in which EMT events occur with a varied generational capacity and fixed probability of EMT; (

**C**) the fraction of tumors exhibiting specific numbers of EMT events as generational capacity is altered; (

**D**) the fraction of tumors exhibiting occurrences of EMT over time, for three different generational capacities; (

**E**) the fraction of tumors in which EMT events occur with a varied generational capacity of 18 and fixed probability of EMT. In (

**A**–

**C**), tumors were initiated from a single precursor M cell; while in (

**D**), tumors were started from a population of 100 E cells. The probability of EMT in (

**B**–

**E**) was set at ${10}^{-7}$.

#### 2.6. Dormant Tumor Recurrence in the Stochastic Model

## 3. Discussion

## Acknowledgments

## Author Contributions

## Abbreviations

## Conflicts of Interest

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**MDPI and ACS Style**

MacLean, A.L.; Harrington, H.A.; Stumpf, M.P.H.; Hansen, M.D.H.
Epithelial-Mesenchymal Transition in Metastatic Cancer Cell Populations Affects Tumor Dormancy in a Simple Mathematical Model. *Biomedicines* **2014**, *2*, 384-402.
https://doi.org/10.3390/biomedicines2040384

**AMA Style**

MacLean AL, Harrington HA, Stumpf MPH, Hansen MDH.
Epithelial-Mesenchymal Transition in Metastatic Cancer Cell Populations Affects Tumor Dormancy in a Simple Mathematical Model. *Biomedicines*. 2014; 2(4):384-402.
https://doi.org/10.3390/biomedicines2040384

**Chicago/Turabian Style**

MacLean, Adam L., Heather A. Harrington, Michael P. H. Stumpf, and Marc D. H. Hansen.
2014. "Epithelial-Mesenchymal Transition in Metastatic Cancer Cell Populations Affects Tumor Dormancy in a Simple Mathematical Model" *Biomedicines* 2, no. 4: 384-402.
https://doi.org/10.3390/biomedicines2040384