# Mathematical Modeling of Metastatic Cancer Migration through a Remodeling Extracellular Matrix

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Cancer Cells

#### 2.2. Extracellular Matrix Collagen Fibers

#### 2.3. Enzyme MMP

#### 2.4. Enzyme LOX

#### 2.5. Nondimensionalization and Parameter Values

#### 2.6. Initial and Boundary Conditions

#### 2.7. Numerical Methods and Code Repository

`pdepe`function, an internal PDE solver in Matlab that discretizes the equations in space to obtain a system of ordinary differential equations in time that is then solved along the discrete grid points. This function can handle solving initial-boundary value problems for systems of parabolic and elliptic PDEs in one spatial variable x and time t [63]. The PDEs that the function

`pdepe`can solve must follow the general form

`pdepe`.

## 3. Results and Discussion

#### 3.1. Case I: LOX Is Absent

#### 3.2. Case II: LOX Is Present But Not Coupled to the Haptotactic Migration of Cancer Cells

#### 3.3. Case III: LOX Is Present and Is Coupled to the Haptotactic Migration of Cancer Cells

#### 3.4. Sensitivity Analysis

## 4. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Dynamics in a metastatic tumor microenvironment: (

**A**) The basement membrane around the edge of the primary tumor (cluster of cancer cells in red) has already been perforated by cancer cells. The surrounding collagen fibers are randomly oriented in the extracellular matrix (ECM). (

**B**) Enzymes metalloproteinase (MMP) and lysyl oxidase (LOX) are secreted by cancer cells to degrade and cross-link collagen fibers. MMP generates spaces for cancer cells to begin detaching away from the primary tumor mass to invade the ECM. (

**C**) Meanwhile, aligned and cross-linked collagen fibers form a fibrous pathway along which cancer cells prefer to travel. (

**D**) Collagen fibers continue to be cross-linked, aiding the maneuvers of cancer cells further through the matrix.

**Figure 2.**One-dimensional numerical results for Case I when there is zero concentration of LOX ($l\equiv 0$) and hence, zero cross-linked ECM collagen fibers ${f}_{cl}$ in the system. Results are snapshots of the system dynamics at four simulation times: (

**A**) $t=0$, (

**B**) $t=1$, (

**C**) $t=10$, and (

**D**) $t=20$. For all four plots, the horizontal axis x indicates the dimensionless spatial position, and the vertical axis y indicates the dimensionless population density or concentration of the species listed in the legend.

**Figure 3.**One-dimensional numerical results for Case II when LOX ($l\not\equiv 0$) and its effect only on the ECM collagen fibers and not on the cancer haptotaxis ($g\equiv 0$) are considered in the modeled system. The results are snapshots of the system dynamics at four simulation times: (

**A**) $t=0$, (

**B**) $t=1$, (

**C**) $t=10$, and (

**D**) $t=20$. For all four plots, the horizontal axis x indicates the dimensionless spatial position, and the vertical axis y indicates the dimensionless population density or concentration of the species listed in the legend.

**Figure 4.**One-dimensional numerical results for Case III when LOX ($l\not\equiv 0$) and its effects both on the ECM collagen fibers and cancer cells motility are considered in the modeled system. Results are snapshots of the system dynamics at four simulation times: (

**A**) $t=0$, (

**B**) $t=1$, (

**C**) $t=10$, and (

**D**) $t=20$. For all four plots, the horizontal axis x indicates the dimensionless spatial position, and the vertical axis y indicates the dimensionless population density or concentration of the species listed in the legend.

**Figure 5.**Local sensitivity index as a function of time assessing the impacts of $10\%$ one-at-a-time increases in dimensionless parameters listed in Table 2 on the following model output variables: (

**A**) the population density of cancer cells, (

**B**) the concentration of regular ECM fibers, (

**C**) the concentration of cross-linked ECM fibers, (

**D**) the concentration of MMP, and (

**E**) the concentration of LOX. (

**F**) Model output profiles for concentration of LOX as a function of position x at at simulation time $t=20$ corresponding to $10\%$ changes in each parameter input. In (

**A**–

**E**), the baseline marks the threshold value of $S=1$. In (

**F**) the baseline marks the nominal LOX profile.

**Table 1.**Parameter values available from the literature used in the model of metastatic invasion of cancer through remodeling ECM.

Term | Description | Value | Unit | Sources |
---|---|---|---|---|

L | Reference length | 1 | cm | [33] |

$\tau $ | Reference time | 32 | hours | [55,56,57,58] |

${c}_{o}$ | Reference number of cancer cells per volume | $6.7\times {10}^{7}$ | cells/cm${}^{3}$ | [33] |

${f}_{o}$ | Reference value for $f,{f}_{cl}$ | ${10}^{-11}$ | M | [61] |

${m}_{o}$, ${l}_{o}$ | Reference value for m, l | $0.1\times {10}^{-9}$ | M | [47] |

${D}_{c}$ | Diffusion coefficient of cancer cells | ${10}^{-9}$ | cm${}^{2}/$s | [59] |

${D}_{m}$ | Diffusion coefficient of MMP | ${10}^{-9}$ | cm${}^{2}/$s | [60] |

D | Reference chemical diffusion coefficient | $\phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ | cm${}^{2}/$s | [59] |

$\rho $ | Haptotaxis coefficient toward f | $\phantom{\rule{3.33333pt}{0ex}}2600$ | cm${}^{2}$ M${}^{-1}$ s${}^{-1}$ | [32] |

**Table 2.**Dimensionless expression and values of parameters used in Matlab simulation of metastatic invasion of cancer through a remodeling ECM.

Term | Description | Value | Sources |
---|---|---|---|

$\widehat{{D}_{c}}={D}_{c}/D$ | Diffusion coefficient of cell | 0.001 | Calculated [59] |

$\widehat{\gamma}=\gamma \tau $ | Rate expression for tumor proliferation | $exp(-{\widehat{x}}^{2}/\widehat{\u03f5}),\phantom{\rule{1.em}{0ex}}\widehat{\u03f5}=0.001$ | Assumed |

$\widehat{\rho}=\rho {f}_{o}\tau /{L}^{2}$ | Haptotaxis toward f | $0.0075$ | Calculated [59,61] |

$\widehat{{\rho}_{cl}}={\rho}_{cl}{f}_{o}\tau /{L}^{2}$ | Haptotaxis toward ${f}_{cl}$ | $0.05$ | Assumed |

$\widehat{{v}_{1}}={v}_{1}{c}_{o}$ | Space fraction per unit $\widehat{c}$ | 1 | By definition from [40] |

$\widehat{{v}_{2}}={v}_{2}{f}_{o}$ | Space fraction per unit $\widehat{f}$ | 1 | By definition from [40] |

$\widehat{{v}_{3}}={v}_{3}{f}_{o}$ | Space fraction per unit ${\widehat{f}}_{cl}$ | 1 | By definition from [40] |

$\widehat{{\alpha}_{f}}={\alpha}_{f}\tau {m}_{o}$ | Rate constant for MMP cleavage of f | 10 | Estimated from [33] |

$\widehat{{\mu}_{f}}={\mu}_{f}\tau /{f}_{o}$ | Rate constant for production of f | 0.15 | Estimated from [47] |

$\widehat{{\beta}_{f}}={\beta}_{f}\tau {l}_{o}$ | Rate constant for LOX remodeling of f | 18 | Assumed |

$\widehat{{D}_{m}}={D}_{m}/D$ | Diffusion of MMP | $0.001$ | Calculated [33,59,60] |

$\widehat{{\alpha}_{m}}={\alpha}_{m}\tau $ | Rate constant for decay of MMP | $0.001$ | Estimated from [33,62] |

$\widehat{{\beta}_{m}}={\beta}_{m}{c}_{o}\tau /{m}_{o}$ | Rate constant for secretion of MMP by cells | $0.1$ | Estimated from [33] |

$\widehat{{D}_{l}}={D}_{l}/D$ | Diffusion coefficient of LOX | $0.002$ | Assumed |

$\widehat{{\alpha}_{l}}={\alpha}_{l}\tau $ | Rate constant for decay of LOX | $0.001$ | Assumed |

$\widehat{{\beta}_{l}}={\beta}_{l}{c}_{o}\tau /{m}_{o}$ | Rate constant for secretion of LOX by cells | $0.1$ | Assumed |

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**MDPI and ACS Style**

Nguyen Edalgo, Y.T.; Ford Versypt, A.N.
Mathematical Modeling of Metastatic Cancer Migration through a Remodeling Extracellular Matrix. *Processes* **2018**, *6*, 58.
https://doi.org/10.3390/pr6050058

**AMA Style**

Nguyen Edalgo YT, Ford Versypt AN.
Mathematical Modeling of Metastatic Cancer Migration through a Remodeling Extracellular Matrix. *Processes*. 2018; 6(5):58.
https://doi.org/10.3390/pr6050058

**Chicago/Turabian Style**

Nguyen Edalgo, Yen T., and Ashlee N. Ford Versypt.
2018. "Mathematical Modeling of Metastatic Cancer Migration through a Remodeling Extracellular Matrix" *Processes* 6, no. 5: 58.
https://doi.org/10.3390/pr6050058