# Exploring Cell Migration Mechanisms in Cancer: From Wound Healing Assays to Cellular Automata Models

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## Abstract

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## Simple Summary

## Abstract

## 1. Introduction

## 2. Materials and Methods

#### 2.1. In Vitro Experiments

_{0}). To accomplish this, we employed custom-made automated image analysis software that relied on image variance analysis to identify the wound’s edge as the boundary of the cell-free area. Wound closure rate (α), obtained as the slope of the linear reduction in A/A

_{0}as a function of time, was used as the key parameter for the comparison between in silico and in vitro experiments. The same approach was used to compare our data to previous in vitro experiments collected from the literature. For further elaboration on this technique, please refer to our previous publication [51].

#### 2.2. Model Development

#### 2.2.1. Domain Building

_{0}) divided by δ. Similarly, the initial number of cells in L and R, randomly distributed, was calculated from the cell density (ρ) in the in vitro experiments by assuming the L and R lengths as half of the dimension of W. As the simulation time went on, cells moved toward the wound from the edges, where a periodic boundary condition was set to guarantee constant cell density. The periodic boundary simulated an infinite space for the surrounding cells; in the script, the first and last columns of the lattice were repopulated with cells at each time step in case any vacancy was generated during the simulation by cell migration steps.

#### 2.2.2. Rules Governing Cellular Dynamics

- Migration and proliferation

_{m}is the time necessary for a cell to move a distance equal to its characteristic size $\mathsf{\delta}$. T

_{m}can be calculated using the random motility coefficient as ${\mathrm{T}}_{\mathrm{m}}={\mathsf{\delta}}^{2}/\mathrm{D}$, where D is the constant diffusion rate of cells. Similarly, proliferation probability is given by ${\mathrm{P}}_{\mathrm{d}}=\frac{1/{\mathrm{T}}_{\mathrm{d}}}{1/{\mathrm{T}}_{\mathrm{d}}+1/{\mathrm{T}}_{\mathrm{m}}}$, where T

_{d}is the doubling time of a given cell line, typically available in the literature. Since, in a single step, each cell may either migrate or proliferate, the sum of these two probabilities is 1.

_{m}. If λ < ${\mathrm{P}}_{\mathrm{m}}$ and a neighboring location is available, the cell will migrate; otherwise, the cell will proliferate.

- 2.
- Quiescence

- (1)
- Spatial domain discretization and initialization of cell positions.
- (2)
- Testing for empty neighbors for every occupied element.
- (3)
- Random number (λ) assignment to the occupied CA elements to decide the actions of cells:
- If λ > ${\mathrm{P}}_{\mathrm{d}}$, the actual site of the cell of interest will remain occupied by the cell, and a daughter cell will be placed in a randomly chosen empty site among the neighbors.
- If λ < ${\mathrm{P}}_{\mathrm{m}}$, the site of the cell of interest will become empty, and a neighboring empty site will become occupied.

- (4)
- Lattice updates according to the selected actions based on probabilities.
- (5)
- Stop if the wound was healed; otherwise, proceed to next time step and return to (1).

#### 2.3. Parameter Sensitivity Analysis

_{m}, T

_{d}, $\mathsf{\rho}$, δ, b

_{0}) were evaluated for their effect on the kinetics of wound closure (v, α, $\mathrm{D}$, and characteristic times of the assay). The characteristic times of wound closure, typically measured experimentally in the literature, were T

_{half}and T

_{closure}, defined as the times required for a closure of 50% and 100% of the original wound area A

_{0}, respectively.

#### Global Sensitivity Analysis

_{0}, and the cells, d, were only geometric parameters for our simulation, they were excluded from this analysis and fixed to the mean value (575 μm and 20 μm, respectively) obtained from the values reported in Table 1. The distribution of regression coefficients ${\mathsf{\beta}}_{\mathrm{I}}$ is reported for α, ${\mathrm{T}}_{\mathrm{closure}}$, and ${\mathrm{T}}_{\mathrm{half}}$ in Figure 2a–c, respectively, related to each input parameter. Through the application of statistical analysis, with one-way ANOVA and Tukey’s test (confident interval 95%), parameters were ranked according to their significance to the output (from 1 (high significance) up to 3 (low significance)), as shown in Figure 2d.

_{m}was ranked as the most significant parameter for all the outputs, implying a stronger dependency of the outputs on it. No significant differences between the effects of $\mathsf{\rho}$ and T

_{d}on the rate of closure a were revealed. On the contrary, $\mathsf{\rho}$ was more significant (rank 2) to T

_{half}than T

_{d}(rank 3), suggesting the effect of the density being higher in the earlier steps of the process, when the domain was poorly occupied by cells, while at later stages, the influence of density was limited. An opposite trend was observed for the effect of T

_{d}, which was more significant at later stages (T

_{closure}) with respect to the initial steps (T

_{half}) of the process.

_{m}and T

_{d}(Table 1), assuming a cell type has the lowest doubling time and the highest migration time, the probability to proliferate each time step was only 4% (see Supplementary Materials). The idea that, in the case of concurring mechanisms, the fastest one (governed by the shortest characteristic time) was also controlling the rate of the entire process was a clear concept from the analysis of the process based on the transport phenomena approach [57]. An electric analogy simplification would consider the two concurring mechanisms as two resistances acting in parallel and driven by the same driving force, which in this case was the difference between cell density at confluence (or at least in the bulk of the tissue, far from the wound) and the density in the wound, which in this case was 0. A wider analysis of the role of transport phenomena in the process was reported in a previous paper [5].

## 3. Results

#### 3.1. Baseline Model Behavior and Model Calibration

_{0}, as reported in Figure 3c. A strong agreement in terms of α was clearly observed.

#### 3.2. Model Predictions

^{−4}–0.5], defined by physiological limits. As regards the minimum physiological value of ${\mathrm{T}}_{\mathrm{d}}$, it is related to the time necessary for the duplication of DNA (S1 phase), which in eukaryotic cells takes typically about 10–12 h [62]. On the other hand, ${\mathrm{T}}_{\mathrm{m}}$ can typically vary from a few seconds to a few hours in the case of poorly motile cells, such as osteoblasts [59].

_{m}/T

_{d}, compares the contributions of migration and proliferation processes to wound healing. To elaborate further, a low Φ value, along with a decreased T

_{m}, indicates that cells require less time to migrate from one site to another. This signifies that these cells display high migratory behavior and, consequently, a greater potential for invasiveness.

_{m}, such as osteoblasts) [59,60]. Consequently, this high Φ region might be of marginal interest in practical applications.

^{−2}, cell proliferation can be neglected. The knowledge of this information, here clearly quantified for the first time, to the best of our knowledge, can simplify most of the experimental protocols.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Initial configuration of domain. At t = t

_{0}, the lattice domain was divided into two lateral domains (L and R, left and right, respectively) occupied by cells (each cell occupies a squared element of size δ) representing the edge of the wound and a central cell-free domain (W, wound) of size b

_{0}along the x-direction. (

**b**) Scheme of migration rule. A total of 3 × 3 lattice squares where automaton rules were applied: cell in the center of the square (${\mathrm{M}}_{0}$) decides to migrate; in the following steps (t

_{1}and t

_{2}), it can move to one of the empty adjacent spaces, such as ${\mathrm{M}}_{1}$ or ${\mathrm{M}}_{2}$. (

**c**) Scheme of proliferation rule. Cell identified with ${\mathrm{P}}_{0}$ decides to proliferate; in the following step (t = t

_{1}), it proliferates, and the daughter cell (referred to as ${\mathrm{D}}_{1}$) occupies a space in the neighborhood. At a later time, t = t

_{2}, the daughter cell may proliferate again, and its daughter cell, referred to as ${\mathrm{D}}_{2}$, occupies another space. (

**d**) Flow chart of the model algorithm.

**Figure 2.**Global sensitivity analysis results. The violin plots show the regression parameter distribution (mean value reported as red line) obtained from MLRA with respect to input parameters (

**a**) α, (

**b**) T

_{closure}, and (

**c**) T

_{half}. The weighted parameter ranking (

**d**) was obtained from the weighted average of rankings from ANOVA and Tukey’s test. Note: Rank 1 (dark green) denotes the parameter of the highest significance and rank 3 (light green) the parameter of the lowest significance.

**Figure 3.**Comparison of results obtained by our CA with experimental measurements. On the left (

**a**), a snapshot of in silico experiments was reported and compared to in vitro images acquired with Time-lapse microscopy during WH assay of HT-1080 at two different times (0 and 9 h) (

**b**). Simulations were run using input parameters estimated from the experimental setup (Id 2 in Table 2). The wound area variation A normalized to the initial wound area A

_{0}was monitored in time (

**c**), providing a direct comparison between experimental data [5] (green circle symbols) and CA outputs (purple solid line).

**Figure 4.**Wound area A, normalized with respect to the initial wound area A

_{0}, was reported as a function of time t. The evolution of the wound area in silico (green and red solid lines) was compared with the wound area variation in vitro (green and orange circles symbols) computed for all cell lines here investigated. For brevity, only four cell lines (HT1080 [5], MDA-MB231 [31,51], HaCaT [51], and MDA-MB468 [49,50]) and 7 densities were reported.

**Figure 5.**Wound area reduction rate, α, as a function of the ratio between the characteristic time of migration and proliferation, Φ. A total of 2000 in silico experiments with input parameters randomly chosen in the ranges reported in (

**a**) provided outputs (black dots in (

**b**)) that show power-law dependency of α with respect to Φ. The fitting curve was reported as a solid line. The inset reports the same data on a log scale. In (

**c**), the trends estimated by our analysis (solid line) were compared with experimental data (our data: green circles; literature data: orange circles; new data: pink circles), reported in Table 2.

**Table 1.**List of parameters relevant to the model.$\mathsf{\rho}$, δ, T

_{m}, T

_{d}, and b

_{0}were input parameters, while the model allows calculating D, v, and α, which have been used to compare in silico and in vitro data. The range of variation in $\mathsf{\rho}$ was chosen to define the degree of coverage of the domain. The minimum corresponds to a single cell in each sub-domain (L or R), and the maximum value was associated with full coverage of the L and R sub-domains. The values of other parameters depend on the specificity of the cell line. Ranges of variation have been defined from information available in the literature [5,31,49,50,51,52] to validate the model. The details of the calculations performed to estimate parameters were reported in the Supplementary Materials.

Parameter | Description | Range of Variation | Dimension |
---|---|---|---|

${\mathrm{T}}_{\mathrm{m}}$ | Characteristic time of migration | 0.005; 0.5 | h |

${\mathrm{T}}_{\mathrm{d}}$ | Characteristic time of proliferation | 12; 40 | h |

$\mathsf{\rho}$ | Density: number of cells in unit area | 10^{−6}; 10^{−3} | cells/µm^{2} |

$\mathsf{\delta}$ | Characteristic dimension of the cell | 15; 25 | µm |

b_{0} | Initial length of the wound | 370; 900 | µm |

$\mathrm{D}$ | Motility: the time necessary to travel a length equal to delta | 10^{3}; 10^{4} | µm^{2}/h |

v | Velocity of the fronts of cells | 5; 60 | µm/h |

$\mathsf{\alpha}$ | Velocity of wound area variation | 0.02; 0.13 | 1/h |

**Table 2.**Characteristic parameters of cell lines used for validation of CA. Ids 1–20 were derived from previous works by some of the co-authors of this paper or were taken from the literature. Id 21 presents new data from a new experiment reported here for the first time. Details about parameter estimates (*) derived from raw data available in the original papers were reported in the Supplementary Materials. N/A= Not available; the values of some parameters were not reported due to a lack of specific information in the original papers.

Cell line | Id | $\mathit{\rho}$ [#cells/μm^{2}] | α [1/h] | b_{0}[μm] | T_{d}[h] | T_{m}[h] | References |
---|---|---|---|---|---|---|---|

HT-1080 | 1 | 2.7 × 10^{−3} | 0.012 | 468 * | 24 | 0.063 | [5] |

2 | 2.9 × 10^{−3} | 0.128 | 371 * | 24 | 0.075 | ||

3 | 1.6 × 10^{−3} | 0.078 | 532 * | 24 | 0.107 | ||

4 | 2.1 × 10^{−3} | 0.078 | 638 * | 24 | 0.075 | ||

5 | 1.5 × 10^{−3} | 0.069 | 548 * | 24 | 0.129 | ||

6 | 1.2 × 10^{−3} | 0.069 | 687 * | 24 | 0.082 | ||

7 | N/A | 0.110 | 288 | 24 | 0.110 | [58] | |

MDA-MB-231 | 8 | 1.0 × 10^{−3} | 0.023 | 800 | 38 | 0.338 | [31,51] |

9 | 1.2 × 10^{−3} | 0.042 | 930 | 38 | 0.075 | ||

10 | 2.3 × 10^{−3} | 0.044 | 800 | 38 | 0.095 | ||

11 | N/A | 0.040 | 288 | 38 | 0.476 | [58] | |

MDA-MB468 | 12 | 1.2 × 10^{−3} | 0.031 | 800 | 47 | 0.154 | [51] |

HaCaT | 13 | 1.2 × 10^{−3} | 0.043 | 900 | 19 | 0.156 | [49,50] |

14 | 1.7 × 10^{−3} | 0.132 | 900 | 19 | 0.017 | ||

15 | 2.5 × 10^{−2} * | 0.029 | N/A | 19 | 0.078 | [21] | |

Saos-2: HTB 85 | 16 | N/A | 0.010 | 800 | 37 | 5.851 | [59,60] |

Caco-2 | 17 | 1.2 × 10^{−3} * | 0.014 | 882 * | 80 | 0.385 | [61] |

BEAS | 18 | N/A | 0.054 | 500 | 26 | 0.188 | [18] |

MCF-7 | 19 | N/A | 0.031 | 500 | 38 | 0.741 | |

20 | N/A | 0.040 | 287 | 38 | 0.54 | [58] | |

NIH/3T3 | 21 | 1.3 × 10^{−4} | 0.062 | 933 | 20 | 0.002 |

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## Share and Cite

**MDPI and ACS Style**

Migliaccio, G.; Ferraro, R.; Wang, Z.; Cristini, V.; Dogra, P.; Caserta, S.
Exploring Cell Migration Mechanisms in Cancer: From Wound Healing Assays to Cellular Automata Models. *Cancers* **2023**, *15*, 5284.
https://doi.org/10.3390/cancers15215284

**AMA Style**

Migliaccio G, Ferraro R, Wang Z, Cristini V, Dogra P, Caserta S.
Exploring Cell Migration Mechanisms in Cancer: From Wound Healing Assays to Cellular Automata Models. *Cancers*. 2023; 15(21):5284.
https://doi.org/10.3390/cancers15215284

**Chicago/Turabian Style**

Migliaccio, Giorgia, Rosalia Ferraro, Zhihui Wang, Vittorio Cristini, Prashant Dogra, and Sergio Caserta.
2023. "Exploring Cell Migration Mechanisms in Cancer: From Wound Healing Assays to Cellular Automata Models" *Cancers* 15, no. 21: 5284.
https://doi.org/10.3390/cancers15215284