# Modeling Interactions among Migration, Growth and Pressure in Tumor Dynamics

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## Abstract

**:**

## 1. Introduction

## 2. Biomechanical Mathematical Modeling

#### 2.1. Internal Pressure and Growth

#### 2.2. Some Mathematical Properties of the Growth Model with Internal Pressure

**Lemma**

**1.**

- $\rho (\overline{t},\overline{x})=0$, for some $\overline{t}\in {I}_{w}$. Then, $\rho (t,\overline{x})=0$, for every $t\in {I}_{w}$,
- $A(\overline{t},\overline{x})=0$, for some $\overline{t}\in {I}_{w}$. Then, $A(t,\overline{x})=0$, for every $t\in {I}_{w}$,

**Proposition**

**1.**

**Proof.**

**Lemma**

**2.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

#### 2.3. Cell Density Evolution

## 3. Numerical Results

#### 3.1. Numerical Treatment of the Model

#### 3.2. Growth and Internal Pressure Dependence without Migration

#### 3.3. Migration, Growth and Pressure Interactions

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Heterogeneous growth. (

**a**) A cluster of cells growing faster than its surroundings could cause the loss of cell–cell adhesion. (

**b**) Rearrangement of growth and pressure affects both mutant cluster cells and adjacent cells.

**Figure 2.**Cytonemes mechanosensing. Outline of the cytoneme–cytoneme or cytoneme–membrane interactions of the adjacent cells, which are assumed to be spatially consistent with the mechanical transmissions due to growth and internal regulation of pressure.

**Figure 3.**Growth and internal pressure dependence without migration. (

**a**) Cluster of cells growing slower than its surroundings ($\alpha =1\xb7{10}^{-1}$ and $\alpha =1\xb7{10}^{-2}\phantom{\rule{4pt}{0ex}}{\mathrm{mm}}^{2}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{cell}}^{-1}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{h}}^{-1}$ in the core and the inner region, respectively). (

**b**) Cluster of cells growing faster than its surroundings ($\alpha =1\xb7{10}^{-3}$ and $\alpha =1\xb7{10}^{-2}\phantom{\rule{4pt}{0ex}}{\mathrm{mm}}^{2}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{cell}}^{-1}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{h}}^{-1}$ in the core and the inner region, respectively). For the four cases, proliferation rate is assumed constant, $\beta =3.45\xb7{10}^{-2}\phantom{\rule{4pt}{0ex}}{\mathrm{h}}^{-1}$. (

**c**) Cluster of cells growing slower than its surroundings ($\alpha =1\xb7{10}^{-1}$ and $\alpha =1\xb7{10}^{-2}\phantom{\rule{4pt}{0ex}}{\mathrm{mm}}^{2}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{cell}}^{-1}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{h}}^{-1}$ in the core and the inner region, respectively). (

**d**) Cluster of cells growing faster than its surroundings ($\alpha =1\xb7{10}^{-3}$ and $\alpha =1\xb7{10}^{-2}\phantom{\rule{4pt}{0ex}}{\mathrm{mm}}^{2}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{cell}}^{-1}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{h}}^{-1}$ in the core and the inner region, respectively). For the four cases, proliferation rate is assumed constant, $\beta =3.45\xb7{10}^{-2}\phantom{\rule{4pt}{0ex}}{\mathrm{h}}^{-1}$ and the figures have the same axis label. The results correspond to a tumor with a radius of 1 cm.

**Figure 4.**Migration, growth and pressure interactions. (

**a**) Effect of constant $\alpha $ parameter on growth and migration at time $t=11.25$ h with $\beta =3.45\xb7{10}^{-2}\phantom{\rule{4pt}{0ex}}{\mathrm{h}}^{-1}$. (

**b**) Cluster of cells with high pressure in the core than its surroundings ($\alpha $ = $1\phantom{\rule{4pt}{0ex}}{\mathrm{mm}}^{2}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{cell}}^{-1}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{h}}^{-1}$ and $\alpha $ = $1\xb7{10}^{-3}\phantom{\rule{4pt}{0ex}}{\mathrm{mm}}^{2}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{cell}}^{-1}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{h}}^{-1}$ respectively). Proliferation rate is assumed constant $\beta =3.45\xb7{10}^{-2}\phantom{\rule{4pt}{0ex}}{\mathrm{h}}^{-1}$. (

**c**) Heterogeneous proliferation rate is considered in space (same $\alpha $ parameters than in Figure 4b) and in time. At $t=30$ h, $\beta =3.45\xb7{10}^{-3}\phantom{\rule{4pt}{0ex}}{\mathrm{h}}^{-1}$ and at $t=50$ h proliferation is considered totally inhibited, $\beta =0\phantom{\rule{4pt}{0ex}}{\mathrm{h}}^{-1}$. In the inner regions, $\beta =3.45\xb7{10}^{-2}\phantom{\rule{4pt}{0ex}}{\mathrm{h}}^{-1}$. (

**d**) Pressure is lifted at time $t=30$ h, where $\alpha =1\xb7{10}^{-3}\phantom{\rule{4pt}{0ex}}{\mathrm{mm}}^{2}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{cell}}^{-1}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{h}}^{-1}$ and $\beta =3.45\xb7{10}^{-2}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{h}}^{-1}$. At $t=50$ h, tendency of $\alpha $ and $\beta $ changes: $\alpha =1\xb7{10}^{-3}\phantom{\rule{4pt}{0ex}}{\mathrm{mm}}^{2}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{cell}}^{-1}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{h}}^{-1}$, $\beta =6.9\xb7{10}^{-2}\phantom{\rule{4pt}{0ex}}{\mathrm{h}}^{-1}$ and $\alpha =5\xb7{10}^{-3}\phantom{\rule{4pt}{0ex}}{\mathrm{mm}}^{2}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{cell}}^{-1}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{h}}^{-1}$, $\beta =3.45\xb7{10}^{-2}\phantom{\rule{4pt}{0ex}}{\mathrm{h}}^{-1}$ for the core and inner region. For the four cases, ${c}_{\rho}=0.87\xb7{10}^{-2}\phantom{\rule{4pt}{0ex}}\mathrm{mm}\phantom{\rule{4pt}{0ex}}{\mathrm{h}}^{-1}$ and $m=1$.

Parameter | Description | Value [Unit] | Source |
---|---|---|---|

${\nu}_{\rho}$ | Tumor viscosity | 0.348 $\xb7{10}^{-2}$$\left[{\mathrm{mm}}^{2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{h}}^{-1}\right]$ | [31] |

${c}_{\rho}$ | Tumor velocity | 0.87 $\xb7{10}^{-2}$$\left[\mathrm{mm}\phantom{\rule{0.166667em}{0ex}}{\mathrm{h}}^{-1}\right]$ | [31] |

$\beta $ | Net proliferation rate ${}^{1}$ | 3.45 $\xb7{10}^{-2}$$\left[{\mathrm{h}}^{-1}\right]$ | [31] |

K | Carrying capacity | ${10}^{6}$$\left[\mathrm{cells}\right]$ | [31] |

$\alpha $ | Growth area rate ${}^{1}$ | $[{10}^{-3},1]$$\left[{\mathrm{mm}}^{2}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{cell}}^{-1}\phantom{\rule{0.166667em}{0ex}}{\mathrm{h}}^{-1}\right]$ | See description |

$\kappa $ | Bulk modulus | ≈3.33 [KPa] | See description |

$\mu $ | Shear modulus | ≈0.34 [KPa] | See description |

$\u03f5$ | Cytonemes | 60 [$\mathsf{\mu}$m] | See description |

^{1}when it is constant. The units refer to the case $m=1$ for the two-dimensional tumor dynamics. All the equations have been solved in the one-dimensional case (or the radial case in 2D) with the corresponding adaptation of parameters.

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**MDPI and ACS Style**

Blanco, B.; Campos, J.; Melchor, J.; Soler, J. Modeling Interactions among Migration, Growth and Pressure in Tumor Dynamics. *Mathematics* **2021**, *9*, 1376.
https://doi.org/10.3390/math9121376

**AMA Style**

Blanco B, Campos J, Melchor J, Soler J. Modeling Interactions among Migration, Growth and Pressure in Tumor Dynamics. *Mathematics*. 2021; 9(12):1376.
https://doi.org/10.3390/math9121376

**Chicago/Turabian Style**

Blanco, Beatriz, Juan Campos, Juan Melchor, and Juan Soler. 2021. "Modeling Interactions among Migration, Growth and Pressure in Tumor Dynamics" *Mathematics* 9, no. 12: 1376.
https://doi.org/10.3390/math9121376