# A Mathematical Model of Breast Tumor Progression Based on Immune Infiltration

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Interaction Network of Cells and Molecules—ODE Model

#### 2.1.1. T-Cells

#### CD4+ Helper T-Cells (${T}_{h}$)

#### Cytotoxic T-Cells (${T}_{c}$)

#### Regulatory T-Cells (${T}_{r}$)

#### Naive T-Cells (${T}_{N}$)

#### 2.1.2. Dendritic Cells (D)

#### 2.1.3. Macrophages (M)

#### 2.1.4. Cancer Cells (C)

#### 2.1.5. Cancer Associated Adipocytes (A)

#### 2.1.6. Necrotic Cells (N)

#### 2.1.7. Molecules

#### HMGB1 (H)

#### IL-12 ($I{L}_{12}$)

#### IL-10 ($I{L}_{10}$)

#### Estrogen (E)

#### IFN-$\gamma $ (${I}_{\gamma}$)

#### IL-6 ($I{L}_{6}$)

#### 2.2. Data of the Model

#### 2.2.1. Breast Cancer Patients’ Data

#### 2.2.2. Patient Data Analysis

#### 2.2.3. Parameter Estimation

#### 2.2.4. Sensitivity Analysis

- First, we define a local sensitivity measure for each parameter ${\theta}_{i}$ in the neighborhood ${\mathsf{\Omega}}^{k}\left({\theta}_{i}\right)$ as$${S}_{i}^{k}={\int}_{{\mathsf{\Omega}}^{k}\left({\theta}_{i}\right)}{s}_{i}\left(\theta \right)d\theta .$$
- Second, we found weights for the aforementioned neighborhoods. Scaling each assumption provides us with a new set of parameters. The weights were then determined by calculating the distance of each resulting parameter set to a fixed base parameter set. We assigned higher weights to the parameters that were closer to the base values. We denote each weight by ${w}_{i}^{k}$ for $i=1,\cdots ,N$ and $k=1,\cdots ,K$ corresponding to the parameter and its neighborhood, respectively.
- Finally, we obtained the global sensitivity level ${\mathbf{S}}_{i}$ to each parameter ${\theta}_{i}$ by$${\mathbf{S}}_{i}=\sum _{k=1}^{K}{w}_{i}^{k}{S}_{i}^{k}.$$

## 3. Results

#### 3.1. Data Analysis of the Clusters

#### 3.2. Dynamics of the Breast Cancer Microenvironment

#### 3.3. Sensitivity Analysis

#### 3.4. Dynamics with Varying Assumptions

#### 3.5. Dynamics with Different Initial Conditions

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

TCGA | The Cancer Genome Atlas |

METABRIC | Molecular Taxonomy of Breast Cancer International Consortium |

HMGB1 | High mobility group box-1 |

LumA | Luminal A |

LumB | Luminal B |

TNBC | Triple negative breast cancer |

IFN-$\gamma $ | Interferon gamma |

HER2 | Human epidermal growth factor 2 |

ER | Estrogen receptor |

DAMP | Damage-associated molecular pattern |

UCSC | University of Santa Cruz |

## Appendix A. Derivation of Sample Parameters

#### Appendix A.1. Steady State and Additional Assumptions

**Table A1.**

**Mean values of the variables used.**The mean values were calculated from the combination of TCGA and METABRIC datasets for all of the clusters.

${\mathit{IL}}_{6}^{\mathbf{mean}}$ | ${\mathit{A}}^{\mathbf{mean}}$ | ${\mathit{T}}_{\mathit{c}}^{\mathbf{mean}}$ | ${\mathit{I}}_{\mathit{\gamma}}^{\mathbf{mean}}$ |
---|---|---|---|

$3.906$ | $7.112\times {10}^{4}$ | $4.129\times {10}^{3}$ | $3.675$ |

**Table A2.**

**Maximum values of the variables.**The maximum values were calculated from the combination of TCGA and METABRIC datasets for all of the clusters.

${\mathit{T}}_{\mathit{N}}^{\mathbf{max}}$ | ${\mathit{T}}_{\mathit{c}}^{\mathbf{max}}$ | ${\mathit{T}}_{\mathit{h}}^{\mathbf{max}}$ | ${\mathit{T}}_{\mathit{r}}^{\mathbf{max}}$ | ${\mathit{D}}_{\mathit{N}}^{\mathbf{max}}$ | ${\mathit{D}}^{\mathbf{max}}$ |
---|---|---|---|---|---|

$4.270\times {10}^{4}$ | $1.599\times {10}^{4}$ | $2.495\times {10}^{4}$ | $1.174\times {10}^{4}$ | $6.507\times {10}^{3}$ | $8.274\times {10}^{3}$ |

${M}_{N}^{\mathrm{max}}$ | ${M}^{\mathrm{max}}$ | ${N}^{\mathrm{max}}$ | ${N}^{\mathrm{max}}$ | ${A}^{\mathrm{max}}$ | ${H}^{\mathrm{max}}$ |

$4.577\times {10}^{4}$ | $5.247\times {10}^{4}$ | $4.258\times {10}^{5}$ | $1.019\times {10}^{5}$ | $2.612\times {10}^{5}$ | $1.160\times {10}^{1}$ |

$I{L}_{12}^{\mathrm{max}}$ | $I{L}_{10}^{\mathrm{max}}$ | ${E}^{\mathrm{max}}$ | ${I}_{\gamma}^{\mathrm{max}}$ | $I{L}_{6}^{\mathrm{max}}$ | |

$2.516\times {10}^{1}$ | $5.962$ | $1.868\times {10}^{1}$ | $8.768$ | $1.073\times {10}^{1}$ |

#### Appendix A.2. Non-Dimensionalization

#### Appendix A.3. Parameter Values

**Table A3.**

**Half-lives and estimated death rate**. Degredation and death rate taken or calculated from the given references.

Parameter | Value | Reference | Parameter | Value | Reference |
---|---|---|---|---|---|

${\delta}_{{T}_{N}}$ | $9.49\times {10}^{-4}$ | [71] | ${\delta}_{H}$ | 18 | [71,121] |

${\delta}_{{T}_{c}}$ | 0.231 | [71] | ${\delta}_{E}$ | 4.16 | [122] |

${\delta}_{{T}_{h}}$ | 0.406 | [71] | ${\delta}_{I{L}_{6}}$ | 1.07 | [71] |

${\delta}_{{T}_{r}}$ | 0.406 | [71] | ${\delta}_{I{L}_{10}}$ | 4.62 | [71] |

${\delta}_{D}$ | 0.277 | [71] | ${\delta}_{I{L}_{12}}$ | 128 | [123] |

${\delta}_{M}$ | 0.0198 | [71] | ${\delta}_{{I}_{\gamma}}$ | 33.3 | [71] |

${\delta}_{A}$ | $2.8\times {10}^{-3}$ | [125] |

Parameter | Cluster 1 | Cluster 2 | Cluster 3 | Cluster 4 | Cluster 5 |
---|---|---|---|---|---|

${\overline{\lambda}}_{{T}_{h}H}$ | $1.791\times {10}^{-1}$ | $4.048\times {10}^{-2}$ | $1.002\times {10}^{-1}$ | $2.478\times {10}^{-1}$ | $3.871\times {10}^{-1}$ |

${\overline{\lambda}}_{{T}_{h}D}$ | $2.933$ | $1.801$ | $2.200$ | $4.027$ | $3.135$ |

${\overline{\lambda}}_{{T}_{h}I{L}_{12}}$ | $1.569\times {10}^{-1}$ | $2.286\times {10}^{-2}$ | $6.850\times {10}^{-2}$ | $2.623\times {10}^{-1}$ | $3.919\times {10}^{-1}$ |

${\overline{\lambda}}_{{T}_{h}E}$ | $1.756\times {10}^{-1}$ | $2.633\times {10}^{-2}$ | $1.063\times {10}^{-1}$ | $2.795\times {10}^{-1}$ | $4.696\times {10}^{-1}$ |

${\overline{\lambda}}_{{T}_{c}E}$ | $6.022$ | $3.297$ | $4.332$ | $8.420$ | $7.669$ |

${\overline{\lambda}}_{{T}_{c}D}$ | $5.028\times {10}^{-3}$ | $1.128\times {10}^{-2}$ | $4.482\times {10}^{-3}$ | $6.064\times {10}^{-3}$ | $2.560\times {10}^{-3}$ |

${\overline{\lambda}}_{{T}_{c}I{L}_{12}}$ | $2.690\times {10}^{-2}$ | $1.431\times {10}^{-2}$ | $1.396\times {10}^{-2}$ | $3.950\times {10}^{-2}$ | $3.200\times {10}^{-2}$ |

${\overline{\lambda}}_{{T}_{r}D}$ | $5.783\times {10}^{-2}$ | $1.334\times {10}^{-1}$ | $6.763\times {10}^{-2}$ | $5.167\times {10}^{-2}$ | $2.721\times {10}^{-2}$ |

${\overline{\lambda}}_{{T}_{r}E}$ | $1.732\times {10}^{-1}$ | $9.755\times {10}^{-2}$ | $1.634\times {10}^{-1}$ | $1.793\times {10}^{-1}$ | $2.038\times {10}^{-1}$ |

${\overline{\lambda}}_{DC}$ | $3.346\times {10}^{-2}$ | $6.768\times {10}^{-2}$ | $4.237\times {10}^{-2}$ | $2.540\times {10}^{-2}$ | $2.505\times {10}^{-2}$ |

${\overline{\lambda}}_{DH}$ | $1.526\times {10}^{-1}$ | $1.709\times {10}^{-1}$ | $1.444\times {10}^{-1}$ | $1.474\times {10}^{-1}$ | $1.430\times {10}^{-1}$ |

${\overline{\lambda}}_{DE}$ | $1.497\times {10}^{-1}$ | $1.112\times {10}^{-1}$ | $1.532\times {10}^{-1}$ | $1.663\times {10}^{-1}$ | $1.734\times {10}^{-1}$ |

${\overline{\lambda}}_{MI{L}_{10}}$ | $2.499\times {10}^{-3}$ | $1.201\times {10}^{-3}$ | $1.978\times {10}^{-3}$ | $2.535\times {10}^{-3}$ | $3.128\times {10}^{-3}$ |

${\overline{\lambda}}_{M{I}_{\gamma}}$ | $1.720\times {10}^{-3}$ | $7.178\times {10}^{-4}$ | $1.244\times {10}^{-3}$ | $1.941\times {10}^{-3}$ | $2.177\times {10}^{-3}$ |

${\overline{\lambda}}_{MI{L}_{12}}$ | $1.991\times {10}^{-3}$ | $9.880\times {10}^{-4}$ | $1.469\times {10}^{-3}$ | $2.050\times {10}^{-3}$ | $2.484\times {10}^{-3}$ |

${\overline{\lambda}}_{M{T}_{h}}$ | $1.136\times {10}^{-2}$ | $1.575\times {10}^{-2}$ | $1.283\times {10}^{-2}$ | $1.109\times {10}^{-2}$ | $9.035\times {10}^{-3}$ |

${\overline{\lambda}}_{ME}$ | $2.229\times {10}^{-3}$ | $1.138\times {10}^{-3}$ | $2.279\times {10}^{-3}$ | $2.184\times {10}^{-3}$ | $2.977\times {10}^{-3}$ |

${\overline{\lambda}}_{C}$ | $5.756\times {10}^{-2}$ | $4.250\times {10}^{-2}$ | $5.669\times {10}^{-2}$ | $3.427\times {10}^{-2}$ | $3.959\times {10}^{-2}$ |

${\overline{\lambda}}_{CI{L}_{6}}$ | $4.345\times {10}^{-4}$ | $8.494\times {10}^{-4}$ | $3.381\times {10}^{-4}$ | $5.530\times {10}^{-3}$ | $4.595\times {10}^{-3}$ |

${\overline{\lambda}}_{CA}$ | $2.357\times {10}^{-4}$ | $1.243\times {10}^{-3}$ | $2.859\times {10}^{-4}$ | $1.807\times {10}^{-3}$ | $1.205\times {10}^{-3}$ |

${\overline{\lambda}}_{A}$ | $3.384\times {10}^{-3}$ | $3.394\times {10}^{-3}$ | $3.367\times {10}^{-3}$ | $3.400\times {10}^{-3}$ | $3.282\times {10}^{-3}$ |

${\overline{\lambda}}_{HD}$ | $1.458$ | $2.113$ | $1.529$ | $1.239$ | $7.319\times {10}^{-1}$ |

${\overline{\lambda}}_{HN}$ | $4.156$ | $2.858$ | $2.861$ | $3.552$ | $4.083$ |

${\overline{\lambda}}_{HM}$ | $9.720\times {10}^{-1}$ | $8.862\times {10}^{-1}$ | $1.421$ | $7.101\times {10}^{-1}$ | $7.883\times {10}^{-1}$ |

${\overline{\lambda}}_{H{T}_{c}}$ | $3.605$ | $4.620$ | $4.014$ | $7.244$ | $6.063$ |

${\overline{\lambda}}_{HC}$ | $7.809$ | $7.523$ | $8.173$ | $5.255$ | $6.334$ |

${\overline{\lambda}}_{I{L}_{12}M}$ | $5.253\times {10}^{1}$ | $4.698\times {10}^{1}$ | $6.480\times {10}^{1}$ | $3.738\times {10}^{1}$ | $4.730\times {10}^{1}$ |

${\overline{\lambda}}_{I{L}_{12}D}$ | $7.880$ | $1.120\times {10}^{1}$ | $6.972$ | $6.521$ | $4.391$ |

${\overline{\lambda}}_{I{L}_{12}{T}_{h}}$ | $4.810\times {10}^{1}$ | $4.533\times {10}^{1}$ | $3.793\times {10}^{1}$ | $4.596\times {10}^{1}$ | $3.994\times {10}^{1}$ |

${\overline{\lambda}}_{I{L}_{12}{T}_{c}}$ | $1.948\times {10}^{1}$ | $2.449\times {10}^{1}$ | $1.830\times {10}^{1}$ | $3.813\times {10}^{1}$ | $3.638\times {10}^{1}$ |

${\overline{\lambda}}_{I{L}_{10}M}$ | $1.197$ | $1.155$ | $1.606$ | $9.978\times {10}^{-1}$ | $1.252$ |

${\overline{\lambda}}_{I{L}_{10}D}$ | $1.795\times {10}^{-1}$ | $2.754\times {10}^{-1}$ | $1.728\times {10}^{-1}$ | $1.741\times {10}^{-1}$ | $1.162\times {10}^{-1}$ |

${\overline{\lambda}}_{I{L}_{10}{T}_{r}}$ | $7.419\times {10}^{-1}$ | $4.921\times {10}^{-1}$ | $5.241\times {10}^{-1}$ | $4.651\times {10}^{-1}$ | $2.271\times {10}^{-1}$ |

${\overline{\lambda}}_{I{L}_{10}{T}_{h}}$ | $1.096$ | $1.115$ | $9.400\times {10}^{-1}$ | $1.227$ | $1.057$ |

${\overline{\lambda}}_{I{L}_{10}{T}_{c}}$ | $4.440\times {10}^{-1}$ | $6.022\times {10}^{-1}$ | $4.536\times {10}^{-1}$ | $1.018$ | $9.626\times {10}^{-1}$ |

${\overline{\lambda}}_{I{L}_{10}C}$ | $9.615\times {10}^{-1}$ | $9.806\times {10}^{-1}$ | $9.235\times {10}^{-1}$ | $7.384\times {10}^{-1}$ | $1.006$ |

${\overline{\lambda}}_{EA}$ | $1.024\times {10}^{-1}$ | $8.334\times {10}^{-2}$ | $9.433\times {10}^{-2}$ | $1.389\times {10}^{-1}$ | $1.278\times {10}^{-1}$ |

${\overline{\lambda}}_{E}$ | $5.935$ | $4.761$ | $5.600$ | $7.870$ | $8.708$ |

${\overline{\lambda}}_{{I}_{\gamma}{T}_{c}}$ | $2.115\times {10}^{1}$ | $2.278\times {10}^{1}$ | $2.234\times {10}^{1}$ | $2.611\times {10}^{1}$ | $2.677\times {10}^{1}$ |

${\overline{\lambda}}_{{I}_{\gamma}{T}_{h}}$ | $1.044\times {10}^{1}$ | $8.434$ | $9.259$ | $6.295$ | $5.879$ |

${\overline{\lambda}}_{{I}_{\gamma}D}$ | $1.711$ | $2.084$ | $1.702$ | $8.931\times {10}^{-1}$ | $6.465\times {10}^{-1}$ |

${\overline{\lambda}}_{I{L}_{6}A}$ | $5.692\times {10}^{-1}$ | $5.110\times {10}^{-1}$ | $4.652\times {10}^{-1}$ | $5.329\times {10}^{-1}$ | $5.150\times {10}^{-1}$ |

${\overline{\lambda}}_{I{L}_{6}M}$ | $4.355\times {10}^{-1}$ | $4.513\times {10}^{-1}$ | $5.460\times {10}^{-1}$ | $4.573\times {10}^{-1}$ | $5.079\times {10}^{-1}$ |

${\overline{\lambda}}_{I{L}_{6}D}$ | $6.532\times {10}^{-2}$ | $1.076\times {10}^{-1}$ | $5.875\times {10}^{-2}$ | $7.977\times {10}^{-2}$ | $4.715\times {10}^{-2}$ |

${\overline{\delta}}_{{T}_{h}{T}_{r}}$ | $7.562\times {10}^{-1}$ | $6.086\times {10}^{-1}$ | $5.961\times {10}^{-1}$ | $6.522\times {10}^{-1}$ | $2.427\times {10}^{-1}$ |

${\overline{\delta}}_{{T}_{h}I{L}_{10}}$ | $2.457$ | $1.051$ | $1.648$ | $3.933$ | $3.910$ |

${\overline{\delta}}_{{T}_{c}I{L}_{10}}$ | $4.319$ | $1.847$ | $2.896$ | $6.913$ | $6.871$ |

${\overline{\delta}}_{{T}_{c}Tr}$ | $1.329$ | $1.070$ | $1.048$ | $1.146$ | $4.266\times {10}^{-1}$ |

${\overline{\delta}}_{DC}$ | $5.876\times {10}^{-2}$ | $7.271\times {10}^{-2}$ | $6.297\times {10}^{-2}$ | $6.209\times {10}^{-2}$ | $6.444\times {10}^{-2}$ |

${\overline{\delta}}_{{D}_{N}}$ | $2.770\times {10}^{-1}$ | $2.770\times {10}^{-1}$ | $2.770\times {10}^{-1}$ | $2.770\times {10}^{-1}$ | $2.770\times {10}^{-1}$ |

${\overline{\delta}}_{{M}_{N}}$ | $1.980\times {10}^{-2}$ | $1.980\times {10}^{-2}$ | $1.980\times {10}^{-2}$ | $1.980\times {10}^{-2}$ | $1.980\times {10}^{-2}$ |

${\overline{\delta}}_{C{T}_{c}}$ | $4.399\times {10}^{-3}$ | $5.276\times {10}^{-3}$ | $4.936\times {10}^{-3}$ | $8.052\times {10}^{-3}$ | $6.762\times {10}^{-3}$ |

${\overline{\delta}}_{C{I}_{\gamma}}$ | $2.741\times {10}^{-3}$ | $7.415\times {10}^{-4}$ | $1.654\times {10}^{-3}$ | $2.832\times {10}^{-3}$ | $2.982\times {10}^{-3}$ |

${\overline{\delta}}_{C}$ | $4.492\times {10}^{-2}$ | $3.273\times {10}^{-2}$ | $4.421\times {10}^{-2}$ | $2.606\times {10}^{-2}$ | $3.037\times {10}^{-2}$ |

${\overline{\delta}}_{N}$ | $2.043\times {10}^{-1}$ | $2.130\times {10}^{-1}$ | $3.031\times {10}^{-1}$ | $1.141\times {10}^{-1}$ | $1.300\times {10}^{-1}$ |

${\overline{A}}_{{T}_{N}}$ | $9.730$ | $5.445$ | $7.057$ | $1.351\times {10}^{1}$ | $1.232\times {10}^{1}$ |

${\overline{A}}_{{D}_{N}}$ | $6.128\times {10}^{-1}$ | $6.267\times {10}^{-1}$ | $6.170\times {10}^{-1}$ | $6.161\times {10}^{-1}$ | $6.184\times {10}^{-1}$ |

${\overline{A}}_{M}$ | $3.960\times {10}^{-2}$ | $3.960\times {10}^{-2}$ | $3.960\times {10}^{-2}$ | $3.960\times {10}^{-2}$ | $3.960\xb7{10}^{-2}$ |

${\overline{\alpha}}_{NC}$ | $3.924$ | $5.497$ | $5.966$ | $3.089$ | $3.240$ |

${\overline{C}}_{0}$ | $9.428$ | $7.619$ | $8.798$ | $8.923$ | $8.597$ |

${\overline{A}}_{0}$ | $5.795$ | $5.713$ | $5.937$ | $5.666$ | $6.813$ |

${\overline{E}}_{0}$ | $3.161$ | $6.958$ | $3.649$ | $2.045$ | $1.862$ |

#### Appendix A.4. Dynamics with Varying Initial Conditions

**Figure A1.**

**The dynamics with varying initial conditions**. Subfigures (

**A**–

**E**) show the dynamics of cells and cytokines with initial conditions from different patients in clusters 1, 2, 3, 4, and 5, respectively.

#### Appendix A.5. Dynamics of the Tumor Microenvironment with Cross-Cluster Initial Conditions

**Figure A2.**

**Dynamics with cross-cluster initial conditions:**(

**A**) parameters from cluster 1 and initial conditions from clusters 2, 3, 4, and 5; (

**B**) parameters from cluster 2 and initial conditions from clusters 1, 3, 4, and 5; (

**C**) parameters from cluster 3 and initial conditions from clusters 1, 2, 4, and 5; (

**D**) parameters from cluster 4 and initial conditions from clusters 1, 2, 3, and 5; and (

**E**) parameters from cluster 5 and initial conditions from clusters 1, 2, 3, and 4.

#### Appendix A.6. Bifurcation and Lyapunov Exponent for the Cancer ODE

**Figure A3.**

**Bifurcation and Lyapunov exponent diagrams for cancer parameters:**Subfigures show the bifurcation on top of the corresponding Lyapunov diagrams for clusters 1–5. Bifurcation was done for parameters ${\lambda}_{C}$, ${\lambda}_{CI{L}_{6}}$, ${\lambda}_{CA}$, ${\delta}_{C{T}_{c}}$, ${\delta}_{C{I}_{\gamma}}$, and ${\delta}_{C}$ from the cancer ODE (9).

#### Appendix A.7. Positivity

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**Figure 1.**

**Interaction network.**The main interaction network of cells and molecules in breast tumors, modeled in this paper. Variables of the model with their descriptions are given in Table 1.

**Figure 2.**

**Immune cell frequencies in each cluster.**Clusters were obtained by applying K-means clustering to the percentages of 22 immune cell types in breast tumors. The most variant cells among clusters are shown in this figure.

**Figure 3.**

**More Clinical features of each cluster.**Subfigures (

**A**,

**B**), respectively, show box plots of patients’ age at diagnosis and survival months in each cluster. Subfigure (

**C**) demonstrates Kaplan–Meier curves of overall survival probability across five clusters. Asterisks in the figures show the significance levels with Mann–Whitney-Wilcoxon (MWW) statistical test where, ns: no significance, ***: $0.0001<p\le 0.001$, ****: $p\le 0.0001$.

**Figure 4.**

**Clinical features of each cluster.**Subfigures (

**A**–

**D**) show the percentage of patients with different subtypes of breast cancer, survival status, HER2 status, and ER status, respectively.

**Figure 5.**

**Dynamics of all variables.**Dynamics of variables of the model over 3000 days. The different color lines describe the dynamics of different clusters.

**Figure 6.**

**Sensitivity analysis.**Sensitivity level of the most sensitive parameters for cancer and total cell population, respectively.

**Figure 8.**

**Dynamics of cancer after the assumptions of the sensitive parameters were modified.**Subfigures (

**A**–

**F**) present dynamics after scaling the assumptions (24)–(29), respectively. The transparent region was the result of 10% perturbation of all the sensitive parameters from Section 3.3.

**Table 1.**

**Patient data correspondence with variables.**Correspondence among the model variables and the gene expression data of the primary tumors and deconvolution results.

Variable | Name | Data Used |
---|---|---|

${T}_{N}$ | Naive T-cells | Combination of CD4 naive and memory resting T-cells and resting NK cells |

${T}_{h}$ | Helper T-cells | Combination of memory activated CD4 T-cells and follicular helper T-cells |

${T}_{C}$ | Cytotoxic cells | Combination of CD8 T-cells and activated NK cells |

${T}_{r}$ | Regulatory T-cells | Regulatory T-cells |

${D}_{N}$ | Naive dendritic cells | Naive dendritic cells |

D | Activated dendritic cells | Activated dendritic cells |

${M}_{N}$ | Naive Macrophages | Combination of Macrophages M0 and Monocytes |

M | Macrophages | Combination of M1 and M2 Macrophages |

C | Cancer cells | Estimated |

N | Necrotic cells | Estimated |

A | Cancer Associated Adipocytes | Assumed to be twice the total number of immune cells |

H | HMGB1 | HMGB1 gene expression |

$I{L}_{12}$ | IL-12 | IL12A and IL12B gene expressions |

$I{L}_{10}$ | IL-10 | IL10 gene expression |

E | Estrogen | ESR1 and ESR2 gene expressions |

${I}_{\gamma}$ | IFN-$\gamma $ | IFNG gene expressions |

$I{L}_{6}$ | IL-6 | IL6 gene expression |

**Table 2.**

**Initial conditions of the model variables.**Values of the initial conditions were obtained from the averages of the smallest tumors in METABRIC and TCGA data.

Cluster | ${\mathbf{T}}_{\mathbf{N}}$ | ${\mathbf{T}}_{\mathbf{h}}$ | ${\mathbf{T}}_{\mathbf{C}}$ | ${\mathbf{T}}_{\mathbf{r}}$ | ${\mathbf{D}}_{\mathbf{N}}$ | D | ${\mathbf{M}}_{\mathbf{N}}$ | M | C |
---|---|---|---|---|---|---|---|---|---|

1 | $2.62\times {10}^{2}$ | $1.93\times {10}^{3}$ | $2.94\times {10}^{3}$ | $5.55\times {10}^{2}$ | $9.90\times {10}^{-3}$ | $9.90\times {10}^{-3}$ | $1.34\times {10}^{4}$ | $5.37\times {10}^{3}$ | $1.56\times {10}^{3}$ |

2 | $3.78\times {10}^{3}$ | $2.16\times {10}^{3}$ | $2.95\times {10}^{3}$ | $5.04\times {10}^{2}$ | $1.07\times {10}^{3}$ | $9.83\times {10}^{3}$ | $2.63\times {10}^{3}$ | $1.12\times {10}^{4}$ | $2.31\times {10}^{3}$ |

3 | $2.93\times {10}^{3}$ | $1.15\times {10}^{3}$ | $1.39\times {10}^{3}$ | $3.68\times {10}^{1}$ | $3.22$ | $3.62\times {10}^{-2}$ | $6.75\times {10}^{3}$ | $9.67\times {10}^{3}$ | $3.53\times {10}^{3}$ |

4 | $4.78\times {10}^{3}$ | $4.60\times {10}^{3}$ | $2.76\times {10}^{3}$ | $1.66\times {10}^{3}$ | $3.51\times {10}^{2}$ | $1.34\times {10}^{3}$ | $2.29\times {10}^{3}$ | $4.69\times {10}^{3}$ | $7.96\times {10}^{3}$ |

5 | $3.78\times {10}^{3}$ | $1.33\times {10}^{3}$ | $3.11\times {10}^{3}$ | $1.05\times {10}^{3}$ | $5.16\times {10}^{2}$ | $1.03\times {10}^{-2}$ | $2.85\times {10}^{3}$ | $6.07\times {10}^{3}$ | $5.57\times {10}^{3}$ |

$\mathbf{N}$ | $\mathbf{A}$ | $\mathbf{H}$ | ${\mathbf{IL}}_{\mathbf{12}}$ | ${\mathbf{IL}}_{\mathbf{10}}$ | $\mathbf{E}$ | ${\mathbf{I}}_{\mathit{\gamma}}$ | ${\mathbf{IL}}_{\mathbf{6}}$ | ||

1 | $1.21\times {10}^{2}$ | $4.89\times {10}^{4}$ | $5.06$ | $5.82$ | $2.90$ | $7.62$ | $3.05$ | $3.67$ | |

2 | $3.16\times {10}^{2}$ | $4.86\times {10}^{4}$ | $5.01$ | $6.40$ | $2.93$ | $8.76$ | $2.93$ | $3.02$ | |

3 | $3.11\times {10}^{2}$ | $4.39\times {10}^{4}$ | $5.29$ | $5.25$ | $2.79$ | $9.16$ | $2.81$ | $4.27$ | |

4 | $2.71\times {10}^{3}$ | $4.49\times {10}^{4}$ | $5.29$ | $6.88$ | $3.27$ | $6.38$ | $4.10$ | $3.40$ | |

5 | $3.73\times {10}^{2}$ | $3.74\times {10}^{4}$ | $6.16$ | $5.67$ | $2.66$ | $8.63$ | $2.70$ | $3.03$ |

**Table 3.**

**Steady state values of the model variables.**Large tumors in each cluster were grouped, and their average values were found for each variable, to be used in the parameter estimation.

Cluster | ${\mathbf{T}}_{\mathbf{N}}^{\mathbf{\infty}}$ | ${\mathbf{T}}_{\mathbf{h}}^{\mathbf{\infty}}$ | ${\mathbf{T}}_{\mathbf{C}}^{\mathbf{\infty}}$ | ${\mathbf{T}}_{\mathbf{r}}^{\mathbf{\infty}}$ | ${\mathbf{D}}_{\mathbf{N}}^{\mathbf{\infty}}$ | ${\mathbf{D}}^{\mathbf{\infty}}$ | ${\mathbf{M}}_{\mathbf{N}}^{\mathbf{\infty}}$ | ${\mathbf{M}}^{\mathbf{\infty}}$ | ${\mathbf{C}}^{\mathbf{\infty}}$ |
---|---|---|---|---|---|---|---|---|---|

1 | $4.55\times {10}^{3}$ | $3.87\times {10}^{3}$ | $2.44\times {10}^{3}$ | $1.92\times {10}^{3}$ | $1.26\times {10}^{2}$ | $3.28\times {10}^{2}$ | $1.80\times {10}^{4}$ | $1.39\times {10}^{4}$ | $9.03\times {10}^{4}$ |

2 | $1.13\times {10}^{4}$ | $4.77\times {10}^{3}$ | $4.02\times {10}^{3}$ | $1.55\times {10}^{3}$ | $3.27\times {10}^{2}$ | $6.10\times {10}^{2}$ | $6.96\times {10}^{3}$ | $1.62\times {10}^{4}$ | $1.12\times {10}^{5}$ |

3 | $5.73\times {10}^{3}$ | $3.70\times {10}^{3}$ | $2.79\times {10}^{3}$ | $1.51\times {10}^{3}$ | $4.00\times {10}^{2}$ | $3.52\times {10}^{2}$ | $8.77\times {10}^{3}$ | $2.07\times {10}^{4}$ | $9.68\times {10}^{4}$ |

4 | $4.14\times {10}^{3}$ | $5.95\times {10}^{3}$ | $7.71\times {10}^{3}$ | $1.66\times {10}^{3}$ | $4.99\times {10}^{2}$ | $4.37\times {10}^{2}$ | $9.81\times {10}^{3}$ | $1.59\times {10}^{4}$ | $9.54\times {10}^{4}$ |

5 | $5.69\times {10}^{3}$ | $3.91\times {10}^{3}$ | $5.56\times {10}^{3}$ | $6.16\times {10}^{2}$ | $7.04\times {10}^{2}$ | $2.22\times {10}^{2}$ | $6.45\times {10}^{3}$ | $1.52\times {10}^{4}$ | $9.90\times {10}^{4}$ |

${\mathbf{N}}^{\infty}$ | ${\mathbf{A}}^{\infty}$ | ${\mathbf{H}}^{\infty}$ | ${\mathbf{IL}}_{\mathbf{12}}^{\infty}$ | ${\mathbf{IL}}_{\mathbf{10}}^{\infty}$ | ${\mathbf{E}}^{\infty}$ | ${\mathbf{I}}_{\mathit{\gamma}}^{\infty}$ | ${\mathbf{IL}}_{\mathbf{6}}^{\infty}$ | ||

1 | $1.15\times {10}^{4}$ | $9.01\times {10}^{4}$ | $5.61$ | $6.42$ | $3.17$ | $8.86$ | $3.21$ | $3.42$ | |

2 | $1.02\times {10}^{4}$ | $9.14\times {10}^{4}$ | $3.84$ | $2.84$ | $1.36$ | $4.03$ | $1.19$ | $1.28$ | |

3 | $8.11\times {10}^{3}$ | $8.80\times {10}^{4}$ | $4.49$ | $4.02$ | $2.13$ | $7.68$ | $1.97$ | $2.14$ | |

4 | $1.54\times {10}^{4}$ | $9.22\times {10}^{4}$ | $7.54$ | $1.04$ | $5.08$ | $1.37$ | $5.72$ | $5.80$ | |

5 | $1.53\times {10}^{4}$ | $7.67\times {10}^{4}$ | $7.70$ | $1.02$ | $5.05$ | $1.50$ | $5.17$ | $6.01$ |

Clusters | Without Scaling | Scale = 0.2 | Scale = 5 |
---|---|---|---|

Cluster 1 | ${\delta}_{C{T}_{c}}=0.00440$ ${\delta}_{C{I}_{\gamma}}=0.00274$ ${\delta}_{C}=0.04492$ | ${\delta}_{C{T}_{c}}=0.00895$ ${\delta}_{C{I}_{\gamma}}=0.00111$ ${\delta}_{C}=0.01827$ | ${\delta}_{C{T}_{c}}=0.00124$ ${\delta}_{C{I}_{\gamma}}=0.00390$ ${\delta}_{C}=0.06341$ |

Cluster 2 | ${\delta}_{C{T}_{c}}=0.00528$ ${\delta}_{C{I}_{\gamma}}=0.00074$ ${\delta}_{C}=0.03273$ | ${\delta}_{C{T}_{c}}=0.00953$ ${\delta}_{C{I}_{\gamma}}=0.00027$ ${\delta}_{C}=0.01182$ | ${\delta}_{C{T}_{c}}=0.00163$ ${\delta}_{C{I}_{\gamma}}=0.00115$ ${\delta}_{C}=0.05063$ |

Cluster 3 | ${\delta}_{C{T}_{c}}=0.00494$ ${\delta}_{C{I}_{\gamma}}=0.00165$ ${\delta}_{C}=0.04421$ | ${\delta}_{C{T}_{c}}=0.01027$ ${\delta}_{C{I}_{\gamma}}=0.00069$ ${\delta}_{C}=0.01839$ | ${\delta}_{C{T}_{c}}=0.00137$ ${\delta}_{C{I}_{\gamma}}=0.00230$ ${\delta}_{C}=0.06150$ |

Cluster 4 | ${\delta}_{C{T}_{c}}=0.00805$ ${\delta}_{C{I}_{\gamma}}=0.00283$ ${\delta}_{C}=0.02606$ | ${\delta}_{C{T}_{c}}=0.01183$ ${\delta}_{C{I}_{\gamma}}=0.00083$ ${\delta}_{C}=0.00766$ | ${\delta}_{C{T}_{c}}=0.00310$ ${\delta}_{C{I}_{\gamma}}=0.00546$ ${\delta}_{C}=0.05018$ |

Cluster 5 | ${\delta}_{C{T}_{c}}=0.00676$ ${\delta}_{C{I}_{\gamma}}=0.00298$ ${\delta}_{C}=0.03037$ | ${\delta}_{C{T}_{c}}=0.01068$ ${\delta}_{C{I}_{\gamma}}=0.00094$ ${\delta}_{C}=0.00960$ | ${\delta}_{C{T}_{c}}=0.00238$ ${\delta}_{C{I}_{\gamma}}=0.00526$ ${\delta}_{C}=0.05355$ |

Cluster | Without Scaling | Scale = 0.2 | Scale = 5 |
---|---|---|---|

Cluster 1 | ${\delta}_{C{T}_{c}}=0.00440$ ${\delta}_{C{I}_{\gamma}}=0.00274$ ${\delta}_{C}=0.04492$ | ${\delta}_{C{T}_{c}}=0.00168$ ${\delta}_{C{I}_{\gamma}}=0.00104$ ${\delta}_{C}=0.08556$ | ${\delta}_{C{T}_{c}}=0.00652$ ${\delta}_{C{I}_{\gamma}}=0.00406$ ${\delta}_{C}=0.013308$ |

Cluster 2 | ${\delta}_{C{T}_{c}}=0.00528$ ${\delta}_{C{I}_{\gamma}}=0.00074$ ${\delta}_{C}=0.03273$ | ${\delta}_{C{T}_{c}}=0.00161$ ${\delta}_{C{I}_{\gamma}}=0.00023$ ${\delta}_{C}=0.05002$ | ${\delta}_{C{T}_{c}}=0.00967$ ${\delta}_{C{I}_{\gamma}}=0.00136$ ${\delta}_{C}=0.01199$ |

Cluster 3 | ${\delta}_{C{T}_{c}}=0.00494$ ${\delta}_{C{I}_{\gamma}}=0.00165$ ${\delta}_{C}=0.04421$ | ${\delta}_{C{T}_{c}}=0.00155$ ${\delta}_{C{I}_{\gamma}}=0.00052$ ${\delta}_{C}=0.06927$ | ${\delta}_{C{T}_{c}}=0.00879$ ${\delta}_{C{I}_{\gamma}}=0.00294$ ${\delta}_{C}=0.01574$ |

Cluster 4 | ${\delta}_{C{T}_{c}}=0.00805$ ${\delta}_{C{I}_{\gamma}}=0.00283$ ${\delta}_{C}=0.02606$ | ${\delta}_{C{T}_{c}}=0.00467$ ${\delta}_{C{I}_{\gamma}}=0.00164$ ${\delta}_{C}=0.07563$ | ${\delta}_{C{T}_{c}}=0.00941$ ${\delta}_{C{I}_{\gamma}}=0.00331$ ${\delta}_{C}=0.00609$ |

Cluster 5 | ${\delta}_{C{T}_{c}}=0.00676$ ${\delta}_{C{I}_{\gamma}}=0.00298$ ${\delta}_{C}=0.03037$ | ${\delta}_{C{T}_{c}}=0.00358$ ${\delta}_{C{I}_{\gamma}}=0.00158$ ${\delta}_{C}=0.08036$ | ${\delta}_{C{T}_{c}}=0.00823$ ${\delta}_{C{I}_{\gamma}}=0.00363$ ${\delta}_{C}=0.00739$ |

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**MDPI and ACS Style**

Mohammad Mirzaei, N.; Su, S.; Sofia, D.; Hegarty, M.; Abdel-Rahman, M.H.; Asadpoure, A.; Cebulla, C.M.; Chang, Y.H.; Hao, W.; Jackson, P.R.; Lee, A.V.; Stover, D.G.; Tatarova, Z.; Zervantonakis, I.K.; Shahriyari, L. A Mathematical Model of Breast Tumor Progression Based on Immune Infiltration. *J. Pers. Med.* **2021**, *11*, 1031.
https://doi.org/10.3390/jpm11101031

**AMA Style**

Mohammad Mirzaei N, Su S, Sofia D, Hegarty M, Abdel-Rahman MH, Asadpoure A, Cebulla CM, Chang YH, Hao W, Jackson PR, Lee AV, Stover DG, Tatarova Z, Zervantonakis IK, Shahriyari L. A Mathematical Model of Breast Tumor Progression Based on Immune Infiltration. *Journal of Personalized Medicine*. 2021; 11(10):1031.
https://doi.org/10.3390/jpm11101031

**Chicago/Turabian Style**

Mohammad Mirzaei, Navid, Sumeyye Su, Dilruba Sofia, Maura Hegarty, Mohamed H. Abdel-Rahman, Alireza Asadpoure, Colleen M. Cebulla, Young Hwan Chang, Wenrui Hao, Pamela R. Jackson, Adrian V. Lee, Daniel G. Stover, Zuzana Tatarova, Ioannis K. Zervantonakis, and Leili Shahriyari. 2021. "A Mathematical Model of Breast Tumor Progression Based on Immune Infiltration" *Journal of Personalized Medicine* 11, no. 10: 1031.
https://doi.org/10.3390/jpm11101031