# A PDE Model of Breast Tumor Progression in MMTV-PyMT Mice

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. PDE System

#### Boundary Conditions

#### 2.2. Mechanical Model

^{3}[98], for simplicity, we assume that the constant in (8) is on average 1. Hence, summing both sides of (1) over $i\in I$ and applying (8) implies:

#### 2.3. Data of the Model

#### 2.3.1. Mouse Model and Experiments

#### 2.3.2. Preparation of Initial Conditions

## 3. Results

#### 3.1. No Influx

#### 3.2. Immune Cell Influx

#### 3.3. Sensitivity Analysis

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Cells and molecule dynamics from the ODE model [75].

## Appendix B

_{2}O

_{2}in methanol for 15 min within one cycle; AEC removal and stripping of antibodies was accomplished by ethanol gradient incubation and heat-mediated antigen retrieval between cycles. After washing, samples were subjected to the subsequent staining round.

## Appendix C

**Figure A2.**Column 1: Discontinuous fields. Column 2: Projection onto a function space with linear Lagrangian elements. Column 3: Smoothened fields via diffusion. Column 4: Non-dimensionalized fields.

## Appendix D

#### Appendix D.1. Weak Formulation and Discretization

#### Appendix D.2. Simulation

## Appendix E

**Table A1.**

**Full sensitivity report.**This table contains the sensitivity value of ${\int}_{\Omega}Cdx$ to all the parameters used in the model. The rows are ordered in a decreasing fashion based on the absolute value of their sensitivity values.

Order | Notation | Sensitivity | Definition | Order | Notation | Sensitivity | Definition |
---|---|---|---|---|---|---|---|

1 | ${D}_{C}$ | 2242.835 | Diffusion coefficient of C | 44 | ${\delta}_{{T}_{h}}$ | −$2.87\times {10}^{-5}$ | Death rate of ${T}_{h}$ |

2 | ${D}_{A}$ | 56.46447 | Diffusion coefficient of A | 45 | ${\lambda}_{M{T}_{h}}$ | $2.10\times {10}^{-5}$ | Activation rate of M by ${T}_{h}$ |

3 | ${\delta}_{C}$ | −3.16066 | Death rate of C | 46 | ${A}_{{D}_{N}}$ | −$1.48\times {10}^{-5}$ | Independent production rate of ${D}_{N}$ |

4 | ${\lambda}_{C}$ | 2.989047 | Proliferation rate of C | 47 | ${\delta}_{{T}_{N}}$ | $1.41\times {10}^{-5}$ | Death rate of ${T}_{N}$ |

5 | ${\alpha}_{{T}_{C}}$ | −0.31011 | Influx rate of ${T}_{C}$ | 48 | ${\lambda}_{{T}_{h}H}$ | $1.03\times {10}^{-5}$ | Activation rate of ${T}_{h}$ by H |

6 | ${D}_{{T}_{C}}$ | −0.2628 | Diffusion coefficient of ${T}_{C}$ | 49 | ${\lambda}_{DC}$ | −$6.42\times {10}^{-6}$ | Activation rate of D by C |

7 | ${\lambda}_{CI{L}_{6}}$ | 0.138431 | Proliferation rate of C by $I{L}_{6}$ | 50 | ${\lambda}_{{T}_{h}I{L}_{12}}$ | $6.27\times {10}^{-6}$ | Activation rate of ${T}_{h}$ by $I{L}_{12}$ |

8 | ${\lambda}_{CA}$ | 0.107539 | Proliferation rate of C by A | 51 | ${A}_{{M}_{N}}$ | $3.50\times {10}^{-6}$ | Independent production rate of ${M}_{N}$ |

9 | ${\delta}_{C{T}_{C}}$ | −0.06665 | Inhibition rate of C by ${T}_{C}$ | 52 | ${\lambda}_{{T}_{C}D}$ | −$3.46\times {10}^{-6}$ | Activation rate of ${T}_{C}$ by D |

10 | ${\delta}_{A}$ | −0.03611 | Death rate of A | 53 | ${\delta}_{{D}_{N}}$ | $3.26\times {10}^{-6}$ | Death rate of ${D}_{N}$ |

11 | ${\lambda}_{A}$ | 0.03531 | Proliferation rate of A | 54 | ${\lambda}_{I{L}_{10}D}$ | $3.14\times {10}^{-6}$ | Production rate of $I{L}_{10}$ by D |

12 | ${\alpha}_{M}$ | 0.025664 | Influx rate of M | 55 | ${\delta}_{{T}_{h}I{L}_{10}}$ | −$1.71\times {10}^{-6}$ | Inhibition rate of ${T}_{h}$ by $I{L}_{10}$ |

13 | ${\delta}_{I{L}_{6}}$ | −0.01152 | Decay rate of $I{L}_{6}$ | 56 | ${\delta}_{{T}_{h}{T}_{r}}$ | −$1.46\times {10}^{-6}$ | Inhibition rate of ${T}_{h}$ by ${T}_{r}$ |

14 | ${\lambda}_{I{L}_{6}A}$ | 0.009239 | Production rate of $I{L}_{6}$ by A | 57 | ${\lambda}_{DH}$ | −$1.37\times {10}^{-6}$ | Activation rate of D by H |

15 | ${\lambda}_{I{L}_{6}M}$ | 0.007813 | Production rate of $I{L}_{6}$ by M | 58 | ${A}_{0}$ | $9.75\times {10}^{-7}$ | Carrying capacity of A |

16 | ${\alpha}_{{T}_{h}}$ | 0.002154 | Influx rate of ${T}_{h}$ | 59 | ${\delta}_{{M}_{N}}$ | −$3.58\times {10}^{-7}$ | Death rate of ${M}_{N}$ |

17 | ${\delta}_{{T}_{C}}$ | 0.001508 | Death rate of ${T}_{C}$ | 60 | ${\lambda}_{I{L}_{12}M}$ | −$3.55\times {10}^{-7}$ | Production rate of $I{L}_{12}$ by M |

18 | ${\alpha}_{{T}_{r}}$ | 0.001441 | Influx rate of ${T}_{r}$ | 61 | ${\delta}_{I{L}_{12}}$ | $2.87\times {10}^{-7}$ | Decay rate of $I{L}_{12}$ |

19 | ${D}_{{D}_{N}}$ | −0.00094 | Diffusion coefficient of ${D}_{N}$ | 62 | ${\lambda}_{I{L}_{12}{T}_{h}}$ | −$2.31\times {10}^{-7}$ | Production rate of $I{L}_{12}$ by ${T}_{h}$ |

20 | ${D}_{N}$ | −0.00094 | Diffusion coefficient of N | 63 | ${\lambda}_{I{L}_{12}{T}_{C}}$ | −$2.18\times {10}^{-7}$ | Production rate of $I{L}_{12}$ by ${T}_{C}$ |

21 | ${C}_{0}$ | 0.000541 | Carrying capacity of C | 64 | ${D}_{I{L}_{12}}$ | $9.50\times 106-8$ | Diffusion coefficient of $I{L}_{12}$ |

22 | ${D}_{{T}_{r}}$ | −0.00044 | Diffusion coefficient of ${T}_{r}$ | 65 | ${\lambda}_{{T}_{h}D}$ | $7.78\times {10}^{-8}$ | Activation rate of ${T}_{h}$ by D |

23 | ${\delta}_{I{L}_{10}}$ | −0.00039 | Decay rate of $I{L}_{10}$ | 66 | ${\lambda}_{{T}_{r}D}$ | $6.93\times {10}^{-8}$ | Activation rate of ${T}_{r}$ by D |

24 | ${\lambda}_{I{L}_{10}C}$ | 0.000386 | Production rate of $I{L}_{10}$ by C | 67 | ${\delta}_{D}$ | −$3.70\times {10}^{-8}$ | Death rate of D |

25 | ${\lambda}_{{T}_{C}I{L}_{12}}$ | −0.00035 | Activation rate of ${T}_{C}$ by $I{L}_{12}$ | 68 | ${\delta}_{H}$ | −$1.49\times {10}^{-8}$ | Decay rate of H |

26 | ${D}_{M}$ | −0.00034 | Diffusion coefficient of M | 69 | ${\lambda}_{HM}$ | $8.05\times {10}^{-9}$ | Production rate of H by M |

27 | ${\lambda}_{I{L}_{10}{T}_{r}}$ | 0.000319 | Production rate of $I{L}_{10}$ by ${T}_{r}$ | 70 | ${\lambda}_{H{T}_{C}}$ | $5.35\times {10}^{-9}$ | Production rate of H by ${T}_{C}$ |

28 | ${\delta}_{M}$ | −0.00029 | Death rate of M | 71 | ${\lambda}_{HC}$ | $5.00\times {10}^{-9}$ | Production rate of H by C |

29 | ${\lambda}_{I{L}_{10}M}$ | 0.00024 | Production rate of $I{L}_{10}$ by M | 72 | ${\delta}_{DC}$ | $3.46\times {10}^{-9}$ | Activation rate of D by C |

30 | ${\lambda}_{I{L}_{10}{T}_{h}}$ | 0.00018 | Production rate of $I{L}_{10}$ by ${T}_{h}$ | 73 | ${\lambda}_{I{L}_{12}D}$ | −$2.50\times {10}^{-9}$ | Production rate of $I{L}_{12}$ by D |

31 | ${\lambda}_{I{L}_{10}{T}_{C}}$ | 0.000153 | Production rate of IL_{10} by
${T}_{C}$ | 74 | ${\lambda}_{HN}$ | $4.23\times {10}^{-10}$ | Production rate of H by N |

32 | ${D}_{{T}_{h}}$ | −0.00015 | Diffusion coefficient of ${T}_{h}$ | 75 | ${\lambda}_{HD}$ | $1.96\times {10}^{-10}$ | Production rate of H by D |

33 | ${D}_{I{L}_{6}}$ | 0.000126 | Diffusion coefficient of $I{L}_{6}$ | 76 | ${\delta}_{N}$ | $1.14\times {10}^{-11}$ | Death rate of N |

34 | ${A}_{{T}_{N}}$ | −$9.73\times {10}^{-5}$ | Independent production rate of ${T}_{N}$ | 77 | ${D}_{H}$ | $1.90\times {10}^{-13}$ | Diffusion coefficient of H |

35 | ${\alpha}_{{D}_{N}}$ | $9.34\times {10}^{-5}$ | Influx rate of ${D}_{N}$ | 78 | ${\alpha}_{NC}$ | $8.36\times {10}^{-15}$ | C to N conversion fraction |

36 | ${\delta}_{{T}_{C}{T}_{r}}$ | $8.92\times {10}^{-5}$ | Inhibition rate of ${T}_{C}$ by ${T}_{r}$ | 79 | ${T}_{C}^{*}$ | −$3.56\times {10}^{-16}$ | ${T}_{C}$ influx source |

37 | ${D}_{D}$ | $8.69\times {10}^{-5}$ | Diffusion coefficient of D | 80 | ${M}^{*}$ | $3.44\times {10}^{-17}$ | M influx source |

38 | ${\delta}_{{T}_{C}I{L}_{10}}$ | $7.88\times {10}^{-5}$ | Inhibition rate of ${T}_{C}$ by $I{L}_{10}$ | 81 | ${T}_{h}^{*}$ | $2.87\times {10}^{-18}$ | ${T}_{h}$ influx source |

39 | ${\lambda}_{I{L}_{6}D}$ | $7.41\times {10}^{-5}$ | Production rate of $I{L}_{6}$ by D | 82 | ${T}_{r}^{*}$ | $2.47\times {10}^{-18}$ | ${T}_{r}$ influx source |

40 | ${D}_{I{L}_{10}}$ | −$5.72\times {10}^{-5}$ | Diffusion coefficient of $I{L}_{10}$ | 83 | ${D}_{N}^{*}$ | −$8.55\times {10}^{-20}$ | ${D}_{N}$ influx source |

41 | ${\lambda}_{MI{L}_{10}}$ | $5.35\times {10}^{-5}$ | Activation rate of M by $I{L}_{10}$ | ||||

42 | ${\delta}_{{T}_{r}}$ | −$4.66\times {10}^{-5}$ | Death rate of ${T}_{r}$ | ||||

43 | ${\lambda}_{MI{L}_{12}}$ | $3.41\times {10}^{-5}$ | Activation rate of M by $I{L}_{12}$ |

## References

- Harbeck, N.; Penault-Llorca, F.; Cortes, J.; Gnant, M.; Houssami, N.; Poortmans, P.; Ruddy, K.; Tsang, J.; Cardoso, F. Breast cancer. Nat. Rev. Dis. Prim.
**2019**, 5, 66. [Google Scholar] [CrossRef] [PubMed] - Ferlay, J.; Soerjomataram, I.; Dikshit, R.; Eser, S.; Mathers, C.; Rebelo, M.; Parkin, D.M.; Forman, D.; Bray, F. Cancer incidence and mortality worldwide: Sources, methods and major patterns in GLOBOCAN 2012. Int. J. Cancer
**2015**, 136, E359–E386. [Google Scholar] [CrossRef] [PubMed] - Siegel, R.L.; Miller, K.D.; Fuchs, H.E.; Jemal, A. Cancer statistics, 2021. CA Cancer J. Clin.
**2021**, 71, 7–33. [Google Scholar] [CrossRef] [PubMed] - Waks, A.G.; Winer, E.P. Breast cancer treatment: A review. JAMA
**2019**, 321, 288–300. [Google Scholar] [CrossRef] [PubMed] - Maughan, K.L.; Lutterbie, M.A.; Ham, P. Treatment of breast cancer. Am. Fam. Phys.
**2010**, 81, 1339–1346. [Google Scholar] - Joyce, J.A. Therapeutic targeting of the tumor microenvironment. Cancer Cell
**2005**, 7, 513–520. [Google Scholar] [CrossRef][Green Version] - Meenakshi Upreti, A.J.; Sethi, P. Tumor microenvironment and nanotherapeutics. Transl. Cancer Res.
**2013**, 2, 309. [Google Scholar] - Natrajan, R.; Sailem, H.; Mardakheh, F.K.; Arias Garcia, M.; Tape, C.J.; Dowsett, M.; Bakal, C.; Yuan, Y. Microenvironmental heterogeneity parallels breast cancer progression: A histology–genomic integration analysis. PLoS Med.
**2016**, 13, e1001961. [Google Scholar] [CrossRef][Green Version] - Yuan, Y. Spatial heterogeneity in the tumor microenvironment. Cold Spring Harb. Perspect. Med.
**2016**, 6, a026583. [Google Scholar] [CrossRef][Green Version] - Burlingame, E.A.; Eng, J.; Thibault, G.; Chin, K.; Gray, J.W.; Chang, Y.H. Toward reproducible, scalable, and robust data analysis across multiplex tissue imaging platforms. Cell Rep. Methods
**2021**, 1, 100053. [Google Scholar] [CrossRef] - Parker, T.M.; Gupta, K.; Palma, A.M.; Yekelchyk, M.; Fisher, P.B.; Grossman, S.R.; Won, K.J.; Madan, E.; Moreno, E.; Gogna, R. Cell competition in intratumoral and tumor microenvironment interactions. EMBO J.
**2021**, 40, e107271. [Google Scholar] [CrossRef] [PubMed] - Lyssiotis, C.A.; Kimmelman, A.C. Metabolic interactions in the tumor microenvironment. Trends Cell Biol.
**2017**, 27, 863–875. [Google Scholar] [CrossRef] [PubMed][Green Version] - Lampreht Tratar, U.; Horvat, S.; Cemazar, M. Transgenic mouse models in cancer research. Front. Oncol.
**2018**, 8, 268. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hursting, S.D.; Slaga, T.J.; Fischer, S.M.; DiGiovanni, J.; Phang, J.M. Mechanism-based cancer prevention approaches: Targets, examples, and the use of transgenic mice. J. Natl. Cancer Inst.
**1999**, 91, 215–225. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hurwitz, A.A.; Foster, B.A.; Kwon, E.D.; Truong, T.; Choi, E.M.; Greenberg, N.M.; Burg, M.B.; Allison, J.P. Combination immunotherapy of primary prostate cancer in a transgenic mouse model using CTLA-4 blockade. Cancer Res.
**2000**, 60, 2444–2448. [Google Scholar] [PubMed] - Gingrich, J.; Greenberg, N. A transgenic mouse prostate cancer model. Toxicol. Pathol.
**1996**, 24, 502–504. [Google Scholar] [CrossRef] - Korac-Prlic, J.; Degoricija, M.; Vilović, K.; Haupt, B.; Ivanišević, T.; Franković, L.; Grivennikov, S.; Terzić, J. Targeting Stat3 signaling impairs the progression of bladder cancer in a mouse model. Cancer Lett.
**2020**, 490, 89–99. [Google Scholar] [CrossRef] - Floc’h, N.; Kinkade, C.W.; Kobayashi, T.; Aytes, A.; Lefebvre, C.; Mitrofanova, A.; Cardiff, R.D.; Califano, A.; Shen, M.M.; Abate-Shen, C. Dual targeting of the Akt/mTOR signaling pathway inhibits castration-resistant prostate cancer in a genetically engineered mouse model. Cancer Res.
**2012**, 72, 4483–4493. [Google Scholar] [CrossRef][Green Version] - Zhao, H.; Richardson, R.; Talebloo, N.; Mukherjee, P.; Wang, P.; Moore, A. uMUC1-targeting magnetic resonance imaging of therapeutic response in an orthotropic mouse model of colon cancer. Mol. Imaging Biol.
**2019**, 21, 852–860. [Google Scholar] [CrossRef] - Zeng, H.; Wei, W.; Xu, X. Chemokine (CXC motif) receptor 4 RNA interference inhibits bone metastasis in breast cancer. Oncol. Lett.
**2014**, 8, 77–81. [Google Scholar] [CrossRef][Green Version] - Chang, A.; Le, C.P.; Walker, A.K.; Creed, S.J.; Pon, C.K.; Albold, S.; Carroll, D.; Halls, M.L.; Lane, J.R.; Riedel, B.; et al. β2-Adrenoceptors on tumor cells play a critical role in stress-enhanced metastasis in a mouse model of breast cancer. Brain Behav. Immun.
**2016**, 57, 106–115. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ling, X.; Arlinghaus, R.B. Knockdown of STAT3 expression by RNA interference inhibits the induction of breast tumors in immunocompetent mice. Cancer Res.
**2005**, 65, 2532–2536. [Google Scholar] [CrossRef] [PubMed][Green Version] - Welm, A.L.; Sneddon, J.B.; Taylor, C.; Nuyten, D.S.; van de Vijver, M.J.; Hasegawa, B.H.; Bishop, J.M. The macrophage-stimulating protein pathway promotes metastasis in a mouse model for breast cancer and predicts poor prognosis in humans. Proc. Natl. Acad. Sci. USA
**2007**, 104, 7570–7575. [Google Scholar] [CrossRef] [PubMed][Green Version] - Pollard, J.W. Macrophages define the invasive microenvironment in breast cancer. J. Leukoc. Biol.
**2008**, 84, 623–630. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kitamura, T.; Doughty-Shenton, D.; Cassetta, L.; Fragkogianni, S.; Brownlie, D.; Kato, Y.; Carragher, N.; Pollard, J.W. Monocytes differentiate to immune suppressive precursors of metastasis-associated macrophages in mouse models of metastatic breast cancer. Front. Immunol.
**2018**, 8, 2004. [Google Scholar] [CrossRef] [PubMed][Green Version] - Coffelt, S.B.; Kersten, K.; Doornebal, C.W.; Weiden, J.; Vrijland, K.; Hau, C.S.; Verstegen, N.J.; Ciampricotti, M.; Hawinkels, L.J.; Jonkers, J.; et al. IL-17-producing γδ T cells and neutrophils conspire to promote breast cancer metastasis. Nature
**2015**, 522, 345–348. [Google Scholar] [CrossRef] - Lin, E.Y.; Li, J.F.; Gnatovskiy, L.; Deng, Y.; Zhu, L.; Grzesik, D.A.; Qian, H.; Xue, X.N.; Pollard, J.W. Macrophages regulate the angiogenic switch in a mouse model of breast cancer. Cancer Res.
**2006**, 66, 11238–11246. [Google Scholar] [CrossRef][Green Version] - He, H.; Wang, X.; Chen, J.; Sun, L.; Sun, H.; Xie, K. High-mobility group box 1 (HMGB1) promotes angiogenesis and tumor migration by regulating hypoxia-inducible factor 1 (HIF-1α) expression via the phosphatidylinositol 3-kinase (PI3K)/AKT signaling pathway in breast cancer cells. Med. Sci. Monit. Int. Med. J. Exp. Clin. Res.
**2019**, 25, 2352. [Google Scholar] [CrossRef] - Lewis, B.C.; Klimstra, D.S.; Varmus, H.E. The c-myc and PyMT oncogenes induce different tumor types in a somatic mouse model for pancreatic cancer. Genes Dev.
**2003**, 17, 3127–3138. [Google Scholar] [CrossRef][Green Version] - Lin, E.Y.; Jones, J.G.; Li, P.; Zhu, L.; Whitney, K.D.; Muller, W.J.; Pollard, J.W. Progression to malignancy in the polyoma middle T oncoprotein mouse breast cancer model provides a reliable model for human diseases. Am. J. Pathol.
**2003**, 163, 2113–2126. [Google Scholar] [CrossRef][Green Version] - Boyle, S.T.; Faulkner, J.W.; McColl, S.R.; Kochetkova, M. The chemokine receptor CCR6 facilitates the onset of mammary neoplasia in the MMTV-PyMT mouse model via recruitment of tumor-promoting macrophages. Mol. Cancer
**2015**, 14, 1–14. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hollern, D.P.; Honeysett, J.; Cardiff, R.D.; Andrechek, E.R. The E2F transcription factors regulate tumor development and metastasis in a mouse model of metastatic breast cancer. Mol. Cell. Biol.
**2014**, 34, 3229–3243. [Google Scholar] [CrossRef] [PubMed][Green Version] - Cowen, S.; McLaughlin, S.L.; Hobbs, G.; Coad, J.; Martin, K.H.; Olfert, I.M.; Vona-Davis, L. High-fat, high-calorie diet enhances mammary carcinogenesis and local inflammation in MMTV-PyMT mouse model of breast cancer. Cancers
**2015**, 7, 1125–1142. [Google Scholar] [CrossRef] [PubMed] - Sancho-Araiz, A.; Mangas-Sanjuan, V.; F Trocóniz, I. The Role of Mathematical Models in Immuno-Oncology: Challenges and Future Perspectives. Pharmaceutics
**2021**, 13, 1016. [Google Scholar] [CrossRef] [PubMed] - de Pillis, L.G.; Gu, W.; Radunskaya, A.E. Mixed immunotherapy and chemotherapy of tumors: Modeling, applications and biological interpretations. J. Theor. Biol.
**2006**, 238, 841–862. [Google Scholar] [CrossRef] - Kareva, I.; Berezovskaya, F. Cancer immunoediting: A process driven by metabolic competition as a predator–prey–shared resource type model. J. Theor. Biol.
**2015**, 380, 463–472. [Google Scholar] [CrossRef][Green Version] - Renardy, M.; Jilkine, A.; Shahriyari, L.; Chou, C.S. Control of cell fraction and population recovery during tissue regeneration in stem cell lineages. J. Theor. Biol.
**2018**, 445, 33–50. [Google Scholar] [CrossRef] - Mehdizadeh, R.; Shariatpanahi, S.P.; Goliaei, B.; Peyvandi, S.; Rüegg, C. Dormant Tumor Cell Vaccination: A Mathematical Model of Immunological Dormancy in Triple-Negative Breast Cancer. Cancers
**2021**, 13, 245. [Google Scholar] [CrossRef] - Oke, S.I.; Matadi, M.B.; Xulu, S.S. Optimal control analysis of a mathematical model for breast cancer. Math. Comput. Appl.
**2018**, 23, 21. [Google Scholar] - 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.; et al. A Mathematical Model of Breast Tumor Progression Based on Immune Infiltration. J. Pers. Med.
**2021**, 11, 1031. [Google Scholar] [CrossRef] - Shahriyari, L.; Komarova, N.L. The role of the bi-compartmental stem cell niche in delaying cancer. Phys. Biol.
**2015**, 12, 055001. [Google Scholar] [CrossRef] [PubMed] - Shahriyari, L.; Mahdipour-Shirayeh, A. Modeling dynamics of mutants in heterogeneous stem cell niche. Phys. Biol.
**2017**, 14. [Google Scholar] [CrossRef] [PubMed] - Shahriyari, L. Cell dynamics in tumour environment after treatments. J. R. Soc. Interface
**2017**, 14, 20160977. [Google Scholar] [CrossRef] - Kirshtein, A.; Akbarinejad, S.; Hao, W.; Le, T.; Su, S.; Aronow, R.A.; Shahriyari, L. Data Driven Mathematical Model of Colon Cancer Progression. J. Clin. Med.
**2020**, 9, 3947. [Google Scholar] [CrossRef] - Budithi, A.; Su, S.; Kirshtein, A.; Shahriyari, L. Data Driven Mathematical Model of FOLFIRI Treatment for Colon Cancer. Cancers
**2021**, 13, 2632. [Google Scholar] [CrossRef] [PubMed] - Le, T.; Su, S.; Kirshtein, A.; Shahriyari, L. Data-Driven Mathematical Model of Osteosarcoma. Cancers
**2021**, 13, 2367. [Google Scholar] [CrossRef] - Le, T.; Su, S.; Shahriyari, L. Investigating Optimal Chemotherapy Options for Osteosarcoma Patients through a Mathematical Model. Cells
**2021**, 10, 2009. [Google Scholar] [CrossRef] - Hao, W.; Friedman, A. The LDL-HDL profile determines the risk of atherosclerosis: A mathematical model. PLoS ONE
**2014**, 9, e90497. [Google Scholar] - Mohammad Mirzaei, N.; Weintraub, W.S.; Fok, P.W. An integrated approach to simulating the vulnerable atherosclerotic plaque. Am. J. Physiol.-Heart Circ. Physiol.
**2020**, 319, H835–H846. [Google Scholar] [CrossRef] - Hao, W.; Friedman, A. Mathematical model on Alzheimer’s disease. BMC Syst. Biol.
**2016**, 10, 108. [Google Scholar] [CrossRef][Green Version] - Weickenmeier, J.; Jucker, M.; Goriely, A.; Kuhl, E. A physics-based model explains the prion-like features of neurodegeneration in Alzheimer’s disease, Parkinson’s disease, and amyotrophic lateral sclerosis. J. Mech. Phys. Solids
**2019**, 124, 264–281. [Google Scholar] [CrossRef] - Viguerie, A.; Veneziani, A.; Lorenzo, G.; Baroli, D.; Aretz-Nellesen, N.; Patton, A.; Yankeelov, T.E.; Reali, A.; Hughes, T.J.; Auricchio, F. Diffusion–reaction compartmental models formulated in a continuum mechanics framework: Application to COVID-19, mathematical analysis, and numerical study. Comput. Mech.
**2020**, 66, 1131–1152. [Google Scholar] [CrossRef] [PubMed] - Iwata, K.; Kawasaki, K.; Shigesada, N. A dynamical model for the growth and size distribution of multiple metastatic tumors. J. Theor. Biol.
**2000**, 203, 177–186. [Google Scholar] [CrossRef][Green Version] - Barbolosi, D.; Benabdallah, A.; Hubert, F.; Verga, F. Mathematical and numerical analysis for a model of growing metastatic tumors. Math. Biosci.
**2009**, 218, 1–14. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kim, Y.; Friedman, A. Interaction of tumor with its micro-environment: A mathematical model. Bull. Math. Biol.
**2010**, 72, 1029–1068. [Google Scholar] [CrossRef] - Friedman, A.; Jain, H.V. A partial differential equation model of metastasized prostatic cancer. Math. Biosci. Eng.
**2013**, 10, 591. [Google Scholar] - Liu, J.; Wang, X.S. Numerical optimal control of a size-structured PDE model for metastatic cancer treatment. Math. Biosci.
**2019**, 314, 28–42. [Google Scholar] [CrossRef] - Knútsdóttir, H.; Pálsson, E.; Edelstein-Keshet, L. Mathematical model of macrophage-facilitated breast cancer cells invasion. J. Theor. Biol.
**2014**, 357, 184–199. [Google Scholar] [CrossRef] - Bretti, G.; De Ninno, A.; Natalini, R.; Peri, D.; Roselli, N. Estimation Algorithm for a Hybrid PDE–ODE Model Inspired by Immunocompetent Cancer-on-Chip Experiment. Axioms
**2021**, 10, 243. [Google Scholar] [CrossRef] - Lai, X.; Stiff, A.; Duggan, M.; Wesolowski, R.; Carson, W.E.; Friedman, A. Modeling combination therapy for breast cancer with BET and immune checkpoint inhibitors. Proc. Natl. Acad. Sci. USA
**2018**, 115, 5534–5539. [Google Scholar] [CrossRef][Green Version] - Fung, Y.C. Biomechanics; Springer: New York, NY, USA, 1993. [Google Scholar] [CrossRef]
- Fung, Y.C. Biomechanics: Motion, Flow, Stress, and Growth; Springer Science & Business Media: Berlin, Germany, 2013. [Google Scholar]
- Fung, Y.C. Biomechanics: Circulation; Springer Science & Business Media: Berlin, Germany, 2013. [Google Scholar]
- Fung, Y.C. Biomechanics: Mechanical Properties of Living Tissues; Springer Science & Business Media: Berlin, Germany, 2013. [Google Scholar]
- Prevost, T.P.; Balakrishnan, A.; Suresh, S.; Socrate, S. Biomechanics of brain tissue. Acta Biomater.
**2011**, 7, 83–95. [Google Scholar] [CrossRef] [PubMed] - Holzapfel, G.A. Biomechanics of soft tissue. Handb. Mater. Behav. Model.
**2001**, 3, 1049–1063. [Google Scholar] - Holzapfel, G.A.; Ogden, R.W. Biomechanics of Soft Tissue in Cardiovascular Systems; Springer: Berlin, Germany, 2014; Volume 441. [Google Scholar]
- Rajagopal, V.; Nielsen, P.M.; Nash, M.P. Modeling breast biomechanics for multi-modal image analysis—Successes and challenges. Wiley Interdiscip. Rev. Syst. Biol. Med.
**2010**, 2, 293–304. [Google Scholar] [CrossRef] [PubMed] - Frieboes, H.B.; Edgerton, M.E.; Fruehauf, J.P.; Rose, F.R.; Worrall, L.K.; Gatenby, R.A.; Ferrari, M.; Cristini, V. Prediction of drug response in breast cancer using integrative experimental/computational modeling. Cancer Res.
**2009**, 69, 4484–4492. [Google Scholar] [CrossRef] [PubMed][Green Version] - Friedman, A.; Hu, B. Bifurcation for a free boundary problem modeling tumor growth by Stokes equation. SIAM J. Math. Anal.
**2007**, 39, 174–194. [Google Scholar] [CrossRef] - Pham, K.; Frieboes, H.B.; Cristini, V.; Lowengrub, J. Predictions of tumour morphological stability and evaluation against experimental observations. J. R. Soc. Interface
**2011**, 8, 16–29. [Google Scholar] [CrossRef] [PubMed] - Hao, W.; Hauenstein, J.D.; Hu, B.; McCoy, T.; Sommese, A.J. Computing steady-state solutions for a free boundary problem modeling tumor growth by Stokes equation. J. Comput. Appl. Math.
**2013**, 237, 326–334. [Google Scholar] [CrossRef][Green Version] - Huang, Y.; Zhang, Z.; Hu, B. Bifurcation for a free-boundary tumor model with angiogenesis. Nonlinear Anal. Real World Appl.
**2017**, 35, 483–502. [Google Scholar] [CrossRef] - Wu, J.; Zhou, F. Asymptotic behavior of solutions of a free boundary problem modeling tumor spheroid with Gibbs–Thomson relation. J. Differ. Equ.
**2017**, 262, 4907–4930. [Google Scholar] [CrossRef][Green Version] - Mohammad Mirzaei, N.; Changizi, N.; Asadpoure, A.; Su, S.; Sofia, D.; Tatarova, Z.; Zervantonakis, I.K.; Chang, Y.H.; Shahriyari, L. Investigating key cell types and molecules dynamics in PyMT mice model of breast cancer through a mathematical model. PLoS Comput. Biol.
**2022**, 18, e1009953. [Google Scholar] [CrossRef] - Buzby, G.P.; Mullen, J.L.; Stein, T.P.; Miller, E.E.; Hobbs, C.L.; Rosato, E.F. Host-tumor interaction and nutrient supply. Cancer
**1980**, 45, 2940–2948. [Google Scholar] [CrossRef] - Blagih, J.; Hennequart, M.; Zani, F. Tissue nutrient environments and their effect on regulatory T cell biology. Front. Immunol.
**2021**, 12, 908. [Google Scholar] [CrossRef] - Zeng, H.; Combs, G.F., Jr. Selenium as an anticancer nutrient: Roles in cell proliferation and tumor cell invasion. J. Nutr. Biochem.
**2008**, 19, 1–7. [Google Scholar] [CrossRef] [PubMed] - Balkwill, F.R.; Capasso, M.; Hagemann, T. The tumor microenvironment at a glance. J. Cell Sci.
**2012**, 125, 5591–5596. [Google Scholar] [CrossRef][Green Version] - Franks, S.; Byrne, H.; King, J.; Underwood, J.; Lewis, C. Modelling the early growth of ductal carcinoma in situ of the breast. J. Math. Biol.
**2003**, 47, 424–452. [Google Scholar] [CrossRef] [PubMed] - Hao, W.; Friedman, A. Serum upar as biomarker in breast cancer recurrence: A mathematical model. PLoS ONE
**2016**, 11, e0153508. [Google Scholar] [CrossRef] - Liao, K.L.; Bai, X.F.; Friedman, A. Mathematical modeling of interleukin-27 induction of anti-tumor T cells response. PLoS ONE
**2014**, 9, e91844. [Google Scholar] [CrossRef][Green Version] - Liao, K.L.; Bai, X.F.; Friedman, A. The role of CD200–CD200R in tumor immune evasion. J. Theor. Biol.
**2013**, 328, 65–76. [Google Scholar] [CrossRef] - Hao, W.; Gong, S.; Wu, S.; Xu, J.; Go, M.R.; Friedman, A.; Zhu, D. A mathematical model of aortic aneurysm formation. PLoS ONE
**2017**, 12, e0170807. [Google Scholar] [CrossRef] - Lee, C.H.; Espinosa, I.; Vrijaldenhoven, S.; Subramanian, S.; Montgomery, K.D.; Zhu, S.; Marinelli, R.J.; Peterse, J.L.; Poulin, N.; Nielsen, T.O.; et al. Prognostic significance of macrophage infiltration in leiomyosarcomas. Clin. Cancer Res.
**2008**, 14, 1423–1430. [Google Scholar] [CrossRef][Green Version] - Zhang, J.; Endres, S.; Kobold, S. Enhancing tumor T cell infiltration to enable cancer immunotherapy. Immunotherapy
**2019**, 11, 201–213. [Google Scholar] [CrossRef] [PubMed] - Teng, M.W.; Ngiow, S.F.; Ribas, A.; Smyth, M.J. Classifying cancers based on T-cell infiltration and PD-L1. Cancer Res.
**2015**, 75, 2139–2145. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ono, M.; Torisu, H.; Fukushi, J.i.; Nishie, A.; Kuwano, M. Biological implications of macrophage infiltration in human tumor angiogenesis. Cancer Chemother. Pharmacol.
**1999**, 43, S69–S71. [Google Scholar] [CrossRef] [PubMed] - Treilleux, I.; Blay, J.Y.; Bendriss-Vermare, N.; Ray-Coquard, I.; Bachelot, T.; Guastalla, J.P.; Bremond, A.; Goddard, S.; Pin, J.J.; Barthelemy-Dubois, C.; et al. Dendritic cell infiltration and prognosis of early stage breast cancer. Clin. Cancer Res.
**2004**, 10, 7466–7474. [Google Scholar] [CrossRef][Green Version] - Friedman, A.; Hao, W. The role of exosomes in pancreatic cancer microenvironment. Bull. Math. Biol.
**2018**, 80, 1111–1133. [Google Scholar] [CrossRef] - Szomolay, B.; Eubank, T.D.; Roberts, R.D.; Marsh, C.B.; Friedman, A. Modeling the inhibition of breast cancer growth by GM-CSF. J. Theor. Biol.
**2012**, 303, 141–151. [Google Scholar] [CrossRef] - Kremheller, J.; Vuong, A.T.; Yoshihara, L.; Wall, W.A.; Schrefler, B.A. A monolithic multiphase porous medium framework for (a-) vascular tumor growth. Comput. Methods Appl. Mech. Eng.
**2018**, 340, 657–683. [Google Scholar] [CrossRef] - Chapman, S.J.; Shipley, R.J.; Jawad, R. Multiscale modeling of fluid transport in tumors. Bull. Math. Biol.
**2008**, 70, 2334–2357. [Google Scholar] [CrossRef] - Sciume, G.; Shelton, S.; Gray, W.G.; Miller, C.T.; Hussain, F.; Ferrari, M.; Decuzzi, P.; Schrefler, B. A multiphase model for three-dimensional tumor growth. New J. Phys.
**2013**, 15, 015005. [Google Scholar] [CrossRef] - Govindaraju, K.; Kamangar, S.; Badruddin, I.A.; Viswanathan, G.N.; Badarudin, A.; Ahmed, N.S. Effect of porous media of the stenosed artery wall to the coronary physiological diagnostic parameter: A computational fluid dynamic analysis. Atherosclerosis
**2014**, 233, 630–635. [Google Scholar] [CrossRef] - Karagiannis, G.S.; Pastoriza, J.M.; Borriello, L.; Jafari, R.; Coste, A.; Condeelis, J.S.; Oktay, M.H.; Entenberg, D. Assessing tumor microenvironment of metastasis doorway-mediated vascular permeability associated with cancer cell dissemination using intravital imaging and fixed tissue analysis. JoVE (J. Vis. Exp.)
**2019**, 148, e59633. [Google Scholar] [CrossRef] [PubMed] - Franks, S.; Byrne, H.; Underwood, J.; Lewis, C. Biological inferences from a mathematical model of comedo ductal carcinoma in situ of the breast. J. Theor. Biol.
**2005**, 232, 523–543. [Google Scholar] [CrossRef] [PubMed] - Johnson, T.; Ding, H.; Le, H.Q.; Ducote, J.L.; Molloi, S. Breast density quantification with cone-beam CT: A post-mortem study. Phys. Med. Biol.
**2013**, 58, 8573. [Google Scholar] [CrossRef][Green Version] - Byrne, H.M.; Chaplain, M.A. Modelling the role of cell-cell adhesion in the growth and development of carcinomas. Math. Comput. Model.
**1996**, 24, 1–17. [Google Scholar] [CrossRef] - Byrne, H. The importance of intercellular adhesion in the development of carcinomas. Math. Med. Biol. J. IMA
**1997**, 14, 305–323. [Google Scholar] [CrossRef] - Friedman, A. A free boundary problem for a coupled system of elliptic, hyperbolic, and Stokes equations modeling tumor growth. Interfaces Free Boundaries
**2006**, 8, 247–261. [Google Scholar] [CrossRef][Green Version] - Rianna, C.; Radmacher, M. Comparison of viscoelastic properties of cancer and normal thyroid cells on different stiffness substrates. Eur. Biophys. J.
**2017**, 46, 309–324. [Google Scholar] [CrossRef] - Sancho, A.; Vandersmissen, I.; Craps, S.; Luttun, A.; Groll, J. A new strategy to measure intercellular adhesion forces in mature cell-cell contacts. Sci. Rep.
**2017**, 7, 1–14. [Google Scholar] [CrossRef][Green Version] - Talari, A.C.S.; Raza, A.; Rehman, S.; Rehman, I.U. Analyzing normal proliferating, hypoxic and necrotic regions of T-47D human breast cancer spheroids using Raman spectroscopy. Appl. Spectrosc. Rev.
**2017**, 52, 909–924. [Google Scholar] [CrossRef] - Gallaher, J.A.; Brown, J.S.; Anderson, A.R. The impact of proliferation-migration tradeoffs on phenotypic evolution in cancer. Sci. Rep.
**2019**, 9, 1–10. [Google Scholar] [CrossRef] - Datta, P.; Dey, M.; Ataie, Z.; Unutmaz, D.; Ozbolat, I.T. 3D bioprinting for reconstituting the cancer microenvironment. NPJ Precis. Oncol.
**2020**, 4, 1–13. [Google Scholar] [CrossRef] [PubMed] - Wang, M.; Zhang, C.; Song, Y.; Wang, Z.; Wang, Y.; Luo, F.; Xu, Y.; Zhao, Y.; Wu, Z.; Xu, Y. Mechanism of immune evasion in breast cancer. OncoTargets Ther.
**2017**, 10, 1561. [Google Scholar] [CrossRef] [PubMed][Green Version] - Vesely, M.D.; Kershaw, M.H.; Schreiber, R.D.; Smyth, M.J. Natural innate and adaptive immunity to cancer. Annu. Rev. Immunol.
**2011**, 29, 235–271. [Google Scholar] [CrossRef] [PubMed][Green Version] - Schreiber, R.D.; Old, L.J.; Smyth, M.J. Cancer immunoediting: Integrating immunity’s roles in cancer suppression and promotion. Science
**2011**, 331, 1565–1570. [Google Scholar] [CrossRef][Green Version] - Soysal, S.D.; Tzankov, A.; Muenst, S.E. Role of the tumor microenvironment in breast cancer. Pathobiology
**2015**, 82, 142–152. [Google Scholar] [CrossRef] - Bertram, C.A.; Aubreville, M.; Gurtner, C.; Bartel, A.; Corner, S.M.; Dettwiler, M.; Kershaw, O.; Noland, E.L.; Schmidt, A.; Sledge, D.G.; et al. Computerized calculation of mitotic count distribution in canine cutaneous mast cell tumor sections: Mitotic count is area dependent. Vet. Pathol.
**2020**, 57, 214–226. [Google Scholar] [CrossRef] - Li, S.; Petzold, L. Adjoint sensitivity analysis for time-dependent partial differential equations with adaptive mesh refinement. J. Comput. Phys.
**2004**, 198, 310–325. [Google Scholar] [CrossRef] - Mitusch, S.K.; Funke, S.W.; Dokken, J.S. dolfin-adjoint 2018.1: Automated adjoints for FEniCS and Firedrake. J. Open Source Softw.
**2019**, 4, 1292. [Google Scholar] [CrossRef] - Qian, B.Z.; Pollard, J.W. Macrophage diversity enhances tumor progression and metastasis. Cell
**2010**, 141, 39–51. [Google Scholar] [CrossRef][Green Version] - Nielsen, S.R.; Schmid, M.C. Macrophages as key drivers of cancer progression and metastasis. Mediat. Inflamm.
**2017**, 2017, 9624760. [Google Scholar] [CrossRef] - Doak, G.R.; Schwertfeger, K.L.; Wood, D.K. Distant relations: Macrophage functions in the metastatic niche. Trends Cancer
**2018**, 4, 445–459. [Google Scholar] [CrossRef] [PubMed] - Ma, R.Y.; Zhang, H.; Li, X.F.; Zhang, C.B.; Selli, C.; Tagliavini, G.; Lam, A.D.; Prost, S.; Sims, A.H.; Hu, H.Y.; et al. Monocyte-derived macrophages promote breast cancer bone metastasis outgrowth. J. Exp. Med.
**2020**, 217, e20191820. [Google Scholar] [CrossRef] [PubMed] - Palmer, T.D.; Ashby, W.J.; Lewis, J.D.; Zijlstra, A. Targeting tumor cell motility to prevent metastasis. Adv. Drug Deliv. Rev.
**2011**, 63, 568–581. [Google Scholar] [CrossRef] [PubMed][Green Version] - Yamazaki, D.; Kurisu, S.; Takenawa, T. Regulation of cancer cell motility through actin reorganization. Cancer Sci.
**2005**, 96, 379–386. [Google Scholar] [CrossRef] [PubMed] - Gregory, A.D.; Houghton, A.M. Tumor-associated neutrophils: New targets for cancer therapy. Cancer Res.
**2011**, 71, 2411–2416. [Google Scholar] [CrossRef][Green Version] - Raskov, H.; Orhan, A.; Christensen, J.P.; Gögenur, I. Cytotoxic CD8+ T cells in cancer and cancer immunotherapy. Br. J. Cancer
**2021**, 124, 359–367. [Google Scholar] [CrossRef] - Farhood, B.; Najafi, M.; Mortezaee, K. CD8+ cytotoxic T lymphocytes in cancer immunotherapy: A review. J. Cell. Physiol.
**2019**, 234, 8509–8521. [Google Scholar] [CrossRef] - Johar, D.; Roth, J.C.; Bay, G.H.; Walker, J.N.; Kroczak, T.J.; Los, M. Inflammatory response, reactive oxygen species, programmed (necrotic-like and apoptotic) cell death and cancer. Rocz. Akad. Med. Bialymst.
**2004**, 49, 31–39. [Google Scholar] - Bredholt, G.; Mannelqvist, M.; Stefansson, I.M.; Birkeland, E.; Bø, T.H.; Øyan, A.M.; Trovik, J.; Kalland, K.H.; Jonassen, I.; Salvesen, H.B.; et al. Tumor necrosis is an important hallmark of aggressive endometrial cancer and associates with hypoxia, angiogenesis and inflammation responses. Oncotarget
**2015**, 6, 39676. [Google Scholar] [CrossRef][Green Version] - Ferreira, S., Jr.; Martins, M.; Vilela, M. Reaction-diffusion model for the growth of avascular tumor. Phys. Rev. E
**2002**, 65, 021907. [Google Scholar] [CrossRef][Green Version] - Cassim, S.; Pouyssegur, J. Tumor microenvironment: A metabolic player that shapes the immune response. Int. J. Mol. Sci.
**2019**, 21, 157. [Google Scholar] [CrossRef] [PubMed][Green Version] - Cai, Y.; Nogales-Cadenas, R.; Zhang, Q.; Lin, J.R.; Zhang, W.; O’Brien, K.; Montagna, C.; Zhang, Z.D. Transcriptomic dynamics of breast cancer progression in the MMTV-PyMT mouse model. BMC Genom.
**2017**, 18, 1–14. [Google Scholar] [CrossRef] [PubMed][Green Version] - Brezzi, F.; Russo, A. Choosing bubbles for advection-diffusion problems. Math. Model. Methods Appl. Sci.
**1994**, 4, 571–587. [Google Scholar] [CrossRef] - Franca, L.P.; Nesliturk, A.; Stynes, M. On the stability of residual-free bubbles for convection-diffusion problems and their approximation by a two-level finite element method. Comput. Methods Appl. Mech. Eng.
**1998**, 166, 35–49. [Google Scholar] [CrossRef] - Sendur, A. A Comparative Study on Stabilized Finite Element Methods for the Convection-Diffusion-Reaction Problems. J. Appl. Math.
**2018**, 2018, 4259634. [Google Scholar] [CrossRef][Green Version] - Logg, A.; Mardal, K.A.; Wells, G. Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book; Springer Science & Business Media: Berlin, Germany, 2012; Volume 84. [Google Scholar]

**Figure 2.**Nine figures showing the position of the chosen elliptical domain compared to each cell type. Blue dots represent a single cell of the corresponding cell type, and gray dots are the rest.

**Figure 3.**Solid red: A discontinuous function. Dashed blue: The projection onto a finite element space with linear Lagrangian bases.

**Figure 4.**Comparison between the dimensions of the tumor at $t=0$ h versus $t=600$ h. The graphs show the time evolution of the bounding box dimensions.

**Figure 5.**Results with no flux of immune cells. (

**A**) Column 1: Spatial distribution of cytokines. Column 2: Maximum, average, and minimum concentration (ng/mL) of each cytokine over the whole domain with respect to time. (

**B**) Evolution of naive T cells and naive macrophages. (

**C**) Column 1: Spatial distribution of cell types. Column 2: Maximum, average, and minimum number of each cell type over the whole domain with respect to time.

**Figure 6.**(

**Left**): Level-curves indicating the mean value of each cell type at $t=600$. (

**Right**): Level-curves indicating the mean value of each molecule at $t=600$. Areas ${A}_{1}$ and ${A}_{2}$ correspond to the regions with the most and second-most immune cell intersections, respectively. Area ${A}_{3}$ corresponds to the region with the highest cytokine intersections.

**Figure 7.**Nine figures showing each cell type in the mouse model. Blue dots represent a single cell of the corresponding cell type, and gray dots represent the rest.

**Figure 8.**Dimensions of the tumor subject to immune cells influx at $t=600$ h. The curves show the time evolution of the bounding box dimensions.

**Figure 9.**Results with flux of immune cells. (

**A**) Column 1: Spatial distribution of cytokines. Column 2: Maximum, average, and minimum concentration (ng/mL) of each cytokine over the whole domain with respect to time. (

**B**) Evolution of naive T cells and naive macrophages. (

**C**) Column 1: Spatial distribution of cell types. Column 2: Maximum, average, and minimum number of each cell type over the whole domain with respect to time.

**Figure 11.**The sensitivity of ${\int}_{\Omega}C\phantom{\rule{4pt}{0ex}}dx$ at $t=600$ to four categories of parameters: diffusion rates, influx rates, influx sources, and the reaction parameters.

**Figure 12.**(

**A**) The variation of the total number of cancer cells as a result of 10% perturbation of the most sensitive parameters. (

**B**) The leftmost circle corresponds to the lower bound, the middle circle corresponds to the thick solid red curve and the right circle corresponds to the upper bound of the graph in (

**A**).

Variable in PDE | Variable in ODE | Name |
---|---|---|

${X}_{1}$ | ${T}_{N}$ | Naive T cells |

${X}_{2}$ | ${T}_{h}$ | Helper T cells |

${X}_{3}$ | ${T}_{C}$ | Cytotoxic cells |

${X}_{4}$ | ${T}_{r}$ | Regulatory T cells |

${X}_{5}$ | ${D}_{N}$ | Naive dendritic cells |

${X}_{6}$ | D | Activated dendritic cells |

${X}_{7}$ | ${M}_{N}$ | Naive macrophages |

${X}_{8}$ | M | Activated macrophages |

${X}_{9}$ | C | Cancer cells |

${X}_{10}$ | N | Necrotic cells |

${X}_{11}$ | A | Cancer associated Adipocytes |

${X}_{12}$ | H | HMGB1 |

${X}_{13}$ | $I{L}_{12}$ | IL-12 |

${X}_{14}$ | $I{L}_{10}$ | IL-10 |

${X}_{15}$ | $I{L}_{6}$ | IL-6 |

**Table 2.**

**Biomarker combinations.**(+) means high expression and (−) means lack of expression of a protein at a certain location.

Cell Type | Biomarker Combination |
---|---|

Helper T cells (${T}_{h}$) | Epcam(−) CD45(+) CD3(+) CD4(+) CD8(−) |

Cytotoxic T cells (${T}_{C}$) | Epcam(−) CD45(+) CD3(+) CD4(−) CD8(+) |

Naive dendritic cells (${D}_{N}$) | Epcam(−) CD45(+) F4/80(−) CD11C(+) |

Dendritic cells (D) | Epcam(−) CD45(+) F4/80(−) CD11C(+) MHC-II(+) |

Activated macrophages (M) | Epcam(−) CD45(+) F4/80(+) CD11C(−) CSF1R(+) or Epcam(−) CD45(+) F4/80(+) CD11C(−) CSF1R(−) MHC-II(+) |

Cancer cells (C) | Epcam(+) CD45(−) |

Necrotic cells (N) | CC3(+) |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mohammad Mirzaei, N.; Tatarova, Z.; Hao, W.; Changizi, N.; Asadpoure, A.; Zervantonakis, I.K.; Hu, Y.; Chang, Y.H.; Shahriyari, L.
A PDE Model of Breast Tumor Progression in MMTV-PyMT Mice. *J. Pers. Med.* **2022**, *12*, 807.
https://doi.org/10.3390/jpm12050807

**AMA Style**

Mohammad Mirzaei N, Tatarova Z, Hao W, Changizi N, Asadpoure A, Zervantonakis IK, Hu Y, Chang YH, Shahriyari L.
A PDE Model of Breast Tumor Progression in MMTV-PyMT Mice. *Journal of Personalized Medicine*. 2022; 12(5):807.
https://doi.org/10.3390/jpm12050807

**Chicago/Turabian Style**

Mohammad Mirzaei, Navid, Zuzana Tatarova, Wenrui Hao, Navid Changizi, Alireza Asadpoure, Ioannis K. Zervantonakis, Yu Hu, Young Hwan Chang, and Leili Shahriyari.
2022. "A PDE Model of Breast Tumor Progression in MMTV-PyMT Mice" *Journal of Personalized Medicine* 12, no. 5: 807.
https://doi.org/10.3390/jpm12050807