# Mathematical Modeling of Breast Cancer Based on the Caputo–Fabrizio Fractal-Fractional Derivative

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Preliminaries of Fractional Calculus

**Definition 1.**

**Definition 2.**

**Definition 3.**

**Definition 4.**

**Definition 5.**

**Definition 6.**

**Definition 7.**

#### 2.2. Mathematical Model with the Caputo–Fabrizio Fractal-Fractional Derivative

## 3. Numerical Solution

## 4. Results and Discussion

#### 4.1. Case I

#### 4.2. Case II

**Figure 1.**(

**a**–

**c**) Dynamics of cancer cells and CTLs at different values of $\xi $ and a fixed value of $\tau =0.95$.

**Figure 2.**(

**a**–

**c**) Dynamics of cancer cells and CTLs at different values of $\tau $ and a fixed value of $\xi =0.85$.

**Figure 3.**(

**a**,

**b**) Dynamics of cancer cells and CTLs at different values of the fractal order and a fixed value of the fractional order.

#### 4.3. Case III

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Fathoni, M.; Gunardi, G.; Kusumo, F.A.; Hutajulu, S.H. Mathematical model analysis of breast cancer stages with side effects on heart in chemotherapy patients. In Proceedings of the AIP Conference Proceedings, Yogyakarta, Indonesia, 29 July–1 August 2019; AIP Publishing: Melville, NY, USA, 2019. [Google Scholar]
- Breast Cancer. Available online: https://www.who.int/news-room/fact-sheets/detail/breast-cancer (accessed on 15 July 2023).
- DeSantis, C.E.; Bray, F.; Ferlay, J.; Lortet-Tieulent, J.; Anderson, B.O.; Jemal, A. International variation in female breast cancer incidence and mortality rates. Cancer Epidemiol. Biomark. Prev.
**2015**, 24, 1495–1506. [Google Scholar] [CrossRef] - Michor, F.; Iwasa, Y.; Nowak, M.A. Dynamics of cancer progression. Nat. Rev. Cancer
**2004**, 4, 197–205. [Google Scholar] [CrossRef] [PubMed] - Bardelli, A.; Cahill, D.P.; Lederer, G.; Speicher, M.R.; Kinzler, K.W.; Vogelstein, B.; Lengauer, C. Carcinogen-specific induction of genetic instability. Proc. Natl. Acad. Sci. USA
**2001**, 98, 5770–5775. [Google Scholar] [CrossRef] [PubMed] - Loeb, L.A. Microsatellite instability: Marker of a mutator phenotype in cancer. Cancer Res.
**1994**, 54, 5059. [Google Scholar] - Tomlinson, I.; Bodmer, W. Selection, the mutation rate and cancer: Ensuring that the tail does not wag the dog. Nat. Med.
**1999**, 5, 11–12. [Google Scholar] [CrossRef] [PubMed] - Zhang, X.; Fang, Y.; Zhao, Y.; Zheng, W. Mathematical modeling the pathway of human breast cancer. Math. Biosci.
**2014**, 253, 25–29. [Google Scholar] [CrossRef] - Enderling, H.; Chaplain, M.A.; Anderson, A.R.; Vaidya, J.S. A mathematical model of breast cancer development, local treatment and recurrence. J. Theor. Biol.
**2007**, 246, 245–259. [Google Scholar] [CrossRef] - Simmons, A.; Burrage, P.M.; Nicolau, D.V., Jr.; Lakhani, S.R.; Burrage, K. Environmental factors in breast cancer invasion: A mathematical modelling review. Pathology
**2017**, 49, 172–180. [Google Scholar] [CrossRef] - Frank, S.A.; Iwasa, Y.; Nowak, M.A. Patterns of cell division and the risk of cancer. Genetics
**2003**, 163, 1527–1532. [Google Scholar] [CrossRef] - Byrne, H.M. Dissecting cancer through mathematics: From the cell to the animal model. Nat. Rev. Cancer
**2010**, 10, 221–230. [Google Scholar] [CrossRef] - Armitage, P.; Doll, R. The age distribution of cancer and a multi-stage theory of carcinogenesis. Br. J. Cancer
**2004**, 91, 1983–1989. [Google Scholar] [CrossRef] [PubMed] - Dixit, D.S.; Kumar, D.; Kumar, S.; Johri, R. A mathematical model of chemotherapy for tumor treatment. Adv. Appl. Math. Biosci.
**2012**, 3, 1–10. [Google Scholar] - Schättler, H.; Ledzewicz, U.; Amini, B. Dynamical properties of a minimally parameterized mathematical model for metronomic chemotherapy. J. Math. Biol.
**2016**, 72, 1255–1280. [Google Scholar] [CrossRef] [PubMed] - Jordão, G.; Tavares, J.N. Mathematical models in cancer therapy. Biosystems
**2017**, 162, 12–23. [Google Scholar] [CrossRef] - Khajanchi, S.; Nieto, J.J. Mathematical modeling of tumor-immune competitive system, considering the role of time delay. Appl. Math. Comput.
**2019**, 340, 180–205. [Google Scholar] [CrossRef] - Mahlbacher, G.E.; Reihmer, K.C.; Frieboes, H.B. Mathematical modeling of tumor-immune cell interactions. J. Theor. Biol.
**2019**, 469, 47–60. [Google Scholar] [CrossRef] - Lorenzo, C.F. Initialized Fractional Calculus; NASA Glenn Research Center: Cleveland, OH, USA, 2000.
- Sun, H.; Chen, W.; Wei, H.; Chen, Y. A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems. Eur. Phys. J. Spec. Top.
**2011**, 193, 185–192. [Google Scholar] [CrossRef] - Sabir, Z.; Munawar, M.; Abdelkawy, M.A.; Raja, M.A.Z.; Ünlü, C.; Jeelani, M.B.; Alnahdi, A.S. Numerical investigations of the fractional-order mathematical model underlying immune-chemotherapeutic treatment for breast cancer using the neural networks. Fractal Fract.
**2022**, 6, 184. [Google Scholar] [CrossRef] - Solís-Pérez, J.; Gómez-Aguilar, J.F.; Atangana, A. A fractional mathematical model of breast cancer competition model. Chaos Solitons Fractals
**2019**, 127, 38–54. [Google Scholar] [CrossRef] - Hassani, H.; Machado, J.T.; Avazzadeh, Z.; Safari, E.; Mehrabi, S. Optimal solution of the fractional order breast cancer competition model. Sci. Rep.
**2021**, 11, 15622. [Google Scholar] [CrossRef] - Abaid Ur Rehman, M.; Ahmad, J.; Hassan, A.; Awrejcewicz, J.; Pawlowski, W.; Karamti, H.; Alharbi, F.M. The dynamics of a fractional-order mathematical model of cancer tumor disease. Symmetry
**2022**, 14, 1694. [Google Scholar] [CrossRef] - Öztürk, I.; Özköse, F. Stability analysis of fractional order mathematical model of tumor-immune system interaction. Chaos Solitons Fractals
**2020**, 133, 109614. [Google Scholar] [CrossRef] - Chen, Y.; Yi, M.; Chen, C.; Yu, C. Bernstein polynomials method for fractional convection-diffusion equation with variable coefficients. Comput. Model. Eng. Sci.
**2012**, 83, 639–653. [Google Scholar] - Ou, C.; Cen, D.; Vong, S.; Wang, Z. Mathematical analysis and numerical methods for Caputo-Hadamard fractional diffusion-wave equations. Appl. Numer. Math.
**2022**, 177, 34–57. [Google Scholar] [CrossRef] - Galue, L.; Kalla, S.L.; Al-Saqabi, B. Fractional extensions of the temperature field problems in oil strata. Appl. Math. Comput.
**2007**, 186, 35–44. [Google Scholar] [CrossRef] - Valentim, C.A.; Rabi, J.A.; David, S.A.; Machado, J.A.T. On multistep tumor growth models of fractional variable-order. Biosystems
**2021**, 199, 104294. [Google Scholar] [CrossRef] - Farayola, M.F.; Shafie, S.; Siam, F.M.; Khan, I. Mathematical modeling of radiotherapy cancer treatment using Caputo fractional derivative. Comput. Methods Programs Biomed.
**2020**, 188, 105306. [Google Scholar] [CrossRef] - Idrees, M.; Sohail, A. Bio-algorithms for the modeling and simulation of cancer cells and the immune response. Bio-Algorithms Med-Syst.
**2021**, 17, 55–63. [Google Scholar] [CrossRef] - Britton, N.F.; Britton, N. Essential Mathematical Biology; Springer: Berlin/Heidelberg, Germany, 2003; Volume 453. [Google Scholar]
- Kawarada, Y.; Ganss, R.; Garbi, N.; Sacher, T.; Arnold, B.; Hammerling, G.J. NK-and CD8+ T cell-mediated eradication of established tumors by peritumoral injection of CpG-containing oligodeoxynucleotides. J. Immunol.
**2001**, 167, 5247–5253. [Google Scholar] [CrossRef] - Dudley, M.E.; Wunderlich, J.R.; Robbins, P.F.; Yang, J.C.; Hwu, P.; Schwartzentruber, D.J.; Topalian, S.L.; Sherry, R.; Restifo, N.P.; Hubicki, A.M.; et al. Cancer regression and autoimmunity in patients after clonal repopulation with antitumor lymphocytes. Science
**2002**, 298, 850–854. [Google Scholar] [CrossRef] - Adam, J.A.; Bellomo, N. A Survey of Models for Tumor-Immune System Dynamics; Springer Science & Business Media: Berlin/Heidelberg, Germany, 1997. [Google Scholar]
- Nawata, H.; Chong, M.; Bronzert, D.; Lippman, M.E. Estradiol-independent growth of a subline of MCF-7 human breast cancer cells in culture. J. Biol. Chem.
**1981**, 256, 6895–6902. [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] [PubMed] - López, Á.G.; Seoane, J.M.; Sanjuán, M.A. A validated mathematical model of tumor growth including tumor–host interaction, cell-mediated immune response and chemotherapy. Bull. Math. Biol.
**2014**, 76, 2884–2906. [Google Scholar] [CrossRef] [PubMed] - Fernandez, M.; Zhou, M.; Soto-Ortiz, L. A computational assessment of the robustness of cancer treatments with respect to immune response strength, tumor size and resistance. Int. J. Tumor. Ther.
**2018**, 7, 1–19. [Google Scholar] - Muller, M.R.; Grunebach, F.; Nencioni, A.; Brossart, P. Transfection of dendritic cells with RNA induces CD4- and CD8-mediated T cell immunity against breast carcinomas and reveals the immunodominance of presented T cell epitopes. J. Immunol.
**2003**, 170, 5892–5896. [Google Scholar] [CrossRef] - Gruber, I.; Landenberger, N.; Staebler, A.; Hahn, M.; Wallwiener, D.; Fehm, T. Relationship between circulating tumor cells and peripheral T-cells in patients with primary breast cancer. Anticancer Res.
**2013**, 33, 2233–2238. [Google Scholar] - Herrmann, R. Fractional Calculus: An Introduction for Physicists; World Scientific: Singapore, 2011. [Google Scholar]
- Qureshi, S.; Yusuf, A.; Aziz, S. On the use of Mohand integral transform for solving fractional-order classical Caputo differential equations. J. Appl. Math. Comput. Mech.
**2020**, 19, 99–109. [Google Scholar] [CrossRef] - Jain, S. Numerical analysis for the fractional diffusion and fractional Buckmaster equation by the two-step Laplace Adam-Bashforth method. Eur. Phys. J. Plus
**2018**, 133, 19. [Google Scholar] [CrossRef] - Caputo, M.; Fabrizio, M. A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl.
**2015**, 1, 73–85. [Google Scholar] - Fatmawati; Khan, M.A.; Alfiniyah, C.; Alzahrani, E. Analysis of dengue model with fractal-fractional Caputo–Fabrizio operator. Adv. Differ. Equ.
**2020**, 2020, 422. [Google Scholar] [CrossRef] - Thomlinson, R. Measurement and management of carcinoma of the breast. Clin. Radiol.
**1982**, 33, 481–493. [Google Scholar] [CrossRef] [PubMed] - Deisboeck, T.S.; Wang, Z. Cancer dissemination: A consequence of limited carrying capacity? Med. Hypotheses
**2007**, 69, 173–177. [Google Scholar] [CrossRef] [PubMed] - Vacca, P.; Munari, E.; Tumino, N.; Moretta, F.; Pietra, G.; Vitale, M.; Del Zotto, G.; Mariotti, F.R.; Mingari, M.C.; Moretta, L. Human natural killer cells and other innate lymphoid cells in cancer: Friends or foes? Immunol. Lett.
**2018**, 201, 14–19. [Google Scholar] [CrossRef] [PubMed] - Fidler, I.J. Metastasis: Quantitative analysis of distribution and fate of tumor emboli labeled with 125I-5-iodo-2′-deoxyuridine. J. Natl. Cancer Inst.
**1970**, 45, 773–782. [Google Scholar] - Folkman, J.; Kalluri, R. Cancer without disease. Nature
**2004**, 427, 787. [Google Scholar] [CrossRef] - Fehm, T.; Mueller, V.; Marches, R.; Klein, G.; Gueckel, B.; Neubauer, H.; Solomayer, E.; Becker, S. Tumor cell dormancy: Implications for the biology and treatment of breast cancer. Apmis
**2008**, 116, 742–753. [Google Scholar] [CrossRef] - Franco, O.E.; Shaw, A.K.; Strand, D.W.; Hayward, S.W. Cancer associated fibroblasts in cancer pathogenesis. Semin. Cell Dev. Biol.
**2010**, 21, 33–39. [Google Scholar] [CrossRef]

**Figure 4.**(

**a**,

**b**) Dynamics of cancer cells and CTLs at different values of ${\alpha}_{2}$ and fixed values of $\xi =0.85$ and $\tau =0.95$.

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**MDPI and ACS Style**

Idrees, M.; Alnahdi, A.S.; Jeelani, M.B.
Mathematical Modeling of Breast Cancer Based on the Caputo–Fabrizio Fractal-Fractional Derivative. *Fractal Fract.* **2023**, *7*, 805.
https://doi.org/10.3390/fractalfract7110805

**AMA Style**

Idrees M, Alnahdi AS, Jeelani MB.
Mathematical Modeling of Breast Cancer Based on the Caputo–Fabrizio Fractal-Fractional Derivative. *Fractal and Fractional*. 2023; 7(11):805.
https://doi.org/10.3390/fractalfract7110805

**Chicago/Turabian Style**

Idrees, Muhammad, Abeer S. Alnahdi, and Mdi Begum Jeelani.
2023. "Mathematical Modeling of Breast Cancer Based on the Caputo–Fabrizio Fractal-Fractional Derivative" *Fractal and Fractional* 7, no. 11: 805.
https://doi.org/10.3390/fractalfract7110805