Cancer Modeling by Fractional Derivative Equation and Chemotherapy Stabilizing

Year 2024, Volume: 7 Issue: 3, 125 - 134, 29.09.2024

Amine Moustafid

https://doi.org/10.33434/cams.1486049

Abstract

This paper discusses the theme of cancer modeling and the control problem of chemotherapy. Cancer spread is modeled by fractional derivative equation and asymptotically stabilized by chemotherapy law. The model is converted by fractional complex transform into a simple partial derivative equation and associated with a viability problem, and the set-valued analysis is used to make the converted model viable by the regulation law of the regulation map. The regulation law is used to give the stabilizing chemotherapy control for a specific model of the glioblastomas multiforme (GBM) tumor concentration.

Keywords

Chemotherapy, Fractional derivative equation, Viability theory

References

• [1] T. Alinei-Poiana, E. Dulf, L. Kovacs, Fractional calculus in mathematical oncology, Sci. Rep., 13 (2023), 10083.
• [2] N. Sweilam, M. Khader, A. Mahdy, Numerical studies for solving fractional-order logistic equation, Int. J. Pure Appl. Math., 78 (2012), 1199-1210.
• [3] N. Djeddi, S. Hasan, M. Al-Smadi, S. Momani, Modified analytical approach for generalized quadratic and cubic logistic models with Caputo-Fabrizio fractional derivative, Alex. Eng. J., 59 (2020), 5111-5122.
• [4] A. Kanth, N. Garg, Computational simulations for solving a class of fractional models via Caputo-Fabrizio fractional derivative, Procedia Comput. Sci., 125 (2018), 476-482.
• [5] S. Arshad, I. Saleem, A. Akg¨ul, J. Huang, Y. Tang, S. Eldin, A novel numerical method for solving the Caputo-Fabrizio fractional differential equation, AIMS Math., 8 (2023), 9535-9556.
• [6] N. Varalta, A. Gomes, R. Camargo, A prelude to the fractional calculus applied to tumor dynamic, TEMA Tend. Mat. Apl. Comput., 15 (2014), 211-221.
• [7] F. Ariza-Hernandez, M. Arciga-Alejandre, J. Sanchez-Ortiz, A. Fleitas-Imbert, Bayesian derivative order estimation for a fractional logistic model, Mathematics, 8 (2020), 109.
• [8] M. Meabed Khader, M. Babatin, Others on approximate solutions for fractional logistic differential equation, Math. Probl. Eng., 2013 (2013).
• [9] S. Khajanchi, M. Sardar, J. Nieto, Application of non-singular kernel in a tumor model with strong Allee effect, Differ. Equ. Dyn. Syst., (2022), 1-6.
• [10] S. Debbouche, Implicit solution for logistic Caputo-Fabrizio fractional differential equation with Allee effect, J. Fract. Calc. Nonlinear Syst., 4 (2023), 1-7.
• [11] C. Jadhav, T. Dale, D. Chinchane, A method to solve ordinary fractional differential equations using Elzaki and Sumudu transform, J. Fract. Calc. Nonlinear Syst., 4 (2023), 8-16.
• [12] M. Etefa, G. Guerekata, P. Ngnepieba, O. Iyiola, On a generalized fractional differential Cauchy problem, Malaya J. Mat., 11 (2023), 80-93.
• [13] Y. Karaca, Computational complexity-based fractional-order neural network models for the diagnostic treatments and predictive transdifferentiability of heterogeneous cancer cell propensity, Chaos Theory Appl., 5 (2023), 34-51.
• [14] Z. Chebana, T. Oussaeif, A. Ouannas, Others solvability of Dirichlet problem for a fractional partial differential equation by using energy inequality and Faedo-Galerkin method, Innovative J. Math. (IJM), 1 (2022), 34-44.
• [15] H. Husni Zureigat, M. Al-Smadi, A. Al-Khateeb, S. Al-Omari, S. Alhazmi, Numerical solution for fuzzy time-fractional cancer tumor model with a time-dependent net killing rate of cancer cells, Int. J. Environ. Res. Public Health., 20 (2023), 3766.
• [16] Z. Körpinar. Inc, E. Hınçal, D. Baleanu, Residual power series algorithm for fractional cancer tumor models, Alex. Eng. J., 59 (2020), 1405-1412.
• [17] R. Saadeh, A. Qazza, K. Amawi, A new approach using integral transform to Solve cancer models, Fractal Fract., 6 (2022), 490.
• [18] R. Singh, V. Gupta, J. Mishra, An investigation of the complexities of a malignant tumor’s fractional-order mathematical model, Ann. Comput. Sci. Inf. Syst., 33 (2022), 207-211.
• [19] B. Batiha, A. Al-khateeb, H. Zureigat, Improving numerical solutions for the generalized Huxley equation: The new iterative method (NIM), Appl. Math., 17 (2023), 423-427.
• [20] N. Ahmed, N. Shah, S. Taherifar, F. Zaman, Memory effects and of the killing rate on the tumor cells concentration for a one-dimensional cancer model, Chaos, Solitons & Fractals, 144 (2021), 110750.
• [21] R. Moallem Ganji, H. Jafari, S. Moshokoa, N. Nkomo, A mathematical model and numerical solution for brain tumor derived using fractional operator, Results Phys., 28 (2021), 104671.
• [22] M. Partohaghighi, A. Akg¨ul, E. Akg¨ul, N. Attia, M. Sen, M. Bayram, Analysis of the fractional differential equations using two different methods, Symmetry, 15 (2023), 65.
• [23] F. Mohd, H. Sulaiman, N. Alias, Modified Swanson’s equation to detect the growth of glioblastomas multiforme (GBM) tumour, Int. J. Adv. Res. Engineering Innovation, 3 (2021), 1-18.
• [24] H. Gandhi, A. Tomar, D. Singh, A predicted mathematical cancer tumor growth model of brain and its analytical solution by reduced differential transform method, TCCE 2019, (2021), 203.
• [25] V. Srivastava, S. Kumar, M. Awasthi, B. Singh, Two-dimensional time fractional-order biological population model and its analytical solution, Egypt. J. Basic Appl. Sci., 1 (2014), 71-76.
• [26] O. Iyiola, F. Zaman, A fractional diffusion equation model for cancer tumor, AIP Adv., 4 (2014), 107121.
• [27] A. Omame, F. Zaman, Analytic solution of a fractional order mathematical model for tumour with polyclonality and cell mutation, Partial Differ. Equ. Appl. Math., (2023), 100545.
• [28] S. Gimnitz, B. Bira, D. Zeidan, Optimal systems, series solutions and conservation laws for a time fractional cancer tumor model, Chaos, Solitons & Fractals, 169 (2023), 113311.
• [29] I. Area, J. Nieto, On the fractional Allee logistic equation in the Caputo sense, Ex. Countex., 4 (2023), 100121.
• [30] K. Kassara, A unified set-valued approach to control immunotherapy, SIAM J. Control Optim., 48 (2009), 909-924.
• [31] K. Kassara, A. Moustafid, Angiogenesis inhibition and tumor-immune interactions with chemotherapy by a control set-valued method, Math. Biosci., 231 (2011), 135-143.
• [32] L. Boujallal, M. Elhia, O. Balatif, A novel control set-valued approach with application to epidemic models, J. Appl. Math. Comput., 65 (2021), 295-319.
• [33] A. Moustafid, General chemotherapy protocols, J. Appl. Dynamic Syst. Control., 4 (2021), 18-25.
• [34] A. Moustafid, Set-valued control of cancer by combination chemotherapy, J. Math. Sci. Model., 6 (2023), 7-16.
• [35] A. Moustafid, Viability control of chemo-immunotherapy and radiotherapy by set-valued analysis, Int. J. Inform. Appl. Math., 6 (2023), 40-56.
• [36] A. Moustafid, Set-valued analysis of anti-angiogenic therapy and radiotherapy, Math. Modelling Numer. Simul. Appl., 2 (2022), 187-196.
• [37] A. Moustafid, General anti-angiogenic therapy protocols with chemotherapy, Int. J. Math. Model. Comput., 11 (2021).
• [38] A. Moustafid, Feedback protocols for anti-angiogenic therapy in the treatment of cancer tumors by chemotherapy, Int. J. Optim. Appl., 2 (2022), 17-24.
• [39] A. Moustafid, Set-valued stabilization of reaction-diffusion model by chemotherapy and or radiotherapy, Fun. J. Math. Appl., 6 (2023), 147-156.
• [40] J. P. Aubin, H. Frankowska, Set-valued Analysis, Springer Science & Business Media, 2009.
• [41] J. H. He, Z. B. Li, Converting fractional differential equations into partial differential equations, Therm. Sci., 16 (2012), 331-334.
• [42] R. Ibrahim, Fractional complex transforms for fractional differential equations, Adv. Difference Equ., 2012 (2012), 1-12.