Investigation of Fractional Order Tumor Cell Concentration Equation Using Finite Difference Method
DOI:
https://doi.org/10.21123/bsj.2024.9246Keywords:
Caputo Derivative, Finite Difference Scheme, Fractional Differential Equation, Python, Tumor cell.Abstract
The fractional differential equation provides a powerful mathematical framework for modeling and understanding a wide range of complex, non-local and memory-dependent phenomena in various scientific, engineering and real-world applications. Liver metastasis is a secondary cancerous tumor that develops in the liver due to the spread of cancer cells from a primary cancer originating in another part of the human body. The primary focus of this study is to understand tumor growth in the human liver, both in the presence and absence of medication therapy. To achieve this, a temporal fractional-order parabolic partial differential equation is utilized, and its analysis is carried out using numerical methods. The Caputo derivative is employed to explore the impact of medication therapy on tumor growth. To numerically solve the mathematical model, the Crank-Nicolson finite difference method has been developed. This method is selected for its remarkable attributes, including unconditional stability and second-order accuracy in both spatial and temporal dimensions. The outcomes and insights derived from this study are effectively communicated through various graphical representations. These visual aids serve as invaluable tools for comprehending the profound impact of medication therapy on the growth of liver tumors. Through the medium of these graphical depictions, one can glean a clearer and more intuitive understanding of the complex dynamics at play. The numeric solution to this intricate problem is achieved through the implementation of algorithms meticulously crafted using versatile and powerful Python programming language. Python’s flexibility, extensive libraries, and robust numerical computing capabilities make it a good choice for handling the complexities of this study.
Received 12/06/2023
Revised 31/12/2023
Accepted 02/01/2024
Published Online First 20/02/2024
References
Mubarak S, Khanday MA, Lone AUH. Mathematical analysis based on eigenvalue approach to study liver metastasis disease with applied drug therapy. Netw Model Anal Health Inform Bioinform. 2020; 9(1). DOI: https://doi.org/10.1007/s13721-020-00231-0
Khanday MA, Nazir K. Mathematical and numerical analysis of thermal distribution in cancerous tissues under the local heat therapy. Int J Biomath. 2017; 10(7): 1–10. https://doi.org/10.1142/S1793524517500991
Swanson KR, Bridge C, Murray JD, Alvord EC. Virtual and real brain tumors: Using mathematical modeling to quantify glioma growth and invasion. J Neurol Sci. 2003; 216(1): 1–10. https://doi.org/10.1016/j.jns.2003.06.001
Filipovic N, Djukic T, Saveljic I, Milenkovic P, Jovicic G, Djuric M. Modeling of liver metastatic disease with applied drug therapy. Comput Methods Programs Biomed. 2014; 115(3): 162–70. Available from: https://dx.doi.org/10.1016/j.cmpb.2014.04.013
Ghode K, Takale K, Gaikwad S. Traveling Wave Solutions of Fractional Differential Equations Arising in Warm Plasma. Baghdad Sci J. 2023; 20(1(SI)): 0318–0318. https://dx.doi.org/10.21123/bsj.2023.8394
Sonawane J, Sontakke B, Takale K. Approximate Solution of Sub diffusion Bio heat Transfer Equation. Baghdad Sci J.; 20(1(SI)): 0394. https://dx.doi.org/10.21123/bsj.2023.8410
Joshi H, Jha BK. Fractional-order mathematical model for calcium distribution in nerve cells. Comput Appl Math. 2020; 39(2): 1–22. https://doi.org/10.1007/s40314-020-1082-3
Khan AA, Amin R, Ullah S, Sumelka W, Altanji M. Numerical simulation of a Caputo fractional epidemic model for the novel coronavirus with the impact of environmental transmission. Alexandria Eng J. 2022; 61(7): 5083–95. https://doi.org/10.1016/j.aej.2021.10.008
Alzubaidi AM, Othman HA, Ullah S, Ahmad N, Alam MM. Analysis of Monkeypox viral infection with human to animal transmission via a fractional and Fractal-fractional operators with power law kernel. Math Biosci Eng. 2023; 20(4): 6666–90. https://doi.org/10.3934/mbe.2023287
Bolton L, Cloot AHJJ, Schoombie SW, Slabbert JP. A proposed fractional-order Gompertz model and its application to tumour growth data. Math Med Biol. 2015; 32(2): 187–207. https://doi.org/10.1093/imammb/dqt024A
Iyiola OS, Zaman FD. A fractional diffusion equation model for cancer tumor. AIP Adv. 2014; 4(10): 107121. http://dx.doi.org/10.1063/1.4898331
Rihan FA, Velmurugan G. Dynamics of fractional-order delay differential model for tumor-immune system. Chaos Solitons Fractals. 2020; 132: 109592. https://doi.org/10.1016/j.chaos.2019.109592
Abaid Ur Rehman M, Ahmad J, Hassan A, Awrejcewicz J, Pawlowski W, Karamti H, et al. The Dynamics of a Fractional-Order Mathematical Model of Cancer Tumor Disease. Symmetry (Basel). 2022; 14(8): 1694. https://doi.org/10.3390/sym14081694
Owolabi KM, Atangana A. Numerical Methods for Fractional Differentiation. Springer Series in Computational Mathematics 54. Singapore. 1st Ed. 2019. 328 p. https://doi.org/10.1007/978-981-15-0098-5
Kadalbajoo MK, Awasthi A. A numerical method based on Crank-Nicolson scheme for Burgers’ equation. Appl Math Comput. 2006; 182(2): 1430–42. https://doi.org/10.1016/j.amc.2006.05.030
Ali AH, Jaber AS, Yaseen MT, Rasheed M, Bazighifan O, Nofal TA. A Comparison of Finite Difference and Finite Volume Methods with Numerical Simulations: Burgers Equation Model. Complexity. 2022: 1–9. https://doi.org/10.1155/2022/9367638
Podlubny I. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Math Sci Eng. 1999; 198: 340.
Kharde U, Takale K, Gaikwad S. Crank-Nicolson Method For Time Fractional Drug Concentration Equation in Central Nervous System. Adv Appl Math Sci. 2022; 22(2): 407–433.
Downloads
Issue
Section
License
Copyright (c) 2024 Baghdad Science Journal
This work is licensed under a Creative Commons Attribution 4.0 International License.
How to Cite
