Mathematics and Computational Sciences

Modelling of the tumor growth under oncolytic virotherapy with piecewise differential operators: The effects of combinations of specialist viruses

Document Type : Original Article

Authors

Seda Igret Araz 1
Elif Denk 2

1 Department of Mathematics Education, Siirt University, Siirt, Turkey

2 Department of Mathematics Education, Siirt University, Siirt, Turkey,

https://doi.org/10.30511/mcs.2024.2033648.1191
Abstract
This study proposes to modify a mathematical model of virotherapy inducing cytokine IL-12 and co-stimulatory molecule 4-1BB ligand release with the concept of piecewise derivatives with the aim of analyzing the effects of treatment combinations on tumor growth. In addition to the equilibrium points for the tumor model, the solutions of the model have been proven to be positive. For the model under investigation, the basic reproduction number has been calculated to examine the transmission potential of oncolytic viruses. A method based on Newton polynomial is presented for the numerical solution of the model with piecewise derivative and the numerical simulations for piecewise model has been depicted for different values of fractional orders.
Simulations show that viral oncolytic plays a crucial role in reducing tumor size but an increase in stimulation of cytotoxic T cells can lead to a short-term reduction followed by a more rapid relapse. Furthermore, thanks to the model modified with the concept of piecewise derivative to examine the effects of using different doses at different times on tumor growth, it has been possible to conclude that the virus dose given at the time when the tumor size started to increase after the first dose caused a decrease in tumor size. Finally, according to the assumptions of the considered model and the outputs of the mathematical tools used, it can be concluded that tumor growth seems to be controllable through treatment combinations in virotherapy.

Keywords

Tumor model
Virotherapy
Fractional differentiation and integration
Numerical analysis
Piecewise derivative

Subjects

[1] Abbas M., Aslam S., Abdullah FA., Riaz MB., Gepreel KA., (2023), An efficient spline technique for solving time-fractional integro-differential equations, Heliyon 9 (9).
[2] Abernathy Z., Abernathy K., Steven J., (2020), A mathematical model for tumor growth and treatment using virotherapy, AIMS Mathematics, 5(5): 4136–4150.
[3] Agarwal M., Bhadauria AS., et al., (2011), Mathematical modeling and analysis of tumor therapy with oncolytic virus. Appl. Math., 2, 131.
[4] Ahmad S., Ullah A., Riaz MB., Ali A., Akgul A., Partohaghighi M., (2021), Complex dynamics of multi strain TB model under nonlocal and nonsingular fractal fractional operator, Results in Physics, 30, 104823.[5] Atangana A., Baleanu D., (2016), New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2), 763-769.
[6] Atangana A., Igret Araz S., (2021), New concept in calculus: Piecewise differential and integral operators, Chaos Solit. Fract., 145.
[7] Atangana A., Igret Araz S., (2021), New Numerical scheme with Newton polynomial: Theory, methods and applications, Elsevier, Academic press, ISBN:9780323854481.
[8] Atangana A., Igret Araz S., (2022), Piecewise derivatives versus short memory concept: Analysis and application, Mathematical Biosciences and Engineering, 19 (8).
[9] Atangana A., Igret Araz S., (2022),Rhythmic behaviors of the human heart with piecewise derivative. Mathematical Biosciences and Engineering, 19(3): 3091-3109.
[10] Barillot, E., Calzone, L., Hupe, P., Vert, J.P., Zinovyev, A., (2013), Computational systems biology of cancer, Boca Raton: CRC Press.
[11] Caputo M., Fabrizio M., (2016), Applications of new time and spatial fractional derivatives with exponential kernels, Progress in Fractional Differentiation and Applications, 2, 1-11.
[12] Chinyoka M., Gashirai TB., Mushayabasa S., (2021), On the Dynamics of a Fractional-Order Ebola Epidemic Model with Nonlinear Incidence Rates, Discrete Dynamics in Nature and Society, 2021.
[13] De Pillis, L.G., Radunskaya, A.E., (2003), The dynamics of optimally controlled tumor model: A case study. Math. Comput. Model. 37, 1221–1244.
[14] Deng WH., (2007), Short memory principle and a predictor-corrector approach for fractional differential equations, J. Comput. Appl. Math., 206, 174–188.
[15] Driessche P., Watmough J., (2002), Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (1), 29-48.
[16] Ganji RM., Jafari H., Moshokoa SP., Nkomo NS., (2021), A mathematical model and numerical solution for brain tumor derived using fractional operator, Results in Physics 28, 104671.
[17] Igret Araz S., (2021), Numerical approximation with Newton polynomial for the solution of a tumor growth model including fractional differential operators, Journal of Science and Technology, 14(1), 249-259 2021, 14(1), 249-259.
[18] Kim, Y., Magdalena, A.S., Othmer, H.G., (2007), A hybrid model for tumor spheroid growth in vitro I: theoretical development and early results, Math Models Methods Appl Sci., 17:1773–98.
[19] Kim PS., Crivelli JJ., Choi I., et al., (2015), Quantitative impact of immunomodulation versus oncolysis with cytokine-expressing virus therapeutics, Math. Biosci. Eng., 12, 841–858.
[20] Malinzi J., Sibanda P., Mambili-Mamboundou H., (2015), Analysis of virotherapy in solid tumor invasion, Math. Biosci., 263, 102–110.
[21] Mast, I., Khosro Sayevand, Jafari, H., (2023), On analyzing two dimensional fractional order brain tumor model based on orthonormal Bernoulli polynomials and Newton’s method. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 14(1), 12–19.
[22] Mast, I., Sayvand K., Jafari H., (2024), On epidemiological transition model of the Ebola virus in fractional sense, Journal of Applied Analysis and Computation, 14 (3), 1625-1647.
[23] Podlubny I., (1999), Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Appli-cations, Academic Press, San Diego -New York London.
[24] Prabhakar TR., (1971), A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J., 19, 7–15.
[25] Reale A., Vitiello A., Conciatori V., et al., (2016), Perspectives on immunotherapy via oncolytic viruses, Infect. agents canc., 14, 5.
[26] Watanabe, Y., Dahlman, E., Hui, S., (2016), A mathematical model of tumor growth and its response to single irradiation, Theoretical Biology and Medical Modeling, 13:6.
[27] World Health Organization(WHO), Cancer, https://www.who.int/news-room/factsheets/detail/cancer.
[28] Wu GC., Deng ZG., Baleanu D., Zeng DQ., (2019), Fractional impulsive differential equations: Exact solutions, integral equations and short memory case, Fract. Calc. Appl. Anal., 22.
Mathematics and Computational Sciences
Volume 5, Issue 4
Autumn 2024
Pages 1-25

  • Receive Date 24 June 2024
  • Revise Date 11 October 2024
  • Accept Date 12 October 2024

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