# Mathematical modeling of cancer treatments with fractional derivatives: An Overview

## DOI:

https://doi.org/10.11113/mjfas.v17n4.2062## Keywords:

Cancer treatments, fractional derivative, hyperthermia, Chemotherapy, Immunotherapy, Radiotherapy,## Abstract

This review article presents fractional derivative cancer treatment models to show the importance of fractional derivatives in modeling cancer treatments. Cancer treatment is a significant research area with many mathematical models developed by mathematicians to represent the cancer treatment processes like hyperthermia, immunotherapy, chemotherapy, and radiotherapy. However, many of these models were based on ordinary derivatives and the use of fractional derivatives is still new to many mathematicians. Therefore, it is imperative to review fractional cancer treatment models. The review was done by first presenting 22 various definitions of fractional derivative. Thereafter, 11 articles were selected from different online databases which included Scopus, EBSCOHost, ScienceDirect Journal, SpringerLink Journal, Wiley Online Library, and Google Scholar. These articles were summarized, and the used fractional derivative models were analyzed. Based on this analysis, the merit of modeling with fractional derivative, the most used fractional derivative definition, and the future direction for cancer treatment modeling were presented. From the results of the analysis, it was shown that fractional derivatives incorporated memory effects which gave it an advantage over ordinary derivative for cancer treatment modeling. Moreover, the fractional derivative is a general definition of all derivatives. Also, the fractional models can be applied to different cancer treatment procedures and the most used fractional derivative is the Caputo as well as its non-singular kernel versions. Finally, it was concluded that the future direction for cancer treatment modeling is the adoption of fractional derivative models corroborated with experimental or clinical data.

## References

B. Ross, "A brief history and exposition of the fundamental theory of fractional calculus," in Fractional calculus and its applications, ed: Springer, 1975, pp. 1-36.

M. Du, Z. Wang, and H. Hu, "Measuring memory with the order of fractional derivative," Scientific reports, vol. 3, pp. 1-3, 2013.

M. Dalir and M. Bashour, "Applications of fractional calculus," Applied Mathematical Sciences, vol. 4, pp. 1021-1032, 2010.

G. Jumarie, "Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results," Computers & Mathematics with Applications, vol. 51, pp. 1367-1376, 2006.

U. N. Katugampola, "A new approach to generalized fractional derivatives," Bull. Math. Anal. Appl, vol. 6, pp. 1-15, 2014.

A. Atangana and A. Secer, "A note on fractional order derivatives and table of fractional derivatives of some special functions," in Abstract and applied analysis, 2013.

J. Hadamard, Essai sur l'étude des fonctions, données par leur développement de Taylor: Gauthier-Villars, 1892.

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations vol. 204: elsevier, 2006.

A. Erdélyi and H. Kober, "Some remarks on Hankel transforms," The Quarterly Journal of Mathematics, pp. 212-221, 1940.

T. J. Osler, "Leibniz rule for fractional derivatives generalized and an application to infinite series," SIAM Journal on Applied Mathematics, vol. 18, pp. 658-674, 1970.

U. N. Katugampola, "A new fractional derivative with classical properties," arXiv preprint arXiv:1410.6535, 2014.

R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, "A new definition of fractional derivative," Journal of Computational and Applied Mathematics, vol. 264, pp. 65-70, 2014.

M. Caputo and M. Fabrizio, "A new definition of fractional derivative without singular kernel," Progr. Fract. Differ. Appl, vol. 1, pp. 1-13, 2015.

J. Losada and J. J. Nieto, "Properties of a new fractional derivative without singular kernel," Progr. Fract. Differ. Appl, vol. 1, pp. 87-92, 2015.

C. Çelik and M. Duman, "Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative," Journal of computational physics, vol. 231, pp. 1743-1750, 2012.

J. V. d. C. Sousa and E. C. de Oliveira, "On the ψ-Hilfer fractional derivative," Communications in Nonlinear Science and Numerical Simulation, vol. 60, pp. 72-91, 2018.

A. Atangana and D. Baleanu, "New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model," arXiv preprint arXiv:1602.03408, 2016.

R. Damor, S. Kumar, and A. Shukla, "Numerical simulation of fractional bioheat equation in hyperthermia treatment," Journal of Mechanics in Medicine and Biology, vol. 14, p. 1450018, 2014.

D. Kumar and K. Rai, "Numerical simulation of time fractional dual-phase-lag model of heat transfer within skin tissue during thermal therapy," Journal of Thermal Biology, vol. 67, pp. 49-58, 2017.

T. Akman Yıldız, S. Arshad, and D. Baleanu, "Optimal chemotherapy and immunotherapy schedules for a cancer‐obesity model with Caputo time fractional derivative," Mathematical Methods in the Applied Sciences, vol. 41, pp. 9390-9407, 2018.

T. A. Yıldız, S. Arshad, and D. Baleanu, "New observations on optimal cancer treatments for a fractional tumor growth model with and without singular kernel," Chaos, Solitons & Fractals, vol. 117, pp. 226-239, 2018.

N. Sweilam and S. AL‐Mekhlafi, "Optimal control for a nonlinear mathematical model of tumor under immune suppression: A numerical approach," Optimal Control Applications and Methods, vol. 39, pp. 1581-1596, 2018.

M. I. Asjad, "Fractional mechanism with power law (singular) and exponential (non-singular) kernels and its applications in bio heat transfer model," International Journal of Heat and Technology, vol. 37, pp. 846-852, 2019.

V. F. Morales‐Delgado, J. F. Gómez‐Aguilar, K. Saad, and R. F. Escobar Jiménez, "Application of the Caputo‐Fabrizio and Atangana‐Baleanu fractional derivatives to mathematical model of cancer chemotherapy effect," Mathematical Methods in the Applied Sciences, vol. 42, pp. 1167-1193, 2019.

D. Baleanu, A. Jajarmi, S. Sajjadi, and D. Mozyrska, "A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator," Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 29, p. 083127, 2019.

K. J. Mahasa, R. Ouifki, A. Eladdadi, and L. de Pillis, "Mathematical model of tumor–immune surveillance," Journal of theoretical biology, vol. 404, pp. 312-330, 2016.

M. A. Dokuyucu, E. Celik, H. Bulut, and H. M. Baskonus, "Cancer treatment model with the Caputo-Fabrizio fractional derivative," The European Physical Journal Plus, vol. 133, p. 92, 2018.

G. Belostotski and H. Freedman, "A control theory model for cancer treatment by radiotherapy," Int. J. Pure Appl. Math, vol. 25, pp. 447-480, 2005.

M. Awadalla, Y. Yameni, and K. Abuassba, "A new Fractional Model for the Cancer Treatment by Radiotherapy Using the Hadamard Fractional Derivative," Online Math., vol. 1, pp. 14-18, 2019.

M. F. Farayola, S. Shafie, F. M. Siam, and I. Khan, "Mathematical modeling of radiotherapy cancer treatment using Caputo fractional derivative," Computer Methods and Programs in Biomedicine, vol. 188, p. 105306, 2020.