Fractional-order mathematical modelling of cancer cells-cancer stem cells-immune system interaction with chemotherapy

Abstract

In this paper, we present a mathematical model of stem cells and chemotherapy for cancer treatment, in which the model is represented by fractional-order differential equations. Local stability of equilibrium points is discussed. Then, the existence and uniqueness of the solution are studied. In addition, in order to point out the advantages of the fractional-order modeling, memory trace and hereditary traits are taken into consideration. Numerical simulations have been used to investigate how the fractional-order derivative and different parameters affect the population dynamics, the graphs have been illustrated according to different values of fractional order $\alpha$ and different parameter values. Moreover, we have examined the effect of chemotherapy on tumor cells and stem cells over time. Furthermore, we concluded that the memory effect occurs as the $\alpha$ decreases from 1 and the chemotherapy drug is quite effective on the populations. We hope that this work will contribute to helping medical scientists take the necessary measures during the screening process and treatment.

Keywords

• Fractional-order differential equations
• cancer stem cells
• immune system
• numerical solutions
• memory effect
• existence and uniqueness

References

1. El-Gohary, A. Chaos and optimal control of cancer self-remission and tumor system steady states. Chaos, Solutions and Fractals, 37(5), 1305-1316, (2008).
2. El-Gohary, A. The chaos and optimal control of cancer model with complete unknown parameters. Chaos, Solutions and Fractals, 42(5), 2865-2874, (2009).
3. Kirschner, D., Panetta, J.C. Modeling immunotherapy of the tumor–immune interaction. Journal of mathematical biology, 37(3), 235-252, (1998).
4. Öztürk, I. & Özköse, F. Stability analysis of fractional order mathematical model of tumor-immune system interaction. Chaos, Solitons & Fractals, 133, 109614, (2020).
5. Hadamard, J. Essai sur l’étude des fonctions données par leur développement de Taylor (PDF). Journal de Mathématiques Pures et Appliquées, 4(8), 101–186, (1892).
6. Li, C. & Tao, C. On the fractional Adams method. Computers & Mathematics with Applications, 58(8), 1573-1588, (2009).
7. Yavuz, M. & Sene, N. Stability analysis and numerical computation of the fractional predator–prey model with the harvesting rate. Fractal and Fractional, 4(3), 35, (2020).
8. Naik, P.A., Owolabi, K.M., Yavuz, M. & Zu, J. Chaotic dynamics of a fractional order HIV-1 model involving AIDS-related cancer cells. Chaos, Solitons & Fractals, 140, 110272, (2020).

9. Özköse, F. & Yavuz, M. Investigation of interactions between COVID-19 and diabetes with hereditary traits using real data: A case study in Turkey. Computers in biology and medicine, 105044, (2021).
10. Hammouch, Z., Yavuz, M. & Özdemir, N. Numerical solutions and synchronization of a variable-order fractional chaotic system. Mathematical Modelling and Numerical Simulation with Applications, 1(1), 11-23, (2021).
11. Naik, P.A., Yavuz, M. & Zu, J. The role of prostitution on HIV transmission with memory: A modeling approach. Alexandria Engineering Journal, 59(4), 2513-2531, (2020).
12. Naik, P.A., Yavuz, M., Qureshi, S., Zu, J., Townley, S. Modeling and analysis of COVID-19 epidemics with treatment in fractional derivatives using real data from Pakistan. The European Physical Journal Plus, 135(10), 1-42, (2020).
13. Thomson, J.A., Itskovitz-Eldor, J., Shapiro, S.S., Waknitz, M.A., Swiergiel, J.J., Marshall, V.S., Jones, J.M. Embryonic stem cell lines derived from human blastocysts. Science, 282(5391), 1145–1147, (1998).
14. Hernigou, P., Beaujean, F. Treatment of osteonecrosis with autologous bone marrow grafting. Clinical Orthopaedics and Related Research, 405, 14–23, (2002).
15. Körbling, M., Estrov, Z. Adult stem cells for tissue repair—a new therapeutic concept?. New England Journal of Medicine, 349(6), 570–582, (2003).
16. Chakrabarty, K., Shetty, R., Ghosh, A. Corneal cell therapy: with iPSCs, it is no more a far-sight. Stem cell research & therapy, 9(1), 1-15, (2018).
17. Alqudah, M.A. Cancer treatment by stem cells and chemotherapy as a mathematical model with numerical simulations. Alexandria Engineering Journal, 59, 1953–1957, (2020).
18. Naik, P.A., Zu, J. & Naik, M. Stability analysis of a fractional-order cancer model with chaotic dynamics. International Journal of Biomathematics, 14(6), 2150046, (2021).
19. Naik, P.A. Global dynamics of a fractional-order SIR epidemic model with memory. International Journal of Biomathematics, 13(8), 2050071, (2020).
20. Podlubny, I. Fractional Differential Equations. Academic Press New York, (1999).
21. Naik, P.A., Zu, J., Owolabi, K.M. Global dynamics of a fractional order model for the transmission of HIV epidemic with optimal control. Chaos, Solitons & Fractals, 138, 109826, (2020).
22. Jin, B., Lazarov, R. & Zhou, Z. An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data. IMA Journal of Numerical Analysis, 36(1), 197-221, (2016).
23. Du, M. & Wang, Z. Correcting the initialization of models with fractional derivatives via history-dependent conditions. Acta Mechanica Sinica, 32(2), 320-325, (2016).
24. Magin, R.L. Fractional Calculus in Bioengineering. Redding: Begell House, (2006).
25. Ahmed, E., El-Sayed, A.M.A., El-Saka, H.A.A. Equilibrium points, stability and numerical solutions of fractional-order predator–prey and rabies models. Journal of Matematical Analysis and Applications, 325(1), 542-553, (2007).
26. El-Sayed, A.M.A., El-Mesiry, A.E.M., El-Saka, H.A.A. On the fractional- order logistic equation. Applied Mathematics Letters, (20), 817-823, (2007).
27. Petras, I. Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Springer Berlin, (2011).
28. Baisad, K., Moonchai, S. Analysis of stability and hopf bifurcation in a fractional Gauss-type predator-prey model with Allee effect and Holling type-III functional response. Advances in difference equations, 2018(1), 1-20, (2018).
29. Daşbaşı, B., Öztürk, İ. & Özköse, F. Çoklu Antibiyotik Tedavisiyle Bakteriyel Rekabetin Matematiksel Modeli ve Kararlılık Analizi. Karaelmas Science and Engineering Journal, 6(2), 299-306, (2016).
30. Bozkurt, F. & Özköse, F. Stability analysis of macrophage-tumor interaction with piecewise constant arguments. In AIP Conference Proceedings, (Vol. 1648, No. 1, p. 850035), AIP Publishing LLC, (2015, March).
31. Özköse, F., Yılmaz, S., Yavuz, M., Öztürk, İ., Şenel, M.T., Bağcı, B.Ş. & Doğan, M., Önal, Ö. A fractional modelling of tumor-immune system interaction related to lung cancer with real data. The European Physical Journal Plus, (2022).
32. Diethelm, K. An algorithm for the numerical solution of differential equations of fractional order. Electronic transactions on numerical analysis, 5(1), 1-6, (1997).
33. Diethelm, K., Ford, N.J., Freed, A.D. A predictor-corrector approch for the numerical solution of fractional differential equations. Nonlinear Dynamics, 29, 3-22, (2002).
34. Garrappa, R. On linear stability of predictor-corrector algorithms for fractional differential equations. International Journal of Computer Mathematics, 87(10), 2281-2290, (2010).