Dynamic analysis of fractal-fractional cancer model under chemotherapy drug with generalized Mittag-Leffler kernel

Background and Objective: Cancer's complex and multifaceted nature makes it challenging to identify unique molecular and pathophysiological signatures, thereby hindering the development of effective therapies. This paper presents a novel fractal-fractional cancer model to study the complex interplay among stem cells, effectors cells, and tumor cells in the presence and absence of chemotherapy. The cancer model with effective treatment through chemotherapy drugs is considered and discussed in detail. Methods: The numerical method for the fractal-fractional cancer model with a generalized Mittag-Leffler kernel is presented. The Routh-Hurwitz stability criteria are applied to confirm the local asymptotically stability of an endemic equilibrium point of the cancer model without treatment and with effective treatment under some conditions. The existence and uniqueness criteria of the fractal-fractional cancer model are derived. Furthermore, the stability analysis of the fractal-fractional cancer model is performed. Results: The temporal concentration pattern of stem cells, effectors cells, tumor cells, and chemotherapy drugs are procured. The usage of chemotherapy drugs kills the tumor cells or decreases their density over time and as a consequence takes a longer time to reach to equilibrium point. The decay rate of stem cells and tumor cells plays a crucial role in cancer dynamics. The notable role of fractal dimensions along with fractional order is observed in capturing the cancer cell concentration. Conclusion: A dynamic analysis of the fractal-fractional cancer model is demonstrated to examine the impact of chemotherapy drugs with a generalized Mittag-Leffler kernel. The model possesses three equilibrium points among them two correspond to the cancer model without treatment namely the tumor-free equilibrium point and endemic equilibrium point. One additional endemic equilibrium point exists in the case of effective treatment through chemotherapy drugs. The Routh-Hurwitz stability criteria are applied to confirm the local asymptotically stability of an endemic equilibrium point of the cancer model with and without treatment under some conditions. The chemotherapy drug plays a crucial role in controlling the growth of tumor cells. The fractal-fractional operator provided robustness to study cancer dynamics by the inclusion of memory and heterogeneity.