Evaluating the stability and efficacy of fractal-fractional models in reproductive cancer apoptosis with ABT-737

Cancer refers to a group of diseases characterized by the uncontrolled growth of abnormal cells. Conventional cancer therapies, such as chemotherapy and radiation therapy, often encounter issues like toxicity and resistance, primarily due to their inability to effectively differentiate between cancerous and normal cells. As a result, these treatments frequently harm healthy cells, underscoring the urgent need for innovative and more effective cancer treatment strategies. This article aims to investigate the changes in the Bcl-2/Bax ratio over time and its stability, using the Atangana–Baleanu fractional derivative operators as a key factor in reproductive cancer. The article also discusses the potential application of this model to investigate the effects of the ABT-737 inhibitor on mitochondrial apoptosis under initial conditions. The methodology entails using a fixed-point approach to investigate existence and uniqueness, as well as exploring Hyers–Ulam stability results for the given model. The numerical schemes utilize the basic concepts of fractional calculus and Lagrange polynomial interpolation. We generate simulated graphical representations of these models for a variety of fractional-order values. Finally, we present the simulation results to confirm the effectiveness and practicality of the theoretical findings.