Modeling and Analysis of Breast Cancer With Variable-Order Fractional and Optimal Control Approach

This study presents a novel variable-order (VO) fractional model (VOFM) to describe breast cancer progression and therapy, incorporating five patient compartments that reflect disease stages and treatment effects, including cardiotoxicity. The model employs the constant proportional Caputo (CPC) VO operator to capture memory effects and time-varying biological behaviors. We rigorously analyze the model’s mathematical properties, proving the existence, uniqueness, positivity, and boundedness of solutions. Stability is assessed using the Ulam–Hyers (UH) and generalized UH approaches under specific conditions. To improve clinical relevance, we formulate an optimal control (OC) problem aimed at increasing remission rates and minimizing cardiotoxicity. The Pontryagin maximum principle (PMP) is applied to derive the optimality conditions, leading to a system of adjoint equations and expressions for OCs. Numerical simulations are carried out using the generalized Runge–Kutta method (GRK4M) and the CPC-Θ finite difference method (CPC-Θ FDM) to solve and validate the model. Results show that applying OC significantly enhances therapeutic outcomes, increasing remission rates while reducing cardiotoxic side effects. This study demonstrates the effectiveness of VOFMs in capturing the complex dynamics of breast cancer and optimizing treatment strategies.