Fractional and stochastic modeling of breast cancer progression with real data validation.

This study presents a novel approach to modeling breast cancer dynamics, one of the most significant health threats to women worldwide. Utilizing a piecewise mathematical framework, we incorporate both deterministic and stochastic elements of cancer progression. The model is divided into three distinct phases: (1) initial growth, characterized by a constant-order Caputo proportional operator (CPC), (2) intermediate growth, modeled by a variable-order CPC, and (3) advanced stages, capturing stochastic fluctuations in cancer cell populations using a stochastic operator. Theoretical analysis, employing fixed-point theory for the fractional-order phases and Ito calculus for the stochastic phase, establishes the existence and uniqueness of solutions. A robust numerical scheme, combining the nonstandard finite difference method for fractional models and the Euler-Maruyama method for the stochastic system, enables simulations of breast cancer progression under various scenarios. Critically, the model is validated against real breast cancer data from Saudi Arabia spanning 2004-2016. Numerical simulations accurately capture observed trends, demonstrating the model's predictive capabilities. Further, we investigate the impact of chemotherapy and its associated cardiotoxicity, illustrating different treatment response scenarios through graphical representations. This piecewise fractional-stochastic model offers a powerful tool for understanding and predicting breast cancer dynamics, potentially informing more effective treatment strategies.