Analysis and numerical investigation of a breast cancer treatment model incorporating ketogenic diet and immune boosters via optimal control theory

This paper develops a mathematical model to investigate breast cancer dynamics by incorporating tumor–immune interactions, ketogenic diet effects, and medical treatment. The model is formulated as a system of nonlinear ordinary differential equations and analyzed within an optimal control framework. Time-dependent control variables are introduced to represent treatment strategies aimed at minimizing tumor progression while reducing therapeutic costs. The model’s well-posedness is established through positivity and boundedness analysis. The necessary conditions for optimality are derived using Pontryagin’s Minimum Principle, resulting in a coupled system of state and adjoint equations. Numerical solutions are obtained using the fourth-order Runge–Kutta method combined with a forward–backward sweep algorithm. Simulation results demonstrate that the proposed control strategy effectively suppresses tumor growth and enhances immune response compared to uncontrolled scenarios. Cost-effectiveness analysis also shows the treatment strategies with the greatest impact at minimum costs. The numerical findings are consistent with the analytical results and highlight the importance of optimized treatment scheduling. The proposed framework provides a reliable and computationally efficient approach for analyzing cancer treatment dynamics and may serve as a basis for further extensions involving parameter estimation or data-driven studies.