A study of a fractional-order cancer biomarker model theoretical analysis and numerical simulation

This paper presents a fractional-order mathematical model for describing the dynamics of cancer biomarkers based on the Caputo fractional derivative. The proposed model incorporates memory effects inherent in biological systems, which provides a more realistic framework than the corresponding classical integer-order model. The existence and uniqueness of solutions are established using fixed-point theory, while the local stability of the equilibrium point is investigated through suitable stability criteria. Numerical solutions are obtained using the Generalized Fractional Taylor Expansion (GFTE), and the effectiveness and accuracy of the proposed numerical scheme are demonstrated. Furthermore, Particle Swarm Optimization (PSO) is employed for determining the optimal model parameters and nanoparticle dosage to improve tumor size estimation. The numerical results reveal the influence of the fractional-order parameter on biomarker dynamics and show that the proposed fractional-order model offers a flexible and robust framework for cancer biomarker analysis, early tumor detection, and optimization-based parameter estimation.