This study proposes compartment models with the Caputo fractional derivative to capture drug dynamics across absorption, central, and peripheral compartments, and to account for memory effects and anomalous kinetics. The first part of the study covers mathematical modeling, which enables analysis of the amount of drug in each compartment using closed-form expressions. The second part is to estimate the pharmacokinetic parameters and compare models using the Akaike information criterion and Bayesian information criterion for the drugs pindolol, fenoprofen, and cyclosporin A. The regression analysis was performed by using the corresponding Mittag-Leffler regression model and the trust-region algorithm with the Adams predictor-corrector method.

This study investigates a time-fractional optimal control problem for the Bergman minimal model of type 1 diabetes by employing the Caputo derivative of order $q\in (0,1)$ to incorporate memory effects into the glucose–insulin dynamics. First, the well-posedness of the fractional-order model is established by proving the existence, uniqueness, boundedness, and positivity of its solutions. Subsequently, an optimal control problem describing a normalized therapeutic intervention is formulated, and the existence of an optimal control is rigorously proved. By applying the fractional Pontryagin maximum principle, the corresponding optimality system is derived, and the optimal control law is explicitly characterized. The obtained results demonstrate that the proposed control strategy effectively stabilizes the glucose dynamics around the desired equilibrium while reducing excessive insulin administration. In comparison with the classical integer-order model, the fractional formulation provides a more realistic representation of delayed physiological responses. Finally, numerical simulations, carried out using a predictor–corrector scheme together with a forward–backward sweep algorithm, confirm the effectiveness and reliability of the proposed approach for blood glucose regulation.