Drug Delivery to the Bloodstream within the Cardiovascular System using Caputo-Fabrizio Fractional Derivatives

We present a fractional-order extension of the third compartmental model of (Khanday et al. 2017) to study drug distribution after oral and intravenous administration. The Caputo–Fabrizio (CF) fractional derivative order (0 < γ < 1) replaces the integer-order time derivatives in the original model, which provides a non-singular memory kernel suitable for modeling biological processes with finite memory. We first establish the existence and uniqueness of solutions for the fractional model using a fixed-point theorem under mild Lipschitz conditions. An analytic representation of the model solution is then obtained via Laplace transform techniques adapted to the CF operator. Numerical results (MATLAB simulations) illustrate how the fractional order γ and key rate parameters shape arterial, tissue, and venous concentrations. In particular, γ < 1 introduces a memory effect that slows concentration decay and may increase residual drug levels, and discusses the implications for dosing and residue accumulation. Finally, we discuss limitations, key model parameters with physical units, and directions for further validation with experimental pharmacokinetic data.