This study develops a reduced-order, physics-informed framework for analyzing and optimizing drug delivery in a vascular environment by coupling hemodynamic flow dynamics with pharmacokinetic transport. The blood flow field is represented using a Hopf-type nonlinear oscillator derived from reduced Navier–Stokes dynamics, capturing the emergence of oscillatory flow states that significantly influence mixing and transport. This fluid subsystem is directly coupled to a two-compartment pharmacokinetic model that describes drug concentration evolution in plasma and surrounding tissue. The resulting system accounts for convection-enhanced dispersion, inter-compartmental mass transfer, and systemic clearance, thereby providing a more realistic representation of in vivo drug transport than classical compartmental approaches. Bifurcation analysis reveals a Hopf bifurcation that governs the transition between steady and oscillatory flow regimes, which in turn modulates drug dispersion and tissue uptake efficiency. The interaction between flow instability and drug transport is shown to play a critical role in determining concentration profiles in both plasma and tissue compartments. Furthermore, the framework enables the study of how flow-induced mixing can enhance or impair therapeutic delivery depending on system parameters. An optimal control formulation is developed to regulate drug concentration and flow activity, with and without bifurcation-aware constraints. The results demonstrate that incorporating stability constraints improves performance and controls drug distribution, reducing undesirable concentration fluctuations while maintaining effective tissue delivery. The proposed approach provides a unified computational framework integrating nonlinear dynamics, transport physics, and control theory, offering new insights into the role of hemodynamic instabilities in drug delivery systems and enabling improved therapeutic design strategies.