Combating liver tumors via yttrium-90 ((90)Y) radioembolization is a viable treatment option of nonresectable liver tumors. Employing clinical (90)Y microparticles (i.e., SIR-Spheres and TheraSpheres) in a computational model of a representative hepatic artery system, laminar transient 3D particle-hemodynamics were simulated. Specifically, optimal particle release positions in the right hepatic (parent) artery as well as the best temporal release window were determined for the microspheres to exit specific outlet daughter vessels, potentially connected to liver tumors. The results illustrate the influence of a curved geometry on the velocity field and the particle trajectory dependence on the spatial and temporal particle injection conditions. The differing physical particle characteristics of the SIR-Spheres and the TheraSpheres had a subtle impact on particle trajectories in the decelerating portion of the arterial pulse, i.e., when the inertial forces on the particles are weaker. Conversely, particle characteristics and inelastic wall collisions had little effect on particles released during the accelerating phase of the arterial pulse, i.e., both types of microspheres followed organized paths to predetermined outlets. Such results begin paving the way towards directing 100% of the released microspheres to specific daughter vessels (e.g., those connected to tumors) under transient flow conditions in realistic geometries via a novel drug-particle targeting methodology.