Based on the hypothesis that aggravating hemodynamic factors play a key role in the onset of arterial diseases, the methodology of "virtual prototyping" of branching blood vessels was applied to diseased external carotid artery (ECA) segments. The goals were to understand the underlying particle-hemodynamics and to provide various geometric design options for improved surgical reconstruction based on the minimization of critical hemodynamic wall parameters (HWPs). First, a representative carotid artery bifurcation (CAB) and then CABs with stenosed ECAs, i.e., a distally occluded ECA and an ECA stump, were analyzed based on transient three-dimensional blood flow solutions, employing a user-enhanced commercial finite volume code. Specifically, the HWPs, i.e., oscillatory shear index, wall shear stress angle gradient, near-wall residence time of monocytes, and near-wall helicity angle difference were evaluated to compare the merits of each design option, including a reconstructed near-optimal junction which generates the lowest HWP-values. The results provide physical insight to the biofluid dynamics of branching blood vessels and guide vascular surgeons as well as stent manufacturers towards interventions leading to high sustained patency rates.