The Circle of Willis is a redundant network of blood vessels that perfuses the brain. The ring-like anatomy mitigates the negative effects of stroke by activating collateral pathways that help maintain physiological perfusion. Previous studies have investigated the activation of these pathways during embolic stroke and internal carotid artery occlusion. However, the role of collateral pathways during cerebral vasospasm - an involuntary constriction of blood vessels after subarachnoid hemorrhage - is not well-documented. This study presents a novel technique to create patient-specific computational fluid dynamics simulations of the Circle of Willis before and during vasospasm. Computed tomographic angiography scans are segmented to model the vasculature, and transcranial Doppler ultrasound measurements of blood flow velocity are applied as boundary conditions. Bayesian analysis leverages information about the uncertainty in the measurements of vessel diameters and velocities to find an optimized parameter set that satisfies mass conservation and that is applied in the final simulation. With this optimized parameter set, the diameters, velocities, and flow rates fall within typical literature values. Virtual angiograms modeled using passive scalar transport agree closely with clinical angiography. A sensitivity analysis quantifies the changes in collateral flow rates with respect to changes in the inlet and outlet flow rates. This analysis can be applied in the future to a cohort of patients to investigate the relationship between the location and severity of vasospasm, the patient-to-patient anatomical variability in the Circle of Willis, and the activation of collateral pathways.