Accurate geometric modeling of blood vessel lumen from 3D images is crucial for vessel quantification as part of the diagnosis, treatment, and monitoring of vascular diseases. Our method, unlike other approaches which assume a circular or elliptical vessel cross-section, employs parametric B-splines combined with image formation system equations to accurately localize the highly curved lumen boundaries. This approach avoids the need for image segmentation, which may reduce the localization accuracy due to spatial discretization. We demonstrate that the model parameters can be reliably identified by a feedforward neural network which, driven by the cross-section images, predicts the parameter values many times faster than a reference least-squares (LS) model fitting algorithm. We present and discuss two example applications, modeling the lower extremities of artery-vein complexes visualized in steady-state contrast-enhanced magnetic resonance images (MRI) and the coronary arteries pictured in computed tomography angiograms (CTA). Beyond applications in medical diagnosis, blood-flow simulation and vessel-phantom design, the method can serve as a tool for automated annotation of image datasets to train machine-learning algorithms.