Unsupervised 4D Flow MRI Velocity Enhancement and Unwrapping Using Divergence-Free Neural Networks

Abstract

This work introduces an unsupervised Divergence and Aliasing-Free neural network (DAF-FlowNet) for 4D Flow Magnetic Resonance Imaging (4D Flow MRI) that jointly enhances noisy velocity fields and corrects phase wrapping artifacts. DAF-FlowNet parameterizes velocities as the curl of a vector potential, enforcing mass conservation by construction and avoiding explicit divergence-penalty tuning. A cosine data-consistency loss enables simultaneous denoising and unwrapping from wrapped phase images. On synthetic aortic 4D Flow MRI generated from computational fluid dynamics, DAF-FlowNet achieved lower errors than existing techniques (up to 11% lower velocity normalized root mean square error, 11% lower directional error, and 44% lower divergence relative to the best-performing alternative across noise levels), with robustness to moderate segmentation perturbations. For unwrapping, at peak velocity/velocity-encoding ratios of 1.4 and 2.1, DAF-FlowNet achieved 0.18% and 5.2% residual wrapped voxels, representing reductions of 72% and 18% relative to the best alternative method, respectively. In scenarios with both noise and aliasing, the proposed single-stage formulation outperformed a state-of-the-art sequential pipeline (up to 15% lower velocity normalized root mean square error, 11% lower directional error, and 28% lower divergence). Across 10 hypertrophic cardiomyopathy patient datasets, DAF-FlowNet preserved fine-scale flow features, corrected aliased regions, and improved internal flow consistency, as indicated by reduced inter-plane flow bias in aortic and pulmonary mass-conservation analyses recommended by the 4D Flow MRI consensus guidelines. These results support DAF-FlowNet as a framework that unifies velocity enhancement and phase unwrapping to improve the reliability of cardiovascular 4D Flow MRI.

Figures (5)

• Figure 1: Overview of DAF-FlowNet: Voxel spatial coordinates are encoded using Fourier features and a multilayer perceptron predicts a vector potential $\mathbf{\Phi}$ whose curl yields divergence-free velocities via automatic differentiation. The network is trained to match the original noisy and wrapped velocities using a cosine-based loss and no-slip boundary condition, enabling simultaneous velocity enhancement and phase unwrapping.
• Figure 2: Synthetic (no-wrap) denoising and divergence minimization benchmark ($\text{VENC}=250$ cm/s). Quantitative comparison across noise levels using velocity NRMSE, direction error, and RMS divergence for DAF-FlowNet and divergence-free/physics-based alternatives.
• Figure 3: Synthetic experiments: (A) Velocity magnitude maps and absolute error maps for reference, noisy input, and denoising/divergence minimization outputs. (B) Foot–head velocity component for a simulation without wraps, wrapped velocity ($\text{VENC} = 100$ cm/s), and unwrapped outputs; bottom row indicates wrapped-voxel locations. (C) Combined velocity enhancement and unwrapping. (D) Fourier scale ($\sigma$) sweep analysis.
• Figure 4: Hyperparameter sensitivity from the grid search. Relative influence of the evaluated architectural parameters on validation error, highlighting that the Fourier scale $\sigma$ dominates performance compared with embedding size, depth, and width.
• Figure 5: In-vivo validation. (A) HNCM example: velocity magnitude (top row) and divergence maps (middle row) for Original, DFW, and DAF-FlowNet; arrows highlight fine structure smoothed by DFW and preserved by DAF-FlowNet. Bottom: absolute differences vs. Original. (B) HOCM example: velocity magnitudes for Original, GC3D + DFW, and DAF-FlowNet; arrows indicate wrapped regions corrected by GC3D and DAF-FlowNet. (C) Flow waveforms at pulmonary and aortic planes; top row corresponds to (A) and bottom row to (B).