Unsupervised 4D Flow MRI Velocity Enhancement and Unwrapping Using Divergence-Free Neural Networks
Authors: Javier Bisbal, Julio Sotelo, Hernán Mella, Oliver Welin Odeback, Joaquín Mura, David Marlevi, Junya Matsuda, Kotomi Iwata, +3 more
Organizations: Biomedical Imaging Center, Pontificia Universidad Católica de Chile (PUC), Santiago, Chile · Department of Electrical Engineering, PUC, Santiago, Chile · Millennium Institute for Intelligent Healthcare Engineering, iHEALTH, Santiago, Chile · Departamento de Informática, Universidad Técnica Federico Santa Maria, Santiago, Chile · School of Electrical Engineering, Pontificia Universidad Católica de Valparaiso (PUCV), Valparaíso, Chile · Center for Interdisciplinary Research in Biomedicine, Biotechnology and Well-Being (CID3B), PUCV, Chile · Department of Medical Imaging and Radiation Sciences, School of Primary and Allied Health Care. Faculty of Medicine, Nursing and Health Sciences. Monash University, Melbourne, Australia · Karolinska Institute, Department of Molecular Medicine and Surgery, Stockholm, Sweden · Departamento de Ingeniería Mecánica, Universidad Técnica Federico Santa Maria, Santiago, Chile · Department of Cardiovascular Medicine, Nippon Medical School, 1-1-5 Sendagi, Bunkyo-ku, Tokyo, Japan · Department of Radiology, Nippon Medical School, 1-1-5 Sendagi, Bunkyo-ku, Tokyo, Japan
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 & tables
Hyperparameter
Grid-search values
Selected / fixed
Depth
{3,4,5,6}
5
Width
{100,200,300,400}
200
Embedding size
{64,128,256}
256
Fourier scale ( σ )
{0.05,0.1,1.0,3.0}
1.0
Learning rate
10−3 with exp. decay
fixed
Epochs
3500
fixed
Table 1: Network hyperparameters: grid-search ranges, selected optimal values, and fixed training hyperparameters are reported.
Figure 1: Overview of DAF-FlowNet: Voxel spatial coordinates are encoded using Fourier features and a multilayer perceptron predicts a vector potential Φ 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.
Imaging parameter
Value
VENC (m/s)
2.5 , 1.0 , 0.7
Matrix size
120×30×50
Voxel size (mm 3 )
2×2×2
Oversampling factor
2
Cardiac phases
29
Time spacing (ms)
32.0
Table 2: Simulation parameters for the generation of synthetic 4D flow MR images. These parameters gave an echo-time of 1.92 ms. Hardware parameters, including maximum gradient amplitude, slew rate, and ADC frequency bandwidth, which dictated sequence timings and image quality, were taken from [ 35 , 36 ] .
PVNR
Metric
DAF-FlowNet
RBF
DFW
27.18 dB
VelNRMSE
0.038 → 0.038
0.044 → 0.045
0.041 → 0.042
DE
0.079 → 0.079
0.100 → 0.101
0.082 → 0.108
Div. (1/s)
0.012 → 0.010
0.020 → 0.013
0.016 → 0.022
25.43 dB
VelNRMSE
0.039 → 0.039
0.048 → 0.047
0.043 → 0.047
DE
0.105 → 0.101
0.154 → 0.143
0.117 → 0.175
Div. (1/s)
0.013 → 0.011
0.029 → 0.019
0.020 → 0.035
Table 3: Sensitivity to segmentation: effect of mask dilation on denoising metrics. Results are reported as original-mask → dilated-mask.
Figure 2: Synthetic (no-wrap) denoising and divergence minimization benchmark ( 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 ( VENC=100 cm/s), and unwrapped outputs; bottom row indicates wrapped-voxel locations. (C) Combined velocity enhancement and unwrapping. (D) Fourier scale ( σ ) sweep analysis.
Method
VENC (cm/s)
Only wraps
Wraps + complex noise
150
100
70
150
100
70
DAF-FlowNet
0.18
5.17
38.66
0.18
4.92
37.39
Lap4D
0.65
11.14
31.97
0.76
12.18
37.81
NPRS
1.12
7.88
7.16
1.17
7.16
10.17
GC3D
0.83
6.31
6.83
0.88
6.59
8.98
Table 4: Remaining wrapped voxels (%) for each method with different VENC values with and without noise ( PVNR≈25dB ). Bold indicates best (lowest) value per column. As reference, the maximum velocity of the CFD simulation is 207 cm/s.
PVNR
Metric
DAF-FlowNet
GC3D + DFW
24.57 dB
VelNRMSE
0.050
0.057
DE
0.060
0.060
Divergence (1/s)
0.013
0.015
Wrapped voxels (%)
5.280
6.303
24.32 dB
VelNRMSE
0.050
0.059
DE
0.067
0.075
Table 5: Combined analysis (enhancement + unwrapping) at VENC=100 cm/s. Results compare DAF-FlowNet against the best sequential baseline (GC3D + DFW). PVNR was computed on the fluid region without wraps.
Figure 4: Hyperparameter sensitivity from the grid search. Relative influence of the evaluated architectural parameters on validation error, highlighting that the Fourier scale σ 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).
Flow (L/min)
HNCM
HOCM
Original
DAF-FlowNet
GC3D+DFW
DFW
Original
DAF-FlowNet
GC3D+DFW
DFW
RPA
2.42±1.07
2.37±1.10
2.39±1.06
2.39±1.06
2.65±0.69
2.57±0.71
2.57±0.69
2.61±0.65
PA
3.68±1.66
3.64±1.60
3.66±1.65
3.65±1.65
4.26±1.04
4.29±1.04
4.23±1.00
4.22±1.03
LPA
1.27±0.58
1.28±0.64
1.23±0.59
1.23±0.59
1.63±0.58
1.68±0.53
1.56±0.57
1.56±0.57
AAoD
3.92±0.86
3.84±0.86
3.89±0.85
3.89±0.85
4.72±1.41
4.51±1.41
4.60±1.37
4.63±1.42
AAoP
3.89±1.03
3.75±0.91
3.83±1.01
3.83±1.01
4.72±1.25
4.54±1.29
4.74±1.23
4.73±1.23
Table 6: In-vivo flow quantification and mass-conservation analysis for the HNCM (n=5) and HOCM (n=5) cohorts. The upper section reports plane-based mean flow rates (L/min; mean ± SD) for the original acquisition and each post-processing method. The lower section reports mass-conservation error as absolute bias (L/min) and relative bias (%) using paired aortic and pulmonary planes.
Dataset
Method
Per TF (s)
Total
Synthetic
DAF-FlowNet
94.00
00:45:26
DivCorr
62.00
00:29:58
4DFlowNet
1.03
00:00:30
RBF
5.89
00:02:51
DFW
0.23
00:00:07
Lap4D
0.140
00:00:04
Table 7: Execution times per timeframe (s) and total runtime (HH:MM:SS). Simulation has 29 timeframes (TF); in-vivo has median 18 timeframes.
4D flow MRI measures time-resolved, three-directional blood velocity but requires long acquisition times, and its diagnostic signal is carried by the phase difference between velocity encodings, not by image magnitude. Recent work has developed a per-encoding variational network to address image reconstruction in this field. In this work, we incorporate a joint supervised and self-supervised training regime and utilize both magnitude and velocity data during supervision. At the same time, we add multiple acquisition-robust and conditioning strategies based on the acceleration factors. On the CMRx4DFlow~2026 aortic dataset (1.5 and 3T), our model lowers RelErr by 38--50% and AngErr by 7.0--8.7∘ against a training-matched baseline across R=10--50, improving on every held-out subject at every acceleration. Our model also shows strong generalization ability to transfer on out-of-distribution data by employing the joint training scheme, with an increase of 7.2% in SSIM and decrease of 38% and 31% in AngErr and RelErr respectively.
Mengyuan Xue, Bochun Mei
Department of Radiology and Biomedical Imaging, University of California, San Francisco, San Francisco, CA, USA
We present FlowMoDL, an unrolled neural network for highly accelerated 4D flow MRI reconstruction that directly optimizes for both anatomical magnitude and phase-derived velocity accuracy. Building on the MoDL framework, FlowMoDL alternates a learned (3+1)D spatiotemporal denoiser with conjugate-gradient data-consistency updates based on the SENSE forward model. A novel dual-pathway conditioning scheme adapts the denoiser features and data-consistency weighting, enabling a single model to handle varying acceleration factors (10× to 50×). To ensure physiological accuracy, the network is trained using a deep-supervision composite loss that explicitly penalizes velocity magnitude and angular errors, stabilized by a curriculum schedule. We evaluate FlowMoDL on the multi-center CMRx4DFlow dataset against classical and deep-learning baselines (CG-SENSE, MoDL, FlowVN, and FlowMRI-Net). A key advantage of FlowMoDL is its superior gradient step efficiency. When evaluated under an equivalent, limited budget of gradient steps, competing flow-specific networks degrade significantly. In contrast, FlowMoDL robustly converges and strictly outperforms all competitors across all acceleration factors in magnitude SSIM, nRMSE, relative velocity error, and angular error, successfully recovering sharp structural details and temporally coherent velocity fields.
Institute of Information Processing (tnt), Leibniz University Hannover, Germany · L3S, Leibniz University Hannover, Hannover, Germany · Department of Data Science, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany +2
Four-dimensional (4D) flow magnetic resonance imaging (MRI) is a powerful non-invasive technique for visualizing and quantifying complex blood flow patterns in vivo. Despite its clinical promise, broader adoption is limited by low spatial resolution and sensitivity to noise, which restrict accurate assessment of critical hemodynamic biomarkers such as wall shear stress, pressure gradients, and turbulent kinetic energy. To overcome these challenges, we propose a deep learning-based super-resolution framework that integrates multi-scale feature extraction and attention mechanisms to enhance the quality of 4D flow MRI data. The model was trained on a dataset of 120 patients with 240 stenosed carotid arteries. High-resolution ground truth data were generated using patient-specific computational fluid dynamics (CFD) simulations based on segmented vascular geometries and physiologically realistic boundary conditions, and the resulting velocity fields served as targets for supervised learning. The proposed architecture uses convolutional block attention modules (CBAM) to guide the network toward clinically relevant spatial features and to suppress noise in low-resolution inputs. Quantitative results show that the attention-guided model substantially reduces the root mean square error (RMSE) compared with a baseline model without attention, and qualitative velocity contour analysis confirms improved reconstruction of intricate flow patterns. These findings highlight the capacity of the model to restore high-fidelity flow fields under noisy conditions and support the use of deep learning to extend the clinical utility of 4D flow MRI for non-invasive hemodynamic assessment.
Ali Mokhtari, Dominik Obrist
ARTORG Center for Biomedical Engineering Research, University of Bern, Bern, Switzerland