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

Javier Bisbal; Julio Sotelo; Hernán Mella; Oliver Welin Odeback; Joaquín Mura; David Marlevi; Junya Matsuda; Kotomi Iwata

cs.LGMar 31, 2026

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

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 & 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 ( $\sigma$ )	$\{0.05,0.1,1.0,3.0\}$	$1.0$
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 $\mathbf{\Phi}$ 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.

Imaging parameter	Value
VENC (m/s)	$2.5$ , $1.0$ , $0.7$
Matrix size	$120\times 30\times 50$
Voxel size (mm 3 )	$2\times 2\times 2$
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 $\rightarrow$ 0.038	0.044 $\rightarrow$ 0.045	0.041 $\rightarrow$ 0.042
	DE	0.079 $\rightarrow$ 0.079	0.100 $\rightarrow$ 0.101	0.082 $\rightarrow$ 0.108
	Div. (1/s)	0.012 $\rightarrow$ 0.010	0.020 $\rightarrow$ 0.013	0.016 $\rightarrow$ 0.022
25.43 dB	VelNRMSE	0.039 $\rightarrow$ 0.039	0.048 $\rightarrow$ 0.047	0.043 $\rightarrow$ 0.047
	DE	0.105 $\rightarrow$ 0.101	0.154 $\rightarrow$ 0.143	0.117 $\rightarrow$ 0.175
	Div. (1/s)	0.013 $\rightarrow$ 0.011	0.029 $\rightarrow$ 0.019	0.020 $\rightarrow$ 0.035

Table 3: Sensitivity to segmentation: effect of mask dilation on denoising metrics. Results are reported as original-mask

\rightarrow

dilated-mask.

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.

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 (

\text{PVNR}\approx 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	$\mathbf{0.050}$	$0.057$
	DE	$\mathbf{0.060}$	$\mathbf{0.060}$
	Divergence (1/s)	$\mathbf{0.013}$	$0.015$
	Wrapped voxels (%)	$\mathbf{5.280}$	$6.303$
24.32 dB	VelNRMSE	$\mathbf{0.050}$	$0.059$
	DE	$\mathbf{0.067}$	$0.075$

Table 5: Combined analysis (enhancement + unwrapping) at

\mathrm{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 $\sigma$ 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).

Flow (L/min)	HNCM				HOCM
Flow (L/min)	Original	DAF-FlowNet	GC3D+DFW	DFW	Original	DAF-FlowNet	GC3D+DFW	DFW
RPA	$2.42\pm 1.07$	$2.37\pm 1.10$	$2.39\pm 1.06$	$2.39\pm 1.06$	$2.65\pm 0.69$	$2.57\pm 0.71$	$2.57\pm 0.69$	$2.61\pm 0.65$
PA	$3.68\pm 1.66$	$3.64\pm 1.60$	$3.66\pm 1.65$	$3.65\pm 1.65$	$4.26\pm 1.04$	$4.29\pm 1.04$	$4.23\pm 1.00$	$4.22\pm 1.03$
LPA	$1.27\pm 0.58$	$1.28\pm 0.64$	$1.23\pm 0.59$	$1.23\pm 0.59$	$1.63\pm 0.58$	$1.68\pm 0.53$	$1.56\pm 0.57$	$1.56\pm 0.57$
AAoD	$3.92\pm 0.86$	$3.84\pm 0.86$	$3.89\pm 0.85$	$3.89\pm 0.85$	$4.72\pm 1.41$	$4.51\pm 1.41$	$4.60\pm 1.37$	$4.63\pm 1.42$
AAoP	$3.89\pm 1.03$	$3.75\pm 0.91$	$3.83\pm 1.01$	$3.83\pm 1.01$	$4.72\pm 1.25$	$4.54\pm 1.29$	$4.74\pm 1.23$	$4.73\pm 1.23$

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.

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Sep 29, 2026J/K move · Enter open · S save

Joint Supervised and Self-Supervised Training with Acquisition-Robust Techniques for Accelerated 4D Flow MRI Reconstruction

Mengyuan Xue, Bochun Mei

Department of Radiology and Biomedical Imaging, University of California, San Francisco, San Francisco, CA, USA

4D Reconstruction Accelerated MRI Reconstruction Magnetic Resonance Imaging Cardiac MRI

Open paper AskPDF arXiv

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^\circ

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.

Abstract

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Joint Supervised and Self-Supervised Training with Acquisition-Robust Techniques for Accelerated 4D Flow MRI Reconstruction

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