FlowMoDL: Model-Based Deep Learning with Conjugate-Gradient Data Consistency for Highly Accelerated 4D Flow MRI Reconstruction

Tristan Gottwald; Michelle Bruch; Mubashir-Ul Hassan; Fatma Alickovic; Milan Kloiber; Daniel Tenbrinck; Torsten Panholzer; Melanie Schaller

cs.CVAug 26, 2026

FlowMoDL: Model-Based Deep Learning with Conjugate-Gradient Data Consistency for Highly Accelerated 4D Flow MRI Reconstruction

Authors: Tristan Gottwald, Michelle Bruch, Mubashir-Ul Hassan, Fatma Alickovic, Milan Kloiber, Daniel Tenbrinck, Torsten Panholzer, Melanie Schaller, +1 more

Organizations: 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 · CAIMED, Leibniz University Hannover, Hannover, Germany · Institute of Medical Biostatistics, Epidemiology and Informatics, University Medical Center of the Johannes Gutenberg University Mainz, Germany

Abstract

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 ( $10\times$ to $50\times$ ). 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.

Figures & tables

Figure 1: Detailed architecture of a single FlowMoDL cascade

k

(top) and the internal structure of its factorized (3+1)D spatiotemporal blocks (bottom). The network maps the previous estimate

\hat{\mathbf{x}}^{(k-1)}

to an intermediate denoised target

\mathbf{z}^{(k)}

using a learned prior

\mathcal{D}_{\theta_{k}}

equipped with a global residual connection. To improve the handling of varying acceleration rates,

R

is routed to both a FiLM generator within the denoiser and an MLP computing the data-consistency penalty weight

\lambda_{k}(R)

. The cascade concludes by solving the linear data-consistency equations via conjugate-gradient iterations.

Model	nRMSE ( $\downarrow$ )	SSIM ( $\uparrow$ )	RelErr ( $\downarrow$ )	AngErr ( $\downarrow$ )
CG-SENSE [ 11 ]	0.1592	0.6831	0.7087	45.6948
FlowVN [ 15 ]	0.2311 $\pm$ 0.0061	0.4687 $\pm$ 0.0200	2.1668 $\pm$ 0.0648	79.8481 $\pm$ 0.4238
FlowMRI-Net [ 6 ]	0.2126 $\pm$ 0.0540	0.6325 $\pm$ 0.0602	0.8660 $\pm$ 0.0741	69.2204 $\pm$ 3.7776
MoDL [ 1 ]	0.1147 $\pm$ 0.0035	0.8062 $\pm$ 0.0095	0.7412 $\pm$ 0.0126	56.9912 $\pm$ 0.6201
FlowMoDL (Ours)	0.0469 $\pm$ 0.0017	0.9409 $\pm$ 0.0026	0.2664 $\pm$ 0.0065	24.5639 $\pm$ 0.2458

Table 1: Mean performance of reconstruction models on the unseen test set averaged across all acceleration factors (

R\in\{10,20,30,40,50\}

). The best results are highlighted in bold , and the second-best results are underlined .

Panel of Figure 2 — Figure 2: Metrics across different $R$ ’s. For deep learning methods, solid lines denote the mean and shaded areas represent standard deviation ( $n=5$ seeds).

Figure 3: Representative qualitative results at $R=50$ comparing CG-SENSE, MoDL, and the proposed FlowMoDL with the fully sampled reference. FlowMoDL reconstructs the volume with the lowest magnitude and velocity errors.

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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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