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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, Jana Hutter
TL;DR
Highly accelerated 4D flow MRI reconstruction must preserve both anatomical magnitude and phase-derived velocity accuracy. FlowMoDL addresses this with an acceleration-conditioned, unrolled MoDL architecture combining spatiotemporal denoising, conjugate-gradient data consistency, and phase-aware deep supervision. On the multi-center CMRx4DFlow dataset, it strictly outperforms the evaluated baselines across accelerations from 10× to 50× and shows superior training efficiency under a normalized gradient-step budget.
Problem
Highly accelerated 4D flow MRI reconstruction is challenging because velocity depends on relative phase, while magnitude-focused methods can produce severe velocity deviations.
Method
FlowMoDL alternates an acceleration-conditioned (3+1)D spatiotemporal denoiser with SENSE-based conjugate-gradient data consistency and trains with a curriculum-stabilized composite loss.
Results
FlowMoDL strictly outperforms CG-SENSE, MoDL, FlowVN, and FlowMRI-Net across acceleration factors from 10× to 50×.
Takeaways & Limitations
FlowMoDL recovers anatomical structures and complex flow dynamics with one trained model while maintaining superior efficiency under a normalized gradient-step budget.
Abstract
from arXiv · showhide
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.
1 Introduction
4D flow MRI provides time-resolved anatomy and three-directional blood-flow information, but highly accelerated reconstruction is difficult because velocity depends on relative phase differences.
- Aggressive k-space undersampling is necessary because 4D flow MRI requires extensive data.
- Small phase errors can produce severe, non-physiological velocity deviations because velocity is derived from relative phase differences.
- Magnitude-focused reconstruction methods are insufficient for preserving accurate velocity fields.
- Existing flow-specific networks either use computationally heavy recurrent mechanisms or omit explicit phase optimization.
2 Method
FlowMoDL combines a learned spatiotemporal prior with SENSE-based conjugate-gradient data consistency, acceleration-conditioned modulation, and phase-aware deep supervision for 4D flow reconstruction.
- 2.1 Problem formulation: FlowMoDL reconstructs undersampled multi-coil k-space data and derives three-directional velocities from phase differences between directional and reference encodings.
- 2.1 Problem formulation: The unrolled model alternates learned spatiotemporal denoising with conjugate-gradient data-consistency updates based on the SENSE forward model.
- 2.2 FlowMoDL: Acceleration factor R conditions both denoiser features and data-consistency weighting, allowing the reconstruction coupling to adapt across acceleration rates.
- 2.2 FlowMoDL: The objective uses deep supervision with image, velocity, relative-error, and angular-error terms across cascade outputs.
- 2.2 FlowMoDL: A curriculum schedule increases phase-related loss multipliers after initially emphasizing magnitude reconstruction.
- 2.2 FlowMoDL: The implementation uses K = 8 untied cascades and J = 10 conjugate-gradient iterations per cascade.
3 Experiments
Experiments evaluate FlowMoDL on multi-center CMRx4DFlow data against classical and learned baselines under a normalized gradient-step budget. Across extreme acceleration, FlowMoDL preserves structural detail, velocity fidelity, and temporal coherence, while flow-specific competitors degrade substantially.
- Dataset and protocol: The study partitions 138 CMRx4DFlow training cases into 96 training, 21 validation, and 21 test cases while preserving center and scanner distributions.Undersampling masks are generated randomly during training to match selected acceleration factors.
- Dataset and protocol: All learned models are compared using an equivalent gradient-step budget, with evaluation based on magnitude SSIM, magnitude nRMSE, velocity relative error, and angular error.The comparison includes CG-SENSE, MoDL, FlowVN, and FlowMRI-Net.
- Reconstruction quality: At R = 50, FlowMoDL recovers sharper spatial structures and smoother velocity transitions than CG-SENSE and MoDL, which exhibit detail loss and velocity artifacts.The qualitative comparison also indicates the lowest magnitude and velocity errors for FlowMoDL.
- Temporal coherence: At R = 50, FlowMoDL achieves ρ1 ≈ 0.72 and η ≈ 0.07, compared with CG-SENSE at ρ1 ≈ 0.00 and η ≈ 0.53 and MoDL at ρ1 ≈ −0.05 and η ≈ 0.55.These temporal metrics indicate that FlowMoDL more closely matches the ground truth than the temporally noisy baselines.
4 Conclusion
FlowMoDL combines a spatiotemporal learned prior, conjugate-gradient data consistency, and acceleration-factor conditioning for highly accelerated 4D flow MRI. On CMRx4DFlow, it outperforms the evaluated baselines from 10× to 50× while recovering sharp, temporally coherent velocity fields.
- Conclusion: FlowMoDL strictly outperforms CG-SENSE, MoDL, FlowVN, and FlowMRI-Net across acceleration factors from 10× to 50×.The reported gains cover magnitude and velocity reconstruction quality under the evaluated normalized gradient-step budget.
- Conclusion: A curriculum-based composite loss explicitly incorporates velocity, relative, and angular error terms to address magnitude bias in generic MRI reconstruction.The conclusion links this phase-aware optimization to physiologically accurate velocity fields.
- Conclusion: FlowMoDL maintains sharp magnitude reconstructions and physiologically accurate, temporally coherent velocity fields even under extreme undersampling.This conclusion is supported by the reported cross-acceleration evaluation.