FlowNet: High-Speed Compressed Sensing 4D Flow MRI Image Reconstruction using Loop Unrolling
Jonas Walheim1, Valery Vishnevskiy 1, and Sebastian Kozerke1

1Institute for Biomedical Engineering, University and ETH Zurich, Zurich, Switzerland


A variational neural network for the reconstruction of compressed sensing 4D flow MRI is presented. Nine iterations of an iterative reconstruction are unfolded in a neural network which was trained using eight retrospectively undersampled datasets. A phase-invariant network architecture was designed with two types of filter operations, one with equal real and imaginary component and the other operating on image magnitude only. The method is shown to outperform spatial regularization in the Wavelet domain. A retrospectively undersampled patient scan demonstrates that the network can reconstruct pathologies based on healthy training samples. Reconstruction of prospectively undersampled 4D flow MRI shows good agreement of peak velocities and peak flow.


Scan times in 4D flow MRI are considerable, hampering the use of the method in clinical routine. With the advent of compressed sensing (CS)1 and low-rank modelling2,3 scan times have been reduced, but only at the cost of an increase in reconstruction times which require clinicians to wait several minutes up to hours for the results. Moreover, to enable high undersampling factors, modern CS methods2,4,5 employ temporal correlations which can lead to underestimation of peak velocities. Unrolling iterative reconstruction methods using neural networks6 enables immediate reconstruction of CS acquisitions once the network is trained and promises improved results over standard methods.

In this work, loop unrolling with adaptive filters was implemented and applied to 4D flow MRI. To this end, a phase-invariant network architecture was designed with two sets of filters, one independently filters real and imaginary component and the other operates on image magnitudes. Only spatial regularization was considered due to limited availability of training datasets, their high heterogeneity in terms of sequence parameters and noise.



Following6,7 we define the reconstruction network layers via unrolling$$$\;k=9\;$$$iterations of the gradient descent (GD) algorithm with$$$\;N_f=15\;$$$filters that are applied on complex-valued and magnitude images:

$$\mathbf{x}^{(k+1)}\leftarrow \mathbf{x}^{(k)}-\alpha^{(k)}\mathbf{F}^{\mathtt{H}}\text{diag}(\mathbf{m})(\mathbf{F}\mathbf{x}^{(k)}-\mathbf{y}) - \sum_{i\leq N_f}\Big[\mathbf{D}_{i,k,1}^{\mathtt{T}}\varphi_{i,k,1}(\mathbf{D}_{i,k,1}\text{Re}(\mathbf{x}^{(k)}))+ \mathbf{D}_{i,k,1}^{\mathtt{T}}\varphi_{i,k,1}(\mathbf{D}_{i,k,1}\text{Im}(\mathbf{x}^{(k)}))+\text{sgn}(\mathbf{x^{(k)}})\odot\mathbf{D}_{i,k,2}^{\mathtt{T}}\varphi_{i,k,2}(\mathbf{D}_{i,k,2}\lvert\mathbf{x}^{(k)}\rvert) \Big]$$

where $$$\mathbf{y}$$$ is the zero-filled k-space for given velocity encoding and cardiac phase, $$$\mathbf{D}_{i,j,k}$$$ are convolution matrices defined by kernels of size $$$n_p=5\times5\times5$$$, $$$\alpha^{(k)}$$$ are the GD step lengths, and $$$\varphi_{i,j,k}$$$ are potential functions parametrized by linear interpolation of $$$n_i=35$$$ reference values $$$\phi_{ijk}\in\mathbb{R}^{n_i}$$$ placed on a uniform grid. The reconstruction model is then defined by the set of parameters

$$\Theta=\left\{\alpha^k,\bf{D}_{i,j,k},\phi_{i,j,k}\right\} _{i,j,k}$$

which is tuned via stochastic optimization of weighted reconstruction errors over the training set $$$\mathcal{T}$$$ and the set of undersampling masks $$$\mathcal{M}$$$ :

$$\min_{\Theta}\mathop{\mathbb{E}}_{\{\mathbf{x}^{\text{gt}},\mathbf{y}\}\in\mathcal{T},\mathbf{m}\in\mathcal{M}}\sum_{k\leq K}\|\mathbf{x}_\Theta^{(k)}(\mathbf{y},\mathbf{m})-\mathbf{x}^\text{gt} \|_1\exp(k-K)\tag{*}$$

During initial experiments we observed that such exponential weighting improves training convergence and effectively constrains the admissible set of reconstruction architectures, thus improving generalization ability of the learned solution.

To minimize (*) we performed$$$\;5*10^{-4}\;$$$iterations of the Adam algorithm8 with learning rate$$$\;10^{-3}\;$$$and batch size 2. Kernels of$$$\;\mathbf{D}_{i,j,k}\;$$$are ensured to be zero-centered unit-norm via reparametrization.


Data processing for retrospectively and prospectively undersampled data is illustrated in Figure 1. The training set consists of flow data acquired in the aorta of 8 healthy subjects using a 3T Philips Ingenia system (Philips Healthcare, Best, the Netherlands) using either parallel imaging9 or CS10 for an accelerated acquisition. Spatial and temporal resolution were 2.5x2.5x2.5 mm3 and 25 cardiac phases and encoding velocity was set to 150 cm/s. Data were interpolated on a grid size of 113x113x21, yielding 800 3D complex-valued images which were retrospectively undersampled with 800 variable density CS sampling masks and an acceleration factor of R=6.

Another dataset of a healthy subject acquired with parallel imaging was retrospectively undersampled with acceleration factors of 4, 6 and 8 and a dataset of a patient with dilation of the ascending aorta and combined aortic stenosis and regurgitation due to a bicuspid aortic valve was retrospectively undersampled with R=4.

Prospectively undersampled flow in the aorta of a healthy volunteer was acquired with a pseudo-radial Cartesian spiral sequence and data-driven respiratory motion detection and R=4. The coil dimension was reduced to 5 channels using geometric coil compression11 and the coil channels were reconstructed separately and combined12. For comparison, 4D flow MRI was also measured and reconstructed with a standard parallel imaging protocol9.

Compressed Sensing Reconstruction

As a baseline for comparison, the data were reconstructed using standard CS reconstruction with spatial regularization and Daubechies-4 wavelets as sparse transform (CS-Wavelet).$$\widehat{x}=\min_x ||F^HF(x)-y||_2^2 + \lambda ||\Psi x||_1.$$


Reconstruction time was 0.12 seconds per 3D volume, leading to 12 seconds reconstruction time for the retrospectively undersampled data and 1 minute for the prospectively undersampled data with 5 separate coil channels.

Reconstruction results for different acceleration factors are compared in Figure 2. FlowNet consistently outperforms CS-Wavelet in terms of RMSE of magnitude and velocity magnitude. Figure 3 shows reconstruction results for the retrospectively undersampled patient dataset. FlowNet accurately depicts the flow field whereas CS-Wavelet shows artifacts and cannot preserve the jet at the inbound section of the aorta.

Reconstruction results for prospectively undersampled data are shown in Figure 4. Assessment of peak velocities and peak flow show good agreement with the reference measurement.


FlowNet provides an architecture for rapid reconstruction of accelerated 4D flow MRI. The network consistently outperforms spatial regularization with Wavelets in terms of reconstruction accuracy and time. Of note, FlowNet was able to reconstruct pathologies in patient data based on training using healthy cases. Future investigations shall focus on exploiting temporal correlations additionally thereby enabling higher acceleration factors.


FlowNet enables very rapid reconstruction of undersampled 4D Flow MRI data while outperforming standard Wavelet CS reconstruction.


No acknowledgement found.


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Figure 1: Processing pipeline for training and test data. Retrospectively undersampled data are interpolated to a grid size of $$$113\times 113 \times 21$$$, k-space data are obtained by Fourier transform and a variable density sampling pattern is applied. For prospectively undersampled data, k-Space is zero-padded to grid size $$$113\times 113 \times 21$$$ and fed into the FlowNet reconstruction.

Figure 2 Comparison of reconstruction results for FlowNet and standard CS-Wavelet for retrospective undersampling with acceleration factors of 4, 6 and 8. For all acceleration factors, FlowNet provides better reconstruction results than standard CS-Wavelet in terms of RMSE. For FlowNet reconstructions with acceleration factors of 6 and 8, details like the subclavian arteries are less pronounced in the magnitude images and velocity magnitude images show artifacts in the descending aorta whereas at an acceleration factor of 4 no artifacts are observed.

Figure 3: Reconstruction results for a retrospectively undersampled dataset of a patient with dilation of the ascending aorta and combined aortic stenosis and regurgitation due to a bicuspid aortic valve. Velocity magnitudes in systole reconstructed with FlowNet show good agreement with the ground truth data whereas CS-Wavelet shows strong artifacts. Magnitude images reconstructed with FlowNet show some loss in contrast whereas CS-Wavelet shows a high amount of residual aliasing artifacts.

Figure 4: Comparison of prospectively undersampled FlowNet reconstructions with a separate reference scan. Magnitude images accurately depict the aorta, including fine details. A bland-Altman analysis of peak velocities shows good agreement on average between the two methods and no systematic underestimation of peak velocities for FlowNet.

Proc. Intl. Soc. Mag. Reson. Med. 27 (2019)