Deep variational network for rapid 4D flow MRI reconstruction

Methods

Compressed Sensing 4D Flow Reconstruction

PC MRI encodes flow velocity $v (r, t) \in R^{3}$ at spatial location $r$ during cardiac phase $t$ ( $1 \leq t \leq N_{t}$ ) according to the following equation:

ρ_{i} (r, t) = ρ_{0} (r, t) exp (j π \frac{{(Φ v (r, t))}_{i}}{v_{enc}}),

(4)

where $v_{enc}$ is the velocity corresponding to a phase of $\pm π$ , $y_{i}, i = 0, \dots, 3$

are the encoded velocity vector components. The four-point velocity encoding matrix is given as

Φ = ⎡ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 0 & 0 & 0 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎦ .

(5)

Therefore, flow velocity $v$ can be calculated from the phase difference of reconstructed PC images $ρ_{i}$ .

Let $ρ_{i t} \in C^{N_{v}}$ be a discretized image on a $N_{x}$ $\times$ $N_{y}$ $\times$ $N_{z}$ $=$ $N_{v}$ grid corresponding to a cardiac phase $t$ and velocity encoding $i$ . Assuming Cartesian sampling on a regular $N_{1}$ $\times$ $N_{2}$ $\times$ $N_{3}$ $=$ $N_{s}$

grid, the Fourier transform

F \in C^{N_{s} \times N_{v}}

, and

N_{c}

coil sensitivity maps

W_{k} d e f = d i a g (c_{k}) \in C^{N_{v} \times N_{v}}

define the spatial encoding operator

E \in C^{N_{s} N_{c} \times N_{v}}

E ρ d e f = {[{(F W_{1} ρ)}_{1}^{T}, \dots, {(F W_{N_{c}} ρ)}_{N_{c}}^{T}]}^{T} \in C^{N_{s} N_{c}} .

(6)

Considering a single velocity-encoded image sequence, let $P \in C^{N_{v} \times N_{t}}$ and $B \in C^{N_{s} N_{c} \times N_{t}}$ be stacked column-vectors of signals $ρ$ and zero-filled k-space samples respectively, while $M \in {0, 1}^{N_{s} N_{c} \times N_{t}}$ defines the undersampling mask. Iterative image reconstruction methods seek for a maximum a posteriori (MAP) solution defined by the following optimization problem:

^PMAP=\operatornamewithlimitsargminP12∥M⊙(EP−B)∥2F+R(P),

(7)

where the regularization term $R$ enforces prior assumptions about image regularities. Herein we consider the local low-rank (LLR) regularization ¹⁴ to leverage image correlations among cardiac phases:

R_{L L R} (P) = λ_{L L R} \sum i \leq N_{p t c h} ∥ T_{i} P ∥_{*},

(8)

where $T_{i} \in {0, 1}^{p^{3} \times N_{v}}$ is the corresponding $p$ $\times$ $p$ $\times$ $p$ patch extraction operator, yielding $N_{p t c h}$ patches, and $∥ \cdot ∥_{*}$ is the nuclear norm. For LLR regularization the optimization problem (7) is convex and can be efficiently solved using operator splitting techniques such as the fast iterative shrinkage-thresholding algorithm (FISTA) ²⁰.

FlowVN Training

We employ a $K$ =10 layer VN and perform 5 $\cdot$ $10^{4}$ iterations of the ADAM algorithm (learning rate $10^{- 3}$ , $β_{1}$ $=$ $0.85$ , $β_{2}$ $=$ $0.98$ , batch size of 3) for training, during which we continually adjust $τ = i_{opt}$ $\cdot$ $10^{- 3}$ with $i_{opt}$ being the iteration number. On every layer, each 3D filter bank contains $N_{f}$ = $8$ filters of size $n_{c}$ = $5$ voxels. Activation functions $φ {.}$ are parametrized by $N_{k n t s}$ = $91$ control knots with spacing $ω$ = $0.17$ :

	$φ {h} =$	$(1 - h / ω + ⌊ h / ω ⌋) ϕ_{⌊ h / ω ⌋} +$		(9)
		$(h / ω - ⌊ h / ω ⌋) ϕ_{⌊ h / ω ⌋ + 1},$		(9)

with gradients provided by the following formulas:

		$\frac{\partial φ}{\partial ϕ_{i}} {h} = 1_{i \leq h \leq i + 1} \cdot (1 - h + ⌊ h ⌋) +$		(10)
		$1_{i - 1 \leq h \leq i} \cdot (h - ⌊ h ⌋),$
		$\frac{\partial φ}{\partial h} {h} = ϕ_{⌊ h ⌋ + 1} - ϕ_{⌊ h ⌋} .$

The acquired zero-filled k-space $B$ with undersampling mask $M$ was normalized by $\frac{∥ M ∥_{1}}{∥ B ∥_{F}}$ .

To enable backpropagation to be carried out with limited GPU memory, we employ spatio-temporal equivariance of the convolution and exploit the fact that k-space is fully sampled in the readout dimension

k_{x}

for Cartesian acquisitions. Therefore, to draw a training sample, we perform random cropping of width

w_{x}

and

w_{t}

in dimensions

x

and

t

respectively and simulate Fourier encoding in

k_{y} k_{z}

dimensions as illustrated in Fig. 2(a). The network was implemented using the Tensorflow framework ³². Fully-sampled and partial Fourier acquisition data from 11 healthy volunteers was used during training.

In Vivo Data Acquisition

As illustrated in Fig. 1, we used 11 subject for network training, and 7 healthy subjects and 1 patient for evaluation. All in-vivo work was performed upon written informed consent of the subjects and according to local ethics regulations.

Training datasets comprised 4D flow data measured in the aorta of 11 healthy subjects, 9 of them fully sampled and 2 acquired with partial Fourier ³⁸ (factor 0.75 $\times$ 0.75).

For evaluation, data in the ascending aorta of 7 healthy subjects were acquired on a 3T Philips Ingenia system (Philips Healthcare, Best, the Netherlands) using a Cartesian 4-point referenced phase-contrast gradient-echo sequence with an encoding velocity $v_{e n c} = 150 c m / s$ , a spatial resolution of $2.5$ $\times$ $2.5$ $\times$ $2.5 m m^{3}$ , $T_{E} = 3.3 m s$ , $T_{R} = 4.9 m s$ , $25$ cardiac phases and flip angle = 8 $^{\circ}$ . Exams for each of the 7 healthy subjects comprised a standard navigator-gated 2-fold accelerated parallel imaging ⁵ exam for reference, and a CS acquisition with an acceleration factor of $R$ = $12.4 - 13.8$ , using Cartesian pseudo-radial golden angle sampling pattern ³⁹ and data driven-respiratory motion detection, as in ¹¹. Only data in expiration were kept for reconstruction as shown in Fig. 1.

To evaluate reconstruction accuracy on pathological anatomy, 4D flow data was acquired in a single patient with dilation of the ascending aorta, combined aortic stenosis and regurgitation due to a bicuspid aortic valve on a 3T Philips Ingenia system (Philips Healthcare, Best, the Netherlands) using a navigator-gated 2-fold accelerated parallel imaging ⁵ scan.

A receiver coil with 28 channels was used for acquisition which were reduced to 5 channels using coil compression ⁴⁰. Coil sensitivity maps were estimated with ESPIRiT ⁴¹. Concomitant field correction was applied to the signal phase according to Bernstein et al. ⁴² and eddy currents were corrected for with a third-order polynomial model fitted to stationary tissue ^{43, 44}.

Evaluation

We compared the proposed FlowVN to the state-of-the-art compressed sensing LLR-regularized (8) reconstruction ¹⁴ and the variational network by Hammernik et al. ¹⁵ which we refer to as HamVN. The LLR implementation from the Berkeley advanced reconstruction toolbox (BART) ⁴⁵ was used with patch size $p$ = $8$ and a maximum number of optimization iterations of 80. The optimal value of regularization parameter $λ_{L L R}$ = $2.06$ was chosen via grid search to minimize the reconstructed flow field residual $∥^v - v^{*} ∥_{2}$ averaged over the manually segmented aorta on the retrospectively 12 $\times$ undersampled acquisition. Since the original VN ¹⁵ was proposed for magnitude reconstruction of 2D and 3D data, we introduced the following modifications for the presented 4D flow evaluation: (i) 3D filters were grouped into 4 banks as in FlowVN (see Supplemental Algorithm 1), (ii) $ℓ_{1}$ -norm of reconstruction was optimized (i.e. equation (3) with $τ$ $\to$ + $\infty$ ). We refer to this architecture as “HamVN”, the number of network layers, filters and control knots were the same as in FlowVN.

Retrospective Study

For simulated retrospective undersampling experiments, we used 2 $\times$ PI data and simulate a pseudo-radial Golden angle sampling pattern ³⁹ with acceleration factors of 6 to 22.

For each undersampling factor we evaluated the nRMSE of the image magnitude, the relative error (RelErr) of velocity magnitudes inside the aorta and the angular error (AngErr) of the estimated velocity vectors:

		$nRMSE (a, a^{*}) = \sqrt{N \sum i \frac{(a_{i} - a_{i}^{⋆})^{2}}{N {max}_{j} (a_{j}^{⋆})^{2}}},$		(11)
		$RelErr (a, a^{}) = \frac{∥ a - a^{} ∥_{2}}{∥ a^{*} ∥_{2}},$
		$AngErr (u, v) = arccos (\frac{⟨ u, v ⟩}{∥ u ∥_{2} ∥ v ∥_{2}}) .$

Additionally, we report the structural similarity index (SSIM) ⁴⁶ with $σ_{SSIM}$ = $1.5$ on the reconstructed magnitude images.

Prospective Study

Using manual aorta segmentations we compute flow over cross sections of the aorta by integrating velocity components projected onto the cross section normal. The peak flow is then defined as the maximal flow over cardiac phases for a given cross section. Moreover, we calculate peak through-plane velocity defined as maximum velocity projection across cross sections of the aorta over cardiac phases.

To quantify agreement with the reference 2 $\times$ PI reconstruction, we performed Bland-Altman analysis ⁴⁷ of peak flow and peak through-plane velocities.

Data and code availability

The code for the network training and inference used in this study and network weights are avaliable on CodeOcean together with volunteer data: https://codeocean.com/capsule/0115983/tree ⁴⁸. The code for analysis is available on CodeOcean: https://codeocean.com/capsule/2587940/tree ⁴⁹.

References

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1 Supplementary Material

Input:

B \in C^{N_{s} \times N_{t}}

— zero-filled k-space samples,

M \in {0, 1}^{N_{s} \times N_{t}}

— undersampling mask

Parameters:

Θ = {ϕ_{ud}^{(k)}, ϕ_{ur%}^{(k)}, ϕ_{d}^{(k)}, ϕ_{r, i n}^{(k)}, D_{i n}^{(k)}, α^{(k)}, α^{(0)}}_{i = 1, \dots, N_{f}, k = 1, \dots, K}, n = 1, \dots, 4

¯ ¯¯¯¯ ¯ M \leftarrow \frac{1}{N_{s} N_{t}} \sum_{i \leq N_{s} j \leq N_{t}} M_{i j}

P^{(0)} \leftarrow α^{(0)} E^{H} B, S^{(0)} \leftarrow 0

for

k := 0

K - 1

S^{(k + 1)} \leftarrow α^{(k + 1)} S^{(k)} + G^{(k)}

P^{(k + 1)} \leftarrow P^{(k)} - S^{(k + 1)}

Output: reconstructed image

V (B, M; Θ) d e f = P^{(K)}

Algorithm 1 Proposed variational reconstruction network model FlowVN.

R

=6 p-val.

R

=8 p-val.

R

=10 p-val.

R

=12 p-val.

R

=14 p-val.

R

=16 p-val.

R

=18 p-val.

R

=20 p-val.

R

=22 p-val. Image Magnitude nRMSE [%] LLR HamVN FlowVN 1.9

\pm

0.3 2.3

\pm

0.2 1.7

\pm

0.2

<

0.0001 0.0017 2.0

\pm

0.3 2.5

\pm

0.2 1.9

\pm

0.2

<

0.0001 0.0076 2.2

\pm

0.3 2.8

\pm

0.3 2.2

\pm

0.2

<

0.0001 0.1418 2.5

\pm

0.3 3.0

\pm

0.3 2.4

\pm

0.2

<

0.0001 0.0256 2.8

\pm

0.3 3.3

\pm

0.3 2.6

\pm

0.3

<

0.0001

<

0.0001 3.3

\pm

0.4 3.6

\pm

0.3 2.9

\pm

0.3

<

0.0001

<

0.0001 3.6

\pm

0.4 3.8

\pm

0.4 3.0

\pm

0.3

<

0.0001

<

0.0001 3.8

\pm

0.4 4.0

\pm

0.4 3.2

\pm

0.3 0.0211

<

0.0001 4.2

\pm

0.5 4.1

\pm

0.4 3.4

\pm

0.3 0.9551

<

0.0001 Velocity Magnitude RelErr [%] LLR HamVN FlowVN 9.4

\pm

1.9 12.8

\pm

2.5 8.7

\pm

2.4

<

0.0001 0.0225 10.5

\pm

2.3 13.5

\pm

2.8 10.0

\pm

2.8

<

0.0001 0.0790 11.8

\pm

2.2 14.9

\pm

3.0 11.4

\pm

2.8

<

0.0001 0.0385 13.2

\pm

2.4 16.4

\pm

2.9 12.4

\pm

2.4

<

0.0001 0.0003 15.1

\pm

2.5 18.0

\pm

3.5 14.0

\pm

2.9

<

0.0001

<

0.0001 17.4

\pm

2.3 20.2

\pm

3.2 15.5

\pm

3.3

<

0.0001

<

0.0001 19.0

\pm

2.7 22.4

\pm

2.9 16.9

\pm

2.9

<

0.0001

<

0.0001 19.9

\pm

3.4 23.1

\pm

4.1 17.5

\pm

3.4

<

0.0001

<

0.0001 21.9

\pm

2.8 25.3

\pm

3.6 19.2

\pm

3.3

<

0.0001

<

0.0001 Mean Flow AngErr [deg] LLR HamVN FlowVN 7.0

\pm

1.8 9.6

\pm

2.6 6.9

\pm

1.9

<

0.0001 0.1956 7.9

\pm

2.0 10.5

\pm

2.9 8.0

\pm

2.3

<

0.0001 0.2228 8.9

\pm

2.2 11.5

\pm

3.1 9.2

\pm

2.5

<

0.0001 0.0204 10.2

\pm

2.7 12.7

\pm

3.4 10.3

\pm

2.8

<

0.0001 0.0225 11.3

\pm

2.9 13.6

\pm

3.8 11.4

\pm

3.2

<

0.0001 0.1166 12.2

\pm

3.1 14.3

\pm

4.0 12.3

\pm

3.4

<

0.0001 0.7785 13.3

\pm

3.4 15.3

\pm

4.2 13.0

\pm

3.6

<

0.0001 0.4799 13.7

\pm

3.7 15.5

\pm

4.4 13.6

\pm

3.9

<

0.0001 0.0073 14.8

\pm

3.6 16.1

\pm

4.5 14.3

\pm

3.8

<

0.0001 0.0005

Table 2: Comparison of reconstruction errors (

\pm

standard deviation) of 7 retrospectively undersampled acquisitions of healthy subjects for different acceleration factors

R

. For each row the best performing method is highlighted in bold. P-values are obtained by comparing error metrics distributions of LLR and corresponding VN architectures using the two-sided Wilcoxon signed-rank test.

Figure 6: Retrospective ablation study. a, Comparison of VN architectures and modifications proposed in this paper, using retrospectively undersampled data from 7 healthy volunteers for different acceleration factors $R$ . HamVN-3D performs 3D filtering of $y z t$ dimensional data (assuming that data is fully sampled in the $x$ dimension). Modifications are accumulated as presented in the legend and, eventually, yield the FlowVN configuration. Undersampling-dependent modulation improves reconstruction quality only for higher acceleration factors. Note that the layer-wise exponential weighting of the reconstruction loss does not increase model complexity and, therefore, does not improve reconstruction accuracy for retrospective acceleration simulations. b, Velocity magnitude error maps for $R$ =16 shown at systolic peak flow.

Method	Peak Velocity Error[%]	p-value	Peak Flow Error[%]	p-value
HamVN-3D, single coil	19.38 $\pm$ 19.01	$<$ 0.0001	21.37 $\pm$ 15.11	$<$ 0.0001
HamVN-3D	6.44 $\pm$ 13.64	$<$ 0.0001	3.73 $\pm$ 12.12	0.0020
HamVN	3.64 $\pm$ 10.10	$<$ 0.0001	1.90 $\pm$ 9.69	0.3345
+linear activation functions	3.18 $\pm$ 10.35	0.0002	1.65 $\pm$ 10.70	0.3833
+unroll GD with momentum	2.95 $\pm$ 9.89	0.0004	1.47 $\pm$ 9.82	0.4749
+data activation functions	2.29 $\pm$ 10.12	0.0213	0.63 $\pm$ 9.74	0.7957
+US-depending modulation	1.98 $\pm$ 8.43	0.0645	0.47 $\pm$ 10.30	0.9272
+exp. weighting (FlowVN)	1.59 $\pm$ 9.65	0.0875	0.05 $\pm$ 9.79	0.9550

Table 3: Quantitative prospective comparison of VN architectures and modifications proposed in this paper using data from 7 healthy volunteers (

12.4 \leq R \leq 13.8

). Peak velocities and peak flow are assessed over manually segmented aorta slices and reported relative to average values of the corresponding reference scan (

\pm

standard deviation). P-values are obtained by comparing error metrics distributions of reference PI data and the corresponding reconstruction method using the two-sided Wilcoxon signed-rank test. HamVN-3D performs 3D filtering of

y z t

dimensional data (assuming that data is fully sampled in the

x

dimension). “HamVN-3D, single coil” was trained to reconstruct each coil image separately and then to be combined using provided sensitivity maps. Modifications are accumulated as presented and, eventually, yield the FlowVN configuration.

Deep variational network for rapid 4D flow MRI reconstruction

Authors

Accurate quantification of blood flow wall shear stress using simulation-based imaging: a synthetic, comparative study

End-to-End Variational Networks for Accelerated MRI Reconstruction

Memory-efficient Learning for High-Dimensional MRI Reconstruction

Scale-Equivariant Unrolled Neural Networks for Data-Efficient Accelerated MRI Reconstruction

Deep J-Sense: Accelerated MRI Reconstruction via Unrolled Alternating Optimization

Compressive MRI quantification using convex spatiotemporal priors and deep auto-encoders

Free-breathing Cardiovascular MRI Using a Plug-and-Play Method with Learned Denoiser

Methods

Compressed Sensing 4D Flow Reconstruction

FlowVN Training

In Vivo Data Acquisition

Evaluation

Retrospective Study

Prospective Study

Data and code availability

References

1 Supplementary Material

Deep variational network for rapid 4D flow MRI reconstruction

Authors

Related Research

Accurate quantification of blood flow wall shear stress using simulation-based imaging: a synthetic, comparative study

End-to-End Variational Networks for Accelerated MRI Reconstruction

Memory-efficient Learning for High-Dimensional MRI Reconstruction

Scale-Equivariant Unrolled Neural Networks for Data-Efficient Accelerated MRI Reconstruction

Deep J-Sense: Accelerated MRI Reconstruction via Unrolled Alternating Optimization

Compressive MRI quantification using convex spatiotemporal priors and deep auto-encoders

Free-breathing Cardiovascular MRI Using a Plug-and-Play Method with Learned Denoiser

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Methods

Compressed Sensing 4D Flow Reconstruction

FlowVN Training

In Vivo Data Acquisition

Evaluation

Retrospective Study

Prospective Study

Data and code availability

References

1 Supplementary Material