Jonas Walheim^{1}, Claudio Santelli^{1}, and Sebastian Kozerke^{1}

An image reconstruction algorithm termed k-t2 ESPIRiT is proposed, which constrains respiratory motion-resolved MRI to a high-dimensional subspace spanned by receiver-coil channels, cardiac phases, and respiratory motion states. Respiratory motion resolved 4D flow MRI data is reconstructed and compared to k-t ESPIRiT reconstruction of end-expiration and a standard parallel imaging protocol. Increased reconstruction accuracy compared to k-t ESPIRiT reveals that k-t2 ESPIRiT can effectively exploit redundancies between respiratory motion states to improve scan efficiency.

**Synopsis**

Respiratory motion is a major source of artifacts in MR imaging and often leads to low scan efficiencies in combination with respiratory gating approaches. Scans which are too long for breathholds are acquired in a respiratory-gated acquisition and only data measured during end-expiration are used for the image reconstruction.

Respiratory
motion resolved magnetic resonance imaging (MRI)^{1,2,3} promises increased scan
efficiency by making use of data from all respiratory motion states. In this
work, we propose an efficient framework for motion resolved image reconstruction,
based on the extension of k-t ESPIRiT^{4} to include the respiratory
motion dimension and apply it for the reconstruction of 4D flow MRI data.

**Theory**

In
self-consistency based parallel imaging (PI) methods, data correlations are
derived from a fully sampled training region to enforce consistency in an
iterative image reconstruction procedure^{5,6}. For dynamic MRI, the
methods were extended by incorporating a spatiotemporal consistency kernel in a
higher-dimensional k-t space^{4,7,8}.

When
data are recorded with a respiratory motion dimension, extending the k-t space
to a higher dimensional k-t^{2} space was shown to improve
reconstruction accuracy^{2}. We therefore propose to
extend k-t ESPIRiT to a higher dimensional k-t2 ESPIRiT
reconstruction according to:

$$ \underset{x}{\operatorname{argmin}} ||\Omega\mathcal{F} {E}_{c,t^2}(x)-y||_2^2 +\lambda||\Psi x||_1\> .$$ where
the k-t^{2} correlations are incorporated in the encoding operator
. $$$\Omega$$$ represents the sampling pattern and $$$\mathcal{F}$$$
is the spatial Fourier transform.
$$$\lambda$$$ is the regularization weight and
$$$\Psi$$$ is the sparsifying transform. Here, $$$\Psi$$$
was set to the Fourier transform along cardiac
phases and respiratory motion states.

Following^{9}, the encoding operator $$$E_{c,t^2}$$$ is calculated via the null space
of an autocalibration matrix built from
a fully sampled center region in k-t^{2} space. However, measuring a fully sampled
k-space region for all motion states and heart phases would severely limit the
acceleration factor. Instead, a low-rank reconstruction with a relatively small
regularization weight is performed on the undersampled data to remove aliasing
noise before computing
from a k-space center region:

$$ \underset{x}{\operatorname{argmin}} ||\Omega\mathcal{F} \mathcal{S}(x)-y||_2^2 +\lambda||x||_*\> , \ X \in \mathcal{C}^{N_x \times N_{hp} N_{kv}} .$$ The procedure is illustrated in Figure 1.

*Acquisition*

* *4D
Flow data in the aortic arch of a healthy volunteer was acquired on a 3T Philips
Ingenia system (Philips Healthcare, Best, the Netherlands) using a Cartesian
four-point phase-contrast gradient-echo sequence with uniform venc of 220 cm/s and
a spatial resolution of 2.5x2.5x2.5 mm^{3}. Data were recorded with a 28 channel receiver
coil and compressed to 5 virtual channels with geometric coil compression^{10}. A pseudo-radial Cartesian sampling pattern with
Golden angle increments^{11} was used to ensure
sampling incoherency in all respiratory states and cardiac phases. The
respiratory motion signal was extracted with self-gating and coil clustering^{12}. The measured data were
binned into 6 different motion states with equal acceleration factors of
approximately 10 for each respiratory state.
As
a reference, a navigator-gated scan was acquired with SENSE^{13} with an acceleration
factor of 2 and using a 5 mm navigator window.

*
Reconstruction and Postprocessing*

Eq.
1 was solved with a projection onto convex sets algorithm^{14} in MATLAB and the low-rank reconstruction was performed with fast iterative shrinkage thresholding^{15}. For comparison, data in end
expiration were reconstructed with k-t ESPIRiT^{4}.

Coil sensitivity maps for SENSE and the low-rank reconstruction were calculated from a separate calibration scan using ESPIRiT^{9}.

Concomitant
field correction was applied to the signal phase according to^{16} and eddy currents were
corrected for with a linear model fitted to stationary tissue^{17}.

**Results**

**Discussion**

This work has provided
advances towards improved reconstruction accuracy from undersampled 4D flow MRI
data by exploiting redundancies between different motion states and thus making
the modality less dependent on respiratory motion. Respiratory motion resolved
data can be reconstructed in a higher-dimensional subspace based on an eigendecomposition
of data extracted from the k-space center. Moreover, a low-rank reconstruction
approach was successfully employed to interpolate missing samples in the
k-space center, thus making the method independent of the k-space trajectory.
In future investigations, more advanced methods will be considered for this
purpose, e.g. structured low-rank completion^{18}.

In summary, k-t^{2} ESPIRiT holds promise for accelerated 4D Flow MRI with high independence
of respiratory motion. In future investigations, the robustness of the method
will be investigated with respect to different respiratory motion patterns in
order to evaluate benefits for clinical practice.

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Figure 1: Illustration of the k-t^{2} ESPIRiT reconstruction
framework. Respiratory motion resolved 4D flow data is sparsely sampled in k-t space.
A low-rank reconstruction is performed on the non-velocity encoded segment to
estimate missing samples in the k-space center. Then k-t^{2} blocks
from the calibration region are vectorized and arranged in columns of the
autocalibration matrix $$$A$$$. A singular value decomposition provides the row space,
which spans the k-t^{2} blocks.

Figure 2: Reconstruction results for SENSE, k-t^{2}-ESPIRiT,
and k-t ESPIRiT in expiration and in end expiration. A
higher noise level can be observed for k-t ESPIRiT compared to k-t^{2} ESPIRiT
(marked by arrows).

Figure 3: Comparison of velocity components during systole in two selected
slices in the ascending aorta. Some locations where deviations of k-t ESPIRiT
from the reference measurement can be observed are marked by arrows.

Figure 4: Example results for respiratory motion resolved 4D flow MRI
reconstructed with k-t2 ESPIRiT.