Malte Steinhoff^{1}, Kay Nehrke^{2}, Peter Börnert^{2,3}, and Alfred Mertins^{1}

EPI trajectories using Half-Fourier achieve shorter echo times and therefore higher SNR, which is especially desirable in low-SNR applications like diffusion-weighted MRI. For the same reason, methods enabling phase-corrected image recovery for multi-shot diffusion acquisitions have been intensively studied for both spiral and EPI trajectories. In this work, two algorithms are presented comprising both half-Fourier and the multi-shot same-magnitude constraints to exploit the advantages of both techniques. The algorithms are shown to robustly recover interleaved half-Fourier datasets from in-vivo brain acquisitions.

POCS-ICE is an
iterative SENSE-based algorithm for multi-shot spiral DWI comprising five basic
steps^{4}. First, data projection is performed in k-space. Then, shots are
reconstructed using SENSE. Next, motion-induced shot phases are estimated followed
by phase-corrected shot combination. Finally, the next shot guesses are formed
by recombining the joint image and the shot-specific phase.

Half-Fourier
techniques assume the complex image to be real-valued except for a
low-resolution phase^{6}. Appropriate reconstruction schemes therefore
comprise two main steps. Firstly, the low-resolution phase is estimated from
the fully-sampled k-space center. Secondly, missing k-space data is recovered
by conjugate symmetry. In practice, POCS-based schemes are used iteratively enforcing
data consistency in k-space and phase consistency in image space

The algorithms in this
work extend POCS-ICE by the half-Fourier phase consistency constraint. After
data projection and coil combination, half-Fourier phase projection^{7}
is performed replacing the shot phases by half-Fourier low-resolution phase estimates.
Next, conventional shot combination is performed using a 2D triangular window^{4}
with the window size set to the available range of the symmetric data. Two
algorithms were tested for the aforementioned half-Fourier phase estimation.
The structures are schematically illustrated in Figure 1.

1) *Sequential-HF POCS-ICE* does the half-Fourier
phase estimation once in advance. Firstly, the symmetric data is reconstructed by
conventional POCS-ICE^{4}, interpolated by zero-filling in k-space,
additional 2D-Hann-window-filtering adapted to the symmetric matrix size and
phase extraction (no unwrapping).

2) *Iterative-HF POCS-ICE *does the half-Fourier
phase estimation within each iteration. The shot images are therefore filtered
in k-space by 2D-Hann-window-filtering adapted to the symmetric matrix size and
subsequent phase extraction (no unwrapping).

The Cartesian EPI undersampling
properties were exploited to accelerate computations. For the HF reconstruction,
data projections are performed in hybrid
$$$x$$$-$$$k_y$$$-space replacing
2D- by 1D-FFTs^{7}. For the symmetric part, the undersampling can be
described in image-space using the point spread function^{2} and phase
ramps for the shot trajectory shifts.

The algorithms were tested in pseudo half-Fourier simulations and in-vivo. In the simulations, full-Fourier multi-shot in-vivo brain data was acquired with $$$N_{shots}=\{3, 4, 6\}$$$ and 5 slices from 4 healthy subjects. This full data was reconstructed by POCS-ICE as reference and then reduced by 5/8th for HF testing. We decided to use in-vivo data for simulating under realistic phase conditions. For in-vivo evaluation, healthy brain data was acquired with $$$N_{shots}=4$$$ and $$$HF-factors=\{0.632, 1\}$$$ with echo times $$$T_E=\{65, 98\} \, ms$$$, respectively. Measurements were performed using a 13-channel head coil (3T Philips Ingenia), $$$b=\{0, 1000\} \, \dfrac{s}{mm^2}$$$ in three directions and $$$1 \times 1 \times 4 \, mm^3$$$ resolution. Informed consent was attained according to the rules of the institution.

Python 3.6.5 was used with a 2.7 GHz Intel Core
i7 4-core CPU and 16 GB RAM. The algorithms were stopped when the residual
error^{4} of subsequent iterations dropped below 10^{-3} or the iteration
number exceeded 400. Motion-induced phase-estimation was disabled for
$$$b=0 \, \dfrac{s}{mm^2}$$$.

Normalized
root-mean-square errors (nRMSE) and durations of the pseudo half-Fourier
results are shown in Figure 2. *Sequential-HF
POCS-ICE* has slightly lower nRMSE and is faster than *Iterative-HF POCS-ICE*.

For the full-Fourier in-vivo case, pseudo
half-Fourier reconstructed diffusion images and fractional anisotropy (FA) maps
are compared to the full POCS-ICE reference in Figure 3. Figure 4 analogously shows
real half-Fourier reconstructions. *Sequential-HF*
and* Iterative-HF POCS-ICE* results
appear consistent with the reference and sharply resolve the anatomical brain structures.

Both half-Fourier algorithms successfully recover in-vivo HF datasets with increased SNR and different contrast caused by the reduced echo time. The methods reveal ventricle microstructures for $$$b_0$$$ and $$$b_y$$$ that are not visible in the fully sampled reference. This is probably caused by the constrained phase within the diffusion-problematic ventricle and, moreover, emphasized by the higher SNR. Sequential-HF outperforms Iterative-HF POCS-ICE in computational speed. The nRMSE is comparable and remains nearly stable (Figure 2a) as the reference equally suffers from higher segmentation.

In conclusion, half-Fourier and multi-shot constraints were successfully combined in two novel algorithms achieving high-quality reconstructions in simulations and in-vivo paving the way for clinical adoption.

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