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

Multi-shot diffusion-weighted imaging offers increased SNR and higher resolution, but makes the acquisition vulnerable to shot-specific phase variations and macroscopic inter-shot motion. A wide range of iterative phase-corrected reconstruction schemes have been proposed to overcome the inter-shot phase inconsistencies, but robust motion estimation is still challenging due to the inherently low SNR of DWI. This work moves forward from initial one-time rigid motion estimation to an alternating optimization balancing the joint image, phase and motion estimation. A novel multi-shot echo-planar diffusion algorithm with iterative motion and phase correction is presented in simulations and in-vivo.

Multi-shot reconstruction mutually relates the shot images $$$x_i$$$ by a consensus constraint $$$x_i=\Phi_i\Omega_i x$$$, which, in case of DWI, contains a shot-specific motion operator $$$\Omega_i$$$ and phase operator $$$\Phi_i$$$. $$$x$$$ is the joint image and $$$i$$$ the shot index. The joint functional is then given as the sum of the shot data consistencies with the data $$$d$$$, the undersampled Fourier operator $$$F$$$ and the SENSE operator $$$S$$$.

$$\underset{x, \Phi_i, \Omega_i}{arg min} \sum_i \lVert F_i S \Phi_i \Omega_i x - d_i \rVert_2^2$$

Three
algorithm variants were implemented alternately optimizing the phase, motion,
and image estimation. The basic scheme is visualized in Figure 1. As motion
correction (MC) is generally a non-convex optimization, all algorithms start
with an initial CG-SENSE^{8} shot guess. Then, shot registration and
alignment are performed, whereby the shot with the highest correlation to all
other shots is chosen as reference. Next, an optional data rejection module
follows to account for through-plane motion. Phase estimation is subsequently
performed by a 2D triangular k-space window^{4} scaled to half the
image size. From this point:

1) *MC-SENSE+CG* performs a final conjugate
gradients (CG) algorithm extending conventional SENSE+CG^{3} by the
estimated motion operator within the forward model.

2) *Iterative-MC *scheme reapplies the
shot-wise phase and motion operators and performs a gradient-based SENSE update
to recover the next shot guesses. Then, the algorithm iteratively repeats this
process until convergence.

3) *Prior-MC *scheme adapts the iterative
procedure of *Iterative-MC*, but skips
the motion estimation relying on the initial motion estimate.

Motion
estimation was achieved by a rigid preregistration based on the fast elastic
image registration^{9} using a normalized gradient field metric. Motion
transformations were performed by k-space operations^{10}. Gradient
operations were performed by image space operations regular Cartesian undersampling
properties. Data rejection was tested by shot correlation threshold after
alignment. Coil compression was performed by PCA with 99%-threshold.

The algorithms were evaluated
in BrainWeb^{11} simulations and in-vivo. For simulations, 12
2D-Gaussian sensitivities were arranged circularly around the head. Phase maps were
created^{12}, rigid motion was sampled uniformly from a range of
$$$\pm 5 \, pix$$$ and
$$$\pm 10°$$$,
$$$N_{shots}=\{2, 3, 4, 5, 6\}$$$
and
$$$SNR=\{10, 20, 30, 40\}$$$. Performance
was averaged over 10 random cases. For in-vivo evaluation, multi-shot EPI DWI brain data was attained from 5 healthy volunteers
using a 13-channel head coil, 3T Philips Ingenia,
$$$b=\{0, 1000\} \, \dfrac{s}{mm^2}$$$ with three orientations and
$$$1 \times 1 \times 4 \, mm^3$$$ resolution. Informed consent was attained according to the rules of the institution.

Computations were
performed using Python 3.6.5 on a 2.7GHz Intel Core i7 4-core CPU and 16 GB
RAM. The algorithms were stopped when the residual error^{5} of
subsequent iterations dropped below 10^{-3}
or the iteration number exceeded 200. Phase estimation was disabled for
$$$b=0 \, \dfrac{s}{mm^2}$$$
.

Normalized
root-mean-square errors and durations of the simulations are shown in Figure 2 for
varying segmentation. The time for iterative reconstruction increases
tremendously with segmentation, but *Iterative-MC
*has best performance.

Figures 3 and 4 both show 6-shot in-vivo results
with shot and global estimates. *MC-SENSE+CG
*comprises strong motion and phase artifacts, whereas* Prior-MC* shows no visible artifacts in Figure 3 but blurring due to
the heavy motion artifacts in Figure 4. *Iterative-MC*
nicely recovers all cases.

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