Malte Steinhoff^{1,2}, Kay Nehrke^{3}, and Peter Börnert^{3,4}

Multi-shot diffusion-weighted MRI allows for higher spatial resolution than single-shot approaches, but suffers from image artifacts due to motion-induced shot-specific phase variations. Advanced parallel imaging reconstructions can mitigate these effects, but a comparison of these approaches is still missing, especially for the widespread EPI. Following the concept of reproducible research, a number of algorithms were implemented, adapted and refined for EPI, and compared in simulations and in-vivo. Results show different performance and applicability. The iterative feedback and the sharing of joint global information in the reconstruction and appropriate constraints were found to be crucial for high-quality high-resolution DWI.

Three multi-shot
DWI algorithms (*SENSE+CG*^{1}, *POCS-ICE*^{3} and the *Newton’s Method* approach^{6}) were
studied. All are estimating the motion-induced phase maps for each uniformly
under-sampled EPI shot using SENSE^{4,5}, assuming their spatial smoothness
and including them into a global reconstruction model^{3,7} as shown in
Fig. 1. Additionally, all shots are assumed to share an equal
magnitude image.

(1)* SENSE+CG*^{1} can be interpreted as a two-step
SENSE approach: first estimating phase
maps $$$\phi_\xi$$$, second estimating the image $$$\mathbf{\rho}$$$ from all data. Each shot is reconstructed in
full resolution, using SENSE, applying median filtering (9x9 kernel) and a
maximum of 12 iterations to estimate the shot-specific phases once. The global
reconstruction performed subsequently uses the phase maps as fixed input and
iterates at most 10 times.

(2) *POCS-ICE*^{3}, conversely, iterates over the
phase estimation in the global reconstruction by including the current global
image and phase maps as a guess for the next phase estimations. This iterative
feedback in *POCS-ICE* was adapted
here to EPI. The corresponding *POCS-ICE*
parameters were set according to Guo et al.^{3}.

(3) *Newton’s Method* alternately optimizes the
shot-wise and global convex functional using Newton iterations^{6}, partly
inspired by *PR-SENSE*^{8}.
In the present work, the inversion of the Hessian matrix is circumvented using
nested *Conjugate Gradient*^{6}
(CG) methods. After a global optimization, the subsequent reconstruction of shot
$$$\xi$$$ is initialized by the current shot image guess $$$\mathbf{\rho}_\xi = \phi_\xi \mathbf{\rho}$$$.
Phase maps
$$$\phi_\xi$$$ were extracted in half resolution using median filtering (kernel 9x9). Maximum
iteration numbers were set to 12 and two for the nested shot-wise and global
CGs, respectively, and the feedback was done at most 50 times.

Algorithms
were evaluated in simulations using a *BrainWeb*^{9} phantom, disturbed
by simulated phase errors^{8}, assuming eight receive coils, radially
distributed around the head. Gaussian noise was added to the coil images before
shot-specific undersampling. Simulations were executed with $$$SNR=\{5,10,15,20\}$$$ and segmentations
$$$N_{Shots}=\{2,3,4,5,6,7,8\}$$$. Performance
was averaged over ten randomly disturbed phantoms. In addition, DWI-EPI
measurements were performed in five healthy volunteers (3T Philips Ingenia). Informed
consent was attained, according to the rules of the institution. The data was
acquired using 1.0 mm² in-plane resolution, a b-value of 1000 s/mm² and between four and six segments
employing 13 receive coils.

All algorithms
were implemented in Python 3.6.0 to run on 2.66 GHz Intel Core2 Duo CPU with 4
GB RAM. Convergence
was assumed when the residual error^{3} of subsequent iterations fell
below 10^{-8} in simulations and 10^{-6} in-vivo or maximum
iteration number was reached.

The present
results for EPI are generally consistent with previous studies^{1,3} for
spiral trajectories. Unlike spiral segmentation, EPI shots, although uniformly
subsample k-space, contain different energy which influences reconstruction stability
and convergence. Therefore, iterative feedback with phase updates and same
magnitude constraint are found to be crucial for good image quality. For low
segmentation, *SENSE+CG* provides a fast
and cheap solution with comparable quality. For higher segmentation, *Newton’s Method* and *POCS-ICE* provide high-performance reconstructions, whereby *Newton’s Method* is faster and *POCS-ICE* is more robust against noise.

Although the performance of these parallel imaging driven approaches is promising, further effort is necessary to ease clinical adoption.

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9) Brainweb: Simulated Brain Database, http://www.bic.mni.mcgill.ca/brainweb/. Accessed July 27, 2017.