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

For multi-shot diffusion-weighted imaging, iterative SENSE-based algorithms like POCSMUSE are boosting SNR allowing for higher image resolution. These advantages are achieved at the cost of higher computational load, thereby narrowing the clinical use case. The Cartesian implementation of such SENSE algorithms iteratively involves time-consuming 1D-Fast Fourier Transforms. In this abstract, the well-known point spread function for regular Cartesian undersampling is exploited to accelerate gradient- and projection-based SENSE updates. Accelerations of approximately 45% were achieved. Furthermore, coil compression is evaluated for these algorithms.

Standard SENSE reconstructions
enforce data consistency between the
model and the undersampled data within an L2-norm^{1}. Two main iterative
approaches exist to minimize SENSE-based functionals. Firstly, gradient-based
methods gradually decrease the functional
by evaluating the gradient of the functional^{5}.
Secondly, projection-based methods approach a solution by enforcing data
consistency in k-space^{6}.

For Cartesian
trajectories, both approaches involve forward and backward 1D-FFTs as well as
regular k-space undersampling which
can be expressed by three image-space operations. Firstly, a phase ramp^{7} $$$\phi$$$ is
multiplied in image space to account for optional shot trajectory offsets before
undersampling. These offsets from k-space center occur, for example, in
interleaved acquisitions. Secondly, regular Cartesian undersampling is achieved
by convolution in image space $$$PSF*$$$. Third, the conjugate phase ramp
$$$\phi^H$$$ is applied to reverse the initial trajectory shift.
Hence, k-space undersampling comprising 1D-FFT $$$F$$$ and the sampling mask
$$$M$$$ can be rewritten as

$$F^HM^HMF=\phi^HPSF*\;\phi.$$

This identity allows us to reformulate gradient-based approaches like CG-SENSE by

$$grad\;J(x)=S^H\phi^HPSF*\; \phi Sx-S^H \phi^Hd,$$

where $$$d$$$ is the undersampled full-FOV image space data, $$$S$$$ the SENSE operator and $$$x$$$ the image.

Projection-based methods like POCSENSE and POCS-MUSE can be rewritten as

$$P_c x=S_cx+d_c-\phi^HPSF*\;\phi S_c x,$$

where $$$P$$$ is the projection operator and $$$c$$$ the coil index. Note that the FFTs have vanished. Furthermore, the image space-based operation requires the image matrix size to be divisible by the reduction factor.

Moreover, coil
compression (CC) can be performed to further reduce computational load^{2}.
For this purpose, the coil set is reduced using the principal component
analysis and the data is transformed to the new basis. All SENSE-related
operations scale linearly with the coil number so that computations are expected
to be accelerated by this factor. Note that this operation is not an identity
but an approximation.

For numerical evaluation, three multi-shot EPI diffusion reconstructions were implemented based on the POCSMUSE algorithm:

1) The *k-space-based* method is the direct
reimplementation of Cartesian POCSMUSE using 1D-FFTs.

2) The *PSF-based *method uses the aforementioned
image space-based projection operator.

3) The *PSF-based
+ CC* method further uses coil compression with a threshold of 99%.

The approaches were compared
in BrainWeb^{8} simulations and in-vivo. For simulations, 12
2D-Gaussian sensitivities were placed circularly around the head. Phase maps were
created^{9},
$$$N_{shots}=\{2, 3, 4, 5, 6\}$$$ 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.
Non-Cartesian EPI ramp sampling was gridded in advance to enable PSF-based
methods.

The algorithms were
executed with a 2.7GHz Intel Core i7 4-core CPU and 16 GB RAM using Python
3.6.5. Convergence was assumed when the residual error^{6} of
subsequent iterations fell below 10^{-3}
or the maximum iteration number of 300 was exceeded.

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2. Buehrer et al. Array compression for MRI with large coil arrays. MRM: An Official Journal of the ISMRM. 2007;57(6):1131-1139.

3. Samsonov AA, Kholmovski EG, Parker DL, and Johnson CR. POCSENSE: POCS-based reconstruction for sensitivity encoded magnetic resonance imaging. MRM. 2004;52(6):1397–1406.

4. Chu et al. POCS-Based Reconstruction of Multiplexed Sensitivity Encoded MRI (POCSMUSE): A General Algorithm for Reducing Motion-Related Artifacts. MRM. 2015;74:1336–1348.

5. Pruessmann et al. Advances in sensitivity encoding with arbitrary k‐space trajectories. MRM. 2001;46(4):638-651.

6. Guo et al. POCS‐enhanced inherent correction of motion‐induced phase errors (POCS‐ICE) for high‐resolution multishot diffusion MRI. MRM. 2016;75(1):169-180.

7. Nielsen T and Boernert P. Iterative Motion Compensated Reconstruction for Parallel Imaging Using an Orbital Navigator. MRM. 2011;66(5):1339–45.

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

9. Hu et al. Phase-Updated Regularized SENSE for Navigator-Free Multishot Diffusion Imaging. MRM. 2017;78(1):172-181.