W Scott Hoge^{1,2,3} and Jonathan R Polimeni^{2,3,4}

This work examines methods to improve the conditioning of the linear system of equations used to compute GRAPPA and Dual-Polarity GRAPPA reconstruction coefficients, and it's effect on temporal SNR in applications that employ accelerated EPI data. We test three methods: (i) system normalization, (ii) simple Tikhonov regularization, and (iii) 2D k-space filters applied to the calibration data prior to the linear system formation. Examples of tSNR improvement are shown, drawing from EPI-based in-vivo functional, diffusion, and perfusion imaging data acquired at 3T and 7T.

Accelerated parallel MR imaging (pMRI) can increase temporal efficiency and reduce geometric distortions in echo planar imaging (EPI). However, these improvements come at a cost of SNR loss due to* g-factor* effects. To date, many investigations to mitigate SNR loss in pMRI methods have focused on SENSE[1,2], perhaps in part because the SENSE linear system of equations map naturally to cost-function minimization approaches[3]. System conditioning in GRAPPA[4], however, has received less attention.

This work examines methods to mitigate SNR loss in GRAPPA-based reconstructions of accelerated 2D EPI data. We examine system normalization/preconditioning, Tikhonov regularization, and 2D filters applied to the k-space calibration data set itself, and their effect on EPI-based functional, diffusion, and perfusion imaging.

GRAPPA reconstructs under-sampled imaging data by estimating missing data from a linear combination of sampled data. This is modeled as $$K g = t,$$ with $$$t$$$ the target k-space data, $$$K$$$ an array composed of neighboring k-space data, and $$$g$$$ a coefficients vector that maps elements of $$$K$$$ to $$$t$$$.

Although $$$K$$$ is typically over-determined, the condition of the system can often be improved. One approach is preconditioning[5], $$\psi K g = \psi t ,$$ with $$$\psi$$$ a diagonal matrix with elements normalized to rows of the system matrix, $$$\psi_{ii} = 1 / \sqrt{\sum_j \left| K_{ij} \right|^2}$$$. With most of the signal energy concentrated in the center region of k-space, preconditioning can normalize the signal-energy balance across all rows of the system. A similar effect can be achieved by applying a 2D k-space filter prior to forming $$$K$$$ [6], to reduce bias from the high-signal central region.

After forming the normal equations $$K^H \psi^2 K g = K^H \psi^2 t $$ to produce a smaller system matrix and reduce computational load, Tikhonov regularization[7] can be applied which results in $$(K^H \psi^2 K + \lambda I) g = K^H \psi^2 t.$$

The effect on temporal SNR (tSNR)[8] of including the preconditioning factor $$$\psi$$$, applying k-space filters, and varying $$$\lambda$$$, in accelerated EPI data is presented below. In-plane accelerated gradient-echo BOLD-weighted EPI data (1.5 mm iso, TR=2 sec, TE=22 ms, 128x128 matrix size, 41 slices, R=3, 75 repetitions) at 7T and PASL-based perfusion-weighted (3.4x3.4x4.0 mm, TR=3.2 sec, TE=13 ms, 64x64 matrix size, 24 slices, R=2, Q2TIPS labeling (TI=1800 ms, 700 ms bolus), 52 control/label pairs) and diffusion-weighted (2.0 mm iso, TR=7.7 sec, TE=84 ms, 128x128 matrix size, R=2, 59 slices, b=1000 s/mm^{2}, 1 diffusion direction, 61 repetitions) data at 3T, respectively, were acquired in-vivo.

Conventional multi-shot EPI pre-scan calibration data was collected using temporal encoding for Dual-Polarity GRAPPA (DPG)[9] calibration. DPG reconstruction was performed to both remove Nyquist ghosts and synthesize missing data. While not shown, standard GRAPPA reconstructions followed similar trends. Calculated tSNR maps along with the corresponding condition number---the ratio of the smallest to largest singular value, σ_{min}/σ_{max}, of the modified system matrix---are shown in the Results.

Figure 1 presents tSNR maps of the BOLD-weighted EPI data. The lowest tSNR is seen where no system regularization or normalization is used, Fig. 1a. As various system conditioning methods are applied, the tSNR improves. Similarly, Figure 2, as the number of phase-encode lines centered around the echo (at k_{y}=0) that contribute to the GRAPPA linear system is increased, both the condition number and the tSNR increases.

Figure 3 illustrates the tSNR of the BOLD-weighted data after applying k-space filters prior to GRAPPA linear system formation. The applied filter is shown below the tSNR image for each case. Filtering boosts tSNR in all cases by emphasizing high-spatial-frequency k-space data, with minimal difference between the various filters.

Figure 4 illustrates tSNR maps and corresponding images for the diffusion-weighted data. Although the effect is subtle, the tSNR maps show that diffusion tSNR can be improved via algorithmic normalization and regularization.

Figure 5 illustrates that improved system conditioning can be critically important in perfusion imaging. Both tSNR and the associated perfusion-weighted images are shown, with substantial tSNR improvement as subsequent system conditioning methods are applied. Notably, with both normalization and regularization, the noise in the low-perfusion regions is reduced.

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Figure 1. Temporal SNR maps from BOLD-weighted EPI data acquired at 7T. The effect of system matrix normalization, via the preconditioning operator ψ, and Tikhonov regularization is show left-to-right. Notably, the condition number, σ_{min}/σ_{max}, of the system matrix improves as the conditioning methods are applied.

Figure 2. Temporal SNR maps from the 7T EPI BOLD-weighted data, reconstructed utilizing different amounts of k-space calibration data in the GRAPPA linear system formation. Notably, the inclusion of high-spatial-frequency data has a regularization effect.

Figure 3. tSNR maps of 7T BOLD-weighted EPI data after k-space filters have been applied. The mesh plots illustrate the filter applied prior to GRAPPA calibration for each tSNR image in the first row. Sub-figure (b) corresponds to a filter that selects 64 high-spatial-frequency k-space lines from the calibration data. Plots (c) and (d) show a narrow-notch (w=1, c=2) and wide-notch (w=16, c=32) filter, generated from Eq. 2 of [5], $$$F_2 = 1 - (1 + e^{(\sqrt{k_x^2 + k_y^2} - c )/w})^{-1} + (1 + e^{(\sqrt{ k_x^2 + k_y^2} + c)/w})^{-1}$$$,

where c and w control the cut-off frequency and the sharpness of the filter boundary.

Figure 4. tSNR maps and images from *R*=2 accelerated 3T diffusion-weighted EPI data. The top row of images show a slight tSNR boost after normalization and regularization.

Figure 5. tSNR maps and images from *R*=2 accelerated 3T PASL-based perfusion-weighted EPI data. The top row shows a dramatic boost in tSNR after normalization and regularization, which correspondingly reduces speckle noise in the low-perfusion regions in images on the second row.