Jong Bum Son^{1}, Colleen Costelloe^{2}, Tao Zhang^{1,3}, and Jingfei Ma^{1}

ESPIRiT is a hybrid-domain parallel imaging method which can estimate the coil-sensitivity information from the k-space calibration matrix. In ESPIRiT, the calibration matrix is constructed by sliding a window through the fully sampled data region of auto-calibrating signals. Presently, the kernel size of the sliding window determining the performance of ESPIRiT reconstruction is empirically chosen, even though an optimal value may vary depending on a combination of scan parameters and scan configurations. In this work, we developed an automatic data-driven method for determining an optimal kernel size in ESPIRiT to reduce the performance variation of ESPIRiT reconstructions.

**INTRODUCTION**

ESPIRiT is a hybrid-domain parallel imaging method that
has the combined advantages of SENSE (which requires explicit coil sensitivities) and GRAPPA (which is based on local kernels in k-space that
encode the coil sensitivities).^{1-3} In ESPIRiT, a
coil calibration matrix is constructed by sliding a window through the fully
sampled data region of auto-calibrating signals (ACS). The calibration matrix intrinsically
has a null space and the k-space local kernels can be determined from the
vectors supporting the null space.^{1} The accuracy in determining the
calibration matrix and k-space local kernels, and therefore the ultimate performance
of ESPIRiT reconstruction may be dependent on the kernel size of the sliding
window. Presently, the actual kernel size used is empirically chosen, even
though an optimal value may vary depending on a combination of various scan
parameters and scan configurations. In this work, we developed an automatic data-driven method for determining an optimal kernel size of the sliding window in ESPIRiT.

**METHODS**

In ESPIRiT, the
kernel size of the sliding window used in determining the calibration matrix is
selected empirically and fixed independent of scan parameters and scan
configurations.^{1} Once selected, the kernel size determines the
number of the weighting coefficients that are used for k-space calibration (N_{Kernel}∙N_{Kernel}∙N_{Coil}), and the number of equations that can be
composed for calculating the weighting coefficients (ACS_{kx}−N_{Kernel}+1)∙(ACS_{ky}−N_{Kernel}+1) where ACS_{kx} and ACS_{ky}
are the numbers of ACS lines in k_{x} and k_{y}, N_{Kernel}
is the kernel size, and N_{Coil} is the number of coils. Increasing the
kernel size helps to estimate the calibration matrix by including more neighbors
in k-space and increasing the number of weighting coefficients. On the other
hand, fewer number of composed equations becomes available, which limits
training accuracy in the required Singular-Value Decomposition (SVD) process.

In the proposed
method for determining an optimal kernel size, we first determine a series of
calibration matrices with different kernel sizes. We then subsample the fully
sampled ACS data regions using the same subsampling factor(R) and pattern that are
used in acquiring the outer k-space regions. The missing k-space lines in the
ACS region are reconstructed according to the ESPIRiT algorithm using the
calibration matrix from each kernel size. These reconstructed k-space lines in
the ACS region are compared with the actually acquired k-space lines to help select
an optimal kernel size. In our method, a simple k-space mean squared error (kMSE)
was used as the quality index in selecting the kernel size. The final images
were reconstructed using the calibration matrix with the selected kernel size
and the Cartesian conjugate gradient ESPIRiT.^{4}

We evaluated
the proposed method with in vivo axial abdomen images that were acquired using
31 channels of a body array coil on a 3.0T whole-body wide bore scanner (GE
Healthcare, Waukesha, Wisconsin, USA). Fully sampled k-space data were acquired
using a dual-echo FLEX pulse sequence with the following scan parameters: TR/TE1/TE2=4056/1180/2380 us, N_{FE}/N_{PE}=256/130, FOV=50x40 cm, RBW=±142.86 kHz, N_{Slice}=112, slice-thickness/slice-gap=4/-2 mm,
and scan-time=22 secs. The proposed algorithm was implemented in MATLAB (MathWorks,
Natick, Massachusetts, USA) and an in-phase echo image was selected from a
slice, then decimated in k-space to simulate subsampled k-space data (R=2, ACS_{kx}=256, and ACS_{ky}=24).

**RESULTS**

**DISCUSSION**

1. Uecker M, Lai P, Murphy MJ, Virtue P, Elad M, Pauly JM, Vasanawala SS, and Lustig M. ESPIRiT - An Eigenvalue Approach to Autocalibrating Parallel MRI: Where SENSE meets GRAPPA. Magn Reson Med. 2014;71(3):990-1001.

2. Pruessmann KP, Weiger M, Scheidegger MB, and Boesiger P. SENSE: Sensitivity encoding for fast MRI. Magn Reson Mad. 1999;42:952-962.

3. Griswold MA, Jakob PM, Heidemann RM, Nittka M, Jellus V, Wang J, Kiefer B, and Hasse A. Generalized autocalibrating partially parallel acquisitions (GRAPPA). Magn Reson Med. 2002;47:1202-1210.

4. Uecker M, Virtue P, Vasanawala SS, and Lustig M. ESPIRiT Reconstruction Using Soft SENSE. Proc Intl Soc Mag Reson Med. 2013:127.

Fig. 1. SVD
coil-correlation estimation using different kernels varying from 2 x 2 to 24 x
24. Only the first 50 largest singular values are displayed.

Fig. 2. EVD
coil-sensitivity estimation and ESPIRiT reconstructions for "3 x 3", "9 x 9", and "15 x 15" kernel sizes. (a) The last two Eigen-value
images acquired from the EVD coil-sensitivity estimation, (b) ESPIRiT
reconstructions from the subsampled k-space (R = 2) and error-maps
compared to the fully sampled k-space reference, and (c) ESPIRiT
reconstructions from the subsampled ACS (R = 2) and error-maps compared to
the fully sampled ACS reference. The signal intensity of error-maps is magnified
by the ten times.

Fig. 3. The
k-space mean squared error (kMSE) between ESPIRiT reconstructions from (a) the
subsampled k-space (R=2) and the fully sampled k-space reference, and (b) the
subsampled ACS (R=2) and the fully sampled ACS reference.