Auto-Calibrating Wave-CS for Motion-Robust Accelerated MRI

Feiyu Chen^{1}, Tao Zhang^{1,2}, Joseph Y. Cheng^{1,2}, John M. Pauly^{1}, and Shreyas S. Vasanawala^{2}

The
concept is to use the wave-encoded center k-space and the known
point-spread-function (PSF) of wave-encoding to reconstruct a Cartesian central
k-space for calibration. In detail, the wave-modulated signal $$$S_{wave}$$$ and the Cartesian
signal $$$S_{Cartesian}$$$ have the following relationship in the kx-y-z domain^{1}:$$S_{wave}[k_x,y,z]=PSF[k_x,y,z]\cdot S_{Cartesian}[k_x,y,z]$$The
PSF can be expressed as: $$\begin{aligned} PSF[k_x,y,z] &=\exp(-i\gamma\int g_y(\tau)y+g_z(\tau)z d\tau) \\ &=\exp(C_1\cdot y+C_2\cdot z) \\ &=\exp(C_1\cdot \Delta y \cdot i_y+C_2\cdot \Delta z \cdot i_z)\end{aligned} $$ where $$$C_1$$$ and $$$C_2$$$ are
constants associated with gradients $$$g_y$$$ and $$$g_z$$$, $$$y$$$ and $$$z$$$ are
positions in image space, $$$\Delta y$$$ and $$$\Delta z$$$ are
the corresponding spatial resolutions, and $$$i_y$$$ and $$$i_z$$$ are
spatial indices. The wave-encoded central k-space can be treated as an
independent low-resolution k-space modulated by the same PSF with lower
resolution. Thus, by replacing $$$\Delta y$$$ and $$$\Delta z$$$ with $$$\Delta y_{calib} = \frac{N_y}{N_{y, calib}}\cdot \Delta y$$$ and $$$\Delta z_{calib} = \frac{N_z}{N_{z, calib}}\cdot \Delta z$$$, where $$$N_y$$$ and $$$N_z$$$
are
the acquisition matrix size, $$$N_{y, calib}$$$ and $$$N_{z, calib}$$$ are the calibration matrix size, and $$$\Delta y_{calib}$$$ and $$$\Delta z_{calib}$$$ are the spatial resolution of the calibration k-space, we can obtain a set of low-resolution PSF
(Fig.1b), and subsequently reconstruct the Cartesian central k-space using
inverse Fourier transform of the PSF. The resulted Cartesian center can then be
used for calibration.

To test the accuracy of this approach, five
phantom experiments with both fully sampled Cartesian acquisition and
wave-encoded acquisition were conducted. The normalized root-mean-square-error
(RMSE) between low-resolution images from the calibration data in Cartesian
acquisition and those reconstructed by the proposed method were calculated as
an indicator of calibration accuracy (Fig. 2). Two volunteer scans (Figs. 3 and
4) were acquired on a 3T GE MR750 scanner (GE Healthcare, Waukesha,
WI) with wave encoding (3 cycles of sinusoids, 4mT/m amplitude) using a
32-channel cardiac coil (Invivo Corp., Gainesville, FL) and a 32-channel torso
array coil (NeoCoil, Pewaukee, WI), respectively. In the second scan, spatial
selective excitations in frequency-encoding directions were used to maintain a
reasonable over-sampling factor of ~1.5. Down sampling using a VDRad trajectory^{3} was simulated (Fig. 3) and implemented (Fig. 4) in a 3D SPGR sequence with a 16×16
calibration region at a reduction factor of 5.1 and 6.2, respectively. Sensitivity
maps were estimated using ESPIRiT^{4} from Cartesian acquisition directly and from
wave-encoded acquisition with the proposed method. CS-SENSE reconstruction using
the same sampling pattern was implemented for the auto-calibrating method and compared
with the conventional Wave-CS method. Acquisition parameters for the 3D SPGR sequence
were: TR/TE 12/2.2ms, FA 15°, and BW $$$\pm$$$125kHz. Acquisition matrices were 308(k_{x})×128(k_{y})×128(k_{z})
(with partial readout factor 0.6) for FOV 420mm (R/L)×210mm (A/P)×210mm (S/I), and
256(k_{x})×256(k_{y})×64(k_{z}) for FOV 400mm (S/I)×400mm (R/L)×256mm (A/P).

1. Bilgic B, et al. Wave-CAIPI for highly accelerated 3D imaging. Magnetic Resonance in Medicine. 2015, 73(6): 2152-2162.

2. Curtis A, et al. Wave-CS: Combining wave encoding and compressed sensing. Proc. Intl. Soc. Mag. Reson. Med. 2015, 23: 0082.

3. Cheng JY, et al. Free-breathing pediatric MRI with nonrigid motion correction and acceleration, Journal of Magnetic Resonance Imaging. 2014, 42(2): 407-420.

4. Uecker M, et al. ESPIRiT-an eigenvalue approach to autocalibrating parallel MRI: where SENSE meets GRAPPA. Magnetic Resonance in Medicine. 2014, 71(3): 990-1001.

Fig. 1
(a) Phase of the full-size PSF displayed in the y-z plane in [-π, π]
for *k*_{x} = 2, 78, 154, 230, and 306; (b) Phase of the corresponding
low-resolution PSF for the 16(*y*)×16(*z*) calibration region.

Fig. 2
Normalized RMSE for low-resolution calibrating data decreases while size of
calibration kspace increases, as shown in: (a) simulated wave-encoded kspace
with an accurate PSF, and (b) five phantom experiments with estimated PSFs.

Fig. 3 Comparison of reconstructed images using full acquisition
(a), Wave-CS (b), and the auto-calibrating Wave-CS (c) during free breathing scans
at a reduction factor of 5.1. 1.5x zoomed-in images are shown in the second
row. Yellow arrows point the major difference between these reconstructions. The VDRad sampling mask with a 16×16 calibration region is shown in (d).

Fig. 4 Comparison of reconstructed images using conventional
Wave-CS (a) and auto-calibrating Wave-CS (b) during breath-held scans at a
reduction factor of 6.2. Conventional Wave-CS (a) uses a separated breath-held
calibration scan to achieve the coil sensitivity maps. 1.5x zoomed-in images
are shown at the bottom right corners. Yellow arrows point the major difference
between these reconstructions. The VDRad sampling mask with a 16×16 calibration region is shown in (c).

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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