Feiyu Chen^{1}, Joseph Y Cheng^{2}, Tao Zhang^{3}, John M Pauly^{1}, and Shreyas S Vasanawala^{2}

The motion-robustness of a 3D wave-encoded SPGR sequence was evaluated by simulating the acquisition of a Gaussian-profile object with periodic motion. Compared with non-wave-encoded sampling, wave-encoding provides better motion property because of wider diffusion of motion artifacts. A motion-correction method was also proposed for wave-encoding based on 3D translational motion estimates. This motion-correction method is demonstrated to effectively reduce motion artifacts in wave-encoded scans.

*Evaluation
of motion-robustness of wave-encoding*

A 3D wave-encoded SPGR acquisition (Fig. 1a) was simulated to
evaluate theoretically the motion property of wave-encoding. The wave-encoding
gradient parameters used for simulation were 8mT/m amplitude and 7 cycles of
sinusoids. A Cartesian acquisition was also simulated for comparison. An object
with a Gaussian profile (Fig. 1b) was simulated with periodic motion in the
frequency-encoding (FE) direction (Fig. 1c). VDRad^{3} Cartesian sampling
ordering was used in the simulation. Images were obtained through inverse Fourier
transform of the simulated fully-sampled k-space with simulated motion, and off-peak
motion artifacts were estimated based on the cross-sectional images.

*Motion correction
of wave-encoded k-space*

As demonstrated previously^{1}, the wave-encoded signal
and the Cartesian signal have the following relation in the *k*_{x}-*y*-*z* domain:
$$S_\text{wave}[k_x,y,z] =PSF[k_x,y,z]\cdot S_\text{Cartesian}[k_x,y,z]$$
and $$PSF[k_x,y,z] = \exp(-i\gamma\int_0^{t[k_x]}(g_y(\tau)\cdot y + g_z(\tau)\cdot z)d\tau)=\exp(C_1\cdot y + C_2\cdot z)$$ where *C*_{1} and *C*_{2} are constants associated with gradients *g*_{y} and *g*_{z}, and y and z are positions in image space.

Linear-phase correction based on
3D translational motion estimates is proposed to correct each readout of the
motion-corrupted wave-encoded k-space respectively. The relation between the
motion-corrupted wave-encoded k-space signal *S*_{m} and the motion-corrected
wave-encoded k-space signal *S*_{0} for a time point *n* can be expressed as: $$S_0[n]=S_m[n]\exp(-i2\pi(k_{x}[n]d_{x}[n]+k_y[n]d_{y}[n]+k_z[n]d_{z}[n]))\cdot\exp(C_1d_{y}[n]+C_2d_{z}[n])$$ where *d*_{x}, *d*_{y},
and *d*_{z} are the estimated
distance of linear translation in three orthogonal directions, and $$$\exp(C_1d_{y}[n]+C_2d_{z}[n])$$$ compensates for phase shifts due to linear
translation of the wave-encoding PSF.
To evaluate the performance of the derived
motion-correction equation, one phantom study and two volunteer studies were
conducted with Institutional Review Board approval and informed consent. Butterfly
navigators^{4} were used to estimate the distance of translational
motion, and VDRad sampling ordering^{3} was used in both scans. The
imaging parameters for phantom study and volunteer scans were shown in Fig. 2. In
volunteer scans, 8mT/m wave-encoding gradients with 7 cycles of sinusoids were
used. Auto-calibrated estimation of coil sensitivity maps via ESPIRiT^{5},
and CS-SENSE image reconstruction^{6} with L1-wavelet regularization
were incorporated to reduce aliasing artifacts. Prior to reconstruction,
acquired signals of each readout were corrected with the proposed
motion-correction equation according to the corresponding motion estimates. Two
free-breathing volunteers were scanned on a 3T scanner (GE MR750, Waukesha, WI)
using a 32-channel torso coil (NeoCoil, Pewaukee, WI) with FE in S/I.
Conventional Cartesian acquisitions with the same sampling pattern, scan time, and
reconstruction framework were performed for comparison in the volunteer scans.

As shown in Fig. 3, wave-encoding diffuses
motion artifacts more widely than conventional Cartesian acquisitions in the 3D
space in the simulation. It also yielded a better object profile with fewer
surrounding artifacts in the PE_{y}-PE_{z} plane (Fig. 3a-c). The
average magnitude of off-peak motion-corrupted pixels was 6.4% of the peak
magnitude in the Cartesian case, and 3.0% in the wave-encoded case. In the 1D
cross-section plots (Fig. 3d), wave-encoding yielded lower motion-induced side-lobes
than the Cartesian approach.

In the phantom study, the derived motion-correction equation reduced blurring from the original images without motion correction (Fig. 4). In volunteer scans, wave-encoding yielded slightly better images with less blurring than Cartesian acquisitions without motion-correction (Fig. 5). Better structural delineation and less motion artifacts were observed using the proposed motion-corrected wave-encoding approach (Fig. 5), compared with the conventional motion-corrected Cartesian approach.

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[2] Chen F, Zhang T, Cheng JY, Pauly JM, and Vasanawala SS. Auto-Calibrating Wave-CS for Motion-Robust Accelerated MRI. Proc. Intl. Soc. Mag. Reson. Med. 2016; 24: 1857.

[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] Cheng JY, Alley MT, Cunningham CH, Vasanawala SS, Pauly JM, Lustig M. Nonrigid motion correction in 3D using autofocusing with localized linear translations. Magnetic resonance in medicine 2012; 68:1785-1797.

[5] Uecker M, Lai P, Murphy MJ, Virtue P, Elad M, Pauly J, Vasanawala SS, Lustig M. ESPIRiT - An Eigenvalue Approach to Autocalibrating Parallel MRI: Where SENSE meets GRAPPA. Magn Reson Med 2014; 71:990-1001.

[6] Curtis A, Bilgic B, Setsompop K, Menon RS, Anand CK. Wave-CS: Combining wave encoding and compressed sensing. Proc. Intl. Soc. Mag. Reson. Med. 2015; 23: 0082.