Zechen Zhou^{1}, Baocheng Chu^{2}, Chun Yuan^{2,3}, and Peter BĂ¶rnert^{4}

Wave encoding is an emerging approach that can take better usage of the three-dimensional (3D) spatial encoding power of multi-channel coils employed in parallel imaging (PI). In this work, a variable frequency (VF) wave encoding approach is proposed to improve the aliasing propagation property and reduce the side lobe amplitude of the transformed point spread function. This VF approach can also induce amplitude modulated wave encoding gradients to reduce eddy currents and improve the slice selection profile. The preliminary results demonstrated its improved PI performance for 3D turbo spin echo imaging over Cartesian and constant frequency wave encoding schemes.

Variable frequency
wave encoding gradient waveform design: The instantaneous frequency $$$f(t)$$$
is designed as
a CF component
modulated by
a nonlinear symmetric VF component within the data acquisition time ($$$T_{acq}$$$) such that the corkscrew shaped readout trajectory
rotates more frequently at the central k-space while behaving oppositely at the
peripheral k-space. So the sampling spacing is varying along the readout
direction which can mimic the variable density sampling to reduce the maximum
side lobes of transformed point spread function (TPSF). In this work, $$$f(t)=f_c-2{\pi}{\beta}/T_{acq}cos(2{\pi}t/T_{acq}), t\in[0,T_{acq}]$$$ is used, where
parameter $$$\beta$$$
can be further
optimized the performance and adapted to the hardware constraints including
maximum gradient strength (G_{max}) and slew rate. The VF wave encoding k-space
trajectories can be scaled by a pair of amplitude parameters ($$$A_y$$$ and $$$A_z$$$) and represented as $$$C_y(t)=A_y(1-cos(2{\pi}\int_0^tf(\tau)d\tau))$$$ and $$$C_z(t)=A_zsin(2{\pi}\int_0^tf(\tau)d\tau))$$$, which will also apply an additional amplitude
modulation for the wave encoding gradient waveform (figure 1a) to reduce eddy
currents and to achieve similar slice/slab selection profiles in comparison to
Cartesian encoding scheme.

Wave encoding PI
reconstruction and TPSF analysis: After CF and VF wave encoding MR scans,
the wave point spread function ($$$Psf$$$) and coil sensitivities can be calibrated with the
previously developed self-calibration method^{7} with different subspace models
for k-space trajectory representation, and SPIRiT^{8} based PI method can be
performed for image reconstruction. The TPSF between two voxels $$$r$$$ and $$$\rho$$$ defined by $$$TPSF(r,\rho)=\delta_r^HF_x^HPsf^HF_{yz}^HP^TPF_{yz}PsfF_x\delta_{\rho}$$$, where $$$P^TP$$$ stands for
the undersampling mask, can be further analyzed and compared between CF and VF
wave encoding schemes.

MR experiments: The fully sampled wave encoded (CF & VF)
and Cartesian encoded whole brain datasets were acquired by a 15-channel head coil on
Philips Ingenia 3.0T scanner using the 3D TSE sequence (FOV = 230x230x198mm^{3},
resolution = 1x1x2mm^{3}, TE/TR = 15ms/800ms, TSE factor = 30) with or
without wave encoding gradients (CF: G_{max}=8mT/m,
4 cycles; VF: G_{max}=11mT/m,
4 cycles). The acquired datasets were retrospectively subsampled with a 25x25
central calibration area and CAIPI undersampling pattern^{9} in peripheral
k-space.

Comparison of TPSF between CF and VF wave encoding: Figure 1b illustrates the calibrated CF and VF wave encoding k-space trajectories and demonstrates the adaptive capacity of previously developed self-calibration method (7). With the calibrated wave encoding k-space trajectories, the simulation results in figure 2 demonstrate that VF wave encoding method can improve the aliasing propagation width along the readout direction and reduce the side lobe amplitudes of TPSF.

Comparison of PI reconstruction performance: For 3x2 CAIPI in vivo brain acceleration experiment, VF wave encoding can provide better aliasing suppression results in both central and lateral slices (figure 3 and 4). In addition, VF wave encoding can significantly reduce the signal loss due to eddy current induced slice profile degradation in lateral slice (figure 4).

1. Bilgic B, Gagoski BA, Cauley SF, Fan AP, Polimeni JR, Grant PE, Wald LL, Setsompop K. Wave-CAIPI for highly accelerated 3D imaging. Magn Reson Med. 2015;73(6):2152-62.

2. Polak D, Setsompop K, Cauley SF, Gagoski BA, Bhat H, Maier F, Bachert P, Wald LL, Bilgic B. Wave-CAIPI for highly accelerated MP-RAGE imaging. Magn Reson Med. 2018 Jan;79(1):401-406.

3. Chen F, Zhang T, Cheng JY, Shi X, Pauly JM, Vasanawala SS. Autocalibrating motion-corrected wave-encoding for highly accelerated free breathing abdominal MRI. Magn Reson Med. 2017;78(5):1757-1766.

4. Gagoski BA, Bilgic B, Eichner C, Bhat H, Grant PE, Wald LL, Setsompop K. RARE/turbo spin echo imaging with Simultaneous Multislice Wave-CAIPI. Magn Reson Med. 2015;73(3):929-38.

5. Poser BA, Bilgic B, Gagoski BA, Uludag K, Stenger VA, Wald LL, Setsompop K. Echo-Planar Imaging with Wave-CAIPI Acquisition and Reconstruction. ISMRM 2017. p1198.

6. Polak D, Cauley S, Huang S, Longo M, Bilgic B, Raithel E, Wald L, Setsompop K. Highly-accelerated volumetric brain protocol using optimized Wave-CAIPI encoding. ISMRM 2018. p0937.

7. Zhou Z, Yuan C, Börnert P. Self-calibrating wave-encoded 3D turbo spin echo imaging using subspace model based autofocus. ISMRM 2018. p0939.

8. Lustig M, Pauly JM. SPIRiT: Iterative self-consistent parallel imaging reconstruction from arbitrary k-space. Magn Reson Med. 2010;64(2):457-71.

9. Breuer FA, Blaimer M, Mueller MF, Seiberlich N, Heidemann RM, Griswold MA, Jakob PM. Controlled aliasing in volumetric parallel imaging (2D CAIPIRINHA). Magn Reson Med. 2006;55(3):549-56.

Figure 1: Comparison of schematic
gradient waveform (a) and self-calibrated k-space
trajectories (b) between constant frequency (CF) and variable frequency (VF)
wave encoding schemes. The sampling spacing is varying on
VF wave encoding k-space trajectories to mimic variable density sampling, and
the additional amplitude modulation on VF wave encoding gradient waveform can
largely reduce the gradient strength at the beginning and end.

Figure 2: Comparison of transformed point spread
function (TPSF) between constant frequency (CF) and variable frequency (VF)
wave encoding schemes. (a) 2D CAIPI undersampling mask
with 2x2 acceleration and 25x25 fully sampled central area. (b) Subsampled
aliasing propagation within the phase and slice plane. (c) Comparison of TPSF
along the readout direction between CF and VF wave encoding methods at four
different locations marked as blue dots in (b).

Figure 3: Comparison of parallel imaging
(PI) reconstruction performance at 3x2 CAIPI accelerated 3D brain scan in the
central slice for different encoding schemes. Top and bottom row correspond to
the fully sampled/wave point
spread function corrected
and reconstructed central-slice images. Columns from left to right represent
Cartesian, constant frequency (CF) and variable frequency (VF) wave encoding
scheme. Note the difference of residual aliasing artifacts as shown by the red arrows at
the bottom row.

Figure 4: Comparison of parallel imaging
(PI) reconstruction performance at 3x2 CAIPI accelerated 3D brain scan in a
lateral slice for different encoding schemes. Top and bottom row correspond to
the fully sampled/wave point
spread function corrected
and reconstructed lateral-slice images. Columns from left to right represent
Cartesian, constant frequency (CF) and variable frequency (VF) wave encoding
scheme. Note the overall signal drop in the middle column as shown by the red
arrows.