Shengzhen Tao^{1}, Yunhong Shu^{1}, Joshua D Trzasko^{1}, Paul T Weavers^{1}, Erin M Gray^{1}, John Huston III^{1}, and Matt A Bernstein^{1}

The 3D Shells trajectory-based MRI acquisition is a non-Cartesian acquisition technique that divides the 3D k-space into a series of concentric shells and samples each one with 3D helical readouts. Using the Shells trajectory, the inner k-space can be efficiently sampled within several interleaves, making it a maximally centric 3D acquisition. Partial Fourier (PF) acquisition is a commonly-used acceleration technique by exploiting the conjugate symmetry of k-space measurement. In this work, we present a new asymmetric 3D Shells trajectory design with PF acceleration to combine the advantages from both techniques, and develop a non-iterative homodyne reconstruction framework for it.

The 3D k-space shells with a radius of $$$k_r(n)=n\Delta$$$$$$k_r$$$ can be sampled using a series of 3D helical readouts expressed as: $$$k_z=k_r(n)\cos(\pi\tau/T)$$$, $$$k_x=k_r(n)\sin(\pi\tau/T)\cos(\omega_n\tau/T+\theta_{n,m})$$$, $$$k_y=k_r(n)\sin(\pi\tau/T)\cos(\omega_n\tau/T+\theta_{n,m})$$$, where $$$T$$$ is the total number of samples per readout, $$$\tau(\kappa)$$$ indices each readout sample, $$$\theta_{n,m}=\frac{2m\pi}{M(n)}$$$ is the initial phase of a interleave, $$$M(n)$$$ is the total interleave number, $$$m\leq$$$$$$M$$$ indices different interleaves, $$$\omega_n$$$ controls the rotational speed in k-space azimuthal direction. The gradient waveforms of each readout/interleave can be generated via a time-optimal waveform design strategy described by Lustig^{9}. The gradient amplitude and slew rate limits are chosen to conform to both hardware limits and Nyquist sampling requirements^{10}. The $$$\omega_n$$$ is iteratively determined so that each readout contains $$$T$$$ samples. The number of interleaves for each shell ($$$M$$$) is then calculated by constraining the maximal k-space distance (i.e., on the equator of each shell) between two adjacent interleaves to satisfy the Nyquist requirement, which yields $$$M=\lceil2\pi n\sin(\arctan(\pi/\omega_n))\rceil$$$. For a fully-sampled shells acquisition, $$$\tau\leq$$$$$$T$$$, and the trajectory samples the entire shell along z-direction (i.e. from $$$k_z=N\Delta k_r$$$ to $$$k_z=-N\Delta k_r$$$). To achieve PF acceleration, $$$\tau$$$ can be modified as $$$\tau=T/\pi\arccos((1-2\kappa)N/n)$$$, so that the trajectory only samples from $$$k_z=N\Delta$$$$$$k_r$$$ to $$$k_z=(1-2\kappa)N\Delta$$$$$$k_r$$$, where $$$\kappa$$$ denotes the PF factor ($$$0.5\leq\kappa\leq1$$$).

Assuming
the proposed trajectory, the real-valued image vector, $$$\bf{u_r}$$$, can be reconstructed
from the measured k-space data, $$$\bf{g}$$$, using a non-iterative gridding-type reconstruction^{5,11,12}:

$$\bf{u_r}=\it{real}\bf{\{\Phi^*A^*D(\Psi_L+2\Psi_H)g\}} \it{(Eq.1)}$$

where $$$\bf{A}$$$ is the forward-encoding
operator including off-resonance effects (presumed known from separate
dual-echo prescan), and $$$\bf{A^*}$$$ its adjoint^{11,12}. $$$\bf{D}$$$ is the sampling density
compensation function determined for the Shells trajectory using Pipe and
Menon’s algorithm^{13}. $$$\bf{\Psi_L}$$$ and $$$\bf{\Psi_H}$$$ are binary
diagonal matrices extracting the low-pass and high-pass regions of k-space along
the PF acceleration direction (i.e., z). $$$\bf{\Phi}$$$ is
a diagonal matrix that approximates the residual spatial phase signal (e.g.,
from eddy-currents, B1^{-} fields) not considered within $$$\bf{A}$$$, which is estimated from the
fully-sampled (z-direction), low-pass k-space data^{5}. Note that Eq. 1
is applicable to Shells trajectory because one k-space hemisphere is fully
sampled, based on our design.

As a test, the ACR phantom
was scanned using both the fully-sampled and PF-accelerated (PF factor=0.7/0.6) Shells (Table 1). The brain of a healthy volunteer was also scanned under an IRB-approved
protocol, using magnetization-prepared shells (MP-Shells) acquisitions^{3}
and MPRAGE with matching parameters (Table 1). In MP-Shells, a series of Shell readouts
are played following a center-out view-ordering after a MP module, i.e., the inner-most
shells are acquired at inversion time (TI), followed by outer shells. The
images were reconstructed (coil-by-coil) using a type-III NUFFT-based
reconstruction with simultaneous (time-segmented) off-resonance and gradient-nonlinearity
correction^{14}, and sum-of-square combined post-reconstruction. Fat saturation
was enabled in MP-Shells to suppress fat-induced blurring.

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3. Shu Y, Tao S, Trzasko JD, Huston J 3rd, Weavers PT, Bernstein MA. Magnetization-prepared shells trajectory with automated gradient waveform design. Med Physics 2016;43:WE-FG-206-1.

4. Mugler JP 3rd, Brookeman JR. Three-dimensional magnetization-prepared rapid gradient-echo imaging (3D MP RAGE). Magn Reson Med 1990;15:152-157.

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6. Block KT, Frahm J. Spiral imaging: a critical appraisal. J Magn Reson Imaging 2005;21:657-668.

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8. Lee JH, Pauly JM, Nishimura DG . Partial k-space reconstruction for radial k-space trajectories in magnetic resonance imaging. US patent 7,277,597. 2003.

9. Lustig M, Kim SJ, Pauly JM. A fast method for designing time-optimal gradient waveforms for arbitrary k-space trajectories. IEEE Trans Med Imaging 2008;27(6):866-873.

10. Shu Y, Riederer SJ, Bernstein MA. Three-dimensional MRI with an undersampled spherical shells trajectory. Magn Reson Med 2006;56:553-562.

11. Fessler JA, Sutton BP. Nonuniform fast Fourier transforms using minmax interpolation. IEEE Trans Signal Process 2003;51:560–574. 12. Beatty PJ, Nishimura DG, Pauly JM. Rapid gridding reconstruction with a minimal oversampling ratio. IEEE Trans Med Imaging 2005;24:799–808.

13. Pipe JG, Menon P. Sampling density compensation in MRI: rationale and an iterative numerical solution. Magn Reson Med 1999;41:179-186.

14. Tao S, Trzasko JD, Shu Y,
Huston J 3rd, Johnson KM, Weavers PT, Gray EM, Bernstein MA. NonCartesian MR
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2015;42:7190– 7201.

Figure 1: Examples
of k-space trajectory design for fully-sampled Shells (a) and partial Fourier-accelerated Shells (b) acquisition. (c) shows the sampled part of k-space using trajectory
designed in (b) (assuming PF factor=0.7), and (d) shows the un-sampled part of
k-space due to partial Fourier acceleration, but assumed in reconstruction.
Note that in order to enable non-iterative homodyne reconstruction, (d) is designed
to be conjugate symmetric with respect to the corresponding part in (c).

Figure 2: Examples of phantom images (reformatted
in sagittal plane) acquired using fully-sampled shells (a), partial Fourier
(PF) accelerated shells (b-e) with a PF factor of 0.7 or 0.6. Images
reconstructed using the proposed non-iterative homodyne reconstruction
(b,c) or using the zero-filled k-space
data (d,e). Note simply zero-filling causes image blurring in (d) and (e).
Image resolution is preserved using the proposed homodyne reconstruction (a vs
b and c). The line profiles across a resolution bar in the z-direction (the
direction partial Fourier acceleration is applied) are also shown, which
confirms these observations.

Figure 3: Phantom
images (reformatted in axial plane) acquired with fully-sampled Shells and PF-accelerated Shells with 512 and 384 samples per readout. Shorter readout length is possible
when the PF acceleration is applied because the sampling only needs to cover
70% (in this example) of the k-space in z-direction. Comparison among a, b and c
shows that the shorter readout (c) reduces the off-resonance effect and is less
susceptible to imperfect off-resonance correction (green arrows). Note that the
resolution dots corresponding to 1 mm resolution (i.e., the nominal resolution
based on trajectory design) were well resolved (red arrows).

Figure 4: Example
of MPRAGE (a) and MP-Shells images with full-sampling (FS) (b) and PF acceleration
(c) (factor=0.7). Fat-saturation was enabled in MP-Shells (b,c) to suppress
fat-induced blurring. Table 1 shows acquisition time for each case. (b) and (c)
show increased contrast between gray and white matters, compared with MPRAGE
(a). The mean pixel values of white matter (WM) and gray matter (GM) in
selected ROIs are measured and shown below each panel, along with the standard
deviation of background noise. The white-matter-to-gray-matter contrast-to-noise
ratio is then calculated as (WM-GM)/(BG std.), and are 11.8 (MPRAGE), 24.6
(MPShells-FS), 17.6 (MPShells-PF), respectively.

Table 1a: Data acquisition settings for spoiled fast gradient echo Shells acquisition. All experiments (phantom and in vivo) were performed on a 3T GE Signa HDxt system (v16.0) with maximum gradient amplitude of 40 mT/m and slew rate of 150 T/m/s using an 8-channel brain coil.

1b: Data acquisition settings for in vivo MP-Shells and MPRAGE acquisitions.