Partial Fourier Shells Trajectory with Non-Iterative Homodyne Reconstruction
Shengzhen Tao1, Yunhong Shu1, Joshua D Trzasko1, Paul T Weavers1, Erin M Gray1, John Huston III1, and Matt A Bernstein1

1Radiology, Mayo Clinic, Rochester, MN, United States


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 Shells trajectory divides the 3D k-space into a series of concentric shells and samples each one using 3D helical readouts1,2. With the Shells trajectory, the inner k-space, which determines the image contrast, can be efficiently sampled within several interleaves, making it a maximally centric 3D acquisition. The acquisition of inner shells can be synchronized with the time when a certain desired image contrast is expected, such as the contrast agent arrival time in gadolinium-enhanced MR angiography1, or when the peak of white-to-gray-matter contrast is achieved after a magnetization-preparation (MP) pulse3,4. Partial Fourier (PF) homodyne acquisition is a commonly-used acceleration technique in Cartesian MRI by exploiting the conjugate symmetry of k-space measurement5. However, unlike Cartesian MRI, utilizing homodyne acquisition in non-Cartesian acquisition is not straightforward, since directly applying homodyne acquisition in a direction sampled with non-Cartesian trajectory does not always yield a physically-achievable trajectory (e.g., 2D spiral)6, or an iterative reconstruction may be required7,8. Here, we present a new asymmetric 3D Shells trajectory design, and develop a non-iterative PF reconstruction 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 Lustig9. The gradient amplitude and slew rate limits are chosen to conform to both hardware limits and Nyquist sampling requirements10. 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 reconstruction5,11,12:

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

where $$$\bf{A}$$$ is the forward-encoding operator including off-resonance effects (presumed known from separate dual-echo prescan), and $$$\bf{A^*}$$$ its adjoint11,12. $$$\bf{D}$$$ is the sampling density compensation function determined for the Shells trajectory using Pipe and Menon’s algorithm13. $$$\bf{\Psi_L}$$$ 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 data5. 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) acquisitions3 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 correction14, and sum-of-square combined post-reconstruction. Fat saturation was enabled in MP-Shells to suppress fat-induced blurring.

Results and Discussion

Figure1 shows an example of the fully-sampled (a) and PF-accelerated (factor=0.7) Shells trajectory (b; see captions). Figure 2 shows phantom images reconstructed from fully-sampled (a) or PF-accelerated Shells using the proposed (b/c) and zero-filling (d/e) reconstruction. Comparing with the fully-sampled case, the proposed PF-acquisition and reconstruction preserves image resolution while reducing acquisition time (Table 1). Figure 3 compares two PF Shells designs with different readout lengths showing reduced off-resonance effect with shorter readout (see captions). Figure 4 compares the MP-Shells images with MPRAGE. Note the improved white-matter-to-gray-matter-contrast in MP-Shells from acquiring the entire 3D k-space center shells at desired TI time.


We developed a 3D Shells trajectory with PF-acceleration and a non-iterative homodyne reconstruction for it. This trajectory inherits the high k-space sampling efficiency from fully-sampled Shells but reduces acquisition time or sensitivity to off-resonance.


This work was supported in part by the NIH grant R01EB010065.


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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.

Proc. Intl. Soc. Mag. Reson. Med. 25 (2017)