Synopsis

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.

Purpose

Methods

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 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 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

Conclusion

Acknowledgements

References

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