0954

Fast volumetric Cartesian MRI with auto-calibrated local B0 coil array – a reproducing kernel Hilbert space perspective
Rui Tian1, Martin Uecker2, Oliver Holder1, Theodor Steffen1, and Klaus Scheffler1,3
1High-Field MR center, Max Planck Institute for Biological Cybernetics, Tuebingen, Germany, 2Institute of Biomedical Imaging, Graz University of Technology, Graz, Austria, 3Department for Biomedical Magnetic Resonance, University of Tuebingen, Tuebingen, Germany

Synopsis

Keywords: Image Reconstruction, Image Reconstruction, nonlinear gradient

Motivation: The local B0 coil array has been shown to speed up 2D Cartesian MRI and provides a platform for investigating the most efficient B0 encoding fields. Nevertheless, optimizing the rapid modulations for accelerating volumetric scans without introducing additional artifacts becomes more challenging.

Goal(s): We explore distinct nonlinear modulation B0 fields and reconstruct artifact-free accelerated images.

Approach: With a recent RKHS framework, the k-space efficiency maps for various modulation fields are analyzed, and a novel auto-calibration reconstruction method is introduced.

Results: Our k-space analysis provides insights validating optimal modulation fields, and the ex-vivo and in-vivo scans demonstrate the robustness of the proposed reconstruction technique.

Impact: We demonstrate the RKHS formalism as a valuable tool for understanding 3D MRI scans encoded with nonlinear modulation fields. Our auto-calibration reconstruction, analogous to GRAPPA in parallel imaging, offers a promising approach for image acceleration with rapid B0 modulation.

Introduction

MRI sequentially samples object signals in Fourier domain1,2, which inherently leads to prolonged scan time, and therefore, has driven long-standing interests in developing image acceleration methods. Nonlinear gradients3–8, in addition to the well-established parallel imaging technique9–11, have demonstrated potential to speed up MRI acquisitions; however, a rigorous analysis comparing the encoding capability of various nonlinear and linear fields requires more in-depth investigations12.

Early nonlinear (i.e., B0) gradient methods spatially encode spins using sinusoidal3, quadrupolar4,6, or quadratic5 magnetic fields for phase and frequency encoding. Later, sinusoidal modulation of 2nd/3rd-order gradient coils7,13 was applied during signal readout to accelerate scans, similar to bunched phase encoding14 or wave-CAIPI15, but with nonlinear modulation fields. To further investigate optimal field shapes for image acceleration, we developed a local B0 coil array to provide more flexible spatial-temporal modulation patterns to enhance the speed of MRI8,12.

For accelerated 2D Cartesian scans12, the optimal modulation field by our 8-channel local coil setup converges to a linear gradient regardless of whether parallel imaging is combined or not, with the k-space sampling efficiency visualized in a novel reproducing kernel Hilbert Space (RKHS) framework16,12. This time, we apply the same principles12 to accelerate volumetric Cartesian MRI, and investigate the optimal modulation field shapes given a specific coil-patient position and readout axis, as in Figure 1(A).

Additionally, to enhance image reconstruction for 3D accelerated scans, we propose a robust auto-calibration algorithm to address field calibration challenges arising from potentially larger phase errors induced by eddy currents in 3D, as well as stronger motion due to longer acquisition time.

Methods

During signal readout of 3D FLASH scans in a Siemens 9T human scanner, sinusoidal currents in 9.26kHz/42Apeak-to-zero (ex. 9.80kHz/37Apeak-to-zero for in-vivo scans) with distinct phase offsets are applied to the 8 local B0 coils to accelerate acquisitions, corresponding to various modulation field shapes. In addition to the G-map, the k-space efficiency maps12 (i.e., cardinal and power functions) for visualizing noise amplifications and approximation errors have been computed and compared for different field patterns.

In Figure 2, a novel algorithm is presented for extracting the additional spin phase evolution caused by local B0 modulations without field mapping scans, as a crucial step for the robust reconstruction of 3D accelerated scans. In section A, we formulate linear equations in k-space. Here, a single data point acquired with B0 modulations can be understood as a linear combination of a local region within low-resolution reference data without modulation (i.e., typically obtained for parallel imaging10,11,17), given the interpolation weights as a cardinal function12,16. By setting the sinusoidal modulation period (e.g., 108 μs) as an integer multiple of the scanner ADC dwell time (e.g., 3 μs), the cardinal functions undergo a periodic change in shape along the readout modulation. This results in independent linear equations obtained by shifting the interpolating position among readout time instants for the identical sinusoidal modulation phase, and across dimensions of phase encoding and RF receivers.

Hence, in section B, the local cardinal functions are estimated by solving the linear system. In section C, the cardinal functions can undergo zero-filling and Fourier transform to directly yield the estimated phase evolution maps in image space. Alternatively, they can be exploited to synthesize low-resolution k-space data with distinct image space modulation, which is used to produce phase evolution maps through ESPIRiT17 for additional subspace filtering.

Results

In Figure 1, the forward model reconstruction of ex-vivo phantom18,19 scans using auto-calibrated maps exhibits no noticeable artifacts, whereas the previous approach12 that involves field mapping scans and current monitors results in slice-dependent residue artifacts due to mis-calibration in three-dimension.

In Figure 3, ex-vivo scans with distinct phase offsets (i.e., modulation field shapes) are jointly reconstructed with SENSE10,15,20, with a retrospective undersampling factor 4x4. Accelerated scans with phase arrangements of “all zero” and “quadrupolar” show no apparent difference except for very mild SNR loss compared to the reference image, which matches the G-map10 calculation.

In Figure 4, the superior encoding efficiency by the “all zero” configuration can be well explained by the k-space efficiency maps12, since this coil phase arrangement leads to strong modulation along the z dimension, which is usually insufficiently encoded by RF receivers' sensitivity10.

In Figure 5, the in-vivo scans with joint reconstruction of SENSE & B0 modulations show negligible artifacts given a retrospective undersampling factor of 3x3, which substantially outperforms the accelerated scans with SENSE alone.

Discussion/Conclusion

Our proposed robust reconstruction algorithm successfully enables 3D image acceleration by the local B0 coils, and the recently developed RKHS framework12 serves as a valuable tool for guiding the exploration of optimal field modulations.

Acknowledgements

This study is supported by ERC Advanced Grant No 834940.

The ex-vivo brain phantom was with courtesy of the Institute of Clinical Anatomy and Cell Analysis, Department of Anatomy, Eberhard Karls University of Tübingen. The first author thanks Dr. Thomas Shiozawa (Institute of Clinical Anatomy and Cell Analysis) for assistance with sample preparation, and Dr. Gisela Hagberg for assistance in scanning this phantom.

The first author would also like to thank Stefan Plappert for guidance in programming the ADwin high-speed processor, and Pavel Povolni & Dr. Georgiy Alekseevich Solomakha for assistance in RF system testing.

References

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13. Dispenza NL, Littin S, Zaitsev M, Constable RT, Galiana G. Clinical Potential of a New Approach to MRI Acceleration. Sci Rep. 2019;9(1):1912. doi:10.1038/s41598-018-36802-5

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Figures

Figure 1. In the 3D Cartesian scan (here, FLASH) with local B0 coil modulations, section A shows the imaging volume relative to the local B0 coil and readout axis. Section B displays reconstructed images from fully sampled k-space data. Previous methods using current monitors and B0 field mapping caused slice-dependent artifacts where errors by eddy currents and motions in 3D scans can become more severe. However, auto-calibrated phase evolution leads to high reconstruction quality comparable to the reference images without local coils, even when patient motion is present.

Figure 2. Auto-calibration of B0 modulation. A. Establishing linear equations to interpolate k-space data with and without modulations, estimating k-space cardinal functions corresponding to the changing phase modulation patterns in image space. B. Solving the linear systems at different sinusoidal modulation phases, where cardinal functions are shift-invariant convolution kernels in the patches shifting across k-space and RF receivers. C. The estimated phase evolution used for reconstruction is obtained based on cardinal functions, with optional subspace filtering.

Figure 3. Comparison of 3D ex-vivo scans accelerated by SENSE & local B0 modulations, SENSE alone, and the reference image without acceleration. Phase offset arrangements for sinusoidal currents in local B0 coils are shown below. The "all zero" phase arrangement, as the most efficient encoding scheme here, introduces a quadratic field modulation in the transverse plane and a nearly linear z-gradient. Images and G-maps confirm the sampling efficiencies for distinct modulation field patterns, while their origins can be observed in the k-space efficiency maps in Figure 4.

Figure 4. Analyzing k-space origins of variations in sampling efficiency with distinct modulation fields in the RKHS framework. The approximation errors reveal potential signal energy loss due to insufficient sampling, while the cardinal function quantifies noise amplification. Given R=4x3, the SENSE method captures a broader k-space range than the non-accelerated reference. Combining SENSE and B0 modulations boosts sampling efficiency, notably with the "all zero" scheme, which enhances the z-direction coverage insufficiently covered by the SENSE-only acceleration.

Figure 5. Comparison of 3D in-vivo scans with acquisitions accelerated by SENSE & local B0 modulations, SENSE alone, and the reference image without acceleration. The "all zero" phase arrangement is employed for the optimized modulation field pattern, given the current patient-coil position and readout axis. The reconstructed images from the joint acceleration of SENSE and B0 modulations achieve a 3x3 faster scan with acceptable image quality, even when patient motion was observed between the calibration datasets (i.e., modulated and non-modulated).

Proc. Intl. Soc. Mag. Reson. Med. 32 (2024)
0954
DOI: https://doi.org/10.58530/2024/0954