Non‐Cartesian parallel imaging reconstruction

Non‐Cartesian parallel imaging has played an important role in reducing data acquisition time in MRI. The use of non‐Cartesian trajectories can enable more efficient coverage of k‐space, which can be leveraged to reduce scan times. These trajectories can be undersampled to achieve even faster scan times, but the resulting images may contain aliasing artifacts. Just as Cartesian parallel imaging can be used to reconstruct images from undersampled Cartesian data, non‐Cartesian parallel imaging methods can mitigate aliasing artifacts by using additional spatial encoding information in the form of the nonhomogeneous sensitivities of multi‐coil phased arrays. This review will begin with an overview of non‐Cartesian k‐space trajectories and their sampling properties, followed by an in‐depth discussion of several selected non‐Cartesian parallel imaging algorithms. Three representative non‐Cartesian parallel imaging methods will be described, including Conjugate Gradient SENSE (CG SENSE), non‐Cartesian generalized autocalibrating partially parallel acquisition (GRAPPA), and Iterative Self‐Consistent Parallel Imaging Reconstruction (SPIRiT). After a discussion of these three techniques, several potential promising clinical applications of non‐Cartesian parallel imaging will be covered. J. Magn. Reson. Imaging 2014;40:1022–1040 . © 2014 Wiley Periodicals, Inc.     

Advances in sensitivity encoding with arbitrary k-space trajectories.

New, efficient reconstruction procedures are proposed for sensitivity encoding (SENSE) with arbitrary k-space trajectories. The presented methods combine gridding principles with so-called conjugate-gradient iteration. In this fashion, the bulk of the work of reconstruction can be performed by fast Fourier transform (FFT), reducing the complexity of data processing to the same order of magnitude as in conventional gridding reconstruction. Using the proposed method, SENSE becomes practical with nonstandard k-space trajectories, enabling considerable scan time reduction with respect to mere gradient encoding. This is illustrated by imaging simulations with spiral, radial, and random k-space patterns. Simulations were also used for investigating the convergence behavior of the proposed algorithm and its dependence on the factor by which gradient encoding is reduced. The in vivo feasibility of non-Cartesian SENSE imaging with iterative reconstruction is demonstrated by examples of brain and cardiac imaging using spiral trajectories. In brain imaging with six receiver coils, the number of spiral interleaves was reduced by factors ranging from 2 to 6. In cardiac real-time imaging with four coils, spiral SENSE permitted reducing the scan time per image from 112 ms to 56 ms, thus doubling the frame-rate.     

SPIRiT: Iterative self‐consistent parallel imaging reconstruction from arbitrary k‐space

A new approach to autocalibrating, coil‐by‐coil parallel imaging reconstruction, is presented. It is a generalized reconstruction framework based on self‐consistency. The reconstruction problem is formulated as an optimization that yields the most consistent solution with the calibration and acquisition data. The approach is general and can accurately reconstruct images from arbitrary k‐space sampling patterns. The formulation can flexibly incorporate additional image priors such as off‐resonance correction and regularization terms that appear in compressed sensing. Several iterative strategies to solve the posed reconstruction problem in both image and k‐space domain are presented. These are based on a projection over convex sets and conjugate gradient algorithms. Phantom and in vivo studies demonstrate efficient reconstructions from undersampled Cartesian and spiral trajectories. Reconstructions that include off‐resonance correction and nonlinear ?₁‐wavelet regularization are also demonstrated. Magn Reson Med, 2010. © 2010 Wiley‐Liss, Inc.     

Radial k‐t FOCUSS for high‐resolution cardiac cine MRI

A compressed sensing dynamic MR technique called k‐t FOCUSS (k‐t FOCal Underdetermined System Solver) has been recently proposed. It outperforms the conventional k‐t BLAST/SENSE (Broad‐use Linear Acquisition Speed‐up Technique/SENSitivity Encoding) technique by exploiting the sparsity of x‐f signals. This paper applies this idea to radial trajectories for high‐resolution cardiac cine imaging. Radial trajectories are more suitable for high‐resolution dynamic MRI than Cartesian trajectories since there is smaller tradeoff between spatial resolution and number of views if streaking artifacts due to limited views can be resolved. As shown for Cartesian trajectories, k‐t FOCUSS algorithm efficiently removes artifacts while preserving high temporal resolution. k‐t FOCUSS algorithm applied to radial trajectories is expected to enhance dynamic MRI quality. Rather than using an explicit gridding method, which transforms radial k‐space sampling data to Cartesian grid prior to applying k‐t FOCUSS algorithms, we use implicit gridding during FOCUSS iterations to prevent k‐space sampling errors from being propagated. In addition, motion estimation and motion compensation after the first FOCUSS iteration were used to further sparsify the residual image. By applying an additional k‐t FOCUSS step to the residual image, improved resolution was achieved. In vivo experimental results show that this new method can provide high spatiotemporal resolution even from a very limited radial data set. Magn Reson Med, 2010. © 2009 Wiley‐Liss, Inc.     

Reconstruction of undersampled non-Cartesian data sets using pseudo-Cartesian GRAPPA in conjunction with GROG.

Most k-space-based parallel imaging reconstruction techniques, such as Generalized Autocalibrating Partially Parallel Acquisitions (GRAPPA), necessitate the acquisition of regularly sampled Cartesian k-space data to reconstruct a nonaliased image efficiently. However, non-Cartesian sampling schemes offer some inherent advantages to the user due to their better coverage of the center of k-space and faster acquisition times. On the other hand, these sampling schemes have the disadvantage that the points acquired generally do not lie on a grid and have complex k-space sampling patterns. Thus, the extension of Cartesian GRAPPA to non-Cartesian sequences is nontrivial. This study introduces a simple, novel method for performing Cartesian GRAPPA reconstructions on undersampled non-Cartesian k-space data gridded using GROG (GRAPPA Operator Gridding) to arrive at a nonaliased image. Because the undersampled non-Cartesian data cannot be reconstructed using a single GRAPPA kernel, several Cartesian patterns are selected for the reconstruction. This flexibility in terms of both the appearance and number of patterns allows this pseudo-Cartesian GRAPPA to be used with undersampled data sets acquired with any non-Cartesian trajectory. The successful implementation of the reconstruction algorithm using several different trajectories, including radial, rosette, spiral, one-dimensional non-Cartesian, and zig-zag trajectories, is demonstrated.     

Estimation of k‐space trajectories in spiral MRI

For non‐Cartesian data acquisition in MRI, k‐space trajectory infidelity due to eddy current effects and other hardware imperfections will blur and distort the reconstructed images. Even with the shielded gradients and eddy current compensation techniques of current scanners, the deviation between the actual k‐space trajectory and the requested trajectory remains a major reason for image artifacts in non‐Cartesian MRI. It is often not practical to measure the k‐space trajectory for each imaging slice. It has been reported that better image quality is achieved in radial scanning by correcting anisotropic delays on different physical gradient axes. In this article the delay model is applied in spiral k‐space trajectory estimation to reduce image artifacts. Then a novel estimation method combining the anisotropic delay model and a simple convolution eddy current model further reduces the artifact level in spiral image reconstruction. The root mean square error and peak error in both phantom and in vivo images reconstructed using the estimated trajectories are reduced substantially compared to the results achieved by only tuning delays. After a one‐time calibration, it is thus possible to get an accurate estimate of the spiral trajectory and a high‐quality image reconstruction for an arbitrary scan plane. Magn Reson Med, 2009. © 2009 Wiley‐Liss, Inc.     

Whole‐heart imaging using undersampled radial phase encoding (RPE) and iterative sensitivity encoding (SENSE) reconstruction

Whole‐heart isotropic nonangulated cardiac magnetic resonance (CMR) is becoming an important protocol in simplifying MRI, since it reduces the need of cumbersome planning of angulations. However the acquisition times of whole‐heart MRI are prohibitive due to the large fields of view (FOVs) and the high spatial resolution required for depicting small structures and vessels. To address this problem, we propose a three‐dimensional (3D) acquisition scheme that combines Cartesian sampling in the readout direction with an undersampled radial scheme in the phase‐encoding plane. Different undersampling patterns were investigated in combination with an iterative sensitivity encoding (SENSE) reconstruction and a 32‐channel cardiac coil. Noise amplification maps were calculated to compare the performance of the different patterns using iterative SENSE reconstruction. The radial phase‐encoding (RPE) scheme was implemented on a clinical MR scanner and tested on phantoms and healthy volunteers. The proposed method exhibits better image quality even for high acceleration factors (up to 12) in comparison to Cartesian acquisitions. Magn Reson Med, 2009. © 2009 Wiley‐Liss, Inc.     

Single‐shot magnetic resonance spectroscopic imaging with partial parallel imaging

A magnetic resonance spectroscopic imaging (MRSI) pulse sequence based on proton–echo‐planar‐spectroscopic‐imaging (PEPSI) is introduced that measures two‐dimensional metabolite maps in a single excitation. Echo‐planar spatial–spectral encoding was combined with interleaved phase encoding and parallel imaging using SENSE to reconstruct absorption mode spectra. The symmetrical k‐space trajectory compensates phase errors due to convolution of spatial and spectral encoding. Single‐shot MRSI at short TE was evaluated in phantoms and in vivo on a 3‐T whole‐body scanner equipped with a 12‐channel array coil. Four‐step interleaved phase encoding and fourfold SENSE acceleration were used to encode a 16 × 16 spatial matrix with a 390‐Hz spectral width. Comparison with conventional PEPSI and PEPSI with fourfold SENSE acceleration demonstrated comparable sensitivity per unit time when taking into account g‐factor–related noise increases and differences in sampling efficiency. LCModel fitting enabled quantification of inositol, choline, creatine, and N‐acetyl‐aspartate (NAA) in vivo with concentration values in the ranges measured with conventional PEPSI and SENSE‐accelerated PEPSI. Cramer–Rao lower bounds were comparable to those obtained with conventional SENSE‐accelerated PEPSI at the same voxel size and measurement time. This single‐shot MRSI method is therefore suitable for applications that require high temporal resolution to monitor temporal dynamics or to reduce sensitivity to tissue movement. Magn Reson Med, 2009. © 2008 Wiley‐Liss, Inc.     

Reconstruction of MRI data encoded with arbitrarily shaped,curvilinear, nonbijective magnetic fields

A basic framework for image reconstruction from spatial encoding by curvilinear, nonbijective magnetic encoding fields in combination with multiple receivers is presented. The theory was developed in the context of the recently introduced parallel imaging technique using localized gradients (PatLoc) approach. In this new imaging modality, the linear gradient fields are generalized to arbitrarily shaped, nonbijective spatial encoding magnetic fields, which lead to ambiguous encoding. Ambiguities are resolved by adaptation of concepts developed for parallel imaging. Based on theoretical considerations, a practical algorithm for Cartesian trajectories is derived in the case that the conventional gradient coils are replaced by coils for PatLoc. The reconstruction method extends Cartesian sensitivity encoding (SENSE) reconstruction with an additional voxelwise intensity‐correction step. Spatially varying resolution, signal‐to‐noise ratio, and truncation artifacts are described and analyzed. Theoretical considerations are validated by two‐dimensional simulations based on multipolar encoding fields and they are confirmed by applying the reconstruction algorithm to initial experimental data. Magn Reson Med, 2010. © 2010 Wiley‐Liss, Inc.     

Using the GRAPPA operator and the generalized sampling theorem to reconstruct undersampled non‐Cartesian data

As expected from the generalized sampling theorem of Papoulis, the use of a bunched sampling acquisition scheme in conjunction with a conjugate gradient (CG) reconstruction algorithm can decrease scan time by reducing the number of phase‐encoding lines needed to generate an unaliased image at a given resolution. However, the acquisition of such bunched data requires both modified pulse sequences and high gradient performance. A novel method of generating the “bunched” data using self‐calibrating GRAPPA operator gridding (GROG), a parallel imaging method that shifts data points by small distances in k‐space (with Δk usually less than 1.0, depending on the receiver coil) using the GRAPPA operator, is presented here. With the CG reconstruction method, these additional “bunched” points can then be used to reconstruct an image with reduced artifacts from undersampled data. This method is referred to as GROG‐facilitated bunched phase encoding (BPE), or GROG‐BPE. To better understand how the patterns of bunched points, maximal blip size, and number of bunched points affect the reconstruction quality, a number of simulations were performed using the GROG‐BPE approach. Finally, to demonstrate that this method can be combined with a variety of trajectories, examples of images with reduced artifacts reconstructed from undersampled in vivo radial, spiral, and rosette data are shown. Magn Reson Med, 2009. © 2009 Wiley‐Liss, Inc.     

Parallel magnetic resonance imaging with adaptive radius in k-space (PARS): constrained image reconstruction using k-space locality in radiofrequency coil encoded data.

A parallel image reconstruction algorithm is presented that exploits the k-space locality in radiofrequency (RF) coil encoded data. In RF coil encoding, information relevant to reconstructing an omitted datum rapidly diminishes as a function of k-space separation between the omitted datum and the acquired signal data. The proposed method, parallel magnetic resonance imaging with adaptive radius in k-space (PARS), harnesses this physical property of RF coil encoding via a sliding-kernel approach. Unlike generalized parallel imaging approaches that might typically involve inverting a prohibitively large matrix for arbitrary sampling trajectories, the PARS sliding-kernel approach creates manageable and distributable independent matrices to be inverted, achieving both computational efficiency and numerical stability. An empirical method designed to measure total error power is described, and the total error power of PARS reconstructions is studied over a range of k-space radii and accelerations, revealing "minimal-error" conditions at comparatively modest k-space radii. PARS reconstructions of undersampled in vivo Cartesian and non-Cartesian data sets are shown and are compared selectively with traditional SENSE reconstructions. Various characteristics of the PARS k-space locality constraint (such as the tradeoff between signal-to-noise ratio and artifact power and the relationship with iterative parallel conjugate gradient approaches or nonparallel gridding approaches) are discussed.     

Reconstruction of undersampled radial PatLoc imaging using total generalized variation

In the case of radial imaging with nonlinear spatial encoding fields, a prominent star‐shaped artifact has been observed if a spin distribution is encoded with an undersampled trajectory. This work presents a new iterative reconstruction method based on the total generalized variation, which reduces this artifact. For this approach, a sampling operator (as well as its adjoint) is needed that maps data from PatLoc k‐space to the final image space. It is shown that this can be realized as a type‐3 nonuniform fast Fourier transform, which is implemented by a combination of a type‐1 and type‐2 nonuniform fast Fourier transform. Using this operator, it is also possible to implement an iterative conjugate gradient SENSE based method for PatLoc reconstruction, which leads to a significant reduction of computation time in comparison to conventional PatLoc image reconstruction methods. Results from numerical simulations and in vivo PatLoc measurements with as few as 16 radial projections are presented, which demonstrate significant improvements in image quality with the total generalized variation‐based approach. Magn Reson Med, 2013. © 2012 Wiley Periodicals, Inc.     

Contrast‐enhanced MR angiography using time resolved interleaved projection sampling with three‐dimensional cartesian phase and slice encoding (TRIPPS)

Undersampled projection reconstruction (PR) techniques provide contrast enhanced MR angiography (CE‐MRA) with high temporal resolution, but sensitivity to eddy current, gradient error and off‐resonance effects. It is desirable to combine the time efficiency of undersampled PR acquisition with the robustness of Cartesian imaging. In this work we present a technique designed to do this termed time resolved projection sampling with three‐dimensional (3D) Cartesian phase and slice encoding (TRIPPS), where 3D Cartesian k‐space is partitioned into multiple half projections in the ky‐kz plane. The phase and slice encoding are performed along predefined center‐out radial trajectories. The whole set of half projections is interleaved into multiple groups of half projections, with each group sparsely but uniformly covering the ky‐kz space. A view sharing sliding window reconstruction algorithm is adapted to reconstruct the dynamic images. The feasibility of the TRIPPS technique for CE‐MRA was demonstrated on the renal, pulmonary, and intracranial vasculatures of healthy volunteers with a high temporal resolution of 2 s/frame. Magn Reson Med, 2009. © 2009 Wiley‐Liss, Inc.     

Accelerating SENSE using compressed sensing

Both parallel MRI and compressed sensing (CS) are emerging techniques to accelerate conventional MRI by reducing the number of acquired data. The combination of parallel MRI and CS for further acceleration is of great interest. In this paper, we propose a novel method to combine sensitivity encoding (SENSE), one of the standard methods for parallel MRI, and compressed sensing for rapid MR imaging (SparseMRI), a recently proposed method for applying CS in MR imaging with Cartesian trajectories. The proposed method, named CS‐SENSE, sequentially reconstructs a set of aliased reduced‐field‐of‐view images in each channel using SparseMRI and then reconstructs the final image from the aliased images using Cartesian SENSE. The results from simulations and phantom and in vivo experiments demonstrate that CS‐SENSE can achieve a reduction factor higher than those achieved by SparseMRI and SENSE individually and outperform the existing method that combines parallel MRI and CS. Magn Reson Med, 2009. © 2009 Wiley‐Liss, Inc.     

High‐resolution human diffusion tensor imaging using 2‐D navigated multishot SENSE EPI at 7 T

The combination of parallel imaging with partial Fourier acquisition has greatly improved the performance of diffusion‐weighted single‐shot EPI and is the preferred method for acquisitions at low to medium magnetic field strength such as 1.5 or 3 T. Increased off‐resonance effects and reduced transverse relaxation times at 7 T, however, generate more significant artifacts than at lower magnetic field strength and limit data acquisition. Additional acceleration of k‐space traversal using a multishot approach, which acquires a subset of k‐space data after each excitation, reduces these artifacts relative to conventional single‐shot acquisitions. However, corrections for motion‐induced phase errors are not straightforward in accelerated, diffusion‐weighted multishot EPI because of phase aliasing. In this study, we introduce a simple acquisition and corresponding reconstruction method for diffusion‐weighted multishot EPI with parallel imaging suitable for use at high field. The reconstruction uses a simple modification of the standard sensitivity‐encoding (SENSE) algorithm to account for shot‐to‐shot phase errors; the method is called image reconstruction using image‐space sampling function (IRIS). Using this approach, reconstruction from highly aliased in vivo image data using 2‐D navigator phase information is demonstrated for human diffusion‐weighted imaging studies at 7 T. The final reconstructed images show submillimeter in‐plane resolution with no ghosts and much reduced blurring and off‐resonance artifacts. Magn Reson Med, 2013. © 2012 Wiley Periodicals, Inc.     

1H spectroscopic imaging of human brain at 3 Tesla: Comparison of fast three‐dimensional magnetic resonance spectroscopic imaging techniques

Purpose

To investigate the signal‐to‐noise‐ratio (SNR) and data quality of time‐reduced three‐dimensional (3D) proton magnetic resonance spectroscopic imaging (¹H MRSI) techniques in the human brain at 3 Tesla.

Materials and Methods

Techniques that were investigated included ellipsoidal k‐space sampling, parallel imaging, and echo‐planar spectroscopic imaging (EPSI). The SNR values for N‐acetyl aspartate, choline, creatine, and lactate or lipid peaks were compared after correcting for effective spatial resolution and acquisition time in a phantom and in the brains of human volunteers. Other factors considered were linewidths, metabolite ratios, partial volume effects, and subcutaneous lipid contamination.

Results

In volunteers, the median normalized SNR for parallel imaging data decreased by 34–42%, but could be significantly improved using regularization. The normalized signal to noise loss in flyback EPSI data was 11–18%. The effective spatial resolutions of the traditional, ellipsoidal, sensitivity encoding (SENSE) sampling scheme, and EPSI data were 1.02, 2.43, 1.03, and 1.01 cm³, respectively. As expected, lipid contamination was variable between subjects but was highest for the SENSE data. Patient data obtained using the flyback EPSI method were of excellent quality.

Conclusion

Data from all ¹H 3D‐MRSI techniques were qualitatively acceptable, based upon SNR, linewidths, and metabolite ratios. The larger field of view obtained with the EPSI methods showed negligible lipid aliasing with acceptable SNR values in less than 9.5 min without compromising the point‐spread function. J. Magn. Reson. Imaging 2009;30:473–480. © 2009 Wiley‐Liss, Inc.     

A rapid and robust numerical algorithm for sensitivity encoding with sparsity constraints: Self‐feeding sparse SENSE

The method of enforcing sparsity during magnetic resonance imaging reconstruction has been successfully applied to partially parallel imaging (PPI) techniques to reduce noise and artifact levels and hence to achieve even higher acceleration factors. However, there are two major problems in the existing sparsity‐constrained PPI techniques: speed and robustness. By introducing an auxiliary variable and decomposing the original minimization problem into two subproblems that are much easier to solve, a fast and robust numerical algorithm for sparsity‐constrained PPI technique is developed in this work. The specific implementation for a conventional Cartesian trajectory data set is named self‐feeding Sparse Sensitivity Encoding (SENSE). The computational cost for the proposed method is two conventional SENSE reconstructions plus one spatially adaptive image denoising procedure. With reconstruction time approximately doubled, images with a much lower root mean square error (RMSE) can be achieved at high acceleration factors. Using a standard eight‐channel head coil, a net acceleration factor of 5 along one dimension can be achieved with low RMSE. Furthermore, the algorithm is insensitive to the choice of parameters. This work improves the clinical applicability of SENSE at high acceleration factors. Magn Reson Med, 2010. © 2010 Wiley‐Liss, Inc.     

Increasing efficiency of parallel imaging for 2D multislice acquisitions

Parallel imaging algorithms require precise knowledge about the spatial sensitivity variation of the receiver coils to reconstruct images with full field of view (FOV) from undersampled Fourier encoded data. Sensitivity information must either be given a priori, or estimated from calibration data acquired along with the actual image data. In this study, two approaches are presented, which require very little or no additional data at all for calibration in two‐dimensional multislice acquisitions. Instead of additional data, information from spatially adjacent slices is used to estimate coil sensitivity information, thereby increasing the efficiency of parallel imaging. The proposed approaches rely on the assumption that over a small range of slices, coil sensitivities vary smoothly in slice direction. Both methods are implemented as variants of the GRAPPA algorithm. For a given effective acceleration, superior image quality is achieved compared to the conventional GRAPPA method. For the latter calibration lines for coil weight computation must be acquired in addition to the undersampled k‐spaces for coil weight computation, thus requiring higher k‐space undersampling, that is, a higher reduction factor to achieve the same effective acceleration. The proposed methods are particularly suitable to speed up parallel imaging for clinical applications where the reduction factor is limited to two or three. Magn Reson Med 2009 © 2009 Wiley‐Liss, Inc.     

Rosette spectroscopic imaging: Optimal parameters for alias‐free,high sensitivity spectroscopic imaging

Purpose

To optimize the Rosette trajectories for fast, high sensitivity spectroscopic imaging experiments and to compare this acquisition technique with other chemical shift imaging (CSI) methods.

Materials and Methods

A framework for comparing the sensitivity of the Rosette Spectroscopic Imaging (RSI) acquisition to other spectroscopic imaging experiments is outlined. Accounting for hardware constraints, trajectory parameters that provide for optimal sampling and minimal artifact production are found. Along with an analytical expression for the number of excitations to be used in an RSI experiment that is provided, the theoretical precompensation weights used for optimal image reconstruction are derived.

Results

The spectral response function for RSI is shown to be approximately the same as the point spread function of standard Fourier reconstructions. While the signal‐to‐noise ratio (SNR) for an RSI experiment is reduced by the inherent nonuniform sampling of these trajectories, their circular k‐space support and speed of spatial encoding leads to greater SNR efficiency and improvements in the total data acquisition time relative to the gold standard CSI approach with square k‐space support and to similar efficiency to spiral CSI acquisitions. Numerical simulations and in vivo experimental data are presented to demonstrate the properties of this data acquisition technique.

Conclusion

This work demonstrates the use of Rosette trajectories and how to achieve improved efficiency for these trajectories in a two‐dimensional spectroscopic imaging experiment. J. Magn. Reson. Imaging 2009;29:1375–1385. © 2009 Wiley‐Liss, Inc.     

Gradient moment compensated magnetic resonance spectroscopic imaging

Spectroscopic imaging applications outside of the brain can suffer from artifacts due to inherent long scan times and susceptibility to motion. A fast spectroscopic imaging sequence has been devised with reduced sensitivity to motion. The sequence uses oscillating readout gradients and acquires k‐space data in a spiral out–in fashion, which allows fast k‐space coverage. We show that a spiral out–in readout acquisition is characterized by small gradient moments, reducing sensitivity to motion‐induced artifacts. Data are acquired comparing the sequence to normal phase encoded spectroscopic imaging and conventional spiral spectroscopic imaging protocols. In addition, in vivo data are acquired from the liver, demonstrating potential usage as a multivoxel fat/water spectroscopic imaging tool. Results indicate that in the presence of motion, ghosting effects are reduced while metabolite signal increases of approximately 10% can be achieved. Magn Reson Med 61:457–461, 2009. © 2009 Wiley‐Liss, Inc.