On optimality of parallel MRI reconstruction in k-space.

Parallel MRI reconstruction in k-space has several advantages, such as tolerance to calibration data errors and efficient non-Cartesian data processing. These benefits largely accrue from the approximation that a given unsampled k-space datum can be synthesized from only a few local samples. In this study, several aspects of parallel MRI reconstruction in k-space are studied: the design of optimized reconstruction kernels, the effect of regularization on image error, and the accuracy of different k-space-based parallel MRI methods. Reconstruction of parallel MRI data in k-space is posed as the problem of approximating the pseudoinverse with a sparse matrix. The error of the approximation is used as an optimization criterion to find reconstruction kernels optimized for the given coil setup. An efficient algorithm for automatic selection of reconstruction kernels is described. Additionally, a total error metric is introduced for validation of the reconstruction kernel and choice of regularization parameters. The new methods yield reduced reconstruction and noise errors in both simulated and real data studies when compared with existing methods. The new methods may be useful for reduction of image errors, faster data processing, and validation of parallel MRI reconstruction design for a given coil system and k-space trajectory.     

2D partially parallel imaging with k-space surrounding neighbors-based data reconstruction.

Partially parallel imaging (PPI) achieves imaging acceleration by replacing partial phase encoding (PE) with the spatially localized sensitivity encoding of a receiver surface coil array. Further accelerations can be achieved through 2D PPI along two PE directions in 3D MRI. This paper is to explore the k-space-based PPI acquisition and reconstruction strategies for 3D MRI. A surrounding neighbors-based autocalibrating PPI (SNAPPI) was first presented by generalizing the 2D multicolumn multiline interpolation method. Several 2D PPI reconstruction methods were then provided by applying SNAPPI to recover the partially skipped k-space data along two PE directions separately or nonseparately, in k-space or in the hybrid k and image space. An optimal 2D PPI sampling-based reconstruction approach was also presented for applying PPI along certain spatial direction along which the array coil has not sufficient sensitivity variation for a valid PPI reconstruction. Both simulated and in vivo 2D PPI data were used to evaluate the proposed methods.     

Detection and elimination of motion artifacts by regeneration of k-space.

A method has been developed using techniques from partially parallel imaging (PPI) to detect localized inconsistencies in k-space that are caused by certain types of motion. The inconsistent data are discarded and consistent data regenerated from the remaining data using PPI techniques. The price is a small decrease in signal-to-noise ratio (SNR) and additional postprocessing. An iterative scheme is presented which does not require separately acquired coil sensitivity information for the PPI reconstructions. This method has been found to reduce artifact levels in phantom and in vivo test studies.     

Functional MRI using regularized parallel imaging acquisition.

Parallel MRI techniques reconstruct full-FOV images from undersampled k-space data by using the uncorrelated information from RF array coil elements. One disadvantage of parallel MRI is that the image signal-to-noise ratio (SNR) is degraded because of the reduced data samples and the spatially correlated nature of multiple RF receivers. Regularization has been proposed to mitigate the SNR loss originating due to the latter reason. Since it is necessary to utilize static prior to regularization, the dynamic contrast-to-noise ratio (CNR) in parallel MRI will be affected. In this paper we investigate the CNR of regularized sensitivity encoding (SENSE) acquisitions. We propose to implement regularized parallel MRI acquisitions in functional MRI (fMRI) experiments by incorporating the prior from combined segmented echo-planar imaging (EPI) acquisition into SENSE reconstructions. We investigated the impact of regularization on the CNR by performing parametric simulations at various BOLD contrasts, acceleration rates, and sizes of the active brain areas. As quantified by receiver operating characteristic (ROC) analysis, the simulations suggest that the detection power of SENSE fMRI can be improved by regularized reconstructions, compared to unregularized reconstructions. Human motor and visual fMRI data acquired at different field strengths and array coils also demonstrate that regularized SENSE improves the detection of functionally active brain regions.     

Artifact and noise suppression in GRAPPA imaging using improved k-space coil calibration and variable density sampling.

A parallel imaging technique, GRAPPA (GeneRalized Auto-calibrating Partially Parallel Acquisitions), has been used to improve temporal or spatial resolution. Coil calibration in GRAPPA is performed in central k-space by fitting a target signal using its adjacent signals. Missing signals in outer k-space are reconstructed. However, coil calibration operates with signals that exhibit large amplitude variation while reconstruction is performed using signals with small amplitude variation. Different signal variations in coil calibration and reconstruction may result in residual image artifact and noise. The purpose of this work was to improve GRAPPA coil calibration and variable density (VD) sampling for suppressing residual artifact and noise. The proposed coil calibration was performed in local k-space along both the phase and frequency encoding directions. Outer k-space was acquired with two different reduction factors. Phantom data were reconstructed by both the conventional GRAPPA and the improved technique for comparison at an acceleration of two. Under the same acceleration, optimal sampling and calibration parameters were determined. An in vivo image was reconstructed in the same way using the predetermined optimal parameters. The performance of GRAPPA was improved by the localized coil calibration and VD sampling scheme.     

Self-calibrating parallel imaging with automatic coil sensitivity extraction.

Calibration of the spatial sensitivity functions of coil arrays is a crucial element in parallel magnetic resonance imaging (PMRI). The most common approach has been to measure coil sensitivities directly using one or more low-resolution images acquired before or after accelerated data acquisition. However, since it is difficult to ensure that the patient and coil array will be in exactly the same positions during both calibration scans and accelerated imaging, this approach can introduce sensitivity miscalibration errors into PMRI reconstructions. This work shows that it is possible to extract sensitivity calibration images directly from a fully sampled central region of a variable-density k-space acquisition. These images have all the features of traditional PMRI sensitivity calibrations and therefore may be used for any PMRI reconstruction technique without modification. Because these calibration data are acquired simultaneously with the data to be reconstructed, errors due to sensitivity miscalibration are eliminated. In vivo implementations of self-calibrating parallel imaging using a flexible coil array are demonstrated in abdominal imaging and in real-time cardiac imaging studies.     

Non-Fourier-encoded parallel MRI using multiple receiver coils.

This paper describes a general theoretical framework that combines non-Fourier (NF) spatially-encoded MRI with multichannel acquisition parallel MRI. The two spatial-encoding mechanisms are physically and analytically separable, which allows NF encoding to be expressed as complementary to the inherent encoding imposed by RF receiver coil sensitivities. Consequently, the number of NF spatial-encoding steps necessary to fully encode an FOV is reduced. Furthermore, by casting the FOV reduction of parallel imaging techniques as a dimensionality reduction of the k-space that is NF-encoded, one can obtain a speed-up of each digital NF spatial excitation in addition to accelerated imaging. Images acquired at speed-up factors of 2x to 8x with a four-element RF receiver coil array demonstrate the utility of this framework and the efficiency afforded by it.     

k-t GRAPPA: a k-space implementation for dynamic MRI with high reduction factor.

A novel technique called "k-t GRAPPA" is introduced for the acceleration of dynamic magnetic resonance imaging. Dynamic magnetic resonance images have significant signal correlations in k-space and time dimension. Hence, it is feasible to acquire only a reduced amount of data and recover the missing portion afterward. Generalized autocalibrating partially parallel acquisitions (GRAPPA), as an important parallel imaging technique, linearly interpolates the missing data in k-space. In this work, it is shown that the idea of GRAPPA can also be applied in k-t space to take advantage of the correlations and interpolate the missing data in k-t space. For this method, no training data, filters, additional parameters, or sensitivity maps are necessary, and it is applicable for either single or multiple receiver coils. The signal correlation is locally derived from the acquired data. In this work, the k-t GRAPPA technique is compared with our implementation of GRAPPA, TGRAPPA, and sliding window reconstructions, as described in Methods. The experimental results manifest that k-t GRAPPA generates high spatial resolution reconstruction without significant loss of temporal resolution when the reduction factor is as high as 4. When the reduction factor becomes higher, there might be a noticeable loss of temporal resolution since k-t GRAPPA uses temporal interpolation. Images reconstructed using k-t GRAPPA have less residue/folding artifacts than those reconstructed by sliding window, much less noise than those reconstructed by GRAPPA, and wider temporal bandwidth than those reconstructed by GRAPPA with residual k-space. k-t GRAPPA is applicable to a wide range of dynamic imaging applications and is not limited to imaging parts with quasi-periodic motion. Since only local information is used for reconstruction, k-t GRAPPA is also preferred for applications requiring real time reconstruction, such as monitoring interventional MRI.     

2D-GRAPPA-operator for faster 3D parallel MRI.

When using parallel MRI (pMRI) methods in combination with three-dimensional (3D) imaging, it is beneficial to subsample the k-space along both phase-encoding directions because one can then take advantage of coil sensitivity variations along two spatial dimensions. This results in an improved reconstruction quality and therefore allows greater scan time reductions as compared to subsampling along one dimension. In this work we present a new approach based on the generalized autocalibrating partially parallel acquisitions (GRAPPA) technique that allows Fourier-domain reconstructions of data sets that are subsampled along two dimensions. The method works by splitting the 2D reconstruction process into two separate 1D reconstructions. This approach is compared with an extension of the conventional GRAPPA method that directly regenerates missing data points of a 2D subsampled k-space by performing a linear combination of acquired data points. In this paper we describe the theoretical background and present computer simulations and in vivo experiments.     

Reduced k-space encoding for dual-contrast TrueFISP imaging.

Dual-contrast TrueFISP imaging relies on the use of two RF pulses with different RF flip-angles for enhancing image contrast and performing automatic tissue classification based on multispectrum clustering. The original technique, however, involves an extended imaging time, which limits its clinical application. The purpose of this study is to compare the applicability of two reduced k-space sampling techniques, reduced imaging using generalized series reconstruction (RIGR) and key-hole imaging, for minimizing the imaging time required. The performance of both approaches was evaluated theoretically and practically using 3D cine datasets acquired from eight asymptomatic subjects.     

Description of parallel imaging in MRI using multiple coils.

A general formulation for parallel imaging using multiple coils is derived from the Fourier transform of coil sensitivity functions. This formulation provides a unified account for developed parallel imaging techniques such as subencoding, simultaneous acquisition of spatial harmonics (SMASH), and sensitivity encoding (SENSE), and indicates a guideline for coil configuration and k-space sampling in parallel imaging. The views that can be acquired simultaneously in parallel imaging have to be contained in the spatial frequency band of coil sensitivity functions.     

Strategies to improve contrast in turboFLASH imaging: reordered phase encoding and k-space segmentation.

TurboFLASH (fast low-angle shot) sequences enable the acquisition of an image in a fraction of a second. However, unique to T1-weighted ultrafast imaging, the magnetization variation during image acquisition can produce artifacts along the phase-encoding direction. In this study, the signal behavior and nature of these artifacts were analyzed with various acquisition schemes to improve image contrast. The magnetization variation during image acquisition and its filtering effect on the image were simulated for three different approaches to T1-weighted turboFLASH imaging: standard turboFLASH with (a) monotonically ascending phase-encoding steps, (b) reordered phase encoding, and (c) k-space segmentation. Each of the modified data acquisition schemes has advantages. However, for subsecond imaging, reordered phase encoding produced improved image contrast over that of standard turboFLASH, and segmented k-space imaging gave superior tissue contrast compared with that of both standard and reordered turboFLASH, with imaging time that permits breath-hold studies.     

k-space undersampling in PROPELLER imaging.

PROPELLER MRI (periodically rotated overlapping parallel lines with enhanced reconstruction) provides images with significantly fewer B(0)-related artifacts than echo-planar imaging (EPI), as well as reduced sensitivity to motion compared to conventional multiple-shot fast spin-echo (FSE). However, the minimum imaging time in PROPELLER is markedly longer than in EPI and 50% longer than in conventional multiple-shot FSE. Often in MRI, imaging time is reduced by undersampling k-space. In the present study, the effects of undersampling on PROPELLER images were evaluated using simulated and in vivo data sets. Undersampling using PROPELLER patterns with reduced number of samples per line, number of lines per blade, or number of blades per acquisition, while maintaining the same k-space field of view (FOV(k)) and uniform sampling at the edges of FOV(k), reduced imaging time but led to severe image artifacts. In contrast, undersampling by means of removing whole blades from a PROPELLER sampling pattern that sufficiently samples k-space produced only minimal image artifacts, mainly manifested as blurring in directions parallel to the blades removed, even when reducing imaging time by as much as 50%. Finally, undersampling using asymmetric blades and taking advantage of Hermitian symmetries to fill-in the missing data significantly reduced imaging time without causing image artifacts.     

Coil-by-coil image reconstruction with SMASH.

The SiMultaneous Acquisition of Spatial Harmonics (SMASH) technique uses linear combinations of undersampled datasets from the component coils of an RF coil array to reconstruct fully sampled composite datasets in reduced imaging times. In previously reported implementations, SMASH reconstructions were designed to reproduce the images that would otherwise be obtained by simple sums of fully gradient encoded component coil images. This strategy has left SMASH images vulnerable to phase cancellation artifacts when the sensitivities of RF coil array elements are not suitably phase-aligned. In fully gradient encoded imaging schemes these artifacts can be eliminated using a variety of methods for combining the individual coil images, including matched filter combinations as well as sum of squares combinations. Until now, these reconstruction schemes have been unavailable to SMASH reconstructions as SMASH produced a final composite image directly from the raw component coil k-space datasets. This article demonstrates a modification to SMASH that allows reconstruction of a full set of accelerated individual component coil images by fitting component coil sensitivity functions to a complete set of spatial harmonics tailored for each coil in the array. Standard component coil combinations applied to the individual reconstructed images produce final composite images free of phase cancellation artifacts.     

Localization by nonlinear phase preparation and k-space trajectory design

A technique is described to localize MR signals from a target volume using nonlinear pulsed magnetic fields and spatial encoding trajectories designed using local k-space theory. The concept of local k-space is outlined theoretically, and this principle is applied to simulated phantom and cardiac MRI data in the presence of surface and quadrupolar gradient coil phase modulation. Phantom and in vivo human brain images are obtained using a custom, high-performance quadrupolar gradient coil integrated with a whole-body 3-T MRI system to demonstrate target localization using three-dimensional T 2*-weighted spoiled gradient echo, two-dimensional segmented, multiple gradient encoded spin echo, and three-dimensional balanced steady-state free precession acquisitions. This method may provide a practical alternative to selective radiofrequency excitation at ultra-high-field, particularly for steady-state applications where repetition time (TR) must be minimized and when the amount of energy deposited in human tissues is prohibitive. There are several limitations to the approach including the spatial variation in resolution, high frequency aliasing artifacts, and spatial variation in echo times and contrast.     

VD-AUTO-SMASH imaging.

Recently a self-calibrating SMASH technique, AUTO-SMASH, was described. This technique is based on PPA with RF coil arrays using auto-calibration signals. In AUTO-SMASH, important coil sensitivity information required for successful SMASH reconstruction is obtained during the actual scan using the correlation between undersampled SMASH signal data and additionally sampled calibration signals with appropriate offsets in k-space. However, AUTO-SMASH is susceptible to noise in the acquired data and to imperfect spatial harmonic generation in the underlying coil array. In this work, a new modified type of internal sensitivity calibration, VD-AUTO-SMASH, is proposed. This method uses a VD k-space sampling approach and shows the ability to improve the image quality without significantly increasing the total scan time. This new k-space adapted calibration approach is based on a k-space-dependent density function. In this scheme, fully sampled low-spatial frequency data are acquired up to a given cutoff-spatial frequency. Above this frequency, only sparse SMASH-type sampling is performed. On top of the VD approach, advanced fitting routines, which allow an improved extraction of coil-weighting factors in the presence of noise, are proposed. It is shown in simulations and in vivo cardiac images that the VD approach significantly increases the potential and flexibility of rapid imaging with AUTO-SMASH.     

Tailored SMASH image reconstructions for robust in vivo parallel MR imaging.

The simultaneous acquisition of spatial harmonics (SMASH) imaging technique uses spatial information from an array of RF coils to substitute for omitted encoding gradient steps and thereby to accelerate MR image acquisition. Since SMASH image reconstructions rely on the accurate generation of sinusoidally varying composite sensitivity functions to emulate the spatial modulations produced by gradients, the technique was originally believed to be limited to certain image planes or coil array configurations which were particularly suited to the generation of spatial harmonics. Several key improvements to the SMASH reconstruction procedure are described, taking advantage of various degrees of freedom in the spatial harmonic fit. The use of tailored fitting procedures, in combination with a numerical conditioning approach based on new observations about noise propagation in the fit, are shown to allow high-quality SMASH image reconstructions in oblique and double-oblique image planes, both in phantoms and in high-resolution cardiac MR images. Magn Reson Med 44:243-251, 2000.     

Ultimate intrinsic signal-to-noise ratio for parallel MRI: electromagnetic field considerations.

A method is described for establishing an upper bound on the spatial encoding capabilities of coil arrays in parallel MRI. Ultimate intrinsic signal-to-noise ratio (SNR), independent of any particular conductor arrangement, is calculated by expressing arbitrary coil sensitivities in terms of a complete set of basis functions that satisfy Maxwell's equations within the sample and performing parallel imaging reconstructions using these basis functions. The dependence of the ultimate intrinsic SNR on a variety of experimental conditions is explored and a physically intuitive explanation for the observed behavior is provided based on a comparison between the electromagnetic wavelength and the distance between aliasing points. Imaging at high field strength, with correspondingly short wavelength, is shown to offer advantages for parallel imaging beyond those already expected due to the larger available spin polarization. One-dimensional undersampling of k-space yields a steep drop in attainable SNR for more than a 5-fold reduction of scan time, while 2D undersampling permits access to much higher degrees of acceleration. Increased tissue conductivity decreases baseline SNR, but improves parallel imaging performance. A procedure is also provided for generating the optimal coil sensitivity pattern for a given acceleration, which will serve as a useful guide for future coil designs.     

Spatial domain method for the design of RF pulses in multicoil parallel excitation.

Parallel excitation has been introduced as a means of accelerating multidimensional, spatially-selective excitation using multiple transmit coils, each driven by a unique RF pulse. Previous approaches to RF pulse design in parallel excitation were either formulated in the frequency domain or restricted to echo-planar trajectories, or both. This paper presents an approach that is formulated as a quadratic optimization problem in the spatial domain and allows the use of arbitrary k-space trajectories. Compared to frequency domain approaches, the new design method has some important advantages. It allows for the specification of a region of interest (ROI), which improves excitation accuracy at high speedup factors. It allows for magnetic field inhomogeneity compensation during excitation. Regularization may be used to control integrated and peak pulse power. The effects of Bloch equation nonlinearity on the large-tip-angle excitation error of RF pulses designed with the method are investigated, and the utility of Tikhonov regularization in mitigating this error is demonstrated.     

Parallel imaging reconstruction for arbitrary trajectories using k-space sparse matrices (kSPA).

Although the concept of receiving MR signal using multiple coils simultaneously has been known for over two decades, the technique has only recently become clinically available as a result of the development of several effective parallel imaging reconstruction algorithms. Despite the success of these algorithms, it remains a challenge in many applications to rapidly and reliably reconstruct an image from partially-acquired general non-Cartesian k-space data. Such applications include, for example, three-dimensional (3D) imaging, functional MRI (fMRI), perfusion-weighted imaging, and diffusion tensor imaging (DTI), in which a large number of images have to be reconstructed. In this work, a systematic k-space-based reconstruction algorithm based on k-space sparse matrices (kSPA) is introduced. This algorithm formulates the image reconstruction problem as a system of sparse linear equations in k-space. The inversion of this system of equations is achieved by computing a sparse approximate inverse matrix. The algorithm is demonstrated using both simulated and in vivo data, and the resulting image quality is comparable to that of the iterative sensitivity encoding (SENSE) algorithm. The kSPA algorithm is noniterative and the computed sparse approximate inverse can be applied repetitively to reconstruct all subsequent images. This algorithm, therefore, is particularly suitable for the aforementioned applications.