Fast method for 1D non-cartesian parallel imaging using GRAPPA.

MRI with non-Cartesian sampling schemes can offer inherent advantages. Radial acquisitions are known to be very robust, even in the case of vast undersampling. This is also true for 1D non-Cartesian MRI, in which the center of k-space is oversampled or at least sampled at the Nyquist rate. There are two main reasons for the more relaxed foldover artifact behavior: First, due to the oversampling of the center, high-energy foldover artifacts originating from the center of k-space are avoided. Second, due to the non-equidistant sampling of k-space, the corresponding field of view (FOV) is no longer well defined. As a result, foldover artifacts are blurred over a broad range and appear less severe. The more relaxed foldover artifact behavior and the densely sampled central k-space make trajectories of this type an ideal complement to autocalibrated parallel MRI (pMRI) techniques, such as generalized autocalibrating partially parallel acquisitions (GRAPPA). Although pMRI can benefit from non-Cartesian trajectories, this combination has not yet entered routine clinical use. One of the main reasons for this is the need for long reconstruction times due to the complex calculations necessary for non-Cartesian pMRI. In this work it is shown that one can significantly reduce the complexity of the calculations by exploiting a few specific properties of k-space-based pMRI.     

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.     

k-t BLAST reconstruction from non-Cartesian k-t space sampling.

Current implementations of k-t Broad-use Linear Acqusition Speed-up Technique (BLAST) require the sampling in k-t space to conform to a lattice. To permit the use of k-t BLAST with non-Cartesian sampling, an iterative reconstruction approach is proposed in this work. This method, which is based on the conjugate gradient (CG) method and gridding reconstruction principles, can efficiently handle data that are sampled along non-Cartesian trajectories in k-t space. The approach is demonstrated on prospectively gated radial and retrospectively gated Cartesian imaging. Compared to a sliding window (SW) reconstruction, the resulting image series exhibit lower artifact levels and improved temporal fidelity. The proposed approach thus allows investigators to combine the specific advantages of non-Cartesian imaging or retrospective gating with the acceleration provided by k-t BLAST.     

Non-Cartesian data reconstruction using GRAPPA operator gridding (GROG).

A novel approach that uses the concepts of parallel imaging to grid data sampled along a non-Cartesian trajectory using GRAPPA operator gridding (GROG) is described. GROG shifts any acquired data point to its nearest Cartesian location, thereby converting non-Cartesian to Cartesian data. Unlike other parallel imaging methods, GROG synthesizes the net weight for a shift in any direction from a single basis set of weights along the logical k-space directions. Given the vastly reduced size of the basis set, GROG calibration and reconstruction requires fewer operations and less calibration data than other parallel imaging methods for gridding. Instead of calculating and applying a density compensation function (DCF), GROG requires only local averaging, as the reconstructed points fall upon the Cartesian grid. Simulations are performed to demonstrate that the root mean square error (RMSE) values of images gridded with GROG are similar to those for images gridded using the gold-standard convolution gridding. Finally, GROG is compared to the convolution gridding technique using data sampled along radial, spiral, rosette, and BLADE (a.k.a. periodically rotated overlapping parallel lines with enhanced reconstruction [PROPELLER]) trajectories.     

Consistent non-cartesian off-axis MRI quality: calibrating and removing multiple sources of demodulation phase errors.

The consistency of off-axis MRI with non-Cartesian sequences across a large number of scanners is highly variable. Improper timing alignment of the gradient fields, data acquisition system, and real-time frequency demodulation reference signal, which are necessary for off-axis imaging, is an important source of this variability. In addition, eddy currents and anisotropic gradient delays cause deviations in k-space trajectories that in turn make the demodulation reference signals inaccurate. A method is presented to quickly measure the timing error in the frequency demodulation reference signal and separate it from anisotropic gradient delays. k-Space deviations, as measured with a previous gradient calibration technique, are shown to be a second source of demodulation phase errors that degrade image quality. Using the timing delay and k-space deviations, a retrospective phase correction is applied to each k-space sample before the data are regridded during reconstruction. The timing delays of four MR scanners were measured to be 4.2-7.5 micros below the manufacturer's suggested delay. Significant degradation in 3D radial (3D projection reconstruction (PR)) knee and breast images are retrospectively corrected while a partial prospective correction is applied for spiral imaging. The method allows for more consistent performance of non-Cartesian sequences across multiple scanners without operator intervention.     

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.     

Decomposed direct matrix inversion for fast non-cartesian SENSE reconstructions.

A new k-space direct matrix inversion (DMI) method is proposed here to accelerate non-Cartesian SENSE reconstructions. In this method a global k-space matrix equation is established on basic MRI principles, and the inverse of the global encoding matrix is found from a set of local matrix equations by taking advantage of the small extension of k-space coil maps. The DMI algorithm's efficiency is achieved by reloading the precalculated global inverse when the coil maps and trajectories remain unchanged, such as in dynamic studies. Phantom and human subject experiments were performed on a 1.5T scanner with a standard four-channel phased-array cardiac coil. Interleaved spiral trajectories were used to collect fully sampled and undersampled 3D raw data. The equivalence of the global k-space matrix equation to its image-space version, was verified via conjugate gradient (CG) iterative algorithms on a 2x undersampled phantom and numerical-model data sets. When applied to the 2x undersampled phantom and human-subject raw data, the decomposed DMI method produced images with small errors (< or = 3.9%) relative to the reference images obtained from the fully-sampled data, at a rate of 2 s per slice (excluding 4 min for precalculating the global inverse at an image size of 256 x 256). The DMI method may be useful for noise evaluations in parallel coil designs, dynamic MRI, and 3D sodium MRI with fixed coils and trajectories.     

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.     

Varying kernel-extent gridding reconstruction for undersampled variable-density spirals.

Nonuniform, non-Cartesian k-space trajectories enable fast scanning with reduced motion and flow artifacts. In such cases, the data are usually convolved with a kernel and resampled onto a Cartesian grid before reconstruction. For trajectories such as undersampled variable-density spirals, the mainlobe width of the kernel for undersampled high spatial frequencies has to be larger to limit the amount of aliasing energy. Continuously varying the kernel extent is time consuming. By dividing k-space into several annuli and using appropriate mainlobe widths for each, the aliasing energy and noise can be reduced at the expense of lower resolution towards the edge of the field of view (FOV). Resolution can instead be preserved at the center of the FOV, which is expected to be free of artifacts, without any artifact reduction. The image reconstructed from each annulus can be deapodized separately. The method can be applied to most k-space trajectories used in MRI.     

Self-calibrating GRAPPA operator gridding for radial and spiral trajectories.

Self-calibrating GRAPPA operator gridding (GROG) is a method by which non-Cartesian MRI data can be gridded using spatial information from a multichannel coil array without the need for an additional calibration dataset. Using self-calibrating GROG, the non-Cartesian datapoints are shifted to nearby k-space locations using parallel imaging weight sets determined from the datapoints themselves. GROG employs the GRAPPA Operator, a special formulation of the general reconstruction method GRAPPA, to perform these shifts. Although GROG can be used to grid undersampled datasets, it is important to note that this method uses parallel imaging only for gridding, and not to reconstruct artifact-free images from undersampled data. The innovation introduced here, namely, self-calibrating GROG, allows the shift operators to be calculated directly out of the non-Cartesian data themselves. This eliminates the need for an additional calibration dataset, which reduces the imaging time and also makes the GROG reconstruction more robust by removing possible inconsistencies between the calibration and non-Cartesian datasets. Simulated and in vivo examples of radial and spiral datasets gridded using self-calibrating GROG are compared to images gridded using the standard method of convolution gridding.     

Inherently self-calibrating non-Cartesian parallel imaging.

The use of self-calibrating techniques in parallel magnetic resonance imaging eliminates the need for coil sensitivity calibration scans and avoids potential mismatches between calibration scans and subsequent accelerated acquisitions (e.g., as a result of patient motion). Most examples of self-calibrating Cartesian parallel imaging techniques have required the use of modified k-space trajectories that are densely sampled at the center and more sparsely sampled in the periphery. However, spiral and radial trajectories offer inherent self-calibrating characteristics because of their densely sampled center. At no additional cost in acquisition time and with no modification in scanning protocols, in vivo coil sensitivity maps may be extracted from the densely sampled central region of k-space. This work demonstrates the feasibility of self-calibrated spiral and radial parallel imaging using a previously described iterative non-Cartesian sensitivity encoding algorithm.     

Motion Artifacts in fMRI: Comparison of 2DFT with PR and Spiral Scan Methods

Activation signals based on BOLD contrast changes consequent to neuronal stimulation typically produce cortical intensity differences of < 10% at 1.5T. Hemodynamically driven pulsation of the brain can cause highly pulsatile phase shifts, which in turn result in motion artifacts whose intensity is larger than the activation signals in 2DFT scan methods. This paper presents a theoretical and experimental comparison of the magnitude of such artifacts for 2DFT and two other methods using non-Cartesian k-space trajectories. It is shown that artifacts increase with TR for 2DFT methods, and that projection reconstruction (PR) and spiral methods have significantly reduced artifact intensities, because these trajectories collect low spatial frequencies with every view. The spiral technique is found to be superior in terms of efficiency and motion insensitivity.     

Reduction of reconstruction time for time-resolved spiral 3D contrast-enhanced magnetic resonance angiography using parallel computing.

Time-resolved 3D MRI with high spatial and temporal resolution can be achieved using spiral sampling and sliding-window reconstruction. Image reconstruction is computationally intensive because of the need for data regridding, a large number of temporal phases, and multiple RF receiver coils. Inhomogeneity blurring correction for spiral sampling further increases the computational work load by an order of magnitude, hindering the clinical utility of spiral trajectories. In this work the reconstruction time is reduced by a factor of >40 compared to reconstruction using a single processor. This is achieved by using a cluster of 32 commercial off-the-shelf computers, commodity networking hardware, and readily available software. The reconstruction system is demonstrated for time-resolved spiral contrast-enhanced (CE) peripheral MR angiography (MRA), and a reduction of reconstruction time from 80 min to 1.8 min is achieved.     

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.     

POCSENSE: POCS-based reconstruction for sensitivity encoded magnetic resonance imaging.

A novel method for iterative reconstruction of images from undersampled MRI data acquired by multiple receiver coil systems is presented. Based on Projection onto Convex Sets (POCS) formalism, the method for SENSitivity Encoded data reconstruction (POCSENSE) can be readily modified to include various linear and nonlinear reconstruction constraints. Such constraints may be beneficial for reconstructing highly and overcritically undersampled data sets to improve image quality. POCSENSE is conceptually simple and numerically efficient and can reconstruct images from data sampled on arbitrary k-space trajectories. The applicability of POCSENSE for image reconstruction with nonlinear constraining was demonstrated using a wide range of simulated and real MRI data.     

A brief review of parallel magnetic resonance imaging

Since the 1980s, the implementation of fast imaging methods and dedicated hardware for MRI scanners has reduced the image acquisition time from nearly an hour down to several seconds and has therefore enabled a widespread use of MRI in clinical diagnosis. Since this development, the greatest incremental gain in imaging speed has been provided by the development of parallel MRI (pMRI) techniques in late 1990s. Within the past 2 years, parallel imaging methods have become commercially available, which means that pMRI is now available for broad clinical use. In the clinical routine, virtually any MRI method can be enhanced by pMRI, allowing faster image acquisitions without any increased gradient system performance. In some cases pMRI can even result in a significant gain in image quality due to this faster acquisition. In this review article, the advantages and the disadvantages of pMRI in clinical applications are discussed and examples from many different daily applications are given.     

Dynamic three-dimensional undersampled data reconstruction employing temporal registration.

Dynamic 3D imaging is needed for many applications such as imaging of the heart, joints, and abdomen. For these, the contrast and resolution that magnetic resonance imaging (MRI) offers are desirable. Unfortunately, the long acquisition time of MRI limits its application. Several techniques have been proposed to shorten the scan time by undersampling the k-space. To recover the missing data they make assumptions about the object's motion, restricting it in space, spatial frequency, temporal frequency, or a combination of space and temporal frequency. These assumptions limit the applicability of each technique. In this work we propose a reconstruction technique based on a weaker complementary assumption that restricts the motion in time. The technique exploits the redundancy of information in the object domain by predicting time frames from frames where there is little motion. The proposed method is well suited for several applications, in particular for cardiac imaging, considering that the heart remains relatively still during an important fraction of the cardiac cycle, or joint imaging where the motion can easily be controlled. This paper presents the new technique and the results of applying it to knee and cardiac imaging. The results show that the new technique can effectively reconstruct dynamic images acquired with an undersampling factor of 5. The resulting images suffer from little temporal and spatial blurring, significantly better than a sliding window reconstruction. An important attraction of the technique is that it combines reconstruction and registration, thus providing not only the 3D images but also its motion quantification. The method can be adapted to non-Cartesian k-space trajectories and nonuniform undersampling patterns.     

3D hyperpolarized He-3 MRI of ventilation using a multi-echo projection acquisition.

A method is presented for high-resolution 3D imaging of the whole lung using inhaled hyperpolarized (HP) He-3 MR with multiple half-echo radial trajectories that can accelerate imaging through undersampling. A multiple half-echo radial trajectory can be used to reduce the level of artifact for undersampled 3D projection reconstruction (PR) imaging by increasing the amount of data acquired per unit time for HP He-3 lung imaging. The point spread functions (PSFs) for breath-held He-3 MRI using multiple half-echo trajectories were evaluated using simulations to predict the effects of T(2)* and gas diffusion on image quality. Results from PSF simulations were consistent with imaging results in volunteer studies showing improved image quality with increasing number of echoes using up to 8 half-echoes. The 8-half-echo acquisition is shown to accommodate lost breath-holds as short as 6 sec using a retrospective reconstruction at reduced resolution and also to allow reduced breath-hold time compared with an equivalent Cartesian trajectory. Furthermore, preliminary results from a 3D dynamic inhalation-exhalation maneuver are demonstrated using the 8-half-echo trajectory. Results demonstrate the first high-resolution 3D PR imaging of ventilation and respiratory dynamics in humans using HP He-3 MR.     

Off-resonance artifacts correction with convolution in k-space (ORACLE)

Off-resonance artifacts hinder the wider applicability of echo-planar imaging and non-Cartesian MRI methods such as radial and spiral. In this work, a general and rapid method is proposed for off-resonance artifacts correction based on data convolution in k-space. The acquired k-space is divided into multiple segments based on their acquisition times. Off-resonance-induced artifact within each segment is removed by applying a convolution kernel, which is the Fourier transform of an off-resonance correcting spatial phase modulation term. The field map is determined from the inverse Fourier transform of a basis kernel, which is calibrated from data fitting in k-space. The technique was demonstrated in phantom and in vivo studies for radial, spiral and echo-planar imaging datasets. For radial acquisitions, the proposed method allows the self-calibration of the field map from the imaging data, when an alternating view-angle ordering scheme is used. An additional advantage for off-resonance artifacts correction based on data convolution in k-space is the reusability of convolution kernels to images acquired with the same sequence but different contrasts.     

Density compensation functions for spiral MRI

In interleaved spiral MRI, an object's Fourier transform is sampled along a set of curved trajectories in the spatial frequency domain (k-space). An image of the object is then reconstructed, usually by interpolating the sampled Fourier data onto a Cartesian grid and applying the fast Fourier transform (FFT) algorithm. To obtain accurate results, it is necessary to account for the nonuniform density with which k-space is sampled. An analytic density compensation function (DCF) for spiral MRI, based on the Jacobian determinant for the transformation between Cartesian coordinates and the spiral sampling parameters of time and interleaf rotation angle, is derived in this paper, and the reconstruction accuracy achieved using this function is compared with that obtained using several previously published expressions. Various non-ideal conditions, including intersecting trajectories, are considered. The new DCF eliminated intensity cupping that was encountered in images reconstructed with other functions, and significantly reduced the level of artifact observed when unevenly spaced sampling trajectories, such as those achieved with trapezoidal gradient waveforms, were employed. Modified forms of this function were found to provide similar improvements when intersecting trajectories made the spiral-Cartesian transformation noninvertible, and when the shape of the spiral trajectory varied between interleaves.