Frequency resolved single-shot MR imaging using stochastic k-space trajectories

A new single-shot stochastic imaging technique with a random k-space path that provides very selective filtering with respect to chemical shift or off-resonance signals of the investigated tissue is proposed. It is demonstrated that in stochastic imaging only on-resonance compartments are visible whereas frequency shifted compartments cancel to noise that is distributed over the whole image. This method can be used as a single-shot chemical shift selective imaging technique and allows to calculate frequency resolved spectra for each spatial position of the image based on a single signal aquisition. The single-shot stochastic imaging sequence makes high demands on the gradient system and the theoretical k-space trajectory is distorted by imperfect gradient performance. Therefore an additional k-space guided imaging technique that uses the true, measured k-space trajectory to correct artifacts generated by eddy currents and delay times of the rapid switched gradients is presented. In vitro and in vivo measurements demonstrate the successful implementation of single-shot stochastic imaging on a conventional MR scanner with unshielded gradient systems.     

Analytical resolution and noise characteristics of linearly reconstructed magnetic resonance data with arbitraryk-space sampling

The effects of time-varying readout gradients and data sampling with variable dwell times in magnetic resonance imaging are examined. General reconstruction formulas are given for linear reconstruction with even k-space weighting. Closed analytic expressions for estimator variance are given for data sampling during arbitrary gradient waveforms with both uniform k_x step size and nonuniform k_x step size. It is shown that estimator variance increases (the signal-to-noise ratio decreases) for nonconstant gradient waveforms. It is also shown that estimator variance is greater for constant k-space sampling strategies than for constant time sampling at the Nyquist rate. Data collected during a triangular readout gradient waveform, with either constant time or constant k-space sampling, versus conventional (constant gradient) collection confirms theoretical predictions for estimator variance. The benefits of collecting data while the readout gradient is ramping up from and down to zero are discussed.     

Compensation of multi-dimensional selective excitation pulses using measured k-space trajectories

Multidimensional spatially selective excitation pulses rely on the accuracy of gradient waveforms to achieve desired excitation volumes. Unfortunately, the high gradient slew-rates and magnitudes required by these pulses often lead to distortion of the waveforms produced by imaging systems resulting in poor selection profiles. In this paper, a k-space calibration procedure, used to determine the actual trajectory produced by the scanner's field gradients, is extended to two spatial dimensions. This measured information is then incorporated in a selective excitation design technique for correcting the RF pulse envelopes to compensate for gradient waveform induced distortion of the excitation volumes.     

Interleaved asymmetric echo-planar imaging

A version of interleaved echo-planar imaging (EPI) is presented in which only one polarity of the readout gradient is used for signal acquisition to avoid ghosting artifacts. Two possible forms of the phase encoding gradient, blipped and constant, are discussed. With the constant phase encoding, interleaving of partial trajectories in the Fourier domain (k-space) is controlled automatically by the echo train delay. The constant phase encoding gradient introduces a shear distortion of the k-space grid. A modification of the reconstruction procedure is given which corrects for this effect. The method provides a 128 × 128 image in 1 s on a clinical system with standard gradients.     

Gradient characterization using a Fourier-transform technique

This paper describes a technique for characterizing the gradient subsystem of a magnetic resonance (MR) system. The technique uses a Fourier-transform analysis to directly measure the k-space trajectory produced by an arbitrary gradient waveform. In addition, the method can be easily extended to multiple dimensions and can be adapted to measuring residual gradient effects such as eddy currents. Several examples of gradient waveform and eddy-current measurements are presented. Also, it is demonstrated how the eddy-current measurements can be parameterized with an impulse-response formalism for later use in system tuning. When compared to a peak-fitting analysis, this technique provides a more direct extraction of the k-space measurements, which reduces the possibility of analysis error. This approach also has several advantages as compared to the conventional eddy-current measurement technique, including the ability to measure very short time constant effects.     

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.     

A Velocity k-Space Analysis of Flow Effects in Echo-Planar and Spiral Imaging

A velocity k-space formalism facilitates the analysis of flow effects for imaging sequences involving time-varying gradients such as echo-planar and spiral. For each sequence, the velocity k-space trajectory can be represented by k_v (k)_r; that is, its velocity-frequency (k_r) position as a function of spatial-frequency (k_r) position. In an echo-planar sequence, k_r is discontinuous and asymmetric. However, in a spiral sequence, k_r is smoothly varying, circularly symmetric, and small near the k_r origin. To compare the effects of these trajectory differences, simulated images were generated by computing the k-space values for an in-plane vessel with parabolic flow. Whereas the resulting echo-planar images demonstrate distortions and ghosting that depend on the vessel orientation, the spiral images exhibit minimal artifacts.     

Excitation of arbitrary shapes by gradient optimized random walk in discrete k-space

A new technique for the excitation of arbitrary shapes is proposed. It is based on a parallel sequence of small tip angle RF pulses and gradient pulses. The small tip angle rotations co-add yielding a 90° excitation pulse within the selected excitation profile while outside the profile, the rotations cancel each other. A full theory of the completely arbitrary regional volume excitation (CARVE) method is presented and experimentally verified. In CARVE, k-space is discrete because the RF is applied in pulses. The discrete character of k-space permits an arbitrary trajectory for the k-space walk. The optimal random trajectory is found by minimizing the gradient load using simulated annealing. It is shown, both theoretically and experimentally, that such a trajectory is much better than any other systematic or random trajectory in k-space.     

Fast Three Dimensional Magnetic Resonance Imaging

To reduce the scan time in three-dimensional (3D) imaging, the authors consider alternative trajectories for traversing k-space. They differ from traditional 3D trajectories, such as 3DFT, in that they employ time-varying gradients allowing longer readouts and in turn a reduced scan time. Some of these trajectories reduce by an order of magnitude the number of excitations compared with 3DFT and provide flexibility for trading off signal-to-noise ratio for scan time. Other concerns are the minimum echo time and flow/motion properties. As examples, the authors show two applications: A 3D data set of the head (field of view of 30 x 30 x 7.5 cm and resolution of 1.5 x 1.5 x 1.5 mm) acquired in 56 s using a stack of spirals in 3D k-space; and a 3D movie of the heart (20 x 20 x 20 cm field of view, 2 x 2 x 2 mm resolution, and 16 time frames per cardiac cycle) acquired in 11 min using a cones trajectory.     

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.     

A new design and rationale for 3D orthogonally oversampled k‐space trajectories

A novel center‐out 3D trajectory for sampling magnetic resonance data is presented. The trajectory set is based on a single Fermat spiral waveform, which is substantially undersampled in the center of k‐space. Multiple trajectories are combined in a “stacked cone” configuration to give very uniform sampling throughout a “hub,” which is very efficient in terms of gradient performance and uniform trajectory spacing. The fermat looped, orthogonally encoded trajectories (FLORET) design produces less gradient‐efficient trajectories near the poles, so multiple orthogonal hub designs are shown. These multihub designs oversample k‐space twice with orthogonal trajectories, which gives unique properties but also doubles the minimum scan time for critical sampling of k‐space. The trajectory is shown to be much more efficient than the conventional stack of cones trajectory, and has nearly the same signal‐to‐noise ratio efficiency (but twice the minimum scan time) as a stack of spirals trajectory. As a center‐out trajectory, it provides a shorter minimum echo time than stack of spirals, and its spherical k‐space coverage can dramatically reduce Gibbs ringing. Magn Reson Med, 2011. © 2011 Wiley Periodicals, Inc.     

The effect of high performance gradients on fast gradient echo imaging

The effect of gradient system performance on segmented k-space gradient echo imaging is presented. Three cases were investigated. First, an ideal system that has infinite slew rates and unlimited maximum gradient strengths was considered. Second, a “high speed” imaging system (2.3 (G/cm), 23 (G/cm)/ms) was considered. These two cases were compared with a “conventional” imaging system (1(G/cm), 1.67 (G/cm)/ms). It was found that substantial increases in SNR can be achieved (≈? 45%) by using high speed versus a conventional gradient system, for a TR of 6 ms. For trapezoidal gradient waveforms, there exists an optimum maximum gradient strength for a given slew rate, and any increase in gradient strength above this optimum will not be utilized by an optimized sequence. These studies have shown that increasing TR without decreasing the bandwidth is not a good way to increase SNR for constant scan time.     

MRI using radiofrequency magnetic field phase gradients

Conventionally, MR images are formed by applying gradients to the main static magnetic field (B₀). However, the B₀ gradient equipment is expensive, power‐hungry, complex, and noisy and can induce eddy currents in nearby conducting structures, including the patient. Here, we describe a new silent, B₀ gradient‐free MRI principle, Transmit Array Spatial Encoding (TRASE), based on phase gradients of the radio‐frequency (RF) field. RF phase gradients offer a new method of k‐space traversal. Echo trains using at least two different RF phase gradients allow spin phase to accumulate, causing k‐space traversal. Two such RF fields provide one‐dimensional imaging and three are sufficient for two‐dimensional imaging. Since TRASE is a k‐space method, analogues of many conventional pulse sequences are possible. Experimental results demonstrate one‐dimensional and two‐dimensional RF MRI and slice selection using a single‐channel, transmit/receive, 0.2 T, permanent magnet, human MR system. The experimentally demonstrated spatial resolution is much higher than that provided by RF receive coil array sensitivity encoding alone but lower than generally achievable with B₀ gradients. Potential applications are those in which one or more of the features of simplified equipment, lower costs, silent MRI, or the different physics of the image formation process are particularly advantageous. Magn Reson Med, 2010. © 2009 Wiley‐Liss, Inc.     

High-Field MR microscopy using fast spin-echoes

Fast spin-echo imaging has been investigated with attention to the requirements and opportunities for high-field MR microscopy. Two-and three-dimensional versions were implemented at 2.0 T, 7.1 T, and 9.4 T. At these fields, at least eight echoes were collectable with a 10 ms TE from fixed tissue specimens and living animals, giving an eightfold improvement in imaging efficiency. To reduce the phase-encoding gradient amplitude and its duty cycle, a modified pulse sequence with phase accumulation was developed. Images obtained using this pulse sequence exhibited comparable signal-to-noise (SNR) to those obtained from the conventional fast spin-echo pulse sequences. Signal losses due to imperfections in RF pulses and lack of phase rewinders were offset in this sequence by reduced diffusion losses incurred with the gradients required for MR microscopy. Image SNR, contrast, edge effects and spatial resolution for three k-space sampling schemes were studied experimentally and theoretically. One method of sampling k-space, 4-GROUP FSE, was found particularly useful in producing varied T₂ contrast at high field. Two-dimensional images of tissue specimens were obtained in a total acquisition time of 1 to 2 min with in-plane resolution between 30 to 70 μm, and 3D images with 256³ arrays were acquired from fixed rat brain tissue (isotropic voxel = 70 μm) and a living rat (isotropic voxel = 117 μm) in∼4.5 h.     

Effects of polar sampling in k-space

Magnetic resonance imaging allows numerous k-space sampling schemes such as cartesian, polar, spherical, and other non-rectilinear trajectories. Non-rectilinear MR acquisitions permit fast scan times and can suppress motion artifacts. Still, these sampling schemes may adversely affect the image characteristics due to aliasing. Here, the Fourier aliasing effects of uniform polar sampling, i.e., equally spaced radial and azimuthal samples, are explained from the principal point spread function (PSF). The principal PSF is determined by assuming equally spaced concentric ring samples in k-space. The radial effects such as replication, smearing, truncation artifacts, and sampling requirements, are characterized based on the PSF.     

Calculation of signal intensities in hybrid sequences for fast NMR imaging

A number of techniques that recently have been used for fast NMR-imaging are based on a hybrid sequence of echo planar imaging (EPI) and FLASH imaging: after each NMR excitation several k-space lines are measured. The complete k-space is covered by performance of several excitations. It has been observed that there is usually an optimal hybrid sequence that maximizes the signal-to-noise ratio. In this work, a method is presented that allows a determination of the optimal sequence as a function of the relaxation times T₁ and T₂^*.     

MRI using piecewise-linear spiral trajectory

A new generation of high power gradient systems which allow much faster MR imaging as well as shorter echo times has recently become available. Some of these high-speed gradient systems impose limits on the percentage of time during which the gradient can change in amplitude (slewing duty cycle). While this limitation may be immaterial to many 2DFT and echo planar imaging methods, a traditional circular spiral trajectory is difficult to use on these systems because its gradient waveforms change during the entire course of the trajectory so that the slewing duty cycle during the readout period is 100%. We describe a piecewise-linear spiral trajectory which is composed of linear segments and rounded corners. This trajectory reduces the slewing duty cycle while maintaining the desirable imaging properties of circular spirals including interleaving by simple gradient rotation. For one representative example, the slewing duty cycle is reduced to 46%. A conventional gridding method was used for image reconstruction, but a new numerical algorithm to calculate the density compensation factor was required. Use of piecewise-linear spiral trajectories reduces the impact imposed by limited gradient slewing duty cycle.     

Volumetric spectroscopic imaging with spiral-based k-space trajectories

Spiral-based k-space trajectories were applied in a spectroscopic imaging sequence with time-varying readout gradients to collect volumetric chemical shift information. In addition to spectroscopic imaging of low signal-to-noise ratio (SNR) brain metabolites, the spiral trajectories were used to rapidly collect reference signals from the high SNR water signal to automatically phase the spectra and to aid the reconstruction of metabolite maps. Spectral-spatial pulses were used for excitation and water suppression. The pulses were designed to achieve stable phase profiles in the presence of up to 20% variation in the radiofrequency field. A gridding algorithm was used to resample the data onto a rectilinear grid before fast Fourier transforms. This method was demonstrated by in vivo imaging of brain metabolites at 1.5 T with 10 slices of 18 × 18 pixels each. Nominal voxel size was 1.1 cc, spectral bandwidth was 400 Hz, scan time was 18 min for the metabolite scan and 3 min for the reference scan.     

Fast three dimensional sodium imaging

An efficient scheme for fast three dimensional acquisition of sodium MR images is described. This scheme relies on the use of three dimensional k-space trajectories with constant sample density to achieve significant (60–70%) reductions in total data acquisition time over conventional projection imaging schemes. The performance of this data acquisition scheme is demonstrated with acquisition of sodium data sets on phantoms and normal human volunteers at 1.5 and 3.0 Tesla. The experimental results demonstrate that high quality three dimensional sodium images (0.2 cc voxel size, 10:1 signal-to-noise ratio) can be acquired at clinical field strengths (1.5 Tesla) in under 10 min.     

Measurement of the point spread function in MRI using constant time imaging

The point spread function is a fundamental property of magnetic resonance imaging methods that affects image quality and spatial resolution. The point spread function is difficult to measure precisely in magnetic resonance even with the use of carefully designed phantoms, and it is difficult to calculate this function for complex sequences such as echo-planar imaging. This report describes a method that measures the point spread function with high spatial resolution at each pixel in samples of uniform intensity distribution. This method uses additional phase encoding gradients before the echo-planar acquisition that are constant in length but vary in amplitude. The additional gradients are applied to image the contents within each individual voxel. This method has been used to measure the point spread function for echo-planar imaging to demonstrate the effects of limited k-space sampling and transverse relaxation, as well as the effects of object motion. By considering the displacement of the point spread function, local distortions due to susceptibility and chemical shift effects have been quantified and corrected. The method allows rapid assessment of the point spread function in echo-planar imaging, in vivo, and may also be applied to other rapid imaging sequences that can be modified to include these additional phase encoding gradients.