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.     

Reduced phase encoding in spectroscopic imaging

The effect of different spatial-encoding (k-space) sampling distributions are evaluated for magnetic resonance spectroscopic imaging (MRSI) using Fourier reconstruction. Previously, most MRSI studies have used square or cubic k-space functions, symmetrically distributed. These studies examine the conventional k-space distribution with spherical distribution, and 1/2 k-space acquisition, using computer simulation studies of the MRSI acquisition for three spatial dimensions and experimental results. Results compare the spatial response function, Gibbs ringing effects, and signal contamination for different spatial-encoding distribution functions. Results indicate that spherical encoding, in comparison with cubic encoding, results in a modest improvement of the re sponse function with approximately equivalent spatial resolution for the same acquisition time. For spin-echo acquired data, reduced acquisition times can readily be obtained using 1/2 k-space methods, with a concomitant reduction in signal to noise ratio.     

Sampling errors in projection reconstruction MRI

Certain kinds of artifacts have been reported when projection reconstruction (PR) techniques are used in magnetic resonance imaging (MRI). These will occur if the spacing between samples in k-space is too large. It has been suggested that PR requires finer k-space sampling than does two-dimensional Fourier reconstruction, and that the usual Nyquist criterion is inadequate. This paper examines this problem, with the conclusion that the Nyquist sampling criterion is adequate to avoid aliasing effects, provided that a band-limited interpolation is used in k-space. This procedure is motivated by analysis of the PR technique as it is commonly implemented in x-ray computed tomography. A related problem is shown to be the construction of a filter function in k-space that gives proper weight to the low spatial frequencies. It is shown that a simple k filter does not satisfy this requirement, and a procedure for deriving a suitable filter is described. The methods are tested in simulated PR profiles of circular disks of two different sizes. It is shown that the combination of the two new methods gives virtually perfect reconstruction for disks up to the size implied by the Nyquist limit.     

A Method to measure arbitrary k-space trajectories for rapid MR imaging

A method to measure arbitrary k-space trajectories was developed to compensate for nonideal gradient performance during rapid magnetic resonance (MR) imaging with actively or nonactively shielded gradients at a magnetic field strength of 4.1 T. Accurate MR image reconstruction requires knowledge of the k-trajectory produced by the gradient waveforms during k-space sampling. Even with shielded gradients, residual eddy currents and imperfections in gradient amplifier performance can cause the true k-space trajectory to deviate from the ideal trajectory. The k-space determination was used for spiral gradient-echo imaging of the human brain. While individual calibrations are needed for new pulse sequences, the method of k-space determination can be used for any sequence of preparation pulses and readout gradient waveforms and should prove useful for other trajectories, including the rastered lines of echo-planar imaging.     

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.     

Motion‐adaptive spatio‐temporal regularization for accelerated dynamic MRI

Accelerated magnetic resonance imaging techniques reduce signal acquisition time by undersampling k‐space. A fundamental problem in accelerated magnetic resonance imaging is the recovery of quality images from undersampled k‐space data. Current state‐of‐the‐art recovery algorithms exploit the spatial and temporal structures in underlying images to improve the reconstruction quality. In recent years, compressed sensing theory has helped formulate mathematical principles and conditions that ensure recovery of (structured) sparse signals from undersampled, incoherent measurements. In this article, a new recovery algorithm, motion‐adaptive spatio‐temporal regularization, is presented that uses spatial and temporal structured sparsity of MR images in the compressed sensing framework to recover dynamic MR images from highly undersampled k‐space data. In contrast to existing algorithms, our proposed algorithm models temporal sparsity using motion‐adaptive linear transformations between neighboring images. The efficiency of motion‐adaptive spatio‐temporal regularization is demonstrated with experiments on cardiac magnetic resonance imaging for a range of reduction factors. Results are also compared with k‐t FOCUSS with motion estimation and compensation—another recently proposed recovery algorithm for dynamic magnetic resonance imaging. Magn Reson Med 70:800–812, 2013. © 2012 Wiley Periodicals, Inc.     

Comparison of k-space sampling schemes for multidimensional MR spectroscopic imaging

For clinical ³¹P MR spectroscopic imaging (MRSI) studies, where signal averaging is necessary, some improvement of sensitivity and spatial response function may be achieved by acquiring data over a spherical k-space volume and varying the number of averages acquired in proportion to the desired spatial filter. Eight different k-space sampling schemes are compared through simulations that provide graphs of the spatial response functions (SRF), and tabulations of voxel volumes, relative signal-to-noise ratios (SNR), and relative data collection efficiencies (SNR per unit volume over the same time). All schemes were based on practical experiments, each of which could be implemented in the same length of time. The results show that in comparison with cubic k-space sampling with the same number of signal averages at each point, spherical and acquisition-weighted k-space sampling can be used to achieve reduced Gibbs ringing along the principal axes directions, and thus reduced contamination from adjacent tissue in these directions, without degradation of voxel volume or SNR.     

The influence of radial undersampling schemes on compressed sensing reconstruction in breast MRI

Fast imaging applications in magnetic resonance imaging (MRI) frequently involve undersampling of k‐space data to achieve the desired temporal resolution. However, high temporal resolution images generated from undersampled data suffer from aliasing artifacts. In radial k‐space sampling, this manifests as undesirable streaks that obscure image detail. Compressed sensing reconstruction has been shown to reduce such streak artifacts, based on the assumption of image sparsity. Here, compressed sensing is implemented with three different radial sampling schemes (golden‐angle, bit‐reversed, and random sampling), which are compared over a range of spatiotemporal resolutions. The sampling methods are implemented in static scenarios where different undersampling patterns could be compared. Results from point spread function studies, simulations, phantom and in vivo experiments show that the choice of radial sampling pattern influences the quality of the final image reconstructed by the compressed sensing algorithm. While evenly undersampled radial trajectories are best for specific temporal resolutions, golden‐angle radial sampling results in the least overall error when various temporal resolutions are considered. Reduced temporal fluctuations from aliasing artifacts in golden‐angle sampling translates to improved compressed sensing reconstructions overall. Magn Reson Med, 2012. © 2011 Wiley Periodicals, Inc.     

Interleaved variable density sampling with a constrained parallel imaging reconstruction for dynamic contrast‐enhanced MR angiography

For MR applications such as contrast‐enhanced MR angiography, it is desirable to achieve simultaneously high spatial and temporal resolution. The current clinical standard uses view‐sharing methods combined with parallel imaging; however, this approach still provides limited spatial and temporal resolution. To improve on the clinical standard, we present an interleaved variable density (IVD) sampling method that pseudorandomly undersamples each individual frame of a 3D Cartesian k_y–k_z plane combined with parallel imaging acceleration. From this dataset, time‐resolved images are reconstructed with a method that combines parallel imaging with a multiplicative constraint. Total acceleration factors on the order of 20 are achieved for contrast‐enhanced MR angiography of the lower extremities, and improvements in temporal fidelity of the depiction of the contrast bolus passage are demonstrated relative to the clinical standard. Magn Reson Med, 2011. © 2011 Wiley‐Liss, Inc.     

Temporal resolution improvement in dynamic imaging

In some dynamic imaging applications, only a fraction, 1/n, of the field of view (FOV) may show considerable change during the motion cycle. A method is presented that improves the temporal resolution for a dynamic region by a factor, n, while maintaining spatial resolution at a cost of √n in signal-to-noise ratio (SNR). Temporal resolution is improved, or alternatively, total imaging time is reduced by reducing the number of phase encodes acquired for each temporal frame by 1/n. To eliminate aliasing, a representation of the signal from the static outer portion of the FOV is constructed using all the raw data. The k-space data derived from this representation is subtracted from the original data sets, and the differences correspond to the dynamic portion of the FOV. Improved resolution results are presented in phantom studies, and in vivo phase contrast quantitative flow imaging.     

Improving the temporal resolution of functional MR imaging using keyhole techniques

Using a keyhole technique, it is shown that the data acquisition rate of gradient-echo imaging for functional MRI (fMRI) studies can be increased substantially. The resulting enhancement of the temporal resolution of fMRIs was accomplished without modifying the hardware of a conventional MRI system. High spatial resolution fMRI images were first collected with conventional full k-space acquisition and image reconstruction. Using the same data set, simulation reconstruction using the keyhole principle and zero-padding were performed for comparison with the full k-space reconstruction. No significant changes were found for fMRI images generated from the keyhole technique with a data sharing profile of 50% of the k-space. As k-space data sharing profiles increased to 75 and 87.5%, the keyhole fMRI images began to show only modest changes in activation intensity and area compared with the standard images. In contrast, zero-padding fMRI images produced a significant disparity both in activation intensity and area relative to the truly high-resolution fMRI images. The keyhole technique's ability to retain the intensity and area of fMRI information, while substantially reducing acquisition time, makes it a promising method for fMRI studies.     

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.     

Variable‐density spiral‐in/out functional magnetic resonance imaging

A variable‐density spiral k‐space trajectory is introduced for brain functional magnetic resonance imaging. The proposed spiral trajectory consists of an Archimedean spiral from the k‐space origin to an arbitrary fraction r of the maximum k‐space radius, extending beyond this point with a variable‐density spiral in which the sampling density decreases as the k‐space radius increases. It, therefore, permits a reduction in readout time at the expense of undersampling only the high spatial frequencies, in which the energy in T₂*‐weighted brain images is low. The trajectory was implemented in a two‐dimensional spiral‐in/out sequence, and single‐shot high‐resolution (1.72 × 1.72 mm² in‐plane) functional magnetic resonance imaging data were acquired from human volunteers. Compared with a two‐shot fully Archimedean spiral sequence with the same spatial coverage and total scan time, the variable‐density sequence yielded greater activation magnitudes with improved temporal efficiency and minor artifacts. Magn Reson Med, 2011. © 2010 Wiley‐Liss, Inc.     

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.     

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 rapid 2D time-resolved variable-rate k-space sampling MR technique for passive catheter tracking during endovascular procedures

A new, fast, 2D MR imaging technique allowing passive catheter visualization adequate for use as a tool for guiding the movement of a catheter during endovascular procedures is described. This imaging technique samples low spatial frequencies more often than high spatial frequencies; it also uses both k-space view sharing and temporal interpolation. Unlike other techniques for passive visualization that exploit magnetic-susceptibility–induced artifacts, we have adopted a strategy that takes advantage of the T₁-shortening effect of paramagnetic contrast agents, such as Gd-DTPA and a projection dephaser. This not only permits visualization of the entire catheter length but also minimizes the risk of intravascular heating. Using this method, a temporal frame rate of up to eight images per second and a tip localization accuracy of ± 1mm (root mean square difference) can be achieved.     

Fast interleaved echo-planar imaging with navigator: High resolution anatomic and functional images at 4 tesla

Echo-planar Imaging (EPI) Is sensitive to magnetic field inhomogeneities, which lead to signal loss and geometric distortions of the image. Magnetic field inhomogeneities induced by susceptibility differences, as encountered in the human body, increase with the magnetic field strength, thus, complicating implementation of high resolution EPI techniques on high magnetic field systems. These problems were overcome by using a fast multishot high resolution EPI method that uses variable flip angles, center-out k-space sampling, and navigator echoes. This approach maximizes signal-to-noise ratio, reduces flow artifacts, and permits correction of intersegment amplitude and phase variations, providing high spatial and temporal resolution. This scheme can be implemented with a single magnetization preparation for contrast that precedes the segments. The utility of this ultrafast segmented EPI technique with navigator is demonstrated for anatomic and functional imaging studies on the human brain at 4 T.     

Two‐dimensional radial acquisition technique with density adaption in sodium MRI

Conventional 2D radial projections suffer from losses in signal‐to‐noise ratio efficiency because of the nonuniform k‐space sampling. In this study, a 2D projection reconstruction method with variable gradient amplitudes is presented to cover the k‐space uniformly. The gradient is designed to keep the average sampling density constant. By this, signal‐to‐noise ratio is increased, and the linear form of the radial trajectory is kept. The simple gradient design and low hardware requirements in respect of slew rate allow an easy implementation at MR scanners. Measurements with the density‐adapted 2D radial trajectory were compared with the conventional projection reconstruction method. It is demonstrated that the density‐adapted 2D radial trajectory technique provides higher signal‐to‐noise ratio (up to 28% in brain tissue), less blurring, and fewer artifacts in the presence of magnetic field inhomogeneities than imaging with the conventional 2D radial trajectory scheme. The presented sequence is well‐suited for electrocardiographically gated sodium heart MRI and other applications with short relaxation times. Magn Reson Med, 2010. © 2010 Wiley‐Liss, Inc.     

A Modified EPI sequence for high‐resolution imaging at ultra‐short echo time

A robust modification of echo‐planar imaging dubbed double‐shot echo‐planar imaging with center‐out trajectories and intrinsic navigation (DEPICTING) is proposed, which permits imaging at ultra‐short echo time. The k‐space data is sampled by two center‐out trajectories with a minimal delay achieving a temporal efficiency similar to conventional single‐shot echo‐planar imaging. Intersegment phase and intensity imperfections are corrected by exploiting the intrinsic navigator information from both central lines, which are subsequently averaged for image reconstruction. Phase errors induced by inhomogeneities of the main magnetic field are corrected in k‐space, recovering the superior point‐spread function achieved with center‐out trajectories. The minimal echo time (<2 msec) is nearly independent of the acquisition matrix permitting applications, which simultaneously require high spatial and temporal resolution. Examples of demonstrated applications include anatomical imaging, BOLD‐based functional brain mapping, and quantitative perfusion imaging. Magn Reson Med, 2010. © 2010 Wiley‐Liss, Inc.     

High efficiency multishot interleaved spiral‐in/out: Acquisition for high‐resolution BOLD fMRI

Growing demand for high spatial resolution blood oxygenation level dependent (BOLD) functional magnetic resonance imaging faces a challenge of the spatial resolution versus coverage or temporal resolution tradeoff, which can be addressed by methods that afford increased acquisition efficiency. Spiral acquisition trajectories have been shown to be superior to currently prevalent echo‐planar imaging in terms of acquisition efficiency, and high spatial resolution can be achieved by employing multiple‐shot spiral acquisition. The interleaved spiral in/out trajectory is preferred over spiral‐in due to increased BOLD signal contrast‐to‐noise ratio (CNR) and higher acquisition efficiency than that of spiral‐out or noninterleaved spiral in/out trajectories (Law & Glover. Magn Reson Med 2009; 62:829–834.), but to date applicability of the multishot interleaved spiral in/out for high spatial resolution imaging has not been studied. Herein we propose multishot interleaved spiral in/out acquisition and investigate its applicability for high spatial resolution BOLD functional magnetic resonance imaging. Images reconstructed from interleaved spiral‐in and ‐out trajectories possess artifacts caused by differences in T2* decay, off‐resonance, and k‐space errors associated with the two trajectories. We analyze the associated errors and demonstrate that application of conjugate phase reconstruction and spectral filtering can substantially mitigate these image artifacts. After applying these processing steps, the multishot interleaved spiral in/out pulse sequence yields high BOLD CNR images at in‐plane resolution below 1 × 1 mm while preserving acceptable temporal resolution (4 s) and brain coverage (15 slices of 2 mm thickness). Moreover, this method yields sufficient BOLD CNR at 1.5 mm isotropic resolution for detection of activation in hippocampus associated with cognitive tasks (Stern memory task). The multishot interleaved spiral in/out acquisition is a promising technique for high spatial resolution BOLD functional magnetic resonance imaging applications. Magn Reson Med 70:420–428, 2013. © 2012 Wiley Periodicals, Inc.