Gradient Nonlinearity Research Articles

Effects of gradient non‐linearity correction and intensity non‐uniformity correction in longitudinal studies using structural image evaluation using normalization of atrophy (SIENA)

To evaluate the effects of gradient nonlinearity correction and intensity nonuniformity correction on longitudinal (two-year) changes in global and regional brain volumes. A total of 208 subjects (70 females and 138 males, age range = 38.1-83.0 years) were included in this study. Each subject was scanned twice, at an interval of approximately two years (range = 1.5-2.3 years). Three-dimensional fast spoiled-gradient recalled acquisition in the steady state (3D-FSPGR) images corrected for gradient nonlinearity and/or intensity nonuniformity were compared with uncorrected 3D-FSPGR images with use of structural image evaluation using normalization of atrophy 2.6 (SIENA). The mean absolute deviations of percentage brain volume change (PBVC) values in the gradient nonlinearity +/- intensity nonuniformity corrected images were significantly less than that in the uncorrected images, and the difference in the mean absolute deviation of PBVC was the most significant between the uncorrected images and the images corrected for both gradient nonlinearity and intensity nonuniformity. Voxel-wise comparisons showed large significant differences between the uncorrected images and the corrected images. Correction for gradient nonlinearity and intensity nonuniformity reduces the variance of measured longitudinal changes in brain volumes and will improve accuracy for detecting subtle brain changes.

Reduction of fast spin echo cusp artifact using a slice‐tilting gradient

A "featherlike" artifact, termed a cusp artifact, is sometimes seen along the phase-encoding direction in sagittal or coronal fast spin echo images. This artifact arises from the spins, at a location distant from the magnet isocenter, that are excited and aliased to the field of view because their precession frequency is similar to those at the isocenter. Such a situation is created due to a combination of excessive gradient nonlinearity and rapid change of the main magnetic field near the edge of the magnet where the artifact-producing spins are located. A novel technique is proposed to reduce this artifact, in which a fast spin echo pulse sequence is modified to slightly tilt the slice selected by the radiofrequency excitation pulse away from the slice selected by the radiofrequency refocusing pulses. At the edge of the field of view, the incomplete overlap between the slices selected by the excitation and refocusing pulses effectively reduces the signals from the artifact-prone region. In contrast, the slices overlap substantially within the field of view so that the signals are largely retained. This slice-tilting technique has been implemented on two commercial MRI scanners operating at 3.0 T and 1.5 T, respectively, and evaluated on phantoms and human spine and extremities using clinical protocols. Both phantom and human results showed that the technique decreased the strength of the cusp artifact by at least 65% and substantially limited the spatial extent of the artifact. This technique, which can be further enhanced by a simple postprocessing step, offers significant advantages over a number of other techniques for reducing the fast spin echo cusp artifact. It can be implemented on virtually any scanner without hardware modification, complicated calibration, sophisticated image reconstruction, or patient-handling alteration.

3-D visualizations of coastal bathymetry by utilization of airborne TOPSAR polarized data

Multi-frequency C and L bands in the TOPSAR data have been utilized to reconstruct three-dimensional (3-D) bathymetry pattern. The main objective of this study is to utilize fuzzy arithmetic to reduce the errors arising from speckle in synthetic aperture radar (SAR) data when constructing ocean bathymetry from polarized SAR data. In doing so, two 3-D surface models, the Volterra algorithm and a fuzzy B-spline (FBS) algorithm, which construct a global topological structure between the data points, were used to support an approximation to the real surface. Volterra algorithm was used to express the non-linearity of TOPSAR data intensity gradient based on the action balance equation (ABC). In this context, a first-order kernel of Volterra algorithm was used to express ABC equation. The inverse of Volterra algorithm then performed to simulate 2-D current velocities from CVV and LHH band. Furthermore, the 2-D continuity equation then used to estimate the water depth. In order to reconstruct 3-D bathymetry pattern, the FBS has been performed to water depth information which was estimated from 2-D continuity equation. The best reconstruction of coastal bathymetry of the test site in Kuala Terengganu, Malaysia, was obtained with polarized L and C bands SAR acquired with HH and VV polarizations, respectively. With 10 m spatial resolution of TOPSAR data, bias of –0.004 m, the standard error mean of 0.023 m, r 2 value of 0.95, and 90% confidence intervals in depth determination was obtained with LHH band.

Voxel-based analysis of the diffusion tensor

Diffusion tensor imaging (DTI) has provided important insights into the neurobiological basis for normal development and aging and various disease processes in the central nervous system. The aim of this article is to review the current protocols for DTI acquisition and preprocessing and statistical testing for a voxelwise analysis of DTI, focused on statistical parametric mapping (SPM) and tract-based spatial statistics (TBSS). We tested the effects of distortion correction induced by gradient nonlinearity on fractional anisotropy (FA) maps or FA skeletons processed via two SPM-based methods (coregistration and FA template methods), or TBSS-based method, respectively. With two SPM-based methods, we found similar results in some points (e.g., significant FA elevation for uncorrected images in anterior-dominant white matter and for corrected images in bilateral middle cerebellar peduncles) and different results in other points (e.g., significantly larger FA for corrected images with coregistration method, but significantly smaller with FA template method in bilateral internal capsules, extending to corona radiata, and semioval centers). In contrast, there was no area with significant difference between uncorrected and corrected FA skeletons with TBSS-based method. The discrepancy among these results was not explained in full, but possible explanations were misregistration and smoothing for the SPM-based methods and insensitivity to FA changes outside the local centers of white matter bundles for TBSS-based method.

Implications of tissue magnetic susceptibility-related distortion on the rotating magnet in an MR-linac design

One of the recently published concepts that combine the soft-tissue imaging capabilities of MRI with external beam radiotherapy involves the rigid coupling of a linac with a rotating biplanar low-field MR imaging system. While such a system would prevent possible image distortion resulting from relative motion between the magnet and the linac, the rotation of the magnet around the patient can itself introduce possibilities for image distortion that need to be addressed. While there are straightforward techniques in the literature for correcting distortions from gradient nonlinearities and nonuniform magnetic fields during image reconstruction, the correction of distortions related to tissue magnetic susceptibility is more complex. This work investigates the extent of this latter distortion type under the regime of a rotating magnetic field. CT images covering patient anatomy in the head, lung, and male pelvic regions were obtained and segmented into components of air, bone, and soft tissue. Each of these three components was assigned bulk magnetic susceptibility values in accordance with those found in the literature. A finite-difference algorithm was then implemented to solve for magnetic field distortion maps should the anatomies be placed in the uniform polarizing field of an MR system. The algorithm was repeated multiple times as the polarizing field was rotated axially about the virtual patient in 15 degrees increments. In this way, a map of maximum distortion, and the range of distortion as the magnetic field is rotated about each anatomical region could be determined. The consequence of these susceptibility distortions in terms of geometric signal shift was calculated for 0.2 T, as well as another low-field system (0.5 T), and a higher field 1.5 T system for comparison, using the assumption of a frequency encoding gradient strength of 5 mT/m. At 0.2 T, the susceptibility-related distortion was limited to less than 0.5 mm given an encoding gradient strength of 5 mT/m or higher. To maintain this same level of geometric accuracy, the 0.5 T system would require a moderately higher minimum gradient strength of 11 mT/m, and at a typical MR field strength of 1.5 T this minimum gradient strength would increase to 33 mT/m. The influence of magnetic susceptibility on mean frequency shift as the field orientation was rotated was also investigated and found to account for less than half a millimeter at 1.5 T, and negligible for low-field systems. A study of three sites (head, lung, and prostate) that are vulnerable to magnetic susceptibility-related distortions were studied, and showed that in the context of a rotating polarizing magnet, low-field systems can maintain geometric accuracy of 0.5 mm with at most moderate limitations on sequence parameters. This conclusion will likely apply only to endogenous tissues, as implanted materials such as titanium can create field distortions much in excess of what may normally be induced in the body. Items containing such materials (hip prostheses, for example) will require individual scrutiny.

Regularity for a fourth-order critical equation with gradient nonlinearity

Given Ω a smooth bounded domain of R n , n ⩾ 3 , we consider functions u ∈ H 2 , 0 2 ( Ω ) that are weak solutions to the equation Δ 2 u + a u = − div ( f | x | s | ∇ u | 2 ⋆ − 2 ∇ u ) in Ω , where 2 ⋆ : = 2 ( n − s ) n − 2 , s ∈ [ 0 , 2 ) and a , f ∈ C ∞ ( Ω ¯ ) . In this article, we prove the maximal regularity of solutions to the above equation, depending on the value of s ∈ [ 0 , 2 ) and the relative position of Ω with respect to the origin. In particular, the solutions are in C 4 ( Ω ¯ ) when s = 0 .

A Priori and A Posteriori Analysis of Mixed Finite Element Methods for Nonlinear Elliptic Equations

We study the mixed finite element approximation of the second order elliptic problem with gradient nonlinearities in two and three space dimensions. Existence and uniqueness of the approximate solution are proved and optimal order a priori error estimates in $L^m(\Omega)$ are obtained. Also, reliable and efficient a posteriori error estimators measured in the $L^m(\Omega)$-norm are derived. Numerical examples are provided to illustrate the performance of the proposed estimator.

Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena

We consider a one-dimensional semilinear parabolic equation with a gradient nonlinearity. We provide a complete classification of large time behavior of the classical solutions u: either the space derivative u., blows up in finite time (with u itself remaining bounded), or u is global and converges in C-1 norm to the unique steady state. The main difficulty is to prove C-1 boundedness of all global solutions. To do so, we explicitly compute a nontrivial Lyapunov functional by carrying out the method of Zelenyak. After deriving precise estimates on the solutions and on the Lyapunov functional, we proceed by contradiction by showing that any C-1 unbounded global solution should converge to a singular stationary solution, which does not exist. As a consequence of our results, we exhibit the following interesting situation: the trajectories starting from some bounded set of initial data in C-1 describe an unbounded set, although each of them is individually bounded and converges to the tame limit; the existence time T* is not a continuous function of the initial data.

A two‐step scheme for distortion rectification of magnetic resonance images

The aim of this work is to demonstrate a complete, robust, and time-efficient method for distortion correction of magnetic resonance (MR) images. It is well known that MR images suffer from both machine-related spatial distortions [gradient nonlinearity and main field (B0) inhomogeneity] and patient-related spatial distortions (susceptibility and chemical shift artifacts), and growing interest in the area of MR-based radiotherapy treatment planning has put new requirements on the geometric accuracy of such images. The authors present a two-step method that combines a phantom-based reverse gradient technique for measurement of gradient nonlinearities and a patient-based phase difference mapping technique for measurement of B0 inhomogeneities, susceptibility, and chemical shift distortions. The phase difference mapping technique adds only minutes to the total patient scan time and can be used to correct a variety of images of the same patient and anatomy. The technique was tested on several different phantoms, each designed to isolate one type of distortion. The mean distortion was reduced to 0.2 +/- 0.1 mm in both gradient echo and spin echo images of a grid phantom. For the more difficult case of a highly distorted echo planar image, residual distortion was reduced to subvoxel dimensions. As a final step, the technique was implemented on patient images. The current technique is effective, time efficient, and robust and provides promise for preparing distortion-rectified MR images for use in MR-based treatment planning.

Numerical Modelling of the Influence of Joint Typologies on the 3D Behaviour of a Steel Sub-Frame Under a Natural Fire

In order to address the complex loading condition of steel joints in fire, and based on the experimental fire tests on steel frames with different joint typologies, performed at the University of Coimbra (Santiago et al. 2008), a detailed three-dimensional model was developed and calibrated to simulate the behaviour of welded and bolted end plate beam-to-column joints. The structural frame is modeled combining 3D shell, solid and joint elements, thereby taking into account the effect of the local failure modes, and the realistic behaviour of the frame exposed to a natural fire. The numerical model accounts for the initial geometrical imperfections, non-linear temperature gradient, geometrical and material nonlinearity and temperature dependent material properties. The results show the performance of each individual structural joint component under heating and cooling conditions and identify the main dependencies of the geometrical and mechanical variables of the components on the joint behaviour. The global behaviour of the frame and the joint failure modes are compared and discussed with the experimental observations.

Novel reconstruction method for three‐dimensional axial continuously moving table whole‐body magnetic resonance imaging featuring autocalibrated parallel imaging GRAPPA

Continuously moving table MR imaging has been successfully evaluated for whole-body tumor staging and metastasis screening. In previous studies it was demonstrated that three-dimensional (3D) slab-selective excitation with lateral readout can provide very efficient k-space coverage when the longitudinal field of view (FOV) is limited. To reduce respiratory artifacts, data acquisition in the thoracoabdominal region of the patient typically must be performed during one single breath hold. This consequently restricts acquisition time and thus spatial resolution. In this work, a novel reconstruction method is introduced for axial 3D moving table data acquisition with lateral readout. The method features table position correction completely in k-space and is compatible with autocalibrated parallel imaging (GRAPPA). Parallel imaging can be applied to increase spatial resolution while maintaining the breath-holding time. A sophisticated protocol for whole-body moving table MRI was developed. The impact of gradient nonlinearity on the featured imaging method was evaluated in phantom and volunteer experiments. Finally, the protocol was optimized toward minimizing residual artifacts. Moving table whole-body MRI with lateral readout was performed in 5 healthy volunteers and was compared with lateral readout data acquired with a GRAPPA accelerated protocol providing increased spatial resolution.

Lower bounds for blow-up time in a nonlinear parabolic problem

For a parabolic problem with a gradient nonlinearity which was introduced by Chipot and Weissler [M. Chipot, F.B. Weissler, Some blow up results for a nonlinear parabolic problem with a gradient term, SIAM J. Math. Anal. 20 (1989) 886–907] (see also [B. Kawohl, L.A. Peletier, Observations on blow up and dead cores for nonlinear parabolic equations, Math. Z. 202 (1989) 207–217]), the question of blow-up is investigated. Specifically, if the solution blows up, a lower bound for the time of blow-up is derived

Sci-Fri AM: YIS-01: Comprehensive MR distortion correction: Phantom validation and in-vivo application.

MR images provide excellent diagnostic information; however, their treatment planning utility is limited due to geometric uncertainties from both system and patient related sources. Despite this concern, interest in developing MR-based treatment planning protocols is on the rise because of the ease with which clinically relevant structures can be identified in MR. Here we present our systematic approach to quantifying both machine (gradient non-linearity and B0 inhomogeneity) and patient (susceptibility and chemical shift) distortions. Gradient non-linearities were previously measured using a 3D grid phantom while the remaining types of distortion were measured using a double gradient echo scan to obtain a B0 distortion map specific to each object/patient. Distortion measurement and correction were validated on phantoms and then implemented on a volunteer. B0 inhomogeneity and susceptibility distortions were simulated by offsetting the x2 -y2 shims; maximum absolute distortion was reduced from 5.4 mm to 1.0 mm and mean (± standard deviation) was reduced from 1.7 ± 1.4 mm to 0.4 ± 0.2 mm. Chemical shift distortion was qualitatively evaluated using a phantom containing fat and water inserts; displacement of the fat signal was much improved following distortion correction. Intensity correction was validated using a uniformity phantom and undistorted image profiles were compared to distorted image profiles and to profiles corrected for geometric and geometric/intensity distortion; the need for intensity correction was clearly demonstrated. Once all types of distortion correction were validated on phantoms, the technique was implemented on a volunteer brain image. Both GE and multi-shot EPI images were corrected.

An alternating direction Galerkin method for elliptic problems with gradient nonlinearity

AbstractSeveral problems in many applications involve the solution of partial differential equations with gradient dependent nonlinearity. The numerical solution of the resulting nonlinear system is rather expensive. We present an alternating direction Galerkin method that allows much faster solution of the nonlinear system. The alternating direction formulation help reduce the problem into a sequence of nonlinear systems that may be solved very efficiently. Theoretical study of the convergence of the method will also be presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Local planar gradients with order-of-magnitude strength and speed advantage.

A three-axis uniplanar gradient coil was designed and built to provide order-of-magnitude increases in gradient strength of up to 500 mT/m on the x- and y-axes, and 1000 mT/m for the z-axis at 640 A input over a limited FOV ( approximately 16 cm) for superficial regions, compared to conventional gradient coils, with significant gradient strengths extending deeper into the body. The gradient set is practically accommodated in the bore of a conventional whole-body, cylindrical-geometry MRI scanner, and operated using standard gradient supplies. The design was optimized for gradient linearity over a restricted volume while accounting for the practical problems of torque and heating. Tests at 320 A demonstrated up to 420-mT/m gradients near the surface at efficiencies of up to 1.4 mT/m/A. A new true 2D gradient-nonlinearity correction algorithm was developed to rectify gradient nonlinearities and considerably expand the imageable volumes. The gradient system and correction algorithm were implemented in a standard 1.5 T scanner and demonstrated by high-resolution imaging of phantoms and humans.

Characterization, prediction, and correction of geometric distortion in MR images

The work presented herein describes our methods and results for predicting, measuring and correcting geometric distortions in a 3 T clinical magnetic resonance (MR) scanner for the purpose of image guidance in radiation treatment planning. Geometric inaccuracies due to both inhomogeneities in the background field and nonlinearities in the applied gradients were easily visualized on the MR images of a regularly structured three-dimensional (3D) grid phantom. From a computed tomography scan, the locations of just under 10 000 control points within the phantom were accurately determined in three dimensions using a MATLAB-based computer program. MR distortion was then determined by measuring the corresponding locations of the control points when the phantom was imaged using the MR scanner. Using a reversed gradient method, distortions due to gradient nonlinearities were separated from distortions due to inhomogeneities in the background B0 field. Because the various sources of machine-related distortions can be individually characterized, distortions present in other imaging sequences (for which 3D distortion cannot accurately be measured using phantom methods) can be predicted negating the need for individual distortion calculation for a variety of other imaging sequences. Distortions were found to be primarily caused by gradient nonlinearities and maximum image distortions were reported to be less than those previously found by other researchers at 1.5 T. Finally, the image slices were corrected for distortion in order to provide geometrically accurate phantom images.

Boundedness vs. blow-up in a degenerate diffusion equation with gradient nonlinearity

This work deals with positive solutions of the Dirichlet problem for the degenerate equation u t = u p Δu + u q + u r |∇u| 2 , p > 0, q > 1, r > -1, in a smoothly bounded domain Q ⊂ R n . It addresses the question in which cases the gradient nonlinearity enforces blow-up as compared to the well-understood equation u t = u p Δu + u q . Particularly, it is shown that each of the parameter regimes where either • all solutions are global and bounded, or • both global bounded and blow-up solutions exist, or • all solutions blow up in finite time is bounded by one of the critical hyperplanes q = p + 1 and r = 2p - q in the (p, q, r)-space. Moreover, unlike for most parameter constellations in the equation without gradient term, the size of Q is seen to play an important role in respect of blow-up.

Artifact reduction in moving‐table acquisitions using parallel imaging and multiple averages

Two-dimensional (2D) axial continuously-moving-table imaging has to deal with artifacts due to gradient nonlinearity and breathing motion, and has to provide the highest scan efficiency. Parallel imaging techniques (e.g., generalized autocalibrating partially parallel acquisition GRAPPA)) are used to reduce such artifacts and avoid ghosting artifacts. The latter occur in T(2)-weighted multi-spin-echo (SE) acquisitions that omit an additional excitation prior to imaging scans for presaturation purposes. Multiple images are reconstructed from subdivisions of a fully sampled k-space data set, each of which is acquired in a single SE train. These images are then averaged. GRAPPA coil weights are estimated without additional measurements. Compared to conventional image reconstruction, inconsistencies between different subsets of k-space induce less artifacts when each k-space part is reconstructed separately and the multiple images are averaged afterwards. These inconsistencies may lead to inaccurate GRAPPA coil weights using the proposed intrinsic GRAPPA calibration. It is shown that aliasing artifacts in single images are canceled out after averaging. Phantom and in vivo studies demonstrate the benefit of the proposed reconstruction scheme for free-breathing axial continuously-moving-table imaging using fast multi-SE sequences.

General algorithm for automated off‐center MRI

A general formula was derived that automatically modifies any MRI pulse sequence to realize arbitrary field-of-view (FOV) shifts. Unlike conventional techniques for implementing off-center MRI, the new method is completely automatic and can therefore be incorporated into the scanner hardware or software, thereby simplifying the development of MRI pulse sequences. The algorithm was incorporated into a visual pulse sequence programming environment, and several pulse sequences were programmed and tested at various off-center locations using the new technique. Unless there is significant background field inhomogeneity or gradient nonlinearity, research sequences employing the automatic technique need only be programmed and tested at the gradient isocenter, whereas with conventional methods, artifacts can sometimes depend on the position of the FOV.

Primal mixed finite-element approximation of elliptic equations with gradient nonlinearities

We study the primal mixed finite-element approximation of the second-order elliptic problem with gradient nonlinearities. Existence and uniqueness of the approximate solution are proved and optimal-order a priori error estimates are obtained. Also, reliable a posteriori error estimators are derived.