Gradient Nonlinearity Research Articles

Vendor-Specific Correction Software for Apparent Diffusion Coefficient Bias Due to Gradient Nonlinearity in Breast Diffusion-Weighted Imaging Using Ice-Water Phantom.

This study aimed to evaluate a vendor-specific correction software for apparent diffusion coefficient (ADC) bias due to gradient nonlinearity in breast diffusion-weighted magnetic resonance imaging using an ice-water phantom. The phantom consists of 5 plastic tubes with a length of 100 mm and a diameter of 15 mm, filled with distilled water and immersed in an ice-water bath. Diffusion-weighted images were acquired by echo-planar imaging sequence on a 3.0-T scanner. ADC maps with and without correction were calculated using 4 b -values (0, 100, 600, and 800 s/mm 2 ). The mean ADCs were measured using a rectangular profile with 5 × 40 pixels in the anterior-posterior (AP) and a square region of interest with 5 × 5 pixels in the right-left (RL) and superior-inferior (SI) directions on the ADC map. ADC was compared with and without correction using a paired t test. Additionally, ADC of the ice-water phantom was measured at the magnet isocenter. ADC increased in the AP and RL directions and decreased in the SI direction with increasing distance from the isocenter before correction. After the correction, ADC at the off-center positions in the AP, RL, and SI directions was reduced to within 5% of the expected value. There were significant differences in the ADC at the off-center positions without and with correction ( P < 0.001); however, ADC at the magnet isocenter did not vary after correction (1.08 ± 0.02 × 10 -3 mm 2 /s). The vendor-specific software corrected the ADC bias due to gradient nonlinearity at the off-center positions in the AP, RL, and SI directions. Therefore, the software will contribute to the accurate ADC assessment in breast DWI.

Impact Assessment of Systemic Geometric Distortion in 1.5T Magnetic Resonance Imaging Simulation through Three-dimensional Geometric Distortion Phantom on Dosimetric Accuracy for Magnetic Resonance Imaging-only Prostate Treatment Planning.

Magnetic resonance imaging (MRI)-only radiotherapy has emerged as a solution to address registration errors that can lead to missed dose delivery. However, the presence of systemic geometric distortion (SGD) stemming from gradient nonlinearity (GNL) and inhomogeneity of the main magnetic field (B0) necessitates consideration. This study aimed to quantitatively assess residual SGD in 1.5T MRI simulation using a three-dimensional (3D) geometric distortion phantom and evaluate its impact on dosimetric accuracy for retrospective prostate cancer patients. Ten retrospective cases of prostate cancer patients treated with volumetric modulated arc radiotherapy (VMAT) were randomly selected. A geometric distortion phantom was scanned on a 1.5T MRI simulation using a 3D T1 volumetric interpolated breath-hold examination sequence, varying bandwidth (BW), and two-phase-encoding directions. Distortion maps were generated and applied to the original computed tomography (oriCT) plan to create a distorted computed tomography plan (dCT), and a dice similarity coefficient (DSC) was observed. Dosimetric accuracy was evaluated by recalculating radiation dose for dCT plans using identical beam parameters as oriCT. The SGD increased with distance from the isocenter in all series. DSC exceeded 0.95 for all plans except the rectum. Regarding GNL's impact on dosimetric accuracy, most mean percentage errors for clinical target volume, planning target volume, and both femurs were under 2% in all plans, except for the bladder and rectum. SGD pre-evaluation is crucial and should be incorporated into a quality assurance program to ensure effective MRI-simulation performance before MRI-only treatment planning for prostate cancer.

The Kuznetsov and Blackstock Equations of Nonlinear Acoustics with Nonlocal-in-Time Dissipation

In ultrasonics, nonlocal quasilinear wave equations arise when taking into account a class of heat flux laws of Gurtin–Pipkin type within the system of governing equations of sound motion. The present study extends previous work by the authors to incorporate nonlocal acoustic wave equations with quadratic gradient nonlinearities which require a new approach in the energy analysis. More precisely, we investigate the Kuznetsov and Blackstock equations with dissipation of fractional type and identify a minimal set of assumptions on the memory kernel needed for each equation. In particular, we discuss the physically relevant examples of Abel and Mittag–Leffler kernels. We perform the well-posedness analysis uniformly with respect to a small parameter on which the kernels depend and which can be interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the limiting behavior of solutions with respect to this parameter, and how it is influenced by the specific class of memory kernels at hand. Through such a limiting study, we relate the considered nonlocal quasilinear equations to their limiting counterparts and establish the convergence rates of the respective solutions in the energy norm.

Nonlinear Gradient Field Estimation in Diffusion MRI Tensor Simulation.

Gradient nonlinearities not only induce spatial distortion in magnetic resonance imaging (MRI), but also introduce discrepancies between intended and acquired diffusion sensitization in diffusion weighted (DW) MRI. Advances in scanner performance have increased the importance of correcting gradient nonlinearities. The most common approaches for gradient nonlinear field estimations rely on phantom calibration field maps which are not always feasible, especially on retrospective data. Here, we derive a quadratic minimization problem for the complete gradient nonlinear field (L(r)). This approach starts with corrupt diffusion signal and estimates the L(r) in two scenarios: (1) the true diffusion tensor known and (2) the true diffusion tensor unknown (i.e., diffusion tensor is estimated). We show the validity of this mathematical approach, both theoretically and through tensor simulation. The estimated field is assessed through diffusion tensor metrics: mean diffusivity (MD), fractional anisotropy (FA), and principal eigenvector (V1). In simulation with 300 diffusion tensors, the study shows the mathematical model is not ill-posed and remains stable. We find when the true diffusion tensor is known (1) the change in determinant of the estimated L(r) field and the true field is near zero and (2) the median difference in estimated L(r) corrected diffusion metrics to true values is near zero. We find the results of L(r) estimation are dependent on the level of L(r) corruption. This work provides an approach to estimate gradient field without the need for additional calibration scans. To the best of our knowledge, the mathematical derivation presented here is novel.

A Liouville-type theorem and one-dimensional symmetry of solutions for elliptic equations with general gradient nonlinearity

In this paper, we are concerned with the following elliptic equation involving a general gradient nonlinearity:−Δv=μ|v|p−1v+G(∇v)inΩ, where n≥1, either Ω=Rn is the whole space or Ω=R+n={x=(x1,…,xn)∈Rn:xn>0} is the half space, μ≥0, p∈R and G:Rn→R satisfies some suitable conditions. One particular case of G is G(γ)=∑i=1kai|γ|pi with ai>0 and pi>1 for any 1≤i≤k. When Ω=Rn, we establish a Bernstein estimate to prove some Liouville-type results of the equation under certain conditions on μ and p. When Ω=R+n, by combining the Bernstein estimates with a moving planes argument, we obtain the one-dimensional symmetry of the solution v with v=0 on ∂R+n. We note that the exponent p is allowed to be negative in our results. Finally, we obtain a gradient estimate in some special cases with μ>0 and p>1.

Global existence and blow-up of weak solutions for a fourth-order parabolic equation with gradient nonlinearity

Global existence and blow-up of weak solutions for a fourth-order parabolic equation with gradient nonlinearity

Local well-posedness results for the nonlinear fractional diffusion equation involving a Erdélyi-Kober operator

&lt;p&gt;In this paper, we study an initial boundary value problem of a nonlinear fractional diffusion equation with the Caputo-type modification of the Erdélyi-Kober fractional derivative. The main tools are the Picard-iteration method, fixed point principle, Mittag-Leffler function, and the embedding theorem between Hilbert scales spaces and Lebesgue spaces. Through careful analysis and precise calculations, the priori estimates of the solution and the smooth effects of the Erdélyi-Kober operator are demonstrated, and then the local existence, uniqueness, and stability of the solution of the nonlinear fractional diffusion equation are established, where the nonlinear source function satisfies the Lipschitz condition or has a gradient nonlinearity.&lt;/p&gt;

A Keller-Segel type taxis model with ecological interpretation and boundedness due to gradient nonlinearities

A Keller-Segel type taxis model with ecological interpretation and boundedness due to gradient nonlinearities

Nonexistence of Nontrivial Weak Solutions of Some Nonlinear Inequalities with Gradient Nonlinearity

Nonexistence of Nontrivial Weak Solutions of Some Nonlinear Inequalities with Gradient Nonlinearity

On the Existence of Radially Symmetric Solutions for the \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$ p $\\end{document}-Laplace Equation with Strong Gradient Nonlinearities

We consider the Dirichlet problem for the \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$ p $\\end{document}-Laplace equationin presence of a gradient not satisfying the Bernstein–Nagumo type condition.We define some class of gradient nonlinearities,for which we prove the existence of a radially symmetric solution with a Hölder continuous derivative.

Design of gradient coils based on the target field method in a high-permeability shield considering sidewall apertures

The combination of high-permeability ferrite and coils is often used to nullify ambient magnetic fields and is widely used in applications such as magnetic resonance imaging and high-sensitivity magnetic sensors. We propose the design method of transverse gradient coils based on the target field method in a high-permeability shield considering sidewall apertures (TFM-HPSCSA), which are installed in the ferrite shield to zero the residual magnetic field gradient. TFM-HPSCSA coils enable an innovative combination of magnetic field gradient linearity and structural luminescence. The coil current density distribution is set to avoid the central sidewall apertures. As a result, the contour distribution of the stream function under the optimized weight coefficients is calculated to obtain the wire alignments of the dBy/dx, dBy/dy and dBy/dz gradient coils. The deviations of the experimentally measured values from the finite element simulations of the three types coils are 0.63%, 1.17% and 0.48%, respectively, which verified the validity and the accuracy of the proposed method. In comparison to the traditional saddle gradient coil designed under the free space condition, the proposed dBy/dx coil produces a 30.5% improvement in the percentage range of regions with 5% magnetic field gradient nonlinearity. The increased linear range of the gradient magnetic field contributes to the improvement of the residual field nulling range in magnetic sensors and magnetic shielding systems in practical applications.

Frequency-dependent diffusion kurtosis imaging in the human brain using an oscillating gradient spin echo sequence and a high-performance head-only gradient

Measuring the time/frequency dependence of diffusion MRI is a promising approach to distinguish between the effects of different tissue microenvironments, such as membrane restriction, tissue heterogeneity, and compartmental water exchange. In this study, we measure the frequency dependence of diffusivity (D) and kurtosis (K) with oscillating gradient diffusion encoding waveforms and a diffusion kurtosis imaging (DKI) model in human brains using a high-performance, head-only MAGNUS gradient system, with a combination of b-values, oscillating frequencies (f), and echo time that has not been achieved in human studies before. Frequency dependence of diffusivity and kurtosis are observed in both global and local white matter (WM) and gray matter (GM) regions and characterized with a power-law model ∼Λ*fθ. The frequency dependences of diffusivity and kurtosis (including changes between fmin and fmax, Λ, and θ) vary over different WM and GM regions, indicating potential microstructural differences between regions. A trend of decreasing kurtosis over frequency in the short-time limit is successfully captured for in vivo human brains. The effects of gradient nonlinearity (GNL) on frequency-dependent diffusivity and kurtosis measurements are investigated and corrected. Our results show that the GNL has prominent scaling effects on the measured diffusivity values (3.5∼5.5% difference in the global WM and 6∼8% difference in the global cortex) and subsequently affects the corresponding power-law parameters (Λ, θ) while having a marginal influence on the measured kurtosis values (<0.05% difference) and power-law parameters (Λ, θ). This study expands previous OGSE studies and further demonstrates the translatability of frequency-dependent diffusivity and kurtosis measurements to human brains, which may provide new opportunities to probe human brain microstructure in health and disease.

On Critical Exponents for Weak Solutions of the Cauchy Problem for a (2 + 1)-Dimensional Nonlinear Composite-Type Equation with Gradient Nonlinearity

The Cauchy problem for a model nonlinear equation with gradient nonlinearity is considered. We prove the existence of two critical exponents, and, such that this problem has no local-in-time weak (in some sense) solution for , while such a solution exists for , but, for , there is no global-in-time weak solution.

On the optimal Lq-regularity for viscous Hamilton–Jacobi equations with subquadratic growth in the gradient

This paper studies a maximal [Formula: see text]-regularity property for nonlinear elliptic equations of second order with a zeroth order term and gradient nonlinearities having superlinear and subquadratic growth, complemented with Dirichlet boundary conditions. The approach is based on the combination of linear elliptic regularity theory and interpolation inequalities, so that the analysis of the maximal regularity estimates boils down to determine lower order integral bounds. The latter are achieved via a [Formula: see text] duality method, which exploits the regularity properties of solutions to stationary Fokker–Planck equations. For the latter problems, we discuss both global and local estimates. Our main novelties for the regularity properties of this class of nonlinear elliptic boundary-value problems are the analysis of the end-point summability threshold [Formula: see text], [Formula: see text] being the dimension of the ambient space and [Formula: see text] the growth of the first-order term in the gradient variable, along with the treatment of the full integrability range [Formula: see text].

Distortion-corrected image reconstruction with deep learning on an MRI-Linac.

MRI is increasingly utilized for image-guided radiotherapy due to its outstanding soft-tissue contrast and lack of ionizing radiation. However, geometric distortions caused by gradient nonlinearities (GNLs) limit anatomical accuracy, potentially compromising the quality of tumor treatments. In addition, slow MR acquisition and reconstruction limit the potential for effective image guidance. Here, we demonstrate a deep learning-based method that rapidly reconstructs distortion-corrected images from raw k-space data for MR-guided radiotherapy applications. We leverage recent advances in interpretable unrolling networks to develop a Distortion-Corrected Reconstruction Network (DCReconNet) that applies convolutional neural networks (CNNs) to learn effective regularizations and nonuniform fast Fourier transforms for GNL-encoding. DCReconNet was trained on a public MR brain dataset from 11 healthy volunteers for fully sampled and accelerated techniques, including parallel imaging (PI) and compressed sensing (CS). The performance of DCReconNet was tested on phantom, brain, pelvis, and lung images acquired on a 1.0T MRI-Linac. The DCReconNet, CS-, PI-and UNet-based reconstructed image quality was measured by structural similarity (SSIM) and RMS error (RMSE) for numerical comparisons. The computation time and residual distortion for each method were also reported. Imaging results demonstrated that DCReconNet better preserves image structures compared to CS- and PI-based reconstruction methods. DCReconNet resulted in the highest SSIM (0.95 median value) and lowest RMSE (<0.04) on simulated brain images with four times acceleration. DCReconNet is over 10-times faster than iterative, regularized reconstruction methods. DCReconNet provides fast and geometrically accurate image reconstruction and has the potential for MRI-guided radiotherapy applications.

Potential errors and improvements in estimating the humidity dependency of vapor permeability using average humidity during cup tests

The cup method is a widely used technique for determining the vapor permeability of porous building materials. Despite the nonlinearity of the humidity gradient in a specimen during testing (due to the humidity dependency of vapor permeability), the measurement results are often regarded as representative of the average humidity values inside and outside the cup. In this study, we aimed to quantitatively clarify the errors associated with creating approximate functions for vapor permeability from cup test results using the average humidity, and we proposed suitable humidity conditions for the test as a countermeasure. We reinterpreted the measurement results in the literature using various building materials. Our findings confirm that the approximate functions using the average humidity of the test conditions significantly overestimate the vapor permeability of materials having strong humidity dependency (especially wooden materials) when a series of measurements were mostly performed using a desiccant or water. Even when intermediate humidity levels were included, errors still exceeded 20%. Our results indicated that a series of measurements with 20% humidity differences inside and outside the cup was necessary to create approximate functions without significant errors (less than 10%) when using average humidity. The findings are significant for preventing the overestimation of vapor permeability in hygrothermal simulations.

Effect of geometric distortion correction on thickness and volume measurements of cortical parcellations in 3D T1w gradient echo sequences.

Automated brain volumetric analysis based on high-resolution T1-weighted MRI datasets is a frequently used tool in neuroimaging for early detection, diagnosis, and monitoring of various neurological diseases. However, image distortions can corrupt and bias the analysis. The aim of this study was to explore the variability of brain volumetric analysis due to gradient distortions and to investigate the effect of distortion correction methods implemented on commercial scanners. 36 healthy volunteers underwent brain imaging using a 3T magnetic resonance imaging (MRI) scanner, including a high-resolution 3D T1-weighted sequence. For all participants, each T1-weighted image was reconstructed directly on the vendor workstation with (DC) and without (nDC) distortion correction. For each participant's set of DC and nDC images, FreeSurfer was used for the determination of regional cortical thickness and volume. Overall, significant differences were found in 12 cortical ROIs comparing the volumes of the DC and nDC data and in 19 cortical ROIs comparing the thickness of the DC and nDC data. The most pronounced differences for cortical thickness were found in the precentral gyrus, the lateral occipital and postcentral ROI (2.69, -2.91% and -2.79%, respectively) while cortical volumes differed most prominently in the paracentral, the pericalcarine and lateral occipital ROI (5.52%, -5.40% and -5.11%, respectively). Correcting for gradient non-linearities can have significant influence on volumetric analysis of cortical thickness and volume. Since the distortion correction is an automatic feature of the MR scanner, it should be stated by each study that applies volumetric analysis which images were used.

Mapping the Impact of Approximate Gradient Nonlinearity Fields Correction on Tractography.

Nonlinear gradients impact diffusion weighted MRI by introducing spatial variation in estimated diffusion tensors. Recent studies have shown that increasing signal-to-noise ratios and the use of ultra-strong gradients may lead to clinically significant impacts on analyses due to these nonlinear gradients in microstructural measures. These effects can potentially bias tractography results and cause misinterpretation of data. Herein, we characterize the impact of an "approximate" gradient nonlinearity correction technique in tractography using empirically derived gradient nonlinear fields. This technique scales the diffusion signal by the change in magnitude due to the gradient nonlinearities, without concomitant correction of gradient direction errors. The impact of this correction on tractography is assessed through white matter bundle segmentation and connectomics via bundle-wise volume, fractional anisotropy, mean diffusivity, radial diffusivity, axial diffusivity, primary eigenvector, and length; as well as the modularity, global efficiency, and characteristic path length connectomics graph measures. We investigate the differences between (1) these measures directly and (2) the within session variability of these measures before and after approximate correction in 61 subjects from the MASiVar pediatric reproducibility dataset. We find approximate correction results is little to no differences on the population level, but large differences on the subject-specific level for both the measures directly and their within session variability. Thus, this study suggests though approximate correction of gradient nonlinearities may not change tractography findings on the population level, subject-specific interpretations may exhibit large fluctuations. A limitation is the lack of comparison with the empirical voxel-wise gradient table correction.

Efficient approximate signal reconstruction for correction of gradient nonlinearities in diffusion-weighted imaging

In diffusion weighted MRI (DW-MRI), hardware nonlinearities lead to spatial variations in the orientation and magnitude of diffusion weighting. While the correction of these spatial distortions has been well established for analyses of DW-MRI, the existing voxel-wise empirical correction for gradient nonlinearities requires reimplementation of existing models, as the resultant gradients vary by voxel. Herein, we propose a two-step signal approximation after voxel-wise correction of gradient nonlinearity effects in DW-MRI. The proposed technique (1) scales the diffusion signal and (2) resamples the gradient orientations. This results in uniform gradients across the corrected image and provides the key advantage of seamless integration into current diffusion workflows. We investigated the validity of our technique by fitting a multi-compartment neurite orientation dispersion and density imaging (NODDI) model to the empirical correction and proposed approximation in five subjects from the MASiVar pediatric dataset. We evaluated intra-cellular volume fraction (iVF), CSF volume fraction (cVF), and orientation dispersion index (ODI) from NODDI. The Cohen's d of iVF, cVF and ODI between the techniques was <0.2 indicating the proposed technique does not exhibit significant differences from the voxel-wise correction technique. Our two-step signal approximation is an efficient representation of the voxel-wise gradient table correction. Using this approximation, correction of gradient nonlinearities can be easily incorporated into existing diffusion preprocessing pipelines and is implemented in “PreQual: An automated pipeline for integrated preprocessing and quality assurance of diffusion weighted MRI images”.

Mapping the impact of nonlinear gradient fields with noise on diffusion MRI

In diffusion MRI, gradient nonlinearities cause spatial variations in the magnitude and direction of diffusion gradients. Studies have shown artifacts from these distortions can results in biased diffusion tensor information and tractography. Here, we investigate the impact of gradient nonlinearity correction in the presence of noise. We introduced empirically derived gradient nonlinear fields at different signal-to-noise ratio (SNR) levels in two experiments: tensor field simulation and simulation of the brain. For each experiment, this work compares two techniques empirically: voxel-wise gradient table correction and approximate correction by scaling the signal directly. The impact was assessed through diffusion metrics including mean diffusivity (MD), fractional anisotropy (FA), axial diffusivity (AD), radial diffusivity (RD), and principal eigen vector (V1). The study shows (1) the correction of gradient nonlinearities will not lead to substantively incorrect estimation of diffusion metrics in a linear system, (2) gradient nonlinearity correction does not interact adversely with noise, (3) nonlinearity correction suppresses the impact of nonlinearities in typical SNR data, (4) for SNR below 30, the performance of both the gradient nonlinearity correction techniques were similar, and (5) larger impacts are seen in regions where the gradient nonlinearities are distinct. Thus, this study suggests that there were greater beneficial effects than adverse effects due to the correction of nonlinearities. Additionally, correction of nonlinearities is recommended when region of interests are in areas with pronounced nonlinearities.