Estimation and application of spatially variable noise fields in diffusion tensor imaging

Characterizing patterns of DTI variance in aging brains

Background

As large analyses merge data across sites, a deeper understanding of variance in statistical assessment across the sources of data becomes critical for valid analyses. Diffusion tensor imaging (DTI) exhibits spatially varying and correlated noise, so care must be taken with distributional assumptions.

Purpose

We characterize the role of physiology, subject compliance, and the interaction of subject with the scanner in the understanding of DTI variability, as modeled in spatial variance of derived metrics in homogeneous regions.

Methods

We analyze DTI data from 1035 subjects in the Baltimore Longitudinal Study of Aging (BLSA), with ages ranging from 22.4 to 103 years old. For each subject, up to 12 longitudinal sessions were conducted. We assess variance of DTI scalars within regions of interest (ROIs) defined by four segmentation methods and investigate the relationships between the variance and covariates, including baseline age, time from the baseline (referred to as "interval"), motion, sex, and whether it is the first scan or the second scan in the session.

Results

Covariate effects are heterogeneous and bilaterally symmetric across ROIs. Inter-session interval is positively related (p ≪ 0.001) to FA variance in the cuneus and occipital gyrus, but negatively (p ≪ 0.001) in the caudate nucleus. Males show significantly (p ≪ 0.001) higher FA variance in the right putamen, thalamus, body of the corpus callosum, and cingulate gyrus. In 62 out of 176 ROIs defined by the Eve type-1 atlas, an increase in motion is associated (p < 0.05) with a decrease in FA variance. Head motion increases during the rescan of DTI (Δμ = 0.045 millimeters per volume).

Conclusions

The effects of each covariate on DTI variance, and their relationships across ROIs are complex. Ultimately, we encourage researchers to include estimates of variance when sharing data and consider models of heteroscedasticity in analysis. This work provides a foundation for study planning to account for regional variations in metric variance.

A Simple Measure for Acuity in Medical Images

An automatic and objective assessment of image quality is important in an era, where large-scale processing of imaging data from multi-center studies becomes commonplace. Based on a comprehensive statistical image model that includes noise and blur, a measure for image acuity is derived here as the ratio of the maximal gradient magnitude and the intensity difference at a boundary. Acuity may be affected by the object under study, the image acquisition, reconstruction processes, and any post-processing steps. The acuity measure presented here is post-hoc, intuitive to understand, simple to compute, and easily integrates with other standard measures of image quality. Three applications in medical imaging are included where our acuity measure is useful in the objective and automatic assessment of image quality.

De-noising of 3D multiple-coil MR images using modified LMMSE estimator

De-noising is a crucial topic in Magnetic Resonance Imaging (MRI) which focuses on less loss of Magnetic Resonance (MR) image information and details preservation during the noise suppression. Nowadays multiple-coil MRI system is preferred to single one due to its acceleration in the imaging process. Due to the fact that the model of noise in single-coil and multiple-coil MRI systems are different, the de-noising methods that mostly are adapted to single-coil MRI systems, do not work appropriately with multiple-coil one. The model of noise in single-coil MRI systems is Rician while in multiple-coil one (if no subsampling occurs in k-space or GRAPPA reconstruction process is being done in the coils), it obeys noncentral Chi (nc-χ). In this paper, a new filtering method based on the Linear Minimum Mean Square Error (LMMSE) estimator is proposed for multiple-coil MR Images ruined by nc-χ noise. In the presented method, to have an optimum similarity selection of voxels, the Bayesian Mean Square Error (BMSE) criterion is used and proved for nc-χ noise model and also a nonlocal voxel selection methodology is proposed for nc-χ distribution. The results illustrate robust and accurate performance compared to the related state-of-the-art methods, either on ideal nc-χ images or GRAPPA reconstructed ones.

Non-Stationary Rician Noise Estimation in Parallel MRI Using a Single Image: A Variance-Stabilizing Approach

Parallel magnetic resonance imaging (pMRI) techniques have gained a great importance both in research and clinical communities recently since they considerably accelerate the image acquisition process. However, the image reconstruction algorithms needed to correct the subsampling artifacts affect the nature of noise, i.e., it becomes non-stationary. Some methods have been proposed in the literature dealing with the non-stationary noise in pMRI. However, their performance depends on information not usually available such as multiple acquisitions, receiver noise matrices, sensitivity coil profiles, reconstruction coefficients, or even biophysical models of the data. Besides, some methods show an undesirable granular pattern on the estimates as a side effect of local estimation. Finally, some methods make strong assumptions that just hold in the case of high signal-to-noise ratio (SNR), which limits their usability in real scenarios. We propose a new automatic noise estimation technique for non-stationary Rician noise that overcomes the aforementioned drawbacks. Its effectiveness is due to the derivation of a variance-stabilizing transformation designed to deal with any SNR. The method was compared to the main state-of-the-art methods in synthetic and real scenarios. Numerical results confirm the robustness of the method and its better performance for the whole range of SNRs.

Distribution-aware estimation of the minimum achievable uncertainty in diffusion-tensor imaging (DTI)

Diffusion tensor imaging (DTI) provides exquisite sensitivity to structural and microstructural characteristics of brain tissue, and is routinely employed in advanced neuroimaging applications. DTI is commonly performed using intrinsically noisy echo-planar imaging techniques and poses high demands both on scanner performance and on in-scanner subject time, which in turn is directly related to the number of diffusion-weighting direction one requires. While DTI-derived indices such as fractional anisotropy (FA), diffusion tensor trace and anisotropy mode have proven extremely useful in characterizing disease-related aberrations, their estimation is commonly performed using fitting routines that do not properly take into account MRI noise distribution. In this paper, we present a distribution-aware maximum likelihood tensor estimation framework which also allows, for the first time, separate local noise estimation in both diffusion weighted and reference images. We validate our framework using multiple water phantom diffusion weighted acquisitions, and demonstrate its feasibility in human data. We then employ our framework within Monte Carlo simulations to show how the minimum achievable uncertainty attainable in DTI depends on signal-to-noise ratio (SNR) and number of diffusion gradient directions, demonstrating that these dependencies could be recast into simple power laws which may serve as guidelines for application-specific DTI protocol design.

Diffusion MRI noise mapping using random matrix theory

PURPOSE

To estimate the spatially varying noise map using a redundant series of magnitude MR images.

METHODS

We exploit redundancy in non-Gaussian distributed multidirectional diffusion MRI data by identifying its noise-only principal components, based on the theory of noisy covariance matrices. The bulk of principal component analysis eigenvalues, arising due to noise, is described by the universal Marchenko-Pastur distribution, parameterized by the noise level. This allows us to estimate noise level in a local neighborhood based on the singular value decomposition of a matrix combining neighborhood voxels and diffusion directions.

RESULTS

We present a model-independent local noise mapping method capable of estimating the noise level down to about 1% error. In contrast to current state-of-the-art techniques, the resultant noise maps do not show artifactual anatomical features that often reflect physiological noise, the presence of sharp edges, or a lack of adequate a priori knowledge of the expected form of MR signal.

CONCLUSIONS

Simulations and experiments show that typical diffusion MRI data exhibit sufficient redundancy that enables accurate, precise, and robust estimation of the local noise level by interpreting the principal component analysis eigenspectrum in terms of the Marchenko-Pastur distribution. Magn Reson Med 76:1582-1593, 2016. © 2015 International Society for Magnetic Resonance in Medicine.

The effect of Gibbs ringing artifacts on measures derived from diffusion MRI

Diffusion-weighted (DW) magnetic resonance imaging (MRI) is a unique method to investigate microstructural tissue properties noninvasively and is one of the most popular methods for studying the brain white matter in vivo. To obtain reliable statistical inferences with diffusion MRI, however, there are still many challenges, such as acquiring high-quality DW-MRI data (e.g., high SNR and high resolution), careful data preprocessing (e.g., correcting for subject motion and eddy current induced geometric distortions), choosing the appropriate diffusion approach (e.g., diffusion tensor imaging (DTI), diffusion kurtosis imaging (DKI), or diffusion spectrum MRI), and applying a robust analysis strategy (e.g., tractography based or voxel based analysis). Notwithstanding the numerous efforts to optimize many steps in this complex and lengthy diffusion analysis pipeline, to date, a well-known artifact in MRI--i.e., Gibbs ringing (GR)--has largely gone unnoticed or deemed insignificant as a potential confound in quantitative DW-MRI analysis. Considering the recent explosion of diffusion MRI applications in biomedical and clinical applications, a systematic and comprehensive investigation is necessary to understand the influence of GR on the estimation of diffusion measures. In this work, we demonstrate with simulations and experimental DW-MRI data that diffusion estimates are significantly affected by GR artifacts and we show that an off-the-shelf GR correction procedure based on total variation already can alleviate this issue substantially.

Region of interest correction factors improve reliability of diffusion imaging measures within and across scanners and field strengths

Diffusion tensor imaging (DTI) measures are commonly used as imaging markers to investigate individual differences in relation to behavioral and health-related characteristics. However, the ability to detect reliable associations in cross-sectional or longitudinal studies is limited by the reliability of the diffusion measures. Several studies have examined the reliability of diffusion measures within (i.e. intra-site) and across (i.e. inter-site) scanners with mixed results. Our study compares the test-retest reliability of diffusion measures within and across scanners and field strengths in cognitively normal older adults with a follow-up interval less than 2.25 years. Intra-class correlation (ICC) and coefficient of variation (CoV) of fractional anisotropy (FA) and mean diffusivity (MD) were evaluated in sixteen white matter and twenty-six gray matter bilateral regions. The ICC for intra-site reliability (0.32 to 0.96 for FA and 0.18 to 0.95 for MD in white matter regions; 0.27 to 0.89 for MD and 0.03 to 0.79 for FA in gray matter regions) and inter-site reliability (0.28 to 0.95 for FA in white matter regions, 0.02 to 0.86 for MD in gray matter regions) with longer follow-up intervals were similar to earlier studies using shorter follow-up intervals. The reliability of across field strengths comparisons was lower than intra- and inter-site reliabilities. Within and across scanner comparisons showed that diffusion measures were more stable in larger white matter regions (>1500 mm(3)). For gray matter regions, the MD measure showed stability in specific regions and was not dependent on region size. Linear correction factor estimated from cross-sectional or longitudinal data improved the reliability across field strengths. Our findings indicate that investigations relating diffusion measures to external variables must consider variable reliability across the distinct regions of interest and that correction factors can be used to improve consistency of measurement across field strengths. An important result of this work is that inter-scanner and field strength effects can be partially mitigated with linear correction factors specific to regions of interest. These data-driven linear correction techniques can be applied in cross-sectional or longitudinal studies.

Detecting statistically significant differences in quantitative MRI experiments, applied to diffusion tensor imaging

In this work we present a framework for reliably detecting significant differences in quantitative magnetic resonance imaging and evaluate it with diffusion tensor imaging (DTI) experiments. As part of this framework we propose a new spatially regularized maximum likelihood estimator that simultaneously estimates the quantitative parameters and the spatially-smoothly-varying noise level from the acquisitions. The noise level estimation method does not require repeated acquisitions. We show that the amount of regularization in this method can be set a priori to achieve a desired coefficient of variation of the estimated noise level. The noise level estimate allows the construction of a Cramér-Rao-lower-bound based test statistic that reliably assesses the significance of differences between voxels within a scan or across different scans. We show that the regularized noise level estimate improves upon existing methods and results in a substantially increased precision of the uncertainty estimates of the DTI parameters. It enables correct specification of the null distribution of the test statistic and with it the test statistic obtains the highest sensitivity and specificity. The source code of the estimation framework, test statistic and experiment scripts are made available to the community.

A simple noise correction scheme for diffusional kurtosis imaging

PURPOSE

Diffusional kurtosis imaging (DKI) is sensitive to the effects of signal noise due to strong diffusion weightings and higher order modeling of the diffusion weighted signal. A simple noise correction scheme is proposed to remove the majority of the noise bias in the estimated diffusional kurtosis.

METHODS

Weighted linear least squares (WLLS) fitting together with a voxel-wise, subtraction-based noise correction from multiple, independent acquisitions are employed to reduce noise bias in DKI data. The method is validated in phantom experiments and demonstrated for in vivo human brain for DKI-derived parameter estimates.

RESULTS

As long as the signal-to-noise ratio (SNR) for the most heavily diffusion weighted images is greater than 2.1, errors in phantom diffusional kurtosis estimates are found to be less than 5 percent with noise correction, but as high as 44 percent for uncorrected estimates. In human brain, noise correction is also shown to improve diffusional kurtosis estimates derived from measurements made with low SNR.

CONCLUSION

The proposed correction technique removes the majority of noise bias from diffusional kurtosis estimates in noisy phantom data and is applicable to DKI of human brain. Features of the method include computational simplicity and ease of integration into standard WLLS DKI post-processing algorithms.

Local estimation of the noise level in MRI using structural adaptation

We present a method for local estimation of the signal-dependent noise level in magnetic resonance images. The procedure uses a multi-scale approach to adaptively infer on local neighborhoods with similar data distribution. It exploits a maximum-likelihood estimator for the local noise level. The validity of the method was evaluated on repeated diffusion data of a phantom and simulated data using T1-data corrupted with artificial noise. Simulation results were compared with a recently proposed estimate. The method was also applied to a high-resolution diffusion dataset to obtain improved diffusion model estimation results and to demonstrate its usefulness in methods for enhancing diffusion data.

Diffusion weighted image denoising using overcomplete local PCA

Diffusion Weighted Images (DWI) normally shows a low Signal to Noise Ratio (SNR) due to the presence of noise from the measurement process that complicates and biases the estimation of quantitative diffusion parameters. In this paper, a new denoising methodology is proposed that takes into consideration the multicomponent nature of multi-directional DWI datasets such as those employed in diffusion imaging. This new filter reduces random noise in multicomponent DWI by locally shrinking less significant Principal Components using an overcomplete approach. The proposed method is compared with state-of-the-art methods using synthetic and real clinical MR images, showing improved performance in terms of denoising quality and estimation of diffusion parameters.

Correcting power and p-value calculations for bias in diffusion tensor imaging

Diffusion tensor imaging (DTI) provides quantitative parametric maps sensitive to tissue microarchitecture (e.g., fractional anisotropy, FA). These maps are estimated through computational processes and subject to random distortions including variance and bias. Traditional statistical procedures commonly used for study planning (including power analyses and p-value/alpha-rate thresholds) specifically model variability, but neglect potential impacts of bias. Herein, we quantitatively investigate the impacts of bias in DTI on hypothesis test properties (power and alpha-rate) using a two-sided hypothesis testing framework. We present theoretical evaluation of bias on hypothesis test properties, evaluate the bias estimation technique SIMEX for DTI hypothesis testing using simulated data, and evaluate the impacts of bias on spatially varying power and alpha rates in an empirical study of 21 subjects. Bias is shown to inflame alpha rates, distort the power curve, and cause significant power loss even in empirical settings where the expected difference in bias between groups is zero. These adverse effects can be attenuated by properly accounting for bias in the calculation of power and p-values.

Comprehensive framework for accurate diffusion MRI parameter estimation

During the last decade, many approaches have been proposed for improving the estimation of diffusion measures. These techniques have already shown an increase in accuracy based on theoretical considerations, such as incorporating prior knowledge of the data distribution. The increased accuracy of diffusion metric estimators is typically observed in well-defined simulations, where the assumptions regarding properties of the data distribution are known to be valid. In practice, however, correcting for subject motion and geometric eddy current deformations alters the data distribution tremendously such that it can no longer be expressed in a closed form. The image processing steps that precede the model fitting will render several assumptions on the data distribution invalid, potentially nullifying the benefit of applying more advanced diffusion estimators. In this work, we present a generic diffusion model fitting framework that considers some statistics of diffusion MRI data. A central role in the framework is played by the conditional least squares estimator. We demonstrate that the accuracy of that particular estimator can generally be preserved, regardless the applied preprocessing steps, if the noise parameter is known a priori. To fulfill that condition, we also propose an approach for the estimation of spatially varying noise levels.

Nonlocal maximum likelihood estimation method for denoising multiple-coil magnetic resonance images

Effective denoising is vital for proper analysis and accurate quantitative measurements from magnetic resonance (MR) images. Even though many methods were proposed to denoise MR images, only few deal with the estimation of true signal from MR images acquired with phased-array coils. If the magnitude data from phased array coils are reconstructed as the root sum of squares, in the absence of noise correlations and subsampling, the data is assumed to follow a non central-χ distribution. However, when the k-space is subsampled to increase the acquisition speed (as in GRAPPA like methods), noise becomes spatially varying. In this note, we propose a method to denoise multiple-coil acquired MR images. Both the non central-χ distribution and the spatially varying nature of the noise is taken into account in the proposed method. Experiments were conducted on both simulated and real data sets to validate and to demonstrate the effectiveness of the proposed method.

Assessment of bias in experimentally measured diffusion tensor imaging parameters using SIMEX

Diffusion tensor imaging enables in vivo investigation of tissue cytoarchitecture through parameter contrasts sensitive to water diffusion barriers at the micrometer level. Parameters are derived through an estimation process that is susceptible to noise and artifacts. Estimated parameters (e.g., fractional anisotropy) exhibit both variability and bias relative to the true parameter value estimated from a hypothetical noise-free acquisition. Herein, we present the use of the simulation and extrapolation (SIMEX) approach for post hoc assessment of bias in a massively univariate imaging setting and evaluate the potential of a SIMEX-based bias correction. Using simulated data with known truth models, spatially varying fractional anisotropy bias error maps are evaluated on two independent and highly differentiated case studies. The stability of SIMEX and its distributional properties are further evaluated on 42 empirical diffusion tensor imaging datasets. Using gradient subsampling, an empirical experiment with a known true outcome is designed and SIMEX performance is compared to the original estimator. With this approach, we find SIMEX bias estimates to be highly accurate offering significant reductions in parameter bias for individual datasets and greater accuracy in averaged population-based estimates.

Informed RESTORE: A method for robust estimation of diffusion tensor from low redundancy datasets in the presence of physiological noise artifacts

Physiological noise artifacts, especially those originating from cardiac pulsation and subject motion, are common in clinical Diffusion tensor-MRI acquisitions. Previous works show that signal perturbations produced by artifacts can be severe and neglecting to account for their contribution can result in erroneous diffusion tensor values. The Robust Estimation of Tensors by Outlier Rejection (RESTORE) method has been shown to be an effective strategy for improving tensor estimation on a voxel-by-voxel basis in the presence of artifactual data points in diffusion-weighted images. In this article, we address potential instabilities that may arise when using RESTORE and propose practical constraints to improve its usability. Moreover, we introduce a method, called informed RESTORE designed to remove physiological noise artifacts in datasets acquired with low redundancy (less than 30-40 diffusion-weighted image volumes)--a condition in which the original RESTORE algorithm may converge to an incorrect solution. This new method is based on the notion that physiological noise is more likely to result in signal dropouts than signal increases. Results from both Monte Carlo simulation and clinical diffusion data indicate that informed RESTORE performs very well in removing physiological noise artifacts for low redundancy diffusion-weighted image datasets.

Spatially variable Rician noise in magnetic resonance imaging

Magnetic resonance images tend to be influenced by various random factors usually referred to as "noise". The principal sources of noise and related artefacts can be divided into two types: arising from hardware (acquisition coil arrays, gradient coils, field inhomogeneity); and arising from the subject (physiological noise including body motion, cardiac pulsation or respiratory motion). These factors negatively affect the resolution and reproducibility of the images. Therefore, a proper noise treatment is important for improving the performance of clinical and research investigations. Noise reduction becomes especially critical for the images with a low signal-to-noise ratio, such as those typically acquired in diffusion tensor imaging at high diffusion weightings. The standard methods of signal correction usually assume a uniform distribution of the standard deviation of the noise across the image and evaluate a single correction parameter for the whole image. We pursue a more advanced approach based on the assumption of an inhomogeneous distribution of noise in space and evaluate correction factors for each voxel individually. The Rician nature of the underlying noise is considered for low and high signal-to-noise ratios. The approach developed here has been examined using numerical simulations and in vivo brain diffusion tensor imaging experiments. The efficacy and usefulness of this approach is demonstrated here and the resultant effective tool is described.

Robust tensor estimation in diffusion tensor imaging

The signal response measured in diffusion tensor imaging is subject to detrimental influences caused by noise. Noise fields arise due to various contributions such as thermal and physiological noise and sources related to the hardware imperfection. As a result, diffusion tensors estimated by different linear and non-linear least squares methods in absence of a proper noise correction tend to be substantially corrupted. In this work, we propose an advanced tensor estimation approach based on the least median squares method of the robust statistics. Both constrained and non-constrained versions of the method are considered. The performance of the developed algorithm is compared to that of the conventional least squares method and of the alternative robust methods proposed in the literature. Two examples of simulated diffusion attenuations and experimental in vivo diffusion data sets were used as a basis for comparison. The robust algorithms were shown to be advantageous compared to the least squares method in the cases where elimination of the outliers is desirable. Additionally, the constraints were applied in order to prevent generation of the non-positive definite tensors and reduce related artefacts in the maps of fractional anisotropy. The developed method can potentially be exploited also by other MR techniques where a robust regression or outlier localisation is required.