Diffusion Tensor Estimation by Maximizing Rician Likelihood

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.

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.

A robust variational approach for simultaneous smoothing and estimation of DTI

Estimating diffusion tensors is an essential step in many applications - such as diffusion tensor image (DTI) registration, segmentation and fiber tractography. Most of the methods proposed in the literature for this task are not simultaneously statistically robust and feature preserving techniques. In this paper, we propose a novel and robust variational framework for simultaneous smoothing and estimation of diffusion tensors from diffusion MRI. Our variational principle makes use of a recently introduced total Kullback-Leibler (tKL) divergence for DTI regularization. tKL is a statistically robust dissimilarity measure for diffusion tensors, and regularization by using tKL ensures the symmetric positive definiteness of tensors automatically. Further, the regularization is weighted by a non-local factor adapted from the conventional non-local means filters. Finally, for the data fidelity, we use the nonlinear least-squares term derived from the Stejskal-Tanner model. We present experimental results depicting the positive performance of our method in comparison to competing methods on synthetic and real data examples.

Assessment of bias for MRI diffusion tensor imaging using SIMEX

Diffusion Tensor Imaging (DTI) is a Magnetic Resonance Imaging method for measuring water diffusion in vivo. One powerful DTI contrast is fractional anisotropy (FA). FA reflects the strength of water's diffusion directional preference and is a primary metric for neuronal fiber tracking. As with other DTI contrasts, FA measurements are obscured by the well established presence of bias. DTI bias has been challenging to assess because it is a multivariable problem including SNR, six tensor parameters, and the DTI collection and processing method used. SIMEX is a modem statistical technique that estimates bias by tracking measurement error as a function of added noise. Here, we use SIMEX to assess bias in FA measurements and show the method provides; i) accurate FA bias estimates, ii) representation of FA bias that is data set specific and accessible to non-statisticians, and iii) a first time possibility for incorporation of bias into DTI data analysis.

Least squares for diffusion tensor estimation revisited: propagation of uncertainty with Rician and non-Rician signals

Least Squares (LS) and its minimum variance counterpart, Weighted Least Squares (WLS), have become very popular when estimating the Diffusion Tensor (DT), to the point that they are the standard in most of the existing software for diffusion MRI. They are based on the linearization of the Stejskal-Tanner equation by means of the logarithmic compression of the diffusion signal. Due to the Rician nature of noise in traditional systems, a certain bias in the estimation is known to exist. This artifact has been made patent through some experimental set-ups, but it is not clear how the distortion translates in the reconstructed DT, and how important it is when compared to the other source of error contributing to the Mean Squared Error (MSE) in the estimate, i.e. the variance. In this paper we propose the analytical characterization of log-Rician noise and its propagation to the components of the DT through power series expansions. We conclude that even in highly noisy scenarios the bias for log-Rician signals remains moderate when compared to the corresponding variance. Yet, with the advent of Parallel Imaging (pMRI), the Rician model is not always valid. We make our analysis extensive to a number of modern acquisition techniques through the study of a more general Non Central-Chi (nc-χ) model. Since WLS techniques were initially designed bearing in mind Rician noise, it is not clear whether or not they still apply to pMRI. An important finding in our work is that the common implementation of WLS is nearly optimal when nc-χ noise is considered. Unfortunately, the bias in the estimation becomes far more important in this case, to the point that it may nearly overwhelm the variance in given situations. Furthermore, we evidence that such bias cannot be removed by increasing the number of acquired gradient directions. A number of experiments have been conducted that corroborate our analytical findings, while in vivo data have been used to test the actual relevance of the bias in the estimation.

Increased regional fractional anisotropy in highly screened attention-deficit hyperactivity disorder (ADHD)

Diffusion tensor imaging data were collected at 3.0 Tesla from 16 children with attention-deficit hyperactivity disorder (ADHD) and 16 typically developing controls, ages 9 to 14 years. Fractional anisotropy images were calculated and normalized by linear transformation. Voxel-wise and atlas-based region-of-interest analyses were performed. Using voxel-wise analysis, fractional anisotropy was found to be significantly increased in the attention-deficit hyperactivity disorder group in the right superior frontal gyrus and posterior thalamic radiation, and left dorsal posterior cingulate gyrus, lingual gyrus, and parahippocampal gyrus. No regions showed significantly decreased fractional anisotropy in attention-deficit hyperactivity disorder. Region-of-interest analysis revealed increased fractional anisotropy in the left sagittal stratum, that is, white matter that connects the temporal lobe to distant cortical regions. Only fractional anisotropy in the left sagittal stratum was significantly associated with attention-deficit hyperactivity disorder symptom severity. Several recent studies have reported pathological increases in fractional anisotropy in other conditions, highlighting the relevance of diffusion tensor imaging in identifying atypical white matter structure associated with neurodevelopmental processes.

Estimation of diffusion properties in crossing fiber bundles

There is an ongoing debate on how to model diffusivity in fiber crossings. We propose an optimization framework for the selection of a dual tensor model and the set of diffusion weighting parameters b, such that both the diffusion shape and orientation parameters can be precisely as well as accurately estimated. For that, we have adopted the Cramér-Rao lower bound (CRLB) on the variance of the model parameters, and performed Monte Carlo simulations. We have found that the axial diffusion lambda(parallel) needs to be constrained, while an isotropic fraction can be modeled by a single parameter f(iso). Under these circumstances, the Fractional Anisotropy (FA) of both tensors can theoretically be independently estimated with a precision of 9% (at SNR = 25). Levenberg-Marquardt optimization of the Maximum Likelihood function with a Rician noise model approached this precision while the bias was insignificant. A two-element b-vector b = [1.0 3.5] x 10(3) mm(-2)s was found to be sufficient for estimating parameters of heterogeneous tissue with low error. This has allowed us to estimate consistent FA-profiles along crossing tracts. This work defines fundamental limits for comparative studies to correctly analyze crossing white matter structures.

White matter microstructural differences linked to left perisylvian language network in children with dyslexia

Studies of dyslexia using diffusion tensor imaging (DTI) have reported fractional anisotropy (FA) differences in left inferior frontal gyrus (LIFG) and left temporo-parietal white matter, suggesting that impaired reading is associated with atypical white matter microstructure in these regions. These anomalies might reflect abnormalities in the left perisylvian language network, long implicated in dyslexia. While DTI investigations frequently report analyses on multiple tensor-derived measures (e.g., FA, orientation, tractography), it is uncommon to integrate analyses to examine the relationships between atypical findings. For the present study, semi-automated techniques were applied to DTI data in an integrated fashion to examine white matter microstructure in 14 children with dyslexia and 17 typically developing readers (ages 7-16 years). Correlations of DTI metrics (FA and fiber orientation) to reading skill (accuracy and speed) and to probabilistic tractography maps of the left perisylvian language tracts were examined. Consistent with previous reports, our findings suggest FA decreases in dyslexia in LIFG and left temporo-parietal white matter. The LIFG FA finding overlaps an area showing differences in fiber orientation in an anterior left perisylvian language pathway. Additionally, a positive correlation of FA to reading speed was found in a posterior circuit previously associated with activation on functional imaging during reading tasks. Overall, integrating results from several complementary semi-automated analyses reveals evidence linking atypical white matter microstructure in dyslexia to atypical fiber orientation in circuits implicated in reading including the left perisylvian language network.

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

Optimal interpretation of magnetic resonance image content often requires an estimate of the underlying image noise, which is typically realized as a spatially invariant estimate of the noise distribution. This is not an ideal practice in diffusion tensor imaging because the noise distribution is usually spatially varying due to the use of fast imaging and noise suppression techniques. A new estimation approach for spatially varying noise fields (NFs) is proposed in this article. The approach is based on a noise invariance property in scenarios in which more than one image, each with potentially different signal levels, is acquired on each slice, as in diffusion-weighted MRI. This technique leads to improved NF estimates in simulations, phantom experiments and in vivo studies when compared to traditional NF estimators that use regional variability or background intensity histograms. The proposed method reduces the NF estimation error by a factor of 100 in simulations, shows a strong linear correlation (R(2)=0.99) between theoretical and estimated noise changes in phantoms and demonstrates consistent (<5% variability) NF estimates in vivo. The advantages of spatially varying NF estimation are demonstrated for power analysis, outlier detection and tensor estimation.

An object-based method for Rician noise estimation in MR images

The estimation of the noise level in MR images is used to assess the consistency of statistical analysis or as an input parameter in some image processing techniques. Most of the existing Rician noise estimation methods are based on background statistics, and as such are sensitive to ghosting artifacts. In this paper, a new object-based method is proposed. This method is based on the adaptation of the Median Absolute Deviation (MAD) estimator in the wavelet domain for Rician noise. The adaptation for Rician noise is performed by using only the wavelet coefficients corresponding to the object and by correcting the estimation with an iterative scheme based on the SNR of the image. A quantitative validation on synthetic phantom with artefacts is presented and a new validation framework is proposed to perform quantitative validation on real data. The results show the accuracy and the robustness of the proposed method.

Real-time MR diffusion tensor and Q-ball imaging using Kalman filtering

Diffusion magnetic resonance imaging (dMRI) has become an established research tool for the investigation of tissue structure and orientation. In this paper, we present a method for real-time processing of diffusion tensor and Q-ball imaging. The basic idea is to use Kalman filtering framework to fit either the linear tensor or Q-ball model. Because the Kalman filter is designed to be an incremental algorithm, it naturally enables updating the model estimate after the acquisition of any new diffusion-weighted volume. Processing diffusion models and maps during ongoing scans provides a new useful tool for clinicians, especially when it is not possible to predict how long a subject may remain still in the magnet. First, we introduce the general linear models corresponding to the two diffusion tensor and analytical Q-ball models of interest. Then, we present the Kalman filtering framework and we focus on the optimization of the diffusion orientation sets in order to speed up the convergence of the online processing. Last, we give some results on a healthy volunteer for the online tensor and the Q-ball model, and we make some comparisons with the conventional offline techniques used in the literature. We could achieve full real-time for diffusion tensor imaging and deferred time for Q-ball imaging, using a single workstation.

ROBUST MAXIMUM LIKELIHOOD ESTIMATION IN Q-SPACE MRI

Q-space imaging is an emerging diffusion weighted MR imaging technique to estimate molecular diffusion probability density functions (PDF's) without the need to assume a Gaussian distribution. We present a robust M-estimator, Q-space Estimation by Maximizing Rician Likelihood (QEMRL), for diffusion PDF's based on maximum likelihood. PDF's are modeled by constrained Gaussian mixtures. In QEMRL, robust likelihood measures mitigate the impacts of imaging artifacts. In simulation and in vivo human spinal cord, the method improves reliability of estimated PDF's and increases tissue contrast. QEMRL enables more detailed exploration of the PDF properties than prior approaches and may allow acquisitions at higher spatial resolution.

Rician noise removal by non-Local Means filtering for low signal-to-noise ratio MRI: applications to DT-MRI

Diffusion-Weighted MRI (DW-MRI) is subject to random noise yielding measures that are different from their real values, and thus biasing the subsequently estimated tensors. The Non-Local Means (NLMeans) filter has recently been proposed to denoise MRI with high signal-to-noise ratio (SNR). This filter has been shown to allow the best restoration of image intensities for the estimation of diffusion tensors (DT) compared to state-of-the-art methods. However, for DW-MR images with high b-values (and thus low SNR), the noise, which is strictly Rician-distributed, can no longer be approximated as additive white Gaussian, as implicitly assumed in the classical formulation of the NLMeans. High b-values are typically used in high angular resolution diffusion imaging (HARDI) or q-space imaging (QSI), for which an optimal restoration is critical. In this paper, we propose to adapt the NLMeans filter to Rician noise corrupted data. Validation is performed on synthetic data and on real data for both conventional MR images and DT images. Our adaptation outperforms the original NLMeans filter in terms of peak-signal-to-noise ratio (PSNR) for DW-MRI.

Rician noise removal by non-Local Means filtering for low signal-to-noise ratio MRI: applications to DT-MRI

Diffusion-Weighted MRI (DW-MRI) is subject to random noise yielding measures that are different from their real values, and thus biasing the subsequently estimated tensors. The Non-Local Means (NLMeans) filter has recently been proposed to denoise MRI with high signal-to-noise ratio (SNR). This filter has been shown to allow the best restoration of image intensities for the estimation of diffusion tensors (DT) compared to state-of-the-art methods. However, for DW-MR images with high b-values (and thus low SNR), the noise, which is strictly Rician-distributed, can no longer be approximated as additive white Gaussian, as implicitly assumed in the classical formulation of the NLMeans. High b-values are typically used in high angular resolution diffusion imaging (HARDI) or q-space imaging (QSI), for which an optimal restoration is critical. In this paper, we propose to adapt the NLMeans filter to Rician noise corrupted data. Validation is performed on synthetic data and on real data for both conventional MR images and DT images. Our adaptation outperforms the original NLMeans filter in terms of peak-signal-to-noise ratio (PSNR) for DW-MRI.