The reduction of the sorting bias in the eigenvalues of the diffusion tensor

Mapping early tumor response to radiotherapy using diffusion kurtosis imaging*

PURPOSE

In this pilot study, DKI measures of diffusivity and kurtosis were compared in active tumor regions and correlated to radiologic response to radiotherapy after completion of 2 weeks of treatment to derive potential early measures of tumor response.

METHODS

MRI and Magnetic Resonance Spectroscopic Imaging (MRSI) data were acquired before the beginning of RT (pre-RT) and 2 weeks after the initiation of treatment (during-RT) in 14 glioblastoma patients. The active tumor region was outlined as the union of the residual contrast-enhancing region and metabolically active tumor region. Average and standard deviation of mean, axial, and radial diffusivity (MD, AD, RD) and mean, axial, and radial kurtosis (MK, AK, RK) values were calculated for the active tumor VOI from images acquired pre-RT and during-RT and paired t-tests were executed to estimate pairwise differences. Receiver operating characteristic (ROC) curve analysis was conducted to evaluate the predictive capabilities of changes in diffusion metrics for progression-free survival (PFS).

RESULTS

Analysis showed significant pairwise differences for AD (p = 0.035; Cohen's d of 0.659) and AK (p = 0.019; Cohen's d of 0.753) in diffusion measures after 2 weeks of RT. ROC curve analysis showed that percentage change differences in AD and AK between pre-RT and during-RT scans provided an Area Under the Curve (AUC) of 0.524 and 0.762, respectively, in discriminating responders (PFS>180 days) and non-responders (PFS<180 days).

CONCLUSION

This pilot study, although preliminary in nature, showed significant changes in AD and AK maps, with kurtosis derived AK maps showing an increased sensitivity in mapping early changes in the active tumor regions.

On the averaging of cardiac diffusion tensor MRI data: the effect of distance function selection

Diffusion tensor magnetic resonance imaging (DT-MRI) allows a unique insight into the microstructure of highly-directional tissues. The selection of the most proper distance function for the space of diffusion tensors is crucial in enhancing the clinical application of this imaging modality. Both linear and nonlinear metrics have been proposed in the literature over the years. The debate on the most appropriate DT-MRI distance function is still ongoing. In this paper, we presented a framework to compare the Euclidean, affine-invariant Riemannian and log-Euclidean metrics using actual high-resolution DT-MRI rat heart data. We employed temporal averaging at the diffusion tensor level of three consecutive and identically-acquired DT-MRI datasets from each of five rat hearts as a means to rectify the background noise-induced loss of myocyte directional regularity. This procedure is applied here for the first time in the context of tensor distance function selection. When compared with previous studies that used a different concrete application to juxtapose the various DT-MRI distance functions, this work is unique in that it combined the following: (i) metrics were judged by quantitative-rather than qualitative-criteria, (ii) the comparison tools were non-biased, (iii) a longitudinal comparison operation was used on a same-voxel basis. The statistical analyses of the comparison showed that the three DT-MRI distance functions tend to provide equivalent results. Hence, we came to the conclusion that the tensor manifold for cardiac DT-MRI studies is a curved space of almost zero curvature. The signal to noise ratio dependence of the operations was investigated through simulations. Finally, the 'swelling effect' occurrence following Euclidean averaging was found to be too unimportant to be worth consideration.

Corpus callosum morphology and microstructure assessed using structural MR imaging and diffusion tensor imaging: initial findings in adults with neurofibromatosis type 1

BACKGROUND AND PURPOSE

Imaging studies have shown that children with NF-1 have increased brain volumes compared with age-matched controls and the CCs are disproportionately large. The purpose of this study was to determine if the CC in adults with NF-1 differed from that in matched controls by using DTI and volumetric imaging.

MATERIALS AND METHODS

MR imaging with DTI was performed in 10 adults with NF-1 and in 10 age-, sex-, and handedness-matched controls by using a 3T system. Total brain volumes and the areas and central lengths of the CC were calculated, along with the radial width of callosal subdivisions, in the 2 groups.

RESULTS

Our results showed that the total brain volume was not significantly different between adults with NF-1 and matched controls. The length and total cross-sectional area of the CC were statistically larger in adults with NF-1 compared with controls (approximately 10% longer and 20% greater area). On DTI we found a preservation of the primary eigenvalue with increases in the minor eigenvalues at the genu.

CONCLUSIONS

We have shown that the increased size of the CC found in children with NF-1 is also present in adults with the syndrome, whereas no difference in total brain volume was found.

About "axial" and "radial" diffusivities

This article presents the potential problems arising from the use of "axial" and "radial" diffusivities, derived from the eigenvalues of the diffusion tensor, and their interpretation in terms of the underlying biophysical properties, such as myelin and axonal density. Simulated and in vivo data are shown. The simulations demonstrate that a change in "radial" diffusivity can cause a fictitious change in "axial" diffusivity and vice versa in voxels characterized by crossing fibers. The in vivo data compare the direction of the principle eigenvector in four different subjects, two healthy and two affected by multiple sclerosis, and show that the angle, alpha, between the principal eigenvectors of corresponding voxels of registered datasets is greater than 45 degrees in areas of low anisotropy, severe pathology, and partial volume. Also, there are areas of white matter pathology where the "radial" diffusivity is 10% greater than that of the corresponding normal tissue and where the direction of the principal eigenvector is altered by more than 45 degrees compared to the healthy case. This should strongly discourage researchers from interpreting changes of the "axial" and "radial" diffusivities on the basis of the underlying tissue structure, unless accompanied by a thorough investigation of their mathematical and geometrical properties in each dataset studied.

Using Perturbation theory to reduce noise in diffusion tensor fields

We propose the use of Perturbation theory to reduce noise in Diffusion Tensor (DT) fields. Diffusion Tensor Imaging (DTI) encodes the diffusion of water molecules along different spatial directions in a positive definite, 3 x 3 symmetric tensor. Eigenvectors and eigenvalues of DTs allow the in vivo visualization and quantitative analysis of white matter fiber bundles across the brain. The validity and reliability of these analyses are limited, however, by the low spatial resolution and low Signal-to-Noise Ratio (SNR) in DTI datasets. Our procedures can be applied to improve the validity and reliability of these quantitative analyses by reducing noise in the tensor fields. We model a tensor field as a three-dimensional Markov Random Field and then compute the likelihood and the prior terms of this model using Perturbation theory. The prior term constrains the tensor field to be smooth, whereas the likelihood term constrains the smoothed tensor field to be similar to the original field. Thus, the proposed method generates a smoothed field that is close in structure to the original tensor field. We evaluate the performance of our method both visually and quantitatively using synthetic and real-world datasets. We quantitatively assess the performance of our method by computing the SNR for eigenvalues and the coherence measures for eigenvectors of DTs across tensor fields. In addition, we quantitatively compare the performance of our procedures with the performance of one method that uses a Riemannian distance to compute the similarity between two tensors, and with another method that reduces noise in tensor fields by anisotropically filtering the diffusion weighted images that are used to estimate diffusion tensors. These experiments demonstrate that our method significantly increases the coherence of the eigenvectors and the SNR of the eigenvalues, while simultaneously preserving the fine structure and boundaries between homogeneous regions, in the smoothed tensor field.

The ellipsoidal area ratio: an alternative anisotropy index for diffusion tensor imaging

In the processing and analysis of diffusion tensor imaging (DTI) data, certain predefined morphological features of diffusion tensors are often represented as simplified scalar indices, termed diffusion anisotropy indices (DAIs). When comparing tensor morphologies across differing voxels of an image, or across corresponding voxels in different images, DAIs are mathematically and statistically more tractable than are the full tensors, which are probabilistic ellipsoids consisting of three orthogonal vectors that each has a direction and an associated scalar magnitude. We have developed a new DAI, the "ellipsoidal area ratio" (EAR), to represent the degree of anisotropy in the morphological features of a diffusion tensor. The EAR is a normalized geometrical measure of surface curvature in the 3D diffusion ellipsoid. Monte Carlo simulations and applications to the study of in vivo human data demonstrate that, at low noise levels, EAR provides a similar contrast-to-noise ratio (CNR) but a higher signal-to-noise ratio (SNR) than does fractional anisotropy (FA), which is currently the most popular anisotropy index in active use. Moreover, at the high noise levels encountered most commonly in real-world DTI datasets, EAR compared with FA is consistently much more robust to perturbations from noise and it provides a higher CNR, features useful for the analysis of DTI data that are inherently noise sensitive.

Towards in vivo diffusion tensor MRI on human heart using edge-preserving regularization

We investigate the noise sensitivity in various Diffusion Tensor MRI acquisition protocols in sixteen human ex vivo hearts. In particular, we compare the accuracy of protocols with various numbers of excitations and diffusion sensitizing directions for estimating the principal diffusion directions in the myocardium. It is observed that noise sensitivity decreases as the number of excitations and the number of sensitizing directions increase (and hence as the acquisition time increases). To reduce the effects of noise and to improve the results obtained with a smaller number of excitations and/or a smaller number of sensitizing directions, we introduce a 3-D edge-preserving regularization method operating on diffusion weighted images. It allows to maintain the quality of the principal diffusion direction field while minimizing the acquisition time, which is a necessary step for in vivo diffusion tensor MR imaging of the human heart.

Voxel-wise comparisons of the morphology of diffusion tensors across groups of experimental subjects

Water molecules in the brain diffuse preferentially along the fiber tracts within white matter that form the anatomical connections across spatially distant brain regions. A diffusion tensor (DT) is a probabilistic ellipsoid composed of three orthogonal vectors, each having a direction and an associated scalar magnitude, that represent the probability of water molecules diffusing in each of those directions. The 3D morphologies of DTs can be compared across groups of subjects to reveal disruptions in structural organization and neuroanatomical connectivity of the brains of persons with various neuropsychiatric illnesses. Comparisons of tensor morphology across groups have typically been performed on scalar measures of diffusivity, such as Fractional Anisotropy (FA) rather than directly on the complex 3D morphologies of DTs. Scalar measures, however, are related in nonlinear ways to the eigenvalues and eigenvectors that create the 3D morphologies of DTs. We present a mathematical framework that permits the direct comparison across groups of mean eigenvalues and eigenvectors of individual DTs. We show that group-mean eigenvalues and eigenvectors are multivariate Gaussian distributed, and we use the Delta method to compute their approximate covariance matrices. Our results show that the theoretically computed mean tensor (MT) eigenvectors and eigenvalues match well with their respective true values. Furthermore, a comparison of synthetically generated groups of DTs highlights the limitations of using FA to detect group differences. Finally, analyses of in vivo DT data using our method reveal significant between-group differences in diffusivity along fiber tracts within white matter, whereas analyses based on FA values failed to detect some of these differences.

Impact of an improved combination of signals from array coils in diffusion tensor imaging

An improved method for the combination of signals from array coils is presented as a way to reduce the influence of the noise floor on the estimation of diffusion tensor imaging (DTI) parameters. By an optimized combination of signals from the array channels and complex averaging of measurements, this method leads to a significant reduction of the noise bias. This combination algorithm allows computation of accurate tensors by using the simple two point method and is shown to provide results similar to the ones obtained using the standard signal combination and a nonlinear regression method with noise parameter estimation. In many applications, the use of this combination method would result in a scan time reduction in comparison to the current standard. The effects of the improved combination on diffusion decay curves, fractional anisotropy maps, and apparent diffusion coefficient (ADC) profiles are demonstrated.

Non-invasive Fiber Tracking on Diffusion Tensor MRI Using High-Temperature Superconducting Tape RF coil

High-temperature superconducting (HTS) coil is one of the best ways to increase the signal-to-noise ratio (SNR). Bi2Sr2Ca2Cu3Ox (Bi-2223) tapes were suitable to use because of the easier fabrications and lower cost. In this study, we built HTS Bi-2223 tape coils and demonstrated that the SNR of using the HTS tape coil was 3 or 4 folds higher than that of the traditional copper coil for a rat brain MR study. Acquisition time of MR diffusion tensor imaging (DTI) can be reduced by factor of 9 for the same signal-to-noise. Accuracy of fiber tracking using DTI is also significantly improved by a factor of 2.5 or so using HTS coil. In summary, with this HTSC system, a 3T MR system could reach the high signal-to-noise of 12 T MR system with the advantage of less T2 shortening effects at high field. Currents researches are focused on brain connectivity and fMRI studies.

A statistical framework for the classification of tensor morphologies in diffusion tensor images

Tractography algorithms for diffusion tensor (DT) images consecutively connect directions of maximal diffusion across neighboring DTs in order to reconstruct the 3-dimensional trajectories of white matter tracts in vivo in the human brain. The performance of these algorithms, however, is strongly influenced by the amount of noise in the images and by the presence of degenerate tensors-- i.e., tensors in which the direction of maximal diffusion is poorly defined. We propose a simple procedure for the classification of tensor morphologies that uses test statistics based on invariant measures of DTs, such as fractional anisotropy, while accounting for the effects of noise on tensor estimates. Examining DT images from seven human subjects, we demonstrate that this procedure validly classifies DTs at each voxel into standard types (nondegenerate DTs, as well as degenerate oblate, prolate or isotropic DTs), and we provide preliminary estimates for the prevalence and spatial distribution of degenerate tensors in these brains. We also show that the P values for test statistics are more sensitive tools for classifying tensor morphologies than are invariant measures of anisotropy alone.

Resolving fiber crossing using advanced fast marching tractography based on diffusion tensor imaging

Magnetic resonance diffusion tensor tractography is a powerful tool for the non-invasive depiction of the white matter architecture in the human brain. However, due to limitations in the underlying tensor model, the technique is often unable to reconstruct correct trajectories in heterogeneous fiber arrangements, such as axonal crossings. A novel tractography method based on fast marching (FM) is proposed which is capable of resolving fiber crossings and also permits trajectories to branch. It detects heterogeneous fiber arrangements by incorporating information from the entire diffusion tensor. The FM speed function is adapted to the local tensor characteristics, allowing in particular to maintain the front evolution direction in crossing situations. In addition, the FM's discretization error is reduced by increasing the number of considered possible front evolution directions. The performance of the technique is demonstrated in artificial data and in the healthy human brain. Comparisons with standard FM tractography and conventional line propagation algorithms show that, in the presence of interfering structures, the proposed method is more accurate in reconstructing trajectories. The in vivo results illustrate that the elucidated major white matter pathways are consistent with known anatomy and that multiple crossings and tract branching are handled correctly.

Parametric and non-parametric statistical analysis of DT-MRI data

In this work parametric and non-parametric statistical methods are proposed to analyze Diffusion Tensor Magnetic Resonance Imaging (DT-MRI) data. A Multivariate Normal Distribution is proposed as a parametric statistical model of diffusion tensor data when magnitude MR images contain no artifacts other than Johnson noise. We test this model using Monte Carlo (MC) simulations of DT-MRI experiments. The non-parametric approach proposed here is an implementation of bootstrap methodology that we call the DT-MRI bootstrap. It is used to estimate an empirical probability distribution of experimental DT-MRI data, and to perform hypothesis tests on them. The DT-MRI bootstrap is also used to obtain various statistics of DT-MRI parameters within a single voxel, and within a region of interest (ROI); we also use the bootstrap to study the intrinsic variability of these parameters in the ROI, independent of background noise. We evaluate the DT-MRI bootstrap using MC simulations and apply it to DT-MRI data acquired on human brain in vivo, and on a phantom with uniform diffusion properties.

Euclidian distance-weighted smoothing for quantitative MRI: application to intervoxel anisotropy index mapping with DTI

During the computation of intervoxel anisotropy features, the inclusion of both eigenvalues and eigenvectors reduces the effect of noise, but spatial averaging blurs the resulting maps. We propose a new adaptive technique that uses data-dependent weights in the averaging process so that the influence of each neighbor in the local window is proportional to the similarity of characteristics of the neighbor considered to those of the reference central voxel. This likeness criterion is based on the multidimensional Euclidian distance using the entire available multispectral information contained in the diffusion-weighted images. This solution is controlled by a single parameter beta that results from a compromise between edge-preserving and noise-smoothing abilities. This Euclidian distance-weighted technique is a generic solution for filtering noise during parametric reconstruction. It was applied to map anisotropy using an intervoxel lattice index (LI) from experimental images of mouse brain in vivo and achieves noise reduction without distorting small anatomical structures. We also show how to employ in the discrimination scheme the images not used in the estimation of the considered feature.

Diffusion-tensor MRI: theory, experimental design and data analysis - a technical review

This article treats the theoretical underpinnings of diffusion-tensor magnetic resonance imaging (DT-MRI), as well as experimental design and data analysis issues. We review the mathematical model underlying DT-MRI, discuss the quantitative parameters that are derived from the measured effective diffusion tensor, and describe artifacts that arise in typical DT-MRI acquisitions. We also discuss difficulties in identifying appropriate models to describe water diffusion in heterogeneous tissues, as well as in interpreting experimental data obtained in such issues. Finally, we describe new statistical methods that have been developed to analyse DT-MRI data, and their potential uses in clinical and multi-site studies.

Anisotropic water transport in the human eye lens studied by diffusion tensor NMR micro-imaging

We report in vitro measurements of effective diffusion tensors characterising the anisotropic transport of water in human eye lenses ranging in age from 13 to 86 years. The measurements were obtained by means of a pulsed field gradient spin echo (PFGSE) magnetic resonance imaging (MRI) technique at a spatial resolution of 218 x 218 x 1000 microm(3). The results show that water diffusion is both spatially inhomogeneous and highly anisotropic on the timescale of the measurements (approximately 15 msec). Diffusion parallel to the long axes of the lens fibre cells is relatively unrestricted, whereas that between cells is substantially inhibited by the cell membranes, particularly in the inner cortex region of the lens. The data confirm the presence of a diffusion barrier surrounding the lens nucleus, which inhibits transport of water and other small molecules into and out of the nucleus. The results shed light on factors that may influence the onset of presbyopia and senile cataract. They also have implications for delivery of drugs to the lens nucleus.

White matter mapping using diffusion tensor MRI

Diffusion tensor MRI is used to define trajectories that reflect the long-range order of in vivo white matter (WM) fiber tracts. Fiber tracking is particularly prone to cumulative error from noise and partial volume along the length of the trajectory paths, but the overall shape of each path is anatomically meaningful. By considering only the long-range similarity of path shapes, a method of constructing 3D maps of specific WM structures has been developed. A trajectory is first computed from an operator-selected seed voxel, located within the anatomical structure of interest (SOI). Voxels from the same structure are then automatically identified based on the similarity of trajectory path shapes, assessed using Pearson's correlation coefficient. The corpus callosum and pyramidal tracts in 14 patients with multiple sclerosis, and in 10 healthy controls were mapped by this method, and the apparent diffusion coefficient (ADC) was measured. The ADC was significantly higher in patients than in controls, and higher in the corpus callosum than in the pyramidal tracts for both groups. Using this method the different functional structures in the WM may be identified and mapped. Within these maps, MRI parameters can be measured for subsequent comparison with relevant clinical data.

Diffusion tensor brain imaging and tractography

Diffusion-tensor MR imaging is a promising tool to evaluate white-matter integrity by quantitative and graphic maps including neural fiber tractogram. Current challenges afoot are to obtain higher quality diffusion-weighted MR images (high SNR, isotropic voxel, and high spatial resolution), to create a robust mathematical framework to process the data, to construct a user-friendly computer-based algorithm, to reveal determinants of diffusion process, and to establish analytical methodology.

Theoretical analysis of the effects of noise on diffusion tensor imaging

A theoretical framework is presented for understanding the effects of noise on estimates of the eigenvalues and eigenvectors of the diffusion tensor at moderate to high signal-to-noise ratios. Image noise produces a random perturbation of the diffusion tensor. Power series solutions to the eigenvalue equation are used to evaluate the effects of the perturbation to second order. It is shown that in anisotropic systems the expectation value of the largest eigenvalue is overestimated and the lowest eigenvalue is underestimated. Hence, diffusion anisotropy is overestimated in general. This result is independent of eigenvalue sorting bias. Furthermore, averaging eigenvalues over a region of interest produces greater bias than averaging tensors prior to diagonalization. Finally, eigenvector noise is shown to depend on the eigenvalue contrast and imposes a theoretical limit on the accuracy of simple fiber tracking schemes. The theoretical results are shown to agree with Monte Carlo simulations.

Improving relative anisotropy measurement using directional correlation of diffusion tensors

A method employing directional correlation of the diffusion tensor, directional-correlation weighted relative anisotropy (DRA), was developed to improve the accuracy of estimated relative anisotropy (RA). The intravoxel directional correlation was established on the same voxel between two identically acquired diffusion tensor images, and the correlation coefficient derived from tensor dot product was employed as the weighting factor applied in the calculation of RA. The effect of noise influence was reduced since the random noise between repeated scans is not directionally correlated. The RA and the inter- and intravoxel DRA estimations were examined on rat brains in vivo. The background noise alters the direction of eigenvectors and the magnitude of eigenvalues. The dispersion angle between repeatedly obtained eigenvectors, representing the extent of directional alteration of eigenvectors, depends on the tissue anisotropy as well as the signal-to-noise ratio (SNR) of the source images. Current results demonstrate that the intravoxel DRA improves the accuracy of RA estimation, increases the relative contrast of gray and white matter, and avoids the partial volume effect commonly seen in the intervoxel operations.

Validation of diffusion tensor magnetic resonance axonal fiber imaging with registered manganese-enhanced optic tracts

Noninvasive mapping of white matter tracts using diffusion tensor magnetic resonance imaging (DTMRI) is potentially useful in revealing anatomical connectivity in the human brain. However, a gold standard for validating DTMRI in defining axonal fiber orientation is still lacking. This study presents the first validation of the principal eigenvector of the diffusion tensor in defining axonal fiber orientation by superimposing DTMRI with manganese-enhanced MRI of optic tracts. A rat model was developed in which optic tracts were enhanced by manganese ions. Manganese ion (Mn(2+)) is a potent T1-shortening agent and can be uptaken and transported actively along the axon. Based on this property, we obtained enhanced optic tracts with a T1-weighted spin-echo sequence 10 h after intravitreal injection of Mn(2+). The images were compared with DTMRI acquired with exact spatial registration. Deviation angles between tangential vectors of the enhanced tracts and the principal eigenvectors of the diffusion tensor were then computed pixel by pixel. We found that under signal-to-noise (SNR) of 30, the variance of deviation angles was (13.27 degrees). In addition, the dependence of this variance on SNR obeys stochastic behavior if SNR is greater than 10. Based on this relation, we estimated that an rms deviation of less than 10 degrees could be achieved with DTMRI when SNR is 40 or greater. In conclusion, our method bypasses technical difficulties in conventional histological approach and provides an in vivo gold standard for validating DTMRI in mapping white matter tracts.

Noise considerations in the determination of diffusion tensor anisotropy

In this study the noise sensitivity of various anisotropy indices has been investigated by Monte-Carlo computer simulations and magnetic resonance imaging (MRI) measurements in a phantom and 5 healthy volunteers. Particularly, we compared the noise performance of indices defined solely in terms of eigenvalues and those based on both the eigenvalues and eigenvectors. It is found that anisotropy indices based on both eigenvalues and eigenvectors are less sensitive to noise, and spatial averaging with neighboring pixels can further reduce the standard deviation. To reduce the partial volume effect caused by the spatial averaging with neighboring voxels, an averaging method in the time domain based on the orientation coherence of eigenvectors in repeated experiments has been proposed.

Nonlinear smoothing for reduction of systematic and random errors in diffusion tensor imaging

Calculation and sorting of the eigenvectors of diffusion using diffusion tensor imaging has previously been shown to be sensitive to noise levels in the acquired data. This sensitivity manifests as random and systematic errors in the diffusion eigenvalues and derived parameters such as indices of anisotropy. An optimized application of nonlinear smoothing techniques to diffusion data prior to calculation of the diffusion tensor is shown to reduce both random and systematic errors, while causing little blurring of anatomical structures. Conversely, filtering applied to calculated images of fractional anisotropy is shown to fail in reducing systematic errors and in recovering anatomical detail. Using both real and simulated brain data sets, it is demonstrated that this approach has the potential to allow acquisition of data that would otherwise be too noisy to be of use.