Characterizing fiber directional uncertainty in diffusion tensor MRI

Abstract

Image noise in diffusion tensor MRI (DT-MRI) causes errors in the measured tensor and hence variance in the estimated fiber orientation. Uncertainty in fiber orientation has been described using a circular "cone of uncertainty" (CU) around the principal eigenvector of the DT. The CU has proved to be a useful construct for quantifying and visualizing the variability of DT-MRI parameters and fiber tractography. The assumption of circularity of the CU has not been tested directly, however. In this work, bootstrap analysis and simple theoretical arguments were used to show that the CU is elliptical and multivariate normal in the vast majority of white matter (WM) voxels for typical measurement conditions. The dependence of the cone angle on the signal-to-noise ratio (SNR) and eigenvalue contrast was established. The major and minor cone axes are shown to be coincident with the second and third eigenvectors of the tensor, respectively, in the limit of many uniformly spaced diffusionencoding directions. The deviation between the major cone axis and the second eigenvector was quantified for typical sets of diffusion-weighting ( Diffusion tensor MRI (DT-MRI) provides information on water molecular diffusion based on a series of images with diffusion weighting applied in at least six noncollinear directions (1,2). Given the effective DT, it is possible to characterize tissue microstructure using the directional dependence of diffusion in vivo (3,4). In addition, this directional information has been exploited to estimate the paths of fiber bundles in the brain in order to infer axonal connectivity (5). In this case, the principal eigenvector (i.e., the estimated direction of maximum diffusivity), v 1 , is taken to be parallel to the local fiber bundle. In the presence of image noise, however, perturbations of the DT field introduce errors in the estimated diffusion anisotropy (6 -8) and fiber direction (9,10).The directional uncertainty in v 1 has been characterized by the "cone of uncertainty" (CU) (11,12). This was defined as a circular cone with axis along the expectation value of v 1 and cone angle equal to the uncertainty (i.e., confidence interval) in the orientation of v 1 . The CU is particularly useful for visualizing the uncertainty in fiber orientation and predicting error in MR fiber tractography. However, Lazar and Alexander (13) and Lazar et al. (14) showed that errors in MR tractography typically have an elliptical distribution, and noted that an elliptical CU is expected in voxels that lack axially symmetric diffusion. They measured the dispersion of fiber-tracking errors in a plane perpendicular to the fiber axis at some distance from the seed points. They noted a strong correlation between the direction of the greatest dispersion of tracking errors and the second eigenvector of the tensor (corresponding to the second-largest diffusivity). The same relationship existed between the direction of the smallest dispersion of tracking errors and the third eigenvector of the tensor (corresponding to the small...

The Elliptical Cone of Uncertainty and Its Normalized Measures in Diffusion Tensor Imaging

“…Although it is interesting to compare the angle of deviation of the two major axes of the elliptical COU to that of the circular COU, a consistent comparison cannot be easily made because the formalism used in the circular COU, (26), is ill-suited for establishing the joint confidence region for the major eigenvector. Specifically, the angle of deviation of the circular COU is derived from a hyper-cuboid region in the space of the diffusion tensor elements determined by δD rather than from a confidence region in the space of the elements of the major eigenvector.…”

“…Note that δD is taken to be the standard deviation of the tensor elements in computing the angle in (26), [17], [20], [29].…”

“…For example, the noninformative prior in angular space used by Behrens et al [23] and the zeroth order directional uncertainty proposed by Parker et al [24] were modeled after directional dispersion that is symmetric or circular. Although we have previously described an analytical method based on error propagation via diffusion tensor representations to account for asymmetry in tract dispersion [25], [47] the experimental observations by Jeong et al [26] and Lazar et al [27], [28] that the cones of uncertainty in the brain are mostly elliptical call for a closer investigation of analytical approaches to support their nonparametric studies, which were based on the bootstrap method. Since the dispersion is actually elliptical, the proposed analytical local measure of tract dispersion has a critical role to play in both deterministic and probabilistic methods of tractography [23], [24], [29]- [32].…”

Principles and Limitations of Computational Algorithms in Clinical Diffusion Tensor MR Tractography

“…54,55 Investigators having noticed the importance of estimation errors in fiber direction have proposed means to graphically depict the directional uncertainty on a voxel-by-voxel basis (Fig 7). 45,56 Models applicable to identifying the limitations of tractography algorithms are also available. 57 While likely useful in checking the reliability of fiber tracking results, the complexity of these tools is perhaps no less than the tractography algorithms themselves, hence substantially lowering the intention of usage for the general clinicians.…”

“…In addition to spending time studying the computational principles and playing around with freely adjustable parameter settings, trying an uncertainty test on major and minor tracts is strongly encour- aged 56,57,69 to have an idea of the algorithm's limitations so that one knows where to start or stop the interpretations of results. Another rough way to evaluate uncertainty is to check the left-right symmetry of reconstructed tracts when selecting a central region of interest (eg, the splenium of the corpus callosum) on images obtained from healthy subjects, 70 provided that the fiber bundles do not play functional roles with known lateralization (Fig 10).…”