Theoretical analysis of the effects of noise on diffusion tensor imaging

Abstract

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. Key words: diffusion tensor; noise; fiber trackingDiffusion tensor imaging (DTI) is a noninvasive method of characterizing tissue microstructure (1). In simple models of water diffusion in tissues, the directional dependence of diffusion is defined by the diffusion tensor. For example, the directions of highest and lowest diffusion are eigenvectors of the tensor and the diffusion coefficients in these directions are the corresponding eigenvalues. The anisotropy of diffusion in tissues is due primarily to cell membranes (2,3). Hence, the diffusion tensor depends on the cellular structure of tissues. DTI has been used to classify tissues on the basis of diffusion anisotropy (4), characterize changes associated with disease (5-7) and aging (8), and construct three-dimensional models of fiber tracts (9 -11).The diffusion tensor is estimated using a set of diffusion weighted images. Image noise produces errors in the calculated tensor and hence in its eigenvalues (principal diffusivities) and eigenvectors (principal axes). Random variations in these quantities complicate the analysis and interpretation of DTI experiments. For example, a basic step in quantifying diffusion is the calculation of mean eigenvalues of the tensor over repeated measures in a pixel, or across pixels in a homogeneous region of interest (ROI). Typically, this is accomplished by diagonalizing each independent measure of the tensor and averaging the corresponding eigenvalues. However, to perform this calculation some rule must be used to identify the corresponding eigenvalues in the independent measurements. At high signal-to-noise ratio (SNR) this is trivial, but otherwise noise perturbs the eigenvalues, making misclassification more likely. Pierpaoli and Basser (12) used Monte Carlo simulations to show that sorting eigenvalues by magnitude leads to an overestimate of the highest eigenvalue and an underestimate of the lower eigenvalue. This in turn leads to an overestimate of diffusion anisotropy. Hence, the difference between eigenvalues in an isotropic system can appear to be statistically significant. Other studies have used im...

Diffusion tensor fiber tracking of human brain connectivity: aquisition methods, reliability analysis and biological results

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“…This square-root relation was also predicted analytically by Anderson under certain assumptions. 68 Variances as a function of length were also assessed analytically in MRI computation of velocity streamlines. 59 Table 1 reports the statistical results in d x and d y for a variety of technical factors, while Table 2 reports the results for different biological factors.…”

Analytical error propagation in diffusion anisotropy calculations

“…The tensor shape invariants were then calculated at each voxel according to Eqs. [15][16][17] and their variances according to Eqs. [18 -20].…”

“…[15][16][17] in the text. The error in the shape invariants can be computed analytically using propagation of error (30) and the variances of the diffusion tensor elements in Eq.…”

Assessment of diffusion tensor image quality across sites and vendors using the American College of Radiology head phantom

“…Many aspects of scanner performance have been reported to affect the measurement results and introduce discrepancies in the measurement (14) . Factors affecting scanner performance include, but are not limited to, SNR, 15 , 16 , 17 susceptibility effects due to poor shimming, (18) and image distortion resulting from eddy currents (19) . In this study, we compared some specific products of three commercial MR vendors.…”