Analytical error propagation in diffusion anisotropy calculations

Abstract

Purpose:To develop an analytical formalism describing how noise and selection of diffusion-weighting scheme propagate through the diffusion tensor imaging (DTI) computational chain into variances of the diffusion tensor elements, and errors in the relative anisotropy (RA) and fractional anisotropy (FA) indices. Materials and Methods:Singular-value decomposition (SVD) was used to determine the tensor variances, with diffusion-weighting scheme and measurement noise incorporated into the design matrix. Anisotropy errors were then derived using propagation of error. To illustrate the applications of the model, 12 data sets were acquired from each human subject, over a range of b-values (500 -2500 seconds/mm 2 ) and diffusion-weighting gradient directions (N ϭ 6 -55). The mean RA and FA values and their respective errors were calculated within a region of interest (ROI) in the splenium. The RA and FA errors as a function of b-value and N were evaluated, and a number of diffusion-weighting schemes were assessed based on a new metric, sum of diffusion tensor variances.Results: When the acquisition time was held constant, the sum of the diffusion tensor variances decreased as N increased. The same trend was also observed for several diffusion-weighting schemes with constant condition number when noise in the diffusion-weighted (DW) images was assumed unity. Errors in both FA and RA increased with b-value and decreased with N. The FA error in the splenium was approximately threefold smaller than RA error, irrespective of b-value or N. Conclusion:The condition number may not adequately characterize the noise sensitivity for a given diffusionweighting scheme. Signal averaging may not be as effective as increasing N, especially when N is small (e.g., N Ͻ 13). Due to its smaller error, FA is preferred over RA for quantitative DTI applications.

A framework for quality control and parameter optimization in diffusion tensor imaging: theoretical analysis and validation

“…(14)(15)(16) are the DTI analogue of some results presented by Xing et al [9]. These important results expressed in terms of Eqs.…”

“…This work has also presented the propagation of error in the axial tensor described recently [21] and used the special case (FÃ 0) to provide an analytical formalism for DTI quality control that can be tested experimentally on widely available spherical water phantoms. The spherical tensor used in our formalism could not be handled asymptotically using analytical or perturbation theoretical methods attempted previously [9][10][11][12][13][14]. Note that Eqs.…”

“…DTI involves the acquisition of large data sets that require the selection of an immense parameter space which has not been clearly investigated with analytical approaches [4][5][6][7][8]. In general, the optimization of the single tensor multi-dimensional parameter problem remains a challenging complex multi-dimensional problem [1,[4][5][6][7][8][9][10][11][12][13][14][15]. The DT-MRI of the entire brain or a certain tissue volume-of-interest is usually sought in some clinically affordable time, T, and a repetition time, T R .…”

“…The DTI experiment seeks to acquire a total of N T =T/T R images to estimate the diffusion tensor D, which is represented as a symmetric 3×3 positive definite matrix (D ij =D ji , i,j=1,2,3) with six independent elements (three diagonal and three off-diagonal). A minimum of seven independent measurements are generally needed to encode the tensor [1,[4][5][6][7][8][9][10][11][12][13][14][15]. For example, we can acquire images in the steady state with fixed repetition time (T R ), echo time, (T E ), diffusion time (τ) at multiple diffusion weighting (b-factors), diffusion averaging (N d ), multiple encoding directions (N e ), and/or multiple reference images (N ref ).…”

Prediction of Functional Outcome in Acute Cerebral Hemorrhage Using Diffusion Tensor Imaging at 3T: A Prospective Study

“…However, the error was found to be approximately 3 times smaller than that in the relative anisotropy. 34 There was no significant difference in the FA from 6 and 13 MPG in the cerebral peduncles according to our preliminary study. The 6-MPG acquisition is more tolerable to the patients, given the very acute phase of the ICH.…”