Analytical error propagation in diffusion anisotropy calculations

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 seeonds/mm(2)) and diffusion-weighting gradient directions (N = 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 diffusion-weighting scheme. Signal averaging may not be as effective as increasing N, especially when N is small (e.g., N < 13). Due to its smaller error, FA is preferred over RA for quantitative DTI applications.