Analytic Expressions for the Uncertainty of DTI-derived Parameters and their Validation Using Monte Carlo Methods

Introduction: Monte Carlo (1) and Bootstrap (2) methods provide powerful statistical tools for determining the effects of background noise in diffusion weighted imaging (DWI) data on DTI-derived parameters, and for optimizing the design of DTI experiments. While these empirical methods do not provide analytical relationships between the variance of the distribution of noise in the DWI data and the variance of DTI-derived parameters, some progress has been made in this area: Skare et al. employed error propagation analysis to determine how noise in DWI data affects the uncertainty in the estimated ADCs (3) and in diffusion anisotropy measures (4). Matrix perturbation methods have also been used to propagate errors in estimated DTs themselves to determine the uncertainty in various quantities derived from the DT, such as its eigenvalues and eigenvectors (5,6), and its Trace (6). Analytic error propagation formulas to estimate uncertainties in DTI-derived quantities, such as the variation in the direction of the largest principal diffusivity, due to background noise have been derived elsewhere (7). Here Monte Carlo simulations of DTI experiments (1) are performed to validate these formulae, and to determine their applicability over a broad range of experimental design parameters (e.g., SNR, number of diffusion gradient directions, number of DWI acquisitions, etc.).