Generalized scalar measures for diffusion MRI using trace, variance, and entropy

Abstract

This paper details the derivation of rotationally invariant scalar measures from higher-rank diffusion tensors (DTs) and functions defined on a unit sphere. This was accomplished with the use of an expression that generalizes the evaluation of the trace operator to tensors of arbitrary rank, and even to functions whose domains are the unit sphere. It is shown that the mean diffusivity is invariant to the selection of tensor rank for the model used. However, this rank invariance does not apply to the anisotropy measures. Therefore, a variance-based, general anisotropy measure is proposed. Also an information theoretical parametrization of anisotropy is introduced that is frequently more consistent with the meaning attributed to anisotropy. We accomplished this by associating anisotropy with the amount of orientational information present in the data Diffusional attenuation of the magnetic resonance (MR) signal as a result of the mixing of phase incoherent spins has been known since Hahn (1) first introduced spin echoes in the early days of MR. When the diffusion process is non-Fickian (i.e., the molecular flux density is not oriented opposite to the concentration gradient), the diffusivity is better quantified with a symmetric, rank-2, positive definite tensor (2). At typical resolutions for MRI and microscopy, it has been shown that a macroscopic effective diffusion tensor (DT; henceforth referred to as the "traditional" rank-2 DT) can be calculated that is assumed to have properties similar to those of the true DT (3). The traditional DT has six distinct components, implying that six independent numbers are needed to fully describe this tensor. These six numbers can be chosen to be quantities that are more meaningful than the tensor components. As an example, the principal eigenvector, which can be expressed in terms of two numbers (such as the azimuthal and polar angles that specify a direction in three-dimensional space), has been hypothesized to give the local fiber orientation within the tissue (4). Two other numbers, the mean diffusivity and a measure of anisotropy, were found to be useful in quantitative studies in which comparative analyses were performed (5).The inability of traditional DT imaging (DTI) to resolve more than one fiber direction in a voxel has prompted recent interest in formulating more sophisticated techniques. Tuch et al. (6) developed a clinically feasible approach called high-angular-resolution diffusion imaging (HARDI), in which apparent diffusion coefficients are measured along many directions distributed almost isotropically on the surface of a sphere. In a recent publication (7), we expressed the diffusivities in terms of Cartesian tensors of rank higher than 2 that enabled a straightforward generalization of the Bloch-Torrey result (8) and led to the formulation of a generalized Stejskal-Tanner (9) equation:where u is a unit vector that specifies the direction of the diffusion gradients, whose components are given bywhere and are the polar and azimuthal angles, respectively. In ...

Fast Orientation Mapping from HARDI

“…In order to test the performance of the DOT method on real tissue, we computed the orientation probabilities on data from three anatomical regions (all from ex- For visualization purposes, we have overlaid the orientation surfaces on generalized anisotropy (GA) maps [17] computed from the displacement probabilities. The directionality of the probability profiles on the image plane is readily available from the surfaces.…”

Disease classification: A probabilistic approach

“…Several scalar measures derived from the single tensor model have been proposed in the literature [8,12,13]. In particular, we use a set of three orthogonal invariants studied in [8], namely the norm N , fractional anisotropy F A and mode M d .…”

Efficient experimental designs for isotropic generalized diffusion tensor MRI (IGDTI)

“…We evaluate the potential of IGDTI to measure, within clinically feasible scan durations, mADCs over a wide range of b-values and various rotation-invariant microstructural parameters, such as HOT-Traces (10,17), or the mean t-kurtosis (14). The proposed IGDTI framework may improve and accelerate experimental designs for clinical assessments with advanced diffusion MRI methods (13,23) and encourage the development of new, biologically specific, rotation-invariant parameters for quantifying tissue water mobilities in clinical applications.…”