Symmetric positive 4th order tensors & their estimation from diffusion weighted MRI

Abstract

In Diffusion Weighted Magnetic Resonance Image (DW-MRI) processing a 2nd order tensor has been commonly used to approximate the diffusivity function at each lattice point of the DW-MRI data. It is now well known that this 2nd-order approximation fails to approximate complex local tissue structures, such as fibers crossings. In this paper we employ a 4th order symmetric positive semi-definite (PSD) tensor approximation to represent the diffusivity function and present a novel technique to estimate these tensors from the DW-MRI data guaranteeing the PSD property. There have been several published articles in literature on higher order tensor approximations of the diffusivity function but none of them guarantee the positive semi-definite constraint, which is a fundamental constraint since negative values of the diffusivity coefficients are not meaningful. In our methods, we parameterize the 4th order tensors as a sum of squares of quadratic forms by using the so called Gram matrix method from linear algebra and its relation to the Hilbert's theorem on ternary quartics. This parametric representation is then used in a nonlinear-least squares formulation to estimate the PSD tensors of order 4 from the data. We define a metric for the higher-order tensors and employ it for regularization across the lattice. Finally, performance of this model is depicted on synthetic data as well as real DW-MRI from an isolated rat hippocampus.

Fig. 1

Comparison of the fiber orientation errors for different amount of noise in the data, obtained by using: a) our parametrization to enforce positivity and b) without enforcing positivity of the estimated tensors

Fig. 2

Fiber orientation errors for different SNR in the data using our method for the estimation of positive 4^th-order tensors and two other existing methods: 1) DOT and 2) ODF. In the experiment we used simulated MR signal of a 2-fiber crossing, whose probability profile is shown in (a).

Fig. 3

Isolated rat hipppocampus. a) S₀, b) FA, c) White pixels indicate locations where the estimated 4^th-order tensor was not positive-definite, d) Manually labeled image based on knowledge of hippocampal anatomy. The index of the labels is: 1) dorsal hippocampal commissure, 2) fimbria, 3) alveus, 4) molecular layer, 5) mixture of CA3 stratum pyramidale and stratum lucidum.

Fig. 4

The estimated 4^th-order tensor field from an isolated rat hippocampus dataset using our method. (a) top:S₀ and bottom:FA, (b) the estimated displacement probability profiles of the 4^th-order tensor field in the region of interest (ROI) indicated by a green rectangle in (a). (c)Comparison of the estimated 2^nd-order tensors (top)and the estimated probability profiles of the 4^th-order tensors without (middle) and with regularization (bottom) in a ROI indicated by a black rectangle in (b).

Fig. 5

Left: The region of interest from the “dorsal hippocampal commissure”, which is magnified in the next plates of this figure. Comparison between the displacement probability profiles computed from non-PSD 4^th-order tensors (middle) and PSD tensors estimated by our method (right).