A novel tensor distribution model for the diffusion-weighted MR signal

Abstract

Diffusion MRI is a non-invasive imaging technique that allows the measurement of water molecule diffusion through tissue in vivo. The directional features of water diffusion allow one to infer the connectivity patterns prevalent in tissue and possibly track changes in this connectivity over time for various clinical applications. In this paper, we present a novel statistical model for diffusion-weighted MR signal attenuation which postulates that the water molecule diffusion can be characterized by a continuous mixture of diffusion tensors. An interesting observation is that this continuous mixture and the MR signal attenuation are related through the Laplace transform of a probability distribution over symmetric positive definite matrices. We then show that when the mixing distribution is a Wishart distribution, the resulting closed form of the Laplace transform leads to a Rigaut-type asymptotic fractal expression, which has been phenomenologically used in the past to explain the MR signal decay but never with a rigorous mathematical justification until now. Our model not only includes the traditional diffusion tensor model as a special instance in the limiting case, but also can be adjusted to describe complex tissue structure involving multiple fiber populations. Using this new model in conjunction with a spherical deconvolution approach, we present an efficient scheme for estimating the water molecule displacement probability functions on a voxel-by-voxel basis. Experimental results on both simulations and real data are presented to demonstrate the robustness and accuracy of the proposed algorithms.

Riemann-Finsler Geometry for Diffusion Weighted Magnetic Resonance Imaging

“…Clinical use of dwMRI is hampered by the fact that dwMRI analysis requires radically new approaches, based on abstract representations, a development still in its infancy. Examples are rank-2 symmetric positive-definite tensor representations in diffusion tensor imaging (DTI), pioneered by Basser, Mattiello and Le Bihan et al [1,2] and explored by many others [3,4,5,6,7,8,9,10,11,12,13,14], higher order symmetric positive-definite tensor representations [15,16,17,18,19,20,21], spherical harmonic representations in high angular resolution diffusion imaging (HARDI) [22,23,24,25,26], and SE(3) Lie group representations [27,28,29]. The latter type of representation, developed by Duits et al, appears to bear a particularly close relationship to the theory outlined below.…”

Efficient and robust computation of PDF features from diffusion MR signal

“…Despite its wide use, it assumes a displacement probability characterized by an oriented Gaussian Probability Diffusion Function (PDF). Consequently, DTI can only map a single orientation inside a voxel and fails in voxels having orientational heterogeneity [6]. Although high order methods have been introduced to better fit the ADC profile with more samples, one important issue remains: the ADC peaks do not necessarily yield the underlying main fiber orientations [7].…”

“…However the result is a convolution of the true ODF with a Bessel function so that each direction undesirably get corrupted by neighbor directions. The Fiber Orientation Distribution (FOD) method and its derivatives [6,24,25] try to compute the whole PDF volume by the deconvolution of the diffusion signal. The considered deconvolution kernel usually represents the signal of a single fiber model and requires a prior on either angular or radial MR signal or both.…”

A New Tensorial Framework for Single-Shell High Angular Resolution Diffusion Imaging

“…The terminology HARDI is used here to collectively denote schemes that employ generic functions on the unit sphere, including Tuch's orientation distribution function (ODF) via classical Q-Ball imaging [44], the higher order diffusion tensor model and the diffusion orientation transform (DOT) by Özarslan et al [36,37], analytical Q-Ball imaging [14,15], and the diffusion tensor distribution model by Jian et al [27], et cetera.…”