Fast and Analytical EAP Approximation from a 4th-Order Tensor.

Aurobrata Ghosh,Rachid Deriche

doi:10.1155/2012/192730

Abstract

Generalized diffusion tensor imaging (GDTI) was developed to model complex apparent diffusivity coefficient (ADC) using higher-order tensors (HOTs) and to overcome the inherent single-peak shortcoming of DTI. However, the geometry of a complex ADC profile does not correspond to the underlying structure of fibers. This tissue geometry can be inferred from the shape of the ensemble average propagator (EAP). Though interesting methods for estimating a positive ADC using 4th-order diffusion tensors were developed, GDTI in general was overtaken by other approaches, for example, the orientation distribution function (ODF), since it is considerably difficult to recuperate the EAP from a HOT model of the ADC in GDTI. In this paper, we present a novel closed-form approximation of the EAP using Hermite polynomials from a modified HOT model of the original GDTI-ADC. Since the solution is analytical, it is fast, differentiable, and the approximation converges well to the true EAP. This method also makes the effort of computing a positive ADC worthwhile, since now both the ADC and the EAP can be used and have closed forms. We demonstrate our approach with 4th-order tensors on synthetic data and in vivo human data.

Highlights

Generalized diffusion tensor imaging (GDTI) [1,2,3], was proposed to model the apparent diffusion coefficient (ADC) recovered by diffusion MRI when imaging the diffusion of water molecules in heterogeneous media like the cerebral white matter
This can be understood from the q-space formalism, where it can be seen that the apparent diffusivity coefficient (ADC) and the diffusion signal are in the Fourier domain of the diffusion ensemble average propagator (EAP), which describes the probability of the diffusing particles
GDTI was developed to model complex ADC profiles which was an inherent shortcoming of DTI

Summary

Introduction

Generalized diffusion tensor imaging (GDTI) [1,2,3], was proposed to model the apparent diffusion coefficient (ADC) recovered by diffusion MRI (dMRI) when imaging the diffusion of water molecules in heterogeneous media like the cerebral white matter. The complex shape of the ADC reflects the complex geometry of the underlying tissue, it is well known that the geometry of the ADC does not correspond to the underlying fiber directions [4] This can be understood from the q-space formalism, where it can be seen that the ADC and the diffusion signal are in the Fourier domain of the diffusion ensemble average propagator (EAP), which describes the probability of the diffusing particles. GDTI was proposed because it overcomes the limitation of diffusion tensor imaging (DTI) [5], which is inadequate for modelling the signal from regions with multiple fiber configurations In such regions, the HOT that is used in GDTI can model the signal and the ADC with greater accuracy than the 2nd-order diffusion tensor. In [10] the authors relied on a parameterization based on tensor decomposition into a sum of squares, and in [11], the authors used conic programming approaches to achieve this

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