Processing and visualization for diffusion tensor MRI

Abstract

This paper presents processing and visualization techniques for Diffusion Tensor Magnetic Resonance Imaging (DT-MRI). In DT-MRI, each voxel is assigned a tensor that describes local water diffusion. The geometric nature of diffusion tensors enables us to quantitatively characterize the local structure in tissues such as bone, muscle, and white matter of the brain. This makes DT-MRI an interesting modality for image analysis. In this paper we present a novel analytical solution to the Stejskal–Tanner diffusion equation system whereby a dual tensor basis, derived from the diffusion sensitizing gradient configuration, eliminates the need to solve this equation for each voxel. We further describe decomposition of the diffusion tensor based on its symmetrical properties, which in turn describe the geometry of the diffusion ellipsoid. A simple anisotropy measure follows naturally from this analysis. We describe how the geometry or shape of the tensor can be visualized using a coloring scheme based on the derived shape measures. In addition, we demonstrate that human brain tensor data when filtered can effectively describe macrostructural diffusion, which is important in the assessment of fiber-tract organization. We also describe how white matter pathways can be monitored with the methods introduced in this paper. DT-MRI tractography is useful for demonstrating neural connectivity (in vivo) in healthy and diseased brain tissue.

Introduction

Diffusion is the process by which matter is transported from one part of a system to another owing to random molecular motions. The transfer of heat by conduction is also due to random molecular motion. The analogous nature of the two processes was first recognized by Fick (1855), who described diffusion quantitatively by adopting the mathematical equation of heat conduction derived some years earlier by Fourier (1822). Fick’s law states that local differences in solute concentration will give rise to a net flux of solute molecules from high concentration regions to low concentration regions. The net amount of material diffusing across a unit cross-section that is perpendicular to a direction is proportional to the concentration gradient. Thus, the phenomenon of diffusion was described scientifically before any systematic development of thermodynamics. This phenomenon, known as Brownian Motion, is named after the botanist, Robert Brown, who observed the movement of plant spores floating in water in 1827. The first satisfactory theoretical treatment of Brownian Motion, however, was not made until much later by Albert Einstein (1905) and provided strong circumstantial evidence for the existence of molecules.

Anisotropic media such as crystals, textile fibers, and polymer films have different diffusion properties depending on direction. Anisotropic diffusion is best described by an ellipsoid where the radius defines the diffusion in a particular direction. The widely accepted analogy between symmetric 3×3 tensors and ellipsoids makes such tensors natural descriptors for diffusion. Moreover, the geometric nature of the diffusion tensors can quantitatively characterize the local structure in tissues such as bone, muscle, and white matter of the brain. Within white matter, the mobility of the water is restricted by the axons that are oriented along the fiber tracts. This anisotropic diffusion is due to tightly packed multiple myelin membranes encompassing the axon. Although myelination is not essential for diffusion anisotropy of nerves [as shown in studies of non-myelinated garfish olfactory nerves (Beaulieu and Allen, 1994); and in studies where anisotropy exists in brains of neonates before the histological appearance of myelin (Wimberger et al., 1995)], myelin is generally assumed to be the major barrier to diffusion in myelinated fiber tracts.

Using conventional MRI, we can easily identify the functional centers of the brain (cortex and nuclei). However, with conventional proton magnetic resonance imaging (MRI) techniques, the white matter of the brain appears to be homogeneous without any suggestion of the complex arrangement of fiber tracts. Hence, the demonstration of anisotropic diffusion in the brain by magnetic resonance has paved the way for non-invasive exploration of the structural anatomy of the white matter in vivo (Moseley et al., 1990, Chenevert et al., 1990, Basser, 1995, Pierpaoli et al., 1996).

The paper is organized as follows. First, we review the basics of DT-MRI (Section 2). Section 3.1 presents a new method for calculating the diffusion tensors from the diffusion gradient raw data. The method is based on an analytic solution of the Stejskal–Tanner formula eliminating the need to solve the Stejskal–Tanner equation system for each voxel of the data set. In Section 4, we expand on our previous quantitative characterization of the geometric nature of the diffusion tensors (Westin et al., 1997, Westin et al., 1999) by capturing macrostructural diffusion properties, and depicting detailed in vivo images of human white matter tracts. Using these methods, we can identify the orientation and distribution of most of the known major fiber tracts. In Section 5, we discuss visualization methods for tensor diffusion data and describe a method relating to the barycentric tensor shape descriptors from Section 4. We conclude by describing a novel white matter tractography method (expanded from Westin et al., 1999) and show some tractography results from axial DT-MRI data of the brain (Section 6).