Diffusion is a random transport phenomenon, which describes the transfer of material (e.g., water molecules) from one spatial location to other locations over time. In three dimensions, the Einstein diffusion equation:6
states that the diffusion coefficient, D (in mm2
/s), is proportional to the mean squared-displacement,
divided by the number of dimensions, n, and the diffusion time, Dt. The diffusion coefficient of pure water at 20°C is roughly 2.0 × 10−3
/s and increases at higher temperatures. In the absence of boundaries, the molecular water displacement is described by a Gaussian probability density
The spread in this distribution increases with the diffusion time, Dt, as illustrated in .
Figure 1 Left: Illustration of the diffusion random-walk for a single water molecule from the green location to the red location. The displacement is shown by the yellow arrow. Right three frames: Diffusion describes the displacement probability with time for (more ...)
The diffusion of water in biological tissues occurs inside, outside, around, and through cellular structures. Water diffusion is primarily caused by random thermal fluctuations. The behavior is further modulated by the interactions with cellular membranes, and subcellular and organelles. Cellular membranes hinder the diffusion of water, causing water to take more tortuous paths, thereby decreasing the mean squared displacement. The diffusion tortuosity and corresponding apparent diffusivity may be increased by either cellular swelling or increased cellular density. Conversely, necrosis, which results in a breakdown of cellular membranes, decreases tortuosity and increases the apparent diffusivity. Intracellular water tends to be more restricted (as opposed to hindered) by cellular membranes. Restricted diffusion also decreases the apparent diffusivity, but plateaus with increasing diffusion time.7
Both hindered and restricted diffusion reduce the apparent diffusivity of water.
In fibrous tissues including white matter, water diffusion is relatively unimpeded in the direction parallel to the fiber orientation. Conversely, water diffusion is highly restricted and hindered in the directions perpendicular to the fibers. Thus, the diffusion in fibrous tissues is anisotropic. Early diffusion imaging experiments used measurements of parallel (D||
) and perpendicular (D
) diffusion components to characterize the diffusion anisotropy.8,9
The application of the diffusion tensor to describe anisotropic diffusion behavior was introduced by Basser et al.1,2
In this elegant model, diffusion is described by a multivariate normal distribution
where the diffusion tensor is a 3×3 covariance matrix
which describes the covariance of diffusion displacements in 3D normalized by the diffusion time. The diagonal elements (Dii > 0) are the diffusion variances along the x, y and z axes, and the off-diagonal elements are the covariance terms and are symmetric about the diagonal (Dij = Dji). Diagonalization of the diffusion tensor yields the eigenvalues (l1, l2, l3) and corresponding eigenvectors (ê1, ê2, ê3) of the diffusion tensor, which describe the directions and apparent diffusivities along the axes of principle diffusion. The diffusion tensor may be visualized using an ellipsoid with the eigenvectors defining the directions of the principle axes and the ellipsoidal radii defined by the eigenvalues (see ). Diffusion is considered isotropic when the eigenvalues are nearly equal (e.g., l1 ~ l2 ~ l3). Conversely, the diffusion tensor is anisotropic when the eigenvalues are significantly different in magnitude (e.g., l1 > l2 > l3). The eigenvalue magnitudes may be affected by changes in local tissue microstructure with many types of tissue injury, disease or normal physiological changes (i.e., aging). Thus, the diffusion tensor is a sensitive probe for characterizing both normal and abnormal tissue microstructure.
Figure 2 Schematic representations of diffusion displacement distributions for the diffusion tensor. Ellipsoids are used to represent diffusion displacements. The diffusion is highly anisotropic in fibrous tissues such as white matter and the direction of greatest (more ...)
Specifically in the CNS, water diffusion is usually more anisotropic in white matter regions, and isotropic in both gray matter and cerebrospinal fluid (CSF). The major diffusion eigenvector (ê1- direction of greatest diffusivity) is assumed to be parallel to the tract orientation in regions of homogenous white matter. This directional relationship is the basis for estimating the trajectories of white matter pathways with tractography algorithms.