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Concept
Fractional anisotropy
Summary
Fractional anisotropy (FA) is a scalar value between zero and one that describes the degree of anisotropy of a diffusion process. A value of zero means that diffusion is isotropic, i.e. it is unrestricted (or equally restricted) in all directions. A value of one means that diffusion occurs only along one axis and is fully restricted along all other directions. FA is a measure often used in diffusion imaging where it is thought to reflect fiber density, axonal diameter, and myelination in white matter. The FA is an extension of the concept of eccentricity of conic sections in 3 dimensions, normalized to the unit range.
A Diffusion Ellipsoid is completely represented by the Diffusion Tensor, D. FA is calculated from the eigenvalues () of the diffusion tensor. The eigenvectors give the directions in which the ellipsoid has major axes, and the corresponding eigenvalues give the magnitude of the peak in that direction.
with being the mean value of the eigenvalues.
An equivalent formula for FA is
which is further equivalent to:
where R is the "normalized" diffusion tensor:
Note that if all the eigenvalues are equal, which happens for isotropic (spherical) diffusion, as in free water, the FA is 0. The FA can reach a maximum value of 1 (this rarely happens in real data), in which case D has only one nonzero eigenvalue and the ellipsoid reduces to a line in the direction of that eigenvector. This means that the diffusion is confined to that direction alone.
This can be visualized with an ellipsoid, which is defined by the eigenvectors and eigenvalues of D. The FA of a sphere is 0 since the diffusion is isotropic, and there is equal probability of diffusion in all directions. The eigenvectors and eigenvalues of the Diffusion Tensor give a complete representation of the diffusion process. FA quantifies the pointedness of the ellipsoid, but does not give information about which direction it is pointing to.
Note that the FA of most liquids, including water, is 0 unless the diffusion process is being constrained by structures such as network of fibers.
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