Fractional anisotropy
Fractional anisotropy (FA) is a
Definition
A Diffusion Ellipsoid is completely represented by the Diffusion Tensor, D. FA is calculated from the
with being the mean value of the eigenvalues.
An equivalent formula for FA is
which is further equivalent to:[2]
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.
Details
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. The measured FA may depend on the effective length scale of the diffusion measurement. If the diffusion process is not constrained on the scale being measured (the constraints are too far apart) or the constraints switch direction on a smaller scale than the measured one, then the measured FA will be attenuated. For example, the brain can be thought of as a fluid permeated by many fibers (nerve axons). However, in most parts the fibers go in all directions, and thus although they constrain the diffusion the FA is 0. In some regions, such as the
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FA value of 0.7698, the DT matrix is diagonal([10 2 2])
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FA value of 0, the DT matrix is diagonal([4 4 4])
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FA value of 0.6030, the DT matrix is diagonal([4 4 2])
One drawback of the Diffusion Tensor model is that it can account only for
References
- ^ Basser, P.J. & Pierpaoli, C. (1996). "Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI". Journal of Magnetic Resonance, Series B, 111, 209-219.
- ^ Özarslan, E. Vemuri, B.C. & Mareci, T. H. (2005). "Generalized scalar measures for diffusion MRI using trace, variance, and entropy". Magnetic Resonance in Medicine, , 53, 866-876.
- ^ J. Cohen-Adad, M. Descoteaux, S. Rossignol, RD Hoge, R. Deriche, and H. Benali (2008). "Detection of multiple pathways in the spinal cord using q-ball imaging". NeuroImage, 42, 739-749.