Fractional anisotropy

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.

Definition

A Diffusion Ellipsoid is completely represented by the Diffusion Tensor, D. FA is calculated from the eigenvalues ( $\lambda_1, \lambda_2, \lambda_3$ ) of the diffusion tensor.^[1]. The eigenvectors $\epsilon$ give the directions in which the ellipsoid has major axes, and the corresponding eigenvalues $\lambda$ give the magnitude of the peak in that direction.

$\text{FA} = \sqrt{\frac{3}{2}} \frac{\sqrt{(\lambda_1 - \hat{\lambda})^2 %2B (\lambda_2 - \hat{\lambda})^2 %2B (\lambda_3 - \hat{\lambda})^2}}{\sqrt{\lambda_1^2 %2B \lambda_2^2 %2B \lambda_3^2}}$

with the trace $\hat{\lambda} = (\lambda_1 %2B \lambda_2 %2B \lambda_3)/3$

Alternatively, FA can be calculated as

$\text{FA} = \sqrt{\frac{1}{2}} \frac{\sqrt{(\lambda_1 - \lambda_2)^2 %2B (\lambda_2 - \lambda_3)^2 %2B (\lambda_3 - \lambda_1)^2}}{\sqrt{\lambda_1^2 %2B \lambda_2^2 %2B \lambda_3^2}}$

It can be noted that if all the eigenvalues are equal, which happens in the case of a sphere, the FA becomes 0. The FA can go to a maximum value of 1, and this rarely happens in real data, and the ellipsoid then 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 well with an ellipsoid, which is defined by the tensor whose eigenvectors and eigenvalues are used for it's numerical calculation. The FA of a sphere is 0 since the diffusion is anisotropic, 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.

A drawback of the Diffusion Tensor model is that it can account only for Gaussian diffusion processes, which has been found to be inadequate in accurately representing the true diffusion process in the human brain. Due to this, higher order models using spherical harmonics and Orientation Diffusion Functions (ODF) have been used to define newer and richer estimates of the anisotropy, called Generalized Fractional Anisotropy. GFA computations use samples of the ODF to evaluate the anisotropy in diffusion. They can also be easily calculated by using the Spherical Harmonic coefficients of the ODF model.

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.