Diffusion Tensor Properties

A diffusion tensor is a common model for representing the orientation distribution function of water diffusion within a voxel of a diffusion MRI dataset. A useful model for a diffusion tensor is a three-dimensional ellipsoid. The rate of water diffusion in a given direction is modeled as being proportional to the radius of the ellipsoid in that direction. Therefore, if one diffusion direction is strongly preferred over all others, the ellipsoid is cigar-shaped; if there are two directions with approximately equal high diffusion rates, it is shaped like an M&M; and if the diffusion is isotropic (no preferred direction), it is shaped like a sphere.

Mathematically, the tensor is represented by a 3x3 symmetric positive definite matrix. Such a matrix has six degrees of freedom.

$D=\left[\begin{array}{ccc}{d}_{xx}& d_{xy}& d_{xz}\\ d_{xy}& d_{yy}& d_{yz}\\ d_{xz}& d_{yz}& d_{zz}\end{array}\right]$

The eigenvalues and eigenvectors of this matrix are, respectively, ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$, $e_1$, $e_2$, and $e_3$. Since the ordering of the eigenvalues and eigenvectors is arbitrary, by convention we define them such that ${\lambda }_1>{\lambda }_2>{\lambda }_3$.

Scalar Metrics

From the eigen decomposition, we may define a number of scalar values for the diffusion tensor that are hypothesized to have correlations with the structural properties of the white matter described by the tensor. Most of these can be computed from an MRIimage DTI with the program mritensor.

Fractional Anisotropy

Also known as "FA", fractional anisotropy is defined as:

$\mathbf{𝐅}\mathbf{𝐀}\equiv \sqrt{\frac{1}{2}}\frac{\sqrt{({\lambda }_1-{\lambda }_2{)}^2+({\lambda }_2-{\lambda }_3{)}^2+({\lambda }_3-{\lambda }_1{)}^2}}{\sqrt{{\lambda }_1^2+{\lambda }_2^2+{\lambda }_3^2}}$

For isotropic diffusion, $FA=0$; for infinitely anisotropic diffusion, $FA=1$. Decreases in FA are believed to correlate with demyelination.

Trace

Also known as "Tr(D)" and essentially equivalent to mean diffusivity, the trace of the diffusion tensor is defined as:

$\mathbf{𝐓}\mathbf{𝐫}\mathbf{(}\mathbf{𝐃}\mathbf{)}\equiv d_{xx}+d_{yy}+d_{zz}={\lambda }_1+{\lambda }_2+{\lambda }_3$

Mean Diffusivity

Also known as "MD" and essentially equivalent to trace, mean diffusivity is defined as:

$\mathbf{𝐌}\mathbf{𝐃}\equiv \mathrm{T}\mathrm{r}(D)/3$

Linear Anisotropy

Also known as "LA"

Planar Anisotropy

Also known as "PA"

Relative Anisotropy

Also known as "RA"

Tensor Magnitude

Also known as "Mag(D)", "$||D||$", and "mD"

Slant Tensor Magnitude

Also known as "magnitude of slant-D" and "msD"

Axial Diffusivity

Also known as "AD", axial diffusivity is defined as

$\mathbf{𝐀}\mathbf{𝐃}\equiv {\lambda }_1$

Decreases in AD are believed to correlate with axon breakage or dropout.

Radial Diffusivity

Also known as "RD", radial diffusivity is defined as

$\mathbf{𝐑}\mathbf{𝐃}\equiv (1/2)({\lambda }_2+{\lambda }_3)$

Increases in RD are believed to correlate with demyelination.

Isotropy

Also known as "I"

Lattice Index

Also known as "LI"