Tensors, Eigenvalues, and Eigenvectors

Next, we combine information from all the diffusion weighted volumes into a tensor. A tensor is a matrix modeling the shape & direction of diffusion in each voxel: there’s a tensor matrix for each DTI voxel. The tensor matrix is always 3x3, no matter how many directions (gradient volumes) you combine. But, the calculations in the matrix are more accurate if you combine more directions (i.e., you have better “angular resolution”).

Shape (relative to x, y, z) is stored along the main diagonal of the matrix.
Eigenvalues In the simplest case, the main diagonal contains the eigenvalues. Eigenvalues (L1, L2, L3) represent the length of diffusion.
- The biggest eigenvalue, L1, defines the direction of greatest anisotropy.
- The 2nd biggest eigenvalue is L2.
- The smallest eigenvalue is L3.
If L1, L2 and L3 are equivalent, we get isotropic diffusion.

DTI metrics (scalars)

Mean Diffusivity (MD) is the average of L1, L2 and L3.
Fractional Anisotropy (FA), the extent of anisotropy, varies from 0 to 1. It is probably the most common measure you will encounter. Good wiring usually has high FA, which shows up bright.
Parallel Diffusivity (PD) = L1. PD is high when anisotropy is high. Changes in PD may be related to axonal loss. (PD is also called "principle diffusivity" or "primary diffusivity")
Mode A single FA value can be realized as a range of different shapes “Mode”. Mode varies from -1 to 1.
- Linear Case (CL) If L1 is big, but L2 & L3 are small, Mode approaches 1.
- Planar Case (CP) If L1 & L2 are big, but L3 is small, Mode approaches -1.
Radial Diffusivity (RD) is the average of L2 and L3. Changes in RD may be related to demyelenation.
Eigenvectors: L1, L2 and L3 must be orthogonal to each other, but might not be directed along x, y or z planes of the image, so, tensors must also have a way to represent diffusion direction:
- Direction is stored as eigenvectors in the off-diagonal parts of the matrix.
- If the tensor orientation deviates from the x,y or z planes, these off-diagonal elements acquire some of the information that the main diagonal no longer contains.

Watch this 8.25 minute video clip.

dMRI: Classic DTI (~55-minutes)