Hi, Giuseppe-

If your vector of interest happens to be one of the eigenvectors of a tensor, then you are in luck, because there is an eigenvalue for each one, that provides detailed information on the diffusivity in that direction.

It might be helpful to check out the AFNI Bootcamp presentations on DTI:

https://afni.nimh.nih.gov/pub/dist/doc/htmldoc/educational/bcamp_2017_10_nih.html#dti-fatcat-videos

… esp. the first video+handout there.

Geometrically, eigenvectors correspond to the major axes (“semiaxes”) of the spheroid—these are unit vectors that point out a direction and have magnitude 1 by construction. For each, there is necessarily an eigenvalue, which in the diffusion tensor model corresponds directly to diffusivity in that direction. When we talk about “first eigenvalue”, “second eigenvalue”, etc., it is because the eigenvalues are always sorted in decreasing magnitude in DWI: the first eigenvalue is the largest, and so its corresponding “first eigenvector” shows the orientation of the largest amount of diffusivity in that voxel. When that eigenvalue is much larger than the other ones (so the diffusion is quite anisotropic and pointy), that first eigenvector direction is typically viewed as a proxy for major white matter bundle directionality. (NB: the eigenvectors are shown as a oneway pointing arrow, but are bi-directional, so could equivalently be drawn pointing in both directions.)

If your direction of interest is not along an eigenvector, you can still calculate the relative diffusitivity there, using the tensor surface equation and the full tensor; it is described in the talk+handouts more (and references below).

If you want to talk about diffusivity along a line of voxels/tensors, then you are probably wanting to do tractography, which is discussed in some of the later talks/videos/handouts from above. It’s a larger topic of approximation.

So, please take a look at those and let us know what you would like to calculate.

–pt

ps: For reference, these are also excellent reference articles on the math/quantities of DTI, by Peter B. Kingsley:

Introduction to diffusion tensor imaging mathematics: Part I. Tensors, rotations, and eigenvectors

https://onlinelibrary.wiley.com/doi/10.1002/cmr.a.20048

Introduction to diffusion tensor imaging mathematics: Part II. Anisotropy, diffusion-weighting factors, and gradient encoding schemes

https://onlinelibrary.wiley.com/doi/abs/10.1002/cmr.a.20049

Introduction to diffusion tensor imaging mathematics: Part III. Tensor calculation, noise, simulations, and optimization

https://onlinelibrary.wiley.com/doi/abs/10.1002/cmr.a.20050