Analytical expression for ODF


Hi MRtrix3 Users,

Is it possible to derive the analytical expressions for the diffusion orientation distribution function (ODF) such as the return-to-origin probability (RTOP), and mean-squared-displacement (MSD) using MRtrix3?




Dear Ping,

First of all, MRtrix3 does not offer facilities for diffusion ODF (dODF) computation (Tuch’s Q-ball imaging etc.). The dODF is the radial projection of the ensemble average propagator (EAP) and is clearly not the same as the fibre ODF (fODF or FOD) that MRtrix estimates with spherical deconvolution methods.

If, however, you already have a spherical harmonics representation of the dODF calculated in some other way, you can easily compute the mean-squared-displacement (MSD) using Parseval’s theorem on the SH coefficients. In the orthonormal SH basis, you would have something along the lines of
mrcalc dODF.mif 2 -pow - | mrmath -axis 3 - sum - | mrcalc - -sqrt MSD.mif
I’m not sure about the return-to-origin probability (RTOP) though. In fact, I think you need the full EAP to get RTOP. Maybe other users can help you out with that?



Yes, indeed. Without the full EAP, one would have to assume something somehow about the radial profile of the EAP to be able to compute an RTOP.

@pinghongyeh: we used to have Q-ball imaging in MRtrix, but it was axed. Rather than a dODF, we currently encourage looking at fODFs for analysis and interpretation.

Any tips for working with Q-Ball data?