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?

Thanks.

Ping

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?

Thanks.

Ping

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?

Cheers,

Daan

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.