In this work, we proposed a novel functional basis to efficiently represent the multi-spherical diffusion signal over both three-dimensional q-space and diffusion time. We regularized this basis by imposing both smoothness in the fitted signal using Laplacian regularization and sparsity in the fitted coefficients. Compared to the work by Fick et al. [4], the main methodological differences are the q-space representation, where we use the MAP basis instead of 3D-SHORE, and the sparsity term. As Fig. 2b shows, using MAP allows us to fit the multi-spherical signal better than [4] using the same number of coefficients. We remark that DTI fits the multispherical signal worst as it cannot describe diffusion restriction over q or x. This limitation becomes more apparent at higher b-values, which is exactly where the diffusion signal is most characterizing of the underlying tissue.

This work is also the first to estimate and study the progress of three-dimensional q-space indices over diffusion time. Our basis is especially well-suited for this exploration. For any evaluated diffusion time the basis reduces to MAP, which allows us to calculate all of its previously proposed indices [13, 14]. For now, we focused on the well-known Mean Squared Displacement (MSD) and Return-to- Origin Probability (RTOP). We found that the recovered trends in synthetic data correspond with what we expect from theory (Fig. 3b, c). As diffusion time increases, spins get more time to diffusive, so MSD increases and RTOP decreases. Decreasing the number of samples did not influence MSD trends so much, but RTOP trends did lower, possibly related to removal of samples along the “restricted” direction in the signal. Overall, a lower bound of reliable index estimation seems to be around 200 samples using random subsampling, as both profiles flatten out at this point.

Applying our method to real multi-spherical data from a mouse produces mostly coherent results with the simulated data. Again we find that RTOP drops as diffusion time increases, and lowering the number of samples decreases the RTOP and leaves MSD mostly unaffected. As fewer samples were used, we found more negative (infeasible) RTOP values. To avoid this, our framework could still be improved by adding a positivity constraint like in Ozarslan et al. [13].

Regardless, our multi-spherical basis is the first of its kind in being specifically designed to represent the four-dimensional EAP and analyzing its properties. Our proposed regularization allows us to significantly reduce the number of measured samples, which may eventually bring multi-spherical diffusion MRI within the reach of clinical application.

Acknowledgements This work was partly supported by ANR “MOSIFAH” under ANR-13- MONU-0009-01, the ERC under the European Union’s Horizon 2020 research and innovation program (ERC Advanced Grant agreement No 694665:CoBCoM), MAXIMS grant funded by ICM’s The Big Brain Theory Program and ANR-10-IAIHU-06.