We first provide the relation between the measured multi-spherical diffusion signal and the four-dimensional ensemble average propagator (EAP) in Sect. 2.1. We then explain the properties that we would like our multi-spherical representation to have, and provide the details on the functional basis representation and regularization which are used to impose the desired properties in Sect. 2.2.

The Four-Dimensional Ensemble Average Propagator

In dMRI, the EAP describes the probability density that a spin diffuses a certain distance in a given diffusion time. The EAP is estimated by obtaining diffusion- weighted images (DWIs). A DWI is obtained by applying two sensitizing diffusion gradients of pulse length 1 to the tissue, separated by separation time A. The resulting signal is ‘weighted’ by the average particle movements along the applied gradient direction. When these gradients are considered infinitely short (1 ! 0), which can only be approximated in practice, the relation between the measured signal S(q, r) and the EAP P(r; r) is given by a Fourier transform [11] as

where E(q, t) = S(q, t)/S_{0} is the normalized signal attenuation measured at diffusion encoding position q, and S_{0} is the baseline image acquired without diffusion sensitization (q = 0). We denote q = |q|, q = qu and R = Rr, where u and r are 3D unit vectors and q, R e R+. The wave vector q on the right side of Eq. (1) is related to pulse length 1, nuclear gyromagnetic ratio у and the applied diffusion gradient vector G.

The four-dimensional EAP has boundary conditions with respect to {q, t}:

- {q, t = 0}: When t = 0 the spins have no time to diffuse and the EAP is a spike function at the origin, i.e., P(R; t = 0/ = 1(R). Following Eq. (1), the signal attenuation will not attenuate for any value of q, i.e., E(q, t = 0) = 1.

- {q, limT!i}: When lim^i E(q, t) the signal attenuation is in the long diffusion time limit and only signal contributions from restricted compartments remain [12]. In this case, given infinite gradient strength and some assumptions on tissue composition [13, 14], q-space indices such as the Return-To-Axis Probability (RTAP) are related to the mean apparent axon diameter.

- {q = 0, t} : When q = 0 there is no diffusion sensitization so E(q = 0, t) = 1. With Fourier relationship in Eq. (1), this point also corresponds to the zeroth harmonic of the EAP, which as a probability density integrates to one.

- {lim_{q!1},T}: lim_{q!1} E(q, t) = 0, as even an infinitesimally small spin movement will attenuate the signal completely.