Mathematical description of q-space in spherical coordinates: exact q-ball imaging

Novel methodologies have been recently developed to characterize the microgeometry of neural tissues and porous structures via diffusion MRI data. In line with these previous works, this article provides a detailed mathematical description of q-space in spherical coordinates that helps to highlight the differences and similarities between various related q-space methodologies proposed to date such as q-ball imaging (QBI), diffusion spectrum imaging (DSI), and diffusion orientation transform imaging (DOT). This formulation provides a direct relationship between the orientation distribution function (ODF) and the diffusion data without using any approximation. Under this relationship, the exact ODF can be computed by means of the Radon transform of the radial projection (in q-space) of the diffusion MRI signal. This new methodology, termed exact q-ball imaging (EQBI), was put into practice using an analytical ODF estimation in terms of spherical harmonics that allows obtaining model-free and model-based reconstructions. This work provides a new framework for combining information coming from diffusion data recorded on multiple spherical shells in q-space (hybrid diffusion imaging encoding scheme), which is capable of mapping ODF to a high accuracy. This represents a step toward a more efficient development of diffusion MRI experiments for obtaining better ODF estimates.