q-space MR propagator in partially-restricted, axially-symmetric and isotropic environments

INTRODUCTION Diffusion of molecules is a powerful indicator of tissue microstructure. The dependence of the MR signal on the acquisition parameters (e.g., the magnitude and orientation of the diffusion gradients) can be exploited to infer valuable information from tissue. One way to elucidate microstructural features of the specimen involves fitting physical models to such q-space data. Alternatively, the diffusion-weighted MR signal profile can be transformed into a probability distribution function quantifying the average probability for molecular displacements. We refer to this function as the apparent propagator. The reconstructed apparent propagator was shown to be a valuable marker of tissue microstructure [1]. For example, in three-dimensional q-space acquisitions, the maxima of the orientation-dependent propagator have been associated with the orientations of white-matter fibers in the brain [2]. In this work, our focus is somewhat different than characterizing the microstructural features of tissue, although our findings are expected to be used in such studies. Instead, in more general terms, we discuss the q-space MR propagators and propose three new ways of obtaining them. We illustrate the utility of these new definitions of the apparent propagator on diffusion taking place in simple geometries such as partially-restricted environments, curving fibers and spherical compartments.