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

The problem of reconstruction of an apparent propagator from a series of diffusion-attenuated magnetic resonance (MR) signals is revisited. In nonimaging acquisitions, the inverse Fourier transform of the MR signal attenuation is consistent with the notion of an ensemble average propagator. However, in image acquisitions where one is interested in quantifying a displacement distribution in every voxel of the image, the propagator derived in the traditional way may lead to a counter-intuitive profile when it is nonsymmetric, which could be a problem in partially restricted environments. By exploiting the reciprocity of the diffusion propagator, an alternative is introduced, which implies a forward Fourier transform of the MR signal attenuations yielding a propagator reflected around the origin. Two simple problems were considered as examples. In the case of diffusion in the proximity of a restricting barrier, the reflected propagator yields a more meaningful result, whereas in the case of curving fibers, the original propagator is more intuitive. In the final section of the article, two more one-dimensional transformations are introduced, which enable the reconstruction of two- and three-dimensional propagators in, respectively, axially symmetric and isotropic environments - in both cases, from one-dimensional q-space MR data.