Diffusion propagator imaging: using Laplace's equation and multiple shell acquisitions to reconstruct the diffusion propagator.

Many recent single-shell high angular resolution diffusion imaging reconstruction techniques have been introduced to reconstruct orientation distribution functions (ODF) that only capture angular information contained in the diffusion process of water molecules. By also considering the radial part of the diffusion signal, the reconstruction of the ensemble average diffusion propagator (EAP) of water molecules can provide much richer information about complex tissue microstructure than the ODF. In this paper, we present diffusion propagator imaging (DPI), a novel technique to reconstruct the EAP from multiple shell acquisitions. The DPI solution is analytical and linear because it is based on a Laplace equation modeling of the diffusion signal. DPI is validated with ex vivo phantoms and also illustrated on an in vivo human brain dataset. DPI is shown to reconstruct EAP from only two b-value shells and approximately 100 diffusion measurements.