A Macroscopic Model for the Diffusion MRI Signal Accounting for Time-Dependent Diffusivity

Diffusion Magnetic Resonance Imaging (dMRI) encodes the displacement of water due to diffusion and is a powerful tool to obtain information on the tissue micro-structure. An important quantity measured in dMRI at the voxel level is the apparent diffusion coefficient (ADC) and it is well-established from imaging experiments that, in the brain, in-vivo, the ADC depends significantly on the diffusion time. To aid in the understanding and interpretation of the ADC, we derived a new asymptotic model for the dMRI signal from the Bloch-Torrey equation that governs the water proton magnetization under the influence of diffusion-encoding magnetic gradient pulses. Our new model was obtained using a particular choice of scalings for the biological cell membrane permeability, the diffusion time, the diffusion-encoding magnetic field gradient strength and a periodicity length of the cellular geometry. The ADC of the resulting model is time-dependent. We numerically validated this model for a wide range of diffusion times for two dimensional geometrical configurations.