A macroscopic model for the diffusion mri signal accounting for time-dependent diffusivity

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