Understanding the concept of diffusion time for finite pulse sequences in diffusion MRI: case of the Karger model

Diffusion magnetic resonance imaging (dMRI) is an imaging modality that probes the diffusion characteristics of a sample via the application of magnetic field gradient pulses. When the duration of the diffusion-encoding gradient pulses is small compared to the delay between the pulses (the narrow pulse assumption), the concept of the probed diffusion time is easy to define. However, when the narrow pulse assumption is not satisfied, the concept of the diffusion time is less straightforward. If the iamging voxel can be spatially divided into different Gaussian diffusion compartments with inter-compartment exchange governed by linear kinetics, then the dMRI signal can be approximated using the macroscopic Karger model, which is a system of coupled ordinary differential equations (ODEs), under the assumption that the duration of the diffusion-encoding gradient pulses is short compared to the diffusion time (the narrow pulse assumption). Recently, a new macroscopic model of the dMRI signal, without the narrow pulse restriction, was derived from the Bloch-Torrey partial differential equation (PDE). When restricted to narrow pulses, this new homogenized model has the same form as the Karger model. Using the FPK formulation, we discuss the concept of the diffusion time for the Karger model and propose a definition of the diffusion time that is physically and mathematically sensible