Definition of displacement probability and diffusion time in q-space magnetic resonance measurements that use finite-duration diffusion-encoding gradients.

In q-space diffusion NMR, the probability P(r,td) of a molecule having a displacement r in a diffusion time td is obtained under the assumption that the diffusion-encoding gradient g has an infinitesimal duration. However, this assumption may not always hold, particularly in human MRI where the diffusion-encoding gradient duration delta is typically of the same order of magnitude as the time offset Delta between encoding gradients. In this case, finite-delta effects complicate the interpretation of displacement probabilities measured in q-space MRI, and the form by which the signal intensity relates to them. By considering the displacement-specific dephasing, , of a set of spins accumulating a constant displacement vector r in the total time Delta+delta during which diffusion is encoded, the probability recovered by a finite-delta q-space experiment can be interpreted. It is shown theoretically that a data analysis using a modified q-space index q=gammadeltaetag, with gamma the gyromagnetic ratio and eta=square root (Delta-delta/3)/(Delta+delta), recovers the correct displacement probability distribution if diffusion is multi-Gaussian free diffusion. With this analysis, we show that the displacement distribution P(r,texp) is measured at the experimental diffusion-encoding time texp=Delta+delta, and not at the reduced diffusion time tr=Delta-delta/3 as is generally assumed in the NMR and MRI literature. It is also shown that, by defining a probability P(y,Delta) that a time tdeltac then eta is not equal to square root (Delta-delta/3)/(Delta+delta) which implies that we can no longer obtain the correct displacement probability from the displacement distribution. In the case that /g/=18 mT/m and Delta-delta=5 ms, the parameter deltac in ms is given by "deltac=0.49a2+0.24" where a is the sphere's radius expressed in microm. Simulation of q-space restricted diffusion MRI experiments indicate that if eta=square root (Delta-delta/3)/(Delta+delta), the recovered displacement probability is always better than the Gaussian approximation, and the measured diffusion coefficient matches the diffusion coefficient at time texp=Delta+delta better than it matches the diffusion coefficient at time tr=Delta-delta/3. These results indicate that q-space MRI measurements of displacement probability distributions are theoretically possible in biological tissues using finite-duration diffusion-encoding gradients provided certain compartment size and diffusion encoding gradient duration constraints are met.