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

摘要

In q-space diffusion NMR, the probability P(r,t_d) of a molecule having a displacement r in a diffusion time t_d 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 is typically of the same order of magnitude as the time offset Δ between encoding gradients. In this case, finite-δ 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 Δ+ δduring which diffusion is encoded, the probability recovered by a finite- q-space experiment can be interpreted. It is shown theoretically that a data analysis using a modified q-space index q=rδηg, with γ the gyromagnetic ratio and η-(Δ-δ/3)/(Δ+δ)~(1/2), recovers the correct displacement probability distribution if diffusion is multi-Gaussian free diffusion. With this analysis, we show that the displacement distribution P(r,t_(exp)) is measured at the experimental diffusion-encoding time t_(exp)=Δ+δ, and not at the reduced diffusion time t_r=Δ-δ/3 as is generally assumed in the NMR and MRI literature. It is also shown that, by defining a probability P(y,Δ) that a time t<δ exists such that a displacement y occurs from time t to t+Δ, it is possible to describe the physical significance of the result obtained when we use the q-space formalism valid for infinitesimal δ when δ is not infinitesimal. These deductions were confirmed by simulations for homogeneous Gaussian diffusion and for heterogeneous diffusion in permeable microscopic Gaussian domains that are homogeneous on the m scale. The results also hold for diffusion inside restricted spherical reflecting domains, but only if the radius of the domain is larger than a critical size. The simulations of the displacement-specific dephasing obtain that if δ>δ_c then η≠(Δ-δ/3)/(Δ+δ)~(1/2) 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 Δ-δ=5 ms, the parameter δ_c in ms is given by "c=0.49a2+0.24" where a is the sphere's radius expressed in m. Simulation of q-space restricted diffusion MRI experiments indicate that if η≠(Δ-δ/3)/(Δ+δ~(1/2), the recovered displacement probability is always better than the Gaussian approximation, and the measured diffusion coefficient matches the diffusion coefficient at time t_(exp)=Δ+δ better than it matches the diffusion coefficient at time t_r=Δ-δ/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.