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

Abstract

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 δ 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, 〈 r | e iφ 〉 , of a set of spins accumulating a constant displacement vector r in the total time Δ+ δ during which diffusion is encoded, the probability recovered by a finite- δ q-space experiment can be interpreted. It is shown theoretically that a data analysis using a modified q-space index q ̃ =γδη g , with γ the gyromagnetic ratio and η= (Δ−δ/3)/(Δ+δ) , 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 μm 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) 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 “ δ c=0.49 a 2+0.24” where a is the sphere’s radius expressed in μm. Simulation of q-space restricted diffusion MRI experiments indicate that if η= (Δ−δ/3)/(Δ+δ) , 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.