The Diffusion Exchange Ratio (DEXR): A minimal sampling of diffusion exchange spectroscopy to probe exchange, restriction, and time-dependence

Water exchange is increasingly recognized as an important biological process that can affect the study of biological tissue using diffusion MR. Methods to measure exchange, however, remain immature as opposed to those used to characterize restriction, with no consensus on the optimal pulse sequence (s) or signal model (s). In general, the trend has been towards data-intensive fitting of highly parameterized models. We take the opposite approach and show that a judicious sub-sample of diffusion exchange spectroscopy (DEXSY) data can be used to robustly quantify exchange, as well as restriction, in a data-efficient manner. This sampling produces a ratio of two points per mixing time: (i) one point with equal diffusion weighting in both encoding periods, which gives maximal exchange contrast, and (ii) one point with the same total diffusion weighting in just the first encoding period, for normalization. We call this quotient the Diffusion EXchange Ratio (DEXR). Furthermore, we show that it can be used to probe time-dependent diffusion by estimating the velocity autocorrelation function (VACF) over intermediate to long times ($\sim 2-500\mathrm{ms}$). We provide a comprehensive theoretical framework for the design of DEXR experiments in the case of static or constant gradients. Data from Monte Carlo simulations and experiments acquired in fixed and viable ex vivo neonatal mouse spinal cord using a permanent magnet system are presented to test and validate this approach. In viable spinal cord, we report the following apparent parameters from just 6 data points: ${\tau }_k=17\pm 4\mathrm{ms}$, $f_{NG}=0.72\pm 0.01$, $R_{\mathrm{eff}}=1.05\pm 0.01\mathrm{\mu m}$, and ${\kappa }_{\mathrm{eff}}=0.19\pm 0.04\mathrm{\mu m/ms}$, which correspond to the exchange time, restricted or non-Gaussian signal fraction, an effective spherical radius, and permeability, respectively. For the VACF, we report a long-time, power-law scaling with $\approx t^{-2.4}$, which is approximately consistent with disordered domains in 3-D. Overall, the DEXR method is shown to be highly efficient, capable of providing valuable quantitative diffusion metrics using minimal MR data.