What Does FEXI Measure in Neurons?

TL;DR

This study interrogates how Filtered Exchange Imaging (FEXI) interprets diffusion in gray-matter neurons by solving the Bloch–Torrey equation in digitized neuron geometries. It demonstrates that the recovery of the diffusion coefficient after filtering is multiexponential and highly dependent on the mixing time $t_m$ and cellular geometry, which can bias the apparent exchange time $\tau_x$ if a single-exponential model is assumed. By calibrating membrane permeability from preexchange lifetime data, the authors estimate $\kappa \approx 0.005~\mu\mathrm{m}/\mathrm{ms}$ and predict an exchange time $\tau_x$ around 140 ms for permeable cells, while short-time exchange (tens of ms) is largely governed by intra-cellular geometric exchange. The work explains the broad literature spread in reported exchange times and suggests that multi-exponential recovery of $D(t_m)$ should be sought in FEXI data to disentangle geometry from permeation, with implications for interpreting diffusion MRI in brain gray matter.

Abstract

Exchange between tissue compartments is crucial for interpretation of diffusion MRI measurements in brain gray matter. Reported values of exchange time are broadly dispersed, about two orders of magnitude. We analyze the measurement technique called Filtered Exchange Imaging (FEXI) using numerical solution of Bloch--Torrey equation in digitalized neurons downloaded from NeuroMorpho.org. The FEXI outcome, the recovery of diffusion coefficient in cells with impermeable membrane is multiexponential, tightly related to the eigenvalues of the Laplace operator. Fitting the commonly used exponential recovery function results in a strong dependence of the apparent exchange time on the involved mixing time interval. For short mixing times, exchange time is of the order of 10 ms. It gets an order of magnitude larger for mixing times of a few hundreds of milliseconds. To obtain an estimate of membrane permeability, we reinterpret previously published data on preexchange lifetime in neuronal cell culture. It results in the permeability 0.005 micrometer/ms. The corresponding exchange time is about 140 ms. We conclude that essentially shorter exchange times are due to fast geometric exchange inside the ramified cells.

Figures (9)

• Figure 1: Left: Schematics of FEXI as implemented in this study. Wide gradient pulses are used as the most realistic approach. Right: The anticipated recovery of diffusion coefficient with the increasing mixing time.
• Figure 2: The three cells used in this study (Table \ref{['tab_eigen']}) and a simple model of a bent cylinder. The colors encode the signal magnitude right after the filter. To cells, the filter gradient was applied along the eigenvectors with the smallest eigenvalues $\lambda_1$, the directions shown with the blue lines. For the cylinder, the direction was along the $y$-axis.
• Figure 3: The recovery of diffusion coefficient as a function of mixing time for all cells. Diffusion-weighting gradients in the filter and detection block were applied along the eigenvectors, ${\bf e}_{1,3}$, with the smallest and largest eigenvalues, $\lambda_1$ and $\lambda_3$, respectively. Top row: The solid lines show $D(t_m)$, Eq. (\ref{['Dapp_tm']}), fitted to data obtained via the solution to the Bloch--Torrey equation (circles). The dashed lines indicate the undisturbed values of $D(\infty)$, which are the corresponding eigenvalues of diffusion tensor. Bottom row: The approach of $D(t_m)$ to its undisturbed values. The ticks at the frame top show the time $\tau_n = 1/\lambda_n$ corresponding to 16 numerically found Li2020 lowest nonzero eigenvalue of Laplace operator $D_0\nabla^2$, Eq. (\ref{['cylinder_sol0']}). The value $\tau_1=14{~\rm s}$ for cell 3 is outside the plot range.
• Figure 4: Fitted apparent exchange time, $\tau_x$, as a function of the mixing time range for one cell. The fitting was performed from the minimum $t_m$ to the value indicated on the axis. The inset shows the practically relevant range of $t_m$. The first value for $t_m=20{~\rm ms}$ is $\tau_x=12{~\rm ms}$.
• Figure 5: Recovery of diffusion coefficient as in in the top row of Fig. \ref{['fig_FEXI_cells']}, here for the bent cylinder shown in Fig. \ref{['fig_cells']} (circles). It is compared with the analytical result in the form of a series over the eigenvalues of the Laplace operator, Eq. (\ref{['Dapp']}) (solid line). The dashed lines show the prediction of the simplified theory for the unperturbed diffusivity and the short-time closed asymptotic form for $\sqrt{D_0 t}\ll L$, Eq. (\ref{['Dapp_short']}). All calculations were performed assuming the approximation of Eq. (\ref{['magnet_ini']}).