Localization-driven exchange contrast in diffusion exchange spectroscopy

Abstract

Diffusion exchange spectroscopy (DEXSY) is a method to probe exchange between domains of varying confinement. Analyses of DEXSY signals typically assume Gaussian diffusion within distinct compartments and first-order exchange kinetics between them. Other situations can yield DEXSY signal contrast with respect to mixing time, however, leading to potentially erroneous interpretation. Here, we demonstrate that a one-dimensional compartment with reflecting boundaries and without relaxation can by itself produce such contrast in certain experimental regimes. The origin of this contrast is the diffusive mixing of spin isochromats initially near versus far from either boundary, as the former can be relatively coherent in an effect known as edge enhancement or signal localization. We consider DEXSY signals in the case of extended field gradients and identical encodings. Signals were generated via a numerical approach that solves the Bloch-Torrey equation in discrete space and time using matrix operators. We find that in the localization regime, an apparent first-order rate constant of exchange, $k$, can be extracted from DEXSY signals even in this minimal system. The measured $k$ is approximately proportional to $D/L^2$, where $D$ is the diffusivity and $L$ is the domain size. Typically, $k \approx π^2 D/L^2$. We attribute this localization-driven exchange to the relaxation of spatial magnetization modes with mixing time, noting that $π^2 D/L^2$ is the first non-zero eigenvalue of the Laplacian basis. These results demonstrate that DEXSY and related methods such as filter exchange spectroscopy (FEXSY) may not be specific to genuine barrier permeation.

Figures (7)

• Figure 1: Absolute magnetization vectors $|\mathbf{m}|$ plotted vs. bin midpoints $\bar{\mathbf{x}}$ (solid lines). Parameters were $L = 20\;\mathrm{\mu m}$, $T = 10\;\mathrm{ms}$, $D = 2\;\mathrm{\mu m^2/ms}$, with varying $g = [0.3,\,0.4,\,0.6] \;\mathrm{T/m}$ (light green to dark blue, respectively), and $\gamma \approx 2.675\times 10^8\;\mathrm{rad/s/T}$. Discretization was $\Delta x = 0.2\;\mathrm{\mu m}$, $\Delta t = 2\;\mathrm{\mu s}$ and $p = 0.1$. Initial condition was $\mathbf{m}(0) = \mathbf{1}$. For $g = 0.4\;\mathrm{T/m}$, MC simulated data is included for comparison, plotted at every other bin (circles). See main text for parameters. (a) Magnetization for the first CGSE encoding block. (b) Magnetization after the second CGSE encoding block, with $t_m = 0$. (c -- d) Keeping the same $y$-axis range as (b), magnetization with $t_m = 10$, $200\;\mathrm{ms}$ between encodings, respectively.
• Figure 2: DEXSY signal $S(t_m)$ for the same parameters as in Fig. \ref{['fig: profiles vs g']}, but just $g = 0.3\;\mathrm{T/m}$. Signals (circles) were generated for 30 values of $t_m$ log-linearly spaced from $10^{-1}$ -- $10^{2.5}\;\mathrm{ms}$, rounded to the nearest multiple of $\Delta t$. A fit of Eq. \ref{['eq: 3 param fit']} is shown (dashed line), yielding $k \approx 64\;\mathrm{s^{-1}}$, $\beta_1 \approx 4.3\times10^{-2}$, and $\beta_3 \approx 0.24$, with root-mean-square error $\approx 1.0\times10^{-6}$. The inset shows early decay behavior for $t_m$ up to $10\;\mathrm{ms}$ with log scaling on the $x$-axis.
• Figure 3: Fits of Eq. \ref{['eq: 3 param fit']} for signals generated at $\ell_D$, $\ell_g$ linearly spaced by $0.05L = 1\;\mathrm{\mu m}$, fixing $L = 20\;\mathrm{\mu m}$, $D = 2\;\mathrm{\mu m^2/ms}$, $\Delta x = 0.2\;\mathrm{\mu m}$, and $\Delta t=2\;\mathrm{\mu s}$. The $t_m$ values were the same as in Fig. \ref{['fig: dexsy fit g = 0.3']}. (a) Total signal variation, $\beta_1$. Detectable signal variation, or $\beta_1 \gtrsim 0.02$, appears in a band lying between the lines $\ell_g = \ell_D$ and $\ell_g = \ell_D/2$ (dashed white lines), up to about $\ell_D \lesssim L$. (b) Corresponding values of $k$ with the same axes as part (a), filtered for $\beta_1 \ge 0.02$. Values are highly uniform $\approx 50 \;\mathrm{s^{-1}}$, see color bar. (c) Values of $k$, focusing only on the region of interest $\ell_D/2 < \ell_g <\ell_D <L$ with finer resolution. The lengthscales $\ell_D$, $\ell_g$ were spaced by $0.01L = 0.2\;\mathrm{\mu m}$. Again, $k$ values shown are filtered by $\beta_1 \ge 0.02$. Note the change in axes range and color scale.
• Figure 4: Fitted $k$ for fixed ratios of $\ell_D/L = 0.5$ and $\ell_g/L = 0.3$, and $L$ spaced linearly by $1$ from $2-20\;\mathrm{\mu m}$ and $D$ by $0.1$ from $0.5 - 3\;\mathrm{\mu m^2/ms}$. Discretization was $\Delta t = 2\;\mathrm{\mu s}$ with $\Delta x$ adjusted to maintain $p \approx 0.1$. (a) Values of $k$ on a log color scale. Dashed lines indicate 1-D cross-sections shown in the next part. (b) 1-D cross sections of (a) at fixed $D = 2\;\mathrm{\mu m^2/ms}$ (left) and $L = 10\;\mathrm{\mu m}$ (right). The plot vs. $L$ is on a log-log axis and the $L^{-2}$ dependence is illustrated.
• Figure 5: Examples of eigen-decomposition of $\mathbf{m}(T)$. The ratio $\ell_g/\ell_D = 0.6$ was fixed, while $\ell_D = [0.1,\,0.2,\,0.3]L$ was varied (light to dark, respectively). (a) Absolute profiles $|\mathbf{m}(T)|$. Note that fixing $\ell_g/\ell_D$ yields profiles with similar maxima as $bD \propto (\ell_D/\ell_g)^6$. (b) Absolute eigen-decomposition coefficients $c_n$ in the basis $\mathbf{u}_n$, plotted on a log $y$-axis. Note the difference in tails, with increasing high-frequency content as $\ell_D$ and $\ell_g$ decrease. The corresponding values of $k$ (fits not shown) are $\approx [310,\,130,\,53]\;\mathrm{s^{-1}}$, in legend order.