A single-shot measurement of time-dependent diffusion over sub-millisecond timescales using static field gradient NMR

Abstract

Time-dependent diffusion behavior is probed over sub-millisecond timescales in a single shot using a nuclear magnetic resonance static gradient time-incremented echo train acquisition (SG-TIETA) framework. The method extends the Carr-Purcell-Meiboom-Gill cycle under a static field gradient by discretely incrementing the π-pulse spacings to simultaneously avoid off-resonance effects and probe a range of timescales (50-500 µs). Pulse spacings are optimized based on a derived ruleset. The remaining effects of pulse inaccuracy are examined and found to be consistent across pure liquids of different diffusivities: water, decane, and octanol-1. A pulse accuracy correction is developed. Instantaneous diffusivity, D_inst(t), curves (i.e., half of the time derivative of the mean-squared displacement in the gradient direction) are recovered from pulse accuracy-corrected SG-TIETA decays using a model-free log-linear least squares inversion method validated by Monte Carlo simulations. A signal-averaged 1-min experiment is described. A flat D_inst(t) is measured on pure dodecamethylcyclohexasiloxane, whereas decreasing D_inst(t) is measured on yeast suspensions, consistent with the expected short-time D_inst(t) behavior for confining microstructural barriers on the order of micrometers.

FIG. 1.

Example $\mathcal{G}\left(t\right)$ calculation. (a) Radiofrequency (RF) pulses in a static gradient of amplitude g. (b) G(t), shifted G(t + s) (red, dashed), and F(t). (c) Corresponding $\mathcal{G}\left(t\right)$. G(t) is assumed to be zero outside of t = [0, 2τ].

FIG. 2.

Example SG-TIETA sequence. (a) Timings: m_j = {1, 3, 1, 2, 1}, τ = 4δ, and δ = 14 µs = 1 dash. π-pulses occur at t_n, and direct echoes (blue lines) form at T_n. Magenta line indicates timing behavior: T_n = t_n + h_n, where h_n is the normalized |F(t)|/γg “height” at t_n given by h₁ = τ and h_n = 2τ + m_nδ − h_n−1 for n > 1. (b) Direct echo F(t) and other coherence pathways that refocus (red dashed-dotted lines) or do not refocus (gray dotted lines). Relative values of h_n are indicated.

FIG. 3.

LLS inversion on Monte Carlo simulated data. (a) Simulated $⟨{\left[\mathbf{r}\left(t\right)\cdot \widehat{\mathbf{g}}\right]}^2⟩$ (D₀ = 2.15 μm²/ms) for free diffusion (black) and restricted geometries (see Fig. S1). $\sum {\mathcal{C}}_n\left(t\right)$ (a.u.) for Eq. (15) is overlaid, omitting the first two echoes. (b) Echo decays simulated from same color curves in (a) for γg = 4.093 μm⁻¹ ms⁻¹. Insets show B with 2W^1/2 for the black curve and A for Δt(k) = {50, 50, 50, 50, 75, 75, 125, 139} μs. (c) D_inst(t) from the gradient of $⟨{\left[\mathbf{r}\left(t\right)\cdot \widehat{\mathbf{g}}\right]}^2⟩$ curves in (a) (solid lines) compared to X inverted from decays in (b) with added Gaussian noise (SNR = 25). Error bars = ±1 SD from 100 replications. Initial X guesses were D₀ for the first point and D_∞ (dashed lines, {0.9, 1.45, 1.48} μm²/ms) for the remaining points. λ = 2 × 10⁻⁶ is selected manually. See Eq. (S1) for Γ.

FIG. 4.

SG-TIETA A_p(n) calibration using pure liquids. (a) Observed SG-TIETA decays compared to exp(−bD₀). D₀ is measured (see Sec. IV of the supplementary material) in the legend order as 2.22 ± 0.01, 1.27 ± 0.01, 0.121 ± 0.002 μm²/ms. Error bars = ±1 SD for 25, 38, and 70 repetitions, respectively. (b) Decay vs exp(−bD₀) ratio truncated at n = 12, 15, 25. Inset shows cubic spline fits. (c) A_p(n) approximated using adjacent fitted ratios.

FIG. 5.

Comparison of the direct echo SG-TIETA (blue) and CPMG (orange) attenuation and echo shape for 1-octanol with 2τ = TE = 98 ms. Echo shapes show the real (solid lines) and imaginary (dotted lines) signals normalized by the area under the real signal curve in a 16 µs window. The CPMG echo width decreases with n and stabilizes around n = 3, consistent with the approach to the asymptotic behavior described by Hürlimann and Griffin. In contrast, the SG-TIETA echo width increases and stabilizes around n = 3, consistent with the direct echo CTP being preferential to the on-resonance signal (see Figs. S7 and S8 for all echo shapes and an exemplar echo decay, respectively)

FIG. 6.

SG-TIETA decays and inverted X for D6, yeast, and water. (a) Decays analyzed as described in the text. See Sec. II of the supplementary material for fitting procedures. Error bars = ±1 SD in the legend order for 38, 3, 4, and 25 repetitions truncated at N = 34, 17, 17, and 15, respectively. Note the erratic early A_p(n) behavior. (b) X solutions. Inversion parameters are identical to Fig. 3 other than Δt(k) for D6. Initial guesses of D₀ = 2.22, D_∞ = {0.9, 1.1} μm²/ms for yeast, and D₀ = D_∞ = 0.114 μm²/ms for D6 are provided. Zoomed-in plot compares short-time D_inst(t) plotted up to t < 0.08 × τ_D.