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Single-shot measurement of sub-millisecond, time-dependent diffusion using optimized, unequal pulse spacings in a static field gradient

Teddy Xuke Cai^1,2, Nathan Hu Williamson^1,3, Velencia Witherspoon¹, Rea Ravin^1,4, and Peter Basser¹
¹Section on Quantitative Imaging and Tissue Sciences, Eunice Kennedy Shriver National Institute of Child Health and Human Development, Bethesda, MD, United States, ²Wellcome Centre for Integrative Neuroimaging, University of Oxford, Oxford, United Kingdom, ³National Institute of General Medical Sciences, Bethesda, MD, United States, ⁴Celoptics, Inc., Rockville, MD, United States

Synopsis

Time-dependent diffusion contains rich information about the tissue microstructure. Conventional methods to measure the time-varying diffusivity probe a single timescale per acquisition, limiting time resolution. Furthermore, access to sub-millisecond timescales is limited by the pulsed gradient hardware. An alternative method is presented here. We extend the static field gradient, Carr-Purcell-Meiboom-Gill cycle by incrementing the $$$\pi$$$-pulse spacings to isolate the on-resonance signal. The resulting spin echo train probes a range of short timescales (50 – 500 microseconds) in one shot and enables a 1-minute time-dependent diffusivity measurement. Proof-of-principle simulations and experimental results on pure liquids and yeast are presented.

Introduction

Time-dependent diffusion is observed in heterogeneous porous media such as biological tissue [1–4]. This time-varying behavior is related to microstructural parameters such as the barrier surface-to-volume ratio [2,5–7], $$$S/V$$$, and permeability [8–10], $$$\kappa$$$. The short-time behavior [2] of the time-dependent diffusion coefficient, $$$D(t)$$$, with a linear permeability correction [10] is $$D(t)\simeq D_0\left[1-\frac{S}{V}\left(\frac{4\sqrt{D_0t}}{9\sqrt{\pi}}-\kappa t\right)\right],\;t \ll \tau_D,$$ where $$$D_0=D(t)|_{t = 0}$$$ is the free diffusivity, $$$\tau_D=\bar{a}^2/(2D_0)$$$ is the time to diffuse across the mean pore size of $$$\bar{a}\equiv6V/S$$$. NMR methods to characterize $$$D(t)$$$ conventionally use a signal representation [11,12] (with Gaussian phase approximation [13–16]) that relates the normalized echo attenuation to the spectrum of the velocity autocorrelation function, $$\boldsymbol{\mathcal{D}}(\omega)=\frac{1}{2}\int_0^{\infty}e^{i\omega t}\langle \textbf{v}(t)\textbf{v}^\mathrm{T}(0)\rangle dt,$$ and the truncated spectrum of the gradient wavevector, $$\textbf{F}(\omega)=\int_0^{T}e^{i\omega t}\left(\gamma\int_0^{t}\textbf{G}(t')dt'\right)dt,$$ where $$$\gamma$$$ is the gyromagnetic ratio, $$$\textbf{G}(t)$$$ is the gradient waveform, and $$$T$$$ is the time of echo formation, via $$\frac{I(T)}{I_0}=\exp{\left(-\frac{1}{\pi}\int_0^{\infty}\textbf{F}^{\mathrm{T}}(\omega)\boldsymbol{\mathcal{D}}(\omega)\textbf{F}(\omega)d\omega\right)}.$$ This signal representation naturally suggests the use of unidirectional oscillating [17] or “modulated” [12,18,19] gradients to concentrate the spectral density of the 1D $$$F(\omega)$$$ near some frequency, $$$\omega_F$$$. The signal becomes well-approximated by $$\frac{I(T)}{I_0}\approx \exp{\left(b\times\hat{\mathbf{g}}^{\mathrm{T}}\boldsymbol{\mathcal{D}}(\omega_F)\hat{\mathbf{g}}\right)},$$ where $$$b=\int_0^{T}|F(t)|^2dt$$$ and $$$\hat{\mathbf{g}}$$$ is the gradient direction. $$$\hat{\mathbf{g}}^{\mathrm{T}}\boldsymbol{\mathcal{D}}(\omega)\hat{\mathbf{g}}$$$ can then be related back to $$$D(t)$$$ [7,20]. This oscillating gradient, “temporal diffusion spectroscopy” [21] approach has limited time resolution because it individually probes $$$\omega_F$$$. Moreover, the largest probe-able $$$\omega_F$$$ is limited by slew rates to $$$\sim$$$100 Hz [4]. The rich, short-time behavior is inaccessible for structures with $$$\bar{a}\lesssim\mu\mathrm{m}$$$. The static gradient (SG), Carr-Purcell-Meiboom-Gill (CPMG) cycle produces a triangle wave $$$F(t)$$$ and can serve as a high-frequency temporal diffusion spectroscopy method [18,22,23] . SG-CPMG diffusion measurements, however, can be difficult to quantitatively interpret due to off-resonance signal contributions from incorrect coherence transfer pathways (CTPs) [24–26].

Theory

We extend the SG-CPMG experiment by discretely incrementing the $$$\pi$$$-pulse spacings in the form: $$$2\tau+m_j\delta$$$, where $$$\tau$$$ is the time between the $$$\pi/2$$$-pulse and the $$$\pi$$$-pulse, $$$j$$$ indexes the pulse-to-pulse spacing, $$$m_j\in\mathbb{N}$$$, and $$$\delta$$$ is a unit time increment. We term this spacing method the SG, time incremented echo train acquisition (SG-TIETA). Timing rules are derived to systematically avoid off-resonance CTPs [27,28]:

$$$h_n$$$ values, i.e., the peaks of $$$|F(t)/\gamma g|$$$, may not be repeated (see Fig. 1)
$$$m_j$$$ and $$$m_{j\pm\Delta j}$$$ with odd $$$\Delta_j$$$ may not be the same
Twice any $$$m_j$$$ may not equal the sum of $$$m_{j+\Delta j}$$$ and $$$m_{j-\Delta j}$$$ for even $$$\Delta j$$$
Any two $$$m_j$$$ with even $$$j$$$ may not equal the sum of any two $$$m_j$$$ with odd $$$j$$$
Twice any $$$m_j$$$ with even $$$j$$$ may not equal the sum of any two $$$m_j$$$ with odd $$$j$$$ and vice versa
$$$\delta$$$ and $$$\tau$$$ satisfy $$$(\tau\bmod\delta)=\delta/2$$$ and $$$\delta >2\tau_p$$$, where $$$\tau_p$$$ is the length of the $$$\pi$$$-pulse.

One satisfactory sequence is $$$\tau=49\;\mu\mathrm{s}$$$, $$$\delta = 14\;\mu\mathrm{s}$$$, and $$m_j =\left\{\begin{matrix}1&3&6&7&10&12&11&15&20&21\\24&26&20&21&33&35&33&34&33& ...\end{matrix}\right\}.$$ Extraneous signal decay due to $$$T_2$$$ relaxation and pulse inaccuracy remains. $$$T_2$$$ may be ignored if $$$2\tau+m_j\delta\ll T_2\forall j$$$. Pulse inaccuracy cannot be ignored and is describable using an $$$n$$$-dependent pulse accuracy factor, $$$A_p(n)$$$ [24]. The bandwidth excited by refocusing $$$\pi$$$-pulses has inconsistent frequency content [26,29] such that $$$A_p(n) < 1$$$. With each pulse, spins which rotate by angles other than $$$\pi$$$ do not refocus until a stable, central slice remains. The signal is then corrected as $$$I(T_n)/I_0\times\left[1/\prod_{l=1}^n A_p(l)\;\right].$$$

An SG-TIETA experiment produces a range of adjacent, inter-echo timescale sensitivities in one shot, but the spin echo $$$F(\omega)$$$ is too broad to be useful. Ning. et al. [30] showed that an equivalent (1D) time-dependent signal representation is $$\frac{I(T)}{I_0}=\exp{\left(-\int_0^{T}\mathcal{C}(t)D_{\mathrm{inst}}(t)dt\right)},$$ where $$$D_{\mathrm{inst}}(t)=d\left[tD(t)\right]/dt$$$, and $$\mathcal{C}(t)=\gamma^2\int_0^{t}\left(\int_0^{t’}G(t’’)G(t’’+s)ds\right)dt.$$ For each SG-TIETA echo, $$\mathcal{C}_n(t)=\gamma^2g^2\begin{cases} t\left(-\frac{3}{2}t+2h_n\right)&0\leq t\leq h_n\\t\left(\frac{1}{2}t-h_n\right)+2h_n^2&h_n\leq t\leq 2h_n\end{cases}.$$ The problem can now be recast into a weighted and regularized log-linear least squares (LLS) form by discretizing the time domain into bins, $$$\Delta t(k)$$$: $$||\textbf{W}^{1/2}\left(\textbf{A}\textbf{X}-\textbf{B}\right)||_2^2+\lambda||\boldsymbol{\Gamma} \textbf{X}||_2^2,$$ with coefficients, $$\textbf{A}=\begin{bmatrix}\int_{0}^{\Delta t(1)}\mathcal{C}_1(t)\,dt&...&\int_{\Delta t(K-1)}^{\Delta t(K)} \mathcal{C}_1(t)\,dt\\\vdots&&\vdots\\\int_{0}^{\Delta t(1)}\mathcal{C}_N(t)\,dt&...&\int_{\Delta t(K-1)}^{\Delta t(K)}\mathcal{C}_N(t)\,dt\\\end{bmatrix},$$ where $$$\textbf{X}$$$ consists of time-interval $$$D_{\mathrm{inst}}(t)$$$ averages, $$$\textbf{B}^{\mathrm{T}}=-\ln\,\begin{bmatrix}I(T_1)/I_0&...&I(T_N)/I(T_{N-1})\end{bmatrix}$$$, $$$\textbf{W}$$$ consists of signal differences, and $$$\boldsymbol{\Gamma}$$$ contains first and second-order finite difference matrices. This LLS inversion is demonstrated on simulated data in Fig. 2.

Results

NMR measurements were performed at $$$B_0=0.3239\;\mathrm{T}$$$ (proton $$$\omega_0=13.79\;\mathrm{MHz}$$$) using a PM-10 NMR MOUSE single-sided permanent magnet [31] (Magritek, Aachen Germany) and a Kea 2 spectrometer (Magritek, Wellington, New Zealand). The decay of the magnetic field produces a strong SG with amplitude $$$g=15.3\;\mathrm{T/m}$$$. Measurements used a home-built test chamber and a $$$13\times2$$$ mm solenoid RF coil and RF circuit. Additional information can be found in Ref. [32]. If, indeed, the derived SG-TIETA rules avoid off-resonance CTPs, then (1) the echo shape should reflect a loss of frequency content, and (2) the observed $$$A_p(n)$$$ should be independent of the diffusion weighting. We validate (1) by observing an increase in echo width (Fig. 3) and we validate (2) by obtaining similar calibration $$$A_p(n)$$$ values for liquids with vastly different diffusivities (Fig. 4). SG-TIETA decays for dodecamethylcyclohexasiloxane (D6) and yeast were corrected using the obtained $$$A_p(n)$$$ values for 1-octanol and water, respectively. Decays were then fit with a piece-wise linear function in the log-$$$b$$$ domain specified to produce a $$$(0,0)$$$-intercept and monotonically decreasing slope. The fit serves as non-negativity constraint. Each repetition consisted of $$$32$$$ scans, totaling to a $$$\sim1$$$-minute experiment. Results are summarized in Fig. 5.

Acknowledgements

TXC, VW, RR, and PJB were supported by the IRP of the NICHD, NIH. NHW was funded by the NIGMS PRAT Fellowship Award #FI2GM133445-01.

References

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Figures

Example SG-TIETA sequence. (a) Timings: $$$m_j = \{1, 3, 1, 2, 1\}$$$, $$$\tau = 4\delta$$$, and $$$\delta = 14\; \mu$$$s $$$= 1$$$ dash. $$$\pi$$$-pulses occur at $$$t_n$$$ and echoes form at $$$T_n$$$. Magenta line indicates timing behavior: $$$T_n = t_{n} + h_n$$$, where $$$h_n$$$ is the $$$|F(t)|/\gamma g$$$ "height" at $$$t_n$$$, given recursively by $$$h_1 = \tau$$$ and $$$h_{n} = 2\tau + m_n\delta -h_{n-1}$$$ for $$$n > 1$$$. (b) Direct echo $$$F(t)$$$ drawn along with other coherence pathways that refocus (red, dash-dot) or do not refocus (gray, dotted).

LLS inversion on Monte Carlo simulated data. (a) Simulated 1D mean-squared displacement, $$$\langle[\textbf{r}(t)\cdot\hat{\textbf{g}}]^2\rangle$$$. (b) Simulated decays from same color curves in (a). (c) Gradient of $$$\langle[\textbf{r}(t)\cdot\hat{\textbf{g}}]^2\rangle$$$ in (a) (solid lines) compared to $$$\textbf{X}$$$ inverted from decays in(b) with added Gaussian noise (SNR = 25). Err. bars $$$=\pm1$$$ SD from 100 replications. Initial $$$\textbf{X}$$$ guesses were $$$D_0$$$ for the first point and $$$D_\infty$$$ (dashed lines) for remaining points.

Comparison of SG-TIETA (blue) and CPMG (orange) attenuation and echo shape for 1-octanol with $$$2\tau$$$ = TE = $$$98 \;\mathrm{ms}$$$. Echo shapes show real (solid lines) imaginary (dotted lines) signal normalized by area under the real signal curve. The CPMG echo width decreases with $$$n$$$ and stabilizes around $$$n = 3$$$, consistent with the approach to asymptotic behavior described in Ref. [25]. In contrast, the SG-TIETA echo width increases, consistent with the direct echo CTP being preferential to on-resonance signal.

SG-TIETA $$$A_p(n)$$$ calibration using pure liquids. (a) Observed SG-TIETA decays compared to $$$\exp{(-bD_0)}$$$. $$$D_0$$$ was measured by spin echo diffusion in legend order as $$$2.22 \pm 0.01, 1.27 \pm 0.01$$$, and $$$0.121 \pm 0.002\; \mu\mathrm{m}^2/\mathrm{ms}$$$. Err. bars $$$=\pm1$$$ SD for $$$25, 38, 70$$$ repetitions, respectively. 1 repetition $$$=32$$$ summed SG-TIETA scans. (b) Decay vs. $$$\exp{(-bD_0)}$$$ ratio truncated at $$$n = 12, \,15, 25$$$. Inset shows cubic spline fits. (c) $$$A_p(n)$$$ values approximated using adjacent fitted ratios in (b).

SG-TIETA decays and inverted $$$\textbf{X}$$$ for D6, yeast, and water. (a) Decays analyzed as described in the text. Err. bars = $$$\pm1$$$ SD for, in legend order, $$$38, 3, 4, 25$$$ repetitions truncated at $$$n = 34, 17, 17, 15$$$, respectively. (b) $$$\textbf{X}$$$ solutions. Inversion parameters were identical to Fig. 2 other than $$$\Delta t(k)$$$ for D6. Again, $$$D_0$$$ and $$$D_\infty$$$ (dashed lines) were given as initial $$$\textbf{X}$$$ guesses. Zoomed plot compares experimental results to the presented theoretical short-time $$$D_{\mathrm{inst}}(t)$$$.

Proc. Intl. Soc. Mag. Reson. Med. 29 (2021)

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