On the Vanishing of the t-term in the Short-Time Expansion of the Diffusion Coefficient for Oscillating Gradients in Diffusion NMR

Introduction

This article builds on and extends a previous article [1], which investigated the term linear in time of the short-time expansion of the diffusion coefficient [2–5] which is given by:

$\begin{array}{ll}D(t)=D_0\left(1-\frac{4}{3d\sqrt{\pi }}\frac{S}{V}\sqrt{D_{0}t}+\left(\frac{1}{d}\frac{S}{V}\kappa +\frac{1}{2d}\frac{S}{V}\varrho -\frac{1}{12}\frac{S}{V}{D}_0\overline{R^{-1}}\right)t+O(t^{3/2})\right),& (1)\end{array}$

where D₀ is the free diffusion coefficient, S/V is the surface-to-volume ratio, κ is the surface permeability, ϱ is the surface relaxivity, $\overline{R^{-1}}$ is a mean curvature term, d is the spatial dimension, and t is the observation time. This universal expansion is valuable, since it connects a measurable quantity, i.e., D(t), to structural parameters of barriers restricting the diffusive motion.

Using magnetic resonance diffusion experiments [6–9], information about the diffusive motion of spin-bearing particles can be encoded into the signal by using diffusion-weighting magnetic field gradient pulses. Regarding the diffusion time t linked to the total duration of the diffusion-weighting gradient profile, one often considers the long-time and short-time limit. In the first case, the limit of long diffusion time, detailed information about the porous structure of the investigated material can be obtained [10–12] such as actual pore shapes [13–15]. On the other hand, D(t) can be measured in the short-time limit to obtain the structural parameters in Equation (1). For this purpose, a pair of bipolar gradient pulses is applied to achieve diffusion encoding [16]. In the short gradient pulse approximation [6, 17], the measured diffusion coefficient in such experiments is D(t) (Equation 1). If the gradient pulses cannot be considered to be short, Equation (1) must be modified to take into account the effect of the gradient pulse duration and of the temporal evolution of the gradients G(t) [18–22]:

$\begin{array}{l}{D}_{\text{app}}\left(t\right)=D_0(1-c_1\frac{4}{3d\sqrt{\pi }}\frac{S}{V}\sqrt{D_{0}t}+c_2(\frac{1}{d}\frac{S}{V}\kappa \\ -\frac{1}{12}\frac{S}{V}{D}_0\overline{R^{-1}})t+O\left(t^{3/2}\right))& (2)\end{array}$

Here, the influence of the temporal gradient profile is expressed solely by the constants c₁ and c₂, which can be computed from G(t), so that an elegant decoupling takes place. Note that surface relaxation is neglected in Equation (2) and in the remainder of the manuscript, thus avoiding the difficulties in the mathematical treatment [23], and that t is the total duration of the diffusion gradients in Equation (2).

In Laun et al. [1], it was shown that c₂ can be tuned to values between 0 and 1. Tuning c₂ to zero can be advantageous, for example, if the aim of the experiment is to measure the $\sqrt{t}$-term without bias from the t-term.

A striking result [24–30] is that the short-time expansion is moreover valid for the diffusion spectrum 𝔇(ω), or ${𝔇}_{\mathrm{\text{app}}}(\tau ):=𝔇(\frac{2\pi }{\tau })$, that can be measured by the use of oscillating gradients. Note that 𝔇_app(τ) and D_app(t) are different functions as outlined in more detail below. Then Equation (1) can be cast in a similar form for the diffusion spectrum, where t is replaced by the duration τ of one gradient oscillation:

$\begin{array}{l}{\mathfrak{D}}_{\text{app}}\left(\tau \right)=D_0(1-C_1\frac{4}{3d\sqrt{\pi }}\frac{S}{V}\sqrt{D_0\tau }+C_2(\frac{1}{d}\frac{S}{V}\kappa \\ -\frac{1}{12}\frac{S}{V}{D}_0\overline{R^{-1}})\tau +O\left({\tau }^{3/2}\right))& (3)\end{array}$

The constants C_n in Equation (3) are printed in capital letters because they differ, in general, from the constants c_n of Equation (2) as will be described below.

Equations (2) and (3) were successfully applied in experiments to obtain information about the first order term and thus about the surface-to-volume ratio [27, 28, 31–37]. The thereto necessary constants C₁ were derived for some gradient waveforms such as the bipolar waveform and oscillating gradients [18, 27, 29, 38].

The aim of the work at hand is to investigate the constant C₂. For this purpose, Equation (3) is derived starting from Equation (1).

Materials and Methods

Numerical simulations were performed as in Laun et al. [1]¹. The diffusion coefficient D_app(t) and the diffusion spectrum 𝔇_app(τ) [using the signal decrease as recalled in Equation (A13) in Appendix A (Supplementary Material)] were computed using the multiple correlation function (MCF) approach [22, 39–45] (using Equation 114 in Grebenkov [42]). The MCF approach decomposes the magnetization into the eigenfunctions of the Laplace operator. One important parameter is the number N_λ of employed eigenfunctions, which should be sufficiently large to ensure numerical accuracy. In the presented results, the accuracy was verified by increasing N_λ and checking whether numerical results remained unchanged. A detailed description of the MCF approach is beyond the scope of this article, but can be found in Grebenkov and Grebenkov [46, 47], for example.

The following closed domains were considered: slab, cylinder, sphere, “bi-slab” (see Figure 1). The bi-slab domain consists of three parallel planes. The inner plane is permeable, while the two outer ones are impermeable. Particles only reside within the volume between the two impermeable slabs. The radii of cylinder and sphere were 5 μm, the separation of the slabs was 10 μm, and the separation of the planes of the bi-slab domain was 10 μm (thus the bi-slab domain was in total 20 μm wide). The free diffusion coefficient D₀ was set to 1 μm²/ms. The boundaries were fully reflecting except for the inner wall of the bi-slab domain, which had a permeability of 50 μm/s. N_λ was 100 for the bi-slab domain, 500 for slab domain and cylinder, and 200 for the sphere. Oscillating cosine gradients were simulated with a total duration T_cos of 0.05, 0.1, and 0.5 s. The number of oscillations N was varied in twenty steps. For bipolar gradients, δ was set to 10⁻³ ms and t was varied between 0.1 and 15 ms.

FIGURE 1

Figure 1. Impact of the t-term on the short-time expansion of the diffusion coefficient. Markers indicate values obtained by numerical simulations. Solid lines represent the short-time expansion to order $\sqrt{t}$ and dotted lines represent it to order t. Red square markers represent bipolar gradients (i.e., D_app(t)) and black circular markers represent oscillating cosine gradients (i.e., 𝔇_app(τ)). (A) Slab domain. The t-term is zero because curvature and permeability of the sample are zero. (B–D) Cylinder, sphere, and bi-slab. In case of oscillating cosine gradients, the t-term is zero, because C₂ is zero. For this reason, the markers stay close to the solid line in contrast to the markers indicating the bipolar gradients, which stay close to the dotted line. T_cos was 500 ms.

Additionally, the difference between simulated diffusion coefficients and first order short time expansion was calculated. This difference is labeled ΔD in the plots and represents $D_{\mathrm{\text{app}},\mathrm{\text{simulated}}}(t)-D_0-M_1{c}_1{t}^{1/2}$ or ${𝔇}_{\mathrm{\text{app}},\mathrm{\text{simulated}}}(\tau )-D_0-M_1{C}_1{\tau }^{1/2}$.

Results

Derivation of the t-Term for Oscillating Gradients

First, a shorthand-notation for Equation (2) is introduced:

$\begin{array}{ll}D\left(t\right)={\displaystyle {\sum }_{n=0}{M}_n{c}_n{t}^{n/2}}& (4)\end{array}$

with the coefficients

$\begin{array}{l}{M}_0=D_0\\ M_1=-\frac{4}{3d\sqrt{\pi }}\frac{S}{V}{D}_0^{3/2}\\ M_2=\frac{1}{d}\frac{S}{V}\kappa D_0-\frac{1}{12}\frac{S}{V}\overline{R^{-1}}{D}_0^2\end{array}$

and so on (with c₀ = 1).

As outlined in Appendix A (Supplementary Material), the short-time expansion for the position correlation function that generates an experimentally detectable signal attenuation reads:

$\begin{array}{ll}\langle x(t_2)x(t_1)\rangle =-D(\left|t_{21}\right|)·\left|t_{21}\right|=-{{\displaystyle \sum }}_{n=0}{M}_n|t_{21}{|}^{1+n/2}& (5)\end{array}$

with t₂₁ = t₂ − t₁, where the brackets 〈…〉 denote the expectation value. Note that the terms $\langle x{(t_2)}^2\rangle$ and $\langle x{(t_1)}^2\rangle$ where neglected in Equation (5) because they do not contribute to the signal attenuation. Equation (5) can be related to the diffusion spectrum 𝔇(ω) (see Appendix A in Supplementary Material) via:

$\begin{array}{l}\frac{\mathfrak{D}\left(\omega \right)}{{\omega }^{\text{2}}}=\frac{1}{2} {\displaystyle {\int }_{-\infty }^{\infty }\langle x\left(t_2\right)x\left(t_1\rangle \right) }{e}^{-i\omega t_{21}}d{t}_{21 }\\ =\frac{1}{2} {\displaystyle {\sum }_{n=0}{\displaystyle {\int }_{-\infty }^{\infty }{M}_n{\left|t_{21}\right|}^{1+n/2}}{e}^{-i\omega t_{21}}d{t}_{21 }.}& (6)\end{array}$

This Fourier integral exists (see Appendix B in Supplementary Material):

$\begin{array}{l}\frac{1}{2}{{\displaystyle \int }}_{-\infty }^{\infty }{\left|t_{21}\right|}^{1+n/2}{e}^{-i\omega t_{21}}d{t}_{21}=-{\omega }^{-2-\frac{n}{2}}\mathrm{\text{cos}}\left(\frac{n\pi }{4}\right)\Gamma \left(2+\frac{n}{2}\right),\\ \omega >0,n\ge 0& (7)\end{array}$

and thus by inserting Equation (7) in Equation (6), one finds:

$\begin{array}{ll}𝔇(\omega )=D_0+M_1\sqrt{\frac{\pi }{2}}\frac{3}{4{\omega }^{1/2}}+0·M_2+0\left({\omega }^{-3/2}\right).& (8)\end{array}$

Note that the gamma function Γ makes the constants c_n increase swiftly at larger n. Defining the time parameter τ = t/n, entailing ω = 2π/τ, one finds:

$\begin{array}{ll}{𝔇}_{\text{app}}\left(\tau \right):=𝔇\left(\frac{2\pi }{\tau }\right)=D_0+M_1\frac{3}{8}{\tau }^{1/2}+0·M_2+O\left({\tau }^{3/2}\right)& (9)\end{array}$

with 𝔇_app(τ) being identical to 𝔇(ω) except for taking a different argument. 𝔇_app(τ) has exactly the form of Equation (3) as desired and one can read off the coefficients C_n directly: C₁ = 3/8 and C₂ = 0. Note that the value of C₁ was reported previously (e.g., in Novikov and Kiselev [29]). The vanishing of C₂ has not been reported so far to our knowledge.

Using the expression for M₁, one finds:

$\begin{array}{ll}𝔇(\omega )=D_0-\frac{{\mathrm{\text{D}}}_0}{d\sqrt{2}}\frac{S}{V}\sqrt{\frac{D_0}{\omega }}+O({\omega }^{-3/2}),& (10)\end{array}$

$\begin{array}{ll}{𝔇}_{\mathrm{\text{app}}}(\tau ):=𝔇\left(\frac{2\pi }{\tau }\right)=D_0-\frac{D_0}{2d\sqrt{\pi }}\frac{S}{V}\sqrt{D_0}{\tau }^{1/2}+O\left({\tau }^{3/2}\right).& (11)\end{array}$

It is interesting to calculate the coefficients c₁ and c₂ for a short-time cosine gradient with one oscillation [with methods as described, e.g., in Laun et al. [1] and references therein]. We find c₁ = 3·(4π FresnelC(2) + 3 FresnelS(2))/16/π ≈ 0.428 and c₂ = 0. These values bear great similarity to C₁ ≈ 0.375 and C₂ = 0. It should be noted that c₂ of oscillating cosine gradients with any number of oscillations equals zero because they are “flow-compensated,” i.e., because their first moment vanishes [1].

Validation with Simulations

Figure 1 displays D_app(t) and 𝔇_app(τ). Markers indicate simulation results using the MCF approach and lines represent the short-time expansion. For D_app(t), solid lines equal $M_0+M_1{c}_1{t}^{1/2}$ and dotted lines equal $M_0+M_1{c}_1{t}^{1/2}+M_2{c}_{2}t$. For 𝔇_app(τ), solid lines equal $M_0+M_1{C}_1{\tau }^{1/2}$ and dotted lines equal $M_0+M_1{C}_1{\tau }^{1/2}+M_2{C}_1^2\tau$. The term $M_2{C}_1^2\tau$ shall represent a reasonable “guess” for the t-term with an effective diffusion time ${\tau }_{\mathrm{\text{eff}}}=C_1^2\tau$, where the coefficient C₂ was set to one. The intention is to visualize a line with C₂ ≠ 0, although this term does not occur in reality. Some remarks on effective diffusion times can be found in Appendix C (Supplementary Material).

For the slab domain (Figure 1A), M_n>1 = 0 holds true (see [42]). Hence, Figure 1A does not display a dotted line and markers stay close to the solid lines.

In Figures 1B,C (cylinder, sphere) and Figure 1D (bi-slab), it is clearly visible that the markers for the bipolar gradients (with c₂ = 1 ≠ 0) stay close to the dotted lines indicating the importance of the t-term. The markers of the oscillating cosine gradients stay close to the solid line indicating that the t-term does not influence 𝔇_app(τ). Owing to higher order terms, deviations between the short-time expansion and markers are present at larger t.

Figure 2 shows ΔD, i.e., the difference between simulated diffusion coefficients and first order short time expansion. The dotted line represents the t-term, i.e., M₂c₂t for the bipolar gradients. For the oscillating gradients, the black dotted line shall represent an educated guess for the t-term, i.e., $M_2{C}_1^2\tau$, as in Figure 1.

FIGURE 2

Figure 2. Difference between simulated diffusion coefficients and first order short time diffusion expansion for slab domain (A), cylinder (B), sphere (C), and bi-slab (D).

First, the bipolar gradients displayed in Figure 2 are discussed (displayed in red color). For the slab domain (Figure 2A), the dotted line is flat because M₂ = 0. However, deviations of ΔD from zero are well visible for t > 10 ms. This does not result from the influence of higher order terms because all higher order terms are zero. It rather indicates the breakdown of the short-time expansion. For cylinder, sphere, and bi-slab (Figures 2B–D), the slope of ΔD is identical to that of the dotted line at t = 0, but starts deviating already roughly at t = 2 ms indicating that either higher order terms are needed or, again, that the short-time expansion breaks down. This deviation is more pronounced for cylinder and sphere than for the bi-slab.

Next, the oscillating gradients in Figure 2 are discussed (displayed in black color). For cylinder, sphere, and bi-slab, ΔD does have zero slope at t = 0 and does not follow the dotted line for any of these domains, which supports the finding that C₂ = 0. This holds true for T_cos = 500 ms, but also for reduced total duration of the oscillating gradients, i.e., for smaller T_cos. The difference of ΔD between T_cos = 500 ms and T_cos = 50 ms is smaller than 0.015 μm²/ms for all domains at τ = 10 ms, which is roughly equally large as the guessed t-term, but an order of magnitude smaller than the $\sqrt{t}$-term. Thus, for the considered domains, T_cos = 50 ms still appears to be well suited for investigations of the $\sqrt{t}$-term, even with as few as five oscillations.

Discussion

The main result of this work is that oscillating cosine gradients are blind with respect to the t-term of the short-time expansion of the apparent diffusion coefficient.

Oscillating gradients and extensions [48–51] have been used in several research studies [27, 28, 32, 35, 38, 52–65], among them applications to human brains in vivo [66]. Comparing oscillating gradients to pulsed gradients, the advantage of the oscillating gradients is that the obtainable b-value is higher allowing the assessment of shorter times. This is particularly useful if strong gradient amplitudes are not available or if the structure of interest is too small. The disadvantage is the need for longer echo times entailing decreased signal-to-noise ratio due to transversal relaxation, which also entails a longer acquisition time.

As oscillating gradients are blind to the t-term, estimates of S/V as in Reynaud et al. [36] are not biased by this term, but, obviously, the membrane permeability, for example, cannot be estimated using the t-term. This is in line with the findings by Li et al. [67], who reported that the membrane permeability has little effect on oscillating gradient derived diffusion coefficients at high frequencies. This is presumably not a major limitation given the smallness of the t-term that is visible in Figures 1, 2, which makes a fit challenging. The permeability information must have some influence on 𝔇_app(τ) at long τ; otherwise diffusion in the bi-slab would have to be identical to that of a single slab domain of double size. Therefore, the estimation of membrane permeability using oscillating gradients might in principle be possible.

As different versions of Equations (10) and (11) can be found in the literature, a comparison is worthwhile. Equation (10) is identical to Equation 10 of the article by Novikov and Kiselev [29]. Except for a small deviation, which may be due to numerics, Equation (10) is also identical to Equation (3) of the article by Xu et al. [35], but, to our understanding, not to the respective equations in an earlier article [27]. In general, care must be taken concerning the definition of τ. For example, Zielinski et al. [38] use the definition τ_Zielinski = τ/2, which is closer to the classical timing definitions of CPMG echo trains than our definition. Considering this difference in definitions, their respective coefficient C₁ for the CPMG condition as stated in their Equation (6) is almost identical to 3/8, which is in agreement with the finding that the difference between C₁ of CPMG and cosine gradients should be almost negligible as stated in section 3.3 of Novikov and Kiselev [29]. Further, we found our coefficient C₁ to be a factor of six smaller than the one stated in Equation 14 in the article by Stepišnik et al. [28]. This difference was noted by the authors themselves and in Novikov and Kiselev [29].

Interestingly, the disappearance of the t-term in the Mitra expansion of Equation (3) using oscillating gradients is due to its disappearance in 𝔇(ω), or 𝔇_app(2π/τ), respectively. Thus, optimizing oscillating gradient profiles instead of using, for example, just cosine gradients, which was a successful approach in other regards [68, 69], does not help to make the t-term reappear in the signal attenuation.

In practice, diffusion measurements use spin echoes and hence two gradients at both sides of the refocusing pulse (as in Baron and Beaulieu [66]). This effectively introduces an extra variable, i.e., the separation of two gradients, which can affect the spectrum of diffusion gradients. When interpreting oscillating gradient experiments, this effect must be taken into account.

A limitation of the presented simulations is that they cannot prove the disappearance of the t-term. In principle, a very small t-term might be present and go unnoticed.

In conclusion, oscillating gradients are blind to the t-term and hence no bias in fits of the surface-to-volume ratio arises from the t-term.

Author Contributions

FL performed the simulations and initial computations. TK and MU were involved in the design of the evaluations. TK, AN, KD, and FL were involved in implementation and testing of the MCF code and of the mathematical derivations.

Funding

Financial support by the DFG (grant numbers SFB TRR 125/2 R01 and KU 3362/1-1) is gratefully acknowledged.

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The handling Editor declared a past co-authorship with the authors TK and FL.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphy.2017.00056/full#supplementary-material

Footnotes

1. ^A considerable overlap of the Methods sections with the corresponding sections of the earlier article is present.

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