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:

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, 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]:

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 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 𝔇app(τ):=𝔇(2πτ), 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:

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).

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.

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 Dapp,simulated(t)-D0-M1c1t1/2 or 𝔇app,simulated(τ)-D0-M1C1τ1/2.

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.

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.