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Edited by: Itamar Ronen, Leiden University, Netherlands

Reviewed by: Silvia Capuani, Consiglio Nazionale Delle Ricerche (CNR), Italy; Christian Herbert Ziener, German Cancer Research Center, Germany

*Correspondence: Evren Özarslan

This article was submitted to Biomedical Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The signature of diffusive motion on the NMR signal has been exploited to characterize the mesoscopic structure of specimens in numerous applications. For compartmentalized specimens comprising isolated subdomains, a representation of individual pores is necessary for describing restricted diffusion within them. When gradient waveforms with long pulse durations are employed, a quadratic potential profile is identified as an effective energy landscape for restricted diffusion. The dependence of the stochastic effective force on the center-of-mass position is indeed found to be approximately linear (Hookean) for restricted diffusion even when the walls are sticky. We outline the theoretical basis and practical advantages of our picture involving effective potentials.

Recovering the mesoscopic structure of porous media and biological tissues via diffusion sensitized NMR methods has been an active area of research since 1960s [

where γ is the gyromagnetic ratio while

Conventional techniques for relating the NMR signal above to microstructural features of the medium vary from “localized” models in which the aggregate signal is envisioned to arise from isolated (e.g., restricted) compartments [

When the gradient waveform,

In this Perspective, we argue that the theory of diffusion under a Hookean restoring force [

Here we demonstrate that, under certain experimental conditions, the influence of restricted diffusion is essentially the same as that for the Hookean potential model, which was studied in-depth recently [

Consider a pulse sequence which consists of _{n} and durations δ_{n}. With the definition

the NMR signal (Equation 1) is given by

where _{n} denotes the leading edge of the

Further, introducing the stochastic center-of-mass of the particle trajectory [

the stochastic signal can be rewritten as

In words, the NMR signal elicited by spatially constant gradient pulses of finite duration is sensitive to the center of mass (average position) of Brownian trajectories, rather than instantaneous coordinates. The duration of the pulses therefore serve to smear out fine spatial features.

Let _{cm}(_{n} → ∞, the dependence on the pulse separation disappears and the signal intensity (Equation 5) factorizes, leading to

where

is the Fourier transform of the center of mass distribution. Due to its construction (Equation 4), the random variable _{cm}(^{1}

The Gaussian distribution is determined simply by its variance matrix (with its mean set to zero for convenience). The relevance of this fact here is that when a train of long pulses is employed, the signal has no means of encoding for fine features of the microscopic environment where diffusion takes place^{2}

While restricted diffusion becomes a difficult problem to tackle in higher dimensions for all but a few special geometries, diffusion under a Hookean force is much more tractable, and it behooves one to adopt the latter model when its features are all that can be observed, as argued above. Hence we consider the case of diffusing particles subject to a (dimensionless) parabolic confining potential ^{⊺}

with the variance matrix

which is just a straightforward generalization of an expression in Mitra and Halperin [

The eigenvalues of the inverse of the confinement tensor

where the subscript stands for restricted diffusion^{3}

In the long pulse duration regime wherein the statistics of the restricted problem approach those of the confinement problem, the variances of the center of mass position for the two problems are found by taking the long time limit of Equations (9) and (10), respectively,

Demanding that the two variances above agree results in the relation

meaning that the confinement model with parameter

While in the above it was the long pulse duration that ensured the Gaussianity of the signal (through the center of mass distribution) for restricted diffusion, an alternative situation is one where the gradient strength is so small that ^{2} and

^{2}. The variation in ^{2} = 100. The restricted diffusion signal was computed using the multiple correlation function method [

Contrary to the two cases above, the signal for restricted diffusion is not Gaussian for other values of

Consider, e.g., a 4 μm pore—roughly the size of a yeast cell. Assuming a bulk diffusivity value of 2 μm^{2}/ms, and for a pulse duration of 20 ms, which is typical for diffusion measurements via clinical scanners, the ^{2} value is 2.5. Thus, the signal response of the confinement picture would be indistinguishable from a much more elaborate theory involving restricted diffusion. The situation is even more favorable for smaller sized compartments. In laboratory spectrometers, however, pulse durations an order of magnitude shorter are feasible; in such scenarios, detectable differences between the restricted diffusion and confinement pictures can be encountered unless the pores are smaller. However, let us remark once more that it is only in this simplified one-dimensional scenario that there exists a single unambiguous size of the region, allowing a one-to-one correspondence to be drawn between

The confinement model is based on a Hookean force assumption, i.e., the presence of a restoring force whose magnitude increases linearly as the particles move away from an attractive center. We shall define an effective force, _{eff}, based on the impulse the particles experience during a time interval of duration δ, i.e.,

Because the time-dependent force

By performing random walk simulations, we investigated whether a similar (Hookean) force model could emerge for the restricted diffusion process as well. In the top panel of Figure ^{−2} just like in Equation (13). These findings further support the idea of employing a Hookean force model as a substitute for restricted diffusion at long durations. Importantly, the Hookean model is valid even for more complicated problems, which are difficult to treat analytically. We illustrate this point by introducing stickiness to the walls of the same restricted geometry. As shown in Figure

^{2}/ms. Different curves represent different time intervals, which can be associated with durations of the pulses in NMR measurements. The distributions of the positions of the center of mass are shown at the top. The average effective force exerted on the diffusing particles plotted vs. the trajectories' center of mass (bottom).

While characterizing a compartmentalized specimen or biological tissue via diffusion NMR, one is faced with the problem of determining a reliable representation of the local compartment. An accurate mathematical description of the pore shape would typically necessitate numerous parameters. However, such parameters are simply unavailable in diffusion NMR measurements featuring long gradients as a result of coarse-graining associated with the diffusion process taking place during the application of the gradients. The confinement tensor model [

Similar to the diffusion tensor, the confinement tensor is real and symmetric, thus is described by 6 independent numbers. However, the confinement tensor

Diffusion tensors have also been employed for representing diffusion in microscopic compartments [

From a practical point of view, treating restricted diffusion is quite difficult even for simple geometries. For example, if we consider an ellipsoidal pore—the simplest geometry with the same number of parameters as a general confinement tensor— the problem would be very difficult to solve analytically. We argue that doing so would also be unnecessary if long pulses are employed. The

We would like to note that, as shown by Bauer et al. [

In a very recent study [^{−U(x)}, which could account for inhomogeneities in the bulk, and incorporate the effects of boundaries if needed, in a mathematically wieldy manner. We demonstrated that pulse sequences involving very short and long pulses [

In conclusion, we have presented a new perspective in which findings of common and relevant NMR signal acquisition scenarios can be interpreted. At the heart of this perspective lies modeling the diffusion as taking place in an effective quadratic potential landscape instead of a restricted domain. We have argued that when probed via waveforms featuring long pulses, the two models become indistinguishable, and the signal should rather be taken to reflect the parameters of such an effective model. Simulations suggest that the stochastic effective force has a linear (Hookean) dependence on the average particle position. The signal for quadratic potential indeed provides a very good approximation to that for restricted diffusion in small (micron-scale) pores when examined via commonly available hardware (see Figure

EÖ conceptualized the problem. EÖ, CY, MH, and HK developed and refined the perspective. HK and EÖ performed the numerical simulations. CY and EÖ wrote the manuscript. C-FW provided guidelines. All authors collaborated in bringing the manuscript to its final state.

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.

^{1}A rigorous proof of this can be found in the mathematics literature [

^{2}Indicated also by the absence of powers of “momentum”

^{3}Coordinates are chosen such that 〈ξ〉 = 〈