In the first case, diffusion is observed for such a short time that the hindrances have not been encountered. This condition implies that the pulse durations are short as well. In fact, when DΔ≪Rc2, spins spread so little that even the most curved point along the projection seems like a straight segment³. Thus, the compartmental signal occurs as the average, over the curve, of the signals originating from the tangents of the curve:

where

is the unit tangent to the curve.

As shown in Appendix A (see the Supplementary Material), the very small-q behavior of the orientationally-averaged signal (q²DΔ ≪ 1) is given by Ē(q) ≈ e^−q2DΔ/3. Thus, the decay rate is determined solely by the (intracellular) diffusivity, and bears no geometric feature of the neural arborization in this regime.

For a more general analysis involving larger q-values, one can still employ Equations (1, 2) as follows. We are interested in the orientational average of the compartmental signal in Equation (3), i.e.,

where the order of integration and orientational averaging, indicated by the angular brackets, was changed. However, the expression within the angular brackets is of the Gaussian form (Equation 1) with a rank-1 decay tensor whose non-vanishing eigenvalue is DΔ. Consequently, the signal for the powdered specimen is given through Equation (2) by setting v_⊥ = 0, and v_∥ = DΔ to be

Clearly, for larger q-values, the orientationally-averaged signal decay in regime A is proportional to q⁻¹ irrespective of the shape of the curve as long as Rc2 ≫ DΔ. It can be observed that Equation (5) has the form of a signal arising from a uniform orientational distribution of “sticks”. Hence the emergence of q⁻¹ at large q-values can be justified alternatively by Veraart et al.'s arguments [23].

When diffusion is probed via a pair of impulses (or, when Dδ≪Rc2) separated from each other by a duration long enough for the molecules to reach all points within the curve, i.e., DΔ ≫ ℓ², the signal is given simply by the expression E(q)=|ρ~(q)|2, where ρ~(q) is the Fourier transform of the density

which is uniform along the curve. The orientationally averaged signal decay is thus obtained by averaging |ρ~(q)|2, and is given by

This expression is referred to as the Debye scattering equation [36], which is widely utilized in studies employing small angle scattering experiments for characterizing the structure of polymers. The essential features of the resulting Ē(q) curve are well-understood [37, 38]. As shown in Appendix A (see the Supplementary Material), the small-q regime (qR_g ≪ 1, the “Guinier regime” of scattering experiments) is described by the relationship Ē(q)≈e-(qRg)2/3. As for larger q-values, such studies indicate that depending on the structure of the polymer, different sections of the curve could be characterized by different power-laws. For example, Gaussian chains undergo q⁻² decay, while fractional exponents are obtained for curves exhibiting fractality. Perhaps the most relevant finding, however, is that wormlike structures (i.e., those characterized by a so-called persistence length over which the polymer is likely to retain its direction) are characterized by a decay ∝ q⁻¹ at q-values about the reciprocal of the chain's persistence length [39]. Thus, even a class of non-straight structures exhibit q⁻¹ decay in Regime B.

For the traditional Stejskal-Tanner measurement utilizing a pair of identical pulses in opposite directions, the compartmental signal has the form

where the averaging is performed over all trajectories, with

being the center of mass coordinate [40] of the fragment of trajectory coinciding with each pulse (t₁ = 0 and t₂ = Δ). Therefore, the MR signal (Equation 15) elicited by flat gradient pulses is not sensitive to the Brownian trajectories instant by instant but only in a time averaged sense.

In the limit Dδ/ℓ² → ∞, the two random variables lose correlation and the explicit dependence on Δ disappears, leading to the signal intensity [40]

where

is the Fourier transform of the center of mass distribution. Moreover, in this limit, the distribution for the random variable ξ approaches a Gaussian due to the central limit theorem [41]⁴. Consequently, the signal (Equation 17) also approaches a quadratic exponential form, encoding no more structural information than the variance of p_cm(ξ, δ). Whether the domain is an irregularly curved fiber or a much more regular shape like a sphere, its fine structural features will find no representation in the signal acquired this way, since the length of the time averaging (pulse) has suppressed all short-scale (high q) variation encoded in the cumulants higher than the second.

Hence, the compartmental signal has the form Equation (1). Here, though, V is the variance tensor for the center of mass coordinate (Equation 16) for a trajectory of (long) duration δ. This variance can be calculated for diffusion along a general continuous curve following Mitra and Halperin's [40] derivation in the case of slab geometry, with slight modifications. One finds

where B_n(·) denotes the nth order Bernoulli polynomial, and Dδ/ℓ² ≫ 1. Here, the exponentially decaying terms are ignored as their contribution is negligible at even moderate durations. The details of the derivation of the above expression is provided in Appendix A (see the Supplementary Material). We note that alternative representations of the final result (Equation 19) can be given in terms of polylogarithmic or Hurwitz zeta functions. We verified that Equation (19) correctly reproduces Mitra and Halperin's expression [40] in the same regime for the slab geometry, i.e., for a straight line.

As were in the previous cases, the signal decay tensor for long pulse duration is not rank-1 for a general curve. However, unlike in previous regimes, the compartmental signal decay is truly Gaussian. Consequently, the orientationally averaged signal in this regime suffers q⁻¹ decay if and only if the fibers are straight.

The above findings can be summarized as follows: For narrow pulses (Dδ≪Rc2,ℓ2) in regimes A and B, the orientationally averaged signal exhibits a slow q⁻¹ tail for large q for diffusion along curved as well as straight 1 dimensional structures. As the pulse duration is prolonged, as we have studied for regime A, the slow q⁻¹ tail gives way to a steeper drop for substantially curved fibers, by which we mean Rc≲Dδ. Indeed, the signal may be expected to bear less features of diffusion along a 1D structure and more of diffusion in a 3D domain (i.e., a steep decay), since a long pulse serves to average the motion of the spin carriers over a length ~Dδ along the path curved in 3D space. Fibers much shorter than the averaging length (ℓ≪Dδ) fall into regime C and may contribute to an exponent of −1 in the orientationally averaged signal only if they are straight (R_c ≫ ℓ).

An important question to ask is: What regime is the most relevant for clinical MR examinations of the brain? There is no clear answer to this question, essentially because of the extremely wide variability in the size and shapes of cells within the brain [44]⁵. Consequently, it is impossible to suggest that diffusion within all neural cells takes place in a single experimental regime. We can, nonetheless, argue that regime B is the least relevant one as it is impossible to meet the narrow pulse condition along with the long diffusion time condition when δ ≈ Δ as in clinical acquisitions performed at larger (in a practical sense) b-values, where b = q²(Δ−δ/3). The diffusion distance 2DΔ is expected to be about 10–20 μm in such acquisitions, which will place longer (ℓ≫DΔ) arborizations toward regime A while the shorter (ℓ≪DΔ) ones will tend to exhibit features of regime C. If the longer structures furthermore exhibit curvature radii safely above the “averaging length” (Rc≫Dδ), the q⁻¹ signature will be possible to observe over a range of large q-values without requiring strict straightness, as demonstrated in Figure 3. For structures whose radii of curvature are small (Rc≲Dδ) the appearance of the slow decay q⁻¹ becomes sensitive to curvature, and is not retained for as wide a range of q-values (see Figure 4). At the extreme end of the spectrum, for structures so short that ℓ≪Dδ, entering into regime C, no curvature is tolerated if the tail q⁻¹ is to appear. These considerations will have to be revisited if one measures the diffusion of molecules other than water, due to differences in their diffusion characteristics [48–50].

In light of the above deliberations, we can revisit the experimental observations of McKinnon et al. [1] who have reported the powder averaged signals stemming from various brain regions dominated by white- or gray-matter. In white-matter regions, they have measured a decay q^−c for large q with c values in the neighborhood c ≈ 1.1 ± 0.1, whereas in gray-matter regions, the decay was significantly faster, with c ≈ 1.8 ± 0.2. Concerning the faster decay observed in gray-matter, they propose an explanation based on permeability differences in white and gray-matter regions. If one assumes impermeable membranes, as we did in this study, the following alternative interpretation seems adequate: In white-matter, the observation of the tail q⁻¹ is compatible with regimes A and C, suggesting a substantial presence of fibers that fall into these regimes. The former implies long fibers (ℓ≫DΔ) that could in fact be modestly curved (as long as Rc≫Dδ remains valid). The latter implies short fibers (ℓ≪Dδ) that are straight. In gray-matter regions, the loss of the slow q⁻¹ decay suggests that the signal originates predominantly from fibers that fall outside these descriptions, i.e., exhibiting strong curvature (Rc≲Dδ). Indeed, gray-matter is rich in dendrites and unmyelinated axons, which will exhibit a fair amount of bending distributed across any voxel. The gray box in Figure 4 depicts roughly the q range used in McKinnon et al.'s study, where it is seen that while modest curvatures do exhibit the exponent −1 for a while, strong curvature yields a steeper decay for most of the range.

Another potential explanation involves glial cells, which constitute a substantial portion of all neural cells. Similar to neurons, glial cells exhibit an extraordinary level of diversity in their size and shape throughout the brain. Their relative number and distribution is the subject of ongoing debate [51]. Thus, an accurate assessment of their influence on the detected diffusion MR signal is infeasible at this time. It is known, however, that the glial cells tend to be smaller than neurons, they lack axons, and many of them are star-shaped [52]. These structural features tend to disfavor the emergence of a q⁻¹ decay. However, recent studies have suggested that the glial cells are significantly more prevalent in cerebral white-matter compared to gray-matter [53]. Thus, it can be argued that their contribution to the overall MR signal must be rather limited. We note that the vast variation in neuronal and glial morphology along with the reported regional differences justify future studies performed at high spatial resolutions, for understanding the influence of compositional variations on the diffusion MR signal.

The detected orientationally-averaged signal would typically include contributions from many different compartments besides the neural projections, including extracellular matrix, cell bodies, and molecules trapped within very small regions (e.g., within certain organelles, between myelin layers, etc.). Among these, molecules diffusing relatively freely (for instance, between the neural processes) are expected to yield a decay rate faster than q⁻¹ so that much of their contribution is expected to disappear at larger q values. Conversely, there would be no significant loss of signal for truly restricted particles. Presence of a substantial portion of signal originating from such compartments as well as noise-induced bias associated with employing magnitude-valued data would make the decay appear slower [54, 55] than q⁻¹. The decay exponent c ≈ 1.1 ± 0.1 reported by McKinnon et al.[1] suggests that contributions from such immobile spins could be negligible in their acquisitions. This can be attributed [23] to the relatively short transverse relaxation times those spins are expected to have, along with the long echo times employed at larger q-value acquisitions via clinical scanners.

One may be tempted to employ Equation (1) for the compartmental signal, as it always has a Gaussian form for sufficiently small q values, and then to attribute the emergence of the q⁻¹ behavior of the orientationally averaged signal through Equation (2) to the rank of the decay tensor V being 1. However, this may be permissible for large q only in regime C. This attribution would therefore be erroneous for large q in regimes A and B. Specifically, in regimes A and B, the q range in which Equation (1) applies is similar to the q range in which the orientational average (Equation 2) exhibits a Gaussian tail no matter the rank of the tensor V (see Appendix A in the Supplementary Material). Thus, the q⁻¹ behavior is not a consequence of the signal decay tensor having rank 1. Importantly, such behavior naturally emerges at larger q-values in regimes A and B for reasonable shapes of neural projections.

The powder averaged signal may be envisioned as having originated from a porous specimen which is macroscopically isotropic, in which case one expects an asymptote of the form q⁻⁴ due to the Debye-Porod law [27] (see Appendix C in the Supplementary Material), while a steeper decay is predicted for longer pulses as the process approaches a Gaussian (Regime C). Since this behavior arises from quite general considerations, the observation of a tail of q⁻¹ appears questionable. Indeed, the observed power is most likely valid only in an intermediate range, as opposed to the strict q → ∞ limit. The asymptotic expansion, whose leading term gives the Debye-Porod law, contains terms that decay faster than any pure power (e.g., exponential). These terms may very well exhibit decays slower than q⁻⁴ over a certain stretch of q-values, before the Debye-Porod asymptote takes over, which happens when q begins to compete with the inverse length of the smallest dimension of the pore geometry. Alas, the q values used in McKinnon et al.'s experiments are of the order 0.1μm⁻¹ which is far from large enough compared to the inverse length scale of 1μm⁻¹ afforded by the axon diameter.

One of the hallmarks of neuronal morphology is the axonal and dendritic branchings [56], which are not accounted for in our treatment of diffusion on curves. Although an accurate assessment of the influence of branchings could be achieved via careful numerical studies [57], the insight gained from our description above can be employed to some extent for making some predictions. For long structures, each branch can be considered a separate segment along which diffusion takes place. Thus, the presence of branchings would not impact the formulation for projections in regime A. However, since the total contour length can be considerably increased in the presence of branchings, it may be more difficult to satisfy the long pulse duration condition of regime C. When this condition is met, however, the rank of the signal decay tensor would almost certainly be greater than one. In other words, detecting q⁻¹ decay in regime C would be nearly impossible for smaller cells unless most arborizations run parallel within a narrow cylindrical region.

The detected MR signal is known to be dependent on factors other than those accounted for in this work. Among these, spatial heterogeneity of magnetic susceptibility within the tissue have been reported to influence the diffusion decay [58, 59] as well. In fact, suppressing [60] or taking advantage [61, 62] of effects due to susceptibility variations is an active area of research. We note that the presence of internal gradients could be investigated as yet another mechanism that could explain the reported features in the orientationally averaged signal; doing so would require an extension of existing studies relating the diffusion MR signal to microscopic perturbations in susceptibility [63, 64].

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.