Oligodendrocytes extend their cell membranes to wrap around axons in WM, creating multiple concentric layers of myelin. Each turn of wrapping adds another bilayer of myelin with a thickness of approximately d _m = 4–5 nm. This process results in a multilayer spiral structure, with gaps of about d _w = 3 nm thick [73] between the layers, filled by myelin water. Figure 1A shows a schematic transverse section of a myelinated axon.

In this study, we approximate the diffusion process along this spiral trajectory as diffusion within a series of impermeable concentric solid cylinders separated by infinitesimal water-filled gaps (see Figure 1B). The rationale for this approximation is as follows: For a given diffusion time, a diffusing water molecule traveling a total displacement of 2πaN (where a is the myelin radius at the starting position and N is an arbitrary number) along the spiral trajectory experiences a net radial displacement of about N (d _w + d _m ) (see cross-sectional plane shown in Figure 1A. This displacement remains negligible, even for molecules traveling long distances. For example, for a = 0.5 µm and N = 10, the path length along the spiral is 31.4 µm, and the net radial displacement is approximately 0.08 µm, hence significantly smaller than the minimum displacement required to attenuate the dMRI signal in state-of-the-art scanners [39, 74]. Moreover, since spin echo dMRI sequences designed to be sensitive to myelin water employ short TEs (equivalently short diffusion times), most molecules will travel relatively short distances along the spiral trajectory, minimizing the net radial displacement.

For this reason, we propose to simplify the spiral trajectory by using concentric cylinders of similar size. As infinitesimal distances separate the cylinders, we assume that the underlying diffusion process is equivalent to random walks confined to the cylinder surfaces. Therefore, we will first derive the diffusion propagator for Brownian motion on the cylinder surface, see Figure 1C, and then extend this model to multiple cylinders. Moreover, to eliminate fiber orientation and dispersion effects (confounding factors), we will derive the spherical mean dMRI signal for this model. This approach will help us to interpret the mean radius estimated by fitting a single-cylinder-surface model to the dMRI signal arising from multiple cylindrical surfaces.

To simplify our model, we will consider an infinitely long cylinder whose main axis is oriented along the z-axis, with its transverse section lying in the x-y plane. An important aspect of this model is that the dMRI signal can be decomposed into contributions from spin particles diffusing parallel and perpendicular to the cylinder’s main axis. In this coordinate frame of reference, these diffusion processes are statistically independent. Therefore, the displacement probability distribution P r , t = P r x y , t P r z , t can be expressed as the product of the distributions for motion in the perpendicular P r x y , t and parallel P r z , t directions. The net displacement vector r = r x y + r z at diffusion time t can be decomposed into the displacement vectors perpendicularly and parallel to the cylinder’s axis. Note that r x y = r x i ^ + r y j ^ and r z = r z k ^ , where r x , r y and r z are the vector’s lengths along the unit vectors i ^ , j ^ , k ^ associated with the x-, y-, and z-axes, respectively.

For this type of decoupled diffusive motion [75], showed that the dMRI signal can be expressed as the product of the dMRI signals arising from displacement parallel and perpendicular to the cylinder’s axis: E q , t = E ⊥ q x y , t E ∥ q z , t , where q x y = q x i ^ + q y j ^ and q z = q z k ^ , q = q x y + q z = γ g δ , γ is the gyromagnetic ratio of the diffusing spin particles (e.g., hydrogen nuclei), g = G g ^ denotes the applied diffusion gradient with magnitude G and unit orientation vector g ^ , and δ is the duration of the diffusion gradient pulses. Note that t should be expressed in terms of the dMRI sequence time parameters. A general detailed derivation of this decoupled signal model is provided in [75].

In this section, we will derive the analytical expressions for E ∥ q z , t and E ⊥ q x y , t necessary to provide the full dMRI signal model. This derivation follows the diffusion propagator formalism under the narrow-pulse (narrow-delta) approximation, which assumes that the duration of the diffusion gradient is very short ( δ → 0 ). Thus, under this formalism and for a PGSE sequence [76], the diffusion time is equal to the time difference between the onset of the two diffusion gradients t = Δ .

The dMRI signal E ∥ q z , t arising from displacements parallel to the cylinder’s main axis r z is related to the 1D displacement probability distribution by the following Fourier-relationship: E ∥ q z , t E ∥ q z = 0 , t = ∫ ‐ ∞ ∞ ∫ ‐ ∞ ∞ P z , z ′ , t P z ′ e i q z z ‐ z ′ d z ′ dz , (1) where P z ′ is the probability for a particle to be at position z ′ = z ′ k ^ at initial time t = 0 , and P z , z ′ , t is the probability that a particle initially located at position z ′ migrate to position z = z k ^ in time t . Assuming that at t = 0 all particles are uniformly distributed along the cylinder’s axis (i.e., P z ′ is constant) and using the change of variables r z = z − z ′ to quantify displacements, Equation 1 can be rewritten as E ∥ q z , t E ∥ q z = 0 , t = ∫ − ∞ ∞ P r z , t e i q z r z d r z , (2)

Since the motion of particles along the cylinder’s main axis is unrestricted (assuming an infinitely long cylinder), we assume 1D Gaussian diffusion with a characteristic myelin water diffusivity D ∥ on the cylinder’s surface: P r z , t = 1 4 π D ∥ t e ‐ r z 2 4 D ∥ t (3)

By inserting Equation 3 into Equation 2, we obtain the familiar dMRI signal expression for Gaussian diffusion, E ∥ q z , t = E ∥ q z = 0 , t e − q z 2 D ∥ t . (4)

Likewise, the dMRI signal arising from displacements perpendicular to the cylinder’s axis E ⊥ q x y , t depends on the 2D displacement probability distribution by the following Fourier-relationship: E ⊥ q x y , t E ⊥ q x y = 0 , t = ∫ R 2 ∫ R 2 P r x y , r x y ′ , t P r x y ′ e i q x y r x y − r x y ′ d r x y ′ d r x y , (5) where P r x y ′ and P r x y , r x y ′ , t are the probability of finding a particle at position r x y ′ in the x-y plane at t = 0 , and the probability of moving from r x y ′ to r x y in time t .

As the particle displacements in the plane perpendicular to the cylinder’s axis are confined on a circle, it is convenient to rewrite the integrals in Equation 5 in polar coordinates due to the polar symmetry of this system, E ⊥ q x y , t E ⊥ q x y = 0 , t = ∫ 0 ∞ ∫ 0 ∞ ∫ 0 2 π ∫ 0 2 π P ρ , θ , ρ ′ , θ ′ , t P ρ ′ , θ ′ × e i q x y ρ ⁡ cos φ + θ e − i q x y ρ ′ ⁡ cos φ + θ ′ ρ ρ ′ d ρ ′ d ρ d θ d θ ′ , (6) where the cartesian components of the 2D vectors, r x y , r x y ′ , q x y , are rewritten in terms of their magnitudes, ρ , ρ ′ , q x y , and angles of orientation, θ , θ ′ , φ , respectively: r x y = ρ ⁡ cos θ , ρ ⁡ sin θ , r x y ′ = ρ ′ ⁡ cos θ ′ , ρ ′ ⁡ sin θ ′ , and q x y = q x y ⁡ cos φ , q x y ⁡ sin φ .

In Supplementary Appendix A, we show that Equation 6 can be simplified to E ⊥ q x y , t E ⊥ q x y = 0 , t = 1 2 π ∫ 0 2 π ∫ 0 2 π P Φ , a , t e i q x y a ⁡ cos ψ e − i q x y a ⁡ cos ψ − Φ d ψ d Φ , (7) where we used the change of variables ψ = φ + θ and Φ = θ − θ ′ , and P Φ , a , t is the probability that the particles’ motion on the circle with radius a covers a polar angle Φ at a time t , see Figure 1C.

We model P Φ , a , t as a wrapped Gaussian distribution [77] with diffusivity D : P Φ , a , t = 1 4 π D t ∑ p = − ∞ ∞ e − a 2 4 D t Φ + 2 π p 2 = 1 2 π 1 + 2 ∑ p = 1 ∞ e − p 2 D t a 2 ⁡ cos p Φ . (8)

This distribution results from wrapping the 1D Gaussian distribution (on the infinite line) around the circle’s circumference. It takes into account that during the diffusion process, a population of particles could travel distances larger than 2 π a p , where 2 π a is the perimeter of the circle, and p = 1 , 2 , … , ∞ . The second expression in Equation 8 provides a helpful alternative representation of this function [77–79]. It is the solution of the diffusion equation of Brownian particles confined in a circle S 1 [80–82]. However, note that in [81], the function was normalized with the circle’s perimeter, whereas our distribution is normalized with the angle, i.e., ∫ 0 2 π P Φ , a , t d Φ = 1 .

Assuming that the translational diffusion parallel to the cylinder’s main axis and along the “unwrapped” circle are equal, then D = D ∥ . After substituting Equation 8 into Equation 7 we obtain E ⊥ q xy , t E ⊥ q xy = 0 , t = J 0 2 a q xy + 2 ∑ p = 1 ∞ J p 2 a q xy e ‐ p 2 D ∥ t a 2 , (9) where J p is the p-th Bessel function of the first kind. The complete derivation is shown in Supplementary Appendix A. This expression does not depend on the orientation φ of vector q x y in the plane perpendicular to the cylinder’s axis due to transverse symmetry, as expected. In the limit D ∥ t ≫ a 2 , Equation 8 becomes a uniform distribution and Equation 9 tends to E ⊥ q x y , t ≫ a 2 / D ∥ E ⊥ q x y = 0 , t ≫ a 2 / D ∥ ≈ J 0 2 a q x y . (10) Note that Equation 10 does not depend on t .

An independent derivation of Equation 9 was reported in [64, 83]. However, the result reported by [64] was obtained by assuming a Gaussian distribution for the angular motion instead of a Wrapped Gaussian, which solution only tends to Equation 9 in the limit case when a 2 ≫ D ∥ t .

By merging results from Equations 4, 9, we obtain the final signal model for a single cylinder: E q , t = E ∥ q z , t E ⊥ q x y , t = E q = 0 , t e − q 2 ⁡ cos β 2 D ∥ t J 0 2 a q ⁡ sin β + 2 ∑ p = 1 ∞ J p 2 a q ⁡ sin β e − p 2 D ∥ t a 2 , (11) where β is the angle between the diffusion gradient orientation and the cylinder’s axis, q x y = q ⁡ sin β and q z = q ⁡ cos β . For practical purposes, the signal can be adequately approximated by the first p = 1 , … , P terms in the series.

When the displacement probability distribution in the x-y plane (perpendicular to the cylinder’s axis) is approximated by an isotropic bivariate Gaussian distribution, the mean-squared displacement of particles r x y 2 is related to the ‘apparent’ radial diffusivity in the 2D plane according to D ⊥ a p p = r x y 2 / 4 t . For such an isotropic Gaussian diffusion process, the corresponding dMRI signal E ⊥ q x y , t is given by E ⊥ q x y , t = E ⊥ q x y = 0 , t e − q x y 2 D ⊥ a p p t (12)

The expression for D ⊥ a p p in Equation 12 depends on the diffusion time and circle radius a as D ⊥ a p p = a 2 2 t 1 − e − D ∥ t a 2 , (13) where we assumed D = D ∥ , like in Equation 9. The full derivation is presented in Supplementary Appendix B. For very short diffusion times, t → 0 , the apparent radial diffusivity does not depend on the circle’s radius, D ⊥ a p p = D ∥ / 2 , because no structural features are probed at such a small time-scale. Conversely, for very long diffusion times, D ⊥ a p p → a 2 / 2 t .

The final dMRI signal, considering both the parallel and radial diffusion components, is given by E q , t = E q = 0 , t e − q 2 ⁡ cos β 2 D ∥ t e − q 2 ⁡ sin β 2 D ⊥ a p p t (14)

This analytical form is equivalent to an axially symmetric diffusion tensor signal, as described in Equation 5 in [84]. However, note that the radial diffusivity depends on the diffusion time and the size of the confining geometry, i.e., the cylinder radius.

Our previous derivations are based on the q-space formalism (see Equations 1, 5). This approach is valid for PGSE sequences [76] using diffusion-encoding gradients with infinitesimal duration δ . Consequently, the proposed signal models are not valid for sequences that do not fulfill this requirement. In this section, we will use the q-space correction approach presented by [85] to provide more general signal approximations beyond this acquisition protocol.

Under the narrow pulse approximation, the dephasing of the spins due to their motion during the application of the diffusion gradients is neglected. Thus, the diffusion time is equal to the time difference between the onset of the two diffusion gradients. However, for finite δ it is unclear what diffusion time derived from the PGSE sequence must be used in the diffusion propagator to evaluate the dMRI model. This problem was tackled by [85], who proposed a general relationship between the signal attenuation e i ϕ Δ , δ , g for the PGSE sequence and the displacement probability e i ϕ Δ , δ , g e i ϕ Δ , δ , g = 0 = ∫ R 3 P r , t exp e i ϕ r Δ , δ , g d r , (15) where the integral in Equation 15 is over the infinite three-dimensional space, t exp is the total diffusion time of the experiment between the onset of the first gradient and the termination of the second gradient, and e i ϕ r Δ , δ , g denotes the average signal attenuation (dephasing) of the population of spins experiencing a net displacement r in time t exp . Note that t exp = Δ + δ for PGSE sequences with rectangular diffusion gradients and t exp = Δ + δ + ξ for trapezoidal diffusion gradients, where ξ is the rise time of the trapezoidal ramp [86].

In Supplementary Appendix C, we provide a compact re-derivation of Lori’s approach, which found the following approximation: e i ϕ Δ , δ , g e i ϕ Δ , δ , g = 0 ≈ ∫ R 3 P r , t exp e i q ′ r d r , (16) where q ′ = q t e f f / t exp is a scaled q-space vector; t e f f denotes the ‘effective’ diffusion time that appears in the b-value definition, i.e., b = q 2 t e f f , which is equal to t e f f = Δ − δ / 3 and t e f f = Δ − δ / 3 + ξ 3 / 30 δ 2 − ξ 2 / 6 δ for rectangular and trapezoidal diffusion gradients, respectively [86]. According to this result, the q-space formalism can still be employed to relate the diffusion propagator and the dMRI signal attenuation produced by a PGSE sequence with finite δ . However, it must be corrected by evaluating the diffusion propagator at the total diffusion encoding time t exp and using a modified q-space vector q ′ . Note that for narrow pulses, the correction converges to the classical q-space formalism with t exp = t e f f = Δ , and e i ϕ r Δ , δ , g = e i qr , as expected.

The theoretical result in Equation 16 was confirmed in [85] by numerical simulations for homogeneous Gaussian diffusion, heterogeneous diffusion in permeable microscopic Gaussian domains, and diffusion inside restricted spherical reflecting domains. In all the analyses, this correction produced better results than using the original q-vector and the relationship t exp = Δ − δ / 3 , for rectangular diffusion gradients. It is important to notice that this approach may only provide a precise correction for displacement distributions that do not deviate significantly from a Gaussian distribution.

In this study, we will use this correction to evaluate our signal models in Equations 11, 14.

The previous signal models, see Equations 11, 14, are based on the assumption of a single cylindrical surface. In the case of a distribution of cylinders with equal radius but multiple orientations, the orientation effect can be removed from Equation 11 by computing the orientation-averaged spherical mean signal E . Following the approach of [33, 87–89], we obtain, E q , Δ , δ , ξ , a E q = 0 = 1 2 [ ∑ k = 0 ∞ ∑ j = 0 k − 1 k k ! 2 2 k k a q 2 t e f f t exp 2 k × k j − 1 j Γ j + 1 2 − Γ j + 1 2 , q 2 D ∥ t e f f q 2 D ∥ t e f f j + 1 / 2 + ∑ p = 1 ∞ ∑ k = 0 ∞ ∑ j = 0 p + k e − p 2 D ∥ t exp a 2 − 1 k k ! 2 p + k ! 2 p + k p + k a q 2 t e f f t exp 2 p + k × p + k j − 1 j Γ j + 1 2 − Γ j + 1 2 , q 2 D ∥ t e f f q 2 D ∥ t e f f j + 1 / 2 ] , (17) where t e f f and t exp depend on the experimental parameters Δ , δ , ξ .

A detailed derivation of this expression is presented in Supplementary Appendix D, which also includes Lori’s q-space correction described in the previous section.

On the other hand, for the Gaussian diffusion model in Equation 14 with time-dependent radial diffusivity, the spherical mean signal is equivalent to that from an axis-symmetric diffusion tensor [33, 37, 43, 84, 87, 90]: E q , Δ , δ , ξ , a E q = 0 = π 4 e − b D ⊥ a p p erf b D ∥ − D ⊥ a p p b D ∥ − D ⊥ a p p , (18) where erf denotes the error function. In our model, the radial diffusivity D ⊥ a p p a , t = Δ + δ depends on the cylinder radius a and the total diffusion time according to the model defined in Equation 13 and incorporating Lori’s correction. Note that this correction does not affect the b-value since b = q 2 t e f f = q ′ 2 t exp for rectangular and trapezoidal diffusion gradients.

In this section, we will derive the spherical mean dMRI signal for a distribution of cylinders with different radii. Specifically, we will consider two cases:

1. Multiple concentric cylinders: This model represents the diffusion process of myelin water within a single axon. Each cylinder corresponds to a layer of the myelin sheath. The diffusion process is confined to these cylindrical surfaces, and the overall dMRI signal is the sum of contributions from each cylindrical layer; see Figure 1B.

We aim to define and estimate the ‘effective’ myelin sheath radius by approximating the signal from multiple cylindrical surfaces with the signal from a single cylindrical surface. The effective myelin sheath radius simplifies the complex distribution into a single representative value. This approach is analogous to axon diameter mapping techniques, which estimate an effective radius from an underlying distribution of inner axon radii [10, 11, 36–41, 45].

Monte Carlo Diffusion Simulations (MCDS) were employed as a benchmark to validate the proposed models. We used an MC simulator developed by our group, available at https://github.com/jonhrafe/Robust-Monte-Carlo-Simulations [92]. This simulator has been validated against analytical models across multiple geometries, including impermeable planes, cylinders, and spheres [92]. For this study, we extended its capabilities to incorporate new myelin water diffusion models, implementing two geometrical structures: 3D infinite, impermeable cylinders and spiral surfaces.

The analytical models were validated by comparing their predicted dMRI signals to those generated by the MC simulations for identical impermeable cylindrical surfaces. Additionally, the dMRI signals from concentric cylinders were compared with those from spiral surfaces to assess the assumption presented in Section 2.1 (Figure 1). This assumption suggests that net radial displacements along the spiral trajectory are negligible, which allows the diffusion process in the more complex spiral geometry to be approximated as that in concentric cylinders.

The diffusion process was simulated for both geometrical models using a total diffusion time of t = 20 ms and N t = 15,000 steps per particle. We conducted a bootstrap-based analysis to ensure convergence of the simulations, as outlined in [92]. A total of 75,000 particles were uniformly distributed on each cylindrical or spiral surface. Three values of parallel diffusivity (D_∥ = 0.3, 0.5, 0.8 μm²/ms) were used to cover the range of myelin water diffusivities reported by [65].

A PGSE sequence with trapezoidal diffusion gradients was used to generate dMRI signals. The sequence was based on the specifications of a Connectome 2.0 scanner, employing a maximum gradient strength of G = 500 mT/m and a maximum slew rate of SR = 600 T/m/s [68], yielding to a trapezoidal ramp rise time ξ = G/SR = 0.833 ms. The protocol included 90° and 180° pulse durations of 2 ms and 4 ms, respectively. Six b-values were selected using the shortest possible TE for each case while maintaining maximum G and SR, following the implementation described in [70, 71]. Table 1 details the experimental parameters.

For each b-value, dMRI signals were generated for 92 gradient orientations uniformly distributed on the unit sphere, along with the signal for b = 0. The subsequent analyses focused on the spherical mean signal normalized by the b = 0 signal.

Figure 3 illustrates the theoretical spherical mean dMRI signal from a cylindrical surface, as generated by the general model presented in Equation 17 using a PGSE sequence with trapezoidal diffusion gradients. The signal is shown for b-values ranging from 0 to 100 ms/μm² and for three cylinders with radii of 0.3 µm, 1.0 µm, and 3.0 µm.

For relatively low b-values (approximately below 3 ms/μm²), the logarithm of the signal approximates a linear relationship. This linearity suggests that a Gaussian model could be valid in this regime. However, as the b-value increases, deviations from Gaussianity become apparent, and signal oscillations, known as diffraction patterns, emerge. These diffraction-like patterns have been reported in other geometries where diffusion is confined, such as planar, cylindrical, and spherical domains [95–97].

To assess the accuracy of the new analytical models proposed in this study, we compared the predicted dMRI signals with those generated by MC simulations. Figure 4 shows the theoretical spherical mean dMRI signals from cylindrical surfaces as a function of the radius, as predicted by both the general analytical model and the Gaussian approximation with time-dependent radial diffusivity (Equations 17, 18, respectively) using a PGSE sequence with trapezoidal diffusion gradients. Additionally, the figure includes the dMRI signals obtained from the MC simulations for validation purposes. This comparison was conducted over a range of parallel diffusivities (D_∥ = 0.3, 0.5, 0.8 μm²/ms) and practical b-values from 0.8 to 3.0 ms/μm², achievable in preclinical and human scanners equipped with strong diffusion gradients.

Increasing the b-value results in greater attenuation of the dMRI signal as a function of the radius across all three diffusivity values. At a b-value of 3.0 ms/μm², the signal exhibits maximum sensitivity to myelin sheath radii in the 0.5–3.0 µm range. However, at this higher b-value, we observe the largest, albeit still minor, deviations between the signals predicted by the analytical models and those generated by the MC simulations. Notably, the agreement between the models and simulations is strongest for the lowest diffusivity (D_∥ = 0.3 μm²/ms, panel A). It diminishes as diffusivity increases, with the largest discrepancy observed at D_∥ = 0.8 μm²/ms (panel C).

For this acquisition protocol, the signal shows minimal sensitivity to myelin radii smaller than 0.5 µm and larger than 3.5–4.0 µm. This result indicates that the method is best suited for detecting myelin sheath sizes in the 0.5–3.5 µm range. Across all b-values, the Gaussian approximation closely follows the analytical model, particularly for radii below 4.0 µm, further confirming the accuracy of the approximation in this parameter range.

The results from the experiment comparing the spherical mean dMRI signals generated by MC simulations for spiral geometries and multiple concentric cylinders are presented in Figure 5. Specifically, Figure 5 shows the dMRI signals as a function of the six b-values employed. The signal from a spiral geometry with inner and outer radii of 0.7 µm and 1.0 µm is compared with the radius-weighted signal from multiple concentric cylinders within the same radius range, calculated using Equation 19. Additionally, we display the signals from individual cylindrical surfaces with radii ranging between 0.7 µm and 1.0 µm, obtained from both MC simulations and the analytical models. Panels A and B correspond to results for diffusivities of D_∥ = 0.3 μm²/ms and D_∥ = 0.8 μm²/ms, respectively.

For both diffusivity values, we observe a strong agreement between the MC-generated signals for the spiral geometry and the radius-weighted aggregation of signals from concentric cylinders with the same range of radii. This result suggests that the spiral geometry can be accurately approximated by multiple concentric cylinders. Notably, for the lower diffusivity (D_∥ = 0.3 μm²/ms, panel A), the analytical model’s predictions for individual cylinders closely match the signals generated by MC simulations. Furthermore, the signal produced by the spiral geometry is very similar to that of a single cylinder with a radius intermediate to the inner and outer radii. This implies that when fitting these signals with a single-radius model, the estimated effective radius would likely correspond to a value close to the average radius of the spiral.

However, for simulations at the higher diffusivity (D_∥ = 0.8 μm²/ms, panel B), the signal decay predicted by the analytical models as a function of the b-value is more pronounced than the decay observed in the MC simulations. This result indicates potential inaccuracies in the analytical model at higher diffusivities and larger b-values. Consequently, the effective radius predicted by the analytical models will likely be biased towards a smaller value than the actual radius.

The results for spirals with other inner and outer radii were consistent with these findings. Specifically, the observed discrepancy for D_∥ = 0.8 μm²/ms was reduced for the spiral with a larger inner radius of 1.0 µm. Conversely, the disagreement increased for the smaller spiral with an inner radius of 0.5 µm (results not shown).

Figure 6 compares the effective radii estimated from simulated dMRI data against three different metrics derived from the distribution of myelin sheath radii in four regions of interest within the Corpus Callosum: axons connecting the prefrontal, motor, parietal, and visual cortices. The inner axon radii for these regions, as reported by [91], were modeled using Gamma distributions. These distributions were subsequently transformed into myelin sheath radii distributions using Equation 25 and a constant g-ratio of 0.7.

We then generated the spherical mean dMRI signals corresponding to these distributions by discretizing Equation 26 and employing the MC simulated signals. We assumed a parallel diffusivity of D_∥ = 0.5 μm²/ms. The generated signals were fitted to the general single-cylinder model in Equation 17 to estimate the effective radius. Figure 6 presents the effective radii a e f f , the mean radii a obtained from the distributions, and the second- and third-moment-based radii metrics a 2 / a and a 3 / a 1 / 2 ​, as defined in Equations 27, 30.

The myelin sheath radii distributions in Figure 6 exhibit slightly longer right-hand tails and lower frequency values for small radii compared to the inner axon radii distributions, as expected. This difference arises because the myelin sheath radii represent all possible layer radii within the range defined by the inner and outer radii for all axons. Hence, it includes contributions from myelin layers near the inner and outer boundaries. These two distributions converge further as the g-ratio increases, as described by Equation 25. This trend is noticeable when comparing the distributions in Figure 2 for a g-ratio of 0.6 with those in Figure 6 employing a g-ratio of 0.7.

The results show that for the distributions with smaller radii (Prefrontal and Parietal regions), the estimated effective radius a e f f closely matches the mean radius a . However, for the Motor and Visual regions, with larger radii distributions, the effective radius aligns more closely with the second-moment-based metric a 2 / a , followed by the third-moment-based metric a 3 / a 1 / 2 . These findings suggest that the appropriate descriptor of the distribution may depend on the range of radii in each region.

To further investigate the relationships between the effective radius and the derived metrics from the myelin sheath radii distributions, we present a correlation analysis in Figure 7. This figure illustrates the correlations between the effective radius and the three descriptive metrics across experiments conducted with three distinct diffusivities.

As shown in Figure 7, although these metrics reflect different aspects of the myelin sheath radii distributions, they exhibit significant correlations with the effective radius. Notably, the second-moment-based radius a 2 / a demonstrated the strongest linear correlation (and smallest p-value) with a e f f ​ across all diffusivity values, indicating its potential as a reliable descriptor of effective radii. The third-moment-based radius a 3 / a 1 / 2 closely followed this trend, while the average radius showed less strong correlations. Interestingly, the analysis reveals a trend where the estimated effective radius tends to decrease with increasing diffusivity, particularly pronounced in distributions characterized by smaller axon radii.