High-gradient diffusion MRI reveals distinct estimates of axon diameter index within different white matter tracts in the in vivo human brain

Abstract

Axon diameter and density are important microstructural metrics that offer valuable insight into the structural organization of white matter throughout the human brain. We report the systematic acquisition and analysis of a comprehensive diffusion MRI data set acquired with 300 mT/m maximum gradient strength in a cohort of 20 healthy human subjects that yields distinct and consistent patterns of axon diameter index in white matter tracts of arbitrary orientation. We use a straightforward, previously validated approach to estimating indices of axon diameter and volume fraction that involves interpolating the diffusion signal perpendicular to the principal fiber orientation and fitting a three-compartment model of intra-axonal, extra-axonal and free water diffusion. The resultant maps confirm the presence of larger diameter indices in the body of corpus callosum compared to the genu and splenium, as previously reported, and show larger axon diameter index in the corticospinal tracts compared to adjacent white matter tracts such as the cingulum. An anterior-to-posterior gradient in axon diameter index is also observed, with smaller diameter indices in the frontal lobes and larger diameter indices in the parieto-occipital white matter. These observations are consistent with known trends from prior histologic studies in humans and non-human primates. Rather than serving as fully quantitative measures of axon diameter and density, our results may be considered as axon diameter- and volume fraction-weighted images that appear to be modulated by the underlying microstructure and may capture broad trends in axonal size and packing density, acknowledging that the precise origin of such modulation requires further investigation that will be facilitated by the availability of high gradient strengths for in vivo human imaging.

Appendices

Appendix 1

The overall signal model is taken to be the sum of the intra-axonal, extra-axonal, and CSF compartment signal models weighted by their respective volume fractions: f_r for the intra-axonal compartment, f_csf for the CSF compartment, and f_h = 1 − f_r − f_csf for the extra-axonal compartment:

$$S = f_{\text{r}} S_{\text{r}} + f_{\text{h}} S_{\text{h}} + f_{\text{csf}} S_{\text{csf}} .$$

(1)

The signal model for intra-axonal restricted diffusion is obtained from the Gaussian phase distribution approximation (Murday and Cotts 1968) of restricted diffusion in impermeable parallel cylinders of diameter a (van Gelderen et al. 1994; Wang et al. 1995), which accounts for diffusion during the gradient pulse when the pulse duration is on the order of the diffusion time Δ:

$$S_{\text{r}} = S_{0} { \exp }\left\{ { - 2\gamma^{2} G^{2} \mathop \sum \limits_{m = 1}^{\infty } \frac{{\left[ {2D_{\text{r}} \alpha_{\text{m}}^{2} \delta - 2 + 2{\text{e}}^{{ - D_{\text{r}} \alpha_{\text{m}}^{2} \delta }} + 2{\text{e}}^{{ - D_{r} \alpha_{\text{m}}^{2} \Delta }} - {\text{e}}^{{ - D_{\text{r}} \alpha_{\text{m}}^{2} \left( {\Delta - \delta } \right)}} - {\text{e}}^{{ - D_{\text{r}} \alpha_{\text{m}}^{2} \left( {\Delta + \delta } \right)}} } \right]}}{{D_{\text{r}}^{2} \alpha_{\text{m}}^{6} \left[ {\left( {a/2} \right)^{2} \alpha_{\text{m}}^{2} - 1} \right]}}} \right\},$$

(2)

where S₀ is the signal obtained at b = 0 without diffusion weighting, γ is the gyromagnetic ratio, G is the gradient strength of the diffusion-encoding gradients, D_r is the diffusion coefficient of water in the restricted compartment, and m are the roots of the equation:

$$J^{\prime}_{1} \left( {\alpha_{\text{m}} \times a/2} \right) = 0,$$

(3)

where $J_{1}^{'}$ is the derivative of the Bessel function of the first kind, order one. The summation in Eq. (1) was taken up to m = 10, with the contribution of terms m > 10 considered negligible. Instead of imposing a gamma distribution of axon diameters (Assaf et al. 2008), we only fitted for a single axon diameter as in Alexander et al. (2010).

The extra-axonal hindered diffusion is modeled by the one-dimensional Stejskal–Tanner equation parameterized by the hindered diffusion coefficient D_h (Stejskal and Tanner 1965):

$$S_{\text{h}} = S_{0} \exp \left[ { - \left( {\gamma G\delta } \right)^{2} \left( {\Delta - \delta /3} \right)D_{\text{h}} } \right]. .$$

(4)

Free diffusion in cerebrospinal fluid (CSF) is modeled as isotropic Gaussian diffusion occurring with diffusion coefficient D_csf (Barazany et al. 2009):

$$S_{\text{csf}} = S_{0} \exp \left[ { - \left( {\gamma G\delta } \right)^{2} \left( {\Delta - \delta /3} \right)D_{\text{csf}} } \right]. .$$

(5)

Appendix 2

To support the sensitivity of our measurements to intra-axonal water diffusion using strong diffusion weighting, we performed the following analysis in two white-matter ROIs with different effective axon diameters. We take advantage of the fact that at high b values, the dominant contribution to the diffusion MRI signal is from the intra-axonal compartment. Here, we plotted the measured average signal perpendicular to the principal fiber orientation for ROIs with distinct axon diameter estimates selected from the superior longitudinal fasciculus (SLF) and uncinate fasciculus in a single representative subject from our cohort. We corroborated the sensitivity of our measurements to intra-axonal water diffusion using diameter distributions obtained from electron microscopy. Toward this end, we digitized the diameter distributions of axons in the SLF and uncinate fasciculus presented in Liewald et al. (2014). To account for tissue shrinkage following fixation, dehydration, and embedding, we multiplied the diameter distributions by a factor of 1.5, as suggested by prior histological (Aboitiz et al. 1992) and MRI studies (Alexander et al. 2010; Veraart et al. 2019). To approximate the effective axonal radius that MRI is sensitive to on the voxel level, we calculated the volume-corrected effective axonal radius, which is dominated by the largest axons within the voxel (Burcaw et al. 2015):

$$r_{\text{eff}} = \sqrt[4]{{\frac{{r^{6} }}{{r^{2} }}}}.$$

We plugged the effective axonal radius estimated from the EM data into the signal model. Figure 6c, f shows the predicted signal decays for restricted diffusion within axons of estimated effective diameter derived from EM in the SLF (a_EM = 3.7 μm) and uncinate fasciculus (a_EM = 2.4 μm), which show greater signal decay for the larger diameter axons in the SLF compared to the uncinate fasciculus. The signal decay in the restricted compartment is also reflected at high q values in the predicted total diffusion-weighted signal (Fig. 6a, d), which is the sum of the restricted signal predicted from EM (Fig. 6c, f) and the estimated hindered diffusion signal obtained from fitting the hindered signal model to the experimental data (Fig. 6b, e). The predicted total diffusion-weighted signal is plotted alongside the measured diffusion MRI signal in regions of interest sampled in the SLF and uncinate fasciculus in a representative subject. Figure 6a, d shows reasonable agreement between the measured and predicted overall diffusion MRI signal S(q,Δ) for both the SLF and uncinate fasciculus ROIs using the effective axonal diameter derived from the EM data, suggesting that our measurements may be sensitive to intra-axonal water diffusion, returning effective axonal radius estimates that are in the ballpark of those expected from electron microscopy.

Fig. 6

Simulated and experimental diffusion signal decays plotted for a–c a region-of-interest (ROI) in the superior longitudinal fasciculus [volume-weighted effective axon diameter of 3.7 μm calculated from histograms derived from electron micrographs in Liewald et al. (2014)and restricted volume fraction of 0.21 calculated from fitting Eq. (1) to the experimental data] compared to d–f an ROI in the uncinate fasciculus [volume-weighted effective axon diameter of 2.4 μm calculated from histograms derived from electron micrographs in Liewald et al. and restricted volume fraction of 0.30 calculated from fitting Eq. (1) to the experimental data]. In a, d, the predicted total diffusion-weighted signal S(q,Δ) is a weighted sum of the signal due to b, e hindered and c, f restricted water. The restricted diffusion signal is calculated from the effective axon diameter derived from electron microscopy. At high q values, the contribution to the tail of S(q,Δ) is dominated by restricted diffusion presumed to arise from the intra-axonal space

We acknowledge the limitations of this approach, not the least of which is the limited empirical evidence that is available from the literature. There are very few studies reporting axon diameter estimates in humans from electron microscopy (Aboitiz et al. 1992; Liewald et al. 2014), and the studies that have been published have only been performed on small numbers of subjects. We note that the approach described above is very similar to the analysis performed by Veraart et al. (2018) using EM data in the corpus callosum from Aboitiz et al. The results reported in Veraart et al. were of a similar order of magnitude to those reported here, which is encouraging as these studies were performed independently with different acquisitions yet converge upon similar estimates and trends in effective axon diameter throughout the brain, suggesting some measure of sensitivity to the underlying microstructure within and across individuals. We fully admit that there may be alternative explanations for the signal behavior, given the complexity of the underlying microstructure, which should be revealed through further investigation.