Including diffusion time dependence in the extra-axonal space improves in vivo estimates of axonal diameter and density in human white matter

Abstract

Axonal density and diameter are two fundamental properties of brain white matter. Recently, advanced diffusion MRI techniques have made these two parameters accessible in vivo. However, the techniques available to estimate such parameters are still under development. For example, current methods to map axonal diameters capture relative trends over different structures, but consistently over-estimate absolute diameters. Axonal density estimates are more accessible experimentally, but different modeling approaches exist and the impact of the experimental parameters has not been thoroughly quantified, potentially leading to incompatibility of results obtained in different studies using different techniques. Here, we characterise the impact of diffusion time on axonal density and diameter estimates using Monte Carlo simulations and STEAM diffusion MRI at 7 T on 9 healthy volunteers. We show that axonal density and diameter estimates strongly depend on diffusion time, with diameters almost invariably overestimated and density both over and underestimated for some commonly used models. Crucially, we also demonstrate that these biases are reduced when the model accounts for diffusion time dependency in the extra-axonal space. For axonal density estimates, both upward and downward bias in different situations are removed by modeling extra-axonal time-dependence, showing increased accuracy in these estimates. For axonal diameter estimates, we report increased accuracy in ground truth simulations and axonal diameter estimates decreased away from high values given by earlier models and towards known values in the human corpus callosum when modeling extra-axonal time-dependence. Axonal diameter feasibility under both advanced and clinical settings is discussed in the light of the proposed advances.

Keywords: Axonal density; Axonal diameters; CHARMED; Diffusion time; STEAM diffusion MRI; White matter microstructure.

Fig. 1

Extra-axonal diffusion coefficient orthogonal to the main fiber orientation as a function of increasing diffusion time for Monte Carlo simulations (a), isotropic phantom (b) and in-vivo data averaged over all the subjects. For the simulations, both ordered (blue) and short-range disordered (red) substrates are used.

Fig. 2

Simulations generated using the signal decay predicted by Novikov et al. (2014), i.e. including the extra-axonal axial diffusion dependency on the diffusion time. The axonal density is plotted as a function of the diffusion time with (red) and without (blue) fixing the orthogonal diffusivity for the extra-axonal compartment. The dotted line is ground truth. Data are reported for different combinations of axonal density and diameter: 0.3/0.5 μm (a), 0.3/3 μm (b), 0.5/0.5 μm (c) and 0.5/3 μm (d).

Fig. 3

Monte Carlo simulations obtained for different microscopic configurations. True axonal diameter versus estimated axonal diameters including (blue) and not including (red) the delta dependency for D₀ = 0.7 ∗ 10^− 3 mm²/s (a–c), D₀ = 1.5 ∗ 10^− 3 mm²/s (d–f) and D₀ = 2.4 ∗ 10^− 3 mm²/s (g–i). For each value of free diffusivity, the results are reported for SNR = 25,40 and 55 respectively. The dotted line is the line of identity.

Fig. 4

Percentage of extra-axonal (light blue) and intra-axonal (orange) signal attenuation for different b-values and Δ = 48 ms, reported for the smaller (a) and the largest (b) axonal diameters and for D₀ = 0.7 ∗ 10^− 3 mm²/s. c) and d) show the same data for D₀ = 1.5 ∗ 10^− 3 mm²/s, while e) and f) show the same data for D₀ = 2.4 ∗ 10^− 3 mm²/s.

Fig. 5

Axonal density maps at varying diffusion times. In a), the results are obtained without using the tortuosity model, while in b) the tortuosity model is used. In c), the difference between the two is shown.

Fig. 6

Mean axonal density and standard error at varying diffusion times for the same subject of Fig. 5. The plots are reported separately for low axonal density (a) and high axonal density (b). The blue fit is obtained without using the tortuosity model, while the red fit is obtained using the tortuosity model. Dashed lines represent the best fit of axonal density estimates over all diffusion times according to the Bayesian information criterion (see text for details).

Fig. 7

Maps of axonal density, axonal diameter, characteristic coefficient A, intra-axonal diffusivity and bulk diffusivity for one representative subject (a). In panel b, the mean profiles along the corpus callosum and the associated standard deviations over all the subjects are reported for the same parameters.

Fig. 8

Maps of axonal diameter with (panel a) and without (panel b) accounting for the dependency on the diffusion time in the extra-axonal compartment.

Fig. 9

Profiles of the axonal density (a) and axonal diameter (b) along the corpus callosum. Data in dark blue are acquired with the high b-value/high Δ/high TE acquisition scheme, while data in light blue are acquired with the low b-value/low Δ/low TE acquisition scheme.

Fig. 10

Scatterplot between axonal diameter and axonal density. The dots are colored according to the location in the corpus callosum (schematically shown in the insert). The dotted line is the linear regression (r² = 0.68).