Time‐dependent diffusion in undulating thin fibers: Impact on axon diameter estimation

Abstract

Diffusion MRI may enable non-invasive mapping of axonal microstructure. Most approaches infer axon diameters from effects of time-dependent diffusion on the diffusion-weighted MR signal by modeling axons as straight cylinders. Axons do not, however, propagate in straight trajectories, and so far the impact of the axonal trajectory on diameter estimation has been insufficiently investigated. Here, we employ a toy model of axons, which we refer to as the undulating thin fiber model, to analyze the impact of undulating trajectories on the time dependence of diffusion. We study time-dependent diffusion in the frequency domain and characterize the diffusion spectrum by its height, width, and low-frequency behavior (power law exponent).Results show that microscopic orientation dispersion of the thin fibers is the main parameter that determines the characteristics of the diffusion spectra. At lower frequencies (longer diffusion times), straight cylinders and undulating thin fibers can have virtually identical spectra. If the straight-cylinder assumption is used to interpret data from undulating thin axons, the diameter is overestimated by an amount proportional to the undulation amplitude and microscopic orientation dispersion of the fibers. At higher frequencies (shorter diffusion times), spectra from cylinders and undulating thin fibers differ. The low-frequency behavior of the spectra from the undulating thin fibers may also differ from that of cylinders, because the power law exponent of undulating fibers can reach values below 2 for experimentally relevant frequency ranges. In conclusion, we argue that the non-straight nature of axonal trajectories should not be overlooked when analyzing and interpreting diffusion MRI data. K E Y W O R D Saxon diameter, axonal trajectories, diffusion MRI, diffusion spectrum, low frequency, restricted diffusion, time dependence, undulation Abbreviations: dMRI, diffusion MRI; MSE, mean squared error; SNR, signal-to-noise ratio; μOD, microscopic orientation dispersion.

Estimating axial diffusivity in the NODDI model

“…Intuitively, the negative correlation could be due to fibre undulations [30, 31, 32, 33, 34] which both act as a form of microscopic dispersion - increasing the amount of dispersion per voxel - and hinder the diffusion of intra-axonal water along the primary fibre axis, decreasing the apparent axial diffusivity. To investigate the impact of fibre undulations, we simulated data for a simple model with particles displacing along a fibre of zero radius and undulations of varying wavelength and amplitude, to find a negative relationship between the estimated orientation dispersion and axial diffusivity as expected (Figure 8).…”

“…Interestingly, in this simplistic model, undulations with an amplitude of only ~ 2 μm could lower the axial diffusivity from 3 to ~ 2.5 μm 2 /ms, as is seen in the data (Figure 7b). In current literature, the impact of undulations often focuses on the measurement of radial diffusion or axon diameter [32, 75, 33], though the effect of undulations on other diffusion characteristics, such as axial diffusion, deserve further investigation. Future work will benefit greatly from more realistic simulations [76, 34, 33] where, for example, the tissue structure is directly inspired by 3D reconstructed surfaces from microscopy images of real tissue [77, 34, 33].…”

“…One possibility is if, rather than fanned sticks, fibre undulations lead to apparent dispersion [30, 31, 32, 33, 34]. Though both fibre fanning and undulations are arguably a form of microscopic dispersion, fibre fanning is an effect of pooling over a number of axons and undulations are a feature of individual axons.…”

The variability of MR axon radii estimates in the human brain

“…Even after eliminating orientational dispersion, it is not understood why we observe such strong along-tract variability of the MR axon radii mapping in certain tracts (e.g., CST). Various additional confounding factors have been discussed, including, but not limited to, the curvature or undulations of the axons (Nilsson et al, 2012;Brabec et al, 2020;Lee et al, 2020a). However, in agreement with our previous hypothesis, this observation might also be explained by the lack of specificity of the high b signal to axons only.…”

Contextual Fibre Growth to Generate Realistic Axonal Packing for Diffusion MRI Simulation

“…A few groups generate WM numerical phantoms with complex fibre configurations for the application to tractography (Close et al, 2009;Neher et al, 2014); however realistic microstructural morphology is not the focus of these approaches. Other studies introduce more microstructural complexity into the numerical phantoms, typically only considering one mode of morphological variation at a time; some examples of this include harmonic beading (Budde and Frank, 2010;Landman et al, 2010), spines (Palombo et al, 2018b), undulation (Brabec et al, 2019;Nilsson et al, 2012) and myelination (Brusini et al, 2019).…”