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.

Keywords: axon diameter; axonal trajectories; diffusion MRI; diffusion spectrum; low frequency; restricted diffusion; time dependence; undulation.

Figure 1

From histology to diffusion‐weighted signals. (A) Images of axons, which inspired our undulating thin fiber model. Axons in the corpus callosum,30 optic nerve,31 and phrenic nerve25 exhibit ubiquitously sinusoidal undulation patterns. (B) Underlying sine wave parameters used in the simulations (amplitude a, wavelength λ, and discretization into segments with variable angle θ). (C) Three cases of our undulating thin fiber model (1‐harmonic, n‐harmonic, and stochastic) and (D) comparison with the cylinder model (non‐dispersed straight and undulating). (E) The diffusion spectra, which were characterized in terms of their spectral width (f_Δ), height (D_hi), and low‐frequency behavior (p). (F) The connection between diffusion spectra, encoding power spectra, and the resulting signal through the first‐order cumulant expansion. Images in a were reproduced with permission from Elsevier, John Wiley & Sons, Inc., and the Institute of Electrical and Electronics Engineers (IEEE)

Figure 2

Comparisons of the Gaussian sampling method and Monte Carlo simulations. Diffusion spectra of the 1‐harmonic case characterized by wavelengths λ = 10, 30, 50 μm (from left to right) and amplitudes a = 1, 2, 3 μm (from top to bottom) computed by the Gaussian sampling method (black lines) and Monte Carlo simulations (red points). The methods agreed well, although the Monte Carlo results (red points) exhibited more noise and required more computational time than the Gaussian sampling method (black lines). This figure indicates that there is no simple relation between the spectral height or spectral width and undulation amplitude or wavelength, which means that a different parameterization of the thin fibers, such as the μOD, may be useful

Figure 3

A study of the undulating thin fiber model: diffusion spectrum examples. The exact (solid black line) and simplified by the single‐Lorentzian approximation (dashed red line) diffusion spectra for the 1‐harmonic case (A), the n‐harmonic case (B), and the stochastic case (C) of our thin fiber model were compared with those of cylinders (D) with diameter d = 10 μm. The bottom row shows the same spectra as the top row but in a log–log plot. The simplified spectra deviate from the simulated ones in the low‐frequency range for the n‐harmonic and stochastic cases; however, in these examples, the differences are not detectable within experimental limitations when SNR ≤ 50 (Table 4)

Figure 4

A study of the undulating thin fiber model: Prediction of diffusion spectra parameters. (A) Predicted and estimated spectral heights D_hi are aligned for the 1‐harmonic case (I), the n‐harmonic case (II), and the stochastic case (III) of the undulating thin fiber model. Values of D_hi were predicted using Equation 9 for the 1‐harmonic and stochastic cases and using Equation 10 for the n‐harmonic case. (B) Predicted and estimated spectral widths f_∆ are aligned again for the 1‐harmonic case (I), the n‐harmonic case (II), and the stochastic case (III). Values of f_∆ were predicted using Equation 15 for the 1‐harmonic case, Equation 16 for the n‐harmonic case, and Equation 17 for the stochastic case

Figure 5

Diffusion spectra of undulating thin fibers and undulating cylinders. Plots show diffusion spectra of undulating cylinders, undulating fibers, and straight cylinders. When d = 1 μm and 2 μm, the spectra of undulating thin fibers closely resemble those of undulating cylinders. For d = 3 μm and 5 μm, the spectra start to deviate above 10 Hz. For d = 10 μm, the spectrum of an undulating cylinder resembles that of a straight cylinder

Figure 6

Limitations of undulating thin fiber model: Short diffusion times. (A) Gradient waveforms and their corresponding encoding spectra for the protocol by Alexander et al8 (Table 3). The four gradient waveforms probe only two frequency regions, [0, 10 Hz] and [0, 50 Hz]. (B) The signals of undulating cylinders (generated by gradients from a and from diffusion spectra from Figure 5) are explained well by undulating thin fibers of the same undulations when d = 1 μm and 3 μm. They start to deviate when d = 5 μm, while at 10 μm the signal from undulating cylinders is nearly identical to that from a straight cylinder. Note that we show the mildest undulation patterns (a/λ = 2 %)

Figure 7

High b‐values. Comparison between signal obtained from Monte Carlo simulation (Equation 28) and from the first‐order approximation (Equation 4). The first‐order approximation is here shown to be valid up to attenuations of approximately 60 %. The maximal b‐values reached in this study were 3 ms/μm² and the maximal signal attenuation 20 % (upper right box)

Figure 8

Implications for the study of the low‐frequency behavior: Estimation of exponent p. (A) Examples of diffusion spectra normalized to the same spectral height. (B) The same data but in a log–log plot. The curves of the 1‐harmonic case (solid and dot‐dashed black line) are aligned with the quadratic frequency curve (dashed red line), whereas the slope of the curves from the n‐harmonic approach the quadratic frequency curve more slowly. The stochastic case (dotted black line) is more aligned with the linear curve (dot‐dashed red line). (C) The exponent p (obtained via Equation 31) is not constant with respect to the frequency. (D) A distribution of exponents p for the simulated cases in the low‐frequency region up to 20 Hz. n‐harmonic fibers had gamma distributed undulation amplitudes and wavelengths further restricted to the range validated by numerical simulations (1 μm ≤ a ≤ 3 μm; 10 μm ≤ λ ≤ 50 μm)

Figure 9

Implications for axon diameter mapping: cylinder diameter estimation. (A) Strong correlation between the undulation amplitude and the estimated diameter was found. (B) Estimated cylinder diameter versus diameters predicted based on a second‐order Taylor expansion (Equation 26). The predictions were more accurate for cases that had higher spectral widths (eg upward‐pointing triangle, corresponding to a = 1 μm, λ = 10 μm, f_Δ = 92 Hz) whereas those that had lower widths (eg downward‐pointing triangle, a = 1 μm, λ = 50 μm, f_Δ = 5 Hz) were not as well predicted. (C) The case marked with the upward‐pointing triangle from (B). the simulated diffusion spectrum (solid black line) was well approximated by its second‐order approximation (dashed black line, Equation 25) and well aligned with the diffusion spectrum of a cylinder with the estimated diameter (solid red line), which is also well approximated (dashed red line, Equation 24). Gray areas show re‐scaled encoding power spectra from Figure 6. (D) The case marked with a downward‐pointing triangle from (B), where the second‐order approximation of the simulated diffusion spectrum failed