  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 modelling axons as straight cylinders. Axons do
not, however, run in straight trajectories and so far, the impact of the axonal
trajectory on diameter estimation has not been systematically investigated.
Here, we employ a toy-model of axons, which we refer to as undulating
thin-fiber model, to analyze the impact of undulating trajectories on the
diffusion-time dependence represented by the diffusion spectrum. We analyze the
spectrum by its height (diffusivity at high frequencies), width (half width at
half maximum), 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.
Straight cylinders and undulating thin-fibers have virtually identical spectra
at lower frequencies. 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 the microscopic orientation
dispersion. At high frequencies (short diffusion times), spectra from cylinders
and undulating thin-fibers differ. The spectra from the undulating thin-fibers
can also differ from that of cylinders by exhibiting power law behaviors with
exponents below two. In conclusion, we argue that the non-straight nature of
axonal trajectories should not be ignored when analyzing dMRI data and that
careful experiments may enable separation of diffusion within straight
cylinders and diffusion in undulating thin-fibers.
