Jan Brabec^{1}, Samo Lasic^{2}, and Markus Nilsson^{1}

We investigated effects on the diffusion MRI signal from harmonic and stochastic variations in the trajectories of thin axon-like structures. Trajectories with different types of variations exhibited characteristically different diffusion spectra (time-dependence) that, for low frequencies, bare similarities with those from restricted diffusion environments. Non-straight trajectories can thus bias axon diameter estimation. Results also indicated that observable effects of structural disorder may not be specific for extracellular diffusion but may also be found for intra-axonal spaces.

*Generation of axonal trajectories: *To generate a toy model that mimics biologically plausible axonal trajectories, axons were represented as thin wires with a position given by

$${\rm y(\mathit{x} )=a\cdot sinā”(\mathit{\phi})=a\cdot sin(2\pi\cdot \mathit{x}/\lambda+r(\mathit{x}))}$$

with amplitudes of $$${\rm a = [2;10]\, \mu m}$$$, wavelengths $$${\rm \lambda=[25;150] \, \mu m}$$$ and a stochastic part, $$${\rm r(\mathit{x})}$$$, generated by cumulative sum of random numbers with variable degree of correlation. For harmonic sine waves $$${\rm r(\mathit{x})=0}$$$. The intrinsic diffusivity within the impermeable paths was set to $$${\rm D_0=1.7\, \mu m^2/ms}$$$.

*Quantification strategies:* Three parameters describing the axonal trajectories were defined from the trajectory, $$${\rm y(\mathit{x})}$$$. First, we defined the microscopic orientation dispersion, $$${\rm \mu OD}$$$, as:

$${\rm \mu OD=\bigg\langle\left(tan^{-1}\left(\frac{d}{d\mathit{x}}y(\mathit{x})\right)\right)^2\bigg\rangle }$$

Second, the dispersion-weighted wavelength was defined by

$${\rm \lambda_{\sigma}=\Bigg(\frac{\langle \mu OD(\mathit{x})\cdot\lambda^{-1}(\mathit{x})\rangle }{\langle\mu OD(\mathit{x})\rangle}\Bigg)^{-1} }$$

where the ‘local’ wavenumber is $$${\rm \lambda^{-1} (\mathit{x})=d/d\mathit{x} \, \phi(\mathit{x}) }$$$. Third, effects of correlation lengths, $$${\rm l_d}$$$, were also studied (Figure 3). This was defined as the decay rate of the envelope of square of autocorrelation function of $$${\rm y(\mathit{x})}$$$.

*Simulating diffusion spectra:* The diffusion spectra, $$${\rm D(\omega)}$$$, is the spectrum of the velocity auto-correlation function^{5},

$${\rm \chi (\mathit{t})= \langle v(\mathit{t})v(0) \rangle}$$

And it has a close relationship with the mean square displacement, according to

$$\langle \Delta x^2 (\mathit{t})\rangle=\int_0^{\mathit{t'}}\int_0^{\mathit{t''}}\chi(\mathit{t'})\,d\mathit{t'}d\mathit{t''}=\frac{4}{\pi}\int_0^{\infty}\frac{D(\mathit{\omega})}{\mathit{\omega^2}}[1-cos(\mathit{\omega t})]\,d\mathit{\omega} $$

Diffusion spectra were numerically obtained by Gaussian sampling of the mean square displacement perpendicularly to the direction of axonal propagation (Figure 2 and 3). Gaussian sampling relies on simulating and averaging over different diffusion particle trajectories weighted by a Gaussian function.

*Characterizing diffusion spectra:* The obtained diffusion spectra were characterized by three parameters: spectral width, spectral height, and low frequency power law behavior (Figure 1A). The height was estimated at high frequencies where the diffusion spectrum remained constant, the width was estimated as the frequency at the half of the height, and the power law behavior at low frequencies was estimated by fitting the spectra in a loglog plot constrained up to half of the width.The descriptive parameters were also predicted from the trajectories themselves. The spectral width was predicted as

$${\rm\omega_{\, predicted}=\bigg(\frac{\langle\mu OD(\mathit{x})\cdot\lambda^{-1}(\mathit{x})\rangle}{\langle\mu OD(\mathit{x}) \rangle}\bigg)^{-2}\cdot D_0}$$

Spectral height was predicted as

$${\rm D_{\infty; \, predicted}=\langle\mu OD(\mathit{x})\rangle}\cdot D_0$$

Power law behavior was not predicted, but we note that the diffusion spectra at low frequencies scales as $$${\rm \omega^p}$$$, with $$${\rm p=2}$$$ for restricted environments and $$${\rm p<2}$$$ for disordered systems.^{6}

Relevant characteristics of the diffusion spectra can be predicted by a small number of structural properties of the trajectories. These properties do not describe well-defined aspects of the trajectories, such as their amplitude and wavelengths, but rather parameters such as microscopic orientation dispersion and a dispersion-weighted wavelength. Custom gradient waveforms, and in particular short diffusion times, may be used to estimate such features^{7} (Fig. 5). Finally, we found a type of power law behavior in these axon-like structures otherwise assigned specifically to the extra-axonal space. This indicates that several emerging methods for axon diameter estimation^{8} may be confounded by axonal trajectories.

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3. Nilsson, Markus, et al. "The importance of axonal undulation in diffusion MR measurements: a Monte Carlo simulation study." NMR in Biomedicine 25.5 (2012): 795-805.

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6. Novikov, Dmitry S., et al. "Revealing mesoscopic structural universality with diffusion." Proceedings of the National Academy of Sciences 111.14 (2014): 5088-5093.

7. Aslund, Ingrid, et al. "Diffusion NMR for determining the homogeneous length-scale in lamellar phases." The Journal of Physical Chemistry B 112.10 (2008): 2782-2794.

8. Burcaw, Lauren M., Els Fieremans, and Dmitry S. Novikov. "Mesoscopic structure of neuronal tracts from time-dependent diffusion." NeuroImage 114 (2015): 18-37.