Alonso Ramirez-Manzanares^{1}, Mario Ocampo-Pineda^{2}, Jonathan Rafael-Patiño^{3}, Giorgio Innocenti^{3,4,5}, Jean-Philippe Thiran^{3}, and Alessandro Daducci^{2,3,6}

This study aims to provide to the research community quantitative measurements about the axon diameter overestimation due to the straight-cylinder assumption usually made in state-of-the-art models for diameter mapping with DW-MRI. Our methodology uses a Monte Carlo diffusion simulator to compute the diffusion weighted magnetic resonance signals of undulating axons. We use in our experiments plausible tissue values, we also explore a broad parameter set that depicts undulation. The results of this study provide useful information to understand the differences between the estimators from histology vs. the estimated diameters when using the model assumption of simple shaped axons.

Introduction and Purpose

The diffusion-weighted (DW) magnetic resonance imaging (MRI) allows estimating White Matter (WM) microstructure parameters such as axon diametersMethods

Axon geometries: We use the helical undulation parameterization of an axon along $$$z$$$ $$$\mathbf{U}(z) = \left(A_x\cos(2\pi z/L) , A_y\sin(2\pi z/L) , z \right)$$$, where $$$L$$$ denotes the wavelength and $$$A_x$$$,$$$A_y$$$ denote the amplitudes in $$$x$$$ and $$$y$$$, respectivelyExamples of used substrates are in Figure 1. General undulation characterization: We performed noise free experiments
with isolated intracellular signals. Table (a)-(c) in Figure 2 reports the diameter overestimation due to undulation for different configurations, (d)-(f) shows the computed $$$D_{||}$$$ values, respectively. Note that Figure 2 indicates
the expected overestimation even in the perfect estimation scenario. Fixing the
bulk diffusivity: Some methodologies^{8,2} fixed the bulk diffusivity of
the cylinder model ($$$D_{||}$$$) and estimate only the diameter, the last row in
Figure 2 shows the effect on the diameter overestimation of that strategy
when $$$D_{||}$$$ is set as the ground-truth value,
interesting, one can see that, by using the right $$$D_{||}$$$ values it is possible to reduce the diameter overestimation. Signal
Fitting Quality: The left and right panels of
Figure 3 show examples of MCDS signals and the signal for the best BF fit when all the
parameters are estimated, and, when only the diameter is estimated; we note that in the second case the signal fitting is poor. Observed axon undulations: The results of the experiments in Figure 4 report the overestimation on some axon geometries observed in actual
tissue samples^{4,12,13} in microscopic and mesoscopic scale. Diameter distributions: Finally, we compute the parameters by the BF approach that best fit the intracellular signal from 46 (amplitude=0.3μm) and 54 (amplitude=0.2μm) undulating axons with wavelengths 12 and 20μm (see Figure 1), and with plausible gamma distributed diameters inside a
periodic voxel with size 10x10x40μm. We computed the ideal-axon-diameter-index α and estimated indices a' from the BF, see^{2}. For instance, on amplitude=0.3μm and wavelength=20μm, a'=3.5μm which is significantly larger than α=1.45μm due to undulation.

Conclusions

Our results indicate that overestimation on the computed diameter due to axon undulation is significant (≈2 times) and could distort the anatomical information on histological validations, this may partially explain (apart from tissue-fixation issues) the differences, for instance, in Fig. 9 in1. Assaf Y., Blumenfeld-Katzir T., et al. Axcaliber: A method for measuring axon diameter distribution from diffusion MRI. Magnetic Resonance in Medicine. 2008; 59(6): 1347-1354.

2. Alexander D.C., Hubbard P.L., et al. Orientationally invariant indices of axon diameter and density from diffusion MRI. NeuroImage. 2010; 52(4): 1374-1389.

3. Jonathan Rafael-Patino. Design and Validation of a Robust Di usion-Weighted-MRI Monte Carlo Simulator. Master Thesis at Centro de Investigation en Matematicas A. C. 2015; http://www.cimat.mx/es/Tesis_digitales .

4. Nilsson M, Lätt J., et al. The importance of axonal undulation in diffusion MR measurements: a Monte Carlo simulation study. NMR in Biomed. 2012; 25(1): 795-805.

5. Nilsson M., van Westen D., et al. The role of tissue microstructure and water exchange in biophysical modelling of diffusion in white matter. Magn. Reson. Mater. Phy. 2013, 26(1), 345-370.

6. Dyrby T B. and Sogaard L.V., et al. Contrast and stability of the axon diameter index from microstructure imaging with diffusion MRI. Magnetic Resonance in Medicine. 2013; 70(3): 711-721.

7. van Gelderen P., DesPers D., et al. Evaluation of Restricted DIffusion in Cylinders. Phosphocreatine in rabbit leg muscle. Magnetic Resonance. 1994; 103: 255-260.

8. Zhang H., Schneider T., et al. NODDI: Practical in vivo neurite orientation dispersion and density imaging of the human brain. Neuroimage. 2012; 61(4): 1000-1016.

9. Zhang H, Hubbard P. L., et al. Axon diameter mapping in the presence of orientation dispersion with diffusion MRI. NeuroImage, 2011, 56(1), 1301-1315.

10. Ocampo-Pineda M., Daducci A., et al. Estimation of a novel set of intra and extracellular diffusivity parameters from modern DW-MRI. ISMRM 2017, No. 1827. Hawaii, U.S.A.

11. Alexander D.C. Modelling, Fitting and Sampling in Diffusion MRI. Springer. Visualization and Processing of Tensor Fields. 2009: 3-20.

12. Jeffery G. PNS features of rodent optic nerve axons.J. Comp. Neurol. 1996; 366(2): 370-378.

13. Haninec P. Undulating course of nerve fibres and bands of Fontana in peripheral nerves of the rat. Anat. Embryol. 1986; 174(3): 407-411.

Figure 1. Examples of the substrates representing undulating axons used in this study. Left panel illustrates the parameters for different configurations. Right panel shows a periodic substrate in a 10x10x40μm voxel from plausible gamma distributed diameters.

Figure 2. Tables of the estimated values of axon diameter $$$d$$$ and axial diffusivity $$$D_{||}$$$ for different amplitudes and wavelengths. The 3 columns correspond to 1, 2, and 3μm diameter undulating axons. Brown and blue color indicates underestimation and overestimation, respectively. (a)-(c) Reports the overestimation on the diameter when the straight cylinder models is assumed, (d)-(f) shows the computed $$$D_{||}$$$ values. Last row, presents the diameter estimation when the $$$D_{||}$$$ is set to the correct value. Note the significant overestimation in diameter for many undulation configurations.

Figure 3. The undulating MCDS DWMR-signal (dot-pattern) for $$$d$$$=2μm (parameters in Figure) and the corresponding straight cylinder signal computed by the BF approach. The continuos line is the solution when both the diameter, $$$d$$$, and the axial diffusivity, $$$D_{||}$$$ are computed, the dashed line is the signal estimated when $$$D_{||}$$$ is fixed to the ground-truth value and only $$$d$$$ is estimated. Note that, the signal fit is poor in the second case, this could be advantageous because it highlights a poor modeling when a good approximation of $$$D_{||}$$$ is available. Similar results are obtained for different undulation configurations (not shown).

Figure 4. Plot of the overestimated diameter for observed undulation axon geometries^{4,12,13}. Horizontal continuous lines denote the ground-truth diameters (each one of different color). Panel in the left contains (among others) diameters estimations for undulation observed in the optic nerve (microscopic scale) with amplitudes between 3-5μm, wavelength around 30μm and $$$d$$$=1μm; right panel contains (among others) the diameter values estimated under undulation configurations observed in the sciatic nerve (mesoscopic scale) with amplitudes around 20-100μm, wavelengths around 100μm and diameters around 4-5μm.