Bibek Dhital^{1}, Marco Reisert^{1}, Elias Kellner^{1}, and Valerij G. Kiselev^{1}

How to measure the intra-axonal diffusion coefficient in the normal human brain? This contribution reports on such a measurement performed with minimum model assumptions. The result is about $$$2\,\rm\mu m^2/ms$$$, which constructively limits the scope of acceptable biophysical models for interpretation of diffusion MRI in terms of tissue microstructure. A side finding is the width of axonal orientation distribution in the most coherent fiber bundles, which is estimated as $$$19^\circ$$$ for the standard deviation. The core of the method is the suppression of signal from water that can move in the plane transverse to the principal fiber direction.

Biophysical modeling is the mediator between the measurable diffusion metrics and the tissue microstructure. While modeling demands knowledge of diverse microstructural parameters, parameter determination is hardly possible without modeling because of NMR-specific parameters (such as intracellular diffusivities) that can be only determined using in-vivo MRI. This makes the problem of self-consistent parameter determination highly challenging. In particular, it is known that straightforward data fitting approach does not work [1,2].

The aim of this study was to measure the intra-axonal water diffusivity in the normal human brain using as few model assumptions as possible.

The core of the method is the suppression of signal from water molecules that can move in the plane orthogonal to the principal fiber direction in voxels with highly coherent axons. This is achieved by the *planar weighting* with the b-matrix $$${\bf b}={\rm diag\,}(b_F,\,b_F,\,0)/2$$$ when the principal fiber direction is selected as the z-axis. The planar weighting can be thought of as shaping the native fiber orientation distribution to a narrower one (Fig.1). For such a distribution, the diffusivity parallel to the fiber is measured using either linear weighting, $$${\bf b}={\rm diag\,}(b,\,0,\,0)$$$, or isotropic weighting, $$${\bf b}={\rm diag\,}(b,\,b,\,b)/3$$$. While the isotropic measurements yields the trace of intra-axonal diffusion tensor, the contribution of its transverse components is negligible (about $$$0.002\,\rm \mu m^2/ms$$$ for $$$1\,\rm \mu m$$$ axons). Note that we rely on the absence of a noticeable amount of fully restricted water in brain white matter [3].

In more detail, the measurement consisted of (i) DTI for selecting voxels with a single principal fiber direction as described in [4] and (ii) performing planar weighting combined with the diffusivity measurement along the normal to the suppression plane, the direction, which is referred to as the *measurement direction* in what follows. Measurements were repeated for multiple measurement directions and strengths of planar weighting, $$$b_F$$$. The isotropic and linear diffusivities were obtained from physically the same measurement by attributing a small portion, $$$b$$$, of the planar weighting to either the diffusivity measurement or suppressing gradients, respectively.

Signal was averaged from voxels with a single principal fiber direction in which this direction was co-linear within $$$15^\circ$$$ with one of sampled measurement directions. Isotropic and linear diffusivities were calculated and represented as functions of $$$b_F$$$. Straightforwardly obtained expressions for the signal were fitted to data with the focus on well-defined fitting parameters as qualified from numerical simulations (not shown).

The values of diffusivities found in this study slightly overestimate their genuine long-time diffusion coefficients due to the wave form of the involved gradients. In more detail, the overestimation results from the finite width of the gradient power spectrum in combination with the frequency dependence of velocity correlation function in proportion to $$$\sqrt{\omega}$$$ for low frequencies, $$$\omega$$$, [5,6]. This gives a correction about $$$-0.25\,\rm \mu m^2/ms$$$ to the above results.

With this small correction, the obtained values of $$$D_0$$$ agree well with the interval $$$[1.9,\,2.2]\,\rm \mu m^2/ms$$$ found by extrapolating data from ultra-high $$$b$$$ to $$$b\to\infty$$$ [7]. The width of the fiber orientation distribution agrees well with histology [8,9] and MRI-based studies [7--9].

The values of $$$D_0$$$ from isotropic measurement is larger than that from the linear measurement by $$$0.17\pm 0.12\,\rm \mu m^2/ms$$$, which remains to be explained, possibly by model imperfections. For example, deviations of axons from straight cylinders over the water diffusion length can create apparent transverse components of intra-axonal diffusion tensor, which are then added to its trace as measured using the isotropic weighting. A simple estimate shows that the curvature with the radius $$$50\,\rm \mu m$$$ of all cylinders would explain the difference. Assigning such a curvature to all axons might be unrealistic, but a relatively large contribution from a small sub-population of cellular processes cannot be currently excluded.

The obtained value of intra-axonal diffusion coefficient constructively limits the scope of acceptable biophysical models for diffusion MRI. Beyond this role, the present results will be subjected to a closer investigation in norm and pathology in future work.

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[2] Dmitry S. Novikov, Jelle Veraart, Ileana O Jelescu, and Els Fieremans. Mapping orientationaland microstructural metrics of neuronal integrity with in vivo diffusion MRI. https://arxiv.org/abs/1609.09144.

[3] Bibek Dhital, Elias Kellner, Marco Reisert, and Valerij G. Kiselev. Isotropic diffusion weighting provides insight on diffusion compartments in human brain white matter in vivo. In Proc. 23st Annual Meeting of ISMRM, page 2788, Toronto, Canada, June 2015.

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FIGURE 1: The principle of the measurement: General and the top views (top and bottom rows, respectively). A: Axons are represented by cylinders positioned at a common origin. Cylinders in the equatorial plane represent processes incoherent with the principal fiber direction. B: Effect of linear weighting: Cells with suppressed signals are shown with the darker color. C: The planar weighting, which is used in this study, suppresses the signal from all cells, but those that are close to the principal fiber direction. Diffusivity in this direction approaches the ground truth. Isotropic diffusion measurement is applied to eliminate the residual effect of the orientation dispersion.

FIGURE 2: Axial diffusion coefficient, $$$D_\parallel$$$, and trace of diffusion tensor, $$$D_{\rm iso}$$$, in single-fiber voxels in all subjects as functions of the strength of the planar weighting, $$$b_F$$$. With increasing $$$b_F$$$, $$$D_{\rm iso}$$$ (circles) rapidly approaches the intra-axonal diffusivity, $$$D_0$$$. The axial diffusivity, $$$D_\parallel$$$, (triangles) is measured in the principal fiber bundle direction and underestimates $$$D_0$$$ due to the axonal orientation dispersion (Fig.1). Its slower approach to $$$D_0$$$ contains information about the width of the distribution, $$$\theta_0$$$. Results of fitting to the pooled data for all subjects are shown for either measurements with black solid lines.