Here we identify a universal power-law scaling behavior of the diffusion MRI signal on a clinical scanner. This specific functional form provides a defining signature of water confined within narrow sticks establishing that exchange between intra- and extra-axonal water is not relevant, and the fraction of fully restricted water is negligible in the clinically accessible regime. The observed scaling for the first time in vivo validates the key ingredient specific to the microstructural models of MRI signal from neuronal tissue and enables the in vivo quantification of intra-axonal properties.
The main experimental observation of this abstract (Figure 2) is the universal power-law form6,10
S(b→∞)≃β⋅b−α+γ(1)
of the dMRI signal in human white matter (WM), with exponent α=1/2. Here we use diffusion time t≈50ms as available on clinical scanners, and diffusion weightings up to b=10ms/μm2.
To understand why this asymptotic power-law originates from the intra-axonal water, consider the signal (normalized to S|b=0≡1):
S(ˆg,b)=f∫dˆnP(ˆn)ψˆn(ˆg,b)+γ+Seas(ˆg,b),(2)
in the unit direction ˆg. The first, intra-axonal term comes from the collection of narrow channels ("sticks") representing axons and possibly glial cell processes, with net water fraction f, and parameterized by the orientational distribution function (ODF) P(ˆn) (Figure 1). If the overarching brain dMRI modeling assumption is correct, the signal (stick response function) ψˆn(ˆg,b) from water confined within a stick pointing in the direction ˆn, can be approximated by a simple Gaussian one-dimensional diffusion propagator ψˆn(ˆg,b)≡e−bD∥a(ˆg⋅ˆn)2. In the limit bD∥a≫1, this response function yields a non-negligible contribution only from axons falling within a thin pancake |\mathbf{\hat{n}}\cdot\mathbf{\hat{g}}\rm|\lesssim(bD_a^\parallel)^{-1/2} nearly transverse to \bf\hat{g}, whose thickness scaling as b^{-1/2} results in the aforementioned asymptotic form (1).
It is essential, for the power-law scaling (1) to hold, and to originate solely from intra-stick water, that the dMRI signal exactly transverse to a stick, \mathbf{\hat{g}\perp\hat{n}}, is not suppressed: \psi_\mathbf{\hat{n}\perp\hat{g}} does not decay at large b, equivalent to a negligible transverse diffusion coefficient D_a^\perp, and a negligible axonal radius compared to the free diffusion length. In contrast, the extra-axonal contribution S^{\mathrm{eas}}(\mathbf{\hat{g}},\,b) coming from water diffusion in a simply-connected space characterized by a finite diffusion coefficient D_e(\mathbf{\hat{g}}) (with D_e(\mathbf{\hat{g}})\geq D_e^\perp), decays exponentially faster than the intra-axonal signal, by virtue of e^{-bD_e^\perp}\ll 1 for large enough b, and can be eventually neglected. Finally, \gamma\equiv\,S|_{b=\infty} is the possible contribution of immobile (fully restricted) water11, which is shown to fall below our detection threshold.
Either a finite axonal radius, or a notable exchange rate between intra- and extra-axonal water, would destroy the b^{-1/2} scaling, see Figure 3 where we investigate finite radius effects.
Fiber ODF quantification: We select voxels characterized by a single fiber population12 (SFP), and focus on the SFP-averaged signal \tilde{S}_b(\theta) as a function of the angle \theta between the gradient direction \mathbf{\hat{g}} and the principal fiber direction \mathbf{\hat{n}}_0. We use the directional signal dependence of \tilde{S}_b(\theta) in order to determine an SFP-averaged fiber ODF, and to obtain estimates of the intra-axonal diffusivity D_a^\parallel and axonal dispersion. We prove that the orientational variance \sigma_b^2 of \tilde{S}_b(\theta) approaches \sigma_b^2\simeq\sigma^2\,+\frac1{2bD_a^\parallel}. In Figure 4, we observe that this variance scales linearly with 1/b in all subjects. This scaling further allows us to determine both the intrinsic fiber orientational dispersion \sigma, and the intra-axonal diffusivity D_a^\parallel separately from the axonal water fraction f.
Our in vivo observation of the power-law exponent \alpha\,=\,1/2 provides direct evidence that axons can be represented as an array of zero-radius sticks, so that the diffusivity inside each stick transverse to its axis can be neglected. In biophysical terms, this picture reveals:
\bullet long water residence time in axons, compared to typical diffusion times;
\bullet the absence of a fully restricted isotropic diffusion compartment; and
\bullet axonal radii do not exceed the shrinkage-corrected values obtained by histology13,14.
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Figure 1: The existing conjecture presents water diffusion in WM (a, b) as restricted diffusion in an array of axons mimicked by sticks (red) embedded in the extra-axonal water (blue), whereas contribution of myelin (yellow) water is considered to be negligible. (c) Schematic representation for water diffusion properties of an individual stick. The diffusion length scale is at least two orders of magnitude smaller than the imaging resolution (d). Hence, the measured intra-axonal signal reflects the averaging over an ensemble of sticks, i.e. a convolution between axonal orientation distribution function and a stick response function, cf. first term of Eq.(2).
Figure 4: (a)\,ODF variance \sigma_b^2 at finite b, computed as the slope of \ln\tilde{S}_b(\theta) vs \cos^2(\theta), decays as 1/b. (b)\,The intercept and slope of the variance as function of 1/b returns an in vivo estimate of the intrinsic axonal orientational dispersion \sigma^2 and intra-axonal diffusivity D_a^\parallel\,=[1.9,\,2.2]\,\mathrm{\mu\,m^2\,/ms}, respectively. The estimated dispersion angle \sin^{-1}\sigma\approx\,17^\circ is in excellent agreement with histological studies yielding dispersion of about 18^\circ15,16. Axonal water fraction f is estimated using D_a^\parallel and the parameter \beta, f ranges between 0.6 and 0.7 amongst the four subjects. Error bars span the 95\% confidence interval; bar plots show reproducibility over 4 subjects.