Synopsis

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.

Introduction

Experiment

Experimental Results and Theoretical Interpretation

The main experimental observation of this abstract (Figure 2) is the universal power-law form^6,10

$${S}(b\rightarrow\infty)\simeq\beta\cdot\,b^{-\alpha}\,+\,\gamma\quad(1)$$

of the dMRI signal in human white matter (WM), with exponent $$$\alpha\,=\,1/2$$$. Here we use diffusion time $$$t\approx\,50\,$$$ms as available on clinical scanners, and diffusion weightings up to $$$b\, =\,10\,\mathrm{ms/\mu m^2}$$$.

To understand why this asymptotic power-law originates from the intra-axonal water, consider the signal (normalized to $$$S|_{b=0}\equiv\,1$$$):

$$S(\mathbf{\hat{g}},\,b)\,=\,f\int\,d\mathbf{\hat{n}} \mathcal{P}(\mathbf{\hat{n}}) \psi_\mathbf{\hat{n}}(\mathbf{\hat{g}},b)+\gamma\,+S^\mathrm{eas}(\mathbf{\hat{g}},b),\quad(2)$$

in the unit direction $$$\mathbf{\hat{g}}$$$. 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) $$$\mathcal{P}(\mathbf{\hat{n}})$$$ (Figure 1). If the overarching brain dMRI modeling assumption is correct, the signal (stick response function) $$$\psi_\mathbf{\hat{n}}(\mathbf{\hat{g}},b)$$$ from water confined within a stick pointing in the direction $$$\mathbf{\hat{n}}$$$, can be approximated by a simple Gaussian one-dimensional diffusion propagator $$$\psi_\mathbf{\hat{n}}(\mathbf{\hat{g}},b)\equiv\,e^{-bD_a^\parallel (\mathbf{\hat{g}}\cdot\mathbf{\hat{n}})^2}$$$. In the limit $$$bD_a^\parallel\gg\,1$$$, 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 $$$f$$$.

Discussion and Conclusion

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 histology^13,14.

Acknowledgements

References

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