Scattering approach to diffusion quantifies axonal damage in brain injury

TL;DR

The study shows that time-dependent diffusion MRI detects subtle axonal changes in brain injury by identifying structural signatures of diffusive dynamics of water along axons, bridging cellular-level alterations with millimeter-scale imaging.

Abstract

Early diagnosis and noninvasive monitoring of neurological disorders require sensitivity to elusive cellular-level alterations that occur much earlier than volumetric changes observable with the millimeter-resolution of medical imaging modalities. Morphological changes in axons, such as axonal varicosities or beadings, are observed in neurological disorders, as well as in development and aging. Here, we reveal the sensitivity of time-dependent diffusion MRI (dMRI) to the structurally disordered axonal morphology at the micrometer scale. Scattering theory uncovers the two parameters that determine the diffusive dynamics of water along axons: the average reciprocal cross-section and the variance of long-range cross-sectional fluctuations. This theoretical development allows us to predict dMRI metrics sensitive to axonal alterations over tens of thousands of axons in seconds rather than months of simulations in a rat model of traumatic brain injury, and is corroborated with ex vivo dMRI. Our approach bridges the gap between micrometers and millimeters in resolution, offering quantitative and objective biomarkers applicable to a broad spectrum of neurological disorders.

Figures (17)

• Figure 1: Axon segmentation and morphology.(a) Representative photomicrographs of 1 mm thick coronal sections, with the cingulum (Cg) and corpus callosum (CC) highlighted. Selected sections for staining encompass parts of the CC, Cg, and cerebral cortex (Cx). (b) A photomicrograph of a semi-thin section stained with toluidine blue, with a block trimmed further for the serial block-face scanning electron microscopy Denk2004 (SBEM) technique. (c) A low-resolution EM image to navigate for the final SBEM imaging. (d) A representative SBEM volume, voxel size $50 \times 50 \times 50$ nm$^3$, from a large field-of-view $200 \times 100 \times 65$ µ m$^3$ that retains two-thirds CC and one-third Cg. (e) DeepACSON Abdollahzadeh2021DeepACSONMicroscopyAbdollahzadeh2021CylindricalObjects, a convolutional neural network (CNN)-based technique (see Methods) segmented tens of thousands of myelinated axons in each SBEM volume; we sampled and visualized myelinated axons at three random positions. (f) Micrometer-scale along-axon shape variations of representative myelinated axons from Cg and CC. Two $10\,\mu$m-fragments of axons within the shaded circles are zoomed in: the corresponding cross-sectional areas $A(x)$ show a substantial variation (e.g., beading) in one axon and a relative uniformity in the other one. Source data are provided as a Source Data file.
• Figure 2: From axon geometry to along-axon diffusivity.(a) Relative cross-sectional variations $\alpha$ for representative SBEM-segmented myelinated axons (sham and TBI), the synthetic axon, and their power spectral densities $\Gamma_\eta(q)$. The finite plateau $\Gamma_0 = \Gamma_{\eta}(q)|_{q \to 0}>0$ signifies the short-range disorder (finite correlation length) in the cross-sections. (b) Monte Carlo simulated $D(t)$ ensemble-averaged over $N_{\text{axon}} = 50$ randomly synthesized, $N_{\text{axon}} = 43$ randomly sampled SBEM myelinated sham and $N_{\text{axon}} = 57$ TBI axons, with colors corresponding to (a). (c)$D(t)$ for all three cases scales asymptotically linearly with $1/\sqrt{t}$, validating the functional form of Eq. (\ref{['eq:diff_time']}). (d) Coarse-graining over the increasing diffusion length $\ell(t)$ makes an axon appear increasingly more uniform, suppressing shape fluctuations $\Gamma_\eta(q)$ with $q\gtrsim 1/\ell(t)$, such that only the $q\to0$ plateau $\Gamma_0$ "survives" for long $t$ and governs the diffusive dynamics (\ref{['eq:diff_time']}). To illustrate the effect, an axon segment is Gaussian-filtered with the standard deviation $\ell(t_i)/\sqrt{2}$ for $\ell(t_i) = 0, 5, 10, 20\,\mu$m. The coarse-graining of the axon segment along its length is color-coded for increasing diffusion times $t_i$ according to the color bar. (e) The exact tortuosity limit (\ref{['eq:tortuosity_res']}) is validated for both synthetic and SBEM individual axons. Axons with larger cross-sectional variations $\hbox{var}\, \alpha$ have higher tortuosity. The center represents the mean, and horizontal error bars reflect errors in estimating $D_\infty$ from Eq. (\ref{['eq:diff_time']}) (see Methods). (f) The predicted amplitude $c_D$ of the $t$-dependent contribution to $D(t)$, Eq. (\ref{['eq:cd_theo_res']}), validated against its MC counterpart estimated from Eq. (\ref{['eq:diff_time']}), for individual synthetic and TBI axons (colors as in (a)). The coefficient $c_D$ is larger for axons with greater cross-sectional variations. The filled circles and error bars reflect means and errors in estimating $c_D$ from MC-simulated $D(t)$ (horizontal) and estimating the plateau $\Gamma_0$ from $\Gamma_\eta(q)$ (vertical), as shown by dashed lines in the power spectral densities of panel (a) (see Methods). The number of samples is indicated in (b). Source data are provided as a Source Data file.
• Figure 3: Effect of chronic TBI on axon morphology and $D(t)$.(a) Geometric tortuosity $\langle 1/\alpha \rangle$, Eq. (\ref{['eq:tortuosity_res']}), and the variance $\Gamma_0$ of long-range cross-sectional fluctuations entering Eq. (\ref{['eq:cd_theo_res']}), are plotted for myelinated axons segmented from the ipsilateral cingulum of sham-operated (shades of green; $N_{\text{axon}}=3,999$) and TBI (shades of red; $N_{\text{axon}}=3,999$) rats. (b) The optimal linear combination $z_G$ of the morphological parameters is derived from a trained support vector machine (SVM). Projecting the points onto the dark blue dashed line in (a) perpendicular to the SVM hyperplane constitutes the maximal separation between the two groups. (c) Predicted individual axon diffusion parameters $D_{\infty,i}$ and $c_{D,i}$ from Eqs. (\ref{['eq:tortuosity_res']})--(\ref{['eq:cd_theo_res']}) plotted for myelinated axons in (a). The size of each point reflects its weight $w_i$ in the net dMRI-accessible $D(t)$, proportional to the axon volume. (d) The optimal SVM-based linear combination $z_D$ of the diffusion parameters is derived by projecting the points onto the dark blue dashed line in (c) perpendicular to the corresponding SVM hyperplane. Dashed lines in (a-d) indicate the medians of the distributions. (e) Macroscopic diffusivity parameters $c_D$ and $D_\infty$ for each animal are obtained by volume-weighting (filled circles; $N=2$ sham-operated and $N=3$ TBI) the individual axonal contributions $D_{\infty, i}$ and $c_{D,i}$. Error bars represent measurement uncertainties in the volume-weighted estimates (see Methods). The SVM hyperplane (cyan dashed line) is the same as that for the diffusion parameters of individual axons in (c). (f) Predicting the along-tract $D(t)/D_0$ as a function of $1/\sqrt{t}$, Eq. (\ref{['eq:diff_time']}), based on the overall $D_\infty$ and $c_D$ in (e). (g) The effect of TBI on the ensemble-averaged geometry (filled circles) is illustrated by transforming the macroscopic ensemble diffusivity in (e,f), as if from an MRI measurement, back onto the space of morphological parameters $\langle 1/\alpha \rangle$ and $\Gamma_0$, via inverting Eqs. (\ref{['eq:tortuosity_res']})--(\ref{['eq:cd_theo_res']}). The SVM hyperplane (cyan dashed line) is the same as that for the morphological parameters of individual axons in (a). Error bars corresponding to standard deviations of $D(t)/D_0$ in (f) and $\langle 1/\alpha \rangle$ and $\Gamma_0$ in (g) are calculated based on errors in (e) (see Methods). Source data are provided as a Source Data file.
• Figure 4: Effect of mild TBI on ex vivo dMRI and axon morphology in rat ipsilateral major white matter tracts.(a) Representative colored fractional anisotropy (FA) maps in sagittal and coronal views, with the cingulum (Cg), splenium of the corpus callosum (Scc), and body of the corpus callosum (Bcc) annotated. (b) Experimental axial DTI diffusivity $D(t)$ plotted as a function of $t$ and $1/\sqrt{t}$, showing a power-law relation in all ipsilateral white matter regions of interest (ROIs). (c) Diffusion parameters $D_{\infty}$ and $c_{D}$ extracted by linear regression of $D(t)$ with respect to $1/\sqrt{t}$ in (b) for voxels within the ipsilateral Scc ROI ($N_{\text{voxel}} = 245$ per group). The optimal SVM-based linear combination $z_D$ of the diffusion parameters is derived by projecting the points onto the dark blue dashed line perpendicular to the corresponding SVM hyperplane. (d) Corresponding geometric parameters $\langle 1/\alpha \rangle$ and $\Gamma_0$, computed by inverting Eqs. (\ref{['eq:tortuosity_res']})–(\ref{['eq:cd_theo_res']}) from the diffusion parameters in (c), plotted for voxels in Scc. The optimal linear combination $z_G$ of the morphological parameters is obtained by projecting the data points onto the dark blue dashed line, which is orthogonal to the SVM hyperplane. In (b), filled triangles with shaded areas indicate the mean and standard deviation across the ROI ($N=2$ sham-operated and $N=3$ TBI). In (c)–(d), each point represents a voxel. Filled circles with error bars indicate the mean and standard deviation across the ROI. Dashed vertical lines overlaid on the distributions denote their medians. Source data are provided in the Source Data file.
• Figure 5: Feynman diagrams for the disorder averaging of the Green's function of Eq. (\ref{['eq:fj_perturb']}). (a) The dashed line represents an elementary scattering act off the static disorder potential (\ref{['eq:perturbation']}) corresponding to the scattering vertex $\mathcal{V}(\cdot) = -D_\infty \partial_x \left( y(x) \, \cdot \, \right)$, Eq. (\ref{['eq:V']}), where for $t\to\infty$, we substitute $D_0\to D_\infty$ (see text after Eq. (\ref{['eq:EMT']})). In the Fourier representation, the scattering momentum (wave vector) is conserved at each scattering event: the sum of incoming momenta ($k_1$ and $k_2-k_1$) equals the outgoing momentum $k_2$. Since the disorder is static, the "energy" (frequency $\omega$) is conserved in all diagrams. (b) The full Green’s function (\ref{['eq:EMT']}), represented by the bold line, is given by the Born series, where propagation between scatterings is described by the free Green’s functions $G^{(0)}$, Eq. (\ref{['eq:G0']}) (thin lines). Averaging over the disorder turns the products $y(x_1) \dots y(x_n)$ into the corresponding $n$-point correlation functions; the sum of all 1-particle-irreducible diagrams (which cannot be split into two parts by cutting a single $G^{(0)}$ line) is by definition the self-energy part $\Sigma(\omega, q)$. (c) To the lowest (second) order, $\Sigma(\omega, q)$ is given by a single Feynman diagram (\ref{['eq:self_energy']}) with the two-point correlation function $\Gamma_y(k)$, Eq. (\ref{['eq:2pointcorr']}).