Due to the limited diffusion MRI (dMRI) resolution and the inherent central symmetry in the measurements, sub-voxel fibre patterns (e.g., fanning/bending/crossing) cannot be distinguished using voxel-wise symmetric FODs (sFODs), leading to tractography errors^1,2,3,4. We present a flexible framework that uses the relative spatial arrangement of such patterns to perform neighbourhood-wise constrained spherical deconvolution (CSD). We estimate sub-voxel asymmetric fibre patterns and propose an improvement in current tractography approaches. We assess performance using real histological fibre patterns and in vivo dMRI data.

Asymmetric FOD

We augment the previous CSD framework⁵ by incorporating information from neighbouring voxels, allowing asymmetry in the estimates. We represent FODs using a linear spherical harmonics (SH) basis, but contrary to previous approaches we use SHs of both even and odd orders. The former are axially symmetric and allow voxel-wise features to be represented. The latter are axially antisymmetric and allow capturing asymmetry, inferred from the spatial arrangements. The unknown SH coefficients $$$\bf{\widehat{\it f_{all}}}$$$ for a voxel V are estimated by minimizing a cost function comprised of a voxel-wise and a neighborhood-wise term:

$${\widehat{{\bf f}_{all}}}={argmin}_{\bf {{f}_{all}}}\left\{\parallel{\bf{Cf}}_{even}-{\bf{y}}\parallel^2+\lambda^2\parallel{\bf{Bf}}_{all}-{\bf z}\parallel^2\right\}{\rm{with}}\ {\bf{Bf}}_{all}\geq0$$

where f_all contains both the odd and even order SH coefficients of the asymmetric FOD (aFOD) and f_even contains only the even order coefficients. The first term minimizes the sum of squared residuals between the signal predicted by voxel-wise symmetric CSD⁵ (Cf_even) and the acquired dMRI signal at V (y). The second term minimizes the difference between the FOD at V (Bf_all) and FODs in a 3x3x3 neighbourhood of V (z). Specifically, we evaluate FODs at N points on the sphere. For each point i and respective orientation u_i (Fig.1A), we select the neighbour X with a vector w_x (connecting its centre to the centre of V) being closest to u_i, and set zi=FOD_x(-u_i). This is based on a fiber continuity assumption⁶: a fiber leaving one voxel with direction ui should enter the next one following the opposite direction -u_i (Fig. 1A). The above function can be minimized iteratively, across all voxels, until convergence. The regularisation parameter λ was determined using cross-validation.

Tractography

Our proposed deterministic and probabilistic approaches take into account the current tracking position to decide which side of the current aFOD to use. A plane perpendicular to the vector connecting the current tracking position to the center of the current voxel divides the local aFOD in two parts (Fig. 1B). The aFOD part on the side of the current tracking position is considered and flipped to obtain an sFOD. To propagate the streamline, the direction of least curvature is selected in the deterministic approach (after trilinear interpolation of the vector field). In the probabilistic approach, the aFOD is sampled using a rejection sampling approach. In this work we have set the step-size to 0.25 (0.2 for deterministic) of the voxel size, a 45° angular threshold (80° for deterministic) and a 0.1 amplitude threshold on the aFOD.

Simulations

Histological sections of a macaque brain, stained for myelin, were processed using structure tensor analysis (STA)⁷. DMRI signal was then simulated using the ball and stick model⁸ and the orientations estimated by STA. B-value was set to 2000 s/mm² (SNR=20). Voxels at histology resolution (50 μm) were then pooled together to simulate voxels at lower MRI resolution (1 mm isotropic) with sub-voxel patterns.

In vivo data

Diffusion MRI data were obtained from the Human Connectome Project^9,10.

Fig. 2 shows reconstructed aFODs obtained from simulated and in-vivo data. Sharp bending, splitting and fanning can be seen in regions where sFODs suggest fibre crossings. Fig. 3 shows a comparison between a “ground-truth” histological FOD (histFOD), sFOD and aFOD for three different configurations. AFODs improve the sub-voxel geometry estimation. The correlations with the histFODs are 0.4–0.78–0.84 for the sFODs and 0.87–0.9–0.95 for the aFODs, for the three different configurations. Fig. 4 shows the benefit in our deterministic tractography approach when using aFODs vs sFODs. When comparing to tractography ran on histology-extracted orientations, it can be seen that aFOD-based tractography improves the accuracy of the reconstructed connections. Fig. 5 shows the advantage of being able to distinguish fanning polarity (spreading vs converging) with our estimates. An improved characterization of sharp bending and fanning fibers can be obtained using aFODs (Fig.5A). Moreover, topological profiles are better preserved when using aFOD-based probabilistic tractography as more cortical end-points correctly project back to the original seed ROI (Fig.5B).The proposed aFODs can reliably represent sub-voxel fiber patterns such as sharp bending and fanning. This added information increases both the accuracy and the precision of fibre tractography.