Surface-Based Probabilistic Fiber Tracking in Superficial White Matter

Abstract

The short association fibers or U-fibers travel in the superficial white matter (SWM) beneath the cortical layer. While the U-fibers play a crucial role in various brain disorders, there is a lack of effective tools to reconstruct their highly curved trajectory from diffusion MRI (dMRI). In this work, we propose a novel surface-based framework for the probabilistic tracking of fibers on the triangular mesh representation of the SWM. By deriving a closed-form solution to transform the spherical harmonics (SPHARM) coefficients of 3D fiber orientation distributions (FODs) to local coordinate systems on each triangle, we develop a novel approach to project the FODs onto the tangent space of the SWM. After that, we utilize parallel transport to realize the intrinsic propagation of streamlines on SWM following probabilistically sampled fiber directions. Our intrinsic and surface-based method eliminates the need to perform the necessary but challenging sharp turns in 3D compared with conventional volume-based tractography methods. Using data from the Human Connectome Project (HCP), we performed quantitative comparisons to demonstrate the proposed algorithm can more effectively reconstruct the U-fibers connecting the precentral and postcentral gyrus than previous methods. Quantitative validations were then performed on post-mortem MRIs to show the reconstructed U-fibers from our method more faithfully follow the SWM than volume-based tractography. Finally, we applied our algorithm to study the parietal U-fiber connectivity changes in autosomal dominant Alzheimer's disease (ADAD) patients and successfully detected significant associations between U-fiber connectivity and disease severity.

Fig. 1.

Accurate placement of the WM cortical surface in the dMRI space. (a) T1-weighted MRI; (b) the extra-axonal diffusivity calculated from dMRI by our compartment models. After the preprocessing by HCP-Pipeline, yellow arrows in (c) highlight the residual misalignment of the WM cortical surface (cyan) and the intra-axonal tissue map calculated by our compartment models. (d) shows the much-improved alignment of the WM cortex (green) with the same tissue map after nonlinear registration between T1-weighted MRI in (a) and the extra-axonal diffusivity map in (b).

Fig. 2.

The projection of 3D FODs onto 2D triangles local coordinate systems. (a) The FOD3D function is represented in a local spherical coordinate system defined on the center of a triangle of the SWM mesh. The reference plane of the system is the triangle plane, meanwhile the zenith direction z is aligned with the normal direction. The x-axis is aligned with an edge of the triangle. (b) The projected FOD2D using the formula (1). The FOD2D function is parametrized by φ in the local coordinate system and can be normalized to a probability distribution function on the unit circle S¹.

Fig. 3.

The local coordinate system Oxyz differs from the physical coordinate system Ox⁰y⁰z⁰ by a rotation R. (θ₀,φ₀) and (θ,φ) represent the same point p in the two systems. The rotation R is decomposed as three successive rotations around three axes: (a) Ox⁰y⁰z⁰ is rotated by an angle α around Oz⁰ → Ox¹y¹z⁰; (b) Ox¹y¹z⁰ is rotated by a second angle β around Oy¹ → Ox²y¹z²; (c) Ox²y¹z² is rotated by a third angle γ around Oz² → Ox³y³z², which is Oxyz.

Fig. 4.

For a vector u_p in the tangent space of point p on a Riemannian manifold, the Levi-Civita connection parallel transports u_p along a geodesic γ(t) to a vector u_q in the tangent space of point q, which allows the comparison of vectors in the tangent spaces of nearby points.

Fig. 5.

The parallel transport of a vector v in the current triangle T₁ to a neighboring triangle T₂. The parallel transport consists of unfolding the triangles to a common plane (a)~(b), translating the red tangent vector v from T₁ to T₂ (b), and folding the triangles back to their original positions (b)~(c). The transported vector v_p can then be compared with any sampling direction w in the triangle T₂.

Fig. 6.

Representative HCP examples of U-fibers between the pre- and post-central gyrus reconstructed by the two surface-based methods (SP: surface-based probabilistic tracking; SD: surface-based deterministic tracking) and the MRtrix software. For better visualization, we downsampled all results to 1000 streamlines. The subject ID is plotted at the top of the results from each HCP subject.

Fig. 7.

Quantitative comparisons of the U-fibers reconstructed by the proposed surface-based probabilistic (SP) fiber tracking, surface-based deterministic (SD) fiber tracking, and volume-based fiber tracking in the MRtrix software on 484 HCP subjects. (a) Box plots of the ‘well-U-connected’ measure of the three methods. Box plots of the ‘well-distributed’ measure on the precentral and postcentral gyrus are shown in (b) and (c), respectively. (d) Box plots of the U-ratio measure across subjects. (e) Box plots of the Procrustes distance to assess the topographic regularity of the reconstructed U-fibers from the three methods.

Fig. 8.

Across the HCP subjects, box plots of the number of touched sections in (a) pre- and (b) post-central gyrus by MRtrix results with varying numbers of seeds from 3000 to 50,000 are shown. For the ease of comparison, box plots of the number of touched sections by results from the proposed surface-based probabilistic (SP) method (30,000 seeds) are also shown.

Fig. 9.

T2-weighted 7T MRIs of two post-mortem brains are shown in (a) Subject 1 (91 years old male) and (b) Subject 2 (97 years old female). SWM with higher iron concentrations is highlighted with blue arrows. Manually delineated SWM masks in the lateral frontal lobe for subjects 1 and 2 are shown in (c) and (d), respectively.

Fig. 10.

Overlay of reconstructed U-fibers with the SWM on T2-weighted post-mortem MRIs. Top row: subject 1; Bottom row: subject 2. (a) and (d) show a zoomed view of the overlay of the U-fibers with the MRI in the red box highlighted in (b) and (e), respectively. (c) and (f) show a zoomed view of the overlay of the U-fibers with the MRI in the blue box highlighted in (b) and (e), respectively.

Fig. 11.

(a) The mean distance from the U-fibers reconstructed by the proposed algorithm to the SWM mask as a function of the deformation distance parameter δ. Boxplots of the distance of the reconstructed U-fibers to the SWM masks in (b) subject 1 and (c) subject 2. MRtrix: iFOD1 algorithm in the MRtrix software. SP: proposed surface-based probabilistic tracking method with δ=0.5mm.

Fig. 12.

The reconstructed U-fibers in the parietal lobe of two ADAD patients are superimposed over their SWM meshes. The left and right hemispheres of subject 1 are shown in (a) and (b), respectively. The left and right hemispheres of subject 2 are shown in (c) and (d), respectively.

Fig. 13.

The linear regression results between the CDR-SOB score, tau SUVR, and the mean FOD2D peak value on the U-fibers of the parietal cortices of the ADAD patients. (a), (c) are results from the left hemisphere and (b), (d) are results from the right hemisphere.