Surface-based Tracking of U-fibers in the Superficial White Matter

Abstract

The superficial white matter (SWM) lies directly underneath the cortical ribbon and contains the short association fibers, or U-fibers, that connect neighboring gyri. Connectivity of these U-fibers is important for various neuroscientific research from the development to the aging of the brain. Nonetheless, conventional tractography methods can only provide a partial representation of these connections. Moreover, previous studies on U-fibers mainly extract tracts based on their shape characteristics without imposing the biologically critical condition that they should tightly follow the cortical surface. In this work we leverage the high resolution diffusion imaging data from the Human Connectome Project (HCP), and develop a novel surface-based framework for reconstructing the U-fibers. Guided by the projected fiber orientation distributions (FODs) on cortical surfaces, our method tracks the U-fibers from sulcal seed regions to neighboring gyrus on the triangular mesh representation of the cortex. Compared to volume-based tractography, the main advantage of our method is that it is intrinsic to the cortical geometry. More specifically, we define a novel approach for measuring the change of angles on the tangent space of the surface and use them to determine the U-fiber passing through a sulcal seed point. In experimental results, we compare our surface-based method with state-of-the-art FOD-based tractography from MRtrix on a large-scale dataset of 484 HCP subjects, and demonstrate that our method clearly achieves superior performance on the reconstruction of U-fibers between the precentral and postcentral gyrus.

1. Introduction

The superficial white matter (SWM) lies directly beneath the cortex and contains the short association fibers, or U-fibers, connecting neighboring gyri [1-3]. These short association fibers or cortico-cortical tracts in the SWM play a crucial role from neurodevelopment of structural integrity during brain maturation [4,5] to neurodegenerative process of aging [6] and brain disorders such as schizophrenia [7], Huntington’s disease [8], and autism spectrum disorder (ASD) [9].

In spite of the significant role of the short association fibers in brain connectivity, only the long association fibers in the deep white matter (DWM) have been heavily studied. The short association fibers are rarely studied relatively because of their small size and structural complexity in the SWM. Hence, conventional tractography methods that focused mainly on the DWM tracts can only provide a partial representation of the short, local connections in the SWM [3,10,11] because existing tractography algorithms tend to have a large number of false negatives [12] in reconstructing highly curved fibers such as the U-fibers. Moreover, these U-fiber studies focused only on extraction of U-fibers based on the U-shape from volume-based tractography using clustering, and ignored U-fibers to be on the tangent space of the SWM surface, which should in fact be the most important biological criteria for U-fiber tracking or reconstruction.

To overcome these fundamental difficulties of volume-based tractography for U-fiber reconstruction, we propose here a novel surface-based framework for tracking the U-fibers in the SWM. Surface-based approaches have been well developed to utilize intrinsic geometry of the cortex instead of the 3D volume space in various cortical mapping research [13, 14]. Once the gray matter and white matter surfaces are reconstructed and the skeletons of the gyral and sulcal regions are computed by preprocessing steps, our surface-based method can start from sulcal seed regions and establish U-fiber connections between neighboring gyri as guided by the surface projection of FODs. On the highly curved cortical surface, our method measures the deviation angles on the tangent space of the surface, so it can easily handle the highly bended U-fibers. In our experimental results on large-scale dataset of 484 subjects from the 500-subject release of the Human Connectome Project (HCP) [15], we compare our surface-based method with FOD-based tractography in MRtrix [16], and show that our method achieves superior performance in reconstructing U-fibers between the precentral and postcentral gyrus.

The rest of our paper is organized as follow. In Section 2, we describe the materials and preprocessing steps, and present the technical details of our surface-based U-fiber tracking algorithm in the SWM. In Section 3, we demonstrate our surface-based method on the large-scale HCP dataset and perform quantitative comparisons with results from the MRtrix software. Finally, conclusions are made in Section 4.

2. Methods

Materials and preprocessing.

We used a large-scale dataset consisting of the T1-weighted and multi-shell diffusion MRI of 484 subjects from the 500-subject release of HCP. We first performed FreeSurfer reconstruction from the T1-weighted MRI to generate the triangular mesh representation and parcellation of the gray matter (GM) and white matter (WM) cortical surface [17, 18]. We also computed the gyral skeletons of the GM surfaces as shown (green) in Fig. 2 (A) using the graph-cut-based method [19]. The sulcal areas between the two neighboring gyri such as the central sulcus shown (red) in Fig. 2 (A) were detected within the 2mm-geodesic distance transforms from the precentral and postcentral ROIs, respectively.

Fig. 2.

Algorithmic details of surface-based tracking of U-fibers between the precentral and postcentral cortex. (A) The gyral skeletons (green) and seed sulcal regions (red) are generated from preprocessing steps. A zoomed view of the boxed region is shown in (B) and (C). (B) A streamline from a seed point (red dot) to the postcentral gyrus is first established. From all the possible fiber directions (brown arrows) at the seed point, a streamline with the minimum average deviation angle is selected. (C) With the predetermined fiber direction (brown arrow), which is opposite to the selected direction in (B) at the seed point, another streamline to the other gyrus (precentral) is generated. (D) A demonstration of the common procedure that controls the smoothness of the pathways within a triangle T_i by the deviation angle θ_in, and at the crossing to a neighboring triangle T_j by the deviation angle θ_xing.

From the multi-shell diffusion MRI data, we reconstructed fiber orientation distributions (FODs) with compartment models [20] as shown in Fig. 1 (A, B). For every vertex of the WM surface, we linearly interpolated FODs from neighboring voxels. We determined the number of peaks (directions) with the largest magnitudes using the spherical harmonics (SPHARM) model of the FODs. We then projected the FOD directions onto the tangent space of the WM surface for surface-based fiber tracking as shown in Fig. 1 (C), where the two fiber directions with the largest magnitudes are plotted. In our experiments, we allowed the use of up to four main fiber directions from the FODs. These projected fiber directions on the surface provided the information to establish the trajectories of the U-fibers.

Fig. 1.

Surface-based reconstruction of U-fibers from FODs. (A) FODs computed from diffusion MRI data. (B) A zoomed view of the ROI (dashed box) in (A) including the superficial white matter (SWM) along the precentral and postcentral cortex. (C) The projection of FOD directions onto the tangent space of the white matter surface in the boxed region. For each vertex, the two fiber directions with the largest magnitudes are plotted. All FODs are color-coded by directions (red: left-right, green: anterior-posterior, blue: inferior-superior). FODs in the SWM show the fiber directions of the U-fiber (yellow curve) along the tangent space of the cortex.

U-fiber tracking on the SWM surface.

To track the U-fiber between two neighboring gyri, we select seeds in the sulcal area as colored in red in Fig. 2 (A). We start the tracking process from a randomly selected seed point on the edge of a triangle in the surface mesh (red dot in Fig. 2 (B)). Since our surface-based tracking approach follows the main peaks of the projected FODs, it is essentially a deterministic tracking algorithm. To generate a complete U-fiber, we track toward each gyrus from the seed point and then merge the result to form the final U-fiber. To determined the next tracking direction, we need to interpolate from all possible combination of the FOD peak directions at neighboring vertices and compare the tracking results in the end to select the most likely trajectory for the U-fiber. As shown in Fig 2 (B), we plotted some of the possible tracking directions as the brown arrows. By taking a step toward a specific direction, we determine the triangle T_i that it enters and compute the barycentric coordinates (x, y, z) of the current point within the triangle as shown in Fig. 2 (D).

Tracking within triangle.

For every combination of the fiber directions at the three vertices of T_i, we compute the tracking directions via linear interpolation based on the barycentric coordinates at each step within T_i. Then we update the position of the point and measure the angle θ_in between the current tracking direction and the incoming direction at the previous step, which is thresholded at THθ_in. The streamline will be terminated back to the starting edge point in T_i whenever θ_in > TH_θin in any step. For every streamline passing through T_i and hitting a point on a different edge of T_i from different combinations of fiber directions at T_i’s vertices, we measure the angle between the current edge-to-edge segment in T_i and the previous one, and select the streamline with the minimum angle.

Tracking across triangles.

With the new edge point in a neighboring triangle T_j, we repeat the same procedure of updating the position within the triangle. After the first step, however, we measure another type of (crossing) angle θ_xing between the current tracking direction and the incoming direction of the last step in the previous triangle T_i, which is projected onto the tangent space of the surface at the current triangle in T_j. This ensures that we remove the tangential component of deviation angle due to the intrinsic geometry of the cortical foldings, and only measure the deviation component from the fiber directions. This is the key factor in our surface-based approach that overcomes the false negative problem in volume-based tractography. The crossing angle θ_xing is also thresholded at THθ_xing, which is set to be higher than TH_θin. The streamline successfully ends when it meets the gyral skeleton (colored in green in Fig. 2).

When we started at the red seed point in Fig. 2 (B), we considered all the possible directions (brown arrows). Out of all the successful streamlines from each direction at the seed point, we select the one with the minimum average deviation angle (blue streamline) along the tract, which then provides the tracking direction toward the other gyrus (precentral gyrus) as shown in Fig. 2 (C). Following the same tracking process, we compute the second half of the U-fiber and establish the complete connection between the two gyri. Note that it is not desirable to start the tracking from the gyri because they typically have more complicated fiber directions than sulcal regions as can be seen in Fig. 1 (C).

Our surface-based method has been implemented in C++. For an HCP subject, it takes about 1-2 hours to generate 2000 U-fiber streamlines between the precentral and postcentral gyrus on a 16-core 2.6-GHz Intel Xeon CPU with maximal memory consumption around 1GB.

3. Results

In this section, we present experimental results to demonstrate our surface-based reconstruction method of U-fibers in the SWM in the precentral and postcentral cortex, and compare its performance with the conventional tractography method on the large-scale dataset of 484 subjects from HCP. Note that our experiments in this paper only focused on the left hemisphere. For our surface-based method, we set the step size to be 0.01, and angle thresholds as TH_θin = 10° and TH_θxing = 10°. We set the minimum and maximum length of any track to be 20 and 80mm for the short association fibers as used in [11]. We randomly sampled seed points (at every new location) on the edges in the sulcal area until we acquired 2000 streamlines.

For the convectional tractography method, we used the probabilistic tracking algorithm in the popular MRtrix software [16]. Mostly the same parameter setups as above were used: the minimum and maximum length as 20 and 80mm; 2000 desired streamlines; step size of 0.1; maximum angle of successive steps as 10°. For a fair comparison, we used a comparable seed region by extending the seed region in our surface-based method to the neighboring voxels within 6mm.

Figure 3 shows the 2000 streamlines between the precentral and postcentral gyrus for one HCP subject (Subject ID: 101006) generated by our surface-based method (A, B) and MRtrix (C, D). Obviously, our surface-based method generated much more streamlines of the U-shape with one end meeting the precentral gyrus and other end meeting the postcentral gyrus. In addition, we also observed that most streamlines generated by the volume-based tractography algorithm in MRtrix did not follow closely the tangent space of the WM surface. More representative examples shown in Fig. 4 further confirm the above observation.

Fig. 3.

An example of U-fiber reconstruction for one subject (101006) using our surface-based method (A, B) and MRtrix (C, D). (A, C) The streamlines superimposed on the WM surface with the same color scheme as in Fig. 1. (B, D) The same streamlines plotted without the meshes in three different views by slightly rotating about the z-axis, where the blue ones (valid U-fibers) meet both the sides of the precentral and postcentral gyrus, but the green ones do not.

Fig. 4.

Four representative examples of U-fiber reconstruction using our surface-based method (A) and MRtrix (B). Each column corresponds to the same HCP subject.

For quantitative comparisons based on the large-scale datasets of 484 HCP subjects, we defined three measures to evaluate the performance of U-fiber reconstruction from different methods: how ‘well-connected,’ ‘well-U-shaped,’ and ‘well-distributed.’ How ‘well-connected’ was measured for each subject’s dataset by the number of streamlines with its end points within 3mm to the gyral skeletons of the precentral and postcentral gyrus as shown in Fig. 2 (A). How ‘well-U-shaped’ of an individual streamline was measured by the ratio of the curve length to the l₂ distance between two end points of the streamline. Lastly, for how ‘well-distributed’ we first partitioned the precentral and postcentral gyral skeletons into 20 equally space sections and counted how many sections were hit by the U-fiber streamlines. For both our surface-based method and MRtrix, we calculated the three measures from their U-fiber reconstruction results on the HCP dataset and plotted their distributions in Figure 5. It is obvious that our surface-based method significantly outperformed the conventional tractography algorithm implemented in MRtrix by generating about 6 times more ‘well-connected’, twice more ‘well-U-shaped’ streamlines that are significantly better ‘well-distributed’ along each gyrus.

Fig. 5.

Quantitative comparisons of the surface-based method and MRtrix using the large-scale datasets of 484 HCP subjects. (A) Box plots of the ‘well-connected’ measure (number of streamlines successfully connecting the precentral and postcentral gyrus out of the total 2000 streamlines generated by each method). (B) Histogram of the ‘well-U-shaped’ measures from all the generated streamlines. (C) Box plots of the ‘well-distributed’ measure (number of the gyral sections hit by the streamlines from each method).

4. Conclusion

In this paper, we proposed a novel surface-based framework for reconstructing the highly curved U-fibers in the SWM. As demonstrated in our experimental results on the large-scale HCP dataset, the proposed method overcomes the inherent limitations of volume-based tractography algorithms and successfully generates U-fiber streamlines precisely following the tangent space of the SWM surface that are ‘well-connected,’ ‘well-U-shaped,’ and ‘well-distributed.’ For future work, we will extend our work to reconstruct the U-fibers of other primary gyri and its application to data acquired with clinical imaging protocols.