Inferring the Localization of White-Matter Tracts using Diffusion Driven Label Fusion

Let $labels=\{l_{i}\}$ be the set of labels representing tracts in the brain. Let ${L_{s}},s\in S$ represent the labeling of a set of template subjects $S$, where each $L_{s}\in labels^{v_{x}\times v_{y}\times v_{z}}$ is a 3D volume with dimension $(v_{x},v_{y},v_{z})$ representing the labeling of template subject s. Majority Voting [8] infers the label of each voxel $x$ in a target subject by computing:

In this case, each subject votes for a label, and the label with the most votes is assigned to the target voxel. It is important to notice that no information from the target subject is being used to infer the label $L^{*}(x)$.

Majority Voting (Eq. 1) decides the label of a target voxel based on the “votes” of template subjects, without using any information from the target subject. However, when inferring the location of white matter pathways, we can profit from the fact that water molecules tend to diffuse in the direction of the pathways. This means that we can use diffusion information to weigh the voting process and help locate the bundles. In particular, votes for tracts aligned with the diffusion of the target subject should get higher weights.

One way to characterize the directionality on a diffusion weighted image (DWI) is by means of fiber orientation distribution functions (fODFs) [12]. Now, we will first explain how to estimate a fODF from both DWI and tracts. Then, we will show how to leverage them in order to compute weights for each vote.

We created three types of diffusion weighted image phantoms using Phantomas [16]. The first phantom possess only one tract, traveling from one side to the other of the image horizontally (Fig. 2 A). The second possess two crossing tracts, forming a 90 degrees angle between them (Fig. 2 B). The last has no fibers on it, representing isotropic diffusion (Fig. 2 C). From each phantom we generated 30 diffusion weighted images (DWIs). All the DWIs were generated using a signal to noise ratio (SNR) of 20, and a resolution of $1mm$ per voxel. The final images are 3-dimensional matrices with 10 voxels in each dimension. Having such small images allows us to test our label fusion on a controlled environment. We estimated fiber orientation distributions (FODs) in each voxels by means of constrained spherical deconvolution (CSD)[17] using Dipy [18]. The fODFs were discretized on a sphere with $n=100$ vertices. A uniform fODF, $U$, was created by assigning to each vertex of the sphere the value $\sqrt{(}n)/n$, making $\langle U,U\rangle=1$.

To test our technique in more realistic scenarios, we randomly selected 50 subjects from the HCP500 dataset from the Human Connectome Project. For each subject, we first estimated FODs by means of CSD from they diffusion data. Then, we leveraged the FODs to perform whole-brain probabilistic tractography. For the tractography we used each voxel in the white-matter as a seed, and simulated 8 particles per seed [18]. Finally, we extracted four tracts from the left hemisphere using the implementation of the white-matter query language (WMQL) [15]. For each subject we computed non linear registrations to the rest using as reference their T1w images [19]. Using the resulting warp transformations, we registered the tracts between every pair of subjects.

In order to study how the tract’s directionality influences its vote weight, we started by reconstructing the tract present in the first phantom (Fig. 2 A). For this, we took one of the 30 generated DWIs, and computed 1000 streamlines by means of probabilistic tracking from the voxels through which the tract passes. If our technique behaves correctly, the tract we reconstructed should obtain a high vote weight in the DWIs derived from phantom A. At the same time, any change in its directionality, should decrease the received weight. To assess this, we computed weights in 30 DWIs for the reconstructed tract, and for planar rotations of it around the central voxel. Figure 3 A shows the obtained weights on the first phantom. Effectively, the weight starts to rapidly decrease as the angle increments and the directionality of the tract moves away from that of the diffusion.

Figure 3 B shows the weights obtained when computing weights of the reconstructed tract in 30 DWIs derived from the second phantom (Fig. 2 B), which have a fiber crossing in the central voxel. In this case, the weight is higher when the reconstructed tract aligns with one of the crossing fibers (at 0 degrees or 90 degrees), while rapidly decaying in between them. Finally, figure 3 C shows the weights when using 30 DWIs with isotropic diffusion (Fig. 2 C). In this case, the weight is always low, driven by the discrepancy between the directionality of the tract and the free diffusion present in the DWIs.

To assess that the proposed model is not over-weighing tracts, we also computed the weight that a “non-tract” label would receive in each of the phantoms. This is, when the user votes that there is no tract present in the voxel. As explained in section 2, equation 3, when a subject is voting for a non-tract label, a uniform fiber ODF is compared against the diffusion fODF. Figure 2 A, B, and C show the weight obtained in the central voxel when a subject is voting for the label “non-tract”. In figure 2 A, we can see that the weight of “non-tract” is low, specially when compared with the high weight of the correctly aligned tracts (low angle rotations). This is driven by the highly directional underlying diffusion data of the phantom. Figure 2 B shows that the weight of a “non-tract” vote is similar to that of an aligned tract. Finally, figure 2 C shows always a higher weight for the “non-tract” than for any tract, consistent with the isotropic diffusion of our third phantom.

To assess that our technique correctly weighed votes, we created three different phantoms, and derived DWIs from them. The phantoms had: (A) a single bundle running straight, (B) a 90 degrees crossing bundles, and (C) no bundles. We reconstructed the tract in the phantom A, and computed the weights obtained by the tract and planar rotations of it in the 3 phantoms. Our results show that our technique assigns high weights when the tract aligns with the bundles represented in the underlying diffusion data. Furthermore, when the directions differ by more than 10 degrees, the weight starts to drop rapidly. Our results also show that in the phantom C, where there is no bundle, the ’no-tract’ vote receives a higher weight than any rotation. These results show that our technique is able to correctly weight each label based on diffusion directionality.

To test our technique in realistic data, we registered tracts between different subjects of the HCP. Using these registered tracts, we inferred the position of individual tracts in each subject. Table 1 shows that in each inferred tract, our technique achieved on average a 13% higher specificity at the trade off of obtaining a 5% lower sensitivity. Hence, our techniques presents less false positives at the cost of more false negatives, making it more conservative than Majority Voting. Furthermore, our specificity is always higher than 0.7. This means that our technique was able to discover less voxels belonging to the tracts, but at least 70% of those labeled are correct.

We further characterized the behaviour of our technique by simulating non-deforming lesions in the white-matter of a HCP subject. Particularly, we defined a spherical region on the path of the Superior Longitudinal Fasciculus, and increased its Fractional Anisotropy until achieving isotropic diffusion. In doing so, we effectively simulated a focal lesions ranging from an infiltration to the interruption of a tract. Under these common scenarios in which the diffusion data is affected, reconstruction techniques such as WMQL are not able to recover the tract. This is because, as the diffusion data losses directionality, the tracking becomes more and more hampered. Indeed, in our experiment WMQL was not able to dissect the affect SLF, since the process of tractography was stopped by the lesion. On the other hand, both majority voting and our technique were able to correctly label the affected tract. However, as shown by our results (Figure 4), only our technique was able to label less voxels as the tract became more interrupted. This is the expected behaviour in the presence of a lesion, since the more damaged an area is, the less it is expected for the tract to be present. When the tract is completely interrupted, our technique is still able to label the voxels surrounding the lesions, thus enabling to detect the affected tract. Therefore, our technique not only allows to detect the affected tracts, but also gives information on how altered the tract is.