Guillermo Gallardo^{1}, Gaston Zanitti^{2}, Samuel Deslauriers-Gauthier^{3}, Matthew Higger^{4}, Sylvain Bouix^{4}, Alfred Anwander^{5}, and Demian Wassermann^{2}

^{1}Neuropsycology, Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany, ^{2}Parietal, Inria Saclay - Ile de France, PAris, France, ^{3}Athena EPI, Universite Cote d’Azur, Inria, Sophia Antipolis, France, ^{4}Psychiatry Neuroimaging Laboratory, Brigham and Womens Hospital, Harvard Medical School, Boston, MA, United States, ^{5}Max Planck Institute for Human Cognitive and Brain Sciences, Leipzig, Germany

White-matter pathologies disrupt the white-matter organization, that manifests as deficits in brain function. When treating such pathologies, it is of great importance to infer which pathways are affected. However, the white-matter lesions hamper the use of tractography to track fiber bundles. In this work, leveraging diffusion imaging, we propose a novel diffusion-driven technique to improve the localization of brain pathways. Aggregating information from few healthy subjects, our technique is able to localize both the affected pathways and the lesion interrupting when tracking is not possible.

\begin{equation*}\label{eq:mvoting_weighted}L^*(x)=argmax_{l\in labels}\sum_{s\in S}p(L(x)=l|L_s(x))p(D(x)|D_{sl}(x))\end{equation*}

\begin{equation*}\begin{aligned}p(L(x)=l|L_s(x))=\begin{cases}1,&\text{if }L_s(x)=l\\0,&\text{otherwise}\end{cases}\end{aligned}\end{equation*}

\begin{equation*}\begin{aligned}p(D(x)|D_{sl}(x))=\begin{cases}\langle F(x),F_{sl}(x)\rangle,&\text{if }L_s(x)=l,\text{and }l\neq0\\\langle F(x),U\rangle,&\text{if }L_s(x)=0\\0,&\text{otherwise.}\end{cases}\end{aligned}\end{equation*}

The term $$$p(L(x)=l|L_s(x))$$$ represents the "vote" of each template[3], being 1 if the bundle $$$l$$$ is present in the template's voxel $$$x$$$ and 0 if not. The term $$$p(D(x)|D_{sl}(x))$$$ weighs the vote based on how much the voted bundle resembles the target's diffusion data. It express the probability of seeing the target's diffusion, $$$D(x)$$$, given that the template's bundle $$$l$$$ is present in the voxel, $$$D_{sl}(x)$$$.

We characterize $$$D(x)$$$ by fitting a Constrained Spherical Deconvolution (CSD) model[4] to the target's diffusion data and estimating a fiber orientation density function (fODF). The fODF $$$F_x(\theta, \phi)$$$ represents the fraction of fibers within the voxel $$$x$$$ aligned along the spherical coordinate $$$(\theta, \phi)$$$. Then, we characterize the within-voxel directionality of the template's bundle by looking at the entry and exit points (Fig. 1.3) of its streamlines. We estimate a fODF from these directions by means of CSD as with the diffusion data. If the template has no streamlines in the voxel, a uniform fODF, $$$U$$$, is used. Finally, we define $$$p(D(x)|D_{sl}(x))$$$ as the inner product between the diffusion-based fODF, $$$F(x)$$$, and the bundle fODF, $$$F_{sl}(x)$$$, with both fODFs normalized such that $$$\langle F(x),F(x)\rangle=1$$$.

To assess the benefits of adding a weight to the votes, we tested our technique in a more realistic scenario. We randomly selected 10 subjects from the Human Connectome Project, computed whole-brain tractography in them and extracted 6 bundles using the white-matter query language (WMQL)[6]. For each bundle we performed a leave-one-out cross-validation experiment. At each step, we inferred the bundle of one subject from the registered bundles of the others using our technique with and without weights. Using as ``ground truth'' the target's bundle obtained with WMQL, we quantified the sensitivity and specificity of both techniques (Table 1). In all bundles adding the weight achieves a lower sensitivity but a greater specificity. Therefore, we label fewer voxels, but those which are labeled can be trusted more.

To compare how the labeling behaves in the presence of lesions, we simulated a disruptive lesion in the Superior Longitudinal Fascicle (SLF) bundle (Fig. 3). We did so by selecting a spherical region of 4mm radius where the SLF passes and mixed the diffusion signal with isotropic diffusion. Figure 3 shows that the more isotropic the signal, the fewer voxels are labeled within the lesion, allowing to identify the lesion within the bundle.

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Figure 1. Pipeline of our technique, we show the simplified case of localizing only one bundle, affected by a disruptive lesion. (1) The bundle of interest is localized in healthy subjects and (2) registered to the target subject. (3) For each template bundle its directionality is represented by means of an fODF, its (4) similarity with the templates diffusion fODF is used as a weight to (5) vote for the presence of the bundle. Repeating this in every voxel allows to localize the bundle in the template's brain and the lesion.

Figure 2. Experiment on synthetic data. A. Phantom with only one fiber bundle. B. Phantom with crossing fibers. C. Phantom with no fiber (isotropic diffusion). D. Weights (blue line) a vote for a specific fiber bundle and its planar rotations would get compared against the vote for "non bundle". The weights are higher when the structure being voted is consistent with the underlying diffusion signal.

Figure 3. Lesions were simulated in a specific region (red circle) by mixing isotropic diffusion signal within the region. The figures show the voxels marked as SLF at different values of signal mixing. (A) 25% of isotropic signal, (B) 50% of isotropic signal, (C) 75% of isotropic signal and (D) 100% of isotropic signal. Results show that fewer voxels with increased tissue lesioning.

Table 1. Sensitivity and specificity of our proposed method (Weighted) and a non-weighted voted (Majority) when inferring single bundles from 9 subjects. The inferred bundles are: Superior Longitudinal Fasciculus (SLF) I, II and III, Inferior Longitudinal Fasciculus (ILF), Cortico Spinal Tract (CST), and Inferior Occipito-Frontal Fascicle (IOFF).