Synopsis

Tractography has proven particularly effective for studying non-invasively the neuronal architecture of the brain, but recent studies have showed that the high incidence of false-positives can significantly bias any connectivity analysis. Last year we presented a method that extended COMMIT framework to consider the prior knowledge that white matter fibers are organized in bundles. Inspired by this, here we propose another extension to further improve the quality of the tractography reconstructions. We introduce a novel regularization term based on the multilevel hierarchy organization of the human brain and we test the results on both synthetic phantom and in vivo data.

Introduction

Tractography employs diffusion MRI (dMRI) data to non-invasively reconstruct the trajectories of major white-matter tracts¹. Thanks to this unique ability it has been widely used to study the structural brain connectivity. However, recent studies have shown that the accuracy of the reconstructions is inherently limited²: existing algorithms suffer from an intrinsic trade-off between sensitivity, i.e. false-negatives connections (FNC), and specificity, i.e. false-positives (FPC), where specificity seems to be the major bottleneck³ when studying the topological properties of brain networks⁴.

To overcome this limitation, a number of solutions have been proposed⁵. A common approach consists of filtering the reconstructed fibers (or streamlines) by using forward models and assessing their contribution using optimization. The Convex Optimization Modeling for Microstructure Informed Tractography^6-7 (COMMIT) is one of them. Last year we extended it by adding a regularization term which allowed us to inject into COMMIT prior knowledge on the anatomical organization of the brain⁸ and which has proved very effective in reducing false positives⁸. A natural extension of this formulation is to further refine this prior by considering that the human brain has a hierarchical organization⁹, i.e. axonal bundles connecting different cortical regions may be composed by sub-bundles (or fascicles). Here, we propose a novel regularization term based on a multilevel-hierarchical organization of the streamlines and we test the benefits of considering such anatomical prior in COMMIT using synthetic and in vivo data.

Methods

Proposed approach. The observation model is $$$\mathbf{y}=\mathbf{Ax}+\eta$$$, where the matrix $$$\mathbf{A}$$$ implements a generic multi-compartment model to characterize the white-matter (WM) tissue, $$$\mathbf{x}$$$ are the contributions of the model’s compartments used for explaining the dMRI data $$$\mathbf{y}$$$, and $$$\eta$$$ represent the noise.

To consider the prior knowledge that WM fibers are organized in bundles, the problem is solved with non-negative least squares (NNLS) coupled with a regularization term that penalizes or promotes groups of streamlines connecting different pairs of cortical regions. The mathematical formulation is then $$$\text{argmin}_{\mathbf{x}\geq0}||\mathbf{Ax}-\mathbf{y}||^2_2+\lambda\sum_{g \in G}||\mathbf{x}^{(g)}||_2$$$, where $$$G$$$ represents a partition in groups of the streamlines in the tractogram. This formulation was presented in ⁸ and we call it gNNLS.

To further refine this prior and consider that axons may have a hierarchical organization⁹, we redefine the partition $$$G$$$ by creating a multilevel-hierarchical structure of the streamlines (Fig.1). The first level uses the same partitioning of gNNLS; then, for each a group we create a second level by clustering the streamlines of each group using QuickBundle¹⁰. This new partitioning may allow filtering out those false-positives fascicles that are part of true-positive bundles (violet in Fig.1), which gNNLS would not be able to distinguish. We call this formulation hNNLS.

For sake of simplicity, in both formulations we implemented a simple forward-model which assigns a signal contribution to each streamline that is proportional to its length inside a voxel. The total amount of streamlines that traverse a voxel must sum to the voxel's total intra-axonal signal fraction. This value can be estimated using standard models such as NODDI¹¹ and SMT¹²; in this work we used the latter.

Experimental settings. We tested gNNLS and hNNLS on both synthetic and in-vivo data. The synthetic phantom is illustrated in Fig.2 and is used for illustrative purposes. In vivo data was acquired on a Siemens Prisma 3T scanner using 5 b-values in the range [300,3000]s/mm², and was used to reconstruct 10M streamlines with iFOD2¹³. A T1-MPRAGE image was acquired to perform gray-matter parcellation using FreeSurfer.

Results and Discussion

Although gNNLS was shown to be very effective for removing FPC⁸, in the ambiguous configuration of Fig.2 it fails to recover the ground-truth configuration, while hNNLS successfully estimates it (Fig.3). To keep the streamline S2 (true-positive) between B-C, which is required to explain the signal in the upper-right voxel, gNNLS has to keep as well S4, which is a false-positive streamline but belongs to the same group of S2. As a consequence, also S1 is removed and, in turn, S3. Because of its finer partition, hNNLS is instead able to decouple their contributions.

In in vivo data, a qualitative look at the connectivity matrices shows that the number of streamlines kept by gNNLS and hNNLS is similar (Fig.4). Yet, by visually inspecting known bundles we can appreciate the different filtering of the two formulations (Fig.5). Indeed, hNNLS is able to filter out streamlines that follow a strange path by keeping intact the structure of the bundle.

Conclusion

We showed that by adding to COMMIT a regularization term to promote a hierarchical structure of the streamlines, it is possible to further improve the quality of tractography reconstructions. We speculate that this improved reconstruction accuracy can have a significant impact on any analysis of structural brain connectivity.

Acknowledgements

References

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