Synopsis

Tractography has proven particularly effective for studying noninvasively the neuronal architecture of the brain but recent studies have showed that the high incidence of false positives can significantly bias any connectivity analysis. We present a novel processing framework that can dramatically reduce these false positives, i.e. improving specificity, without affecting the sensitivity, by considering two very basic observations about white-matter anatomy. Our results may have profound implications for the use of tractography to study brain connectivity.

Context and motivation

Over the years, tractography has proven particularly effective for studying noninvasively the neuronal architecture of the brain. However, recent studies have challenged its use for connectivity analyses, notably showing that the presence of false positive connections in the reconstructions can significantly bias the topological properties of the estimated brain networks^1,2,3. In this study, we present a novel processing framework that can dramatically reduce the incidence of these false positives, i.e. improving specificity, without affecting the true ones, i.e. sensitivity. Our formulation combines basic prior knowledge about brain anatomy with group-sparsity regularization into an efficient convex optimization problem. In fact, tractography algorithms usually exploit only the directional information estimated with diffusion MRI: we speculate that this information is not enough and we advocate the need for additional data to help tractography producing more accurate reconstructions. We evaluated quantitatively the effectiveness of our formulation on different tractography reconstructions using a realistic digital phantom with known ground-truth.

Methods

COMMIT. Convex Optimization Modeling for Microstructure Informed Tractography^4,5 (COMMIT) is a framework to complement tractography with additional microstructural information about the neuronal tissue. The originality of this solution lies in the possibility to express tractography and tissue microstructure in a unified formulation using convex optimization. The observation model is $$$\boldsymbol{y} = \boldsymbol{A} \boldsymbol{x} + \eta$$$, where $$$\boldsymbol{A}$$$ is the linear operator implementing the specific multi-compartment model adopted to characterize the neuronal tissue, $$$\boldsymbol{x}$$$ are the contributions of all compartments of the model to explain the DWI data $$$\boldsymbol{y}$$$ and $$$\eta$$$ accounts for noise and modeling errors. The system is solved as a least-squares problem: $$$\operatorname{argmin}_{\boldsymbol{x} \ge 0} \; || \boldsymbol{A} \boldsymbol{x} - \boldsymbol{y} ||_2^2$$$.

Adding anatomical priors. We extended COMMIT by considering two very basic observations about WM anatomy: (i) streamlines are not "just lines" but represent neuronal fibers, and (ii) they are organized in bundles. To enforce the first assumption, we implemented a simple model in $$$\boldsymbol{A}$$$ that assigns a contribution, i.e. volume or cross-sectional area, to each streamline proportionally to its length inside each voxel. Then, the total amount of streamlines that traverse a voxel must sum up to the intra-axonal volume fraction estimated in that voxel, which can be estimated e.g. using standard models like NODDI⁶ or SMT⁷. To implement the second prior, we clustered all streamlines connecting the same pair of ROIs into groups and then added a regularization term to try explaining the data using the minimum number of bundles; this is achieved by solving the problem using the group lasso: $$\operatorname{argmin}_{\boldsymbol{x} \ge 0} \; || \boldsymbol{A} \boldsymbol{x} - \boldsymbol{y} ||_2^2 \; + \; \lambda \, \sum_{g \in \mathcal{G}} || \boldsymbol{x}^{(g)} ||_2$$ where $$$\mathcal{G}$$$ represents a partition of the streamlines into groups.

Experimental settings. We tested different tractography algorithms on the ISBI Challenge 2013 dataset, which consists of 27 fiber bundles mimicking the major WM tracts of the brain (Fig. 1A). The DWI signal was simulated using Phantomas⁸ according to the CHARMED model⁹, 64 directions, b-value=3000 s/mm², and Rician noise (SNR=30). The ground-truth connectivity of the dataset is known (Fig. 1C), so we could quantitatively evaluate the reconstructions in terms of True Positive (TP) and False Positive (FP) bundles (Fig. 1B). Due to space limitations, we report only the results for deterministic tracking and the following parameters: 1 million fibers, angle-threshold $$$45^{\circ}$$$; results were consistent for any combination of parameters and tracking method.

Results

Fig. 2A reports the number of True Positives (TP) and False Positives (FP) in the raw tractogram. We can easily see that the tracking algorithm was able to reconstruct all 27 TP, i.e. high sensitivity, but at the price of recovering 235 FP, i.e. very low specificity. This result agrees with previous literature^1,2,3. On the other hand, Fig. 2B clearly shows that using COMMIT the number of FP can be dramatically reduced (from 235 to 18) without affecting the sensitivity (the tractogram still contains all 27 TP).

Discussion and Conclusion

Recent studies have shown that current tractography algorithms are quite good at reconstructing the major WM bundles, i.e. high sensitivity, but at the price of also recovering a high incidence of false positive ones, i.e. low specificity. In particular, those false positive streamlines can dramatically bias any subsequent analysis, as non-existent structures are mixed with real ones. In this work, we showed that adding basic prior information on the organization of the neuronal tissue can help tractography in dramatically improving the quality of reconstructions. Our results represent an important step forward to improve the accuracy of tractography and may have profound implications for the use of tractography to study brain connectivity.

Acknowledgements

References

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