Juan Luis Villarreal-Haro^{1}, Alonso Ramirez-Manzanares^{1}, and Juan Antonio Pichardo-Corpus^{2}

We study the impact of the brain tractography false positives in the brain connectivity graphs. The representative input database for the analysis is the set of tractograms from the participants on the ISMRM-2015 Tractography Challenge. We propose 2 novel metrics to rank the quality of a tractogram when it is compared with a known ground truth. The results of this study indicate that the the estimation of graph communities is robust to high levels of overestimation in the connectivity.

The numerous problems we face when using Magnetic Resonance (MR) diffusion tractography to estimate brain connectivity has been reported in research literature.^{1,2} One of its major disadvantages is the overwhelming number in the estimation of false-positive connections.^{1} Solving these problems is challenging due to the partial volume effects, noise, and trajectory-uncertainty in the MR images.^{1,2,3} This work analyzes brain connectivity in terms of graph structure and indicates how this structure is affected by the tractography problems. We develop methods that allow us to characterize the estimated brain connectivity in terms of graph comparisons. Our case-of-study is the state-of-the-art database tractograms stemming from the ISMRM 2015 Tractography Challenge (ISMRM2015-TC).^{4} We explore how to properly rank the tractograms’ performance regarding a known Ground-Truth (GT). Our approach provides a novel quality metric based on graph-communities features.

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2. Neher P F, Descoteaux M, et al. Strengths and weaknesses of state of the art fiber tractography pipelines-A comprehensive in-vivo and phantom evaluation study using Tractometer. Med. Image Anal. 2015, 26(1), 287-305.

3. Yamada K, Sakai K, et al. MR tractography: a review of its clinical applications. Magn. Reson. Med Sci. 2009;8(4), 165-74.

4. ISMRM 2015 Tractography Challenge Results. http://www.tractometer.org/ismrm_2015_challenge/results/. Accessed August 02, 2018.

5. Barabasi A-L. Network science. 2016. Cambridge University Press.

6. Newman M. Networks: an introduction. 2010, Oxford University Press.

7. Estrada E, Knight P A. A first course in network theory. 2015, Oxford University Press.

8. Blondel V D, Guillaume J-L et al. Fast unfolding of communities in large networks. Journal of statistical mechanics: theory and experiment. 2008, 10, 10008.

9. Real R, Vargas J M. The probabilistic basis of Jaccard's index of similarity. Systematic biology. 1996, 45(3), 380-385.

Figure 1. Centrality Measurements (CM) of some participants of the ISMRM2015-TC^{4} and some well-known Random Graphs (Barabasi, Erdos-Rengy and Watts-Strogatzs). The participants in this table were the winners of the metrics coming from the ISMRM2015-TC (Valid Bundles, Valid Connections, Overlap, F1 metric, see^{4}) A blue color indicates similitude to the GT properties (a high-quality graph), while red indicates the opposite. We highlight that by using CMs some random graphs are closer to the GT than some winners of the ISMRM2015-TC, hence, the CM are not useful to compare similitude. Then, more informative metrics should be developed.

Figure 2. Graphical representation of our JIG proposal. LEFT: Comparison of the real communities belonging to the brain-data GT (rows) vs. the id=9_2 participant communities in the ISMRM2015-TC (columns). Darker entries indicate large JI coefficients (higher similarity). For a high-quality graph, we expect a single dark entry per row. RIGHT: The visualization of the 4 darkest entries in the matrix in LEFT in the brain-space. Green and Red stand for GT and estimated communities, respectively, Blue denotes their intersection. This visualization allows to appreciate the communities' overlap, and the common structure they share, thus, allows to appreciate the partially-recovered structure.

Figure 3. Our novel 2D metric composed of MD and JIG distances computed for all the participants in the ISMRM2015-TC. The closer a graph (a point) is to (0,1) the more similar to the GT is. For each Panel, we colored the participants from better (dark) to worse (light) according to the indicated ISMRM2015-TC ranking metric (indicated in each panel title).

Figure 4. Our proposed metric (as in Figure 3) for the whole graph database. Black triangles have a subset of the GT edges. Blue pentagons denote perturbations of the GT where false-positives were added. Magenta stars denote graphs with almost the minimum GT connectivity structure (before becoming completely random) plus some false-positive edges. Cyan squares are well-known random graphs. Red diamonds denote combinations of the properties above. The area bounded by triangles-pentagons-stars denotes the possibilities of the metric for the estimated graphs. The position inside this area indicates connectivity features. Note that ISMRM2015-TC participants (green circles) are inside this area.