Aymeric Stamm^{1,2,3}, Alessandro Zito^{4}, Valeria Callioni^{4}, Ilaria Sartori^{4}, Luca Torriani^{4}, and Simone Vantini^{1}

The corticospinal tract is a critical white matter pathway as it connects the primary motor cortex to the spinal cord and handles voluntary motion. Atlases of major brain connections do exist, but, surprisingly, atlases that depict the actual localisation of a specific pathway are missing. In this work, we propose a comprehensive statistical methodology for generating an atlas of the healthy CST. This should help designing efficient statistical tests for detecting damaged tissue along the CST and consequently improving patient outcome in a number of brain pathologies (tumors, strokes, Parkinson and related disorders, etc.).

Diffusion MR tractography enables the reconstruction of the CST as a set of streamlines, which are sequences of 3D points. Assuming we have a collection of $$$n$$$ healthy CSTs (left and right), our scope is to provide an atlas of the healthy CST, accounting for its challenging tree structure. We propose two solutions based on MDA and FDA. The general pipeline includes 4 steps: (S1) streamline parametrisation using the arc length over a uniform grid of size 50, (S2) streamline representation, (S3) outlier detection and (S4) streamline clustering for recovering the tree structure. S1 is common to both MDA and FDA approaches. Also, we performed the clustering mirroring the right CST onto the left CST and pulling all CSTs together. This avoids dealing with a computationally expensive label matching problem and offers a natural way of defining the optimal number of clusters as the minimal value above which a cluster sample size is less than a given threshold $$$n_\min$$$ in at least one subject and one hemisphere. MDA and FDA differ in the following aspects:

- (S2) Streamline are represented as a vector of shape features (including barycentre, spatial median, curvature, torsion, length and so on) in MDA while, in FDA, as a sequence of 3D points truncated to the length of the smallest one after removing length outliers.
- (S3) Streamline outliers are detected via the distance-based approach described in [3] applied to both barycenters and spatial medians in MDA while, in FDA, via functional boxplot [4] (amplitude outliers) and outliergram [5] (shape outliers).
- (S4) Clustering is performed using the clara algorithm [6] applied to the first principal component of the initial points (on the cortex) of the streamlines in MDA while, in FDA, using the k-medoid alignment algorithm [7] with a weighted $$$L^2$$$ distance that puts more weights on points near the cortex. Subsequently, within-cluster means of streamlines for generating atlases are obtained by first computing the mean feature vector and then identifying the streamline with feature vector at minimal Euclidean distance to the mean vector in MDA and by pointwise mean of aligned streamlines in FDA.

[1] Pujol, S., Wells, W., Pierpaoli, C., Brun, C., Gee, J., Cheng, G., Vemuri, B., Commowick, O., Prima, S., Stamm, A., Goubran, M., Khan, A., Peters, T., Neher, P., Maier-Hein, K. H., Shi, Y., Tristan-Vega, A., Veni, G., Whitaker, R., Styner, M., Westin, C.-F., Gouttard, S., Norton, I., Chauvin, L., Mamata, H., Gerig, G., Nabavi, A., Golby, A. and Kikinis, R. (2015) ‘The DTI Challenge: Toward Standardized Evaluation of Diffusion Tensor Imaging Tractography for Neurosurgery’, Journal of Neuroimaging, 25(6), pp. 875-82.

[2] Schott, G. D. (1993). Penfield's homunculus: a note on cerebral cartography. Journal of Neurology, Neurosurgery & Psychiatry, 56(4), pp. 329-33.

[3] Knorr, E. M. and Ng, R. (1997) ‘A Unified Notion of Outliers: Properties and Computation.’, KDD, pp. 219-22.

[4] Sun, Y. and Genton, M. G. (2012) ‘Adjusted functional boxplots for spatio-temporal data visualization and outlier detection’, Environmetrics, 23(1), pp. 54-64.

[5] Arribas-Gil, A. and Romo, J. (2014) ‘Shape outlier detection and visualization for functional data: the outliergram.’, Biostatistics (Oxford, England), 15(4), pp. 603-19.

[6] Kaufman, L. and Rousseeuw, P.J. (2008) ‘Clustering Large Applications (Program CLARA)’, In: Finding Groups in Data. John Wiley & Sons, Inc, pp. 126-63.

[7] Sangalli, L. M., Secchi, P., Vantini, S. and Vitelli, V. (2010) ‘-mean alignment for curve clustering’, Computational Statistics & Data Analysis. Elsevier B.V., 54(5), pp. 1219-33.

[8] Commowick, O., Wiest-Daessle, N. and Prima, S. (2012) ‘Block-matching strategies for rigid registration of multimodal medical images’, in 2012 9th IEEE International Symposium on Biomedical Imaging (ISBI). IEEE, pp. 700-3.

[9] Commowick, O., Wiest-Daesslé, N. and Prima, S. (2012) ‘Automated Diffeomorphic Registration of Anatomical Structures with Rigid Parts: Application to Dynamic Cervical MRI’, in Ayache, N., Delingette, H., Golland, P., and Mori, K. (eds) Medical Image Computing and Computer-Assisted Intervention – MICCAI 2012. Berlin, Heidelberg: Springer Berlin Heidelberg (Lecture Notes in Computer Science), pp. 163-70.

[10] Stamm, A., Commowick, O., Barillot, C. and Pérez, P. (2013) ‘Adaptive Multi-modal Particle Filtering for Probabilistic White Matter Tractography’, in Gee, J. C., Joshi, S., Pohl, K. M., Wells, W. M., and Zöllei, L. (eds) Information Processing in Medical Imaging. Berlin, Heidelberg: Springer Berlin Heidelberg (Lecture Notes in Computer Science), pp. 594-606.

Figure 1: Example of tractography CST reconstruction. Input streamlines of subject #1 obtained using a multi-modal particle filtering tractography algorithm with the single tensor diffusion model, in MNI space, superimposed on T1-weighted MNI template.

Figure 2: Atlas of the healthy CST using the MDA approach. Colors identify clusters. Left image provides the atlas representation where cluster representatives of the 20 subjects have been averaged while right image displays cluster representatives of each subject (hence 20 per cluster and per hemisphere).

Figure 3: Atlas of the healthy CST using the FDA approach. Colors identify clusters. Left image provides the atlas representation where cluster representatives of the 20 subjects have been averaged with diffusion tensors overlaid while right image displays cluster representatives of each subject (hence 20 per cluster and per hemisphere). Some portions of streamlines can be missing due to averaging of aligned streamlines which is restricted to the part of arc length domain common to all averaged streamlines.