Abstract
We propose a novel connectivity measure between brain regions using diffusion-weighted MRI. This connectivity measure is based on optimal sub-Finslerian geodesic front propagation on the 5D base manifold of (3D) positions and (2D) directions, the so-called sphere bundle. The advantage over spatial front propagations is that it prevents leakage at omnipresent crossings. Our optimal fronts on the sphere bundle are geodesically equidistant w.r.t. an asymmetric Finsler metric, and can be computed with existing anisotropic fast-marching methods. Comparisons to ground truth connectivities provided by the ISBI-HARDI challenge reveal promising results, both quantitatively and qualitatively. We also apply the connectivity measures to real data from the Human Connectome Project.
Keywords
Jorg Portegies, Stephan Meesters and Remco Duits: Joint main authors.
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‘Quasi’ pertains to its asymmetrical nature.
References
Barron, D.S., Tandon, N., Lancaster, J.L., Fox, P.T.: Thalamic structural connectivity in medial temporal lobe epilepsy. Epilepsia 55(6), e50–5 (2014)
Bekkers, E., Duits, R., Mashtakov, A., Sanguinetti, G.: A PDE approach to data-driven sub-riemannian geodesics in SE(2). SIAM J. Imaging Sci. 8(4), 2740–2770 (2015)
Chen, D.: New minimal paths models for tubular structure extraction and image segmentation. Ph.D. thesis, Université Paris Dauphine (2016)
Daducci, A., Caruyer, E., Descoteaux, M., Thiran, J.P.: HARDI reconstruction challenge 2013. In: IEEE International of Symposium on Biomedical Imaging (2013)
Dela Haije, T.C.J., Duits, R., Tax, C.M.W.: Sharpening fibers in diffusion weighted MRI via erosion. In: Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data, pp. 97–126. Springer (2014)
Descoteaux, M., Deriche, R., Knosche, T.R., Anwander, A.: Deterministic and probabilistic tractography based on complex fibre orientation distributions. IEEE Trans. Med. Imaging 28(2), 269–286 (2009)
Desikan, R.S., Segonne, F., et al.: An automated labeling system for subdividing the human cerebral cortex on MRI scans into gyral based regions of interest. Neuroimage 31(3), 968–980 (2006)
Duits, R., Franken, E.M.: Left-invariant diffusions on the space of positions and orientations and their application to crossing-preserving smoothing of HARDI images. Int. J. Comput. Vis. (IJCV) 92, 231–264 (2011)
Duits, R., Meesters, S.P.L., Mirebeau, J.M., Portegies, J.M.: Optimal paths for variants of the 2D and 3D reeds-shepp car with applications in image analysis, JMIV S.I. Differ. Geom. Orientat. Anal. 60(6), 818–846 (2018)
Fisher, R., Salanova, V., et al.: Electrical stimulation of the anterior nucleus of thalamus for treatment of refractory epilepsy. Epilepsia 51(5), 899–908 (2010)
Fletcher, P., Joshi, S.: Riemannian geometry for the statistical analysis of diffusion tensor data. Signal Process. 87(2), 250–262 (2007)
Fuster, A., Dela Haije, T., Tristán-Vega, A., Plantinga, B., Westin, C.F., Florack, L.: Adjugate diffusion tensors for geodesic tractography in white matter. J. Math. Imaging Vis. 54(1), 1–14 (2016)
Glasser, M.F., Sotiropoulos, S.N., Wilson, J.A., Coalson, T.S., Fischl, B., Andersson, J.L., Xu, J., Jbabdi, S., Webster, M., Polimeni, J.R., Van Essen, D.C., Jenkinson, M.: The minimal preprocessing pipelines for the human connectome project. Neuroimage 80, 105–124 (2013)
Gologorsky, Y., Alterman, R.: Chapter 3—cerebral-deep. In: Arle, J.E., Shils, J.L. (eds.) Essential Neuromodulation, pp. 47–72. Academic Press, San Diego (2011)
Granziera, C., Hadjikhani, N., Arzy, S., Seeck, M., Meuli, R., Krueger, G.: In-vivo magnetic resonance imaging of the structural core of the papez circuit in humans. Neuroreport 22(5), 227–31 (2011)
Hodaie, M., Cordella, R., Lozano, A.M., Wennberg, R., Dostrovsky, J.O.: Bursting activity of neurons in the human ATN. Brain Res. 1115(1), 1–8 (2006)
Jakab, A., Blanc, R., Berényi, E.L., Székely, G.: Generation of individualized thalamus target maps by using statistical shape models and thalamocortical tractography. Am. J. Neuroradiol. 33(11), 2110–2116 (2012)
Jankowski, M.M., Ronnqvist, K.C., Tsanov, l., O’Mara, S.M.: The anterior thalamus provides a subcortical circuit supporting memory and spatial navigation. Front. Syst. Neurosci. 7, 45 (2013)
Jbabdi, S., Bellec, P., Toro, R., Daunizeau, J., Pélégrini-Issac, M., Benali, H.: Accurate anisotropic fast marching for diffusion-based geodesic tractography. Int. J. Biomed. Imaging 2008 (2008)
Jenkinson, M., Beckmann, C.F., Behrens, T.E.J., Woolrich, M.W., Smith, S.M.: FSL. Neuroimage 62(2), 782–790 (2012)
Jeurissen, B., Tournier, J.D., Dhollander, T., Connelly, A., Sijbers, J.: Multi-tissue constrained spherical deconvolution for improved analysis of multi-shell diffusion MRI data. Neuroimage 103(0), 411–426 (2014)
Kerrigan, J.F., Litt, B., Fisher, R.S., Cranstoun, S., French, J.A., Blum, D.E., Dichter, M., Shetter, A., Baltuch, G., Jaggi, J., Krone, S., Brodie, M., Rise, M., Graves, N.: Electrical stimulation of the anterior nucleus of the thalamus for the treatment of intractable epilepsy. Epilepsia 45(4), 346–354 (2004)
Krauth, A., Blanc, R., Poveda, A., Jeanmonod, D., Morel, A., Székely, G.: A mean three-dimensional atlas of the human thalamus: generation from multiple histological data. Neuroimage 49(3), 2053–2062 (2010)
Melonakos, J., Mohan, V., Niethammer, M., Smith, K., Kubicki, M., Tannenbaum, A.: Finsler tractography for white matter connectivity analysis of the cingulum bundle. Med. Image Comput. Comput. Assist. Interv. 10(01), 36–43 (2007)
Melonakos, J., Pichon, E., Angenent, S., Tannenbaum, A.: Finsler active contours. IEEE Trans. Pattern Anal. Mach. Intell. 30(3), 412–423 (2008)
Mirebeau, J.: Anisotropic fast-marching on cartesian grids using lattice basis reduction. SIAM J. Numer. Anal. 52(4), 1573–1599 (2014)
Mirebeau, J.M.: Fast-marching methods for curvature penalized shortest paths. JMIV 60, 784–815 (2018)
MomayyezSiahkal, P., Siddiqi, K.: 3D stochastic completion fields for mapping connectivity in diffusion MRI. IEEE Trans. Pattern Anal. Mach. Intell. 35(4), 983–995 (2013)
Niemann, K., Mennicken, V.R., Jeanmonod, D., Morel, A.: The morel stereotactic atlas of the human thalamus: atlas-to-MR registration of internally consistent canonical model. Neuroimage 12(6), 601–616 (2000)
O’Donnell, L., Haker, S., Westin, C.F.: New Approaches to Estimation of White Matter Connectivity in Diffusion Tensor MRI: Elliptic PDEs and Geodesics in a Tensor-Warped Space, pp. 459–466. Springer, Berlin (2002)
Péchaud, M., Descoteaux, M., Keriven, R.: Brain connectivity using geodesics in HARDI. Med. Image Comput. Comput. Assist. Interv. 12(2), 482–489 (2009)
Portegies, J.M., Fick, R.H.J., Sanguinetti, G.R., Meesters, S.P.L., Girard, G., Duits, R.: Improving fiber alignment in HARDI by combining contextual PDE flow with constrained spherical deconvolution. PLoS One 10(10) (2015)
Portegies, J.: PDEs on the Lie Group SE(3) and their applications in diffusion-weighted MRI. Ph.D. thesis, Department of Mathematics and Computer Science, TU/e (2018)
Prckovska, V., Rodrigues, P., Duits, R., Vilanova, A., ter Haar Romeny, B.: Extrapolating fiber crossings from DTI data. Can we infer similar fiber crossings as in HARDI? In: CDMRI 2010. vol. 1, pp. 26–37. Springer, Beijing (2010)
Reisert, M., Skibbe, H.: Fiber continuity based spherical deconvolution in spherical harmonic domain. In: Mori, K.e.a. (ed.) MICCAI, pp. 493–500. Springer (2013)
Sanguinetti, G., Bekkers, E., Duits, R., Janssen, M.H.J., Mashtakov, A., Mirebeau, J.M.: Sub-Riemannian Fast Marching in SE(2). Springer (2015)
Sepasian, N., ten Thije Boonkkamp, J.H.M., ter Haar Romeny, B.M., Vilanova, A.: Multivalued geodesic ray-tracing for computing brain connections using diffusion tensor imaging. SIAM-JIS 5(2), 483–504 (2012)
Shah, A., Jhawar, S.S., Goel, A.: Analysis of the anatomy of the Papez circuit and adjoining limbic system by fiber dissection techniques. J. Clin. Neurosci. 19(2), 289–98 (2012)
Tax, C.M.W., Duits, R., Vilanova, A., ter Haar Romeny, B.M., Hofman, P., Wagner, L., Leemans, A., Ossenblok, P.: Evaluating contextual processing in diffusion MRI: application to optic radiation reconstruction for epilepsy surgery. PLoS One 9(7), 1–19 (2014)
Tournier, J.D., Calamante, F., Connelly, A.: MRtrix: diffusion tractography in crossing fiber regions. Int. J. Imaging Syst. Technol. 22(1), 53–66 (2012)
Tournier, J.D., Yeh, C.H., Calamante, F., Cho, K.H., Connelly, A., Lin, C.P.: Resolving crossing fibres using constrained spherical deconvolution: validation using diffusion-weighted imaging phantom data. NeuroImage 42(2), 617–625 (2008)
Van Essen, D.C., Smith, S.M., Barch, D.M., Behrens, T.E.J., Yacoub, E., Ugurbil, K.: The WU-Minn human connectome project. NeuroImage 80, 62–79 (2013)
Vogt, T., Lellmann, J.: Measure-valued variational models with applications to diffusion-weighted imaging. J. Math. Imaging Vis. (2018)
Acknowledgements
Data provided in part by the HCP, WU-Minn Consortium (PI’s: D. Van Essen and K. Ugurbil; 1U54MH091657). We thank S. Mariën for co-developing the rching Tool, available at https://goo.gl/D5Q7dE (Downloads section). We thank J.M. Mirebeau for his Hamiltonian fast-marching C++-code, available at https://www.math.u-psud.fr/~mirebeau. The research leading to these results has received funding from the European Research Council under the European Community’s 7-th Framework Programme (FP7/2007-2013) / ERC grant Lie Analysis, agr. nr. 335555.
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Portegies, J., Meesters, S., Ossenblok, P., Fuster, A., Florack, L., Duits, R. (2019). Brain Connectivity Measures via Direct Sub-Finslerian Front Propagation on the 5D Sphere Bundle of Positions and Directions. In: Bonet-Carne, E., Grussu, F., Ning, L., Sepehrband, F., Tax, C. (eds) Computational Diffusion MRI. MICCAI 2019. Mathematics and Visualization. Springer, Cham. https://doi.org/10.1007/978-3-030-05831-9_24
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