Brain Connectivity Measures via Direct Sub-Finslerian Front Propagation on the 5D Sphere Bundle of Positions and Directions

  • Jorg Portegies
  • Stephan Meesters
  • Pauly Ossenblok
  • Andrea Fuster
  • Luc Florack
  • Remco DuitsEmail author
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)


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.


Diffusion MRI Brain connectivity Fast-marching Sphere bundle Geodesic fronts Finsler geometry 



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 (Downloads section). We thank J.M. Mirebeau for his Hamiltonian fast-marching C++-code, available at 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.


  1. 1.
    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)CrossRefGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chen, D.: New minimal paths models for tubular structure extraction and image segmentation. Ph.D. thesis, Université Paris Dauphine (2016)Google Scholar
  4. 4.
    Daducci, A., Caruyer, E., Descoteaux, M., Thiran, J.P.: HARDI reconstruction challenge 2013. In: IEEE International of Symposium on Biomedical Imaging (2013)Google Scholar
  5. 5.
    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)Google Scholar
  6. 6.
    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)CrossRefGoogle Scholar
  7. 7.
    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)CrossRefGoogle Scholar
  8. 8.
    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)MathSciNetCrossRefGoogle Scholar
  9. 9.
    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)Google Scholar
  10. 10.
    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)CrossRefGoogle Scholar
  11. 11.
    Fletcher, P., Joshi, S.: Riemannian geometry for the statistical analysis of diffusion tensor data. Signal Process. 87(2), 250–262 (2007)CrossRefGoogle Scholar
  12. 12.
    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)MathSciNetCrossRefGoogle Scholar
  13. 13.
    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)CrossRefGoogle Scholar
  14. 14.
    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)CrossRefGoogle Scholar
  15. 15.
    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)CrossRefGoogle Scholar
  16. 16.
    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)CrossRefGoogle Scholar
  17. 17.
    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)CrossRefGoogle Scholar
  18. 18.
    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)Google Scholar
  19. 19.
    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)CrossRefGoogle Scholar
  20. 20.
    Jenkinson, M., Beckmann, C.F., Behrens, T.E.J., Woolrich, M.W., Smith, S.M.: FSL. Neuroimage 62(2), 782–790 (2012)CrossRefGoogle Scholar
  21. 21.
    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)CrossRefGoogle Scholar
  22. 22.
    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)CrossRefGoogle Scholar
  23. 23.
    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)CrossRefGoogle Scholar
  24. 24.
    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)Google Scholar
  25. 25.
    Melonakos, J., Pichon, E., Angenent, S., Tannenbaum, A.: Finsler active contours. IEEE Trans. Pattern Anal. Mach. Intell. 30(3), 412–423 (2008)CrossRefGoogle Scholar
  26. 26.
    Mirebeau, J.: Anisotropic fast-marching on cartesian grids using lattice basis reduction. SIAM J. Numer. Anal. 52(4), 1573–1599 (2014)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Mirebeau, J.M.: Fast-marching methods for curvature penalized shortest paths. JMIV 60, 784–815 (2018)MathSciNetCrossRefGoogle Scholar
  28. 28.
    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)CrossRefGoogle Scholar
  29. 29.
    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)CrossRefGoogle Scholar
  30. 30.
    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)CrossRefGoogle Scholar
  31. 31.
    Péchaud, M., Descoteaux, M., Keriven, R.: Brain connectivity using geodesics in HARDI. Med. Image Comput. Comput. Assist. Interv. 12(2), 482–489 (2009)Google Scholar
  32. 32.
    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)CrossRefGoogle Scholar
  33. 33.
    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)Google Scholar
  34. 34.
    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)Google Scholar
  35. 35.
    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)Google Scholar
  36. 36.
    Sanguinetti, G., Bekkers, E., Duits, R., Janssen, M.H.J., Mashtakov, A., Mirebeau, J.M.: Sub-Riemannian Fast Marching in SE(2). Springer (2015)Google Scholar
  37. 37.
    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)MathSciNetzbMATHGoogle Scholar
  38. 38.
    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)CrossRefGoogle Scholar
  39. 39.
    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)CrossRefGoogle Scholar
  40. 40.
    Tournier, J.D., Calamante, F., Connelly, A.: MRtrix: diffusion tractography in crossing fiber regions. Int. J. Imaging Syst. Technol. 22(1), 53–66 (2012)CrossRefGoogle Scholar
  41. 41.
    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)CrossRefGoogle Scholar
  42. 42.
    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)CrossRefGoogle Scholar
  43. 43.
    Vogt, T., Lellmann, J.: Measure-valued variational models with applications to diffusion-weighted imaging. J. Math. Imaging Vis. (2018)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Jorg Portegies
    • 1
  • Stephan Meesters
    • 1
    • 2
  • Pauly Ossenblok
    • 1
  • Andrea Fuster
    • 1
  • Luc Florack
    • 1
  • Remco Duits
    • 1
    Email author
  1. 1.Department of Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Academic Center for Epileptology Kempenhaeghe and Maastricht University Medical CenterMaastrichtThe Netherlands

Personalised recommendations