Riemann-Finsler Multi-valued Geodesic Tractography for HARDI

  • Neda Sepasian
  • Jan H. M. ten Thije Boonkkamp
  • Luc M. J. Florack
  • Bart M. Ter Haar Romeny
  • Anna Vilanova
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)

Abstract

We introduce a geodesic based tractography method for High Angular Resolution Diffusion Imaging (HARDI). The concepts used are similar to the ones in geodesic based tractography for Diffusion Tensor Imaging (DTI). In DTI, the inverse of the second-order diffusion tensor is used to define the manifold where the geodesics are traced. HARDI models have been developed to resolve complex fiber populations within a voxel, and higher order tensors (HOT) are possible representations for HARDI data. In our framework, we apply Finsler geometry, which extends Riemannian geometry to a directionally dependent metric. A Finsler metric is defined in terms of HARDI higher order tensors. Furthermore, the Euler-Lagrange geodesic equations are derived based on the Finsler geometry. In contrast to other geodesic based tractography algorithms, the multi-valued numerical solution of the geodesic equations can be obtained. This gives the possibility to capture all geodesics arriving at a single voxel instead of only computing the shortest one. Results are analyzed to show the potential and characteristics of our algorithm.

Keywords

Manifold 

References

  1. 1.
    Aganj, I., Lenglet, C., Sapiro, G.: ODF reconstruction in q-ball imaging with solid angle consideration. In: The 6th IEEE International Symposium on Biomedical Imaging (ISBI), Boston, pp. 1398–1401 (2009)Google Scholar
  2. 2.
    Alexander, D.C., Barker, G., Arridge, S.: Detection and modeling of non-Gaussian apparent diffusion coefficient profiles in human brain data. Magn. Reson. Med. 48(2), 331–340 (2002)CrossRefGoogle Scholar
  3. 3.
    Astola, L., Florack, L.: Finsler geometry on higher order tensor fields and applications to high angular resolution diffusion imaging. Int. J. Comput. Vis. 92, 325–336 (2011)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Basser, P.J., Pierpaoli, C.: Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI. J. Magn. Reson. 111(1), 209–219 (1996)CrossRefGoogle Scholar
  5. 5.
    Behrens, T.E.J., Berg, H.J., Jbabdi, S., Rushworth, M.F.S., Woolrich, M.W.: Probabilistic diffusion tractography with multiple fibre orientations: what can we gain? NeuroImage 34(1), 144–155 (2007)CrossRefGoogle Scholar
  6. 6.
    Blair, D.E.: Inversion Theory and Conformal Mapping. American Mathematical Society, Providence (2000)MATHGoogle Scholar
  7. 7.
    Lenglet, C., Deriche, R., Faugeras, O.D.: Inferring white matter geometry from diffusion tensor MRI: application to connectivity mapping. In: Proceedings of 8th European Conference on Computer Vision, Prague, pp. 127–140 (2004)Google Scholar
  8. 8.
    Campbell, J.S.: Diffusion imaging of white matter fibre tracts. PhD thesis, McGill University, Montreal (2004)Google Scholar
  9. 9.
    Chern, S., Shen, Z.: Lectures on Finsler geometry. Nankai Tracts Math. 6 (2003)Google Scholar
  10. 10.
    Morris, D.M., Embleton, K.V., Parker, G.J.: Probabilistic fibre tracking: differentiation of connections from chance events. NeuroImage 42(4), 1329–1339 (2008)CrossRefGoogle Scholar
  11. 11.
    Descoteaux, M.: High angular resolution diffusion MRI: from local estimation to segmentation and tractography. Ph.D. Thesis, Universite de Nice, Sophia Antipolis (2008)Google Scholar
  12. 12.
    Descoteaux, M., Angelino, E., Fitzgibbons, S., Deriche, R.: Regularized, fast and robust analytical q-ball imaging. Magn. Reson. Med. 58(3) (2007)Google Scholar
  13. 13.
    Descoteaux, M., Deriche, R., Knoesche, T., Anwander, A.: Deterministic and probabilistic tractography based on complex fiber orientation distributions. IEEE Trans. Med. Imaging 2(28), 269–286 (2008)Google Scholar
  14. 14.
    Donnell, L.O., 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. In: International Conference on Medical Image Computing and Computer-Assisted Intervention – MICCAI’02, Tokyo, vol. 2488, pp. 459–466 (2002)Google Scholar
  15. 15.
    Hagmann, P., Reese, T., Tseng, W., Meuli, R., Thiran, J., Wedeen, V.: Diffusion spectrum imaging tractography in complex cerebral white matter: an investigation of the centrum semiovale. In: ISMRM, Kyoto, vol. 12. International Society for Magnetic Resonance in Medicine (2004)Google Scholar
  16. 16.
    Lenglet, C., Prados, E., Pons, J.P., Deriche, R., Faugeras, O.: Brain connectivity mapping using Riemannian geometry, control theory and PDEs. SIAM J. Imaging Sci. (SIIMS) 2(2), 285–322 (2009)Google Scholar
  17. 17.
    Melonakos, J., Pichon, E., Angenent, S., Tannenbaum, A.: Finsler active contours. IEEE Trans. Pattern Anal. Mach. Intell. 30(3), 412–423 (2008)CrossRefGoogle Scholar
  18. 18.
    Mo, X.: An Introduction to Finsler Geometry. Volume 1 of Peking University Series in Mathematics. World Scientific, Singapore (2006)Google Scholar
  19. 19.
    Ozarslan, E., Shepherd, T., Vemuri, B., Blackband, S., Mareci, T.: A nonparametric reconstruction and its matrix implementation for the diffusion orientation transform (dot). In: The 3rd IEEE International Symposium on Biomedical Imaging: Nano to Macro (ISBI), Arlington, pp. 85–88 (2006)Google Scholar
  20. 20.
    Parker, G.J.M., Alexander, D.C.: Probabilistic anatomical connectivity derived from the microscopic persistent angular structure of cerebral tissue. Philos. Trans. R. Soc. Lond. B: Biol. Sci. 360(1457), 893–902 (2005)CrossRefGoogle Scholar
  21. 21.
    Péchaud, M., Descoteaux, M., Keriven, R.: Brain connectivity using geodesics in HARDI. In: International Conference on Medical Image Computing and Computer-Assisted Intervention – MICCAI’09, London, pp. 482–489 (2009)Google Scholar
  22. 22.
    Pichon, E., Westin, C., Tannenbaum, A.: A Hamilton-Jacobi-Bellman approach to high angular resolution diffusion tractography. In: International Conference on Medical Image Computing and Computer-Assisted Intervention – MICCAI’05, Palm Springs. Lecture Notes in Computer Science, pp. 180–187 (2005)Google Scholar
  23. 23.
    Rund, H.: The Hamilton-Jacobi Theory in the Calculus of Variations. Robert E. Krieger Publishing, Huntington (1973)MATHGoogle Scholar
  24. 24.
    Sepasian, N., ten Thije Boonkkamp, J., ter Haar Romeny, B., Vilanova, A.: Multivalued geodesic ray-tracing for computing brain connections using diffusion tensor imaging. SIAM J. Imaging Sci. 5(2), 483–504 (2012)MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Sepasian, N., ten Thije Boonkkamp, J., Vilanova, A., ter Haar Romeny, B.: Multi-valued geodesic based fiber tracking for diffusion tensor imaging. In: MICCAI’09, Diffusion Modeling and the Fiber Cup Workshop, London, vol. 1, pp. 6–13 (2009)Google Scholar
  26. 26.
    Shen, Z.: Lectures on Finsler Geometry. World Scientific, Singapore (2001)MATHCrossRefGoogle Scholar
  27. 27.
    Tournier, J.D., Calamante, F., Connelly, A.: Robust determination of the fibre orientation distribution in diffusion mri: non-negativity constrained super-resolved spherical deconvolution. NeuroImage 35(4), 1459–1472 (2007)CrossRefGoogle Scholar
  28. 28.
    Tristán-Vega, A., Westin, C.F., Aja-Fernández, S.: Estimation of fiber orientation probability density functions in high angular resolution diffusion imaging. NeuroImage 47(2), 638–650 (2009)CrossRefGoogle Scholar
  29. 29.
    Tuch, D., Reese, T., Wiegell, M., Makris, N., Belliveau, J.W., Wedeen, V.: High angular resolution diffusion imaging reveals intravoxel white matter fiber heterogeneity. Magn. Reson. Med. 48, 577–582 (2002)CrossRefGoogle Scholar
  30. 30.
    Tuch, D.S.: Q-ball imaging. Magn. Reson. Med. 52(6), 1358–1372 (2004)CrossRefGoogle Scholar
  31. 31.
    Wedeen, V.J., Hagmann, P., Tseng, W.Y.I., Reese, T.G., Weisskoff, R.M.: Mapping complex tissue architecture with diffusion spectrum magnetic resonance imaging. Magn. Reson. Med. 54(6), 1377–1386 (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Neda Sepasian
    • 1
  • Jan H. M. ten Thije Boonkkamp
    • 2
  • Luc M. J. Florack
    • 1
  • Bart M. Ter Haar Romeny
    • 3
  • Anna Vilanova
    • 4
  1. 1.Department of Biomedical Engineering, Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands
  3. 3.Department of Biomedical EngineeringEindhoven University of TechnologyEindhovenThe Netherlands
  4. 4.Faculty Electrical Engineering, Mathematics and Computer Science, Computer Graphics and VisualizationDelft University of TechnologyDelftThe Netherlands

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