Multivalued Geodesic Ray-Tracing for Computing Brain Connections Using Diffusion Tensor Imaging

SIAM Journal on Imaging Sciences

Volumn 5, Issue 2, 2012, Pages 483-504

  Sepasian, N ^a   ten Thije Boonkkamp, J H M ^a   Ter Haar Romeny, B M ^a   Vilanova, A ^a  

^a EINDHOVEN UNIVERSITY OF TECHNOLOGY   (Netherlands)

Author keywords

Diffusion tensor imaging; Euler Lagrange equations; Fiber tracking; Geodesics; Hamilton Jacobi equation; Riemannian geometry

Indexed keywords

EULER-LAGRANGE EQUATIONS; FIBER TRACKING; GEODESICS; HAMILTON JACOBI EQUATIONS; RIEMANNIAN GEOMETRY;

ALGORITHMS; DIFFERENTIAL EQUATIONS; DIFFUSION TENSOR IMAGING; EULER EQUATIONS; FIBERS; GEODESY; GEOMETRICAL OPTICS; GEOMETRY; MAGNETIC RESONANCE; MAGNETIC RESONANCE IMAGING;

TENSORS;

EID: 84863765641     PISSN: None     EISSN: 19364954     Source Type: Journal    
DOI: 10.1137/110824395     Document Type: Article

References (61)

1. L. J. Astola, L. M. J. Florack, and B. M. ter Haar Romeny, Measures for pathway analysis in brain white matter using diffusion tensor imaging, in Proceedings of the 20th International Conference on Information Processing in Medical Imaging, Lecture Notes in Comput. Sci. 4584, Springer, New York, 2007, pp. 642-649.
2. P. J. Basser and C. Pierpaoli, Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI, J. Magnetic Res., 111 (1996), pp. 209-219.
3. T. E. Behrens, H. J. Berg, S. Jbabdi, M. F. Rushworth, and M. W. Woolrich, Probabilistic diffusion tractography with multiple fibre orientations: What can we gain?, Neuroimage, 34 (2007), pp. 144-155.
4. J. Berman, S. Chung, P. Mukherjee, C. P. Hess, E. T. Han, and R. G. Henry, Probabilistic streamline q-ball tractography using the residual bootstrap, Neuroimage, 39 (2008), pp. 215-222.
5. M. Bj̈ornemo, A. Brun, R. Kikinis, and C. F. Westin, Regularized stochastic white matter tractography using diffusion tensor MRI, (2002), pp. 435-442.
6. P. Bulant and L. Kliměs, Interpolation of ray theory traveltimes within ray cells, Geophys. J. Internat., 139 (1999), pp. 273-282.
7. V. Červeńy, I. A. Molotkov, and I. Psencik, Ray Method in Seismology, University of Karlova Press, 1977.
8. R. C. Y. Chin, G. Hedstrom, and L. Thigpen, Numerical methods in seismology, J. Comput. Phys., 54 (1984), pp. 18-56.
9. P. A. Cook, D. C. Alexander, and G. J. M. Parker, Modelling noise-induced fibre orientation error in diffusion-tensor MRI, Proc. ISBI, 1 (2004), pp. 332-336.
10. P. A. Cook, H. Zhang, B. B. Avants, P. Yushkevich, D. C. Alexander, J. C. Gee, O. Ciccarelli, and A. J. Thompson, An automated approach to connectivity-based partitioning of brain structures, in Proceedings of MICCAI 2005, Lecture Notes in Comput. Sci. 3749, Springer, New York, 2005, pp. 164-171.
11. M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), pp. 1-42.
12. M. Descoteaux, C. Lenglet, and R. Deriche, Diffusion tensor sharpening improves white matter tractography, in Proceedings of SPIE, 6512 (2007), pp. 65121J-65121J.
13. L. O' Donnell, S. Haker, and C. F. Westin, New approaches to estimation of white matter connectivity in diffusion tensor MRI: Elliptic PDEs and geodesics in a tensor-warped space, in Proceedings of MICCAI 2002, Lecture Notes in Comput. Sci. 2488, Springer, New York, 2002, pp. 459-466.
14. L. C. Evans, Partial Differential Equations, 19, American Mathematical Society, Providence, RI, 1998.
15. B. Geyer, D. M. Gitman, and I. V. Tyutin, Canonical form of Euler-Lagrange equations and gauge symmetries, J. Phys. A, 36 (2003).
16. P. A. Gremaud and C. M. Kuster, Computational study of fast methods for the eikonal equation, SIAM J. Sci. Comput., 27 (2006), pp. 1803-1816.
17. H. A. Haroon and G. J. Parker, Using the wild bootstrap to quantify uncertainty in fibre orientations from q-ball analysis, in Proc. Intl. Soc. Mag. Reson. Med., 15 (2007).
18. T. Hosey, G. Williams, and R. Ansorge, Inference of multiple fiber orientations in high angular resolution diffusion imaging, Magn. Reson. Med., 54 (2005), pp. 1480-1489.
19. M. Jackowski, C. Y. Kao, M. Qui, R. T. Constable, and L. H. Staib, White matter tractography by anisotropic wavefront evolution and diffusion tensor imaging, Medical Image Anal., 9 (2005), pp. 424-440.
20. S. Jbabdi, P. Bellec, R. Toro, J. Daunizeau, M. Ṕeĺegrini-Issac, and H. Benali, Accurate anisotropic fast marching for diffusion-based geodesic tractography, Int. J. Biomed. Imaging, 2008 (2008), pp. 1-12.
21. B. Jeurissen, A. Leemans, D. K. Jones, J. Tournier, and J. Sijbers, Probabilistic fiber tracking using the residual bootstrap with constrained spherical deconvolution, Human Brain Mapping, (2010).
22. B. R. Julian and D. Gubbins, Three-dimensional seismic ray tracing, J. Geophys. Res., 43 (1977), pp. 95-114.
23. C. Y. Kao, S. Osher, and J. Qian, Lax-Friedrichs sweeping scheme for static Hamilton-Jacobi equations, J. Comput. Phys., 196 (2004), pp. 367-391.
24. C.-Y. Kao, S. Osher, and Y.-H. Tsai, Fast sweeping methods for static Hamilton-Jacobi equations, SIAM J. Numer. Anal., 42 (2005), pp. 2612-2632.
25. S. Kim and R. Cook, 3-D traveltime computation using second-order ENO scheme, Geophys., 64 (1999), pp. 1867-1876.
26. R. T. Langan, I. Lerche, and R. T. Cutler, Tracing of rays through heterogeneous media: An accurate and efficient procedure, Geophys., 50 (1985), pp. 1456-1465.
27. A. Lenglet, R. Deriche, and O. Faugeras, Inferring white matter geometry from diffusion tensor MRI: Application to connectivity mapping, in Proceedings of the Eighth European Conference on Computer Vision, Lecture Notes in Comput. Sci. 3021-3024, Springer, New York, 2004, pp. 127-140.
28. C. Lenglet, E. Prados, J.-P. Pons, R. Deriche, and O. Faugeras, Brain connectivity mapping using Riemannian geometry, control theory, and PDEs, SIAM J. Imaging Sci., 2 (2009), pp. 285-322.
29. T. Ma and H. D. Tagare, Consistency and stability of active contours with Euclidean and non-Euclidean arc-lengths, IEEE Trans. Image Process., 8 (1999), pp. 1549-1559.
30. C. Mantegazza and A. C. Mennucci, Hamilton-Jacobi equations and distance functions on Riemannian manifolds, Appl. Math. Optim., 47 (2002), pp. 1-25.
31. R. M. M. Mattheij, S. W. Rienstra, and J. H. M. ten Thije Boonkkamp, Partial Differential Equations: Modeling, Analysis, Computation, Math. Model. Comput. 10, SIAM, Philadelphia, 2005.
32. J. Melonakos, V. Mohan, M. Niethammer, K. Smith, M. Kubicki, and A. Tannenbaum, Finsler tracktography for white matter connectivity analysis of the singulum bundle, in Proceedings of MICCAI 2005, Lecture Notes in Comput. Sci. 3749, Springer, New York, 2007.
33. E. N. Miller, Icosahedra constructed from congruent triangles, Discrete Comput. Geom., 24 (2006), pp. 437-451.
34. S. Mori, B. J. Crain, V. P. Chacko, and P. C. M. van Zijl, Three-dimensional tracking of axonal projections in the brain by magnetic resonance imaging, Ann. Neurol., 45 (1999), pp. 265-269.
35. S. Mori, S. Wakana, P. van Zijl, and L. M. Nagae-Poetscher, MRI Atlas of Human White Matter, Elsevier, Berlin, 2005.
36. M. Motamed and O. Runborg, A fast phase space method for computing creeping rays, J. Comput. Phys., 219 (2006), pp. 276-295.
37. NVIDIA, CUDA Programming Guide, v. 4.1, 2011.
38. S. Osher, A level set formulation for the solution of the Dirichlet problem for Hamilton-Jacobi equations, SIAM J. Math. Anal., 24 (1993), pp. 1145-1152.
39. G. J. M. Parker and D. C. Alexander, Probabilistic Monte Carlo based mapping of cerebral connections utilising whole-brain crossing fibre information, Inf. Proc. Med. Imaging, 18 (2003), pp. 684-695.
40. G. J. M. Parker and D. C. Alexander, Probabilistic anatomical connectivity derived from the microscopic persistent angular structure of cerebral tissue, Philos. Trans. R. Soc. Lond. Ser. B Biolog. Sci., 360 (2005), pp. 893-902.
41. G. J. Parker, D. M. Morris, and K. V. Embleton, Probabilistic fibre tracking: Differentiation of connections from chance events, NeuroImage, 42 (2008), pp. 1329-1339.
42. G. J. M. Parker, C. A. M. Wheeler-Kingshott, and G. J. Barker, Distributed anatomical brain connectivity derived from diffusion tensor imaging, in Proceedings of the 17th International Conference on Information Processing in Medical Imaging, Lecture Notes in Comput. Sci. 2482, Springer, New York, 2001, pp. 106-120.
43. V. Pereyra, W. H. K. Lee, and H. B. Keller, Solving two-point seismicray tracing problems in a heterogeneous medium, Bull. Seismol. Soc. Amer., 70 (1980), pp. 79-99.
44. E. Prados, S. Soatto, C. Lenglet, J. P. Pons, N. Wotawa, R. Deriche, O. Faugeras, and S. Soatto, Control theory and fast marching techniques for brain connectivity mapping, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, IEEE Computer Society Press, Piscataway, NJ, 2006, pp. 1076-1083.
45. O. Runborg, Mathematical models and numerical methods for high frequency waves, Comm. Comput. Phys., 2 (2007), pp. 827-880.
46. H. Rund, The Hamilton-Jacobi Theory in the Calculus of Variations: Its Role in Mathematics and Physics, D. Van Nostrand, London, New York, 1966.
47. A. Rutherford, Vectors, Tensors and the Basic Equations of Fluid Mechanics, 1990.
48. N. Sepasian, J. H. M. ten Thije Boonkkamp, A. Vilanova, and B. M. ter Haar Romeny, Multivalued geodesic based fiber tracking for diffusion tensor imaging, in Proceedings of MICCAI 2009, Diffusion Modeling and the Fiber Cup Workshop, Lecture Notes in Comput. Sci. 5761, Springer, New York, 2009, pp. 6-13.
49. N. Sepasian, A. Vilanova, L. M. J. Florack, and B. M. ter Haar Romeny, A ray tracing method for geodesic based tractography in diffusion tensor, in Proceedings of the Workshop on Mathematical Methods in Biomedical Image Analysis (MMBIA), IEEE Computer Society Press, Piscataway, NJ, 2008, pp. 1-6.
50. J. A. Sethian and A. Vladimirsky, Ordered upwind methods for static Hamilton-Jacobi equation, Proc. Natl. Acad. Sci. USA, 98 (2001), pp. 11069-11074.
51. K. K. Seunarine, P. A. Cook, M. G. Hall, K. Embleton, G. J. M. Parker, and D. C. Alexander, Exploiting peak anisotropy for tracking through complex structures, in Proceedings of the Workshop on Mathematical Methods in Biomedical Image Analysis (ICCV2007), Lecture Notes in Comput. Sci. 3749, Springer, New York, 2007, pp. 1-8.
52. E. O. Stejskal and J. E. Tanner, Spin diffusion measurements: Spin echoes in the presence of a time-dependent field gradient, J. Chem. Phys., 42 (1965), pp. 288-292.
53. C. H. Thurber and W. L. Ellsworth, Rapid solution of ray tracing problems in hetereogeneous media, Bull. Seismol. Soc. Amer., 70 (1980), pp. 1137-1148.
54. J. D. Tournier, F. Calamante, D. G. Gadian, and A. Connelly, Diffusion-weighted magnetic resonance imaging fibre tracking using a front evolution algorithm, Neuroimage, 20 (2003), pp. 276-288.
55. E. van Aart, GPU-Based Acceleration of a Novel Fiber Tracking Algorithm for Diffusion Tensor Imaging Using NVIDIA's CUDA, Master's Thesis, Eindhoven University of Technology, 2010; http://alexandria.tue.nl/extra1/afstversl/wsk-i/aart2010.pdf.
56. V. J. Wedeen, T. L. Davis, R. M. Weisskoff, R. Tootell, B. R. Rosen, and J. W. Belliveau, White matter connectivity explored by MRI, in Proceedings of the First International Conference for Functional Mapping of the Human Brain, 1995.
57. T. Xiong, M. Zhang, Y. Zhang, and C. W. Shu, Fast sweeping fifth order WENO scheme for static Hamilton-Jacobi equations with accurate boundary treatment, J. Sci. Comput., 45 (2010), pp. 514-536.
58. L. Yatziv, A. Bartesaghi, and G. Sapiro, O(n) implementation of the fast marching algorithm, J. Comput. Phys., 212 (2001), pp. 393-399.
59. Y. T. Zhang, H. K. Zhao, and S. Chen, Fixed-point iterative sweeping methods for static Hamilton-Jacobi equations, Methods Appl. Anal., 13 (2006), pp. 299-320.
60. H.-K. Zhao, A fast sweeping method for eikonal equation, Math. Comput., 74 (2003), pp. 603-627.
61. P. Zhu and S. Zhou, Relaxation Lax-Friedrichs sweeping scheme for static Hamilton-Jacobi equations, Numer. Algorithms, 54 (2010), pp. 325-342.