Brain connectivity mapping using riemannian geometry, control theory, and PDEs

SIAM Journal on Imaging Sciences

Volumn 2, Issue 2, 2009, Pages 285-322

  Lenglet, Christophe ^a   Prados, Emmanuel ^b   Pons, Jean Philippe ^c   Deriche, Rachid ^d   Faugeras, Olivier ^d  

^a UNIVERSITY OF MINNESOTA MEDICAL SCHOOL   (United States)

^b INRIA RHÔNE ALPES   (France)

^c MOD EVE   (France)

^d INRIA   (France)

Author keywords

Anisotropic Eikonal equation; Brain connectivity mapping; Brownian motion; Control theory; Diffusion process; Diffusion tensor imaging; Fast marching methods; Hamilton jacobi bellman equations; Intrinsic distance function; Level set; Partial differential equations; Riemannian manifolds

Indexed keywords

BRAIN MAPPING; BROWNIAN MOVEMENT; CONTROL THEORY; DIFFUSION; DYNAMIC PROGRAMMING; GEODESY; GEOMETRICAL OPTICS; GEOMETRY; INITIAL VALUE PROBLEMS; MAGNETIC RESONANCE IMAGING; MAPPING; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS; TENSORS;

BRAIN CONNECTIVITY; DIFFUSION PROCESS; EIKONAL EQUATION; FAST MARCHING METHODS; HAMILTON JACOBI BELLMAN EQUATION; INTRINSIC DISTANCES; LEVEL SET; RIEMANNIAN MANIFOLD;

DIFFUSION TENSOR IMAGING;

EID: 77952244937     PISSN: None     EISSN: 19364954     Source Type: Journal    
DOI: 10.1137/070710986     Document Type: Article

References (92)

1. D. Adalsteinsson and J.A. Sethian (1999), The fast construction of extension velocities in level set methods, J. Comput. Phys., 148, pp. 2-22.
2. A.L. Alexander, K.M. Hasan, M. Lazar, J.S. Tsuruda, and D.L. Parker (2001), Analysis of partial volume effects in diffusion-tensor MRI, Magn. Reson. Med., 45, pp. 770-780.
3. M. Bardi and I. Capuzzo-Dolcetta (1997), Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser Boston, Boston.
4. P.J. Basser, J. Mattiello, and D. Le Bihan (1994), Estimation of the effective self-diffusion tensor from the NMR spin echo, J. Magn. Reson. B, 103, pp. 247-254.
5. P.J. Basser, S. Pajevic, C. Pierpaoli, J. Duda, and A. Aldroubi (2000), In vivo fiber tractography using DT-MRI data, Magn. Reson. Med., 44, pp. 625-632.
6. P.G. Batchelor, D.L.G. Hill, F. Calamante, and D. Atkinson (2001), Study of connectivity in the brain using the full diffusion tensor from MRI, in Proceedings of the 17th International Conference on Information Processing in Medical Imaging, Lecture Notes in Comput. Sci. 2082, Springer-Verlag, London, pp. 121-133.
7. T.E.J. Behrens, M.W. Woolrich, M. Jenkinson, H. Johansen-Berg, R.G. Nunes, S. Clare, P.M. Matthews, J.M. Brady, and S.M. Smith (2003), Characterization and propagation of uncertainty in diffusion-weighted MR imaging, Magn. Reson. Med., 50, pp. 1077-1088.
8. M. Bjornemo, A. Brun, R. Kikinis, and C.F. Westin (2002), Regularized stochastic white matter tractography using diffusion tensor MRI, in Proceedings of the 5th International Conference on Medical Image Computing and Computer-Assisted Intervention, Springer-Verlag, London, pp. 435-442.
9. J.S.W. Campbell, K. Siddiqi, V.V. Rymar, A. Sadikot, and B.G. Pike (2005), Flow-based fiber tracking with diffusion tensor q-ball data: Validation and comparison to principal diffusion direction techniques, NeuroImage, 27, pp. 725-736.
10. Y.G. Chen, Y. Giga, and S. Goto (1991), Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33, pp. 749-786.
11. D.L. Chopp (1993), Computing minimal surfaces via level set curvature flow, J. Comput. Phys., 106, pp. 77-91.
12. M.K. Chung, M. Lazar, A.L. Alexander, Y. Lu, and R. Davidson (2003), Probabilistic Connectivity Measure in Diffusion Tensor Imaging via Anisotropic Kernel Smoothing, Technical report 1081, Department of Statistics, University of Wisconsin-Madison, Madison, WI.
13. L. Cohen (2006), Minimal paths and fast marching methods for image analysis, in Handbook of Mathematical Models in Computer Vision, Springer-Verlag, New York, pp. 97-111.
14. R.W.R. Darling (2002), Intrinsic location parameter of a diffusion process, Electron. J. Probab., 7, pp. 1-23.
15. M. de Lara (1995), Geometric and symmetry properties of a nondegenerate diffusion process, Ann. Probab., 23, pp. 1557-1604.
16. M. Descoteaux, C. Lenglet, and R. Deriche (2007), Diffusion tensor sharpening improves white matter tractography, in Medical Imaging 2007: Image Processing, Proceedings of the SPIE, Vol. 6512, International Society for Optical Engineering, Bellingham, WA, p. 65121J.
17. M.P. Do Carmo (1992), Riemannian Geometry, Birkhäuser Boston, Boston.
18. P. Dupuis and J. Oliensis (1994), An optimal control formulation and related numerical methods for a problem in shape reconstruction, Ann. Appl. Probab., 4, pp. 287-346.
19. A. Einstein (1956), Investigations on the Theory of the Brownian Movement, Dover, New York.
20. I. Ekeland and R. Temam (1974), Analyse Convexe et Problèmes Variationnels, Études Mathématiques, Dunod, Gauthier-Villars, Paris, Brussels, Montreal.
21. P.T. Fletcher, R. Tao, W.K. Jeong, and R.T. Whitaker (2007), A volumetric approach to quantifying region-to-region white matter connectivity in diffusion tensor MRI, in Information Processing in Medical Imaging, Lecture Notes in Comput. Sci. 4584, Springer-Verlag, Berlin, Heidelberg, pp. 346-358.
22. O. Friman and C.F. Westin (2005), Uncertainty in white matter fiber tractography, in Proceedings of the International Conference on Medical Image Computing and Computer-Assisted Intervention, Springer-Verlag, Berlin, pp. 107-114.
23. O. Friman, G. Farneback, and C.F. Westin (2006), A Bayesian approach for stochastic white matter tractography, IEEE Trans. Med. Imaging, 25, pp. 965-978.
24. N. Hageman, D. Shattuck, K. Narr, and A.W. Toga (2006), A diffusion tensor imaging tractography method based on Navier-Stokes fluid mechanics, in Proceedings of the IEEE International Symposium on Biomedical Imaging: From Nano to Macro, pp. 798-801.
25. P. Hagmann, J.P. Thiran, L. Jonasson, P. Vandergheynst, S. Clarke, P. Maeder, and R. Meuli (2003), DTI mapping of human brain connectivity: Statistical fiber tracking and virtual dissection, Neuro-Image, 19, pp. 545-554.
26. A. Harten, B. Engquist, S. Osher, and S. Chakravarthy (1987), Uniformly high-order accurate essentially nonoscillatory schemes. III, J. Comput. Phys., 71, pp. 231-303.
27. R. Iyer, R. Holsapple, and D. Doman (2006), Optimal control problems on parallelizable Riemannian manifolds: Theory and applications, ESAIM Control Optim. Calc. Var., 12, pp. 1-11.
28. M. Jackowski, C.Y. Kao, M. Qiu, R.T. Constable, and L.H. Staib (2005), White matter tractography by anisotropic wavefront evolution and diffusion tensor imaging, Med. Image Anal., 9, pp. 427-440.
29. S. Jbabdi, P. Bellec, R. Toro, J. Daunizeau, M. Pélégrini-Isaac, and H. Benali (2008), Accurate anisotropic fast marching for diffusion-based geodesic tractography, J. Biomed. Imaging, 2008, pp. 1-12.
30. W.-K. Jeong and R.T. Whitaker (2008), A fast iterative method for eikonal equations, SIAM J. Sci. Comput., 30, pp. 2512-2534.
31. D.K. Jones, A. Simmons, S.C.R. Williams, and M.A. Horsfield (1999), Noninvasive assessment of axonal fiber connectivity in the human brain via diffusion tensor MRI, Magn. Reson. Med., 42, pp. 37-41.
32. N. Kang, J. Zhang, E.S. Carlson, and D. Gembris (2005), White matter fiber tractography via anisotropic diffusion simulation in the human brain, IEEE Trans. Med. Imaging, 24, pp. 1127-1137.
33. R. Kimmel and J.A. Sethian (1998), Computing geodesic paths on manifolds, Proc. Natl. Acad. Sci. USA, 95, pp. 8431-8435.
34. R. Kimmel, A. Amir, and A.F. Bruckstein (1995), Finding shortest paths on surfaces using level set propagation, IEEE Trans. Pattern Anal. Mach. Intell., 17, pp. 635-640.
35. M. Koch, V. Glauche, J. Finsterbusch, U. Nolte, J. Frahm, and C. Büchel (2001), Estimation of anatomical connectivity from diffusion tensor data, in Proceedings of the 7th Annual Meeting of the Organization for Human Brain Mapping, p. 1974.
36. E. Konukoglu, M. Sermesant, O. Clatz, J.-M. Peyrat, H. Delingette, and N. Ayache (2007), A recursive anisotropic fast marching approach to reaction diffusion equation: Application to tumor growth modeling, in Information Processing in Medical Imaging, Lecture Notes in Comput. Sci. 4584, Springer-Verlag, Berlin, pp. 687-699.
37. M. Lazar and A. Alexander (2005), Bootstrap white matter tractography (BOOT-TRAC), NeuroImage, 24, pp. 524-532.
38. M. Lazar and A.L. Alexander (2002), White matter tractography using random vector (RAVE) perturbation, in Proceedings of the 10th International Society for Magnetic Resonance in Medicine (ISMRM) Annual Meeting, p. 539.
39. M. Lazar, D.M. Weinstein, J.S. Tsuruda, K.M. Hasan, K. Arfanakis, M.E. Meyerand, B. Badie, H.A. Rowley, V. Haughton, A. Field, and A.L. Alexander (2003), White matter tractography using diffusion tensor deflection, Hum. Brain Map., 18, pp. 306-321.
40. D. Le Bihan and E. Breton (1985), Imagerie de diffusion in vivo par résonance magnétique nucléaire, C. R. Acad. Sci. Paris, 301, pp. 1109-1112.
41. S. Lehericy, C. Lenglet, J. Doyon, H. Benali, P.F. Van de Moortele, G. Sapiro, O. Faugeras, R. Deriche, and K. Ugurbil (2005), Activation shifts from the premotor to the sensorimotor territory of the striatum during the course of motor sequence learning, in Proceedings of the 11th Annual Meeting of the Organization for Human Brain Mapping, Toronto, p. 443.
42. C. Lenglet (2006), Geometric and Variational Methods for Diffusion MRI Processing, Ph.D. thesis, INRIA/ University of Nice-Sophia Antipolis, Sophia Antipolis, France.
43. C. Lenglet, R. Deriche, and O. Faugeras (2004), Inferring white matter geometry from diffusion tensor MRI: Application to connectivity mapping, in Proceedings of the European Conference on Computer Vision, Springer-Verlag, Berlin, pp. 127-140.
44. C. Lenglet, M. Rousson, R. Deriche, and O. Faugeras (2006), Statistics on the manifold of multivariate normal distributions: Theory and application to diffusion tensor MRI processing, J. Math. Imaging Vision, 25, pp. 423-444.
45. R.J. LeVeque (1992), Numerical Methods for Conservation Laws, Birkhäuser Verlag, Basel.
46. P.-L. Lions (1982), Generalized Solutions of Hamilton-Jacobi Equations, Res. Notes in Math. 69, Pitman (Advanced Publishing Program), Boston, London.
47. X.D. Liu, S. Osher, and T. Chan (1994), Weighted essentially nonoscillatory schemes, J. Comput. Phys., 115, pp. 200-212.
48. J.-F. Mangin, C. Poupon, Y. Cointepas, D. Rivière, D. Papadopoulos-Orfanos, C. A. Clark, J. Régis, and D. Le Bihan (2002), A framework based on spin glass models for the inference of anatomical connectivity from diffusion-weighted MR data, NMR Biomed., 15, pp. 481-492.
49. C. Mantegazza and A.C. Mennucci (2002), Hamilton-Jacobi equations and distance functions on Riemannian manifolds, Appl. Math. Optim., 47, pp. 1-25.
50. T. McGraw and M. Nadar (2007), Stochastic DT-MRI connectivity mapping on the GPU, IEEE Trans. Visualization Comput. Graphics, 13, pp. 1504-1511.
51. J. Melonakos, V. Mohan, M. Niethammer, K. Smith, M. Kubicki, and A. Tannenbaum (2007), Finsler tractography for white matter connectivity analysis of the cingulum bundle, in Proceedings of the International Conference on Medical Image Computing and Computer Assisted Intervention, Springer-Verlag, Berlin, Heidelberg, pp. 36-43.
52. K.D. Merboldt, W. Hanicke, and J. Frahm (1985), Self-diffusion NMR imaging using stimulated echoes, J. Magn. Reson., 64, pp. 479-486.
53. S. Mori and P. van Zijl (2003), Fiber tracking: Principles and strategies-A technical review, NMR Biomed., 15, pp. 468-480.
54. S. Mori, B.J. Crain, V.P. Chacko, and P.C.M. Van Zijl (1999), Three-dimensional tracking of axonal projections in the brain by magnetic resonance imaging, Ann. Neurol., 45, pp. 265-269.
55. M.E. Moseley, Y. Cohen, J. Mintorovitch, J. Kucharczyk, J. Tsuruda, P. Weinstein, and D. Norman (1990), Diffusion-weighted MR imaging of anisotropic water diffusion in cat central nervous system, Radiol., 176, pp. 439-445.
56. L. O’Donnell, S. Haker, and C.F. Westin (2002), New approaches to estimation of white matter connectivity in diffusion tensor MRI: Elliptic PDEs and geodesics in a tensor-warped space, in Proceedings of the 5th International Conference on Medical Image Computing and Computer-Assisted Intervention, Springer-Verlag, Berlin, Heidelberg, pp. 459-466.
57. S. Osher (1993), A level set formulation for the solution of the Dirichlet problem for Hamilton-Jacobi equations, SIAM J. Math. Anal., 24, pp. 1145-1152.
58. S. Osher and J.A. Sethian (1988), Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79, pp. 12-49.
59. P.A. Osment, K.J. Packer, M.J. Taylor, J.J. Attard, T.A. Carpenter, L.D. Hall, S.J. Doran, and N.J. Herrod (1990), NMR imaging of fluids in porous solids, Phil. Trans. Roy. Soc. London Ser. A, 333, pp. 441-452.
60. G.J Parker, H.A. Haroon, and C.A Wheeler-Kingshott (2003), A framework for a streamline-based probabilistic index of connectivity (PICo) using a structural interpretation of MRI diffusion measurements, J. Magn. Reson. Imaging, 18, pp. 242-254.
61. G.J.M. Parker and D.C. Alexander (2003), Probabilistic Monte Carlo based mapping of cerebral connections utilising whole-brain crossing fibre information, in Information Processing in Medical Imaging, Springer-Verlag, Berlin, Heidelberg, pp. 684-695.
62. G.J.M. Parker, K.E. Stephan, G.J. Barker, J.B. Rowe, D.G. MacManus, C.A.M. Wheeler-Kingshott, O. Ciccarelli, R.E. Passingham, R.L. Spinks, R.N. Lemon, and R. Turner (2002a), Initial demonstration of in vivo tracing of axonal projections in the macaque brain and comparison with the human brain using diffusion tensor imaging and fast marching tractography, NeuroImage, 15, pp. 797-809.
63. G.J.M. Parker, C.A.M. Wheeler-Kingshott, and G.J. Barker (2002b), Estimating distributed anatomical connectivity using fast marching methods and diffusion tensor imaging, IEEE Trans. Med. Imaging, 21, pp. 505-512.
64. M. Perrin, C. Poupon, Y. Cointepas, B. Rieul, N. Golestani, C. Pallier, D. Rivière, D. Le Bihan, and J.F. Mangin (2005), Fiber tracking in q-ball fields using regularized particle trajectories, in Proceedings of the 19th International Conference on Information Processing in Medical Imaging, Lecture Notes in Comput. Sci. 3565, Springer-Verlag, Berlin, Heidelberg, pp. 52-63.
65. E. Pichon, G. Sapiro, and A. Tannenbaum (2003), Segmentation of diffusion tensor imagery, in Directions in Mathematical Systems Theory and Optimization, Lecture Notes in Control and Inform. Sci. 286, Springer-Verlag, Heidelberg, pp. 239-247.
66. E. Pichon, C.F. Westin, and A. A. Tannenbaum (2005), A Hamilton-Jacobi-Bellman approach to high angular resolution diffusion tractography, in Proceedings of MICCAI’05, Springer-Verlag, Berlin, pp. 180-187.
67. E. Prados (2004), Application of the Theory of the Viscosity Solutions to the Shape from Shading Problem, Ph.D. thesis, INRIA/University of Nice-Sophia Antipolis, Sophia Antipolis, France.
68. E. Prados and S. Soatto (2005), Fast marching method for generic shape from shading, in Proceedings of the IEEE Workshop on Variational and Level Set Methods, pp. 320-331.
69. E. Prados, O. Faugeras, and E. Rouy (2002), Shape from Shading and Viscosity Solutions, Research report 4638, INRIA, Sophia Antipolis, France.
70. E. Prados, C. Lenglet, J.P. Pons, N. Wotawa, E. Deriche, O. Faugeras, and S. Soatto (2006a), Control theory and fast marching methods for brain connectivity mapping, in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 1076-1083.
71. E. Prados, C. Lenglet, J.P. Pons, N. Wotawa, R. Deriche, O. Faugeras, and S. Soatto (2006b), Control Theory and Fast Marching Techniques for Brain Connectivity Mapping, Research report 5845, INRIA, Sophia Antipolis, France.
72. E. Prados, F. Camilli, and O. Faugeras (2006c), A viscosity solution method for shape-from-shading without image boundary data, M2AN Math. Model. Numer. Anal., 40, pp. 393-412.
73. J. Qian, Y.-T. Zhang, and H.-K. Zhao (2007), A fast sweeping method for static convex Hamilton-Jacobi equations, J. Sci. Comput., 31, pp. 237-271.
74. L. Qingfen (2003), Enhancement, Extraction, and Visualization of 3D Volume Data, Ph.D. thesis, Linköping University, Linköping, Sweden.
75. E. Rouy and A. Tourin (1992), A viscosity solutions approach to shape-from-shading, SIAMJ. Numer. Anal., 29, pp. 867-884.
76. J.A. Sethian (1996a), A fast marching level set method for monotonically advancing fronts, Proc. Natl. Acad. Sci. USA, 93, pp. 1591-1694.
77. J.A. Sethian (1996b), Level Set Methods, Cambridge University Press, Cambridge, UK.
78. J.A. Sethian and A. Vladimirsky (2003), Ordered upwind methods for static Hamilton-Jacobi equations: Theory and algorithms, SIAM J. Numer. Anal., 41, pp. 325-363.
79. H.M. Soner (1986a), Optimal control with state-space constraint I, SIAM J. Control Optim., 24, pp. 552-561.
80. H.M. Soner (1986b), Optimal control with state-space constraint II, SIAM J. Control Optim., 24, pp. 1110-1122.
81. D.G. Taylor and M.C. Bushell (1985), The spatial mapping of translational diffusion coefficients by the NMR imaging technique, Phys. Med. Bio., 30, pp. 345-349.
82. J.D. Tournier, F. Calamante, D.G. Gadian, and A. Connelly (2003), Diffusion-weighted magnetic resonance imaging fibre tracking using a front evolution algorithm, NeuroImage, 20, pp. 276-288.
83. Y.-H.R. Tsai, L.-T. Cheng, S. Osher, and H.-K. Zhao (2003a), Fast sweeping algorithms for a class of Hamilton-Jacobi equations, SIAM J. Numer. Anal., 41, pp. 673-694.
84. Y.-H. Tsai, Y. Giga, and S. Osher (2003b), A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations, Math. Comp., 72, pp. 159-181.
85. J.N. Tsitsiklis (1995), Efficient algorithms for globally optimal trajectories, IEEE Trans. Automat. Control, 40, pp. 1528-1538.
86. D. Tuch (2002), Diffusion MRI of Complex Tissue Structure, Ph.D. thesis, Harvard University, Massachusetts Institute of Technology, Cambridge, MA.
87. B. Vemuri, Y. Chen, M. Rao, T. McGraw, T. Mareci, and Z. Wang (2001), Fiber tract mapping from diffusion tensor MRI, in Proceedings of the IEEE Workshop on Variational and Level Set Methods, p. 81.
88. R. Venkataraman, R. Holsapple, and D. Doman (2002), Optimal Control Problems on Riemannian Manifolds: Theory and Applications, Technical report A433804, Air Force Research Lab Wright-Patterson AFB OH Air Vehicles Directorate, Wright-Patterson AFB, OH.
89. V.J. Wedeen, T.L. Davis, R.M. Weisskoff, R. Tootell, B.R. Rosen, and J.W. Belliveau (1995), White matter connectivity explored by MRI, in Proceedings of the First International Conference for Functional Mapping of the Human Brain, p. P1.36.
90. L. Yatziv, A. Bartesaghi, and G. Sapiro (2006), O(N) implementation of the fast marching algorithm, J. Comput. Phys., 212, pp. 393-399.
91. E. Yörük, B. Acar, and R. Bammer (2005), A physical model for DT-MRI based connectivity map computation, in Medical Image Computing and Computer-Assisted Intervention, Springer-Verlag, Berlin, Heidelberg, pp. 213-220.
92. F. Zhang, C. Goodlett, E. Hancock, and G. Gerig (2007), Probabilistic fiber tracking using particle filtering, in Medical Image Computing and Computer-Assisted Intervention, Springer-Verlag, Berlin, Heidelberg, pp. 144-152.