Previous |  Up |  Next


Allen–Cahn equation; anisotropic diffusion; finite volume method; MR–DTI; MR tractography; medical visualization
Magnetic Resonance Diffusion Tensor Imaging (MR–DTI) is a noninvasive in vivo method capable of examining the structure of human brain, providing information about the position and orientation of the neural tracts. After a short introduction to the principles of MR–DTI, this paper describes the steps of the proposed neural tract visualization technique based on the DTI data. The cornerstone of the algorithm is a texture diffusion procedure modeled mathematically by the problem for the Allen–Cahn equation with diffusion anisotropy controlled by a tensor field. Focus is put on the issues of the numerical solution of the given problem, using the finite volume method for spatial domain discretization. Several numerical schemes are compared with the aim of reducing the artificial (numerical) isotropic diffusion. The remaining steps of the algorithm are commented on as well, including the acquisition of the tensor field before the actual computation begins and the postprocessing used to obtain the final images. Finally, the visualization results are presented.
[1] M. Beneš: Mathematical analysis of phase-field equations with numerically efficient coupling terms. Interfaces and Free Boundaries 3 (2001), 201–221. MR 1825658
[2] M. Beneš: Diffuse-interface treatment of the anisotropic mean-curvature flow. Appl. Math. 48 (2003), 6, 437–453. MR 2025297
[3] M. Beneš, V. Chalupecký, and K. Mikula: Geometrical image segmentation by the Allen–Cahn equation. Appl. Numer. Math. 52 (2004), 2, 187–205. MR 2091399
[4] D. L. Bihan et al.: Diffusion tensor imaging: Concepts and applications. J. Magnetic Resonance Imaging 13 (2001), 534–546.
[5] W. F. Block et al.: Encyclopedia of Medical Devices and Instrumentation, chapter Magnetic Resonance Imaging. Second edition. Wiley, New York 2006, pp. 283–298.
[6] A. F. M. Dasilva et al.: A primer on diffusion tensor imaging of anatomical substructures. Neurosurgical Focus 15 (2003), 1–4.
[7] R. Eymard, T. Gallouët, and R. Herbin: Finite volume methods. In: Handbook of Numerical Analysis (P. G. Ciarlet and J. L. Lions, eds.), volume 7, Elsevier, 2000, pp. 715–1022. MR 1804748
[8] P. Fillard and G. Gerig: Analysis tool for diffusion tensor MRI. In: Proc. Medical Image Computing and Computer-Assisted Intervention (MICCAI), Springer–Verlag, Berlin 2003, pp. 967–968.
[9] K. M. Hasan, D. L. Parker, and A. L. Alexander: Comparison of gradient encoding schemes for diffusion-tensor MRI. J. Magnetic Resonance Imaging 13 (2001), 769–780.
[10] W. Kahle, H. Leonhardt, and W. Platzer: Color Atlas and Textbook of Human Anatomy in 3 Volumes, volume 1: Locomotor System. Third edition. Georg Thieme Verlag, Stuttgart 1986.
[11] Z.-P. Liang and P. C. Lauterbur: Principles of Magnetic Resonance Imaging: A Signal Processing Perspective. Wiley–IEEE Press, 1999.
[12] H. Lomax, T. H. Pulliam, and D. W. Zingg: Fundamentals of Computational Fluid Dynamics. Springer–Verlag, Berlin 2001. MR 1843296
[13] J. Mach: Application of non-linear diffusion in algorithms of mathematical visualization. In: Proc. Czech–Japanese Seminar in Applied Mathematics 2006 (M. Beneš, M. Kimura, and T. Nakaki, eds.), volume 6 of COE Lecture Note, Faculty of Mathematics, Kyushu University Fukuoka, 2007, pp. 156–164.
[14] S. Mori and J. Zhang: Principles of diffusion tensor imaging and its applications to basic neuroscience research. Neuron 51 (2006), 527–539.
[15] T. Oberhuber: Finite difference scheme for the Willmore flow of graphs. Kybernetika 43 (2007), 6, 855–867. MR 2388399 | Zbl 1140.53032
[16] W. E. Schiesser: The Numerical Method of Lines: Integration of Partial Differential Equations. Academic Press, San Diego 1991. MR 1117905 | Zbl 0763.65076
[17] R. Sierra: Nonrigid Registration of Diffusion Tensor Images. Master’s Thesis, Swiss Federal Institute of Technology, Zurich 2001.
[18] P. Strachota: Anisotropic Diffusion in Mathematical Visualization. In: Science and Supercomputing in Europe – Report 2007, Bologna 2008, CINECA Consorzio Interuniversitario, pp. 826–831,
[19] J. S. Suri, S. K. Setarehdan, and S. Singh, eds.: Advanced Algorithmic Approaches to Medical Image Segmentation: State-of-the-art Application in Cardiology, Neurology, Mammography and Pathology. Springer–Verlag, New York 2002.
[20] D. Tschumperlé and R. Deriche: Variational frameworks for DT-MRI estimation, regularization and visualization. In: Ninth IEEE Internat. Conference on Computer Vision (ICCV’03), volume 1, 2003, p. 116.
[21] D. Tschumperlé and R. Deriche: Tensor Field Visualization with PDE’s and Application to DT-MRI Fiber Visualization. INRIA Sophia-Antipolis, Odyssée Lab. 2004.
[22] C. F. Westin et al.: Processing and visualization for diffusion tensor MRI. Medical Image Analysis 6 (2002), 93–108.
Partner of
EuDML logo