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Computational and Mathematical Methods in Medicine
Volume 2014, Article ID 426902, 12 pages
http://dx.doi.org/10.1155/2014/426902
Research Article

A 3D Finite-Difference BiCG Iterative Solver with the Fourier-Jacobi Preconditioner for the Anisotropic EIT/EEG Forward Problem

1Electrical Geodesics, Inc., Eugene, OR 97403, USA
2Department of Computer and Information Science, 1202 University of Oregon, Eugene, OR 97403, USA
3Department of Mathematics and Mechanics, Belarusian State University, 220050 Minsk, Belarus

Received 28 September 2013; Revised 25 November 2013; Accepted 26 November 2013; Published 12 January 2014

Academic Editor: Jianlong Qiu

Copyright © 2014 Sergei Turovets et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems, vol. 15, no. 2, pp. R41–R49, 1999. View at Publisher · View at Google Scholar · View at Scopus
  2. R. Gulrajani, Bioelectricity and Biomagnetism, John Wiley & Sons, New York, NY, USA, 1998.
  3. H. Hallez, B. Vanrumste, R. Grech et al., “Review on solving the forward problem in EEG source analysis,” Journal of NeuroEngineering and Rehabilitation, vol. 4, article 46, 2007. View at Publisher · View at Google Scholar · View at Scopus
  4. J.-F. P. J. Abascal, S. R. Arridge, W. R. B. Lionheart, R. H. Bayford, and D. S. Holder, “Validation of a finite-element solution for electrical impedance tomography in an anisotropic medium,” Physiological Measurement, vol. 28, no. 7, pp. S129–S140, 2007. View at Publisher · View at Google Scholar · View at Scopus
  5. C. H. Wolters, A. Anwander, X. Tricoche, D. Weinstein, M. A. Koch, and R. S. MacLeod, “Influence of tissue conductivity anisotropy on EEG/MEG field and return current computation in a realistic head model: a simulation and visualization study using high-resolution finite element modeling,” NeuroImage, vol. 30, no. 3, pp. 813–826, 2006. View at Publisher · View at Google Scholar · View at Scopus
  6. R. Bashar, Y. Li, and P. Wen, “Influence of white matter inhomogeneous anisotropy on EEG forward computing,” Australasian Physical and Engineering Sciences in Medicine, vol. 31, no. 2, pp. 122–130, 2008. View at Google Scholar · View at Scopus
  7. W. H. Lee, Z. Liu, B. A. Mueller, K. Lim, and B. He, “Influence of white matter anisotropic conductivity on EEG source localization: comparison to fMRI in human primary visual cortex,” Clinical Neurophysiology, vol. 120, no. 12, pp. 2071–2081, 2009. View at Publisher · View at Google Scholar · View at Scopus
  8. T. de Marco, F. Ries, M. Guermandi, and R. Guerrieri, “EIT forward problem parallel simulation environment with anisotropic tissue and realistic electrode models,” IEEE Transactions on Biomedical Engineering, vol. 59, no. 5, pp. 1229–1239, 2012. View at Publisher · View at Google Scholar · View at Scopus
  9. C. G. Bénar and J. Gotman, “Modeling of post-surgical brain and skull defects in the EEG inverse problem with the boundary element method,” Clinical Neurophysiology, vol. 113, no. 1, pp. 48–56, 2002. View at Publisher · View at Google Scholar · View at Scopus
  10. W. Hackbusch, Multi-Grid Methods and Applications, vol. 4, Springer, Berlin, Germany, 1985.
  11. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, The Numerical Recipes in C: The Art of Scientific Computing, Cambridge University Press, New York, NY, USA, 2nd edition, 1992.
  12. L. A. Hageman and D. M. Young, Applied Iterative Methods, Academic Press, New York, NY, USA, 1981.
  13. R. Barrett, M. Berry, T. F. Chan et al., Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, Pa, USA, 1994.
  14. H. A. van der Vorst, “BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems,” SIAM Journal on Scientific and Statistical Computing, vol. 13, no. 2, pp. 631–644, 1992. View at Publisher · View at Google Scholar
  15. M. Mohr and B. Vanrumste, “Comparing iterative solvers for linear systems associated with the finite difference discretisation of the forward problem in electro-encephalographic source analysis,” Medical and Biological Engineering and Computing, vol. 41, no. 1, pp. 75–84, 2003. View at Publisher · View at Google Scholar · View at Scopus
  16. V. M. Volkov, A. A. Zherdetskij, S. I. Turovets, and A. D. Malony, “A fast BiCG solver for the isotropic Poisson equation in the forward EIT problem in cylinder phantoms,” Journal of Physics: Conference Series, vol. 224, no. 1, Article ID 012153, 2010. View at Publisher · View at Google Scholar · View at Scopus
  17. http://www.mathworks.com/help/matlab/ref/bicgstab.html.
  18. General-purpose computation using graphics hardware, http://gpgpu.org/.
  19. A. Salman, S. Turovets, A. Malony, and V. Volkov, “Multi-cluster, mixed-mode computational modeling of human head conductivity,” in OpenMP Shared Memory Parallel Programming, vol. 4315 of Lecture Notes in Computer Science, pp. 119–130, Springer, Berlin, Germany, 2008. View at Google Scholar
  20. V. Volkov, A. Zherdetsky, S. Turovets, and A. Malony, “A 3D vector-additive iterative solver for the anisotropic inhomogeneous poisson equation in the forward EEG problem,” in Computational Science—ICCS 2009, vol. 5544 of Lecture Notes in Computer Science, pp. 511–520, Springer, Berlin, Germany, 2009. View at Publisher · View at Google Scholar · View at Scopus
  21. D. Ozog, A. Salman, A. Malony, S. Turovets, V. Volkov, and D. Tucker, “Next-generation human brain neuroimaging and the role of high-performance computing,” in Proceedings of the High Performance Computing & Simulation Conference (HPCS '13), Helsinki, Finland, July 2013.
  22. D. N. Barnes, J. S. George, and K. T. Ng, “Finite difference iterative solvers for electroencephalography: serial and parallel performance analysis,” Medical & Biological Engineering & Computing, vol. 46, no. 9, pp. 901–910, 2008. View at Publisher · View at Google Scholar · View at Scopus
  23. R. Glowinski, T.-W. Pan, and J. Periaux, “A fictitious domain method for Dirichlet problem and applications,” Computer Methods in Applied Mechanics and Engineering, vol. 111, no. 3-4, pp. 283–303, 1994. View at Google Scholar · View at Scopus
  24. V. K. Saul'ev, “On the solution of some boundary value problems on high performance computers by fictitious domain method,” Siberian Mathematical Journal, vol. 4, no. 4, pp. 912–925, 1963 (Russian). View at Google Scholar
  25. A. A. Samarskii and A. V. Gulin, Numerical Methods of Mathematical Physics, Nauchnyi Mir, Moscow, Russia, 2000 (Russian).
  26. V. N. Abrashin, A. A. Egorov, and N. G. Zhadaeva, “On a class of additive iterative methods,” Differential Equations, vol. 37, no. 12, pp. 1751–1760, 2001. View at Publisher · View at Google Scholar · View at Scopus
  27. P.-Y. Bondiau, O. Clatz, M. Sermesant et al., “Biocomputing: numerical simulation of glioblastoma growth using diffusion tensor imaging,” Physics in Medicine and Biology, vol. 53, no. 4, pp. 879–893, 2008. View at Publisher · View at Google Scholar · View at Scopus
  28. C. Qin, N. Kang, and N. Cao, “Performance evaluation of anisotropic diffusion simulation based tractography on phantom images,” in Proceedings of the 45th Annual ACM Southeast Conference (ACMSE '07), pp. 521–522, Winston-Salem, NC, USA, July 2007. View at Publisher · View at Google Scholar · View at Scopus
  29. J. C. de Munck and M. J. Peters, “A fast method to compute the potential in the multisphere model,” IEEE Transactions on Biomedical Engineering, vol. 40, no. 11, pp. 1166–1174, 1993. View at Publisher · View at Google Scholar · View at Scopus
  30. T. C. Ferree, K. J. Eriksen, and D. M. Tucker, “Regional head tissue conductivity estimation for improved EEG analysis,” IEEE Transactions on Biomedical Engineering, vol. 47, no. 12, pp. 1584–1592, 2000. View at Publisher · View at Google Scholar · View at Scopus
  31. T. F. Oostendorp, J. Delbeke, and D. F. Stegeman, “The conductivity of the human skull: results of in vivo and in vitro measurements,” IEEE Transactions on Biomedical Engineering, vol. 47, no. 11, pp. 1487–1492, 2000. View at Publisher · View at Google Scholar · View at Scopus
  32. S. I. Gonçalves, J. C. de Munck, J. P. A. Verbunt, F. Bijma, R. M. Heethaar, and F. Lopes da Silva, “In vivo measurement of the brain and skull resistivities using an EIT-based method and realistic models for the head,” IEEE Transactions on Biomedical Engineering, vol. 50, no. 6, pp. 754–767, 2003. View at Publisher · View at Google Scholar · View at Scopus
  33. Y. Zhang, W. van Drongelen, and B. He, “Estimation of in vivo brain-to-skull conductivity ratio in humans,” Applied Physics Letters, vol. 89, no. 22, Article ID 223903, 3 pages, 2006. View at Publisher · View at Google Scholar · View at Scopus
  34. S. Rush and D. A. Driscoll, “EEG electrode sensitivity—an application of reciprocity,” IEEE Transactions on Biomedical Engineering, vol. 16, no. 1, pp. 15–22, 1969. View at Google Scholar · View at Scopus
  35. S. Rush and D. A. Driscoll, “Current distribution in the brain from surface electrodes,” Anesthesia and Analgesia, vol. 47, no. 6, pp. 717–723, 1968. View at Google Scholar · View at Scopus
  36. L. R. Frank, “Characterization of anisotropy in high angular resolution diffusion-weighted MRI,” Magnetic Resonance in Medicine, vol. 47, no. 6, pp. 1083–1099, 2002. View at Publisher · View at Google Scholar · View at Scopus
  37. K. Li, Neuroanatomical segmentation in MRI exploiting a priori knowledge [Ph.D. thesis], Department of Computer and Information Science, University of Oregon, 2007.
  38. K. Li, A. D. Malony, and D. M. Tucker, “Image segmentation method,” US8478011 B2, 2006.
  39. http://teem.sourceforge.net/.
  40. D. K. Hammond, B. Scherrer, and A. Maloney, “Incorporating anatomical connectivity into EEG source estimation via sparse approximation with cortical graph wavelets,” in Proceedings of the IEEE International Conference on Acoustics, Signals and Signal Processing (ICASSP '12), Kyoto, Japan, March 2012.
  41. D. K. Hammond, B. Scherrer, and S. K. Warfield, “Cortical graph smoothing: a novel method for exploiting DWI-derived anatomical brain connectivity to improve EEG source estimation,” IEEE Transactions on Medical Imaging, vol. 32, no. 10, pp. 1952–1963, 2013. View at Publisher · View at Google Scholar
  42. D. Mattes, D. R. Haynor, H. Vesselle, T. K. Lewellen, and W. Eubank, “PET-CT image registration in the chest using free-form deformations,” IEEE Transactions on Medical Imaging, vol. 22, no. 1, pp. 120–128, 2003. View at Publisher · View at Google Scholar · View at Scopus
  43. F. Maes, A. Collignon, D. Vandermeulen, G. Marchal, and P. Suetens, “Multimodality image registration by maximization of mutual information,” IEEE Transactions on Medical Imaging, vol. 16, no. 2, pp. 187–198, 1997. View at Google Scholar · View at Scopus
  44. V. Arsigny, P. Fillard, X. Pennec, and N. Ayache, “Log-Euclidean metrics for fast and simple calculus on diffusion tensors,” Magnetic Resonance in Medicine, vol. 56, no. 2, pp. 411–421, 2006. View at Publisher · View at Google Scholar · View at Scopus
  45. R. Eymard, T. Gallouët, and R. Herbin, “Finite volume methods,” Handbook of Numerical Analysis, vol. 7, pp. 713–1018, 2000. View at Publisher · View at Google Scholar · View at Scopus
  46. I. V. Rybak, “Monotone and conservative difference schemes for elliptic equations with mixed derivatives 1,” Mathematical Modelling and Analysis, vol. 9, no. 2, pp. 169–178, 2004. View at Publisher · View at Google Scholar
  47. H. S. Suh, W. H. Lee, and T.-S. Kim, “Influence of anisotropic conductivity in the skull and white matter on transcranial direct current stimulation via an anatomically realistic finite element head model,” Physics in Medicine and Biology, vol. 57, no. 21, pp. 6961–6980, 2012. View at Publisher · View at Google Scholar
  48. S. Shahid, P. Wen, and T. Ahfock, “Numerical investigation of white matter anisotropic conductivity in defining current distribution under tDCS,” Computer Methods and Programs in Biomedicine, vol. 109, no. 1, pp. 48–64, 2012. View at Publisher · View at Google Scholar