Abstract

Image reconstruction in fluorescence optical tomography is a three-dimensional nonlinear ill-posed problem governed by a system of partial differential equations. In this paper we demonstrate that a combination of state of the art numerical algorithms and a careful hardware optimized implementation allows to solve this large-scale inverse problem in a few seconds on standard desktop PCs with modern graphics hardware. In particular, we present methods to solve not only the forward but also the non-linear inverse problem by massively parallel programming on graphics processors. A comparison of optimized CPU and GPU implementations shows that the reconstruction can be accelerated by factors of about 15 through the use of the graphics hardware without compromising the accuracy in the reconstructed images.

© 2011 OSA

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  1. S. Mordon, V. Maunoury, J. M. Devoisselle, Y. Abbas, and D. Coustaud, “Characterization of tumorous and normal tissue using a pH-sensitive fluorescence indicator (5,6-carboxyfluorescein) in vivo,” J. Photochem. Photobiol. B13, 307–314 (1992).
    [CrossRef] [PubMed]
  2. I. Gannot, I. Ron, F. Hekmat, V. Chernomordik, and A. Gandjbakhche, “Functional optical detection based on pH dependent fluorescence lifetime,” Lasers Surg. Med.35, 342–348 (2004).
    [CrossRef] [PubMed]
  3. E. Shives, Y. Xu, and H. Jiang, “Fluorescence lifetime tomography of turbid media based on an oxygen-sensitive dye,” Opt. Express10, 1557–1562 (2002).
    [PubMed]
  4. B. Zhang, X. Yang, F. Yang, X. Yang, C. Qin, D. Han, X. Ma, K. Liu, and J. Tian, “The CUBLAS and CULA based GPU acceleration of adaptive finite element framework for bioluminescence tomography,” Opt. Express18, 20201–20214 (2010).
    [CrossRef] [PubMed]
  5. J. Prakash, V. Chandrasekharan, V. Upendra, and P. K. Yalavarthy, “Accelerating frequency-domain diffuse optical tomographic image reconstruction using graphics processing units,” J. Biomed. Opt.15, 066009 (2010).
    [CrossRef]
  6. F. Knoll, M. Freiberger, K. Bredies, and R. Stollberger, “AGILE: An open source library for image reconstruction using graphics card hardware acceleration,” in “Proceedings of the 19th Scientific Meeting and Exhibition of ISMRM, Montreal, CA,” (2011), p. 2554.
  7. S. R. Arridge, “Optical tomography in medical imaging,” Inv. Probl.15, R41–R93 (1999).
    [CrossRef]
  8. A. Joshi, W. Bangerth, and W. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express12, 5402–5417 (2004).
    [CrossRef] [PubMed]
  9. S. R. Arridge and M. Schweiger, “The use of multiple data types in time-resolved optical absorption and scattering tomography (TOAST),” in “Mathematical Methods in Medical Imaging II,”, D. C. Wilson and J. N. Wilson, eds. (1993), Proc. SPIE2035, pp. 218–229.
    [CrossRef]
  10. E. M. Sevick and B. Chance, “Photon migration in a model of the head measured using time and frequency domain techniques: potentials of spectroscopy and imaging,” in “Time-Resolved Spectroscopy and Imaging of Tissues,”, B. Chance and A. Katzir, eds. (1991), Proc. SPIE1431, pp. 84–96.
    [CrossRef]
  11. S. R. Arridge, “Photon-measurement density functions. part I: Analytical forms,” Appl. Opt.34, 7395–7409 (1995).
    [CrossRef] [PubMed]
  12. A. Bakushinsky, “The problem of the convergence of the iteratively regularized Gauss-Newton method,” Comput. Math. Math. Phys.32, 1353–1359 (1992).
  13. M. Hanke, “Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems,” Numer. Func. Anal. Optim.18, 971–993 (1997).
    [CrossRef]
  14. D. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” SIAM J. Appl. Math.11, 431–441 (1963).
    [CrossRef]
  15. B. Kaltenbacher, “Some Newton-type methods for the regularization of nonlinear ill-posed problems,” Inv. Probl.13, 729–753 (1997).
    [CrossRef]
  16. H. Jiang, “Frequency-domain fluorescent diffusion tomography: A finite-element-based algorithm and simulations,” Appl. Opt.37, 5337–5343 (1998).
    [CrossRef]
  17. R. Roy and E. M. Sevick-Muraca, “Truncated Newtons optimization scheme for absorption and fluorescence optical tomography: Part I theory and formulation,” Opt. Express4, 353–371 (1999).
    [CrossRef] [PubMed]
  18. M. Schweiger, S. R. Arridge, and I. Nissilä, “Gauss–Newton method of image reconstruction in diffuse optical tomography,” Phys. Med. Biol.50, 2365–2386 (2005).
    [CrossRef] [PubMed]
  19. H. Egger and M. Schlottbom, “Efficient reliable image reconstruction schemes for diffuse optical tomography,” Inv. Probl. Sci. Eng.19, 155–180 (2011).
    [CrossRef]
  20. A. Godavarty, E. M. Sevick-Muraca, and M. J. Eppstein, “Three-dimensional fluorescence lifetime tomography,” Med. Phys.32, 992–1000 (2005).
    [CrossRef] [PubMed]
  21. M. Freiberger, H. Egger, and H. Scharfetter, “Nonlinear inversion schemes for fluorescence optical tomography,” IEEE Trans. Biomed. Eng.57, 2723–2729 (2010).
    [CrossRef]
  22. D. Göddeke, C. Becker, and S. Turek, “Integrating GPUs as fast co–processors into the parallel FE package FEAST,” in “19th Symposium Simulations Technique,”, M. Becker and H. Szczerbicka, eds., Frontiers in Simulation (SCS Publishing House, 2006), pp. 277–282.
  23. M. M. Baskaran and R. Bordawekar, “Optimizing sparse matrix-vector multiplication on GPUs,” IBM Technical Report RC24704, IBM Ltd. (2009).
  24. M. Liebmann, “Efficient PDE solvers on modern hardware with applications in medical and technical sciences,” Ph.D. thesis (University of Graz, 2009).
  25. W. L. Briggs, V. E. Henson, and S. F. McCormick, A multigrid tutorial (SIAM, 2000), 2nd ed.
    [CrossRef]
  26. E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, 1999), 3rd ed.
    [CrossRef]
  27. B. Dogdas, D. Stout, A. F. Chatziiouannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol.52, 577–587 (2007).
    [CrossRef] [PubMed]
  28. G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.50, 4225–4241 (2005).
    [CrossRef] [PubMed]
  29. M. Keijzer, W. M. Star, and P. R. M. Storchi, “Optical diffusion in layered media,” Appl. Opt.27, 1820–1824 (1988).
    [CrossRef] [PubMed]
  30. A. Joshi, “Adaptive finite element methods for fluorescence enhanced optical tomography,” Ph.D. thesis (Texas A&M University, 2005).
  31. M. L. Landsman, G. Kwant, G. A. Mook, and W. G. Zijlstra, “Light-absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol.40, 575–583 (1976).
    [PubMed]
  32. J. Schöberl, “NETGEN - an advancing front 2D/3D-mesh generator based on abstract rules,” Comput. Vis. Sci.1, 41–52 (1997).
    [CrossRef]
  33. NVIDIA, NVIDIA CUDA Programming Guide 2.0 (NVIDIA Cooperation, 2008).
  34. N. Bell and M. Garland, “Implementing sparse matrix-vector multiplication on throughput-oriented processors,” in “SC ’09: Proceedings of the Conference on High Performance Computing Networking, Storage and Analysis,” (ACM, 2009), pp. 1–11.
  35. D. Göddeke, H. Wobker, R. Strzodka, J. Mohd-Yusof, P. S. McCormick, and S. Turek, “Co-processor acceleration of an unmodified parallel solid mechanics code with FEASTGPU,” Int. J. Comput. Sci. Eng.4, 254–269 (2009).
    [CrossRef]
  36. M. Köster, D. Göddeke, H. Wobker, and S. Turek, “How to gain speedups of 1000 on single processors with fast FEM solvers – benchmarking numerical and computational efficiency,” Tech. rep., Fakultät für Mathematik, TU Dortmund (2008). Ergebnisberichte des Instituts für Angewandte Mathematik, Nummer 382.
  37. V. A. Morozov, “On the solution of functional equations by the method of regularization,” Soviet Math. Dokl.7, 414–417 (1966).
  38. H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, 1996).
  39. P. C. Hansen, Rank-Defficient and Discrete Ill-Posed Problems (SIAM, 1998).
    [CrossRef]
  40. G. Wahba, Spline Models for Observational Data (SIAM, 1990).

2011

H. Egger and M. Schlottbom, “Efficient reliable image reconstruction schemes for diffuse optical tomography,” Inv. Probl. Sci. Eng.19, 155–180 (2011).
[CrossRef]

2010

B. Zhang, X. Yang, F. Yang, X. Yang, C. Qin, D. Han, X. Ma, K. Liu, and J. Tian, “The CUBLAS and CULA based GPU acceleration of adaptive finite element framework for bioluminescence tomography,” Opt. Express18, 20201–20214 (2010).
[CrossRef] [PubMed]

J. Prakash, V. Chandrasekharan, V. Upendra, and P. K. Yalavarthy, “Accelerating frequency-domain diffuse optical tomographic image reconstruction using graphics processing units,” J. Biomed. Opt.15, 066009 (2010).
[CrossRef]

M. Freiberger, H. Egger, and H. Scharfetter, “Nonlinear inversion schemes for fluorescence optical tomography,” IEEE Trans. Biomed. Eng.57, 2723–2729 (2010).
[CrossRef]

2009

D. Göddeke, H. Wobker, R. Strzodka, J. Mohd-Yusof, P. S. McCormick, and S. Turek, “Co-processor acceleration of an unmodified parallel solid mechanics code with FEASTGPU,” Int. J. Comput. Sci. Eng.4, 254–269 (2009).
[CrossRef]

2007

B. Dogdas, D. Stout, A. F. Chatziiouannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol.52, 577–587 (2007).
[CrossRef] [PubMed]

2005

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.50, 4225–4241 (2005).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, and I. Nissilä, “Gauss–Newton method of image reconstruction in diffuse optical tomography,” Phys. Med. Biol.50, 2365–2386 (2005).
[CrossRef] [PubMed]

A. Godavarty, E. M. Sevick-Muraca, and M. J. Eppstein, “Three-dimensional fluorescence lifetime tomography,” Med. Phys.32, 992–1000 (2005).
[CrossRef] [PubMed]

2004

I. Gannot, I. Ron, F. Hekmat, V. Chernomordik, and A. Gandjbakhche, “Functional optical detection based on pH dependent fluorescence lifetime,” Lasers Surg. Med.35, 342–348 (2004).
[CrossRef] [PubMed]

A. Joshi, W. Bangerth, and W. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express12, 5402–5417 (2004).
[CrossRef] [PubMed]

2002

1999

1998

1997

M. Hanke, “Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems,” Numer. Func. Anal. Optim.18, 971–993 (1997).
[CrossRef]

B. Kaltenbacher, “Some Newton-type methods for the regularization of nonlinear ill-posed problems,” Inv. Probl.13, 729–753 (1997).
[CrossRef]

J. Schöberl, “NETGEN - an advancing front 2D/3D-mesh generator based on abstract rules,” Comput. Vis. Sci.1, 41–52 (1997).
[CrossRef]

1995

1992

A. Bakushinsky, “The problem of the convergence of the iteratively regularized Gauss-Newton method,” Comput. Math. Math. Phys.32, 1353–1359 (1992).

S. Mordon, V. Maunoury, J. M. Devoisselle, Y. Abbas, and D. Coustaud, “Characterization of tumorous and normal tissue using a pH-sensitive fluorescence indicator (5,6-carboxyfluorescein) in vivo,” J. Photochem. Photobiol. B13, 307–314 (1992).
[CrossRef] [PubMed]

1991

E. M. Sevick and B. Chance, “Photon migration in a model of the head measured using time and frequency domain techniques: potentials of spectroscopy and imaging,” in “Time-Resolved Spectroscopy and Imaging of Tissues,”, B. Chance and A. Katzir, eds. (1991), Proc. SPIE1431, pp. 84–96.
[CrossRef]

1988

1976

M. L. Landsman, G. Kwant, G. A. Mook, and W. G. Zijlstra, “Light-absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol.40, 575–583 (1976).
[PubMed]

1966

V. A. Morozov, “On the solution of functional equations by the method of regularization,” Soviet Math. Dokl.7, 414–417 (1966).

1963

D. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” SIAM J. Appl. Math.11, 431–441 (1963).
[CrossRef]

Abbas, Y.

S. Mordon, V. Maunoury, J. M. Devoisselle, Y. Abbas, and D. Coustaud, “Characterization of tumorous and normal tissue using a pH-sensitive fluorescence indicator (5,6-carboxyfluorescein) in vivo,” J. Photochem. Photobiol. B13, 307–314 (1992).
[CrossRef] [PubMed]

Alexandrakis, G.

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.50, 4225–4241 (2005).
[CrossRef] [PubMed]

Anderson, E.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, 1999), 3rd ed.
[CrossRef]

Arridge, S. R.

M. Schweiger, S. R. Arridge, and I. Nissilä, “Gauss–Newton method of image reconstruction in diffuse optical tomography,” Phys. Med. Biol.50, 2365–2386 (2005).
[CrossRef] [PubMed]

S. R. Arridge, “Optical tomography in medical imaging,” Inv. Probl.15, R41–R93 (1999).
[CrossRef]

S. R. Arridge, “Photon-measurement density functions. part I: Analytical forms,” Appl. Opt.34, 7395–7409 (1995).
[CrossRef] [PubMed]

S. R. Arridge and M. Schweiger, “The use of multiple data types in time-resolved optical absorption and scattering tomography (TOAST),” in “Mathematical Methods in Medical Imaging II,”, D. C. Wilson and J. N. Wilson, eds. (1993), Proc. SPIE2035, pp. 218–229.
[CrossRef]

Bai, Z.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, 1999), 3rd ed.
[CrossRef]

Bakushinsky, A.

A. Bakushinsky, “The problem of the convergence of the iteratively regularized Gauss-Newton method,” Comput. Math. Math. Phys.32, 1353–1359 (1992).

Bangerth, W.

Baskaran, M. M.

M. M. Baskaran and R. Bordawekar, “Optimizing sparse matrix-vector multiplication on GPUs,” IBM Technical Report RC24704, IBM Ltd. (2009).

Becker, C.

D. Göddeke, C. Becker, and S. Turek, “Integrating GPUs as fast co–processors into the parallel FE package FEAST,” in “19th Symposium Simulations Technique,”, M. Becker and H. Szczerbicka, eds., Frontiers in Simulation (SCS Publishing House, 2006), pp. 277–282.

Bell, N.

N. Bell and M. Garland, “Implementing sparse matrix-vector multiplication on throughput-oriented processors,” in “SC ’09: Proceedings of the Conference on High Performance Computing Networking, Storage and Analysis,” (ACM, 2009), pp. 1–11.

Bischof, C.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, 1999), 3rd ed.
[CrossRef]

Blackford, S.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, 1999), 3rd ed.
[CrossRef]

Bordawekar, R.

M. M. Baskaran and R. Bordawekar, “Optimizing sparse matrix-vector multiplication on GPUs,” IBM Technical Report RC24704, IBM Ltd. (2009).

Bredies, K.

F. Knoll, M. Freiberger, K. Bredies, and R. Stollberger, “AGILE: An open source library for image reconstruction using graphics card hardware acceleration,” in “Proceedings of the 19th Scientific Meeting and Exhibition of ISMRM, Montreal, CA,” (2011), p. 2554.

Briggs, W. L.

W. L. Briggs, V. E. Henson, and S. F. McCormick, A multigrid tutorial (SIAM, 2000), 2nd ed.
[CrossRef]

Chance, B.

E. M. Sevick and B. Chance, “Photon migration in a model of the head measured using time and frequency domain techniques: potentials of spectroscopy and imaging,” in “Time-Resolved Spectroscopy and Imaging of Tissues,”, B. Chance and A. Katzir, eds. (1991), Proc. SPIE1431, pp. 84–96.
[CrossRef]

Chandrasekharan, V.

J. Prakash, V. Chandrasekharan, V. Upendra, and P. K. Yalavarthy, “Accelerating frequency-domain diffuse optical tomographic image reconstruction using graphics processing units,” J. Biomed. Opt.15, 066009 (2010).
[CrossRef]

Chatziioannou, A. F.

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.50, 4225–4241 (2005).
[CrossRef] [PubMed]

Chatziiouannou, A. F.

B. Dogdas, D. Stout, A. F. Chatziiouannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol.52, 577–587 (2007).
[CrossRef] [PubMed]

Chernomordik, V.

I. Gannot, I. Ron, F. Hekmat, V. Chernomordik, and A. Gandjbakhche, “Functional optical detection based on pH dependent fluorescence lifetime,” Lasers Surg. Med.35, 342–348 (2004).
[CrossRef] [PubMed]

Coustaud, D.

S. Mordon, V. Maunoury, J. M. Devoisselle, Y. Abbas, and D. Coustaud, “Characterization of tumorous and normal tissue using a pH-sensitive fluorescence indicator (5,6-carboxyfluorescein) in vivo,” J. Photochem. Photobiol. B13, 307–314 (1992).
[CrossRef] [PubMed]

Demmel, J.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, 1999), 3rd ed.
[CrossRef]

Devoisselle, J. M.

S. Mordon, V. Maunoury, J. M. Devoisselle, Y. Abbas, and D. Coustaud, “Characterization of tumorous and normal tissue using a pH-sensitive fluorescence indicator (5,6-carboxyfluorescein) in vivo,” J. Photochem. Photobiol. B13, 307–314 (1992).
[CrossRef] [PubMed]

Dogdas, B.

B. Dogdas, D. Stout, A. F. Chatziiouannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol.52, 577–587 (2007).
[CrossRef] [PubMed]

Dongarra, J.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, 1999), 3rd ed.
[CrossRef]

Du Croz, J.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, 1999), 3rd ed.
[CrossRef]

Egger, H.

H. Egger and M. Schlottbom, “Efficient reliable image reconstruction schemes for diffuse optical tomography,” Inv. Probl. Sci. Eng.19, 155–180 (2011).
[CrossRef]

M. Freiberger, H. Egger, and H. Scharfetter, “Nonlinear inversion schemes for fluorescence optical tomography,” IEEE Trans. Biomed. Eng.57, 2723–2729 (2010).
[CrossRef]

Engl, H. W.

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, 1996).

Eppstein, M. J.

A. Godavarty, E. M. Sevick-Muraca, and M. J. Eppstein, “Three-dimensional fluorescence lifetime tomography,” Med. Phys.32, 992–1000 (2005).
[CrossRef] [PubMed]

Freiberger, M.

M. Freiberger, H. Egger, and H. Scharfetter, “Nonlinear inversion schemes for fluorescence optical tomography,” IEEE Trans. Biomed. Eng.57, 2723–2729 (2010).
[CrossRef]

F. Knoll, M. Freiberger, K. Bredies, and R. Stollberger, “AGILE: An open source library for image reconstruction using graphics card hardware acceleration,” in “Proceedings of the 19th Scientific Meeting and Exhibition of ISMRM, Montreal, CA,” (2011), p. 2554.

Gandjbakhche, A.

I. Gannot, I. Ron, F. Hekmat, V. Chernomordik, and A. Gandjbakhche, “Functional optical detection based on pH dependent fluorescence lifetime,” Lasers Surg. Med.35, 342–348 (2004).
[CrossRef] [PubMed]

Gannot, I.

I. Gannot, I. Ron, F. Hekmat, V. Chernomordik, and A. Gandjbakhche, “Functional optical detection based on pH dependent fluorescence lifetime,” Lasers Surg. Med.35, 342–348 (2004).
[CrossRef] [PubMed]

Garland, M.

N. Bell and M. Garland, “Implementing sparse matrix-vector multiplication on throughput-oriented processors,” in “SC ’09: Proceedings of the Conference on High Performance Computing Networking, Storage and Analysis,” (ACM, 2009), pp. 1–11.

Godavarty, A.

A. Godavarty, E. M. Sevick-Muraca, and M. J. Eppstein, “Three-dimensional fluorescence lifetime tomography,” Med. Phys.32, 992–1000 (2005).
[CrossRef] [PubMed]

Göddeke, D.

D. Göddeke, H. Wobker, R. Strzodka, J. Mohd-Yusof, P. S. McCormick, and S. Turek, “Co-processor acceleration of an unmodified parallel solid mechanics code with FEASTGPU,” Int. J. Comput. Sci. Eng.4, 254–269 (2009).
[CrossRef]

D. Göddeke, C. Becker, and S. Turek, “Integrating GPUs as fast co–processors into the parallel FE package FEAST,” in “19th Symposium Simulations Technique,”, M. Becker and H. Szczerbicka, eds., Frontiers in Simulation (SCS Publishing House, 2006), pp. 277–282.

M. Köster, D. Göddeke, H. Wobker, and S. Turek, “How to gain speedups of 1000 on single processors with fast FEM solvers – benchmarking numerical and computational efficiency,” Tech. rep., Fakultät für Mathematik, TU Dortmund (2008). Ergebnisberichte des Instituts für Angewandte Mathematik, Nummer 382.

Greenbaum, A.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, 1999), 3rd ed.
[CrossRef]

Hammarling, S.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, 1999), 3rd ed.
[CrossRef]

Han, D.

Hanke, M.

M. Hanke, “Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems,” Numer. Func. Anal. Optim.18, 971–993 (1997).
[CrossRef]

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, 1996).

Hansen, P. C.

P. C. Hansen, Rank-Defficient and Discrete Ill-Posed Problems (SIAM, 1998).
[CrossRef]

Hekmat, F.

I. Gannot, I. Ron, F. Hekmat, V. Chernomordik, and A. Gandjbakhche, “Functional optical detection based on pH dependent fluorescence lifetime,” Lasers Surg. Med.35, 342–348 (2004).
[CrossRef] [PubMed]

Henson, V. E.

W. L. Briggs, V. E. Henson, and S. F. McCormick, A multigrid tutorial (SIAM, 2000), 2nd ed.
[CrossRef]

Jiang, H.

Joshi, A.

A. Joshi, W. Bangerth, and W. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express12, 5402–5417 (2004).
[CrossRef] [PubMed]

A. Joshi, “Adaptive finite element methods for fluorescence enhanced optical tomography,” Ph.D. thesis (Texas A&M University, 2005).

Kaltenbacher, B.

B. Kaltenbacher, “Some Newton-type methods for the regularization of nonlinear ill-posed problems,” Inv. Probl.13, 729–753 (1997).
[CrossRef]

Keijzer, M.

Knoll, F.

F. Knoll, M. Freiberger, K. Bredies, and R. Stollberger, “AGILE: An open source library for image reconstruction using graphics card hardware acceleration,” in “Proceedings of the 19th Scientific Meeting and Exhibition of ISMRM, Montreal, CA,” (2011), p. 2554.

Köster, M.

M. Köster, D. Göddeke, H. Wobker, and S. Turek, “How to gain speedups of 1000 on single processors with fast FEM solvers – benchmarking numerical and computational efficiency,” Tech. rep., Fakultät für Mathematik, TU Dortmund (2008). Ergebnisberichte des Instituts für Angewandte Mathematik, Nummer 382.

Kwant, G.

M. L. Landsman, G. Kwant, G. A. Mook, and W. G. Zijlstra, “Light-absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol.40, 575–583 (1976).
[PubMed]

Landsman, M. L.

M. L. Landsman, G. Kwant, G. A. Mook, and W. G. Zijlstra, “Light-absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol.40, 575–583 (1976).
[PubMed]

Leahy, R. M.

B. Dogdas, D. Stout, A. F. Chatziiouannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol.52, 577–587 (2007).
[CrossRef] [PubMed]

Liebmann, M.

M. Liebmann, “Efficient PDE solvers on modern hardware with applications in medical and technical sciences,” Ph.D. thesis (University of Graz, 2009).

Liu, K.

Ma, X.

Marquardt, D.

D. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” SIAM J. Appl. Math.11, 431–441 (1963).
[CrossRef]

Maunoury, V.

S. Mordon, V. Maunoury, J. M. Devoisselle, Y. Abbas, and D. Coustaud, “Characterization of tumorous and normal tissue using a pH-sensitive fluorescence indicator (5,6-carboxyfluorescein) in vivo,” J. Photochem. Photobiol. B13, 307–314 (1992).
[CrossRef] [PubMed]

McCormick, P. S.

D. Göddeke, H. Wobker, R. Strzodka, J. Mohd-Yusof, P. S. McCormick, and S. Turek, “Co-processor acceleration of an unmodified parallel solid mechanics code with FEASTGPU,” Int. J. Comput. Sci. Eng.4, 254–269 (2009).
[CrossRef]

McCormick, S. F.

W. L. Briggs, V. E. Henson, and S. F. McCormick, A multigrid tutorial (SIAM, 2000), 2nd ed.
[CrossRef]

McKenney, A.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, 1999), 3rd ed.
[CrossRef]

Mohd-Yusof, J.

D. Göddeke, H. Wobker, R. Strzodka, J. Mohd-Yusof, P. S. McCormick, and S. Turek, “Co-processor acceleration of an unmodified parallel solid mechanics code with FEASTGPU,” Int. J. Comput. Sci. Eng.4, 254–269 (2009).
[CrossRef]

Mook, G. A.

M. L. Landsman, G. Kwant, G. A. Mook, and W. G. Zijlstra, “Light-absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol.40, 575–583 (1976).
[PubMed]

Mordon, S.

S. Mordon, V. Maunoury, J. M. Devoisselle, Y. Abbas, and D. Coustaud, “Characterization of tumorous and normal tissue using a pH-sensitive fluorescence indicator (5,6-carboxyfluorescein) in vivo,” J. Photochem. Photobiol. B13, 307–314 (1992).
[CrossRef] [PubMed]

Morozov, V. A.

V. A. Morozov, “On the solution of functional equations by the method of regularization,” Soviet Math. Dokl.7, 414–417 (1966).

Neubauer, A.

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, 1996).

Nissilä, I.

M. Schweiger, S. R. Arridge, and I. Nissilä, “Gauss–Newton method of image reconstruction in diffuse optical tomography,” Phys. Med. Biol.50, 2365–2386 (2005).
[CrossRef] [PubMed]

Prakash, J.

J. Prakash, V. Chandrasekharan, V. Upendra, and P. K. Yalavarthy, “Accelerating frequency-domain diffuse optical tomographic image reconstruction using graphics processing units,” J. Biomed. Opt.15, 066009 (2010).
[CrossRef]

Qin, C.

Rannou, F. R.

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.50, 4225–4241 (2005).
[CrossRef] [PubMed]

Ron, I.

I. Gannot, I. Ron, F. Hekmat, V. Chernomordik, and A. Gandjbakhche, “Functional optical detection based on pH dependent fluorescence lifetime,” Lasers Surg. Med.35, 342–348 (2004).
[CrossRef] [PubMed]

Roy, R.

Scharfetter, H.

M. Freiberger, H. Egger, and H. Scharfetter, “Nonlinear inversion schemes for fluorescence optical tomography,” IEEE Trans. Biomed. Eng.57, 2723–2729 (2010).
[CrossRef]

Schlottbom, M.

H. Egger and M. Schlottbom, “Efficient reliable image reconstruction schemes for diffuse optical tomography,” Inv. Probl. Sci. Eng.19, 155–180 (2011).
[CrossRef]

Schöberl, J.

J. Schöberl, “NETGEN - an advancing front 2D/3D-mesh generator based on abstract rules,” Comput. Vis. Sci.1, 41–52 (1997).
[CrossRef]

Schweiger, M.

M. Schweiger, S. R. Arridge, and I. Nissilä, “Gauss–Newton method of image reconstruction in diffuse optical tomography,” Phys. Med. Biol.50, 2365–2386 (2005).
[CrossRef] [PubMed]

S. R. Arridge and M. Schweiger, “The use of multiple data types in time-resolved optical absorption and scattering tomography (TOAST),” in “Mathematical Methods in Medical Imaging II,”, D. C. Wilson and J. N. Wilson, eds. (1993), Proc. SPIE2035, pp. 218–229.
[CrossRef]

Sevick, E. M.

E. M. Sevick and B. Chance, “Photon migration in a model of the head measured using time and frequency domain techniques: potentials of spectroscopy and imaging,” in “Time-Resolved Spectroscopy and Imaging of Tissues,”, B. Chance and A. Katzir, eds. (1991), Proc. SPIE1431, pp. 84–96.
[CrossRef]

Sevick-Muraca, E. M.

Sevick-Muraca, W. M.

Shives, E.

Sorensen, D.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, 1999), 3rd ed.
[CrossRef]

Star, W. M.

Stollberger, R.

F. Knoll, M. Freiberger, K. Bredies, and R. Stollberger, “AGILE: An open source library for image reconstruction using graphics card hardware acceleration,” in “Proceedings of the 19th Scientific Meeting and Exhibition of ISMRM, Montreal, CA,” (2011), p. 2554.

Storchi, P. R. M.

Stout, D.

B. Dogdas, D. Stout, A. F. Chatziiouannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol.52, 577–587 (2007).
[CrossRef] [PubMed]

Strzodka, R.

D. Göddeke, H. Wobker, R. Strzodka, J. Mohd-Yusof, P. S. McCormick, and S. Turek, “Co-processor acceleration of an unmodified parallel solid mechanics code with FEASTGPU,” Int. J. Comput. Sci. Eng.4, 254–269 (2009).
[CrossRef]

Tian, J.

Turek, S.

D. Göddeke, H. Wobker, R. Strzodka, J. Mohd-Yusof, P. S. McCormick, and S. Turek, “Co-processor acceleration of an unmodified parallel solid mechanics code with FEASTGPU,” Int. J. Comput. Sci. Eng.4, 254–269 (2009).
[CrossRef]

M. Köster, D. Göddeke, H. Wobker, and S. Turek, “How to gain speedups of 1000 on single processors with fast FEM solvers – benchmarking numerical and computational efficiency,” Tech. rep., Fakultät für Mathematik, TU Dortmund (2008). Ergebnisberichte des Instituts für Angewandte Mathematik, Nummer 382.

D. Göddeke, C. Becker, and S. Turek, “Integrating GPUs as fast co–processors into the parallel FE package FEAST,” in “19th Symposium Simulations Technique,”, M. Becker and H. Szczerbicka, eds., Frontiers in Simulation (SCS Publishing House, 2006), pp. 277–282.

Upendra, V.

J. Prakash, V. Chandrasekharan, V. Upendra, and P. K. Yalavarthy, “Accelerating frequency-domain diffuse optical tomographic image reconstruction using graphics processing units,” J. Biomed. Opt.15, 066009 (2010).
[CrossRef]

Wahba, G.

G. Wahba, Spline Models for Observational Data (SIAM, 1990).

Wobker, H.

D. Göddeke, H. Wobker, R. Strzodka, J. Mohd-Yusof, P. S. McCormick, and S. Turek, “Co-processor acceleration of an unmodified parallel solid mechanics code with FEASTGPU,” Int. J. Comput. Sci. Eng.4, 254–269 (2009).
[CrossRef]

M. Köster, D. Göddeke, H. Wobker, and S. Turek, “How to gain speedups of 1000 on single processors with fast FEM solvers – benchmarking numerical and computational efficiency,” Tech. rep., Fakultät für Mathematik, TU Dortmund (2008). Ergebnisberichte des Instituts für Angewandte Mathematik, Nummer 382.

Xu, Y.

Yalavarthy, P. K.

J. Prakash, V. Chandrasekharan, V. Upendra, and P. K. Yalavarthy, “Accelerating frequency-domain diffuse optical tomographic image reconstruction using graphics processing units,” J. Biomed. Opt.15, 066009 (2010).
[CrossRef]

Yang, F.

Yang, X.

Zhang, B.

Zijlstra, W. G.

M. L. Landsman, G. Kwant, G. A. Mook, and W. G. Zijlstra, “Light-absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol.40, 575–583 (1976).
[PubMed]

Appl. Opt.

Comput. Math. Math. Phys.

A. Bakushinsky, “The problem of the convergence of the iteratively regularized Gauss-Newton method,” Comput. Math. Math. Phys.32, 1353–1359 (1992).

Comput. Vis. Sci.

J. Schöberl, “NETGEN - an advancing front 2D/3D-mesh generator based on abstract rules,” Comput. Vis. Sci.1, 41–52 (1997).
[CrossRef]

IEEE Trans. Biomed. Eng.

M. Freiberger, H. Egger, and H. Scharfetter, “Nonlinear inversion schemes for fluorescence optical tomography,” IEEE Trans. Biomed. Eng.57, 2723–2729 (2010).
[CrossRef]

Int. J. Comput. Sci. Eng.

D. Göddeke, H. Wobker, R. Strzodka, J. Mohd-Yusof, P. S. McCormick, and S. Turek, “Co-processor acceleration of an unmodified parallel solid mechanics code with FEASTGPU,” Int. J. Comput. Sci. Eng.4, 254–269 (2009).
[CrossRef]

Inv. Probl.

B. Kaltenbacher, “Some Newton-type methods for the regularization of nonlinear ill-posed problems,” Inv. Probl.13, 729–753 (1997).
[CrossRef]

S. R. Arridge, “Optical tomography in medical imaging,” Inv. Probl.15, R41–R93 (1999).
[CrossRef]

Inv. Probl. Sci. Eng.

H. Egger and M. Schlottbom, “Efficient reliable image reconstruction schemes for diffuse optical tomography,” Inv. Probl. Sci. Eng.19, 155–180 (2011).
[CrossRef]

J. Appl. Physiol.

M. L. Landsman, G. Kwant, G. A. Mook, and W. G. Zijlstra, “Light-absorbing properties, stability, and spectral stabilization of indocyanine green,” J. Appl. Physiol.40, 575–583 (1976).
[PubMed]

J. Biomed. Opt.

J. Prakash, V. Chandrasekharan, V. Upendra, and P. K. Yalavarthy, “Accelerating frequency-domain diffuse optical tomographic image reconstruction using graphics processing units,” J. Biomed. Opt.15, 066009 (2010).
[CrossRef]

J. Photochem. Photobiol. B

S. Mordon, V. Maunoury, J. M. Devoisselle, Y. Abbas, and D. Coustaud, “Characterization of tumorous and normal tissue using a pH-sensitive fluorescence indicator (5,6-carboxyfluorescein) in vivo,” J. Photochem. Photobiol. B13, 307–314 (1992).
[CrossRef] [PubMed]

Lasers Surg. Med.

I. Gannot, I. Ron, F. Hekmat, V. Chernomordik, and A. Gandjbakhche, “Functional optical detection based on pH dependent fluorescence lifetime,” Lasers Surg. Med.35, 342–348 (2004).
[CrossRef] [PubMed]

Med. Phys.

A. Godavarty, E. M. Sevick-Muraca, and M. J. Eppstein, “Three-dimensional fluorescence lifetime tomography,” Med. Phys.32, 992–1000 (2005).
[CrossRef] [PubMed]

Numer. Func. Anal. Optim.

M. Hanke, “Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems,” Numer. Func. Anal. Optim.18, 971–993 (1997).
[CrossRef]

Opt. Express

Phys. Med. Biol.

M. Schweiger, S. R. Arridge, and I. Nissilä, “Gauss–Newton method of image reconstruction in diffuse optical tomography,” Phys. Med. Biol.50, 2365–2386 (2005).
[CrossRef] [PubMed]

B. Dogdas, D. Stout, A. F. Chatziiouannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol.52, 577–587 (2007).
[CrossRef] [PubMed]

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.50, 4225–4241 (2005).
[CrossRef] [PubMed]

Proc. SPIE

E. M. Sevick and B. Chance, “Photon migration in a model of the head measured using time and frequency domain techniques: potentials of spectroscopy and imaging,” in “Time-Resolved Spectroscopy and Imaging of Tissues,”, B. Chance and A. Katzir, eds. (1991), Proc. SPIE1431, pp. 84–96.
[CrossRef]

SIAM J. Appl. Math.

D. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” SIAM J. Appl. Math.11, 431–441 (1963).
[CrossRef]

Soviet Math. Dokl.

V. A. Morozov, “On the solution of functional equations by the method of regularization,” Soviet Math. Dokl.7, 414–417 (1966).

Other

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, 1996).

P. C. Hansen, Rank-Defficient and Discrete Ill-Posed Problems (SIAM, 1998).
[CrossRef]

G. Wahba, Spline Models for Observational Data (SIAM, 1990).

NVIDIA, NVIDIA CUDA Programming Guide 2.0 (NVIDIA Cooperation, 2008).

N. Bell and M. Garland, “Implementing sparse matrix-vector multiplication on throughput-oriented processors,” in “SC ’09: Proceedings of the Conference on High Performance Computing Networking, Storage and Analysis,” (ACM, 2009), pp. 1–11.

A. Joshi, “Adaptive finite element methods for fluorescence enhanced optical tomography,” Ph.D. thesis (Texas A&M University, 2005).

D. Göddeke, C. Becker, and S. Turek, “Integrating GPUs as fast co–processors into the parallel FE package FEAST,” in “19th Symposium Simulations Technique,”, M. Becker and H. Szczerbicka, eds., Frontiers in Simulation (SCS Publishing House, 2006), pp. 277–282.

M. M. Baskaran and R. Bordawekar, “Optimizing sparse matrix-vector multiplication on GPUs,” IBM Technical Report RC24704, IBM Ltd. (2009).

M. Liebmann, “Efficient PDE solvers on modern hardware with applications in medical and technical sciences,” Ph.D. thesis (University of Graz, 2009).

W. L. Briggs, V. E. Henson, and S. F. McCormick, A multigrid tutorial (SIAM, 2000), 2nd ed.
[CrossRef]

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users’ Guide (SIAM, 1999), 3rd ed.
[CrossRef]

S. R. Arridge and M. Schweiger, “The use of multiple data types in time-resolved optical absorption and scattering tomography (TOAST),” in “Mathematical Methods in Medical Imaging II,”, D. C. Wilson and J. N. Wilson, eds. (1993), Proc. SPIE2035, pp. 218–229.
[CrossRef]

F. Knoll, M. Freiberger, K. Bredies, and R. Stollberger, “AGILE: An open source library for image reconstruction using graphics card hardware acceleration,” in “Proceedings of the 19th Scientific Meeting and Exhibition of ISMRM, Montreal, CA,” (2011), p. 2554.

M. Köster, D. Göddeke, H. Wobker, and S. Turek, “How to gain speedups of 1000 on single processors with fast FEM solvers – benchmarking numerical and computational efficiency,” Tech. rep., Fakultät für Mathematik, TU Dortmund (2008). Ergebnisberichte des Instituts für Angewandte Mathematik, Nummer 382.

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Figures (4)

Fig. 1
Fig. 1

(a) Sparse matrix which is to be stored in compressed row-storage (CRS) format. (b) The conventional memory layout on the CPU consists of the linearized data elements together with their column indices and a vector holding the offset of each row in the data vector. The number of elements in a row is given implicitly by the difference in the row offsets. (c) The adapted layout for the GPU stores the rows in an interleaved manner which requires memory padding marked with × and an additional vector holding the number of non-zero entries per row.

Fig. 2
Fig. 2

Assembly of the system matrix: The 4 × 4 stiffness-, mass- and boundary mass-matrices K, M, and R of every element are scaled with the optical parameters κ, μ and ρ, summed and stored temporarily into a vector T. In a second step, these elements are compressed into a sparse matrix A by summing the elements specified by the vertex-to-element connectivity which is given as look-up table (LUT).

Fig. 3
Fig. 3

Simulation setup showing the mouse phantom, the excitation sources (cylinders) and the photon-detectors (cones). The cross-section at the height of the central optode ring shows the assumed concentration distribution which consists of two 5 mm spheres with a fluorophore concentration of 10μM.

Fig. 4
Fig. 4

Reconstruction of the two fluorescent inclusions shown in Fig. 3 performed with graphics hardware acceleration. The cross-section is again taken at the height of the middle optode ring. 1% noise relative to the largest measurement datum was added for to the reconstruction.

Tables (5)

Tables Icon

Algorithm 1 Iteratively regularized Gauß-Newton algorithm

Tables Icon

Algorithm 2 GPU kernel for assembling the sensitivity matrix

Tables Icon

Table 1 Optical Parameters Used for the Simulations

Tables Icon

Table 2 Comparison of Single-Threaded CPU and Parallelized GPU Reconstruction Times for the Digimouse Phantom with 174080 Elements

Tables Icon

Table 3 Estimated and True Memory Required (in Mega Bytes; MB) for Selected Data Structures of the Reconstruction Algorithm with Single-Precision Floating-Point Numbers

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

div ( κ x φ x ) + μ x φ x = q , in Ω ,
div ( κ m φ m ) + μ m φ m = γ c φ x , in Ω .
ρ φ i + κ i φ i n = 0 , on Ω , i = x , m .
m ( φ m ) = Ω d ( s ) φ m ( s ) d s ,
argmin c ( ( c ) δ 2 + α 2 | | c c 0 | | 2 ) .
( S k * S k + α k I ) ( c k + 1 c k ) = S k * ( δ ( c k ) ) + α k ( c 0 c k ) k = 0 , 1 ,
[ K x ( c ) + M x ( c ) + R x ] V x = Q
[ K m ( c ) + M m ( c ) + R m ] V m = G ( c ) V x .
S h : = W m K m ( h ) V m W m M m ( h ) V m W x K x ( h ) V x W x M x ( h ) V x + W m G ( h ) V x ,
[ K m ( c ) + M m ( c ) + R m ] W m = D ,
[ K x ( c ) + M x ( c ) + R x ] W x = G ( c ) W m .
K i j T : = T ψ g T ( i ) ψ g T ( j ) d x , M i j T : = T ψ g T ( i ) ψ g T ( j ) d x and R i j T : = T Ω ψ g T ( i ) ψ g T ( j ) d x 1 i , j 4.
A x T = κ x ( c k T ) K T + μ x ( c k T ) M T + ρ x R T ,
A m T = κ m ( c k T ) K T + μ m ( c k T ) M T + ρ m R T .
C D C S max ( C D ) = 4.71 × 10 5
C D G S max ( C D ) = 7.63 × 10 7 .
| | c C D c C S | | / | | c C D | | = 6.94 × 10 3 and | | c C D c G S | | / | | c C D | | = 2.39 × 10 3 .

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