Abstract

We consider the problem of fluorescence lifetime optical tomographic imaging in a weakly scattering medium in the presence of highly scattering inclusions. We suggest an approximation to the radiative transfer equation, which results from the assumption that the transport coefficient of the scattering media differs by an order of magnitude for weakly and highly scattering regions. The image reconstruction algorithm is based on the variational framework and employs angularly selective intensity measurements. We present numerical simulation of light scattering in a weakly scattering medium that embeds highly scattering objects. Our reconstruction algorithm is verified by recovering optical and fluorescent parameters from numerically simulated datasets.

© 2011 Optical Society of America

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  1. P. J. Shaw, D. A. Agard, Y. Hiraoko, and J. W. Sedat, “Tilted view reconstruction in optical microscopy, three-dimensional reconstruction of Drosophila melanogaster embryo nuclei,” Biophys. J. 55, 101–110 (1989).
    [CrossRef] [PubMed]
  2. C. S. Brown, D. H. Burns, F. A. Spelman, and A. C. Nelson, “Computed tomography from optical projections for three-dimensional reconstruction of thick objects,” Appl. Opt. 31, 6247–6254 (1992).
    [CrossRef] [PubMed]
  3. J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
    [CrossRef] [PubMed]
  4. J. Sharpe, “Optical projection tomography as a new tool for studying embryo anatomy,” J. Anat. 202, 175–81 (2003).
    [CrossRef] [PubMed]
  5. M. Fauver, E. J. Seibel, J. R. Rahn, M. G. Meyer, F. W. Patten, T. Neumann, and A. C. Nelson, “Three-dimensional imaging of single isolated cell nuclei using optical projection tomography,” Opt. Express 13, 4210–4223 (2005).
    [CrossRef] [PubMed]
  6. J. McGinty, H. B. Taylor, L. Chen, L. Bugeon, J. R. Lamb, M. J. Dallman, and P. M. W. French, “In vivo fluorescence lifetime optical projection tomography,” Biomed. Opt. Express 2, 1340–1350 (2011).
    [CrossRef] [PubMed]
  7. A. H. Andersen and A. C. Kak, “Digital ray tracing in two-dimensional refractive fields,” J. Acoust. Soc. Am. 72, 1593–1606(1982).
    [CrossRef]
  8. C. M. Vest, “Tomography for properties of materials that bend rays,” Appl. Opt. 24, 4089–4094 (1985).
    [CrossRef] [PubMed]
  9. A. H. Andersen, “Tomography transform and inverse in geometrical optics,” J. Opt. Soc. Am. A 4, 1385–1395 (1987).
    [CrossRef]
  10. A. D. Kim and J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A 20, 92–98 (2003).
    [CrossRef]
  11. A. D. Kim and M. Moscoso, “Beam propagation in sharply peaked forward scattering media,” J. Opt. Soc. Am. A 21, 797–803 (2004).
    [CrossRef]
  12. C. Vinegoni, C. Pitsouli, D. Razansky, N. Perrimon, and V. Ntziachristos, “In vivo imaging of Drosophila melanogaster pupae with mesoscopic fluorescence tomography,” Nat. Methods 5, 45–47 (2007).
    [CrossRef] [PubMed]
  13. O. Lehtikangas, T. Tarvainen, V. Kolehmainen, A. Pulkkinen, S. R. Arridge, and J. P. Kaipio, “Finite element approximation of the Fokker–Planck equation for diffuse optical tomography,” J. Quant. Spectrosc. Radiat. Transfer 111, 1406–1417 (2010).
    [CrossRef]
  14. L. Florescu, V. A. Markel, and J. C. Schotland, “Single-scattering optical tomography,” Phys. Rev. E 79, 036607 (2009).
    [CrossRef]
  15. L. Florescu, V. A. Markel, and J. C. Schotland, “Single-scattering optical tomography: Simultaneous reconstruction of scattering and absorption,” Phys. Rev. E 81, 016602(2010).
    [CrossRef]
  16. J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J. Biophoton. 1, 390–394 (2008).
    [CrossRef]
  17. A. Bassi, D. Brida, C. D’Andrea, G. Valentini, R. Cubeddu, S. De Silvestri, and G. Cerullo, “Time-gated optical projection tomography,” Opt. Lett. 35, 2732–2734 (2010).
    [CrossRef] [PubMed]
  18. V. Y. Soloviev and S. R. Arridge, “Optical tomography in weakly scattering media in the presence of highly scattering inclusions,” Biomed. Opt. Express 2, 440–451 (2011).
    [CrossRef] [PubMed]
  19. V. V. Sobolev, A Treatise on Radiative Transfer (Van Nostrand, 1963).
  20. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
    [CrossRef]
  21. S. R. Arridge and J. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
    [CrossRef]
  22. H. C. van de Hulst, Light Scattering by Small Particles(Dover, 1981).
  23. M. Born and E. Wolf, Principles of Optics (Pergamon, 1968).
  24. R. L. Siddon, “Fast calculation of the exact radiological path for a three-dimensional CT array,” Med. Phys. 12, 252–255 (1985).
    [CrossRef] [PubMed]
  25. S. Kaczmarz, “Approximate solution of system of linear equations,” Int. J. Control 57, 1269–1271 (1993).
    [CrossRef]
  26. V. Y. Soloviev, C. D’Andrea, P. S. Mohan, G. Valentini, R. Cubeddu, and S. R. Arridge, “Fluorescence lifetime optical tomography with discontinuous Galerkin discretisation scheme,” Biomed. Opt. Express 1, 998–1013 (2010).
    [CrossRef]
  27. M. Choulli and P. Stefanov, “Inverse scattering and inverse boundary value problems for the linear Boltzmann equation,” Commun. Partial Diff. Equ. 21, 763–785 (1996).
    [CrossRef]
  28. G. Bal, “Inverse transport theory and applications,” Inverse Probl. 25, 053001, (2009).
    [CrossRef]
  29. S. Arridge, V. Kolehmainen, and M. Schweiger, “Reconstruction and regularisation in optical tomography,” in Proceedings of the Interdisciplinary Workshop on Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT), Y.Censor, M.Jiang, and A.Louis, eds. (Pubblicazioni del Centro De Giorgi, 2008), pp. 1–18.
  30. P.S. Mohan, T. Tarvainen, M. J. Schweiger, A. Pulkkineand, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comp. Phys. , (to be published).

2011 (2)

2010 (4)

A. Bassi, D. Brida, C. D’Andrea, G. Valentini, R. Cubeddu, S. De Silvestri, and G. Cerullo, “Time-gated optical projection tomography,” Opt. Lett. 35, 2732–2734 (2010).
[CrossRef] [PubMed]

V. Y. Soloviev, C. D’Andrea, P. S. Mohan, G. Valentini, R. Cubeddu, and S. R. Arridge, “Fluorescence lifetime optical tomography with discontinuous Galerkin discretisation scheme,” Biomed. Opt. Express 1, 998–1013 (2010).
[CrossRef]

O. Lehtikangas, T. Tarvainen, V. Kolehmainen, A. Pulkkinen, S. R. Arridge, and J. P. Kaipio, “Finite element approximation of the Fokker–Planck equation for diffuse optical tomography,” J. Quant. Spectrosc. Radiat. Transfer 111, 1406–1417 (2010).
[CrossRef]

L. Florescu, V. A. Markel, and J. C. Schotland, “Single-scattering optical tomography: Simultaneous reconstruction of scattering and absorption,” Phys. Rev. E 81, 016602(2010).
[CrossRef]

2009 (3)

L. Florescu, V. A. Markel, and J. C. Schotland, “Single-scattering optical tomography,” Phys. Rev. E 79, 036607 (2009).
[CrossRef]

S. R. Arridge and J. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[CrossRef]

G. Bal, “Inverse transport theory and applications,” Inverse Probl. 25, 053001, (2009).
[CrossRef]

2008 (1)

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J. Biophoton. 1, 390–394 (2008).
[CrossRef]

2007 (1)

C. Vinegoni, C. Pitsouli, D. Razansky, N. Perrimon, and V. Ntziachristos, “In vivo imaging of Drosophila melanogaster pupae with mesoscopic fluorescence tomography,” Nat. Methods 5, 45–47 (2007).
[CrossRef] [PubMed]

2005 (1)

2004 (1)

2003 (2)

J. Sharpe, “Optical projection tomography as a new tool for studying embryo anatomy,” J. Anat. 202, 175–81 (2003).
[CrossRef] [PubMed]

A. D. Kim and J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A 20, 92–98 (2003).
[CrossRef]

2002 (1)

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[CrossRef] [PubMed]

1999 (1)

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

1996 (1)

M. Choulli and P. Stefanov, “Inverse scattering and inverse boundary value problems for the linear Boltzmann equation,” Commun. Partial Diff. Equ. 21, 763–785 (1996).
[CrossRef]

1993 (1)

S. Kaczmarz, “Approximate solution of system of linear equations,” Int. J. Control 57, 1269–1271 (1993).
[CrossRef]

1992 (1)

1989 (1)

P. J. Shaw, D. A. Agard, Y. Hiraoko, and J. W. Sedat, “Tilted view reconstruction in optical microscopy, three-dimensional reconstruction of Drosophila melanogaster embryo nuclei,” Biophys. J. 55, 101–110 (1989).
[CrossRef] [PubMed]

1987 (1)

1985 (2)

C. M. Vest, “Tomography for properties of materials that bend rays,” Appl. Opt. 24, 4089–4094 (1985).
[CrossRef] [PubMed]

R. L. Siddon, “Fast calculation of the exact radiological path for a three-dimensional CT array,” Med. Phys. 12, 252–255 (1985).
[CrossRef] [PubMed]

1982 (1)

A. H. Andersen and A. C. Kak, “Digital ray tracing in two-dimensional refractive fields,” J. Acoust. Soc. Am. 72, 1593–1606(1982).
[CrossRef]

Agard, D. A.

P. J. Shaw, D. A. Agard, Y. Hiraoko, and J. W. Sedat, “Tilted view reconstruction in optical microscopy, three-dimensional reconstruction of Drosophila melanogaster embryo nuclei,” Biophys. J. 55, 101–110 (1989).
[CrossRef] [PubMed]

Ahlgren, U.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[CrossRef] [PubMed]

Andersen, A. H.

A. H. Andersen, “Tomography transform and inverse in geometrical optics,” J. Opt. Soc. Am. A 4, 1385–1395 (1987).
[CrossRef]

A. H. Andersen and A. C. Kak, “Digital ray tracing in two-dimensional refractive fields,” J. Acoust. Soc. Am. 72, 1593–1606(1982).
[CrossRef]

Arridge, S.

S. Arridge, V. Kolehmainen, and M. Schweiger, “Reconstruction and regularisation in optical tomography,” in Proceedings of the Interdisciplinary Workshop on Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT), Y.Censor, M.Jiang, and A.Louis, eds. (Pubblicazioni del Centro De Giorgi, 2008), pp. 1–18.

Arridge, S. R.

V. Y. Soloviev and S. R. Arridge, “Optical tomography in weakly scattering media in the presence of highly scattering inclusions,” Biomed. Opt. Express 2, 440–451 (2011).
[CrossRef] [PubMed]

V. Y. Soloviev, C. D’Andrea, P. S. Mohan, G. Valentini, R. Cubeddu, and S. R. Arridge, “Fluorescence lifetime optical tomography with discontinuous Galerkin discretisation scheme,” Biomed. Opt. Express 1, 998–1013 (2010).
[CrossRef]

O. Lehtikangas, T. Tarvainen, V. Kolehmainen, A. Pulkkinen, S. R. Arridge, and J. P. Kaipio, “Finite element approximation of the Fokker–Planck equation for diffuse optical tomography,” J. Quant. Spectrosc. Radiat. Transfer 111, 1406–1417 (2010).
[CrossRef]

S. R. Arridge and J. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[CrossRef]

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

P.S. Mohan, T. Tarvainen, M. J. Schweiger, A. Pulkkineand, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comp. Phys. , (to be published).

Bal, G.

G. Bal, “Inverse transport theory and applications,” Inverse Probl. 25, 053001, (2009).
[CrossRef]

Baldock, R.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[CrossRef] [PubMed]

Bassi, A.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1968).

Brida, D.

Brown, C. S.

Bugeon, L.

Burns, D. H.

Cerullo, G.

Chen, L.

Choulli, M.

M. Choulli and P. Stefanov, “Inverse scattering and inverse boundary value problems for the linear Boltzmann equation,” Commun. Partial Diff. Equ. 21, 763–785 (1996).
[CrossRef]

Cubeddu, R.

D’Andrea, C.

Dallman, M. J.

Davidson, D.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[CrossRef] [PubMed]

De Silvestri, S.

Dunsby, C.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J. Biophoton. 1, 390–394 (2008).
[CrossRef]

Fauver, M.

Florescu, L.

L. Florescu, V. A. Markel, and J. C. Schotland, “Single-scattering optical tomography: Simultaneous reconstruction of scattering and absorption,” Phys. Rev. E 81, 016602(2010).
[CrossRef]

L. Florescu, V. A. Markel, and J. C. Schotland, “Single-scattering optical tomography,” Phys. Rev. E 79, 036607 (2009).
[CrossRef]

French, P. M. W.

J. McGinty, H. B. Taylor, L. Chen, L. Bugeon, J. R. Lamb, M. J. Dallman, and P. M. W. French, “In vivo fluorescence lifetime optical projection tomography,” Biomed. Opt. Express 2, 1340–1350 (2011).
[CrossRef] [PubMed]

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J. Biophoton. 1, 390–394 (2008).
[CrossRef]

Hecksher-Sorensen, J.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[CrossRef] [PubMed]

Hill, B.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[CrossRef] [PubMed]

Hiraoko, Y.

P. J. Shaw, D. A. Agard, Y. Hiraoko, and J. W. Sedat, “Tilted view reconstruction in optical microscopy, three-dimensional reconstruction of Drosophila melanogaster embryo nuclei,” Biophys. J. 55, 101–110 (1989).
[CrossRef] [PubMed]

Kaczmarz, S.

S. Kaczmarz, “Approximate solution of system of linear equations,” Int. J. Control 57, 1269–1271 (1993).
[CrossRef]

Kaipio, J. P.

O. Lehtikangas, T. Tarvainen, V. Kolehmainen, A. Pulkkinen, S. R. Arridge, and J. P. Kaipio, “Finite element approximation of the Fokker–Planck equation for diffuse optical tomography,” J. Quant. Spectrosc. Radiat. Transfer 111, 1406–1417 (2010).
[CrossRef]

Kak, A. C.

A. H. Andersen and A. C. Kak, “Digital ray tracing in two-dimensional refractive fields,” J. Acoust. Soc. Am. 72, 1593–1606(1982).
[CrossRef]

Keller, J. B.

Kim, A. D.

Kolehmainen, V.

O. Lehtikangas, T. Tarvainen, V. Kolehmainen, A. Pulkkinen, S. R. Arridge, and J. P. Kaipio, “Finite element approximation of the Fokker–Planck equation for diffuse optical tomography,” J. Quant. Spectrosc. Radiat. Transfer 111, 1406–1417 (2010).
[CrossRef]

S. Arridge, V. Kolehmainen, and M. Schweiger, “Reconstruction and regularisation in optical tomography,” in Proceedings of the Interdisciplinary Workshop on Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT), Y.Censor, M.Jiang, and A.Louis, eds. (Pubblicazioni del Centro De Giorgi, 2008), pp. 1–18.

Laine, R.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J. Biophoton. 1, 390–394 (2008).
[CrossRef]

Lamb, J. R.

Lehtikangas, O.

O. Lehtikangas, T. Tarvainen, V. Kolehmainen, A. Pulkkinen, S. R. Arridge, and J. P. Kaipio, “Finite element approximation of the Fokker–Planck equation for diffuse optical tomography,” J. Quant. Spectrosc. Radiat. Transfer 111, 1406–1417 (2010).
[CrossRef]

Markel, V. A.

L. Florescu, V. A. Markel, and J. C. Schotland, “Single-scattering optical tomography: Simultaneous reconstruction of scattering and absorption,” Phys. Rev. E 81, 016602(2010).
[CrossRef]

L. Florescu, V. A. Markel, and J. C. Schotland, “Single-scattering optical tomography,” Phys. Rev. E 79, 036607 (2009).
[CrossRef]

McGinty, J.

J. McGinty, H. B. Taylor, L. Chen, L. Bugeon, J. R. Lamb, M. J. Dallman, and P. M. W. French, “In vivo fluorescence lifetime optical projection tomography,” Biomed. Opt. Express 2, 1340–1350 (2011).
[CrossRef] [PubMed]

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J. Biophoton. 1, 390–394 (2008).
[CrossRef]

Meyer, M. G.

Mohan, P. S.

V. Y. Soloviev, C. D’Andrea, P. S. Mohan, G. Valentini, R. Cubeddu, and S. R. Arridge, “Fluorescence lifetime optical tomography with discontinuous Galerkin discretisation scheme,” Biomed. Opt. Express 1, 998–1013 (2010).
[CrossRef]

P.S. Mohan, T. Tarvainen, M. J. Schweiger, A. Pulkkineand, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comp. Phys. , (to be published).

Moscoso, M.

Neil, M. A. A.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J. Biophoton. 1, 390–394 (2008).
[CrossRef]

Nelson, A. C.

Neumann, T.

Ntziachristos, V.

C. Vinegoni, C. Pitsouli, D. Razansky, N. Perrimon, and V. Ntziachristos, “In vivo imaging of Drosophila melanogaster pupae with mesoscopic fluorescence tomography,” Nat. Methods 5, 45–47 (2007).
[CrossRef] [PubMed]

Patten, F. W.

Perrimon, N.

C. Vinegoni, C. Pitsouli, D. Razansky, N. Perrimon, and V. Ntziachristos, “In vivo imaging of Drosophila melanogaster pupae with mesoscopic fluorescence tomography,” Nat. Methods 5, 45–47 (2007).
[CrossRef] [PubMed]

Perry, P.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[CrossRef] [PubMed]

Pitsouli, C.

C. Vinegoni, C. Pitsouli, D. Razansky, N. Perrimon, and V. Ntziachristos, “In vivo imaging of Drosophila melanogaster pupae with mesoscopic fluorescence tomography,” Nat. Methods 5, 45–47 (2007).
[CrossRef] [PubMed]

Pulkkineand, A.

P.S. Mohan, T. Tarvainen, M. J. Schweiger, A. Pulkkineand, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comp. Phys. , (to be published).

Pulkkinen, A.

O. Lehtikangas, T. Tarvainen, V. Kolehmainen, A. Pulkkinen, S. R. Arridge, and J. P. Kaipio, “Finite element approximation of the Fokker–Planck equation for diffuse optical tomography,” J. Quant. Spectrosc. Radiat. Transfer 111, 1406–1417 (2010).
[CrossRef]

Quintana, L.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J. Biophoton. 1, 390–394 (2008).
[CrossRef]

Rahn, J. R.

Razansky, D.

C. Vinegoni, C. Pitsouli, D. Razansky, N. Perrimon, and V. Ntziachristos, “In vivo imaging of Drosophila melanogaster pupae with mesoscopic fluorescence tomography,” Nat. Methods 5, 45–47 (2007).
[CrossRef] [PubMed]

Ross, A.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[CrossRef] [PubMed]

Schotland, J.

S. R. Arridge and J. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[CrossRef]

Schotland, J. C.

L. Florescu, V. A. Markel, and J. C. Schotland, “Single-scattering optical tomography: Simultaneous reconstruction of scattering and absorption,” Phys. Rev. E 81, 016602(2010).
[CrossRef]

L. Florescu, V. A. Markel, and J. C. Schotland, “Single-scattering optical tomography,” Phys. Rev. E 79, 036607 (2009).
[CrossRef]

Schweiger, M.

S. Arridge, V. Kolehmainen, and M. Schweiger, “Reconstruction and regularisation in optical tomography,” in Proceedings of the Interdisciplinary Workshop on Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT), Y.Censor, M.Jiang, and A.Louis, eds. (Pubblicazioni del Centro De Giorgi, 2008), pp. 1–18.

Schweiger, M. J.

P.S. Mohan, T. Tarvainen, M. J. Schweiger, A. Pulkkineand, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comp. Phys. , (to be published).

Sedat, J. W.

P. J. Shaw, D. A. Agard, Y. Hiraoko, and J. W. Sedat, “Tilted view reconstruction in optical microscopy, three-dimensional reconstruction of Drosophila melanogaster embryo nuclei,” Biophys. J. 55, 101–110 (1989).
[CrossRef] [PubMed]

Seibel, E. J.

Sharpe, J.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J. Biophoton. 1, 390–394 (2008).
[CrossRef]

J. Sharpe, “Optical projection tomography as a new tool for studying embryo anatomy,” J. Anat. 202, 175–81 (2003).
[CrossRef] [PubMed]

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sorensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296, 541–545 (2002).
[CrossRef] [PubMed]

Shaw, P. J.

P. J. Shaw, D. A. Agard, Y. Hiraoko, and J. W. Sedat, “Tilted view reconstruction in optical microscopy, three-dimensional reconstruction of Drosophila melanogaster embryo nuclei,” Biophys. J. 55, 101–110 (1989).
[CrossRef] [PubMed]

Siddon, R. L.

R. L. Siddon, “Fast calculation of the exact radiological path for a three-dimensional CT array,” Med. Phys. 12, 252–255 (1985).
[CrossRef] [PubMed]

Sobolev, V. V.

V. V. Sobolev, A Treatise on Radiative Transfer (Van Nostrand, 1963).

Soloviev, V. Y.

Spelman, F. A.

Stefanov, P.

M. Choulli and P. Stefanov, “Inverse scattering and inverse boundary value problems for the linear Boltzmann equation,” Commun. Partial Diff. Equ. 21, 763–785 (1996).
[CrossRef]

Swoger, J.

J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J. Biophoton. 1, 390–394 (2008).
[CrossRef]

Tahir, K. B.

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Supplementary Material (8)

» Media 1: AVI (3091 KB)     
» Media 2: AVI (3054 KB)     
» Media 3: AVI (959 KB)     
» Media 4: AVI (981 KB)     
» Media 5: AVI (2847 KB)     
» Media 6: AVI (2810 KB)     
» Media 7: AVI (1381 KB)     
» Media 8: AVI (1408 KB)     

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Figures (11)

Fig. 1
Fig. 1

Camera’s images displaying Fourier transformed excitation intensity scattered on an object embedded within a weakly scattering cylinder. The object is a knot made of two scattering tangled tori, whose values of the transport coefficient and albedo are set to μ = 0.75 mm 1 and λ = 0.9 , respectively. Embedding weakly scattering cylinder has the background transport coefficient μ = 0.1 mm 1 and albedo λ = 0.9 . In the middle of the knot two bent fluorescent rods with reduced values of albedo, λ = 0.5 , are inserted. The direct light enters the domain along the direction s 0 = 2 1 / 2 ( 1 , 0 , 1 ) T . Camera rotates around the weakly scattering cylinder starting from its initial position with n = ( 1 , 0 , 0 ) T , where n is the outward camera’s normal. The upper row, (a)–(e), shows real parts of camera’s images for rotation angles { 0 , 72 , 144 , 216 , 288 } degrees, as it is indicated below each image (Media_1). The lower row, (f)–(j), displays imaginary parts of images (Media_2). Images are taken at ω = 500 MHz . Free space propagation is assumed outside the weakly scattering medium with values of the transport coefficient and the albedo set to 0 and 1, respectively.

Fig. 2
Fig. 2

Camera’s images displaying fluorescent intensity resonantly scattered by two bent fluorescent rods embedded inside a highly scattering knot. Values of the quantum yield and the lifetime are set to 0.1 and 1.0 ns , respectively. The same values of optical parameters governing fluorescent and excitation light transport are assumed. The upper row, (a)–(e), shows real parts of camera’s images for rotation angles { 0 , 72 , 144 , 216 , 288 } degrees (Media_3). The lower row, (f)–(j), displays imaginary parts of images (Media_4). Images are taken at the same frequency as above, ω = 500 MHz .

Fig. 3
Fig. 3

Cross section of the scattering knot taken at z = 10 mm displaying distribution of optical and fluorescent parameters. Slices (a)–(b) show the transport coefficient μ in mm 1 and albedo λ, respectively. Slices (c)–(d) display the quantum yield η and the lifetime τ in nanoseconds, correspondingly.

Fig. 4
Fig. 4

Camera’s images displaying Fourier transformed excitation intensity scattered on three cylinders embedded inside a weakly scattering medium. Embedding weakly scattering cylinder has the same optical properties as above. Highly scattering cylinders are such that one thicker cylinder embeds two thinner ones. The thicker cylinder has the transport coefficient μ = 0.75 mm 1 and the albedo λ = 0.9 . Two embedded cylinders have μ = 0.75 mm 1 and λ = 0.5 . The thinnest cylinder is fluorescing. Two nonfluorescing highly scattering cylinders have ellipses in their cross sections as it is seen on slices below. Camera’s views are computed in the same way as for the case of the knot above. The upper row, (a)–(e), shows real parts of images while the lower row, (f)–(j) (Media_5), displays imaginary parts (Media_6). Images are taken at ω = 500 MHz .

Fig. 5
Fig. 5

Camera’s images displaying Fourier transformed fluorescent intensity scattered on three cylinders embedded inside a weakly scattering cylinder. The upper row, (a)–(e), shows real parts (Media_7) while the lower row, (f)–(j), displays imaginary parts of images (Media_8) acquired by the CCD camera.

Fig. 6
Fig. 6

Cross section of scattering cylinders taken at z = 10 mm displaying distribution of optical and fluorescent parameters. Slices (a)–(b) show the transport coefficient μ in mm 1 and albedo λ, respectively. Slices (c)–(d) display the quantum yield η and the lifetime τ in nanoseconds, correspondingly.

Fig. 7
Fig. 7

Measurement system using parallel rays excitation and the CCD array.

Fig. 8
Fig. 8

Slices showing reconstruction results of the knot embedded in the weakly scattering cylinder at three different heights. The first, (a)–(d), second, (e)–(h), and third, (i)–(l), rows display the transport coefficient μ, albedo λ, quantum yield η, and lifetime τ at z = { 9 , 10 , 11 } mm , respectively.

Fig. 9
Fig. 9

The case of the scattering knot embedded in the weakly scattering cylinder. (a) Isosurface of the transport coefficient μ threshold at 0.7 mm 1 . (b) Isosurface of the albedo λ. The threshold value was set to 0.55 . (c) Isosurface of the quantum yield η computed for 0.095 threshold value. (d) Isosurface of the lifetime τ for 1.0 ns threshold.

Fig. 10
Fig. 10

Slices showing reconstruction results of highly scattering cylinders embedded in the weakly scattering medium at z = 10 mm . (a) Reconstructed transport coefficient μ in mm 1 , (b) reconstructed albedo, (c) quantum yields, and (d) lifetime in nanoseconds.

Fig. 11
Fig. 11

The case of scattering cylinders embedded in the weakly scattering medium. (a) Isosurface of the transport coefficient μ at 0.72 mm 1 . (b) Isosurface of the albedo λ at 0.45. (c) Isosurface of the quantum yield η at 0.12. (d) Isosurface of the lifetime τ at 1.05 ns .

Equations (40)

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s · I + μ ˜ I = λ μ B ,
B ( r , s ) = ( 4 π ) p ( s · s ) I ( r , s ) d 2 s + p ( s · s 0 ) I 0 ( r , s 0 ) .
p ( s · s ) = ( 1 / 4 π ) ( 1 + ϵ s · s ) ,
s · I 0 + μ ˜ I 0 = Q 0 δ ( r r 0 ) δ ( s s 0 ) ,
I 0 ( r , s 0 ) = Q 0 exp ( 0 l μ ˜ ( r s 0 l ) d l ) ,
I ( r , s ) = 0 l max λ ( r s l ) μ ( r s l ) B ( r s l , s ) × exp ( 0 l μ ˜ ( r s l ) d l ) d l ,
I = u 3 κ s · u ,
u = 1 4 π ( 4 π ) I ( s ) d 2 s ,
κ = ( 1 / 3 ) [ μ ( 1 λ ϵ / 3 ) + i ω / c ] 1 .
Λ u = λ μ u 0 ,
Λ = · κ + ( 1 λ ) μ + i ω / c ,
B ( r , s ) = u ϵ κ s · u + p ( s · s 0 ) I 0 .
s · I 0 + μ ˜ I 0 = 1 4 π B 0 ,
B 0 ( r ) = 1 λ 1 + i ω τ μ η ( u + I 0 ) .
I 0 ( r , s ) = 1 4 π 0 l max B 0 ( r s l ) exp ( 0 l μ ˜ ( r s l ) d l ) d l .
s · I + μ ˜ I = λ μ B ,
B ( r , s ) = u ϵ κ s · u .
Λ u = B 0 .
I ( r , s ) = I 0 ( r , s ) + 0 l max λ ( r s l ) μ ( r s l ) × B ( r s l , s ) exp ( 0 l μ ˜ ( r s l ) d l ) d l ,
F = ς ( ω ) ( E + L ) d ω + ϒ .
E = ξ ( s ) d 2 s V χ ( r ) ( | I E I | 2 + | I F I | 2 ) d 3 r ,
ξ ( s ) = 0 n < N δ ( s s n ) ,
χ ( r ) = 0 m < M σ m δ ( r r m ) , ς ( ω ) = 0 s < S δ ( ω ω s ) ,
L = Re ξ ( s ) J , s · I + μ ˜ I λ μ B d 2 s + Re ξ ( s ) H , s · I + μ ˜ I λ μ B 1 4 π B 0 d 2 s ,
x = ( μ , λ , η , τ ) T ,
ϒ = j = 1 4 α j Δ x j 2 .
s n · J + μ ˜ J = 2 χ ( r ) ( I E I ) ,
s n · H + μ ˜ H = 2 χ ( r ) ( I F I ) .
Δ x j = α j 1 f x j ,
x k + 1 = x k + Δ x ,
f μ λ Ψ Re ξ ( s ) J * I d 2 s ,
f λ μ Ψ 3 ϵ Re { ( μ κ ) 2 ξ ( s ) J * s · u d 2 s } ,
f η ( 1 λ ) μ Re { Θ 1 + i ω τ } ,
f τ ω ( 1 λ ) μ η Im { Θ ( 1 + i ω τ ) 2 } ,
Ψ = Re ξ ( s ) [ u + p ( s · s 0 ) I 0 ] J * d 2 s , Θ = 1 4 π ξ ( s ) ( u + I 0 ) H * d 2 s .
A i j Δ x j = α i 1 f x k i ,
x k + 1 i = [ x s , k i + ( 1 / α i ) f x k i ( s s ) ] + ( 1 / α i ) n = s + 1 N 1 f x k i ( s n ) ,
x s + 1 , k i = x s , k i + ( 1 / α i ) f x s , k i ( s s ) ,
α j 1 = Δ x j f x j 1 ,
α j 1 = C j ( k ) E 1 / 2 f x j 1 ,

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