Abstract

Optical and near-IR spectroscopy and imaging of highly scattering tissues require information about the distribution of photon-migration paths. We introduce the concept of the photon hitting density, which describes the expected local time spent by photons traveling between a source and a detector. For systems in which photon transport is diffusive we show that the hitting density can be calculated in terms of diffusion Green’s functions. We report calculations of the hitting density in model systems.

© 1993 Optical Society of America

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  1. L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
    [CrossRef] [PubMed]
  2. J. C. Hebden, R. A. Kruger, “Transillumination imaging performance: spatial resolution simulation studies,” Med. Phys. 17, 41–47 (1990).
    [CrossRef] [PubMed]
  3. J. C. Hebden, R. A. Kruger, “Time of flight breast imaging system,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 225–231 (1991).
  4. J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
    [CrossRef] [PubMed]
  5. B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
    [CrossRef] [PubMed]
  6. F. F. Jobsis, J. H. Keizer, J. C. LaManna, M. Rosenthal, “Reflectance spectrophotometry of cytochrome aa3 inυivo,” J. Appl. Physiol. 113, 858–872 (1977).
  7. Y. Yamada Hasegawa, “Simulation of time-resolved optical CT imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 73–83 (1991).
  8. I. Oda, Y. Ito, H. Eda, T. Tamura, M. Takada, R. Abvumi, K. Nagai, H. Nakagawa, M. Tamura, “Noninvasive hemoglobin oxygenation monitor and computed tomography by NIR spectrophotometry,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 73–83 (1991).
  9. R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 192–203 (1991).
  10. J. C. Haselgrove, N. G. Wang, B. Chance, “Investigation of the nonlinear aspects of imaging through a highly scattering medium,” Med. Phys. 19, 17–23 (1992).
    [CrossRef] [PubMed]
  11. A. Ishimaru, Waυe Propagation and Scattering in Random Media (Academic, New York, 1978).
  12. R. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
  13. W. Magnus, “Operator expansion,” Commun. Pure Appl. Math. 7, 649–646 (1954).
    [CrossRef]
  14. J. C. Haselgrove, J. C. Schotland, J. S. Leigh, “Long-time behavior of photon diffusion in an absorbing medium: application to time-resolved spectroscopy,” Appl. Opt. 31, 2678–2683 (1992).
    [CrossRef] [PubMed]

1992 (2)

J. C. Haselgrove, N. G. Wang, B. Chance, “Investigation of the nonlinear aspects of imaging through a highly scattering medium,” Med. Phys. 19, 17–23 (1992).
[CrossRef] [PubMed]

J. C. Haselgrove, J. C. Schotland, J. S. Leigh, “Long-time behavior of photon diffusion in an absorbing medium: application to time-resolved spectroscopy,” Appl. Opt. 31, 2678–2683 (1992).
[CrossRef] [PubMed]

1991 (1)

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

1990 (2)

J. C. Hebden, R. A. Kruger, “Transillumination imaging performance: spatial resolution simulation studies,” Med. Phys. 17, 41–47 (1990).
[CrossRef] [PubMed]

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

1988 (1)

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

1977 (1)

F. F. Jobsis, J. H. Keizer, J. C. LaManna, M. Rosenthal, “Reflectance spectrophotometry of cytochrome aa3 inυivo,” J. Appl. Physiol. 113, 858–872 (1977).

1954 (1)

W. Magnus, “Operator expansion,” Commun. Pure Appl. Math. 7, 649–646 (1954).
[CrossRef]

Abvumi, R.

I. Oda, Y. Ito, H. Eda, T. Tamura, M. Takada, R. Abvumi, K. Nagai, H. Nakagawa, M. Tamura, “Noninvasive hemoglobin oxygenation monitor and computed tomography by NIR spectrophotometry,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 73–83 (1991).

Alfano, R. R.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Aronson, R.

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 192–203 (1991).

Barbour, R. L.

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 192–203 (1991).

Chance, B.

J. C. Haselgrove, N. G. Wang, B. Chance, “Investigation of the nonlinear aspects of imaging through a highly scattering medium,” Med. Phys. 19, 17–23 (1992).
[CrossRef] [PubMed]

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Eda, H.

I. Oda, Y. Ito, H. Eda, T. Tamura, M. Takada, R. Abvumi, K. Nagai, H. Nakagawa, M. Tamura, “Noninvasive hemoglobin oxygenation monitor and computed tomography by NIR spectrophotometry,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 73–83 (1991).

Feynman, R.

R. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

Finander, M.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Graber, H. L.

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 192–203 (1991).

Greenfeld, R.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Grunbaum, F. A.

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Haselgrove, J. C.

J. C. Haselgrove, N. G. Wang, B. Chance, “Investigation of the nonlinear aspects of imaging through a highly scattering medium,” Med. Phys. 19, 17–23 (1992).
[CrossRef] [PubMed]

J. C. Haselgrove, J. C. Schotland, J. S. Leigh, “Long-time behavior of photon diffusion in an absorbing medium: application to time-resolved spectroscopy,” Appl. Opt. 31, 2678–2683 (1992).
[CrossRef] [PubMed]

Hebden, J. C.

J. C. Hebden, R. A. Kruger, “Transillumination imaging performance: spatial resolution simulation studies,” Med. Phys. 17, 41–47 (1990).
[CrossRef] [PubMed]

J. C. Hebden, R. A. Kruger, “Time of flight breast imaging system,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 225–231 (1991).

Hibbs, A. R.

R. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

Ho, P. P.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Ishimaru, A.

A. Ishimaru, Waυe Propagation and Scattering in Random Media (Academic, New York, 1978).

Ito, Y.

I. Oda, Y. Ito, H. Eda, T. Tamura, M. Takada, R. Abvumi, K. Nagai, H. Nakagawa, M. Tamura, “Noninvasive hemoglobin oxygenation monitor and computed tomography by NIR spectrophotometry,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 73–83 (1991).

Jobsis, F. F.

F. F. Jobsis, J. H. Keizer, J. C. LaManna, M. Rosenthal, “Reflectance spectrophotometry of cytochrome aa3 inυivo,” J. Appl. Physiol. 113, 858–872 (1977).

Kaufmann, K.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Keizer, J. H.

F. F. Jobsis, J. H. Keizer, J. C. LaManna, M. Rosenthal, “Reflectance spectrophotometry of cytochrome aa3 inυivo,” J. Appl. Physiol. 113, 858–872 (1977).

Kohn, P.

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Kruger, R. A.

J. C. Hebden, R. A. Kruger, “Transillumination imaging performance: spatial resolution simulation studies,” Med. Phys. 17, 41–47 (1990).
[CrossRef] [PubMed]

J. C. Hebden, R. A. Kruger, “Time of flight breast imaging system,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 225–231 (1991).

LaManna, J. C.

F. F. Jobsis, J. H. Keizer, J. C. LaManna, M. Rosenthal, “Reflectance spectrophotometry of cytochrome aa3 inυivo,” J. Appl. Physiol. 113, 858–872 (1977).

Leigh, J. S.

J. C. Haselgrove, J. C. Schotland, J. S. Leigh, “Long-time behavior of photon diffusion in an absorbing medium: application to time-resolved spectroscopy,” Appl. Opt. 31, 2678–2683 (1992).
[CrossRef] [PubMed]

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Levy, W.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Liu, C.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Lubowsky, J.

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 192–203 (1991).

Magnus, W.

W. Magnus, “Operator expansion,” Commun. Pure Appl. Math. 7, 649–646 (1954).
[CrossRef]

Miyake, H.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Nagai, K.

I. Oda, Y. Ito, H. Eda, T. Tamura, M. Takada, R. Abvumi, K. Nagai, H. Nakagawa, M. Tamura, “Noninvasive hemoglobin oxygenation monitor and computed tomography by NIR spectrophotometry,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 73–83 (1991).

Nakagawa, H.

I. Oda, Y. Ito, H. Eda, T. Tamura, M. Takada, R. Abvumi, K. Nagai, H. Nakagawa, M. Tamura, “Noninvasive hemoglobin oxygenation monitor and computed tomography by NIR spectrophotometry,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 73–83 (1991).

Nioka, S.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Oda, I.

I. Oda, Y. Ito, H. Eda, T. Tamura, M. Takada, R. Abvumi, K. Nagai, H. Nakagawa, M. Tamura, “Noninvasive hemoglobin oxygenation monitor and computed tomography by NIR spectrophotometry,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 73–83 (1991).

Rosenthal, M.

F. F. Jobsis, J. H. Keizer, J. C. LaManna, M. Rosenthal, “Reflectance spectrophotometry of cytochrome aa3 inυivo,” J. Appl. Physiol. 113, 858–872 (1977).

Schotland, J. C.

Singer, J. R.

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Smith, D. S.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Takada, M.

I. Oda, Y. Ito, H. Eda, T. Tamura, M. Takada, R. Abvumi, K. Nagai, H. Nakagawa, M. Tamura, “Noninvasive hemoglobin oxygenation monitor and computed tomography by NIR spectrophotometry,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 73–83 (1991).

Tamura, M.

I. Oda, Y. Ito, H. Eda, T. Tamura, M. Takada, R. Abvumi, K. Nagai, H. Nakagawa, M. Tamura, “Noninvasive hemoglobin oxygenation monitor and computed tomography by NIR spectrophotometry,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 73–83 (1991).

Tamura, T.

I. Oda, Y. Ito, H. Eda, T. Tamura, M. Takada, R. Abvumi, K. Nagai, H. Nakagawa, M. Tamura, “Noninvasive hemoglobin oxygenation monitor and computed tomography by NIR spectrophotometry,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 73–83 (1991).

Wang, L.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Wang, N. G.

J. C. Haselgrove, N. G. Wang, B. Chance, “Investigation of the nonlinear aspects of imaging through a highly scattering medium,” Med. Phys. 19, 17–23 (1992).
[CrossRef] [PubMed]

Yamada Hasegawa, Y.

Y. Yamada Hasegawa, “Simulation of time-resolved optical CT imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 73–83 (1991).

Young, M.

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Zhang, G.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Zubelli, J. P.

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Appl. Opt. (1)

Commun. Pure Appl. Math. (1)

W. Magnus, “Operator expansion,” Commun. Pure Appl. Math. 7, 649–646 (1954).
[CrossRef]

J. Appl. Physiol. (1)

F. F. Jobsis, J. H. Keizer, J. C. LaManna, M. Rosenthal, “Reflectance spectrophotometry of cytochrome aa3 inυivo,” J. Appl. Physiol. 113, 858–872 (1977).

Med. Phys. (2)

J. C. Hebden, R. A. Kruger, “Transillumination imaging performance: spatial resolution simulation studies,” Med. Phys. 17, 41–47 (1990).
[CrossRef] [PubMed]

J. C. Haselgrove, N. G. Wang, B. Chance, “Investigation of the nonlinear aspects of imaging through a highly scattering medium,” Med. Phys. 19, 17–23 (1992).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Young “Comparison of time-resolved and unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
[CrossRef] [PubMed]

Science (2)

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

J. R. Singer, F. A. Grunbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Other (6)

J. C. Hebden, R. A. Kruger, “Time of flight breast imaging system,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 225–231 (1991).

Y. Yamada Hasegawa, “Simulation of time-resolved optical CT imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 73–83 (1991).

I. Oda, Y. Ito, H. Eda, T. Tamura, M. Takada, R. Abvumi, K. Nagai, H. Nakagawa, M. Tamura, “Noninvasive hemoglobin oxygenation monitor and computed tomography by NIR spectrophotometry,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 73–83 (1991).

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1431, 192–203 (1991).

A. Ishimaru, Waυe Propagation and Scattering in Random Media (Academic, New York, 1978).

R. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).

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Figures (7)

Fig. 1
Fig. 1

Typical photon path and a region X with nonvanishing hitting density.

Fig. 2
Fig. 2

Typical photon path that arises in the computation of the three-point Green’s function G(r1, r2, r3; t, t′). The path starts at (r1, 0), ends at (r2, t), and passes through (r3, t′).

Fig. 3
Fig. 3

Time dependence of the diffusion Green’s function for four source–detector pairs in a square box, 10 cm along an edge with D = 0.8 cm2/ns and μ0 = 0. The positions of the source and detectors are indicated in the hitting-density maps of Fig. 4. We calculated the Green’s function by using the eigenfunction expansion of Eq. (5.3).

Fig. 4
Fig. 4

Density plots of the hitting density in a box as in Fig. 3. The source is positioned along the midpoint of the vertical axis; 14 detectors are positioned equidistantly along the horizontal axis. The source and detector positions are highlighted as white squares. The results when the eigenfunction expansion [Eq. (4.2)] is used for four source–detector pairs are displayed. Each row represents a different detector: from the bottom, detectors 2, 6, 10, 14 are shown. Each column represents a different time of flight for the photons: from the left, they measure 5, 10, 30, 50 ns. Each map is scaled separately to its own maximum value.

Fig. 5
Fig. 5

Schematic diagram of the central plane of the cubic chamber used to model the hitting density of photons migrating from a source S in the middle of one face and detected by 14 detectors positioned along the adjacent face. The box is 10 cm along an edge where D = 0.8 cm2/ns.

Fig. 6
Fig. 6

Time-course signals for photons migrating between the source and four of the detectors of the box indicated in Fig. 5. We performed the calculations using Eq. (4.9).

Fig. 7
Fig. 7

Hitting-density maps calculated by using Eqs. (3.9) and (4.9) for the midplane of the migration chamber shown in Fig. 5. The source and detector positions are highlighted as white squares. Each row represents a different detector: from the bottom, detectors 2, 6, 10, 14 are shown. Each column represents a different time of flight for the photons: from the left they measure 5, 10, 30, 50 ns. Each map is separately scaled to its own maximum value.

Equations (35)

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

t n ( r , t ) = D 2 n ( r , t ) γ ( r ) n ( r , t ) ,
n ( t ) = exp ( H t ) n ( 0 ) ,
H ( r , r ) = [ D r 2 + γ ( r ) ] δ ( r r ) .
n ( r , t ) = d 3 r G ( r , r ) n ( r , 0 ) ,
G ( r , r ; t ) = D r exp { S [ r ( t ) ] } ,
S [ r ( t ) ] = 0 t { 1 4 D r ˙ 2 ( t ) + γ [ r ( t ) ] } d t ,
d μ [ r 1 , r 2 ] = D r exp { S [ r ( t ) ] } D r exp { S [ r ( t ) ] } .
d μ [ r 1 , r 2 ] = 1 G ( r 1 , r 2 ; t ) D r exp { S [ r ( t ) ] } .
d μ [ r 1 , r 2 ] = 1 .
ν ( r ; r 1 , r 2 , t ) = d μ [ r 1 , r 2 ] 0 t d t δ [ r r ( t ) ] .
h ( X ) = X d 3 r ν ( r ; r 1 , r 2 , t )
v ( r ; r 1 , r 2 , t ) = 1 G ( r 1 , r 2 ; t ) 0 t d t D r exp { S [ r ( t ) ] } × δ [ r r ( t ) ] .
G ( r 1 , r 2 , r 3 ; t , t ) = d μ ( r 1 , r 2 ) δ [ r 3 r ( t ) ]
ν ( r ; r 1 , r 2 , t ) = 1 G ( r 1 , r 2 ; t ) 0 t d t G ( r 1 , r 2 , r 3 ; t , t ) .
G ( r 1 , r 2 ; t ) = d 3 r 3 G ( r 1 , r 2 , r 3 ; t , t ) .
G ( r 1 , r 2 , r 3 ; t , t ) = G ( r 1 , r 3 ; t ) G ( r 3 , r 2 ; t t ) .
G ( r 1 , r 2 ; t ) = d 3 r 3 G ( r 1 , r 3 ; t ) G ( r 3 , r 2 ; t t ) .
ν ( r ; r 1 , r 2 , t ) = 1 G ( r 1 , r 2 ; t ) 0 t d t G ( r 1 , r ; t ) G ( r , r 2 ; t t ) .
d 3 r ν ( r ; r 1 , r 2 , t ) = t ,
d 3 r ν ( r ; r 1 , r 2 , t ) = 1 G ( r 1 , r 2 ; t ) d 3 r 0 t d t G ( r 1 , r ; t ) G ( r , r 2 ; t t ) = 1 G ( r 1 , r 2 ; t ) 0 t d t G ( r 1 , r 2 ; t ) = t ,
G ( r 1 , r 2 ; t ) = n exp ( λ n t ) ϕ n ( r 1 ) ϕ n ( r 2 ) .
ν ( r ; r 1 , r 2 , t ) = 1 G ( r 1 , r 2 ; t ) n , n 0 t d t ϕ n ( r 1 ) ϕ n ( r ) ϕ n ( r ) ϕ n ( r 2 ) × exp ( λ n t ) exp [ ( λ n λ n ) t ] = 1 G ( r 1 , r 2 ; t ) n , n { t exp ( λ n t ) Δ n n + 1 λ n λ n [ exp ( λ n t ) exp ( λ n t ) ] ( 1 Δ n n ) } × ϕ n ( r 1 ) ϕ n ( r ) ϕ n ' ( r ) ϕ n ( r 2 ) ,
Δ n n = { 1 λ n = λ n 0 λ n λ n .
n ( t + δ t ) = exp ( H δ t ) n ( t ) ,
H ( r 1 , r 2 ) = H 0 ( r 1 , r 2 ) + H 1 ( r 1 , r 2 )
H 0 ( r 1 , r 2 ) = ( D r 1 2 ) δ ( r 1 r 2 ) ,
H 1 ( r 1 , r 2 ) = γ ( r 1 ) δ ( r 1 r 2 ) ,
exp ( H δ t ) = exp ( H 0 δ t ) exp ( H 1 δ t ) × exp [ 1 2 ( δ t ) 2 [ H 0 , H 1 ] ] .
n ( r , t + δ t ) = d 3 r G 0 ( r r ; δ t ) exp [ γ ( r ) δ t ] n ( r , t ) ,
G 0 ( r 1 r 2 ; t ) = 1 ( 4 π D t ) 3 / 2 exp [ ( r 1 r 2 ) 2 4 D t ] .
n ( r , t + δ t ) = r K ( r r ; δ t ) υ ( r ) n ( r , t ) ,
υ ( r ) = { ( Δ L ) 3 exp [ γ ( r ) δ t ] r Ω 0 r Ω .
ϕ k ( r ) = ( 2 L ) 3 / 2 sin ( k x x ) sin ( k y y ) sin ( k z z ) ,
λ k = D k 2 + μ 0 c = π 2 D L 2 ( n x 2 + n y 2 + n z 2 ) + μ 0 c .
G ( r 1 , r 2 ; t ) = k exp [ ( D k 2 + μ 0 c ) t ] ϕ k ( r 1 ) ϕ k ( r 2 ) .

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