Abstract

We investigate the controversy over the precise form of the photon diffusion coefficient and suggest that it is largely independent of absorption, i.e., D0=v/3µs. After presentation of the general theoretical arguments underlying this assertion, Monte Carlo simulations are performed and explicitly reveal that the absorption-independent diffusion coefficient gives better agreement with theory than the traditionally accepted photon diffusion coefficient, Dμa=v/3(μs+μa). The importance of resolving this controversy for the proper characterization of the material optical properties is discussed.

© 1997 Optical Society of America

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  1. M. S. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
    [CrossRef] [PubMed]
  2. S. Fantini, M. A. Franceschini, E. Gratton, “Semi-infinite-geometry boundary problem for light migration in highly scattering media: a frequency-domain study in the diffusion approximation,” J. Opt. Soc. Am. B 11, 2128–2138 (1994).
    [CrossRef]
  3. C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
    [CrossRef]
  4. J. B. Fishkin, E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite place bounded by a straight edge,” J. Opt. Soc. Am. A 10, 127–140 (1993).
    [CrossRef] [PubMed]
  5. T. Durduran, D. A. Boas, B. Chance, A. G. Yodh, “Validity of the diffusion equation for small heterogeneities,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, G. Fujimto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, DC, 1996), pp. 60–63.
  6. A. G. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light, Phys. Today 48, (March), 34–40 (1995).
    [CrossRef]
  7. B. Chance, ed., Photon Migration in Tissues (Plenum, New York, 1989).
  8. D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
    [CrossRef] [PubMed]
  9. P. N. den Outer, T. M. Nieuwenhuizen, A. Lagendijk, “Location of objects in multiple-scattering media,” J. Opt. Soc. Am. A 10, 1209–1218 (1993).
    [CrossRef]
  10. S. Feng, F. Zeng, B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt. 34, 3826–3837 (1995).
    [CrossRef] [PubMed]
  11. X. D. Li, T. Durduran, B. Chance, A. G. Yodh, “Diffraction tomography for biomedical imaging with diffuse photon density waves,” Opt. Lett. 32, 573–575 (1997); errata, 32, 1198 (1997).
    [CrossRef]
  12. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.
  13. S. Glasstone, M. C. Edlund, The Elements of Nuclear Reactor Theory (Van Nostrand, New York, 1952), Chaps. 5 and 14.
  14. W. M. Star, J. P. A. Marijnissen, M. J. C. van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Biol. 33, 437–454 (1988).
    [CrossRef] [PubMed]
  15. B. Davidson, Neutron Transport Theory (Clarendon, Oxford, 1957).
  16. J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979).
  17. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 8.
  18. K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
    [CrossRef]
  19. K. Furutsu, “Pulse wave scattering by an absorber and integrated attenuation in the diffusion approximation,” J. Opt. Soc. Am. A 14, 267–274 (1997).
    [CrossRef]
  20. A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, B. C. Wilson, “Spatially-resolved absolute diffuse reflectance measurements for non-invasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996).
    [CrossRef] [PubMed]
  21. H. B. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, M. S. Patterson, “Optical-image reconstruction using frequency-domain data simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996).
    [CrossRef]
  22. D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–92 (1997).
    [CrossRef] [PubMed]
  23. K. Katayama, G. Nishimura, M. Kinjo, M. Tamura, “Absorbance measurements in turbid media by the photon correlation method,” Appl. Opt. 34, 7419–7427 (1996).
    [CrossRef]
  24. G. Nishimura, K. Katamaya, M. Kinjo, M. Tamura, “Diffusing-wave absorption in the homogeneous turbid media,” Opt. Commun. 128, 99–107 (1996).
    [CrossRef]
  25. A. Pifferi, R. Berg, P. Taroni, S. Andersson-Engels, “Fitting of time-resolved reflectance curves with a Monte Carlo model,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 311–314.
  26. Y. Tsuchiya, T. Urakami, “Photon migration model for turbid biological medium having various shapes,” Jpn. J. Appl. Phys. Part 2 34, 79–81 (1995).
    [CrossRef]
  27. D. A. Boas, “Diffuse photon probes of structural and dynamical properties of turbid media: theory and biomedical applications,” Ph.d. dissertation (University of Pennsylvania, Philadelphia, 1996).
  28. J. Masoliver, G. H. Weiss, “Finite-velocity diffusion,” Eur. J. Phys. 17, 190–196 (1996).
    [CrossRef]
  29. A. H. Gandjbakhche, G. H. Weiss, Random Walk and Diffusion-like Models of Photon Migration in Turbid Media, Vol. XXXIV of Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1995).
  30. L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  31. S. L. Jacques, L. Wang, “Monte Carlo modeling of light transport in tissues,” in Optical–Thermal Response of Laser-irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995).
  32. L. Wang, S. L. Jacques, L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
    [CrossRef] [PubMed]
  33. The photon migration imaging software and the Monte Carlo program are available upon request from the authors.
  34. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  35. M. Bassani, F. Martelli, G. Zaccanti, D. Contini, “Independence of the diffusion coefficient from absorption: experimental and numerical evidence,” Opt. Lett. 22, 853–855 (1997).
    [CrossRef] [PubMed]
  36. Information on independence of the diffusion coefficient from absorption is available from G. Nishimura, Biophysics Laboratory, Research Institute for Hokkaido University, Sapporo 060 Japan; e-mail: gnishi@imdes.hokudai.ac.jp.

1997

1996

1995

Y. Tsuchiya, T. Urakami, “Photon migration model for turbid biological medium having various shapes,” Jpn. J. Appl. Phys. Part 2 34, 79–81 (1995).
[CrossRef]

S. Feng, F. Zeng, B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt. 34, 3826–3837 (1995).
[CrossRef] [PubMed]

A. G. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light, Phys. Today 48, (March), 34–40 (1995).
[CrossRef]

L. Wang, S. L. Jacques, L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

1994

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

S. Fantini, M. A. Franceschini, E. Gratton, “Semi-infinite-geometry boundary problem for light migration in highly scattering media: a frequency-domain study in the diffusion approximation,” J. Opt. Soc. Am. B 11, 2128–2138 (1994).
[CrossRef]

1993

1989

1988

W. M. Star, J. P. A. Marijnissen, M. J. C. van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Biol. 33, 437–454 (1988).
[CrossRef] [PubMed]

1941

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Andersson-Engels, S.

A. Pifferi, R. Berg, P. Taroni, S. Andersson-Engels, “Fitting of time-resolved reflectance curves with a Monte Carlo model,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 311–314.

Bassani, M.

Berg, R.

A. Pifferi, R. Berg, P. Taroni, S. Andersson-Engels, “Fitting of time-resolved reflectance curves with a Monte Carlo model,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 311–314.

Boas, D. A.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–92 (1997).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

D. A. Boas, “Diffuse photon probes of structural and dynamical properties of turbid media: theory and biomedical applications,” Ph.d. dissertation (University of Pennsylvania, Philadelphia, 1996).

T. Durduran, D. A. Boas, B. Chance, A. G. Yodh, “Validity of the diffusion equation for small heterogeneities,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, G. Fujimto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, DC, 1996), pp. 60–63.

Case, K. M.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 8.

Chance, B.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–92 (1997).
[CrossRef] [PubMed]

X. D. Li, T. Durduran, B. Chance, A. G. Yodh, “Diffraction tomography for biomedical imaging with diffuse photon density waves,” Opt. Lett. 32, 573–575 (1997); errata, 32, 1198 (1997).
[CrossRef]

S. Feng, F. Zeng, B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt. 34, 3826–3837 (1995).
[CrossRef] [PubMed]

A. G. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light, Phys. Today 48, (March), 34–40 (1995).
[CrossRef]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

M. S. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

T. Durduran, D. A. Boas, B. Chance, A. G. Yodh, “Validity of the diffusion equation for small heterogeneities,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, G. Fujimto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, DC, 1996), pp. 60–63.

Contini, D.

Davidson, B.

B. Davidson, Neutron Transport Theory (Clarendon, Oxford, 1957).

den Outer, P. N.

Duderstadt, J. J.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979).

Durduran, T.

X. D. Li, T. Durduran, B. Chance, A. G. Yodh, “Diffraction tomography for biomedical imaging with diffuse photon density waves,” Opt. Lett. 32, 573–575 (1997); errata, 32, 1198 (1997).
[CrossRef]

T. Durduran, D. A. Boas, B. Chance, A. G. Yodh, “Validity of the diffusion equation for small heterogeneities,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, G. Fujimto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, DC, 1996), pp. 60–63.

Edlund, M. C.

S. Glasstone, M. C. Edlund, The Elements of Nuclear Reactor Theory (Van Nostrand, New York, 1952), Chaps. 5 and 14.

Fantini, S.

Feng, S.

Fishkin, J. B.

Franceschini, M. A.

Furutsu, K.

K. Furutsu, “Pulse wave scattering by an absorber and integrated attenuation in the diffusion approximation,” J. Opt. Soc. Am. A 14, 267–274 (1997).
[CrossRef]

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

Gandjbakhche, A. H.

A. H. Gandjbakhche, G. H. Weiss, Random Walk and Diffusion-like Models of Photon Migration in Turbid Media, Vol. XXXIV of Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1995).

Glasstone, S.

S. Glasstone, M. C. Edlund, The Elements of Nuclear Reactor Theory (Van Nostrand, New York, 1952), Chaps. 5 and 14.

Gonatas, C. P.

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

Gratton, E.

Greenstein, J. L.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Henyey, L. G.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hibst, R.

Ishii, M.

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

Jacques, S. L.

L. Wang, S. L. Jacques, L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

S. L. Jacques, L. Wang, “Monte Carlo modeling of light transport in tissues,” in Optical–Thermal Response of Laser-irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995).

Jiang, H. B.

Katamaya, K.

G. Nishimura, K. Katamaya, M. Kinjo, M. Tamura, “Diffusing-wave absorption in the homogeneous turbid media,” Opt. Commun. 128, 99–107 (1996).
[CrossRef]

Katayama, K.

Kienle, A.

Kinjo, M.

K. Katayama, G. Nishimura, M. Kinjo, M. Tamura, “Absorbance measurements in turbid media by the photon correlation method,” Appl. Opt. 34, 7419–7427 (1996).
[CrossRef]

G. Nishimura, K. Katamaya, M. Kinjo, M. Tamura, “Diffusing-wave absorption in the homogeneous turbid media,” Opt. Commun. 128, 99–107 (1996).
[CrossRef]

Lagendijk, A.

Leigh, J. S.

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

Li, X. D.

X. D. Li, T. Durduran, B. Chance, A. G. Yodh, “Diffraction tomography for biomedical imaging with diffuse photon density waves,” Opt. Lett. 32, 573–575 (1997); errata, 32, 1198 (1997).
[CrossRef]

Lilge, L.

Marijnissen, J. P. A.

W. M. Star, J. P. A. Marijnissen, M. J. C. van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Biol. 33, 437–454 (1988).
[CrossRef] [PubMed]

Martelli, F.

Martin, W. R.

J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley, New York, 1979).

Masoliver, J.

J. Masoliver, G. H. Weiss, “Finite-velocity diffusion,” Eur. J. Phys. 17, 190–196 (1996).
[CrossRef]

Miwa, M.

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

Nieuwenhuizen, T. M.

Nishimura, G.

K. Katayama, G. Nishimura, M. Kinjo, M. Tamura, “Absorbance measurements in turbid media by the photon correlation method,” Appl. Opt. 34, 7419–7427 (1996).
[CrossRef]

G. Nishimura, K. Katamaya, M. Kinjo, M. Tamura, “Diffusing-wave absorption in the homogeneous turbid media,” Opt. Commun. 128, 99–107 (1996).
[CrossRef]

O’Leary, M. A.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–92 (1997).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

Osterberg, U. L.

Patterson, M. S.

Paulsen, K. D.

Pifferi, A.

A. Pifferi, R. Berg, P. Taroni, S. Andersson-Engels, “Fitting of time-resolved reflectance curves with a Monte Carlo model,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 311–314.

Pogue, B. W.

Schotland, J.

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

Star, W. M.

W. M. Star, J. P. A. Marijnissen, M. J. C. van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Biol. 33, 437–454 (1988).
[CrossRef] [PubMed]

Steiner, R.

Tamura, M.

K. Katayama, G. Nishimura, M. Kinjo, M. Tamura, “Absorbance measurements in turbid media by the photon correlation method,” Appl. Opt. 34, 7419–7427 (1996).
[CrossRef]

G. Nishimura, K. Katamaya, M. Kinjo, M. Tamura, “Diffusing-wave absorption in the homogeneous turbid media,” Opt. Commun. 128, 99–107 (1996).
[CrossRef]

Taroni, P.

A. Pifferi, R. Berg, P. Taroni, S. Andersson-Engels, “Fitting of time-resolved reflectance curves with a Monte Carlo model,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 311–314.

Tsuchiya, Y.

Y. Tsuchiya, T. Urakami, “Photon migration model for turbid biological medium having various shapes,” Jpn. J. Appl. Phys. Part 2 34, 79–81 (1995).
[CrossRef]

Urakami, T.

Y. Tsuchiya, T. Urakami, “Photon migration model for turbid biological medium having various shapes,” Jpn. J. Appl. Phys. Part 2 34, 79–81 (1995).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

van Gemert, M. J. C.

W. M. Star, J. P. A. Marijnissen, M. J. C. van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Biol. 33, 437–454 (1988).
[CrossRef] [PubMed]

Wang, L.

L. Wang, S. L. Jacques, L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

S. L. Jacques, L. Wang, “Monte Carlo modeling of light transport in tissues,” in Optical–Thermal Response of Laser-irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995).

Weiss, G. H.

J. Masoliver, G. H. Weiss, “Finite-velocity diffusion,” Eur. J. Phys. 17, 190–196 (1996).
[CrossRef]

A. H. Gandjbakhche, G. H. Weiss, Random Walk and Diffusion-like Models of Photon Migration in Turbid Media, Vol. XXXIV of Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1995).

Wilson, B. C.

Yamada, Y.

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

Yodh, A. G.

X. D. Li, T. Durduran, B. Chance, A. G. Yodh, “Diffraction tomography for biomedical imaging with diffuse photon density waves,” Opt. Lett. 32, 573–575 (1997); errata, 32, 1198 (1997).
[CrossRef]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–92 (1997).
[CrossRef] [PubMed]

A. G. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light, Phys. Today 48, (March), 34–40 (1995).
[CrossRef]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

T. Durduran, D. A. Boas, B. Chance, A. G. Yodh, “Validity of the diffusion equation for small heterogeneities,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, G. Fujimto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, DC, 1996), pp. 60–63.

Zaccanti, G.

Zeng, F.

Zheng, L.

L. Wang, S. L. Jacques, L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 8.

Appl. Opt.

Astrophys. J.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Comput. Methods Programs Biomed.

L. Wang, S. L. Jacques, L. Zheng, “MCML—Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Eur. J. Phys.

J. Masoliver, G. H. Weiss, “Finite-velocity diffusion,” Eur. J. Phys. 17, 190–196 (1996).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Jpn. J. Appl. Phys. Part 2

Y. Tsuchiya, T. Urakami, “Photon migration model for turbid biological medium having various shapes,” Jpn. J. Appl. Phys. Part 2 34, 79–81 (1995).
[CrossRef]

Opt. Commun.

G. Nishimura, K. Katamaya, M. Kinjo, M. Tamura, “Diffusing-wave absorption in the homogeneous turbid media,” Opt. Commun. 128, 99–107 (1996).
[CrossRef]

Opt. Lett.

X. D. Li, T. Durduran, B. Chance, A. G. Yodh, “Diffraction tomography for biomedical imaging with diffuse photon density waves,” Opt. Lett. 32, 573–575 (1997); errata, 32, 1198 (1997).
[CrossRef]

M. Bassani, F. Martelli, G. Zaccanti, D. Contini, “Independence of the diffusion coefficient from absorption: experimental and numerical evidence,” Opt. Lett. 22, 853–855 (1997).
[CrossRef] [PubMed]

Phys. Med. Biol.

W. M. Star, J. P. A. Marijnissen, M. J. C. van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Biol. 33, 437–454 (1988).
[CrossRef] [PubMed]

Phys. Rev. E

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
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K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

Phys. Today

A. G. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light, Phys. Today 48, (March), 34–40 (1995).
[CrossRef]

Proc. Natl. Acad. Sci. USA

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

Other

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

S. Glasstone, M. C. Edlund, The Elements of Nuclear Reactor Theory (Van Nostrand, New York, 1952), Chaps. 5 and 14.

A. Pifferi, R. Berg, P. Taroni, S. Andersson-Engels, “Fitting of time-resolved reflectance curves with a Monte Carlo model,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 311–314.

D. A. Boas, “Diffuse photon probes of structural and dynamical properties of turbid media: theory and biomedical applications,” Ph.d. dissertation (University of Pennsylvania, Philadelphia, 1996).

B. Chance, ed., Photon Migration in Tissues (Plenum, New York, 1989).

T. Durduran, D. A. Boas, B. Chance, A. G. Yodh, “Validity of the diffusion equation for small heterogeneities,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, G. Fujimto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, DC, 1996), pp. 60–63.

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The photon migration imaging software and the Monte Carlo program are available upon request from the authors.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Information on independence of the diffusion coefficient from absorption is available from G. Nishimura, Biophysics Laboratory, Research Institute for Hokkaido University, Sapporo 060 Japan; e-mail: gnishi@imdes.hokudai.ac.jp.

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Figures (6)

Fig. 1
Fig. 1

Simple two-dimensional sketch of the Monte Carlo approach used in counting and propagating photons. Photons are emitted at the origin from an isotropic source. The thick solid curve indicates the path of a single photon that crosses the concentric spherical detectors (indicated by dashed circles) several times. When the photon crosses the detectors (at points A), the events are scored. This gives us the inward and outward flux at each detector. The shown photon is absorbed at the point B. Further details are explained in the text, and the Monte Carlo code is available upon request from the authors.

Fig. 2
Fig. 2

ln[U1(r, t)/U2(r, t)] is plotted for D=D0 (solid line) and for D=Dμa (dotted curves). Here μa=0 cm-1 for U0(r, t), and μa=0.05 cm-1 for U(r, t). Two different source–detector separations, 2 cm and 3 cm, are shown. As explained in the text, the plots for D=Dμa are curved and dependent on the source–detector separation, whereas for D=D0 the plots are straight and are not dependent on source–detector separation. μs=10 cm-1 for all cases.

Fig. 3
Fig. 3

ln[U0(r, t)/U(r, t)] is plotted for Monte Carlo results. Here U0(r, t) is for μa=0.005 cm-1, and U(r, t) has μa=0.05, 0.1, and 0.5 cm-1, as indicated in the figure. As expected from the theoretical argument for the diffusion coefficient being D=D0, Monte Carlo results produced the expected lines. This clearly indicates that Monte Carlo simulations are indicative of D=D0. μs=10 cm-1, and source–detector separation is 1 cm for all cases shown.

Fig. 4
Fig. 4

Fractional amplitude residuals [Amp(Monte Carlo)/Amp(Theory)] versus modulation frequency is shown for different infinite, homogeneous media: (a) results for D=Dμa, (b) results for D=D0. μs=10 cm-1, and source–detector separation is 1 cm for all cases shown. Media with μa=0.05, 0.1, 0.5, and 1.0 cm-1 are shown as indicated in the figure. Comparison of (a) and (b) shows that the agreement is much better when D=D0, which also allows for the possibility of weighting the absorption coefficient, as discussed in the text.

Fig. 5
Fig. 5

Fractional amplitude residuals [Amp(Monte Carlo)/Amp(Theory)] versus source–detector separation is shown for different infinite homogeneous media. Solid (dotted) curves indicate results for D=D0 (D=Dμa). μs=10 cm-1, and modulation frequency=200 MHz for all cases. μa=0.05, 0.5, and 1.0 cm-1 in (a), (b), and (c), respectively. Fractional amplitude residuals are expected to be approximately 1.0 for good agreement. Comparison of the dotted and solid curves show that D=D0 provides much better agreement, as discussed in the text.

Fig. 6
Fig. 6

μs fitting results through a range of source–detector separations (1 cm to 2 cm) versus modulation frequency is shown for different absorption coefficients. Both μs and μa are fitted simultaneously at each modulation frequency, except for w=0 (i.e., dc case). In the latter case, μa is assumed to be known, and we fit for μs. Solid (dotted) lines indicate results for D=D0 (D=Dμa). μs=10 cm-1; μa=0.05 and 1.0 cm-1 in (a) and (b), respectively. Dashed lines are used to indicate the expected μs. We see that for μa=0.5 and 1.0 cm-1 the accuracy for D=D0 is within 2%, whereas when D=Dμa, the discrepancy is as large as more than 10%. We also note that, even for the time-independent case, D=D0 gives better accuracy. For low μa values investigated, such as μa=0.05 and 0.1 cm-1, we obtain results accurate within 1%, which is under the expected noise level. The discrepancies and the comparison of the results for two different D’s are not significant. Here, once again, the details of D do not play a role, and hence the plots for the μa fits are not shown here. The details are discussed in the text.

Equations (30)

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1vL(r, Ωˆ, t)t+·L(r, Ωˆ, t)Ωˆ+μtL(r, Ωˆ, t)
=μsL(r, Ωˆ, t)f(Ωˆ, Ωˆ)dΩˆ+S(r, Ωˆ, t),
L(r, Ωˆ, t)=exp(-vμat)L0(r, Ωˆ, t).
1vL0(r, Ωˆ, t)t+·L0(r, Ωˆ, t)Ωˆ+μsL0(r, Ωˆ, t)
=μsL0(r, Ωˆ, t)f(Ωˆ, Ωˆ)dΩˆ.
2U(r, t)-vμaDμaU(r, t)-1DμaU(r, t)t
=-1DμaS(r, t),
Dμa=v/3(μs+μa),
U(r, t)=U0(r, t)exp(-vμat),
J(r, t)=J0(r, t)exp(-vμat),
2U0(r, t)-1DμaU0(r, t)t=0.
tU(r, t)+μavU(r, t)+·J(r, t)=S0(r, t),
1vtJ(r, t)+(μs+μa)J(r, t)+v3U(r, t)
=S1(r, t).
L(r, Ωˆ, t)=v4πU(r, t)+34πJ(r, t)·Ωˆ,
S(r, Ωˆ, t)=14πS0(r, t)+34πS1(r, t)·Ωˆ.
tU0(r, t)+·J0(r, t)=0,
1vtJ0(r, t)+μsJ0(r, t)+v3U0(r, t)=0.
-Dμa2U(r, t)+vμaU(r, t)+U(r, t)t+3Dμavμa U(r, t)t+1v2U(r, t)t2̲=S0(r, t)+3Dμav2S0t-3Dμav·S1(r, t)̲.
-2U0(r, t)+3µsvU0(r, t)t+3v22U(r, t)t2=0.
3wDv2w/μsv1.
μs/(μs+μa)1,
μa/(μs+μa)1.
D0=v/3µs.
J-=v4U(r, t)+D2U(r, t)r,
J+=v4U(r, t)-D2U(r, t)r,
cos θ=12g1+g2-1-g21-g+2gζ2,
U(r, t)=2(J-+J+)/vAΔt,
U(r, t)=(4πDt)-3/2 exp(-r2/4Dt-μavt).
lnU1(r, t)U2(r, t)=r24t1D2-1D1+(μa,1-μa,2)vt+lnD2D13/2,

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