Abstract

Monte Carlo simulations are used to discern scaling relationships for photon migration occurring within homogeneous, anisotropic scattering media of semi-infinite extent. Special attention is given to events associated with short path lengths. Empirical scaling relationships for path lengths and surface intensities are shown to agree with a consistency equation derived in an earlier study of anisotropic random walks. They are augmented here by a procedure that accounts for concomitant scaling of optical absorption coefficients. Results then are used to transform expressions that were obtained previously by analytical random-walk theory developed for an isotropic scattering model of photon migration. Quantities that are studied include the diffuse surface reflectance, the depth distribution of the fluence, and the time-resolved intensity of backreflected photons.

© 1993 Optical Society of America

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  1. R. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for photon migration in turbid biological tissue,” J. Opt. Soc. Am. A 4, 423–432 (1987).
    [CrossRef] [PubMed]
  2. R. Nossal, J. Kiefer, G. H. Weiss, R. Bonner, H. Taitelbaum, S. Havlin, “Photon migration in layered media,” Appl. Opt. 27, 3382–3391 (1988).
    [CrossRef] [PubMed]
  3. R. Nossal, R. F. Bonner, G. H. Weiss, “The influence of path length on remote optical sensing of properties of biological tissue,” Appl. Opt. 28, 2238–2244 (1989).
    [CrossRef] [PubMed]
  4. G. H. Weiss, R. Nossal, R. F. Bonner, “Statistics of penetration depth of photons re-emitted from irradiated tissue,” J. Mod. Opt. 36, 349–359 (1989).
    [CrossRef]
  5. R. Nossal, R. F. Bonner, “Differential time-resolved detection of absorbance changes in composite structures,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Soc. Photo-Opt. Instrum. Eng.1431, 21–28 (1991).
  6. M. Stern, “In vivo evaluation of microcirculation by coherent light scattering,” Nature (London) 254, 56–58 (1975).
    [CrossRef]
  7. R. Bonner, R. Nossal, “Model for laser Doppler measurements of blood flow in tissue,” Appl. Opt. 20, 2097–2107 (1981); R. Bonner, R. Nossal, “Principles of laser-Doppler flowmetry,” in Laser-Doppler Blood Flowmetry, A. P. Shepherd, P. Å. Öberg, ed. (Kluwer, Boston, Mass., 1990), Chap. 2, p. 17.
    [CrossRef] [PubMed]
  8. D. T. Delpy, M. Cope, P. van de Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time-of-flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
    [CrossRef] [PubMed]
  9. B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Yound, P. Cohen, H. Yoshioka, R. Boretsky, “Comparison of time-resolved and -unresolved measurements of deoxyhemoglobin in brain,” Proc. Natl. Acad. Sci. USA 85, 4971–4975 (1988).
    [CrossRef] [PubMed]
  10. J. M. Schmitt, G. X. Zhou, E. C. Walker, R. T. Wall, “A multilayer model of photon diffusion in skin,” J. Opt. Soc. Am. A 14, 2141–2153 (1990).
    [CrossRef]
  11. C. C. Johnson, “Optical diffusion in blood,” IEEE Trans. Biomed. Eng. BME-17, 129–133 (1970).
    [CrossRef]
  12. A. Ishimaru, “Theory and application of wave propagation and scattering in random media,” Proc. IEEE 65, 1030–1061 (1977).
    [CrossRef]
  13. M. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  18. K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
    [CrossRef] [PubMed]
  19. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 8, p. 194.
  20. J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976), Chap. 4, p. 217.
  21. L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  22. H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vol. 2, Chap. 14, p. 477.
  23. D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity relations for the interaction parameters in radiation transport,” Appl. Opt. 28, 5243–5249 (1989).
    [CrossRef] [PubMed]
  24. G. Yoon, S. A. Prahl, A. J. Welch, “Accuracies of the diffusion approximation and its similarity relations for laser irradiated biological media,” Appl. Opt. 28, 2250–2261 (1989).
    [CrossRef] [PubMed]
  25. M. P. Arnfield, J. Tulip, M. S. McPhee, “Optical propagation in tissue with anisotropic scattering,” IEEE Trans. Biomed. Eng. 35, 372–380 (1988).
    [CrossRef] [PubMed]
  26. G. E. Roberts, H. Kaufman, Table of Laplace Transforms (Saunders, Philadelphia, Pa., 1966), p. 22.

1992 (1)

A. H. Gandjbakhche, R. F. Bonner, R. Nossal, “Scaling relationships for anisotropic random walks,” J. Stat. Phys. 69, 35–53 (1992).
[CrossRef]

1990 (3)

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

J. M. Schmitt, G. X. Zhou, E. C. Walker, R. T. Wall, “A multilayer model of photon diffusion in skin,” J. Opt. Soc. Am. A 14, 2141–2153 (1990).
[CrossRef]

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (Paris) 51, 2101–2127 (1990).
[CrossRef]

1989 (5)

1988 (5)

D. J. Pine, D. A. Weitz, P. M. Chaikin, E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef] [PubMed]

M. P. Arnfield, J. Tulip, M. S. McPhee, “Optical propagation in tissue with anisotropic scattering,” IEEE Trans. Biomed. Eng. 35, 372–380 (1988).
[CrossRef] [PubMed]

R. Nossal, J. Kiefer, G. H. Weiss, R. Bonner, H. Taitelbaum, S. Havlin, “Photon migration in layered media,” Appl. Opt. 27, 3382–3391 (1988).
[CrossRef] [PubMed]

D. T. Delpy, M. Cope, P. van de Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time-of-flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

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

1987 (1)

1981 (1)

1977 (1)

A. Ishimaru, “Theory and application of wave propagation and scattering in random media,” Proc. IEEE 65, 1030–1061 (1977).
[CrossRef]

1975 (1)

M. Stern, “In vivo evaluation of microcirculation by coherent light scattering,” Nature (London) 254, 56–58 (1975).
[CrossRef]

1970 (1)

C. C. Johnson, “Optical diffusion in blood,” IEEE Trans. Biomed. Eng. BME-17, 129–133 (1970).
[CrossRef]

1941 (1)

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

Alfano, R. R.

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

Arnfield, M. P.

M. P. Arnfield, J. Tulip, M. S. McPhee, “Optical propagation in tissue with anisotropic scattering,” IEEE Trans. Biomed. Eng. 35, 372–380 (1988).
[CrossRef] [PubMed]

Arridge, S.

D. T. Delpy, M. Cope, P. van de Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time-of-flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Bonner, R.

Bonner, R. F.

A. H. Gandjbakhche, R. F. Bonner, R. Nossal, “Scaling relationships for anisotropic random walks,” J. Stat. Phys. 69, 35–53 (1992).
[CrossRef]

R. Nossal, R. F. Bonner, G. H. Weiss, “The influence of path length on remote optical sensing of properties of biological tissue,” Appl. Opt. 28, 2238–2244 (1989).
[CrossRef] [PubMed]

G. H. Weiss, R. Nossal, R. F. Bonner, “Statistics of penetration depth of photons re-emitted from irradiated tissue,” J. Mod. Opt. 36, 349–359 (1989).
[CrossRef]

R. Nossal, R. F. Bonner, “Differential time-resolved detection of absorbance changes in composite structures,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Soc. Photo-Opt. Instrum. Eng.1431, 21–28 (1991).

Boretsky, R.

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

Case, K. M.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 8, p. 194.

Chaikin, P. M.

D. J. Pine, D. A. Weitz, P. M. Chaikin, E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef] [PubMed]

Chance, B.

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

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

Cohen, P.

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

Cope, M.

D. T. Delpy, M. Cope, P. van de Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time-of-flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Delpy, D. T.

D. T. Delpy, M. Cope, P. van de Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time-of-flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Duderstadt, J. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976), Chap. 4, p. 217.

Finander, M.

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

Gandjbakhche, A. H.

A. H. Gandjbakhche, R. F. Bonner, R. Nossal, “Scaling relationships for anisotropic random walks,” J. Stat. Phys. 69, 35–53 (1992).
[CrossRef]

Greenfeld, R.

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

Greenstein, J. L.

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

Hamilton, L. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976), Chap. 4, p. 217.

Havlin, S.

Henyey, L. G.

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

Herbolzheimer, E.

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (Paris) 51, 2101–2127 (1990).
[CrossRef]

D. J. Pine, D. A. Weitz, P. M. Chaikin, E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef] [PubMed]

Ishimaru, A.

A. Ishimaru, “Theory and application of wave propagation and scattering in random media,” Proc. IEEE 65, 1030–1061 (1977).
[CrossRef]

Johnson, C. C.

C. C. Johnson, “Optical diffusion in blood,” IEEE Trans. Biomed. Eng. BME-17, 129–133 (1970).
[CrossRef]

Kaufman, H.

G. E. Roberts, H. Kaufman, Table of Laplace Transforms (Saunders, Philadelphia, Pa., 1966), p. 22.

Kaufmann, K.

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

Kiefer, J.

Leigh, J. S.

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

Liu, F.

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

McPhee, M. S.

M. P. Arnfield, J. Tulip, M. S. McPhee, “Optical propagation in tissue with anisotropic scattering,” IEEE Trans. Biomed. Eng. 35, 372–380 (1988).
[CrossRef] [PubMed]

Miyake, H.

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

Nioka, S.

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

Nossal, R.

A. H. Gandjbakhche, R. F. Bonner, R. Nossal, “Scaling relationships for anisotropic random walks,” J. Stat. Phys. 69, 35–53 (1992).
[CrossRef]

R. Nossal, R. F. Bonner, G. H. Weiss, “The influence of path length on remote optical sensing of properties of biological tissue,” Appl. Opt. 28, 2238–2244 (1989).
[CrossRef] [PubMed]

G. H. Weiss, R. Nossal, R. F. Bonner, “Statistics of penetration depth of photons re-emitted from irradiated tissue,” J. Mod. Opt. 36, 349–359 (1989).
[CrossRef]

R. Nossal, J. Kiefer, G. H. Weiss, R. Bonner, H. Taitelbaum, S. Havlin, “Photon migration in layered media,” Appl. Opt. 27, 3382–3391 (1988).
[CrossRef] [PubMed]

R. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for photon migration in turbid biological tissue,” J. Opt. Soc. Am. A 4, 423–432 (1987).
[CrossRef] [PubMed]

R. Bonner, R. Nossal, “Model for laser Doppler measurements of blood flow in tissue,” Appl. Opt. 20, 2097–2107 (1981); R. Bonner, R. Nossal, “Principles of laser-Doppler flowmetry,” in Laser-Doppler Blood Flowmetry, A. P. Shepherd, P. Å. Öberg, ed. (Kluwer, Boston, Mass., 1990), Chap. 2, p. 17.
[CrossRef] [PubMed]

R. Nossal, R. F. Bonner, “Differential time-resolved detection of absorbance changes in composite structures,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Soc. Photo-Opt. Instrum. Eng.1431, 21–28 (1991).

R. Nossal, J. M. Schmitt, “Measuring photon pathlengths by quasielastic light scattering in a multiply scattering medium,” in Photon Correlation Spectroscopy: Multicomponent System, K. S. Schmitt, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1430, 37–47 (1991).

Patterson, M.

Patterson, M. S.

Pine, D. J.

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (Paris) 51, 2101–2127 (1990).
[CrossRef]

D. J. Pine, D. A. Weitz, P. M. Chaikin, E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef] [PubMed]

Prahl, S. A.

Roberts, G. E.

G. E. Roberts, H. Kaufman, Table of Laplace Transforms (Saunders, Philadelphia, Pa., 1966), p. 22.

Schmitt, J. M.

J. M. Schmitt, G. X. Zhou, E. C. Walker, R. T. Wall, “A multilayer model of photon diffusion in skin,” J. Opt. Soc. Am. A 14, 2141–2153 (1990).
[CrossRef]

R. Nossal, J. M. Schmitt, “Measuring photon pathlengths by quasielastic light scattering in a multiply scattering medium,” in Photon Correlation Spectroscopy: Multicomponent System, K. S. Schmitt, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1430, 37–47 (1991).

Smith, D. S.

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

Stern, M.

M. Stern, “In vivo evaluation of microcirculation by coherent light scattering,” Nature (London) 254, 56–58 (1975).
[CrossRef]

Taitelbaum, H.

Tulip, J.

M. P. Arnfield, J. Tulip, M. S. McPhee, “Optical propagation in tissue with anisotropic scattering,” IEEE Trans. Biomed. Eng. 35, 372–380 (1988).
[CrossRef] [PubMed]

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vol. 2, Chap. 14, p. 477.

van de Zee, P.

D. T. Delpy, M. Cope, P. van de Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time-of-flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Walker, E. C.

J. M. Schmitt, G. X. Zhou, E. C. Walker, R. T. Wall, “A multilayer model of photon diffusion in skin,” J. Opt. Soc. Am. A 14, 2141–2153 (1990).
[CrossRef]

Wall, R. T.

J. M. Schmitt, G. X. Zhou, E. C. Walker, R. T. Wall, “A multilayer model of photon diffusion in skin,” J. Opt. Soc. Am. A 14, 2141–2153 (1990).
[CrossRef]

Weiss, G. H.

Weitz, D. A.

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (Paris) 51, 2101–2127 (1990).
[CrossRef]

D. J. Pine, D. A. Weitz, P. M. Chaikin, E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134–1137 (1988).
[CrossRef] [PubMed]

Welch, A. J.

Wilson, B. C.

Wray, S.

D. T. Delpy, M. Cope, P. van de Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time-of-flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Wyatt, J.

D. T. Delpy, M. Cope, P. van de Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of optical pathlength through tissue from direct time-of-flight measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Wyman, D. R.

Yoo, K. M.

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

Yoon, G.

Yoshioka, H.

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

Yound, M.

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

Zhou, G. X.

J. M. Schmitt, G. X. Zhou, E. C. Walker, R. T. Wall, “A multilayer model of photon diffusion in skin,” J. Opt. Soc. Am. A 14, 2141–2153 (1990).
[CrossRef]

Zhu, J. X.

D. J. Pine, D. A. Weitz, J. X. Zhu, E. Herbolzheimer, “Diffusing-wave spectroscopy: dynamic light scattering in the multiple scattering limit,” J. Phys. (Paris) 51, 2101–2127 (1990).
[CrossRef]

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 8, p. 194.

Appl. Opt. (6)

Astrophys. J. (1)

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

IEEE Trans. Biomed. Eng. (2)

M. P. Arnfield, J. Tulip, M. S. McPhee, “Optical propagation in tissue with anisotropic scattering,” IEEE Trans. Biomed. Eng. 35, 372–380 (1988).
[CrossRef] [PubMed]

C. C. Johnson, “Optical diffusion in blood,” IEEE Trans. Biomed. Eng. BME-17, 129–133 (1970).
[CrossRef]

J. Mod. Opt. (1)

G. H. Weiss, R. Nossal, R. F. Bonner, “Statistics of penetration depth of photons re-emitted from irradiated tissue,” J. Mod. Opt. 36, 349–359 (1989).
[CrossRef]

J. Opt. Soc. Am. A (2)

R. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for photon migration in turbid biological tissue,” J. Opt. Soc. Am. A 4, 423–432 (1987).
[CrossRef] [PubMed]

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[CrossRef]

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[CrossRef]

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J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976), Chap. 4, p. 217.

R. Nossal, J. M. Schmitt, “Measuring photon pathlengths by quasielastic light scattering in a multiply scattering medium,” in Photon Correlation Spectroscopy: Multicomponent System, K. S. Schmitt, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1430, 37–47 (1991).

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Figures (7)

Fig. 1
Fig. 1

Normalized path-length distribution as a function of the scaled path length lT*. (a) In P(lT*) for g = 0 (□), g = 0.2 (+), g = 0.5 (Δ), g = 0.8 (⋄), and g = 0.9 (○), plotted as a function of lT* = lT(1 − g)/(1 + g) for μ* = 0.1; scattering lengths are distributed according to an exponential distribution. For comparison, results for a constant-step distribution, g = 0.9 (●), also are shown. (b) Similar data for various values of μ* for g = 0 and g = 0.9. (c) Similar data as shown in (a) [for g = 0(□), g = 0.5(△), and g = 0.9 (○)], but l* is defined as lT* = lT(1 − g). (d) Comparison of short path-length behavior plotted as P(lT*) versus lT* for μ* = 0.1: [g = 0 (□) and g = 0.9 (○) for an exponential scattering-length distribution, and g = 0.9 (●) for a constant step plotted as a function of lT* = lT(1 − g)/(1 + g); g = 0.9 (×) for an exponential scattering-length distribution plotted as a function of lT* = lT(1 − g)] The solid curves were calculated from random-walk theory according to Eq. (20). [The dashed curve in (d) highlights the plot for (1 − g) scaling.]

Fig. 2
Fig. 2

Intensity of surface reemitted photons: (a) Γ(ρ*) plotted as a function of the scaled variable ρ* = ρ(1 − g)/(1 + g)1/2 = (r/2)[(1 − g)/(1+ g)1/2] Σs (for exponentially distributed scattering lengths): □, g = 0; △, g = 0.5; ⋄, g = 0.8). The solid curves were calculated according to Eq. (21). (b) Comparison for μ* = 0.1 of the scaling of γ(ρ*) = ρ*Γ(ρ*) at small values of ρ* for g = 0 (□), g = 0.9 scaled according to ρ* = ρ(1 − g)/(1 + g)1/2 (○), and g = 0.9 scaled according to ρ* = ρ(1 − g) (×).

Fig. 3
Fig. 3

Reflectance as a function of absorption per unit scattering length for various values of g plotted as a function of the scaled variable μ*. Data corresponding to the open symbols were obtained by scaling μ according to μ* = μ(1 + g)/(1 − g) (□, g = 0; △, g = 0.5; ○, g = 0.9). The filled symbols represent the same data rescaled so that μ* is defined as μ* = μ/(1 − g). The solid curve is calculated according to Eq. (22).

Fig. 4
Fig. 4

Diffuse irradiance Q(ζ) as a function of the scaled depth ζ* = ζ(1 − g)/(1 + ζ)1/2 for various values of g(□, g = 0; △, g = 0.5; ○, g = 0.9). The three groups of data correspond to different values of the absorption coefficient, scaled according to the relation μ* = [(1 + g)/(1 − g)]μ. (Some data points for μ* = 0.1 are not shown to enable examination of the scaling at small values of ζ*.) The solid lines were computed from Eq. (28).

Fig. 5
Fig. 5

Time-resolved intensity Γ(ρ, l) [represented here as Γ(lT*, ρ*) to stress that ρ* is fixed] plotted as a function of the scaled variable lT*, for different values of scaled ρ* and values of μ so that μ* ≡ μ(1 + g)/(1 − g) = 0.2. The symbols correspond to values of g as indicated in the captions of Figs. 2 and 3. The curves were computed for the appropriate values of ρ* and μ* according to the expression given in Eq. (5) (with the substitutions ρ → ρ*, nlT*, and μ → μ*).

Fig. 6
Fig. 6

Expected value of path length 〈lT*, ρ*〉 for photons reemitted at point ρ plotted as a value of the scaled surface distance ρ*. The points correspond to simulations performed for different values of g (□, g = 0; ⋄, g = 0.8; ○, g = 0.9). For each value of g, μ was chosen so that μ* = 0.1. The solid curve was calculated according to Eq. (24).

Fig. 7
Fig. 7

Time resolved data for g = 0: (a) Γ(ρ, lT) versus lT for various values of ρ where Γ(ρ, lT) is represented as Γ(lT*|ρ*) to stess that ρ* is fixed. The solid curves correspond to Eq. (5); the dashed curves are given by Eq. (7). (b) 〈lT|ρ〉 as a function of ρ. The dashed–dot line corresponds to Eq. (24); the dashed curve is derived from Eq. (A8). The absorption per unit scattering length here was taken to be μ = 0.2 in (a) and μ = 0.1 in (b).

Equations (54)

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γ ( ρ ) = 0 P ( ρ | n ) P ( n ) d n ,
P s c ( l ) = λ 1 exp ( l / λ ) ,
P ( n ) = P 0 ( n ) exp ( μ n ) ,
ρ* = [ h ( g ) / f ( g ) ] 1 / 2 ρ w ( g ) ρ , n * = h ( g ) n , μ* = [ h ( g ) ] 1 μ .
Γ ( ρ , n ) = 3 2 [ 2 π ( n 2 ) ] 3 / 2 { 1 exp [ 6 / ( n 2 ) ] } × exp [ 3 ρ 2 / 2 ( n 2 ) ] exp ( μ n ) .
Γ ( r , t ) = 3 2 [ 2 π Σ s c ( t t 0 ) ] 3 / 2 { 1 exp [ 6 / Σ s c ( t t 0 ) ] } × exp [ 3 r 2 / Σ s 4 c ( t t 0 ) ] exp ( Σ a c t ) ,
R ( r , t ) = ( 4 π D 0 c ) 3 / 2 z 0 t 5 / 2 × exp [ ( r 2 + z 0 2 ) 4 D 0 c t ] exp ( Σ a c t ) .
D 0 = [ 3 ( Σ a + Σ s ) ] 1 ,
Γ ( r , t ) R ( r , t ) t 5 / 2 exp ( 3 Σ s r 2 4 c t ) exp ( Σ a c t ) .
Γ ( r , t ) = 3 2 [ 2 π h ( g ) Σ s c ( t t 0 ) ] 3 / 2 × { 1 exp [ 6 / h ( g ) Σ s c ( t t 0 ) ] } × exp [ 3 r 2 ( 1 g ) Σ s / 4 c ( t t 0 ) ] exp ( Σ a c t ) ,
D = { 3 [ Σ a + ( 1 g ) Σ s ] } 1 .
R ( r , t ) t 5 / 2 exp [ 3 ( 1 g ) Σ s r 2 4 c t ] exp ( Σ a c t ) .
F ( θ ) = 1 g 2 2 g [ 1 1 g 1 ( 1 + g 2 2 g cos θ ) 1 / 2 ] 0 θ π .
F ( θ ) | g 0 = 1 cos θ 2 ,
n * = n ( 1 g ) / ( 1 + g ) , ρ * ρ ( 1 g ) / ( 1 + g ) 1 / 2 ,
n * = n ( 1 g ) , ρ * ρ ( 1 g ) .
P ( l T * ) = ρ* H ρ*, l T * ,
Γ ( ρ * ) = ( 2 πρ* ) 1 l T * H ρ*, l T * ,
l T * | ρ * = l T * l T * H ρ*, l T * , l T * H ρ*, l T * , .
P ( n ) ( n 2 ) 1 / 2 { 1 exp [ 6 / ( n 2 ) ] } exp ( μ n ) .
Γ ( ρ ) 1 4 πρ × ( exp ρ ( 6 μ ) 1 / 2 ρ ( ρ 2 + 4 ) 1 / 2 exp { [ 6 μ ( ρ 2 + 4 ) ] 1 / 2 } ) .
R ( μ* ) 1 ( 24 μ* ) 1 / 2 { 1 exp [ ( 24 μ* ) 1 / 2 ] } exp ( 2 μ* ) .
Q ( ζ* ) exp [ ζ* ( 6 μ* ) 1 / 2 ] .
l * | ρ = 2 + ρ ( 3 2 μ ) 1 / 2 × ( 1 exp { ( 6 μ ) 1 / 2 [ ρ ( ρ 2 + 4 ) 1 / 2 ] } 1 ρ ( ρ 2 + 4 ) 1 / 2 exp { ( 6 μ ) 1 / 2 [ ρ ( ρ 2 + 4 ) 1 / 2 ] } ) ,
( 1 g ) ( 1 + g ) 1 / 2 Σ s = const . ,
( 1 + g ) 1 / 2 Σ a = const .
( 1 g ) Σ s = const . ,
Σ a = const . ,
Γ ( r , t ) = 3 2 ( [ 2 π ( 1 g 1 + g ) c ( t t 0 ) ] 3 / 2 × { 1 exp [ 6 ( 1 + g ) / ( 1 g ) Σ s c ( t t 0 ) ] } ) × exp [ 3 r 2 ( 1 g ) Σ s / 4 c ( t t 0 ) ] exp ( Σ a c t ) ,
Q ( z ) exp { z [ 3 Σ a Σ s ( 1 g ) ] 1 / 2 } .
Γ ( r ) exp { r [ 3 Σ a Σ s ( 1 g ) ] 1 / 2 } r 2 .
t | r 1 2 c [ 3 Σ s Σ a 1 ( 1 g ) ] 1 / 2 r .
R r . w . = 1 { 24 Σ a Σ s 1 [ h ( g ) ] 1 } 1 / 2 × [ 1 exp ( { 24 Σ a Σ s 1 [ h ( g ) ] 1 } 1 / 2 ) ] × exp { 2 Σ a Σ s 1 [ h ( g ) ] 1 } .
R diff = exp { [ 3 Σ a Σ s 1 / ( 1 g ) ] 1 / 2 } .
R ( ρ , n ) n 5 / 2 exp [ 3 ( ρ 2 + 1 2 ) 2 n ] exp ( μ n ) .
Γ ( ρ ) = 0 R ( ρ , n ) d n 0 t 1 / 2 exp ( μ / t ) exp [ 3 2 ( ρ 2 + 1 2 ) t ] d t ,
Γ ( ρ ) L [ t 1 / 2 exp ( μ / t ) ] = π 1 / 2 2 s 3 / 2 ( 1 + 2 μ 1 / 2 s 1 / 2 ) exp ( 2 μ 1 / 2 s 1 / 2 ) ,
Γ ( ρ ) 1 + [ 6 μ ( ρ 2 + 1 2 ) ] 1 / 2 ( ρ 2 + 1 2 ) 3 / 2 exp { [ 6 ( ρ 2 + 1 2 ) μ ] 1 / 2 } ,
Γ ( ρ ) 6 μ ρ 2 exp [ ρ ( 6 μ ) 1 / 2 ] .
n | ρ = 0 n 3 / 2 exp [ 3 ( ρ 2 + 1 2 ) 2 n ] exp ( μ n ) d n 0 n 5 / 2 [ 3 ( ρ 2 + 1 2 ) 2 n ] exp ( μ n ) d n .
L [ t 1 / 2 exp ( μ / t ) ] = π 1 / 2 s 1 / 2 1 exp ( 2 μ 1 / 2 s 1 / 2 ) .
n | ρ = 3 ( ρ 2 + 1 2 ) 1 + [ 6 μ ( ρ 2 + 1 2 ) ] 1 / 2 ,
n | ρ ρ ( 3 2 μ ) 1 / 2 .
P ( n ) = 0 R ( ρ , n ) ρ d ρ n 3 / 2 exp ( μ n ) exp ( 3 / 4 n ) .
R ( μ ) = 0 P ( n ) d n = exp [ ( 3 μ ) 1 / 2 ] .
R diff ( μ ) 1 ( 3 μ ) 1 / 2 .
R r . w . ( μ ) 1 ( 6 μ ) 1 / 2 .
Γ ( r ) r . w . = 0 Γ ( r , t ) r . w . d t 1 r ( exp { r [ 3 Σ a Σ s ( 1 g ) ] 1 / 2 } r Σ s ( 1 g ) [ r 2 Σ s 2 ( 1 g ) 2 + 8 ( 1 + g ) ] 1 / 2 × exp { [ 3 r 2 Σ a Σ s ( 1 g ) + 24 Σ a Σ s 1 ( 1 + g ) / ( 1 g ) ] } 1 / 2 ) .
Γ ( r ) r . w . [ 6 Σ a Σ s 1 ( 1 + g ) / ( 1 g ) ] 1 / 2 r 2 × exp { r [ 3 Σ a Σ s ( 1 g ) ] 1 / 2 } .
P ( t ) r . w . = 2 π 0 Γ ( r , t ) r . w . r d r [ 2 π Σ s c ( t t 0 ) ] 1 / 2 × { 1 exp [ 6 ( 1 + g ) / ( 1 g ) Σ s c ( t t 0 ) ] } × exp ( Σ a c t ) ,
t | r = 0 t Γ ( r , t ) d t 0 Γ ( r , t ) d t .
t | r r . w . 1 c Σ s { 2 ( 1 + g ) ( 1 g ) + [ 3 4 Σ s ( 1 g ) Σ a ] 1 / 2 F ( r ) } ,
F ( r ) = [ ( exp { r [ 3 Σ a Σ s ( 1 g ) ] 1 / 2 exp { [ 3 r 2 Σ a Σ s ( 1 g ) + 24 Σ a Σ s 1 ( 1 + g ) / ( 1 g ) ] 1 / 2 } ) / ( 1 r Σ s exp { r [ 3 Σ a Σ s ( 1 g ) ] 1 / 2 } 1 [ r 2 Σ s 2 + 8 ( 1 + g ) ( 1 g ) 2 ] 1 / 2 × exp { [ 3 r 2 Σ a Σ s ( 1 g ) + 24 Σ a Σ a 1 ( 1 + g ) / ( 1 g ) ] 1 / 2 } ) ] .
P ( t ) diff t 3 / 2 exp ( Σ a c t ) exp [ 3 / 4 c ( 1 g ) Σ s t ] .

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