Abstract

Various characteristics of photon diffusion in turbid biological media are examined. Applications include the interpretation of data acquired with laser Doppler blood-flow monitors and the design of protocols for therapeutic excitation of tissue chromophores. Incident radiation is assumed to be applied at an interface between a turbid tissue and a transparent medium, and the reemission of photons from that interface is analyzed. Making use of a discrete lattice model, we derive an expression for the joint probability Γ(n, ρ)d2ρ that a photon will be emitted in the infinitesimal area d2ρ centered at surface point ρ = (x, y), having made n collisions with the tissue. Mathematical expressions are obtained for the intensity distribution of diffuse surface emission, the probability of photon absorption in the interior as a function of depth, and the mean path length of detected photons as a function of the distance between the site of the incident radiation and the location of the detector. We show that the depth dependence of the distribution of photon absorption events can be inferred from measured parameters of the surface emission profile. Results of relevant computer simulations are presented, and illustrative experimental data are shown to be in accord with the theory.

© 1987 Optical Society of America

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  26. A. Ishimaru, “Theory and application of wave propagation and scattering in random media,” Proc. IEEE 65, 1030–1061 (1977).
    [CrossRef]
  27. G. D. Pedersen, N. J. McCormick, L. O. Reynolds, “Transport calculations for light scattering in blood,” Biophys. J. 16, 199–207 (1976).
    [CrossRef] [PubMed]
  28. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).
  29. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.
  30. J. J. Duderstadt, W. R. Martin, Transport Theory (Wiley-Interscience, New York, 1979).
  31. G. H. Weiss, R. J. Rubin, “Random walks, theory and selected applications,” Adv. Chem. Phys. 52, 363–505 (1983).
    [CrossRef]
  32. R. R. Anderson, J. A. Parrish, “The optics of human skin,”J. Invest. Dermatol. 77, 13–19 (1981).
    [CrossRef] [PubMed]
  33. P. J. Kolari, “Penetration of unfocused laser light into the skin,” Arch. Dermatol. Res. 277, 342–344 (1985).
    [CrossRef] [PubMed]

1985

P. J. Kolari, “Penetration of unfocused laser light into the skin,” Arch. Dermatol. Res. 277, 342–344 (1985).
[CrossRef] [PubMed]

M. D. Stern, “Laser-Doppler velocimetry in blood and multiple scattering fluids: theory,” Appl. Opt. 24, 1968–1986 (1985).
[CrossRef] [PubMed]

1984

G. P. Rodgers, A. N. Schechter, C. T. Noguchi, H. G. Klein, A. W. Nienhuis, R. F. Bonner, “Periodic microcirculatory flow in patients with sickle-cell disease,” New Engl. J. Med. 311, 1534–1538 (1984).
[CrossRef] [PubMed]

H. M. Druce, R. F. Bonner, P. Choo, C. Patow, R. Summers, M. A. Kaliner, “Response of nasal blood flow to neurohormones as measured by laser-Doppler velocimetry,”J. Appl. Physiol. 57, 1276–1283 (1984).
[PubMed]

1983

W. F. Larrabee, G. D. Sutton, A. Holloway, G. Tolentino, “Laser Doppler-velocimetry and fluorescein dye in the prediction of skin flap viability, a comparison,” Arch. Otolaryngol. 109, 454–456 (1983).
[CrossRef] [PubMed]

A. J. Tahmoush, P. D. Bowen, R. F. Bonner, T. J. Mancini, W. K. Engel, “Laser Doppler blood flow studies during open-muscle biopsy in patients with neuromuscular diseases,” Neurology 33, 547–551 (1983).
[CrossRef] [PubMed]

G. H. Weiss, R. J. Rubin, “Random walks, theory and selected applications,” Adv. Chem. Phys. 52, 363–505 (1983).
[CrossRef]

R. A. J. Groenhuis, H. A. Ferweda, J. J. Ten Bosch, “Scattering and absorption of turbid materials determined from reflection measurements. 1: Theory.” Appl. Opt. 22, 2456–2462 (1983).
[CrossRef] [PubMed]

R. A. J. Groenhuis, J. J. Ten Bosch, H. A. Ferwerda, “Scattering and absorption of turbid materials determined from reflection measurements. 2: Measuring method and calibration,” Appl. Opt. 22, 2463–2467 (1983).
[CrossRef] [PubMed]

1982

A. Mayevsky, B. Chance, “Intracellular oxidation-reduction state measured in situ by a multichannel fiber-optic surface fluorometer,” Science 217, 537–540 (1982).
[CrossRef] [PubMed]

1981

1978

T. J. Dougherty, J. E. Kaufman, A. Goldfarb, K. R. Weishaupt, D. G. Boyle, A. Mittelman, “Photoradiation therapy for the treatment of malignant tumors,” Cancer Res. 38, 2628–2635 (1978).
[PubMed]

1977

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

1976

1975

R. N. Pittman, B. R. Duling, “Measurement of percent oxyhemoglobin in the microvasculature,”J. Appl. Physiol. 38, 321–327 (1975).
[PubMed]

1973

B. Chance, N. Oshino, T. Sugano, A. Mayevsky, “Basic principles of tissue oxygen determination from mitochondrial signals,” Adv. Exp. Med. Biol. 37, 277–292 (1973).
[CrossRef]

1970

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

R. J. Zdrojkowski, N. R. Pisharoty, “Optical transmission and reflection by blood,”IEEE Trans. Biomed. Eng. BME-17, 122–128 (1970).
[CrossRef]

V. Twersky, “Interface effects in multiple scattering by large, low-refracting, absorbing particles,”J. Opt. Soc. Am. 60, 908–914 (1970).
[CrossRef]

V. Twersky, “Absorption and multiple scattering by biological suspensions,”J. Opt. Soc. Am. 60, 1084–1089 (1970).
[CrossRef] [PubMed]

1968

R. L. Longini, R. Zdrojkowski, “A note on the theory of backscattering of light by living tissue,”IEEE Trans. Biomed. Eng. BME-15, 4–10 (1968).
[CrossRef]

1948

Anderson, R. R.

R. R. Anderson, J. A. Parrish, “The optics of human skin,”J. Invest. Dermatol. 77, 13–19 (1981).
[CrossRef] [PubMed]

Atkins, J. T.

J. T. Atkins, “Optical properties of turbid materials,” in The Biological Effects of Ultraviolet Radiation, F. Urbach, ed. (Pergamon, Oxford, 1968), pp. 141–149.

Bonner, R.

Bonner, R. F.

G. P. Rodgers, A. N. Schechter, C. T. Noguchi, H. G. Klein, A. W. Nienhuis, R. F. Bonner, “Periodic microcirculatory flow in patients with sickle-cell disease,” New Engl. J. Med. 311, 1534–1538 (1984).
[CrossRef] [PubMed]

H. M. Druce, R. F. Bonner, P. Choo, C. Patow, R. Summers, M. A. Kaliner, “Response of nasal blood flow to neurohormones as measured by laser-Doppler velocimetry,”J. Appl. Physiol. 57, 1276–1283 (1984).
[PubMed]

A. J. Tahmoush, P. D. Bowen, R. F. Bonner, T. J. Mancini, W. K. Engel, “Laser Doppler blood flow studies during open-muscle biopsy in patients with neuromuscular diseases,” Neurology 33, 547–551 (1983).
[CrossRef] [PubMed]

R. F. Bonner, T. R. Clem, P. D. Bowen, R. L. Bowman, “Laser-Doppler real-time monitor of pulsatile and mean blood flow in tissue microcirculation,” in Scattering Techniques Applied to Supramolecular and Nonequilibrium Systems, S. H. Chen, B. Chu, R. Nossal, eds, Volume 3 of NATO ASI Series B (Plenum, New York, 1981), pp. 685–701.
[CrossRef]

Bowen, P. D.

A. J. Tahmoush, P. D. Bowen, R. F. Bonner, T. J. Mancini, W. K. Engel, “Laser Doppler blood flow studies during open-muscle biopsy in patients with neuromuscular diseases,” Neurology 33, 547–551 (1983).
[CrossRef] [PubMed]

R. F. Bonner, T. R. Clem, P. D. Bowen, R. L. Bowman, “Laser-Doppler real-time monitor of pulsatile and mean blood flow in tissue microcirculation,” in Scattering Techniques Applied to Supramolecular and Nonequilibrium Systems, S. H. Chen, B. Chu, R. Nossal, eds, Volume 3 of NATO ASI Series B (Plenum, New York, 1981), pp. 685–701.
[CrossRef]

Bowman, R. L.

R. F. Bonner, T. R. Clem, P. D. Bowen, R. L. Bowman, “Laser-Doppler real-time monitor of pulsatile and mean blood flow in tissue microcirculation,” in Scattering Techniques Applied to Supramolecular and Nonequilibrium Systems, S. H. Chen, B. Chu, R. Nossal, eds, Volume 3 of NATO ASI Series B (Plenum, New York, 1981), pp. 685–701.
[CrossRef]

Boyle, D. G.

T. J. Dougherty, J. E. Kaufman, A. Goldfarb, K. R. Weishaupt, D. G. Boyle, A. Mittelman, “Photoradiation therapy for the treatment of malignant tumors,” Cancer Res. 38, 2628–2635 (1978).
[PubMed]

Case, K. M.

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

Chance, B.

A. Mayevsky, B. Chance, “Intracellular oxidation-reduction state measured in situ by a multichannel fiber-optic surface fluorometer,” Science 217, 537–540 (1982).
[CrossRef] [PubMed]

B. Chance, N. Oshino, T. Sugano, A. Mayevsky, “Basic principles of tissue oxygen determination from mitochondrial signals,” Adv. Exp. Med. Biol. 37, 277–292 (1973).
[CrossRef]

Cheung, P. W.

P. W. Cheung, S. Takatani, E. A. Ernst, “Multiple wavelength reflectance oximetry in peripheral tissues,” in Oxygen Transport to Tissue, I. A. Silver, M. Erecinska, H. I. Bicher, eds. (Plenum, New York, 1978), pp. 69–75.

Choo, P.

H. M. Druce, R. F. Bonner, P. Choo, C. Patow, R. Summers, M. A. Kaliner, “Response of nasal blood flow to neurohormones as measured by laser-Doppler velocimetry,”J. Appl. Physiol. 57, 1276–1283 (1984).
[PubMed]

Clem, T. R.

R. F. Bonner, T. R. Clem, P. D. Bowen, R. L. Bowman, “Laser-Doppler real-time monitor of pulsatile and mean blood flow in tissue microcirculation,” in Scattering Techniques Applied to Supramolecular and Nonequilibrium Systems, S. H. Chen, B. Chu, R. Nossal, eds, Volume 3 of NATO ASI Series B (Plenum, New York, 1981), pp. 685–701.
[CrossRef]

Dorion, D. R.

D. R. Dorion, L. O. Svaasand, A. E. Profio, “Light dosimetry in tissue, application to photoradiation therapy,” in Porphyrin Photosensitization, D. Kessel, T. J. Dougherty, eds. (Plenum, New York, 1983), pp. 63–76.
[CrossRef]

Dougherty, T. J.

T. J. Dougherty, J. E. Kaufman, A. Goldfarb, K. R. Weishaupt, D. G. Boyle, A. Mittelman, “Photoradiation therapy for the treatment of malignant tumors,” Cancer Res. 38, 2628–2635 (1978).
[PubMed]

Druce, H. M.

H. M. Druce, R. F. Bonner, P. Choo, C. Patow, R. Summers, M. A. Kaliner, “Response of nasal blood flow to neurohormones as measured by laser-Doppler velocimetry,”J. Appl. Physiol. 57, 1276–1283 (1984).
[PubMed]

Duderstadt, J. J.

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

Duling, B. R.

R. N. Pittman, B. R. Duling, “Measurement of percent oxyhemoglobin in the microvasculature,”J. Appl. Physiol. 38, 321–327 (1975).
[PubMed]

Engel, W. K.

A. J. Tahmoush, P. D. Bowen, R. F. Bonner, T. J. Mancini, W. K. Engel, “Laser Doppler blood flow studies during open-muscle biopsy in patients with neuromuscular diseases,” Neurology 33, 547–551 (1983).
[CrossRef] [PubMed]

Ernst, E. A.

P. W. Cheung, S. Takatani, E. A. Ernst, “Multiple wavelength reflectance oximetry in peripheral tissues,” in Oxygen Transport to Tissue, I. A. Silver, M. Erecinska, H. I. Bicher, eds. (Plenum, New York, 1978), pp. 69–75.

Ferweda, H. A.

Ferwerda, H. A.

Goldfarb, A.

T. J. Dougherty, J. E. Kaufman, A. Goldfarb, K. R. Weishaupt, D. G. Boyle, A. Mittelman, “Photoradiation therapy for the treatment of malignant tumors,” Cancer Res. 38, 2628–2635 (1978).
[PubMed]

Groenhuis, R. A. J.

Holloway, A.

W. F. Larrabee, G. D. Sutton, A. Holloway, G. Tolentino, “Laser Doppler-velocimetry and fluorescein dye in the prediction of skin flap viability, a comparison,” Arch. Otolaryngol. 109, 454–456 (1983).
[CrossRef] [PubMed]

Ishimaru, A.

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

L. Reynolds, C. Johnson, A. Ishimaru, “Diffuse reflectance from a finite blood medium: applications to the modeling of fiber optic catheters,” Appl. Opt. 15, 2059–2067 (1976).
[CrossRef] [PubMed]

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

Johnson, C.

Johnson, C. C.

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

Kaliner, M. A.

H. M. Druce, R. F. Bonner, P. Choo, C. Patow, R. Summers, M. A. Kaliner, “Response of nasal blood flow to neurohormones as measured by laser-Doppler velocimetry,”J. Appl. Physiol. 57, 1276–1283 (1984).
[PubMed]

Kaufman, J. E.

T. J. Dougherty, J. E. Kaufman, A. Goldfarb, K. R. Weishaupt, D. G. Boyle, A. Mittelman, “Photoradiation therapy for the treatment of malignant tumors,” Cancer Res. 38, 2628–2635 (1978).
[PubMed]

Klein, H. G.

G. P. Rodgers, A. N. Schechter, C. T. Noguchi, H. G. Klein, A. W. Nienhuis, R. F. Bonner, “Periodic microcirculatory flow in patients with sickle-cell disease,” New Engl. J. Med. 311, 1534–1538 (1984).
[CrossRef] [PubMed]

Kolari, P. J.

P. J. Kolari, “Penetration of unfocused laser light into the skin,” Arch. Dermatol. Res. 277, 342–344 (1985).
[CrossRef] [PubMed]

Kubelka, P.

Larrabee, W. F.

W. F. Larrabee, G. D. Sutton, A. Holloway, G. Tolentino, “Laser Doppler-velocimetry and fluorescein dye in the prediction of skin flap viability, a comparison,” Arch. Otolaryngol. 109, 454–456 (1983).
[CrossRef] [PubMed]

Longini, R. L.

R. L. Longini, R. Zdrojkowski, “A note on the theory of backscattering of light by living tissue,”IEEE Trans. Biomed. Eng. BME-15, 4–10 (1968).
[CrossRef]

Lubbers, D. W.

D. W. Lubbers, “Spectroscopic examination of tissue oxygenation,” in Oxygen Tranport to Tissue, H. I. Bicher, D. Bruley, eds. (Plenum, New York, 1973), pp. 45–54.
[CrossRef]

Mancini, T. J.

A. J. Tahmoush, P. D. Bowen, R. F. Bonner, T. J. Mancini, W. K. Engel, “Laser Doppler blood flow studies during open-muscle biopsy in patients with neuromuscular diseases,” Neurology 33, 547–551 (1983).
[CrossRef] [PubMed]

Martin, W. R.

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

Mayevsky, A.

A. Mayevsky, B. Chance, “Intracellular oxidation-reduction state measured in situ by a multichannel fiber-optic surface fluorometer,” Science 217, 537–540 (1982).
[CrossRef] [PubMed]

B. Chance, N. Oshino, T. Sugano, A. Mayevsky, “Basic principles of tissue oxygen determination from mitochondrial signals,” Adv. Exp. Med. Biol. 37, 277–292 (1973).
[CrossRef]

McCormick, N. J.

G. D. Pedersen, N. J. McCormick, L. O. Reynolds, “Transport calculations for light scattering in blood,” Biophys. J. 16, 199–207 (1976).
[CrossRef] [PubMed]

Mittelman, A.

T. J. Dougherty, J. E. Kaufman, A. Goldfarb, K. R. Weishaupt, D. G. Boyle, A. Mittelman, “Photoradiation therapy for the treatment of malignant tumors,” Cancer Res. 38, 2628–2635 (1978).
[PubMed]

Nienhuis, A. W.

G. P. Rodgers, A. N. Schechter, C. T. Noguchi, H. G. Klein, A. W. Nienhuis, R. F. Bonner, “Periodic microcirculatory flow in patients with sickle-cell disease,” New Engl. J. Med. 311, 1534–1538 (1984).
[CrossRef] [PubMed]

Noguchi, C. T.

G. P. Rodgers, A. N. Schechter, C. T. Noguchi, H. G. Klein, A. W. Nienhuis, R. F. Bonner, “Periodic microcirculatory flow in patients with sickle-cell disease,” New Engl. J. Med. 311, 1534–1538 (1984).
[CrossRef] [PubMed]

Nossal, R.

Oshino, N.

B. Chance, N. Oshino, T. Sugano, A. Mayevsky, “Basic principles of tissue oxygen determination from mitochondrial signals,” Adv. Exp. Med. Biol. 37, 277–292 (1973).
[CrossRef]

Parrish, J. A.

R. R. Anderson, J. A. Parrish, “The optics of human skin,”J. Invest. Dermatol. 77, 13–19 (1981).
[CrossRef] [PubMed]

Patow, C.

H. M. Druce, R. F. Bonner, P. Choo, C. Patow, R. Summers, M. A. Kaliner, “Response of nasal blood flow to neurohormones as measured by laser-Doppler velocimetry,”J. Appl. Physiol. 57, 1276–1283 (1984).
[PubMed]

Pedersen, G. D.

G. D. Pedersen, N. J. McCormick, L. O. Reynolds, “Transport calculations for light scattering in blood,” Biophys. J. 16, 199–207 (1976).
[CrossRef] [PubMed]

Pisharoty, N. R.

R. J. Zdrojkowski, N. R. Pisharoty, “Optical transmission and reflection by blood,”IEEE Trans. Biomed. Eng. BME-17, 122–128 (1970).
[CrossRef]

Pittman, R. N.

R. N. Pittman, B. R. Duling, “Measurement of percent oxyhemoglobin in the microvasculature,”J. Appl. Physiol. 38, 321–327 (1975).
[PubMed]

Profio, A. E.

D. R. Dorion, L. O. Svaasand, A. E. Profio, “Light dosimetry in tissue, application to photoradiation therapy,” in Porphyrin Photosensitization, D. Kessel, T. J. Dougherty, eds. (Plenum, New York, 1983), pp. 63–76.
[CrossRef]

Reynolds, L.

Reynolds, L. O.

G. D. Pedersen, N. J. McCormick, L. O. Reynolds, “Transport calculations for light scattering in blood,” Biophys. J. 16, 199–207 (1976).
[CrossRef] [PubMed]

Rodgers, G. P.

G. P. Rodgers, A. N. Schechter, C. T. Noguchi, H. G. Klein, A. W. Nienhuis, R. F. Bonner, “Periodic microcirculatory flow in patients with sickle-cell disease,” New Engl. J. Med. 311, 1534–1538 (1984).
[CrossRef] [PubMed]

Rubin, R. J.

G. H. Weiss, R. J. Rubin, “Random walks, theory and selected applications,” Adv. Chem. Phys. 52, 363–505 (1983).
[CrossRef]

Schechter, A. N.

G. P. Rodgers, A. N. Schechter, C. T. Noguchi, H. G. Klein, A. W. Nienhuis, R. F. Bonner, “Periodic microcirculatory flow in patients with sickle-cell disease,” New Engl. J. Med. 311, 1534–1538 (1984).
[CrossRef] [PubMed]

Stern, M. D.

Sugano, T.

B. Chance, N. Oshino, T. Sugano, A. Mayevsky, “Basic principles of tissue oxygen determination from mitochondrial signals,” Adv. Exp. Med. Biol. 37, 277–292 (1973).
[CrossRef]

Summers, R.

H. M. Druce, R. F. Bonner, P. Choo, C. Patow, R. Summers, M. A. Kaliner, “Response of nasal blood flow to neurohormones as measured by laser-Doppler velocimetry,”J. Appl. Physiol. 57, 1276–1283 (1984).
[PubMed]

Sutton, G. D.

W. F. Larrabee, G. D. Sutton, A. Holloway, G. Tolentino, “Laser Doppler-velocimetry and fluorescein dye in the prediction of skin flap viability, a comparison,” Arch. Otolaryngol. 109, 454–456 (1983).
[CrossRef] [PubMed]

Svaasand, L. O.

D. R. Dorion, L. O. Svaasand, A. E. Profio, “Light dosimetry in tissue, application to photoradiation therapy,” in Porphyrin Photosensitization, D. Kessel, T. J. Dougherty, eds. (Plenum, New York, 1983), pp. 63–76.
[CrossRef]

Tahmoush, A. J.

A. J. Tahmoush, P. D. Bowen, R. F. Bonner, T. J. Mancini, W. K. Engel, “Laser Doppler blood flow studies during open-muscle biopsy in patients with neuromuscular diseases,” Neurology 33, 547–551 (1983).
[CrossRef] [PubMed]

Takatani, S.

P. W. Cheung, S. Takatani, E. A. Ernst, “Multiple wavelength reflectance oximetry in peripheral tissues,” in Oxygen Transport to Tissue, I. A. Silver, M. Erecinska, H. I. Bicher, eds. (Plenum, New York, 1978), pp. 69–75.

Ten Bosch, J. J.

Tolentino, G.

W. F. Larrabee, G. D. Sutton, A. Holloway, G. Tolentino, “Laser Doppler-velocimetry and fluorescein dye in the prediction of skin flap viability, a comparison,” Arch. Otolaryngol. 109, 454–456 (1983).
[CrossRef] [PubMed]

Twersky, V.

van de Hulst, H. C.

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

Weishaupt, K. R.

T. J. Dougherty, J. E. Kaufman, A. Goldfarb, K. R. Weishaupt, D. G. Boyle, A. Mittelman, “Photoradiation therapy for the treatment of malignant tumors,” Cancer Res. 38, 2628–2635 (1978).
[PubMed]

Weiss, G. H.

G. H. Weiss, R. J. Rubin, “Random walks, theory and selected applications,” Adv. Chem. Phys. 52, 363–505 (1983).
[CrossRef]

Zdrojkowski, R.

R. L. Longini, R. Zdrojkowski, “A note on the theory of backscattering of light by living tissue,”IEEE Trans. Biomed. Eng. BME-15, 4–10 (1968).
[CrossRef]

Zdrojkowski, R. J.

R. J. Zdrojkowski, N. R. Pisharoty, “Optical transmission and reflection by blood,”IEEE Trans. Biomed. Eng. BME-17, 122–128 (1970).
[CrossRef]

Zweifel, P. F.

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

Adv. Chem. Phys.

G. H. Weiss, R. J. Rubin, “Random walks, theory and selected applications,” Adv. Chem. Phys. 52, 363–505 (1983).
[CrossRef]

Adv. Exp. Med. Biol.

B. Chance, N. Oshino, T. Sugano, A. Mayevsky, “Basic principles of tissue oxygen determination from mitochondrial signals,” Adv. Exp. Med. Biol. 37, 277–292 (1973).
[CrossRef]

Appl. Opt.

Arch. Dermatol. Res.

P. J. Kolari, “Penetration of unfocused laser light into the skin,” Arch. Dermatol. Res. 277, 342–344 (1985).
[CrossRef] [PubMed]

Arch. Otolaryngol.

W. F. Larrabee, G. D. Sutton, A. Holloway, G. Tolentino, “Laser Doppler-velocimetry and fluorescein dye in the prediction of skin flap viability, a comparison,” Arch. Otolaryngol. 109, 454–456 (1983).
[CrossRef] [PubMed]

Biophys. J.

G. D. Pedersen, N. J. McCormick, L. O. Reynolds, “Transport calculations for light scattering in blood,” Biophys. J. 16, 199–207 (1976).
[CrossRef] [PubMed]

Cancer Res.

T. J. Dougherty, J. E. Kaufman, A. Goldfarb, K. R. Weishaupt, D. G. Boyle, A. Mittelman, “Photoradiation therapy for the treatment of malignant tumors,” Cancer Res. 38, 2628–2635 (1978).
[PubMed]

IEEE Trans. Biomed. Eng.

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

R. J. Zdrojkowski, N. R. Pisharoty, “Optical transmission and reflection by blood,”IEEE Trans. Biomed. Eng. BME-17, 122–128 (1970).
[CrossRef]

R. L. Longini, R. Zdrojkowski, “A note on the theory of backscattering of light by living tissue,”IEEE Trans. Biomed. Eng. BME-15, 4–10 (1968).
[CrossRef]

J. Appl. Physiol.

H. M. Druce, R. F. Bonner, P. Choo, C. Patow, R. Summers, M. A. Kaliner, “Response of nasal blood flow to neurohormones as measured by laser-Doppler velocimetry,”J. Appl. Physiol. 57, 1276–1283 (1984).
[PubMed]

R. N. Pittman, B. R. Duling, “Measurement of percent oxyhemoglobin in the microvasculature,”J. Appl. Physiol. 38, 321–327 (1975).
[PubMed]

J. Invest. Dermatol.

R. R. Anderson, J. A. Parrish, “The optics of human skin,”J. Invest. Dermatol. 77, 13–19 (1981).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

Neurology

A. J. Tahmoush, P. D. Bowen, R. F. Bonner, T. J. Mancini, W. K. Engel, “Laser Doppler blood flow studies during open-muscle biopsy in patients with neuromuscular diseases,” Neurology 33, 547–551 (1983).
[CrossRef] [PubMed]

New Engl. J. Med.

G. P. Rodgers, A. N. Schechter, C. T. Noguchi, H. G. Klein, A. W. Nienhuis, R. F. Bonner, “Periodic microcirculatory flow in patients with sickle-cell disease,” New Engl. J. Med. 311, 1534–1538 (1984).
[CrossRef] [PubMed]

Proc. IEEE

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

Science

A. Mayevsky, B. Chance, “Intracellular oxidation-reduction state measured in situ by a multichannel fiber-optic surface fluorometer,” Science 217, 537–540 (1982).
[CrossRef] [PubMed]

Other

J. T. Atkins, “Optical properties of turbid materials,” in The Biological Effects of Ultraviolet Radiation, F. Urbach, ed. (Pergamon, Oxford, 1968), pp. 141–149.

P. W. Cheung, S. Takatani, E. A. Ernst, “Multiple wavelength reflectance oximetry in peripheral tissues,” in Oxygen Transport to Tissue, I. A. Silver, M. Erecinska, H. I. Bicher, eds. (Plenum, New York, 1978), pp. 69–75.

R. F. Bonner, T. R. Clem, P. D. Bowen, R. L. Bowman, “Laser-Doppler real-time monitor of pulsatile and mean blood flow in tissue microcirculation,” in Scattering Techniques Applied to Supramolecular and Nonequilibrium Systems, S. H. Chen, B. Chu, R. Nossal, eds, Volume 3 of NATO ASI Series B (Plenum, New York, 1981), pp. 685–701.
[CrossRef]

D. W. Lubbers, “Spectroscopic examination of tissue oxygenation,” in Oxygen Tranport to Tissue, H. I. Bicher, D. Bruley, eds. (Plenum, New York, 1973), pp. 45–54.
[CrossRef]

D. R. Dorion, L. O. Svaasand, A. E. Profio, “Light dosimetry in tissue, application to photoradiation therapy,” in Porphyrin Photosensitization, D. Kessel, T. J. Dougherty, eds. (Plenum, New York, 1983), pp. 63–76.
[CrossRef]

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

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

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

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

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Figures (10)

Fig. 1
Fig. 1

The logarithm of the surface emission intensity Γ(ρ)e2μ for various values of absorption coefficient, calculated according to Eq. (12). The dashed lines are corresponding curves calculated according to the approximation given in expression (14).

Fig. 2
Fig. 2

Expected number of steps taken by a photon before it is reemitted at the surface, given as a function of the distance along the surface between the point of insertion and the point of emission. The solid line has been calculated by using Eq. (15), and the dashed line has been obtained from the approximate form given in expression (16). The agreement is particularly good for higher values of the absorption coefficient μ.

Fig. 3
Fig. 3

Coefficient of variation C(ρ) ≡ (〈n2|ρ〉 − 〈n|ρ2)1/2/〈n|ρ〉 of photons emerging at a distance ρ from the point of insertion, calculated by using Eqs. (10) and (15) and a similar equation for 〈n2|ρ〉.

Fig. 4
Fig. 4

(a) The expected value of maximum depth 〈Z〉; (b) the variance σ(Z), given as a function of the exit distance ρ. No discernible difference is noted between results calculated from Eq. (28) and data points obtained by fitting according to the forms given in Eq. (29).

Fig. 5
Fig. 5

Emission intensity as a function of ρ. Comparison between analytical results given by Eq. (12) (solid lines) and results of Monte Carlo calculations for the continuum model.

Fig. 6
Fig. 6

Total reflectance, given in terms of the inverse absorption coefficient 1/μ. Values shown as a solid line were calculated from Eq. (13). The open circles are the results of Monte Carlo calculations for a cubic lattice, and the filled circles are the results of Monte Carlo calculations for the isotropic continuum model.

Fig. 7
Fig. 7

Emission intensity measurements, as a function of the distance between the incident source and the detector, of light diffusing within human forearm skin, as determined with a laser Doppler blood-flow monitor. The solid lines are least-squares fits to the data (R2 > 0.99), obtained by using Eq. (12). (Parameters are given in Table 1.) Note that the intensity decreases more rapidly, as a function of ρ, for shorter wavelength radiation (for which tissue absorption is greater).

Fig. 8
Fig. 8

(a) Values of the asymptotic slope, 〈n|ρ〉/ρ, determined from plots of the expected value of path lengths of photons emitted at surface distance ρ, ascertained for different values of μ. Symbols: □, Monte Carlo (cubic lattice); ○, Monte Carlo (continuum model); ●, slope determined from expression (16) when calculated from values of ρ ≲ 10. The dotted line is the limiting slope at very high ρ, as determined by Eq. (17). (b) Typical results of Monte Carlo calculations for the isotropic continuum model. Note that in this case 〈n|ρ〉 is linear with ρ, even for vanishingly small values of ρ. The lines are obtained from expression (16).

Fig. 9
Fig. 9

Average number of times that a photon collides with moving red cells versus distance along the surface, measured in human skin with a laser blood-flow monitor. Data were obtained with the probe that enabled the distances between points of photon injection and detection to be varied (○, resting skin; □, vasodilated skin).

Fig. 10
Fig. 10

Comparison between expected values of maximum 〈Z〉 and average 〈z〉 depths experienced by photons that emerge at distance ρ. Points are obtained from Monte Carlo simulations for isotropic continuum model. Curves are fitted according to the analytical form shown in Eq. (29). (μ = 0.047.)

Tables (1)

Tables Icon

Table 1 Predictions, from in Situ Data, of Transmission through Skin, Compared with Literature Values Determined from in Vitro Measurements in Excised Tissue

Equations (29)

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r 2 ( t ) ~ σ 2 t / T ,             t T ,
exp ( - μ ) = exp ( - ν L )
Q n ( x , y , 0 ) = 0.
Q 1 ( r ) = e - μ δ x , 0 δ y , 0 δ z , 1 ,
Q n ( r ) = [ P n - 1 ( x , y , z - 1 ) - P n - 1 ( x , y , z + 1 ) ] e - μ n ,
λ ( θ ) ( cos θ 1 + cos θ 2 + cos θ 3 )
P n ( r ) = 1 ( 2 π ) 3 - π π λ n ( θ ) exp [ - i ( x θ 1 + y θ 2 + z θ 3 ) ] d 3 θ .
P n ( r ) ( 3 2 π n ) 3 / 2 exp [ - 3 2 n ( x 2 + y 2 + z 2 ) ] .
Γ ( n , ρ ) = Q n - 1 ( x , y , 1 ) exp ( - μ ) ,
Γ ( n , ρ ) = 3 2 ( 1 2 π ( n - 2 ) ) 3 / 2 ( 1 - e - 6 / ( n - 2 ) ) × exp ( - 3 ρ 2 2 ( n - 2 ) - μ n ) ,
γ ( ρ ) d ρ = 2 π ρ Γ ( ρ ) d ρ 2 π ρ n = 2 Γ ( n , ρ ) d ρ 2 π ρ d ρ 0 Γ ( n + 2 , ρ ) d n .
Γ ( ρ ) = ( 2 π ρ ) - 1 γ ( ρ ) = 1 4 π ρ { exp ( - ρ 6 μ ) - ρ ρ 2 + 4 exp [ - 6 μ ( ρ 2 + 4 ) ] } e - 2 μ ,
0 γ ( ρ ) d ρ = 1 24 μ [ 1 - exp ( - 24 μ ) ] e - 2 μ
Γ ( ρ ) 6 μ 4 π ρ 2 e - 2 μ exp ( - ρ 6 μ ) .
n ρ n = 2 n Γ ( n , ρ ) / n = 2 Γ ( n , ρ )
0 ( n + 2 ) Γ ( n + 2 , ρ ) d n 0 Γ ( n + 2 , ρ ) d n = 2 + ρ 3 2 μ × ( 1 - exp { 6 μ [ ρ - ( ρ 2 + 4 ) 1 / 2 ] } 1 - ρ ( ρ 2 + 4 ) 1 / 2 exp { 6 μ [ ρ - ( ρ 2 + 4 ) 1 / 2 ] } ) .
n ρ 2 + 3 ρ / 6 μ ,             ρ 2 4.
C = σ ( n ρ ) / n ρ ,
g n ( r ) = Q n ( r ) · ( 1 - e - μ ) = ( 1 - e - μ ) [ P n - 1 ( x , y , z - 1 ) - P n - 1 ( x , y , z + 1 ) ] e - μ n .
g ( z ) ~ 1 d n - d x d y g n ( r ) ~ exp ( - z 6 μ ) ,
z a z = 1 z g ( z ) / z = 1 g ( z ) = 1 / [ 1 - exp ( - 6 μ ) ] ,
n a n = 1 z = 1 n - g n ( r ) d x d y / n = 1 z = 1 - g n ( r ) d x d y 1 + 0 n 1 / 2 ( 1 + e - 3 / ( 2 n ) ) e - μ n d n / 0 n - 1 / 2 × ( 1 + e - 3 / ( 2 n ) ) e - μ n d n = 1 + 1 2 μ [ 1 + ( 1 + 6 μ ) exp ( - 6 μ ) 1 + exp ( - 6 μ ) ] .
Γ ( ρ Z ) = e - μ n = 1 Q n ( ρ , 1 Z ) / 6 ,
Q n ( ρ , 1 Z ) = l = - { P n - 1 ( ρ , 2 l Z ) - P n - 1 ( ρ , 2 + 2 l Z ) } e - μ n .
4 π Γ ( ρ Z ) l = - ( exp [ - 6 μ ( ρ 2 + 4 l 2 Z 2 ) ] ρ 2 + 4 l 2 Z 2 - exp { - 6 μ [ ρ 2 + 4 ( 1 + l Z ) 2 ] } ρ 2 + 4 ( 1 + l Z ) 2 ) e - 2 μ ,
R 0 ( Z ) 2 π 0 ρ Γ ( ρ Z ) d ρ = 1 24 μ [ 1 + 2 l = 1 exp ( - l Z 24 μ ) - l = - exp ( - 1 + l Z 24 μ ) ] e - 2 μ = 1 24 μ { 1 - exp ( - 24 μ ) + 2 [ 1 - cosh ( 24 μ ) ] exp ( Z 24 μ ) - 1 } e - 2 μ .
U ( Z ρ ) = [ Γ ( ρ Z + 1 ) - Γ ( ρ Z ) ] / Γ ( ρ ) .
Z = j = 1 [ 1 - Γ ( ρ j ) / Γ ( ρ ) ] , Z 2 = j = 1 ( 2 j - 1 ) [ 1 - Γ ( ρ j ) / Γ ( ρ ) ] .
Z = a + b ρ 2 / 3 ,             σ ( Z ) = a + b ρ 1 / 4 ,

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