Abstract

A method is described for finding the optical properties (scattering, absorption, and scattering anisotropy) of a slab of turbid material by using total reflection, unscattered transmission, and total transmission measurements. This method is applicable to homogeneous turbid slabs with any optical thickness, albedo, or phase function. The slab may have a different index of refraction from its surroundings and may or may not be bounded by glass. The optical properties are obtained by iterating an adding–doubling solution of the radiative transport equation until the calculated values of the reflection and transmission match the measured ones. Exhaustive numerical tests show that the intrinsic error in the method is <3% when four quadrature points are used.

© 1993 Optical Society of America

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    [CrossRef]
  14. A. Vogel, C. Dlugos, R. Nuffer, R. Birngruber, “Optical properties of human sclera, and their consequences for transscleral laser applications,” Lasers Surg. Med. 11, 331–340 (1991).
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  20. M. S. Patterson, E. Schwartz, B. C. Wilson, “Quantitative reflectance spectrophotometry for the noninvasive measurement of photosensitizer concentration in tissue during photodynamic therapy,” in Photodynamic Therapy: Mechanisms, T. J. Dougherty, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1065, 115–122 (1989).
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  23. 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, 2250–2255 (1989).
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  30. S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in high scattering tissue—I: Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. BME-36, 1162–1168 (1989).
    [CrossRef]
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    [CrossRef]
  33. S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).
  34. G. Yoon, A. J. Welch, M. Motamedi, M. C. J. V. Gemert, “Development and application of three-dimensional light distribution model for laser irradiated tissue,” IEEE J. Quantum Electron. QE-23, 1721–1733 (1987).
    [CrossRef]
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    [CrossRef]

1993 (1)

1992 (2)

J. W. Pickering, C. J. M. Moes, H. J. C. M. Sterenborg, S. A. Prahl, M. J. C. van Gemert, “Two integrating spheres with an intervening scattering sample,” J. Opt. Soc. Am. A 9, 621–631 (1992).
[CrossRef]

S. A. Prahl, I. A. Vitkin, B. C. Wilson, R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203–1217 (1992).
[CrossRef] [PubMed]

1991 (2)

M. S. Patterson, B. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue II. Optical properties of tissues and resulting fluence distributions,” Lasers Med. Sci. 6, 379–390 (1991).
[CrossRef]

A. Vogel, C. Dlugos, R. Nuffer, R. Birngruber, “Optical properties of human sclera, and their consequences for transscleral laser applications,” Lasers Surg. Med. 11, 331–340 (1991).
[CrossRef] [PubMed]

1990 (2)

G. M. LaMuraglia, M. R. Prince, N. S. Nishioka, S. Obremski, R. Birngruber, “Optical properties of human arterial thrombus, vascular grafts, and sutures: implications for selective laser thrombus ablation,” IEEE J. Quantum Electron. 26, 2200–2206 (1990).
[CrossRef]

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

1989 (4)

1987 (6)

M. J. C. van Gemert, W. M. Star, “Relations between the Kubelka–Munk and the transport equation models for anisotropic scattering,” Lasers Life Sci. 1, 287–298 (1987).

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

G. Yoon, A. J. Welch, M. Motamedi, M. C. J. V. Gemert, “Development and application of three-dimensional light distribution model for laser irradiated tissue,” IEEE J. Quantum Electron. QE-23, 1721–1733 (1987).
[CrossRef]

S. L. Jacques, S. A. Prahl, “Modeling optical and thermal distributions in tissue during laser irradiation,” Lasers Surg. Med. 6, 494–503 (1987).
[CrossRef] [PubMed]

B. C. Wilson, M. S. Patterson, S. T. Flock, “Indirect versus direct techniques for the measurement of the optical properties of tissues,” Photochem. Photobiol. 46, 601–608 (1987).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

1983 (2)

B. L. Diffey, “A mathematical model for ultraviolet optics in skin,” Phys. Med. Biol. 28, 647–657 (1983).
[CrossRef] [PubMed]

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

1977 (2)

W. J. Wiscombe, “Doubling initialization revisited,” J. Quant. Spectrosc. Radiat. Transfer 18, 245–248 (1977).
[CrossRef]

W. J. Wiscombe, “The delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. 34, 1408–1422 (1977).
[CrossRef]

1976 (2)

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

W. J. Wiscombe, “On initialization, error and flux conservation in the doubling method,” J. Quant. Spectrosc. Radiative Transfer 16, 637–658 (1976).
[CrossRef]

1975 (1)

W. M. Irvine, “Multiple scattering in planetary atmospheres,” Icarus 25, 175–204 (1975).
[CrossRef]

1973 (3)

1971 (1)

1969 (1)

1965 (2)

1954 (1)

1948 (2)

P. Kubelka, “Errata: new contributions to the optics of intensely light-scattering materials,” J. Opt. Soc. Am. 38, 1067 (1948).
[CrossRef]

P. Kubelka, “New contributions to the optics of intensely light-scattering materials. Part I,” J. Opt. Am. 38, 448–457 (1948).
[CrossRef]

1942 (1)

Adam, G.

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Alfano, R. R.

K. M. Yoo, F. Liu, R. R. Alfano, “Angle and time resolved studies of backscattering of light from biological tissues,” in Laser–Tissue Interaction, S. L. Jacques, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1202, 260–271 (1990).

Alter, C. A.

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

Anderson, R. R.

S. A. Prahl, I. A. Vitkin, B. C. Wilson, R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203–1217 (1992).
[CrossRef] [PubMed]

Atkins, J. T.

J. T. Atkins, “Optical properties of turbid materials,” in The Biologic Effects of Ultraviolet Radiation (With Emphasis on the Skin), F. Urbach, ed. (Pergamon, London, 1969), pp. 141–149.

Beek, J. B.

Bellman, R.

R. Bellman, G. M. Wing, An Introduction to Invariant Imbedding (Wiley, New York, 1975).

Birngruber, R.

A. Vogel, C. Dlugos, R. Nuffer, R. Birngruber, “Optical properties of human sclera, and their consequences for transscleral laser applications,” Lasers Surg. Med. 11, 331–340 (1991).
[CrossRef] [PubMed]

G. M. LaMuraglia, M. R. Prince, N. S. Nishioka, S. Obremski, R. Birngruber, “Optical properties of human arterial thrombus, vascular grafts, and sutures: implications for selective laser thrombus ablation,” IEEE J. Quantum Electron. 26, 2200–2206 (1990).
[CrossRef]

Bolin, F. P.

Case, K. M.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 8, p. 226.

Catchings, F. E.

Chance, B.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Chap. 8.

Cheong, W. F.

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Diffey, B. L.

B. L. Diffey, “A mathematical model for ultraviolet optics in skin,” Phys. Med. Biol. 28, 647–657 (1983).
[CrossRef] [PubMed]

Dlugos, C.

A. Vogel, C. Dlugos, R. Nuffer, R. Birngruber, “Optical properties of human sclera, and their consequences for transscleral laser applications,” Lasers Surg. Med. 11, 331–340 (1991).
[CrossRef] [PubMed]

Duntley, S. Q.

Egan, W. G.

Ference, R. J.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1986), Chap. 10, p. 289.

Flock, S. T.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in high scattering tissue—I: Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. BME-36, 1162–1168 (1989).
[CrossRef]

B. C. Wilson, M. S. Patterson, S. T. Flock, “Indirect versus direct techniques for the measurement of the optical properties of tissues,” Photochem. Photobiol. 46, 601–608 (1987).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

Gemert, M. C. J. V.

G. Yoon, A. J. Welch, M. Motamedi, M. C. J. V. Gemert, “Development and application of three-dimensional light distribution model for laser irradiated tissue,” IEEE J. Quantum Electron. QE-23, 1721–1733 (1987).
[CrossRef]

Hildebrand, F. B.

F. B. Hildebrand, Introduction to Numerical Analysis (Dover, New York, 1974), Chap. 8.

Hilgeman, T. W.

Irvine, W. M.

W. M. Irvine, “Multiple scattering in planetary atmospheres,” Icarus 25, 175–204 (1975).
[CrossRef]

Jacques, S. L.

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

S. L. Jacques, S. A. Prahl, “Modeling optical and thermal distributions in tissue during laser irradiation,” Lasers Surg. Med. 6, 494–503 (1987).
[CrossRef] [PubMed]

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller, D. H. Sliney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.ISO 5, 102–111 (1989).

Joseph, J. H.

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Kattawar, G. W.

Keijzer, M.

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller, D. H. Sliney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.ISO 5, 102–111 (1989).

Kubelka, P.

LaMuraglia, G. M.

G. M. LaMuraglia, M. R. Prince, N. S. Nishioka, S. Obremski, R. Birngruber, “Optical properties of human arterial thrombus, vascular grafts, and sutures: implications for selective laser thrombus ablation,” IEEE J. Quantum Electron. 26, 2200–2206 (1990).
[CrossRef]

Lathrop, A. L.

Liu, F.

K. M. Yoo, F. Liu, R. R. Alfano, “Angle and time resolved studies of backscattering of light from biological tissues,” in Laser–Tissue Interaction, S. L. Jacques, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1202, 260–271 (1990).

Mead, R.

J. A. Nelder, R. Mead, Comput. J. 7, 380 (1965).

Moes, C. J. M.

Motamedi, M.

G. Yoon, A. J. Welch, M. Motamedi, M. C. J. V. Gemert, “Development and application of three-dimensional light distribution model for laser irradiated tissue,” IEEE J. Quantum Electron. QE-23, 1721–1733 (1987).
[CrossRef]

Mudgett, P. S.

Nelder, J. A.

J. A. Nelder, R. Mead, Comput. J. 7, 380 (1965).

Nishioka, N. S.

G. M. LaMuraglia, M. R. Prince, N. S. Nishioka, S. Obremski, R. Birngruber, “Optical properties of human arterial thrombus, vascular grafts, and sutures: implications for selective laser thrombus ablation,” IEEE J. Quantum Electron. 26, 2200–2206 (1990).
[CrossRef]

Nuffer, R.

A. Vogel, C. Dlugos, R. Nuffer, R. Birngruber, “Optical properties of human sclera, and their consequences for transscleral laser applications,” Lasers Surg. Med. 11, 331–340 (1991).
[CrossRef] [PubMed]

Obremski, S.

G. M. LaMuraglia, M. R. Prince, N. S. Nishioka, S. Obremski, R. Birngruber, “Optical properties of human arterial thrombus, vascular grafts, and sutures: implications for selective laser thrombus ablation,” IEEE J. Quantum Electron. 26, 2200–2206 (1990).
[CrossRef]

Orchard, S. E.

Patterson, M. S.

M. S. Patterson, B. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue II. Optical properties of tissues and resulting fluence distributions,” Lasers Med. Sci. 6, 379–390 (1991).
[CrossRef]

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in high scattering tissue—I: Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. BME-36, 1162–1168 (1989).
[CrossRef]

M. S. 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. C. Wilson, M. S. Patterson, S. T. Flock, “Indirect versus direct techniques for the measurement of the optical properties of tissues,” Photochem. Photobiol. 46, 601–608 (1987).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

M. S. Patterson, E. Schwartz, B. C. Wilson, “Quantitative reflectance spectrophotometry for the noninvasive measurement of photosensitizer concentration in tissue during photodynamic therapy,” in Photodynamic Therapy: Mechanisms, T. J. Dougherty, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1065, 115–122 (1989).

Pickering, J. W.

Plass, G. N.

Prahl, S. A.

J. W. Pickering, S. A. Prahl, N. van Wieringen, J. B. Beek, H. J. C. M. Sterenborg, M. J. C. van Gemert, “Double-integrating-sphere system for measuring optical properties of tissue,” Appl. Opt. 32, 399–410 (1993).
[CrossRef] [PubMed]

J. W. Pickering, C. J. M. Moes, H. J. C. M. Sterenborg, S. A. Prahl, M. J. C. van Gemert, “Two integrating spheres with an intervening scattering sample,” J. Opt. Soc. Am. A 9, 621–631 (1992).
[CrossRef]

S. A. Prahl, I. A. Vitkin, B. C. Wilson, R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203–1217 (1992).
[CrossRef] [PubMed]

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

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, 2250–2255 (1989).
[CrossRef] [PubMed]

S. L. Jacques, S. A. Prahl, “Modeling optical and thermal distributions in tissue during laser irradiation,” Lasers Surg. Med. 6, 494–503 (1987).
[CrossRef] [PubMed]

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller, D. H. Sliney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.ISO 5, 102–111 (1989).

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1986), Chap. 10, p. 289.

Preuss, L. E.

Priesendorfer, R.

R. Priesendorfer, Hydrologie Optics (U.S. Department of Commerce, Washington, D.C., 1976), Vol. 1.

Prince, M. R.

G. M. LaMuraglia, M. R. Prince, N. S. Nishioka, S. Obremski, R. Birngruber, “Optical properties of human arterial thrombus, vascular grafts, and sutures: implications for selective laser thrombus ablation,” IEEE J. Quantum Electron. 26, 2200–2206 (1990).
[CrossRef]

Reichman, J.

Richards, L. W.

Schwartz, E.

M. S. Patterson, E. Schwartz, B. C. Wilson, “Quantitative reflectance spectrophotometry for the noninvasive measurement of photosensitizer concentration in tissue during photodynamic therapy,” in Photodynamic Therapy: Mechanisms, T. J. Dougherty, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1065, 115–122 (1989).

Star, W. M.

M. J. C. van Gemert, W. M. Star, “Relations between the Kubelka–Munk and the transport equation models for anisotropic scattering,” Lasers Life Sci. 1, 287–298 (1987).

Sterenborg, H. J. C. M.

Taylor, R. C.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1986), Chap. 10, p. 289.

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vol. 1.

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

H. C. van de Hulst, A New Look at Multiple Scattering, Tech. Rep. (NASA Institute for Space Studies, New York, 1962).

van Gemert, M. J. C.

van Wieringen, N.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1986), Chap. 10, p. 289.

Vitkin, I. A.

S. A. Prahl, I. A. Vitkin, B. C. Wilson, R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203–1217 (1992).
[CrossRef] [PubMed]

Vogel, A.

A. Vogel, C. Dlugos, R. Nuffer, R. Birngruber, “Optical properties of human sclera, and their consequences for transscleral laser applications,” Lasers Surg. Med. 11, 331–340 (1991).
[CrossRef] [PubMed]

Weinman, J. A.

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Welch, A. J.

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

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, 2250–2255 (1989).
[CrossRef] [PubMed]

G. Yoon, A. J. Welch, M. Motamedi, M. C. J. V. Gemert, “Development and application of three-dimensional light distribution model for laser irradiated tissue,” IEEE J. Quantum Electron. QE-23, 1721–1733 (1987).
[CrossRef]

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller, D. H. Sliney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.ISO 5, 102–111 (1989).

Wilson, B. C.

S. A. Prahl, I. A. Vitkin, B. C. Wilson, R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203–1217 (1992).
[CrossRef] [PubMed]

M. S. Patterson, B. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue II. Optical properties of tissues and resulting fluence distributions,” Lasers Med. Sci. 6, 379–390 (1991).
[CrossRef]

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in high scattering tissue—I: Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. BME-36, 1162–1168 (1989).
[CrossRef]

M. S. 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. C. Wilson, M. S. Patterson, S. T. Flock, “Indirect versus direct techniques for the measurement of the optical properties of tissues,” Photochem. Photobiol. 46, 601–608 (1987).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

M. S. Patterson, E. Schwartz, B. C. Wilson, “Quantitative reflectance spectrophotometry for the noninvasive measurement of photosensitizer concentration in tissue during photodynamic therapy,” in Photodynamic Therapy: Mechanisms, T. J. Dougherty, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1065, 115–122 (1989).

Wing, G. M.

R. Bellman, G. M. Wing, An Introduction to Invariant Imbedding (Wiley, New York, 1975).

Wiscombe, W. J.

W. J. Wiscombe, “Doubling initialization revisited,” J. Quant. Spectrosc. Radiat. Transfer 18, 245–248 (1977).
[CrossRef]

W. J. Wiscombe, “The delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. 34, 1408–1422 (1977).
[CrossRef]

W. J. Wiscombe, “On initialization, error and flux conservation in the doubling method,” J. Quant. Spectrosc. Radiative Transfer 16, 637–658 (1976).
[CrossRef]

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

Wyman, D. R.

M. S. Patterson, B. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue II. Optical properties of tissues and resulting fluence distributions,” Lasers Med. Sci. 6, 379–390 (1991).
[CrossRef]

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in high scattering tissue—I: Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. BME-36, 1162–1168 (1989).
[CrossRef]

Yoo, K. M.

K. M. Yoo, F. Liu, R. R. Alfano, “Angle and time resolved studies of backscattering of light from biological tissues,” in Laser–Tissue Interaction, S. L. Jacques, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1202, 260–271 (1990).

Yoon, G.

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, 2250–2255 (1989).
[CrossRef] [PubMed]

G. Yoon, A. J. Welch, M. Motamedi, M. C. J. V. Gemert, “Development and application of three-dimensional light distribution model for laser irradiated tissue,” IEEE J. Quantum Electron. QE-23, 1721–1733 (1987).
[CrossRef]

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 8, p. 226.

Appl. Opt. (8)

G. N. Plass, G. W. Kattawar, F. E. Catchings, “Matrix operator theory of radiative transfer. 1: Rayleigh scattering,” Appl. Opt. 12, 314–329 (1973).
[CrossRef] [PubMed]

J. Reichman, “Determination of absorption and scattering coefficients for nonhomogeneous media. 1: Theory,” Appl. Opt. 12, 1811–1815 (1973).
[CrossRef] [PubMed]

W. G. Egan, T. W. Hilgeman, J. Reichman, “Determination of absorption and scattering coefficients for nonhomogeneous media. 2: Experiment,” Appl. Opt. 12, 1816–1823 (1973).
[CrossRef] [PubMed]

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, 2250–2255 (1989).
[CrossRef] [PubMed]

F. P. Bolin, L. E. Preuss, R. C. Taylor, R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” Appl. Opt. 28, 2297–2303 (1989).
[CrossRef] [PubMed]

M. S. 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]

J. W. Pickering, S. A. Prahl, N. van Wieringen, J. B. Beek, H. J. C. M. Sterenborg, M. J. C. van Gemert, “Double-integrating-sphere system for measuring optical properties of tissue,” Appl. Opt. 32, 399–410 (1993).
[CrossRef] [PubMed]

P. S. Mudgett, L. W. Richards, “Multiple scattering calculations for technology,” Appl. Opt. 10, 1485–1502 (1971).
[CrossRef] [PubMed]

Comput. J. (1)

J. A. Nelder, R. Mead, Comput. J. 7, 380 (1965).

Icarus (1)

W. M. Irvine, “Multiple scattering in planetary atmospheres,” Icarus 25, 175–204 (1975).
[CrossRef]

IEEE J. Quantum Electron. (3)

G. Yoon, A. J. Welch, M. Motamedi, M. C. J. V. Gemert, “Development and application of three-dimensional light distribution model for laser irradiated tissue,” IEEE J. Quantum Electron. QE-23, 1721–1733 (1987).
[CrossRef]

W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

G. M. LaMuraglia, M. R. Prince, N. S. Nishioka, S. Obremski, R. Birngruber, “Optical properties of human arterial thrombus, vascular grafts, and sutures: implications for selective laser thrombus ablation,” IEEE J. Quantum Electron. 26, 2200–2206 (1990).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modeling of light propagation in high scattering tissue—I: Model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. BME-36, 1162–1168 (1989).
[CrossRef]

J. Atmos. Sci. (2)

W. J. Wiscombe, “The delta-M method: rapid yet accurate radiative flux calculations for strongly asymmetric phase functions,” J. Atmos. Sci. 34, 1408–1422 (1977).
[CrossRef]

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The delta-Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
[CrossRef]

J. Opt. Am. (1)

P. Kubelka, “New contributions to the optics of intensely light-scattering materials. Part I,” J. Opt. Am. 38, 448–457 (1948).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Radiat. Transfer (1)

W. J. Wiscombe, “Doubling initialization revisited,” J. Quant. Spectrosc. Radiat. Transfer 18, 245–248 (1977).
[CrossRef]

J. Quant. Spectrosc. Radiative Transfer (1)

W. J. Wiscombe, “On initialization, error and flux conservation in the doubling method,” J. Quant. Spectrosc. Radiative Transfer 16, 637–658 (1976).
[CrossRef]

Lasers Life Sci. (2)

M. J. C. van Gemert, W. M. Star, “Relations between the Kubelka–Munk and the transport equation models for anisotropic scattering,” Lasers Life Sci. 1, 287–298 (1987).

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

Lasers Med. Sci. (1)

M. S. Patterson, B. C. Wilson, D. R. Wyman, “The propagation of optical radiation in tissue II. Optical properties of tissues and resulting fluence distributions,” Lasers Med. Sci. 6, 379–390 (1991).
[CrossRef]

Lasers Surg. Med. (2)

A. Vogel, C. Dlugos, R. Nuffer, R. Birngruber, “Optical properties of human sclera, and their consequences for transscleral laser applications,” Lasers Surg. Med. 11, 331–340 (1991).
[CrossRef] [PubMed]

S. L. Jacques, S. A. Prahl, “Modeling optical and thermal distributions in tissue during laser irradiation,” Lasers Surg. Med. 6, 494–503 (1987).
[CrossRef] [PubMed]

Med. Phys. (2)

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

Photochem. Photobiol. (1)

B. C. Wilson, M. S. Patterson, S. T. Flock, “Indirect versus direct techniques for the measurement of the optical properties of tissues,” Photochem. Photobiol. 46, 601–608 (1987).
[CrossRef] [PubMed]

Phys. Med. Biol. (2)

B. L. Diffey, “A mathematical model for ultraviolet optics in skin,” Phys. Med. Biol. 28, 647–657 (1983).
[CrossRef] [PubMed]

S. A. Prahl, I. A. Vitkin, B. C. Wilson, R. R. Anderson, “Determination of optical properties of turbid media using pulsed photothermal radiometry,” Phys. Med. Biol. 37, 1203–1217 (1992).
[CrossRef] [PubMed]

Other (13)

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980), Vol. 1.

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller, D. H. Sliney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.ISO 5, 102–111 (1989).

R. Bellman, G. M. Wing, An Introduction to Invariant Imbedding (Wiley, New York, 1975).

J. T. Atkins, “Optical properties of turbid materials,” in The Biologic Effects of Ultraviolet Radiation (With Emphasis on the Skin), F. Urbach, ed. (Pergamon, London, 1969), pp. 141–149.

M. S. Patterson, E. Schwartz, B. C. Wilson, “Quantitative reflectance spectrophotometry for the noninvasive measurement of photosensitizer concentration in tissue during photodynamic therapy,” in Photodynamic Therapy: Mechanisms, T. J. Dougherty, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1065, 115–122 (1989).

K. M. Yoo, F. Liu, R. R. Alfano, “Angle and time resolved studies of backscattering of light from biological tissues,” in Laser–Tissue Interaction, S. L. Jacques, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1202, 260–271 (1990).

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960), Chap. 8.

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

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 8, p. 226.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, New York, 1986), Chap. 10, p. 289.

F. B. Hildebrand, Introduction to Numerical Analysis (Dover, New York, 1974), Chap. 8.

H. C. van de Hulst, A New Look at Multiple Scattering, Tech. Rep. (NASA Institute for Space Studies, New York, 1962).

R. Priesendorfer, Hydrologie Optics (U.S. Department of Commerce, Washington, D.C., 1976), Vol. 1.

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Figures (7)

Fig. 1
Fig. 1

Total reflection and total transmission of an index-matched slab (n = 1) as a function of the albedo a and anisotropy g for a fixed unscattered transmission value of 10%. Each point on the bold (a, g) grid corresponds to a unique (RT, TT) pair.

Fig. 2
Fig. 2

Total reflection and total transmission of a slab as a function of the reduced albedo a′ and reduced optical thickness τ′. Isotropic scattering is assumed as well as an index of refraction mismatch (n = 1.4). Each point on the (a′, τ′) grid corresponds to a unique (RT, TT) pair.

Fig. 3
Fig. 3

Maximum error in the calculated reduced optical thickness τ′ for any anisotropy and any albedo by using the diffusion approximation and the IAD method with four and eight quadrature points.

Fig. 4
Fig. 4

Maximum relative error in the calculated scattering coefficient as a function of the reduced albedo a′, the reduced optical thickness τ′, and the scattering anisotropy g by using the IAD method with four quadrature points and mismatched boundaries n = 1.4.

Fig. 5
Fig. 5

Relative error in the calculated absorption coefficient as a function of reflection and transmission. The IAD method was used with four quadrature points, and the slab had mismatched boundaries.

Fig. 6
Fig. 6

Relative error in the calculated scattering coefficient as a function of reflection and transmission. The IAD method was used with four quadrature points, and the slab had mismatched boundaries.

Fig. 7
Fig. 7

Relative error in the calculated scattering anisotropy as a function of reflection and transmission. The IAD method was used with four quadrature points, and the slab had mismatched boundaries.

Tables (1)

Tables Icon

Table 1 Maximum Errors in the Calculated Reduced Optical Thickness τ′ for any Albedo or Anisotropy as a Function of the True Reduced Optical Thicknessa

Equations (21)

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

a = μ s μ s + μ a , τ = d ( μ s + μ a ) ,
4 π p ( ν ) d ω = 2 π 1 1 p ( ν ) d ν = 1 ,
p ( ν ) = 1 4 π 1 g 2 ( 1 + g 2 2 g ν ) 3 / 2 .
g = 4 π p ( ν ) ν d ω = 2 π 1 1 p ( ν ) ν d ν .
R bndry ( ν i , ν j ) = r ( ν i ) 2 ν i δ i j , T bndry ( ν i , ν j ) = 1 r ( ν i ) 2 ν i ( n 1 n 0 ) 2 δ i j ,
n g ( 1 ν g 2 ) 1 / 2 = n s ( 1 ν i 2 ) 1 / 2 , ν i < ν c ,
r ( ν ) = { r 1 ( ν i ) + r g ( ν g ) 2 r 1 ( ν i ) r g ( ν g ) 1 r 1 ( ν i ) r g ( ν g ) if ν i < ν c 1 if ν i ν c .
0 1 f ( ν , ν ) d ν = k = 1 N H k f ( x k ) .
0 1 A ( ν , ν ) B ( ν , ν ) d ν = 0 ν c A ( ν , ν ) B ( ν , ν ) d ν + ν c 1 A ( ν , ν ) B ( ν , ν ) d ν .
T C = ( 1 r s 1 ) ( 1 r s 2 ) exp ( τ ) 1 r s 1 r s 2 exp ( 2 τ ) ,
M ( a , τ , g ) = | R calc R meas | R meas + 10 6 + | T calc T meas | T meas + 10 6 .
a comp = 2 a 1 a ( 1 a ) .
g comp = g 1 + | g | .
b comp = ln ( τ ) .
a = a ( 1 g ) 1 a g , τ = ( 1 a g ) τ .
a = a 1 g + a g , τ = τ + a τ g 1 g .
a = { 1 ( 1 4 R d T T 1 T T ) 2 if R d 1 T T < 0 . 1 1 4 9 ( 1 R d T T 1 T T ) 2 if R d 1 T T 0 . 1 .
τ = { ln T T ln ( 0 . 05 ) ln R T if R d 0 . 1 2 1 + 5 ( R d + T T ) if R d > 0 . 1 .
P r P = γ 1 [ R c ( 1 γ 2 R d ) + γ 3 R c d ] ( 1 γ 4 R d ) + γ 1 γ 3 γ 4 T d ( T c d + γ 5 T C ) ( 1 γ 4 R d ) ( 1 γ 6 R d ) γ 4 γ 6 T d 2 ,
Δ a = | a true a calc | .
Δ μ a μ a true = 100 [ 1 ( 1 a calc ) τ calc ( 1 a true ) τ true ] , Δ μ s μ s true = 100 ( 1 a calc τ calc a true τ true ) .

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