Abstract

The rigorous method of discrete ordinates was used to evaluate the effects of anisotropic scattering and optical discontinuity at the boundaries on light and temperature distribution in tissue. The influence of optical parameters of tissue on its thermal response was examined by using a finite element solution of the heat conduction equation. Calculations were performed for wide ranges of scattering albedo, the anisotropy factor, as well as interface reflectivities. This study shows that the presence of an optical discontinuity due to an air–tissue interface forces the maximum peak intensity to move from subsurface to the surface for tissue with high scattering albedo, which leads to a higher fluence rate in the near surface region. Temperature field calculations show a higher subsurface temperature for a highly scattering medium during tissue coagulation. Neglecting the anisotropic properties of tissue as well as the optical discontinuity at the boundaries would result in considerable error in the calculated temperature rises. Additionally the accuracy of the photon diffusion theory for predicting light and temperature distribution near the tissue surface is examined.

© 1989 Optical Society of America

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References

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  1. S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).
  2. M.C.J. van Gemert, J.P.H. Henning, “Model Approach to Laser Coagulation of Dermal Vascular Lesions,” Arch. Dermatol. Res. 270, 429–439 (1981).
    [CrossRef] [PubMed]
  3. S. Wan, R. R. Anderson, X. X. Parrish, “Analytical Modeling for the Optical Properties of the Skin with In Vitro and In Vivo Applications,” Photochem. Photobiol. 34, 493–499 (1981).
    [PubMed]
  4. P. Kubelka, “New Contributions to the Optics of Intensely Light Scattering Materials, Part I,” J. Opt. Soc. Am. 38, 448–457 (1948).
    [CrossRef] [PubMed]
  5. L. O. Svaasand, D. R. Doiron, A. E. Profio, “Light Distribution in Tissue During Photoradiation Therapy,” Institute for Physics and Imaging Sci. Rep. USC-IPIS 900-02, U. Southern California (1981).
  6. A. J. Welch, G. Yoon, M. J. C. van Gemert, “Practical Models for Light Distribution in Laser-Irradiated Tissue,” Lasers Surg. Med. 6, 488–493 (1987).
    [CrossRef] [PubMed]
  7. S. L. Jacques, S. A. Prahl, “Modeling Optical and Thermal Distributions in Tissue During Laser Irradiation,” Laser Surg. Med. 6, 494–503 (1987).
    [CrossRef]
  8. S. Takatani, M. D. Graham, “Theoretical Analysis of Diffuse Reflectance from a 2-Layer Tissue Model,” IEEE Trans. Biomed. Eng. BME-26, 656–664 (1979).
    [CrossRef]
  9. G. Yoon, A. J. Welch, M. Motamedia, M. J. C. van Gemert, “Development and Application of Three Dimensional Light Distribution Model for Laser Irradiated Tissue,” IEEE J. Quantum Electron. QE-23, 1721–1733 (1987).
    [CrossRef]
  10. A. Zardecki, S. A. W. Gestel, J. F. Embury, “Application of the 2-D Discrete-Ordinates Method to Multiple Scattering of Laser Radiation,” Appl. Opt. 22, 1346–1353 (1983).
    [CrossRef] [PubMed]
  11. K. Stamnes, P. Conklin, “A New Multi-Layer Discrete Ordinate Approach to Radiative Transfer in Vertically Inhomogeneous Atmosphere,” J. Quant. Spectrosc Radiat. Transfer 31, 273–282. (1984).
    [CrossRef]
  12. K. Liou, “A Numerical Experiment on Chandrasekhar’s Discrete Ordinate Method for Radiative Heat Transfer: Applications to Cloudy and Hazy Atmospheres,” J. Atmos. Sci. 31, 1303–1326 (1973).
    [CrossRef]
  13. W. G. Houf, E. P. Incropera, “An Assessment of Techniques for Predicting Radiation Transfer in Aqueous Media,” J. Quant. Spectrosc. Radiat. Transfer 23, 101–115 (1980).
    [CrossRef]
  14. G. A. Korn, M. T. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961).
  15. H. H. Michels, “Abscissas and Weight Coefficients for Lobatto Quadrature,” Math. Comput. 17, 237–244 (1963).
    [CrossRef]
  16. L. G. Henyey, J. L. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. 88, 70–83 (1941).
    [CrossRef]
  17. S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular Dependence of He-Ne Laser Light Scattering by Human Dermis,” Lasers Life Sci. 4, 309–333 (1987).
  18. W. J. Wiscombe, “On Initialization, Error, and Flux Conservation in the Doubling Method,” J. Quant. Spectrosc. Radiat. Transfer 15, 637–658 (1975).
  19. F. P. Incropera, W. G. Houf, “A Three Flux Method for Predicting Radiative Transfer in Aqueous Suspensions,” J. Heat Transfer 101, 496–501 (1979).
    [CrossRef]
  20. E. H. Wissler, “An Analysis of Chorioretinal Thermal Response to Intense Light Exposure,” IEEE Trans. Biomed. Eng. BME-XX, 207–215 (1976).
    [CrossRef]
  21. A. N. Takata, L. Zaneveld, W. Richter, “Laser-Induced Thermal Damage in Skin,” Report SAM-TR-77-38, USAF School of Aerospace Medicine, Brooks AFB, TX (1977).
  22. M. I. C. van Gemert, G. Schets, E. G. Stassen, J. J. Bonnier, “Modeling (Coronary) Laser Angioplasty,” Lasers Surg. Med. 5, 219–234 (1985).
    [CrossRef] [PubMed]
  23. I. Gladwell, R. Wait, A Survey of Numerical Methods for Partial Differential Equations (Clarendon, Oxford, 1985).
  24. J. W. Valvano, B. Chitsabesan, “Thermal Conductivity and Diffusivity of Arterial Wall and Atherosclerotic Plaque,” Lasers Life Sci. 1, 3,219–229 (1987).

1987 (5)

A. J. Welch, G. Yoon, M. J. C. van Gemert, “Practical Models for Light Distribution in Laser-Irradiated Tissue,” Lasers Surg. Med. 6, 488–493 (1987).
[CrossRef] [PubMed]

S. L. Jacques, S. A. Prahl, “Modeling Optical and Thermal Distributions in Tissue During Laser Irradiation,” Laser Surg. Med. 6, 494–503 (1987).
[CrossRef]

G. Yoon, A. J. Welch, M. Motamedia, M. J. C. van 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, C. A. Alter, S. A. Prahl, “Angular Dependence of He-Ne Laser Light Scattering by Human Dermis,” Lasers Life Sci. 4, 309–333 (1987).

J. W. Valvano, B. Chitsabesan, “Thermal Conductivity and Diffusivity of Arterial Wall and Atherosclerotic Plaque,” Lasers Life Sci. 1, 3,219–229 (1987).

1985 (1)

M. I. C. van Gemert, G. Schets, E. G. Stassen, J. J. Bonnier, “Modeling (Coronary) Laser Angioplasty,” Lasers Surg. Med. 5, 219–234 (1985).
[CrossRef] [PubMed]

1984 (1)

K. Stamnes, P. Conklin, “A New Multi-Layer Discrete Ordinate Approach to Radiative Transfer in Vertically Inhomogeneous Atmosphere,” J. Quant. Spectrosc Radiat. Transfer 31, 273–282. (1984).
[CrossRef]

1983 (1)

1981 (2)

M.C.J. van Gemert, J.P.H. Henning, “Model Approach to Laser Coagulation of Dermal Vascular Lesions,” Arch. Dermatol. Res. 270, 429–439 (1981).
[CrossRef] [PubMed]

S. Wan, R. R. Anderson, X. X. Parrish, “Analytical Modeling for the Optical Properties of the Skin with In Vitro and In Vivo Applications,” Photochem. Photobiol. 34, 493–499 (1981).
[PubMed]

1980 (1)

W. G. Houf, E. P. Incropera, “An Assessment of Techniques for Predicting Radiation Transfer in Aqueous Media,” J. Quant. Spectrosc. Radiat. Transfer 23, 101–115 (1980).
[CrossRef]

1979 (2)

F. P. Incropera, W. G. Houf, “A Three Flux Method for Predicting Radiative Transfer in Aqueous Suspensions,” J. Heat Transfer 101, 496–501 (1979).
[CrossRef]

S. Takatani, M. D. Graham, “Theoretical Analysis of Diffuse Reflectance from a 2-Layer Tissue Model,” IEEE Trans. Biomed. Eng. BME-26, 656–664 (1979).
[CrossRef]

1976 (1)

E. H. Wissler, “An Analysis of Chorioretinal Thermal Response to Intense Light Exposure,” IEEE Trans. Biomed. Eng. BME-XX, 207–215 (1976).
[CrossRef]

1975 (1)

W. J. Wiscombe, “On Initialization, Error, and Flux Conservation in the Doubling Method,” J. Quant. Spectrosc. Radiat. Transfer 15, 637–658 (1975).

1973 (1)

K. Liou, “A Numerical Experiment on Chandrasekhar’s Discrete Ordinate Method for Radiative Heat Transfer: Applications to Cloudy and Hazy Atmospheres,” J. Atmos. Sci. 31, 1303–1326 (1973).
[CrossRef]

1963 (1)

H. H. Michels, “Abscissas and Weight Coefficients for Lobatto Quadrature,” Math. Comput. 17, 237–244 (1963).
[CrossRef]

1948 (1)

1941 (1)

L. G. Henyey, J. L. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. 88, 70–83 (1941).
[CrossRef]

Alter, C. A.

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular Dependence of He-Ne Laser Light Scattering by Human Dermis,” Lasers Life Sci. 4, 309–333 (1987).

Anderson, R. R.

S. Wan, R. R. Anderson, X. X. Parrish, “Analytical Modeling for the Optical Properties of the Skin with In Vitro and In Vivo Applications,” Photochem. Photobiol. 34, 493–499 (1981).
[PubMed]

Bonnier, J. J.

M. I. C. van Gemert, G. Schets, E. G. Stassen, J. J. Bonnier, “Modeling (Coronary) Laser Angioplasty,” Lasers Surg. Med. 5, 219–234 (1985).
[CrossRef] [PubMed]

Chandrasekhar, S.

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

Chitsabesan, B.

J. W. Valvano, B. Chitsabesan, “Thermal Conductivity and Diffusivity of Arterial Wall and Atherosclerotic Plaque,” Lasers Life Sci. 1, 3,219–229 (1987).

Conklin, P.

K. Stamnes, P. Conklin, “A New Multi-Layer Discrete Ordinate Approach to Radiative Transfer in Vertically Inhomogeneous Atmosphere,” J. Quant. Spectrosc Radiat. Transfer 31, 273–282. (1984).
[CrossRef]

Doiron, D. R.

L. O. Svaasand, D. R. Doiron, A. E. Profio, “Light Distribution in Tissue During Photoradiation Therapy,” Institute for Physics and Imaging Sci. Rep. USC-IPIS 900-02, U. Southern California (1981).

Embury, J. F.

Gestel, S. A. W.

Gladwell, I.

I. Gladwell, R. Wait, A Survey of Numerical Methods for Partial Differential Equations (Clarendon, Oxford, 1985).

Graham, M. D.

S. Takatani, M. D. Graham, “Theoretical Analysis of Diffuse Reflectance from a 2-Layer Tissue Model,” IEEE Trans. Biomed. Eng. BME-26, 656–664 (1979).
[CrossRef]

Greenstein, J. L.

L. G. Henyey, J. L. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. 88, 70–83 (1941).
[CrossRef]

Henning, J.P.H.

M.C.J. van Gemert, J.P.H. Henning, “Model Approach to Laser Coagulation of Dermal Vascular Lesions,” Arch. Dermatol. Res. 270, 429–439 (1981).
[CrossRef] [PubMed]

Henyey, L. G.

L. G. Henyey, J. L. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. 88, 70–83 (1941).
[CrossRef]

Houf, W. G.

W. G. Houf, E. P. Incropera, “An Assessment of Techniques for Predicting Radiation Transfer in Aqueous Media,” J. Quant. Spectrosc. Radiat. Transfer 23, 101–115 (1980).
[CrossRef]

F. P. Incropera, W. G. Houf, “A Three Flux Method for Predicting Radiative Transfer in Aqueous Suspensions,” J. Heat Transfer 101, 496–501 (1979).
[CrossRef]

Incropera, E. P.

W. G. Houf, E. P. Incropera, “An Assessment of Techniques for Predicting Radiation Transfer in Aqueous Media,” J. Quant. Spectrosc. Radiat. Transfer 23, 101–115 (1980).
[CrossRef]

Incropera, F. P.

F. P. Incropera, W. G. Houf, “A Three Flux Method for Predicting Radiative Transfer in Aqueous Suspensions,” J. Heat Transfer 101, 496–501 (1979).
[CrossRef]

Jacques, S. L.

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular Dependence of He-Ne Laser Light Scattering by Human Dermis,” Lasers Life Sci. 4, 309–333 (1987).

S. L. Jacques, S. A. Prahl, “Modeling Optical and Thermal Distributions in Tissue During Laser Irradiation,” Laser Surg. Med. 6, 494–503 (1987).
[CrossRef]

Korn, G. A.

G. A. Korn, M. T. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961).

Korn, M. T.

G. A. Korn, M. T. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961).

Kubelka, P.

Liou, K.

K. Liou, “A Numerical Experiment on Chandrasekhar’s Discrete Ordinate Method for Radiative Heat Transfer: Applications to Cloudy and Hazy Atmospheres,” J. Atmos. Sci. 31, 1303–1326 (1973).
[CrossRef]

Michels, H. H.

H. H. Michels, “Abscissas and Weight Coefficients for Lobatto Quadrature,” Math. Comput. 17, 237–244 (1963).
[CrossRef]

Motamedia, M.

G. Yoon, A. J. Welch, M. Motamedia, M. J. C. van Gemert, “Development and Application of Three Dimensional Light Distribution Model for Laser Irradiated Tissue,” IEEE J. Quantum Electron. QE-23, 1721–1733 (1987).
[CrossRef]

Parrish, X. X.

S. Wan, R. R. Anderson, X. X. Parrish, “Analytical Modeling for the Optical Properties of the Skin with In Vitro and In Vivo Applications,” Photochem. Photobiol. 34, 493–499 (1981).
[PubMed]

Prahl, S. A.

S. L. Jacques, S. A. Prahl, “Modeling Optical and Thermal Distributions in Tissue During Laser Irradiation,” Laser Surg. Med. 6, 494–503 (1987).
[CrossRef]

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular Dependence of He-Ne Laser Light Scattering by Human Dermis,” Lasers Life Sci. 4, 309–333 (1987).

Profio, A. E.

L. O. Svaasand, D. R. Doiron, A. E. Profio, “Light Distribution in Tissue During Photoradiation Therapy,” Institute for Physics and Imaging Sci. Rep. USC-IPIS 900-02, U. Southern California (1981).

Richter, W.

A. N. Takata, L. Zaneveld, W. Richter, “Laser-Induced Thermal Damage in Skin,” Report SAM-TR-77-38, USAF School of Aerospace Medicine, Brooks AFB, TX (1977).

Schets, G.

M. I. C. van Gemert, G. Schets, E. G. Stassen, J. J. Bonnier, “Modeling (Coronary) Laser Angioplasty,” Lasers Surg. Med. 5, 219–234 (1985).
[CrossRef] [PubMed]

Stamnes, K.

K. Stamnes, P. Conklin, “A New Multi-Layer Discrete Ordinate Approach to Radiative Transfer in Vertically Inhomogeneous Atmosphere,” J. Quant. Spectrosc Radiat. Transfer 31, 273–282. (1984).
[CrossRef]

Stassen, E. G.

M. I. C. van Gemert, G. Schets, E. G. Stassen, J. J. Bonnier, “Modeling (Coronary) Laser Angioplasty,” Lasers Surg. Med. 5, 219–234 (1985).
[CrossRef] [PubMed]

Svaasand, L. O.

L. O. Svaasand, D. R. Doiron, A. E. Profio, “Light Distribution in Tissue During Photoradiation Therapy,” Institute for Physics and Imaging Sci. Rep. USC-IPIS 900-02, U. Southern California (1981).

Takata, A. N.

A. N. Takata, L. Zaneveld, W. Richter, “Laser-Induced Thermal Damage in Skin,” Report SAM-TR-77-38, USAF School of Aerospace Medicine, Brooks AFB, TX (1977).

Takatani, S.

S. Takatani, M. D. Graham, “Theoretical Analysis of Diffuse Reflectance from a 2-Layer Tissue Model,” IEEE Trans. Biomed. Eng. BME-26, 656–664 (1979).
[CrossRef]

Valvano, J. W.

J. W. Valvano, B. Chitsabesan, “Thermal Conductivity and Diffusivity of Arterial Wall and Atherosclerotic Plaque,” Lasers Life Sci. 1, 3,219–229 (1987).

van Gemert, M. I. C.

M. I. C. van Gemert, G. Schets, E. G. Stassen, J. J. Bonnier, “Modeling (Coronary) Laser Angioplasty,” Lasers Surg. Med. 5, 219–234 (1985).
[CrossRef] [PubMed]

van Gemert, M. J. C.

G. Yoon, A. J. Welch, M. Motamedia, M. J. C. van Gemert, “Development and Application of Three Dimensional Light Distribution Model for Laser Irradiated Tissue,” IEEE J. Quantum Electron. QE-23, 1721–1733 (1987).
[CrossRef]

A. J. Welch, G. Yoon, M. J. C. van Gemert, “Practical Models for Light Distribution in Laser-Irradiated Tissue,” Lasers Surg. Med. 6, 488–493 (1987).
[CrossRef] [PubMed]

van Gemert, M.C.J.

M.C.J. van Gemert, J.P.H. Henning, “Model Approach to Laser Coagulation of Dermal Vascular Lesions,” Arch. Dermatol. Res. 270, 429–439 (1981).
[CrossRef] [PubMed]

Wait, R.

I. Gladwell, R. Wait, A Survey of Numerical Methods for Partial Differential Equations (Clarendon, Oxford, 1985).

Wan, S.

S. Wan, R. R. Anderson, X. X. Parrish, “Analytical Modeling for the Optical Properties of the Skin with In Vitro and In Vivo Applications,” Photochem. Photobiol. 34, 493–499 (1981).
[PubMed]

Welch, A. J.

A. J. Welch, G. Yoon, M. J. C. van Gemert, “Practical Models for Light Distribution in Laser-Irradiated Tissue,” Lasers Surg. Med. 6, 488–493 (1987).
[CrossRef] [PubMed]

G. Yoon, A. J. Welch, M. Motamedia, M. J. C. van Gemert, “Development and Application of Three Dimensional Light Distribution Model for Laser Irradiated Tissue,” IEEE J. Quantum Electron. QE-23, 1721–1733 (1987).
[CrossRef]

Wiscombe, W. J.

W. J. Wiscombe, “On Initialization, Error, and Flux Conservation in the Doubling Method,” J. Quant. Spectrosc. Radiat. Transfer 15, 637–658 (1975).

Wissler, E. H.

E. H. Wissler, “An Analysis of Chorioretinal Thermal Response to Intense Light Exposure,” IEEE Trans. Biomed. Eng. BME-XX, 207–215 (1976).
[CrossRef]

Yoon, G.

A. J. Welch, G. Yoon, M. J. C. van Gemert, “Practical Models for Light Distribution in Laser-Irradiated Tissue,” Lasers Surg. Med. 6, 488–493 (1987).
[CrossRef] [PubMed]

G. Yoon, A. J. Welch, M. Motamedia, M. J. C. van Gemert, “Development and Application of Three Dimensional Light Distribution Model for Laser Irradiated Tissue,” IEEE J. Quantum Electron. QE-23, 1721–1733 (1987).
[CrossRef]

Zaneveld, L.

A. N. Takata, L. Zaneveld, W. Richter, “Laser-Induced Thermal Damage in Skin,” Report SAM-TR-77-38, USAF School of Aerospace Medicine, Brooks AFB, TX (1977).

Zardecki, A.

Appl. Opt. (1)

Arch. Dermatol. Res. (1)

M.C.J. van Gemert, J.P.H. Henning, “Model Approach to Laser Coagulation of Dermal Vascular Lesions,” Arch. Dermatol. Res. 270, 429–439 (1981).
[CrossRef] [PubMed]

Astrophys. J. (1)

L. G. Henyey, J. L. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. 88, 70–83 (1941).
[CrossRef]

IEEE J. Quantum Electron. (1)

G. Yoon, A. J. Welch, M. Motamedia, M. J. C. van Gemert, “Development and Application of Three Dimensional Light Distribution Model for Laser Irradiated Tissue,” IEEE J. Quantum Electron. QE-23, 1721–1733 (1987).
[CrossRef]

IEEE Trans. Biomed. Eng. (2)

S. Takatani, M. D. Graham, “Theoretical Analysis of Diffuse Reflectance from a 2-Layer Tissue Model,” IEEE Trans. Biomed. Eng. BME-26, 656–664 (1979).
[CrossRef]

E. H. Wissler, “An Analysis of Chorioretinal Thermal Response to Intense Light Exposure,” IEEE Trans. Biomed. Eng. BME-XX, 207–215 (1976).
[CrossRef]

J. Atmos. Sci. (1)

K. Liou, “A Numerical Experiment on Chandrasekhar’s Discrete Ordinate Method for Radiative Heat Transfer: Applications to Cloudy and Hazy Atmospheres,” J. Atmos. Sci. 31, 1303–1326 (1973).
[CrossRef]

J. Heat Transfer (1)

F. P. Incropera, W. G. Houf, “A Three Flux Method for Predicting Radiative Transfer in Aqueous Suspensions,” J. Heat Transfer 101, 496–501 (1979).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Quant. Spectrosc Radiat. Transfer (1)

K. Stamnes, P. Conklin, “A New Multi-Layer Discrete Ordinate Approach to Radiative Transfer in Vertically Inhomogeneous Atmosphere,” J. Quant. Spectrosc Radiat. Transfer 31, 273–282. (1984).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (2)

W. G. Houf, E. P. Incropera, “An Assessment of Techniques for Predicting Radiation Transfer in Aqueous Media,” J. Quant. Spectrosc. Radiat. Transfer 23, 101–115 (1980).
[CrossRef]

W. J. Wiscombe, “On Initialization, Error, and Flux Conservation in the Doubling Method,” J. Quant. Spectrosc. Radiat. Transfer 15, 637–658 (1975).

Laser Surg. Med. (1)

S. L. Jacques, S. A. Prahl, “Modeling Optical and Thermal Distributions in Tissue During Laser Irradiation,” Laser Surg. Med. 6, 494–503 (1987).
[CrossRef]

Lasers Life Sci. (2)

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular Dependence of He-Ne Laser Light Scattering by Human Dermis,” Lasers Life Sci. 4, 309–333 (1987).

J. W. Valvano, B. Chitsabesan, “Thermal Conductivity and Diffusivity of Arterial Wall and Atherosclerotic Plaque,” Lasers Life Sci. 1, 3,219–229 (1987).

Lasers Surg. Med. (2)

M. I. C. van Gemert, G. Schets, E. G. Stassen, J. J. Bonnier, “Modeling (Coronary) Laser Angioplasty,” Lasers Surg. Med. 5, 219–234 (1985).
[CrossRef] [PubMed]

A. J. Welch, G. Yoon, M. J. C. van Gemert, “Practical Models for Light Distribution in Laser-Irradiated Tissue,” Lasers Surg. Med. 6, 488–493 (1987).
[CrossRef] [PubMed]

Math. Comput. (1)

H. H. Michels, “Abscissas and Weight Coefficients for Lobatto Quadrature,” Math. Comput. 17, 237–244 (1963).
[CrossRef]

Photochem. Photobiol. (1)

S. Wan, R. R. Anderson, X. X. Parrish, “Analytical Modeling for the Optical Properties of the Skin with In Vitro and In Vivo Applications,” Photochem. Photobiol. 34, 493–499 (1981).
[PubMed]

Other (5)

L. O. Svaasand, D. R. Doiron, A. E. Profio, “Light Distribution in Tissue During Photoradiation Therapy,” Institute for Physics and Imaging Sci. Rep. USC-IPIS 900-02, U. Southern California (1981).

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

G. A. Korn, M. T. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961).

I. Gladwell, R. Wait, A Survey of Numerical Methods for Partial Differential Equations (Clarendon, Oxford, 1985).

A. N. Takata, L. Zaneveld, W. Richter, “Laser-Induced Thermal Damage in Skin,” Report SAM-TR-77-38, USAF School of Aerospace Medicine, Brooks AFB, TX (1977).

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Figures (14)

Fig. 1
Fig. 1

Computed total fluence rate as a function of anisotropy g for a moderately absorbing medium. Incident irradiance of 100 W/cm2. Index matched at surface.

Fig. 2
Fig. 2

Computed total fluence rate as a function of anisotropy factor g for a highly scattering medium. Incident irradiance of 100 W/cm2. Index matched at surface.

Fig. 3
Fig. 3

Total fluence rate as a function of boundary conditions and anisotropy factor. Incident irradiance of 100 W/cm2.

Fig. 4
Fig. 4

Total fluence rate as a function of anisotropy factor g for matched and mismatched boundaries. Incident irradiance of 100 W/cm2.

Fig. 5
Fig. 5

Total fluence rate as a function of index of refraction for low albedo. Incident irradiance of 100 W/cm2.

Fig. 6
Fig. 6

Total fluence rate as a function of index of refraction for high albedo. Incident irradiance of 100 W/cm2.

Fig. 7
Fig. 7

Total fluence rate as a function of bottom reflection.

Fig. 8
Fig. 8

Comparison of total fluence rates computed using discrete ordinate and diffusion approximation.

Fig. 9
Fig. 9

Relative volumetric power density as a function of scattering albedo for isotropic scattering.

Fig. 10
Fig. 10

Relative volumetric power density as a function of scattering albedo for anisotropic scattering.

Fig. 11
Fig. 11

Predicted temperature rise in tissue as a function of tissue anisotropicity. Incident power density of 65 kW/cm2.

Fig. 12
Fig. 12

Predicted temperature rise in tissue with matched and mismatched boundary conditions (isotropic scattering). Incident power density of 65 kW/cm2.

Fig. 13
Fig. 13

Predicted temperature rise in tissue with matched and mismatched boundary conditions (anisotropic scattering). Incident power density of 65 kW/cm2.

Fig. 14
Fig. 14

Predicted temperature rise in tissue on discrete ordinate and diffusion models. Incident power density of 65 kW/cm2.

Equations (16)

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

s · L ( r , s ) = μ t L ( r , s ) + μ t / 4 π 4 π P ( s , s ) L ( r , s ) d ω
d L c ( z ) / d z = μ t L c ( z ) .
μ [ d L ( τ , μ ) / d τ ] = L ( τ , μ ) + ω 0 / 2 1 + 1 P ( μ , μ ) L ( τ , μ ) d μ ;
μ [ d L ( τ , μ ) / d τ ] = L ( τ , μ ) + ω 0 / 2 1 + 1 L ( τ , μ ) × K = 0 N ω k P k ( μ ) P k ( μ ) d μ .
μ i [ d L i ( τ , μ i ) / d τ ] = L i ( τ , μ i ) + ω 0 / 2 i = 1 M a i L i ( τ , μ i ) × k = 0 N ω k P k ( μ i ) P k ( μ k ) 1 i M ,
P ( μ ) = ω 0 ( 1 g 2 ) ( 1 + g 2 2 g μ ) 1 . 5 ,
P ( θ ) = k = 0 N ( 2 k + 1 ) g k P k ( cos θ ) ,
L ( z ) = i = 1 M a i L i ( τ , μ i ) μ i + L c ( z ) .
Q L ( r , z ) = μ a ( r , z ) 4 π L ( r , z ) d ω .
ρ c ( T / t ) = · ( k T ) + Q L ,
T ( z , 0 ) = T i = 37 ° C .
k ( d T / d z ) = 0 .
U ( z , t ) = j U j ( t ) ϕ j ( z ) ,
ρ c U / t , ϕ j = L U , ϕ i + Q L , ϕ i , i = 1 , 2 , , m ,
A d U / d t = B U + c ,
A ( U n U n 1 ) / Δ t = ν [ B U n + c n ] + ( 1 ν ) [ B U n 1 + c n 1 ] .

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