Abstract

We present computationally efficient and accurate semiempirical models of light transfer for real-time analysis of multilayer fluorescing media exposed to normally incident excitation light. The model accounts for absorption and strong forward scattering as well as for internal reflection at the interface between the medium and the surrounding air. The absorption and scattering coefficients are assumed to be constant with depth; the fluorophore concentration is considered piecewise constant. The refractive index ranges from 1.0 to 2.0, and the transport single scattering albedo between 0.50 and 0.99. First, simple analytical expressions for local excitation fluence rate within the medium and surface fluorescence intensity emerging from its surface were derived from the two-flux approximation. Then, parameters appearing in the analytical expression previously derived were fitted to match results from more accurate Monte Carlo simulations. A single semiempirical parameter was sufficient to relate the diffuse reflectance of the medium at the excitation wavelength to the local excitation fluence rate within the medium and to the surface fluorescence emission intensity. The model predictions were compared with Monte Carlo simulations for local fluence rate and total surface fluorescence emission from (i) homogeneous semi-infinite fluorescing media, (ii) media with a semi-infinite fluorescing layer beneath a nonfluorescing layer, and (iii) media with a finite fluorescing layer embedded in a nonfluorescing semi-infinite layer. The model predictions of the local excitation fluence rate and of the total surface fluorescence emission fell to within 5% of predictions by Monte Carlo simulations for single scattering albedo greater than 0.90. The current model can be used for a wide range of applications, including noninvasive diagnosis of biological tissue.

© 2010 Optical Society of America

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2009

2006

E. Salomatina, B. Jiang, J. Novak, and A. N. Yaroslavsky, “Optical properties of normal and cancerous human skin in the visible and near-infrared spectral range,” J. Biomed. Opt. 11, 064026 (2006).
[CrossRef]

K. M. Katika and L. Pilon, “Steady-state directional diffuse reflectance and fluorescence of human skin,” Appl. Opt. 45, 4174–4183 (2006).
[CrossRef] [PubMed]

2005

2004

Y. Wu, P. Xi, J. Y. Qu, T. H. Cheung, and M. Y. Yu, “Depth-resolved fluorescence spectroscopy reveals layered structure of tissue,” Opt. Express 12, 3218–3223 (2004).
[CrossRef] [PubMed]

S. K. Chang, D. Arifler, R. Drezek, M. Follen, and R. Richards-Kortum, “Analytical model to describe fluorescence spectra of normal and preneoplastic epithelial tissue: comparison with Monte Carlo simulations and clinical measurements,” J. Biomed. Opt. 9, 511–522 (2004).
[CrossRef] [PubMed]

J. C. Finlay and T. H. Foster, “Hemoglobin oxygen saturations in phantoms and in vivo from measurements of steady-state diffuse reflectance at a single, short source-detector separation,” Med. Phys. 31, 1949–1959 (2004).
[CrossRef] [PubMed]

M. E. Ramos and M. G. Lagorio, “True fluorescence spectra of leaves,” Photochem. Photobiol. Sci. 3, 1063–1066 (2004).
[CrossRef] [PubMed]

S. K. Chang, D. Arifler, R. Drezek, M. Follen, and R. Richards-Kortum, “Analytical model to describe fluorescence spectra of normal and preneoplastic epithelial tissue: comparison with Monte Carlo simulations and clinical measurements,” J. Biomed. Opt. 9, 511–522 (2004).
[CrossRef] [PubMed]

2003

T. Shakespeare and J. Shakespeare, “A fluorescent extension to the Kubelka–Munk model,” Color Res. Appl. 28, 4–14 (2003).
[CrossRef]

2002

N. Kollias, G. Zonios, and G. N. Stamatas, “Fluorescence spectroscopy of skin,” Vib. Spectrosc. 28, 17–23 (2002).
[CrossRef]

I. Georgakoudi, B. C. Jacobson, M. G. Mu¨ller, E. E. Sheets, K. Badizadegan, D. L. Carr-Locke, C. P. Crum, C. W. Boone, R. R. Dasari, and J. Van Dam, and M. S. Feld, “NAD(P)H and collagen as in vivo quantitative fluorescent biomarkers of epithelial precancerous changes,” Cancer Res. 62, 682–687 (2002).
[PubMed]

2001

I. Georgakoudi, B. C. Jacobson, J. Van Dam, V. Backman, M. B. Wallace, M. G. Müller, Q. Zhang, K. Badizadegan, D. Sun, and G. A. Thomas, L. T. Perelman, and M. S. Feld, “Fluorescence, reflectance, and light-scattering spectroscopy for evaluating dysplasia in patients with Barrett’s esophagus,” Gastroenterology 120, 1620–1629 (2001).
[CrossRef] [PubMed]

2000

S. Achilefu, R. B. Dorshow, J. E. Bugaj, and R. Rajagopalan, “Novel receptor-targeted fluorescent contrast agents for in vivo tumor imaging,” Invest. Radiol. 35, 479–485 (2000).
[CrossRef] [PubMed]

1998

C. R. Simpson, M. Kohl, M. Essenpreis, and M. Cope, “Near-infrared optical properties of ex vivo human skin and subcutaneous tissues measured using the Monte Carlo inversion technique,” Phys. Med. Biol. 43, 2465–2478 (1998).
[CrossRef] [PubMed]

S. A. French, P. R. Territo, and R. S. Balaban, “Correction for inner filter effects in turbid samples: fluorescence assays of mitochondrial NADH,” Am. J. Physiol. Cell Physiol. 275, C900–C909 (1998).

N. N. Zhadin and R. R. Alfano, “Correction of the internal absorption effect in fluorescence emission and excitation spectra from absorbing and highly scattering media: theory and experiment,” J. Biomed. Opt. 3, 171–186 (1998).
[CrossRef]

1997

A. J. Welch, C. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, and S. Warren, “Propagation of fluorescent light,” Lasers Surg. Med. 21, 166–178 (1997).
[CrossRef] [PubMed]

L. Wang, S. L. Jacques, and L. Zheng, “CONV—Convolution for responses to a finite diameter photon beam incident on multi-layered tissues,” Comput. Methods Programs Biomed. 54, 141–150 (1997).
[CrossRef]

W. E. Vargas and G. A. Niklasson, “Applicability conditions of the Kubelka–Munk theory,” Appl. Opt. 36, 5580–5586 (1997).
[CrossRef] [PubMed]

1996

C. M. Gardner, S. L. Jacques, and A. J. Welch, “Fluorescence spectroscopy of tissue: recovery of intrinsic fluorescence from measured fluorescence,” Appl. Opt. 35, 1780–1792 (1996).
[CrossRef] [PubMed]

C. M. Gardner, S. L. Jacques, and A. J. Welch, “Light transport in tissue: accurate expressions for one-dimensional fluence rate and escape function based upon Monte Carlo simulation,” Lasers Surg. Med. 18, 129–138 (1996).
[CrossRef] [PubMed]

1995

R. L. Sheridan, K. T. Schomaker, L. C. Lucchina, J. Hurley, L. M. Yin, R. G. Tompkins, M. Jerath, A. Torri, K. W. Greaves, and D. P. Bua, “Burn depth estimation by use of indocyanine green fluorescence: initial human trial,” J. Burn Care Res. 16, 602–607 (1995).
[CrossRef]

1994

1993

D. A. Hansen, A. M. Spence, T. Carski, and M. S. Berger, “Indocyanine green (ICG) staining and demarcation of tumor margins in a rat glioma model,” Surg. Neurol. 40, 451–456(1993).
[CrossRef] [PubMed]

J. Wu, M. S. Feld, and R. P. Rava, “Analytical model for extracting intrinsic fluorescence in turbid media,” Appl. Opt. 32, 3585–3595 (1993).
[CrossRef] [PubMed]

1990

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

1989

1987

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

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

1981

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

1979

K. J. Daniel, N. M. Laurendeau, and F. P. Incropera, “Prediction of radiation absorption and scattering in turbid water bodies,” ASME J. Heat Transfer 101, 63–67 (1979).
[CrossRef]

1976

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

1945

1940

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Ann. Astrophys. 93, 70–83 (1940).

1931

P. Kubelka and F. Munk, “A contribution to the optics of pigments,” Z. Tech. Phys. 12, 593–599 (1931).

Achilefu, S.

S. Achilefu, R. B. Dorshow, J. E. Bugaj, and R. Rajagopalan, “Novel receptor-targeted fluorescent contrast agents for in vivo tumor imaging,” Invest. Radiol. 35, 479–485 (2000).
[CrossRef] [PubMed]

Alfano, R. R.

N. N. Zhadin and R. R. Alfano, “Correction of the internal absorption effect in fluorescence emission and excitation spectra from absorbing and highly scattering media: theory and experiment,” J. Biomed. Opt. 3, 171–186 (1998).
[CrossRef]

Alter, C. A.

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

Anderson, R. R.

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

Andersson-Engels, S.

Arifler, D.

S. K. Chang, D. Arifler, R. Drezek, M. Follen, and R. Richards-Kortum, “Analytical model to describe fluorescence spectra of normal and preneoplastic epithelial tissue: comparison with Monte Carlo simulations and clinical measurements,” J. Biomed. Opt. 9, 511–522 (2004).
[CrossRef] [PubMed]

S. K. Chang, D. Arifler, R. Drezek, M. Follen, and R. Richards-Kortum, “Analytical model to describe fluorescence spectra of normal and preneoplastic epithelial tissue: comparison with Monte Carlo simulations and clinical measurements,” J. Biomed. Opt. 9, 511–522 (2004).
[CrossRef] [PubMed]

Backman, V.

I. Georgakoudi, B. C. Jacobson, J. Van Dam, V. Backman, M. B. Wallace, M. G. Müller, Q. Zhang, K. Badizadegan, D. Sun, and G. A. Thomas, L. T. Perelman, and M. S. Feld, “Fluorescence, reflectance, and light-scattering spectroscopy for evaluating dysplasia in patients with Barrett’s esophagus,” Gastroenterology 120, 1620–1629 (2001).
[CrossRef] [PubMed]

Badizadegan, K.

I. Georgakoudi, B. C. Jacobson, M. G. Mu¨ller, E. E. Sheets, K. Badizadegan, D. L. Carr-Locke, C. P. Crum, C. W. Boone, R. R. Dasari, and J. Van Dam, and M. S. Feld, “NAD(P)H and collagen as in vivo quantitative fluorescent biomarkers of epithelial precancerous changes,” Cancer Res. 62, 682–687 (2002).
[PubMed]

I. Georgakoudi, B. C. Jacobson, J. Van Dam, V. Backman, M. B. Wallace, M. G. Müller, Q. Zhang, K. Badizadegan, D. Sun, and G. A. Thomas, L. T. Perelman, and M. S. Feld, “Fluorescence, reflectance, and light-scattering spectroscopy for evaluating dysplasia in patients with Barrett’s esophagus,” Gastroenterology 120, 1620–1629 (2001).
[CrossRef] [PubMed]

Balaban, R. S.

S. A. French, P. R. Territo, and R. S. Balaban, “Correction for inner filter effects in turbid samples: fluorescence assays of mitochondrial NADH,” Am. J. Physiol. Cell Physiol. 275, C900–C909 (1998).

Bengtsson, D.

Berger, M. S.

D. A. Hansen, A. M. Spence, T. Carski, and M. S. Berger, “Indocyanine green (ICG) staining and demarcation of tumor margins in a rat glioma model,” Surg. Neurol. 40, 451–456(1993).
[CrossRef] [PubMed]

Boone, C. W.

I. Georgakoudi, B. C. Jacobson, M. G. Mu¨ller, E. E. Sheets, K. Badizadegan, D. L. Carr-Locke, C. P. Crum, C. W. Boone, R. R. Dasari, and J. Van Dam, and M. S. Feld, “NAD(P)H and collagen as in vivo quantitative fluorescent biomarkers of epithelial precancerous changes,” Cancer Res. 62, 682–687 (2002).
[PubMed]

Bua, D. P.

R. L. Sheridan, K. T. Schomaker, L. C. Lucchina, J. Hurley, L. M. Yin, R. G. Tompkins, M. Jerath, A. Torri, K. W. Greaves, and D. P. Bua, “Burn depth estimation by use of indocyanine green fluorescence: initial human trial,” J. Burn Care Res. 16, 602–607 (1995).
[CrossRef]

Bugaj, J. E.

S. Achilefu, R. B. Dorshow, J. E. Bugaj, and R. Rajagopalan, “Novel receptor-targeted fluorescent contrast agents for in vivo tumor imaging,” Invest. Radiol. 35, 479–485 (2000).
[CrossRef] [PubMed]

Carr-Locke, D. L.

I. Georgakoudi, B. C. Jacobson, M. G. Mu¨ller, E. E. Sheets, K. Badizadegan, D. L. Carr-Locke, C. P. Crum, C. W. Boone, R. R. Dasari, and J. Van Dam, and M. S. Feld, “NAD(P)H and collagen as in vivo quantitative fluorescent biomarkers of epithelial precancerous changes,” Cancer Res. 62, 682–687 (2002).
[PubMed]

Carski, T.

D. A. Hansen, A. M. Spence, T. Carski, and M. S. Berger, “Indocyanine green (ICG) staining and demarcation of tumor margins in a rat glioma model,” Surg. Neurol. 40, 451–456(1993).
[CrossRef] [PubMed]

Chan, E.

A. J. Welch, C. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, and S. Warren, “Propagation of fluorescent light,” Lasers Surg. Med. 21, 166–178 (1997).
[CrossRef] [PubMed]

Chandrasekhar, S.

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

Chang, S. K.

S. K. Chang, D. Arifler, R. Drezek, M. Follen, and R. Richards-Kortum, “Analytical model to describe fluorescence spectra of normal and preneoplastic epithelial tissue: comparison with Monte Carlo simulations and clinical measurements,” J. Biomed. Opt. 9, 511–522 (2004).
[CrossRef] [PubMed]

S. K. Chang, D. Arifler, R. Drezek, M. Follen, and R. Richards-Kortum, “Analytical model to describe fluorescence spectra of normal and preneoplastic epithelial tissue: comparison with Monte Carlo simulations and clinical measurements,” J. Biomed. Opt. 9, 511–522 (2004).
[CrossRef] [PubMed]

Cheong, W. F.

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

Cheung, T. H.

Cope, M.

C. R. Simpson, M. Kohl, M. Essenpreis, and M. Cope, “Near-infrared optical properties of ex vivo human skin and subcutaneous tissues measured using the Monte Carlo inversion technique,” Phys. Med. Biol. 43, 2465–2478 (1998).
[CrossRef] [PubMed]

Criswell, G.

A. J. Welch, C. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, and S. Warren, “Propagation of fluorescent light,” Lasers Surg. Med. 21, 166–178 (1997).
[CrossRef] [PubMed]

Crum, C. P.

I. Georgakoudi, B. C. Jacobson, M. G. Mu¨ller, E. E. Sheets, K. Badizadegan, D. L. Carr-Locke, C. P. Crum, C. W. Boone, R. R. Dasari, and J. Van Dam, and M. S. Feld, “NAD(P)H and collagen as in vivo quantitative fluorescent biomarkers of epithelial precancerous changes,” Cancer Res. 62, 682–687 (2002).
[PubMed]

Daniel, K. J.

K. J. Daniel, N. M. Laurendeau, and F. P. Incropera, “Prediction of radiation absorption and scattering in turbid water bodies,” ASME J. Heat Transfer 101, 63–67 (1979).
[CrossRef]

Dasari, R. R.

I. Georgakoudi, B. C. Jacobson, M. G. Mu¨ller, E. E. Sheets, K. Badizadegan, D. L. Carr-Locke, C. P. Crum, C. W. Boone, R. R. Dasari, and J. Van Dam, and M. S. Feld, “NAD(P)H and collagen as in vivo quantitative fluorescent biomarkers of epithelial precancerous changes,” Cancer Res. 62, 682–687 (2002).
[PubMed]

Dorshow, R. B.

S. Achilefu, R. B. Dorshow, J. E. Bugaj, and R. Rajagopalan, “Novel receptor-targeted fluorescent contrast agents for in vivo tumor imaging,” Invest. Radiol. 35, 479–485 (2000).
[CrossRef] [PubMed]

Drezek, R.

S. K. Chang, D. Arifler, R. Drezek, M. Follen, and R. Richards-Kortum, “Analytical model to describe fluorescence spectra of normal and preneoplastic epithelial tissue: comparison with Monte Carlo simulations and clinical measurements,” J. Biomed. Opt. 9, 511–522 (2004).
[CrossRef] [PubMed]

S. K. Chang, D. Arifler, R. Drezek, M. Follen, and R. Richards-Kortum, “Analytical model to describe fluorescence spectra of normal and preneoplastic epithelial tissue: comparison with Monte Carlo simulations and clinical measurements,” J. Biomed. Opt. 9, 511–522 (2004).
[CrossRef] [PubMed]

Durkin, A. J.

Essenpreis, M.

C. R. Simpson, M. Kohl, M. Essenpreis, and M. Cope, “Near-infrared optical properties of ex vivo human skin and subcutaneous tissues measured using the Monte Carlo inversion technique,” Phys. Med. Biol. 43, 2465–2478 (1998).
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D. Q. Nguyen, R. Fedkiw, and H. W. Jensen, “Physically based modeling and animation of fire,” in Proceedings of the 29th Annual Conference on Computer Graphics and Interactive Techniques (ACM New York, 2002), pp. 721–728.

Feld, M. S.

I. Georgakoudi, B. C. Jacobson, M. G. Mu¨ller, E. E. Sheets, K. Badizadegan, D. L. Carr-Locke, C. P. Crum, C. W. Boone, R. R. Dasari, and J. Van Dam, and M. S. Feld, “NAD(P)H and collagen as in vivo quantitative fluorescent biomarkers of epithelial precancerous changes,” Cancer Res. 62, 682–687 (2002).
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S. K. Chang, D. Arifler, R. Drezek, M. Follen, and R. Richards-Kortum, “Analytical model to describe fluorescence spectra of normal and preneoplastic epithelial tissue: comparison with Monte Carlo simulations and clinical measurements,” J. Biomed. Opt. 9, 511–522 (2004).
[CrossRef] [PubMed]

S. K. Chang, D. Arifler, R. Drezek, M. Follen, and R. Richards-Kortum, “Analytical model to describe fluorescence spectra of normal and preneoplastic epithelial tissue: comparison with Monte Carlo simulations and clinical measurements,” J. Biomed. Opt. 9, 511–522 (2004).
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J. C. Finlay and T. H. Foster, “Hemoglobin oxygen saturations in phantoms and in vivo from measurements of steady-state diffuse reflectance at a single, short source-detector separation,” Med. Phys. 31, 1949–1959 (2004).
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S. A. French, P. R. Territo, and R. S. Balaban, “Correction for inner filter effects in turbid samples: fluorescence assays of mitochondrial NADH,” Am. J. Physiol. Cell Physiol. 275, C900–C909 (1998).

Gardner, C.

A. J. Welch, C. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, and S. Warren, “Propagation of fluorescent light,” Lasers Surg. Med. 21, 166–178 (1997).
[CrossRef] [PubMed]

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C. M. Gardner, S. L. Jacques, and A. J. Welch, “Light transport in tissue: accurate expressions for one-dimensional fluence rate and escape function based upon Monte Carlo simulation,” Lasers Surg. Med. 18, 129–138 (1996).
[CrossRef] [PubMed]

C. M. Gardner, S. L. Jacques, and A. J. Welch, “Fluorescence spectroscopy of tissue: recovery of intrinsic fluorescence from measured fluorescence,” Appl. Opt. 35, 1780–1792 (1996).
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I. Georgakoudi, B. C. Jacobson, M. G. Mu¨ller, E. E. Sheets, K. Badizadegan, D. L. Carr-Locke, C. P. Crum, C. W. Boone, R. R. Dasari, and J. Van Dam, and M. S. Feld, “NAD(P)H and collagen as in vivo quantitative fluorescent biomarkers of epithelial precancerous changes,” Cancer Res. 62, 682–687 (2002).
[PubMed]

I. Georgakoudi, B. C. Jacobson, J. Van Dam, V. Backman, M. B. Wallace, M. G. Müller, Q. Zhang, K. Badizadegan, D. Sun, and G. A. Thomas, L. T. Perelman, and M. S. Feld, “Fluorescence, reflectance, and light-scattering spectroscopy for evaluating dysplasia in patients with Barrett’s esophagus,” Gastroenterology 120, 1620–1629 (2001).
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Ghosh, N.

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R. L. Sheridan, K. T. Schomaker, L. C. Lucchina, J. Hurley, L. M. Yin, R. G. Tompkins, M. Jerath, A. Torri, K. W. Greaves, and D. P. Bua, “Burn depth estimation by use of indocyanine green fluorescence: initial human trial,” J. Burn Care Res. 16, 602–607 (1995).
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D. A. Hansen, A. M. Spence, T. Carski, and M. S. Berger, “Indocyanine green (ICG) staining and demarcation of tumor margins in a rat glioma model,” Surg. Neurol. 40, 451–456(1993).
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R. L. Sheridan, K. T. Schomaker, L. C. Lucchina, J. Hurley, L. M. Yin, R. G. Tompkins, M. Jerath, A. Torri, K. W. Greaves, and D. P. Bua, “Burn depth estimation by use of indocyanine green fluorescence: initial human trial,” J. Burn Care Res. 16, 602–607 (1995).
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I. Georgakoudi, B. C. Jacobson, J. Van Dam, V. Backman, M. B. Wallace, M. G. Müller, Q. Zhang, K. Badizadegan, D. Sun, and G. A. Thomas, L. T. Perelman, and M. S. Feld, “Fluorescence, reflectance, and light-scattering spectroscopy for evaluating dysplasia in patients with Barrett’s esophagus,” Gastroenterology 120, 1620–1629 (2001).
[CrossRef] [PubMed]

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L. Wang, S. L. Jacques, and L. Zheng, “CONV—Convolution for responses to a finite diameter photon beam incident on multi-layered tissues,” Comput. Methods Programs Biomed. 54, 141–150 (1997).
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C. M. Gardner, S. L. Jacques, and A. J. Welch, “Fluorescence spectroscopy of tissue: recovery of intrinsic fluorescence from measured fluorescence,” Appl. Opt. 35, 1780–1792 (1996).
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C. M. Gardner, S. L. Jacques, and A. J. Welch, “Light transport in tissue: accurate expressions for one-dimensional fluence rate and escape function based upon Monte Carlo simulation,” Lasers Surg. Med. 18, 129–138 (1996).
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M. J. C. Van Gemert, S. L. Jacques, H. Sterenborg, and W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154(1989).
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S. L. Jacques, C. A. Alter, and S. A. Prahl, “Angular dependence of He–Ne laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

L. Wang and S. L. Jacques, “Monte Carlo modeling of light transport in multi-layered tissues in standard C,” last accessed 3/31/2009, http://labs.seas.wustl.edu/bme/Wang/mcr5/Mcman.pdf.

Jaikumar, S.

Jensen, H. W.

D. Q. Nguyen, R. Fedkiw, and H. W. Jensen, “Physically based modeling and animation of fire,” in Proceedings of the 29th Annual Conference on Computer Graphics and Interactive Techniques (ACM New York, 2002), pp. 721–728.

Jerath, M.

R. L. Sheridan, K. T. Schomaker, L. C. Lucchina, J. Hurley, L. M. Yin, R. G. Tompkins, M. Jerath, A. Torri, K. W. Greaves, and D. P. Bua, “Burn depth estimation by use of indocyanine green fluorescence: initial human trial,” J. Burn Care Res. 16, 602–607 (1995).
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E. Salomatina, B. Jiang, J. Novak, and A. N. Yaroslavsky, “Optical properties of normal and cancerous human skin in the visible and near-infrared spectral range,” J. Biomed. Opt. 11, 064026 (2006).
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J. H. Joseph, W. J. Wiscombe, and J. A. Weinman, “The Delta–Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
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Katika, K. M.

Kohl, M.

C. R. Simpson, M. Kohl, M. Essenpreis, and M. Cope, “Near-infrared optical properties of ex vivo human skin and subcutaneous tissues measured using the Monte Carlo inversion technique,” Phys. Med. Biol. 43, 2465–2478 (1998).
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Kollias, N.

N. Kollias, G. Zonios, and G. N. Stamatas, “Fluorescence spectroscopy of skin,” Vib. Spectrosc. 28, 17–23 (2002).
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P. Kubelka and F. Munk, “A contribution to the optics of pigments,” Z. Tech. Phys. 12, 593–599 (1931).

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M. E. Ramos and M. G. Lagorio, “True fluorescence spectra of leaves,” Photochem. Photobiol. Sci. 3, 1063–1066 (2004).
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K. J. Daniel, N. M. Laurendeau, and F. P. Incropera, “Prediction of radiation absorption and scattering in turbid water bodies,” ASME J. Heat Transfer 101, 63–67 (1979).
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R. L. Sheridan, K. T. Schomaker, L. C. Lucchina, J. Hurley, L. M. Yin, R. G. Tompkins, M. Jerath, A. Torri, K. W. Greaves, and D. P. Bua, “Burn depth estimation by use of indocyanine green fluorescence: initial human trial,” J. Burn Care Res. 16, 602–607 (1995).
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I. Georgakoudi, B. C. Jacobson, M. G. Mu¨ller, E. E. Sheets, K. Badizadegan, D. L. Carr-Locke, C. P. Crum, C. W. Boone, R. R. Dasari, and J. Van Dam, and M. S. Feld, “NAD(P)H and collagen as in vivo quantitative fluorescent biomarkers of epithelial precancerous changes,” Cancer Res. 62, 682–687 (2002).
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Müller, M. G.

I. Georgakoudi, B. C. Jacobson, J. Van Dam, V. Backman, M. B. Wallace, M. G. Müller, Q. Zhang, K. Badizadegan, D. Sun, and G. A. Thomas, L. T. Perelman, and M. S. Feld, “Fluorescence, reflectance, and light-scattering spectroscopy for evaluating dysplasia in patients with Barrett’s esophagus,” Gastroenterology 120, 1620–1629 (2001).
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P. Kubelka and F. Munk, “A contribution to the optics of pigments,” Z. Tech. Phys. 12, 593–599 (1931).

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M. A. Mycek and B. W. Pogue, Handbook of Biomedical Fluorescence (Marcel Dekker, 2003).

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D. Q. Nguyen, R. Fedkiw, and H. W. Jensen, “Physically based modeling and animation of fire,” in Proceedings of the 29th Annual Conference on Computer Graphics and Interactive Techniques (ACM New York, 2002), pp. 721–728.

Niklasson, G. A.

Novak, J.

E. Salomatina, B. Jiang, J. Novak, and A. N. Yaroslavsky, “Optical properties of normal and cancerous human skin in the visible and near-infrared spectral range,” J. Biomed. Opt. 11, 064026 (2006).
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Perelman, L. T.

I. Georgakoudi, B. C. Jacobson, J. Van Dam, V. Backman, M. B. Wallace, M. G. Müller, Q. Zhang, K. Badizadegan, D. Sun, and G. A. Thomas, L. T. Perelman, and M. S. Feld, “Fluorescence, reflectance, and light-scattering spectroscopy for evaluating dysplasia in patients with Barrett’s esophagus,” Gastroenterology 120, 1620–1629 (2001).
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Pfefer, J.

A. J. Welch, C. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, and S. Warren, “Propagation of fluorescent light,” Lasers Surg. Med. 21, 166–178 (1997).
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Pogue, B. W.

M. A. Mycek and B. W. Pogue, Handbook of Biomedical Fluorescence (Marcel Dekker, 2003).

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W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
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G. Yoon, S. A. Prahl, and 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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S. L. Jacques, C. A. Alter, and S. A. Prahl, “Angular dependence of He–Ne laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

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Rajagopalan, R.

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Ramos, M. E.

M. E. Ramos and M. G. Lagorio, “True fluorescence spectra of leaves,” Photochem. Photobiol. Sci. 3, 1063–1066 (2004).
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Rava, R. P.

Richards-Kortum, R.

S. K. Chang, D. Arifler, R. Drezek, M. Follen, and R. Richards-Kortum, “Analytical model to describe fluorescence spectra of normal and preneoplastic epithelial tissue: comparison with Monte Carlo simulations and clinical measurements,” J. Biomed. Opt. 9, 511–522 (2004).
[CrossRef] [PubMed]

S. K. Chang, D. Arifler, R. Drezek, M. Follen, and R. Richards-Kortum, “Analytical model to describe fluorescence spectra of normal and preneoplastic epithelial tissue: comparison with Monte Carlo simulations and clinical measurements,” J. Biomed. Opt. 9, 511–522 (2004).
[CrossRef] [PubMed]

A. J. Welch, C. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, and S. Warren, “Propagation of fluorescent light,” Lasers Surg. Med. 21, 166–178 (1997).
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E. Salomatina, B. Jiang, J. Novak, and A. N. Yaroslavsky, “Optical properties of normal and cancerous human skin in the visible and near-infrared spectral range,” J. Biomed. Opt. 11, 064026 (2006).
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R. L. Sheridan, K. T. Schomaker, L. C. Lucchina, J. Hurley, L. M. Yin, R. G. Tompkins, M. Jerath, A. Torri, K. W. Greaves, and D. P. Bua, “Burn depth estimation by use of indocyanine green fluorescence: initial human trial,” J. Burn Care Res. 16, 602–607 (1995).
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I. Georgakoudi, B. C. Jacobson, M. G. Mu¨ller, E. E. Sheets, K. Badizadegan, D. L. Carr-Locke, C. P. Crum, C. W. Boone, R. R. Dasari, and J. Van Dam, and M. S. Feld, “NAD(P)H and collagen as in vivo quantitative fluorescent biomarkers of epithelial precancerous changes,” Cancer Res. 62, 682–687 (2002).
[PubMed]

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R. L. Sheridan, K. T. Schomaker, L. C. Lucchina, J. Hurley, L. M. Yin, R. G. Tompkins, M. Jerath, A. Torri, K. W. Greaves, and D. P. Bua, “Burn depth estimation by use of indocyanine green fluorescence: initial human trial,” J. Burn Care Res. 16, 602–607 (1995).
[CrossRef]

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C. R. Simpson, M. Kohl, M. Essenpreis, and M. Cope, “Near-infrared optical properties of ex vivo human skin and subcutaneous tissues measured using the Monte Carlo inversion technique,” Phys. Med. Biol. 43, 2465–2478 (1998).
[CrossRef] [PubMed]

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D. A. Hansen, A. M. Spence, T. Carski, and M. S. Berger, “Indocyanine green (ICG) staining and demarcation of tumor margins in a rat glioma model,” Surg. Neurol. 40, 451–456(1993).
[CrossRef] [PubMed]

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N. Kollias, G. Zonios, and G. N. Stamatas, “Fluorescence spectroscopy of skin,” Vib. Spectrosc. 28, 17–23 (2002).
[CrossRef]

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M. J. C. Van Gemert, S. L. Jacques, H. Sterenborg, and W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154(1989).
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M. J. C. van Gemert and W. M. Star, “Relations between the Kubelka–Munk and the transport equation models for anisotropic scattering,” Lasers Life Sci. 1, 287–298 (1987).

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M. J. C. Van Gemert, S. L. Jacques, H. Sterenborg, and W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154(1989).
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I. Georgakoudi, B. C. Jacobson, J. Van Dam, V. Backman, M. B. Wallace, M. G. Müller, Q. Zhang, K. Badizadegan, D. Sun, and G. A. Thomas, L. T. Perelman, and M. S. Feld, “Fluorescence, reflectance, and light-scattering spectroscopy for evaluating dysplasia in patients with Barrett’s esophagus,” Gastroenterology 120, 1620–1629 (2001).
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Swartling, J.

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S. A. French, P. R. Territo, and R. S. Balaban, “Correction for inner filter effects in turbid samples: fluorescence assays of mitochondrial NADH,” Am. J. Physiol. Cell Physiol. 275, C900–C909 (1998).

Thomas, G. A.

I. Georgakoudi, B. C. Jacobson, J. Van Dam, V. Backman, M. B. Wallace, M. G. Müller, Q. Zhang, K. Badizadegan, D. Sun, and G. A. Thomas, L. T. Perelman, and M. S. Feld, “Fluorescence, reflectance, and light-scattering spectroscopy for evaluating dysplasia in patients with Barrett’s esophagus,” Gastroenterology 120, 1620–1629 (2001).
[CrossRef] [PubMed]

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R. L. Sheridan, K. T. Schomaker, L. C. Lucchina, J. Hurley, L. M. Yin, R. G. Tompkins, M. Jerath, A. Torri, K. W. Greaves, and D. P. Bua, “Burn depth estimation by use of indocyanine green fluorescence: initial human trial,” J. Burn Care Res. 16, 602–607 (1995).
[CrossRef]

Torri, A.

R. L. Sheridan, K. T. Schomaker, L. C. Lucchina, J. Hurley, L. M. Yin, R. G. Tompkins, M. Jerath, A. Torri, K. W. Greaves, and D. P. Bua, “Burn depth estimation by use of indocyanine green fluorescence: initial human trial,” J. Burn Care Res. 16, 602–607 (1995).
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V. V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis (SPIE Press, 2007).

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I. Georgakoudi, B. C. Jacobson, M. G. Mu¨ller, E. E. Sheets, K. Badizadegan, D. L. Carr-Locke, C. P. Crum, C. W. Boone, R. R. Dasari, and J. Van Dam, and M. S. Feld, “NAD(P)H and collagen as in vivo quantitative fluorescent biomarkers of epithelial precancerous changes,” Cancer Res. 62, 682–687 (2002).
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I. Georgakoudi, B. C. Jacobson, J. Van Dam, V. Backman, M. B. Wallace, M. G. Müller, Q. Zhang, K. Badizadegan, D. Sun, and G. A. Thomas, L. T. Perelman, and M. S. Feld, “Fluorescence, reflectance, and light-scattering spectroscopy for evaluating dysplasia in patients with Barrett’s esophagus,” Gastroenterology 120, 1620–1629 (2001).
[CrossRef] [PubMed]

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M. J. C. Van Gemert, S. L. Jacques, H. Sterenborg, and W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154(1989).
[CrossRef] [PubMed]

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

Vargas, W. E.

Wallace, M. B.

I. Georgakoudi, B. C. Jacobson, J. Van Dam, V. Backman, M. B. Wallace, M. G. Müller, Q. Zhang, K. Badizadegan, D. Sun, and G. A. Thomas, L. T. Perelman, and M. S. Feld, “Fluorescence, reflectance, and light-scattering spectroscopy for evaluating dysplasia in patients with Barrett’s esophagus,” Gastroenterology 120, 1620–1629 (2001).
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L. Wang, S. L. Jacques, and L. Zheng, “CONV—Convolution for responses to a finite diameter photon beam incident on multi-layered tissues,” Comput. Methods Programs Biomed. 54, 141–150 (1997).
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L. Wang and S. L. Jacques, “Monte Carlo modeling of light transport in multi-layered tissues in standard C,” last accessed 3/31/2009, http://labs.seas.wustl.edu/bme/Wang/mcr5/Mcman.pdf.

Warren, S.

A. J. Welch, C. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, and S. Warren, “Propagation of fluorescent light,” Lasers Surg. Med. 21, 166–178 (1997).
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J. H. Joseph, W. J. Wiscombe, and J. A. Weinman, “The Delta–Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
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A. J. Welch, C. Gardner, R. Richards-Kortum, E. Chan, G. Criswell, J. Pfefer, and S. Warren, “Propagation of fluorescent light,” Lasers Surg. Med. 21, 166–178 (1997).
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C. M. Gardner, S. L. Jacques, and A. J. Welch, “Light transport in tissue: accurate expressions for one-dimensional fluence rate and escape function based upon Monte Carlo simulation,” Lasers Surg. Med. 18, 129–138 (1996).
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C. M. Gardner, S. L. Jacques, and A. J. Welch, “Fluorescence spectroscopy of tissue: recovery of intrinsic fluorescence from measured fluorescence,” Appl. Opt. 35, 1780–1792 (1996).
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W. F. Cheong, S. A. Prahl, and A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
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G. Yoon, S. A. Prahl, and 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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J. H. Joseph, W. J. Wiscombe, and J. A. Weinman, “The Delta–Eddington approximation for radiative flux transfer,” J. Atmos. Sci. 33, 2452–2459 (1976).
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Wu, Y.

Xi, P.

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E. Salomatina, B. Jiang, J. Novak, and A. N. Yaroslavsky, “Optical properties of normal and cancerous human skin in the visible and near-infrared spectral range,” J. Biomed. Opt. 11, 064026 (2006).
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R. L. Sheridan, K. T. Schomaker, L. C. Lucchina, J. Hurley, L. M. Yin, R. G. Tompkins, M. Jerath, A. Torri, K. W. Greaves, and D. P. Bua, “Burn depth estimation by use of indocyanine green fluorescence: initial human trial,” J. Burn Care Res. 16, 602–607 (1995).
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I. Georgakoudi, B. C. Jacobson, J. Van Dam, V. Backman, M. B. Wallace, M. G. Müller, Q. Zhang, K. Badizadegan, D. Sun, and G. A. Thomas, L. T. Perelman, and M. S. Feld, “Fluorescence, reflectance, and light-scattering spectroscopy for evaluating dysplasia in patients with Barrett’s esophagus,” Gastroenterology 120, 1620–1629 (2001).
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Figures (10)

Fig. 1
Fig. 1

Schematic of the geometry considered along with co ordinate system, boundary conditions, and optical properties for the (a) excitation wavelength λ x and (b) fluorescence emission wavelength λ f .

Fig. 2
Fig. 2

(a) Transport single scattering albedo as a function of wavelength for human dermis [43, 44, 45]. (b) Ratio of extinction coefficients r x f as a function of excitation wavelength λ x and emission wavelength λ f equal to 400, 500, 600, and 700 nm for the human dermis [43, 44].

Fig. 3
Fig. 3

Schematic of (a) step, (b) homogeneous, and (c) layered fluorophore concentration profiles M S ( τ t r , λ x ) , M H ( τ t r , λ x ) , and M L ( τ t r , λ x ) , respectively. Light gray represents the medium without fluorophore [ M ( τ t r , λ x ) = 0   mole / cm 3 ], while dark gray represents regions with fluorophore [ M ( τ t r , λ x ) = A   mole / cm 3 ].

Fig. 4
Fig. 4

Fitting parameters k 1 and k 2 retrieved using Monte Carlo simulations as a function of R d , λ x along with predictions by Eqs. (32, 40) for n 1 between 1.33 and 2.00.

Fig. 5
Fig. 5

(a) Normalized local excitation fluence rate as a function of the optical depth ζ d , λ x τ t r , λ x predicted by Monte Carlo simulations and Eq. (30) for ω t r , λ x between 0.50 and 0.99 and n 1 = 1.44 . (b) Root-mean-square (CF) [Eq. (39)] error as a function of the transport single scattering albedo ω t r , λ x between 0.50 and 0.99 and n 1 between 1.33 and 2.00.

Fig. 6
Fig. 6

Transfer function T x f S for step concentration profile M S ( τ t r , λ x , 1 ) versus the optical depth calculated by Monte Carlo simulations and predicted by Eq. (36) for A = 1   mole / cm 2 , ω t r , λ x = 0.70 , n = 1.44 , r x f between 0.2 and 5.0, and ω t r , λ f equal to (a) 0.750 and (b) 0.900.

Fig. 7
Fig. 7

Relative error between prediction of T x f S by Monte Carlo simulations and predicted by Eq. (36) averaged over τ t r , λ x , 1 between 0 and 1 for A = 1   mole / cm 2 , ω t r , λ x = 0.70 , n = 1.44 , and r x f between 0.20 and 5.00.

Fig. 8
Fig. 8

Relative error between predictions of T x f H by Monte Carlo simulations and by Eq. (37) as a function of ω t r , λ f averaged over ω t r , λ x between 0.50 and 1.00 for r x f between 0.20 and 5.00, A = 1   mole / cm 2 , and n 1 = 1.44 .

Fig. 9
Fig. 9

Transfer function T x f L versus the optical depth predicted by Monte Carlo simulations and by Eq. (38) with ω t r , λ x = 0.700 , A = 1   mole / cm 2 , n 1 = 1.44 , r x f between 0.20 and 5.00, and τ t r , λ x , 2 τ t r , λ x , 1 = L β t r , λ x with L = 1 cm for ω t r , λ f equal to (a) 0.750 and (b) 0.900.

Fig. 10
Fig. 10

Relative error between prediction of T x f L by Monte Carlo simulations and by Eq. (38) averaged over τ t r , λ x between 0 and 1 for ω t r , λ x = 0.70 , A = 1   mole / cm 2 , n 1 = 1.44 , and r x f between 0.20 and 5.00.

Equations (41)

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s ^ · I λ x ( r ^ , s ^ ) = μ a , λ x I λ x ( r ^ , s ^ ) μ s , λ x I λ x ( r ^ , s ^ ) + μ s , λ x 4 π × 4 π I λ x ( r ^ , s ^ i ) Φ λ x ( s ^ i , s ^ ) d Ω i ,
s ^ · I λ f ( r ^ , s ^ ) = μ a , λ f I λ f ( r ^ , s ^ ) μ s , λ f I λ f ( r ^ , s ^ ) + μ s , λ f 4 π × 4 π I λ f ( r ^ , s ^ i ) Φ λ f ( s ^ i , s ^ ) d Ω i + γ x f ( r ^ ) G λ x ( r ^ ) 4 π .
G λ x ( r ^ ) = 4 π I λ x ( r ^ , s ^ ) d Ω .
Φ λ ( s ^ i , s ^ ) = 1 g λ [ 1 + g λ 2 2 g λ cos Θ ] 3 / 2 ,
ω t r , λ = μ s , t r , λ μ s , t r , λ + μ a , λ = μ s , λ ( 1 g λ ) μ s , λ ( 1 g λ ) + μ a , λ , β t r , λ = μ s , λ ( 1 g λ ) + μ a , λ .
d F λ x + d z = a λ x S λ x F λ x + + S λ x F λ x + S 1 , λ x F c , λ x ,
d F λ x d z = S λ x F λ x + + a λ x S λ x F λ x S 2 , λ x F c , λ x ,
F λ + ( z ) = 2 π 0 π / 2 I λ ( z , θ ) cos θ sin θ d θ , F λ ( z ) = 2 π π / 2 π I λ ( z , θ ) cos θ sin θ d θ .
F c , λ x ( z ) = ( 1 ρ 01 ) F 0 , λ x e K c , λ x z ,
ρ 01 = ( n 1 n 0 n 1 + n 0 ) 2 ,
F λ x + ( 0 ) = ρ 10 F λ x ( 0 ) ,
ρ 10 = 0 π / 2 ρ ( θ i ) sin 2 θ i d θ i .
ρ ( θ i ) = { 1 2 [ sin 2 ( θ i θ t ) sin 2 ( θ i + θ t ) + tan 2 ( θ i θ t ) tan 2 ( θ i + θ t ) ] for     θ i θ c 1 for     θ i > θ c .
F λ x ( z ) = 0.
G λ x ( z ) = 2 π [ F λ x + ( z ) + F λ x ( z ) + F c , λ x ( z ) ] = ( 1 ρ 01 ) F 0 , λ x ( k 1 e b λ x S λ x z + k 2 e K c , λ x z ) ,
k 1 = 2 π ( a λ x b λ x + 1 ) { K c , λ x ( ρ 10 S 1 , λ x + S 2 , λ x ) + S λ x [ ( a λ x ρ 10 1 ) S 1 , λ x + ( ρ 10 a λ x ) S 2 , λ x ] } ( ( a λ x + b λ x ) ρ 10 1 ) ( K c , λ x 2 b λ x 2 S λ x 2 ) ,
k 2 = 2 π { K c , λ x 2 + ( S 2 , λ x S 1 , λ x ) K c , λ x S λ x [ S λ x b λ x 2 + ( a λ x + 1 ) ( S 1 , λ x + S 2 , λ x ) ] } K c , λ x 2 b λ x 2 S λ x 2 .
d F λ f + d z = a λ f S λ f F λ f + + S λ f F λ f + 1 2 γ x f ( z ) G λ x ( z ) ,
d F λ f d z = S λ f F λ f + + a λ f S λ f F λ f 1 2 γ x f ( z ) G λ x ( z ) ,
F λ f + ( 0 ) = ρ 10 F λ f ( 0 ) , F λ f ( z ) = 0.
K c , λ = μ a , λ + ( 1 g λ 2 ) μ s , λ .
μ a , λ = η λ K λ , μ s , λ ( 1 g λ ) = χ λ S λ ,
η λ = ( ϕ λ 1 ) ( 1 ω t r , λ ) / ζ d , λ ( ϕ λ + 1 ) ,           χ λ = ω t r , λ ( ϕ λ ϕ λ 1 ) / ( 2 ζ d , λ ) .
ϕ λ = ζ d , λ + ln ( 1 ζ d , λ ) ζ d , λ ln ( 1 + ζ d , λ ) ,
ω t r , λ = 2 ζ d , λ ln [ ( 1 + ζ d , λ ) / ( 1 ζ d , λ ) ] .
ζ d , λ 2 = 47 52 + 31 49 ω t r , λ 49 54 ω t r , λ 2 17 27 ω t r , λ 3 .
γ x f ( τ t r , λ x ) = ϵ f , λ x QY x f ( τ t r , λ x ) M ( τ t r , λ x ) .
u ( τ t r , λ x τ t r , λ x , 1 ) = { 1 if     τ t r , λ x τ t r , λ x , 1 0 if     τ t r , λ x < τ t r , λ x , 1 .
K c , λ x μ a , λ x + 2 ( 1 g λ x ) μ s , λ x = β t r , λ x ( 1 + ω t r , λ x ) .
G λ x ( τ t r , λ x ) = ( 1 ρ 01 ) F 0 , λ x ( k 1 e ζ d , λ x τ t r , λ x + k 2 e ζ c , λ x τ t r , λ x ) .
F 0 , λ x = 0 μ a , λ x G λ x ( z ) d z + F 0 , λ x ( R d , λ x + ρ 01 ) ,
k 1 = ζ d , λ x ( 1 R d , λ x ρ 01 1 ω t r , λ x k 2 1 + ω t r , λ x ) .
( 1 ρ 10 ) F λ f ( 0 ) = ( 1 ρ 10 ) F 0 , λ x ϵ f , λ x QY x f AT x f .
T x f δ ( τ t r , λ x , 0 ) = ( 1 ρ 10 ) ( a λ f + b λ f + 1 ) e ζ d , λ f r x f τ t r , λ x , 0 2 ( a λ f + b λ f ρ 10 ) .
T x f = 1 β t r , λ x 0 M ( τ t r , λ x ) G λ x ( τ t r , λ x ) T x f δ ( τ t r , λ x ) d τ t r , λ x .
T x f S ( τ t r , λ x , 1 ) = ( k 1 e ( ζ d , λ f r x f + ζ d , λ x ) τ t r , λ x , 1 ζ d , λ f r x f + ζ d , λ x + k 2 e ( ζ d , λ f r x f + ζ c , λ x ) τ t r , λ x , 1 ζ d , λ f r x f + ζ c , λ x ) T x f δ ( 0 ) β t r , λ x .
T x f H = T x f S ( 0 ) = ( k 1 ζ d , λ f r x f + ζ d , λ x + k 2 ζ d , λ f r x f + ζ c , λ x ) T x f δ ( 0 ) β t r , λ x .
T x f L ( τ t r , λ x , 1 , τ t r , λ x , 2 ) = T x f S ( τ t r , λ x , 1 ) T x f S ( τ t r , λ x , 2 ) ,
CF = 1 N z i = 1 N z [ G λ x MC ( i Δ z ) G λ x ( i Δ z ) ] 2 ,
k 2 0.137 log 10 ( R d , λ x ) 1.357.
E ( T x f ) = | T x f T x f MC T x f MC | ,

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