Abstract

We present computationally efficient and accurate semiempirical models of light transfer suitable for real-time diffuse reflectance spectroscopy. The models predict the diffuse reflectance of both (i) semi- infinite homogeneous and (ii) two-layer media exposed to normal and collimated light. The two-layer medium consisted of a plane-parallel slab of finite thickness over a semi-infinite layer with identical index of refraction but different absorption and scattering properties. The model accounted for absorption and anisotropic scattering, as well as for internal reflection at the medium/air interface. All media were assumed to be nonemitting, strongly forward scattering, with indices of refraction between 1.00 and 1.44 and transport single-scattering albedos between 0.50 and 0.99. First, simple analytical expressions for the diffuse reflectance of the semi-infinite and two-layer media considered were derived using the two-flux approximation. Then, parameters appearing in the analytical expression previously derived were instead fitted to match results from more accurate Monte Carlo simulations. A single semiempirical parameter was sufficient to relate the diffuse reflectance to the radiative properties and thickness of the semi- infinite and two-layer media. The present model can be used for a wide range of applications including noninvasive diagnosis of biological tissue.

© 2009 Optical Society of America

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Corrections

Laurent Pilon, Arka Bhowmik, Ri-Liang Heng, and Dmitry Yudovsky, "Simple and accurate expressions for diffuse reflectance of semi-infinite and two-layer absorbing and scattering media: erratum," Appl. Opt. 54, 6116-6117 (2015)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-54-19-6116

References

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2008 (7)

S. H. Tseng, A. Grant, and A. J. Durkin, “In-vivo determination of skin near-infrared optical properties using diffuse optical spectroscopy,” J. Biomed. Opt. 13, 014016(2008).
[CrossRef]

W. M. Kuebler, “How NIR is the future in blood flow monitoring?,” J. Appl. Physiol. 104, 905-906 (2008).
[CrossRef]

A. Torricelli, D. Contini, A. Pifferi, L. Spinelli, and R. Cubeddu, “Functional brain imaging by multi-wavelength time-resolved near infrared spectroscopy,” Opto-Electron. Rev. 16, 131-135(2008).
[CrossRef]

G. Zonios, A. Dimou, I. Bassukas, D. Galaris, A. Tsolakidis, and E. Kaxiras, “Melanin absorption spectroscopy: new method for noninvasive skin investigation and melanoma detection,” J. Biomed. Opt. 13, 014017 (2008).
[CrossRef]

H. Huang, H. Yu, H. Xu, and Y. Ying, “Near infrared spectroscopy for on/in-line monitoring of quality in foods and beverages: a review,” J. Food Eng. 87, 303-313 (2008).

S. L. Jacques, “Modeling tissue optics using Monte Carlo modeling: a tutorial,” Proc. SPIE 6854, 68540T (2008).
[CrossRef]

J. M. Carvano, “Compositional interpretation of the geometric albedo of asteroids,” Astron. Astrophys. 486, 1031-1038(2008).
[CrossRef]

2007 (4)

P. Edström, “Examination of the revised Kubelka-Munk theory: considerations of modeling strategies,” J. Opt. Soc. Am. 24), 548-556 (2007).
[CrossRef]

A. A. Gowen, C. P. O'Donnell, P. J. Cullen, G. Downey, and J. M. Frias, “Hyperspectral imaging-an emerging process analytical tool for food quality and safety control,” Trends Food Sci. Technol. 18, 590-598 (2007).
[CrossRef]

G. W. Heitschmidt, B. Park, K. C. Lawrence, W. R. Windham, and D. P. Smith, “Improved hyperspectral imaging system for fecal detection on poultry carcasses,” Trans. Am. Soc. Agric. Biol. Eng. 50, 1427-1432 (2007).

L. Khaodhiar, T. Dinh, K. T. Schomacker, S. V. Panasyuk, J. E. Freeman, R. Lew, T. Vo, A. A. Panasyuk, C. Lima, J. M. Giurini, T. E. Lyons, and A. Veves, “The use of medical hyperspectral technology to evaluate microcirculatory changes in diabetic foot ulcers and to predict clinical outcomes,” Diabetes Care 30, 903-910 (2007).
[CrossRef]

2006 (1)

2005 (1)

R. L. P. van Veen, A. Amelink, M. Menke-Pluymers, C. van der Pol, and H. Sterenborg, “Optical biopsy of breast tissue using differential path-length spectroscopy,” Phys. Med. Biol. 50, 2573-2581 (2005).
[CrossRef]

2004 (1)

I. Kim, M. S. Kim, Y. R. Chen, and S. G. Kong, “Detection of skin tumors on chicken carcasses using hyperspectral fluorescence imaging,” Trans. Am. Soc. Agric. Eng. 47, 1785-1792(2004).

2003 (1)

U. Utzinger and R. R. Richards-Kortum, “Fiber optic probes for biomedical optical spectroscopy,” J. Biomed. Opt. 8, 121-147 (2003).
[CrossRef]

2002 (4)

K. J. Zuzak, M. D. Schaeberle, E. N. Lewis, and I. W. Levin, “Visible reflectance hyperspectral imaging: characterization of a noninvasive, in-vivo system for determining tissue perfusion,” Anal. Chem. 74, 2021-2028 (2002).
[CrossRef]

D. Landgrebe, “Hyperspectral image data analysis,” IEEE Signal Processi. Mag. 19, 17-28 (2002).
[CrossRef]

S. L. Jacques, J. C. Ramella-Roman, and K. Lee, “Imaging skin pathology with polarized light,” J. Biomed. Opt. 7, 329-340 (2002).
[CrossRef]

Y. Lee and K. Hwang, “Skin thickness of Korean adults,” Surg. Radiol. Anat. 24, 183-189 (2002).
[CrossRef]

2001 (4)

I. V. Meglinski and S. J. Matcher, “Modeling of skin reflectance spectra,” Proc. SPIE 4241, 78-87 (2001).
[CrossRef]

E. S. Chalhoub and H. F. Campos Velho, “Simultaneous estimation of radiation phase function and albedo in natural waters,” J. Quant. Spectrosc. Radiat. Transfer 69, 137-149(2001).
[CrossRef]

R. Drezek, K. Sokolov, U. Utzinger, I. Boiko, A. Malpica, M. Follen, and R. Richards-Kortum, “Understanding the contributions of NADH and collagen to cervical tissue fluorescence spectra: Modeling, measurements, and implications,” J. Biomed. Opt. 6, 385-396 (2001).
[CrossRef]

D. Calzetti, “The dust opacity of star-forming galaxies,” Publ. Astron. Soc. Pac. 113, 1449-1485 (2001).
[CrossRef]

1998 (2)

1997 (1)

1996 (3)

G. I. Zonios, R. M. Cothren, J. T. Arendt, J. Wu, J. Van Dam, J. M. Crawford, R. Manoharan, and M. S. Feld, “Morphological model of human colon tissue fluorescence,” IEEE Trans. Biomed. Eng. 43, 113-122 (1996).
[CrossRef]

L. L. Richardson, “Remote sensing of algal bloom dynamics,” BioScience 46, 492-501 (1996).
[CrossRef]

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]

1995 (1)

L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef]

1994 (1)

H. Zeng, C. E. MacAulay, B. Palcic, and D. I. McLean, “Monte Carlo modeling of tissue autofluorescence measurement and imaging,” Proc. SPIE 2135, 94-104 (1994).
[CrossRef]

1993 (1)

1991 (2)

N. Yamada and S. Fujimura, “Nondestructive measurement of chlorophyll pigment content in plant leaves from three-color reflectance and transmittance,” Appl. Opt. 30, 3964-3973(1991).
[CrossRef]

S. L. Jacques and D. J. McAuliffe, “The melanosome: threshold temperature for explosive vaporization and internal absorption coefficient during pulsed laser irradiation,” Photochem. Photobiol. 53, 769-775 (1991).

1990 (1)

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 (3)

M. Keijzer, S. L. Jacques, S. A. Prahl, and A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148-154(1989).
[CrossRef]

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]

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).
[CrossRef]

1988 (2)

S. Wray, M. Cope, D. T. Delpy, J. S. Wyatt, and E. O. Reynolds, “Characterization of the near infrared absorption spectra of cytochrome aa3 and haemoglobin for the non-invasive monitoring of cerebral oxygenation.,” Biochim. Biophys. Acta Bioenergetics 933, 184-192 (1988).
[CrossRef]

A. P. Harris, M. J. Sendak, R. T. Donham, M. Thomas, and D. Duncan, “Absorption characteristics of human fetal hemoglobin at wavelengths used in pulse oximetry,” J. Clin. Monit. Comput. 4, 175-177 (1988).
[CrossRef]

1987 (4)

M. J. C. Van Gemert, A. J. Welch, W. M. Star, M. Motamedi, and W. F. Cheong, “Tissue optics for a slab geometry in the diffusion approximation,” Lasers Med. Sci. 2, 295-302(1987).
[CrossRef]

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

M. J. D. 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).
[CrossRef]

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, 98 (1987).

1981 (1)

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

1979 (1)

S. Takatani and M. D. Graham, “Theoretical analysis of diffuse reflectance from a two-layer tissue model,” IEEE Trans. Biomed. Eng. bme-26, 656-664 (1979).
[CrossRef]

1973 (1)

1963 (1)

D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” J. SIAM Control 11, 431-441(1963).

1945 (1)

1942 (1)

1941 (1)

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

1931 (1)

P. Kubelka and F. Munk, “A contribution to the optics of pigments,” Zeit. Technol. Phys. 12, 593-599 (1931).

1905 (1)

A. Schuster, “Radiation through a foggy atmosphere,” Astrophys. J. 21, 1-22 (1905).
[CrossRef]

Alter, C. A.

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

Amelink, A.

R. L. P. van Veen, A. Amelink, M. Menke-Pluymers, C. van der Pol, and H. Sterenborg, “Optical biopsy of breast tissue using differential path-length spectroscopy,” Phys. Med. Biol. 50, 2573-2581 (2005).
[CrossRef]

Anderson, R. R.

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

Arendt, J. T.

G. I. Zonios, R. M. Cothren, J. T. Arendt, J. Wu, J. Van Dam, J. M. Crawford, R. Manoharan, and M. S. Feld, “Morphological model of human colon tissue fluorescence,” IEEE Trans. Biomed. Eng. 43, 113-122 (1996).
[CrossRef]

Bao, H.

K. Zhou, Z. Ren, S. Lin, H. Bao, B. Guo, and H. Y. Shum, “Real-time smoke rendering using compensated ray marching,” in Proceedings of the Association for Computing Machinery's Special Interest Group on Graphics and Interactive Techniques, Vol. 1, pp. 1-12 (2008).

Baranoski, G. V. G.

A. Krishnaswamy and G. V. G. Baranoski, “A biophysically-based spectral model of light interaction with human skin,” in Computer Graphics Forum (Blackwell, 2004), Vol. 23, pp. 331-340.

Bassukas, I.

G. Zonios, A. Dimou, I. Bassukas, D. Galaris, A. Tsolakidis, and E. Kaxiras, “Melanin absorption spectroscopy: new method for noninvasive skin investigation and melanoma detection,” J. Biomed. Opt. 13, 014017 (2008).
[CrossRef]

Battarbee, H.

A. N. Yaroslavsky, A. V. Priezzhev, J. R. I. V. Yaroslavsky, and H. Battarbee, “Optics of blood,” in Handbook of Optical Biomedical Diagnostics, V. V. Tuchin, ed. (SPIE, 2002), pp. 169-216.

Boiko, I.

R. Drezek, K. Sokolov, U. Utzinger, I. Boiko, A. Malpica, M. Follen, and R. Richards-Kortum, “Understanding the contributions of NADH and collagen to cervical tissue fluorescence spectra: Modeling, measurements, and implications,” J. Biomed. Opt. 6, 385-396 (2001).
[CrossRef]

Calzetti, D.

D. Calzetti, “The dust opacity of star-forming galaxies,” Publ. Astron. Soc. Pac. 113, 1449-1485 (2001).
[CrossRef]

Campos Velho, H. F.

E. S. Chalhoub and H. F. Campos Velho, “Simultaneous estimation of radiation phase function and albedo in natural waters,” J. Quant. Spectrosc. Radiat. Transfer 69, 137-149(2001).
[CrossRef]

Carvano, J. M.

J. M. Carvano, “Compositional interpretation of the geometric albedo of asteroids,” Astron. Astrophys. 486, 1031-1038(2008).
[CrossRef]

Chalhoub, E. S.

E. S. Chalhoub and H. F. Campos Velho, “Simultaneous estimation of radiation phase function and albedo in natural waters,” J. Quant. Spectrosc. Radiat. Transfer 69, 137-149(2001).
[CrossRef]

Chandrasekhar, S.

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Chen, Y. R.

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A. P. Harris, M. J. Sendak, R. T. Donham, M. Thomas, and D. Duncan, “Absorption characteristics of human fetal hemoglobin at wavelengths used in pulse oximetry,” J. Clin. Monit. Comput. 4, 175-177 (1988).
[CrossRef]

Shen, D.

Shum, H. Y.

K. Zhou, Z. Ren, S. Lin, H. Bao, B. Guo, and H. Y. Shum, “Real-time smoke rendering using compensated ray marching,” in Proceedings of the Association for Computing Machinery's Special Interest Group on Graphics and Interactive Techniques, Vol. 1, pp. 1-12 (2008).

Smith, D. P.

G. W. Heitschmidt, B. Park, K. C. Lawrence, W. R. Windham, and D. P. Smith, “Improved hyperspectral imaging system for fecal detection on poultry carcasses,” Trans. Am. Soc. Agric. Biol. Eng. 50, 1427-1432 (2007).

Sokolov, K.

R. Drezek, K. Sokolov, U. Utzinger, I. Boiko, A. Malpica, M. Follen, and R. Richards-Kortum, “Understanding the contributions of NADH and collagen to cervical tissue fluorescence spectra: Modeling, measurements, and implications,” J. Biomed. Opt. 6, 385-396 (2001).
[CrossRef]

Spinelli, L.

A. Torricelli, D. Contini, A. Pifferi, L. Spinelli, and R. Cubeddu, “Functional brain imaging by multi-wavelength time-resolved near infrared spectroscopy,” Opto-Electron. Rev. 16, 131-135(2008).
[CrossRef]

Star, W. M.

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]

M. J. C. Van Gemert, A. J. Welch, W. M. Star, M. Motamedi, and W. F. Cheong, “Tissue optics for a slab geometry in the diffusion approximation,” Lasers Med. Sci. 2, 295-302(1987).
[CrossRef]

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, 98 (1987).

M. J. D. 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).
[CrossRef]

Sterenborg, H.

R. L. P. van Veen, A. Amelink, M. Menke-Pluymers, C. van der Pol, and H. Sterenborg, “Optical biopsy of breast tissue using differential path-length spectroscopy,” Phys. Med. Biol. 50, 2573-2581 (2005).
[CrossRef]

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]

Takatani, S.

S. Takatani and M. D. Graham, “Theoretical analysis of diffuse reflectance from a two-layer tissue model,” IEEE Trans. Biomed. Eng. bme-26, 656-664 (1979).
[CrossRef]

Thomas, M.

A. P. Harris, M. J. Sendak, R. T. Donham, M. Thomas, and D. Duncan, “Absorption characteristics of human fetal hemoglobin at wavelengths used in pulse oximetry,” J. Clin. Monit. Comput. 4, 175-177 (1988).
[CrossRef]

Torricelli, A.

A. Torricelli, D. Contini, A. Pifferi, L. Spinelli, and R. Cubeddu, “Functional brain imaging by multi-wavelength time-resolved near infrared spectroscopy,” Opto-Electron. Rev. 16, 131-135(2008).
[CrossRef]

Tseng, S. H.

S. H. Tseng, A. Grant, and A. J. Durkin, “In-vivo determination of skin near-infrared optical properties using diffuse optical spectroscopy,” J. Biomed. Opt. 13, 014016(2008).
[CrossRef]

Tsolakidis, A.

G. Zonios, A. Dimou, I. Bassukas, D. Galaris, A. Tsolakidis, and E. Kaxiras, “Melanin absorption spectroscopy: new method for noninvasive skin investigation and melanoma detection,” J. Biomed. Opt. 13, 014017 (2008).
[CrossRef]

Tuchin, V. V.

V. V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis (SPIE, 2007).

Utzinger, U.

U. Utzinger and R. R. Richards-Kortum, “Fiber optic probes for biomedical optical spectroscopy,” J. Biomed. Opt. 8, 121-147 (2003).
[CrossRef]

R. Drezek, K. Sokolov, U. Utzinger, I. Boiko, A. Malpica, M. Follen, and R. Richards-Kortum, “Understanding the contributions of NADH and collagen to cervical tissue fluorescence spectra: Modeling, measurements, and implications,” J. Biomed. Opt. 6, 385-396 (2001).
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O. W. Van Assendelft, Spectrophotometry of Haemoglobin Derivatives (Thomas, Springfield, 1970).

Van Dam, J.

G. I. Zonios, R. M. Cothren, J. T. Arendt, J. Wu, J. Van Dam, J. M. Crawford, R. Manoharan, and M. S. Feld, “Morphological model of human colon tissue fluorescence,” IEEE Trans. Biomed. Eng. 43, 113-122 (1996).
[CrossRef]

van der Pol, C.

R. L. P. van Veen, A. Amelink, M. Menke-Pluymers, C. van der Pol, and H. Sterenborg, “Optical biopsy of breast tissue using differential path-length spectroscopy,” Phys. Med. Biol. 50, 2573-2581 (2005).
[CrossRef]

Van Gemert, M. J. C.

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]

M. J. C. Van Gemert, A. J. Welch, W. M. Star, M. Motamedi, and W. F. Cheong, “Tissue optics for a slab geometry in the diffusion approximation,” Lasers Med. Sci. 2, 295-302(1987).
[CrossRef]

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, 98 (1987).

van Gemert, M. J. D.

M. J. D. 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).
[CrossRef]

van Veen, R. L. P.

R. L. P. van Veen, A. Amelink, M. Menke-Pluymers, C. van der Pol, and H. Sterenborg, “Optical biopsy of breast tissue using differential path-length spectroscopy,” Phys. Med. Biol. 50, 2573-2581 (2005).
[CrossRef]

Vargas, W. E.

Veves, A.

L. Khaodhiar, T. Dinh, K. T. Schomacker, S. V. Panasyuk, J. E. Freeman, R. Lew, T. Vo, A. A. Panasyuk, C. Lima, J. M. Giurini, T. E. Lyons, and A. Veves, “The use of medical hyperspectral technology to evaluate microcirculatory changes in diabetic foot ulcers and to predict clinical outcomes,” Diabetes Care 30, 903-910 (2007).
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Vo, T.

L. Khaodhiar, T. Dinh, K. T. Schomacker, S. V. Panasyuk, J. E. Freeman, R. Lew, T. Vo, A. A. Panasyuk, C. Lima, J. M. Giurini, T. E. Lyons, and A. Veves, “The use of medical hyperspectral technology to evaluate microcirculatory changes in diabetic foot ulcers and to predict clinical outcomes,” Diabetes Care 30, 903-910 (2007).
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Wang, L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef]

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

Welch, A. J.

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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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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M. Keijzer, S. L. Jacques, S. A. Prahl, and A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148-154(1989).
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[CrossRef]

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G. W. Heitschmidt, B. Park, K. C. Lawrence, W. R. Windham, and D. P. Smith, “Improved hyperspectral imaging system for fecal detection on poultry carcasses,” Trans. Am. Soc. Agric. Biol. Eng. 50, 1427-1432 (2007).

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S. Wray, M. Cope, D. T. Delpy, J. S. Wyatt, and E. O. Reynolds, “Characterization of the near infrared absorption spectra of cytochrome aa3 and haemoglobin for the non-invasive monitoring of cerebral oxygenation.,” Biochim. Biophys. Acta Bioenergetics 933, 184-192 (1988).
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S. Wray, M. Cope, D. T. Delpy, J. S. Wyatt, and E. O. Reynolds, “Characterization of the near infrared absorption spectra of cytochrome aa3 and haemoglobin for the non-invasive monitoring of cerebral oxygenation.,” Biochim. Biophys. Acta Bioenergetics 933, 184-192 (1988).
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Xu, H.

H. Huang, H. Yu, H. Xu, and Y. Ying, “Near infrared spectroscopy for on/in-line monitoring of quality in foods and beverages: a review,” J. Food Eng. 87, 303-313 (2008).

Yamada, N.

Yaroslavsky, A. N.

A. N. Yaroslavsky, A. V. Priezzhev, J. R. I. V. Yaroslavsky, and H. Battarbee, “Optics of blood,” in Handbook of Optical Biomedical Diagnostics, V. V. Tuchin, ed. (SPIE, 2002), pp. 169-216.

Yaroslavsky, J. R. I. V.

A. N. Yaroslavsky, A. V. Priezzhev, J. R. I. V. Yaroslavsky, and H. Battarbee, “Optics of blood,” in Handbook of Optical Biomedical Diagnostics, V. V. Tuchin, ed. (SPIE, 2002), pp. 169-216.

Ying, Y.

H. Huang, H. Yu, H. Xu, and Y. Ying, “Near infrared spectroscopy for on/in-line monitoring of quality in foods and beverages: a review,” J. Food Eng. 87, 303-313 (2008).

Yoon, G.

Yu, H.

H. Huang, H. Yu, H. Xu, and Y. Ying, “Near infrared spectroscopy for on/in-line monitoring of quality in foods and beverages: a review,” J. Food Eng. 87, 303-313 (2008).

Zeng, H.

H. Zeng, C. E. MacAulay, B. Palcic, and D. I. McLean, “Monte Carlo modeling of tissue autofluorescence measurement and imaging,” Proc. SPIE 2135, 94-104 (1994).
[CrossRef]

Zheng, L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef]

Zhou, K.

K. Zhou, Z. Ren, S. Lin, H. Bao, B. Guo, and H. Y. Shum, “Real-time smoke rendering using compensated ray marching,” in Proceedings of the Association for Computing Machinery's Special Interest Group on Graphics and Interactive Techniques, Vol. 1, pp. 1-12 (2008).

Zonios, G.

G. Zonios, A. Dimou, I. Bassukas, D. Galaris, A. Tsolakidis, and E. Kaxiras, “Melanin absorption spectroscopy: new method for noninvasive skin investigation and melanoma detection,” J. Biomed. Opt. 13, 014017 (2008).
[CrossRef]

Zonios, G. I.

G. I. Zonios, R. M. Cothren, J. T. Arendt, J. Wu, J. Van Dam, J. M. Crawford, R. Manoharan, and M. S. Feld, “Morphological model of human colon tissue fluorescence,” IEEE Trans. Biomed. Eng. 43, 113-122 (1996).
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K. J. Zuzak, M. D. Schaeberle, E. N. Lewis, and I. W. Levin, “Visible reflectance hyperspectral imaging: characterization of a noninvasive, in-vivo system for determining tissue perfusion,” Anal. Chem. 74, 2021-2028 (2002).
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BioScience (1)

L. L. Richardson, “Remote sensing of algal bloom dynamics,” BioScience 46, 492-501 (1996).
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Comput. Methods Programs Biomed. (1)

L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef]

Diabetes Care (1)

L. Khaodhiar, T. Dinh, K. T. Schomacker, S. V. Panasyuk, J. E. Freeman, R. Lew, T. Vo, A. A. Panasyuk, C. Lima, J. M. Giurini, T. E. Lyons, and A. Veves, “The use of medical hyperspectral technology to evaluate microcirculatory changes in diabetic foot ulcers and to predict clinical outcomes,” Diabetes Care 30, 903-910 (2007).
[CrossRef]

IEEE J. Quantum Electron. (1)

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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IEEE Signal Processi. Mag. (1)

D. Landgrebe, “Hyperspectral image data analysis,” IEEE Signal Processi. Mag. 19, 17-28 (2002).
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IEEE Trans. Biomed. Eng. (3)

G. I. Zonios, R. M. Cothren, J. T. Arendt, J. Wu, J. Van Dam, J. M. Crawford, R. Manoharan, and M. S. Feld, “Morphological model of human colon tissue fluorescence,” IEEE Trans. Biomed. Eng. 43, 113-122 (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).
[CrossRef]

S. Takatani and M. D. Graham, “Theoretical analysis of diffuse reflectance from a two-layer tissue model,” IEEE Trans. Biomed. Eng. bme-26, 656-664 (1979).
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W. M. Kuebler, “How NIR is the future in blood flow monitoring?,” J. Appl. Physiol. 104, 905-906 (2008).
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G. Zonios, A. Dimou, I. Bassukas, D. Galaris, A. Tsolakidis, and E. Kaxiras, “Melanin absorption spectroscopy: new method for noninvasive skin investigation and melanoma detection,” J. Biomed. Opt. 13, 014017 (2008).
[CrossRef]

U. Utzinger and R. R. Richards-Kortum, “Fiber optic probes for biomedical optical spectroscopy,” J. Biomed. Opt. 8, 121-147 (2003).
[CrossRef]

S. H. Tseng, A. Grant, and A. J. Durkin, “In-vivo determination of skin near-infrared optical properties using diffuse optical spectroscopy,” J. Biomed. Opt. 13, 014016(2008).
[CrossRef]

R. Drezek, K. Sokolov, U. Utzinger, I. Boiko, A. Malpica, M. Follen, and R. Richards-Kortum, “Understanding the contributions of NADH and collagen to cervical tissue fluorescence spectra: Modeling, measurements, and implications,” J. Biomed. Opt. 6, 385-396 (2001).
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J. Food Eng. (1)

H. Huang, H. Yu, H. Xu, and Y. Ying, “Near infrared spectroscopy for on/in-line monitoring of quality in foods and beverages: a review,” J. Food Eng. 87, 303-313 (2008).

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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, 98 (1987).

Lasers Med. Sci. (1)

M. J. C. Van Gemert, A. J. Welch, W. M. Star, M. Motamedi, and W. F. Cheong, “Tissue optics for a slab geometry in the diffusion approximation,” Lasers Med. Sci. 2, 295-302(1987).
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M. Keijzer, S. L. Jacques, S. A. Prahl, and A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148-154(1989).
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[CrossRef]

Opto-Electron. Rev. (1)

A. Torricelli, D. Contini, A. Pifferi, L. Spinelli, and R. Cubeddu, “Functional brain imaging by multi-wavelength time-resolved near infrared spectroscopy,” Opto-Electron. Rev. 16, 131-135(2008).
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S. L. Jacques, “Skin optics,” Oregon Medical Laser Center News 1998, pp. 1-9 (1998).

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S. L. Jacques and D. J. McAuliffe, “The melanosome: threshold temperature for explosive vaporization and internal absorption coefficient during pulsed laser irradiation,” Photochem. Photobiol. 53, 769-775 (1991).

Phys. Med. Biol. (1)

R. L. P. van Veen, A. Amelink, M. Menke-Pluymers, C. van der Pol, and H. Sterenborg, “Optical biopsy of breast tissue using differential path-length spectroscopy,” Phys. Med. Biol. 50, 2573-2581 (2005).
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S. L. Jacques, “Modeling tissue optics using Monte Carlo modeling: a tutorial,” Proc. SPIE 6854, 68540T (2008).
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I. Kim, M. S. Kim, Y. R. Chen, and S. G. Kong, “Detection of skin tumors on chicken carcasses using hyperspectral fluorescence imaging,” Trans. Am. Soc. Agric. Eng. 47, 1785-1792(2004).

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I. Lux and L. Koblinger, Monte Carlo Particle Transport Methods: Neutron and Photon Calculations (CRC Press, 1991), p. 650.

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Figures (10)

Fig. 1
Fig. 1

Schematic of the semi-infinite and two-layer media considered along with coordinate system and boundary conditions.

Fig. 2
Fig. 2

Diffuse reflectance of a homogeneous semi-infinite medium predicted by Monte Carlo simulations (symbols) and the two-flux approximation [Eqs. (9, 19)] (dashed curve) for n 1 = 1.00 , 0.3 ω t r 1.00 , and 0.70 g 0.90 .

Fig. 3
Fig. 3

Diffuse reflectance of a semi-infinite homogeneous medium predicted by Monte Carlo simulations and Eqs. (25) for n 1 = 1.00 , 1.33, 1.44, 1.77, and 2.00, 0.3 ω t r 1.0 , and 0.7 g 0.9 .

Fig. 4
Fig. 4

(a) Diffuse reflectance of two-layer optical medium R ˜ = ( a 1 , a 2 , Y 1 ) as a function of Y 1 predicted by Eq. (13). Asymptotic values of R ( a ) were computed from Eq. (9). (b) Reduced diffuse reflectance R * ˜ given by Eq. (29) as a function of Y 1 for different values of a 1 and a 2 ; ρ 01 = 0.033 and ρ 10 = 0.56 . The legend applies to both figures.

Fig. 5
Fig. 5

Diffuse reflectance R = predicted by Monte Carlo simulations as a function of Y 1 and corresponding reduced reflectance R * [Eq. (32)] for n 1 = 1.44 and (a), (b)  ω t r , 2 = 0.479 and (c), (d)  ω t r , 2 = 0.958 . Predictions of R = by Eqs. (33, 34) are also shown (solid curve).

Fig. 6
Fig. 6

(a) Relationship between ω t r , 2 and 1 / α determined for n 1 = n 2 = 1.44 and 0.70 g 1 = g 2 0.90 . (b) Relationship between ω t r , 2 and 1 / α [Eq. (35) and Table 2] for n 1 = n 2 = 1.00 , 1.33, and 1.44, and 0.70 g 1 = g 2 0.90 .

Fig. 7
Fig. 7

(a) Relative error between predictions by Eq. (34) and Monte Carlo simulations for n 1 = n 2 = 1.44 . (b) Histogram of the relative error for 10,000 simulations for n 1 = n 2 = 1.00 , 1.33, and 1.44.

Fig. 8
Fig. 8

Spectral molar extinction coefficient of human oxyhemoglobin and deoxyhemoglobin in the visible range (480 to 700 nm ) [51].

Fig. 9
Fig. 9

Comparisons of diffuse reflectance of skin predicted by Monte Carlo simulations and by Eq. (34) as a function of wavelength for f m e l = 1.0 % , f b l o o d = 2.5 % , L 1 = 100 μm , and S O 2 = 0 and 100%.

Fig. 10
Fig. 10

Experimentally measured diffuse reflectance from the top of the index finger of a healthy, Caucasian subject along with reconstructed reflectance predicted by Eqs. (33, 34, 35) and Eqs. (36, 37, 38, 39, 40, 41, 42) with best-fit parameters f m e l = 1.03 % , f b l o o d = 2.79 % , S O 2 = 29.0 % and L 1 = 66 μm .

Tables (2)

Tables Icon

Table 1 Regression Coefficients ( A i ) 0 i 3 and ( B i ) 0 i 3 Used in Eqs. (26, 27) to Estimate the Diffuse Reflectance R of a Semi-Infinite Homogeneous Medium with Index of Refraction n 1 = 1.33 , 1.44, 1.77, and 2.00

Tables Icon

Table 2 Regression Coefficients in the Expression of 1 / α Given by Eq. (35)

Equations (43)

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s ^ · I λ ( r ^ , s ^ ) = μ a , λ I λ ( r ^ , s ^ ) μ s , λ I λ ( r ^ , s ^ ) + μ s , λ 4 π 4 π I λ ( r ^ , s ^ i ) Φ λ ( s ^ i , s ^ ) d Ω i ,
ω λ = μ s , λ μ a , λ + μ s , λ .
Φ λ ( s ^ i , s ^ ) = 1 g λ ( 1 + g λ 2 2 g λ cos Θ ) 3 / 2 .
g λ = 1 4 π 4 π Φ λ ( s ^ i , s ^ ) cos Θ d Ω i .
ω t r = μ s , t r μ s , t r + μ a = μ s ( 1 g ) μ s ( 1 g ) + μ a .
F + ( z ) = 2 π 0 π / 2 I ( z , θ ) cos θ sin θ d θ , F ( z ) = 2 π π / 2 π I ( z , θ ) cos θ sin θ d θ ,
1 S d F + d z = a F + + F ,
1 S d F d z = F + + a F ,
R ˜ ( a ) = ρ 01 + ( 1 ρ 01 ) ( 1 ρ 10 ) R ˜ d ( a ) 1 ρ 10 R ˜ d ( a ) ,
ρ 01 = ( n 1 n 0 n 1 + n 0 ) 2 .
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 ,
R ˜ = ( a 1 , a 2 , Y 1 ) = ρ 01 + ( 1 ρ 01 ) ( 1 ρ 10 ) [ b 1 R ˜ d ( a 2 ) + ( 1 a 1 R ˜ d ( a 2 ) ) tanh ( Y 1 ) ] b 1 ( 1 ρ 10 R ˜ d ( a 2 ) ) + [ a 1 ( ρ 10 R ˜ d ( a 2 ) + 1 ) ρ 10 2 R ˜ d ( a 2 ) ] tanh ( Y 1 ) ,
Φ λ ( s ^ i , s ^ ) = [ 4 g δ ( 1 cos Θ ) + ( 1 g ) ] .
μ a = η K , μ s , tr = χ S ,
η = ( ϕ 1 ) ( 1 ω tr ) / ζ ( ϕ + 1 ) , χ = ω tr ( ϕ ϕ 1 ) / ( 2 ζ ) .
ϕ = ζ + ln ( 1 ζ ) ζ ln ( 1 + ζ ) ,
ω t r = 2 ζ ln [ ( 1 + ζ ) / ( 1 ζ ) ] .
ζ 2 = 47 52 + 31 49 ω t r 49 54 ω t r 2 17 27 ω t r 3 .
Y 1 = ζ ( μ a + μ s , t r ) L 1 = ζ τ t r , 1 ,
I ( 0 , θ ) = ( 1 ρ 01 ) q 0 δ ( θ ) + ρ ( θ ) I ( 0 , π θ ) for     0 θ π / 2.
I ( z , θ ) = 0 for     π θ π .
I r ( θ t ) = ρ 01 q 0 δ ( π θ t ) + [ 1 ρ ( θ i ) ] I ( 0 , π θ i ) for     0 θ i π / 2 ,
q r = 2 π π / 2 π [ 1 ρ ( θ t ) ] I ( 0 , π θ t ) cos θ t sin θ t d θ t .
R ( ω t r ) = [ 1 ρ 01 ] [ 1 ρ ^ 10 ( ω t r ) ] R ^ d ( ω t r ) 1 ρ ^ 10 ( ω t r ) R ^ d ( ω t r ) ,
ρ ^ 10 ( ω t r ) = ρ 10 + i = 0 i = N A i [ a ( ω t r ) ] i ,
R ^ d ( ω t r ) = R ˜ d ( a ( ω t r ) ) + i = 0 i = N B i [ a ( ω t r ) ] i ,
R ˜ = ( a 1 , a 2 , Y 1 ) Y 1 0 R ˜ ( a 2 ) , R ˜ = ( a 1 , a 2 , Y 1 ) Y 1 R ˜ ( a 1 ) ,
R * ˜ = R ˜ = ( a 1 , a 2 , Y 1 ) R ˜ ( a 2 ) R ˜ ( a 1 ) R ˜ ( a 2 ) ,
R * ˜ ( α ˜ , Y 1 ) = tanh ( Y 1 ) 1 / α ˜ + ( 1 1 / α ˜ ) tanh ( Y 1 ) ,
α ˜ = 1 + ρ 10 + R ˜ d ( a 2 ) a 1 [ ρ 10 R ˜ d ( a 2 ) + 1 ] a 1 2 1 [ ρ 10 R ˜ d ( a 2 ) 1 ] .
R * = R = R ( ω t r , 2 ) R ( ω t r , 1 ) R ( ω t r , 2 ) .
R * = tanh ( Y 1 ) 1 / α + ( 1 1 / α ) tanh ( Y 1 ) ,
R = = R * [ R ( n 1 , ω t r ,1 ) R ( n 1 , ω t r ,2 ) ] + R ( n 1 , ω t r ,2 ) ,
1 / α = C ( n 1 ) ω t r ,2 2 + D ( n 1 ) ω t r ,2 + E ( n 1 ) ,
R * Y 1 ( Y 1 = 0 , ω t r , 1 , ω t r , 2 ) = α .
μ a , 1 ( λ ) = μ a , m e l f m e l + ( 1 f m e l ) μ a , b a c k ,
μ a , b a c k ( λ ) = 7.84 × 10 8 λ 3.255 .
μ a , m e l ( λ ) = 6.60 × 10 11 λ 3.33 .
μ a , 2 ( λ ) = f b l o o d μ a , b l o o d ( λ ) + ( 1 f b l o o d ) μ a , b a c k ( λ ) ,
μ a , o x y ( λ ) = ϵ o x y ( λ ) C h e m e S O 2 / 66 , 500 ,
μ a , d e o x y ( λ ) = ϵ d e o x y ( λ ) C h e m e ( 1 S O 2 ) / 66 , 500 ,
μ s , t r = C t r λ k ,

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