Abstract

Accurate and rapid estimation of fluence, reflectance, and absorbance in multilayered biological media has been essential in many biophotonics applications that aim to diagnose, cure, or model in vivo tissue. The radiative transfer equation (RTE) rigorously models light transfer in absorbing and scattering media. However, analytical solutions to the RTE are limited even in simple homogeneous or plane media. Monte Carlo simulation has been used extensively to solve the RTE. However, Monte Carlo simulation is computationally intensive and may not be practical for applications that demand real-time results. Instead, the diffusion approximation has been shown to provide accurate estimates of light transport in strongly scattering tissue. The diffusion approximation is a greatly simplified model and produces analytical solutions for the reflectance and absorbance in tissue. However, the diffusion approximation breaks down if tissue is strongly absorbing, which is common in the visible part of the spectrum or in applications that involve darkly pigmented skin and/or high local volumes of blood such as port-wine stain therapy or reconstructive flap monitoring. In these cases, a model of light transfer that can accommodate both strongly and weakly absorbing regimes is required. Here we present a model of light transfer through layered biological media that represents skin with two strongly scattering and one strongly absorbing layer.

© 2011 Optical Society of America

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    [CrossRef] [PubMed]

2010 (2)

M. R. Pharaon, T. Scholz, S. Bogdanoff, D. Cuccia, A. J. Durkin, D. B. Hoyt, and G. R. D. Evans, “Early detection of complete vascular occlusion in a pedicle flap model using quantitation spectral imaging,” Plast. Reconstr. Surg. 126, 1924–1935 (2010).
[CrossRef] [PubMed]

R. B. Saager, D. J. Cuccia, and A. J. Durkin, “Determination of optical properties of turbid media spanning visible and near-infrared regimes via spatially modulated quantitative spectroscopy,” J. Biomed. Opt. 15, 017012 (2010).
[CrossRef] [PubMed]

2009 (1)

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
[CrossRef] [PubMed]

2007 (2)

2006 (4)

K. P. Nielsen, L. Zhao, J. J. Stamnes, K. Stamnes, and J. Moan, “The importance of the depth distribution of melanin in skin for DNA protection and other photobiological processes,” J. Photochem. Photobiol., B 82, 194–198 (2006).
[CrossRef]

T. Gambichler, R. Matip, G. Moussa, P. Altmeyer, and K. Hoffmann, “In vivo data of epidermal thickness evaluated by optical coherence tomography: effects of age, gender, skin type, and anatomic site,” J. Dermatol. Sci. 44, 145–152 (2006).
[CrossRef] [PubMed]

L. Kocsis, P. Herman, and A. Eke, “The modified Beer–Lambert law revisited,” Phys. Med. Biol. 51, N91 (2006).
[CrossRef] [PubMed]

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]

2004 (3)

A. Sassaroli and S. Fantini, “Comment on the modified Beer–Lambert law for scattering media,” Phys. Med. Biol. 49, N255 (2004).
[CrossRef] [PubMed]

A. Krishnaswamy and G. V. G. Baranoski, “A biophysically based spectral model of light interaction with human skin,” Comput. Graph. Forum 23, 331–340 (2004).
[CrossRef]

B. Jung, B. Choi, A. J. Durkin, K. M. Kelly, and J. S. Nelson, “Characterization of port wine stain skin erythema and melanin content using cross polarized diffuse reflectance imaging,” Lasers Surg. Med. 34, 174–181 (2004).
[CrossRef] [PubMed]

2003 (3)

T. Tadokoro, N. Kobayashi, B. Z. Zmudzka, S. Ito, K. Wakamatsu, Y. Yamaguchi, K. S. Korossy, S. A. Miller, J. Z. Beer, and V. J. Hearing, “UV-induced DNA damage and melanin content in human skin differing in racial/ethnic origin,” FASEB J. 17, 1177–1179 (2003).
[CrossRef] [PubMed]

J. Sandby-Moller, T. Poulsen, and H. C. Wulf, “Epidermal thickness at different body sites: relationship to age, gender, pigmentation, blood content, skin type and smoking habits,” Acta Derm.-Venereol. 83, 410–413 (2003).
[CrossRef] [PubMed]

M. F. Modest, Radiative Heat Transfer (Academic, 2003).

2002 (1)

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

2000 (2)

V. V. Tuchin, Tissue Optics (SPIE, 2000).

T. Dwyer, G. Prota, L. Blizzard, R. Ashbolt, and M. R. Vincensi, “Melanin density and melanin type predict melanocytic naevi in 19–20 year olds of northern European ancestry,” Melanoma Res. 10, 387–394 (2000).
[CrossRef] [PubMed]

1999 (1)

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, and M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

1998 (2)

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]

G. Alexandrakis, T. J. Farrell, and M. S. Patterson, “Accuracy of the diffusion approximation in determining the optical properties of a two-layer turbid medium,” Appl. Opt. 37, 7401–7409 (1998).
[CrossRef]

1997 (2)

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]

A. R. Young, “Chromophores in human skin,” Phys. Med. Biol. 42, 789–802 (1997).
[CrossRef] [PubMed]

1996 (1)

S. L. Jacques, “Origins of tissue optical properties in the UVA, visible, and NIR regions,” in OSA TOPS on Advances in Optical Imaging and Photon Migration, R.R.Alfano and J.G.Fujimoto, eds. (Optical Society of America, 1996), Vol.  2, pp. 364–369.

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] [PubMed]

1994 (1)

1993 (1)

G. Reindert, G. A. Jan, F. M. d. M. Frits, and W. J. Henk, “Similarity relations for anisotropic scattering in absorbing media,” Opt. Eng. 32, 244–252 (1993).
[CrossRef]

1992 (1)

R. Marchesini, C. Clemente, E. Pignoli, and M. Brambilla, “Optical properties of in vitro epidermis and their possible relationship with optical properties of in vivo skin,” J. Photochem. Photobiol., B 16, 127–140 (1992).
[CrossRef]

1989 (1)

1981 (2)

S. Wan, R. Anderson, and J. A. Parrish, “Analytical modeling for the optical properties of the skin with in vitro and in vivo applications,” Photochem. Photobiol. 34, 493–499 (1981).
[CrossRef] [PubMed]

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

1960 (1)

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

1955 (1)

W. F. W. Southwood, “The thickness of the skin,” Plast. Reconstr. Surg. 15, 423–429 (1955).
[CrossRef]

Alexandrakis, G.

Altmeyer, P.

T. Gambichler, R. Matip, G. Moussa, P. Altmeyer, and K. Hoffmann, “In vivo data of epidermal thickness evaluated by optical coherence tomography: effects of age, gender, skin type, and anatomic site,” J. Dermatol. Sci. 44, 145–152 (2006).
[CrossRef] [PubMed]

Anderson, R.

S. Wan, R. Anderson, and J. A. Parrish, “Analytical modeling for the optical properties of the skin with in vitro and in vivo applications,” Photochem. Photobiol. 34, 493–499 (1981).
[CrossRef] [PubMed]

Anderson, R. R.

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

Ashbolt, R.

T. Dwyer, G. Prota, L. Blizzard, R. Ashbolt, and M. R. Vincensi, “Melanin density and melanin type predict melanocytic naevi in 19–20 year olds of northern European ancestry,” Melanoma Res. 10, 387–394 (2000).
[CrossRef] [PubMed]

Ayers, F. R.

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
[CrossRef] [PubMed]

Baranoski, G. V. G.

A. Krishnaswamy and G. V. G. Baranoski, “A biophysically based spectral model of light interaction with human skin,” Comput. Graph. Forum 23, 331–340 (2004).
[CrossRef]

Beer, J. Z.

T. Tadokoro, N. Kobayashi, B. Z. Zmudzka, S. Ito, K. Wakamatsu, Y. Yamaguchi, K. S. Korossy, S. A. Miller, J. Z. Beer, and V. J. Hearing, “UV-induced DNA damage and melanin content in human skin differing in racial/ethnic origin,” FASEB J. 17, 1177–1179 (2003).
[CrossRef] [PubMed]

Berns, M. W.

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, and M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

Bevilacqua, F.

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
[CrossRef] [PubMed]

Blizzard, L.

T. Dwyer, G. Prota, L. Blizzard, R. Ashbolt, and M. R. Vincensi, “Melanin density and melanin type predict melanocytic naevi in 19–20 year olds of northern European ancestry,” Melanoma Res. 10, 387–394 (2000).
[CrossRef] [PubMed]

Bogdanoff, S.

M. R. Pharaon, T. Scholz, S. Bogdanoff, D. Cuccia, A. J. Durkin, D. B. Hoyt, and G. R. D. Evans, “Early detection of complete vascular occlusion in a pedicle flap model using quantitation spectral imaging,” Plast. Reconstr. Surg. 126, 1924–1935 (2010).
[CrossRef] [PubMed]

Brambilla, M.

R. Marchesini, C. Clemente, E. Pignoli, and M. Brambilla, “Optical properties of in vitro epidermis and their possible relationship with optical properties of in vivo skin,” J. Photochem. Photobiol., B 16, 127–140 (1992).
[CrossRef]

Chandrasekhar, S.

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

Choi, B.

B. Jung, B. Choi, A. J. Durkin, K. M. Kelly, and J. S. Nelson, “Characterization of port wine stain skin erythema and melanin content using cross polarized diffuse reflectance imaging,” Lasers Surg. Med. 34, 174–181 (2004).
[CrossRef] [PubMed]

Clemente, C.

R. Marchesini, C. Clemente, E. Pignoli, and M. Brambilla, “Optical properties of in vitro epidermis and their possible relationship with optical properties of in vivo skin,” J. Photochem. Photobiol., B 16, 127–140 (1992).
[CrossRef]

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]

Cuccia, D.

M. R. Pharaon, T. Scholz, S. Bogdanoff, D. Cuccia, A. J. Durkin, D. B. Hoyt, and G. R. D. Evans, “Early detection of complete vascular occlusion in a pedicle flap model using quantitation spectral imaging,” Plast. Reconstr. Surg. 126, 1924–1935 (2010).
[CrossRef] [PubMed]

Cuccia, D. J.

R. B. Saager, D. J. Cuccia, and A. J. Durkin, “Determination of optical properties of turbid media spanning visible and near-infrared regimes via spatially modulated quantitative spectroscopy,” J. Biomed. Opt. 15, 017012 (2010).
[CrossRef] [PubMed]

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
[CrossRef] [PubMed]

A. Yafi, T. S. Vetter, T. Scholz, S. Patel, R. B. Saager, D. J. Cuccia, G. R. Evans, and A. J. Durkin, “Postoperative quantitative assessment of reconstructive tissue status in a cutaneous flap model using spatial frequency domain imaging,” Plast. Reconstr. Surg. 127, 117–130 (2011).
[CrossRef] [PubMed]

Durkin, A. J.

M. R. Pharaon, T. Scholz, S. Bogdanoff, D. Cuccia, A. J. Durkin, D. B. Hoyt, and G. R. D. Evans, “Early detection of complete vascular occlusion in a pedicle flap model using quantitation spectral imaging,” Plast. Reconstr. Surg. 126, 1924–1935 (2010).
[CrossRef] [PubMed]

R. B. Saager, D. J. Cuccia, and A. J. Durkin, “Determination of optical properties of turbid media spanning visible and near-infrared regimes via spatially modulated quantitative spectroscopy,” J. Biomed. Opt. 15, 017012 (2010).
[CrossRef] [PubMed]

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
[CrossRef] [PubMed]

B. Jung, B. Choi, A. J. Durkin, K. M. Kelly, and J. S. Nelson, “Characterization of port wine stain skin erythema and melanin content using cross polarized diffuse reflectance imaging,” Lasers Surg. Med. 34, 174–181 (2004).
[CrossRef] [PubMed]

A. Yafi, T. S. Vetter, T. Scholz, S. Patel, R. B. Saager, D. J. Cuccia, G. R. Evans, and A. J. Durkin, “Postoperative quantitative assessment of reconstructive tissue status in a cutaneous flap model using spatial frequency domain imaging,” Plast. Reconstr. Surg. 127, 117–130 (2011).
[CrossRef] [PubMed]

Dwyer, T.

T. Dwyer, G. Prota, L. Blizzard, R. Ashbolt, and M. R. Vincensi, “Melanin density and melanin type predict melanocytic naevi in 19–20 year olds of northern European ancestry,” Melanoma Res. 10, 387–394 (2000).
[CrossRef] [PubMed]

Dykes, P. J.

P. J. Matts, P. J. Dykes, and R. Marks, “The distribution of melanin in skin determined in vivo,” Br. J. Dermatol. 156, 620–628 (2007).
[CrossRef] [PubMed]

Eke, A.

L. Kocsis, P. Herman, and A. Eke, “The modified Beer–Lambert law revisited,” Phys. Med. Biol. 51, N91 (2006).
[CrossRef] [PubMed]

Elias, M.

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).
[CrossRef] [PubMed]

Evans, G. R.

A. Yafi, T. S. Vetter, T. Scholz, S. Patel, R. B. Saager, D. J. Cuccia, G. R. Evans, and A. J. Durkin, “Postoperative quantitative assessment of reconstructive tissue status in a cutaneous flap model using spatial frequency domain imaging,” Plast. Reconstr. Surg. 127, 117–130 (2011).
[CrossRef] [PubMed]

Evans, G. R. D.

M. R. Pharaon, T. Scholz, S. Bogdanoff, D. Cuccia, A. J. Durkin, D. B. Hoyt, and G. R. D. Evans, “Early detection of complete vascular occlusion in a pedicle flap model using quantitation spectral imaging,” Plast. Reconstr. Surg. 126, 1924–1935 (2010).
[CrossRef] [PubMed]

Fantini, S.

A. Sassaroli and S. Fantini, “Comment on the modified Beer–Lambert law for scattering media,” Phys. Med. Biol. 49, N255 (2004).
[CrossRef] [PubMed]

Farrell, T. J.

Feng, T.-C.

Fishkin, J. B.

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, and M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

Frigerio, J.

Frits, F. M. d. M.

G. Reindert, G. A. Jan, F. M. d. M. Frits, and W. J. Henk, “Similarity relations for anisotropic scattering in absorbing media,” Opt. Eng. 32, 244–252 (1993).
[CrossRef]

Gambichler, T.

T. Gambichler, R. Matip, G. Moussa, P. Altmeyer, and K. Hoffmann, “In vivo data of epidermal thickness evaluated by optical coherence tomography: effects of age, gender, skin type, and anatomic site,” J. Dermatol. Sci. 44, 145–152 (2006).
[CrossRef] [PubMed]

Haskell, R. C.

Hearing, V. J.

T. Tadokoro, N. Kobayashi, B. Z. Zmudzka, S. Ito, K. Wakamatsu, Y. Yamaguchi, K. S. Korossy, S. A. Miller, J. Z. Beer, and V. J. Hearing, “UV-induced DNA damage and melanin content in human skin differing in racial/ethnic origin,” FASEB J. 17, 1177–1179 (2003).
[CrossRef] [PubMed]

Henk, W. J.

G. Reindert, G. A. Jan, F. M. d. M. Frits, and W. J. Henk, “Similarity relations for anisotropic scattering in absorbing media,” Opt. Eng. 32, 244–252 (1993).
[CrossRef]

Herman, P.

L. Kocsis, P. Herman, and A. Eke, “The modified Beer–Lambert law revisited,” Phys. Med. Biol. 51, N91 (2006).
[CrossRef] [PubMed]

Hoffmann, K.

T. Gambichler, R. Matip, G. Moussa, P. Altmeyer, and K. Hoffmann, “In vivo data of epidermal thickness evaluated by optical coherence tomography: effects of age, gender, skin type, and anatomic site,” J. Dermatol. Sci. 44, 145–152 (2006).
[CrossRef] [PubMed]

Hoyt, D. B.

M. R. Pharaon, T. Scholz, S. Bogdanoff, D. Cuccia, A. J. Durkin, D. B. Hoyt, and G. R. D. Evans, “Early detection of complete vascular occlusion in a pedicle flap model using quantitation spectral imaging,” Plast. Reconstr. Surg. 126, 1924–1935 (2010).
[CrossRef] [PubMed]

Hwang, K.

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

Ito, S.

T. Tadokoro, N. Kobayashi, B. Z. Zmudzka, S. Ito, K. Wakamatsu, Y. Yamaguchi, K. S. Korossy, S. A. Miller, J. Z. Beer, and V. J. Hearing, “UV-induced DNA damage and melanin content in human skin differing in racial/ethnic origin,” FASEB J. 17, 1177–1179 (2003).
[CrossRef] [PubMed]

Jacques, S. L.

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]

S. L. Jacques, “Origins of tissue optical properties in the UVA, visible, and NIR regions,” in OSA TOPS on Advances in Optical Imaging and Photon Migration, R.R.Alfano and J.G.Fujimoto, eds. (Optical Society of America, 1996), Vol.  2, pp. 364–369.

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] [PubMed]

Jan, G. A.

G. Reindert, G. A. Jan, F. M. d. M. Frits, and W. J. Henk, “Similarity relations for anisotropic scattering in absorbing media,” Opt. Eng. 32, 244–252 (1993).
[CrossRef]

Jiang, B.

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]

Jung, B.

B. Jung, B. Choi, A. J. Durkin, K. M. Kelly, and J. S. Nelson, “Characterization of port wine stain skin erythema and melanin content using cross polarized diffuse reflectance imaging,” Lasers Surg. Med. 34, 174–181 (2004).
[CrossRef] [PubMed]

Kelly, K. M.

B. Jung, B. Choi, A. J. Durkin, K. M. Kelly, and J. S. Nelson, “Characterization of port wine stain skin erythema and melanin content using cross polarized diffuse reflectance imaging,” Lasers Surg. Med. 34, 174–181 (2004).
[CrossRef] [PubMed]

Kobayashi, N.

T. Tadokoro, N. Kobayashi, B. Z. Zmudzka, S. Ito, K. Wakamatsu, Y. Yamaguchi, K. S. Korossy, S. A. Miller, J. Z. Beer, and V. J. Hearing, “UV-induced DNA damage and melanin content in human skin differing in racial/ethnic origin,” FASEB J. 17, 1177–1179 (2003).
[CrossRef] [PubMed]

Kocsis, L.

L. Kocsis, P. Herman, and A. Eke, “The modified Beer–Lambert law revisited,” Phys. Med. Biol. 51, N91 (2006).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

Korossy, K. S.

T. Tadokoro, N. Kobayashi, B. Z. Zmudzka, S. Ito, K. Wakamatsu, Y. Yamaguchi, K. S. Korossy, S. A. Miller, J. Z. Beer, and V. J. Hearing, “UV-induced DNA damage and melanin content in human skin differing in racial/ethnic origin,” FASEB J. 17, 1177–1179 (2003).
[CrossRef] [PubMed]

Krishnaswamy, A.

A. Krishnaswamy and G. V. G. Baranoski, “A biophysically based spectral model of light interaction with human skin,” Comput. Graph. Forum 23, 331–340 (2004).
[CrossRef]

Lee, Y.

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

Magnain, C.

Marchesini, R.

R. Marchesini, C. Clemente, E. Pignoli, and M. Brambilla, “Optical properties of in vitro epidermis and their possible relationship with optical properties of in vivo skin,” J. Photochem. Photobiol., B 16, 127–140 (1992).
[CrossRef]

Marks, R.

P. J. Matts, P. J. Dykes, and R. Marks, “The distribution of melanin in skin determined in vivo,” Br. J. Dermatol. 156, 620–628 (2007).
[CrossRef] [PubMed]

Matip, R.

T. Gambichler, R. Matip, G. Moussa, P. Altmeyer, and K. Hoffmann, “In vivo data of epidermal thickness evaluated by optical coherence tomography: effects of age, gender, skin type, and anatomic site,” J. Dermatol. Sci. 44, 145–152 (2006).
[CrossRef] [PubMed]

Matts, P. J.

P. J. Matts, P. J. Dykes, and R. Marks, “The distribution of melanin in skin determined in vivo,” Br. J. Dermatol. 156, 620–628 (2007).
[CrossRef] [PubMed]

McAdams, M. S.

Miller, S. A.

T. Tadokoro, N. Kobayashi, B. Z. Zmudzka, S. Ito, K. Wakamatsu, Y. Yamaguchi, K. S. Korossy, S. A. Miller, J. Z. Beer, and V. J. Hearing, “UV-induced DNA damage and melanin content in human skin differing in racial/ethnic origin,” FASEB J. 17, 1177–1179 (2003).
[CrossRef] [PubMed]

Moan, J.

K. P. Nielsen, L. Zhao, J. J. Stamnes, K. Stamnes, and J. Moan, “The importance of the depth distribution of melanin in skin for DNA protection and other photobiological processes,” J. Photochem. Photobiol., B 82, 194–198 (2006).
[CrossRef]

Modest, M. F.

M. F. Modest, Radiative Heat Transfer (Academic, 2003).

Moussa, G.

T. Gambichler, R. Matip, G. Moussa, P. Altmeyer, and K. Hoffmann, “In vivo data of epidermal thickness evaluated by optical coherence tomography: effects of age, gender, skin type, and anatomic site,” J. Dermatol. Sci. 44, 145–152 (2006).
[CrossRef] [PubMed]

Nelson, J. S.

B. Jung, B. Choi, A. J. Durkin, K. M. Kelly, and J. S. Nelson, “Characterization of port wine stain skin erythema and melanin content using cross polarized diffuse reflectance imaging,” Lasers Surg. Med. 34, 174–181 (2004).
[CrossRef] [PubMed]

Nielsen, K. P.

K. P. Nielsen, L. Zhao, J. J. Stamnes, K. Stamnes, and J. Moan, “The importance of the depth distribution of melanin in skin for DNA protection and other photobiological processes,” J. Photochem. Photobiol., B 82, 194–198 (2006).
[CrossRef]

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).
[CrossRef]

Parrish, J. A.

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

S. Wan, R. Anderson, and J. A. Parrish, “Analytical modeling for the optical properties of the skin with in vitro and in vivo applications,” Photochem. Photobiol. 34, 493–499 (1981).
[CrossRef] [PubMed]

Patel, S.

A. Yafi, T. S. Vetter, T. Scholz, S. Patel, R. B. Saager, D. J. Cuccia, G. R. Evans, and A. J. Durkin, “Postoperative quantitative assessment of reconstructive tissue status in a cutaneous flap model using spatial frequency domain imaging,” Plast. Reconstr. Surg. 127, 117–130 (2011).
[CrossRef] [PubMed]

Patterson, M. S.

Pham, T.

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, and M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

Pharaon, M. R.

M. R. Pharaon, T. Scholz, S. Bogdanoff, D. Cuccia, A. J. Durkin, D. B. Hoyt, and G. R. D. Evans, “Early detection of complete vascular occlusion in a pedicle flap model using quantitation spectral imaging,” Plast. Reconstr. Surg. 126, 1924–1935 (2010).
[CrossRef] [PubMed]

Pignoli, E.

R. Marchesini, C. Clemente, E. Pignoli, and M. Brambilla, “Optical properties of in vitro epidermis and their possible relationship with optical properties of in vivo skin,” J. Photochem. Photobiol., B 16, 127–140 (1992).
[CrossRef]

Poulsen, T.

J. Sandby-Moller, T. Poulsen, and H. C. Wulf, “Epidermal thickness at different body sites: relationship to age, gender, pigmentation, blood content, skin type and smoking habits,” Acta Derm.-Venereol. 83, 410–413 (2003).
[CrossRef] [PubMed]

Prahl, S. A.

Prota, G.

T. Dwyer, G. Prota, L. Blizzard, R. Ashbolt, and M. R. Vincensi, “Melanin density and melanin type predict melanocytic naevi in 19–20 year olds of northern European ancestry,” Melanoma Res. 10, 387–394 (2000).
[CrossRef] [PubMed]

Reindert, G.

G. Reindert, G. A. Jan, F. M. d. M. Frits, and W. J. Henk, “Similarity relations for anisotropic scattering in absorbing media,” Opt. Eng. 32, 244–252 (1993).
[CrossRef]

Saager, R. B.

R. B. Saager, D. J. Cuccia, and A. J. Durkin, “Determination of optical properties of turbid media spanning visible and near-infrared regimes via spatially modulated quantitative spectroscopy,” J. Biomed. Opt. 15, 017012 (2010).
[CrossRef] [PubMed]

A. Yafi, T. S. Vetter, T. Scholz, S. Patel, R. B. Saager, D. J. Cuccia, G. R. Evans, and A. J. Durkin, “Postoperative quantitative assessment of reconstructive tissue status in a cutaneous flap model using spatial frequency domain imaging,” Plast. Reconstr. Surg. 127, 117–130 (2011).
[CrossRef] [PubMed]

Salomatina, E.

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]

Sandby-Moller, J.

J. Sandby-Moller, T. Poulsen, and H. C. Wulf, “Epidermal thickness at different body sites: relationship to age, gender, pigmentation, blood content, skin type and smoking habits,” Acta Derm.-Venereol. 83, 410–413 (2003).
[CrossRef] [PubMed]

Sassaroli, A.

A. Sassaroli and S. Fantini, “Comment on the modified Beer–Lambert law for scattering media,” Phys. Med. Biol. 49, N255 (2004).
[CrossRef] [PubMed]

Scholz, T.

M. R. Pharaon, T. Scholz, S. Bogdanoff, D. Cuccia, A. J. Durkin, D. B. Hoyt, and G. R. D. Evans, “Early detection of complete vascular occlusion in a pedicle flap model using quantitation spectral imaging,” Plast. Reconstr. Surg. 126, 1924–1935 (2010).
[CrossRef] [PubMed]

A. Yafi, T. S. Vetter, T. Scholz, S. Patel, R. B. Saager, D. J. Cuccia, G. R. Evans, and A. J. Durkin, “Postoperative quantitative assessment of reconstructive tissue status in a cutaneous flap model using spatial frequency domain imaging,” Plast. Reconstr. Surg. 127, 117–130 (2011).
[CrossRef] [PubMed]

Simpson, C. R.

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]

Southwood, W. F. W.

W. F. W. Southwood, “The thickness of the skin,” Plast. Reconstr. Surg. 15, 423–429 (1955).
[CrossRef]

Spott, T.

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, and M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

Stamnes, J. J.

K. P. Nielsen, L. Zhao, J. J. Stamnes, K. Stamnes, and J. Moan, “The importance of the depth distribution of melanin in skin for DNA protection and other photobiological processes,” J. Photochem. Photobiol., B 82, 194–198 (2006).
[CrossRef]

Stamnes, K.

K. P. Nielsen, L. Zhao, J. J. Stamnes, K. Stamnes, and J. Moan, “The importance of the depth distribution of melanin in skin for DNA protection and other photobiological processes,” J. Photochem. Photobiol., B 82, 194–198 (2006).
[CrossRef]

Svaasand, L. O.

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, and M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

R. C. Haskell, L. O. Svaasand, T.-T. Tsay, T.-C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
[CrossRef]

Tadokoro, T.

T. Tadokoro, N. Kobayashi, B. Z. Zmudzka, S. Ito, K. Wakamatsu, Y. Yamaguchi, K. S. Korossy, S. A. Miller, J. Z. Beer, and V. J. Hearing, “UV-induced DNA damage and melanin content in human skin differing in racial/ethnic origin,” FASEB J. 17, 1177–1179 (2003).
[CrossRef] [PubMed]

Tromberg, B. J.

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
[CrossRef] [PubMed]

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, and M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

R. C. Haskell, L. O. Svaasand, T.-T. Tsay, T.-C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
[CrossRef]

Tsay, T.-T.

Tuchin, V. V.

V. V. Tuchin, Tissue Optics (SPIE, 2000).

Vetter, T. S.

A. Yafi, T. S. Vetter, T. Scholz, S. Patel, R. B. Saager, D. J. Cuccia, G. R. Evans, and A. J. Durkin, “Postoperative quantitative assessment of reconstructive tissue status in a cutaneous flap model using spatial frequency domain imaging,” Plast. Reconstr. Surg. 127, 117–130 (2011).
[CrossRef] [PubMed]

Vincensi, M. R.

T. Dwyer, G. Prota, L. Blizzard, R. Ashbolt, and M. R. Vincensi, “Melanin density and melanin type predict melanocytic naevi in 19–20 year olds of northern European ancestry,” Melanoma Res. 10, 387–394 (2000).
[CrossRef] [PubMed]

Wakamatsu, K.

T. Tadokoro, N. Kobayashi, B. Z. Zmudzka, S. Ito, K. Wakamatsu, Y. Yamaguchi, K. S. Korossy, S. A. Miller, J. Z. Beer, and V. J. Hearing, “UV-induced DNA damage and melanin content in human skin differing in racial/ethnic origin,” FASEB J. 17, 1177–1179 (2003).
[CrossRef] [PubMed]

Wan, S.

S. Wan, R. Anderson, and J. A. Parrish, “Analytical modeling for the optical properties of the skin with in vitro and in vivo applications,” Photochem. Photobiol. 34, 493–499 (1981).
[CrossRef] [PubMed]

Wang, L.

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]

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] [PubMed]

Welch, A. J.

Wulf, H. C.

J. Sandby-Moller, T. Poulsen, and H. C. Wulf, “Epidermal thickness at different body sites: relationship to age, gender, pigmentation, blood content, skin type and smoking habits,” Acta Derm.-Venereol. 83, 410–413 (2003).
[CrossRef] [PubMed]

Yafi, A.

A. Yafi, T. S. Vetter, T. Scholz, S. Patel, R. B. Saager, D. J. Cuccia, G. R. Evans, and A. J. Durkin, “Postoperative quantitative assessment of reconstructive tissue status in a cutaneous flap model using spatial frequency domain imaging,” Plast. Reconstr. Surg. 127, 117–130 (2011).
[CrossRef] [PubMed]

Yamaguchi, Y.

T. Tadokoro, N. Kobayashi, B. Z. Zmudzka, S. Ito, K. Wakamatsu, Y. Yamaguchi, K. S. Korossy, S. A. Miller, J. Z. Beer, and V. J. Hearing, “UV-induced DNA damage and melanin content in human skin differing in racial/ethnic origin,” FASEB J. 17, 1177–1179 (2003).
[CrossRef] [PubMed]

Yaroslavsky, A. N.

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]

Yoon, G.

Young, A. R.

A. R. Young, “Chromophores in human skin,” Phys. Med. Biol. 42, 789–802 (1997).
[CrossRef] [PubMed]

Zhao, L.

K. P. Nielsen, L. Zhao, J. J. Stamnes, K. Stamnes, and J. Moan, “The importance of the depth distribution of melanin in skin for DNA protection and other photobiological processes,” J. Photochem. Photobiol., B 82, 194–198 (2006).
[CrossRef]

Zheng, L.

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]

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] [PubMed]

Zmudzka, B. Z.

T. Tadokoro, N. Kobayashi, B. Z. Zmudzka, S. Ito, K. Wakamatsu, Y. Yamaguchi, K. S. Korossy, S. A. Miller, J. Z. Beer, and V. J. Hearing, “UV-induced DNA damage and melanin content in human skin differing in racial/ethnic origin,” FASEB J. 17, 1177–1179 (2003).
[CrossRef] [PubMed]

Acta Derm.-Venereol. (1)

J. Sandby-Moller, T. Poulsen, and H. C. Wulf, “Epidermal thickness at different body sites: relationship to age, gender, pigmentation, blood content, skin type and smoking habits,” Acta Derm.-Venereol. 83, 410–413 (2003).
[CrossRef] [PubMed]

Appl. Opt. (2)

Br. J. Dermatol. (1)

P. J. Matts, P. J. Dykes, and R. Marks, “The distribution of melanin in skin determined in vivo,” Br. J. Dermatol. 156, 620–628 (2007).
[CrossRef] [PubMed]

Comput. Graph. Forum (1)

A. Krishnaswamy and G. V. G. Baranoski, “A biophysically based spectral model of light interaction with human skin,” Comput. Graph. Forum 23, 331–340 (2004).
[CrossRef]

Comput. Methods Programs Biomed. (2)

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]

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] [PubMed]

FASEB J. (1)

T. Tadokoro, N. Kobayashi, B. Z. Zmudzka, S. Ito, K. Wakamatsu, Y. Yamaguchi, K. S. Korossy, S. A. Miller, J. Z. Beer, and V. J. Hearing, “UV-induced DNA damage and melanin content in human skin differing in racial/ethnic origin,” FASEB J. 17, 1177–1179 (2003).
[CrossRef] [PubMed]

J. Biomed. Opt. (3)

R. B. Saager, D. J. Cuccia, and A. J. Durkin, “Determination of optical properties of turbid media spanning visible and near-infrared regimes via spatially modulated quantitative spectroscopy,” J. Biomed. Opt. 15, 017012 (2010).
[CrossRef] [PubMed]

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, F. R. Ayers, and B. J. Tromberg, “Quantitation and mapping of tissue optical properties using modulated imaging,” J. Biomed. Opt. 14, 024012 (2009).
[CrossRef] [PubMed]

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]

J. Dermatol. Sci. (1)

T. Gambichler, R. Matip, G. Moussa, P. Altmeyer, and K. Hoffmann, “In vivo data of epidermal thickness evaluated by optical coherence tomography: effects of age, gender, skin type, and anatomic site,” J. Dermatol. Sci. 44, 145–152 (2006).
[CrossRef] [PubMed]

J. Invest. Dermatol. (1)

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

J. Opt. Soc. Am. A (2)

J. Photochem. Photobiol., B (2)

R. Marchesini, C. Clemente, E. Pignoli, and M. Brambilla, “Optical properties of in vitro epidermis and their possible relationship with optical properties of in vivo skin,” J. Photochem. Photobiol., B 16, 127–140 (1992).
[CrossRef]

K. P. Nielsen, L. Zhao, J. J. Stamnes, K. Stamnes, and J. Moan, “The importance of the depth distribution of melanin in skin for DNA protection and other photobiological processes,” J. Photochem. Photobiol., B 82, 194–198 (2006).
[CrossRef]

Lasers Surg. Med. (1)

B. Jung, B. Choi, A. J. Durkin, K. M. Kelly, and J. S. Nelson, “Characterization of port wine stain skin erythema and melanin content using cross polarized diffuse reflectance imaging,” Lasers Surg. Med. 34, 174–181 (2004).
[CrossRef] [PubMed]

Melanoma Res. (1)

T. Dwyer, G. Prota, L. Blizzard, R. Ashbolt, and M. R. Vincensi, “Melanin density and melanin type predict melanocytic naevi in 19–20 year olds of northern European ancestry,” Melanoma Res. 10, 387–394 (2000).
[CrossRef] [PubMed]

Opt. Eng. (1)

G. Reindert, G. A. Jan, F. M. d. M. Frits, and W. J. Henk, “Similarity relations for anisotropic scattering in absorbing media,” Opt. Eng. 32, 244–252 (1993).
[CrossRef]

Photochem. Photobiol. (1)

S. Wan, R. Anderson, and J. A. Parrish, “Analytical modeling for the optical properties of the skin with in vitro and in vivo applications,” Photochem. Photobiol. 34, 493–499 (1981).
[CrossRef] [PubMed]

Phys. Med. Biol. (5)

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]

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, and M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

A. Sassaroli and S. Fantini, “Comment on the modified Beer–Lambert law for scattering media,” Phys. Med. Biol. 49, N255 (2004).
[CrossRef] [PubMed]

L. Kocsis, P. Herman, and A. Eke, “The modified Beer–Lambert law revisited,” Phys. Med. Biol. 51, N91 (2006).
[CrossRef] [PubMed]

A. R. Young, “Chromophores in human skin,” Phys. Med. Biol. 42, 789–802 (1997).
[CrossRef] [PubMed]

Plast. Reconstr. Surg. (3)

M. R. Pharaon, T. Scholz, S. Bogdanoff, D. Cuccia, A. J. Durkin, D. B. Hoyt, and G. R. D. Evans, “Early detection of complete vascular occlusion in a pedicle flap model using quantitation spectral imaging,” Plast. Reconstr. Surg. 126, 1924–1935 (2010).
[CrossRef] [PubMed]

A. Yafi, T. S. Vetter, T. Scholz, S. Patel, R. B. Saager, D. J. Cuccia, G. R. Evans, and A. J. Durkin, “Postoperative quantitative assessment of reconstructive tissue status in a cutaneous flap model using spatial frequency domain imaging,” Plast. Reconstr. Surg. 127, 117–130 (2011).
[CrossRef] [PubMed]

W. F. W. Southwood, “The thickness of the skin,” Plast. Reconstr. Surg. 15, 423–429 (1955).
[CrossRef]

Surg. Radiol. Anat. (1)

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

Other (4)

V. V. Tuchin, Tissue Optics (SPIE, 2000).

S. L. Jacques, “Origins of tissue optical properties in the UVA, visible, and NIR regions,” in OSA TOPS on Advances in Optical Imaging and Photon Migration, R.R.Alfano and J.G.Fujimoto, eds. (Optical Society of America, 1996), Vol.  2, pp. 364–369.

M. F. Modest, Radiative Heat Transfer (Academic, 2003).

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

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Figures (7)

Fig. 1
Fig. 1

Schematic of the three-layer geometry considered.

Fig. 2
Fig. 2

Estimates of the diffuse reflectance of the three-layer medium R d and absorption in each of three layers A 1 , A 2 , and A 3 as a function of the absorption coefficient μ a , 2 for n 1 = 1.4 , d 1 = 150 μm , d 2 = 50 μm , μ a , 1 = 0 mm 1 , μ a , 3 = 0.02 mm 1 , μ s , 1 = 1 mm 1 , μ s , 2 = 0 mm 1 , μ a , 3 = 2 mm 1 , and μ a , 2 is between 0 and 10 mm 1 , estimated by Monte Carlo (symbols) simulations and the three-layer analytic model (solid line). The relative percent error for each estimate is also shown.

Fig. 3
Fig. 3

Estimates of the diffuse reflectance of the three-layer medium R d and absorption in each of three layers A 1 , A 2 , and A 3 as a function of the layer thickness d 2 for n 1 = 1.4 , d 1 = 150 μm , μ a , 1 = 0 mm 1 , μ a , 2 = 1 mm 1 , μ a , 3 = 0.02 mm 1 , μ s , 1 = 1 mm 1 , μ s , 2 = 0 mm 1 , μ s , 3 = 2 mm 1 , and d 2 is between 10 and 150 μm , estimated by Monte Carlo simulations (symbols) and the three-layer analytic model (solid line). The relative percent error for each estimate is also shown.

Fig. 4
Fig. 4

Estimates of the diffuse reflectance of the three-layer medium R d and absorption in each of three layers A 1 , A 2 , and A 3 as a function of the absorption coefficient μ a , 3 for n 1 = 1.4 , d 2 = 150 μm , d 2 = 50 μm , μ a , 1 = 0 mm 1 , μ a , 2 = 1 mm 1 , μ s , 1 = 1 mm 1 , μ s , 2 = 0 mm 1 , μ s , 3 = 2 mm 1 , and μ a , 3 is between 0.001 and 0.1 mm 1 , estimated by Monte Carlo simulations (symbols) and the three-layer analytic model (solid line). The relative percent error for each estimate is also shown.

Fig. 5
Fig. 5

Estimates of the diffuse reflectance of the three-layer medium R d and absorption in each of three layers A 1 , A 2 , and A 3 as a function of the absorption and scattering coefficients μ a , 1 = μ s , 2 for n 1 = 1.4 , d 1 = 150 μm , d 2 = 50 μm , μ a , 2 = 0.5 mm 1 , μ a , 3 = 0.01 mm 1 , μ s , 1 = 1 mm 1 , μ s , 3 = 2 mm 1 , and μ a , 1 = μ s , 2 is between 0 and 1 mm 1 , estimated by Monte Carlo simulations (symbols) and the three-layer analytic model (solid line). The relative percent error for each estimate is also shown.

Fig. 6
Fig. 6

The fluence φ ( z ) estimated by Monte Carlo simulations and the three-layer analytic model for n 1 = 1.4 , d 1 = 150 μm , d 2 = 50 μm , μ a , 2 = 0.5 mm 1 , μ a , 3 = 0.01 mm 1 , μ s , 1 = 1 mm 1 , μ s , 3 = 2 mm 1 , and μ a , 1 = μ s , 2 = 0.121 mm 1 .

Fig. 7
Fig. 7

Estimates of the diffuse reflectance of the three-layer medium R d and absorption in each of three layers A 1 , A 2 , and A 3 as a function of tissue refractive index n 1 for d 1 = 150 μm , d 2 = 50 μm , μ a , 1 = 0.05 mm 1 , μ a , 2 = 1 mm 1 , μ a , 3 = 0.01 mm 1 , μ s , 1 = 1 mm 1 , μ s , 2 = 0.5 mm 1 , μ s , 3 = 2 mm 1 , and n 1 is between 1 and 2, estimated by Monte Carlo simulations (symbols) and the three-layer analytic model (solid line). The relative percent error for each estimate is also shown.

Equations (29)

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s ^ · I ( r ^ , s ^ ) = ( μ a + μ s ) I ( r ^ , s ^ ) + μ s 4 π I ( r ^ , s ^ i ) Φ ( s ^ i , s ^ ) d Ω i + Q ( r ^ , s ^ ) ,
ω = μ s μ a + μ s .
I ( r ^ , s ^ ) = 1 4 π φ ( r ^ ) + 3 4 π j ( r ^ ) · s ^ .
2 φ ( r ^ ) 3 μ a ( μ a + μ s ( 1 g ) ) φ ( r ^ ) = 3 ( μ a + μ s ( 1 g ) ) q ( r ^ ) ,
d 2 φ ( z ) d z 2 3 μ a μ tr φ ( z ) = 3 μ tr q ( z ) ,
ω tr = μ s μ a + μ s .
j = 1 3 μ tr d φ d z .
1 μ tr d φ ( z ) d z = 4 j ( z ) ,
1 μ tr d j ( z ) d z = ( 1 ω tr ) ( q ( z ) + φ ( z ) ) .
d 2 φ 1 ( z ) d z 2 3 μ a , 1 μ tr , 1 φ 1 ( z ) = 3 μ tr , 1 q 1 ( z ) ,
q 1 ( z ) = μ s , 1 q 0 e μ tr , 1 z ,
1 μ tr , 2 d φ 2 ( z ) d z = 4 j 2 ( z ) ,
1 μ tr , 2 d j 2 ( z ) d z = ( 1 ω tr , 2 ) ( q 2 ( z ) + φ 2 ( z ) ) .
q 2 ( z ) = μ s , 2 μ s , 1 q 1 ( z 1 ) e μ tr , 2 ( z z 1 ) ,
d 2 φ 3 ( z ) d z 2 3 μ a , 3 μ tr , 3 φ 3 ( z ) = 3 μ tr , 3 q 3 ( z ) ,
q 3 ( z ) = μ s , 3 μ s , 2 q 2 ( z 2 ) e μ tr , 3 ( z z 2 ) .
φ 1 ( z ) = e z μ eff , 1 ( C 1 e 2 z μ eff , 1 + C 2 3 e z ( μ eff , 1 + μ tr , 1 ) μ s , 1 2 μ a , 1 q 0 ) ,
φ 2 ( z ) = 1 2 e 2 z μ eff , 2 ( C 3 2 3 μ tr , 2 μ eff , 2 ( 1 e 4 z μ eff , 2 3 ) + C 4 ( 1 + e 4 z μ eff , 2 3 ) ) + q 0 μ a , 2 4 e μ tr , 2 ( z 1 z ) μ tr , 1 z 1 3 μ a , 2 + μ s , 1 ,
φ 3 ( z ) = C 5 e z μ eff , 3 + 3 μ s , 1 q 0 2 μ a , 3 μ s , 3 e μ eff , 3 ( z z 2 ) μ eff , 2 ( z 1 z 2 ) μ eff , 1 z 1 ,
φ 1 ( z ) | z = z 1 = φ 2 ( z ) | z = z 1 + ,
φ 2 ( z ) | z = z 2 = φ 3 ( z ) | z = z 2 + ,
j 1 ( z ) | z = z 1 = j 2 ( z ) | z = z 1 + ,
j 2 ( z ) | z = z 2 = j 3 ( z ) | z = z 2 + .
1 4 φ 1 ( z ) | z = 0 + + 1 2 j 1 ( z ) | z = 0 + = 1 4 ρ φ φ 1 ( z ) | z = 0 + + 1 2 ρ j j 1 ( z ) | z = 0 + ,
R d = j 1 ( z ) | z = 0 q 0 .
A 1 = μ a , 1 0 z 1 ( q 1 ( z ) + φ 1 ( z ) ) d z ,
A 2 = μ a , 2 z 1 z 2 ( q 2 ( z ) + φ 2 ( z ) ) d z ,
A 3 = μ a , 3 z 2 ( q 3 ( z ) + φ 3 ( z ) ) d z .
φ ( z ) = { φ 1 ( z ) for     z < z 1 φ 2 ( z ) for     z 1 z < z 2 φ 3 ( z ) for     z z 2 .

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