Abstract

The auxiliary function method is an efficient technique for solving the radiative tranfer equation without adding any assumption and was applied until now only for theoretical stratified media. The first application (to our knowledge) of the method to a real case, the human skin, is presented. This makes it possible to validate the method by comparing model results with experimental reflectance spectra of real skin. An excellent agreement is obtained for a multilayer model of the skin made of 22 sublayers and taking into account the anisotropic phase function of the scatterers. Thus there is the opportunity to develop interest in such models by quantitatively evaluating the influence of the parameters commonly used in the literature that modify skin color, such as the concentration of the scatterers and the thickness of each sublayer.

© 2007 Optical Society of America

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References

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  1. L. A. Goldsmith, Physiology, Biochemistry, and Molecular Biology of the Skin (Oxford U. Press, 1998).
  2. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  3. G. Poirier, "Human skin modelling and rendering," Master of Mathematics in Computer Science thesis (University of Waterloo, 2004).
  4. P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).
  5. R. R. Anderson and J. A. Parrish, "Optical properties of human skin," in The Science of Photomedicine (Plenum, 1982), pp. 147-194.
  6. K. P. Nielsen, L. Zhao, P. Juzenas, J. J. Stamnes, K. Stamnes, and J. Moan, "Reflectance spectra of pigmented and nonpigmented skin in the UV spectral region," Photochem. Photobiol. 80, 450-455 (2004).
    [PubMed]
  7. K. Stamnes, S. Chee Tsay, W. Wiscombe, and K. Jayaweera, "Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layer media," Appl. Opt. 27, 2502-2510 (1988).
    [CrossRef] [PubMed]
  8. C. F. Bohren and D. R. Hoffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 1983).
  9. P. S. Mudgett and L. W. Richards, "Multiple scattering calculations for technology," Appl. Opt. 10, 1495-1502 (1971).
    [CrossRef]
  10. M. Elias and G. Elias, "New and fast calculation for incoherent multiple scattering," J. Opt. Soc. Am. A 19, 894-901 (2002).
    [CrossRef]
  11. L. Simonot and M. Elias, "Special visual effect of art glazes explained by the radiative transfer equation," Appl. Opt. 43, 2580-2587 (2004).
    [CrossRef] [PubMed]
  12. M. Elias and G. Elias, "Radiative transfer in inhomogeneous stratified scattering media with use of the auxiliary function method," J. Opt. Soc. Am. A 21, 580-589 (2004).
    [CrossRef]
  13. W. J. Wiscombe, "Improved Mie scattering algorithms," Appl. Opt. 19, 1505-1509 (1980).
    [CrossRef] [PubMed]
  14. D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, "Oxygen saturation-dependent absorption and scattering of blood," Phys. Rev. Lett. 93, 028102 (2004).
    [CrossRef] [PubMed]
  15. S. L. Jacques, http://omlc.ogi.edu/spectra/hemoglobin/index.html (1999).
  16. S. L. Jacques, http://omlc.ogi.edu/news/jan98/skinoptics.html (1998).
  17. M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, and W. M. Star, "Skin optics," IEEE Trans. Biomed. Eng. 36, 1146-1154 (1989).
    [CrossRef] [PubMed]
  18. G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley Interscience, 1982).

2004 (4)

K. P. Nielsen, L. Zhao, P. Juzenas, J. J. Stamnes, K. Stamnes, and J. Moan, "Reflectance spectra of pigmented and nonpigmented skin in the UV spectral region," Photochem. Photobiol. 80, 450-455 (2004).
[PubMed]

D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, "Oxygen saturation-dependent absorption and scattering of blood," Phys. Rev. Lett. 93, 028102 (2004).
[CrossRef] [PubMed]

M. Elias and G. Elias, "Radiative transfer in inhomogeneous stratified scattering media with use of the auxiliary function method," J. Opt. Soc. Am. A 21, 580-589 (2004).
[CrossRef]

L. Simonot and M. Elias, "Special visual effect of art glazes explained by the radiative transfer equation," Appl. Opt. 43, 2580-2587 (2004).
[CrossRef] [PubMed]

2002 (1)

1989 (1)

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

1988 (1)

1980 (1)

1971 (1)

P. S. Mudgett and L. W. Richards, "Multiple scattering calculations for technology," Appl. Opt. 10, 1495-1502 (1971).
[CrossRef]

1931 (1)

P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).

Aalders, M. C. G.

D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, "Oxygen saturation-dependent absorption and scattering of blood," Phys. Rev. Lett. 93, 028102 (2004).
[CrossRef] [PubMed]

Anderson, R. R.

R. R. Anderson and J. A. Parrish, "Optical properties of human skin," in The Science of Photomedicine (Plenum, 1982), pp. 147-194.

Bohren, C. F.

C. F. Bohren and D. R. Hoffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 1983).

Chandrasekhar, S.

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

Elias, G.

Elias, M.

Faber, D. J.

D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, "Oxygen saturation-dependent absorption and scattering of blood," Phys. Rev. Lett. 93, 028102 (2004).
[CrossRef] [PubMed]

Goldsmith, L. A.

L. A. Goldsmith, Physiology, Biochemistry, and Molecular Biology of the Skin (Oxford U. Press, 1998).

Hoffman, D. R.

C. F. Bohren and D. R. Hoffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 1983).

Hooper, B. A.

D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, "Oxygen saturation-dependent absorption and scattering of blood," Phys. Rev. Lett. 93, 028102 (2004).
[CrossRef] [PubMed]

Jacques, S. L.

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

S. L. Jacques, http://omlc.ogi.edu/spectra/hemoglobin/index.html (1999).

S. L. Jacques, http://omlc.ogi.edu/news/jan98/skinoptics.html (1998).

Jayaweera, K.

Juzenas, P.

K. P. Nielsen, L. Zhao, P. Juzenas, J. J. Stamnes, K. Stamnes, and J. Moan, "Reflectance spectra of pigmented and nonpigmented skin in the UV spectral region," Photochem. Photobiol. 80, 450-455 (2004).
[PubMed]

Kubelka, P.

P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).

Mik, E. G.

D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, "Oxygen saturation-dependent absorption and scattering of blood," Phys. Rev. Lett. 93, 028102 (2004).
[CrossRef] [PubMed]

Moan, J.

K. P. Nielsen, L. Zhao, P. Juzenas, J. J. Stamnes, K. Stamnes, and J. Moan, "Reflectance spectra of pigmented and nonpigmented skin in the UV spectral region," Photochem. Photobiol. 80, 450-455 (2004).
[PubMed]

Mudgett, P. S.

P. S. Mudgett and L. W. Richards, "Multiple scattering calculations for technology," Appl. Opt. 10, 1495-1502 (1971).
[CrossRef]

Munk, F.

P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).

Nielsen, K. P.

K. P. Nielsen, L. Zhao, P. Juzenas, J. J. Stamnes, K. Stamnes, and J. Moan, "Reflectance spectra of pigmented and nonpigmented skin in the UV spectral region," Photochem. Photobiol. 80, 450-455 (2004).
[PubMed]

Parrish, J. A.

R. R. Anderson and J. A. Parrish, "Optical properties of human skin," in The Science of Photomedicine (Plenum, 1982), pp. 147-194.

Poirier, G.

G. Poirier, "Human skin modelling and rendering," Master of Mathematics in Computer Science thesis (University of Waterloo, 2004).

Richards, L. W.

P. S. Mudgett and L. W. Richards, "Multiple scattering calculations for technology," Appl. Opt. 10, 1495-1502 (1971).
[CrossRef]

Simonot, L.

Stamnes, J. J.

K. P. Nielsen, L. Zhao, P. Juzenas, J. J. Stamnes, K. Stamnes, and J. Moan, "Reflectance spectra of pigmented and nonpigmented skin in the UV spectral region," Photochem. Photobiol. 80, 450-455 (2004).
[PubMed]

Stamnes, K.

K. P. Nielsen, L. Zhao, P. Juzenas, J. J. Stamnes, K. Stamnes, and J. Moan, "Reflectance spectra of pigmented and nonpigmented skin in the UV spectral region," Photochem. Photobiol. 80, 450-455 (2004).
[PubMed]

K. Stamnes, S. Chee Tsay, W. Wiscombe, and K. Jayaweera, "Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layer media," Appl. Opt. 27, 2502-2510 (1988).
[CrossRef] [PubMed]

Star, W. M.

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

Sterenborg, H. J. C. M.

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

Stiles, W. S.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley Interscience, 1982).

Tsay, S. Chee

van Gemert, M. J. C.

D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, "Oxygen saturation-dependent absorption and scattering of blood," Phys. Rev. Lett. 93, 028102 (2004).
[CrossRef] [PubMed]

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

van Leeuwen, T. G.

D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, "Oxygen saturation-dependent absorption and scattering of blood," Phys. Rev. Lett. 93, 028102 (2004).
[CrossRef] [PubMed]

Wiscombe, W.

Wiscombe, W. J.

Wyszecki, G.

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley Interscience, 1982).

Zhao, L.

K. P. Nielsen, L. Zhao, P. Juzenas, J. J. Stamnes, K. Stamnes, and J. Moan, "Reflectance spectra of pigmented and nonpigmented skin in the UV spectral region," Photochem. Photobiol. 80, 450-455 (2004).
[PubMed]

Appl. Opt. (4)

IEEE Trans. Biomed. Eng. (1)

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

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

Photochem. Photobiol. (1)

K. P. Nielsen, L. Zhao, P. Juzenas, J. J. Stamnes, K. Stamnes, and J. Moan, "Reflectance spectra of pigmented and nonpigmented skin in the UV spectral region," Photochem. Photobiol. 80, 450-455 (2004).
[PubMed]

Phys. Rev. Lett. (1)

D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. van Gemert, and T. G. van Leeuwen, "Oxygen saturation-dependent absorption and scattering of blood," Phys. Rev. Lett. 93, 028102 (2004).
[CrossRef] [PubMed]

Z. Tech. Phys. (Leipzig) (1)

P. Kubelka and F. Munk, "Ein Beitrag zur Optik der Farbanstriche," Z. Tech. Phys. (Leipzig) 12, 593-601 (1931).

Other (8)

R. R. Anderson and J. A. Parrish, "Optical properties of human skin," in The Science of Photomedicine (Plenum, 1982), pp. 147-194.

L. A. Goldsmith, Physiology, Biochemistry, and Molecular Biology of the Skin (Oxford U. Press, 1998).

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

G. Poirier, "Human skin modelling and rendering," Master of Mathematics in Computer Science thesis (University of Waterloo, 2004).

S. L. Jacques, http://omlc.ogi.edu/spectra/hemoglobin/index.html (1999).

S. L. Jacques, http://omlc.ogi.edu/news/jan98/skinoptics.html (1998).

C. F. Bohren and D. R. Hoffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 1983).

G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley Interscience, 1982).

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Figures (13)

Fig. 1
Fig. 1

Collimated flux and diffuse flux created by single and multiple scattering inside the medium.

Fig. 2
Fig. 2

Skin structure.

Fig. 3
Fig. 3

Variation of the absorption μ a of melanin with λ.

Fig. 4
Fig. 4

Variations of the absorption and scattering coefficients k and s with λ, for r = 125 nm .

Fig. 5
Fig. 5

Angular variation of the phase function at λ = 380 nm for two melanosomes r = 31.25 nm (left) and r = 500 nm (right).

Fig. 6
Fig. 6

Variations of the real parts of the refractive indexes (top) and of the absorption (bottom) of Hb O 2 and Hb with λ.

Fig. 7
Fig. 7

Variation of the albedo (top) and of the optical depth (bottom) for C S = 2 % and S = 30 % according to λ.

Fig. 8
Fig. 8

Variation of the absorption coefficient k (top) and of the reduced scattering coefficients s (bottom) of keratin and collagen fibers accordng to λ.

Fig. 9
Fig. 9

Different reflectance spectra of real Caucasian skins.

Fig. 10
Fig. 10

Box 1, 5-layer model, box 2, 7-layer model, and box 3, 22-layer model.

Fig. 11
Fig. 11

Comparison between reflectance spectra for skin models of 5, 7, and 22 layers and a measured spectrum.

Fig. 12
Fig. 12

Reflectance spectra of real skins (solid curves 1–4) and associated modeling (1, pluses; 2, triangles; 3, diamonds; 4, squares).

Fig. 13
Fig. 13

Influence of each parameter on the reflectance spectra: (a) volume concentration C of melanosomes, (b) oxygen saturation S, (c) volume concentration C S of blood, (d) epidermal thickness z e , (e) dermal thickness z d .

Tables (1)

Tables Icon

Table 1 Standard Deviation ϵ and Color Change Δ E Sets of Fig. 12

Equations (22)

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d f ( u , τ ) d τ = f ( u , τ ) μ + ϖ 4 π μ μ F ( u , τ ) μ p ( u , u ) + ϖ 4 π μ μ 0 4 π f ( u 1 , τ ) μ 1 p ( u , u 1 ) d Ω 1 ,
d f ( m ) ( μ , τ ) d τ = f ( m ) ( μ , τ ) μ + l = m ν max r l ( m ) ( τ ) P l m ( μ ) [ F ( τ ) P l m ( μ ) π η m μ + 1 1 f ( m ) ( μ 1 , τ ) μ 1 P l m ( μ 1 ) d μ 1 ] ,
A l ( m ) ( τ ) = 1 1 f ( m ) ( μ 1 , τ ) μ 1 P l m ( μ 1 ) d μ 1 .
A l ( m ) ( τ ) = l 1 = m ν max 0 τ h [ H ( m ) ( l , l 1 , τ , y ) + U ( m ) ( l , l 1 , τ , y ) ] ( A l 1 ( m ) ( y ) + s l 1 ( m ) ( y ) ) d y .
s l ( m ) ( τ ) = F k ( τ ) P l m ( μ k ) ( π η m μ k ) .
n ̃ mel = 1.57 + i μ a λ 4 π ,
Δ E = [ ( L meas * L mod * ) 2 + ( a meas * a mod * ) 2 + ( b meas * b mod * ) 2 ] 1 2 .
ϵ = N ( R meas R mod ) 2 N 1 ,
A l ( m ) ( τ ) = l 1 = m ν max 0 τ h [ H ( m ) ( l , l 1 , τ , y ) + U ( m ) ( l , l 1 , τ , y ) ] [ A l 1 ( m ) ( y ) + s l 1 ( m ) ( y ) ] d y .
for τ = 0 , f + ( m ) ( μ , 0 ) = R 10 f ( m ) ( μ , 0 ) ,
for τ = τ h , f ( m ) ( μ , τ h ) = R 12 f + ( m ) ( μ , τ h ) ,
H ( 0 ) ( l , l 1 , τ , y ) = C r l 1 ( 0 ) ( y ) 0 1 P l 0 ( μ ) P l 1 0 ( μ ) μ exp ( x μ ) d μ ,
U ( 0 ) ( l , l 1 , τ , y ) = r l 1 ( 0 ) ( y ) 0 1 d μ P l ( 0 ) ( μ ) P l 1 ( 0 ) ( μ ) μ 1 exp ( 2 τ μ ) R 10 ( μ ) R 12 ( μ ) × { R 10 ( μ ) R 12 ( μ ) [ exp ( y τ μ ) + ( 1 ) l + l 1 exp ( y τ μ ) ] + ( 1 ) l R 12 ( μ ) exp ( y + τ μ ) + ( 1 ) l 1 R 10 ( μ ) exp ( y + τ μ ) exp ( 2 τ μ ) } .
X = K S ( λ ) x ¯ ( λ ) R ( λ ) d λ ,
Y = K S ( λ ) y ¯ ( λ ) R ( λ ) d λ ,
Z = K S ( λ ) z ¯ ( λ ) R ( λ ) d λ ,
K = 100 S ( λ ) y ¯ ( λ ) d λ .
L * = f ( Y Y B ) ,
a * = 500 116 [ f ( X X B ) f ( Y Y B ) ] ,
b * = 200 116 [ f ( Y Y B ) f ( Z Z B ) ] ,
f ( A ) = 116 A 1 3 16 if f ( A ) 8 ,
f ( A ) = ( 29 3 ) 3 A if f ( A ) 8 ( Pauli correction ) .

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