Abstract

We investigate why vessels that contain blood, which has a red or a dark red color, may look bluish in human tissue. A CCD camera was used to make images of diffusely reflected light at different wavelengths. Measurements of reflectance that are due to model blood vessels in scattering media and of human skin containing a prominent vein are presented. Monte Carlo simulations were used to calculate the spatially resolved diffuse reflectance for both situations. We show that the color of blood vessels is determined by the following factors: (i) the scattering and absorption characteristics of skin at different wavelengths, (ii) the oxygenation state of blood, which affects its absorption properties, (iii) the diameter and the depth of the vessels, and (iv) the visual perception process.

© 1996 Optical Society of America

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References

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  1. R. R. Anderson, “The physical basis of brown skin colors (melanoderma),” in Brown Melanoderma, T. B. Fitzpatrick, M. M. Wick, K. Toda, eds. (University of Tokyo, Tokyo, 1986), pp. 4–7.
  2. W. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
  3. A. Kienle, L. Lilge, M. S. Paterson, B. C. Wilson, R. Hibst, R. Steiner, “Investigation of multi-layered tissue with in vivo reflectance measurements,” in Photon Transport in Highly Scattering Tissue, S. Avrillier, B. Chance, G. J. Mueller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2326, 212–221 (1994).
  4. V. Twersky, “Absorption and multiple scattering by biological suspensions,” J. Opt. Soc. Am. 60, 1084–1093 (1970).
  5. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Chap. 7 and Chap. 3, p. 66.
  6. Y. Kuga, A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. A 1, 831–835 (1984).
  7. R. L. Gregory, Eye and Brain: the Physiology of Seeing (PrincetonU. Press, Princeton, N.J., 1990), Chap. 8.
  8. F. P. Bolin, L. E. Preuss, R. C. Taylor, R. J. Ference, “Refractive index of some mammalian tissue using a fiber optic cladding method,” Appl. Opt. 28, 2297–2303 (1989).
  9. B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distribution of light in tissue,” Med. Phys. 10, 824–830 (1983).
  10. L. Wang, S. L. Jacques, Monte Carlo Modeling of Light Transport in Multi-layered Tissues in Standard C (University of Texas M. D. Anderson Cancer Center, Houston, Tex., 1992).
  11. A. Kienle, “Lichtausbreitung in biologischem Gewebe,” thesis (University of Ulm, Ulm, Germany, 1994).
  12. D. L. MacAdam, Color Measurements: Themes and Variations (Springer-Verlag, Berlin, 1985), Chap. 3.
  13. E.H. Land, J. J. McCann, “Lightness and retinex theory,” J. Opt. Soc. Am. 61, 1–11 (1971).
  14. E. H. Land, “Recent advances in retinex theory,” Vision Res. 26, 7–21 (1986).
  15. E. H. Land, “An alternative technique for the computation of the designator in the retinex theory of the color vision,” Proc. Natl. Acad. Sci. U.S.A. 83, 3078–3080 (1986).
  16. E. H. Land, “The retinex theory of color vision,” Sci. Am. 237, 108–128 (1977).
  17. A. Kienle, L. Lilge, M. S. Paterson, B. C. Wilson, R. Hibst, R. Steiner, “Spatially resolved absolute diffuse reflectance measurements for remote determination of the optical properties of biological tissue,” Appl. Opt.35, (to be published).
  18. B. C. Wilson, M. S. Patterson, B. W. Pogue, “Instrumentation for in vivo tissue spectroscopy and imaging,” in Medical Lasers and Systems II, D. M. Harris, CM. Penney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1892, 132–147 (1993).
  19. H. J. van Staveren, C. J. M. Moes, J. van Marie, S. A. Prahl, M. J. C. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400–1100 nm,” Appl. Opt. 31, 4507–4514 (1991).
  20. S. Jacques, M. Keijzer, “Dosimetry for lasers and light in dermatology: Monte Carlo simulations of 577 nm pulsed laser penetration into cutaneous vessels,” in Lasers in Dermatology and Tissue Welding, O. T. Tan, R. A. White, J. V. White, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1422, 2–13 (1991).
  21. M. J. C. van Gemert, J. W Pickering, A. J. Welch, “Modelling laser treatment of port-wine stains,” in Management and Treatment of Benign Cutaneous Vascular Lesions, O. T. Tan, ed. (Lea & Febiger, Philadelphia, 1992), pp. 24–47.
  22. J. M. Steinke, A. P. Shepherd, “Comparison of Mie theory and the light scattering of red blood cells,” Appl. Opt. 27, 4027–4033 (1988).
  23. M. Richter, “Farbmetrik,” in Optik, L. Bergmann, C. Schäfer, eds. (Walter de Gruyter, Berlin, Germany, 1978), pp. 641–699.

1991 (1)

H. J. van Staveren, C. J. M. Moes, J. van Marie, S. A. Prahl, M. J. C. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400–1100 nm,” Appl. Opt. 31, 4507–4514 (1991).

1990 (1)

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

1989 (1)

1988 (1)

1986 (2)

E. H. Land, “Recent advances in retinex theory,” Vision Res. 26, 7–21 (1986).

E. H. Land, “An alternative technique for the computation of the designator in the retinex theory of the color vision,” Proc. Natl. Acad. Sci. U.S.A. 83, 3078–3080 (1986).

1984 (1)

1983 (1)

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distribution of light in tissue,” Med. Phys. 10, 824–830 (1983).

1977 (1)

E. H. Land, “The retinex theory of color vision,” Sci. Am. 237, 108–128 (1977).

1971 (1)

E.H. Land, J. J. McCann, “Lightness and retinex theory,” J. Opt. Soc. Am. 61, 1–11 (1971).

1970 (1)

V. Twersky, “Absorption and multiple scattering by biological suspensions,” J. Opt. Soc. Am. 60, 1084–1093 (1970).

Adam, G.

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distribution of light in tissue,” Med. Phys. 10, 824–830 (1983).

Anderson, R. R.

R. R. Anderson, “The physical basis of brown skin colors (melanoderma),” in Brown Melanoderma, T. B. Fitzpatrick, M. M. Wick, K. Toda, eds. (University of Tokyo, Tokyo, 1986), pp. 4–7.

Bolin, F. P.

Cheong, W.

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

Ference, R. J.

Gregory, R. L.

R. L. Gregory, Eye and Brain: the Physiology of Seeing (PrincetonU. Press, Princeton, N.J., 1990), Chap. 8.

Hibst, R.

A. Kienle, L. Lilge, M. S. Paterson, B. C. Wilson, R. Hibst, R. Steiner, “Spatially resolved absolute diffuse reflectance measurements for remote determination of the optical properties of biological tissue,” Appl. Opt.35, (to be published).

A. Kienle, L. Lilge, M. S. Paterson, B. C. Wilson, R. Hibst, R. Steiner, “Investigation of multi-layered tissue with in vivo reflectance measurements,” in Photon Transport in Highly Scattering Tissue, S. Avrillier, B. Chance, G. J. Mueller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2326, 212–221 (1994).

Ishimaru, A.

Y. Kuga, A. Ishimaru, “Retroreflectance from a dense distribution of spherical particles,” J. Opt. Soc. Am. A 1, 831–835 (1984).

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Chap. 7 and Chap. 3, p. 66.

Jacques, S.

S. Jacques, M. Keijzer, “Dosimetry for lasers and light in dermatology: Monte Carlo simulations of 577 nm pulsed laser penetration into cutaneous vessels,” in Lasers in Dermatology and Tissue Welding, O. T. Tan, R. A. White, J. V. White, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1422, 2–13 (1991).

Jacques, S. L.

L. Wang, S. L. Jacques, Monte Carlo Modeling of Light Transport in Multi-layered Tissues in Standard C (University of Texas M. D. Anderson Cancer Center, Houston, Tex., 1992).

Keijzer, M.

S. Jacques, M. Keijzer, “Dosimetry for lasers and light in dermatology: Monte Carlo simulations of 577 nm pulsed laser penetration into cutaneous vessels,” in Lasers in Dermatology and Tissue Welding, O. T. Tan, R. A. White, J. V. White, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1422, 2–13 (1991).

Kienle, A.

A. Kienle, L. Lilge, M. S. Paterson, B. C. Wilson, R. Hibst, R. Steiner, “Spatially resolved absolute diffuse reflectance measurements for remote determination of the optical properties of biological tissue,” Appl. Opt.35, (to be published).

A. Kienle, L. Lilge, M. S. Paterson, B. C. Wilson, R. Hibst, R. Steiner, “Investigation of multi-layered tissue with in vivo reflectance measurements,” in Photon Transport in Highly Scattering Tissue, S. Avrillier, B. Chance, G. J. Mueller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2326, 212–221 (1994).

A. Kienle, “Lichtausbreitung in biologischem Gewebe,” thesis (University of Ulm, Ulm, Germany, 1994).

Kuga, Y.

Land, E. H.

E. H. Land, “Recent advances in retinex theory,” Vision Res. 26, 7–21 (1986).

E. H. Land, “An alternative technique for the computation of the designator in the retinex theory of the color vision,” Proc. Natl. Acad. Sci. U.S.A. 83, 3078–3080 (1986).

E. H. Land, “The retinex theory of color vision,” Sci. Am. 237, 108–128 (1977).

Land, E.H.

E.H. Land, J. J. McCann, “Lightness and retinex theory,” J. Opt. Soc. Am. 61, 1–11 (1971).

Lilge, L.

A. Kienle, L. Lilge, M. S. Paterson, B. C. Wilson, R. Hibst, R. Steiner, “Investigation of multi-layered tissue with in vivo reflectance measurements,” in Photon Transport in Highly Scattering Tissue, S. Avrillier, B. Chance, G. J. Mueller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2326, 212–221 (1994).

A. Kienle, L. Lilge, M. S. Paterson, B. C. Wilson, R. Hibst, R. Steiner, “Spatially resolved absolute diffuse reflectance measurements for remote determination of the optical properties of biological tissue,” Appl. Opt.35, (to be published).

MacAdam, D. L.

D. L. MacAdam, Color Measurements: Themes and Variations (Springer-Verlag, Berlin, 1985), Chap. 3.

McCann, J. J.

E.H. Land, J. J. McCann, “Lightness and retinex theory,” J. Opt. Soc. Am. 61, 1–11 (1971).

Moes, C. J. M.

H. J. van Staveren, C. J. M. Moes, J. van Marie, S. A. Prahl, M. J. C. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400–1100 nm,” Appl. Opt. 31, 4507–4514 (1991).

Paterson, M. S.

A. Kienle, L. Lilge, M. S. Paterson, B. C. Wilson, R. Hibst, R. Steiner, “Spatially resolved absolute diffuse reflectance measurements for remote determination of the optical properties of biological tissue,” Appl. Opt.35, (to be published).

A. Kienle, L. Lilge, M. S. Paterson, B. C. Wilson, R. Hibst, R. Steiner, “Investigation of multi-layered tissue with in vivo reflectance measurements,” in Photon Transport in Highly Scattering Tissue, S. Avrillier, B. Chance, G. J. Mueller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2326, 212–221 (1994).

Patterson, M. S.

B. C. Wilson, M. S. Patterson, B. W. Pogue, “Instrumentation for in vivo tissue spectroscopy and imaging,” in Medical Lasers and Systems II, D. M. Harris, CM. Penney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1892, 132–147 (1993).

Pickering, J. W

M. J. C. van Gemert, J. W Pickering, A. J. Welch, “Modelling laser treatment of port-wine stains,” in Management and Treatment of Benign Cutaneous Vascular Lesions, O. T. Tan, ed. (Lea & Febiger, Philadelphia, 1992), pp. 24–47.

Pogue, B. W.

B. C. Wilson, M. S. Patterson, B. W. Pogue, “Instrumentation for in vivo tissue spectroscopy and imaging,” in Medical Lasers and Systems II, D. M. Harris, CM. Penney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1892, 132–147 (1993).

Prahl, S. A.

H. J. van Staveren, C. J. M. Moes, J. van Marie, S. A. Prahl, M. J. C. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400–1100 nm,” Appl. Opt. 31, 4507–4514 (1991).

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

Preuss, L. E.

Richter, M.

M. Richter, “Farbmetrik,” in Optik, L. Bergmann, C. Schäfer, eds. (Walter de Gruyter, Berlin, Germany, 1978), pp. 641–699.

Shepherd, A. P.

Steiner, R.

A. Kienle, L. Lilge, M. S. Paterson, B. C. Wilson, R. Hibst, R. Steiner, “Investigation of multi-layered tissue with in vivo reflectance measurements,” in Photon Transport in Highly Scattering Tissue, S. Avrillier, B. Chance, G. J. Mueller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2326, 212–221 (1994).

A. Kienle, L. Lilge, M. S. Paterson, B. C. Wilson, R. Hibst, R. Steiner, “Spatially resolved absolute diffuse reflectance measurements for remote determination of the optical properties of biological tissue,” Appl. Opt.35, (to be published).

Steinke, J. M.

Taylor, R. C.

Twersky, V.

V. Twersky, “Absorption and multiple scattering by biological suspensions,” J. Opt. Soc. Am. 60, 1084–1093 (1970).

van Gemert, M. J. C.

H. J. van Staveren, C. J. M. Moes, J. van Marie, S. A. Prahl, M. J. C. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400–1100 nm,” Appl. Opt. 31, 4507–4514 (1991).

M. J. C. van Gemert, J. W Pickering, A. J. Welch, “Modelling laser treatment of port-wine stains,” in Management and Treatment of Benign Cutaneous Vascular Lesions, O. T. Tan, ed. (Lea & Febiger, Philadelphia, 1992), pp. 24–47.

van Marie, J.

H. J. van Staveren, C. J. M. Moes, J. van Marie, S. A. Prahl, M. J. C. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400–1100 nm,” Appl. Opt. 31, 4507–4514 (1991).

van Staveren, H. J.

H. J. van Staveren, C. J. M. Moes, J. van Marie, S. A. Prahl, M. J. C. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400–1100 nm,” Appl. Opt. 31, 4507–4514 (1991).

Wang, L.

L. Wang, S. L. Jacques, Monte Carlo Modeling of Light Transport in Multi-layered Tissues in Standard C (University of Texas M. D. Anderson Cancer Center, Houston, Tex., 1992).

Welch, A. J.

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

M. J. C. van Gemert, J. W Pickering, A. J. Welch, “Modelling laser treatment of port-wine stains,” in Management and Treatment of Benign Cutaneous Vascular Lesions, O. T. Tan, ed. (Lea & Febiger, Philadelphia, 1992), pp. 24–47.

Wilson, B. C.

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distribution of light in tissue,” Med. Phys. 10, 824–830 (1983).

A. Kienle, L. Lilge, M. S. Paterson, B. C. Wilson, R. Hibst, R. Steiner, “Investigation of multi-layered tissue with in vivo reflectance measurements,” in Photon Transport in Highly Scattering Tissue, S. Avrillier, B. Chance, G. J. Mueller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2326, 212–221 (1994).

A. Kienle, L. Lilge, M. S. Paterson, B. C. Wilson, R. Hibst, R. Steiner, “Spatially resolved absolute diffuse reflectance measurements for remote determination of the optical properties of biological tissue,” Appl. Opt.35, (to be published).

B. C. Wilson, M. S. Patterson, B. W. Pogue, “Instrumentation for in vivo tissue spectroscopy and imaging,” in Medical Lasers and Systems II, D. M. Harris, CM. Penney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1892, 132–147 (1993).

Appl. Opt. (3)

IEEE J. Quantum Electron. (1)

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

J. Opt. Soc. Am (2)

E.H. Land, J. J. McCann, “Lightness and retinex theory,” J. Opt. Soc. Am. 61, 1–11 (1971).

V. Twersky, “Absorption and multiple scattering by biological suspensions,” J. Opt. Soc. Am. 60, 1084–1093 (1970).

J. Opt. Soc. Am. A (1)

Med. Phys. (1)

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distribution of light in tissue,” Med. Phys. 10, 824–830 (1983).

Proc. Natl. Acad. Sci. U.S.A. (1)

E. H. Land, “An alternative technique for the computation of the designator in the retinex theory of the color vision,” Proc. Natl. Acad. Sci. U.S.A. 83, 3078–3080 (1986).

Sci. Am. (1)

E. H. Land, “The retinex theory of color vision,” Sci. Am. 237, 108–128 (1977).

Vision Res. (1)

E. H. Land, “Recent advances in retinex theory,” Vision Res. 26, 7–21 (1986).

Other (12)

A. Kienle, L. Lilge, M. S. Paterson, B. C. Wilson, R. Hibst, R. Steiner, “Investigation of multi-layered tissue with in vivo reflectance measurements,” in Photon Transport in Highly Scattering Tissue, S. Avrillier, B. Chance, G. J. Mueller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2326, 212–221 (1994).

S. Jacques, M. Keijzer, “Dosimetry for lasers and light in dermatology: Monte Carlo simulations of 577 nm pulsed laser penetration into cutaneous vessels,” in Lasers in Dermatology and Tissue Welding, O. T. Tan, R. A. White, J. V. White, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1422, 2–13 (1991).

M. J. C. van Gemert, J. W Pickering, A. J. Welch, “Modelling laser treatment of port-wine stains,” in Management and Treatment of Benign Cutaneous Vascular Lesions, O. T. Tan, ed. (Lea & Febiger, Philadelphia, 1992), pp. 24–47.

R. R. Anderson, “The physical basis of brown skin colors (melanoderma),” in Brown Melanoderma, T. B. Fitzpatrick, M. M. Wick, K. Toda, eds. (University of Tokyo, Tokyo, 1986), pp. 4–7.

A. Kienle, L. Lilge, M. S. Paterson, B. C. Wilson, R. Hibst, R. Steiner, “Spatially resolved absolute diffuse reflectance measurements for remote determination of the optical properties of biological tissue,” Appl. Opt.35, (to be published).

B. C. Wilson, M. S. Patterson, B. W. Pogue, “Instrumentation for in vivo tissue spectroscopy and imaging,” in Medical Lasers and Systems II, D. M. Harris, CM. Penney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1892, 132–147 (1993).

L. Wang, S. L. Jacques, Monte Carlo Modeling of Light Transport in Multi-layered Tissues in Standard C (University of Texas M. D. Anderson Cancer Center, Houston, Tex., 1992).

A. Kienle, “Lichtausbreitung in biologischem Gewebe,” thesis (University of Ulm, Ulm, Germany, 1994).

D. L. MacAdam, Color Measurements: Themes and Variations (Springer-Verlag, Berlin, 1985), Chap. 3.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Chap. 7 and Chap. 3, p. 66.

R. L. Gregory, Eye and Brain: the Physiology of Seeing (PrincetonU. Press, Princeton, N.J., 1990), Chap. 8.

M. Richter, “Farbmetrik,” in Optik, L. Bergmann, C. Schäfer, eds. (Walter de Gruyter, Berlin, Germany, 1978), pp. 641–699.

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Figures (11)

Fig. 1
Fig. 1

Geometry of the model that was used in the Monte Carlo simulation.

Fig. 2
Fig. 2

Schematic representation of the summations in (a) Eq. (3) and (b) Eq. (5).

Fig. 3
Fig. 3

Experimental arrangement for video measurement of the spatially resolved reflectance. The components are M, mirror; IF, interference filter; OF, optical fiber; O, objective; A, aperture.

Fig. 4
Fig. 4

Measurements of diffuse reflectance at 450, 500, 550, and 633 nm for the phantom incorporating a vessel that contains oxygenated blood.

Fig. 5
Fig. 5

Measurements of diffuse reflectance at 700 nm for the phantom incorporating a vessel. The solid curve corresponds to oxygenated arterial blood, the dashed curve corresponds to venous deoxygenated blood in the model vessel.

Fig. 6
Fig. 6

Monte Carlo calculations of diffuse reflectance at 450, 500, 550, and 633 nm.

Fig. 7
Fig. 7

Calculation of diffuse reflectance at 700 nm. The solid curve corresponds to oxygenated blood, the dashed curve corresponds to venous blood in the model vessel.

Fig. 8
Fig. 8

Measurements of diffuse reflectance in vivo at 450, 500, 550, 633, and 700 nm.

Fig. 9
Fig. 9

Simulations of the in vivo vessel at 450, 550, and 633 nm. At 633 nm the dashed curve represents arterial blood and the solid curve represents venous blood. Depth a of the blood vessel is 0.5 mm, and the diameter is also 0.5 mm.

Fig. 10
Fig. 10

Monte Carlo simulations for diffuse reflectance of a vein with a = 0.2 mm and d = 0.5 mm for different wavelengths.

Fig. 11
Fig. 11

Monte Carlo simulations for diffuse reflectance of a vein with a = 0.04 mm and d = 0.5 mm for different wavelengths. Data for 100% oxygenation saturation at 633 nm are also shown (dashed curve).

Tables (3)

Tables Icon

Table 1 Optical Parameters for the Phantom Model a

Tables Icon

Table 2 Optical Parameters used in the Monte Carlo Simulations of Diffusely Reflected Light from Skin above the Vein in vivo a

Tables Icon

Table 3 Parameters of the in vivo Vein Measurement and the Vessel Simulations a

Equations (7)

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

R ( x , y ) = S ( y y ) χ( x , y ) d y .
R ( x , y = 0 ) = S 0 b / 2 b / 2 χ( x , y )d y .
R Λ ( i , j ) = k δ log ( I k + 1 Λ I k Λ ) ,
δ log ( I k + 1 Λ I k Λ ) = { log ( I k + 1 Λ I k Λ ) , log ( I k + 1 Λ I k Λ ) > 0 , log ( I k + 1 Λ I k Λ ) <
R ¯ Λ ( i ) = 1 N j = 1 N R Λ ( i , j ) .
R Λ ( i , j ) = log ( I υ Λ I s Λ ) ,
R ¯ ( i ) = R Λ ( i , j ) = log ( I υ Λ / I s Λ ) .

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