Abstract

We present a new, to our knowledge, method for extracting optical properties from integrating sphere measurements on thin biological samples. The method is based on multivariate calibration techniques involving Monte Carlo simulations, multiple polynomial regression, and a Newton–Raphson algorithm for solving nonlinear equation systems. Prediction tests with simulated data showed that the mean relative prediction error of the absorption and the reduced scattering coefficients within typical biological ranges were less than 0.3%. Similar tests with data from integrating sphere measurements on 20 dye–polystyrene microsphere phantoms led to mean errors less than 1.7% between predicted and theoretically calculated values. Comparisons showed that our method was more robust and typically 5–10 times as fast and accurate as two other established methods, i.e., the inverse adding–doubling method and the Monte Carlo spline interpolation method.

© 2000 Optical Society of America

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References

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  1. N. M. Anderson, P. Sekelj, “Light-absorbed and scattering properties of non-haemolysed blood,” Phys. Med. Biol. 12, 173–184 (1967).
    [CrossRef] [PubMed]
  2. A. M. K. Nilsson, G. W. Lucassen, W. Verkruysse, S. Andersson-Engels, M. J. C. van Gemert, “Changes in optical properties of human whole blood in vitro due to slow heating,” Photopchem. Photobiol. 65, 366–373 (1997).
    [CrossRef]
  3. A. Roggan, M. Friebel, K. Dörschel, A. Hahn, G. Müller, “Optical properties of circulating human blood in the wavelength range 400–2500 nm,” J. Biomed. Opt. 4, 36–46 (1998).
    [CrossRef]
  4. A. N. Yaroslavsky, I. V. Yaroslavsky, T. Goldback, H.-J. Schwarzmaier, “Influence of the scattering phase function approximation on the optical properties of blood determined from the integrating sphere measurements,” J. Biomed. Opt. 4, 47–53 (1998).
    [CrossRef]
  5. A. M. K. Nilsson, R. Berg, S. Andersson-Engels, “Measurements of the optical properties of tissue in conjunction with photodynamic therapy,” Appl. Opt. 34, 4609–4619 (1995).
    [CrossRef] [PubMed]
  6. W.-C. Lin, M. Motamedi, A. J. Welch, “Dynamics of tissue optics during laser heating of turbid media,” Appl. Opt. 35, 3413–3420 (1996).
    [CrossRef] [PubMed]
  7. A. J. Welch, M. J. C. van Gemert, W. M. Star, B. C. Wilson, “Overview of tissue optics,” Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995), Chap. 2.
  8. H. C. van de Hulst, Multiple Light Scattering, Vols. I and II, (Academic, New York, 1980).
  9. R. Graff, J. G. Aarnoudse, F. F. M. de MulHenk, W. Jentink, “Similarity relations for anisotropic scattering in absorbing media,” Opt. Eng. 32, 244–252 (1993).
    [CrossRef]
  10. D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity relations for anisotropic scattering in Monte Carlo Simulations of deeply penetrating neutral particles,” J. Comp. Physiol. 81, 137–150 (1989).
  11. J. W. Pickering, S. A. Prahl, N. van Wieringen, J. B. Beek, H. J. C. M. Sterenborg, M. J. C. van Gemert, “A double integrating sphere system for measuring the optical properties of tissue,” Appl. Opt. 32, 399–410 (1993).
    [CrossRef] [PubMed]
  12. P. Kubelka, “New contributions to the optics of intensely light scattering materials. Part I,” J. Opt. Soc. Am. A 4, 448–457 (1948).
  13. J. Reichmann, “Determination of absorption and scattering coefficients for nonhomogeneous media. 1. Theory,” Appl. Opt. 12, 1811–1815 (1973).
    [CrossRef]
  14. S. A. Prahl, M. J. C. van Gemert, A. J. Welch, “Determining the optical properties of turbid media by using the adding–doubling method,” Appl. Opt. 32, 559–568 (1993).
    [CrossRef] [PubMed]
  15. C. R. Simpson, M. Kohl, M. Essenpreis, M. Cope, “Near Infrared optical properties of ex-vivo human skin and sub-cutaneous tissues measured using the Monte Carlo inversion technique,” Phys. Med. Biol. 43, 2465–2478 (1998).
    [CrossRef] [PubMed]
  16. L.-H. Wang, S. L. Jacques, L.-Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
    [CrossRef] [PubMed]
  17. S. V. Chapra, R. P. Canale, Numerical Methods for Engineers (McGraw-Hill, New York, 1997), Chap. 6.
  18. W. F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissue,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
    [CrossRef]
  19. J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, M. J. C. van Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol. 42, 2255–2261 (1997).
    [CrossRef] [PubMed]
  20. A. Pifferi, P. Taroni, G. Valentini, S. Andersson-Engels, “Real-time method for fitting time-resolved reflectance and transmittance measurements with a Monte Carlo model,” Appl. Opt. 37, 2774–2780 (1998).
    [CrossRef]
  21. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

1998 (4)

A. Roggan, M. Friebel, K. Dörschel, A. Hahn, G. Müller, “Optical properties of circulating human blood in the wavelength range 400–2500 nm,” J. Biomed. Opt. 4, 36–46 (1998).
[CrossRef]

A. N. Yaroslavsky, I. V. Yaroslavsky, T. Goldback, H.-J. Schwarzmaier, “Influence of the scattering phase function approximation on the optical properties of blood determined from the integrating sphere measurements,” J. Biomed. Opt. 4, 47–53 (1998).
[CrossRef]

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

A. Pifferi, P. Taroni, G. Valentini, S. Andersson-Engels, “Real-time method for fitting time-resolved reflectance and transmittance measurements with a Monte Carlo model,” Appl. Opt. 37, 2774–2780 (1998).
[CrossRef]

1997 (2)

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, M. J. C. van Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol. 42, 2255–2261 (1997).
[CrossRef] [PubMed]

A. M. K. Nilsson, G. W. Lucassen, W. Verkruysse, S. Andersson-Engels, M. J. C. van Gemert, “Changes in optical properties of human whole blood in vitro due to slow heating,” Photopchem. Photobiol. 65, 366–373 (1997).
[CrossRef]

1996 (1)

1995 (2)

A. M. K. Nilsson, R. Berg, S. Andersson-Engels, “Measurements of the optical properties of tissue in conjunction with photodynamic therapy,” Appl. Opt. 34, 4609–4619 (1995).
[CrossRef] [PubMed]

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

1993 (3)

1990 (1)

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

1989 (1)

D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity relations for anisotropic scattering in Monte Carlo Simulations of deeply penetrating neutral particles,” J. Comp. Physiol. 81, 137–150 (1989).

1973 (1)

1967 (1)

N. M. Anderson, P. Sekelj, “Light-absorbed and scattering properties of non-haemolysed blood,” Phys. Med. Biol. 12, 173–184 (1967).
[CrossRef] [PubMed]

1948 (1)

P. Kubelka, “New contributions to the optics of intensely light scattering materials. Part I,” J. Opt. Soc. Am. A 4, 448–457 (1948).

Aalders, M.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, M. J. C. van Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol. 42, 2255–2261 (1997).
[CrossRef] [PubMed]

Aarnoudse, J. G.

R. Graff, J. G. Aarnoudse, F. F. M. de MulHenk, W. Jentink, “Similarity relations for anisotropic scattering in absorbing media,” Opt. Eng. 32, 244–252 (1993).
[CrossRef]

Anderson, N. M.

N. M. Anderson, P. Sekelj, “Light-absorbed and scattering properties of non-haemolysed blood,” Phys. Med. Biol. 12, 173–184 (1967).
[CrossRef] [PubMed]

Andersson-Engels, S.

Beek, J. B.

Beek, J. F.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, M. J. C. van Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol. 42, 2255–2261 (1997).
[CrossRef] [PubMed]

Berg, R.

Blokland, P.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, M. J. C. van Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol. 42, 2255–2261 (1997).
[CrossRef] [PubMed]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Canale, R. P.

S. V. Chapra, R. P. Canale, Numerical Methods for Engineers (McGraw-Hill, New York, 1997), Chap. 6.

Chapra, S. V.

S. V. Chapra, R. P. Canale, Numerical Methods for Engineers (McGraw-Hill, New York, 1997), Chap. 6.

Cheong, W. F.

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

Cope, M.

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

de MulHenk, F. F. M.

R. Graff, J. G. Aarnoudse, F. F. M. de MulHenk, W. Jentink, “Similarity relations for anisotropic scattering in absorbing media,” Opt. Eng. 32, 244–252 (1993).
[CrossRef]

Dörschel, K.

A. Roggan, M. Friebel, K. Dörschel, A. Hahn, G. Müller, “Optical properties of circulating human blood in the wavelength range 400–2500 nm,” J. Biomed. Opt. 4, 36–46 (1998).
[CrossRef]

Essenpreis, M.

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

Friebel, M.

A. Roggan, M. Friebel, K. Dörschel, A. Hahn, G. Müller, “Optical properties of circulating human blood in the wavelength range 400–2500 nm,” J. Biomed. Opt. 4, 36–46 (1998).
[CrossRef]

Goldback, T.

A. N. Yaroslavsky, I. V. Yaroslavsky, T. Goldback, H.-J. Schwarzmaier, “Influence of the scattering phase function approximation on the optical properties of blood determined from the integrating sphere measurements,” J. Biomed. Opt. 4, 47–53 (1998).
[CrossRef]

Graff, R.

R. Graff, J. G. Aarnoudse, F. F. M. de MulHenk, W. Jentink, “Similarity relations for anisotropic scattering in absorbing media,” Opt. Eng. 32, 244–252 (1993).
[CrossRef]

Hahn, A.

A. Roggan, M. Friebel, K. Dörschel, A. Hahn, G. Müller, “Optical properties of circulating human blood in the wavelength range 400–2500 nm,” J. Biomed. Opt. 4, 36–46 (1998).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Jacques, S. L.

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

Jentink, W.

R. Graff, J. G. Aarnoudse, F. F. M. de MulHenk, W. Jentink, “Similarity relations for anisotropic scattering in absorbing media,” Opt. Eng. 32, 244–252 (1993).
[CrossRef]

Kohl, M.

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

Kubelka, P.

P. Kubelka, “New contributions to the optics of intensely light scattering materials. Part I,” J. Opt. Soc. Am. A 4, 448–457 (1948).

Lin, W.-C.

Lucassen, G. W.

A. M. K. Nilsson, G. W. Lucassen, W. Verkruysse, S. Andersson-Engels, M. J. C. van Gemert, “Changes in optical properties of human whole blood in vitro due to slow heating,” Photopchem. Photobiol. 65, 366–373 (1997).
[CrossRef]

Motamedi, M.

Müller, G.

A. Roggan, M. Friebel, K. Dörschel, A. Hahn, G. Müller, “Optical properties of circulating human blood in the wavelength range 400–2500 nm,” J. Biomed. Opt. 4, 36–46 (1998).
[CrossRef]

Nilsson, A. M. K.

A. M. K. Nilsson, G. W. Lucassen, W. Verkruysse, S. Andersson-Engels, M. J. C. van Gemert, “Changes in optical properties of human whole blood in vitro due to slow heating,” Photopchem. Photobiol. 65, 366–373 (1997).
[CrossRef]

A. M. K. Nilsson, R. Berg, S. Andersson-Engels, “Measurements of the optical properties of tissue in conjunction with photodynamic therapy,” Appl. Opt. 34, 4609–4619 (1995).
[CrossRef] [PubMed]

Patterson, M. S.

D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity relations for anisotropic scattering in Monte Carlo Simulations of deeply penetrating neutral particles,” J. Comp. Physiol. 81, 137–150 (1989).

Pickering, J. W.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, M. J. C. van Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol. 42, 2255–2261 (1997).
[CrossRef] [PubMed]

J. W. Pickering, S. A. Prahl, N. van Wieringen, J. B. Beek, H. J. C. M. Sterenborg, M. J. C. van Gemert, “A double integrating sphere system for measuring the optical properties of tissue,” Appl. Opt. 32, 399–410 (1993).
[CrossRef] [PubMed]

Pifferi, A.

Posthumus, P.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, M. J. C. van Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol. 42, 2255–2261 (1997).
[CrossRef] [PubMed]

Prahl, S. A.

Reichmann, J.

Roggan, A.

A. Roggan, M. Friebel, K. Dörschel, A. Hahn, G. Müller, “Optical properties of circulating human blood in the wavelength range 400–2500 nm,” J. Biomed. Opt. 4, 36–46 (1998).
[CrossRef]

Schwarzmaier, H.-J.

A. N. Yaroslavsky, I. V. Yaroslavsky, T. Goldback, H.-J. Schwarzmaier, “Influence of the scattering phase function approximation on the optical properties of blood determined from the integrating sphere measurements,” J. Biomed. Opt. 4, 47–53 (1998).
[CrossRef]

Sekelj, P.

N. M. Anderson, P. Sekelj, “Light-absorbed and scattering properties of non-haemolysed blood,” Phys. Med. Biol. 12, 173–184 (1967).
[CrossRef] [PubMed]

Simpson, C. R.

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

Star, W. M.

A. J. Welch, M. J. C. van Gemert, W. M. Star, B. C. Wilson, “Overview of tissue optics,” Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995), Chap. 2.

Sterenborg, H. J. C. M.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, M. J. C. van Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol. 42, 2255–2261 (1997).
[CrossRef] [PubMed]

J. W. Pickering, S. A. Prahl, N. van Wieringen, J. B. Beek, H. J. C. M. Sterenborg, M. J. C. van Gemert, “A double integrating sphere system for measuring the optical properties of tissue,” Appl. Opt. 32, 399–410 (1993).
[CrossRef] [PubMed]

Taroni, P.

Valentini, G.

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering, Vols. I and II, (Academic, New York, 1980).

van Gemert, M. J. C.

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, M. J. C. van Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol. 42, 2255–2261 (1997).
[CrossRef] [PubMed]

A. M. K. Nilsson, G. W. Lucassen, W. Verkruysse, S. Andersson-Engels, M. J. C. van Gemert, “Changes in optical properties of human whole blood in vitro due to slow heating,” Photopchem. Photobiol. 65, 366–373 (1997).
[CrossRef]

J. W. Pickering, S. A. Prahl, N. van Wieringen, J. B. Beek, H. J. C. M. Sterenborg, M. J. C. van Gemert, “A double integrating sphere system for measuring the optical properties of tissue,” Appl. Opt. 32, 399–410 (1993).
[CrossRef] [PubMed]

S. A. Prahl, M. J. C. van Gemert, A. J. Welch, “Determining the optical properties of turbid media by using the adding–doubling method,” Appl. Opt. 32, 559–568 (1993).
[CrossRef] [PubMed]

A. J. Welch, M. J. C. van Gemert, W. M. Star, B. C. Wilson, “Overview of tissue optics,” Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995), Chap. 2.

van Wieringen, N.

Verkruysse, W.

A. M. K. Nilsson, G. W. Lucassen, W. Verkruysse, S. Andersson-Engels, M. J. C. van Gemert, “Changes in optical properties of human whole blood in vitro due to slow heating,” Photopchem. Photobiol. 65, 366–373 (1997).
[CrossRef]

Wang, L.-H.

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

Welch, A. J.

W.-C. Lin, M. Motamedi, A. J. Welch, “Dynamics of tissue optics during laser heating of turbid media,” Appl. Opt. 35, 3413–3420 (1996).
[CrossRef] [PubMed]

S. A. Prahl, M. J. C. van Gemert, A. J. Welch, “Determining the optical properties of turbid media by using the adding–doubling method,” Appl. Opt. 32, 559–568 (1993).
[CrossRef] [PubMed]

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

A. J. Welch, M. J. C. van Gemert, W. M. Star, B. C. Wilson, “Overview of tissue optics,” Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995), Chap. 2.

Wilson, B. C.

D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity relations for anisotropic scattering in Monte Carlo Simulations of deeply penetrating neutral particles,” J. Comp. Physiol. 81, 137–150 (1989).

A. J. Welch, M. J. C. van Gemert, W. M. Star, B. C. Wilson, “Overview of tissue optics,” Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995), Chap. 2.

Wyman, D. R.

D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity relations for anisotropic scattering in Monte Carlo Simulations of deeply penetrating neutral particles,” J. Comp. Physiol. 81, 137–150 (1989).

Yaroslavsky, A. N.

A. N. Yaroslavsky, I. V. Yaroslavsky, T. Goldback, H.-J. Schwarzmaier, “Influence of the scattering phase function approximation on the optical properties of blood determined from the integrating sphere measurements,” J. Biomed. Opt. 4, 47–53 (1998).
[CrossRef]

Yaroslavsky, I. V.

A. N. Yaroslavsky, I. V. Yaroslavsky, T. Goldback, H.-J. Schwarzmaier, “Influence of the scattering phase function approximation on the optical properties of blood determined from the integrating sphere measurements,” J. Biomed. Opt. 4, 47–53 (1998).
[CrossRef]

Zheng, L.-Q.

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

Appl. Opt. (6)

Comput. Methods Programs Biomed. (1)

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

IEEE J. Quantum Electron. (1)

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

J. Biomed. Opt. (2)

A. Roggan, M. Friebel, K. Dörschel, A. Hahn, G. Müller, “Optical properties of circulating human blood in the wavelength range 400–2500 nm,” J. Biomed. Opt. 4, 36–46 (1998).
[CrossRef]

A. N. Yaroslavsky, I. V. Yaroslavsky, T. Goldback, H.-J. Schwarzmaier, “Influence of the scattering phase function approximation on the optical properties of blood determined from the integrating sphere measurements,” J. Biomed. Opt. 4, 47–53 (1998).
[CrossRef]

J. Comp. Physiol. (1)

D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity relations for anisotropic scattering in Monte Carlo Simulations of deeply penetrating neutral particles,” J. Comp. Physiol. 81, 137–150 (1989).

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

P. Kubelka, “New contributions to the optics of intensely light scattering materials. Part I,” J. Opt. Soc. Am. A 4, 448–457 (1948).

Opt. Eng. (1)

R. Graff, J. G. Aarnoudse, F. F. M. de MulHenk, W. Jentink, “Similarity relations for anisotropic scattering in absorbing media,” Opt. Eng. 32, 244–252 (1993).
[CrossRef]

Photopchem. Photobiol. (1)

A. M. K. Nilsson, G. W. Lucassen, W. Verkruysse, S. Andersson-Engels, M. J. C. van Gemert, “Changes in optical properties of human whole blood in vitro due to slow heating,” Photopchem. Photobiol. 65, 366–373 (1997).
[CrossRef]

Phys. Med. Biol. (3)

N. M. Anderson, P. Sekelj, “Light-absorbed and scattering properties of non-haemolysed blood,” Phys. Med. Biol. 12, 173–184 (1967).
[CrossRef] [PubMed]

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

J. F. Beek, P. Blokland, P. Posthumus, M. Aalders, J. W. Pickering, H. J. C. M. Sterenborg, M. J. C. van Gemert, “In vitro double-integrating-sphere optical properties of tissues between 630 and 1064 nm,” Phys. Med. Biol. 42, 2255–2261 (1997).
[CrossRef] [PubMed]

Other (4)

S. V. Chapra, R. P. Canale, Numerical Methods for Engineers (McGraw-Hill, New York, 1997), Chap. 6.

A. J. Welch, M. J. C. van Gemert, W. M. Star, B. C. Wilson, “Overview of tissue optics,” Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995), Chap. 2.

H. C. van de Hulst, Multiple Light Scattering, Vols. I and II, (Academic, New York, 1980).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

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Figures (6)

Fig. 1
Fig. 1

Setup for R meas and T meas phantom measurements. The sphere is an 8-in. (∼20.3 cm) IS 080 SF from Labsphere, and the parameters are r beam = 1 mm, d sample = 2.2 mm, d slide = 1 mm, r sample = 23 mm, r detector = 12.5 mm, and λ = 633 nm. Note, during R meas measurements, the sample is placed at the port to the right-hand side.

Fig. 2
Fig. 2

Total diffuse reflectance R (a) and transmittance T (b) as a function of the absorption coefficient μ a and the reduced scattering coefficient μs for a thin slab geometry. The R and T data for the plots were generated with Monte Carlo simulations.

Fig. 3
Fig. 3

[(a) and (b)] Absolute and [(c) and (d)] fitting errors of R fit and T fit.

Fig. 4
Fig. 4

Solid curves, contour plots of constant R sim and T sim values as a function of μ a and μs. The curves with positive slopes are R sim plots, and the curves with negative slopes are T sim plots. The markers depict the random distribution of μ a and μs values in the simulated prediction set. The gray dots indicate cases with prediction errors less than 0.5%. The open circles are cases in which Err s exceeds 0.5%, and the triangles are cases in which both Err a and Err s exceed 0.5%.

Fig. 5
Fig. 5

Analysis of prediction errors greater than 0.5%. The upper graph (a) shows Err a , and the lower (b) shows the corresponding Err s . In each single case the three bars indicate the following: left, maximum deviation of ten identical simulations from the true value; middle, average deviation from the mean of the ten simulations; right, deviation of the mean of the ten simulations from the true value.

Fig. 6
Fig. 6

Correlation plots of theoretical calculations of μ a (a) and μs (b) versus μ a and μs values predicted by the MPR method from phantom measurements.

Tables (3)

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Table 1 Prediction Errors, Number of Iterations, and Prediction Calculation Times for Polynomial Fits of Orders 3, 4, and 5

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Table 2 Prediction Errors with Fifth-Order Polynomial Fits and a Reduced Number of Photon Packets and/or Simulations for the Calibration Model

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Table 3 Prediction Errors, Prediction Calculation Times, and Number of Outliers

Equations (9)

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Pμa, μs, m=a0+a1μa+a2μa2++amμam×b0+b1μs+b2μs2++bmμsm,
Rfit=PRμa, μs, m,Tfit=PTμa, μs, m.
Fμa, μs=Rfit-Rmeas,Gμa, μs=Tfit-Tmeas.
-Fμa,k, μs,kGμa,k, μs,k=FμaFμsGμaGμsha,khs,kμa,k+1μs,k+1=μa,kμs,k+ha,khs,k,k=0, 1, 2, 3,  ,
0.1 cm-1μa5 cm-1,1 cm-1μs20 cm-1,g=0.9,n=1.4,
0 cm-1μa3 cm-1,4.4 cm-1μs21.8 cm-1,g=0.92,n=1.33.
Pout=P0δdrwn=0αwrw+αsrs+αdrdn=P0δdrw1-αwrw-αsrs-αdrd,
Err=100% μpred-μrefμref,
Err=100% μpred-μrefμref,max-μref,min.

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