Abstract

We have studied light migration in highly scattering media theoretically and experimentally, using the We approximation in a semi-infinite-geometry boundary condition. Both the light source and the diffusion detector were located on the surface of a semi-infinite medium. Working with frequency-domain spectroscopy, we approached the problem in three areas: (1) we derived theoretical expressions for the measured quantities spectroscopy by applying appropriate boundary conditions to the diffusion equation; (2) in frequency-domain we experimentally verified the theoretical expressions by performing measurements on a macroscopically medium in quasi-semi-infinite-geometry conditions; (3) we applied Monte Carlo methods to homogeneous the semi-infinite-geometry boundary problem. The experimental results and the confirming Monte simulate Carlo simulation show that the diffusion approximation, under the appropriate boundary conditions, accurately estimates the optical parameters of the medium.

© 1994 Optical Society of America

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References

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  1. S. Chandrasekhar, Radiative Transfer (Oxford U. Press, New York, 1960).
  2. V. V. Sobolev, A Treatise on Radiative Transfer (Van Nostrand-Reinhold, Princeton, N.J., 1963).
  3. K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Heading, Mass., 1967), pp. 196–199.
  4. J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).
  5. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. I.
  6. K. Furutsu, "Diffusion equation derived from space-time transport equation," J. Opt. Soc. Am. 70, 360–366 (1980).
    [CrossRef]
  7. W. F. Cheong, S. A. Prahl, and A. J. Welch, "A review of the optical properties of biological tissues," IEEE J. Quantum Electron. 26, 2166–2185 (1990).
    [CrossRef]
  8. J. L. Karagiannes, Z. Zhang, B. Grossweiner, and L. I. Gross-weiner, "Applications of the 1-D diffusion approximation to the optics of tissues and tissue phantoms," Appl. Opt. 28, 2311–2317 (1989).
    [CrossRef] [PubMed]
  9. P. Parsa, S. L. Jacques, and N. S. Nishioka, "Optical properties of rat liver between 350 and 2200 nm," Appl. Opt. 28, 2325–2330 (1989).
    [CrossRef] [PubMed]
  10. T. J. Farrel, M. S. Patterson, and B. Wilson, "A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo," Med. Phys. 19, 879–888 (1992).
    [CrossRef]
  11. E. M. Sevick, B. Chance, J. Leigh, S. Nioka, and M. Maris, "Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxgenation," Anal. Biochem. 195, 330–351 (1991).
    [CrossRef] [PubMed]
  12. M. S. Patterson, B. Chance, and B. C. Wilson, "Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties," Appl. Opt. 28, 2331–2336 (1989).
    [CrossRef] [PubMed]
  13. M. Cope, P. van der Zee, M. Essenpreis, S. R. Arridge, and D. T. Delpy, "Data analysis methods for near infrared spectroscopy of tissues: problems in determining the relative cytochrome aa3 concentration," in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance and A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 251–263 (1991).
    [CrossRef]
  14. D. A. Benaron and D. K. Stevenson, "Optical time-of-flight and absorbance imaging of biologic media," Science 259, 1463–1466 (1993).
    [CrossRef] [PubMed]
  15. D. A. Benaron, C. D. Kurth, J. Steven, L. C. Wagerle, B. Chance, and M. Delivoira-Papadopoulos, "Non-invasive estimation of cerebral oxygenation and oxygen consumption using phase-shift spectrophotometry," Proc. IEEE Eng. Biol. Soc. 12, 2004–2006 (1990).
  16. W. M. Star, J. P. A. Marijnissen, and M. J. C. van Gemert, "Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory," Phys. Med. Biol. 33, 437–454 (1988).
    [CrossRef] [PubMed]
  17. M. S. Patterson, J. D. Moulton, B. C. Wilson, K. W. Berndt, and J. R. Lakowicz, "Frequency-domain reflectance for the determination of the scattering and absorption properties of tissue," Appl. Opt. 30, 4474–4476 (1991).
    [CrossRef] [PubMed]
  18. B. J. Tromberg, L. O. Svaasand, T. T. Tsay, and R. C. Haskell, "Properties of photon density waves in multiple-scattering media," Appl. Opt. 32, 607–616 (1993).
    [CrossRef] [PubMed]
  19. S. Fantini, M. A. Franceschini, J. B. Fishkin, B. Barbieri, and E. Gratton, "Quantitative determination of the absorption spectra of chromophores in strongly scattering media: a novel LED based technique," Appl. Opt. 33, 5204–5213 (1994).
    [CrossRef] [PubMed]
  20. J. B. Fishkin and E. Gratton, "Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge," J. Opt. Soc. Am. A 10, 127–140 (1993).
    [CrossRef] [PubMed]
  21. B. J. Tromberg, L. O. Svaasand, T. T. Tsay, R. C. Haskell, and M. W. Berns, "Optical property measurement in turbid media using frequency-domain photon migration," in Future Trends in Biomedical Applications of Lasers, L. O. Svaasand, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1525, 52–58 (1991).
    [CrossRef]
  22. M. Keijzer, W. M. Star, and P. R. M. Storchi, "Optical diffusion in layered media," Appl. Opt. 27, 1820–1824 (1988).
    [CrossRef] [PubMed]
  23. R. A. J. Groenhuis, H. A. Ferwerda, and J. J. Ten Bosch, "Scattering and absorption of turbid materials determined from reflection measurements. 1: Theory," Appl. Opt. 22, 2456–2462 (1983).
    [CrossRef] [PubMed]
  24. F. P. Bolin, L. E. Preuss, R. C. Taylor, and R. J. Ference, "Refractive index of some mammalian tissues using a fiber optic cladding method," Appl. Opt. 28, 2297–2302 (1989).
    [CrossRef] [PubMed]
  25. G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, "The theory of the back scattering of light by blood," J. Phys. D 11, 1463–1479 (1978).
    [CrossRef]
  26. B. A. Feddersen, D. W. Piston, and E. Gratton, "Digital parallel acquisition in frequency domain fluorometry," Rev. Sci. Instrum. 60, 2929–2936 (1989).
    [CrossRef]
  27. B. C. Wilson, M. S. Patterson, and B. W. Pogue, "Instrumentation for in vivo tissue spectroscopy and imaging," in Medical Lasers and Systems II, D. M. Harris and C. M. Penney, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1892, 132–147 (1993).
    [CrossRef]
  28. G. M. Hale and M. R. Querry, "Optical constants of water in the 200-nm to 200-µm wavelength region," Appl. Opt. 12, 555–563 (1973).
    [CrossRef] [PubMed]

1994 (1)

1993 (3)

1992 (1)

T. J. Farrel, M. S. Patterson, and B. Wilson, "A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo," Med. Phys. 19, 879–888 (1992).
[CrossRef]

1991 (2)

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, and M. Maris, "Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxgenation," Anal. Biochem. 195, 330–351 (1991).
[CrossRef] [PubMed]

M. S. Patterson, J. D. Moulton, B. C. Wilson, K. W. Berndt, and J. R. Lakowicz, "Frequency-domain reflectance for the determination of the scattering and absorption properties of tissue," Appl. Opt. 30, 4474–4476 (1991).
[CrossRef] [PubMed]

1990 (1)

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

1989 (5)

1988 (2)

M. Keijzer, W. M. Star, and P. R. M. Storchi, "Optical diffusion in layered media," Appl. Opt. 27, 1820–1824 (1988).
[CrossRef] [PubMed]

W. M. Star, J. P. A. Marijnissen, and M. J. C. van Gemert, "Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory," Phys. Med. Biol. 33, 437–454 (1988).
[CrossRef] [PubMed]

1983 (1)

1980 (1)

1978 (1)

G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, "The theory of the back scattering of light by blood," J. Phys. D 11, 1463–1479 (1978).
[CrossRef]

1973 (1)

van der Zee, P.

M. Cope, P. van der Zee, M. Essenpreis, S. R. Arridge, and D. T. Delpy, "Data analysis methods for near infrared spectroscopy of tissues: problems in determining the relative cytochrome aa3 concentration," in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance and A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 251–263 (1991).
[CrossRef]

van Gemert, M. J. C.

W. M. Star, J. P. A. Marijnissen, and M. J. C. van Gemert, "Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory," Phys. Med. Biol. 33, 437–454 (1988).
[CrossRef] [PubMed]

Arridge, S. R.

M. Cope, P. van der Zee, M. Essenpreis, S. R. Arridge, and D. T. Delpy, "Data analysis methods for near infrared spectroscopy of tissues: problems in determining the relative cytochrome aa3 concentration," in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance and A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 251–263 (1991).
[CrossRef]

Barbieri, B.

Benaron, D. A.

D. A. Benaron and D. K. Stevenson, "Optical time-of-flight and absorbance imaging of biologic media," Science 259, 1463–1466 (1993).
[CrossRef] [PubMed]

D. A. Benaron, C. D. Kurth, J. Steven, L. C. Wagerle, B. Chance, and M. Delivoira-Papadopoulos, "Non-invasive estimation of cerebral oxygenation and oxygen consumption using phase-shift spectrophotometry," Proc. IEEE Eng. Biol. Soc. 12, 2004–2006 (1990).

Berndt, K. W.

Berns, M. W.

B. J. Tromberg, L. O. Svaasand, T. T. Tsay, R. C. Haskell, and M. W. Berns, "Optical property measurement in turbid media using frequency-domain photon migration," in Future Trends in Biomedical Applications of Lasers, L. O. Svaasand, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1525, 52–58 (1991).
[CrossRef]

Bolin, F. P.

Bosch, J. J. Ten

Case, K. M.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Heading, Mass., 1967), pp. 196–199.

Chance, B.

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, and M. Maris, "Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxgenation," Anal. Biochem. 195, 330–351 (1991).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, and B. C. Wilson, "Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties," Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

D. A. Benaron, C. D. Kurth, J. Steven, L. C. Wagerle, B. Chance, and M. Delivoira-Papadopoulos, "Non-invasive estimation of cerebral oxygenation and oxygen consumption using phase-shift spectrophotometry," Proc. IEEE Eng. Biol. Soc. 12, 2004–2006 (1990).

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, New York, 1960).

Cheong, W. F.

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

Cope, M.

M. Cope, P. van der Zee, M. Essenpreis, S. R. Arridge, and D. T. Delpy, "Data analysis methods for near infrared spectroscopy of tissues: problems in determining the relative cytochrome aa3 concentration," in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance and A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 251–263 (1991).
[CrossRef]

Delivoira-Papadopoulos, M.

D. A. Benaron, C. D. Kurth, J. Steven, L. C. Wagerle, B. Chance, and M. Delivoira-Papadopoulos, "Non-invasive estimation of cerebral oxygenation and oxygen consumption using phase-shift spectrophotometry," Proc. IEEE Eng. Biol. Soc. 12, 2004–2006 (1990).

Delpy, D. T.

M. Cope, P. van der Zee, M. Essenpreis, S. R. Arridge, and D. T. Delpy, "Data analysis methods for near infrared spectroscopy of tissues: problems in determining the relative cytochrome aa3 concentration," in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance and A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 251–263 (1991).
[CrossRef]

Duderstadt, J. J.

J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

Eason, G.

G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, "The theory of the back scattering of light by blood," J. Phys. D 11, 1463–1479 (1978).
[CrossRef]

Essenpreis, M.

M. Cope, P. van der Zee, M. Essenpreis, S. R. Arridge, and D. T. Delpy, "Data analysis methods for near infrared spectroscopy of tissues: problems in determining the relative cytochrome aa3 concentration," in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance and A. Katzir, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 251–263 (1991).
[CrossRef]

Fantini, S.

Farrel, T. J.

T. J. Farrel, M. S. Patterson, and B. Wilson, "A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo," Med. Phys. 19, 879–888 (1992).
[CrossRef]

Feddersen, B. A.

B. A. Feddersen, D. W. Piston, and E. Gratton, "Digital parallel acquisition in frequency domain fluorometry," Rev. Sci. Instrum. 60, 2929–2936 (1989).
[CrossRef]

Ference, R. J.

Ferwerda, H. A.

Fishkin, J. B.

Franceschini, M. A.

Furutsu, K.

Gratton, E.

Groenhuis, R. A. J.

Grossweiner, B.

Gross-weiner, L. I.

Hale, G. M.

Hamilton, L. J.

J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

Haskell, R. C.

B. J. Tromberg, L. O. Svaasand, T. T. Tsay, and R. C. Haskell, "Properties of photon density waves in multiple-scattering media," Appl. Opt. 32, 607–616 (1993).
[CrossRef] [PubMed]

B. J. Tromberg, L. O. Svaasand, T. T. Tsay, R. C. Haskell, and M. W. Berns, "Optical property measurement in turbid media using frequency-domain photon migration," in Future Trends in Biomedical Applications of Lasers, L. O. Svaasand, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1525, 52–58 (1991).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. I.

Jacques, S. L.

Karagiannes, J. L.

Keijzer, M.

Kurth, C. D.

D. A. Benaron, C. D. Kurth, J. Steven, L. C. Wagerle, B. Chance, and M. Delivoira-Papadopoulos, "Non-invasive estimation of cerebral oxygenation and oxygen consumption using phase-shift spectrophotometry," Proc. IEEE Eng. Biol. Soc. 12, 2004–2006 (1990).

Lakowicz, J. R.

Leigh, J.

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, and M. Maris, "Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxgenation," Anal. Biochem. 195, 330–351 (1991).
[CrossRef] [PubMed]

Marijnissen, J. P. A.

W. M. Star, J. P. A. Marijnissen, and M. J. C. van Gemert, "Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory," Phys. Med. Biol. 33, 437–454 (1988).
[CrossRef] [PubMed]

Maris, M.

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, and M. Maris, "Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxgenation," Anal. Biochem. 195, 330–351 (1991).
[CrossRef] [PubMed]

Moulton, J. D.

Nioka, S.

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, and M. Maris, "Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxgenation," Anal. Biochem. 195, 330–351 (1991).
[CrossRef] [PubMed]

Nisbet, R. M.

G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, "The theory of the back scattering of light by blood," J. Phys. D 11, 1463–1479 (1978).
[CrossRef]

Nishioka, N. S.

Parsa, P.

Patterson, M. S.

T. J. Farrel, M. S. Patterson, and B. Wilson, "A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo," Med. Phys. 19, 879–888 (1992).
[CrossRef]

M. S. Patterson, J. D. Moulton, B. C. Wilson, K. W. Berndt, and J. R. Lakowicz, "Frequency-domain reflectance for the determination of the scattering and absorption properties of tissue," Appl. Opt. 30, 4474–4476 (1991).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, and B. C. Wilson, "Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties," Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

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

Piston, D. W.

B. A. Feddersen, D. W. Piston, and E. Gratton, "Digital parallel acquisition in frequency domain fluorometry," Rev. Sci. Instrum. 60, 2929–2936 (1989).
[CrossRef]

Pogue, B. W.

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

Prahl, S. A.

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

Preuss, L. E.

Querry, M. R.

Sevick, E. M.

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, and M. Maris, "Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxgenation," Anal. Biochem. 195, 330–351 (1991).
[CrossRef] [PubMed]

Sobolev, V. V.

V. V. Sobolev, A Treatise on Radiative Transfer (Van Nostrand-Reinhold, Princeton, N.J., 1963).

Star, W. M.

M. Keijzer, W. M. Star, and P. R. M. Storchi, "Optical diffusion in layered media," Appl. Opt. 27, 1820–1824 (1988).
[CrossRef] [PubMed]

W. M. Star, J. P. A. Marijnissen, and M. J. C. van Gemert, "Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory," Phys. Med. Biol. 33, 437–454 (1988).
[CrossRef] [PubMed]

Steven, J.

D. A. Benaron, C. D. Kurth, J. Steven, L. C. Wagerle, B. Chance, and M. Delivoira-Papadopoulos, "Non-invasive estimation of cerebral oxygenation and oxygen consumption using phase-shift spectrophotometry," Proc. IEEE Eng. Biol. Soc. 12, 2004–2006 (1990).

Stevenson, D. K.

D. A. Benaron and D. K. Stevenson, "Optical time-of-flight and absorbance imaging of biologic media," Science 259, 1463–1466 (1993).
[CrossRef] [PubMed]

Storchi, P. R. M.

Svaasand, L. O.

B. J. Tromberg, L. O. Svaasand, T. T. Tsay, and R. C. Haskell, "Properties of photon density waves in multiple-scattering media," Appl. Opt. 32, 607–616 (1993).
[CrossRef] [PubMed]

B. J. Tromberg, L. O. Svaasand, T. T. Tsay, R. C. Haskell, and M. W. Berns, "Optical property measurement in turbid media using frequency-domain photon migration," in Future Trends in Biomedical Applications of Lasers, L. O. Svaasand, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1525, 52–58 (1991).
[CrossRef]

Taylor, R. C.

Tromberg, B. J.

B. J. Tromberg, L. O. Svaasand, T. T. Tsay, and R. C. Haskell, "Properties of photon density waves in multiple-scattering media," Appl. Opt. 32, 607–616 (1993).
[CrossRef] [PubMed]

B. J. Tromberg, L. O. Svaasand, T. T. Tsay, R. C. Haskell, and M. W. Berns, "Optical property measurement in turbid media using frequency-domain photon migration," in Future Trends in Biomedical Applications of Lasers, L. O. Svaasand, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1525, 52–58 (1991).
[CrossRef]

Tsay, T. T.

B. J. Tromberg, L. O. Svaasand, T. T. Tsay, and R. C. Haskell, "Properties of photon density waves in multiple-scattering media," Appl. Opt. 32, 607–616 (1993).
[CrossRef] [PubMed]

B. J. Tromberg, L. O. Svaasand, T. T. Tsay, R. C. Haskell, and M. W. Berns, "Optical property measurement in turbid media using frequency-domain photon migration," in Future Trends in Biomedical Applications of Lasers, L. O. Svaasand, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 1525, 52–58 (1991).
[CrossRef]

Turnbull, F. W.

G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, "The theory of the back scattering of light by blood," J. Phys. D 11, 1463–1479 (1978).
[CrossRef]

Veitch, A. R.

G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, "The theory of the back scattering of light by blood," J. Phys. D 11, 1463–1479 (1978).
[CrossRef]

Wagerle, L. C.

D. A. Benaron, C. D. Kurth, J. Steven, L. C. Wagerle, B. Chance, and M. Delivoira-Papadopoulos, "Non-invasive estimation of cerebral oxygenation and oxygen consumption using phase-shift spectrophotometry," Proc. IEEE Eng. Biol. Soc. 12, 2004–2006 (1990).

Welch, A. J.

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

Wilson, B.

T. J. Farrel, M. S. Patterson, and B. Wilson, "A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo," Med. Phys. 19, 879–888 (1992).
[CrossRef]

Wilson, B. C.

Zhang, Z.

Zweifel, P. F.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Heading, Mass., 1967), pp. 196–199.

Anal. Biochem. (1)

E. M. Sevick, B. Chance, J. Leigh, S. Nioka, and M. Maris, "Quantitation of time- and frequency-resolved optical spectra for the determination of tissue oxgenation," Anal. Biochem. 195, 330–351 (1991).
[CrossRef] [PubMed]

Appl. Opt. (10)

G. M. Hale and M. R. Querry, "Optical constants of water in the 200-nm to 200-µm wavelength region," Appl. Opt. 12, 555–563 (1973).
[CrossRef] [PubMed]

R. A. J. Groenhuis, H. A. Ferwerda, and J. J. Ten Bosch, "Scattering and absorption of turbid materials determined from reflection measurements. 1: Theory," Appl. Opt. 22, 2456–2462 (1983).
[CrossRef] [PubMed]

M. Keijzer, W. M. Star, and P. R. M. Storchi, "Optical diffusion in layered media," Appl. Opt. 27, 1820–1824 (1988).
[CrossRef] [PubMed]

F. P. Bolin, L. E. Preuss, R. C. Taylor, and R. J. Ference, "Refractive index of some mammalian tissues using a fiber optic cladding method," Appl. Opt. 28, 2297–2302 (1989).
[CrossRef] [PubMed]

J. L. Karagiannes, Z. Zhang, B. Grossweiner, and L. I. Gross-weiner, "Applications of the 1-D diffusion approximation to the optics of tissues and tissue phantoms," Appl. Opt. 28, 2311–2317 (1989).
[CrossRef] [PubMed]

P. Parsa, S. L. Jacques, and N. S. Nishioka, "Optical properties of rat liver between 350 and 2200 nm," Appl. Opt. 28, 2325–2330 (1989).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, and B. C. Wilson, "Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties," Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

B. J. Tromberg, L. O. Svaasand, T. T. Tsay, and R. C. Haskell, "Properties of photon density waves in multiple-scattering media," Appl. Opt. 32, 607–616 (1993).
[CrossRef] [PubMed]

S. Fantini, M. A. Franceschini, J. B. Fishkin, B. Barbieri, and E. Gratton, "Quantitative determination of the absorption spectra of chromophores in strongly scattering media: a novel LED based technique," Appl. Opt. 33, 5204–5213 (1994).
[CrossRef] [PubMed]

M. S. Patterson, J. D. Moulton, B. C. Wilson, K. W. Berndt, and J. R. Lakowicz, "Frequency-domain reflectance for the determination of the scattering and absorption properties of tissue," Appl. Opt. 30, 4474–4476 (1991).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

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

J. Opt. Soc. Am. (1)

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

J. Phys. D (1)

G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, "The theory of the back scattering of light by blood," J. Phys. D 11, 1463–1479 (1978).
[CrossRef]

Med. Phys. (1)

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Figures (7)

Fig. 1
Fig. 1

Semi-infinite-geometry model: zb is the distance between the extrapolated boundary and the surface of the medium and z0 is the depth of the effective single scatter source inside the scattering medium. The strongly scattering medium extends in the space z > zb. The detector optical fiber, which is parallel to the z axis, is immersed in the scattering medium at a depth z ranging from zb to zb + z0. The distance between the effective (image) photon source and the tip of the optical fiber is ra(ri). The projection of the source–detector distance onto the plane z = 0 is ρ.

Fig. 2
Fig. 2

Experimental arrangement showing the two source–detector configurations used in (a) the semi-infinite geometry and (b) the infinite geometry. In (a) the source and the detector are at the surface, and in (b) they are immersed in the medium. Diode laser DL is modulated by synthesizer Synth.1 (frequency f = 120 MHz) through rf amplifier A1. The detector light is collected by a bundle of optical fibers coupled to a PMT, whose gain function is modulated at frequency f + Δf = 120 MHz + 80 Hz by synthesizer synth.2 via amplifier A2. The output signal of the PMT (frequency Δf = 80 Hz) is sent to a computer card for processing. Synth.1, synth.2, and the computer card are all synchronized. Sync.’s, reference signals (synchronous clock).

Fig. 3
Fig. 3

Results of the infinite-geometry measurements relative to the various concentrations of Liposyn and black India ink. In (a1) and (b1) the x axis indicates the Liposyn solids content (%) at constant black-India-ink concentration (0 mL/L), and in (a2) and (b2) the x axis shows black-India-ink concentrations (mL/L) at constant Liposyn solids content (1.8%). In all the panels the error bars are of the order of the symbol dimensions or smaller. (a1), (a2) μs′, the straight line through the points relative to different Liposyn solids content is obtained by a linear least-squares fit. (bi), (b2) μa, the straight line, obtained by a linear least-squares fit, has been calculated with the points relative to ink concentrations smaller than 2 mL/L (see Section 6).

Fig. 4
Fig. 4

Comparison of the values of (a1), (a2) μs′ and (b1), (b2) μa measured in the three cases considered: circles, infinite geometry; squares, semi-infinite geometry with boundary conditions; triangles, semi-infinite geometry without boundary conditions. The conditions for the x axis are described in the caption of Fig. 3. The error bars are of the order of the symbol dimensions or smaller.

Fig. 5
Fig. 5

Comparison of the slopes associated with (a1), (a2) dc intensity; (b1), (b2) ac intensity; and (c1), (c2) phase, measured for different Liposyn and ink concentrations in the three cases considered: circles, infinite geometry; squares, semi-infinite geometry with boundary conditions; triangles, semi-infinite geometry without boundary conditions. The conditions for the x axis are described in the caption of Fig. 3.

Fig. 6
Fig. 6

Straight lines associated with (a) dc, (b) ac, and (c) phase as a function of r (infinite geometry) or ρ (semi-infinite geometry), obtained from the Monte Carlo simulation. The different symbols refer to the three conditions examined (dc* and ac* refer to values relative to the maximum source–detector distance and Φ* refers to a value relative to the minimum source–detector distance). Circles, infinite-medium simulation, infinite-geometry equations: dc* = ln(rUdc), ac* = ln(rUac), Φ* = Φ Squares, semi-infinite-medium simulation, semi-infinite-geometry equations: dc*, ac*, and Φ* given by the left-hand sides of Eqs. (20)(22). Triangles, semi-infinite-medium simulation, infinite-geometry equations: dc* = ln(ρψdcs), ac* = ln(ρψacs), Φ* = Φs.

Fig. 7
Fig. 7

Differences between the values of (a1), (a2) μs′ and (b1), (b2) μa measured in the semi-infinite geometry an; those obtained in the infinite geometry: squares, with boundary conditions; triangles, without boundary conditions. The conditions for the x axis are described in the caption of Fig. 3.

Tables (2)

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Table 1 Sensitivity to Source-Detector Positioning on the Surface

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Table 2 Monte Carlo Simulation Results

Equations (26)

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u t = - v · u - v ( μ a + μ s ) u + 4 π d Ω v μ s p s ( Ω Ω ) × u ( r , Ω , t ) + q ( r , Ω , t ) ,
ψ ( r , Ω , t ) = 1 4 π Ψ + 3 4 π J · Ω ,             | Ψ 3 J · Ω | 1 ,
U t + · J + v μ a U = q 0 ,
1 v J t + v 3 U + ( μ a + μ s ) J = q 1 ,
q ( r , Ω , t ) = 1 4 π q 0 ( r , t ) + 3 4 π q 1 ( r , t ) · Ω .
J = - v D U ,
U t - v D 2 U + v μ a U = q 0 .
U = U dc + U ac exp [ - i ( ω t - ϕ ) ] ,
U dc = S 4 π v D exp [ - r ( μ a D ) 1 / 2 ] r ,
U ac = S A 4 π v D exp { - r ( μ a 2 D ) 1 / 2 [ ( 1 + x 2 ) 1 / 2 + 1 ] 1 / 2 } r ,
ϕ = r ( μ a 2 D ) 1 / 2 [ ( 1 + x 2 ) 1 / 2 - 1 ] 1 / 2 ,
| Ψ 3 J · Ω | | v U 3 v D U · Ω | > U 3 D U 8.
U s = S 4 π v D [ exp [ - r a ( μ a D ) 1 / 2 ] r a - exp [ - r i ( μ a D ) 1 / 2 ] r i ] + S A 4 π v D [ exp [ - r a ( μ a 2 D ) 1 / 2 ] { [ ( 1 + x 2 ) 1 / 2 + 1 ] 1 / 2 - i [ ( 1 + x 2 ) 1 / 2 - 1 ] 1 / 2 } r a [ - exp [ - r i ( μ a 2 D ) 1 / 2 ] { [ ( 1 + x 2 ) 1 / 2 + 1 ] 1 / 2 - i [ ( 1 + x 2 ) 1 / 2 - 1 ] 1 / 2 } r i ] exp ( - i ω t ) ,
r a = ρ [ 1 + ( z b + z 0 - z ρ ) 2 ] 1 / 2 , r i = ρ [ 1 + ( z b + z 0 + z ρ ) 2 ] 1 / 2 .
ψ dc s = 2 S ( 4 π ) 2 D exp [ - ρ ( μ a D ) 1 / 2 ] ρ 3 × [ 1 + ρ ( μ a D ) 1 / 2 ] ( z b + z 0 ) × ( z + 3 D { 1 - ( z b + z 0 ) 2 + 3 z 2 2 ρ 2 × { [ 3 + ρ 2 μ a D 1 + ρ ( μ a D ) 1 / 2 ] } ) ,
ψ ac s = 2 S A ( 4 π ) 2 D exp [ - ρ ( μ a 2 D ) 1 / 2 V + ] ρ 3 × { 1 + ρ ( 2 μ a D ) 1 / 2 V + + ρ 2 μ a D ( 1 + x 2 ) 1 / 2 } 1 / 2 × ( z b + z 0 ) ( z + 3 D { 1 - ( z b + z 0 ) 2 + 3 z 2 2 ρ 2 × [ 2 + 1 + ρ ( μ a 2 D ) 1 / 2 V + 1 + ρ ( 2 μ a D ) 1 / 2 V + + ρ 2 μ a D ( 1 + x 2 ) 1 / 2 + { [ ρ ( μ a 2 D ) 1 / 2 V + ] } ( ) ,
Φ s = ρ ( μ a 2 D ) 1 / 2 V - - arctan [ ρ ( μ a 2 D ) 1 / 2 V - 1 + ρ ( μ a 2 D ) 1 / 2 V + ] ,
V + = [ ( 1 + x 2 ) + 1 ] 1 / 2 ,             V - = [ ( 1 + x 2 ) - 1 ] 1 / 2 ,
x = ω v μ a .
1 8 ρ 2 μ a D ( z b + z 0 + z ρ ) 4 1 ,
1 8 ρ 2 ( V + ) 2 μ a 2 D ( z b + z 0 + z ρ ) 4 1 ,
1 2 ρ 2 ( μ a D ) 1 / 2 V - ( z b + z 0 ) 2 + 3 z 2 ρ 2 1.
- + J ( ρ , t ) exp ( i ω t ) d t = J ˜ ( ρ , ω ) .
ln { ρ 3 ψ dc s [ 1 + ρ ( μ a D ) 1 / 2 ] F dc ( ρ , μ a , D , z b , z 0 , z } = - ρ ( μ a D ) 1 / 2 + G dc ( D , S , z b , z 0 ) ,
ln { ρ 3 ψ dc s [ 1 + ρ ( 2 μ a D ) 1 / 2 V + + ρ 2 μ a D ( 1 + x 2 ) 1 / 2 ] 1 / 2 F ac ( ρ , μ a , D , ω , v , z b , z 0 , z ) } = - ρ ( z a 2 D ) 1 / 2 V + + G ac ( D , S , A , z b , z 0 ) ,
Φ s + arctan [ F Φ ( ρ , μ a , D , ω , v ) ] = ρ ( μ a 2 D ) 1 / 2 V - ,

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