Abstract

We modified the diffusion approximation of the time-dependent radiative transfer equation to account for a finite scattering delay time. Under the usual assumptions of the diffusion approximation, the effect of the scattering delay leads to a simple renormalization of the light velocity that appears in the diffusion equation. Accuracy of the model was evaluated by comparison with Monte Carlo simulations in the frequency domain for a semi-infinite geometry. A good agreement is demonstrated for both matched and mismatched boundary conditions when the distance from the source is sufficiently large. The modified diffusion model predicts that the neglect of the scattering delay when the optical properties of the turbid material are derived from normalized frequency- or time-domain measurements should result in an underestimation of the absorption coefficient and an overestimation of the transport coefficient. These observations are consistent with the published experimental data.

© 1997 Optical Society of America

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References

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  1. G. J. Müller, ed., Medical Optical Tomography: Functional Imaging and Monitoring (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp.395–549.
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  3. M. Essenpreis, C. E. Elwell, M. Cope, P. van der Zee, S. R. Arridge, D. T. Delpy, “Spectral dependence of temporal point spread functions in human tissues,” Appl. Opt. 32, 418–425 (1993).
    [CrossRef] [PubMed]
  4. J. B. Fishkin, P. T. C. So, A. E. Cerusci, S. Fantini, M. A. Franceschini, E. Gratton, “Frequency-domain method for measuring spectral properties in multiple-scattering media: methemoglobin absorption spectrum in a tissuelike phantom,” Appl. Opt. 34, 1143–1155 (1995).
    [CrossRef] [PubMed]
  5. B. J. Tromberg, L. O. Svaas, T. T. Tsay, R. C. Haskell, “Properties of photon density waves in multiple-scattering media,” Appl. Opt. 32, 607–616 (1993).
    [CrossRef] [PubMed]
  6. S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of the optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
    [CrossRef] [PubMed]
  7. V. V. Sobolev, A Treatise on Radiative Transfer (Van Nostrand-Reinhold, Princeton, N.J., 1963), pp.240–244.
  8. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1, pp. 175–186.
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    [CrossRef]
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    [CrossRef] [PubMed]
  11. M. S. Patterson, J. D. Moulton, B. C. Wilson, K. W. Berndt, 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]
  12. K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
    [CrossRef] [PubMed]
  13. S. Fantini, M. A. Franceschini, E. Gratton, “Semi-infinite geometry boundary problem for light migration in highly scattering media: a frequency-domain study in the diffusion approximation,” J. Opt. Soc. Am. B 11, 2128–2138 (1994).
    [CrossRef]
  14. R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in the radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
    [CrossRef]
  15. I. V. Yaroslavsky, H.-J. Schwarzmaier, A. N. Yaroslavsky, V. V. Tuchin, “The radiative transfer equation and its diffusion approximation in the frequency-domain technique: a comparison,” in Photon Transport in Highly Scattering Tissue, S. Avriller, B. Chance, G. J. Müller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 465–474 (1994).
    [CrossRef]
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    [CrossRef]
  17. W.-F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
    [CrossRef]
  18. K. S. Shifrin, I. G. Zolotov, “Nonstationary scattering of electromagnetic pulses by spherical particles,” Appl. Opt. 34, 552–558 (1995).
    [CrossRef] [PubMed]
  19. E. J. Heilweil, R. M. Hochstrasser, “Nonlinear spectroscopy and picosecond transient grating study of colloidal gold,” J. Chem. Phys. 82, 4762–4770 (1985).
    [CrossRef]
  20. S. J. Strickler, R. A. Berg, “Relationship between absorption intensity and fluorescence lifetime of molecules,” J. Chem. Phys. 37, 814–822 (1962).
    [CrossRef]
  21. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), pp.174–175.
  22. M. Keijzer, W. M. Star, P. R. M. Storchi, “Optical diffusion in layered media,” Appl. Opt. 27, 1820–1824 (1988).
    [CrossRef] [PubMed]
  23. S. J. Madsen, B. C. Wilson, M. S. Patterson, Y. D. Park, S. L. Jacques, Y. Hefetz, “Experimental tests of a simple diffusion model for the estimation of scattering and absorption coefficients of turbid media from time-resolved diffuse reflectance measurements,” Appl. Opt. 31, 3509–3517 (1992).
    [CrossRef] [PubMed]
  24. A. H. Hielscher, H. Liu, B. Chance, F. K. Tittel, S. L. Jacques, “Time-resolved photon emission from layered turbid media,” in Appl. Opt. 35, 719–728 (1996).
    [CrossRef] [PubMed]
  25. I. V. Yaroslavsky, A. N. Yaroslavsky, H.-J. Schwarzmaier, G. G. Akchurin, V. V. Tuchin, “New approach to Monte Carlo simulation of photon transport in the frequency-domain,” in Photon Propagation in Tissues, B. Chance, D. T. Delpy, G. J. Müller, eds., Proc. SPIE2626, 45–55 (1995).
    [CrossRef]
  26. M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. 66, 3132–3135 (1991).
    [CrossRef] [PubMed]
  27. A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, B. C. Wilson, “Spatially resolved absolute measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996).
    [CrossRef] [PubMed]

1996 (3)

1995 (3)

1994 (2)

1993 (2)

1992 (2)

1991 (2)

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. 66, 3132–3135 (1991).
[CrossRef] [PubMed]

M. S. Patterson, J. D. Moulton, B. C. Wilson, K. W. Berndt, 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 (2)

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

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

1989 (1)

1988 (1)

1985 (1)

E. J. Heilweil, R. M. Hochstrasser, “Nonlinear spectroscopy and picosecond transient grating study of colloidal gold,” J. Chem. Phys. 82, 4762–4770 (1985).
[CrossRef]

1980 (1)

1962 (1)

S. J. Strickler, R. A. Berg, “Relationship between absorption intensity and fluorescence lifetime of molecules,” J. Chem. Phys. 37, 814–822 (1962).
[CrossRef]

Akchurin, G. G.

I. V. Yaroslavsky, A. N. Yaroslavsky, H.-J. Schwarzmaier, G. G. Akchurin, V. V. Tuchin, “New approach to Monte Carlo simulation of photon transport in the frequency-domain,” in Photon Propagation in Tissues, B. Chance, D. T. Delpy, G. J. Müller, eds., Proc. SPIE2626, 45–55 (1995).
[CrossRef]

Alfano, R. R.

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

Arridge, S. R.

M. Essenpreis, C. E. Elwell, M. Cope, P. van der Zee, S. R. Arridge, D. T. Delpy, “Spectral dependence of temporal point spread functions in human tissues,” Appl. Opt. 32, 418–425 (1993).
[CrossRef] [PubMed]

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of the optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Berg, R. A.

S. J. Strickler, R. A. Berg, “Relationship between absorption intensity and fluorescence lifetime of molecules,” J. Chem. Phys. 37, 814–822 (1962).
[CrossRef]

Berndt, K. W.

Boas, D. A.

Case, K. M.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), pp.174–175.

Cerusci, A. E.

Chance, B.

Cheong, W.-F.

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

Cope, M.

M. Essenpreis, C. E. Elwell, M. Cope, P. van der Zee, S. R. Arridge, D. T. Delpy, “Spectral dependence of temporal point spread functions in human tissues,” Appl. Opt. 32, 418–425 (1993).
[CrossRef] [PubMed]

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of the optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Delpy, D. T.

M. Essenpreis, C. E. Elwell, M. Cope, P. van der Zee, S. R. Arridge, D. T. Delpy, “Spectral dependence of temporal point spread functions in human tissues,” Appl. Opt. 32, 418–425 (1993).
[CrossRef] [PubMed]

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of the optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Elwell, C. E.

Essenpreis, M.

Fantini, S.

Feng, T. C.

Fishkin, J. B.

Franceschini, M. A.

Furutsu, K.

Gratton, E.

Haskell, R. C.

Hefetz, Y.

Heilweil, E. J.

E. J. Heilweil, R. M. Hochstrasser, “Nonlinear spectroscopy and picosecond transient grating study of colloidal gold,” J. Chem. Phys. 82, 4762–4770 (1985).
[CrossRef]

Hibst, R.

Hielscher, A. H.

Hochstrasser, R. M.

E. J. Heilweil, R. M. Hochstrasser, “Nonlinear spectroscopy and picosecond transient grating study of colloidal gold,” J. Chem. Phys. 82, 4762–4770 (1985).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1, pp. 175–186.

Jacques, S. L.

Keijzer, M.

Kienle, A.

Lagendijk, A.

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. 66, 3132–3135 (1991).
[CrossRef] [PubMed]

Lakowicz, J. R.

Lilge, L.

Lisyansky, A. A.

Liu, F.

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

Liu, H.

Livdan, D.

Madsen, S. J.

McAdams, M. S.

Moulton, J. D.

O’Leary, M. A.

Park, Y. D.

Patterson, M. S.

Prahl, S. A.

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

Schwarzmaier, H.-J.

I. V. Yaroslavsky, A. N. Yaroslavsky, H.-J. Schwarzmaier, G. G. Akchurin, V. V. Tuchin, “New approach to Monte Carlo simulation of photon transport in the frequency-domain,” in Photon Propagation in Tissues, B. Chance, D. T. Delpy, G. J. Müller, eds., Proc. SPIE2626, 45–55 (1995).
[CrossRef]

I. V. Yaroslavsky, H.-J. Schwarzmaier, A. N. Yaroslavsky, V. V. Tuchin, “The radiative transfer equation and its diffusion approximation in the frequency-domain technique: a comparison,” in Photon Transport in Highly Scattering Tissue, S. Avriller, B. Chance, G. J. Müller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 465–474 (1994).
[CrossRef]

Shifrin, K. S.

So, P. T. C.

Sobolev, V. V.

V. V. Sobolev, A Treatise on Radiative Transfer (Van Nostrand-Reinhold, Princeton, N.J., 1963), pp.240–244.

Star, W. M.

Steiner, R.

Storchi, P. R. M.

Strickler, S. J.

S. J. Strickler, R. A. Berg, “Relationship between absorption intensity and fluorescence lifetime of molecules,” J. Chem. Phys. 37, 814–822 (1962).
[CrossRef]

Svaas, L. O.

Svaasand, L. O.

Tip, A.

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. 66, 3132–3135 (1991).
[CrossRef] [PubMed]

Tittel, F. K.

Tromberg, B. J.

Tsay, T. T.

Tuchin, V. V.

I. V. Yaroslavsky, H.-J. Schwarzmaier, A. N. Yaroslavsky, V. V. Tuchin, “The radiative transfer equation and its diffusion approximation in the frequency-domain technique: a comparison,” in Photon Transport in Highly Scattering Tissue, S. Avriller, B. Chance, G. J. Müller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 465–474 (1994).
[CrossRef]

I. V. Yaroslavsky, A. N. Yaroslavsky, H.-J. Schwarzmaier, G. G. Akchurin, V. V. Tuchin, “New approach to Monte Carlo simulation of photon transport in the frequency-domain,” in Photon Propagation in Tissues, B. Chance, D. T. Delpy, G. J. Müller, eds., Proc. SPIE2626, 45–55 (1995).
[CrossRef]

van Albada, M. P.

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. 66, 3132–3135 (1991).
[CrossRef] [PubMed]

van der Zee, P.

van Tiggelen, B. A.

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. 66, 3132–3135 (1991).
[CrossRef] [PubMed]

Welch, A. J.

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

Wilson, B. C.

Yaroslavsky, A. N.

I. V. Yaroslavsky, A. N. Yaroslavsky, H.-J. Schwarzmaier, G. G. Akchurin, V. V. Tuchin, “New approach to Monte Carlo simulation of photon transport in the frequency-domain,” in Photon Propagation in Tissues, B. Chance, D. T. Delpy, G. J. Müller, eds., Proc. SPIE2626, 45–55 (1995).
[CrossRef]

I. V. Yaroslavsky, H.-J. Schwarzmaier, A. N. Yaroslavsky, V. V. Tuchin, “The radiative transfer equation and its diffusion approximation in the frequency-domain technique: a comparison,” in Photon Transport in Highly Scattering Tissue, S. Avriller, B. Chance, G. J. Müller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 465–474 (1994).
[CrossRef]

Yaroslavsky, I. V.

I. V. Yaroslavsky, H.-J. Schwarzmaier, A. N. Yaroslavsky, V. V. Tuchin, “The radiative transfer equation and its diffusion approximation in the frequency-domain technique: a comparison,” in Photon Transport in Highly Scattering Tissue, S. Avriller, B. Chance, G. J. Müller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 465–474 (1994).
[CrossRef]

I. V. Yaroslavsky, A. N. Yaroslavsky, H.-J. Schwarzmaier, G. G. Akchurin, V. V. Tuchin, “New approach to Monte Carlo simulation of photon transport in the frequency-domain,” in Photon Propagation in Tissues, B. Chance, D. T. Delpy, G. J. Müller, eds., Proc. SPIE2626, 45–55 (1995).
[CrossRef]

Yodh, A. G.

Yoo, K. M.

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

Zolotov, I. G.

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), pp.174–175.

Appl. Opt. (10)

S. J. Madsen, B. C. Wilson, M. S. Patterson, Y. D. Park, S. L. Jacques, Y. Hefetz, “Experimental tests of a simple diffusion model for the estimation of scattering and absorption coefficients of turbid media from time-resolved diffuse reflectance measurements,” Appl. Opt. 31, 3509–3517 (1992).
[CrossRef] [PubMed]

M. Essenpreis, C. E. Elwell, M. Cope, P. van der Zee, S. R. Arridge, D. T. Delpy, “Spectral dependence of temporal point spread functions in human tissues,” Appl. Opt. 32, 418–425 (1993).
[CrossRef] [PubMed]

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

K. S. Shifrin, I. G. Zolotov, “Nonstationary scattering of electromagnetic pulses by spherical particles,” Appl. Opt. 34, 552–558 (1995).
[CrossRef] [PubMed]

J. B. Fishkin, P. T. C. So, A. E. Cerusci, S. Fantini, M. A. Franceschini, E. Gratton, “Frequency-domain method for measuring spectral properties in multiple-scattering media: methemoglobin absorption spectrum in a tissuelike phantom,” Appl. Opt. 34, 1143–1155 (1995).
[CrossRef] [PubMed]

A. H. Hielscher, H. Liu, B. Chance, F. K. Tittel, S. L. Jacques, “Time-resolved photon emission from layered turbid media,” in Appl. Opt. 35, 719–728 (1996).
[CrossRef] [PubMed]

A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, B. C. Wilson, “Spatially resolved absolute measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996).
[CrossRef] [PubMed]

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

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

M. S. Patterson, J. D. Moulton, B. C. Wilson, K. W. Berndt, 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, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

J. Chem. Phys. (2)

E. J. Heilweil, R. M. Hochstrasser, “Nonlinear spectroscopy and picosecond transient grating study of colloidal gold,” J. Chem. Phys. 82, 4762–4770 (1985).
[CrossRef]

S. J. Strickler, R. A. Berg, “Relationship between absorption intensity and fluorescence lifetime of molecules,” J. Chem. Phys. 37, 814–822 (1962).
[CrossRef]

J. Opt. Soc. Am. (1)

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

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

Opt. Lett. (1)

Phys. Med. Biol. (1)

S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of the optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

M. P. van Albada, B. A. van Tiggelen, A. Lagendijk, A. Tip, “Speed of propagation of classical waves in strongly scattering media,” Phys. Rev. Lett. 66, 3132–3135 (1991).
[CrossRef] [PubMed]

Other (6)

G. J. Müller, ed., Medical Optical Tomography: Functional Imaging and Monitoring (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp.395–549.

I. V. Yaroslavsky, H.-J. Schwarzmaier, A. N. Yaroslavsky, V. V. Tuchin, “The radiative transfer equation and its diffusion approximation in the frequency-domain technique: a comparison,” in Photon Transport in Highly Scattering Tissue, S. Avriller, B. Chance, G. J. Müller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 465–474 (1994).
[CrossRef]

V. V. Sobolev, A Treatise on Radiative Transfer (Van Nostrand-Reinhold, Princeton, N.J., 1963), pp.240–244.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1, pp. 175–186.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), pp.174–175.

I. V. Yaroslavsky, A. N. Yaroslavsky, H.-J. Schwarzmaier, G. G. Akchurin, V. V. Tuchin, “New approach to Monte Carlo simulation of photon transport in the frequency-domain,” in Photon Propagation in Tissues, B. Chance, D. T. Delpy, G. J. Müller, eds., Proc. SPIE2626, 45–55 (1995).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Semi-infinite geometry assumed for Monte Carlo simulations in the frequency domain. The extrapolated boundary and effective source used in the diffusion approximation are also shown: zb, distance between the extrapolated boundary and the real boundary; z0, depth of the effective source; ρ, source–detector separation;θsd, half-aperture angles of source and detector, respectively.

Fig. 2
Fig. 2

(a) Modulation M and (b) phase lag Φ of the reflected signal as a function of the source–detector separation (modulation frequency, f = 800 MHz). The data were calculated with the modified diffusion approximation (curves)and Monte Carlo simulations in the frequency domain (symbols) for the geometry of Fig. 1 and the matched boundary conditions(n = 1). Assumed optical properties:μa = 0.001mm-1s = 10mm-1, g = 0.9,t1 = 0 (squares and solid curve); 100 fs (circles and dashed curve), 500 fs (triangles and dotted–dashed curve).

Fig. 3
Fig. 3

(a) Modulation M and (b) phase lag Φ of the reflected signal as a function of the modulation frequency(source–detector separation, ρ = 15.1 mm). The data were calculated with the modified diffusion approximation (curves) and Monte Carlo simulations in the frequency domain (symbols) for the geometry of Fig.1 and the matched boundary conditions(n = 1). See Fig. 2caption for the parameters.

Fig. 4
Fig. 4

(a) Modulation M and (b) phase lag Φ of the reflected signal as a function of the source-detector separation (modulation frequency f = 800 MHz). The data were calculated with the modified diffusion approximation (curves) and Monte Carlo simulations in the frequency domain (symbols) for the geometry of Fig. 1 and the mismatched boundary conditions(n = 1.4). Assumed optical properties:μs = 10 mm-1, g = 0.9,t1 = 0,μa = 0.01 mm-1 (squares and solid curve);t1 = 200 fs,μa = 0.01 mm-1 (circles and dashed curve);t1 = 0,μa = 0.001mm-1 (up triangles and dotted-dashed curve);t1 = 200fs, μa = 0.001mm-1 (down triangles and double-dotted-dashed curve).

Fig. 5
Fig. 5

(a) Modulation M and (b) phase lag Φ of the reflected signal as a function of the modulation frequency(source–detector separation, ρ = 15.1 mm). The data were calculated with the modified diffusion approximation (curves) and Monte Carlo simulations in the frequency domain (symbols) for the geometry of Fig.1 and the mismatched boundary conditions(n = 1.4). See Fig. 4caption for the parameters.

Fig. 6
Fig. 6

(a) Modulation m and (b) phase lag Φ data generated by the modified diffusion approximation (solid symbols) and the best fit by use of the standard diffusion approximation(curves) for the infinite geometry. The data were generated forμa = 0.01 mm-1s = 10 mm-1, g = 0.9, the scattering delay time t1 = 500 fs, and n = 1.4.The best-fit parameters of the standard diffusion approximation are μaapp = 0.0048mm-1 andμsapp = 2.09 mm-1. Two percent noise was added to the generated data. Also displayed are the modulation and phase lag calculated for the intrinsic optical properties, but with t1 = 0 (open symbols). The results are shown for two source–detector separations: 9 mm (squares) and 27 mm(circles).

Fig. 7
Fig. 7

Apparent absorption coefficient μaapp (solid circles) and apparent reduced scattering coefficient μsapp (open squares) obtained with the standard diffusion approximation from the data generated by the modified diffusion approximation. The infinite geometry was assumed. The data were generated for the following intrinsic optical properties:μa = 0.01 mm-1s = 10 mm-1, g = 0.9,n = 1.4.

Fig. 8
Fig. 8

Effect of the scattering delay on the time-resolved reflectance of a semi-infinite medium for two source–detector separations. Matched boundary conditions (n = 1)were assumed. The time-domain reflectance was calculated with Eq. (28). Intrinsic optical properties:μa = 0.01 mm-1, μs = 10 mm-1,g = 0.9. Apparent optical properties, μaapp = 0.004 mm-1and μsapp = 2.52mm-1. Solid curve, intrinsic optical properties and t1 = 500 fs at ρ= 2 mm; crosses, apparent optical properties and t1 = 0 at ρ = 2 mm; dashed curve, intrinsic optical properties and t1 = 500 fs at ρ = 20 mm; stars, apparent optical properties and t1= 0 at ρ = 20 mm; squares, intrinsic optical properties andt1 = 0 at ρ = 2mm; circles, intrinsic optical properties and t1 = 0 at ρ = 20 mm. All curves are normalized on their maximal values.

Tables (1)

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Table 1 Relative DiscrepanciesδDA-MC (%) in the Phase LagΦ between the Modified Diffusion Approximation and the Monte CarloSimulationsa

Equations (32)

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s · I r ,   s ,   t μ t + t 2 I r ,   s ,   t t = - I r ,   s ,   t + c 4 π   4 π - t I r ,   s ,   t f t - t d t p s · s d Ω ,
t 2 t 1 ,
t 1 = 0   tf t d t 0 f t d t .
f t - t = 1 t 1   exp - t - t t 1 .
I r ,   s ,   t | s · e r < 0 = S r ,   s ,   t + I r ,   s ,   t | s · e r > 0 ,   r G ,
J r ,   s ,   ω = - I r ,   s ,   t   exp - i ω t d t ,
I r ,   s ,   t = 1 2 π   - J r ,   s ,   ω   exp i ω t d ω .
s · J r ,   s ,   ω μ t = - 1 + i α β 2 J r ,   s ,   ω + c 4 π   4 π J r ,   s ,   ω 1 + i α β 1   p s · s d Ω ,
β 1 = t 1 t 1 + t 2 ,   β 2 = t 2 t 1 + t 2 ,   α = ω t 1 + t 2 .
η t = μ t 1 + i β 2 α ,     d = c 1 + i β 1 α 1 + i β 2 α .
J r ,   s ,   ω = J ri r ,   s ,   ω + J d r ,   s ,   ω .
J d r ,   s ,   ω U d r ,   ω + 3 4 π   F d r ,   ω s ,
2 U d r ,   ω - γ ω U d r ,   ω = - Q r ,   ω , .
Q r ,   ω = 3 η tr c μ t U ri r ,   ω 1 + i α β 1 - 3 4 π K ri r ,   s ,   ω s d Ω ,     K ri r ,   s ,   ω = c μ t 4 π   4 π J ri r ,   s ,   ω 1 + i α β 1   p s · s d Ω ,     γ ω = 3 η a η tr .
η a = μ a - μ t α 2 β 1 β 2 + i μ t α 1 + i α β 1 ,     η tr = μ s 1 + i α β 1 + η a ,
ω 2 t 1 2 1 ,
ω 2 t 1 n μ a ν 0 1 ,
η a = μ a + i ω 1 ν + t 1 μ s ,     η tr = μ tr + i ω 1 ν + t 1 μ s g ,
ω 2 1 + t 1 μ s ν 2 μ a μ tr ν 2 1 .
γ = 3 μ a μ tr + 3 i ω μ tr 1 ν + t 1 μ s .
2 U d r ,   ω - 3 μ a μ tr U d r ,   ω - 3 i ω μ tr 1 ν + t 1 μ s U d r ,   ω = - Q r ,   ω .
Q r ,   ω = 3 P ω μ tr δ r ,
U d inf   r ,   ω = 3 P ω μ tr exp - γ ω 1 / 2 r 4 π r ,
mod U d inf = 3 mod P μ tr 4 π r   exp 9 μ tr 2 μ a 2 + ω 2 n + t 1 μ s ν 0 2 ν 0 2 1 / 4   cos φ 2 r ,   arg U d inf = arg P - 9 μ tr 2 μ a 2 + ω 2 n + t 1 μ s ν 0 2 ν 0 2 1 / 4   sin φ 2 r ,
φ = arctan ω n + t 1 μ s ν 0 μ a ν 0 .
n eff = n + t 1 μ s ν 0 .
mod F d s - inf = 6 P μ tr   exp - ρ Y 1 4 π 2 ρ 3   1 + ρ Y 2 + ρ 2 Y 4 2 1 / 2   z 0 + z b × z + 1 μ tr 1 - z b + z 0 2 + 3 z 2 2 ρ 2 × 2 + 1 + ρ Y 1 1 + ρ Y 2 + ρ 2 Y 4 2 + ρ Y 1 ,   arg F d s - inf = arctan ρ Y 3 1 + ρ Y 1 - ρ Y 3 ,
Y 1 = k d V + / 2 ,   Y 2 = k d V + 2 ,     Y 3 = k d V - / 2 ,   Y 4 = k d 1 + x 2 1 / 4 ,     V + = 1 + x 2 1 / 2 + 1 1 / 2 ,   V - = 1 + x 2 1 / 2 - 1 1 / 2 ,     x = ω n + t 1 μ s ν 0 ν 0 μ a ,     k d = 3 μ a μ tr 1 / 2 ,
M r ,   ω = mod AC r ,   ω m 0 DC r ,     Ω r ,   ω = arg AC r ,   ω - arg AC r 0 ,   ω ,
μ a app = n n eff   μ a ,     μ tr app = n eff n   μ tr .
m k = cm k - 1 α 2 + χ 2 1 / 2     tan   φ k = χ   sin   φ k - 1 - α   cos   φ k - 1 χ   cos   φ k - 1 + α   sin   φ k - 1 ,
R ρ ,   t = 4 π ν eff 3 μ tr - 3 / 2   z 0 t - 5 / 2 × exp - μ a ν eff t exp - ρ 2 + z 0 2 3 μ tr 4 ν eff t ,

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