Abstract

Light propagation in two-layered turbid media having an infinitely thick second layer is investigated in the steady-state, frequency, and time domains. A solution of the diffusion approximation to the transport equation is derived by employing the extrapolated boundary condition. We compare the reflectance calculated from this solution with that computed with Monte Carlo simulations and show good agreement. To investigate if it is possible to determine the optical coefficients of the two layers and the thickness of the first layer, the solution of the diffusion equation is fitted to reflectance data obtained from both the diffusion equation and the Monte Carlo simulations. Although it is found that it is, in principle, possible to derive the optical coefficients of the two layers and the thickness of the first layer, we concentrate on the determination of the optical coefficients, knowing the thickness of the first layer. In the frequency domain, for example, it is shown that it is sufficient to make relative measurements of the phase and the steady-state reflectance at three distances from the illumination point to obtain useful estimates of the optical coefficients. Measurements of the absolute steady-state spatially resolved reflectance performed on two-layered solid phantoms confirm the theoretical results.

© 1998 Optical Society of America

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    [CrossRef] [PubMed]
  29. R. Bays, G. Wagnières, D. Robert, J.-F. Theumann, A. Vitkin, J.-F. Savary, P. Monnier, H. van den Bergh, “Three-dimensional optical phantom and its application in photodynamic therapy,” Laser Surg. Med. 21, 227–234 (1997).
    [CrossRef]
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    [CrossRef]
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1997 (4)

A. Kienle, M. S. Patterson, “Improved solutions of the steady-state and time-resolved diffusion equations for reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14, 246–254 (1997).
[CrossRef]

J. T. Bruulsema, J. E. Hayward, T. J. Farrell, M. S. Patterson, L. Heinemann, M. Berger, T. Koschinsky, J. Sandahl-Christiansen, H. Orskov, M. Essenpreis, G. Schmelzeisen-Redeker, D. Böcker, “Correlation between blood glucose concentration in diabetics and noninvasively measured tissue optical scattering coefficient,” Opt. Lett. 22, 190–192 (1997).
[CrossRef] [PubMed]

A. Kienle, M. S. Patterson, “Determination of the optical properties of semi-infinite turbid media from frequency-domain reflectance close to source,” Phys. Med. Biol. 42, 1801–1819 (1997).
[CrossRef] [PubMed]

R. Bays, G. Wagnières, D. Robert, J.-F. Theumann, A. Vitkin, J.-F. Savary, P. Monnier, H. van den Bergh, “Three-dimensional optical phantom and its application in photodynamic therapy,” Laser Surg. Med. 21, 227–234 (1997).
[CrossRef]

1996 (6)

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

B. W. Pogue, M. S. Patterson, “Error assessment of a wavelength tunable frequency domain system for noninvasive tissue spectroscopy,” J. Biomed. Opt. 1, 311–323 (1996).
[CrossRef] [PubMed]

A. Kienle, M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. Biol. 41, 2221–2227 (1996).
[CrossRef] [PubMed]

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

S. Homma, T. Fukunaga, A. Kagaya, “Influence of adipose tissue thickness on near infrared spectroscopic signals in the measurement of human muscle,” J. Biomed. Opt. 1, 418–424 (1996).
[CrossRef] [PubMed]

P. J. Kirkpatrick, P. Smielewski, J. M. K. Lam, P. Al-Rawi, “Use of near infrared spectroscopy for the clinical monitoring of adult brain,” J. Biomed. Opt. 1, 363–372 (1996).
[CrossRef] [PubMed]

1995 (6)

A. Kienle, R. Hibst, “New optimal wavelength for treatment of portwine stains?,” Phys. Med. Biol. 40, 1559–1576 (1995).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element model for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef]

S. Fantini, M. A. Franceschini-Fantini, J. S. Maier, S. A. Walker, B. Barbieri, E. Gratton, “Frequency-domain multichannel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng. 34, 32–42 (1995).
[CrossRef]

S. R. Arridge, M. Schweiger, “Photon-measurement density functions. Part 2: Finite-element-method calculations,” Appl. Opt. 34, 8026–8037 (1995).
[CrossRef] [PubMed]

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

E. Okada, M. Firbank, D. T. Delpy, “The effect of overlying tissue on the spatial sensitivity profile of near-infrared spectroscopy,” Phys. Med. Biol. 40, 2093–2108 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (1)

1992 (2)

W. Cui, L. E. Ostrander, “The relationship of surface reflectance measurements to optical properties of layered biological media,” IEEE Trans. Biomed. Eng. 39, 194–201 (1992).
[CrossRef] [PubMed]

I. Dayan, S. Havlin, G. H. Weiss, “Photon migration in a two-layer turbid medium. A diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[CrossRef]

1990 (1)

1989 (1)

1988 (2)

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).
[CrossRef] [PubMed]

1979 (1)

S. Takatani, M. D. Graham, “Theoretical analysis of diffuse reflectance from a two-layer tissue model,” IEEE Trans. Biomed. Eng. BME-26, 656–664 (1979).
[CrossRef]

1941 (1)

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Aarnoudse, J. G.

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).
[CrossRef] [PubMed]

Al-Rawi, P.

P. J. Kirkpatrick, P. Smielewski, J. M. K. Lam, P. Al-Rawi, “Use of near infrared spectroscopy for the clinical monitoring of adult brain,” J. Biomed. Opt. 1, 363–372 (1996).
[CrossRef] [PubMed]

Arridge, S. R.

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element model for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef]

S. R. Arridge, M. Schweiger, “Photon-measurement density functions. Part 2: Finite-element-method calculations,” Appl. Opt. 34, 8026–8037 (1995).
[CrossRef] [PubMed]

Barbieri, B.

S. Fantini, M. A. Franceschini-Fantini, J. S. Maier, S. A. Walker, B. Barbieri, E. Gratton, “Frequency-domain multichannel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng. 34, 32–42 (1995).
[CrossRef]

Bays, R.

R. Bays, G. Wagnières, D. Robert, J.-F. Theumann, A. Vitkin, J.-F. Savary, P. Monnier, H. van den Bergh, “Three-dimensional optical phantom and its application in photodynamic therapy,” Laser Surg. Med. 21, 227–234 (1997).
[CrossRef]

Berger, M.

Bevington, P. R.

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1983), Chap. 11.

Böcker, D.

Bonner, R.

Bruulsema, J. T.

Chance, B.

Cui, W.

W. Cui, L. E. Ostrander, “The relationship of surface reflectance measurements to optical properties of layered biological media,” IEEE Trans. Biomed. Eng. 39, 194–201 (1992).
[CrossRef] [PubMed]

Dassel, A. C. M.

Dayan, I.

I. Dayan, S. Havlin, G. H. Weiss, “Photon migration in a two-layer turbid medium. A diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[CrossRef]

de Mul, F. F. M.

Delpy, D. T.

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element model for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef]

E. Okada, M. Firbank, D. T. Delpy, “The effect of overlying tissue on the spatial sensitivity profile of near-infrared spectroscopy,” Phys. Med. Biol. 40, 2093–2108 (1995).
[CrossRef] [PubMed]

Essenpreis, M.

Fantini, S.

S. Fantini, M. A. Franceschini-Fantini, J. S. Maier, S. A. Walker, B. Barbieri, E. Gratton, “Frequency-domain multichannel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng. 34, 32–42 (1995).
[CrossRef]

Farrell, T. J.

Feng, T. C.

Firbank, M.

E. Okada, M. Firbank, D. T. Delpy, “The effect of overlying tissue on the spatial sensitivity profile of near-infrared spectroscopy,” Phys. Med. Biol. 40, 2093–2108 (1995).
[CrossRef] [PubMed]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Pascal (Cambridge University, Cambridge, England, 1990).

Franceschini-Fantini, M. A.

S. Fantini, M. A. Franceschini-Fantini, J. S. Maier, S. A. Walker, B. Barbieri, E. Gratton, “Frequency-domain multichannel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng. 34, 32–42 (1995).
[CrossRef]

Fukunaga, T.

S. Homma, T. Fukunaga, A. Kagaya, “Influence of adipose tissue thickness on near infrared spectroscopic signals in the measurement of human muscle,” J. Biomed. Opt. 1, 418–424 (1996).
[CrossRef] [PubMed]

Graaff, R.

Graham, M. D.

S. Takatani, M. D. Graham, “Theoretical analysis of diffuse reflectance from a two-layer tissue model,” IEEE Trans. Biomed. Eng. BME-26, 656–664 (1979).
[CrossRef]

Gratton, E.

S. Fantini, M. A. Franceschini-Fantini, J. S. Maier, S. A. Walker, B. Barbieri, E. Gratton, “Frequency-domain multichannel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng. 34, 32–42 (1995).
[CrossRef]

Greenstein, J. L.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Haskell, R. C.

Havlin, S.

I. Dayan, S. Havlin, G. H. Weiss, “Photon migration in a two-layer turbid medium. A diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
[CrossRef]

R. Nossal, J. Kiefer, G. H. Weiss, R. Bonner, H. Taitelbaum, S. Havlin, “Photon migration in layered media,” Appl. Opt. 27, 3382–3391 (1988).
[CrossRef] [PubMed]

Hayward, J. E.

Heinemann, L.

Henyey, L. G.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Hibst, R.

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

A. Kienle, R. Hibst, “New optimal wavelength for treatment of portwine stains?,” Phys. Med. Biol. 40, 1559–1576 (1995).
[CrossRef] [PubMed]

A. Kienle, L. Lilge, M. S. Patterson, 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. Müller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 212–221 (1994).
[CrossRef]

Hielscher, A. H.

Hiraoka, M.

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element model for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef]

Homma, S.

S. Homma, T. Fukunaga, A. Kagaya, “Influence of adipose tissue thickness on near infrared spectroscopic signals in the measurement of human muscle,” J. Biomed. Opt. 1, 418–424 (1996).
[CrossRef] [PubMed]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Chaps. 7 and 9.

Jacques, S. L.

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

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

Kagaya, A.

S. Homma, T. Fukunaga, A. Kagaya, “Influence of adipose tissue thickness on near infrared spectroscopic signals in the measurement of human muscle,” J. Biomed. Opt. 1, 418–424 (1996).
[CrossRef] [PubMed]

Keijzer, M.

Kiefer, J.

Kienle, A.

A. Kienle, M. S. Patterson, “Improved solutions of the steady-state and time-resolved diffusion equations for reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14, 246–254 (1997).
[CrossRef]

A. Kienle, M. S. Patterson, “Determination of the optical properties of semi-infinite turbid media from frequency-domain reflectance close to source,” Phys. Med. Biol. 42, 1801–1819 (1997).
[CrossRef] [PubMed]

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

A. Kienle, M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. Biol. 41, 2221–2227 (1996).
[CrossRef] [PubMed]

A. Kienle, R. Hibst, “New optimal wavelength for treatment of portwine stains?,” Phys. Med. Biol. 40, 1559–1576 (1995).
[CrossRef] [PubMed]

A. Kienle, L. Lilge, M. S. Patterson, 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. Müller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 212–221 (1994).
[CrossRef]

Kirkpatrick, P. J.

P. J. Kirkpatrick, P. Smielewski, J. M. K. Lam, P. Al-Rawi, “Use of near infrared spectroscopy for the clinical monitoring of adult brain,” J. Biomed. Opt. 1, 363–372 (1996).
[CrossRef] [PubMed]

Koelink, M. H.

Koschinsky, T.

Lam, J. M. K.

P. J. Kirkpatrick, P. Smielewski, J. M. K. Lam, P. Al-Rawi, “Use of near infrared spectroscopy for the clinical monitoring of adult brain,” J. Biomed. Opt. 1, 363–372 (1996).
[CrossRef] [PubMed]

Lilge, L.

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

A. Kienle, L. Lilge, M. S. Patterson, 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. Müller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 212–221 (1994).
[CrossRef]

Liu, H.

Maier, J. S.

S. Fantini, M. A. Franceschini-Fantini, J. S. Maier, S. A. Walker, B. Barbieri, E. Gratton, “Frequency-domain multichannel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng. 34, 32–42 (1995).
[CrossRef]

McAdams, M.

Monnier, P.

R. Bays, G. Wagnières, D. Robert, J.-F. Theumann, A. Vitkin, J.-F. Savary, P. Monnier, H. van den Bergh, “Three-dimensional optical phantom and its application in photodynamic therapy,” Laser Surg. Med. 21, 227–234 (1997).
[CrossRef]

Nossal, R.

Okada, E.

E. Okada, M. Firbank, D. T. Delpy, “The effect of overlying tissue on the spatial sensitivity profile of near-infrared spectroscopy,” Phys. Med. Biol. 40, 2093–2108 (1995).
[CrossRef] [PubMed]

Orskov, H.

Ostrander, L. E.

W. Cui, L. E. Ostrander, “The relationship of surface reflectance measurements to optical properties of layered biological media,” IEEE Trans. Biomed. Eng. 39, 194–201 (1992).
[CrossRef] [PubMed]

Patterson, M. S.

J. T. Bruulsema, J. E. Hayward, T. J. Farrell, M. S. Patterson, L. Heinemann, M. Berger, T. Koschinsky, J. Sandahl-Christiansen, H. Orskov, M. Essenpreis, G. Schmelzeisen-Redeker, D. Böcker, “Correlation between blood glucose concentration in diabetics and noninvasively measured tissue optical scattering coefficient,” Opt. Lett. 22, 190–192 (1997).
[CrossRef] [PubMed]

A. Kienle, M. S. Patterson, “Improved solutions of the steady-state and time-resolved diffusion equations for reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14, 246–254 (1997).
[CrossRef]

A. Kienle, M. S. Patterson, “Determination of the optical properties of semi-infinite turbid media from frequency-domain reflectance close to source,” Phys. Med. Biol. 42, 1801–1819 (1997).
[CrossRef] [PubMed]

B. W. Pogue, M. S. Patterson, “Error assessment of a wavelength tunable frequency domain system for noninvasive tissue spectroscopy,” J. Biomed. Opt. 1, 311–323 (1996).
[CrossRef] [PubMed]

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

A. Kienle, M. S. Patterson, “Determination of the optical properties of turbid media from a single Monte Carlo simulation,” Phys. Med. Biol. 41, 2221–2227 (1996).
[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]

A. Kienle, L. Lilge, M. S. Patterson, 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. Müller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 212–221 (1994).
[CrossRef]

Pogue, B. W.

B. W. Pogue, M. S. Patterson, “Error assessment of a wavelength tunable frequency domain system for noninvasive tissue spectroscopy,” J. Biomed. Opt. 1, 311–323 (1996).
[CrossRef] [PubMed]

Press, W. H.

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R. Bays, G. Wagnières, D. Robert, J.-F. Theumann, A. Vitkin, J.-F. Savary, P. Monnier, H. van den Bergh, “Three-dimensional optical phantom and its application in photodynamic therapy,” Laser Surg. Med. 21, 227–234 (1997).
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R. Bays, G. Wagnières, D. Robert, J.-F. Theumann, A. Vitkin, J.-F. Savary, P. Monnier, H. van den Bergh, “Three-dimensional optical phantom and its application in photodynamic therapy,” Laser Surg. Med. 21, 227–234 (1997).
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A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, B. C. Wilson, “Spatially-resolved absolute diffuse reflectance measurements for non-invasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996).
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A. Kienle, L. Lilge, M. S. Patterson, 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. Müller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 212–221 (1994).
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R. Bays, G. Wagnières, D. Robert, J.-F. Theumann, A. Vitkin, J.-F. Savary, P. Monnier, H. van den Bergh, “Three-dimensional optical phantom and its application in photodynamic therapy,” Laser Surg. Med. 21, 227–234 (1997).
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R. Bays, G. Wagnières, D. Robert, J.-F. Theumann, A. Vitkin, J.-F. Savary, P. Monnier, H. van den Bergh, “Three-dimensional optical phantom and its application in photodynamic therapy,” Laser Surg. Med. 21, 227–234 (1997).
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I. Dayan, S. Havlin, G. H. Weiss, “Photon migration in a two-layer turbid medium. A diffusion analysis,” J. Mod. Opt. 39, 1567–1582 (1992).
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J. Opt. Soc. Am. A (3)

Laser Surg. Med. (1)

R. Bays, G. Wagnières, D. Robert, J.-F. Theumann, A. Vitkin, J.-F. Savary, P. Monnier, H. van den Bergh, “Three-dimensional optical phantom and its application in photodynamic therapy,” Laser Surg. Med. 21, 227–234 (1997).
[CrossRef]

Med. Phys. (2)

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).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, D. T. Delpy, “The finite element model for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22, 1779–1792 (1995).
[CrossRef]

Opt. Eng. (1)

S. Fantini, M. A. Franceschini-Fantini, J. S. Maier, S. A. Walker, B. Barbieri, E. Gratton, “Frequency-domain multichannel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng. 34, 32–42 (1995).
[CrossRef]

Opt. Lett. (1)

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A. Kienle, M. S. Patterson, “Determination of the optical properties of semi-infinite turbid media from frequency-domain reflectance close to source,” Phys. Med. Biol. 42, 1801–1819 (1997).
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Other (4)

A. Kienle, L. Lilge, M. S. Patterson, 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. Müller, A. V. Priezzhev, V. V. Tuchin, eds., Proc. SPIE2326, 212–221 (1994).
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Chaps. 7 and 9.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Pascal (Cambridge University, Cambridge, England, 1990).

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1983), Chap. 11.

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Figures (14)

Fig. 1
Fig. 1

Scheme of the manufactured phantoms.

Fig. 2
Fig. 2

Comparison of the steady-state spatially resolved reflectance calculated with Eq. (16) (lines) to Monte Carlo simulations (symbols). The optical parameters of the two-layered turbid media are μ a1 = 0.02 mm-1, μ s1′ = 1.3 mm-1, μ a2 = 0.01 mm-1, and μ s2′ = 1.2 mm-1 (solid curve, crosses) or μ s2′ = 0.7 mm-1 (dashed curve, circles). The thickness of the first layer is l = 2 mm and n = 1.4.

Fig. 3
Fig. 3

Comparison of the steady-state spatially resolved reflectance calculated with Eq. (16) (lines) to Monte Carlo simulations (symbols). The optical parameters of the two-layered turbid media are μ s1′ = 1.3 mm-1, μ a1 = 0.005 mm-1, μ s2′ = 1.0 mm-1, and μ a2 = 0.01 mm-1 (solid curve, circles) or μ a2= 0.022 mm-1 (dashed curve, crosses). The thickness of the first layer is l = 6 mm and n = 1.4.

Fig. 4
Fig. 4

Comparison of the steady-state spatially resolved reflectance calculated with Eq. (16) (curves) to Monte Carlo simulations (symbols). The optical parameters of the two-layered turbid media are μ s1′ = 1.3 mm-1, μ a1 = 0.005 mm-1, μ s2′ = 1.0 mm-1, and μ a2 = 0.01 mm-1 (solid curve, circles) or μ a2= 0.022 mm-1 (dashed curve, crosses). The thickness of the first layer is l = 10 mm and n = 1.4.

Fig. 5
Fig. 5

Comparison of the phase versus distance calculated with Eq. (17) (curves) to Monte Carlo simulations (symbols). The optical parameters of the two-layered turbid media are n = 1.4, μ a1 = 0.02 mm-1, μ s1′ = 1.3 mm-1, μ a2 = 0.01 mm-1, and μ s2′ = 1.2 mm-1 (solid curve, crosses) or μ s2′ = 0.7 mm-1 (dashed curve, circles). The thickness of the first layer is l = 2 mm and the modulation frequency is f = 195 MHz.

Fig. 6
Fig. 6

Comparison of the time-resolved reflectance calculated with the diffusion theory (curves) to Monte Carlo simulations (symbols). The optical parameters of the two-layered turbid media are n = 1.4, μ a1 = 0.02 mm-1, μ s1′ = 1.3 mm-1, μ a2 = 0.01 mm-1, and μ s2′ = 1.2 mm-1 (solid curve, crosses) or μ s2′ = 0.7 mm-1 (dashed curve, circles). The thickness of the first layer is l = 2 mm, and the distance is ρ = 9.75 mm. The time-resolved reflectance calculated with the solution proposed by Dayan et al. is also shown for the medium with μ s2′ = 0.7 mm-1 (long dashed curve).

Fig. 7
Fig. 7

Estimated absorption coefficients of the first (μ a1*, crosses) and second layer (μ a2*, open circles) determined by nonlinear regressions of Eq. (16) to Monte Carlo data are shown versus the true absorption coefficient of the second layer used in the Monte Carlo simulations (μ a2). The optical parameters of the Monte Carlo simulations are μ s1′ = 1.3 mm-1, μ a1 = 0.02 mm-1, μ s2′ = 1.0 mm-1, and μ a2 is varied between μ a2= 0.0025 mm-1 and μ a2 = 0.02 mm-1. The thickness of the first layer is l = 2 mm. The lines indicate the correct values. Also shown are the absorption coefficients (μ a *) obtained from nonlinear regressions to the two-layer Monte Carlo data using a diffusion solution that assumes a semi-infinite medium (solid circles). Reflectance data at distances ρ = 1.25, 1.75, … , 17.75 mm were used in the nonlinear regression.

Fig. 8
Fig. 8

Estimated reduced scattering coefficients of the first (μ s1′*, crosses) and second layer (μ s2′*, open circles) determined by nonlinear regressions of Eq. (16) to Monte Carlo data are shown versus the absorption coefficient of the second layer used in the Monte Carlo simulations (μ a2). The optical parameters of the Monte Carlo simulations are μ s1′ = 1.3 mm-1, μ a1 = 0.02 mm-1, μ s2′ = 1.0 mm-1, and μ a2 is varied between μ a2 = 0.0025 mm-1 and μ a2 = 0.02 mm-1. The thickness of the first layer is l = 2 mm. The lines indicate the correct values. Also shown are the reduced scattering coefficients (μ s ′*) obtained from nonlinear regressions to the two-layer Monte Carlo data using a diffusion solution that assumes a semi-infinite medium (solid circles). Reflectance data at distances ρ = 1.25, 1.75, … , 17.75 mm were used in the nonlinear regression.

Fig. 9
Fig. 9

Estimated reduced scattering coefficients of the first (μ s1′*, crosses) and second layer (μ s2′*, open circles) determined by nonlinear regressions of Eq. (16) to Monte Carlo data are shown versus the reduced scattering coefficient of the second layer used in the Monte Carlo simulations (μ s2′). The optical parameters of the Monte Carlo simulations are μ a1 = 0.02 mm-1, μ s1′ = 1.3 mm-1, μ a2 = 0.01 mm-1, and μ s2′ is varied between μ s2′ = 0.7 mm-1 and μ s2′ = 1.2 mm-1. The thickness of the first layer is l = 2 mm. The lines indicate the correct values. Also shown are the reduced scattering coefficients (μ s ′*) obtained from nonlinear regressions to the two-layer Monte Carlo data using a diffusion solution that assumes a semi-infinite medium (solid circles). Reflectance data at distances ρ = 1.25, 1.75, … , 19.75 mm were used in the nonlinear regression.

Fig. 10
Fig. 10

Estimated reduced scattering coefficients of the first (μ s1′*, crosses) and second layer (μ s2′*, open circles) determined by nonlinear regressions of Eqs. (16) and (17) to Monte Carlo data are shown versus the reduced scattering coefficient of the second layer used in the Monte Carlo simulations (μ s2′). The optical parameters of the Monte Carlo simulations are μ a1 = 0.02 mm-1, μ s1′ = 1.3 mm-1, μ a2 = 0.01 mm-1, and μ s2′ is varied between μ s2′ = 0.7 mm-1 and μ s2′ = 1.2 mm-1. The thickness of the first layer is l = 2 mm. The lines indicate the correct values. The frequency domain reflectance at distances ρ = 3.75, 6.75, 9.75 mm were used in the nonlinear regression and the modulation frequency is f = 195 MHz. Also shown are the reduced scattering coefficients (μ s *) obtained from nonlinear regressions to the two-layer Monte Carlo data using a diffusion solution that assumes a semi-infinite medium (solid circles).

Fig. 11
Fig. 11

Estimated absorption coefficients of the second layer (μ a2*, open circles, pluses) determined by nonlinear regressions of Eqs. (16) and (17) to Monte Carlo data are shown versus the absorption coefficient of the second layer used in the Monte Carlo simulations (μ a2). The optical parameters of the Monte Carlo simulations are μ s1′ = 1.3 mm-1, μ a1 = 0.005 mm-1, μ s2′ = 1.0 mm-1, and μ a2 is varied between μ a2= 0.01 mm-1 and μ a2 = 0.025 mm-1. The thickness of the first layer is l = 6 mm. The line indicates the correct values. Also shown is the absorption coefficient (μ a *, crosses, boxes) obtained from nonlinear regressions to the two-layer Monte Carlo data when a diffusion solution that assumes a semi-infinite medium is used. The frequency domain reflectance at distances ρ = 7.5, 13.5, 19.5 mm were used in the nonlinear regression and the modulation frequency is f = 195 MHz. Two independent Monte Carlo simulations were performed.

Fig. 12
Fig. 12

Estimated absorption coefficients of the second layer (μ a2*, solid circles) determined by nonlinear regressions of Eqs. (16) and (17) to Monte Carlo data are shown versus the absorption coefficient of the second layer used in the Monte Carlo simulations (μ a2). The optical parameters of the Monte Carlo simulations are μ s1′ = 1.3 mm-1, μ a1 = 0.005 mm-1, μ s2′ = 1.0 mm-1, and μ a2 is varied between μ a2= 0.01 mm-1 and μ a2 = 0.022 mm-1. The thickness of the first layer is l = 10 mm. The line indicates the correct values. Also shown is the absorption coefficient (μ a *, open circles) obtained from nonlinear regressions to the two-layer Monte Carlo data when a diffusion solution that assumes a semi-infinite medium is used. The frequency domain reflectance at distances ρ = 7.5, 18.5, 29.5 mm were used in the nonlinear regression and the modulation frequency is f = 195 MHz.

Fig. 13
Fig. 13

Measurements of the spatially resolved absolute reflectance at the top of phantom 1 (two-layered, upper dashed curve) and at the side of phantom 2 (semi-infinite, lower dashed curve). Also shown are the nonlinear regression to the semi-infinite measurement (lower solid curve) and the theoretical curve of the two-layer measurement calculated from the known optical coefficients (upper solid curve). A He–Ne laser at λ = 543 nm was used as a light source.

Fig. 14
Fig. 14

Measurements of the spatially resolved absolute reflectance at the top of phantom 2 (two-layered, lower dashed curve) and at the side of phantom 1 (semi-infinite, upper dashed curve). Also shown are the nonlinear regression to the semi-infinite measurement (lower solid curve) and the theoretical curve of the two-layer measurement calculated from the known optical coefficients (upper solid curve). A He–Ne laser at λ = 543 nm was used as a light source.

Tables (1)

Tables Icon

Table 1 Optical Coefficients of the Two-Layered Medium (Phantom 1) Having a First Layer Thickness of l = 2 mm Derived from Measurements on the Side of Phantom 1 and 2 (Semi-Infinite Media) and on the Top (Two-Layered Media) at 543 and 612 nm

Equations (18)

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D 1 Δ Φ 1 r - μ a 1 Φ 1 r = - δ x ,   y ,   z - z 0 ,     0 z < l ,
D 2 Δ Φ 2 r - μ a 2 Φ 2 r = 0 ,     l z ,
ϕ i z ,   s 1 ,   s 2 = - -   Φ i x ,   y ,   z exp i s 1 x + s 2 y d x d y .
2 z 2   ϕ 1 z ,   s - α 1 2 ϕ 1 z ,   s = - 1 D 1   δ z - z 0 ,     0 z < l ,
2 z 2   ϕ 2 z ,   s - α 2 2 ϕ 2 z ,   s = 0 ,     l z ,
ϕ 1 - z b ,   s = 0 ,
ϕ 2 ,   s = 0 ,
ϕ 1 l ,   s ϕ 2 l ,   s = n 1 2 n 2 2 = 1 ,
D 1 ϕ 1 z ,   s z | z = l = D 2 ϕ 2 z ,   s z | z = l .
z b = 1 + R eff 1 - R eff   2 D 1 .
ϕ 1 z ,   s = sinh α 1 z b + z 0 D 1 α 1 × D 1 α 1 cosh α 1 l - z + D 2 α 2 sinh α 1 l - z D 1 α 1 cosh α 1 l + z b + D 2 α 2 sinh α 1 l + z b - sinh α 1 z 0 - z D 1 α 1 ,     0 z < z 0 ,
ϕ 1 z ,   s = sinh α 1 z b + z 0 D 1 α 1 × D 1 α 1 cosh α 1 l - z + D 2 α 2 sinh α 1 l - z D 1 α 1 cosh α 1 l + z b + D 2 α 2 sinh α 1 l + z b ,     z 0 < z < l ,
ϕ 2 z ,   s = sinh α 1 z b + z 0 exp α 2 l - z D 1 α 1 cosh α 1 l + z b + D 2 α 2 sinh α 1 l + z b .
Φ i ρ ,   z = 1 2 π 2   ϕ i z ,   s exp - i s 1 x + s 2 y d s 1 d s 2 = 1 2 π 0   ϕ i z ,   s sJ 0 s ρ d s ,
R ρ = 2 π   d Ω 1 - R fres θ × 1 4 π Φ 1 ρ ,   z = 0 + 3 D 1 Φ 1 ρ ,   z = 0 z cos   θ × cos   θ ,
R ρ = 0.118 Φ 1 ρ ,   z = 0 + 0.306 D 1 z   Φ 1 ρ ,   z | z = 0 .
θ = tan - 1 Im R ρ ,   ω Re R ρ ,   ω ,
M = Im R ρ ,   ω 2 + Re R ρ ,   ω 2 Re R ρ ,   ω = 0 2 1 / 2 ,

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