Abstract

We propose a hybrid Monte Carlo (MC) diffusion model for calculating the spatially resolved reflectance amplitude and phase delay resulting from an intensity-modulated pencil beam vertically incident on a two-layer turbid medium. The model combines the accuracy of MC at radial distances near the incident beam with the computational efficiency afforded by a diffusion calculation at further distances. This results in a single forward calculation several hundred times faster than pure MC, depending primarily on model parameters. Model predictions are compared with MC data for two cases that span the extremes of physiologically relevant optical properties: skin overlying fat and skin overlying muscle, both in the presence of an exogenous absorber. It is shown that good agreement can be achieved for radial distances from 0.5 to 20 mm in both cases. However, in the skin-on-muscle case the choice of model parameters and the definition of the diffusion coefficient can lead to some interesting discrepancies.

© 2000 Optical Society of America

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    [CrossRef]

1999 (4)

J. R. Mourant, T. M. Johnson, G. Los, I. Bigio, “Non-invasive measurements of chemotherapy drug concentrations in tissue: preliminary demonstrations of in vivo measurements,” Phys. Med. Biol. 44, 1397–1417 (1999).
[CrossRef] [PubMed]

A. Kienle, T. Glanzmann, “In vivo determination of the optical properties of muscle with time-resolved reflectance using a layered model,” Phys. Med. Biol. 44, 2689–2702 (1999).
[CrossRef] [PubMed]

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

R. Aronson, N. Corngold, “Photon diffusion coefficient in an absorbing medium,” J. Opt. Soc. Am. A 16, 1066–1071 (1999).
[CrossRef]

1998 (7)

1997 (4)

1996 (3)

1995 (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]

1994 (2)

1993 (1)

1992 (2)

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

T. J. Farrell, M. S. Patterson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the non-invasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

1991 (1)

1990 (1)

1989 (2)

1987 (1)

M. S. Patterson, B. C. Wilson, J. W. Feather, D. M. Burns, W. Pushka, “The measurement of dihematoporphyrin ether concentration in tissue by reflectance spectrophotometry,” Photochem. Photobiol. 46, 337–343 (1987).
[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]

Alexandrakis, G.

G. Alexandrakis, T. J. Farrell, M. S. Patterson, “Accuracy of the diffusion approximation in determining the optical properties of a two-layer turbid medium,” Appl. Opt. 37, 7401–7410 (1998).
[CrossRef]

G. Alexandrakis, R. A. Weersink, J. T. Bruulsema, M. S. Patterson, “Estimation of optical properties of two-layer tissue simulating phantoms from spatially resolved frequency-domain reflectance,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. R. Alfano, B. J. Tromberg, eds., Proc. SPIE3597, 155–163 (1999).
[CrossRef]

Alfano, R. R.

Andreola, S.

Aronson, R.

Arridge, S. R.

S. R. Arridge, “Why optical tomography is hard,” in Biomedical Optics: New Concepts in Therapeutic Laser Applications, Novel Biomedical Optical Spectroscopy, Imaging, and Diagnostics, Advances in Optical Imaging, Photon Migration, and Tissue Optics, OSA 1999 Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper AMB1–1.

Avrillier, S.

J.-M. Tualle, E. Tinet, J. Prat, B. Gelebart, S. Avrillier, “Light propagation in layered turbid media: a new analytical model for ultrafast calculation of the direct problem,” in Biomedical Optics: New Concepts in Therapeutic Laser Applications, Novel Biomedical Optical Spectroscopy, Imaging, and Diagnostics, Advances in Optical Imaging, Photon Migration, and Tissue Optics, OSA 1999 Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper AMA3–1.

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]

Bassani, M.

Bays, R.

Berndt, K. W.

Berns, M. W.

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

Bertoni, A.

Bigio, I.

J. R. Mourant, T. M. Johnson, G. Los, I. Bigio, “Non-invasive measurements of chemotherapy drug concentrations in tissue: preliminary demonstrations of in vivo measurements,” Phys. Med. Biol. 44, 1397–1417 (1999).
[CrossRef] [PubMed]

Boas, D. A.

Bruulsema, J. T.

G. Alexandrakis, R. A. Weersink, J. T. Bruulsema, M. S. Patterson, “Estimation of optical properties of two-layer tissue simulating phantoms from spatially resolved frequency-domain reflectance,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. R. Alfano, B. J. Tromberg, eds., Proc. SPIE3597, 155–163 (1999).
[CrossRef]

Burns, D. M.

M. S. Patterson, B. C. Wilson, J. W. Feather, D. M. Burns, W. Pushka, “The measurement of dihematoporphyrin ether concentration in tissue by reflectance spectrophotometry,” Photochem. Photobiol. 46, 337–343 (1987).
[CrossRef] [PubMed]

Chance, B.

Contini, D.

Corngold, N.

Diamond, K. R.

R. A. Weersink, J. E. Hayward, K. R. Diamond, M. S. Patterson, “Accuracy of non-invasive in vivo measurements of photosensitizer uptake based on a diffusion model of reflectance spectroscopy,” Photochem. Photobiol. 66, 326–335 (1997).
[CrossRef] [PubMed]

Durduran, T.

Durian, D. J.

Dyan, I.

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

Essenpreis, M.

Fantini, S.

Farrell, T. J.

Feather, J. W.

M. S. Patterson, B. C. Wilson, J. W. Feather, D. M. Burns, W. Pushka, “The measurement of dihematoporphyrin ether concentration in tissue by reflectance spectrophotometry,” Photochem. Photobiol. 46, 337–343 (1987).
[CrossRef] [PubMed]

Feng, T.-C.

Fishkin, J. B.

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes—The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, N.Y., 1996).

Franceschini, M. A.

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]

Gelebart, B.

J.-M. Tualle, E. Tinet, J. Prat, B. Gelebart, S. Avrillier, “Light propagation in layered turbid media: a new analytical model for ultrafast calculation of the direct problem,” in Biomedical Optics: New Concepts in Therapeutic Laser Applications, Novel Biomedical Optical Spectroscopy, Imaging, and Diagnostics, Advances in Optical Imaging, Photon Migration, and Tissue Optics, OSA 1999 Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper AMA3–1.

Glanzmann, T.

A. Kienle, T. Glanzmann, “In vivo determination of the optical properties of muscle with time-resolved reflectance using a layered model,” Phys. Med. Biol. 44, 2689–2702 (1999).
[CrossRef] [PubMed]

A. Kienle, T. Glanzmann, G. Wagnières, H. van den Bergh, “Investigation of two-layered turbid media with time-resolved reflectance,” Appl. Opt. 37, 6852–6862 (1998).
[CrossRef]

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.

Haskell, R. C.

Havlin, S.

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

H. Taitelbaum, S. Havlin, G. Weiss, “Approximate theory of photon migration in a two-layer medium,” Appl. Opt. 28, 2245–2249 (1989).
[CrossRef] [PubMed]

Hayward, J. E.

R. A. Weersink, J. E. Hayward, K. R. Diamond, M. S. Patterson, “Accuracy of non-invasive in vivo measurements of photosensitizer uptake based on a diffusion model of reflectance spectroscopy,” Photochem. Photobiol. 66, 326–335 (1997).
[CrossRef] [PubMed]

Hibst, R.

Jacques, S. L.

Johnson, T. M.

J. R. Mourant, T. M. Johnson, G. Los, I. Bigio, “Non-invasive measurements of chemotherapy drug concentrations in tissue: preliminary demonstrations of in vivo measurements,” Phys. Med. Biol. 44, 1397–1417 (1999).
[CrossRef] [PubMed]

Kienle, A.

Lakowicz, J. R.

Lilge, L.

Los, G.

J. R. Mourant, T. M. Johnson, G. Los, I. Bigio, “Non-invasive measurements of chemotherapy drug concentrations in tissue: preliminary demonstrations of in vivo measurements,” Phys. Med. Biol. 44, 1397–1417 (1999).
[CrossRef] [PubMed]

Maier, J. S.

M. A. Franceschini, S. Fantini, L. A. Paunescu, J. S. Maier, E. Gratton, “Influence of a superficial layer in the quantitative spectroscopic study of strongly scattering media,” Appl. Opt. 37, 7447–7458 (1998).
[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]

Marchesini, R.

Martelli, F.

McAdams, M. S.

Melloni, E.

Moulton, J. D.

Mourant, J. R.

J. R. Mourant, T. M. Johnson, G. Los, I. Bigio, “Non-invasive measurements of chemotherapy drug concentrations in tissue: preliminary demonstrations of in vivo measurements,” Phys. Med. Biol. 44, 1397–1417 (1999).
[CrossRef] [PubMed]

Patterson, M. S.

T. J. Farrell, M. S. Patterson, M. Essenpreis, “Influence of layered tissue architecture on estimates of tissue optical properties obtained from spatially resolved diffuse reflectometry,” Appl. Opt. 37, 1958–1972 (1998).
[CrossRef]

A. Kienle, M. S. Patterson, N. Utke, R. Bays, G. Wagnières, H. van den Bergh, “Determination of the optical properties of two-layer turbid media,” Appl. Opt. 37, 779–791 (1998).
[CrossRef]

G. Alexandrakis, T. J. Farrell, M. S. Patterson, “Accuracy of the diffusion approximation in determining the optical properties of a two-layer turbid medium,” Appl. Opt. 37, 7401–7410 (1998).
[CrossRef]

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

R. A. Weersink, J. E. Hayward, K. R. Diamond, M. S. Patterson, “Accuracy of non-invasive in vivo measurements of photosensitizer uptake based on a diffusion model of reflectance spectroscopy,” Photochem. Photobiol. 66, 326–335 (1997).
[CrossRef] [PubMed]

A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, B. C. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive 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]

T. J. Farrell, M. S. Patterson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the non-invasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[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]

M. S. Patterson, B. C. Wilson, J. W. Feather, D. M. Burns, W. Pushka, “The measurement of dihematoporphyrin ether concentration in tissue by reflectance spectrophotometry,” Photochem. Photobiol. 46, 337–343 (1987).
[CrossRef] [PubMed]

G. Alexandrakis, R. A. Weersink, J. T. Bruulsema, M. S. Patterson, “Estimation of optical properties of two-layer tissue simulating phantoms from spatially resolved frequency-domain reflectance,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. R. Alfano, B. J. Tromberg, eds., Proc. SPIE3597, 155–163 (1999).
[CrossRef]

Paunescu, L. A.

Pham, T.

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

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]

Polishchuk, A. Ya

Prat, J.

J.-M. Tualle, E. Tinet, J. Prat, B. Gelebart, S. Avrillier, “Light propagation in layered turbid media: a new analytical model for ultrafast calculation of the direct problem,” in Biomedical Optics: New Concepts in Therapeutic Laser Applications, Novel Biomedical Optical Spectroscopy, Imaging, and Diagnostics, Advances in Optical Imaging, Photon Migration, and Tissue Optics, OSA 1999 Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper AMA3–1.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes—The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, N.Y., 1996).

Pushka, W.

M. S. Patterson, B. C. Wilson, J. W. Feather, D. M. Burns, W. Pushka, “The measurement of dihematoporphyrin ether concentration in tissue by reflectance spectrophotometry,” Photochem. Photobiol. 46, 337–343 (1987).
[CrossRef] [PubMed]

Schmitt, J. M.

Sichirollo, A. E.

Spott, T.

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

Steiner, R.

Svaasand, L. O.

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

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 radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
[CrossRef]

Taitelbaum, H.

Takatani, S.

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]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes—The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, N.Y., 1996).

Tinet, E.

J.-M. Tualle, E. Tinet, J. Prat, B. Gelebart, S. Avrillier, “Light propagation in layered turbid media: a new analytical model for ultrafast calculation of the direct problem,” in Biomedical Optics: New Concepts in Therapeutic Laser Applications, Novel Biomedical Optical Spectroscopy, Imaging, and Diagnostics, Advances in Optical Imaging, Photon Migration, and Tissue Optics, OSA 1999 Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper AMA3–1.

Tromberg, B. J.

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

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 radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
[CrossRef]

Tsay, T.-T.

Tualle, J.-M.

J.-M. Tualle, E. Tinet, J. Prat, B. Gelebart, S. Avrillier, “Light propagation in layered turbid media: a new analytical model for ultrafast calculation of the direct problem,” in Biomedical Optics: New Concepts in Therapeutic Laser Applications, Novel Biomedical Optical Spectroscopy, Imaging, and Diagnostics, Advances in Optical Imaging, Photon Migration, and Tissue Optics, OSA 1999 Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper AMA3–1.

Utke, N.

van den Bergh, H.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes—The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, N.Y., 1996).

Wagnières, G.

Walker, E. C.

Walker, S. 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]

Wall, R. T.

Wang, L.

Warwick, R.

P. L. Williams, R. Warwick, “The integument,” in Gray’s Anatomy, 36th ed. (Churchill Livingstone, Edinburgh, UK, 1986), pp. 1216–1222.

Weersink, R. A.

R. A. Weersink, J. E. Hayward, K. R. Diamond, M. S. Patterson, “Accuracy of non-invasive in vivo measurements of photosensitizer uptake based on a diffusion model of reflectance spectroscopy,” Photochem. Photobiol. 66, 326–335 (1997).
[CrossRef] [PubMed]

G. Alexandrakis, R. A. Weersink, J. T. Bruulsema, M. S. Patterson, “Estimation of optical properties of two-layer tissue simulating phantoms from spatially resolved frequency-domain reflectance,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. R. Alfano, B. J. Tromberg, eds., Proc. SPIE3597, 155–163 (1999).
[CrossRef]

Weiss, G.

Weiss, G. H.

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

Williams, P. L.

P. L. Williams, R. Warwick, “The integument,” in Gray’s Anatomy, 36th ed. (Churchill Livingstone, Edinburgh, UK, 1986), pp. 1216–1222.

Wilson, B. C.

Yodh, A. G.

Zaccanti, G.

Zhou, G. X.

Appl. Opt. (9)

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

R. Marchesini, A. Bertoni, S. Andreola, E. Melloni, A. E. Sichirollo, “Extinction and absorption coefficients and scattering phase functions of human tissues in vitro,” Appl. Opt. 28, 2318–2324 (1989).
[CrossRef] [PubMed]

T. J. Farrell, M. S. Patterson, M. Essenpreis, “Influence of layered tissue architecture on estimates of tissue optical properties obtained from spatially resolved diffuse reflectometry,” Appl. Opt. 37, 1958–1972 (1998).
[CrossRef]

M. A. Franceschini, S. Fantini, L. A. Paunescu, J. S. Maier, E. Gratton, “Influence of a superficial layer in the quantitative spectroscopic study of strongly scattering media,” Appl. Opt. 37, 7447–7458 (1998).
[CrossRef]

A. Kienle, M. S. Patterson, N. Utke, R. Bays, G. Wagnières, H. van den Bergh, “Determination of the optical properties of two-layer turbid media,” Appl. Opt. 37, 779–791 (1998).
[CrossRef]

A. Kienle, T. Glanzmann, G. Wagnières, H. van den Bergh, “Investigation of two-layered turbid media with time-resolved reflectance,” Appl. Opt. 37, 6852–6862 (1998).
[CrossRef]

G. Alexandrakis, T. J. Farrell, M. S. Patterson, “Accuracy of the diffusion approximation in determining the optical properties of a two-layer turbid medium,” Appl. Opt. 37, 7401–7410 (1998).
[CrossRef]

H. Taitelbaum, S. Havlin, G. Weiss, “Approximate theory of photon migration in a two-layer medium,” Appl. Opt. 28, 2245–2249 (1989).
[CrossRef] [PubMed]

IEEE Trans. Biomed. Eng. (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]

J. Biomed. Opt. (1)

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]

J. Mod. Opt. (1)

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

J. Opt. Soc. Am. A (7)

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

Med. Phys. (1)

T. J. Farrell, M. S. Patterson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the non-invasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

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. (3)

Photochem. Photobiol. (2)

M. S. Patterson, B. C. Wilson, J. W. Feather, D. M. Burns, W. Pushka, “The measurement of dihematoporphyrin ether concentration in tissue by reflectance spectrophotometry,” Photochem. Photobiol. 46, 337–343 (1987).
[CrossRef] [PubMed]

R. A. Weersink, J. E. Hayward, K. R. Diamond, M. S. Patterson, “Accuracy of non-invasive in vivo measurements of photosensitizer uptake based on a diffusion model of reflectance spectroscopy,” Photochem. Photobiol. 66, 326–335 (1997).
[CrossRef] [PubMed]

Phys. Med. Biol. (3)

J. R. Mourant, T. M. Johnson, G. Los, I. Bigio, “Non-invasive measurements of chemotherapy drug concentrations in tissue: preliminary demonstrations of in vivo measurements,” Phys. Med. Biol. 44, 1397–1417 (1999).
[CrossRef] [PubMed]

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

A. Kienle, T. Glanzmann, “In vivo determination of the optical properties of muscle with time-resolved reflectance using a layered model,” Phys. Med. Biol. 44, 2689–2702 (1999).
[CrossRef] [PubMed]

Other (5)

J.-M. Tualle, E. Tinet, J. Prat, B. Gelebart, S. Avrillier, “Light propagation in layered turbid media: a new analytical model for ultrafast calculation of the direct problem,” in Biomedical Optics: New Concepts in Therapeutic Laser Applications, Novel Biomedical Optical Spectroscopy, Imaging, and Diagnostics, Advances in Optical Imaging, Photon Migration, and Tissue Optics, OSA 1999 Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper AMA3–1.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes—The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, N.Y., 1996).

G. Alexandrakis, R. A. Weersink, J. T. Bruulsema, M. S. Patterson, “Estimation of optical properties of two-layer tissue simulating phantoms from spatially resolved frequency-domain reflectance,” in Optical Tomography and Spectroscopy of Tissue III, B. Chance, R. R. Alfano, B. J. Tromberg, eds., Proc. SPIE3597, 155–163 (1999).
[CrossRef]

P. L. Williams, R. Warwick, “The integument,” in Gray’s Anatomy, 36th ed. (Churchill Livingstone, Edinburgh, UK, 1986), pp. 1216–1222.

S. R. Arridge, “Why optical tomography is hard,” in Biomedical Optics: New Concepts in Therapeutic Laser Applications, Novel Biomedical Optical Spectroscopy, Imaging, and Diagnostics, Advances in Optical Imaging, Photon Migration, and Tissue Optics, OSA 1999 Technical Digest Series (Optical Society of America, Washington, D.C., 1999), paper AMB1–1.

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Figures (9)

Fig. 1
Fig. 1

(a) In the single-source hybrid model, photon histories are scored only if photons exit the medium at ρ ≤ ρ c . The reflectance calculation for ρ > ρ c is given by the two-layer diffusion theory which assumes that all incident photons scatter at a depth z o = 1/(μ a1 + μ s1′). (b) The distributed-source hybrid model requires that if a photon migrates below a critical depth z c (≥l or < l) and continues to move in the downward direction, its history is terminated. The photon’s final location is defined to be one transport mean free path (mfp′) along the direction of its last scatter. It is then assumed to have become an isotropic photon source (asterisk). All other photons are scored in a normal MC fashion.

Fig. 2
Fig. 2

Predicted frequency-domain reflectance amplitude from the single-source hybrid model (curves) are compared with the corresponding MC data (symbols) for the optical property sets shown in Table 1. Good agreement is observed with the skin-on-fat MC data (circles) where the mini-MC contribution (dash–dot–dot curve) connects seamlessly with the diffusion contribution (solid curve). On the contrary, the agreement is not good with skin-on-muscle MC data (triangles) where there is an abrupt gap between the mini-MC (dash–dot curve) and the diffusion contributions (D a , dashed curve; D w/ a , dotted curve).

Fig. 3
Fig. 3

Good agreement is observed between MC data (circles) and the distributed-source hybrid model (solid curve) in the skin-on-fat case. Pure diffusion (dashed curve) also performs well except for ρ < 5 mm (inset). Simulation parameters are f = 100 MHz, z c = 1 mm, 100,000 photon histories.

Fig. 4
Fig. 4

Distributed-source hybrid model appears to be less sensitive to the definition of the diffusion coefficient (D a , solid curve; D w/ a , dash–dot–dot curve) relative to pure diffusion (D a , dashed curve; D w/ a , dotted curve) in the skin-on-muscle case. Simulation parameters are f = 100 MHz, z c = 1 mm, 100,000 photon histories.

Fig. 5
Fig. 5

Model predictions of phase delay for the skin-on-fat case. Symbol representation follows that of Fig. 3.

Fig. 6
Fig. 6

Model predictions for phase delay for the skin-on-muscle case. Symbol representation follows that of Fig. 4.

Fig. 7
Fig. 7

When z c is increased to 4 mm, agreement with MC data (fat, circles; muscle, triangles) improves relative to the z c = 1-mm case. Good agreement is observed for the skin-on-fat predictions of the hybrid model (solid curve). Differences between D a (solid curve) and D w/ a (dash–dot–dot curve) diminish relative to Fig. 4 and are now closer to MC data (inset). Simulation parameters are f = 100 MHz, 800,000 photon histories.

Fig. 8
Fig. 8

Phase delay model predictions for z c = 4 mm. The symbol representation follows that of Fig. 7.

Fig. 9
Fig. 9

Reflectance amplitude predicted by the hybrid model (curves) at f = 1 GHz (D w/ a , z c = 4 mm) agrees well with the corresponding MC data (symbols) for both skin-on-fat (circles, dashed curve) and skin-on-muscle (triangles, dash–dot–dot curve). The agreement appears to be good both at small ρ (inset) and at large ρ.

Tables (1)

Tables Icon

Table 1 Two Sets of Tissue Optical Propertiesa

Equations (19)

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PL=-lnRNμai+μsi, i=1, 2,
W=a1NSCAT1a2NSCAT2,
z0=1μa1+μs1=1mfp1,
ϕ1s, z, ω=sinhα1zb+z0D1α1D1α1 coshα1l-z+D2α2 sinhα1l-zD1α1 coshα1l+zb+D2α2 sinhα1l+zb-sinhα1z0-zD1α1,
Φ1ρ, z, ω=12π0 ϕ1s, z, ωsJ0sρds,
Rρ, w=14 Φ1ρ, z=0, ω+12 D1Φ1ρ, z, ωzz=0.
ACRρ, w=ImRρ, ω2+ReRρ, ω21/2,
θρ, ω=tan-1ImRρ, ωReRρ, ω.
ρi=i-0.5Δρ, i=1, Nρ,
zj=j-0.5Δz, j=1, Nz,
RMCρ=WMCρ2πρΔρMCNph.
ϕ2s, z, ω=Y2Y1 coshα1l-z+sinhα1l-z,
Y1=-tanhα1l+zb,
Y2=1Y111Y1-D2α2D1α11Y1α2D2sinhα2z0j-l-1α1D1coshα2z0j-l-1α2D2sinhα2z0j-l,
Y2=1Y1α2D2-α1D1.
WDIFρi, zj=SRCρi, zj2πρiΔρΔzNph, i=1, Nρ, j=1, Nz.
WDIFs, z0j=0 WDIFρ, zjρJ0sρdρ, j=1, Nz.
RDIFρ=j=1Nz0 WDIFs, z0jϕis, z0j, ωsJ0sρds,
Rρ=RMCρ+RDIFρ.

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