Abstract

Frequency-domain photon migration (FDPM) is a widely used technique for measuring the optical properties (i.e., absorption, μa, and reduced scattering, μs′, coefficients) of turbid samples. Typically, FDPM data analysis is performed with models based on a photon diffusion equation; however, analytical solutions are difficult to obtain for many realistic geometries. Here, we describe the use of models based instead on representative samples and multivariate calibration (chemometrics).

FDPM data at seven wavelengths (ranging from 674 to 956 nm) and multiple modulation frequencies (ranging from 50 to 600 MHz) were gathered from turbid samples containing mixtures of three absorbing dyes. Values for μa and μs′ were extracted from the FDPM data in different ways, first with the diffusion theory and then with the chemometric technique of partial least squares. Dye concentrations were determined from the FDPM data by three methods, first by least-squares fits to the diffusion results and then by two chemometric approaches. The accuracy of the chemometric predictions was comparable or superior for all three dyes. Our results indicate that chemometrics can recover optical properties and dye concentrations from the frequency-dependent behavior of photon density waves, without the need for diffusion-based models. Future applications to more complicated geometries, lower-scattering samples, and simpler FDPM instrumentation are discussed.

© 2000 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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1999 (1)

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]

1998 (3)

1997 (2)

B. W. Pogue, M. Testorf, T. McBride, U. Österberg, K. Paulsen, “Instrumentation and design of a frequency-domain diffuse optical tomography imager for breast cancer detection,” Opt. Express 1, 391–403 (1997).
[CrossRef] [PubMed]

B. J. Tromberg, O. Coquoz, J. B. Fishkin, T. Pham, E. R. Anderson, J. Butler, M. Cahn, J. D. Gross, V. Venugopalan, D. Pham, “Non-invasive measurements of breast tissue optical properties using frequency-domain photon migration,” Philos. Trans. R. Soc. Lond. 352, 661–668 (1997).
[CrossRef]

1996 (2)

1994 (4)

E. V. Thomas, “A primer on multivariate calibration,” Anal. Chem. 66, 795A–804A (1994).

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. 11, 2727–2741 (1994).
[CrossRef]

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

B. W. Pogue, M. S. Patterson, “Frequency-domain optical absorption spectroscopy of finite tissue volumes using diffusion theory,” Phys. Med. Biol. 39, 1157–1180 (1994).
[CrossRef] [PubMed]

1993 (2)

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

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering and wavelength transduction of diffuse photon density waves,” Phys. Rev. E 47, R2999–R3002 (1993).
[CrossRef]

1992 (1)

T. J. Farrell, B. C. Wilson, M. S. Patterson, “The use of a neural network to determine tissue optical properties from spatially resolved diffuse reflectance measurements,” Phys. Med. Biol. 37, 2281–2286 (1992).
[CrossRef] [PubMed]

1991 (2)

H. J. van Staveren, C. J. M. Moes, J. van Marle, S. A. Prahl, M. J. C. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400–1100 nm,” Appl. Opt. 30, 4507–4514 (1991).
[CrossRef] [PubMed]

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

1988 (1)

D. M. Haaland, E. V. Thomas, “Partial least-squares methods for spectral analyses. 1. Relation to other quantitative calibration methods and the extraction of qualitative information,” Anal. Chem. 60, 1193–1202 (1988).
[CrossRef]

1986 (1)

P. Geladi, B. R. Kowalski, “Partial least-squares regression: a tutorial,” Anal. Chim. Acta 185, 1–17 (1986).
[CrossRef]

1973 (1)

Andersen, P. E.

Anderson, E.

T. H. Pham, O. Coquoz, J. B. Fishkin, E. Anderson, B. J. Tromberg are preparing a manuscript to be called, “A broad bandwidth frequency domain instrument for quantitative tissue optical spectroscopy.”

Anderson, E. R.

B. J. Tromberg, O. Coquoz, J. B. Fishkin, T. Pham, E. R. Anderson, J. Butler, M. Cahn, J. D. Gross, V. Venugopalan, D. Pham, “Non-invasive measurements of breast tissue optical properties using frequency-domain photon migration,” Philos. Trans. R. Soc. Lond. 352, 661–668 (1997).
[CrossRef]

Barbieri, B.

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]

Boas, D. A.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering and wavelength transduction of diffuse photon density waves,” Phys. Rev. E 47, R2999–R3002 (1993).
[CrossRef]

Brown, S. D.

S. D. Brown, S. T. Sum, F. Despagne, B. K. Lavine, “Chemometrics,” Anal. Chem. 68, R21–R61 (1996).
[CrossRef]

Butler, J.

B. J. Tromberg, O. Coquoz, J. B. Fishkin, T. Pham, E. R. Anderson, J. Butler, M. Cahn, J. D. Gross, V. Venugopalan, D. Pham, “Non-invasive measurements of breast tissue optical properties using frequency-domain photon migration,” Philos. Trans. R. Soc. Lond. 352, 661–668 (1997).
[CrossRef]

Cahn, M.

B. J. Tromberg, O. Coquoz, J. B. Fishkin, T. Pham, E. R. Anderson, J. Butler, M. Cahn, J. D. Gross, V. Venugopalan, D. Pham, “Non-invasive measurements of breast tissue optical properties using frequency-domain photon migration,” Philos. Trans. R. Soc. Lond. 352, 661–668 (1997).
[CrossRef]

Chance, B.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering and wavelength transduction of diffuse photon density waves,” Phys. Rev. E 47, R2999–R3002 (1993).
[CrossRef]

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

Coquoz, O.

B. J. Tromberg, O. Coquoz, J. B. Fishkin, T. Pham, E. R. Anderson, J. Butler, M. Cahn, J. D. Gross, V. Venugopalan, D. Pham, “Non-invasive measurements of breast tissue optical properties using frequency-domain photon migration,” Philos. Trans. R. Soc. Lond. 352, 661–668 (1997).
[CrossRef]

T. H. Pham, O. Coquoz, J. B. Fishkin, E. Anderson, B. J. Tromberg are preparing a manuscript to be called, “A broad bandwidth frequency domain instrument for quantitative tissue optical spectroscopy.”

Dalgaard, T.

Dam, J. S.

Despagne, F.

S. D. Brown, S. T. Sum, F. Despagne, B. K. Lavine, “Chemometrics,” Anal. Chem. 68, R21–R61 (1996).
[CrossRef]

Fabricius, P. E.

Fantini, S.

Farrell, T. J.

T. J. Farrell, B. C. Wilson, M. S. Patterson, “The use of a neural network to determine tissue optical properties from spatially resolved diffuse reflectance measurements,” Phys. Med. Biol. 37, 2281–2286 (1992).
[CrossRef] [PubMed]

Feng, T.-C.

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. 11, 2727–2741 (1994).
[CrossRef]

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]

B. J. Tromberg, O. Coquoz, J. B. Fishkin, T. Pham, E. R. Anderson, J. Butler, M. Cahn, J. D. Gross, V. Venugopalan, D. Pham, “Non-invasive measurements of breast tissue optical properties using frequency-domain photon migration,” Philos. Trans. R. Soc. Lond. 352, 661–668 (1997).
[CrossRef]

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

J. B. Fishkin, E. Gratton, M. J. Vandeven, W. W. Mantulin, “Diffusion of intensity modulated near-IR light in turbid media,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 122–135 (1991).
[CrossRef]

T. H. Pham, O. Coquoz, J. B. Fishkin, E. Anderson, B. J. Tromberg are preparing a manuscript to be called, “A broad bandwidth frequency domain instrument for quantitative tissue optical spectroscopy.”

Franceschini, M. A.

Geladi, P.

P. Geladi, B. R. Kowalski, “Partial least-squares regression: a tutorial,” Anal. Chim. Acta 185, 1–17 (1986).
[CrossRef]

Gratton, E.

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

J. B. Fishkin, E. Gratton, M. J. Vandeven, W. W. Mantulin, “Diffusion of intensity modulated near-IR light in turbid media,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 122–135 (1991).
[CrossRef]

Gross, J. D.

B. J. Tromberg, O. Coquoz, J. B. Fishkin, T. Pham, E. R. Anderson, J. Butler, M. Cahn, J. D. Gross, V. Venugopalan, D. Pham, “Non-invasive measurements of breast tissue optical properties using frequency-domain photon migration,” Philos. Trans. R. Soc. Lond. 352, 661–668 (1997).
[CrossRef]

Haaland, D. M.

D. M. Haaland, E. V. Thomas, “Partial least-squares methods for spectral analyses. 1. Relation to other quantitative calibration methods and the extraction of qualitative information,” Anal. Chem. 60, 1193–1202 (1988).
[CrossRef]

Hale, G. M.

Haskell, R. C.

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. 11, 2727–2741 (1994).
[CrossRef]

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

Hibst, R.

Kaschle, M.

Kienle, A.

Kowalski, B. R.

P. Geladi, B. R. Kowalski, “Partial least-squares regression: a tutorial,” Anal. Chim. Acta 185, 1–17 (1986).
[CrossRef]

Lavine, B. K.

S. D. Brown, S. T. Sum, F. Despagne, B. K. Lavine, “Chemometrics,” Anal. Chem. 68, R21–R61 (1996).
[CrossRef]

Leigh, J.

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

Lilge, L.

Mantulin, W. W.

J. B. Fishkin, E. Gratton, M. J. Vandeven, W. W. Mantulin, “Diffusion of intensity modulated near-IR light in turbid media,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 122–135 (1991).
[CrossRef]

Maris, M.

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

Martens, H.

H. Martens, T. Næs, Multivariate Calibration (Wiley, New York, 1989).

McAdams, M. S.

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. 11, 2727–2741 (1994).
[CrossRef]

McBride, T.

Moes, C. J. M.

Moesta, K. T.

Næs, T.

H. Martens, T. Næs, Multivariate Calibration (Wiley, New York, 1989).

Nioka, S.

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

O’Leary, M. A.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering and wavelength transduction of diffuse photon density waves,” Phys. Rev. E 47, R2999–R3002 (1993).
[CrossRef]

Österberg, U.

Patterson, M. S.

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, “Frequency-domain optical absorption spectroscopy of finite tissue volumes using diffusion theory,” Phys. Med. Biol. 39, 1157–1180 (1994).
[CrossRef] [PubMed]

T. J. Farrell, B. C. Wilson, M. S. Patterson, “The use of a neural network to determine tissue optical properties from spatially resolved diffuse reflectance measurements,” Phys. Med. Biol. 37, 2281–2286 (1992).
[CrossRef] [PubMed]

Paulsen, K.

Pham, D.

B. J. Tromberg, O. Coquoz, J. B. Fishkin, T. Pham, E. R. Anderson, J. Butler, M. Cahn, J. D. Gross, V. Venugopalan, D. Pham, “Non-invasive measurements of breast tissue optical properties using frequency-domain photon migration,” Philos. Trans. R. Soc. Lond. 352, 661–668 (1997).
[CrossRef]

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]

B. J. Tromberg, O. Coquoz, J. B. Fishkin, T. Pham, E. R. Anderson, J. Butler, M. Cahn, J. D. Gross, V. Venugopalan, D. Pham, “Non-invasive measurements of breast tissue optical properties using frequency-domain photon migration,” Philos. Trans. R. Soc. Lond. 352, 661–668 (1997).
[CrossRef]

Pham, T. H.

T. H. Pham, O. Coquoz, J. B. Fishkin, E. Anderson, B. J. Tromberg are preparing a manuscript to be called, “A broad bandwidth frequency domain instrument for quantitative tissue optical spectroscopy.”

Pogue, B. W.

B. W. Pogue, M. Testorf, T. McBride, U. Österberg, K. Paulsen, “Instrumentation and design of a frequency-domain diffuse optical tomography imager for breast cancer detection,” Opt. Express 1, 391–403 (1997).
[CrossRef] [PubMed]

B. W. Pogue, M. S. Patterson, “Frequency-domain optical absorption spectroscopy of finite tissue volumes using diffusion theory,” Phys. Med. Biol. 39, 1157–1180 (1994).
[CrossRef] [PubMed]

Prahl, S. A.

Querry, M. R.

Schlag, P. M.

Sevick, E. M.

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

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.

Sum, S. T.

S. D. Brown, S. T. Sum, F. Despagne, B. K. Lavine, “Chemometrics,” Anal. Chem. 68, R21–R61 (1996).
[CrossRef]

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. 11, 2727–2741 (1994).
[CrossRef]

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

Testorf, M.

Thomas, E. V.

E. V. Thomas, “A primer on multivariate calibration,” Anal. Chem. 66, 795A–804A (1994).

D. M. Haaland, E. V. Thomas, “Partial least-squares methods for spectral analyses. 1. Relation to other quantitative calibration methods and the extraction of qualitative information,” Anal. Chem. 60, 1193–1202 (1988).
[CrossRef]

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]

V. Venugopalan, J. S. You, B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source-detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
[CrossRef]

B. J. Tromberg, O. Coquoz, J. B. Fishkin, T. Pham, E. R. Anderson, J. Butler, M. Cahn, J. D. Gross, V. Venugopalan, D. Pham, “Non-invasive measurements of breast tissue optical properties using frequency-domain photon migration,” Philos. Trans. R. Soc. Lond. 352, 661–668 (1997).
[CrossRef]

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. 11, 2727–2741 (1994).
[CrossRef]

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

T. H. Pham, O. Coquoz, J. B. Fishkin, E. Anderson, B. J. Tromberg are preparing a manuscript to be called, “A broad bandwidth frequency domain instrument for quantitative tissue optical spectroscopy.”

Tsay, T.-T.

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. 11, 2727–2741 (1994).
[CrossRef]

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

van Gemert, M. J. C.

van Marle, J.

van Staveren, H. J.

Vandeven, M. J.

J. B. Fishkin, E. Gratton, M. J. Vandeven, W. W. Mantulin, “Diffusion of intensity modulated near-IR light in turbid media,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 122–135 (1991).
[CrossRef]

Venugopalan, V.

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B. J. Tromberg, O. Coquoz, J. B. Fishkin, T. Pham, E. R. Anderson, J. Butler, M. Cahn, J. D. Gross, V. Venugopalan, D. Pham, “Non-invasive measurements of breast tissue optical properties using frequency-domain photon migration,” Philos. Trans. R. Soc. Lond. 352, 661–668 (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 noninvasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996).
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D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering and wavelength transduction of diffuse photon density waves,” Phys. Rev. E 47, R2999–R3002 (1993).
[CrossRef]

You, J. S.

V. Venugopalan, J. S. You, B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source-detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
[CrossRef]

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J. S. Dam, P. E. Andersen, T. Dalgaard, P. E. Fabricius, “Determination of tissue optical properties from diffuse reflectance profiles by multivariate calibration,” Appl. Opt. 37, 772–778 (1998).
[CrossRef]

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Opt. Express (1)

Philos. Trans. R. Soc. Lond. (1)

B. J. Tromberg, O. Coquoz, J. B. Fishkin, T. Pham, E. R. Anderson, J. Butler, M. Cahn, J. D. Gross, V. Venugopalan, D. Pham, “Non-invasive measurements of breast tissue optical properties using frequency-domain photon migration,” Philos. Trans. R. Soc. Lond. 352, 661–668 (1997).
[CrossRef]

Phys. Med. Biol. (3)

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).
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[CrossRef] [PubMed]

Phys. Rev. E (2)

V. Venugopalan, J. S. You, B. J. Tromberg, “Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source-detector separations,” Phys. Rev. E 58, 2395–2407 (1998).
[CrossRef]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering and wavelength transduction of diffuse photon density waves,” Phys. Rev. E 47, R2999–R3002 (1993).
[CrossRef]

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T. H. Pham, O. Coquoz, J. B. Fishkin, E. Anderson, B. J. Tromberg are preparing a manuscript to be called, “A broad bandwidth frequency domain instrument for quantitative tissue optical spectroscopy.”

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Figures (14)

Fig. 1
Fig. 1

Sample preparation sequence for creation of 40 mixtures with varying μ a and μ s ′ values. NG, nigrosin; NP, naphthol; CP, CPTA; W, water; IL, intralipid; SPM, spectrophotometer.

Fig. 2
Fig. 2

Plot of μ a versus μ s ′ values for the 40 samples in this study, showing the lack of correlation between the two optical parameters.

Fig. 3
Fig. 3

Infinite geometry for FDPM data acquisition in phantoms.

Fig. 4
Fig. 4

Logic of transforming original FDPM data so that it can be analyzed by PLS.

Fig. 5
Fig. 5

Data-processing sequence for chemometric prediction of μ a and μ s ′ from FDPM data.

Fig. 6
Fig. 6

Schematic of different ways to compute dye concentrations from FDPM data.

Fig. 7
Fig. 7

Typical FDPM data for the samples used in this study. All data were taken with the 674-nm diode laser at modulation frequencies between 50 and 600 MHz. Top, phase difference between r = 10 and r = 6 mm [see Eq. (8)]. Bottom, amplitude ratio between r = 10 and r = 6 mm [see Eq. (9)].

Fig. 8
Fig. 8

Predictions of μ a values at 674 nm from FDPM data of 40 turbid mixtures. (a) Plot obtained with the best fit to the phase and amplitude data for each sample independently. (b) Plot derived with chemometric cross validation as described in the text. The r 2 values for the plots are 0.98 for the top and 0.99 for the bottom, respectively.

Fig. 9
Fig. 9

Comparison of data vectors analyzed for extraction of dye concentrations from a typical sample. Note that the x axis is not linear with respect to wavelength. Circles represent reference μ a values as determined by spectrophotometer. Squares with error bars represent μ a predictions from diffusion theory analysis of FDPM data. Upwards triangles represent μ a predictions from chemometric analysis of FDPM data. Connected downwards triangles represent μˆaω vector created from FDPM data. Note that the μˆaω vector contains 20 data points plotted per laser wavelength, corresponding to frequencies from 50 to 600 MHz. Error bars on the diffusion predictions are ±8 × 10-4 mm-1, as measured in an earlier study.19

Fig. 10
Fig. 10

Predictions of μ s ′ values at 674 nm from FDPM data of 40 turbid mixtures. (a) Plot obtained with the best fit to the phase and amplitude data for each sample independently. (b) Plot derived from the chemometric cross validation as described in the text. The r 2 value for both plots is 0.99.

Fig. 11
Fig. 11

Predictions of nigrosin concentrations from the FDPM data.

Fig. 12
Fig. 12

Predictions of naphthol concentrations from the FDPM data.

Fig. 13
Fig. 13

Predictions of CPTA concentrations from the FDPM data.

Fig. 14
Fig. 14

Summary of dye concentration predictions with the three strategies depicted in Fig. 6.

Tables (1)

Tables Icon

Table 1 Slopes of Diffusion-Predicted μa and μs Values at Each Wavelength in the Studya

Equations (13)

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μa=A ln1010 mm.
μa674=d=13μad674VdVtot,
μaλ=μa674AλA674.
ϕ=kir+ϕI,
A=CIexp-krrr,
kr=32 μaμa+μs1/21+ωμac21/2+11/2,
ki=32 μaμa+μs1/21+ωμac21/2-11/2,
Φϕ1-ϕ2=kir1-r2,
AA1/A2=exp-krr1-r2r1/r2
kr=-lnr1r2Ar1-r2,
ki=Φr1-r2,
μa=ω2ckr2-ki2krki,
μs=kr2-ki23μa-μa.

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