Abstract

A technique for measuring broadband near-infrared absorption spectra of turbid media that uses a combination of frequency-domain (FD) and steady-state (SS) reflectance methods is presented. Most of the wavelength coverage is provided by a white-light SS measurement, whereas the FD data are acquired at a few selected wavelengths. Coefficients of absorption (μa) and reduced scattering (μs) derived from the FD data are used to calibrate the intensity of the SS measurements and to estimate μs at all wavelengths in the spectral window of interest. After these steps are performed, one can determine μa by comparing the SS reflectance values with the predictions of diffusion theory, wavelength by wavelength. Absorption spectra of a turbid phantom and of human breast tissue in vivo, derived with the combined SSFD technique, agree well with expected reference values. All measurements can be performed at a single source–detector separation distance, reducing the variations in sampling volume that exist in multidistance methods. The technique uses relatively inexpensive light sources and detectors and is easily implemented on an existing multiwavelength FD system.

© 2000 Optical Society of America

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  1. 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]
  2. A. Kienle, M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from a semi-infinite medium,” J. Opt. Soc. Am. 14, 246–254 (1997).
    [CrossRef]
  3. R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, “Noninvasive absorption and scattering spectroscopy of bulk diffusive media: an application to the optical characterization of human breast,” Appl. Phys. Lett. 74, 874–876 (1999).
    [CrossRef]
  4. 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]
  5. 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]
  6. J. B. Fishkin, O. Coquoz, E. R. Anderson, M. Brenner, B. J. Tromberg, “Frequency-domain photon migration measurements of normal and malignant tissue optical properties in a human subject,” Appl. Opt. 36, 10–20 (1997).
    [CrossRef] [PubMed]
  7. T. H. Pham, O. Coquoz, J. B. Fishkin, E. Anderson, B. J. Tromberg, “Broad bandwidth frequency domain instrument for quantitative tissue optical spectroscopy,” Rev. Sci. Instrum. 71, 2500–2513 (2000).
    [CrossRef]
  8. L. Reynolds, C. Johnson, A. Ishimaru, “Diffuse reflectance from a finite blood medium: applications to the modeling of fiber optic catheters,” Appl. Opt. 15, 2059–2067 (1976).
    [CrossRef] [PubMed]
  9. T. J. Farrell, M. S. Patterson, B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
    [CrossRef] [PubMed]
  10. 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]
  11. R. Bays, G. Wagnières, D. Robert, D. Braichotte, J.-F. Savary, P. Monnier, H. van den Bergh, “Clinical determination of tissue optical properties by endoscopic spatially resolved reflectometry,” Appl. Opt. 35, 1756–1766 (1996).
    [CrossRef] [PubMed]
  12. R. A. Weersink, J. Hayward, K. Diamond, M. Patterson, “Accuracy of noninvasive in vivo measurements of photosensitizer uptake based on a diffusion model of reflectance spectroscopy,” Photochem. Photobiol. 66, 326–335 (1997).
    [CrossRef] [PubMed]
  13. F. Bevilacqua, D. Piguet, P. Marquet, J. Gross, B. Tromberg, C. Depeursinge, “In vivo local determination of tissue optical properties: applications to human brain,” Appl. Opt. 38, 4939–4950 (1999).
    [CrossRef]
  14. E. L. Hull, M. G. Nichols, T. H. Foster, “Quantitative broadband near-infrared spectroscopy of tissue-stimulating phantoms containing erythrocytes,” Phys. Med. Biol. 43, 3381–3404 (1998).
    [CrossRef] [PubMed]
  15. R. M. P. Doornbos, R. Lang, M. C. Aalders, F. W. Cross, H. J. C. M. Sterenborg, “The determination of in vivo human tissue optical properties and absolute chromophore concentrations using spatially resolved steady-state diffuse reflectance spectroscopy,” Phys. Med. Biol. 44, 967–981 (1999).
    [CrossRef] [PubMed]
  16. 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,” Phil. Trans. R. Soc. London 352, 661–668 (1997).
  17. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, Orlando, Fla., 1978).
  18. J. B. Fishkin, E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge,” J. Opt. Soc. Am. 10, 127–140 (1993).
    [CrossRef]
  19. 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]
  20. 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]
  21. 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]
  22. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, in Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1993), Chap. 15, pp. 683–688.
  23. R. Graaff, J. G. Aarnoose, J. R. Zijp, P. M. A. Sloot, F. F. M. de Mul, J. Greve, M. H. Koelink, “Reduced light-scattering properties for mixtures of spherical particles: a simple approximation derived from Mie calculations,” Appl. Opt. 31, 1370–1376 (1992).
    [CrossRef] [PubMed]
  24. J. R. Mourant, T. Fuselier, J. Boyer, T. Johnson, I. J. Bigio, “Predictions and measurements of scattering and absorption over broad wavelength ranges in tissue phantoms,” Appl. Opt. 36, 949–957 (1997).
    [CrossRef] [PubMed]
  25. J. M. Schmitt, G. Kumar, “Optical scattering properties of soft tissue: a discrete particle model,” Appl. Opt. 37, 2788–2797 (1998).
    [CrossRef]
  26. L. H. Kou, D. Labrie, P. Chylek, “Refractive indices of water and ice in the 0.65- to 2.5-µm spectral range,” Appl. Opt. 32, 3531–3540 (1993).
    [CrossRef] [PubMed]
  27. S. Wray, M. Cope, D. T. Delpy, J. S. Wyatt, E. O. R. Reynolds, “Characterization of the near-infrared absorption spectra of cytochrome-AA3 and hemoglobin for the non-invasive monitoring of cerebral oxygenation,” Biochim. Biophys. Acta 933, 184–192 (1988).
    [CrossRef] [PubMed]
  28. C. Eker, Optical Characterization of Tissue for Medical Diagnostics, Ph.D. dissertation (Lund University, Lund, Sweden, 1999).
  29. G. M. Hale, M. R. Querry, “Optical constants of water in the 200-nm to 200-µm wavelength region,” Appl. Opt. 12, 555–563 (1973).
    [CrossRef] [PubMed]
  30. A. J. Berger, T.-W. Koo, I. Itzkan, M. S. Feld, “An enhanced algorithm for linear multivariate calibration,” Anal. Chem. 70, 623–628 (1998).
    [CrossRef] [PubMed]
  31. 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, 7401–7409 (1998).
    [CrossRef]
  32. 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]

2000 (1)

T. H. Pham, O. Coquoz, J. B. Fishkin, E. Anderson, B. J. Tromberg, “Broad bandwidth frequency domain instrument for quantitative tissue optical spectroscopy,” Rev. Sci. Instrum. 71, 2500–2513 (2000).
[CrossRef]

1999 (3)

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, “Noninvasive absorption and scattering spectroscopy of bulk diffusive media: an application to the optical characterization of human breast,” Appl. Phys. Lett. 74, 874–876 (1999).
[CrossRef]

R. M. P. Doornbos, R. Lang, M. C. Aalders, F. W. Cross, H. J. C. M. Sterenborg, “The determination of in vivo human tissue optical properties and absolute chromophore concentrations using spatially resolved steady-state diffuse reflectance spectroscopy,” Phys. Med. Biol. 44, 967–981 (1999).
[CrossRef] [PubMed]

F. Bevilacqua, D. Piguet, P. Marquet, J. Gross, B. Tromberg, C. Depeursinge, “In vivo local determination of tissue optical properties: applications to human brain,” Appl. Opt. 38, 4939–4950 (1999).
[CrossRef]

1998 (5)

1997 (5)

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

A. Kienle, M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from a semi-infinite medium,” J. Opt. Soc. Am. 14, 246–254 (1997).
[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,” Phil. Trans. R. Soc. London 352, 661–668 (1997).

J. B. Fishkin, O. Coquoz, E. R. Anderson, M. Brenner, B. J. Tromberg, “Frequency-domain photon migration measurements of normal and malignant tissue optical properties in a human subject,” Appl. Opt. 36, 10–20 (1997).
[CrossRef] [PubMed]

J. R. Mourant, T. Fuselier, J. Boyer, T. Johnson, I. J. Bigio, “Predictions and measurements of scattering and absorption over broad wavelength ranges in tissue phantoms,” Appl. Opt. 36, 949–957 (1997).
[CrossRef] [PubMed]

1996 (2)

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 (3)

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]

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]

1993 (3)

1992 (2)

R. Graaff, J. G. Aarnoose, J. R. Zijp, P. M. A. Sloot, F. F. M. de Mul, J. Greve, M. H. Koelink, “Reduced light-scattering properties for mixtures of spherical particles: a simple approximation derived from Mie calculations,” Appl. Opt. 31, 1370–1376 (1992).
[CrossRef] [PubMed]

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

1989 (1)

1988 (1)

S. Wray, M. Cope, D. T. Delpy, J. S. Wyatt, E. O. R. Reynolds, “Characterization of the near-infrared absorption spectra of cytochrome-AA3 and hemoglobin for the non-invasive monitoring of cerebral oxygenation,” Biochim. Biophys. Acta 933, 184–192 (1988).
[CrossRef] [PubMed]

1976 (1)

1973 (1)

Aalders, M. C.

R. M. P. Doornbos, R. Lang, M. C. Aalders, F. W. Cross, H. J. C. M. Sterenborg, “The determination of in vivo human tissue optical properties and absolute chromophore concentrations using spatially resolved steady-state diffuse reflectance spectroscopy,” Phys. Med. Biol. 44, 967–981 (1999).
[CrossRef] [PubMed]

Aarnoose, J. G.

Alexandrakis, G.

Anderson, E.

T. H. Pham, O. Coquoz, J. B. Fishkin, E. Anderson, B. J. Tromberg, “Broad bandwidth frequency domain instrument for quantitative tissue optical spectroscopy,” Rev. Sci. Instrum. 71, 2500–2513 (2000).
[CrossRef]

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,” Phil. Trans. R. Soc. London 352, 661–668 (1997).

J. B. Fishkin, O. Coquoz, E. R. Anderson, M. Brenner, B. J. Tromberg, “Frequency-domain photon migration measurements of normal and malignant tissue optical properties in a human subject,” Appl. Opt. 36, 10–20 (1997).
[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]

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]

Bays, R.

Berger, A. J.

A. J. Berger, T.-W. Koo, I. Itzkan, M. S. Feld, “An enhanced algorithm for linear multivariate calibration,” Anal. Chem. 70, 623–628 (1998).
[CrossRef] [PubMed]

Bevilacqua, F.

Bigio, I. J.

Boyer, J.

Braichotte, D.

Brenner, M.

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,” Phil. Trans. R. Soc. London 352, 661–668 (1997).

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,” Phil. Trans. R. Soc. London 352, 661–668 (1997).

Chance, B.

Chylek, P.

Cope, M.

S. Wray, M. Cope, D. T. Delpy, J. S. Wyatt, E. O. R. Reynolds, “Characterization of the near-infrared absorption spectra of cytochrome-AA3 and hemoglobin for the non-invasive monitoring of cerebral oxygenation,” Biochim. Biophys. Acta 933, 184–192 (1988).
[CrossRef] [PubMed]

Coquoz, O.

T. H. Pham, O. Coquoz, J. B. Fishkin, E. Anderson, B. J. Tromberg, “Broad bandwidth frequency domain instrument for quantitative tissue optical spectroscopy,” Rev. Sci. Instrum. 71, 2500–2513 (2000).
[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,” Phil. Trans. R. Soc. London 352, 661–668 (1997).

J. B. Fishkin, O. Coquoz, E. R. Anderson, M. Brenner, B. J. Tromberg, “Frequency-domain photon migration measurements of normal and malignant tissue optical properties in a human subject,” Appl. Opt. 36, 10–20 (1997).
[CrossRef] [PubMed]

Cross, F. W.

R. M. P. Doornbos, R. Lang, M. C. Aalders, F. W. Cross, H. J. C. M. Sterenborg, “The determination of in vivo human tissue optical properties and absolute chromophore concentrations using spatially resolved steady-state diffuse reflectance spectroscopy,” Phys. Med. Biol. 44, 967–981 (1999).
[CrossRef] [PubMed]

Cubeddu, R.

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, “Noninvasive absorption and scattering spectroscopy of bulk diffusive media: an application to the optical characterization of human breast,” Appl. Phys. Lett. 74, 874–876 (1999).
[CrossRef]

de Mul, F. F. M.

Delpy, D. T.

S. Wray, M. Cope, D. T. Delpy, J. S. Wyatt, E. O. R. Reynolds, “Characterization of the near-infrared absorption spectra of cytochrome-AA3 and hemoglobin for the non-invasive monitoring of cerebral oxygenation,” Biochim. Biophys. Acta 933, 184–192 (1988).
[CrossRef] [PubMed]

Depeursinge, C.

Diamond, K.

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

Doornbos, R. M. P.

R. M. P. Doornbos, R. Lang, M. C. Aalders, F. W. Cross, H. J. C. M. Sterenborg, “The determination of in vivo human tissue optical properties and absolute chromophore concentrations using spatially resolved steady-state diffuse reflectance spectroscopy,” Phys. Med. Biol. 44, 967–981 (1999).
[CrossRef] [PubMed]

Eker, C.

C. Eker, Optical Characterization of Tissue for Medical Diagnostics, Ph.D. dissertation (Lund University, Lund, Sweden, 1999).

Fantini, S.

Farrell, T. J.

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, 7401–7409 (1998).
[CrossRef]

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

Feld, M. S.

A. J. Berger, T.-W. Koo, I. Itzkan, M. S. Feld, “An enhanced algorithm for linear multivariate calibration,” Anal. Chem. 70, 623–628 (1998).
[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.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, in Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1993), Chap. 15, pp. 683–688.

Foster, T. H.

E. L. Hull, M. G. Nichols, T. H. Foster, “Quantitative broadband near-infrared spectroscopy of tissue-stimulating phantoms containing erythrocytes,” Phys. Med. Biol. 43, 3381–3404 (1998).
[CrossRef] [PubMed]

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]

Fuselier, T.

Graaff, R.

Gratton, E.

Greve, J.

Gross, J.

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,” Phil. Trans. R. Soc. London 352, 661–668 (1997).

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]

Hayward, J.

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

Hibst, R.

Hull, E. L.

E. L. Hull, M. G. Nichols, T. H. Foster, “Quantitative broadband near-infrared spectroscopy of tissue-stimulating phantoms containing erythrocytes,” Phys. Med. Biol. 43, 3381–3404 (1998).
[CrossRef] [PubMed]

Ishimaru, A.

Itzkan, I.

A. J. Berger, T.-W. Koo, I. Itzkan, M. S. Feld, “An enhanced algorithm for linear multivariate calibration,” Anal. Chem. 70, 623–628 (1998).
[CrossRef] [PubMed]

Johnson, C.

Johnson, T.

Kienle, A.

Koelink, M. H.

Koo, T.-W.

A. J. Berger, T.-W. Koo, I. Itzkan, M. S. Feld, “An enhanced algorithm for linear multivariate calibration,” Anal. Chem. 70, 623–628 (1998).
[CrossRef] [PubMed]

Kou, L. H.

Kumar, G.

Labrie, D.

Lang, R.

R. M. P. Doornbos, R. Lang, M. C. Aalders, F. W. Cross, H. J. C. M. Sterenborg, “The determination of in vivo human tissue optical properties and absolute chromophore concentrations using spatially resolved steady-state diffuse reflectance spectroscopy,” Phys. Med. Biol. 44, 967–981 (1999).
[CrossRef] [PubMed]

Lilge, L.

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]

Marquet, P.

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]

Monnier, P.

Mourant, J. R.

Nichols, M. G.

E. L. Hull, M. G. Nichols, T. H. Foster, “Quantitative broadband near-infrared spectroscopy of tissue-stimulating phantoms containing erythrocytes,” Phys. Med. Biol. 43, 3381–3404 (1998).
[CrossRef] [PubMed]

Patterson, M.

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

Patterson, M. S.

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, 7401–7409 (1998).
[CrossRef]

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

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

Paunescu, L. A.

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,” Phil. Trans. R. Soc. London 352, 661–668 (1997).

Pham, T.

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,” Phil. Trans. R. Soc. London 352, 661–668 (1997).

Pham, T. H.

T. H. Pham, O. Coquoz, J. B. Fishkin, E. Anderson, B. J. Tromberg, “Broad bandwidth frequency domain instrument for quantitative tissue optical spectroscopy,” Rev. Sci. Instrum. 71, 2500–2513 (2000).
[CrossRef]

Pifferi, A.

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, “Noninvasive absorption and scattering spectroscopy of bulk diffusive media: an application to the optical characterization of human breast,” Appl. Phys. Lett. 74, 874–876 (1999).
[CrossRef]

Piguet, D.

Pogue, B. W.

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]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, in Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1993), Chap. 15, pp. 683–688.

Querry, M. R.

Reynolds, E. O. R.

S. Wray, M. Cope, D. T. Delpy, J. S. Wyatt, E. O. R. Reynolds, “Characterization of the near-infrared absorption spectra of cytochrome-AA3 and hemoglobin for the non-invasive monitoring of cerebral oxygenation,” Biochim. Biophys. Acta 933, 184–192 (1988).
[CrossRef] [PubMed]

Reynolds, L.

Robert, D.

Savary, J.-F.

Schmitt, J. M.

Sloot, P. M. A.

Steiner, R.

Sterenborg, H. J. C. M.

R. M. P. Doornbos, R. Lang, M. C. Aalders, F. W. Cross, H. J. C. M. Sterenborg, “The determination of in vivo human tissue optical properties and absolute chromophore concentrations using spatially resolved steady-state diffuse reflectance spectroscopy,” Phys. Med. Biol. 44, 967–981 (1999).
[CrossRef] [PubMed]

Svaasand, L. O.

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]

Taroni, P.

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, “Noninvasive absorption and scattering spectroscopy of bulk diffusive media: an application to the optical characterization of human breast,” Appl. Phys. Lett. 74, 874–876 (1999).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, in Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1993), Chap. 15, pp. 683–688.

Torricelli, A.

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, “Noninvasive absorption and scattering spectroscopy of bulk diffusive media: an application to the optical characterization of human breast,” Appl. Phys. Lett. 74, 874–876 (1999).
[CrossRef]

Tromberg, B.

Tromberg, B. J.

T. H. Pham, O. Coquoz, J. B. Fishkin, E. Anderson, B. J. Tromberg, “Broad bandwidth frequency domain instrument for quantitative tissue optical spectroscopy,” Rev. Sci. Instrum. 71, 2500–2513 (2000).
[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,” Phil. Trans. R. Soc. London 352, 661–668 (1997).

J. B. Fishkin, O. Coquoz, E. R. Anderson, M. Brenner, B. J. Tromberg, “Frequency-domain photon migration measurements of normal and malignant tissue optical properties in a human subject,” Appl. Opt. 36, 10–20 (1997).
[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]

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]

Valentini, G.

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, “Noninvasive absorption and scattering spectroscopy of bulk diffusive media: an application to the optical characterization of human breast,” Appl. Phys. Lett. 74, 874–876 (1999).
[CrossRef]

van den Bergh, H.

Venugopalan, V.

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,” Phil. Trans. R. Soc. London 352, 661–668 (1997).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, in Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1993), Chap. 15, pp. 683–688.

Wagnières, G.

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]

Weersink, R. A.

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

Wilson, B. C.

Wray, S.

S. Wray, M. Cope, D. T. Delpy, J. S. Wyatt, E. O. R. Reynolds, “Characterization of the near-infrared absorption spectra of cytochrome-AA3 and hemoglobin for the non-invasive monitoring of cerebral oxygenation,” Biochim. Biophys. Acta 933, 184–192 (1988).
[CrossRef] [PubMed]

Wyatt, J. S.

S. Wray, M. Cope, D. T. Delpy, J. S. Wyatt, E. O. R. Reynolds, “Characterization of the near-infrared absorption spectra of cytochrome-AA3 and hemoglobin for the non-invasive monitoring of cerebral oxygenation,” Biochim. Biophys. Acta 933, 184–192 (1988).
[CrossRef] [PubMed]

Zijp, J. R.

Anal. Chem. (1)

A. J. Berger, T.-W. Koo, I. Itzkan, M. S. Feld, “An enhanced algorithm for linear multivariate calibration,” Anal. Chem. 70, 623–628 (1998).
[CrossRef] [PubMed]

Appl. Opt. (15)

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

L. Reynolds, C. Johnson, A. Ishimaru, “Diffuse reflectance from a finite blood medium: applications to the modeling of fiber optic catheters,” Appl. Opt. 15, 2059–2067 (1976).
[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]

R. Graaff, J. G. Aarnoose, J. R. Zijp, P. M. A. Sloot, F. F. M. de Mul, J. Greve, M. H. Koelink, “Reduced light-scattering properties for mixtures of spherical particles: a simple approximation derived from Mie calculations,” Appl. Opt. 31, 1370–1376 (1992).
[CrossRef] [PubMed]

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]

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, O. Coquoz, E. R. Anderson, M. Brenner, B. J. Tromberg, “Frequency-domain photon migration measurements of normal and malignant tissue optical properties in a human subject,” Appl. Opt. 36, 10–20 (1997).
[CrossRef] [PubMed]

J. M. Schmitt, G. Kumar, “Optical scattering properties of soft tissue: a discrete particle model,” Appl. Opt. 37, 2788–2797 (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, 7401–7409 (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]

R. Bays, G. Wagnières, D. Robert, D. Braichotte, J.-F. Savary, P. Monnier, H. van den Bergh, “Clinical determination of tissue optical properties by endoscopic spatially resolved reflectometry,” Appl. Opt. 35, 1756–1766 (1996).
[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]

L. H. Kou, D. Labrie, P. Chylek, “Refractive indices of water and ice in the 0.65- to 2.5-µm spectral range,” Appl. Opt. 32, 3531–3540 (1993).
[CrossRef] [PubMed]

F. Bevilacqua, D. Piguet, P. Marquet, J. Gross, B. Tromberg, C. Depeursinge, “In vivo local determination of tissue optical properties: applications to human brain,” Appl. Opt. 38, 4939–4950 (1999).
[CrossRef]

J. R. Mourant, T. Fuselier, J. Boyer, T. Johnson, I. J. Bigio, “Predictions and measurements of scattering and absorption over broad wavelength ranges in tissue phantoms,” Appl. Opt. 36, 949–957 (1997).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, “Noninvasive absorption and scattering spectroscopy of bulk diffusive media: an application to the optical characterization of human breast,” Appl. Phys. Lett. 74, 874–876 (1999).
[CrossRef]

Biochim. Biophys. Acta (1)

S. Wray, M. Cope, D. T. Delpy, J. S. Wyatt, E. O. R. Reynolds, “Characterization of the near-infrared absorption spectra of cytochrome-AA3 and hemoglobin for the non-invasive monitoring of cerebral oxygenation,” Biochim. Biophys. Acta 933, 184–192 (1988).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (3)

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]

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

J. B. Fishkin, E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge,” J. Opt. Soc. Am. 10, 127–140 (1993).
[CrossRef]

Med. Phys. (1)

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

Phil. Trans. R. Soc. London (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,” Phil. Trans. R. Soc. London 352, 661–668 (1997).

Photochem. Photobiol. (1)

R. A. Weersink, J. Hayward, K. Diamond, M. Patterson, “Accuracy of noninvasive 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)

E. L. Hull, M. G. Nichols, T. H. Foster, “Quantitative broadband near-infrared spectroscopy of tissue-stimulating phantoms containing erythrocytes,” Phys. Med. Biol. 43, 3381–3404 (1998).
[CrossRef] [PubMed]

R. M. P. Doornbos, R. Lang, M. C. Aalders, F. W. Cross, H. J. C. M. Sterenborg, “The determination of in vivo human tissue optical properties and absolute chromophore concentrations using spatially resolved steady-state diffuse reflectance spectroscopy,” Phys. Med. Biol. 44, 967–981 (1999).
[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]

Rev. Sci. Instrum. (1)

T. H. Pham, O. Coquoz, J. B. Fishkin, E. Anderson, B. J. Tromberg, “Broad bandwidth frequency domain instrument for quantitative tissue optical spectroscopy,” Rev. Sci. Instrum. 71, 2500–2513 (2000).
[CrossRef]

Other (3)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, Orlando, Fla., 1978).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, in Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1993), Chap. 15, pp. 683–688.

C. Eker, Optical Characterization of Tissue for Medical Diagnostics, Ph.D. dissertation (Lund University, Lund, Sweden, 1999).

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Figures (10)

Fig. 1
Fig. 1

Configuration of light sources, optical fibers, and detectors: APD, avalanche photodiode. The dashed rectangles denote components that belong to the FD and SS systems.

Fig. 2
Fig. 2

SS reflectance spectrum acquired from a dye–Intralipid phantom, scaled to fit the discrete reflectance values (filled circles) predicted by Eqs. (1) and (2) with FD-derived values of μs and μ a . This scaling causes the entire spectrum to be converted into absolute reflectance units; see text for a discussion. The error in FD reflectance is estimated to be ±3 × 10-7/mm-2; in SS reflectance, ±1 × 10-7/mm-2.

Fig. 3
Fig. 3

SS reflectance spectra acquired from three locations in breast tissue of female volunteers. The spectra have been scaled to fit reflectance values (filled circles) calculated from FD data in the same manner as for Fig. 2. Errors are the same as for Fig. 2.

Fig. 4
Fig. 4

Determination of the broadband μs spectrum for the dye–Intralipid phantom. Open circles, discrete μs values measured by the FD technique; solid curve, best power-law fit to Eq. (4). Fitting parameters are A = 2200 and B = -0.82. Error bars on μs are 3%.7

Fig. 5
Fig. 5

Determination of broadband μs spectra for the three breast samples. Circles, the discrete μs values measured by the FD technique; solid curves, best power-law fits to Eq. (4). Fitting parameters (A, B) from top to bottom are (240, -0.86), (2700, -12), and (250, -0.85). Error bars on μs are 3%.7

Fig. 6
Fig. 6

Comparison of μ a values generated by FD (open circles) and by SSFD (thicker curve) for the dye–Intralipid phantom. Also shown is the best fit (thinner curve) to the SSFD data by use of the spectra of naphthol (measured with a spectrophotometer) and of water (from Kou et al.26). Open circles, discrete μ a values from FD alone. Error bars, ±0.0005 mm-1 or 5%, whichever is larger.

Fig. 7
Fig. 7

μ a predictions for the first breast measurement (inner breast; 37-year-old volunteer). Thicker curve, the SSFD data; thinner curve, least-squares fit for a superposition of Hb, HbO2, water, and fat spectra; filled circles, the FD values (error bars, ±0.0005 mm-1 or 5%, whichever is larger); dotted curve the least-squares fit when only the FD values are weighted. Physiological parameters from the SSFD spectral fit: total hemoglobin concentration, 22 µM; oxygen saturation, 73%; water, 15 g/cm3; fat, 0.75 g/cm3. See Table 1 for more details. Note that the fat peak between 900 and 950 nm is significantly underfitted by the FD calculation.

Fig. 8
Fig. 8

μ a predictions for the second breast measurement (areolar border; 21-year-old volunteer). Curves and circles have the same assignments as for Fig. 7. Physiological parameters: total hemoglobin concentration, 30 µM; oxygen saturation, 70%; water, 0.51 g/cm3; fat, 0.42 g/cm3. See Table 1 for more details.

Fig. 9
Fig. 9

μ a predictions for the third breast measurement (inner breast; 21-year-old volunteer). Curves and circles have the same assignments as for Fig. 7. Physiological parameters: total hemoglobin concentration, 19 µM; oxygen saturation, 72%; water, 28 g/cm3; fat, 0.56 g/cm3. See Table 1 for more details.

Fig. 10
Fig. 10

Absorption spectra of major absorbers in breast tissue. Upper left, oxyhemoglobin, 1 µM; upper right, deoxyhemoglobin, 1 µM; lower left, water, 1 g/cm3; lower right, fat (soybean oil), 0.9 g/cm3. The hemoglobin spectra are from Wray et al.,27 the water is from Kou et al.,26 and the fat is from the doctoral dissertation of Eker.28

Tables (1)

Tables Icon

Table 1 Results of Physiological Predictions for Breast Samples 1–3a

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

kreal=32 μaμa+μs1/21+ωμac21/2+11/2,
kimag=32 μaμa+μs1/21+ωμac21/2-11/2,
R=c1Φ-c2DΦ·-z,
Φ=P4πDexp-krsrs-exp-kriri,  DΦ·z=P4πzok+1rsexp-krsrs2+zo+2zb×k+1riexp-kriri2,
R=CoA exp-iϕ+ϕo,
μsλ=Aλ-B

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