Abstract

The Green’s function for the diffusion equation is widely used to describe photon transport in turbid media. We have performed aseries of spectroscopy experiments on a number of uniform turbid media with different optical properties (absorption coefficient in the range 0.03–0.14 cm-1, reduced scattering coefficient in the range 5–22 cm-1). Our experiments have been conducted in the frequency domain, where the measured parameters are the dc intensity (Idc), ac amplitude (Iac), and phase (Φ) of the light intensity wave. In an infinite medium, the Green’s function predicts a linear dependence of ln(rIdc) and Φ on the source–detector separation r. Our measurements show that the intercepts of these straight lines predicted by the Green’s function do not agree with the experimental results. To reproduce the experimental results, we have introduced an effective photon source whose spatial extent and source strength depend on the optical properties of the medium. This effective source term has no effect on the slopes of the straight lines predicted by the Green’sfunction at large values of r.

© 1997 Optical Society of America

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References

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  1. MUSCLE4, Fourth International Workshop on Multiple Scattering Lidar Experiments, Florence, Italy, 29–31 October 1990.
  2. J. A. Weinman, “Effects of multiple scattering on light pulses reflected by turbid atmosphere,” J. Atmos. Sci. 33, 1763–1771 (1976).
    [CrossRef]
  3. B. D. Savage, J. S. Mathis, “Observed properties of interstellar dust,” Ann. Rev. Astron. Astrophys. 17, 73–111 (1979).
    [CrossRef]
  4. R. R. Alfano, X. Liang, L. Wang, P. P. Ho, “Time-resolved imaging of oil droplets in highly scattering soil solution,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 2–5 (1995).
  5. G. J. Müller, ed., Medical Optical Tomography: Functional Imaging and Monitoring, Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993).
  6. B. Chance, R. R. Alfano, eds., Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, Proc. SPIE2389 (1995).
  7. G. A. Millikan, “The oximeter, an instrument for measuring continuously the oxygen saturation of arterial blood in man,” Rev. Sci. Instrum. 13, 434–444 (1942).
    [CrossRef]
  8. B. Beauvoit, T. Kitai, B. Chance, “Contribution of the mitochondrial compartment to the optical properties of the rat liver: a theoretical and practical approach,” Biophys. J. 67, 2501–2510 (1994).
    [CrossRef] [PubMed]
  9. M. Kohl, M. Cope, M. Essenpreis, D. Böcker, “Influence of glucose concentration on light scattering in tissue-simulating phantoms,” Opt. Lett. 19, 2170–2172 (1994).
    [CrossRef] [PubMed]
  10. J. S. Maier, S. A. Walker, S. Fantini, M. A. Franceschini, E. Gratton, “Possible correlation between blood glucose concentration and the reduced scattering coefficient of tissues in the near infrared,” Opt. Lett. 19, 2062–2064 (1994).
    [CrossRef] [PubMed]
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    [CrossRef]
  12. S. R. Arridge, M. Cope, D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
    [CrossRef] [PubMed]
  13. J.-M. Kaltenbach, M. Kaschke, “Frequency- and time-domain modelling of light transport in random media,” in Ref. 5, pp. 65–86.
  14. F. Liu, K. M. Yoo, R. R. Alfano, “Should the photon flux or the photon density be used to describe the temporal profiles of scattered ultrashort laser pulses in random media?,” Opt. Lett. 18, 432–434 (1993).
    [CrossRef] [PubMed]
  15. J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976), p. 137.
  16. 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. A 10, 127–140 (1993).
    [CrossRef] [PubMed]
  17. T. J. Farrel, M. S. Patterson, B. 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]
  18. B. A. Feddersen, D. W. Piston, E. Gratton, “Digital parallel acquisition in frequency domain fluorometry,” Rev. Sci. Instrum. 60, 2929–2936 (1989).
    [CrossRef]
  19. 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]
  20. M. A. Franceschini, S. Fantini, S. A. Walker, J. S. Maier, W. W. Mantulin, E. Gratton, “Multi-channel optical instrument for near-infrared imaging of tissue,” in Ref. 6, pp. 264–273.
  21. H. Liu, D. A. Boas, Y. Zhang, A. G. Yodh, B. Chance, “A simplified approach to characterize optical properties and blood oxygenation in tissue using continuous near infrared light,” in Ref. 6, pp. 496–502.

1994 (5)

1993 (2)

1992 (2)

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

T. J. Farrel, M. S. Patterson, B. 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]

1989 (1)

B. A. Feddersen, D. W. Piston, E. Gratton, “Digital parallel acquisition in frequency domain fluorometry,” Rev. Sci. Instrum. 60, 2929–2936 (1989).
[CrossRef]

1979 (1)

B. D. Savage, J. S. Mathis, “Observed properties of interstellar dust,” Ann. Rev. Astron. Astrophys. 17, 73–111 (1979).
[CrossRef]

1976 (1)

J. A. Weinman, “Effects of multiple scattering on light pulses reflected by turbid atmosphere,” J. Atmos. Sci. 33, 1763–1771 (1976).
[CrossRef]

1942 (1)

G. A. Millikan, “The oximeter, an instrument for measuring continuously the oxygen saturation of arterial blood in man,” Rev. Sci. Instrum. 13, 434–444 (1942).
[CrossRef]

Alfano, R. R.

F. Liu, K. M. Yoo, R. R. Alfano, “Should the photon flux or the photon density be used to describe the temporal profiles of scattered ultrashort laser pulses in random media?,” Opt. Lett. 18, 432–434 (1993).
[CrossRef] [PubMed]

R. R. Alfano, X. Liang, L. Wang, P. P. Ho, “Time-resolved imaging of oil droplets in highly scattering soil solution,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 2–5 (1995).

Arridge, S. R.

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

Barbieri, B.

Beauvoit, B.

B. Beauvoit, T. Kitai, B. Chance, “Contribution of the mitochondrial compartment to the optical properties of the rat liver: a theoretical and practical approach,” Biophys. J. 67, 2501–2510 (1994).
[CrossRef] [PubMed]

Böcker, D.

Chance, B.

B. Beauvoit, T. Kitai, B. Chance, “Contribution of the mitochondrial compartment to the optical properties of the rat liver: a theoretical and practical approach,” Biophys. J. 67, 2501–2510 (1994).
[CrossRef] [PubMed]

Cope, M.

M. Kohl, M. Cope, M. Essenpreis, D. Böcker, “Influence of glucose concentration on light scattering in tissue-simulating phantoms,” Opt. Lett. 19, 2170–2172 (1994).
[CrossRef] [PubMed]

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

Delpy, D. T.

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

Duderstadt, J. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976), p. 137.

Essenpreis, M.

Fantini, S.

Farrel, T. J.

T. J. Farrel, M. S. Patterson, B. 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]

Feddersen, B. A.

B. A. Feddersen, D. W. Piston, E. Gratton, “Digital parallel acquisition in frequency domain fluorometry,” Rev. Sci. Instrum. 60, 2929–2936 (1989).
[CrossRef]

Feng, T. C.

Fishkin, J. B.

Franceschini, M. A.

Gratton, E.

Hamilton, L. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976), p. 137.

Haskell, R. C.

Ho, P. P.

R. R. Alfano, X. Liang, L. Wang, P. P. Ho, “Time-resolved imaging of oil droplets in highly scattering soil solution,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 2–5 (1995).

Kitai, T.

B. Beauvoit, T. Kitai, B. Chance, “Contribution of the mitochondrial compartment to the optical properties of the rat liver: a theoretical and practical approach,” Biophys. J. 67, 2501–2510 (1994).
[CrossRef] [PubMed]

Kohl, M.

Liang, X.

R. R. Alfano, X. Liang, L. Wang, P. P. Ho, “Time-resolved imaging of oil droplets in highly scattering soil solution,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 2–5 (1995).

Liu, F.

Maier, J. S.

Mathis, J. S.

B. D. Savage, J. S. Mathis, “Observed properties of interstellar dust,” Ann. Rev. Astron. Astrophys. 17, 73–111 (1979).
[CrossRef]

McAdams, M. S.

Millikan, G. A.

G. A. Millikan, “The oximeter, an instrument for measuring continuously the oxygen saturation of arterial blood in man,” Rev. Sci. Instrum. 13, 434–444 (1942).
[CrossRef]

Patterson, M. S.

T. J. Farrel, M. S. Patterson, B. 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]

Piston, D. W.

B. A. Feddersen, D. W. Piston, E. Gratton, “Digital parallel acquisition in frequency domain fluorometry,” Rev. Sci. Instrum. 60, 2929–2936 (1989).
[CrossRef]

Savage, B. D.

B. D. Savage, J. S. Mathis, “Observed properties of interstellar dust,” Ann. Rev. Astron. Astrophys. 17, 73–111 (1979).
[CrossRef]

Svaasand, L. O.

Tromberg, B. J.

Tsay, T. T.

Walker, S. A.

Wang, L.

R. R. Alfano, X. Liang, L. Wang, P. P. Ho, “Time-resolved imaging of oil droplets in highly scattering soil solution,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 2–5 (1995).

Weinman, J. A.

J. A. Weinman, “Effects of multiple scattering on light pulses reflected by turbid atmosphere,” J. Atmos. Sci. 33, 1763–1771 (1976).
[CrossRef]

Wilson, B.

T. J. Farrel, M. S. Patterson, B. 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]

Yoo, K. M.

Ann. Rev. Astron. Astrophys. (1)

B. D. Savage, J. S. Mathis, “Observed properties of interstellar dust,” Ann. Rev. Astron. Astrophys. 17, 73–111 (1979).
[CrossRef]

Appl. Opt. (1)

Biophys. J. (1)

B. Beauvoit, T. Kitai, B. Chance, “Contribution of the mitochondrial compartment to the optical properties of the rat liver: a theoretical and practical approach,” Biophys. J. 67, 2501–2510 (1994).
[CrossRef] [PubMed]

J. Atmos. Sci. (1)

J. A. Weinman, “Effects of multiple scattering on light pulses reflected by turbid atmosphere,” J. Atmos. Sci. 33, 1763–1771 (1976).
[CrossRef]

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

Med. Phys. (1)

T. J. Farrel, M. S. Patterson, B. 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]

Opt. Lett. (3)

Phys. Med. Biol. (1)

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

Rev. Sci. Instrum. (2)

B. A. Feddersen, D. W. Piston, E. Gratton, “Digital parallel acquisition in frequency domain fluorometry,” Rev. Sci. Instrum. 60, 2929–2936 (1989).
[CrossRef]

G. A. Millikan, “The oximeter, an instrument for measuring continuously the oxygen saturation of arterial blood in man,” Rev. Sci. Instrum. 13, 434–444 (1942).
[CrossRef]

Other (8)

MUSCLE4, Fourth International Workshop on Multiple Scattering Lidar Experiments, Florence, Italy, 29–31 October 1990.

M. A. Franceschini, S. Fantini, S. A. Walker, J. S. Maier, W. W. Mantulin, E. Gratton, “Multi-channel optical instrument for near-infrared imaging of tissue,” in Ref. 6, pp. 264–273.

H. Liu, D. A. Boas, Y. Zhang, A. G. Yodh, B. Chance, “A simplified approach to characterize optical properties and blood oxygenation in tissue using continuous near infrared light,” in Ref. 6, pp. 496–502.

J.-M. Kaltenbach, M. Kaschke, “Frequency- and time-domain modelling of light transport in random media,” in Ref. 5, pp. 65–86.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976), p. 137.

R. R. Alfano, X. Liang, L. Wang, P. P. Ho, “Time-resolved imaging of oil droplets in highly scattering soil solution,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 2–5 (1995).

G. J. Müller, ed., Medical Optical Tomography: Functional Imaging and Monitoring, Vol. IS11 of SPIE Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993).

B. Chance, R. R. Alfano, eds., Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, Proc. SPIE2389 (1995).

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Figures (5)

Fig. 1
Fig. 1

Source fiber that delivers the light to the sample and the detector fiber that collects the optical signal. The effective photon source is distributed in a hemisphere (shaded) centered at the tip of the source fiber and has a radius γ.

Fig. 2
Fig. 2

Experimental arrangement. The laser diode (LD) emitting at 780 nm is intensity modulated at 120 MHz by a frequency synthesizer (Synth 1) by means of amplifier A1. The DC Bias selects a working point for the laser diode just above threshold. The tips of the source and the detector optical fibers are immersed in the sample to mimic an infinite geometry. The two fibers face each other, and the volume of the sample is ∼9 L. The gain of the PMT detector is modulated at 120 MHz + 400 Hz by synthesizer 2 (Synth 2) by means of amplifier A2. Neutral-density filters (F) are used to attenuate the detected signal at smaller source–detector separations. The data acquisition card in the computer and the two frequency synthesizers are phase locked (Synch).

Fig. 3
Fig. 3

Intercept of the phase-versus-r straight line as a function of (a) μa, (b) μs′. In (a), μs′ is 16 cm-1, whereas in (b), μa is 0.035 cm-1. The symbols are the experimental values obtained from the phase data taken at r ranging between 1.5 and 2 cm. The curves are the phase intercepts predicted by diffusion theory for a point source (dashed curve) and for the effective extended source described in the text (solid curve).

Fig. 4
Fig. 4

Intercept of the ln (rIdc)-versus-r straight line as a function of (a) μa, (b) μs′. In (a), μs′ is 16 cm-1, whereas in (b), μ a is 0.035 cm-1. The symbols are the experimental values obtained from the intensity data taken at r ranging between 1.5 and 2 cm. The curves are the intensity intercepts predicted by diffusion theory for a point source (dashed curve) and for the effective extended source described in the text (solid curve).

Fig. 5
Fig. 5

Symbols represent (a) the measured phase, (b) ln (rIdc) as functions of source–detector separation r. The medium has an absorption coefficient of μa = 0.035 cm-1 and a reduced scattering coefficient of μs′ = 18 cm-1. Consequently the radius of the hemispheric effective photon source is γ = 0.71 cm. The continuous curves have the slopes predicted by diffusion theory and are shifted by an offset to match the experimental data at r > γ. At r < γ the experimental points deviate significantly from the linear behavior predicted by diffusion theory.

Equations (24)

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dI=vur, t, ΩˆΩˆ·nˆdΩΔA,
I=ΔA2π 0arcsinN vur, t, Ωˆcos θ sin θ dθ,
hνvur, t, Ωˆ=hνvUr, t4π-3hνvD4πU · Ωˆ,
I=ΔAvU2×141-cos 2θF+D1+krr1-cos3 θF.
hνvur, t, rˆ=hνvUr, t4π-3hνvD4πU,
hνvur, t, θˆ=hνvur, t, ϕˆ=hνvUr, t4π
Ur, tt-vD2Ur, t+vμaUr, t=qr, t,
UGr, ω=14πvDexp-krr,
Ur, ω=qr*UGr, ω= qrUGr-r, ωd3r.
qr*UGr, ω=02πdϕ0πdθ sin θ 0drr2qr, θ, ϕ14πvDexp-kr2+r2-2rr cos θ1/2r2+r2-2rr cos θ1/2.
r, θ, ϕ=3Seffω/2πγ3,for 0rγ and 0θπ/20,for r>γ or π/2<θπ,
Uhsr, ω=SeffωUGr, ω3k3γ3×1+kr1+γ2r21/2expkr1--1+γ2r21/2-kr+kγ-1expkγ.
γ=1Rekμa, μs0=132μaμs01/21+ω2v2μa21/2+11/2,
SeffSμaμs,
Ir, ωSΔAμak3γ31+kγ-1expkγ×141-cos 2θF+Dk1-cos3 θFexp-krr.
qr=3Sω4πγ3r/γ,
Ur, ω=SUGr, ω3k3γ3kγ coshkγ-sinhkγ,
qΛr=3Sωπγ3Λr/γ,
UΛr, ω=SUGr, ω12k4γ4kγ sinhkγ-2 coshkγ+2.
qexpr=Sω4πγ2exp-r/γr.
Uexpr, ω=SUGr, ωexp-1-kγrγ1-k2γ2.
qshellr=Sω4πγ2δr-γ,
Ushellr, ω=SUGr, ωsinhkγkγ.
limγ0U=limγ0UΛ=limγ0Uexp=limγ0 Ushell=SωUGr, ω.

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