Abstract

A unified analysis is presented for the sensitivities of reflectance and path length to scattering variations in a diffusive medium, using an improved solution of the steady-state diffusion equation. This approach enables one to (1) explain theoretically two kinds of dependency of near-infrared reflectance on source–detector separations and (2) obtain an analytical expression for optical path lengths. The results shown here are consistent with those of Kumar and Schmitt [Appl. Opt. 36, 2286 (1997)] and Mourant et al. [Appl. Opt. 36, 5655 (1997)]. Also, discussions are given on (1) possible reasons for some inconsistency between the conclusion drawn by Mourant et al. and results given here and (2) the usefulness of making reflectance measurements while minimizing the sensitivity of reflectance and path length to scattering variations.

© 2001 Optical Society of America

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References

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  1. G. Kumar, J. M. Schmitt, “Optimal probe geometry for near-infrared spectroscopy of biological tissue,” Appl. Opt. 36, 2286–2293 (1997).
    [CrossRef] [PubMed]
  2. J. R. Mourant, I. J. Bigio, D. A. Jack, T. M. Johnson, H. D. Miller, “Measuring absorption coefficients in small volumes of highly scattering media: source–detector separations for which path lengths do not depend on scattering properties,” Appl. Opt. 36, 5655–5661 (1997).
    [CrossRef] [PubMed]
  3. A. Kienle, M. S. Patterson, “Improved solutions of 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]
  4. T. J. Farrell, M. S. Patterson, “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]
  5. H. Liu, B. Beauvoit, M. Kimura, B. Chance, “Dependence of tissue optical properties on solute-induced changes in refractive index and osmolarity,” J. Biomed. Opt. 1, 200–211 (1996).
    [CrossRef] [PubMed]
  6. 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]
  7. W. M. Star, “Diffusion theory of light transport,” in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch, M. J. C. Van Gemert, eds. (Plenum, New York, 1995).
    [CrossRef]
  8. H. Liu, Y. Song, K. L. Worden, X. Jiang, A. Constantinescu, R. P. Mason, “Noninvasive investigation of blood oxygenation dynamics of tumors by near-infrared spectroscopy,” Appl. Opt. 39, 5231–5243 (2000).
    [CrossRef]
  9. J. R. Mourant, T. M. Johnson, G. Los, I. J. Bigio, “Non-invasive measurement of chemotherapy drug concentrations in tissue: preliminary demonstrations of in vivo measurements,” Phys. Med. Biol. 44, 1397–1417 (1999).
    [CrossRef] [PubMed]

2000 (1)

1999 (1)

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

1997 (3)

1996 (1)

H. Liu, B. Beauvoit, M. Kimura, B. Chance, “Dependence of tissue optical properties on solute-induced changes in refractive index and osmolarity,” J. Biomed. Opt. 1, 200–211 (1996).
[CrossRef] [PubMed]

1992 (1)

T. J. Farrell, M. S. Patterson, “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]

1991 (1)

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]

Beauvoit, B.

H. Liu, B. Beauvoit, M. Kimura, B. Chance, “Dependence of tissue optical properties on solute-induced changes in refractive index and osmolarity,” J. Biomed. Opt. 1, 200–211 (1996).
[CrossRef] [PubMed]

Bigio, I. J.

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

J. R. Mourant, I. J. Bigio, D. A. Jack, T. M. Johnson, H. D. Miller, “Measuring absorption coefficients in small volumes of highly scattering media: source–detector separations for which path lengths do not depend on scattering properties,” Appl. Opt. 36, 5655–5661 (1997).
[CrossRef] [PubMed]

Chance, B.

H. Liu, B. Beauvoit, M. Kimura, B. Chance, “Dependence of tissue optical properties on solute-induced changes in refractive index and osmolarity,” J. Biomed. Opt. 1, 200–211 (1996).
[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]

Constantinescu, A.

Farrell, T. J.

T. J. Farrell, M. S. Patterson, “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]

Jack, D. A.

Jiang, X.

Johnson, T. M.

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

J. R. Mourant, I. J. Bigio, D. A. Jack, T. M. Johnson, H. D. Miller, “Measuring absorption coefficients in small volumes of highly scattering media: source–detector separations for which path lengths do not depend on scattering properties,” Appl. Opt. 36, 5655–5661 (1997).
[CrossRef] [PubMed]

Kienle, A.

Kimura, M.

H. Liu, B. Beauvoit, M. Kimura, B. Chance, “Dependence of tissue optical properties on solute-induced changes in refractive index and osmolarity,” J. Biomed. Opt. 1, 200–211 (1996).
[CrossRef] [PubMed]

Kumar, G.

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]

Liu, H.

H. Liu, Y. Song, K. L. Worden, X. Jiang, A. Constantinescu, R. P. Mason, “Noninvasive investigation of blood oxygenation dynamics of tumors by near-infrared spectroscopy,” Appl. Opt. 39, 5231–5243 (2000).
[CrossRef]

H. Liu, B. Beauvoit, M. Kimura, B. Chance, “Dependence of tissue optical properties on solute-induced changes in refractive index and osmolarity,” J. Biomed. Opt. 1, 200–211 (1996).
[CrossRef] [PubMed]

Los, G.

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

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]

Mason, R. P.

Miller, H. D.

Mourant, J. R.

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

J. R. Mourant, I. J. Bigio, D. A. Jack, T. M. Johnson, H. D. Miller, “Measuring absorption coefficients in small volumes of highly scattering media: source–detector separations for which path lengths do not depend on scattering properties,” Appl. Opt. 36, 5655–5661 (1997).
[CrossRef] [PubMed]

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]

Patterson, M. S.

A. Kienle, M. S. Patterson, “Improved solutions of 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]

T. J. Farrell, M. S. Patterson, “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]

Schmitt, J. 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]

Song, Y.

Star, W. M.

W. M. Star, “Diffusion theory of light transport,” in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch, M. J. C. Van Gemert, eds. (Plenum, New York, 1995).
[CrossRef]

Worden, K. L.

Anal. Biochem. (1)

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]

Appl. Opt. (3)

J. Biomed. Opt. (1)

H. Liu, B. Beauvoit, M. Kimura, B. Chance, “Dependence of tissue optical properties on solute-induced changes in refractive index and osmolarity,” J. Biomed. Opt. 1, 200–211 (1996).
[CrossRef] [PubMed]

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

Med. Phys. (1)

T. J. Farrell, M. S. Patterson, “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]

Phys. Med. Biol. (1)

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

Other (1)

W. M. Star, “Diffusion theory of light transport,” in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch, M. J. C. Van Gemert, eds. (Plenum, New York, 1995).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Dependence of optical reflectance on μ s ′, calculated with Eq. (1), at (a) ρ = 1.0 mm, (b) 2.0 mm, and (c) 10.0 mm with a fixed μ a value of 0.1 cm-1. The Y-axis unit is cm-2. In each panel the curve plotted with triangles results from the isotropic light source at z 0, the curve with circles results from the virtual image source at z 0′, and the solid curve is the summation of the other two curves that are due to the two different light sources.

Fig. 2
Fig. 2

Comparison of sensitivity of R with μ s ′. The thick curve was calculated with Eq. (4), the thin curve was obtained with the expression given by Kumar and Schmitt1 with their empirically determined coefficient (K = 2.5), and the open circles were selectively replotted from the Monte Carlo simulations given by Kumar and Schmitt in Ref. 1. The absorption and reduced scattering coefficients used here are μ a = 0.5 cm-1 and μ s ′ = 10 cm-1, respectively.

Fig. 3
Fig. 3

Contour plot of sensitivity of R to μ s ′, S s , versus source–detector separation, ρ, and reduced scattering coefficient, μ s ′, with μ a = 0.1 cm-1 (solid curves) and 0.5 cm-1 (dotted curves). The Y-axis unit is cm-1. For both μ a values the plotted contour curves have three constant values of S s : -0.02, 0.00, and 0.02 cm. The dotted contour curves do not have labels because of space limitation, but they have the same varying tendency and pattern as the solid curves.

Fig. 4
Fig. 4

Contour plot of optical pathlength, L, versus source–detector separation, ρ, and reduced scattering coefficient, μ s ′, with μ a = 0.1 cm-1 (solid curves) and 0.5 cm-1 (dotted curves). The Y-axis unit is cm-1. For both μ a values the plotted contour curves have four constant values of L: 0.2, 0.5, 1.0, and 2.0 cm.

Equations (15)

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Rimpρ, z0=0.118Φρ, z0+0.306Rρ, z0,
Φρ, z0=14πDexp-μeffr1r1-exp-μeffr2r2,
Rρ, z0=14πz0μeff+1r1exp-μeffr1r12+z0+4AD×μeff+1r2exp-μeffr2r22,
Ss=-ln Rimpμs,  Sa=-ln Rimpμa,
Ss=-1RimpRimpμs=-1Rimpyiso+yima,
yiso=μt4πx12exp-kx10.3063x13+3kx12+k2-1x1-k-k22x1+0.3541x1+k+x1-k2x12,
yima=μt4πx22exp-kx20.306α3α2x23+3kα2x22+k2α2-1x2-k-k22x2-0.354α2x2+kα2+x2-k2x22,
Rimp=I0 exp-μaL,
L=Sa=- lnRimpμa=-1RimpRimpμa=1Rimpziso+zima,
ziso=μt4πx12exp-kx10.3063x13+3kx12+k2-1x1-k-32x1+0.3541x1+k+x1-32kx12,
zima=μt4πx22exp-kx20.306α3α2x23+3kα2x22+k2α2-1x2-k-32x2-0.354α2x2+kα2+x2-32kx22.
area=A=-i=575 nm595 nmlnIλiI0λi=i=575 nm595 nmO.D.λi=i=575 nm595 nmμaλiLiλikμa585 nmL,
L32μsμa1/2ρ  if ρ  z0.
O.D.=μaL=logI0/Rimp.
c1c2=μa1μa2=O.D.1O.D.2=logRimp2Rimp1,

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