Abstract

Recent studies have shown that the slope of logarithmic scattering spectroscopy of a turbid medium is related to the sizes of the scattering particles within the turbid medium. Mie theory can be used to generate a logarithmic plot of the reduced-scattering coefficient versus wavelength. According to Nilsson et al. [Appl. Opt. 37, 1256 (1998)], the slope value of a linear fit of the logarithmic scattering spectroscopy between 600 and 1050 nm can be used for direct determination of particle size. We performed similar calculations using the Rayleigh-Gans approximation and obtained an analogous overall shape with additional sinusoidal features. Our calculations indicate a possible relationship between the slope and the particle size when the size is used to calculate the slope, namely, in the forward calculation. However, because of the sinusoidal pattern, the inverse calculation to obtain the particle size from the slope may be applied only for particles with a radius of <0.13 μm in combination with 650–1050-nm light. Caution should be exercised when inverse calculation is performed to determine the scattering particle sizes in the range of radii >0.13 μm, with the slope of logarithmic scattering spectroscopy within 650–1050 nm.

© 2003 Optical Society of America

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References

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  1. R. Graaff, J. G. Aarnoudse, 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]
  2. J. R. Mourant, T. Fuselier, J. Boyer, T. M. Johnson, I. Bigio, “Predictions and measurements of scattering and absorption over broad wavelength ranges in tissue phantoms,” Appl. Opt. 36, 949–957 (1997).
    [CrossRef]
  3. J. R. Mourant, J. P. Freyer, A. H. Hielscher, A. A. Eick, D. Shen, T. M. Johnson, “Mechanisms of light scattering from biological cells relevant to noninvasive optical-tissue diagnostics,” Appl. Opt. 37, 3586–3593 (1998).
    [CrossRef]
  4. J. R. Mourant, A. H. Hielscher, A. A. Eick, T. M. Johnson, J. P. Freyer, “Evidence of intrinsic differences in the light scattering properties of tumorigenic and nontumorigenic cells,” Cancer (Cancer Cytopathol) 84, 366–374 (1998).
  5. A. M. K. Nilsson, C. Sturesson, D. L. Liu, S. Andersson-Engels, “Changes in spectral shape of tissue optical properties in conjunction with laser-induced thermotherapy,” Appl. Opt. 37, 1256–1267 (1998).
    [CrossRef]
  6. H. Liu, B. Beauvoit, M. Kimura, B. Chance, “Dependence of tissue optical properties on solute-induced changes on refractive index and osmolarity,” J. Biomed. Opt. 1, 200–211 (1996).
    [CrossRef] [PubMed]
  7. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  8. J. R. Zijp, J. J. ten Bosch, “Pascal program to perform Mie calculations,” Opt. Eng. 32, 1691–1695 (1993).
    [CrossRef]
  9. H. Jiang, G. Marquez, L. V. Wang, “Particle sizing in concentrated suspensions by use of steady-state, continuous-wave photon-migration techniques,” Opt. Lett. 23, 394–396 (1998).
    [CrossRef]
  10. H. Jiang, J. Pierce, J. Kao, E. Sevick-Muraca, “Measurement of particle-size distribution and volume fraction in concentrated suspensions with photon migration techniques,” Appl. Opt. 36, 3310–3318 (1997).
    [CrossRef] [PubMed]

1998 (4)

1997 (2)

1996 (1)

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

1993 (1)

J. R. Zijp, J. J. ten Bosch, “Pascal program to perform Mie calculations,” Opt. Eng. 32, 1691–1695 (1993).
[CrossRef]

1992 (1)

Aarnoudse, J. G.

Andersson-Engels, S.

Beauvoit, B.

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

Bigio, I.

Boyer, J.

Chance, B.

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

de Mul, F. F. M.

Eick, A. A.

J. R. Mourant, A. H. Hielscher, A. A. Eick, T. M. Johnson, J. P. Freyer, “Evidence of intrinsic differences in the light scattering properties of tumorigenic and nontumorigenic cells,” Cancer (Cancer Cytopathol) 84, 366–374 (1998).

J. R. Mourant, J. P. Freyer, A. H. Hielscher, A. A. Eick, D. Shen, T. M. Johnson, “Mechanisms of light scattering from biological cells relevant to noninvasive optical-tissue diagnostics,” Appl. Opt. 37, 3586–3593 (1998).
[CrossRef]

Freyer, J. P.

J. R. Mourant, J. P. Freyer, A. H. Hielscher, A. A. Eick, D. Shen, T. M. Johnson, “Mechanisms of light scattering from biological cells relevant to noninvasive optical-tissue diagnostics,” Appl. Opt. 37, 3586–3593 (1998).
[CrossRef]

J. R. Mourant, A. H. Hielscher, A. A. Eick, T. M. Johnson, J. P. Freyer, “Evidence of intrinsic differences in the light scattering properties of tumorigenic and nontumorigenic cells,” Cancer (Cancer Cytopathol) 84, 366–374 (1998).

Fuselier, T.

Graaff, R.

Greve, J.

Hielscher, A. H.

J. R. Mourant, A. H. Hielscher, A. A. Eick, T. M. Johnson, J. P. Freyer, “Evidence of intrinsic differences in the light scattering properties of tumorigenic and nontumorigenic cells,” Cancer (Cancer Cytopathol) 84, 366–374 (1998).

J. R. Mourant, J. P. Freyer, A. H. Hielscher, A. A. Eick, D. Shen, T. M. Johnson, “Mechanisms of light scattering from biological cells relevant to noninvasive optical-tissue diagnostics,” Appl. Opt. 37, 3586–3593 (1998).
[CrossRef]

Jiang, H.

Johnson, T. M.

Kao, J.

Kimura, M.

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

Koelink, M. H.

Liu, D. L.

Liu, H.

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

Marquez, G.

Mourant, J. R.

Nilsson, A. M. K.

Pierce, J.

Sevick-Muraca, E.

Shen, D.

Sloot, P. M. A.

Sturesson, C.

ten Bosch, J. J.

J. R. Zijp, J. J. ten Bosch, “Pascal program to perform Mie calculations,” Opt. Eng. 32, 1691–1695 (1993).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Wang, L. V.

Zijp, J. R.

Appl. Opt. (5)

Cancer (Cancer Cytopathol) (1)

J. R. Mourant, A. H. Hielscher, A. A. Eick, T. M. Johnson, J. P. Freyer, “Evidence of intrinsic differences in the light scattering properties of tumorigenic and nontumorigenic cells,” Cancer (Cancer Cytopathol) 84, 366–374 (1998).

J. Biomed. Opt. (1)

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

Opt. Eng. (1)

J. R. Zijp, J. J. ten Bosch, “Pascal program to perform Mie calculations,” Opt. Eng. 32, 1691–1695 (1993).
[CrossRef]

Opt. Lett. (1)

Other (1)

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

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Figures (4)

Fig. 1
Fig. 1

Relationship between slope, n, and particle radius in the range of 0.05–1 μm with 96 data points. The slope values were obtained from plots of ln(Qsca) versus ln(wavelength). Consistent results between the R-G approximation (open circles) and Mie theory (filled squares) are shown.

Fig. 2
Fig. 2

Example of the ln(Qsca) versus ln(wavelength) is plotted by the solid curve for r = 0.6 μm with two linearly fitted lines in the two wavelength ranges of 600–850 nm (solid gray line) and 600–1012 nm (dashed black line), respectively. For clarity, a second x axis was added at the bottom of the figure to show the corresponding wavelength values in nanometers.

Fig. 3
Fig. 3

Enlarged portion of Fig. 1, emphasizing the sinusoidal feature. The dark horizontal line demonstrates that one slope value can lead to nine different radii.

Fig. 4
Fig. 4

Relationship between slope (n) versus particle radius (r), comparing data calculated from Eq. (5) (solid black line), our R-G approximation (open diamonds), and Mie theory (black circles).

Equations (8)

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μs=ρQsca1-gπr2=ρQscaπr2.
μsλμsλ0=ρQscaλπr2ρQscaλ0πr2=QscaλQscaλ0.
lnμs=lnk+n lnλ.
n=-1109.5r3+341.67r2-9.369r-3.9359, for r<0.23 μm,
n=23.909r3-37.218r2+19.534r-3.965, for 0.23r0.6 μm.
σsλ=9πr264x2m2-1m2+220πsin u-u cos u2×1+cos2 θsin θ1-cos θsin6θ/2dθ =9256πλnex2m2-1m2+220πsin u-u cos u2×1+cos2 θsin θ1-cos θsin6θ/2dθ,
u=2x sinθ2=22πrnexλsinθ2,
Qscaλ=σsλπr2.

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