Abstract

The noninvasive measurement of variations in absorption that are due to changes in concentrations of biochemically relevant compounds in tissue is important in many clinical settings. One problem with such measurements is that the path length traveled by the collected light through the tissue depends on the scattering properties of the tissue. We demonstrate, using both Monte Carlo simulations and experimental measurements, that for an appropriate separation between light-delivery and light-collection fibers the path length of the collected photons does not depend on scattering parameters for the range of parameters typically found in tissue. This is important for developing rapid, noninvasive, and inexpensive methods for measuring absorption changes in tissue.

© 1997 Optical Society of America

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References

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  1. J. L. Pujol, D. Cupissol, C. Gestinboyer, J. Bres, B. Serrou, F. B. Michel, “Tumor-tissue and plasma concentrations of platinum during chemotherapy of non-small-cell lung cancer patients,” Cancer Chemother. Pharmacol. 27, 72–75 (1990).
    [Crossref] [PubMed]
  2. M. G. Donelli, M. Zucchetti, M. Dincalci, “Do anticancer agents reach the tumor target in the human brain?” Cancer Chemother. Pharmacol. 30, 251–260 (1992).
    [Crossref] [PubMed]
  3. L. Stahle, “Microdialysis in pharmocokinetics,” Eur. J. Drug Metab. Pharmokinet. 18, 89–96 (1993).
    [Crossref]
  4. D. J. Kerr, G. Los, “Pharmacokinetic principles of locoregional chemotherapy,” Cancer Surv. 17, 105–122 (1993).
    [PubMed]
  5. S. Fantini, M. A. Francescini, J. B. Fishkin, B. Barbieri, “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]
  6. 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]
  7. B. C. Wilson, “Measurement of tissue optical properties: methods and theories,” in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995), pp. 233–274.
    [Crossref]
  8. 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]
  9. A. Kienle, L. Ligle, 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]
  10. P. Marquet, F. Bevilacqua, C. Depeursinge, E. B. Haller, “Determination of reduced scattering and absorption coefficients by a single charge-coupled-device array measurement, part I: comparison between experiments and simulations,” Opt. Eng. 34, 2055–2063 (1995).
    [Crossref]
  11. J. R. Mourant, J. Boyer, A. H. Hielscher, I. J. Bigio, “Influence of the scattering phase function on light transport measurements in turbid media performed with small source–detector separations,” Opt. Lett. 21, 546–548 (1996).
    [Crossref] [PubMed]
  12. J. R. Mourant, T. Fuselier, J. Boyer, T. M. 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]
  13. H. J. van Staveren, C. J. M. Moes, J. van Marle, S. A. Prahl, M. J. C. Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400–1100 nm,” Appl. Opt. 30, 4507–4514 (1991).
    [Crossref] [PubMed]
  14. M. S. Patterson, B. C. Wilson, J. W. Feather, D. M. Burns, W. Pushka, “The measurement of dihemaoporphyrin ether concentration in tissue by reflectance spectrophotometry,” Photochem. Photobiol. 46, 337–343 (1987).
    [Crossref] [PubMed]
  15. C. E. Cooper, S. J. Matcher, J. S. Wyatt, M. Cope, G. C. Brown, E. M. Nemota, D. P. Delpy, “Near-infrared spectroscopy of the brain: relevance to cytochrome oxidase bioenergetics,” Biochem. Soc. Trans. 22, 974–980 (1994).
    [PubMed]

1997 (1)

1996 (2)

1995 (1)

P. Marquet, F. Bevilacqua, C. Depeursinge, E. B. Haller, “Determination of reduced scattering and absorption coefficients by a single charge-coupled-device array measurement, part I: comparison between experiments and simulations,” Opt. Eng. 34, 2055–2063 (1995).
[Crossref]

1994 (3)

S. Fantini, M. A. Francescini, J. B. Fishkin, B. Barbieri, “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]

C. E. Cooper, S. J. Matcher, J. S. Wyatt, M. Cope, G. C. Brown, E. M. Nemota, D. P. Delpy, “Near-infrared spectroscopy of the brain: relevance to cytochrome oxidase bioenergetics,” Biochem. Soc. Trans. 22, 974–980 (1994).
[PubMed]

1993 (2)

L. Stahle, “Microdialysis in pharmocokinetics,” Eur. J. Drug Metab. Pharmokinet. 18, 89–96 (1993).
[Crossref]

D. J. Kerr, G. Los, “Pharmacokinetic principles of locoregional chemotherapy,” Cancer Surv. 17, 105–122 (1993).
[PubMed]

1992 (2)

M. G. Donelli, M. Zucchetti, M. Dincalci, “Do anticancer agents reach the tumor target in the human brain?” Cancer Chemother. Pharmacol. 30, 251–260 (1992).
[Crossref] [PubMed]

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)

1990 (1)

J. L. Pujol, D. Cupissol, C. Gestinboyer, J. Bres, B. Serrou, F. B. Michel, “Tumor-tissue and plasma concentrations of platinum during chemotherapy of non-small-cell lung cancer patients,” Cancer Chemother. Pharmacol. 27, 72–75 (1990).
[Crossref] [PubMed]

1987 (1)

M. S. Patterson, B. C. Wilson, J. W. Feather, D. M. Burns, W. Pushka, “The measurement of dihemaoporphyrin ether concentration in tissue by reflectance spectrophotometry,” Photochem. Photobiol. 46, 337–343 (1987).
[Crossref] [PubMed]

Barbieri, B.

Bevilacqua, F.

P. Marquet, F. Bevilacqua, C. Depeursinge, E. B. Haller, “Determination of reduced scattering and absorption coefficients by a single charge-coupled-device array measurement, part I: comparison between experiments and simulations,” Opt. Eng. 34, 2055–2063 (1995).
[Crossref]

Bigio, I. J.

Boyer, J.

Bres, J.

J. L. Pujol, D. Cupissol, C. Gestinboyer, J. Bres, B. Serrou, F. B. Michel, “Tumor-tissue and plasma concentrations of platinum during chemotherapy of non-small-cell lung cancer patients,” Cancer Chemother. Pharmacol. 27, 72–75 (1990).
[Crossref] [PubMed]

Brown, G. C.

C. E. Cooper, S. J. Matcher, J. S. Wyatt, M. Cope, G. C. Brown, E. M. Nemota, D. P. Delpy, “Near-infrared spectroscopy of the brain: relevance to cytochrome oxidase bioenergetics,” Biochem. Soc. Trans. 22, 974–980 (1994).
[PubMed]

Burns, D. M.

M. S. Patterson, B. C. Wilson, J. W. Feather, D. M. Burns, W. Pushka, “The measurement of dihemaoporphyrin ether concentration in tissue by reflectance spectrophotometry,” Photochem. Photobiol. 46, 337–343 (1987).
[Crossref] [PubMed]

Cooper, C. E.

C. E. Cooper, S. J. Matcher, J. S. Wyatt, M. Cope, G. C. Brown, E. M. Nemota, D. P. Delpy, “Near-infrared spectroscopy of the brain: relevance to cytochrome oxidase bioenergetics,” Biochem. Soc. Trans. 22, 974–980 (1994).
[PubMed]

Cope, M.

C. E. Cooper, S. J. Matcher, J. S. Wyatt, M. Cope, G. C. Brown, E. M. Nemota, D. P. Delpy, “Near-infrared spectroscopy of the brain: relevance to cytochrome oxidase bioenergetics,” Biochem. Soc. Trans. 22, 974–980 (1994).
[PubMed]

Cupissol, D.

J. L. Pujol, D. Cupissol, C. Gestinboyer, J. Bres, B. Serrou, F. B. Michel, “Tumor-tissue and plasma concentrations of platinum during chemotherapy of non-small-cell lung cancer patients,” Cancer Chemother. Pharmacol. 27, 72–75 (1990).
[Crossref] [PubMed]

Delpy, D. P.

C. E. Cooper, S. J. Matcher, J. S. Wyatt, M. Cope, G. C. Brown, E. M. Nemota, D. P. Delpy, “Near-infrared spectroscopy of the brain: relevance to cytochrome oxidase bioenergetics,” Biochem. Soc. Trans. 22, 974–980 (1994).
[PubMed]

Depeursinge, C.

P. Marquet, F. Bevilacqua, C. Depeursinge, E. B. Haller, “Determination of reduced scattering and absorption coefficients by a single charge-coupled-device array measurement, part I: comparison between experiments and simulations,” Opt. Eng. 34, 2055–2063 (1995).
[Crossref]

Dincalci, M.

M. G. Donelli, M. Zucchetti, M. Dincalci, “Do anticancer agents reach the tumor target in the human brain?” Cancer Chemother. Pharmacol. 30, 251–260 (1992).
[Crossref] [PubMed]

Donelli, M. G.

M. G. Donelli, M. Zucchetti, M. Dincalci, “Do anticancer agents reach the tumor target in the human brain?” Cancer Chemother. Pharmacol. 30, 251–260 (1992).
[Crossref] [PubMed]

Fantini, S.

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]

Feather, J. W.

M. S. Patterson, B. C. Wilson, J. W. Feather, D. M. Burns, W. Pushka, “The measurement of dihemaoporphyrin ether concentration in tissue by reflectance spectrophotometry,” Photochem. Photobiol. 46, 337–343 (1987).
[Crossref] [PubMed]

Fishkin, J. B.

Francescini, M. A.

Fuselier, T.

Gemert, M. J. C.

Gestinboyer, C.

J. L. Pujol, D. Cupissol, C. Gestinboyer, J. Bres, B. Serrou, F. B. Michel, “Tumor-tissue and plasma concentrations of platinum during chemotherapy of non-small-cell lung cancer patients,” Cancer Chemother. Pharmacol. 27, 72–75 (1990).
[Crossref] [PubMed]

Haller, E. B.

P. Marquet, F. Bevilacqua, C. Depeursinge, E. B. Haller, “Determination of reduced scattering and absorption coefficients by a single charge-coupled-device array measurement, part I: comparison between experiments and simulations,” Opt. Eng. 34, 2055–2063 (1995).
[Crossref]

Hibst, R.

Hielscher, A. H.

Johnson, T. M.

Kerr, D. J.

D. J. Kerr, G. Los, “Pharmacokinetic principles of locoregional chemotherapy,” Cancer Surv. 17, 105–122 (1993).
[PubMed]

Kienle, A.

Ligle, L.

Los, G.

D. J. Kerr, G. Los, “Pharmacokinetic principles of locoregional chemotherapy,” Cancer Surv. 17, 105–122 (1993).
[PubMed]

Marquet, P.

P. Marquet, F. Bevilacqua, C. Depeursinge, E. B. Haller, “Determination of reduced scattering and absorption coefficients by a single charge-coupled-device array measurement, part I: comparison between experiments and simulations,” Opt. Eng. 34, 2055–2063 (1995).
[Crossref]

Matcher, S. J.

C. E. Cooper, S. J. Matcher, J. S. Wyatt, M. Cope, G. C. Brown, E. M. Nemota, D. P. Delpy, “Near-infrared spectroscopy of the brain: relevance to cytochrome oxidase bioenergetics,” Biochem. Soc. Trans. 22, 974–980 (1994).
[PubMed]

Michel, F. B.

J. L. Pujol, D. Cupissol, C. Gestinboyer, J. Bres, B. Serrou, F. B. Michel, “Tumor-tissue and plasma concentrations of platinum during chemotherapy of non-small-cell lung cancer patients,” Cancer Chemother. Pharmacol. 27, 72–75 (1990).
[Crossref] [PubMed]

Moes, C. J. M.

Mourant, J. R.

Nemota, E. M.

C. E. Cooper, S. J. Matcher, J. S. Wyatt, M. Cope, G. C. Brown, E. M. Nemota, D. P. Delpy, “Near-infrared spectroscopy of the brain: relevance to cytochrome oxidase bioenergetics,” Biochem. Soc. Trans. 22, 974–980 (1994).
[PubMed]

Patterson, M. S.

A. Kienle, L. Ligle, 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, “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. C. Wilson, J. W. Feather, D. M. Burns, W. Pushka, “The measurement of dihemaoporphyrin ether concentration in tissue by reflectance spectrophotometry,” Photochem. Photobiol. 46, 337–343 (1987).
[Crossref] [PubMed]

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]

Prahl, S. A.

Pujol, J. L.

J. L. Pujol, D. Cupissol, C. Gestinboyer, J. Bres, B. Serrou, F. B. Michel, “Tumor-tissue and plasma concentrations of platinum during chemotherapy of non-small-cell lung cancer patients,” Cancer Chemother. Pharmacol. 27, 72–75 (1990).
[Crossref] [PubMed]

Pushka, W.

M. S. Patterson, B. C. Wilson, J. W. Feather, D. M. Burns, W. Pushka, “The measurement of dihemaoporphyrin ether concentration in tissue by reflectance spectrophotometry,” Photochem. Photobiol. 46, 337–343 (1987).
[Crossref] [PubMed]

Serrou, B.

J. L. Pujol, D. Cupissol, C. Gestinboyer, J. Bres, B. Serrou, F. B. Michel, “Tumor-tissue and plasma concentrations of platinum during chemotherapy of non-small-cell lung cancer patients,” Cancer Chemother. Pharmacol. 27, 72–75 (1990).
[Crossref] [PubMed]

Stahle, L.

L. Stahle, “Microdialysis in pharmocokinetics,” Eur. J. Drug Metab. Pharmokinet. 18, 89–96 (1993).
[Crossref]

Steiner, R.

van Marle, J.

van Staveren, H. J.

Wilson, B. C.

A. Kienle, L. Ligle, 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]

M. S. Patterson, B. C. Wilson, J. W. Feather, D. M. Burns, W. Pushka, “The measurement of dihemaoporphyrin ether concentration in tissue by reflectance spectrophotometry,” Photochem. Photobiol. 46, 337–343 (1987).
[Crossref] [PubMed]

B. C. Wilson, “Measurement of tissue optical properties: methods and theories,” in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995), pp. 233–274.
[Crossref]

Wyatt, J. S.

C. E. Cooper, S. J. Matcher, J. S. Wyatt, M. Cope, G. C. Brown, E. M. Nemota, D. P. Delpy, “Near-infrared spectroscopy of the brain: relevance to cytochrome oxidase bioenergetics,” Biochem. Soc. Trans. 22, 974–980 (1994).
[PubMed]

Zucchetti, M.

M. G. Donelli, M. Zucchetti, M. Dincalci, “Do anticancer agents reach the tumor target in the human brain?” Cancer Chemother. Pharmacol. 30, 251–260 (1992).
[Crossref] [PubMed]

Appl. Opt. (4)

Biochem. Soc. Trans. (1)

C. E. Cooper, S. J. Matcher, J. S. Wyatt, M. Cope, G. C. Brown, E. M. Nemota, D. P. Delpy, “Near-infrared spectroscopy of the brain: relevance to cytochrome oxidase bioenergetics,” Biochem. Soc. Trans. 22, 974–980 (1994).
[PubMed]

Cancer Chemother. Pharmacol. (2)

J. L. Pujol, D. Cupissol, C. Gestinboyer, J. Bres, B. Serrou, F. B. Michel, “Tumor-tissue and plasma concentrations of platinum during chemotherapy of non-small-cell lung cancer patients,” Cancer Chemother. Pharmacol. 27, 72–75 (1990).
[Crossref] [PubMed]

M. G. Donelli, M. Zucchetti, M. Dincalci, “Do anticancer agents reach the tumor target in the human brain?” Cancer Chemother. Pharmacol. 30, 251–260 (1992).
[Crossref] [PubMed]

Cancer Surv. (1)

D. J. Kerr, G. Los, “Pharmacokinetic principles of locoregional chemotherapy,” Cancer Surv. 17, 105–122 (1993).
[PubMed]

Eur. J. Drug Metab. Pharmokinet. (1)

L. Stahle, “Microdialysis in pharmocokinetics,” Eur. J. Drug Metab. Pharmokinet. 18, 89–96 (1993).
[Crossref]

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]

Opt. Eng. (1)

P. Marquet, F. Bevilacqua, C. Depeursinge, E. B. Haller, “Determination of reduced scattering and absorption coefficients by a single charge-coupled-device array measurement, part I: comparison between experiments and simulations,” Opt. Eng. 34, 2055–2063 (1995).
[Crossref]

Opt. Lett. (1)

Photochem. Photobiol. (1)

M. S. Patterson, B. C. Wilson, J. W. Feather, D. M. Burns, W. Pushka, “The measurement of dihemaoporphyrin ether concentration in tissue by reflectance spectrophotometry,” Photochem. Photobiol. 46, 337–343 (1987).
[Crossref] [PubMed]

Phys. Med. Biol. (1)

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]

Other (1)

B. C. Wilson, “Measurement of tissue optical properties: methods and theories,” in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995), pp. 233–274.
[Crossref]

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Figures (9)

Fig. 1
Fig. 1

Average path lengths of photons traveling from a delivery fiber to a collection fiber (in a backscatter geometry such as that shown in Fig. 1) as calculated by Monte Carlo simulations. Results are shown for three sets of source- and detector–fiber center-to-center separations, d, and values of μa. For d = 1.75 and μa = 0.5 cm−1 the percent difference in path lengths is 9.5%. For d = 1.75 and μa = 0.1 cm−1 the difference in path lengths is ~16%. For d = 3.0 and μa = 0.1 cm−1 the percent difference in path lengths is 30%. The error bars are smaller than the symbols and were determined by several simulations with different numbers of incident photons. Simulations in which more than 400 photons were collected all resulted in the same value (to within 0.1%) of the path length. Deviations from this value when smaller numbers of photons were collected were used to estimate the errors.

Fig. 2
Fig. 2

Schematic of the experimental setup.

Fig. 3
Fig. 3

Spectra taken after the addition of Direct Blue dye, which have been divided by a spectrum with no blue dye, normalized to 1 at 800 nm, and with their negative natural log taken. (The source–detector separation was 1.42 mm. The scattering media was 10% Intralipid-10%.) The values of μa given in the figure caption are the average absorption coefficient of the scattering solution from 575 to 595 nm that is due to the addition of the Direct Blue dye.

Fig. 4
Fig. 4

(a) Areas under the curve from 575 to 595 nm of analyzed spectra plotted versus the average absorption coefficient of the blue dye from 575 to 595 nm; dye was added to the scattering solutions for a source–detector separation of 1.42 mm. (b) Same for a fiber separation of 2.98 mm.

Fig. 5
Fig. 5

Absorption spectra of India ink and blue dye.

Fig. 6
Fig. 6

Percent difference in the largest and smallest areas under the curve from 575 to 595 nm of the analyzed spectra plotted versus the source–detector fiber separation. Analyzed spectra were used for which the amount of added Direct Blue dye had an absorption of 0.15 cm−1 at 585 nm. The three scattering suspensions composed of 0.2%, 0.4%, and 0.6% 0.890-µm-diameter spheres had reduced scattering coefficients of ~6, ~12, and ~18 cm−1 at 650 nm, where g = 0.91. The three scattering suspensions composed of 5%, 10%, and 15% Intralipid-10% had reduced scattering coefficients of 5, 10, and 16 cm−1 at 650 nm, where g = 0.79.

Fig. 7
Fig. 7

Percent difference in the largest and the smallest areas under the curve from 575 to 595 nm of the analyzed spectra for the three scattering solutions composed of 0.2%, 0.4%, and 0.6% by weight 0.913-µm-diameter polystyrene spheres plotted versus the source–detector fiber separation. Analyzed spectra were used for which the amount of added Direct Blue dye had an absorption of 0.15 cm−1 at 585 nm. Data are given for three different values of μa.

Fig. 8
Fig. 8

Best values of the area of analyzed spectra from 575 to 595 nm versus amount of added absorber for a fiber separation of 1.7 mm for three different values of background absorption. These curves were obtained by fitting raw data [similar to that shown in Fig. 4(a)] to a third-order polynomial.

Fig. 9
Fig. 9

Areas under the curve from 575 to 595 nm of analyzed spectra plotted versus the average absorption coefficient of the blue dye from 575 to 595 nm; dye was added to the scattering solutions for a source–detector separation of 1.7 mm. The scattering media was 10% Intralipid-10% and the background μa ~ 0. The non-linearity of the curve for small values of μa and the linearity of the curve at larger values of μa have important consequences for how accurately the background absorbance must be known.

Tables (1)

Tables Icon

Table 1 Results of Measurements of Absorption Changes due to the Addition of Direct Blue Dye

Equations (4)

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

I = I o exp ( μ a L ) .
μ a ( λ 1 ) μ a ( λ 2 ) = ln I ( λ 1 ) / I o ( λ 1 ) I ( λ 2 ) / I o ( λ 2 ) L .
percent variation = A max A min A min × 100.
ln I I o = ln i = 1 I i   exp ( μ a L i ) I o c μ a L ,   i = 1 I i < I o .

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