Abstract

In spite of many progresses achieved both with theories and with experiments in studying light propagation through diffusive media, a reliable method for accurate measurements of the optical properties of diffusive media at NIR wavelengths is, in our opinion, still missing. It is therefore difficult to create a diffusive medium with well known optical properties to be used as a reference. In this paper we describe a method to calibrate the reduced scattering coefficient, μ́s, of a liquid diffusive medium and the absorption coefficient, μa, of an absorbing medium with a standard error smaller than 2% both on μ́s and on μa. The method is based on multidistance measurements of fluence into an infinite medium illuminated by a CW source. The optical properties are retrieved with simple inversion procedures (linear fits) exploiting the knowledge of the absorption coefficient of the liquid into which the diffuser and the absorber are dispersed. In this study Intralipid diluted in water has been used as diffusive medium and Indian ink as absorber. For a full characterization of these media measurements of collimated transmittance have also been carried out, from which the asymmetry factor of the scattering function of Intralipid and the single scattering albedo of Indian ink have been determined.

© 2007 Optical Society of America

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  1. B. W. Pogue and M. S. Patterson, "Review of tissue simulating phantoms for optical spectroscopy, imaging and dosimetry," J. Biomed. Opt. 11,041102 (2006).
  2. A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, R. Cubeddu, H. Wabnitz, D. Grosenick, M. Mller, R. Macdonald, J. Swartling, T. Svensson, S. Andersson-Engels, R. L. P. van Veen, H. J. C. M. Sterenborg, J. -M. Tualle, H. L. Nghiem, S. Avrillier, M. Whelan, and H. Stamm, "Performance assessment of photon migration instruments: the MEDPHOT protocol," Appl. Opt. 44,2104-2114 (2005).
    [CrossRef] [PubMed]
  3. A. Dimofte, J. C. Finlay, and T. C. Zhu, "A method for determination of the absorption and scattering properties interstitially in turbid media," Phys. Med. Biol. 50,2291-2311 (2005).
    [CrossRef] [PubMed]
  4. H. Xu and M. S. Patterson, "Determination of the optical properties of tissue-simulating phantoms from interstitial frequency domain measurements of relative fluence and phase difference," Opt. Express 14,6485-6501 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-14-6485.
    [CrossRef] [PubMed]
  5. C. Chen, J. Q. Lu, H. Ding, K. M. Jacobs, Y. Du, and X. -H. Hu, "A primary method for determination of optical parameters of turbid samples and application to intralipid between 550 and 1630nm, " Opt. Express 14,7420-7435 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-16-7420.
    [CrossRef] [PubMed]
  6. F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, "Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation," Phys. Med. Biol. 45,1359-1373 (2000).
    [CrossRef] [PubMed]
  7. B. C. Wilson, M. S. Patterson, and D. M. Burns, "Effect of photosensitizer concentration in tissue on the penetration depth of photoactivating light," Lasers Med. Sci. 1,235-244 (1986).
    [CrossRef]
  8. H. J. van Staveren, C. J. M. Moes, J. van Marle, S. A. Prahl, and M. J. C. van Gemert, "Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm," Appl. Opt. 31,4507-1514 (1991).
    [CrossRef]
  9. G. Zaccanti, S. Del Bianco, and F. Martelli, "Measurements of optical properties of high density media," Appl. Opt. 42,4023-4030 (2003).
    [CrossRef] [PubMed]
  10. J. C. Hebden, R. Arridge, and D. T. Delpy, "Optical imaging in medicine: I.Experimental techniques," Phys. Med. Biol. 42,825-840 (1997).
    [CrossRef] [PubMed]
  11. R. Cubeddu, C. D’andrea, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, C. Dovar, D. Johnson, M. Ruiz-Altisent, and C. Valero, "Nondestructive quantification of chemical and physical properties of fruits by time-resolved reflectance spectroscopy in the wavelength range 650-1000 nm," Appl. Opt. 40,538-543 (2001).
    [CrossRef]
  12. S. Prahl, "Optical absorption of water," http://omlc.ogi.edu/spectra/water/index.html.
  13. R. M. Pope, "Optical absorption of pure water and sea water using the integrating cavity absorption meter," Phd Thesis, 1993, Texas A&M University.
  14. H. Buiteveld, J. M. H. Hakvoort, M. Donze, "The optical properties of pure water," in Ocean Optics XII, J. S. Jaffe ed.,Proc. SPIE 2258,174-183 (1994).
    [CrossRef]
  15. L. Kou, D. Labrie, and P. Chylek, "Refractive indices of water and ice in the 0.65-2.5 m spectral range," Appl. Opt. 32,3531-3540 (1993).
    [CrossRef] [PubMed]
  16. S. Fantini, M. A. Franceschini, and E. Gratton, "Effective source term in the diffusion equation for photon transport in turbid media," Appl. Opt. 36,156-163 (1997).
    [CrossRef] [PubMed]
  17. M. Bassani, Limits of validity of the diffusion equation and methodologies for measuring optical properties of highly scattering media, (in Italian) M. S. Thesis, University of Florence, Italy (1997).
    [PubMed]
  18. R.L.P. van Veen and H.J.C.M. Sterenborg, A. Pifferi, A. Torricelli, and R. Cubeddu, "Determination of VIS- NIR absorption coefficients of mammalian fat, with time-and spatially resolved diffuse reflectance and transmission spectroscopy," OSA Annual BIOMED Topical Meeting, 2004.
  19. S. Jacques, "Optical properties of Intralipid, an aqueous suspension of lipid droplets," http://omlc.ogi.edu/spectra/intralipid/index.html
  20. L. Spinelli, F. Martelli, A. Pifferi, A. Torricelli, R. Cubeddu, and G. Zaccanti, "Calibration of scattering and absorption properties of a liquid diffusive medium at NIR wavelengths. Time resolved method," In preparation.
  21. S. J. Madsen, M. S. Patterson, and B. C. Wilson, "The use of India ink as an optical absorber in tissue-simulating phantoms," Phys. Med. Biol. 37,985-993 (1992).
    [CrossRef] [PubMed]
  22. D. Contini, F. Martelli, and G. Zaccanti, "Photon migration through a turbid slab described by a model based on diffusion approximation. I) Theory," Appl. Opt. 36,4587-4599 (1997).
    [CrossRef] [PubMed]
  23. L. Rovati, A. Bandera, M. Donini, G. Salvatori, and L. Pollonini, "Design and performance of a wide-bandwidth and sensitive instrument for near-infrared spectroscopic measurements on human tissue," Rev. Sci. Instrum. 75,5315-5325 (2004).
    [CrossRef]

2006

2005

2004

L. Rovati, A. Bandera, M. Donini, G. Salvatori, and L. Pollonini, "Design and performance of a wide-bandwidth and sensitive instrument for near-infrared spectroscopic measurements on human tissue," Rev. Sci. Instrum. 75,5315-5325 (2004).
[CrossRef]

2003

2001

2000

F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, "Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation," Phys. Med. Biol. 45,1359-1373 (2000).
[CrossRef] [PubMed]

1997

1994

H. Buiteveld, J. M. H. Hakvoort, M. Donze, "The optical properties of pure water," in Ocean Optics XII, J. S. Jaffe ed.,Proc. SPIE 2258,174-183 (1994).
[CrossRef]

1993

1992

S. J. Madsen, M. S. Patterson, and B. C. Wilson, "The use of India ink as an optical absorber in tissue-simulating phantoms," Phys. Med. Biol. 37,985-993 (1992).
[CrossRef] [PubMed]

1991

H. J. van Staveren, C. J. M. Moes, J. van Marle, S. A. Prahl, and M. J. C. van Gemert, "Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm," Appl. Opt. 31,4507-1514 (1991).
[CrossRef]

1986

B. C. Wilson, M. S. Patterson, and D. M. Burns, "Effect of photosensitizer concentration in tissue on the penetration depth of photoactivating light," Lasers Med. Sci. 1,235-244 (1986).
[CrossRef]

Alianelli, L.

F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, "Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation," Phys. Med. Biol. 45,1359-1373 (2000).
[CrossRef] [PubMed]

Andersson-Engels, S.

Arridge, R.

J. C. Hebden, R. Arridge, and D. T. Delpy, "Optical imaging in medicine: I.Experimental techniques," Phys. Med. Biol. 42,825-840 (1997).
[CrossRef] [PubMed]

Avrillier, S.

Bandera, A.

L. Rovati, A. Bandera, M. Donini, G. Salvatori, and L. Pollonini, "Design and performance of a wide-bandwidth and sensitive instrument for near-infrared spectroscopic measurements on human tissue," Rev. Sci. Instrum. 75,5315-5325 (2004).
[CrossRef]

Bassani, M.

F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, "Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation," Phys. Med. Biol. 45,1359-1373 (2000).
[CrossRef] [PubMed]

Bassi, A.

Buiteveld, H.

H. Buiteveld, J. M. H. Hakvoort, M. Donze, "The optical properties of pure water," in Ocean Optics XII, J. S. Jaffe ed.,Proc. SPIE 2258,174-183 (1994).
[CrossRef]

Burns, D. M.

B. C. Wilson, M. S. Patterson, and D. M. Burns, "Effect of photosensitizer concentration in tissue on the penetration depth of photoactivating light," Lasers Med. Sci. 1,235-244 (1986).
[CrossRef]

Chen, C.

Chylek, P.

Contini, D.

Cubeddu, R.

Del Bianco, S.

Delpy, D. T.

J. C. Hebden, R. Arridge, and D. T. Delpy, "Optical imaging in medicine: I.Experimental techniques," Phys. Med. Biol. 42,825-840 (1997).
[CrossRef] [PubMed]

Dimofte, A.

A. Dimofte, J. C. Finlay, and T. C. Zhu, "A method for determination of the absorption and scattering properties interstitially in turbid media," Phys. Med. Biol. 50,2291-2311 (2005).
[CrossRef] [PubMed]

Ding, H.

Donini, M.

L. Rovati, A. Bandera, M. Donini, G. Salvatori, and L. Pollonini, "Design and performance of a wide-bandwidth and sensitive instrument for near-infrared spectroscopic measurements on human tissue," Rev. Sci. Instrum. 75,5315-5325 (2004).
[CrossRef]

Donze, M.

H. Buiteveld, J. M. H. Hakvoort, M. Donze, "The optical properties of pure water," in Ocean Optics XII, J. S. Jaffe ed.,Proc. SPIE 2258,174-183 (1994).
[CrossRef]

Du, Y.

Fantini, S.

Finlay, J. C.

A. Dimofte, J. C. Finlay, and T. C. Zhu, "A method for determination of the absorption and scattering properties interstitially in turbid media," Phys. Med. Biol. 50,2291-2311 (2005).
[CrossRef] [PubMed]

Franceschini, M. A.

Gratton, E.

Grosenick, D.

Hakvoort, J. M. H.

H. Buiteveld, J. M. H. Hakvoort, M. Donze, "The optical properties of pure water," in Ocean Optics XII, J. S. Jaffe ed.,Proc. SPIE 2258,174-183 (1994).
[CrossRef]

Hebden, J. C.

J. C. Hebden, R. Arridge, and D. T. Delpy, "Optical imaging in medicine: I.Experimental techniques," Phys. Med. Biol. 42,825-840 (1997).
[CrossRef] [PubMed]

Hu, X. -H.

Jacobs, K. M.

Kou, L.

Labrie, D.

Lu, J. Q.

Macdonald, R.

Madsen, S. J.

S. J. Madsen, M. S. Patterson, and B. C. Wilson, "The use of India ink as an optical absorber in tissue-simulating phantoms," Phys. Med. Biol. 37,985-993 (1992).
[CrossRef] [PubMed]

Martelli, F.

Mller, M.

Moes, C. J. M.

H. J. van Staveren, C. J. M. Moes, J. van Marle, S. A. Prahl, and M. J. C. van Gemert, "Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm," Appl. Opt. 31,4507-1514 (1991).
[CrossRef]

Nghiem, H. L.

Patterson, M. S.

H. Xu and M. S. Patterson, "Determination of the optical properties of tissue-simulating phantoms from interstitial frequency domain measurements of relative fluence and phase difference," Opt. Express 14,6485-6501 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-14-6485.
[CrossRef] [PubMed]

B. W. Pogue and M. S. Patterson, "Review of tissue simulating phantoms for optical spectroscopy, imaging and dosimetry," J. Biomed. Opt. 11,041102 (2006).

S. J. Madsen, M. S. Patterson, and B. C. Wilson, "The use of India ink as an optical absorber in tissue-simulating phantoms," Phys. Med. Biol. 37,985-993 (1992).
[CrossRef] [PubMed]

B. C. Wilson, M. S. Patterson, and D. M. Burns, "Effect of photosensitizer concentration in tissue on the penetration depth of photoactivating light," Lasers Med. Sci. 1,235-244 (1986).
[CrossRef]

Pifferi, A.

Pogue, B. W.

B. W. Pogue and M. S. Patterson, "Review of tissue simulating phantoms for optical spectroscopy, imaging and dosimetry," J. Biomed. Opt. 11,041102 (2006).

Pollonini, L.

L. Rovati, A. Bandera, M. Donini, G. Salvatori, and L. Pollonini, "Design and performance of a wide-bandwidth and sensitive instrument for near-infrared spectroscopic measurements on human tissue," Rev. Sci. Instrum. 75,5315-5325 (2004).
[CrossRef]

Prahl, S. A.

H. J. van Staveren, C. J. M. Moes, J. van Marle, S. A. Prahl, and M. J. C. van Gemert, "Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm," Appl. Opt. 31,4507-1514 (1991).
[CrossRef]

Rovati, L.

L. Rovati, A. Bandera, M. Donini, G. Salvatori, and L. Pollonini, "Design and performance of a wide-bandwidth and sensitive instrument for near-infrared spectroscopic measurements on human tissue," Rev. Sci. Instrum. 75,5315-5325 (2004).
[CrossRef]

Salvatori, G.

L. Rovati, A. Bandera, M. Donini, G. Salvatori, and L. Pollonini, "Design and performance of a wide-bandwidth and sensitive instrument for near-infrared spectroscopic measurements on human tissue," Rev. Sci. Instrum. 75,5315-5325 (2004).
[CrossRef]

Stamm, H.

Sterenborg, H. J. C. M.

Svensson, T.

Swartling, J.

Taroni, P.

Torricelli, A.

Tualle, J. -M.

van Gemert, M. J. C.

H. J. van Staveren, C. J. M. Moes, J. van Marle, S. A. Prahl, and M. J. C. van Gemert, "Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm," Appl. Opt. 31,4507-1514 (1991).
[CrossRef]

van Marle, J.

H. J. van Staveren, C. J. M. Moes, J. van Marle, S. A. Prahl, and M. J. C. van Gemert, "Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm," Appl. Opt. 31,4507-1514 (1991).
[CrossRef]

van Staveren, H. J.

H. J. van Staveren, C. J. M. Moes, J. van Marle, S. A. Prahl, and M. J. C. van Gemert, "Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm," Appl. Opt. 31,4507-1514 (1991).
[CrossRef]

van Veen, R. L. P.

Wabnitz, H.

Whelan, M.

Wilson, B. C.

S. J. Madsen, M. S. Patterson, and B. C. Wilson, "The use of India ink as an optical absorber in tissue-simulating phantoms," Phys. Med. Biol. 37,985-993 (1992).
[CrossRef] [PubMed]

B. C. Wilson, M. S. Patterson, and D. M. Burns, "Effect of photosensitizer concentration in tissue on the penetration depth of photoactivating light," Lasers Med. Sci. 1,235-244 (1986).
[CrossRef]

Xu, H.

Zaccanti, G.

Zangheri, L.

F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, "Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation," Phys. Med. Biol. 45,1359-1373 (2000).
[CrossRef] [PubMed]

Zhu, T. C.

A. Dimofte, J. C. Finlay, and T. C. Zhu, "A method for determination of the absorption and scattering properties interstitially in turbid media," Phys. Med. Biol. 50,2291-2311 (2005).
[CrossRef] [PubMed]

Appl. Opt.

H. J. van Staveren, C. J. M. Moes, J. van Marle, S. A. Prahl, and M. J. C. van Gemert, "Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm," Appl. Opt. 31,4507-1514 (1991).
[CrossRef]

S. Fantini, M. A. Franceschini, and E. Gratton, "Effective source term in the diffusion equation for photon transport in turbid media," Appl. Opt. 36,156-163 (1997).
[CrossRef] [PubMed]

D. Contini, F. Martelli, and G. Zaccanti, "Photon migration through a turbid slab described by a model based on diffusion approximation. I) Theory," Appl. Opt. 36,4587-4599 (1997).
[CrossRef] [PubMed]

L. Kou, D. Labrie, and P. Chylek, "Refractive indices of water and ice in the 0.65-2.5 m spectral range," Appl. Opt. 32,3531-3540 (1993).
[CrossRef] [PubMed]

R. Cubeddu, C. D’andrea, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, C. Dovar, D. Johnson, M. Ruiz-Altisent, and C. Valero, "Nondestructive quantification of chemical and physical properties of fruits by time-resolved reflectance spectroscopy in the wavelength range 650-1000 nm," Appl. Opt. 40,538-543 (2001).
[CrossRef]

G. Zaccanti, S. Del Bianco, and F. Martelli, "Measurements of optical properties of high density media," Appl. Opt. 42,4023-4030 (2003).
[CrossRef] [PubMed]

A. Pifferi, A. Torricelli, A. Bassi, P. Taroni, R. Cubeddu, H. Wabnitz, D. Grosenick, M. Mller, R. Macdonald, J. Swartling, T. Svensson, S. Andersson-Engels, R. L. P. van Veen, H. J. C. M. Sterenborg, J. -M. Tualle, H. L. Nghiem, S. Avrillier, M. Whelan, and H. Stamm, "Performance assessment of photon migration instruments: the MEDPHOT protocol," Appl. Opt. 44,2104-2114 (2005).
[CrossRef] [PubMed]

J. Biomed. Opt.

B. W. Pogue and M. S. Patterson, "Review of tissue simulating phantoms for optical spectroscopy, imaging and dosimetry," J. Biomed. Opt. 11,041102 (2006).

Lasers Med. Sci.

B. C. Wilson, M. S. Patterson, and D. M. Burns, "Effect of photosensitizer concentration in tissue on the penetration depth of photoactivating light," Lasers Med. Sci. 1,235-244 (1986).
[CrossRef]

Opt. Express

Phys. Med. Biol.

A. Dimofte, J. C. Finlay, and T. C. Zhu, "A method for determination of the absorption and scattering properties interstitially in turbid media," Phys. Med. Biol. 50,2291-2311 (2005).
[CrossRef] [PubMed]

F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, "Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation," Phys. Med. Biol. 45,1359-1373 (2000).
[CrossRef] [PubMed]

S. J. Madsen, M. S. Patterson, and B. C. Wilson, "The use of India ink as an optical absorber in tissue-simulating phantoms," Phys. Med. Biol. 37,985-993 (1992).
[CrossRef] [PubMed]

J. C. Hebden, R. Arridge, and D. T. Delpy, "Optical imaging in medicine: I.Experimental techniques," Phys. Med. Biol. 42,825-840 (1997).
[CrossRef] [PubMed]

Proc. SPIE

H. Buiteveld, J. M. H. Hakvoort, M. Donze, "The optical properties of pure water," in Ocean Optics XII, J. S. Jaffe ed.,Proc. SPIE 2258,174-183 (1994).
[CrossRef]

Rev. Sci. Instrum.

L. Rovati, A. Bandera, M. Donini, G. Salvatori, and L. Pollonini, "Design and performance of a wide-bandwidth and sensitive instrument for near-infrared spectroscopic measurements on human tissue," Rev. Sci. Instrum. 75,5315-5325 (2004).
[CrossRef]

Other

M. Bassani, Limits of validity of the diffusion equation and methodologies for measuring optical properties of highly scattering media, (in Italian) M. S. Thesis, University of Florence, Italy (1997).
[PubMed]

R.L.P. van Veen and H.J.C.M. Sterenborg, A. Pifferi, A. Torricelli, and R. Cubeddu, "Determination of VIS- NIR absorption coefficients of mammalian fat, with time-and spatially resolved diffuse reflectance and transmission spectroscopy," OSA Annual BIOMED Topical Meeting, 2004.

S. Jacques, "Optical properties of Intralipid, an aqueous suspension of lipid droplets," http://omlc.ogi.edu/spectra/intralipid/index.html

L. Spinelli, F. Martelli, A. Pifferi, A. Torricelli, R. Cubeddu, and G. Zaccanti, "Calibration of scattering and absorption properties of a liquid diffusive medium at NIR wavelengths. Time resolved method," In preparation.

S. Prahl, "Optical absorption of water," http://omlc.ogi.edu/spectra/water/index.html.

R. M. Pope, "Optical absorption of pure water and sea water using the integrating cavity absorption meter," Phd Thesis, 1993, Texas A&M University.

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Figures (4)

Fig. 1.
Fig. 1.

Measurements of μeff : The figure reports ln[rΦ(r)] as a function of the source receiver distance r for three values of Intralipid concentration. Marks: experimental results; continuous lines: linear fits of the results.

Fig. 2.
Fig. 2.

Measurement of ε aH2O: The figure reports the logarithm of the received power as a function of the depth.

Fig. 3.
Fig. 3.

Calibration of Intralipid: The figure reports μ 2 eff (ρ il)/ρ il as a function of volume concentration ρ il. The coefficients ε ail and έ sil are obtained from the slope and the intercept of the straight line that best fits the results.

Fig. 4.
Fig. 4.

Calibration of Indian ink: The figure reports μ 2 eff (ρ ink) as a function of ρ ink. The coefficient ε aink is obtained from the slope of the straight line that best fits the results.

Tables (1)

Tables Icon

Table 1. Summary of the optical properties of Intralipid-20% and of non diluted Indian ink

Equations (15)

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

Φ ( r ) = 3 μ s ́ 4 πr exp ( μ effr r )
ln [ ( r ) ] = μ eff r + ln 3 μ s ́ 4 π
μ a ( ρ il ) = ε a il ρ il + ε a H 2 O ( 1 ρ il )
μ s ́ ( ρ il ) = ε s il ́ ρ il
μ eff 2 ( ρ il ) = 3 ε s il ́ ε a H 2 O ρ il + 3 ε s il ́ ( ε a il ε a H 2 O ) ρ il 2
μ eff 2 ( ρ il ) ρ il = 3 ε s il ́ ε a H 2 O + 3 ε s il ́ ( ε a il ε a H2O ) ρ il
ε s il ́ = I il 3 ε a H 2 O
ε a il = S il 3 ε s il ́ + ε a H2O
μ eff 2 ( ρ ink ) = 3 μ s 0 ́ μ a 0 + 3 μ s 0 ́ ε a ink ρ ink
ε a ink = S ink 3 μ s 0 ́
μ a 0 = I ink 3 μ s 0 ́ .
δ Φ a ( r ) = δ μ a ( 3 μ s ́ 4 π ) 2 V i 1 r 1 r 2 exp [ μ eff ( r 1 + r 2 ) ] d V i
δ Φ a ( r ) Φ ( r ) 3 2 a 2 μ s ́ δ μ a .
δ Φ s ( r ) = 3 ( 4 π ) 2 δ μ s ́ V i 1 + μ eff r 1 r 1 2 1 + μ eff r 2 r 2 2 exp [ μ eff ( r 1 + r 2 ) ] cos ϑ d V i
μ s ́ ( ρ̃ il ) = 127.0 ρ̃ il 205.3 ρ̃ il 2 .

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