Abstract

In this paper, a general method to calibrate the absorption coefficient of an absorber and the reduced scattering coefficient of a liquid diffusive medium, based on time-resolved measurements, is reported. An exhaustive analysis of the error sources affecting the estimation is also performed. The method has been applied with a state-of-the-art time-resolved instrumentation to determine the intrinsic absorption coefficient of Indian ink and the reduced scattering coefficient of Intralipid-20%, with a standard error smaller than 1% and 2%, respectively. Finally, the results have been compared to those retrieved for the same compounds by applying a continuous wave method recently published, obtaining an agreement within the error bars. This fact represents a cross validation of the two independent calibration methods.

©2007 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]

2007 (1)

2006 (4)

2005 (1)

2000 (1)

1999 (1)

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “Noninvasive absorption and scattering spectroscopy of bulk diffusive media: An application to the optical characterization of human breast,” Appl. Phys. Lett. 74, 874 (1999).
[Crossref]

1998 (1)

1997 (2)

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 (1997).
[Crossref] [PubMed]

Y. Nomura, O. Hazeki, and M. Tamura, “Relationship between time-resolved and non-time-resolved Beer-Lambert law in turbid media,” Phys. Med. Biol. 42, 1009 (1997).
[Crossref] [PubMed]

1991 (1)

Andersson-Engels, S.

Avrillier, S.

Bassi, A.

Becker, W.

W. Becker, Advanced Time-Correlated Single Photon Counting Techniques (Spinger-Verlag, Berlin Heidelberg, 2005).
[Crossref]

Chen, C.

Contini, D.

Cubeddu, R.

Ding, H.

Du, Y.

Grosenick, D.

Hasegawa, Y.

Hazeki, O.

Y. Nomura, O. Hazeki, and M. Tamura, “Relationship between time-resolved and non-time-resolved Beer-Lambert law in turbid media,” Phys. Med. Biol. 42, 1009 (1997).
[Crossref] [PubMed]

Hu, X. -H.

Jacobs, K. M.

Lu, J. Q.

Macdonald, R.

Martelli, F.

Möller, M.

Nghiem, H. L.

Nomura, Y.

Y. Nomura, O. Hazeki, and M. Tamura, “Relationship between time-resolved and non-time-resolved Beer-Lambert law in turbid media,” Phys. Med. Biol. 42, 1009 (1997).
[Crossref] [PubMed]

Paglia, F.

Patterson, M. S.

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).
[Crossref] [PubMed]

Sassaroli, A.

Spinelli, L.

Stamm, H.

Sterenborg, H. J. C. M.

Svensson, T.

Swartling, J.

Tamura, M.

Y. Nomura, O. Hazeki, and M. Tamura, “Relationship between time-resolved and non-time-resolved Beer-Lambert law in turbid media,” Phys. Med. Biol. 42, 1009 (1997).
[Crossref] [PubMed]

Taroni, P.

Taylor, J. R.

J. R. Taylor, An Introduction to Error Analysis, The Study of Uncertainties in Physical Measurements (University Science Books, 1982).

Tinet, E.

Torricelli, A.

Tualle, J. M.

Valentini, G.

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “Noninvasive absorption and scattering spectroscopy of bulk diffusive media: An application to the optical characterization of human breast,” Appl. Phys. Lett. 74, 874 (1999).
[Crossref]

Veen, R. L. P. van

Wabnitz, H.

Wang, R. K.

Whelan, M.

Wickramasinghe, Y. A.

Xu, H.

Yamada, Y.

Zaccanti, G.

Appl. Opt. (4)

Appl. Phys. Lett. (1)

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “Noninvasive absorption and scattering spectroscopy of bulk diffusive media: An application to the optical characterization of human breast,” Appl. Phys. Lett. 74, 874 (1999).
[Crossref]

J. Biomed. Opt. (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).
[Crossref] [PubMed]

Opt. Express (4)

Opt. Lett. (1)

Phys. Med. Biol. (1)

Y. Nomura, O. Hazeki, and M. Tamura, “Relationship between time-resolved and non-time-resolved Beer-Lambert law in turbid media,” Phys. Med. Biol. 42, 1009 (1997).
[Crossref] [PubMed]

Other (2)

W. Becker, Advanced Time-Correlated Single Photon Counting Techniques (Spinger-Verlag, Berlin Heidelberg, 2005).
[Crossref]

J. R. Taylor, An Introduction to Error Analysis, The Study of Uncertainties in Physical Measurements (University Science Books, 1982).

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Figures (6)

Fig. 1.
Fig. 1. (a) two instrumental response functions (IRF_1 and IRF_2) and three examples of Temporal Point-Spread Function (TPSF) for different values of absorption. The ratio (Δμa )est/Δμa is shown for IRF_1 (b) and IRF_2 (c) as a function of time. The curves are for the light transmitted through a 40 mm thick slab, with μ′ s = 0.5 mm-1, μ a0 = 0 and for three values of Δμ a . (d) examples of TPSF for different values of source-detector lateral distances and μ′ s . The ratio (μ′ s )est/μ′ s is shown for IRF_1 (e) and IRF_2 (f) as a function of time. The curves are for the light transmitted through a non-absorbing slab, 20 mm thick, for μ′ s = 1, 1.5, 2 mm-1 and for r 0 = 0 and r 1 = 25 mm-1.

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Fig. 2.
Fig. 2. Plot of the IRF (green) and a typical diffused time-resolved curve (red) recorded by our system as a function of the MCA channel of the time-correlated single-photon counting board. The time scale is 2.035 ps per channel. Other parameters for the diffused time-resolved curve are: d = 40.8 mm; μa = 0.00278mm-1; μ′ s = 0.64mm-1; count rate ≈ 500kcps; acquisition time 60 s.

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Fig. 3.
Fig. 3. Plot of numerical simulations showing the ratio (Δμa )est/Δμa as a function of the initial time of the time range adopted for the linear regression in Eq. (5), expressed in percentage of the curve peak, for different values of the absorption increment (mm-1 in the legend). The maximum time is fixed at 0.5% of the curve peak. Other parameters are: d = 40.8 mm, μ a0 = 0 , μ′ s = 0.64 mm-1, r = 0.

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Fig. 4.
Fig. 4. (a) Natural logarithm of the ratio of the measured curves Y ≡ ln(M 0/M), before and after the Indian ink adding, as function of X ≡ t (symbol) and resulting linear best fit (line) according to Eq. (4), for the first five absorption addings. (b) Plot of the estimated variation of absorption as a function of the added Indian ink concentration (symbol). Linear best fit for the first five concentrations of the Indian ink (line).

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Fig. 5.
Fig. 5. Ratio (μ′ s )est/μ′ s as a function of the source-detector lateral distance r, for the values of the reduced scattering coefficients used in the measurements (reported in mm-1 in the legend). The simulations are convolved with the measured IRF. We consider different time range for the linear regression (8), expressed in percentage of the curve peak: (a) 1% in the rising edge, 1% in the tail; (b) 10% in the rising edge, 1% in the tail; (c) 50% in the rising edge, 1% in the tail; (d) 10% in the rising edge, 10% in the tail. Other parameters are: d = 20 mm, μa = 0.01mm-1, r 0 = 0.

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Fig. 6.
Fig. 6. (a) Natural logarithm of the ratio of the measured curves Y = ln(M 0/M) corresponding to r 0 = 0 and r = 25 mm , as function of X ≡ 1/t (symbol) and resulting linear best fit (line) according to Eq. (8), for four values of Intralipid-20% concentration. (b) Plot of the estimated reduced scattering coefficient as a function of the Intralipid-20% concentration (symbol). Also the error bars of μ′ s and the linear best fit (line) are reported.

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Tables (1)

Tables Icon

Table 1. Intrinsic absorption and reduced scattering coefficients, ε aink and ε′ sil for Indian ink and Intralipid-20%, obtained with both the CW [4] and time-resolved procedure at 750 nm.

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Equations (21)

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1 v ∂ t I ( x → , t , s̑ ) + s̑ ∙ ∇ I ( x → , t , s̑ ) = − ( μ s + μ a ) I ( x → , t , s̑ ) + μ s ∫ 4 π d ω ′ p ( s̑ , s̑′ ) I ( x → , t , s̑′ ) + q ( x → , t , s̑ ) ,
M 0 ( x → d , t ) = A 0 S ( x → d , t , μ s ) exp ⁡ ( − μ a 0 vt ) ,
M ( x → d , t ) = A S ( x → d , t , μ s ) exp ⁡ ( − μ a vt ) ,
ln ( M 0 M ) = Δ μ a vt + ln ( A 0 A ) ,
Δ μ a = ε a ρ a .
( 1 v ∂ ∂ t + μ a − D ∇ 2 ) U ( x → , t ) = Q ( x → , t ) ,
M ( r , t ) = A exp ⁡ ( − μ a vt ) exp ⁡ ( − 3 μ s ′ r 2 4 vt ) S ( d , t , μ s ′ ) ,
ln ( M 0 M ) = 3 μ s ′ ( r 2 − r 0 2 ) 4 v 1 t + ln ( A 0 A ) ,
μ s ′ = ε s ′ ρ s ,
M 0 ( x → d , t ) = A 0 S ( x → d , t , μ s ) exp ⁡ [ − μ a 0 v ( t − t 0 ) ] ,
M ( x → d , t ) = A S ( x → d , t , μ s ) exp ⁡ [ − μ a v ( t − t′ 0 ) ] .
ln ( M 0 M ) = Δ μ a vt + ln ( A 0 A ) + v ( μ a 0 t 0 − μ a t 0 ′ ) .
ln ( M 0 M ) = 3 μ s ′ ( r 2 − r 0 2 ) 4 v 1 t + 3 μ s ′ ( r 2 t 0 ′ − r 0 2 t 0 ) 4 v 1 t 2 + ln ( A 0 A ) ,
( Δ μ a ) est ( t ) = 1 vt ln ( T 0 ( r = 0 , t ; μ a 0 , μ s ′ ) T ( r = 0 , t ; μ a 0 + Δ μ a , μ s ′ ) ) .
( μ s ′ ) est ( t ) = 4 vt 3 ( r 1 2 − r 0 2 ) ln ( T 0 ( r 0 , t ; μ a , μ s ′ ) T ( r 1 , t ; μ a , μ s ′ ) ) .
μ s ′ = 4 vs 3 ( r 2 − r 0 2 ) ,
( ε μ s ′ ) r = 2 r 2 + r 0 2 r 2 − r 0 2 σ r .
ε Δ μ a = ε s 2 + ε α 2 ,
ε μ s ′ = ε s 2 + ε α 2 + ε t 0 2 + ( ε′ μ′ s ) r 2 ,
ε a ink = ( 649 ± 3 ) mm − 1 .
ε s il ′ = ( 19.8 ± 0.3 ) mm − 1 .

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