Abstract

The main challenge of noninvasive optical biopsy is to obtain an accurate value of the optical coefficients of an encapsulated organ (muscle, brain, etc.). The idea developed by us is that some interesting information could be deduced from the long-time behavior of the reflectance function. This asymptotic behavior is analyzed for layered media in the framework of the diffusion approximation. A new method is derived to obtain accurate values for the optical parameters of the deepest layers. This method is designed to work in a specific long-time regime that is still within the scope of standard time-of-flight experiments but far from being included in the mathematically defined asymptotic region. The limits of this method, linked to the cases where the asymptotic behavior is no longer governed by the deepest layer, are then discussed.

© 2004 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  6. J.-M. Tualle, J. Prat, E. Tinet, S. Avrillier, “Real-space Green’s function calculation for the solution of diffusion equation in stratified turbid media,” J. Opt. Soc. Am. A 17, 2046–2055 (2000).
    [Crossref]
  7. A. Pifferi, A. Torricelli, P. Taroni, R. Cubeddu, “Reconstruction of absorber concentrations in a two-layer structure by use of multidistance time-resolved reflectance spectroscopy,” Opt. Lett. 26, 1963–1965 (2001).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  12. R. C. Haskell, L. O. Svaasand, T.-T. Tsay, T.-C. Feng, M. S. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
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  14. M. Mochi, G. Pacelli, M. C. Recchioni, F. Zirilli, “Inverse problem for a class of two-dimensional diffusion equations with piecewise constant coefficients,” J. Optim. Theory Appl. 100, 29–57 (1999).
    [Crossref]
  15. E. Tinet, S. Avrillier, J.-M. Tualle, “Fast semi-analytical Monte Carlo simulation for time resolved light propagation in turbid media,” J. Opt. Soc. Am. A 13, 1903–1915 (1996).
    [Crossref]
  16. See http://www.medphys.ucl.ac.uk/research/borg/research/NIR_topics/phantoms.htm .
  17. S. Avrillier, E. Tinet, J.-M. Tualle, F. Costes, F. Revel, J.-P. Ollivier, “Real-time inversion using Monte Carlo results for the determination of absorption coefficients in multilayered tissues: application to noninvasive muscle oximetry (Part 2),” in Diagnostic Optical Spectroscopy in Biomedicine, T. G. Papazoglou, G. A. Wagnieres, eds., Proc. SPIE4432, 75–84 (2001).
    [Crossref]
  18. C. R. Simpson, M. Kohl, M. Essenpreis, M. Cope, “Near-infrared optical properties of ex vivo human skin and subcutaneous tissues measured using the Monte Carlo inversion technique,” Phys. Med. Biol. 43, 2465–2478 (1998).
    [Crossref] [PubMed]

2001 (2)

2000 (3)

1999 (2)

M. Mochi, G. Pacelli, M. C. Recchioni, F. Zirilli, “Inverse problem for a class of two-dimensional diffusion equations with piecewise constant coefficients,” J. Optim. Theory Appl. 100, 29–57 (1999).
[Crossref]

J. Ripoll, M. Nieto-Vesperinas, “Index mismatch for diffuse photon density waves at both flat and rough diffuse–diffuse interfaces,” J. Opt. Soc. Am. A 16, 1947–1957 (1999).
[Crossref]

1998 (3)

1997 (1)

1996 (1)

1995 (1)

1994 (1)

1989 (1)

1986 (1)

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[Crossref] [PubMed]

Akkermans, E.

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[Crossref] [PubMed]

Alexandrakis, G.

Aronson, R.

Avrillier, S.

J.-M. Tualle, E. Tinet, J. Prat, S. Avrillier, “Light propagation near turbid–turbid planar interfaces,” Opt. Commun. 183, 337–346 (2000).
[Crossref]

J.-M. Tualle, J. Prat, E. Tinet, S. Avrillier, “Real-space Green’s function calculation for the solution of diffusion equation in stratified turbid media,” J. Opt. Soc. Am. A 17, 2046–2055 (2000).
[Crossref]

E. Tinet, S. Avrillier, J.-M. Tualle, “Fast semi-analytical Monte Carlo simulation for time resolved light propagation in turbid media,” J. Opt. Soc. Am. A 13, 1903–1915 (1996).
[Crossref]

S. Avrillier, E. Tinet, J.-M. Tualle, F. Costes, F. Revel, J.-P. Ollivier, “Real-time inversion using Monte Carlo results for the determination of absorption coefficients in multilayered tissues: application to noninvasive muscle oximetry (Part 2),” in Diagnostic Optical Spectroscopy in Biomedicine, T. G. Papazoglou, G. A. Wagnieres, eds., Proc. SPIE4432, 75–84 (2001).
[Crossref]

Bays, R.

Busch, D. R.

Chance, B.

Cope, M.

C. R. Simpson, M. Kohl, M. Essenpreis, M. Cope, “Near-infrared optical properties of ex vivo human skin and subcutaneous tissues measured using the Monte Carlo inversion technique,” Phys. Med. Biol. 43, 2465–2478 (1998).
[Crossref] [PubMed]

Costes, F.

S. Avrillier, E. Tinet, J.-M. Tualle, F. Costes, F. Revel, J.-P. Ollivier, “Real-time inversion using Monte Carlo results for the determination of absorption coefficients in multilayered tissues: application to noninvasive muscle oximetry (Part 2),” in Diagnostic Optical Spectroscopy in Biomedicine, T. G. Papazoglou, G. A. Wagnieres, eds., Proc. SPIE4432, 75–84 (2001).
[Crossref]

Cubeddu, R.

Dögnitz, N.

Essenpreis, M.

C. R. Simpson, M. Kohl, M. Essenpreis, M. Cope, “Near-infrared optical properties of ex vivo human skin and subcutaneous tissues measured using the Monte Carlo inversion technique,” Phys. Med. Biol. 43, 2465–2478 (1998).
[Crossref] [PubMed]

Faris, G. W.

Farrell, T. J.

Feng, T.-C.

Glanzmann, T.

Haskell, R. C.

Kienle, A.

Kohl, M.

C. R. Simpson, M. Kohl, M. Essenpreis, M. Cope, “Near-infrared optical properties of ex vivo human skin and subcutaneous tissues measured using the Monte Carlo inversion technique,” Phys. Med. Biol. 43, 2465–2478 (1998).
[Crossref] [PubMed]

Maynard, R.

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[Crossref] [PubMed]

McAdams, M. S.

Mochi, M.

M. Mochi, G. Pacelli, M. C. Recchioni, F. Zirilli, “Inverse problem for a class of two-dimensional diffusion equations with piecewise constant coefficients,” J. Optim. Theory Appl. 100, 29–57 (1999).
[Crossref]

Nieto-Vesperinas, M.

Ollivier, J.-P.

S. Avrillier, E. Tinet, J.-M. Tualle, F. Costes, F. Revel, J.-P. Ollivier, “Real-time inversion using Monte Carlo results for the determination of absorption coefficients in multilayered tissues: application to noninvasive muscle oximetry (Part 2),” in Diagnostic Optical Spectroscopy in Biomedicine, T. G. Papazoglou, G. A. Wagnieres, eds., Proc. SPIE4432, 75–84 (2001).
[Crossref]

Pacelli, G.

M. Mochi, G. Pacelli, M. C. Recchioni, F. Zirilli, “Inverse problem for a class of two-dimensional diffusion equations with piecewise constant coefficients,” J. Optim. Theory Appl. 100, 29–57 (1999).
[Crossref]

Patterson, M. S.

Pifferi, A.

Prat, J.

Recchioni, M. C.

M. Mochi, G. Pacelli, M. C. Recchioni, F. Zirilli, “Inverse problem for a class of two-dimensional diffusion equations with piecewise constant coefficients,” J. Optim. Theory Appl. 100, 29–57 (1999).
[Crossref]

Revel, F.

S. Avrillier, E. Tinet, J.-M. Tualle, F. Costes, F. Revel, J.-P. Ollivier, “Real-time inversion using Monte Carlo results for the determination of absorption coefficients in multilayered tissues: application to noninvasive muscle oximetry (Part 2),” in Diagnostic Optical Spectroscopy in Biomedicine, T. G. Papazoglou, G. A. Wagnieres, eds., Proc. SPIE4432, 75–84 (2001).
[Crossref]

Ripoll, J.

Simpson, C. R.

C. R. Simpson, M. Kohl, M. Essenpreis, M. Cope, “Near-infrared optical properties of ex vivo human skin and subcutaneous tissues measured using the Monte Carlo inversion technique,” Phys. Med. Biol. 43, 2465–2478 (1998).
[Crossref] [PubMed]

Svaasand, L. O.

Taroni, P.

Tinet, E.

J.-M. Tualle, E. Tinet, J. Prat, S. Avrillier, “Light propagation near turbid–turbid planar interfaces,” Opt. Commun. 183, 337–346 (2000).
[Crossref]

J.-M. Tualle, J. Prat, E. Tinet, S. Avrillier, “Real-space Green’s function calculation for the solution of diffusion equation in stratified turbid media,” J. Opt. Soc. Am. A 17, 2046–2055 (2000).
[Crossref]

E. Tinet, S. Avrillier, J.-M. Tualle, “Fast semi-analytical Monte Carlo simulation for time resolved light propagation in turbid media,” J. Opt. Soc. Am. A 13, 1903–1915 (1996).
[Crossref]

S. Avrillier, E. Tinet, J.-M. Tualle, F. Costes, F. Revel, J.-P. Ollivier, “Real-time inversion using Monte Carlo results for the determination of absorption coefficients in multilayered tissues: application to noninvasive muscle oximetry (Part 2),” in Diagnostic Optical Spectroscopy in Biomedicine, T. G. Papazoglou, G. A. Wagnieres, eds., Proc. SPIE4432, 75–84 (2001).
[Crossref]

Torricelli, A.

Tromberg, B. J.

Tsay, T.-T.

Tualle, J.-M.

J.-M. Tualle, J. Prat, E. Tinet, S. Avrillier, “Real-space Green’s function calculation for the solution of diffusion equation in stratified turbid media,” J. Opt. Soc. Am. A 17, 2046–2055 (2000).
[Crossref]

J.-M. Tualle, E. Tinet, J. Prat, S. Avrillier, “Light propagation near turbid–turbid planar interfaces,” Opt. Commun. 183, 337–346 (2000).
[Crossref]

E. Tinet, S. Avrillier, J.-M. Tualle, “Fast semi-analytical Monte Carlo simulation for time resolved light propagation in turbid media,” J. Opt. Soc. Am. A 13, 1903–1915 (1996).
[Crossref]

S. Avrillier, E. Tinet, J.-M. Tualle, F. Costes, F. Revel, J.-P. Ollivier, “Real-time inversion using Monte Carlo results for the determination of absorption coefficients in multilayered tissues: application to noninvasive muscle oximetry (Part 2),” in Diagnostic Optical Spectroscopy in Biomedicine, T. G. Papazoglou, G. A. Wagnieres, eds., Proc. SPIE4432, 75–84 (2001).
[Crossref]

Van den Bergh, H.

Wagnières, G.

Wilson, B. C.

Wolf, P. E.

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[Crossref] [PubMed]

Zirilli, F.

M. Mochi, G. Pacelli, M. C. Recchioni, F. Zirilli, “Inverse problem for a class of two-dimensional diffusion equations with piecewise constant coefficients,” J. Optim. Theory Appl. 100, 29–57 (1999).
[Crossref]

Appl. Opt. (5)

J. Opt. Soc. Am. A (6)

J. Optim. Theory Appl. (1)

M. Mochi, G. Pacelli, M. C. Recchioni, F. Zirilli, “Inverse problem for a class of two-dimensional diffusion equations with piecewise constant coefficients,” J. Optim. Theory Appl. 100, 29–57 (1999).
[Crossref]

Opt. Commun. (1)

J.-M. Tualle, E. Tinet, J. Prat, S. Avrillier, “Light propagation near turbid–turbid planar interfaces,” Opt. Commun. 183, 337–346 (2000).
[Crossref]

Opt. Lett. (1)

Phys. Med. Biol. (1)

C. R. Simpson, M. Kohl, M. Essenpreis, M. Cope, “Near-infrared optical properties of ex vivo human skin and subcutaneous tissues measured using the Monte Carlo inversion technique,” Phys. Med. Biol. 43, 2465–2478 (1998).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

E. Akkermans, P. E. Wolf, R. Maynard, “Coherent backscattering of light by disordered media: analysis of the peak line shape,” Phys. Rev. Lett. 56, 1471–1474 (1986).
[Crossref] [PubMed]

Other (2)

See http://www.medphys.ucl.ac.uk/research/borg/research/NIR_topics/phantoms.htm .

S. Avrillier, E. Tinet, J.-M. Tualle, F. Costes, F. Revel, J.-P. Ollivier, “Real-time inversion using Monte Carlo results for the determination of absorption coefficients in multilayered tissues: application to noninvasive muscle oximetry (Part 2),” in Diagnostic Optical Spectroscopy in Biomedicine, T. G. Papazoglou, G. A. Wagnieres, eds., Proc. SPIE4432, 75–84 (2001).
[Crossref]

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Figures (10)

Fig. 1
Fig. 1

Layered medium.

Fig. 2
Fig. 2

Integration contour. The right curve corresponds to hyperbolic contour Γ(k), that is, to Re(s2)=k2+μa/D; the left part is the imaginary axis ]0+-j, 0++j[. The integration contour is closed by arbitrary upper and lower curves that have no contribution when extended to infinity.

Fig. 3
Fig. 3

Convergence of the development in Eq. (12), using the medium described in Table 1 with l=1 cm, at ρ=0. ln R-ln Rn is plotted versus time for different values of the order n of the development and with a logarithmic scale.

Fig. 4
Fig. 4

Parameters (a) E and (b) F calculated from Eq. (15) versus t0=1/(4Dcα0) in the case of the medium of Table 1 with l=1 cm. The values C1,0 and C2,1 of the development (13) are shown as dashed lines.

Fig. 5
Fig. 5

Logarithm of the time-resolved reflectance function obtained from a Monte Carlo simulation with the medium of Table 1 and l=5 mm for six detectors chosen with an equal spacing in [1.14 cm, 2.49 cm]. The fit, performed for 1500 ps<t<2000 ps, is presented together with a zoom of this time interval.

Fig. 6
Fig. 6

Results obtained for (a) μa and (b) μs for different values of l using the medium described in Table 1. The squares correspond to the results obtained with a simple semi-infinite fit (E=F=0), and the circles correspond to the results obtained with relation (16).

Fig. 7
Fig. 7

Results obtained (circles) for (a) E and (b) F, assuming correct values for μa and μs, for different values of l. The curves with open circles correspond to the results obtained with relation (16).

Fig. 8
Fig. 8

Results obtained for (a) μa and (b) μs for different values of l using the medium described in Table 2. The squares correspond to the results obtained with a simple semi-infinite fit (E=F=0), and the circles correspond to the results obtained with relation (16).

Fig. 9
Fig. 9

Results obtained (circles) for (a) E and (b) F, assuming correct values for μa and μs, for different values of l. The curves with open circles correspond to the results obtained with relation (16).

Fig. 10
Fig. 10

Results obtained (circles) for (a) μeff and (b) μs,eff using Eq. (21) on the medium of Table 2 for different values of l, together with their values corrected by using Eqs. (22) (squares).

Tables (4)

Tables Icon

Table 1 Optical Coefficients of a Two-Layered Medium for the Simulation of a Neonate Heada

Tables Icon

Table 2 Optical Coefficients of a Two-Layered Medium for the Simulation of the Fat/Muscle Succession

Tables Icon

Table 3 Optical Coefficients of a Four-Layered Medium for the Simulation of a Neonate Head

Tables Icon

Table 4 Results Obtained from Simulations Performed with the Data of Table 3

Equations (59)

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1ci ϕt-DiΔϕ+μaiϕ=S,
ni+12ϕ(li-)=ni2ϕ(li+),
Di ϕz(li-)=Di+1 ϕz(li+).
ϕ(-zb)=0,
ϕ˜(z)=Ai exp(-siz)+Bi exp(siz),
si=k2+μaiDi+jωDici1/2.
ni+12[Ai exp(-sili)+Bi exp(sili)]=ni2[Ai+1 exp(-si+1li)+Bi+1 exp(si+1li)],
Disi[-Ai exp(-sili)+Bi exp(sili)]=Di+1si+1[-Ai+1 exp(-si+1li)+Bi+1 exp(si+1li)],
AiBi=PiAi+1Bi+1,
A1B1=P1P2P3PN-1ANBN,
ϕ˜(z0+)=ϕ˜(z0-),
ϕ˜z(z0+)-ϕ˜z(z0-)=-1/D1,
R˜(ω, k2)=-jz(0)=D1 ϕ˜z(0)
R(ρ, t)=14π2 0kdkdωR˜(ω, k2)J0(kρ)exp(jωt).
R˜=cosh(s1zb)×D1s1 cosh[s1(l-z0)]+D2s2 sinh[s1(l-z0)]D1s1 cosh[s1(l+zb)]+D2s2 sinh[s1(l+zb)].
R(ρ, t)=-jDNcN2π2 exp(-μaNcNt)×0kdkΓ(k)sdsR˜(s, k2)J0(kρ)×exp[DNcN(s2-k2)t],
R˜(s, k2)=p,q0R˜p,qspk2q.
R(ρ, t)=-Dc 2π2π2 n2p+1,q0R˜2p+1,q(2p+1)!!×(2Dct)-p(-1)p+qΔTq+O(t-n-1)(2Dct)-5/2 exp-ρ24Dct-μact,
R(ρ, t)=np2q0Rp,qρ2qt-p+O(t-n-1)t-5/2 exp-ρ24Dct-μact
ln R(ρ, t)=-52 ln t-ρ24Dct-μact+np2q0Cp,qρ2qt-p+O(t-n-1).
R(ρ, t)=-α5/2π-5/2 exp(-αρ2-μact)×0RdR-dZRˆ(Z, R)(1-2αZ2)×J0(2jαρR)exp[-α(R2+Z2)],
Rˆ(Z, R)=dθd2k Dcjπθ[R˜(s, k2)-R˜(0, k2)]×exp(jk· R+jZθ).
R(ρ, t)=-α5/2π-5/2 exp(-αρ2-μact)×RdRdZ Rˆ(Z, R)exp[-α0(R2+Z2)]×{(1-2α0Z2)-[2Z2+(1-2α0Z2)×(R2+Z2)]δα+R2(1-2α0Z2)α2ρ2-R2[2Z2+(1-2α0Z2)(R2+Z2)]α02ρ2δα}.
ln R(ρ, t)R0-52 ln t-ρ24Dct-μact+Et+F ρ2t2.
σ2=M-1y-Jp2
p=[tJJ]-1 tJy.
δptδp=σ2[tJJ]-1
Γ(k)sds R˜(s, k2)exp(Dcs2)=-j+0+j+0+sds R˜(s, k2)exp(Dcs2)+2πjsPiRe s[R˜(sPi, k2)]exp(DcsPi2t),
lc+zb=π2 D1μa2-μa11/2.
sP=s0+βk2+o(k2)orsP2=s02+2s0βk2+o(k2).
s0π2D14(lc+zb)2D2(l-lc),βD2-D12D2(lc+zb).
τksds R˜(s, k2)exp(Dcs2t)2πj Re s[R˜(sP, k2)]exp(DcsP2t).
R(ρ, t)=Dcπ exp(-μact)0kdk Re s[R˜(k2)]×exp[Dco(k2)t]J0(kρ)×exp{Dc[s02+(2βs0-1)k2]t}.
R(ρ, t)1t exp-(μa-Ds02)ct-ρ24(1-2βs0)Dct.
ln R(ρ, t)=R0-ln t-μeff ct-ρ24Deff ct,
μeff=μa+Ds02,Deff=D(1-2βs0).
P1P2P3PN=abcd.
A1=aAN,B1=cAN.
A0=A1-exp(s1z0)2s1D1=aAN-exp(s1z0)2s1D1,
B0=B1+exp(-s1z0)2s1D1=cAN+exp(-s1z0)2s1D1.
[a exp(s1zb)+c exp(-s1zb)]ANDen(s, k2)AN=sinh[s1(z0+zb)]s1D1,
R˜(s, k2)=-s1D1(A0-B0)=a exp(-s1z0)+c exp(s1z0)Den(s, k2) cosh(s1zb).
R(ρ, t)=-j2γπ2 0kdkΓ(η)sds R˜(s, k2)J0(kρ)exp[(s2-η)t/γ].
R(ρ, t)=Oρ(t-n)exp[-(μa-δμ)t](δμ>0).
R(ρ, t)=-Dc4π3 exp(-μact)d2kd2kdθdθs-1×[R˜(s, k2)-R˜(0, k2)]×δ(θ-θ)δ(k-k)θ2×exp[-Dc(θ2+k2)t-jk·ρ];
δ(θ-θ)δ(k-k)=18π3 exp[jZ(θ-θ)+jR·(k-k)]dZd2R,
d2kdθ 1jθ[R˜(s, k2)-R˜(0, k2)]exp[jZθ+jR·k]πDcRˆ(Z, R),
d2kdθ θ2×exp[-Dc(θ2+k2)t-jk·(ρ+R)-jθZ].
dθ exp(-Dcθ2t-jθZ)=πDct1/2 exp-Z24Dct,
dθθ2 exp(-Dcθ2t-jθZ)=-πDct1/2 d2dZ2 exp-Z24Dct=-πDct1/2Z2(Dct)2-12Dctexp-Z24Dct,
dk2 exp[-Dck2t-jk·(R+ρ)]=πDct exp-R2+ρ2+2R ·ρ4Dct.
dϕ expj 2jRρ4Dct cos ϕ=2πJ02jRρ4Dct.
R˜=cosh(s1zb)×D1s1 cosh[s1(l-z0)]+D2s2 sinh[s1(l-z0)]D1s1 cosh[s1(l+zb)]+D2s2 sinh[s1(l+zb)].
coth[s1(sP)l*]=-D2sPD1s1(sP),
s1(sP)=D2D1sp2+k21-D2D1+μa1-μa2D11/2.
cothjlc*μa2-μa1D11/2=-j cotlc*μa2-μa1D11/2=0,
-sinh-2[s1(0)lc*]s1(0)δl=-D2sPD1s1(0)
sP=D1D2s12(0)δl=π2D14lc*2D2(l-lc).
s1(sp)s1(0)+k22s1(0) 1-D2D1, -sinh-2[s1(0)lc*] k22s1(0) 1-D2D1lc*=-D2sPD1s1(0) sP=k22 1-D1D2lc*.

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