Abstract

Based on a previous theory of diffuse photon density waves by Furutsu [J. Opt. Soc. Am. A 15, 1371 (1998)], several sets of figures are prepared to detect a fixed scatterer (object) embedded in a turbid layer, such as a tumor in tissue, with a source and a detector placed independently along the boundaries on different sides. The relative total intensity of the wave is introduced such that it is reduced to 1 in the case of no scatterer and usually less than that, owing to a shadowing by the scatterer. Sets of curves are presented to demonstrate shadow images of the scatterer observed along the layer boundaries depending on the scatterer’s location.

© 2000 Optical Society of America

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References

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  1. D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and application,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
    [CrossRef]
  2. D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–91 (1997).
    [CrossRef] [PubMed]
  3. K. Furutsu, “Boundary conditions of the diffusion equation and applications,” Phys. Rev. A 39, 1386–1401 (1989).
    [CrossRef] [PubMed]
  4. J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
    [CrossRef] [PubMed]
  5. P. N. den Outer, Th. M. Nieuwenhuizen, Ad Lagendijk, “Location of objects in multiple-scattering media,” J. Opt. Soc. Am. A 10, 1209–1218 (1993).
    [CrossRef]
  6. C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
    [CrossRef]
  7. K. Furutsu, “Theory of a fixed scatterer embedded in a turbid medium,” J. Opt. Soc. Am. A 15, 1371–1382 (1998).
    [CrossRef]
  8. K. Furutsu, “Transport theory and boundary-value solutions. I. The Bethe–Salpeter equation and scattering matrices,” J. Opt. Soc. Am. A 2, 913–931 (1985); “II. Addition theorem of scattering matrices and applications,” J. Opt. Soc. Am. A 2, 932–944 (1985); see also K. Furutsu, Random Media and Boundaries—Unified Theory, Two-Scale Method, and Applications, Vol. 14 of Springer Series on Wave Phenomena (Springer-Verlag, New York, 1993).
    [CrossRef]
  9. K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
    [CrossRef]

1998 (1)

1997 (1)

1994 (2)

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and application,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

1993 (2)

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

P. N. den Outer, Th. M. Nieuwenhuizen, Ad Lagendijk, “Location of objects in multiple-scattering media,” J. Opt. Soc. Am. A 10, 1209–1218 (1993).
[CrossRef]

1991 (1)

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

1989 (1)

K. Furutsu, “Boundary conditions of the diffusion equation and applications,” Phys. Rev. A 39, 1386–1401 (1989).
[CrossRef] [PubMed]

1985 (1)

Boas, D. A.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–91 (1997).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and application,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

Chance, B.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–91 (1997).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and application,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

den Outer, P. N.

Furutsu, K.

Gonatas, C. P.

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

Ishii, M.

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

Lagendijk, Ad

Leigh, J. S.

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

Miwa, M.

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

Nieuwenhuizen, Th. M.

O’Leary, M. A.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–91 (1997).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and application,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

Pine, D. J.

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

Schotland, J.

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

Weitz, D. A.

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

Yamada, Y.

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

Yodh, A. G.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–91 (1997).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and application,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

Zhu, J. X.

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

Appl. Opt. (1)

J. Opt. Soc. Am. A (3)

Phys. Rev. A (2)

K. Furutsu, “Boundary conditions of the diffusion equation and applications,” Phys. Rev. A 39, 1386–1401 (1989).
[CrossRef] [PubMed]

J. X. Zhu, D. J. Pine, D. A. Weitz, “Internal reflection of diffusive light in random media,” Phys. Rev. A 44, 3948–3959 (1991).
[CrossRef] [PubMed]

Phys. Rev. E (2)

C. P. Gonatas, M. Miwa, M. Ishii, J. Schotland, B. Chance, J. S. Leigh, “Effects due to geometry and boundary conditions in multiple light scattering,” Phys. Rev. E 48, 2212–2216 (1993).
[CrossRef]

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and application,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

DPDW’s are generated by the injection of light from a sinusoidally modulated source at 200 MHz. The turbid layer is 6.0 cm thick with a reduced scattering coefficient, μs, of 10.0 cm-1 and absorption coefficient μa=0.05 cm-1. A spherical object is embedded at distance zV from upper boundary (12) and is located at R=0 with the horizontal coordinates ρ=(ρ1, ρ2), R=|ρ|. For simulations, the source at R=R1 and a detector at R=R3 are scanned together along the boundaries, or the source is held fixed close to the object and the detector is scanned. Two different objects are studied for the case of attaching the superscript (α): a scattering object with μs(α)=15.0 cm-1 and μa(α)=0.05 cm-1 and an absorbing object with μs(α)=10.0 cm-1 and μa(α)=0.15 cm-1.

Fig. 2
Fig. 2

Absolute value of the relative total intensity of a wave as defined by Eqs. (3.1) and (3.2) is shown against the distance R=|ρ| along the layer boundaries. With layer width L=6.0 cm and the other notation in Fig. 1, the curve numbers 1, 2, and 3 denote whether R1=0, R=R30; R=R1, R3=0; or R=R1=R3, respectively, for the embedded spherical object with diameter dv1.0 cm, scattering coefficient μs(α)=15 cm-1, absorption coefficient μa(α)=0.05 cm-1, and object distance from upper layer boundary, zV, of (a) 1.5 cm, (b) 1.0 cm, (c) 0.7 cm.

Fig. 3
Fig. 3

Absolute value of the relative total intensity of a wave in the same turbid layer as in Fig. 2 is shown for a dissipative object with scattering coefficient μs(α)=10 cm-1 and absorption coefficient μa(α)=0.15 cm-1 and placed at a distance zV of (a) 1.5 cm, (b) 1.0 cm, (c) 0.7 cm.

Equations (65)

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I11(q+12+α)=I11(q+12)+I11(α/q+12),
I11(α/q+12)[ρˆ(z=0)|ρˆ(z=0)]=(4πμs)-1(1+3Z)2×R21(12)4αdρˆSA(12)[ρˆ(z=0)|ρˆ]××V(α)(-ρ|ρˆ|ρ)ASA(12+α)[ρˆ|ρˆ(z=0)].
μs-1(μa+iω/c-Dρ2)SA(12)(ρˆ|ρˆ)=δ(ρˆ-ρˆ),
D=(3μs)-1,μs=(1-μ¯1)μs,
V(α)(-ρ|ρˆ|ρ)A=dΩˆdΩˆϕ¯A(Ωˆ,-iρ)×V(α)(Ωˆ|ρˆ|Ωˆ)ϕA(Ωˆ, iρ),
ϕA(Ωˆ, iρ)=μs-1(1-3DΩˆ·ρ),
ϕ¯A(Ωˆ,-iρ)=(4π)-1(1+3DΩˆ·ρ),
V(α)(Ωˆ|ρˆ|Ωˆ)=(4π)-1μs(α)(ρˆ)-(μs(α)+μa(α))(ρˆ)×δ(Ωˆ-Ωˆ),
dΩˆV(α)(Ωˆ|ρˆ|Ωˆ)=-μa(α)(ρˆ)0.
V(α)(-ρ|ρˆ|ρ)A=μs-1[-μa(α)(ρˆ)+3D2ρ(μs(α)+μa(α))(ρˆ)·ρ],
IA(q/12)(Ωˆ, ρˆ|Ωˆ, ρˆ)=ϕA(Ωˆ, iρ)SA(12)(ρˆ|ρˆ)ϕ¯A(Ωˆ,-iρ)=IA(q/12)(-Ωˆ, ρˆ|-Ωˆ, ρˆ).
SA(12+α)(ρˆ|ρˆ)=SA(12)(ρˆ|ρˆ)+dρˆSA(12)(ρˆ|ρˆ)×V(α)(-ρ|ρˆ|ρ)ASA(12+α)(ρˆ|ρˆ).
ρ·ρ=Ωˆs·Ωˆs,
Ωˆs=ρˆlog {SA(12)[ρˆ(z=0)|ρˆ]}
|μa+iω/c|μs,
-D zSA(12)[ρˆ(z)|ρˆ]z=0=ZSA(12)[ρˆ(z=0)|ρˆ].
Z=2-11-(2π)-12π dΩˆ2Ωz(2)R22(12)(Ωˆ)21+(2π)-12πdΩˆ3(Ωz(2))2R22(12)(Ωˆ)2.
I31(q+12+23+α)=I31(q+12+23)+I31(α/q+12+23),
I31(α/q+12+23)[ρˆ(z=-L)|ρˆ(z=0)]=(4πμs)-1(1+3Z)2×R21(12)2R32(23)2dρˆSA(12+23)×[ρˆ(z=-L)|ρˆ]×V(α)(-p|ρˆ|ρ)A×SA(12+23+α)[ρˆ|ρˆ(z=0)].
I31(q+12+23+α)[ρˆ(z=-L)|ρˆ(z=0)]
=(4πμS)-1(1+3Z)2R21(12)2R32(23)2
×SA(12+23+α)[ρˆ(z=-L)|ρˆ(z=0)].
SA(12+23+α)=SA(12+23)+SA(α/12+23),
SA(α/12+23)=dpˆSA(12+23)[ρˆ(z=-L)|ρˆ]×V(α)(-ρ|ρˆ|ρ)A×SA(12+23+α)[ρˆ|ρˆ(z=0)].
SA(12+23+α)/SA(12+23)[ρˆ(z=-L)|ρˆ(z=0)]
=[SA(12+23)+SA(α/12+23)]/SA(12+23)[ρˆ(z=-L)|ρˆ(z=0)],
SA(α/12+23)[ρˆ(z=-L)|ρˆ(z=0)]
dρˆSA(23)[ρˆ(z=-L)|ρˆ]V(α)(-ρ|ρˆ|ρ)A
×SA(12)[ρˆ|ρˆ(z=0)],
SA(12+α)=SA(12)+SA(12)V(α/q+12)ASA(12),
V(α)ASA(12+α)=V(α/q+12)ASA(12).
V(α/q+12)A=VA(α)[1+SA(12)V(α/q+12)A]
=[1-V(α)ASA(12)]-1V(α)A
=V(α)A+V(α)ASA(12)V(α)A+V(α)ASA(12)V(α)ASA(12)V(α)A+ .
V(α)ASA(12+23+α)=V(α/q+12+23)ASA(12+23),
SA(12+23+α)=SA(12+23)[1+V(α)ASA(12+23+α)]
=SA(12+23)+SA(12+23)×V(α/q+12+23)ASA(12+23),
V(α/q+12+23)A=V(α)A[1+SA(12+23)V(α/q+12+23)A]=[1-V(α)ASA(12+23)]-1V(α)A.
SA(12+23)=SA(12)[1+σ(23)ASA(12+23)]
=SA(12)+SA(12)σ(23/q+12)ASA(12),
V(α/q+12+23)A=V(α/q+12)A×[1+SA(12)σ(23/q+12)A×SA(12)V(α/q+12+23)A],
V(α/q+12+23)A=[1-V(α/q+12)A×SA(12)σ(23/q+12)ASA(12)]-1V(α/q+12)A,
V(α/q+12+23)AV(α/q+12)AV(α)A
SA(12)(ρˆ|ρˆ)=(2π)-2dλ exp(-iλ·ρ)SA(12)(z|z),
SA(12)(z|z)=μs(κλD+Z)-1ϕ(12)(z>)exp(κλz<),
κλ=(κ2+λ2)1/2,κ=κλ|λ=0,
κ=[(μa+iω/c)D-1]1/2.
φ(12)(z)=cosh(κλz)-(Z/κλD)sinh(κλz)
(1+κλD/Z)-1=0dx exp[-(1+κλD/Z)x],
SA(12)[ρˆ(z=0)|ρˆ]=0dx e-x(2π)-2dλ(μs/Z)
×exp{-iλ·(ρ-ρ)-κλ[(D/Z)x-z<]}.
SA(ρˆ)=(2π)-2dλμs(2Dκλ)-1 exp(-iλ·ρ-κλ|z|)=(μs/D)|4πρˆ|-1 exp(-κ|ρˆ|),
zSA[ρˆ(z<0)]=(2π)-2dλ(μs/2D)exp(-iλ·ρ+κλz).
SA(12)[ρˆ(z=0)|ρˆ(z<0)]=2(D/Z)0dxe-xzSA[ρ-ρ,(D/Z)x-z],
SA(12+23)(z|z)=CAφ(12)(z>)φ(23)(z<).
φ(23)(z)=cosh[κλ(z+L)]+(Z/κλD)sinh[κλ(z+L)],
CA=μsD[φ(12) zφ(23)-φ(23) zφ(12)]-1,
CA=SA(12+23)(z=0|z=-L)=μs{Zφ(23)(z=0)+D(/z)φ(23)[z(=0)]}-1,
SA(12+23)(z|z)=SA(12+23)(z=0|z=-L)×φ(12)(z>)φ(23)(z<).
SA(12+23)(z=0|z=-L)
2µsDκλZ-2(1+Dκλ/Z)-2 exp(-κλL)=2µsDκλZ-20dxx exp[-(1+κλD/Z)x-κλL],
SA(12+23)[ρˆ(z=0)|ρˆ(z=-L)]
4(D/Z)0dx e-x(x-1)L
×SA[ρ-ρ,(D/Z)x+L],
0dx e-xF(x),

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