Abstract

Recently it has been shown that clear regions within diffusive media can be accurately modeled within the diffusion approximation by means of a novel boundary condition [J. Opt. Soc. Am. A 17, 1671 (2000)] or by an approximation to it [Phys. Med. Biol. 41, 767 (1996); Med. Phys. 27, 252 (2000)]. This can be directly applied to the study of light propagation in brain tissue, in which there exist clear regions, and in particular in the cerebrospinal fluid (CSF) layer under the skull. In this work we present the effect that roughness in the boundary of nondiffusive regions has on the measured average intensity, since, in practice, the CSF layer is quite rough. The same conclusions can be extended to any diffusive medium that encloses rough nondiffusive regions. We will demonstrate with numerical calculations that the roughness statistics of the interfaces (although not their actual profiles) must be known a priori to correctly predict the shape of the average intensity. We show that as the roughness increases, the effect of the nondiffusive region diminishes until it disappears, thus yielding data similar to those of a fully diffusive region. We also present a numerical study of the diffuse light scattered in the presence of both diffusive and nondiffusive regions and the interaction between the two, showing that when the nondiffusive region is rough, the spatial-intensity distribution produced by the two regions can be very similar.

© 2001 Optical Society of America

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2000 (3)

1999 (9)

J. Ripoll, M. Nieto-Vesperinas, “Scattering integral equations for diffusive waves. Detection of objects buried in diffusive media in the presence of rough interfaces,” J. Opt. Soc. Am. A 16, 1453–1465 (1999).
[CrossRef]

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

J. Ripoll, M. Nieto-Vesperinas, R. Carminati, “Spatial resolution of diffuse photon density waves,” J. Opt. Soc. Am. A 16, 1466–1476 (1999).
[CrossRef]

J. C. Hebden, F. E. W. Schmidt, M. E. Fry, M. Schweiger, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, “Simultaneous reconstruction of absorption and scattering images by multichannel measurement of purely temporal data,” Opt. Lett. 24, 534–536 (1999).
[CrossRef]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

T. Durduran, J. P. Culver, M. J. Holboke, X. D. Li, L. Zubkov, B. Chance, D. N. Pattanayak, A. G. Yodh, “Algorithms for 3D localization and imaging using near-field diffraction tomography with diffuse light,” Opt. Exp. 4, pp. 247–262 (1999).
[CrossRef]

V. Ntziachristos, B. Chance, A. G. Yodh, “Differential diffuse optical tomography,” Opt. Exp. 5, 230–242 (1999).
[CrossRef]

D. J. Durian, J. Rudnick, “Spatially resolved backscattering: implementation of extrapolation boundary condition and exponential source,” J. Opt. Soc. Am. A 16, 837–844 (1999).
[CrossRef]

F. Martinelli, A. Sassaroli, G. Zaccanti, Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. Biol. 44, 1257–1275 (1999).
[CrossRef]

1998 (5)

1997 (7)

1996 (2)

E. B. de Haller, “Time-resolved transillumination and optical tomography,” J. Biomed. Opt. 1, 7–17 (1996).
[CrossRef] [PubMed]

M. Firbank, S. R. Arridge, M. Schweiger, D. T. Delpy, “An investigation of light transport through scattering bodies with nonscattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

1995 (7)

1994 (1)

1993 (3)

1992 (1)

I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
[CrossRef] [PubMed]

1991 (1)

1990 (2)

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

J. M. Soto-Crespo, M. Nieto-Vesperinas, A. T. Friberg, “Scattering from slightly rough random surfaces: a detailed study on the validity of the small perturbation method,” J. Opt. Soc. Am. A 7, 1185–1201 (1990).
[CrossRef]

1989 (2)

1971 (1)

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface. II,” Proc. Cambridge Philos. Soc. 65, 217–224 (1971).
[CrossRef]

’t Hooft, G. W.

Arfken, G. B.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, New York, 1995).

Aronson, R.

Arridge, S. R.

H. Dehghani, S. R. Arridge, M. Schweiger, D. T. Delpy, “Optical tomography in the presence of void regions,” J. Opt. Soc. Am. A 17, 1659–1670 (2000).
[CrossRef]

J. Ripoll, S. R. Arridge, H. Dehghani, M. Nieto-Vesperinas, “Boundary conditions for light propagation in diffusive media with nonscattering regions,” J. Opt. Soc. Am. A 17, 1671–1681 (2000).
[CrossRef]

S. R. Arridge, H. Dehghani, M. Schweiger, E. Okada, “The finite-element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions,” Med. Phys. 27, 252–264 (2000).
[CrossRef] [PubMed]

J. C. Hebden, F. E. W. Schmidt, M. E. Fry, M. Schweiger, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, “Simultaneous reconstruction of absorption and scattering images by multichannel measurement of purely temporal data,” Opt. Lett. 24, 534–536 (1999).
[CrossRef]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

E. Okada, M. Firbank, M. Schweiger, S. R. Arridge, M. Cope, D. T. Delpy, “Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head,” Appl. Opt. 36, 21–31 (1997).
[CrossRef] [PubMed]

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modeling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[CrossRef] [PubMed]

M. Firbank, S. R. Arridge, M. Schweiger, D. T. Delpy, “An investigation of light transport through scattering bodies with nonscattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for near infrared absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
[CrossRef]

Bai, J.

N. G. Chen, J. Bai, “Monte Carlo approach to modeling of boundary conditions for the diffusion equation,” Phys. Rev. Lett. 80, 5321–4324 (1998).
[CrossRef]

Boas, D. A.

Carminati, R.

Chance, B.

T. Durduran, J. P. Culver, M. J. Holboke, X. D. Li, L. Zubkov, B. Chance, D. N. Pattanayak, A. G. Yodh, “Algorithms for 3D localization and imaging using near-field diffraction tomography with diffuse light,” Opt. Exp. 4, pp. 247–262 (1999).
[CrossRef]

V. Ntziachristos, B. Chance, A. G. Yodh, “Differential diffuse optical tomography,” Opt. Exp. 5, 230–242 (1999).
[CrossRef]

X. de Li, T. Durduran, A. G. Yodh, B. Chance, D. N. Pattanayak, “Diffraction tomography for biochemical imaging with diffuse-photon density waves,” Opt. Lett. 22, 573–575 (1997).
[CrossRef]

S. Feng, F. Zeng, B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt. 34, 3826–3837 (1995).
[CrossRef] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. 20, 426–428 (1995).
[CrossRef] [PubMed]

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48, 38–40 (1995).
[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]

M. S. Patterson, B. Chance, B. C. Wilson, “Time-resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

Chen, N. G.

N. G. Chen, J. Bai, “Monte Carlo approach to modeling of boundary conditions for the diffusion equation,” Phys. Rev. Lett. 80, 5321–4324 (1998).
[CrossRef]

Clark, N.

Colak, S. B.

Cope, M.

E. Okada, M. Firbank, M. Schweiger, S. R. Arridge, M. Cope, D. T. Delpy, “Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head,” Appl. Opt. 36, 21–31 (1997).
[CrossRef] [PubMed]

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, “Reconstruction methods for near infrared absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 204–215 (1991).
[CrossRef]

Culver, J. P.

T. Durduran, J. P. Culver, M. J. Holboke, X. D. Li, L. Zubkov, B. Chance, D. N. Pattanayak, A. G. Yodh, “Algorithms for 3D localization and imaging using near-field diffraction tomography with diffuse light,” Opt. Exp. 4, pp. 247–262 (1999).
[CrossRef]

de Haller, E. B.

E. B. de Haller, “Time-resolved transillumination and optical tomography,” J. Biomed. Opt. 1, 7–17 (1996).
[CrossRef] [PubMed]

de Li, X.

Dehghani, H.

Delpy, D. T.

den Outer, P. N.

Durduran, T.

T. Durduran, J. P. Culver, M. J. Holboke, X. D. Li, L. Zubkov, B. Chance, D. N. Pattanayak, A. G. Yodh, “Algorithms for 3D localization and imaging using near-field diffraction tomography with diffuse light,” Opt. Exp. 4, pp. 247–262 (1999).
[CrossRef]

X. de Li, T. Durduran, A. G. Yodh, B. Chance, D. N. Pattanayak, “Diffraction tomography for biochemical imaging with diffuse-photon density waves,” Opt. Lett. 22, 573–575 (1997).
[CrossRef]

Durian, D. J.

Fantini, S.

Fender, J. S.

Feng, S.

Feng, T.

Field, M. S.

Firbank, M.

E. Okada, M. Firbank, M. Schweiger, S. R. Arridge, M. Cope, D. T. Delpy, “Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head,” Appl. Opt. 36, 21–31 (1997).
[CrossRef] [PubMed]

M. Firbank, S. R. Arridge, M. Schweiger, D. T. Delpy, “An investigation of light transport through scattering bodies with nonscattering regions,” Phys. Med. Biol. 41, 767–783 (1996).
[CrossRef] [PubMed]

Franceschini, M. A.

Freund, I.

I. Freund, “Surface reflections and boundary conditions for diffusive photon transport,” Phys. Rev. A 45, 8854–8858 (1992).
[CrossRef] [PubMed]

Friberg, A. T.

Fry, M. E.

Furutsu, K.

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

Gonatas, C. P.

C. P. Gonatas, M. Ishuii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct-inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[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]

Gratton, E.

Haskell, R. C.

Hebden, J. C.

Hillman, E. M. C.

Holboke, M. J.

T. Durduran, J. P. Culver, M. J. Holboke, X. D. Li, L. Zubkov, B. Chance, D. N. Pattanayak, A. G. Yodh, “Algorithms for 3D localization and imaging using near-field diffraction tomography with diffuse light,” Opt. Exp. 4, pp. 247–262 (1999).
[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]

Ishimaru, A.

Ishuii, M.

C. P. Gonatas, M. Ishuii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct-inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[CrossRef]

Ito, S.

Jiang, H.

Kaschke, M.

Kim, A. D.

Lagendijk, A.

Leigh, J. S.

C. P. Gonatas, M. Ishuii, J. S. Leigh, J. C. Schotland, “Optical diffusion imaging using a direct-inversion method,” Phys. Rev. E 52, 4361–4365 (1995).
[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]

Li, X. D.

T. Durduran, J. P. Culver, M. J. Holboke, X. D. Li, L. Zubkov, B. Chance, D. N. Pattanayak, A. G. Yodh, “Algorithms for 3D localization and imaging using near-field diffraction tomography with diffuse light,” Opt. Exp. 4, pp. 247–262 (1999).
[CrossRef]

Maradudin, A. A.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Martinelli, F.

F. Martinelli, A. Sassaroli, G. Zaccanti, Y. Yamada, “Properties of the light emerging from a diffusive medium: angular dependence and flux at the external boundary,” Phys. Med. Biol. 44, 1257–1275 (1999).
[CrossRef]

Matson, C. L.

McAdams, M. S.

McGurn, A. R.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

McMackin, L.

Melissen, J. B. M.

Mendez, E. R.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Michel, T.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Mendez, “Enhanced backscattering of light from a random grating,” Ann. Phys. (N.Y.) 203, 255–307 (1990).
[CrossRef]

Millar, R. F.

R. F. Millar, “On the Rayleigh assumption in scattering by a periodic surface. II,” Proc. Cambridge Philos. Soc. 65, 217–224 (1971).
[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]

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Figures (10)

Fig. 1
Fig. 1

Scattering geometry consisting of a rough nondiffusive cylinder with statistical parameters σ and T, embedded in a diffusive cylinder of radius Rcyl.

Fig. 2
Fig. 2

Scattered wave U(sc) represented in logarithmic scale for the cases (a) no nonscattering cylinder present, (b) a nondiffusive cylinder of R=1.0 cm, (c) a nondiffusive gap of Rin=1.0 cm, Rout=1.5 cm. In all cases Rcyl=2.5 cm (see Section 3 for the optical parameters).

Fig. 3
Fig. 3

Scattered wave U(sc) represented in logarithmic scale for different statistical parameters σ and T for the case of a nondiffusive cylinder of R=1.0 cm embedded in a diffusive cylinder Rcyl=2.5 cm (see Section 3 for the optical parameters).

Fig. 4
Fig. 4

Total average intensity U measured at Rcyl for the case of a nondiffusive rough cylinder with cosine profile with the following amplitude h and period d. Solid curve: smooth, h=0, d=; open circles: h=0.05 cm, d=0.32 cm; solid circles: h=0.05 cm, d=0.16 cm; open triangles: h=0.05 cm, d=0.1 cm; dotted curve: absence of the nondiffusive object. In all cases Rcyl=2.5 cm (see Section 3 for the optical parameters).

Fig. 5
Fig. 5

Total transmitted intensity U¯trans versus roughness [see Eq. (13)] for a rough nondiffusive cylinder with random rough profile and radii of lengths are given in the legend. U¯trans in the absence of the nondiffusive volume is represented by a solid line (see Section 3 for the optical parameters).

Fig. 6
Fig. 6

Same as Fig. 5 but for the case of a rough cylinder with cosine profile [see Eq. (15)].

Fig. 7
Fig. 7

Scattered wave U(sc) represented in logarithmic scale for different random profiles of statistical parameters σ and T for the case of a nondiffusive gap of inner radius Rin=1.0 cm and outer radius Rout=1.5 cm embedded in a diffusive cylinder Rcyl=2.5 cm (see Section 3 for the optical parameters).

Fig. 8
Fig. 8

Scattering configuration for a nondiffusive rough cylinder and a diffusive cylinder separated by a distance d from their centers, embedded in an otherwise-infinite homogeneous diffusive medium.

Fig. 9
Fig. 9

Scattered wave U(sc) measured at zd=1.0 cm for the case of a diffusive cylinder of radius R=0.5 cm placed at (0, 1.0 cm) and a random rough nondiffusive cylinder of mean radius R=0.5 cm placed at (0, -1.0 cm) with statistical parameters as given in the legend. Both objects are embedded in an infinite homogeneous diffusive medium (see Section 4 for the optical parameters).

Fig. 10
Fig. 10

Same as Fig. 9 but for determinations performed at zd=-1.0 cm.

Equations (18)

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

ρ(θ)=[R+D(θ)]uˆr,
ρ(θ)ρ(θ+ζ)=σ2 exp(-ζ2/Tθ2),
σ=[ρ2(θ)-R2]1/2.
U(r)=U(i)(r)-14πRcyl(r)-14π×SU(r) G(κ0|r-r|)nˆ-G(κ0|r-r|) U(r)nˆdSrV˜,
Rcyl(r)=RcylU(r)G(κ0|r-r|)mˆ+1CcylD0G(κ0|r-r|)dR,
U(r)|Rcyl=-CcylD0 U(r)mˆRcyl,
RUcyl=01[1-|R0air(θ)|2]cos θ d(cos θ),
RJcyl=301[1-|R0air(θ)|2]cos2 θd(cos θ).
G2D(r-r)=π2 exp[-μclr|r-r|]|r-r| V(r-r)cos θ cos θ,
cos θ=nˆ(r)·(r-r)|r-r|,cos θ=nˆ(r)·(r-r)|r-r|,
U(r)=2Jn(r)+1π S[U(r)+2Jn(r)]G2D(r-r)dS×rS.
U(r)=CclrJn(r)rS.
U(sc)(r)=U(r)-U(i)(r),
L(i)(rs-r)=L(rs-rv)+L(rv-r),
Roughness=σ/RT/R exp(σ/R),
Roughness1.020.410.202.631.050.535.422.171.08.
ρ(θ)=R+h cos2πdθ.
U¯trans=1π π/23π/2U(Rcyl, θ)dθ.

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