Abstract

Biological tissue scatters light mainly in the forward direction where the scattering phase function has a narrow peak. This peak makes it difficult to solve the radiative transport equation. However, it is just for forward-peaked scattering that the Fokker–Planck equation provides a good approximation, and it is easier to solve than the transport equation. Furthermore, the modification of the Fokker–Planck equation by Leakeas and Larsen provides an even better approximation and is also easier to solve. We demonstrate the accuracy of these two approximations by solving the problem of reflection and transmission of a plane wave normally incident on a slab composed of a uniform scattering medium.

© 2003 Optical Society of America

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References

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  1. W-F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
    [CrossRef]
  2. C. L. Leakeas, E. W. Larsen, “Generalized Fokker–Planck approximations of particle transport with highly forward-peaked scattering,” Nucl. Sci. Eng. 137, 236–250 (2001).
  3. G. C. Pomraning, “The Fokker–Planck operator as an asymptotic limit,” Math. Models Methods Appl. Sci. 2, 21–36 (1992).
  4. E. W. Larsen, “The linear Boltzmann equation in optically thick systems with forward-peaked scattering,” Prog. Nucl. Energy 34, 413–423 (1999).
    [CrossRef]
  5. A. D. Kim, J. B. Keller, M. Moscoso, Department of Mathematics, Stanford University, Stanford, Calif. 94305-2125 are preparing a manuscript to be called “Point-spread functions in scattering media.”
  6. A. D. Kim, M. Moscoso, “Radiative transfer computations for optical beams,” J. Comput. Phys. (to be published).
  7. M. Moscoso, J. B. Keller, G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. Am. A 18, 948–960 (2001).
    [CrossRef]
  8. A. Y. Polishchuk, M. Zevallos, F. Liu, R. R. Alfano, “Generalization of Fermat’s principle for photons in random media: the least mean square curvature of paths and photon diffusion on the velocity sphere,” Phys. Rev. E 53, 5523–5526 (1996).
    [CrossRef]
  9. J. B. Keller, H. F. Weinberger, “Boundary and initial boundary-value problems for separable backward–forward parabolic problems,” J. Math. Phys. 38, 4343–4353 (1997).
    [CrossRef]
  10. L. M. Delves, J. L. Mohamed, Computational Methods for Integral Equations (Cambridge U. Press, Cambridge, UK, 1985).
  11. L. N. Trefethen, Spectral Methods in Matlab (Society for Industrial and Applied Mathematics, Philadephia, Pa., 2000).
  12. A. D. Kim, A. Ishimaru, “Optical diffusion of continuous-wave, pulsed, and density waves in scattering media and comparisons with radiative transfer,” Appl. Opt. 37, 5313–5319 (1998).
    [CrossRef]

2001 (2)

C. L. Leakeas, E. W. Larsen, “Generalized Fokker–Planck approximations of particle transport with highly forward-peaked scattering,” Nucl. Sci. Eng. 137, 236–250 (2001).

M. Moscoso, J. B. Keller, G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. Am. A 18, 948–960 (2001).
[CrossRef]

1999 (1)

E. W. Larsen, “The linear Boltzmann equation in optically thick systems with forward-peaked scattering,” Prog. Nucl. Energy 34, 413–423 (1999).
[CrossRef]

1998 (1)

1997 (1)

J. B. Keller, H. F. Weinberger, “Boundary and initial boundary-value problems for separable backward–forward parabolic problems,” J. Math. Phys. 38, 4343–4353 (1997).
[CrossRef]

1996 (1)

A. Y. Polishchuk, M. Zevallos, F. Liu, R. R. Alfano, “Generalization of Fermat’s principle for photons in random media: the least mean square curvature of paths and photon diffusion on the velocity sphere,” Phys. Rev. E 53, 5523–5526 (1996).
[CrossRef]

1992 (1)

G. C. Pomraning, “The Fokker–Planck operator as an asymptotic limit,” Math. Models Methods Appl. Sci. 2, 21–36 (1992).

1990 (1)

W-F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Alfano, R. R.

A. Y. Polishchuk, M. Zevallos, F. Liu, R. R. Alfano, “Generalization of Fermat’s principle for photons in random media: the least mean square curvature of paths and photon diffusion on the velocity sphere,” Phys. Rev. E 53, 5523–5526 (1996).
[CrossRef]

Cheong, W-F.

W-F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Delves, L. M.

L. M. Delves, J. L. Mohamed, Computational Methods for Integral Equations (Cambridge U. Press, Cambridge, UK, 1985).

Ishimaru, A.

Keller, J. B.

M. Moscoso, J. B. Keller, G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. Am. A 18, 948–960 (2001).
[CrossRef]

J. B. Keller, H. F. Weinberger, “Boundary and initial boundary-value problems for separable backward–forward parabolic problems,” J. Math. Phys. 38, 4343–4353 (1997).
[CrossRef]

A. D. Kim, J. B. Keller, M. Moscoso, Department of Mathematics, Stanford University, Stanford, Calif. 94305-2125 are preparing a manuscript to be called “Point-spread functions in scattering media.”

Kim, A. D.

A. D. Kim, A. Ishimaru, “Optical diffusion of continuous-wave, pulsed, and density waves in scattering media and comparisons with radiative transfer,” Appl. Opt. 37, 5313–5319 (1998).
[CrossRef]

A. D. Kim, M. Moscoso, “Radiative transfer computations for optical beams,” J. Comput. Phys. (to be published).

A. D. Kim, J. B. Keller, M. Moscoso, Department of Mathematics, Stanford University, Stanford, Calif. 94305-2125 are preparing a manuscript to be called “Point-spread functions in scattering media.”

Larsen, E. W.

C. L. Leakeas, E. W. Larsen, “Generalized Fokker–Planck approximations of particle transport with highly forward-peaked scattering,” Nucl. Sci. Eng. 137, 236–250 (2001).

E. W. Larsen, “The linear Boltzmann equation in optically thick systems with forward-peaked scattering,” Prog. Nucl. Energy 34, 413–423 (1999).
[CrossRef]

Leakeas, C. L.

C. L. Leakeas, E. W. Larsen, “Generalized Fokker–Planck approximations of particle transport with highly forward-peaked scattering,” Nucl. Sci. Eng. 137, 236–250 (2001).

Liu, F.

A. Y. Polishchuk, M. Zevallos, F. Liu, R. R. Alfano, “Generalization of Fermat’s principle for photons in random media: the least mean square curvature of paths and photon diffusion on the velocity sphere,” Phys. Rev. E 53, 5523–5526 (1996).
[CrossRef]

Mohamed, J. L.

L. M. Delves, J. L. Mohamed, Computational Methods for Integral Equations (Cambridge U. Press, Cambridge, UK, 1985).

Moscoso, M.

M. Moscoso, J. B. Keller, G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. Am. A 18, 948–960 (2001).
[CrossRef]

A. D. Kim, M. Moscoso, “Radiative transfer computations for optical beams,” J. Comput. Phys. (to be published).

A. D. Kim, J. B. Keller, M. Moscoso, Department of Mathematics, Stanford University, Stanford, Calif. 94305-2125 are preparing a manuscript to be called “Point-spread functions in scattering media.”

Papanicolaou, G.

Polishchuk, A. Y.

A. Y. Polishchuk, M. Zevallos, F. Liu, R. R. Alfano, “Generalization of Fermat’s principle for photons in random media: the least mean square curvature of paths and photon diffusion on the velocity sphere,” Phys. Rev. E 53, 5523–5526 (1996).
[CrossRef]

Pomraning, G. C.

G. C. Pomraning, “The Fokker–Planck operator as an asymptotic limit,” Math. Models Methods Appl. Sci. 2, 21–36 (1992).

Prahl, S. A.

W-F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Trefethen, L. N.

L. N. Trefethen, Spectral Methods in Matlab (Society for Industrial and Applied Mathematics, Philadephia, Pa., 2000).

Weinberger, H. F.

J. B. Keller, H. F. Weinberger, “Boundary and initial boundary-value problems for separable backward–forward parabolic problems,” J. Math. Phys. 38, 4343–4353 (1997).
[CrossRef]

Welch, A. J.

W-F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Zevallos, M.

A. Y. Polishchuk, M. Zevallos, F. Liu, R. R. Alfano, “Generalization of Fermat’s principle for photons in random media: the least mean square curvature of paths and photon diffusion on the velocity sphere,” Phys. Rev. E 53, 5523–5526 (1996).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

W-F. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

J. Math. Phys. (1)

J. B. Keller, H. F. Weinberger, “Boundary and initial boundary-value problems for separable backward–forward parabolic problems,” J. Math. Phys. 38, 4343–4353 (1997).
[CrossRef]

J. Opt. Soc. Am. A (1)

Math. Models Methods Appl. Sci. (1)

G. C. Pomraning, “The Fokker–Planck operator as an asymptotic limit,” Math. Models Methods Appl. Sci. 2, 21–36 (1992).

Nucl. Sci. Eng. (1)

C. L. Leakeas, E. W. Larsen, “Generalized Fokker–Planck approximations of particle transport with highly forward-peaked scattering,” Nucl. Sci. Eng. 137, 236–250 (2001).

Phys. Rev. E (1)

A. Y. Polishchuk, M. Zevallos, F. Liu, R. R. Alfano, “Generalization of Fermat’s principle for photons in random media: the least mean square curvature of paths and photon diffusion on the velocity sphere,” Phys. Rev. E 53, 5523–5526 (1996).
[CrossRef]

Prog. Nucl. Energy (1)

E. W. Larsen, “The linear Boltzmann equation in optically thick systems with forward-peaked scattering,” Prog. Nucl. Energy 34, 413–423 (1999).
[CrossRef]

Other (4)

A. D. Kim, J. B. Keller, M. Moscoso, Department of Mathematics, Stanford University, Stanford, Calif. 94305-2125 are preparing a manuscript to be called “Point-spread functions in scattering media.”

A. D. Kim, M. Moscoso, “Radiative transfer computations for optical beams,” J. Comput. Phys. (to be published).

L. M. Delves, J. L. Mohamed, Computational Methods for Integral Equations (Cambridge U. Press, Cambridge, UK, 1985).

L. N. Trefethen, Spectral Methods in Matlab (Society for Industrial and Applied Mathematics, Philadephia, Pa., 2000).

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Figures (5)

Fig. 1
Fig. 1

Eigenvalues of the scattering operator in the radiative transport equation, the Fokker–Planck operator, the Leakeas–Larsen operator, and the modified Leakeas–Larsen operator. The Henyey–Greenstein phase function is used with g=0.90.

Fig. 2
Fig. 2

Backscattered radiance Ψ(μ, 0) (upper figure) and transmitted radiance Ψ(μ, z0) (lower figure) for a slab of thickness z0=1cm, with σs=100cm-1 and σa=1cm-1. The exponential phase function with ϵ=0.10 is used.

Fig. 3
Fig. 3

Same as Fig. 2 with ϵ=0.01.

Fig. 4
Fig. 4

Same as Fig. 2 with z0=0.05cm.

Fig. 5
Fig. 5

Backscattered radiance (upper figure) and transmitted radiance (lower figure) for the Henyey–Greenstein phase function with g=0.90 in a slab of thickness z0=0.1cm, with σs=100cm-1 and σa=1cm-1.

Equations (54)

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(c-1t+ω·+σa)Ψ(ω, r, t)=Ωf(ω·ω)Ψ(ω, r, t)dω-σsΨ(ω, r, t).
σs=Ωf(ω·ω)dω.
g=1σs 2π0π cos Θf(cos Θ)sin ΘdΘ.
Ψ(ω, r, t)=Ψ(ω, r, t)+(ω-ω)·ωΨ(ω, r, t)+12 [(ω-ω)·ω]2Ψ(ω, r, t)+O[(ω-ω)3].
L=µ(1-μ2)μ+(1-μ2)-1φ2.
(c-1t+ω·+σa)Ψ(μ, φ, r, t)=12σtrLΨ(μ, φ, r, t).
SYnm(ω)ωf(ω·ω)Ynm(ω)dω-σsYnm(ω)=(fn-σs)Ynm(ω).
fn=2π-11f(μ)Pn(μ)dμ.
LYnm(ω)=-n(n+1)Ynm(ω).
S=F(L).
fn-σs=F[-n(n+1)].
SαL(I-βL)-1.
α=12 (σs-f1)(1+2β),β=2σs-3f1+f26(f1-f2).
(c-1t+ω·+σa)Ψ=αL(I-βL)-1Ψ.
α=12 (σs-f1)(1+2β),β=σs-f12f1-4σs.
Ψ(ω, r, t)=v(ω, t)exp(λr·zˆ),
(c-1t+λω·zˆ+σa)v(ω, t)=Ωf(ω·ω)v(ω, t)dω-σsv(ω, t),
(c-1t+λω·zˆ+σa)v(ω, t)=(σtr/2)Lv(ω, t),
(c-1t+λω·zˆ+σa)v(ω, t)=αL(I-βL)-1v(ω, t).
(μz+σa)Ψ(μ, z)=-11h(μ, μ)Ψ(μ, z)dμ-σsΨ(μ, z).
h(μ, μ)=02πf[μμ+(1-μ2)1/2(1-μ2)1/2×cos(φ-φ)]d(φ-φ).
Ψ(μ, 0)=Fδ(μ-1),0<μ1.
Ψ(μ, z0)=0,-1μ<0.
(μz+σa)Ψ=σtr2 LΨ,
(μz+σa)Ψ=αL(I-βL)-1Ψ.
Ψ(μ, z)=v(μ)exp(λz).
(λμ+σa)v(μ)=-11h(μ, μ)v(μ)dμ-σsv(μ),
(λμ+σa)v(μ)=σtr2 Lv(μ),
(λμ+σa)v(μ)=αL(I-βL)-1v(μ).
Ψ(μ, z)=n=1{cnvn(μ)exp[λn(z-z0)]+c-nvn(-μ)exp(-λnz)}.
n=1[cnvn(μ)exp(-λnz0)+c-nvn(-μ)]=Fδ(μ-1),0<μ1,
n=1[cnvn(μ)+c-nvn(-μ)exp(-λnz0)]=0,-1μ<0.
Ψ(μ, z)=n=1c-nvn(-μ)exp(-λnz).
n=1c-nvn(-μ)=Fδ(μ-1),0<μ1.
Ψ(μ, z)c-1v1(-μ)exp(-λ1z),z1.
λμjVj+σaVj-k=1Nh(μj, μk)Vkwk=0forj=1, , N.
[λA+(σa+σs)I-H]v=0.
Ajk=δjkμk,j, k=1, , N,
Hjk=h(μj, μk)wk.
LLN=DN[diag(1-μj2)]DN.
λA+σaI-σtr2 LNv=0.
(λμ+σa)v(μ)=αLw(μ),
(I-βL)w(μ)=v(μ).
(λA+σaI)v=αLNw,
(I-βLN)w=v.
(λA+σaI)(I-βLN)w=αLNw.
Vik=V˜i,N-k+1,k=1, , N.
Ψ(μk, z)=n=1N/2{cn exp[λn(z-z0)]Vnk+c-n exp(-λnz)Vn,N-k+1}fork=1, , N.
n=1N/2[cn exp(-λnz0)Vnj+c-nVn,N-j+1]=Fjforj=N/2+1, , N,
n=1N/2[cnVnj+c-n exp(-λnz0)Vn,N-j+1]=0,forj=1, , N/2.
f(cos Θ)=σs exp[-(1-cos Θ)/ϵ]2πϵ[1-exp(-2/ϵ)].
h(μ, μ)=σsϵ exp[-(1-μμ)/ϵ]1-exp(-2/ϵ)×I0(ϵ-11-μ21-μ2).
f(cos Θ)=σs4π 1-g2(1+g2-2g cos Θ)3/2.
h(μ, μ)=2(1-g2)π(a-b)[(a+b)1/2] E2ba+b.

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