Abstract

The diffusion approximation to the radiative transport equation applies for light that has propagated deeply into an optically thick medium, such as biological tissue. It does not accurately model light near boundaries where measurements of scattered light are often taken. Here, we compute a correction to the diffusion approximation at the boundary. This correction requires only small modifications to the standard diffusion approximation used in biomedical optics. In particular, one needs only to compute the coefficients in the boundary condition for the diffusion approximation and an additive correction. We give explicit procedures for these computations. Using numerical results for the steady-state plane–parallel slab problem, we show that this corrected diffusion approximation is a much better approximation than the standard diffusion approximation for modeling the reflectance and transmittance.

© 2011 Optical Society of America

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References

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  2. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).
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2009

S. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[CrossRef]

2008

2007

2005

A. D. Kim, “A boundary integral method to compute Green’s functions for the radiative transport equation,” Waves Random Complex Media 15, 17–42 (2005).
[CrossRef]

2004

1999

1998

1996

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[CrossRef]

1995

1994

1993

R. Aronson, “Extrapolation distance for diffusion of light,” Proc. SPIE 1888, 297–305 (1993).
[CrossRef]

1991

F. Malvagi and G. C. Pomraning, “Initial and boundary conditions for diffusive linear transport problems,” J. Math. Phys. 32, 805–820 (1991).
[CrossRef]

1989

1977

E. W. Larsen, “Asymptotic theory of the linear transport equation for small mean free paths. II,” SIAM J. Appl. Math. 33, 427–445 (1977).
[CrossRef]

1976

E. W. Larsen and J. D’Arruda, “Asymptotic theory of the linear transport equation for small mean free paths. I,” Phys. Rev. A 13, 1933–1939 (1976).
[CrossRef]

1975

G. J. Habetler and B. J. Matkowsky, “Uniform asymptotic expansions in transport theory with small mean free paths, and the diffusion approximation,” J. Math. Phys. 16, 846–854 (1975).
[CrossRef]

1974

E. W. Larsen and J. B. Keller, “Asymptotic solution of neutron transport problems for small mean free paths,” J. Math. Phys. 15, 75–81 (1974).
[CrossRef]

Aronson, R.

Arridge, S.

S. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[CrossRef]

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

Case, K. M.

K. M. Case and P. F. Zweifel, Linear Transport Theory(Addison-Wesley, 1967).

Corngold, N.

D’Arruda, J.

E. W. Larsen and J. D’Arruda, “Asymptotic theory of the linear transport equation for small mean free paths. I,” Phys. Rev. A 13, 1933–1939 (1976).
[CrossRef]

Feng, T.-C.

González-Rodríguez, P.

Habetler, G. J.

G. J. Habetler and B. J. Matkowsky, “Uniform asymptotic expansions in transport theory with small mean free paths, and the diffusion approximation,” J. Math. Phys. 16, 846–854 (1975).
[CrossRef]

Haskell, R. C.

Ishimaru, A.

Keller, J. B.

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[CrossRef]

E. W. Larsen and J. B. Keller, “Asymptotic solution of neutron transport problems for small mean free paths,” J. Math. Phys. 15, 75–81 (1974).
[CrossRef]

Kim, A. D.

Larsen, E. W.

E. W. Larsen, “Asymptotic theory of the linear transport equation for small mean free paths. II,” SIAM J. Appl. Math. 33, 427–445 (1977).
[CrossRef]

E. W. Larsen and J. D’Arruda, “Asymptotic theory of the linear transport equation for small mean free paths. I,” Phys. Rev. A 13, 1933–1939 (1976).
[CrossRef]

E. W. Larsen and J. B. Keller, “Asymptotic solution of neutron transport problems for small mean free paths,” J. Math. Phys. 15, 75–81 (1974).
[CrossRef]

Malvagi, F.

F. Malvagi and G. C. Pomraning, “Initial and boundary conditions for diffusive linear transport problems,” J. Math. Phys. 32, 805–820 (1991).
[CrossRef]

Matkowsky, B. J.

G. J. Habetler and B. J. Matkowsky, “Uniform asymptotic expansions in transport theory with small mean free paths, and the diffusion approximation,” J. Math. Phys. 16, 846–854 (1975).
[CrossRef]

McAdams, M. S.

Moscoso, M.

Nieto-Vesperinas, M.

Papanicolaou, G.

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[CrossRef]

Pomraning, G. C.

F. Malvagi and G. C. Pomraning, “Initial and boundary conditions for diffusive linear transport problems,” J. Math. Phys. 32, 805–820 (1991).
[CrossRef]

Ripoll, J.

Ryzhik, L.

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[CrossRef]

Schotland, J. C.

S. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[CrossRef]

Svaasand, L. O.

Tromberg, B. J.

Tsay, T.-T.

Wang, L. V.

L. V. Wang and H. Wu, Biomedical Optics: Principles and Imaging (Wiley, 2007).

Wu, H.

L. V. Wang and H. Wu, Biomedical Optics: Principles and Imaging (Wiley, 2007).

Zweifel, P. F.

K. M. Case and P. F. Zweifel, Linear Transport Theory(Addison-Wesley, 1967).

Appl. Opt.

Inverse Probl.

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

S. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25, 123010 (2009).
[CrossRef]

J. Math. Phys.

E. W. Larsen and J. B. Keller, “Asymptotic solution of neutron transport problems for small mean free paths,” J. Math. Phys. 15, 75–81 (1974).
[CrossRef]

G. J. Habetler and B. J. Matkowsky, “Uniform asymptotic expansions in transport theory with small mean free paths, and the diffusion approximation,” J. Math. Phys. 16, 846–854 (1975).
[CrossRef]

F. Malvagi and G. C. Pomraning, “Initial and boundary conditions for diffusive linear transport problems,” J. Math. Phys. 32, 805–820 (1991).
[CrossRef]

J. Opt. Soc. Am. A

Phys. Rev. A

E. W. Larsen and J. D’Arruda, “Asymptotic theory of the linear transport equation for small mean free paths. I,” Phys. Rev. A 13, 1933–1939 (1976).
[CrossRef]

Proc. SPIE

R. Aronson, “Extrapolation distance for diffusion of light,” Proc. SPIE 1888, 297–305 (1993).
[CrossRef]

SIAM J. Appl. Math.

E. W. Larsen, “Asymptotic theory of the linear transport equation for small mean free paths. II,” SIAM J. Appl. Math. 33, 427–445 (1977).
[CrossRef]

Wave Motion

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327–370 (1996).
[CrossRef]

Waves Random Complex Media

A. D. Kim, “A boundary integral method to compute Green’s functions for the radiative transport equation,” Waves Random Complex Media 15, 17–42 (2005).
[CrossRef]

Other

K. M. Case and P. F. Zweifel, Linear Transport Theory(Addison-Wesley, 1967).

L. V. Wang and H. Wu, Biomedical Optics: Principles and Imaging (Wiley, 2007).

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE, 1997).

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Figures (2)

Fig. 1
Fig. 1

Sketch showing the coordinate system defined with respect to the boundary point r b . The vector ρ lies on the tangent plane, and the coordinate ζ is orthogonal to the tangent plane directed in the domain D.

Fig. 2
Fig. 2

Plots of the diffusion approximation (dashed curves), the corrected diffusion approximation (solid curves), and the numerical solution (circles) at (a)  z = z 0 = 0 and (b)  z = z 1 = 1 . The parameters for this problem are ϵ = 0.01 , α = σ = 1 , g = 0.8 , and z 0 = 1 . The ratio of the refractive index inside the slab over the refractive index outside of the slab on either side is set to m = 1.40 .

Tables (2)

Tables Icon

Table 1 Percent Relative Errors for the Reflectance and Transmittance with ϵ = 0.01 and Varying Values of g

Tables Icon

Table 2 Percent Relative Errors for the Reflectance and Transmittance with g = 0.8 and Varying Values of ϵ

Equations (79)

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Ω · I + μ a I + μ s L I = 0 ,
L I = I S 2 p ( Ω · Ω ) I ( Ω , r ) d Ω .
I = f + R I on     Ω · n ^ ( r b ) > 0 , r b D ,
I Φ Ω · ( 3 κ Φ ) ,
· ( κ Φ ) α Φ = 0 in     D ,
a Φ b n ^ · ( κ Φ ) = f 0 on     D ,
I Φ c Ω · ( κ Φ ) + Ψ ,
μ a = ϵ α and μ s = ϵ 1 σ ,
ϵ Ω · I + ϵ 2 α + σ L I = 0 .
I = Φ + Ψ .
Φ k = 0 ϵ k ϕ k .
σ L ϕ k = Ω · ϕ k 1 α ϕ k 2 for     k = 0 , 1 , ,
S 2 p ( Ω · Ω ) d Ω = 1 .
S 2 p ( Ω · Ω ) Ω · F d Ω = g Ω · F ,
L ϕ 2 = Ω · [ Ω · ϕ 0 ( r ) σ ( 1 g ) ] α ϕ 0 .
· ( κ ϕ 0 ) α ϕ 0 = 0 ,
κ = 1 / [ 3 σ ( 1 g ) ] .
Φ ϕ 0 ϵ 3 Ω · ( κ ϕ 0 ) ,
ζ = ϵ h ( ϵ ρ ) , h ( 0 ) = 0 .
Ψ ( Ω , ρ , 0 ) = f ( Ω , ρ , 0 ) + R Ψ ( Ω , ρ , 0 ) Φ ( Ω , ρ , 0 ) + R Φ ( Ω , ρ , 0 ) + O ( ϵ 2 ) on     Ω · n ^ ( r b ) > 0 .
R I ( μ , Ω , ρ , 0 ) = r ( μ ) I ( μ , Ω , ρ , 0 ) on     0 < μ 1 ,
Ψ ( μ , Ω , ρ , 0 ) = f ( μ , ρ , 0 ) + r ( μ ) Ψ ( μ , Ω , ρ , 0 ) Φ ( μ , Ω , ρ , 0 ) + r ( μ ) Φ ( μ , Ω , ρ , 0 ) + O ( ϵ 2 ) on     0 < μ 1 .
μ ζ * ψ + ϵ Ω · ψ + ϵ 2 α ψ + σ L ψ = 0 ,
ψ 0 , as     ζ * .
ψ k = 0 ϵ k ψ k .
μ z * ψ k + σ L ψ k = Ω · ψ k 1 α ψ k 2 for     k = 0 , 1 , ,
ψ 0 ( μ , Ω , ρ , 0 ) = f ( μ , ρ , 0 ) + r ( μ ) ψ 0 ( μ , Ω , ρ , 0 ) [ 1 r ( μ ) ] ϕ 0 ( ρ , 0 ) on     0 < μ 1 ,
ψ 1 ( μ , Ω , ρ , 0 ) = r ( μ ) ψ 1 ( μ , Ω , ρ , 0 ) + 3 [ 1 + r ( μ ) ] μ κ ζ ϕ 0 ( ρ , 0 ) + 3 [ 1 r ( μ ) ] Ω · [ κ ϕ 0 ( ρ , 0 ) ] on     0 < μ 1 .
μ ζ * ψ 0 + σ L 0 ψ 0 = 0 in     ζ * > 0 ,
ψ 0 ( μ , ρ , 0 ) = f ( μ , ρ , 0 ) + r ( μ ) ψ 0 ( μ , ρ , 0 ) [ 1 r ( μ ) ] ϕ 0 ( ρ , 0 ) on     0 < μ 1 .
L 0 ψ 0 = ψ 0 1 1 p 0 ( μ , μ ) ψ 0 ( μ , ρ , ζ * ) d μ ,
p 0 ( μ , μ ) = 1 2 π π π p ( μ μ + ( 1 μ 2 ) 1 / 2 ( 1 μ 2 ) 1 / 2 cos ( φ φ ) ) d ( φ φ ) .
μ ζ * ψ 1 + σ L 0 ψ 1 = 0 in     ζ * > 0 ,
ψ 1 ( μ , ρ , 0 ) = r ( μ ) ψ 1 ( μ , ρ , 0 ) + 3 [ 1 + r ( μ ) ] μ κ ζ ϕ 0 ( ρ , 0 ) on     0 < μ 1 .
μ ζ * ψ + σ L 0 ψ = 0 in     ζ * > 0 ,
ψ ( μ , ρ , 0 ) = f ( μ , ρ , 0 ) + r ( μ ) ψ ( μ , ρ , 0 ) [ 1 r ( μ ) ] ϕ 0 ( ρ , 0 ) + 3 ϵ [ 1 + r ( μ ) ] μ κ ζ ϕ 0 ( ρ , 0 ) on     0 < μ 1 .
μ ζ * ψ + σ L 0 = 0 in     ζ * > 0 ,
ψ ( μ , 0 ) = s ( μ ) + r ( μ ) ψ ( μ , 0 ) on     0 < μ 1 .
μ ζ * H + σ L 0 H = δ ( μ μ ) δ ( ζ * ζ ) in     ζ * > 0 ,
H ( μ , 0 ; μ , ζ ) = r ( μ ) H ( μ , 0 ; μ , ζ ) on     0 < μ 1 .
ψ ( μ , ζ * ) = 0 1 H ( μ , ζ * ; μ , 0 ) s ( μ ) μ d μ .
ψ ( μ , ζ * ) = j > 0 W j ( μ ) exp ( λ j ζ * ) 0 1 [ W j ( μ ) + k > 0 y j k V k ( μ ) ] s ( μ ) μ d μ .
P s = 0 1 [ W 1 ( μ ) + k > 0 y 1 k V k ( μ ) ] s ( μ ) μ d μ = 0 ,
P [ f ( μ , ρ , 0 ) [ 1 r ( μ ) ] ϕ 0 ( ρ , 0 ) + μ [ 1 + r ( μ ) ] 3 ϵ κ ζ ϕ 0 ( ρ , 0 ) ] = 0 .
f 0 = P [ f ( μ , ρ , 0 ) ] ,
a = P [ 1 r ( μ ) ] ,
b = ϵ 3 κ P [ μ ( 1 + r ( μ ) ) ] .
a ϕ 0 b n ^ · ϕ 0 = f 0 on     D .
ψ ( μ , ρ , ζ * ) = 0 1 H ( μ , 0 ; μ , ζ ) [ f ( μ , ρ , 0 ) [ 1 r ( μ ) ] ϕ 0 ( ρ , 0 ) + μ [ 1 + r ( μ ) ] ϵ 3 κ ζ ϕ 0 ( ρ , 0 ) ] μ d μ .
m ( r b ) = NA I ( Ω , r b ) Ω · n ^ ( r b ) d Ω ,
I ( Ω , r b ) ϕ 0 ( r b ) ϵ 3 Ω · [ κ ϕ 0 ( r b ) ] + ψ ( μ , r b ) .
R = 1 0 t 0 ( μ ) I ( μ , z 0 ) μ d μ ,
T = 0 1 t 1 ( μ ) I ( μ , z 1 ) μ d μ .
μ z I + ϵ α I + ϵ 1 σ L 0 I = 0 in     z 0 < z < z 1 ,
I ( μ , z 0 ) = δ ( μ 1 ) + r 0 ( μ ) I ( μ , z 0 ) on     0 < μ 1 ,
I ( μ , z 1 ) = r 1 ( μ ) I ( μ , z 1 ) on 1 μ < 0 .
κ z 2 ϕ 0 α ϕ 0 = 0 in     z 0 < z < z 1 ,
a 0 ϕ 0 b 0 z ϕ 0 = f 0 on     z = z 0 ,
a 1 ϕ 0 + b 1 z ϕ 0 = 0 on     z = z 1 .
f 0 = P [ δ ( μ 1 ) ] ,
a i = P [ 1 r i ( μ ) ] , i = 0 , 1 ,
b i = ϵ 3 κ P [ μ ( 1 r i ( μ ) ) ] , i = 0 , 1 .
[ a 0 cos h ( k z 0 ) k b 0 sin h ( k z 0 ) ] c 1 + [ a 0 sin h ( k z 0 ) k b 0 cos h ( k z 0 ) ] c 2 = f 0 ,
[ a 1 cos h ( k z 1 ) + k b 1 sin h ( k z 1 ) ] c 1 + [ a 1 sin h ( k z 1 ) + k b 1 cos h ( k z 1 ) ] c 2 = 0 .
I DA ( μ , z ) = [ c 1 μ ϵ 3 κ k c 2 ] cos h ( k z ) + [ c 2 μ ϵ 3 κ k c 1 ] sin h ( k z ) .
I cDA ( μ , z 0 ) = I DA ( μ , z 0 ) + ψ ( 0 ) ( μ ) ,
ψ ( 0 ) ( μ ) = H ( μ , 0 ; 1 , 0 + ) ϕ 0 ( z 0 ) 0 1 H ( μ , 0 ; μ , 0 + ) [ 1 r 0 ( μ ) ] μ d μ + ϵ 3 κ d ϕ 0 ( z 0 ) d z 0 1 H ( μ , 0 ; μ , 0 + ) [ 1 + r 0 ( μ ) ] μ 2 d μ .
H ( μ , 0 + ; μ , 0 ) = j > 0 W j ( μ ) [ W j ( μ ) + k > 0 y j k V k ( μ ) ] .
I cDA ( μ , z 1 ) = I DA ( μ , z 1 ) + ψ ( 1 ) ( μ ) ,
ψ ( 1 ) ( μ ) = ϕ 0 ( z 1 ) 1 0 H ( μ , 0 ; μ , 0 + ) [ 1 r 1 ( μ ) ] μ d μ ϵ 3 κ d ϕ 0 ( z 1 ) d z 1 0 H ( μ , 0 ; μ , 0 + ) [ 1 + r 1 ( μ ) ] μ 2 d μ .
μ ζ * ψ + α ¯ ψ + σ ¯ L 0 ψ = 0
λ μ V + α ¯ V + σ ¯ L 0 V = 0 .
( λ λ ) 1 1 V ( μ ) V ( μ ) μ d μ = 0 .
< λ j < < λ 1 < λ 1 < < λ j < .
λ j = λ j , V j ( μ ) = V j ( μ ) , j = 1 , 2 , .
1 1 V j 2 ( μ ) μ d μ = sgn ( j ) .
H ( μ , ζ * ; μ , ζ ) = G ( μ , ζ * ; μ , ζ ) j > 0 W j ( μ ) e λ j ζ * k > 0 y j k V k ( μ ) e λ k ζ ,
G ( μ , ζ * ; μ , z ) = { j > 0 V j ( μ ) e λ j ( z z ) V j ( μ ) ζ * < ζ , j > 0 W j ( μ ) e λ j ( z z ) W j ( μ ) ζ * > ζ .
j > 0 [ W j ( μ ) r ( μ ) V j ( μ ) ] y j k = [ V k ( μ ) r ( μ ) W k ( μ ) ] on     0 < μ 1 for     k > 0 .

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