Abstract

We study light propagation in biological tissue using the radiative transport equation. The Green’s function is the fundamental solution to the radiative transport equation from which all other solutions can be computed. We compute the Green’s function as an expansion in plane-wave modes. We calculate these plane-wave modes numerically using the discrete-ordinate method. When scattering is sharply peaked, calculating the plane-wave modes for the transport equation is difficult. For that case we replace it with the Fokker–Planck equation since the latter gives a good approximation to the transport equation and requires less work to solve. We calculate the plane-wave modes for the Fokker–Planck equation numerically using a finite-difference approximation. The method of computing the Green’s function for it is the same as for the transport equation. We demonstrate the use of the Green’s function for the transport and Fokker–Planck equations by computing the point-spread function in a half-space composed of a uniform scattering and absorbing medium.

© 2004 Optical Society of America

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Corrections

Arnold D. Kim, "Transport theory for light propagation in biological tissue: erratum," J. Opt. Soc. Am. A 21, 1585-1585 (2004)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-21-8-1585

References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, Piscataway, N.J., 1996).
  2. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).
  3. K. M. Case, “On boundary value problems of linear transport theory,” in Proceedings of the Symposium in Applied Mathematics, Vol. 1, R. Bellman, G. Birkhoff, I. Abu-Shumays, eds. (American Mathematical Society, Providence, R.I., 1969), pp. 17–36.
  4. A. D. Kim, J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A 20, 92–98 (2003).
    [CrossRef]
  5. J. E. Morel, “An improved Fokker–Planck angular differencing scheme,” Nucl. Sci. Eng. 89, 131–136 (1985).
  6. 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).

2003 (1)

2001 (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).

1985 (1)

J. E. Morel, “An improved Fokker–Planck angular differencing scheme,” Nucl. Sci. Eng. 89, 131–136 (1985).

Case, K. M.

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

K. M. Case, “On boundary value problems of linear transport theory,” in Proceedings of the Symposium in Applied Mathematics, Vol. 1, R. Bellman, G. Birkhoff, I. Abu-Shumays, eds. (American Mathematical Society, Providence, R.I., 1969), pp. 17–36.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, Piscataway, N.J., 1996).

Keller, J. B.

Kim, A. D.

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).

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).

Morel, J. E.

J. E. Morel, “An improved Fokker–Planck angular differencing scheme,” Nucl. Sci. Eng. 89, 131–136 (1985).

Zweifel, P. F.

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

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

Nucl. Sci. Eng. (2)

J. E. Morel, “An improved Fokker–Planck angular differencing scheme,” Nucl. Sci. Eng. 89, 131–136 (1985).

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).

Other (3)

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, Piscataway, N.J., 1996).

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

K. M. Case, “On boundary value problems of linear transport theory,” in Proceedings of the Symposium in Applied Mathematics, Vol. 1, R. Bellman, G. Birkhoff, I. Abu-Shumays, eds. (American Mathematical Society, Providence, R.I., 1969), pp. 17–36.

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Figures (5)

Fig. 1
Fig. 1

Contour plot of the point-spread function for the radiative transport equation evaluated in direction ( μ ,   ϕ ) = ( - 0.9894 ,   π ) . The half-space is an isotropic scattering medium with σ a / σ s = 0.01 . It is for a source at depth z = 4 l s in direction ( μ ,   ϕ ) = ( - 0.9894 ,   0 ) . The x and y axes are normalized by the scattering mean free path l s . The gray scale of the contours is given in decibels.

Fig. 2
Fig. 2

Same as Fig. 1, but evaluated in the direction ( μ ,   ϕ ) = ( - 0.8565 ,   π ) .

Fig. 3
Fig. 3

Comparison of the results shown in Figs. 1 and 2. The top plot shows the point-spread function as a function of x normalized by l s for y = 0 , 2.5 l s , and 5.0 l s . The bottom plot shows the point-spread function as a function of y normalized by l s for x = 0 , 2.5 l s , and 5.0 l s . Dark curves, ( μ ,   ϕ ) = ( - 0.9894 ,   π ) ; light curves, ( μ ,   ϕ ) = ( - 0.8565 ,   π ) .

Fig. 4
Fig. 4

Contour plot of the point-spread function for the Fokker–Planck equation with g = 0.95 . All other parameters are the same as in Fig. 1.

Fig. 5
Fig. 5

Comparison of the results shown in Figs. 1 and 4. Dark curves, results from the Fokker–Planck equation; light curves, results from the radiative transport equation. All other parameters are the same as in Fig. 3.

Equations (72)

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ω Ψ + σ a Ψ + σ s L Ψ = Q ,
L Ψ = Ψ ( ω ,   r ) - Ω f ( ω ω ) Ψ ( ω ,   r ) d ω .
Ψ ( ω ,   r s ) = h ( ω ,   r s ) , ω Ω in ( r s ) , r s S .
Ω in ( r s ) = { ω : ω n ˆ ( r s ) > 0 } ,
Ψ ( ω ,   r ) = D Ω G ( ω ,   r ;   ω ,   r ) Q ( ω ,   r ) d ω d r + S Ω G ( ω ,   r ;   ω ,   r s ) [ ω n ˆ ( r s ) ]× Ψ ( ω ,   r s ) d ω d r s .
Ψ ( ω ,   r s ) = D Ω G ( ω ,   r s ;   ω ,   r ) Q ( ω ,   r ) d ω d r + S Ω G ( ω ,   r s ;   ω ,   r s ) [ ω n ˆ ( r s ) ]× Ψ ( ω ,   r s ) d ω d r s .
Ψ ( ω ,   r s ) = Ψ in ( ω ,   r s ) ω Ω in ( r s ) Ψ out ( ω ,   r s ) ω Ω out ( r s ) ,
Ω out ( r s ) = { ω : ω n ˆ ( r s ) < 0 } .
Ψ ( ν ,   μ ,   ρ ,   z ) = ( 2 π ) - 2 R 2 V ( ν ,   μ ;   κ ) exp [ λ ( κ ) z ]× exp ( i κ ρ ) d κ .
λ μ V + i ν κ V + σ a V + σ s LV = 0 .
λ - j = - λ j , V - j ( ν ,   μ ;   κ ) = V j ( ν ,   - μ ;   κ ) ,
j = 1 , 2 ,   .
λ j μ V j + i ν κ V j + σ a V j + σ s LV j = 0 ,
λ k μ V k + i ν κ V k + σ a V k + σ s LV k = 0 .
( λ j - λ k ) Ω V j ( ν ,   μ ) V k ( ν ,   μ ) μ d ω = 0 .
Ω V j ( ν ,   μ ) V j ( ν ,   μ ) μ d ω = c .
Ω V - j ( ν ,   μ ) V - j ( ν ,   μ ) μ d ω = - c .
Ω V j ( ν ,   μ ) V j ( ν ,   μ ) μ d ω = + 1 for j < 0 - 1 for j > 0 .
μ z Ψ ( ω ,   z ) + σ a Ψ ( ω ,   z ) + σ s L Ψ ( ω ,   z ) = 0
Ψ ( ω ,   0 ) = h ( ω ) , ω z ˆ > 0 .
Ψ ( ω ,   z ) = j a j   exp ( λ j z ) V j ( ω ) .
j < 0 a j V j ( ω ) = h ( ω ) , ω z ˆ > 0 .
ω = [ ( 1 - μ 2 ) 1 / 2   cos   ϕ ,   ( 1 - μ 2 ) 1 / 2   sin   ϕ ,   μ ] .
LV ( μ m ,   ϕ m ) V ( μ m ,   ϕ n ) - m = 1 M n = 1 N f ( μ m ,   ϕ n ;   μ m ,   ϕ n )× V ( μ m ,   ϕ n ) w m Δ ϕ .
λ μ m V ( μ m ,   ϕ n ) + i ( 1 - μ m 2 ) 1 / 2 ( κ x   cos   ϕ n
+ κ y   sin   ϕ n ) V ( μ m ,   ϕ n ) + σ a V ( μ m ,   ϕ n )
+ σ s LV ( μ m ,   ϕ n ) = 0 ,
m = 1 , , M , n = 1 , , N .
γ j = m = 1 M n = 1 N V j ( μ m ,   ϕ n ) V j ( μ m ,   ϕ n ) μ m w m Δ ϕ .
Re ( λ - MN / 2 ) < < Re ( λ - 1 ) < Re ( λ + 1 ) < < Re ( λ + MN / 2 ) .
ω G + σ a G + σ s LG = δ ( ω - ω ) δ ( r - r )
G ( ω ,   r ;   ω ,   r ) = 1 ( 2 π ) 2 R 2 G ˆ ( ω ,   z ;   ω ,   z ,   κ )× exp [ i κ ( ρ - ρ ) ] d κ ,
μ z G ˆ + i ν κ G ˆ + σ a G ˆ + σ s LG ˆ = δ ( ω - ω ) δ ( z - z ) .
μ G ˆ ( ω ,   z + 0 ;   ω ,   z ,   κ ) - μ G ˆ ( ω ,   z - 0 ;   ω ,   z ,   κ )
= δ ( ω - ω ) .
G ˆ ( ω ,   z ;   ω ,   z ,   κ ) = j g j ( z ;   ω ,   z ,   κ ) V j ( ω ) .
j { [ z - λ j ( κ ) ] g j ( z ;   ω ,   z ,   κ ) μ V j ( ω ;   κ ) }
= δ ( ω - ω ) δ ( z - z ) .
z g j ( z ;   ω ,   z ,   κ ) - λ j ( κ ) g j ( z ;   ω ,   z ,   κ )
= - sgn ( j ) V j ( ω ;   κ ) δ ( z - z ) ,
z C j ( z ;   z ,   κ ) - λ j ( κ ) C j ( z ;   z ,   κ ) = δ ( z - z ) .
C j ( z + 0 ;   z ,   κ ) - C j ( z - 0 ;   z ,   κ ) = 1 .
C j ( z ;   z ,   κ ) = - exp [ λ j ( κ ) ( z - z ) ] , z < z , j > 0 0 , z < z , j < 0 0 , z > z , j > 0 + exp [ λ j ( κ ) ( z - z ) ] , z > z , j < 0 .
G ˆ ( ω ,   z ;   ω ,   z ,   κ )
= j > 0 exp [ λ j ( κ ) ( z - z ) ] V j ( ω ;   κ ) V j ( ω ;   κ ) , z < z j < 0 exp [ λ j ( κ ) ( z - z ) ] V j ( ω ;   κ ) V j ( ω ;   κ ) , z > z .
L Ψ = - 1 2 ( 1 - g ) Δ Ψ .
g = 2 π - 1 + 1 ω ω f ( ω ω ) d ( ω ω ) .
α m + 1 / 2 = α m - 1 / 2 - 2 μ m w m , m = 1 , , M
Δ V ( μ m ,   ϕ n )
1 w m α m + 1 / 2   V ( μ m + 1 ,   ϕ n ) - V ( μ m ,   ϕ n ) μ m + 1 - μ m - α m - 1 / 2   V ( μ m ,   ϕ n ) - V ( μ m - 1 ,   ϕ n ) μ m - μ m - 1 + 1 1 - μ m 2× V ( μ m ,   ϕ n + 1 ) - 2 V ( μ m ,   ϕ n ) + V ( μ m ,   ϕ n - 1 ) ( Δ ϕ ) 2 .
V ( μ ,   ϕ ) = V ( μ ,   ϕ + 2 π ) .
λ μ m V ( μ m ,   ϕ n ) + i ( 1 - μ m 2 ) 1 / 2 ( κ x cos   ϕ n
+ κ y sin   ϕ n ) V ( μ m ,   ϕ n ) + σ a V ( μ m ,   ϕ n )
- 1 2 σ s ( 1 - g ) Δ V ( μ m ,   ϕ n ) = 0 ,
m = 1 , , M , n = 1 , , N .
ω Ψ + σ a Ψ + σ s L Ψ = δ ( ρ ) δ ( z - z ) δ ( ω - ω )
Ψ ( ω ,   ρ ,   z = 0 ) = 0 , ω z ˆ > 0 , ρ R 2 .
Ψ = G - Y
Y ( ω ,   ρ ,   0 ;   ω ,   0 ,   z ) = G ( ω ,   ρ ,   0 ;   ω ,   0 ,   z ) ,
ω z ˆ > 0 .
Y ( ω ,   r ;   ω ,   r ) = 1 ( 2 π ) 2 R 2 Y ˆ ( ω ,   z ;   ω ,   z ,   κ )× exp ( i κ ρ ) d κ ,
Y ˆ ( ω ,   z ;   ω ,   z , κ ) = j < 0 y j ( ω ,   z ;   κ )× exp [ λ j ( κ ) z ] V j ( ω ;   κ ) .
j < 0 y j ( ω ,   z ;   κ ) V j ( ω ) = G ˆ ( ω ,   0 ;   ω ,   z ,   κ ) ,
ω z ˆ > 0 .
G ˆ ( ω ,   0 ;   ω ,   z ,   κ ) = j > 0 exp [ - λ j ( κ ) z ]× V j ( ω ;   κ ) V j ( ω ;   κ ) .
y j ( ω ,   z ;   κ ) = k d jk ( κ ) exp [ - λ k ( κ ) z ] V k ( ω ;   κ ) .
j < 0 d jk ( κ ) V j ( ω ;   κ ) = V k ( ω ;   κ ) ,
ω z ˆ > 0 , k > 0 .
Y ˆ ( ω ,   z ;   ω ,   z , κ ) = j < 0 k > 0 d jk ( κ )× exp [ - λ k ( κ ) z ] V k ( ω ;   κ )× exp [ λ j ( κ ) z ] V j ( ω ;   κ ) .
Ψ ˆ ( ω ,   0 ;   κ ) = R 2 Ψ ( ω ,   ρ ,   0 ) exp ( - i κ ρ ) d ρ .
Ψ ˆ ( ω ,   0 ;   κ ) exp [ - λ 1 ( κ ) z ] V 1 ( ω ;   κ )× V 1 ( ω ;   κ ) - j < 0 d j , 1 ( κ ) V j ( ω ;   κ ) ,
as z .

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