Abstract

We consider a concise method based on recurrent relations that permit rigorous computing of the first and the second moments of the components of the vector locating a randomly walking photon in an infinite homogeneous light-scattering medium. On assumption that the components obey a three-dimensional Gaussian distribution a probability density for the photon locations at the Nth scattering event can readily be written down and the light-intensity distribution in the medium may be calculated. The results from theoretical analyses are compared with the solution of a light-diffusion equation and with results of Monte Carlo simulations and show a better fit with simulated data than the diffusion approximation.

© 1996 Optical Society of America

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  2. K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 1531–1560 (1994).
    [CrossRef]
  3. M. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
    [CrossRef] [PubMed]
  4. R. Graaff, J. G. Aarnoudse, F. F. M. de Mul, H. W. Jentink, “Light propagation parameters for anisotropically scattering media based on a rigorous solution of the transport equation,” Appl. Opt. 28, 2273–2279 (1989).
    [CrossRef] [PubMed]
  5. R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
    [CrossRef] [PubMed]
  6. A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Scaling relationships for theories of anisotropic random walks applied to tissue optics,” Appl. Opt. 32, 504–516 (1993).
    [CrossRef] [PubMed]
  7. R. F. Lutomirski, A. P. Ciervo, G. J. Hall, “Moments of multiple scattering,” Appl. Opt. 34, 7125–7136 (1995).
    [CrossRef] [PubMed]

1995 (1)

1994 (1)

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 1531–1560 (1994).
[CrossRef]

1993 (1)

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Scaling relationships for theories of anisotropic random walks applied to tissue optics,” Appl. Opt. 32, 504–516 (1993).
[CrossRef] [PubMed]

1989 (2)

1987 (1)

Aarnoudse, J. G.

Bonner, R. F.

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Scaling relationships for theories of anisotropic random walks applied to tissue optics,” Appl. Opt. 32, 504–516 (1993).
[CrossRef] [PubMed]

R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
[CrossRef] [PubMed]

Chance, B.

Ciervo, A. P.

de Mul, F. F. M.

Furutsu, K.

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 1531–1560 (1994).
[CrossRef]

Gandjbakhche, A. H.

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Scaling relationships for theories of anisotropic random walks applied to tissue optics,” Appl. Opt. 32, 504–516 (1993).
[CrossRef] [PubMed]

Graaff, R.

Hall, G. J.

Havlin, S.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).

Jentink, H. W.

Lutomirski, R. F.

Nossal, R.

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Scaling relationships for theories of anisotropic random walks applied to tissue optics,” Appl. Opt. 32, 504–516 (1993).
[CrossRef] [PubMed]

R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
[CrossRef] [PubMed]

Patterson, M.

Weiss, G. H.

Wilson, B. C.

Yamada, Y.

K. Furutsu, Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 1531–1560 (1994).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Geometry of a scattering event. The scattering particle is at the origin of the coordinate system.

Fig. 2
Fig. 2

Trajectory traveled by a photon by the instance of the Nth scattering event.

Fig. 3
Fig. 3

Relative average Oz component of the vector R N as a function of the number of scattering events N for different values of asymmetry factor g. Circles, squares, and plusses represent Monte Carlo simulated values.

Fig. 4
Fig. 4

Relative second moment of the Ox component of the vector R N as a function of the number of scattering events N for different values of asymmetry factor g. Circles, squares, and plusses represent Monte Carlo simulated values.

Fig. 5
Fig. 5

Relative second moment of the Oz component of the vector R N as a function of the number of scattering events N for different values of asymmetry factor g. Circles, squares, and plusses represent Monte Carlo simulated values.

Fig. 6
Fig. 6

Propagation of the front of the photon’s cloud injected simultaneously at the origin of the coordinate system in the Oz direction. (The origin is the intersection of the x and the z axes.) The cloud propagates in the infinite light-scattering medium with the time represented in terms of the number of scattering events N. All fronts are plotted on the same space scale for the H–G phase function of the scattering particles.

Fig. 7
Fig. 7

Relative position (〈R z 〉/λ) of the center and the radius [std(R z )/λ] of the photon’s cloud with respect to the Oz axis as a function of the number of scattering events N. All the photons enter the infinite scattering medium at the same time in the Oz direction. The crossing point of the curves corresponds to the moment N = N del when the front of the cloud reaches the XOY plane. Asymmetry factor g = 0.9 and N del ≈ 25.

Fig. 8
Fig. 8

Time delay represented in terms of the number of scattering events N del as a function of asymmetry factor g.

Fig. 9
Fig. 9

Intensity profiles of light I(r) propagating in the infinite scattering medium as measured in the xOy plane with the photons entering in the Oz direction. The solid curves correspond to the approximation suggested in this paper, the dashed curve is the diffusion approximation. Results of the Monte Carlo simulations are shown as open squares. The results are presented for two values of asymmetry factor (a) g = 0.9 and (b) g = 0.5.

Fig. 10
Fig. 10

Probability densities W(r|N) of the photon coordinates as measured in the xOy plane for asymmetry factor g = 0.9 at two distances, (a) r = 3 mm and (b) r = 1 mm, from the origin at which photons enter the medium in the Oz direction. Solid curve, the approximation suggested in this paper; dashed curve, the diffusion approximation; open squares, results of the Monte Carlo simulation.

Fig. 11
Fig. 11

Probability densities W(r|N) of the photon coordinates as measured in the xOy plane for the asymmetry factor g = 0.5 at two distances, (a) r = 3 mm and (b) r = 1 mm, from the origin at which the photons enter the medium in the Oz direction. Solid curves, the approximation suggested in this paper; dashed curves, the diffusion approximation; open squares, results of the Monte Carlo simulation.

Fig. 12
Fig. 12

Geometry of a scattering event in the laboratory and local (primed) systems of coordinates. (The origin is at the point of intersection of the axes.)

Tables (1)

Tables Icon

Table 1 Characteristic Numbers of Scattering Events as a Function of the Asymmetry Factor

Equations (51)

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W ( l ) = 1 λ exp ( l / λ ) ,
W ( φ s ) = 1 2 π .
W ( θ s ) = f ( θ s ) sin θ s .
f ( θ s ) = ( 1 g 2 ) 2 [ 1 + g 2 2 g cos ( θ s ) ] 3 / 2 ,
g = cos θ s = 0 π cos θ s W ( θ s ) s .
W a ( s ) = μ a exp ( μ a s ) ,
R N = k = 1 N r k = k = 1 N l k · n k ,
R N z = λ 1 g N 1 g , R N x = R N y = 0 ;
R N z 2 = 2 λ 2 3 ( 1 g ) [ N 1 g N 1 g 2 × ( 2 + g + g 2 + 2 g N + 1 ) ] , R N x 2 = R N y 2 = 2 λ 2 3 ( 1 g ) × [ N 1 g N 1 g 2 ( 1 + g + g 2 g N + 1 ) ] , R N x R N y = R N x R N z = R N y R N z = 0 .
R N z = λ , R N x = R N y = 0 , R N x 2 = R N y 2 = R N z 2 = 2 3 λ 2 N , R N x R N y = R N x R N z = R N y R N z = 0 .
R N z = λ 1 g , R N x = R N y = 0 , R N x 2 = R N y 2 = R N z 2 = 2 λ 2 3 ( 1 g ) N , R N x R N y = R N x R N z = R N y R N z = 0 .
rms ( R x ) [ 2 c t 3 ( 1 g ) ] 1 / 2 = V t ,
I ( z ) = 1 λ exp ( z / λ )
W ( x , y , z | N ) = ( 2 πσ 2 ) 3 / 2 exp [ x 2 + y 2 + ( z z ¯ ) 2 2 σ 2 ] ,
W ( x , y , z | N ) μ a λ ( 2 πσ 2 ) 3 / 2 × exp [ x 2 + y 2 + ( z z ¯ ) 2 2 σ 2 ] × exp ( μ a λ N ) .
Φ ( x , y , z | t ) = c ( 4 π D c t ) 3 / 2 × exp ( x 2 + y 2 + z 2 4 D c t ) exp ( μ a c t ) .
D = [ 3 μ s ( 1 g ) ] 1 .
I ( x , y , z ) = const N = 1 W ( x , y , z | N ) ,
D = [ 3 μ s ( 1 g ) ] 1
n s = { n s x , n s y , n s z } ,
n s x = sin θ s cos φ s , n s y = sin θ s sin φ s n s z = cos θ s .
n s = { n s x , n s y , n s z } ,
n s x = n s r cos φ i + n s x sin φ i , n s y = n s r sin φ i n s x cos φ i , n s z = n s z cos θ i n s y sin θ i , n s r = n s y cos θ i + n s z sin θ i .
R N = k = 1 N l k · n k ,
R N λ = k = 1 N l k · n k ,
R N λ = λ k = 1 N n k .
n k = n k 1 · g .
R N = λ k = 1 N 1 n k + g · n N 1 ,
R N = λ k = 1 N 2 n k + g ( 1 + g ) · n N 2 ,
R N = λ { ( 1 + g ) [ 1 + + g ( 1 + g ) ] } · n 1 .
R N = λ 1 g N 1 g · n 1 .
n N = g N 1 · n 1 ,
R = lim N R N = λ 1 g · n 1 , with g 0 ( isotropic scattering ) R N λ · n 1 , with g 1 ( strictly forwardscattering ) R N λ N · n 1 .
R N z = λ 1 g N 1 g , R N x = R N y = 0 .
R N x = k = 1 N l k n k x , R N y = k = 1 N l k n k y , R N z = k = 1 N l k n k z ,
R N a R N b = k = 1 N m = 1 N l k l m n k a n m b ,
R N a R N b = 2 λ 2 k = 1 N n k a n k b + λ 2 k , m = 1 ( k m ) N n k a n m b .
n k a n ( k + i ) b = g n k a n ( k + i 1 ) b ,
n k a n m b = g | m k | n k a n k b .
R N a R N b = 2 λ 2 k = 1 N n k a n k b 1 g N k + 1 1 g .
n k x n k x = n s n s k = [ ( n s y cos θ i + n s z sin θ i ) cos φ i + n s x sin φ i ] 2 = [ ( sin θ s sin φ s cos θ i + cos θ s sin θ i ) cos φ i + sin θ s cos φ s sin φ i ] 2 = sin 2 θ s 2 cos 2 θ i cos 2 φ i + cos 2 θ s sin 2 θ i cos 2 φ i + sin 2 θ s 2 sin 2 φ i = sin 2 θ s 2 ( 1 sin 2 θ i cos 2 φ i ) + cos 2 θ s sin 2 θ i cos 2 φ i = sin 2 θ s 2 + ( 1 3 sin 2 θ s 2 ) cos 2 θ i cos 2 φ i ,
n k x n k x = α + β n ( k 1 ) x n ( k 1 ) x ,
n k x n k x = α + β n ( k 1 ) x n ( k 1 ) x .
n k a n k b = αδ a b + β n ( k 1 ) a n ( k 1 ) b ,
n k a n k b = 1 3 δ a b + β k 1 ( n 1 a n 1 b 1 3 δ a b ) ,
R N a R N b = 2 λ 2 3 ( 1 g ) { N δ a b + ( 3 n 1 a n 1 b δ a b ) × 1 β N 1 β g [ 1 g N 1 g δ a b + ( 3 n 1 a n 1 b δ a b ) β N g N β g ] } .
R N 2 = 2 λ 2 ( 1 g ) ( N g 1 g N 1 g ) .
with   N R N 2 N 2 λ 2 ( 1 g ) , with g 0 ( isotropic scattering ) R N 2 2 λ 2 N , with g 1 ( strictly forward scattering ) R N 2 λ 2 N ( N + 1 ) .
R N z R N z = 2 λ 2 3 ( 1 g ) [ N + 2 1 β N 1 β g ( 1 g N 1 g + 2 β N g N β g ) ] , R N x R N x = R N y R N y = 2 λ 2 3 ( 1 g ) [ N 1 β N 1 β g ( 1 g N 1 g β N g N β g ) , R N x R N y = R N x R N z = R N y R N z = 0 .
α = ( 1 g 2 ) 3 , β = g 2 ,
R N z 2 = 2 λ 2 3 ( 1 g ) [ N 1 g N 1 g 2 × ( 2 + g + g 2 + 2 g N + 1 ) ] , R N x 2 = R N y 2 = 2 λ 2 3 ( 1 g ) × [ N 1 g N 1 g 2 ( 1 + g + g 2 + g N + 1 ) ] , R N x R N y = R N x R N z = R N y R N z = 0 .

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