Abstract

In this study, the third-order simplified spherical harmonics equations (SP3), an approximation of the radiative transfer equation, are solved for a semi-infinite geometry considering the exact simplified spherical harmonics boundary conditions. The obtained Green’s function is compared to radiative transfer calculations and the diffusion theory. In general, it is shown that the SP3 equations provide better results than the diffusion approximation in media with high absorption coefficient values but no improvement is found for small distances to the source.

© 2011 Optical Society of America

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  2. J. Tian, K. Liu, Y. Lu, C. Chin, X. Yang, S. Zhu, D. Han, J. Feng, X. Ma, and Z. Chang, Opt. Express 18, 20988 (2010).
    [CrossRef] [PubMed]
  3. L. D. Montejo, H. K. K. Kim, and A. H. Hielscher, Proc. SPIE 7896, 78960J (2011).
    [CrossRef]
  4. K. M. Case and P. F. Zweifel, Linear Transport Theory(Addison-Wesley, 1967).
  5. M. Frank, A. Klar, E. W. Larsen, and S. Yasuda, J. Comput. Phys. 226, 2289 (2007).
    [CrossRef]
  6. M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, Phys. Med. Biol. 54, 2493 (2009).
    [CrossRef] [PubMed]
  7. Y. Lu, B. Zuh, H. Shen, J. C. Rasmussen, G. Wang, and E. M. Sevick-Muraca, Phys. Med. Biol. 55, 4625 (2010).
    [CrossRef] [PubMed]
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    [CrossRef]
  11. A. Kienle and M. S. Patterson, J. Opt. Soc. Am. A 14, 246 (1997).
    [CrossRef]

2011 (1)

L. D. Montejo, H. K. K. Kim, and A. H. Hielscher, Proc. SPIE 7896, 78960J (2011).
[CrossRef]

2010 (4)

2009 (1)

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, Phys. Med. Biol. 54, 2493 (2009).
[CrossRef] [PubMed]

2007 (1)

M. Frank, A. Klar, E. W. Larsen, and S. Yasuda, J. Comput. Phys. 226, 2289 (2007).
[CrossRef]

2006 (1)

A. D. Klose and E. W. Larsen, J. Comput. Phys. 220, 441 (2006).
[CrossRef]

1997 (1)

Bérubé-Lauzière, Y.

Bouza-Domínguez, J.

Case, K. M.

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

Chang, Z.

Chin, C.

Chu, M.

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, Phys. Med. Biol. 54, 2493 (2009).
[CrossRef] [PubMed]

Dehghani, H.

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, Phys. Med. Biol. 54, 2493 (2009).
[CrossRef] [PubMed]

Del Bianco, S.

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation Through Biological Tissue (SPIE, 2010).
[CrossRef]

Feng, J.

Frank, M.

M. Frank, A. Klar, E. W. Larsen, and S. Yasuda, J. Comput. Phys. 226, 2289 (2007).
[CrossRef]

Han, D.

Hielscher, A. H.

L. D. Montejo, H. K. K. Kim, and A. H. Hielscher, Proc. SPIE 7896, 78960J (2011).
[CrossRef]

Ismaelli, A.

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation Through Biological Tissue (SPIE, 2010).
[CrossRef]

Kienle, A.

Kim, H. K. K.

L. D. Montejo, H. K. K. Kim, and A. H. Hielscher, Proc. SPIE 7896, 78960J (2011).
[CrossRef]

Klar, A.

M. Frank, A. Klar, E. W. Larsen, and S. Yasuda, J. Comput. Phys. 226, 2289 (2007).
[CrossRef]

Klose, A. D.

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, Phys. Med. Biol. 54, 2493 (2009).
[CrossRef] [PubMed]

A. D. Klose and E. W. Larsen, J. Comput. Phys. 220, 441 (2006).
[CrossRef]

Larsen, E. W.

M. Frank, A. Klar, E. W. Larsen, and S. Yasuda, J. Comput. Phys. 226, 2289 (2007).
[CrossRef]

A. D. Klose and E. W. Larsen, J. Comput. Phys. 220, 441 (2006).
[CrossRef]

Liemert, A.

Liu, K.

Lu, Y.

J. Tian, K. Liu, Y. Lu, C. Chin, X. Yang, S. Zhu, D. Han, J. Feng, X. Ma, and Z. Chang, Opt. Express 18, 20988 (2010).
[CrossRef] [PubMed]

Y. Lu, B. Zuh, H. Shen, J. C. Rasmussen, G. Wang, and E. M. Sevick-Muraca, Phys. Med. Biol. 55, 4625 (2010).
[CrossRef] [PubMed]

Ma, X.

Martelli, F.

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation Through Biological Tissue (SPIE, 2010).
[CrossRef]

Montejo, L. D.

L. D. Montejo, H. K. K. Kim, and A. H. Hielscher, Proc. SPIE 7896, 78960J (2011).
[CrossRef]

Patterson, M. S.

Rasmussen, J. C.

Y. Lu, B. Zuh, H. Shen, J. C. Rasmussen, G. Wang, and E. M. Sevick-Muraca, Phys. Med. Biol. 55, 4625 (2010).
[CrossRef] [PubMed]

Sevick-Muraca, E. M.

Y. Lu, B. Zuh, H. Shen, J. C. Rasmussen, G. Wang, and E. M. Sevick-Muraca, Phys. Med. Biol. 55, 4625 (2010).
[CrossRef] [PubMed]

Shen, H.

Y. Lu, B. Zuh, H. Shen, J. C. Rasmussen, G. Wang, and E. M. Sevick-Muraca, Phys. Med. Biol. 55, 4625 (2010).
[CrossRef] [PubMed]

Tian, J.

Vishwanath, K.

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, Phys. Med. Biol. 54, 2493 (2009).
[CrossRef] [PubMed]

Wang, G.

Y. Lu, B. Zuh, H. Shen, J. C. Rasmussen, G. Wang, and E. M. Sevick-Muraca, Phys. Med. Biol. 55, 4625 (2010).
[CrossRef] [PubMed]

Yang, X.

Yasuda, S.

M. Frank, A. Klar, E. W. Larsen, and S. Yasuda, J. Comput. Phys. 226, 2289 (2007).
[CrossRef]

Zaccanti, G.

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation Through Biological Tissue (SPIE, 2010).
[CrossRef]

Zhu, S.

Zuh, B.

Y. Lu, B. Zuh, H. Shen, J. C. Rasmussen, G. Wang, and E. M. Sevick-Muraca, Phys. Med. Biol. 55, 4625 (2010).
[CrossRef] [PubMed]

Zweifel, P. F.

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

Appl. Opt. (1)

J. Comput. Phys. (2)

M. Frank, A. Klar, E. W. Larsen, and S. Yasuda, J. Comput. Phys. 226, 2289 (2007).
[CrossRef]

A. D. Klose and E. W. Larsen, J. Comput. Phys. 220, 441 (2006).
[CrossRef]

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

Opt. Express (1)

Opt. Lett. (1)

Phys. Med. Biol. (2)

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, Phys. Med. Biol. 54, 2493 (2009).
[CrossRef] [PubMed]

Y. Lu, B. Zuh, H. Shen, J. C. Rasmussen, G. Wang, and E. M. Sevick-Muraca, Phys. Med. Biol. 55, 4625 (2010).
[CrossRef] [PubMed]

Proc. SPIE (1)

L. D. Montejo, H. K. K. Kim, and A. H. Hielscher, Proc. SPIE 7896, 78960J (2011).
[CrossRef]

Other (2)

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

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation Through Biological Tissue (SPIE, 2010).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Comparison of the steady-state reflectance versus radial distance ρ for a semi-infinite geometry with optical properties of μ a = 0.01 m m 1 and μ s = 10 m m 1 .

Fig. 2
Fig. 2

Comparison of the steady-state reflectance versus radial distance ρ for a semi-infinite geometry with optical properties of μ a = 1 m m 1 and μ s = 10 m m 1 .

Equations (28)

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1 3 σ 1 φ 1 ( r ) + σ 0 φ 1 ( r ) = S ( r ) + 2 3 σ 0 φ 2 ( r ) , 1 7 σ 3 φ 2 ( r ) + ( 4 9 σ 0 + 5 9 σ 2 ) φ 2 ( r ) = 2 3 S ( r ) + 2 3 σ 0 φ 1 ( r ) ,
f n = 2 π 1 1 f ( μ ) P n ( μ ) d μ .
φ i ( r ) = 1 2 π 0 φ i ( q , z ) J 0 ( q ρ ) q d q ,
δ ( r r ) = δ ( z z ) 2 π 0 J 0 ( q ρ ) q d q ,
d 2 ψ d z 2 = ( A + q 2 I 2 ) ψ + δ ( z z ) ε ,
A = ( 3 σ 0 σ 1 2 σ 0 σ 1 14 3 σ 0 σ 3 28 9 σ 0 σ 3 + 35 9 σ 2 σ 3 )
ε = 1 3 ( 9 σ 1 14 σ 3 ) .
ψ ( q , z ) = e λ z | ν ,
ξ 1 / 2 2 = α ± α 2 β ,
α = 3 2 σ 0 σ 1 + 28 18 σ 0 σ 3 + 35 18 σ 2 σ 3 , β = 35 18 3 σ 0 σ 1 σ 2 σ 3 .
[ D ( λ 2 q 2 ) I 2 ] · B 1 | ν = 0 ,
λ 1 ( q ) = q 2 + ξ 1 2 , λ 2 ( q ) = q 2 + ξ 2 2 .
| ν i = ( 2 σ 0 σ 1 3 σ 0 σ 1 ξ i 2 ) .
ψ ( h ) ( q , z ) = C 1 ( q ) | ν 1 e λ 1 ( q ) z + C 2 ( q ) | ν 2 e λ 2 ( q ) z .
φ i ( q , z ) = 1 2 π φ i ( q , k ) e j k z e j k z d k , δ ( z z ) = 1 2 π e j k z e j k z d k
[ A + ( q 2 + k 2 ) I 2 ] ψ ( p ) ( q , k ) = ε .
ψ ( p ) ( q , k ) = H 1 | ν 1 k 2 + λ 1 2 ( q ) H 2 | ν 2 k 2 + λ 2 2 ( q ) ,
H 1 = 14 σ 3 1 | ν 2 + 9 σ 1 2 | ν 2 3 | A | , H 2 = 14 σ 3 1 | ν 1 + 9 σ 1 2 | ν 1 3 | A | .
ψ ( p ) ( q , z ) = H 1 | ν 1 2 exp ( λ 1 ( q ) | z z | ) λ 1 ( q ) ,
H 2 | ν 2 2 exp ( λ 2 ( q ) | z z | ) λ 2 ( q ) .
ψ ( q , z ) = ψ ( h ) ( q , z ) + ψ ( p ) ( q , z ) .
( 1 2 + A 1 ) φ 1 ( q , 0 ) 1 + B 1 3 σ 1 d φ 1 ( q , z ) d z | z = 0 = ( 1 8 + C 1 ) φ 2 ( q , 0 ) D 1 σ 3 d φ 2 ( q , z ) d z | z = 0 ,
( 7 24 + A 2 ) φ 2 ( q , 0 ) 1 + B 2 7 σ 3 d φ 2 ( q , z ) d z | z = 0 = ( 1 8 + C 2 ) φ 1 ( q , 0 ) D 2 σ 1 d φ 1 ( q , z ) d z | z = 0
ψ ( r ) = H 1 | ν 1 exp ( ξ 1 ρ 2 + ( z z ) 2 ) 4 π ρ 2 + ( z z ) 2 ,
H 2 | ν 2 exp ( ξ 2 ρ 2 + ( z z ) 2 ) 4 π ρ 2 + ( z z ) 2 ,
+ 1 2 π 0 ψ ( h ) ( q , z ) J 0 ( q ρ ) q d q .
Φ ( r ) = φ 1 ( r ) 2 3 φ 2 ( r ) ,
R ( ρ ) = ( 1 4 + J 0 ) ( φ 1 2 3 φ 2 ) + 1 + 2 J 1 6 σ 1 d φ 1 d z + ( 5 16 + J 2 ) φ 2 3 + J 3 7 σ 3 d φ 2 d z ,

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