Abstract

We derived analytical solutions of the simplified spherical harmonics equations, an approximation of the radiative transfer equation, for infinitely extended scattering media. The derived equations are simple (sum of exponential functions) and quickly evaluated. We compared the solutions with Monte Carlo simulations in the steady-state and time domains and found much better agreement compared to solutions of the diffusion equation, especially for large absorption coefficients, short time values, and small distances from the source.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. D. Klose and E. W. Larsen, J. Comput. Phys. 220, 441 (2006).
    [CrossRef]
  2. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).
  3. H. Xu and M. S. Patterson, Phys. Med. Biol. 51, N247 (2006).
    [CrossRef] [PubMed]
  4. P. S. Brantley and E. W. Larsen, Nucl. Sci. Eng. 134, 1 (2000).
  5. M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, Phys. Med. Biol. 54, 2493 (2009).
    [CrossRef] [PubMed]
  6. J. Bouza-Domínguez, Y. Bérubé-Lauzière, and V. Issa, Appl. Opt. 49, 1414 (2010).
    [CrossRef] [PubMed]
  7. T. Frey and M. Bossert, Signal- und Systemtheorie (B. G. Teubner, 2004).
  8. A. Kienle and M. S. Patterson, J. Opt. Soc. Am. A 14, 246 (1997).
    [CrossRef]
  9. A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, Phys. Rev. Lett. 100, 138101 (2008).
    [CrossRef] [PubMed]

2010 (1)

2009 (1)

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

2008 (1)

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, Phys. Rev. Lett. 100, 138101 (2008).
[CrossRef] [PubMed]

2006 (2)

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

H. Xu and M. S. Patterson, Phys. Med. Biol. 51, N247 (2006).
[CrossRef] [PubMed]

2004 (1)

T. Frey and M. Bossert, Signal- und Systemtheorie (B. G. Teubner, 2004).

2000 (1)

P. S. Brantley and E. W. Larsen, Nucl. Sci. Eng. 134, 1 (2000).

1997 (1)

1978 (1)

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

Bérubé-Lauzière, Y.

Bossert, M.

T. Frey and M. Bossert, Signal- und Systemtheorie (B. G. Teubner, 2004).

Bouza-Domínguez, J.

Brantley, P. S.

P. S. Brantley and E. W. Larsen, Nucl. Sci. Eng. 134, 1 (2000).

Chu, M.

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

Contini, D.

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, Phys. Rev. Lett. 100, 138101 (2008).
[CrossRef] [PubMed]

Cova, S.

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, Phys. Rev. Lett. 100, 138101 (2008).
[CrossRef] [PubMed]

Cubeddu, R.

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, Phys. Rev. Lett. 100, 138101 (2008).
[CrossRef] [PubMed]

Dehghani, H.

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

Frey, T.

T. Frey and M. Bossert, Signal- und Systemtheorie (B. G. Teubner, 2004).

Ishimaru, A.

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

Issa, V.

Kienle, A.

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.

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

P. S. Brantley and E. W. Larsen, Nucl. Sci. Eng. 134, 1 (2000).

Martelli, F.

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, Phys. Rev. Lett. 100, 138101 (2008).
[CrossRef] [PubMed]

Mora, A. D.

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, Phys. Rev. Lett. 100, 138101 (2008).
[CrossRef] [PubMed]

Patterson, M. S.

Pifferi, A.

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, Phys. Rev. Lett. 100, 138101 (2008).
[CrossRef] [PubMed]

Spinelli, L.

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, Phys. Rev. Lett. 100, 138101 (2008).
[CrossRef] [PubMed]

Torricelli, A.

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, Phys. Rev. Lett. 100, 138101 (2008).
[CrossRef] [PubMed]

Tosi, A.

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, Phys. Rev. Lett. 100, 138101 (2008).
[CrossRef] [PubMed]

Vishwanath, K.

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

Xu, H.

H. Xu and M. S. Patterson, Phys. Med. Biol. 51, N247 (2006).
[CrossRef] [PubMed]

Zaccanti, G.

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, Phys. Rev. Lett. 100, 138101 (2008).
[CrossRef] [PubMed]

Zappa, F.

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, Phys. Rev. Lett. 100, 138101 (2008).
[CrossRef] [PubMed]

Appl. Opt. (1)

J. Comput. Phys. (1)

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

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

Nucl. Sci. Eng. (1)

P. S. Brantley and E. W. Larsen, Nucl. Sci. Eng. 134, 1 (2000).

Phys. Med. Biol. (2)

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

H. Xu and M. S. Patterson, Phys. Med. Biol. 51, N247 (2006).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, Phys. Rev. Lett. 100, 138101 (2008).
[CrossRef] [PubMed]

Other (2)

T. Frey and M. Bossert, Signal- und Systemtheorie (B. G. Teubner, 2004).

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Steady-state fluence rate versus distance from the isotropic source for an infinite scattering medium. The optical parameters are μ s = 1 mm 1 , g = 0.9 , μ a = 0.2 mm 1 (upper curves), and μ a = 2 mm 1 (lower curves).

Fig. 2
Fig. 2

Time resolved reflectance from a semi-infinite scattering medium. The optical properties are μ s = 1 mm 1 , g = 0.9 , and μ a = 0.1 mm 1 . A refractive index of 1.0 and 1.4 was used outside and inside the scattering medium.

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

1 3 μ a 1 φ 1 ( r ) + μ a φ 1 ( r ) = S ( r ) + 2 3 μ a φ 2 ( r ) , 1 7 μ a 3 φ 2 ( r ) + ( 4 9 μ a + 5 9 μ a 2 ) φ 2 ( r ) = 2 3 S ( r ) + 2 3 μ a φ 1 ( r ) ,
φ ( r ) = 1 2 π 2 r 0 k φ ( k ) sin ( k r ) d k ,
S ( r ) = 1 2 π 2 r 0 k sin ( k r ) d k .
( k 2 3 μ a 1 + μ a 2 3 μ a 2 3 μ a k 2 7 μ a 3 + 4 9 μ a + 5 9 μ a 2 ) ( φ 1 ( k ) φ 2 ( k ) ) = ( 1 2 3 ) .
φ i ( k ) = F i ( 1 ) ( k 2 ) k 4 + α k 2 + β , F i ( m ) ( x ) = j = i 1 m a i j x j ,
a 10 = 35 3 μ a 1 μ a 2 μ a 3 , a 11 = 3 μ a 1 , a 21 = 14 3 μ a 3 ,
α = 3 μ a μ a 1 + 28 9 μ a μ a 3 + 35 9 μ a 2 μ a 3 , β = 35 3 μ a μ a 1 μ a 2 μ a 3 .
φ i ( k ) = F i ( 1 ) ( k 2 ) ( k 2 + k 1 2 ) ( k 2 + k 2 2 ) = A i k 2 + k 1 2 + B i k 2 + k 2 2 ,
A i = F i ( 1 ) ( λ 1 ) k 2 2 k 1 2 , B i = F i ( 1 ) ( λ 2 ) k 1 2 k 2 2 .
G ( r ) = e μ eff r 4 π D r = 1 2 π 2 D r 0 k sin ( k r ) k 2 + μ eff 2 d k ,
φ i ( r ) = A i e k 1 r 4 π r + B i e k 2 r 4 π r , i = 1 , 2.
ψ ( r ) = φ 1 ( r ) 2 3 φ 2 ( r ) .
φ i ( k ) = F i ( 2 ) ( k 2 ) k 6 + α k 4 + β k 2 + γ ,
a 10 = 231 5 μ a 1 μ a 2 μ a 3 μ a 4 μ a 5 , a 11 = 35 3 μ a 1 μ a 2 μ a 3 + 33 μ a 1 μ a 5 ( 16 45 μ a 2 + 9 25 μ a 4 ) , a 12 = 3 μ a 1 , a 21 = 462 25 μ a 3 μ a 4 μ a 5 , a 22 = 14 3 μ a 3 , a 32 = 88 15 μ a 5 ,
α = 3 μ a μ a 1 + 28 9 μ a μ a 3 + 35 9 μ a 2 μ a 3 + 11 μ a 5 ( 64 225 μ a + 16 45 μ a 2 + 9 25 μ a 4 ) , β = μ a μ a 1 ( 35 3 μ a 2 μ a 3 + 176 15 μ a 2 μ a 5 + 297 25 μ a 4 μ a 5 ) + μ a 3 μ a 4 μ a 5 ( 308 25 μ a + 77 5 μ a 2 ) , γ = 231 5 μ a μ a 1 μ a 2 μ a 3 μ a 4 μ a 5 .
p = 1 3 α 2 β , q = 2 27 α 3 1 3 α β + γ ,
λ k + 1 = 2 p 3 cos ( φ + 2 k π 3 ) α 3 ,
φ = arccos ( 3 q 2 p 3 p ) ,
φ i ( r ) = A i e k 1 r 4 π r + B i e k 2 r 4 π r + C i e k 3 r 4 π r , i = 1 , 2 , 3 ,
A i = F i ( 2 ) ( λ 1 ) ( k 2 2 k 1 2 ) ( k 3 2 k 1 2 ) , B i = F i ( 2 ) ( λ 2 ) ( k 1 2 k 2 2 ) ( k 3 2 k 2 2 ) , C i = F i ( 2 ) ( λ 3 ) ( k 1 2 k 3 2 ) ( k 2 2 k 3 2 ) .
ψ ( r ) = φ 1 ( r ) 2 3 φ 2 ( r ) + 8 15 φ 3 ( r ) .

Metrics