Abstract

Using the algebra transformation method, we develop and demonstrate the use of the δP1 approximation to improve steady-state radiative-transfer estimates on spatial scales comparable to the mean free path. We show that the δP1 approximation agrees well with Monte Carlo simulation from source to infinity when we choose an appropriate parameter f (fractional portion that scatters directly forward) in the δ–Eddington phase function. We also provide the empirical formula to determine the parameter f.

© 2007 Optical Society of America

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References

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  1. W. Lihong, "Rapid modeling of diffuse reflectance of light in turbid slabs," J. Opt. Soc. Am. A 15, 936-944 (1998).
    [CrossRef]
  2. M. C. Kenneth and F. Z. Paul, Linear Transport Theory (Addison-Wesley, 1967).
  3. A Ishimaru, Wave Propagation and Scattering in Random Media. Vol. 1. Single Scattering and Transport Theory (Academic, 1978).
  4. F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, "Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation," Phys. Med. Biol. 45, 1359-1373 (2000).
    [CrossRef] [PubMed]
  5. L. Marti-Lopez and J. Bouza-Dominguez, "Validity conditions for the radiative transfer equation," J. Opt. Soc. Am. A 20, 2046-2056 (2003).
    [CrossRef]
  6. K. Furutsu and Y. Yamada, "Diffusion approximation for a dissipative random medium and the applications," Phys. Rev. E 50, 3634-3640 (1994).
    [CrossRef]
  7. T. Nakai, G. Nishimura, K. Yamamoto, and M. Tamura, "Expression of optical diffusion coefficient in high-absorption turbid media," Phys. Med. Biol. 42, 2541-2549 (1997).
    [CrossRef]
  8. M. Bassani, F. Martelli, G. Zaccanti, and D. Contini, "Independence of the diffusion coefficient from absorption: experimental and numerical evidence," Opt. Lett. 22, 853-855 (1997).
    [CrossRef] [PubMed]
  9. T. Durduran, A. G. Yodh, B. Chance, and D. A. Boas, "Does the photon-diffusion coefficient depend on absorption?," J. Opt. Soc. Am. A 14, 3358-3365 (1997).
    [CrossRef]
  10. D. J. Durian, "The diffusion coefficient depends on absorption," Opt. Lett. 23, 1502-1504 (1998).
    [CrossRef]
  11. K. Rinzema, L. H. P. Murrer, and W. M. Star, "Direct experimental verification of light transport theory in an optical phantom," J. Opt. Soc. Am. A 15, 2078-2088 (1998).
    [CrossRef]
  12. R. Aronson and N. Corngold, "Photon diffusion coefficient in an absorbing medium," J. Opt. Soc. Am. A 16, 1066-1071 (1999).
    [CrossRef]
  13. A. M. Weinberg and E. P. Wigner, The Physical Theory of Neutron Chain Reactors (Chicago Press, 1958).
  14. C. C. Grosjean, "A high accuracy approximation for solving multiple scattering problems in infinite homogeneous media," Nuovo Cim. 3, 1262-1275 (1956).
    [CrossRef]
  15. R. Graaff and K. Rinzema, "Practical improvements on photon diffusion theory: application to isotropic scattering," Phys. Med. Biol. 46, 3043-3050 (2001).
    [CrossRef] [PubMed]
  16. V. Venugopalan, J. S. You, and B. J. Tromberg, "Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source-detector separations," Phys. Rev. E 58, 2395-2406 (1998).
    [CrossRef]
  17. W. M. Star, "Comparing the P3-approximation with diffusion theory and with Monte Carlo calculations of light propagation in a slab geometry," in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller and D. H. Sliney, eds., Proc. SPIE IS5, 146-154 (1989).
  18. S. L. Jacques, "Light distributions from point, line, and plane sources for photochemical reactions and fluorescence in turbid biological tissues," Photochem. Photobiol. 67, 23-32 (1998).
    [CrossRef] [PubMed]
  19. Wai-Fung Cheong, "Summary of optical properties," in Optical-Thermal Response of Laser Irradiated Tissue, A. J. Welch and M. J. C. van Gemert, eds. (Plenum, 1995), pp. 275-301.
  20. G. Yoon, "Absorption and scattering of laser light in biological media-mathematical modeling and methods for determining optical properties," Ph.D. dissertation (University of Texas at Austin, 1988).
  21. P. A. Wilksch, F. Jacka, and A. J. Blake, "Studies of light propagation in tissue," in Porphyrin Localization and Treatment of Tumors, D. R. Doiron and C. J. Gomer eds. (Alan R. Liss, 1984), pp. 149-161.

2003 (1)

2001 (1)

R. Graaff and K. Rinzema, "Practical improvements on photon diffusion theory: application to isotropic scattering," Phys. Med. Biol. 46, 3043-3050 (2001).
[CrossRef] [PubMed]

2000 (1)

F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, "Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation," Phys. Med. Biol. 45, 1359-1373 (2000).
[CrossRef] [PubMed]

1999 (1)

1998 (5)

W. Lihong, "Rapid modeling of diffuse reflectance of light in turbid slabs," J. Opt. Soc. Am. A 15, 936-944 (1998).
[CrossRef]

K. Rinzema, L. H. P. Murrer, and W. M. Star, "Direct experimental verification of light transport theory in an optical phantom," J. Opt. Soc. Am. A 15, 2078-2088 (1998).
[CrossRef]

V. Venugopalan, J. S. You, and B. J. Tromberg, "Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source-detector separations," Phys. Rev. E 58, 2395-2406 (1998).
[CrossRef]

S. L. Jacques, "Light distributions from point, line, and plane sources for photochemical reactions and fluorescence in turbid biological tissues," Photochem. Photobiol. 67, 23-32 (1998).
[CrossRef] [PubMed]

D. J. Durian, "The diffusion coefficient depends on absorption," Opt. Lett. 23, 1502-1504 (1998).
[CrossRef]

1997 (3)

1994 (1)

K. Furutsu and Y. Yamada, "Diffusion approximation for a dissipative random medium and the applications," Phys. Rev. E 50, 3634-3640 (1994).
[CrossRef]

1989 (1)

W. M. Star, "Comparing the P3-approximation with diffusion theory and with Monte Carlo calculations of light propagation in a slab geometry," in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller and D. H. Sliney, eds., Proc. SPIE IS5, 146-154 (1989).

1956 (1)

C. C. Grosjean, "A high accuracy approximation for solving multiple scattering problems in infinite homogeneous media," Nuovo Cim. 3, 1262-1275 (1956).
[CrossRef]

J. Opt. Soc. Am. A (5)

Nuovo Cim. (1)

C. C. Grosjean, "A high accuracy approximation for solving multiple scattering problems in infinite homogeneous media," Nuovo Cim. 3, 1262-1275 (1956).
[CrossRef]

Opt. Lett. (2)

Photochem. Photobiol. (1)

S. L. Jacques, "Light distributions from point, line, and plane sources for photochemical reactions and fluorescence in turbid biological tissues," Photochem. Photobiol. 67, 23-32 (1998).
[CrossRef] [PubMed]

Phys. Med. Biol. (3)

R. Graaff and K. Rinzema, "Practical improvements on photon diffusion theory: application to isotropic scattering," Phys. Med. Biol. 46, 3043-3050 (2001).
[CrossRef] [PubMed]

F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, "Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation," Phys. Med. Biol. 45, 1359-1373 (2000).
[CrossRef] [PubMed]

T. Nakai, G. Nishimura, K. Yamamoto, and M. Tamura, "Expression of optical diffusion coefficient in high-absorption turbid media," Phys. Med. Biol. 42, 2541-2549 (1997).
[CrossRef]

Phys. Rev. E (2)

K. Furutsu and Y. Yamada, "Diffusion approximation for a dissipative random medium and the applications," Phys. Rev. E 50, 3634-3640 (1994).
[CrossRef]

V. Venugopalan, J. S. You, and B. J. Tromberg, "Radiative transport in the diffusion approximation: an extension for highly absorbing media and small source-detector separations," Phys. Rev. E 58, 2395-2406 (1998).
[CrossRef]

Proc. SPIE (1)

W. M. Star, "Comparing the P3-approximation with diffusion theory and with Monte Carlo calculations of light propagation in a slab geometry," in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller and D. H. Sliney, eds., Proc. SPIE IS5, 146-154 (1989).

Other (6)

M. C. Kenneth and F. Z. Paul, Linear Transport Theory (Addison-Wesley, 1967).

A Ishimaru, Wave Propagation and Scattering in Random Media. Vol. 1. Single Scattering and Transport Theory (Academic, 1978).

Wai-Fung Cheong, "Summary of optical properties," in Optical-Thermal Response of Laser Irradiated Tissue, A. J. Welch and M. J. C. van Gemert, eds. (Plenum, 1995), pp. 275-301.

G. Yoon, "Absorption and scattering of laser light in biological media-mathematical modeling and methods for determining optical properties," Ph.D. dissertation (University of Texas at Austin, 1988).

P. A. Wilksch, F. Jacka, and A. J. Blake, "Studies of light propagation in tissue," in Porphyrin Localization and Treatment of Tumors, D. R. Doiron and C. J. Gomer eds. (Alan R. Liss, 1984), pp. 149-161.

A. M. Weinberg and E. P. Wigner, The Physical Theory of Neutron Chain Reactors (Chicago Press, 1958).

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Figures (7)

Fig. 1
Fig. 1

(a) Photon fluence rates and (b) the relative errors versus dimensionless radial position obtained from diffusion approximation, adjustable f δ P 1 approximation and MC simulation for Intralipid, an absorber (NiSPC).

Fig. 2
Fig. 2

(a) Photon fluence rates and (b) the relative errors versus dimensionless radial position obtained from diffusion approximation, adjustable f δ P 1 approximation and MC simulation for normal aorta at λ = 633   nm .

Fig. 3
Fig. 3

(a) Photon fluence rates and (b) the relative errors versus dimensionless radial position obtained from diffusion approximation, adjustable f δ P 1 approximation and MC simulation for normal aorta at λ = 1320 n m .

Fig. 4
Fig. 4

(a) Photon fluence rates and (b) the relative errors versus dimensionless radial position obtained from diffusion approximation, adjustable f δ P 1 approximation and MC simulation for pig epidermis.

Fig. 5
Fig. 5

Fluence rate of an adjustable f δ P 1 approximation at various parameters f and MC simulation for Intralipid.

Fig. 6
Fig. 6

(a) Fluence rates versus dimensionless radial position obtained from an adjustable f δ P 1 approximation, the generalized diffusion approximation by venugopalan et al. and Monte Carlo. (b) Error of an adjustable f δ P 1 approximation and the generalized diffusion approximation relative to Monte Carlo.

Fig. 7
Fig. 7

Fitted curve for f and g.

Tables (2)

Tables Icon

Table 1 Typical Optical Parameters of Biological Tissue and Tissuelike Phantom

Tables Icon

Table 2 Appropriate f and g ∗ for Biological Tissues and Tissuelike Phantom

Equations (41)

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ε ( r , s ) = μ s * P 0 4 π r 2 p ( s , s 0 ) exp ( μ t * r ) .
2 ϕ d ( r ) μ eff 2 ϕ d ( r ) = 3 μ s * ( μ t * + g * μ a ) P 0 4 π r 2 × exp ( μ t * r ) .
ϕ d ( r ) = B exp ( μ eff r ) r + 3 μ s * ( μ t * + g * μ a ) P 0 8 π μ eff r × [ exp ( μ eff r ) E 1 ( μ t * r + μ eff r ) exp ( μ eff r ) E 1 ( μ t * r μ eff r ) ] ,
F d ( r ) = μ s * g * P 0 exp ( μ t * r ) 4 π μ t r * r 2 + B 3 μ t r * exp ( μ eff r ) × ( μ eff r + 1 r 2 ) μ s * ( μ t * + g * μ a ) P 0 8 π μ eff μ t r * [ ( μ eff r 1 r 2 ) × exp ( μ eff r ) E 1 ( μ t * r + μ eff r ) + ( μ eff r + 1 r 2 ) × exp ( μ eff r ) E 1 ( μ t * r μ eff r ) ] ,
B = 3 μ t r * μ s * P 0 4 π μ t * 3 μ s * ( μ t * + g * μ a ) P 0 8 π × ( 1 μ eff ln μ t * μ eff μ t * + μ eff + 2 μ t * ) .
ϕ d ( r ) = 3 μ t r P 0 exp ( μ eff r ) 4 π r ,
F d ( r ) = r [ exp ( μ eff r ) 4 π r ] P 0 r = P 0 4 π ( μ eff r + 1 r 2 ) exp ( μ eff r ) r .
g * = g f 1 f .
ln ( μ t * μ eff μ t * + μ eff ) 2 μ eff μ t * ,
B = 3 μ t r * μ s * P 0 4 π μ t * 3 μ s * ( μ t * + g * μ a ) P 0 8 π × ( 1 μ eff ln μ t * μ eff μ t * + μ eff + 2 μ t * ) 3 μ t r * P 0 4 π ,
ϕ d ( r ) 3 μ t r * P 0 4 π exp ( μ eff r ) r 3 P 0 4 π r 2 exp ( μ t * r ) ,
F d ( r ) [ μ s * g * P 0 exp ( μ t * r ) 4 π μ tr * r 2 + P 0 4 π exp ( μ eff r ) × ( μ eff r + 1 r 2 ) P 0 exp ( μ t * r ) 4 π μ t r * r 2 ( μ t * r + 1 ) ] r .
ϕ [ i r ] / P 0 = A [ i r ] / ( N Δ V [ i r ] μ a ) .
ϕ ( r ) = ϕ c ( r ) + ϕ d ( r ) .
ϕ c ( r ) = P 0 4 π r 2 exp ( μ t * r ) ,
f = 0.026094 g 3 + 0.23597 g 2 + 0.13572 g + 0.60366.
ln ( μ t * μ eff μ t * + μ eff ) .
s L ( r , s ) = μ t L ( r , s ) + μ s s p ( s s ) L ( r , s ) d Ω + S ( r , s ) .
p δ E ( s s ) = 1 4 π [ ( 1 f ) ( 1 + 3 g * s s ) + 2 f δ ( 1 s s ) ] .
s L ( r , s ) = μ t * L ( r , s ) + μ s * 4 π L ( r , s ) × p E ( s s ) d Ω + S ( r , s ) ,
L ( r , s ) = L c ( r , s ) + L d ( r , s ) .
s ( L c ( r , s ) + L d ( r , s ) ) = μ t * ( L c ( r , s ) + L d ( r , s ) ) + μ s * 4 π ( L c ( r , s ) + L d ( r , s ) ) p E ( s , s ) d Ω .
L c ( r , s ) = P ( r ) δ ( 1 s s 0 ) / ( 2 π ) .
ϕ c ( r , s ) = 4 π L c ( r , s ) d Ω = P ( r ) .
s L c ( r , s ) = μ t * L c ( r , s ) .
s L d ( r , s ) = μ t * L d ( r , s ) + μ s * 4 π L d ( r , s ) × p E ( s , s ^ ) d Ω + μ s * p ( s , s 0 ) P ( r ) .
F ( r ) + μ a ϕ d ( r ) = S 0 ( r ) ,
ϕ d ( r ) + 3 μ t r * F ( r ) = 3 S 1 ( r ) s 0 ,
    S 0 ( r ) = μ s * P 0 4 π r 2 exp [ μ t * ( r r 0 ) ] ,
S 1 ( r ) = μ s * g * P 0 4 π r 2 exp [ μ t * ( r r 0 ) ] ,
ϕ d ( r ) = 4 π L d ( r , s ) d Ω ,
F ( r ) = 4 π L d ( r , s ) s d Ω .
d 2 d r 2 [ r ϕ d ( r ) ] μ eff [ r ϕ d ( r ) ] = S ( r ) ,
S ( r ) = 3 μ s * ( μ t * + g * μ a ) P 0 4 π exp [ μ t * ( r r 0 ) ] ,
μ eff = 3 μ a μ t r * .
r ϕ d o c ( r ) = B + exp ( + μ eff r ) + B exp ( μ eff r ) .
r ϕ d o p ( r ) = 1 2 μ eff r r 1 [ exp ( + μ eff r ) exp ( μ eff r ) exp ( μ eff r ) exp ( μ eff r ) ] S ( r ) d r = 3 μ s * ( μ t * + g * μ a ) P 0 8 μ eff π { exp ( μ eff r ) E 1 [ ( μ t * + μ eff ) r , ( μ t * + μ eff ) r 1 ] exp ( μ eff r ) E 1 [ ( μ t * μ eff ) r , ( μ t * μ eff ) r 1 ] } ,
E 1 ( a , b ) = a b exp ( u ) u d u .
ϕ d ( r ) = B + exp ( + μ eff r ) r + B exp ( μ eff r ) r + 3 μ s * ( μ t * + g * μ a ) P 0 8 μ eff π r { exp ( μ eff r ) × E 1 [ ( μ t * + μ eff ) r , ( μ t * + μ eff ) r 1 ] exp ( μ eff r ) × E 1 [ ( μ t * μ eff ) r , ( μ t * μ eff ) r 1 ] } .
μ a 0 ϕ ( r ) 4 π r 2 d r = P 0 ,
B = 3 μ t r * μ s * P 0 4 π μ t * 3 μ s * ( μ t * + g * μ a ) P 0 8 π × [ 1 μ eff ln μ t * μ eff μ t * + μ eff + 2 μ t * ] .

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