Abstract

The accuracy of the diffusion approximation is compared with more accurate solutions for describing light interaction with biological tissues. Generally the diffusion approximation underestimates the light distribution in the surface region, and, for high albedos, it significantly underestimates the fluence rate. This difference is only a few percent for albedos of less than 0.5 due to the dominance of collimated light. As the anisotropy of scattering increases, deviations increase. In general, fluxes can be computed more accurately with the diffusion approximation than fluence rates. For anisotropic scattering, better results can be obtained by simple transforms of optical coefficients using the similarity relations. The similarity relations improve flux calculations, but computed fluence rates have substantial errors for high albedo and the large index of refraction differences at the surface.

© 1989 Optical Society of America

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References

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  1. R. A. J. Groenhuis, H. A. Ferwerda, J. J. T. Bosch, “Scattering and Absorption of Turbid Materials Determined from Reflection Measurements,” Appl. Opt. 22, 2456–2467 (1983).
    [CrossRef] [PubMed]
  2. S. L. Jacques, S. A. Prahl, “Modeling Optical and Thermal Distribution in Tissue During Laser Irradiation,” Lasers Surg. Med. 6, 494–503 (1987).
    [CrossRef] [PubMed]
  3. J. M. Steinke, A. P. Shepherd, “Diffusion Model of the Optical Absorbance of Whole Blood,” J. Opt. Soc. Am. A 5, 813–822 (1988).
    [CrossRef] [PubMed]
  4. W. M. Star, J. P. A. Marijnissen, “Calculating the Response of Isotropic Light Dosimetry Probes as a Function of the Tissue Refractive Index,” Appl. Opt. 28, 2288–2291 (1989).
    [CrossRef] [PubMed]
  5. W. G. Houf, F. P. Incropera, “An Assessment of Techniques for Predicting Radiation Transfer in Aqueous Media,” J. Quant. Spectrosc. Radiat. Transfer 23, 101–115 (1980).
    [CrossRef]
  6. H. C. van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications,” (Academic, New York, 1980), Vols. 1, 2.
  7. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.
  8. F. P. Bolin, L. E. Preuss, R. C. Taylor, R. J. Ference, “Measurement of the Index of Refraction of Mammalian Tissue,” OSA Annual Meeting, 1988 Technical Digest Series, Vol. 11 (Optical Society of America, Washington, DC, 1988), paper TH04.
  9. M. Keijzer, W. M. Star, P. R. M. Storchi, “Optical Diffusion in Layered Media,” Appl. Opt. 27, 1820–1824 (1988).
    [CrossRef] [PubMed]
  10. J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The Delta-Eddington Approximation for Radiative Flux Transfer,” J. Atm. Sci. 33, 2452–2459 (1976).
    [CrossRef]
  11. A. J. Welch, J. A. Pearce, K. R. Diller, G. Yoon, W. F. Cheong, “Heat Generation in Laser Irradiated Tissue,” ASME J. Biomech. Eng. 111, 62–68 (1989).
    [CrossRef]
  12. W. E. Meador, W. R. Weaver, “Diffusion Approximation for Large Absorption in Radiative Transfer,” Appl. Opt. 18, 1204–1208 (1979).
    [CrossRef] [PubMed]

1989 (2)

A. J. Welch, J. A. Pearce, K. R. Diller, G. Yoon, W. F. Cheong, “Heat Generation in Laser Irradiated Tissue,” ASME J. Biomech. Eng. 111, 62–68 (1989).
[CrossRef]

W. M. Star, J. P. A. Marijnissen, “Calculating the Response of Isotropic Light Dosimetry Probes as a Function of the Tissue Refractive Index,” Appl. Opt. 28, 2288–2291 (1989).
[CrossRef] [PubMed]

1988 (2)

1987 (1)

S. L. Jacques, S. A. Prahl, “Modeling Optical and Thermal Distribution in Tissue During Laser Irradiation,” Lasers Surg. Med. 6, 494–503 (1987).
[CrossRef] [PubMed]

1983 (1)

1980 (1)

W. G. Houf, F. P. Incropera, “An Assessment of Techniques for Predicting Radiation Transfer in Aqueous Media,” J. Quant. Spectrosc. Radiat. Transfer 23, 101–115 (1980).
[CrossRef]

1979 (1)

1976 (1)

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The Delta-Eddington Approximation for Radiative Flux Transfer,” J. Atm. Sci. 33, 2452–2459 (1976).
[CrossRef]

Bolin, F. P.

F. P. Bolin, L. E. Preuss, R. C. Taylor, R. J. Ference, “Measurement of the Index of Refraction of Mammalian Tissue,” OSA Annual Meeting, 1988 Technical Digest Series, Vol. 11 (Optical Society of America, Washington, DC, 1988), paper TH04.

Bosch, J. J. T.

Cheong, W. F.

A. J. Welch, J. A. Pearce, K. R. Diller, G. Yoon, W. F. Cheong, “Heat Generation in Laser Irradiated Tissue,” ASME J. Biomech. Eng. 111, 62–68 (1989).
[CrossRef]

Diller, K. R.

A. J. Welch, J. A. Pearce, K. R. Diller, G. Yoon, W. F. Cheong, “Heat Generation in Laser Irradiated Tissue,” ASME J. Biomech. Eng. 111, 62–68 (1989).
[CrossRef]

Ference, R. J.

F. P. Bolin, L. E. Preuss, R. C. Taylor, R. J. Ference, “Measurement of the Index of Refraction of Mammalian Tissue,” OSA Annual Meeting, 1988 Technical Digest Series, Vol. 11 (Optical Society of America, Washington, DC, 1988), paper TH04.

Ferwerda, H. A.

Groenhuis, R. A. J.

Houf, W. G.

W. G. Houf, F. P. Incropera, “An Assessment of Techniques for Predicting Radiation Transfer in Aqueous Media,” J. Quant. Spectrosc. Radiat. Transfer 23, 101–115 (1980).
[CrossRef]

Incropera, F. P.

W. G. Houf, F. P. Incropera, “An Assessment of Techniques for Predicting Radiation Transfer in Aqueous Media,” J. Quant. Spectrosc. Radiat. Transfer 23, 101–115 (1980).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

Jacques, S. L.

S. L. Jacques, S. A. Prahl, “Modeling Optical and Thermal Distribution in Tissue During Laser Irradiation,” Lasers Surg. Med. 6, 494–503 (1987).
[CrossRef] [PubMed]

Joseph, J. H.

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The Delta-Eddington Approximation for Radiative Flux Transfer,” J. Atm. Sci. 33, 2452–2459 (1976).
[CrossRef]

Keijzer, M.

Marijnissen, J. P. A.

Meador, W. E.

Pearce, J. A.

A. J. Welch, J. A. Pearce, K. R. Diller, G. Yoon, W. F. Cheong, “Heat Generation in Laser Irradiated Tissue,” ASME J. Biomech. Eng. 111, 62–68 (1989).
[CrossRef]

Prahl, S. A.

S. L. Jacques, S. A. Prahl, “Modeling Optical and Thermal Distribution in Tissue During Laser Irradiation,” Lasers Surg. Med. 6, 494–503 (1987).
[CrossRef] [PubMed]

Preuss, L. E.

F. P. Bolin, L. E. Preuss, R. C. Taylor, R. J. Ference, “Measurement of the Index of Refraction of Mammalian Tissue,” OSA Annual Meeting, 1988 Technical Digest Series, Vol. 11 (Optical Society of America, Washington, DC, 1988), paper TH04.

Shepherd, A. P.

Star, W. M.

Steinke, J. M.

Storchi, P. R. M.

Taylor, R. C.

F. P. Bolin, L. E. Preuss, R. C. Taylor, R. J. Ference, “Measurement of the Index of Refraction of Mammalian Tissue,” OSA Annual Meeting, 1988 Technical Digest Series, Vol. 11 (Optical Society of America, Washington, DC, 1988), paper TH04.

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications,” (Academic, New York, 1980), Vols. 1, 2.

Weaver, W. R.

Weinman, J. A.

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The Delta-Eddington Approximation for Radiative Flux Transfer,” J. Atm. Sci. 33, 2452–2459 (1976).
[CrossRef]

Welch, A. J.

A. J. Welch, J. A. Pearce, K. R. Diller, G. Yoon, W. F. Cheong, “Heat Generation in Laser Irradiated Tissue,” ASME J. Biomech. Eng. 111, 62–68 (1989).
[CrossRef]

Wiscombe, W. J.

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The Delta-Eddington Approximation for Radiative Flux Transfer,” J. Atm. Sci. 33, 2452–2459 (1976).
[CrossRef]

Yoon, G.

A. J. Welch, J. A. Pearce, K. R. Diller, G. Yoon, W. F. Cheong, “Heat Generation in Laser Irradiated Tissue,” ASME J. Biomech. Eng. 111, 62–68 (1989).
[CrossRef]

Appl. Opt. (4)

ASME J. Biomech. Eng. (1)

A. J. Welch, J. A. Pearce, K. R. Diller, G. Yoon, W. F. Cheong, “Heat Generation in Laser Irradiated Tissue,” ASME J. Biomech. Eng. 111, 62–68 (1989).
[CrossRef]

J. Atm. Sci. (1)

J. H. Joseph, W. J. Wiscombe, J. A. Weinman, “The Delta-Eddington Approximation for Radiative Flux Transfer,” J. Atm. Sci. 33, 2452–2459 (1976).
[CrossRef]

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

J. Quant. Spectrosc. Radiat. Transfer (1)

W. G. Houf, F. P. Incropera, “An Assessment of Techniques for Predicting Radiation Transfer in Aqueous Media,” J. Quant. Spectrosc. Radiat. Transfer 23, 101–115 (1980).
[CrossRef]

Lasers Surg. Med. (1)

S. L. Jacques, S. A. Prahl, “Modeling Optical and Thermal Distribution in Tissue During Laser Irradiation,” Lasers Surg. Med. 6, 494–503 (1987).
[CrossRef] [PubMed]

Other (3)

H. C. van de Hulst, Multiple Light Scattering: Tables, Formulas, and Applications,” (Academic, New York, 1980), Vols. 1, 2.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), Vol. 1.

F. P. Bolin, L. E. Preuss, R. C. Taylor, R. J. Ference, “Measurement of the Index of Refraction of Mammalian Tissue,” OSA Annual Meeting, 1988 Technical Digest Series, Vol. 11 (Optical Society of America, Washington, DC, 1988), paper TH04.

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Figures (6)

Fig. 1
Fig. 1

Forward and backward diffuse fluxes for a uniform collimated irradiance of 100 W/cm2 are shown as a function of depth. Albedo = 0.99, g = 0, sample thickness = 4.0, and matched boundary conditions are assumed. For high albedo and isotropic scattering, there are little differences between diffusion and discrete ordinate solutions.

Fig. 2
Fig. 2

Total fluence rates computed with the diffusion approximation and the discrete ordinate method with respect to different albedos. Differences are larger for high albedos, g = 0, sample thickness = 4.0, matched boundary conditions, and uniformly collimated irradiance of 100 W/cm2 are assumed.

Fig. 3
Fig. 3

Reflection computed using the diffusion approximation with respect to the average cosine angle of phase function g. Albedo = 0.99, sample thickness = 4.0, matched boundaries, and the Henyey-Greenstein phase function are assumed. [1] Original coefficients (μs= 0.99, g) are converted into, [2] μ s = ( 1 g ) μ s and g′ = 0 using Eq. (11), [3] μ s = ( 1 g 2 ) μ s and g′ = g/(1 + g) using Eq. (14). μa = 0.01 is the same for all cases. The values of van de Hulst are used as references.

Fig. 4
Fig. 4

Total fluence rates computed using [1] original coefficients (μa = 0.01, μs = 0.99, g = 0.8), [2] μ a = 0 . 01 , μ s = 0 . 198, and g′ = 0 using Eq. (11), [3] μ s = 0 . 01 , μ s = 0 . 356, and g′ = 0.444 using Eq. (14) are compared with discrete ordinate solutions. Sample thickness = 4.0 and matched boundary are assumed.

Fig. 5
Fig. 5

Total fluence rates for a tissue index of 1.4 are computed from diffusion and discrete ordinate models. Sample thickness = 4.0, albedo = 0.99, and a uniform irradiance of 100 W/cm2 are assumed. For g = 0.8 Eq. (14) is used to transform the coefficients. For diffuse light, the boundary conditions in Eq. (9) are implemented. For collimated light, the external reflection of 2.8% is considered only at the front surface.

Fig. 6
Fig. 6

Diffuse radiance (W/cm2 sr) and collimated flux (W/cm2) at an optical depth of 3.0. Albedo = 0.99, g = 0.8, sample thickness = 4.0, and matched boundary conditions are assumed: [1] orginal coefficients (μa = 0.01, μs = 0.99, g = 0.8); [2] μ a = 0 . 01 , μ s = 0 . 198, and g′ = 0 using Eq. (11); [3] μ a = 0 . 01 , μ s = 0 . 356, and g′ = 0.444 using Eq. (14) are compared with discrete ordinate solutions.

Equations (17)

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s · L ( r , s ) + μ t L ( r , s ) = μ s 4 π 4 π p ( s , s ) L ( r , s ) d ω ,
( 1 4 π ) 4 π p ( s , s ) d ω = 1 ,
d L c ( r , s ) / d s = μ t L c ( r , s ) .
L d ( r , s ) U d ( r ) + F d ( r ) · s ( 3 / 4 π ) ,
2 U d ( z ) / z 2 3 μ a μ tr U d ( z ) = ( 3 μ s μ tr + 3 g μ s μ t ) F c ( z ) / 4 π ,
L d ( z , μ ) = U d ( z ) + ( 3 / 4 π ) g μ s F c ( z ) μ / μ tr U d ( z ) μ / μ tr .
2 π, n L d ( r , s ) s · n d ω = 0 ,
2 π , n L d ( r , s ) s · n d ω = 2 π , n R ( s · n ) L d ( r , s ) ( s · n ) d ω ,
R ( θ i ) = 1 2 [ R 2 + R 2 ] = 1 2 [ sin 2 ( θ i θ t ) sin 2 ( θ i + θ t ) + tan 2 ( θ i θ t ) tan 2 ( θ i + θ t ) ] ,
R ( 0 ) = ( n i n t ) 2 ( n i + n t ) 2 if θ i = 0 , R ( θ i ) = 1 . 0 if θ i > θ c = critical angle .
U d ( z ) A h U d ( z ) / z = A ( g μ s / μ tr ) F c ( z ) / 2 π at z = 0 , U d ( z ) + A h U d ( z ) / z = A ( g μ s / μ tr ) F c ( z ) / 2 π at z = d ,
R 1 = 2 μ c 1 R ( μ ) μ d μ + μ c 2 ; R 2 = 3 μ c 1 R ( μ ) μ 2 d μ + μ c 3 ; μ c = cos θ c .
μ a = μ a ; ( 1 g ) μ s = ( 1 g ) μ s .
μ a = μ a μ s = ( 1 g ) μ s g = 0 .
p ( s , s ) = f δ ( s s ) + ( 1 f ) p 1 ( s , s ) ,
p hg ( s , s ) = ( 1 4 π ) n = 0 ( 2 n + 1 ) g n P n ( s , s ) ,
μ a = μ a , μ s = ( 1 g 2 ) μ s , g = g / ( 1 + g ) .

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