Abstract

A method for improving the accuracy of the optical diffusion theory for a multilayer scattering medium is presented. An infinitesimally narrow incident light beam is replaced by multiple isotropic point sources of different strengths that are placed in the scattering medium along the incident beam. The multiple sources are then used to develop a multilayer diffusion theory. Diffuse reflectance is then computed using the multilayer diffusion theory and compared with accurate data computed by the Monte Carlo method. This multisource method is found to be significantly more accurate than the previous single-source method.

© 2007 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  12. L.-H. Wang, S. L. Jacques, and L.-Q. Zheng, "MCML-Monte Carlo modeling of photon transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
    [CrossRef] [PubMed]
  13. S. T. Flock, B. C. Wilson, and M. S. Patterson, "Monte Carlo modeling of light propagation in highly scattering tissues--II: comparison with measurements in phantoms," IEEE Trans. Biomed. Eng. 36, 1169-1173 (1989).
    [CrossRef] [PubMed]

2004 (1)

2000 (1)

L.-H. Wang and S. L. Jacques, "Source of error in calculation of optical diffuse reflectance from turbid media using diffusion theory," Computer Methods and Programs in Biomedicine 61, 163-170 (2000).
[CrossRef] [PubMed]

1999 (1)

A. Bono, S. Tomatis, and C. Bartoli, et al., "The ABCD system of melanoma detection: A spectrophotometric analysis of the asymmetry, border, color, and dimension," Cancer 85, 72-77 (1999).
[CrossRef] [PubMed]

1998 (2)

1997 (1)

1996 (1)

R. Richards-Kortum and E. Sevick-Muraca, "Quantitative optical spectroscopy for tissue diagnosis," Annu. Rev. Phys. Chem. 555-606 (1996).
[CrossRef] [PubMed]

1995 (1)

L.-H. Wang, S. L. Jacques, and L.-Q. Zheng, "MCML-Monte Carlo modeling of photon transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

1994 (1)

1992 (1)

T. J. Farrell, M. S. Patterson, and B. Wilson, "A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo," Med. Phys. 19, 879-888 (1992).
[CrossRef] [PubMed]

1989 (2)

S. T. Flock, B. C. Wilson, and M. S. Patterson, "Monte Carlo modeling of light propagation in highly scattering tissues--II: comparison with measurements in phantoms," IEEE Trans. Biomed. Eng. 36, 1169-1173 (1989).
[CrossRef] [PubMed]

G. Yoon, S. A. Prahl, and A. J. Welch, "Accuracies of the diffusion approximation and its similarity relations for laser irradiated biological media," Appl. Opt. 28, 2250-2255 (1989).
[CrossRef] [PubMed]

Annu. Rev. Phys. Chem. (1)

R. Richards-Kortum and E. Sevick-Muraca, "Quantitative optical spectroscopy for tissue diagnosis," Annu. Rev. Phys. Chem. 555-606 (1996).
[CrossRef] [PubMed]

Appl. Opt. (4)

Cancer (1)

A. Bono, S. Tomatis, and C. Bartoli, et al., "The ABCD system of melanoma detection: A spectrophotometric analysis of the asymmetry, border, color, and dimension," Cancer 85, 72-77 (1999).
[CrossRef] [PubMed]

Comput. Methods Programs Biomed. (1)

L.-H. Wang, S. L. Jacques, and L.-Q. Zheng, "MCML-Monte Carlo modeling of photon transport in multi-layered tissues," Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

Computer Methods and Programs in Biomedicine (1)

L.-H. Wang and S. L. Jacques, "Source of error in calculation of optical diffuse reflectance from turbid media using diffusion theory," Computer Methods and Programs in Biomedicine 61, 163-170 (2000).
[CrossRef] [PubMed]

IEEE Trans. Biomed. Eng. (1)

S. T. Flock, B. C. Wilson, and M. S. Patterson, "Monte Carlo modeling of light propagation in highly scattering tissues--II: comparison with measurements in phantoms," IEEE Trans. Biomed. Eng. 36, 1169-1173 (1989).
[CrossRef] [PubMed]

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

Med. Phys. (1)

T. J. Farrell, M. S. Patterson, and B. Wilson, "A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo," Med. Phys. 19, 879-888 (1992).
[CrossRef] [PubMed]

Opt. Express (1)

Other (1)

M. N. O. Sadiku, "Finite difference methods," in Numerical Techniques in Electromagnetics, 2nd Ed. (CRC Press, 2001), pp. 197-199.

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Figures (4)

Fig. 1
Fig. 1

Positions of isotropic point sources in a two-layer medium. The first layer has a thickness of L.

Fig. 2
Fig. 2

(a) Comparison of diffuse reflectance calculated with single-source (dotted line), double-source approximations (solid line) and with the Monte Carlo method (circles). (b) Absolute value of the relative error for the single-source (dotted line) and double-source (solid line) diffusion approximations.

Fig. 3
Fig. 3

(a) Absolute value of the relative error for the single-source (dotted line) and double-source (solid line) diffusion approximations for g = 0.5 . (b) Absolute value of the relative error for the single-source (dotted line) and double-source (solid line) diffusion approximations for g = 0 . The reduced scattering and absorption coefficients are the same as those used for Fig. 2.

Fig. 4
Fig. 4

(a) Comparison of diffuse reflectance calculated with the double-source approximation (solid line) and with the Monte Carlo method (circles). The first layer is optically thin. (b) Relative error for the double-source diffusion approximation (line) and the semi-infinite diffusion equation solution utilizing the optical properties of the second layer (dots).

Equations (20)

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2 Φ ( x , y , z ) μ a D Φ ( x , y , z ) = w D δ ( x , y , z z 0 ) ,
L t = 1 μ t ,
2 Φ i ( x , y , z ) μ a i D i Φ i ( x , y , z ) = w i D i δ ( x , y , z z i ) ,
i = 1 , 2
z 1 = 0 L z exp ( μ t 1 z ) d z 0 L exp ( μ t 1 z ) d z = 1 ( μ t 1 L + 1 ) exp ( μ t 1 L ) μ t 1 ( 1 exp ( μ t 1 L ) ) ,
z 2 = L z exp ( μ t 1 L μ t 2 ( z L ) ) d z L exp ( μ t 1 L μ t 2 ( z L ) ) d z = L + 1 μ t 2 .
w 1 ( z ) = μ s 1 0 L exp ( μ t 1 z ) d z = a 1 ( 1 exp ( μ t 1 L ) ) ,
w 2 ( z ) = μ s 2 L exp ( μ t 1 L μ t 2 ( z L ) ) d z = a 2 exp ( μ t 1 L ) .
2 z 2 ϕ i ( z , s a , s b ) α i 2 ϕ i ( z , s a , s b ) = w i D i δ ( z z i ) ,
i = 1 , 2 ,
ϕ 1 ( z b , s ) = 0 ,
ϕ 2 ( + , s ) = 0.
ϕ 1 ( L , s ) ϕ 2 ( L , s ) = 1 ,
D 1 ϕ 1 ( z , s ) z | z = L = D 2 ϕ 2 ( z , s ) z | z = L .
ϕ 1 ( z , s ) = w 1 / 2 α 1 D 1 × [ ( α 2 D 2 α 1 D 1 ) ( α 2 D 2 + α 1 D 1 ) exp ( α 1 ( z + 2 L + 2 z b z 1 ) ) + ( α 1 D 1 α 2 D 2 ) exp ( α 1 ( z z 1 ) ) + exp ( α 1 | z z 1 | ) ( α 1 D 1 + α 2 D 2 ) ( α 1 D 1 + α 2 D 2 ) exp ( α 1 ( z + z 1 + 2 z b ) ) + ( α 1 D 1 α 2 D 2 ) exp ( α 1 ( z + z 1 2 L ) ) + ( α 1 D 1 α 2 D 2 ) ( α 1 D 1 + α 2 D 2 ) exp ( α 1 ( 2 L z 1 z ) ) + ( α 1 D 1 α 2 D 2 ) exp ( α 1 ( z + z 1 + 2 z b ) ) + ( α 2 D 2 α 1 D 1 ) ( α 1 D 1 + α 2 D 2 ) exp ( α 1 ( z 1 + 2 z b + 2 L z ) ) + ( α 1 D 1 α 2 D 2 ) exp ( α 1 ( z 1 z ) ) ] + w 2 exp ( α 2 ( z 2 L ) ) sinh ( α 1 ( z b + z ) ) α 2 D 2 sinh ( α 1 ( z b + L ) ) + α 1 D 1 cosh ( α 1 ( z b + L ) ) , , z L ,
ϕ 2 ( z , s ) = ( α 2 D 2 w 1 exp ( α 1 ( L z 1 ) ) α 1 D 1 w 2 exp ( α 2 ( z 2 L ) ) ) 2 ( α 2 D 2 tanh ( α 1 ( z b + L ) ) + α 1 D 1 ) exp ( α 2 ( z L ) ) α 2 D 2 + ( w 2 exp ( α 2 ( z 2 L ) ) + w 1 exp ( α 1 ( L z 1 ) ) ) sinh ( α 1 ( L + z b ) ) w 1 exp ( α 1 ( z b + z 1 ) ) 2 ( α 2 D 2 sinh ( α 1 ( z b + L ) ) + α 1 D 1 cosh ( α 1 ( z b + L ) ) ) × exp ( α 2 ( z L ) ) + w 2 exp ( α 2 | z 2 z | ) 2 α 2 D 2 , z > L .
Φ i ( x , y , z ) = 1 4 π 2 ϕ i ( z , s ) × exp ( i ( s a x + s b y ) ) d s a d s b .
Φ i ( ρ , z ) = 1 2 π 0 s ϕ i ( z , s ) J 0 ( s ρ ) d s ,
R ( ρ ) = 1 4 Φ 1 ( ρ , z = 0 ) + 1 2 D 1 Φ 1 ( ρ , z ) z | z = 0 .
μ s i = μ s i 1 g i ,

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