Abstract

An efficient and accurate hybrid model of the Monte Carlo technique and the diffusion theory was developed to simulate the diffuse reflectance of light in a turbid slab due to an infinitely narrow light beam. The narrow beam was normally incident on the top surface of the slab. The hybrid model was accurate in modeling the diffuse reflectance near the light source, where the diffusion theory was most inaccurate. The hybrid model was much faster than a pure Monte Carlo method by a factor as great as several hundred, depending on the optical properties, the thickness of the slab, and the settings of the hybrid and the Monte Carlo computations. The computation speed of the hybrid model was insensitive to the optical properties of the medium, in contrast to the pure Monte Carlo technique. The diffusion theory was accurate in modeling both the diffuse reflectance far from the source and the diffuse transmittance. The hybrid model and the diffusion theory should be used in conjunction for efficient and accurate computation of diffuse reflectance and diffuse transmittance.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. See related studies in S. L. Jacques, ed., Laser-Tissue Interaction VIII, Proc. SPIE2975 (1997).
  2. See related studies in B. Chance, R. R. Alfano, eds., Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model, and Human Studies II, Proc. SPIE2979 (1997).
  3. L. -H. Wang, S. L. Jacques, “Hybrid model of Monte Carlo simulation diffusion theory for light reflectance by turbid media,” J. Opt. Soc. Am. A 10, 1746–1752 (1993).
    [CrossRef]
  4. S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo diffusion theory modeling of light distributions in tissue,” in Laser Interaction with Tissue, M. W. Berns, ed., Proc. SPIE908, 20–28 (1988).
    [CrossRef]
  5. S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).
  6. L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  7. L. -H. Wang, X. -M. Zhao, S. L. Jacques, “Computation of the optical properties of tissues from light reflectance using a neural network,” in Laser-Tissue Interaction V, S. L. Jacques, ed., Proc. SPIE2134, 391–399 (1994).
  8. T. J. Farrell, M. S. Patterson, B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the non-invasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
    [CrossRef] [PubMed]
  9. M. S. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
    [CrossRef] [PubMed]
  10. R. A. J. Groenhuis, H. A. Ferwerda, J. J. Ten Bosch, “Scattering and absorption of turbid materials determined from reflection measurements. I: Theory,” Appl. Opt. 22, 2456–2462 (1983).
    [CrossRef] [PubMed]
  11. L. -H. Wang, S. L. Jacques, “Analysis of diffusion theory and similarity relations,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 107–116 (1993).
    [CrossRef]
  12. S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller, D. H. Sliney, eds., Proc. SPIEIS 5, 102–111 (1989).
  13. L. -H. Wang, S. L. Jacques, L. -Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995). The software package is downloadable from the web page at http://biomed.tamu.edu/∼lw .
    [CrossRef] [PubMed]
  14. H. Kahn, T. E. Harris, “Estimation of particle transmission by random sampling,” Monte Carlo Method, National Bureau of Standards Applied Mathematics Series No. 12 (U.S. Government Printing Office, Washington, D.C., (1951).
  15. G. Yoon, S. A. Prahl, 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]
  16. D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity relations for anisotropic scattering in Monte Carlo simulations of deeply penetrating neutral particles,” J. Comput. Phys. 81, 137–150 (1989).
    [CrossRef]
  17. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).
  18. L. -H. Wang, S. L. Jacques, L. -Q. Zheng, “CONV—Convolution for responses to a finite diameter photon beam incident on multi-layered tissues,” Comput. Methods Programs Biomed. 54, 141–150 (1997).
    [CrossRef]
  19. A. Kienle, M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14, 246–254 (1997).
    [CrossRef]

1997

L. -H. Wang, S. L. Jacques, L. -Q. Zheng, “CONV—Convolution for responses to a finite diameter photon beam incident on multi-layered tissues,” Comput. Methods Programs Biomed. 54, 141–150 (1997).
[CrossRef]

A. Kienle, M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14, 246–254 (1997).
[CrossRef]

1995

L. -H. Wang, S. L. Jacques, L. -Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995). The software package is downloadable from the web page at http://biomed.tamu.edu/∼lw .
[CrossRef] [PubMed]

1993

1992

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

1989

1987

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

1983

1941

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Alter, C. A.

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

Chance, B.

Farrell, T. J.

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

Ferwerda, H. A.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Flock, S. T.

S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo diffusion theory modeling of light distributions in tissue,” in Laser Interaction with Tissue, M. W. Berns, ed., Proc. SPIE908, 20–28 (1988).
[CrossRef]

Greenstein, J. L.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Groenhuis, R. A. J.

Harris, T. E.

H. Kahn, T. E. Harris, “Estimation of particle transmission by random sampling,” Monte Carlo Method, National Bureau of Standards Applied Mathematics Series No. 12 (U.S. Government Printing Office, Washington, D.C., (1951).

Henyey, L. G.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Jacques, S. L.

L. -H. Wang, S. L. Jacques, L. -Q. Zheng, “CONV—Convolution for responses to a finite diameter photon beam incident on multi-layered tissues,” Comput. Methods Programs Biomed. 54, 141–150 (1997).
[CrossRef]

L. -H. Wang, S. L. Jacques, L. -Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995). The software package is downloadable from the web page at http://biomed.tamu.edu/∼lw .
[CrossRef] [PubMed]

L. -H. Wang, S. L. Jacques, “Hybrid model of Monte Carlo simulation diffusion theory for light reflectance by turbid media,” J. Opt. Soc. Am. A 10, 1746–1752 (1993).
[CrossRef]

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller, D. H. Sliney, eds., Proc. SPIEIS 5, 102–111 (1989).

L. -H. Wang, X. -M. Zhao, S. L. Jacques, “Computation of the optical properties of tissues from light reflectance using a neural network,” in Laser-Tissue Interaction V, S. L. Jacques, ed., Proc. SPIE2134, 391–399 (1994).

L. -H. Wang, S. L. Jacques, “Analysis of diffusion theory and similarity relations,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 107–116 (1993).
[CrossRef]

Kahn, H.

H. Kahn, T. E. Harris, “Estimation of particle transmission by random sampling,” Monte Carlo Method, National Bureau of Standards Applied Mathematics Series No. 12 (U.S. Government Printing Office, Washington, D.C., (1951).

Keijzer, M.

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller, D. H. Sliney, eds., Proc. SPIEIS 5, 102–111 (1989).

Kienle, A.

Patterson, M. S.

A. Kienle, M. S. Patterson, “Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from a semi-infinite turbid medium,” J. Opt. Soc. Am. A 14, 246–254 (1997).
[CrossRef]

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

M. S. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity relations for anisotropic scattering in Monte Carlo simulations of deeply penetrating neutral particles,” J. Comput. Phys. 81, 137–150 (1989).
[CrossRef]

S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo diffusion theory modeling of light distributions in tissue,” in Laser Interaction with Tissue, M. W. Berns, ed., Proc. SPIE908, 20–28 (1988).
[CrossRef]

Prahl, S. A.

G. Yoon, S. A. Prahl, 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]

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller, D. H. Sliney, eds., Proc. SPIEIS 5, 102–111 (1989).

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Ten Bosch, J. J.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Wang, L. -H.

L. -H. Wang, S. L. Jacques, L. -Q. Zheng, “CONV—Convolution for responses to a finite diameter photon beam incident on multi-layered tissues,” Comput. Methods Programs Biomed. 54, 141–150 (1997).
[CrossRef]

L. -H. Wang, S. L. Jacques, L. -Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995). The software package is downloadable from the web page at http://biomed.tamu.edu/∼lw .
[CrossRef] [PubMed]

L. -H. Wang, S. L. Jacques, “Hybrid model of Monte Carlo simulation diffusion theory for light reflectance by turbid media,” J. Opt. Soc. Am. A 10, 1746–1752 (1993).
[CrossRef]

L. -H. Wang, S. L. Jacques, “Analysis of diffusion theory and similarity relations,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 107–116 (1993).
[CrossRef]

L. -H. Wang, X. -M. Zhao, S. L. Jacques, “Computation of the optical properties of tissues from light reflectance using a neural network,” in Laser-Tissue Interaction V, S. L. Jacques, ed., Proc. SPIE2134, 391–399 (1994).

Welch, A. J.

G. Yoon, S. A. Prahl, 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]

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller, D. H. Sliney, eds., Proc. SPIEIS 5, 102–111 (1989).

Wilson, B. C.

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

M. S. Patterson, B. Chance, B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity relations for anisotropic scattering in Monte Carlo simulations of deeply penetrating neutral particles,” J. Comput. Phys. 81, 137–150 (1989).
[CrossRef]

S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo diffusion theory modeling of light distributions in tissue,” in Laser Interaction with Tissue, M. W. Berns, ed., Proc. SPIE908, 20–28 (1988).
[CrossRef]

Wyman, D. R.

D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity relations for anisotropic scattering in Monte Carlo simulations of deeply penetrating neutral particles,” J. Comput. Phys. 81, 137–150 (1989).
[CrossRef]

Yoon, G.

Zhao, X. -M.

L. -H. Wang, X. -M. Zhao, S. L. Jacques, “Computation of the optical properties of tissues from light reflectance using a neural network,” in Laser-Tissue Interaction V, S. L. Jacques, ed., Proc. SPIE2134, 391–399 (1994).

Zheng, L. -Q.

L. -H. Wang, S. L. Jacques, L. -Q. Zheng, “CONV—Convolution for responses to a finite diameter photon beam incident on multi-layered tissues,” Comput. Methods Programs Biomed. 54, 141–150 (1997).
[CrossRef]

L. -H. Wang, S. L. Jacques, L. -Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995). The software package is downloadable from the web page at http://biomed.tamu.edu/∼lw .
[CrossRef] [PubMed]

Appl. Opt.

Astrophys. J.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Comput. Methods Programs Biomed.

L. -H. Wang, S. L. Jacques, L. -Q. Zheng, “MCML—Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995). The software package is downloadable from the web page at http://biomed.tamu.edu/∼lw .
[CrossRef] [PubMed]

L. -H. Wang, S. L. Jacques, L. -Q. Zheng, “CONV—Convolution for responses to a finite diameter photon beam incident on multi-layered tissues,” Comput. Methods Programs Biomed. 54, 141–150 (1997).
[CrossRef]

J. Comput. Phys.

D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity relations for anisotropic scattering in Monte Carlo simulations of deeply penetrating neutral particles,” J. Comput. Phys. 81, 137–150 (1989).
[CrossRef]

J. Opt. Soc. Am. A

Lasers Life Sci.

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

Med. Phys.

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

Other

L. -H. Wang, S. L. Jacques, “Analysis of diffusion theory and similarity relations,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 107–116 (1993).
[CrossRef]

S. A. Prahl, M. Keijzer, S. L. Jacques, A. J. Welch, “A Monte Carlo model of light propagation in tissue,” in Dosimetry of Laser Radiation in Medicine and Biology, G. J. Muller, D. H. Sliney, eds., Proc. SPIEIS 5, 102–111 (1989).

H. Kahn, T. E. Harris, “Estimation of particle transmission by random sampling,” Monte Carlo Method, National Bureau of Standards Applied Mathematics Series No. 12 (U.S. Government Printing Office, Washington, D.C., (1951).

S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo diffusion theory modeling of light distributions in tissue,” in Laser Interaction with Tissue, M. W. Berns, ed., Proc. SPIE908, 20–28 (1988).
[CrossRef]

L. -H. Wang, X. -M. Zhao, S. L. Jacques, “Computation of the optical properties of tissues from light reflectance using a neural network,” in Laser-Tissue Interaction V, S. L. Jacques, ed., Proc. SPIE2134, 391–399 (1994).

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

See related studies in S. L. Jacques, ed., Laser-Tissue Interaction VIII, Proc. SPIE2975 (1997).

See related studies in B. Chance, R. R. Alfano, eds., Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model, and Human Studies II, Proc. SPIE2979 (1997).

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 (8)

Fig. 1
Fig. 1

Arrangement of the original and image point sources in a turbid slab, where the signs indicate whether a source is positive or negative.

Fig. 2
Fig. 2

Illustration of the conversion from an infinitely narrow light beam to an isotropic point source in the Monte Carlo step of the hybrid model. The last step of length Lt converts the light into an isotropic point source.

Fig. 3
Fig. 3

Comparison between the pure Monte Carlo method and the diffusion theory in terms of (a) the diffuse reflectance and (b) the diffuse transmittance. The properties of the turbid slab included the relative index of refraction nrel=1, the absorption coefficient μa=0.1 cm-1, the scattering coefficient μs=100 cm-1, the scattering anisotropy g=0.9, and the thickness d=1 cm.

Fig. 4
Fig. 4

Comparison between the pure Monte Carlo method and the diffusion theory in terms of (a) the diffuse reflectance and (b) the diffuse transmittance, and (c) the relative errors between the results. The properties of the turbid slab were described in Fig. 3.

Fig. 5
Fig. 5

(a) Comparisons between the pure Monte Carlo method and the diffusion theory in terms of the diffuse reflectance when an isotropic point source was placed at z=0.3 Lt and (b) the relative errors between the results when an isotropic point source was placed at z=0.1, 0.3, and 0.5 Lt. The properties of the turbid slab were described in Fig. 3.

Fig. 6
Fig. 6

Distribution of the source density generated by the initial Monte Carlo step in the hybrid model, where the critical depth zc was set to 0.05 cm. The unit of the source density was in cm-3 for a unity input and would be in W/cm3 for a 1-W input. The properties of the turbid slab were described in Fig. 3.

Fig. 7
Fig. 7

(a) Comparisons between the pure Monte Carlo method and the hybrid model in terms of the diffuse reflectance and (b) the relative error between the results. The properties of the turbid slab were described in Fig. 3.

Fig. 8
Fig. 8

Comparisons between the pure Monte Carlo method and the hybrid model in terms of the diffuse reflectance when the critical depth was set to (a) 0.05 cm and (b) 0.1 cm. The absorption coefficient of the turbid slab was varied from 0.1, 1, 10 cm-1, while the other properties were held constant, including the relative index of refraction nrel=1.37, the scattering coefficient μs=100 cm-1, the scattering anisotropy g=0.9, and the thickness d=1 cm.

Tables (1)

Tables Icon

Table 1 User Times for the Pure Monte Carlo Method and the Hybrid Model and the Ratio between the User Times (Monte Carlo/Hybrid) under Various Conditions

Equations (21)

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

ϕ1(r, θ, z; r, θ, z)=14πDexp(-μeffρ)ρ,
ρ=[r2+r2-2rr cos(θ-θ)+(z-z)2]1/2
D=13[μa+μs(1-g)],
μeff=μa/D.
R1(r, θ, 0; r, θ, z)=D ϕzz=0=zs(1+μeffρ)exp(-μeffρ)4πρ3,
T1(r, θ, d; r, θ, z)=-D ϕzz=d=(d-zs)(1+μeffρ)exp(-μeffρ)4πρ3,
zb=2AD,
A=(1+ri)/(1-ri),
ri=-1.440nrel-2+0.710nrel-1+0.668+0.0636nrel.
zsi±=-zb+2i(d+2zb)±(zs+zb),
ϕ(r, θ, z; rs, θs, zs)=i=mn[ϕ1(r, θ, z; rs, θs, zsi+)-ϕ1(r, θ, z; rs, θs, zsi-)],
R(r, θ, 0; rs, θs, zs)=i=mn[R1(r, θ, 0; rs, θs, zsi+)-R1(r, θ, 0; rs, θs, zsi-)],
T(r, θ, d; rs, θs, zs)=i=mn[T1(r, θ, d; rs, θs, zsi+)-T1(r, θ, d; rs, θs, zsi-)].
s=-ln(ξ)μa+μs,
Rdiff(r)=0002πSd(r, z)R(r, 0, 0; r, θ, z)r×dθ dr dz.
Rd(r)=Rmc(r)+Rdiff(r).
ri=(i+0.5)Δr,
zj=(j+0.5)Δz.
Sd(ri, zi)=S(ri, zi)/(ΔViN),
ΔVi=2πriΔrΔz.
Rdiff(r)=i=0Nr-2j=0Nz-2Sd(ri, zj)riΔrΔz×20πR(r, 0, 0; ri, θ, zj)dθ.

Metrics