Abstract

Light reflectance by semi-infinite turbid media is modeled by a hybrid of Monte Carlo simulation and diffusion theory, which combines the accuracy of Monte Carlo simulation near the source and the speed of diffusion theory distant from the source. For example, when the turbid medium has the following optical properties—absorption coefficient 1 cm−1, scattering coefficient 100 cm−1, anisotropy 0.9, and refractive-index-matched boundary—the hybrid simulation is 7 times faster than the pure Monte Carlo simulation (100,000 photon packets were traced), and the difference between the two simulations is within 2 standard deviations of the Monte Carlo simulation.

© 1993 Optical Society of America

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References

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  1. J. L. Bulnois, “Photophysical processes in recent medical laser developments: a review,” Lasers Med. Sci. 1, 47–66 (1986).
    [CrossRef]
  2. B. C. Wilson, M. S. Patterson, “The physics of photo-dynamic therapy,” Phys. Med. Biol. 31, 327–360 (1986).
    [CrossRef] [PubMed]
  3. M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
    [CrossRef] [PubMed]
  4. S. A. Prahl, “Calculation of light distributions and optical properties of tissue,” Ph.D. dissertation (University of Texas at Austin, Austin, Tex., 1988).
  5. R. A. J. Groenhuis, J. J. Ten Bosch, H. A. Ferwerda, “Scattering and absorption of turbid materials determined from reflection measurements. 1: Theory,” Appl. Opt. 22, 2456–2462 (1983).
    [CrossRef] [PubMed]
  6. R. A. J. Groenhuis, J. J. Ten Bosch, H. A. Ferwerda, “Scattering and absorption of turbid materials determined from reflection measurements. 2: Measuring method and calibration,” Appl. Opt. 22, 2463–2467 (1983).
    [CrossRef] [PubMed]
  7. 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]
  8. B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
    [CrossRef] [PubMed]
  9. 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. Miller, D. H. Sliney, eds., Proc. Soc. Photo-Opt. Instrum. Eng. Institute Ser.IS5, 102–111 (1989). [Note the typo in Eq. (10), where the denominator should be 1 − g0+ 2g0x.]
  10. A. N. Witt, “Multiple scattering in reflection nebulae I. A Monte Carlo approach,” Astrophys. J. Suppl. Ser. 35, 1–6 (1977).
    [CrossRef]
  11. S. T. Flock, B. C. Wilson, 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]
  12. 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. Soc. Photo-Opt. Instrum. Eng.908, 20–28 (1988).
    [CrossRef]
  13. 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).
  14. L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  15. D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity relations for the interaction parameters in radiation transport,” Appl. Opt. 28, 5243–5249 (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. H. Kahn, T. E. Harris, “Estimation of particle transmission by random sampling,” in Monte Carlo Method, Institute for Numerical Analysis, ed., Natl. Bur. Stand. (US) Appl. Math. Ser.12 (U.S. Government Printing Office, Washington, D.C., 1951).
  18. Lihong Wang, S. L. Jacques, Monte Carlo Modeling of Light Transport in Multi-layered Tissues in Standard C (The University of Texas M.D. Anderson Cancer Center, Houston, Tex., 1992).
  19. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).
  20. B. C. Wilson, T. J. Farrell, M. S. Patterson, “An optical fiber-based diffuse reflectance spectrometer for non-invasive investigation of photodynamic sensitizers in vivo,” in Future Directions and Applications in Photodynamic Therapy, C. J. Gomer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.IS6, 219–232 (1990).

1992 (1)

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 (4)

S. T. Flock, B. C. Wilson, 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]

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef] [PubMed]

D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity relations for the interaction parameters in radiation transport,” Appl. Opt. 28, 5243–5249 (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]

1987 (1)

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).

1986 (2)

J. L. Bulnois, “Photophysical processes in recent medical laser developments: a review,” Lasers Med. Sci. 1, 47–66 (1986).
[CrossRef]

B. C. Wilson, M. S. Patterson, “The physics of photo-dynamic therapy,” Phys. Med. Biol. 31, 327–360 (1986).
[CrossRef] [PubMed]

1983 (3)

1977 (1)

A. N. Witt, “Multiple scattering in reflection nebulae I. A Monte Carlo approach,” Astrophys. J. Suppl. Ser. 35, 1–6 (1977).
[CrossRef]

1941 (1)

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

Adam, G.

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

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).

Bulnois, J. L.

J. L. Bulnois, “Photophysical processes in recent medical laser developments: a review,” Lasers Med. Sci. 1, 47–66 (1986).
[CrossRef]

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]

B. C. Wilson, T. J. Farrell, M. S. Patterson, “An optical fiber-based diffuse reflectance spectrometer for non-invasive investigation of photodynamic sensitizers in vivo,” in Future Directions and Applications in Photodynamic Therapy, C. J. Gomer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.IS6, 219–232 (1990).

Ferwerda, H. A.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

Flock, S. T.

S. T. Flock, B. C. Wilson, 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]

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. Soc. Photo-Opt. Instrum. Eng.908, 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,” in Monte Carlo Method, Institute for Numerical Analysis, ed., Natl. Bur. Stand. (US) Appl. Math. Ser.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.

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (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. Miller, D. H. Sliney, eds., Proc. Soc. Photo-Opt. Instrum. Eng. Institute Ser.IS5, 102–111 (1989). [Note the typo in Eq. (10), where the denominator should be 1 − g0+ 2g0x.]

Lihong Wang, S. L. Jacques, Monte Carlo Modeling of Light Transport in Multi-layered Tissues in Standard C (The University of Texas M.D. Anderson Cancer Center, Houston, Tex., 1992).

Kahn, H.

H. Kahn, T. E. Harris, “Estimation of particle transmission by random sampling,” in Monte Carlo Method, Institute for Numerical Analysis, ed., Natl. Bur. Stand. (US) Appl. Math. Ser.12 (U.S. Government Printing Office, Washington, D.C., 1951).

Keijzer, M.

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (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. Miller, D. H. Sliney, eds., Proc. Soc. Photo-Opt. Instrum. Eng. Institute Ser.IS5, 102–111 (1989). [Note the typo in Eq. (10), where the denominator should be 1 − g0+ 2g0x.]

Patterson, M. S.

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]

S. T. Flock, B. C. Wilson, 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]

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]

D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity relations for the interaction parameters in radiation transport,” Appl. Opt. 28, 5243–5249 (1989).
[CrossRef] [PubMed]

B. C. Wilson, M. S. Patterson, “The physics of photo-dynamic therapy,” Phys. Med. Biol. 31, 327–360 (1986).
[CrossRef] [PubMed]

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. Soc. Photo-Opt. Instrum. Eng.908, 20–28 (1988).
[CrossRef]

B. C. Wilson, T. J. Farrell, M. S. Patterson, “An optical fiber-based diffuse reflectance spectrometer for non-invasive investigation of photodynamic sensitizers in vivo,” in Future Directions and Applications in Photodynamic Therapy, C. J. Gomer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.IS6, 219–232 (1990).

Prahl, S. A.

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (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, “Calculation of light distributions and optical properties of tissue,” Ph.D. dissertation (University of Texas at Austin, Austin, Tex., 1988).

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. Miller, D. H. Sliney, eds., Proc. Soc. Photo-Opt. Instrum. Eng. Institute Ser.IS5, 102–111 (1989). [Note the typo in Eq. (10), where the denominator should be 1 − g0+ 2g0x.]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

Ten Bosch, J. J.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

Wang, Lihong

Lihong Wang, S. L. Jacques, Monte Carlo Modeling of Light Transport in Multi-layered Tissues in Standard C (The University of Texas M.D. Anderson Cancer Center, Houston, Tex., 1992).

Welch, A. J.

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (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. Miller, D. H. Sliney, eds., Proc. Soc. Photo-Opt. Instrum. Eng. Institute Ser.IS5, 102–111 (1989). [Note the typo in Eq. (10), where the denominator should be 1 − g0+ 2g0x.]

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]

S. T. Flock, B. C. Wilson, 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]

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]

D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity relations for the interaction parameters in radiation transport,” Appl. Opt. 28, 5243–5249 (1989).
[CrossRef] [PubMed]

B. C. Wilson, M. S. Patterson, “The physics of photo-dynamic therapy,” Phys. Med. Biol. 31, 327–360 (1986).
[CrossRef] [PubMed]

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

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. Soc. Photo-Opt. Instrum. Eng.908, 20–28 (1988).
[CrossRef]

B. C. Wilson, T. J. Farrell, M. S. Patterson, “An optical fiber-based diffuse reflectance spectrometer for non-invasive investigation of photodynamic sensitizers in vivo,” in Future Directions and Applications in Photodynamic Therapy, C. J. Gomer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.IS6, 219–232 (1990).

Witt, A. N.

A. N. Witt, “Multiple scattering in reflection nebulae I. A Monte Carlo approach,” Astrophys. J. Suppl. Ser. 35, 1–6 (1977).
[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]

D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity relations for the interaction parameters in radiation transport,” Appl. Opt. 28, 5243–5249 (1989).
[CrossRef] [PubMed]

Appl. Opt. (3)

Astrophys. J. (1)

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

Astrophys. J. Suppl. Ser. (1)

A. N. Witt, “Multiple scattering in reflection nebulae I. A Monte Carlo approach,” Astrophys. J. Suppl. Ser. 35, 1–6 (1977).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

S. T. Flock, B. C. Wilson, 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. Comput. Phys. (1)

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]

Lasers Life Sci. (1)

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).

Lasers Med. Sci. (1)

J. L. Bulnois, “Photophysical processes in recent medical laser developments: a review,” Lasers Med. Sci. 1, 47–66 (1986).
[CrossRef]

Lasers Surg. Med. (1)

M. Keijzer, S. L. Jacques, S. A. Prahl, A. J. Welch, “Light distributions in artery tissue: Monte Carlo simulations for finite-diameter laser beams,” Lasers Surg. Med. 9, 148–154 (1989).
[CrossRef] [PubMed]

Med. Phys. (2)

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]

B. C. Wilson, G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Phys. Med. Biol. (1)

B. C. Wilson, M. S. Patterson, “The physics of photo-dynamic therapy,” Phys. Med. Biol. 31, 327–360 (1986).
[CrossRef] [PubMed]

Other (7)

S. A. Prahl, “Calculation of light distributions and optical properties of tissue,” Ph.D. dissertation (University of Texas at Austin, Austin, Tex., 1988).

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. Miller, D. H. Sliney, eds., Proc. Soc. Photo-Opt. Instrum. Eng. Institute Ser.IS5, 102–111 (1989). [Note the typo in Eq. (10), where the denominator should be 1 − g0+ 2g0x.]

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. Soc. Photo-Opt. Instrum. Eng.908, 20–28 (1988).
[CrossRef]

H. Kahn, T. E. Harris, “Estimation of particle transmission by random sampling,” in Monte Carlo Method, Institute for Numerical Analysis, ed., Natl. Bur. Stand. (US) Appl. Math. Ser.12 (U.S. Government Printing Office, Washington, D.C., 1951).

Lihong Wang, S. L. Jacques, Monte Carlo Modeling of Light Transport in Multi-layered Tissues in Standard C (The University of Texas M.D. Anderson Cancer Center, Houston, Tex., 1992).

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, 1988).

B. C. Wilson, T. J. Farrell, M. S. Patterson, “An optical fiber-based diffuse reflectance spectrometer for non-invasive investigation of photodynamic sensitizers in vivo,” in Future Directions and Applications in Photodynamic Therapy, C. J. Gomer, ed., Proc. Soc. Photo-Opt. Instrum. Eng.IS6, 219–232 (1990).

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Figures (6)

Fig. 1
Fig. 1

Illustration of the hybrid model. Zc is the critical depth. The last step size is 1 mfp′.

Fig. 2
Fig. 2

Contour plot of the source term Sd(r, z) (in cm−3) for diffusion theory after the Monte Carlo simulation step of 100,000 photon packets in the hybrid model. The optical parameters of the medium are μa = 1 cm−1, μs = 100 cm−1, g = 0.9, nrel = 1. The grid separations for the hybrid model in the r and the z directions are 5 × 10−3 cm and 3 × 10−3 cm, respectively; and the number of grid elements is 100 in both directions. Note that we intentionally plot versus r in both directions symmetrically, although r is always greater than or equal to zero in the coordinate system.

Fig. 3
Fig. 3

Comparison between diffuse reflectances from hybrid model and pure Monte Carlo simulation (both with 100,000 photon packets) for an index-matched boundary. The solid curve with open circles is the contribution from the Monte Carlo simulation part of the hybrid model, Rmc(r). The optical parameters and the computation settings are described in the caption for Fig. 2.

Fig. 4
Fig. 4

Error of hybrid model with respect to pure Monte Carlo simulation. Ten runs of 100,000 photon packets were completed for each model. The optical parameters and the grid system are given in the caption for Fig. 2. The average diffuse reflectances of 10 runs are denoted by Mavg and Havg for the pure Monte Carlo simulation and for that of the hybrid model, respectively, and the corresponding standard deviations are denoted Msd and Hsd. (a) Relative error of the pure Monte Carlo simulation, (b) relative error of the hybrid model, (c) difference between the average values of the hybrid model and of the pure Monte Carlo model divided by the standard deviation of the pure Monte Carlo simulation, (d) difference between the average values of the hybrid model and of the pure Monte Carlo model divided by the average values of the pure Monte Carlo simulation.

Fig. 5
Fig. 5

Comparison of diffuse reflectances from the hybrid model and from the pure Monte Carlo simulation (both with 100,000 photon packets) for an index-mismatched boundary. The solid curve with open circles is the contribution from the Monte Carlo simulation part of the hybrid model, Rmc(r). The optical parameters of the medium are μa = 1 cm−1, μs = 100 cm−1, g = 0.9, and nrel = 1.37. The grid system is the same as in Fig. 2.

Fig. 6
Fig. 6

Comparison of diffuse reflectances from the hybrid model (solid diamonds) and from the pure Monte Carlo simulation (open squares) with varied absorption coefficients while other optical parameters are kept constant (100,000 photon packets are used for both simulations). The optical parameters of the medium are μs = 100 cm−1, g = 0.9, and nrel = 1; and varied μa = 0.1, 1, and 10 cm−1, respectively. The grid system is the same as in Fig. 2.

Equations (18)

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

s = - ln ( ξ ) / ( μ a + μ s ) ,
R ( r - r , θ - θ , z ) = 1 4 π [ z ( μ eff + 1 d 1 ) exp ( - μ eff d 1 ) d 1 2 + ( z + 2 z b ) ( μ eff + 1 d 2 ) exp ( - μ eff d 2 ) d 2 2 ] .
z b = 2 A D ,
A = ( 1 + r i ) / ( 1 - r i ) ,
r i = - 1.440 n rel - 2 + 0.710 n rel - 1 + 0.668 + 0.0636 n rel ,
D = 1 / { 3 [ μ a + μ s ( 1 - g ) ] } .
μ eff = { 3 μ a [ μ a + μ s ( 1 - g ) ] } 1 / 2 .
d 1 = [ r 2 + r 2 - 2 r r cos ( θ - θ ) + z 2 ] 1 / 2 ,
d 2 = [ r 2 + r 2 - 2 r r cos ( θ - θ ) + ( z + 2 z b ) 2 ] 1 / 2 .
R diff ( r ) = 0 0 0 2 π S d ( r , z ) R ( r - r , - θ , z ) r d θ d r d z .
R d ( r ) = R m c ( r ) + R diff ( r ) .
r i = ( i + 0.5 ) Δ r ,
z j = ( j + 0.5 ) Δ z .
S d ( r i , z j ) = S ( r i , z j ) / ( Δ V i N ) ,
Δ V i = 2 π r i Δ r Δ z .
S d ( r i , z j ) = S ( r i , z j ) / ( 2 π r i Δ r Δ z N ) .
R diff ( r ) = i = 0 N r - 2 j = 0 N z - 2 S d ( r i , z j ) r i Δ r Δ z 2 × 0 π R ( r - r i , - θ , z j ) d θ .
R diff ( r ) = i = 0 N r - 2 j = 0 N z - 2 S ( r i , z j ) / ( π N ) 0 π R ( r - r i , - θ , z j ) d θ .

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