Abstract

A new method of Monte Carlo simulation has been developed to simulate the spatial distribution of photon density of converging laser beams propagating in a turbid medium such as the phantom of biological tissue. This method can be used to obtain steady-state light distribution in the tissue phantom for a continuous-wave laser beam. We have calculated the steady-state distribution of the photon density and found important features that are uniquely related to the propagation of the converging beams in the tissue phantom.

© 1999 Optical Society of America

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  1. S. Chandrasekhar, Radiative Transfer (Oxford University Press, London, 1950).
  2. B. C. Wilson, G. Adams, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10, 824–830 (1983).
    [CrossRef] [PubMed]
  3. M. Keijzer, S. T. 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. R. Marchesini, C. Clemente, E. Pignoli, M. Brambilla, “Optical properties of in vitro epidermis and their possible relationship with optical properties of in vivo skin,” J. Photochem. Photobiol. B 16, 127–140 (1992).
    [CrossRef] [PubMed]
  5. R. Graaff, A. C. M. Dassel, M. H. Koelink, F. F. M. de Mul, J. G. Aarnoudse, W. G. Zijlstra, “Optical properties of human dermis in vitro and in vivo,” Appl. Opt. 32, 435–447 (1993).
    [CrossRef] [PubMed]
  6. S. L. Jacques, “Time resolved propagation of ultrashort laser pulses within turbid tissues,” Appl. Opt. 28, 2223–2229 (1989).
    [CrossRef] [PubMed]
  7. A. Ishimara, “Diffusion of light in turbid material,” Appl. Opt. 28, 2210–2215 (1989).
    [CrossRef]
  8. S. Wan, R. R. Anderson, J. A. Parrish, “Analytical modeling for the optical properties of the skin with in vitro and in vivo applications,” Photochem. Photobiol. 34, 493–499 (1981).
    [PubMed]
  9. L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astroph. J. 93, 70–83 (1941).
    [CrossRef]
  10. M. de Belder, J. de Kerf, J. Jespers, R. Verbrugge, “Light diffusion in photographic layers: its influence on sensitivity and modulation transfer,” J. Opt. Soc. Am. 55, 1261–1268 (1965).
    [CrossRef]
  11. M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
    [CrossRef] [PubMed]
  12. X. H. Hu, “Efficient use of Q-switched lasers in the treatment of cutaneous lesions,” in Lasers in Surgery: Advanced Characterization, Therapeutics, and SystemsV. R. R. Anderson, ed., Proc. SPIE2395, 586–591 (1995).
  13. A. H. Hielscher, R. E. Alcouffe, K. M. Hanson, J. S. George, “Comparison of finite difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimuto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 55–59.

1993 (1)

1992 (1)

R. Marchesini, C. Clemente, E. Pignoli, M. Brambilla, “Optical properties of in vitro epidermis and their possible relationship with optical properties of in vivo skin,” J. Photochem. Photobiol. B 16, 127–140 (1992).
[CrossRef] [PubMed]

1989 (4)

M. Keijzer, S. T. 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]

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

A. Ishimara, “Diffusion of light in turbid material,” Appl. Opt. 28, 2210–2215 (1989).
[CrossRef]

S. L. Jacques, “Time resolved propagation of ultrashort laser pulses within turbid tissues,” Appl. Opt. 28, 2223–2229 (1989).
[CrossRef] [PubMed]

1983 (1)

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

1981 (1)

S. Wan, R. R. Anderson, J. A. Parrish, “Analytical modeling for the optical properties of the skin with in vitro and in vivo applications,” Photochem. Photobiol. 34, 493–499 (1981).
[PubMed]

1965 (1)

1941 (1)

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

Aarnoudse, J. G.

Adams, G.

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

Alcouffe, R. E.

A. H. Hielscher, R. E. Alcouffe, K. M. Hanson, J. S. George, “Comparison of finite difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimuto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 55–59.

Anderson, R. R.

S. Wan, R. R. Anderson, J. A. Parrish, “Analytical modeling for the optical properties of the skin with in vitro and in vivo applications,” Photochem. Photobiol. 34, 493–499 (1981).
[PubMed]

Brambilla, M.

R. Marchesini, C. Clemente, E. Pignoli, M. Brambilla, “Optical properties of in vitro epidermis and their possible relationship with optical properties of in vivo skin,” J. Photochem. Photobiol. B 16, 127–140 (1992).
[CrossRef] [PubMed]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford University Press, London, 1950).

Clemente, C.

R. Marchesini, C. Clemente, E. Pignoli, M. Brambilla, “Optical properties of in vitro epidermis and their possible relationship with optical properties of in vivo skin,” J. Photochem. Photobiol. B 16, 127–140 (1992).
[CrossRef] [PubMed]

Dassel, A. C. M.

de Belder, M.

de Kerf, J.

de Mul, F. F. M.

George, J. S.

A. H. Hielscher, R. E. Alcouffe, K. M. Hanson, J. S. George, “Comparison of finite difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimuto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 55–59.

Graaff, R.

Greenstein, J. L.

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

Hanson, K. M.

A. H. Hielscher, R. E. Alcouffe, K. M. Hanson, J. S. George, “Comparison of finite difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimuto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 55–59.

Henyey, L. G.

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

Hielscher, A. H.

A. H. Hielscher, R. E. Alcouffe, K. M. Hanson, J. S. George, “Comparison of finite difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimuto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 55–59.

Hu, X. H.

X. H. Hu, “Efficient use of Q-switched lasers in the treatment of cutaneous lesions,” in Lasers in Surgery: Advanced Characterization, Therapeutics, and SystemsV. R. R. Anderson, ed., Proc. SPIE2395, 586–591 (1995).

Ishimara, A.

Jacques, S. L.

S. L. Jacques, “Time resolved propagation of ultrashort laser pulses within turbid tissues,” Appl. Opt. 28, 2223–2229 (1989).
[CrossRef] [PubMed]

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

Jacques, S. T.

M. Keijzer, S. T. 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]

Jespers, J.

Keijzer, M.

M. Keijzer, S. T. 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]

Koelink, M. H.

Marchesini, R.

R. Marchesini, C. Clemente, E. Pignoli, M. Brambilla, “Optical properties of in vitro epidermis and their possible relationship with optical properties of in vivo skin,” J. Photochem. Photobiol. B 16, 127–140 (1992).
[CrossRef] [PubMed]

Parrish, J. A.

S. Wan, R. R. Anderson, J. A. Parrish, “Analytical modeling for the optical properties of the skin with in vitro and in vivo applications,” Photochem. Photobiol. 34, 493–499 (1981).
[PubMed]

Pignoli, E.

R. Marchesini, C. Clemente, E. Pignoli, M. Brambilla, “Optical properties of in vitro epidermis and their possible relationship with optical properties of in vivo skin,” J. Photochem. Photobiol. B 16, 127–140 (1992).
[CrossRef] [PubMed]

Prahl, S. A.

M. Keijzer, S. T. 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]

Star, W. M.

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

Sterenborg, H. J. C. M.

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

van Gemert, M. J. C.

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

Verbrugge, R.

Wan, S.

S. Wan, R. R. Anderson, J. A. Parrish, “Analytical modeling for the optical properties of the skin with in vitro and in vivo applications,” Photochem. Photobiol. 34, 493–499 (1981).
[PubMed]

Welch, A. J.

M. Keijzer, S. T. 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]

Wilson, B. C.

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

Zijlstra, W. G.

Appl. Opt. (3)

Astroph. J. (1)

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

IEEE Trans. Biomed. Eng. (1)

M. J. C. van Gemert, S. L. Jacques, H. J. C. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Photochem. Photobiol. B (1)

R. Marchesini, C. Clemente, E. Pignoli, M. Brambilla, “Optical properties of in vitro epidermis and their possible relationship with optical properties of in vivo skin,” J. Photochem. Photobiol. B 16, 127–140 (1992).
[CrossRef] [PubMed]

Lasers Surg. Med. (1)

M. Keijzer, S. T. 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. (1)

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

Photochem. Photobiol. (1)

S. Wan, R. R. Anderson, J. A. Parrish, “Analytical modeling for the optical properties of the skin with in vitro and in vivo applications,” Photochem. Photobiol. 34, 493–499 (1981).
[PubMed]

Other (3)

S. Chandrasekhar, Radiative Transfer (Oxford University Press, London, 1950).

X. H. Hu, “Efficient use of Q-switched lasers in the treatment of cutaneous lesions,” in Lasers in Surgery: Advanced Characterization, Therapeutics, and SystemsV. R. R. Anderson, ed., Proc. SPIE2395, 586–591 (1995).

A. H. Hielscher, R. E. Alcouffe, K. M. Hanson, J. S. George, “Comparison of finite difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. G. Fujimuto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 55–59.

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Figures (7)

Fig. 1
Fig. 1

Geometric configuration of a converging laser beam incident on a phantom of the biological tissue from air where α = 30°, d = 1.0 mm, and the refractive index of the tissue n = 1.34.

Fig. 2
Fig. 2

Steady-state distribution of photon density in the yz plane of the tissue for a converging incident laser beam. The profile of the incident laser beam at the air–tissue boundary is Gaussian with a radius r = 0.268 mm defined at (1/e 2) of the central peak. Parameters for the tissue are μ s = 10.0 (mm-1), μ a = 0.5 (mm-1), and g = 0.9.

Fig. 3
Fig. 3

Same as Fig. 2, except for a laser beam of top-hat profile with the same radius r = 0.268 mm.

Fig. 4
Fig. 4

Comparison of the photon density on the z axis for the two cases shown in Figs. 2 and 3.

Fig. 5
Fig. 5

Dependence of photon density on the asymmetry parameter g: (a) contour diagram of Fig. 2 where g = 0.9; (b) same as part (a), except g = 0.8; (c) same as part (a), except g = 0.6. The photon density of each contour is indicated by the attached number with a standard deviation of ±0.3 in the unit of (1011/mm3).

Fig. 6
Fig. 6

Comparison of the photon density on the z axis for the three cases shown in Fig. 5.

Fig. 7
Fig. 7

Same as Fig. 2, except μ s = 5.0 (mm-1).

Equations (6)

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N0Fρδt-t0δz=0+,
Δt0=ti+1-ti=av,
Wjtib3Δt,
Wjtib3Δt
N=N0×m
Wjb3=i=0mmaxWjtib3.

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