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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 31, Iss. 2 — Feb. 1, 2014
  • pp: 301–311

Comparative analysis of discrete and continuous absorption weighting estimators used in Monte Carlo simulations of radiative transport in turbid media

Carole K. Hayakawa, Jerome Spanier, and Vasan Venugopalan  »View Author Affiliations

JOSA A, Vol. 31, Issue 2, pp. 301-311 (2014)

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We examine the relative error of Monte Carlo simulations of radiative transport that employ two commonly used estimators that account for absorption differently, either discretely, at interaction points, or continuously, between interaction points. We provide a rigorous derivation of these discrete and continuous absorption weighting estimators within a stochastic model that we show to be equivalent to an analytic model, based on the radiative transport equation (RTE). We establish that both absorption weighting estimators are unbiased and, therefore, converge to the solution of the RTE. An analysis of spatially resolved reflectance predictions provided by these two estimators reveals no advantage to either in cases of highly scattering and highly anisotropic media. However, for moderate to highly absorbing media or isotropically scattering media, the discrete estimator provides smaller errors at proximal source locations while the continuous estimator provides smaller errors at distal locations. The origin of these differing variance characteristics can be understood through examination of the distribution of exiting photon weights.

© 2014 Optical Society of America

OCIS Codes
(170.3660) Medical optics and biotechnology : Light propagation in tissues
(170.5280) Medical optics and biotechnology : Photon migration
(290.7050) Scattering : Turbid media
(300.1030) Spectroscopy : Absorption

ToC Category:
Medical Optics and Biotechnology

Original Manuscript: October 8, 2013
Revised Manuscript: November 26, 2013
Manuscript Accepted: December 13, 2013
Published: January 20, 2014

Virtual Issues
Vol. 9, Iss. 4 Virtual Journal for Biomedical Optics

Carole K. Hayakawa, Jerome Spanier, and Vasan Venugopalan, "Comparative analysis of discrete and continuous absorption weighting estimators used in Monte Carlo simulations of radiative transport in turbid media," J. Opt. Soc. Am. A 31, 301-311 (2014)

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  1. S. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” Proc. SPIE 5, 102–111 (1989).
  2. M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, and D. T. Delpy, “A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,” Phys. Med. Biol. 38, 1859–1876 (1993). [CrossRef]
  3. L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995). [CrossRef]
  4. D. A. Boas, J. P. Culver, J. J. Stott, and A. K. Dunn, “Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head,” Opt. Express 10, 159–170 (2002). [CrossRef]
  5. J. Ramella-Roman, S. Prahl, and S. Jacques, “Three Monte Carlo programs of polarized light transport into scattering media: part I,” Opt. Express 13, 4420–4438 (2005). [CrossRef]
  6. E. Margallo-Balbás and P. J. French, “Shape based Monte Carlo code for light transport in complex heterogeneous tissues,” Opt. Express 15, 14086–14098 (2007). [CrossRef]
  7. E. Alerstam, T. Svensson, and S. Andersson-Engels, “Parallel computing with graphics processing units for high-speed Monte Carlo simulation of photon migration,” J. Biomed. Opt. 13, 060504 (2008). [CrossRef]
  8. Q. Fang and D. A. Boas, “Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units,” Opt. Express 17, 20178–20190 (2009). [CrossRef]
  9. H. Shen and G. Wang, “A tetrahedron-based inhomogeneous Monte Carlo optical simulator,” Phys. Med. Biol. 55, 947–962 (2010). [CrossRef]
  10. A. Doronin and I. Meglinski, “Online object oriented Monte Carlo computational tool for the needs of biomedical optics,” Biomed. Opt. Express 2, 2461–2469 (2011). [CrossRef]
  11. X-5 Monte Carlo Team, “MCNP, a general Monte Carlo N-particle transport code, version 5, Report LA-UR-03-1987,” Technical Report (Los Alamos National Laboratory, 2003).
  12. B. T. Wong and M. P. Mengüç, “Comparison of Monte Carlo techniques to predict the propagation of a collimated beam in participating media,” Numer. Heat Transfer 42, 119–140 (2002). [CrossRef]
  13. A. Sassaroli and F. Martelli, “Equivalence of four Monte Carlo methods for photon migration in turbid media,” J. Opt. Soc. Am. A 29, 2110–2116 (2012). [CrossRef]
  14. G. I. Bell and S. Glasstone, Nuclear Reactor Theory (Krieger, 1970).
  15. J. Spanier and E. Gelbard, Monte Carlo Principles and Neutron Transport Problems (Addison-Wesley, 1969), reprinted by Dover, 2008.
  16. K. M. Case and P. F. Zweifel, “Existence and uniqueness theorems for the neutron transport equation,” J. Math. Phys. 4, 1376–1386 (1963). [CrossRef]
  17. G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, 1980).
  18. L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations (Cambridge University, 1985).
  19. A. Dubi, Monte Carlo Calculations for Nuclear Reactors, Vol. 2 (CRC Press, 1986).
  20. P. Hoel, S. Port, and C. Stone, Introduction to Probability Theory (Houghton Mifflin, 1971).
  21. L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941). [CrossRef]
  22. P. Billingsley, Probability and Measure (Wiley, 1979).
  23. J. R. Taylor, An Introduction to Error Analysis (University Science Books, 1982).

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