Advertisement

Towards Real-Time Monte Carlo for Biomedicine

  • Shuang Zhao
  • Rong Kong
  • Jerome Spanier
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 241)

Abstract

Monte Carlo methods provide the “gold standard” computational technique for solving biomedical problems but their use is hindered by the slow convergence of the sample means. An exponential increase in the convergence rate can be obtained by adaptively modifying the sampling and weighting strategy employed. However, if the radiance is represented globally by a truncated expansion of basis functions, or locally by a region-wise constant or low degree polynomial, a bias is introduced by the truncation and/or the number of subregions. The sheer number of expansion coefficients or geometric subdivisions created by the biased representation then partly or entirely offsets the geometric acceleration of the convergence rate. As well, the (unknown amount of) bias is unacceptable for a gold standard numerical method. We introduce a new unbiased estimator of the solution of radiative transfer equation (RTE) that constrains the radiance to obey the transport equation. We provide numerical evidence of the superiority of this Transport-Constrained Unbiased Radiance Estimator (T-CURE) in various transport problems and indicate its promise for general heterogeneous problems.

Keywords

Monte Carlo simulations Transport-constrained radiance estimators 

Notes

Acknowledgements

The third author gratefully acknowledges partial support from award numbers: P41RR001192 from the National Center for Research Resources and P41EB015890 from the National Institute of Biomedical Imaging and Bioengineering.

The content of this paper is solely the responsibility of the authors and does not necessarily represent the official views of the National Center For Research Resources, National Institute of Biomedical Imaging and Bioengineering, or the National Institutes of Health.

References

  1. 1.
    Baggerly, K., Cox, D., Picard, R.: Exponential convergence of adaptive importance sampling for Markov chains. J. Appl. Probab. 37(2), 342–358 (2000)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Booth, T.: Exponential convergence on a continuous Monte Carlo transport problem. Nucl. Sci. Eng. 127(3), 338–345 (1997)CrossRefGoogle Scholar
  3. 3.
    Booth, T.: Adaptive importance sampling with a rapidly varying importance function. Nucl. Sci. Eng. 136(3), 399–408 (2000)CrossRefGoogle Scholar
  4. 4.
    Chang, V.T.C., Cartwright, P.S., Bean, S.M., Palmer, G.M., Bentley, R.C., Ramanujam, N.: Quantitative physiology of the precancerous cervix in vivo through optical spectroscopy. Neoplasia 11(4), 325–332 (2009)CrossRefGoogle Scholar
  5. 5.
    Collier, T., Arifler, D., Malpica, A., Follen, M., Richards-Kortum, R.: Determination of epithelial tissue scattering coefficient using confocal microscopy. IEEE J. Sel. Top. Quantum Electron. 9(2), 307–313 (2003)CrossRefGoogle Scholar
  6. 6.
    Devroye, L.: A Course in Density Estimation. Progress in Probability. Birkhauser, Boston (1987)zbMATHGoogle Scholar
  7. 7.
    Jensen, H.W.: Realistic Image Synthesis Using Photon Mapping. AK Peters Ltd, Wellesley (2001)CrossRefGoogle Scholar
  8. 8.
    Kong, R.: Transport problems and Monte Carlo methods. Ph.D. thesis, Claremont Graduate University (1999)Google Scholar
  9. 9.
    Kong, R., Ambrose, M., Spanier, J.: Efficient, automated Monte Carlo methods for radiation transport. J. Comput. Phys. 227(22), 9463–9476 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kong, R., Spanier, J.: A new proof of geometric convergence for general transport problems based on sequential correlated sampling methods. J. Comput. Phys. 227(23), 9762–9777 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kong, R., Spanier, J.: Geometric convergence adaptive Monte Carlo algorithms for radiative transport problems based on importance sampling methods. Nucl. Sci. Eng. 168(3), 197–225 (2011)CrossRefGoogle Scholar
  12. 12.
    Kong, R., Spanier, J.: Transport-constrained extensions of collision and track length estimators for solutions of radiative transport problems. J. Comput. Phys. 242(0), 682–695 (2013).  https://doi.org/10.1016/j.jcp.2013.02.023, http://www.sciencedirect.com/science/article/pii/S0021999113001423MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kong, R., Spanier, J.: A new proof of geometric convergence for the adaptive generalized weighted analog sampling (GWAS) method. Monte Carlo Methods Appl. 22(3), 161–196 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lai, Y., Spanier, J.: Adaptive importance sampling algorithms for transport problems. Monte Carlo and Quasi-MonteCarlo Methods 1998, pp. 273–283. Springer, Berlin (1999)CrossRefGoogle Scholar
  15. 15.
    Rao, C.R.: Linear Statistical Inference and Its Applications. Wiley, New York (1973)CrossRefGoogle Scholar
  16. 16.
    Scott, D.: Multivariate Density Estimation: Theory. Practice and Visualization. Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1990)Google Scholar
  17. 17.
    Silverman, B.: Density Estimation for Statistics and Data Analysis. Chapman and Hall, London (1986)CrossRefGoogle Scholar
  18. 18.
    Spanier, J., Gelbard, E.: Monte Carlo Principles and Neutron Transport Problems. Wesley, New York (1969). (reprinted by Dover Publications, Inc. 2008)Google Scholar
  19. 19.
    Spanier, J., Kong, R.: A new adaptive method for geometric convergence. Monte Carlo and Quasi-MonteCarlo Methods 2002, pp. 439–449. Springer, Berlin (2004)Google Scholar
  20. 20.
    Veach, E.: Robust Monte Carlo methods for light transport simulation. Ph.D. thesis, Stanford, CA, USA (1998). AAI9837162Google Scholar
  21. 21.
    Walker, D., Brown, B., Blackett, A., Tidy, J., Smallwood, R.: A study of the morphological parameters of cervical squamous epithelium. Physiol. Meas. 24(1), 121 (2003)CrossRefGoogle Scholar
  22. 22.
    Wilson, B.C., Adam, G.: A Monte Carlo model for the absorption and flux distributions of light in tissue. Med. Phys. 10(6), 824–830 (1983)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Donald Bren School of Information and Computer SciencesUniversity of California @ IrvineIrvineUSA
  2. 2.Hyundai Capital AmericaIrvineUSA
  3. 3.Beckman Laser Institute and Medical ClinicUniversity of California @ IrvineIrvineUSA

Personalised recommendations