Skip to main content

Monte Carlo Geometry Modeling for Particle Transport Using Conformal Geometric Algebra

Advances in Applied Clifford Algebras volume 29, Article number: 26 (2019) Cite this article

Abstract

The Monte Carlo (MC) simulations are considered the gold-standard method for calculating the transport and interaction of radiation with the matter. A fundamental component of any MC simulation is the geometrical modeling. Current implementations of the geometrical modeling are based only on the Euclidean representations. However, Euclidean representations may not be the best option for speed up the geometric debugging-modeling computations of radiation transport, due to the number of operations involved in the estimation of position and direction of particles within each geometry shape. In this work, it is proposed for the first time, the use of Conformal Geometric Algebra (CGA), for geometric modeling in MC simulation for radiation transport. In this context, we present some elemental CGA equations for the microscopic modeling of positions and rotations of a radiation particle and the macroscopic modeling of geometrical shapes. It is shown that it is possible to take advantage of the expression power of CGA to create and debug geometry modeling with a triboelectric X-ray application. Additionally, some advantages of the CGA for the microscopic geometric computations are explored.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. Almansa, J., Salvat-Pujol, F., Díaz-Londoño, G., Carnicer, A., Lallena, A.M., Salvat, F.: Pengeoma general-purpose geometry package for Monte Carlo simulation of radiation transport in material systems defined by quadric surfaces. Comput. Phys. Commun. 199, 102–113 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  2. Badal, A., Kyprianou, I.S., Banh, D.P., Badano, A., Sempau, J.: penmesh-monte carlo radiation transport simulation in a triangle mesh geometry. IEEE Trans. Med. Imaging 28(12), 1894–1901 (2009)

    Article  Google Scholar 

  3. Bayro-Corrochano, E.: Geometric Computing: For Wavelet Transforms, Robot Vision, Learning, Control and Action. Springer, Berlin (2010)

    Book  Google Scholar 

  4. Bayro-Corrochano, E., Rivera-Rovelo, J.: The use of geometric algebra for 3d modeling and registration of medical data. J. Math. Imaging Vis. 34(1), 48–60 (2009)

    Article  MathSciNet  Google Scholar 

  5. Bielajew, A.F.: Fundamentals of the monte carlo method for neutral and charged particle transport. The University of Michigan (2001)

  6. Bushberg, J.T., Boone, J.M.: The essential physics of medical imaging. Lippincott Williams & Wilkins, Philadelphia (2011)

    Google Scholar 

  7. Colapinto, P.: Spatial computing with conformal geometric algebra. PhD thesis, University of California Santa Barbara (2011)

  8. Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups, vol. 290. Springer, Berlin (2013)

    Google Scholar 

  9. Deul, C., Burger, M., Hildenbrand, D., Koch, A.: Raytracing point clouds using geometric algebra. In: Submitted to the Proceedings of the GraVisMa workshop, Plzen (2010)

  10. Doran, C., Lasenby, A., Lasenby, J.: Geometric Algebra for Physicists. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  11. Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry. Morgan Kaufmann Publishers Inc, Burlington (2009)

    Google Scholar 

  12. Fontijne, D., Dorst, L.: Modeling 3d euclidean geometry. IEEE Comput. Graph. Appl. 23(2), 68–78 (2003)

    Article  Google Scholar 

  13. Franchini, S., Gentile, A., Sorbello, F., Vassallo, G., Vitabile, S.: Conformalalu: a conformal geometric algebra coprocessor for medical image processing. IEEE Trans. Comput. 64(4), 955–970 (2015)

    Article  MathSciNet  Google Scholar 

  14. Hestenes, D., Sobczyk, G.: Clifford algebra to geometric calculus: a unified language for mathematics and physics, vol. 5. Springer (2012)

  15. Hildenbrand, D.: Using gaalop for high-performance geometric algebra computing. In: Foundations of Geometric Algebra Computing, pp. 141–154, Springer, Berlin (2013)

    Google Scholar 

  16. Hildenbrand, D., Franchini, S., Gentile, A., Vassallo, G., Vitabile, S.: Gappco: an easy to configure geometric algebra coprocessor based on gapp programs. Adv. Appl. Clifford Algebras 27(3), 2115–2132 (2017)

    Article  MathSciNet  Google Scholar 

  17. Hird, J.R., Camara, C.G., Putterman, S.J.: A triboelectric X-ray source. Appl. Phys. Lett. 98(13), 891–921 (2011). (no. 133501)

    Article  Google Scholar 

  18. Lange, H., Stock, F., Koch, A., Hildenbrand, D.: Acceleration and energy efficiency of a geometric algebra computation using reconfigurable computers and gpus. In: 17th IEEE Symposium on Field Programmable Custom Computing Machines, 2009. FCCM’09. pp. 255–258, IEEE (2009)

  19. Lavor, C., Xambó-Descamps, S., Zaplana, I.: A Geometric Algebra Invitation to Space–Time Physics, Robotics and Molecular Geometry. SBMA/Springerbrief. Springer, Berlin (2018)

    Book  Google Scholar 

  20. Lee, C.: Monte carlo calculations in nuclear medicine second edition: applications in diagnostic imaging (2014)

    Article  Google Scholar 

  21. Nelson, W., Jenkins, T.: Writing subroutine howfar for egs4, tech. rep., SLAC TN-87-4 (1987)

  22. Perl, J., Shin, J., Schümann, J., Faddegon, B., Paganetti, H.: Topas: an innovative proton monte carlo platform for research and clinical applications. Med. Phys. 39(11), 6818–6837 (2012)

    Article  Google Scholar 

  23. Perwass, C., Edelsbrunner, H., Kobbelt, L., Polthier, K.: Geometric Algebra with Applications in Engineering, vol. 20. Springer, Berlin (2009)

    Google Scholar 

  24. Pia, M.G., Weidenspointner, G.: Monte carlo simulation for particle detectors. arXiv preprint arXiv:1208.0047 (2012)

  25. Popescu, L.M.: A geometry modeling system for ray tracing or particle transport monte carlo simulation. Comput. Phys. Commun. 150(1), 21–30 (2003)

    Article  ADS  Google Scholar 

  26. Seltzer, S.M.: X-ray mass attenuation coefficients. https://www.nist.gov/pml/x-ray-mass-attenuation-coefficients. Accessed 04 Aug 2017

  27. Sveier, A., Kleppe, A.L., Tingelstad, L., Egeland, O.: Object detection in point clouds using conformal geometric algebra. Adv. Appl. Clifford Algebras, 1–16 (2017)

  28. Tseung, H Wan Chan, Ma, J., Beltran, C.: A fast gpu-based monte carlo simulation of proton transport with detailed modeling of nonelastic interactions. Med. Phys. 42(6), 2967–2978 (2015)

    Article  Google Scholar 

  29. Weller, R., Zachmann, G.: A unified approach for physically-based simulations and haptic rendering. In: Proceedings of the 2009 ACM SIGGRAPH Symposium on Video Games, pp. 151–159, ACM (2009)

Download references

Acknowledgements

This research has been supported by the CONACYT SNI Grant.

Author information

Authors and Affiliations

  1. Universidad Autónoma de Guadalajara and Barcelona Supercomputing Center, Av Patria 1201, Zapopan, Jalisco, Mexico

    E. Ulises Moya-Sánchez

  2. Nexus II, Building c/Jordi Girona, 29, 08034, Barcelona, Spain

    E. Ulises Moya-Sánchez

  3. Universidad Autónoma de Guadalajara, Av Patria 1201, Zapopan, Jalisco, Mexico

    A. Moisés Maciel-Hernández & Adrian S. Niebla

  4. University of California San Francisco, 1600 Divisadero, San Francisco, CA, USA

    José Ramos-Méndez

  5. ITESM Campus Guadalajara, Av. General Ramón Corona 2514, Zapopan, Jalisco, Mexico

    Oscar Carbajal-Espinosa

Authors
  1. E. Ulises Moya-Sánchez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Moisés Maciel-Hernández
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Adrian S. Niebla
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. José Ramos-Méndez
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Oscar Carbajal-Espinosa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Ulises Moya-Sánchez.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We gratefully acknowledge the help provided by Prof. Sebatià Xambó and the constructive comments of the anonymous referees

This article is part of the Topical Collection on Proceedings ICCA 11, Ghent, 2017, edited by Hennie De Schepper, Fred Brackx, Joris van der Jeugt, Frank Sommen, and Hendrik De Bie.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Moya-Sánchez, E.U., Maciel-Hernández, A.M., Niebla, A.S. et al. Monte Carlo Geometry Modeling for Particle Transport Using Conformal Geometric Algebra. Adv. Appl. Clifford Algebras 29, 26 (2019). https://doi.org/10.1007/s00006-019-0939-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00006-019-0939-2

Keywords

  • Monte Carlo geometry
  • Ray tracing
  • Conformal Geometry Algebra