Solution of the Fokker–Planck Pencil Beam Equation for Electrons by the Laplace Transform Technique

  • B. Rodriguez
  • M. T. Vilhena


While many medical physicists understand the basic principles underlying Monte Carlo codes such as EGS [Ka00], Geant [Wr01], and MCNP [Br93], there is less appreciation of the capabilities of deterministic methods which in principle can provide comparable accuracies to Monte Carlo. Only within the last years have serious studies been made on the appliance of deterministic calculations to medical physics applications. The most versatile and widely used deterministic methods are the P N approximation [Da57]; [SeViPa00], the S N method (discrete ordinates method) [ViBa95]; [ViSeBa95], and their variants [SeVi94]; [RoViVo06]. The method of discrete ordinates has been used successfully in neutral particle applications [D096]; [Da92] and gamma ray transport calculations for many years. The calculations for these two types of radiation are done very similarly, since they are both neutral particles. On the other hand, to our knowledge, the P N approximation has not yet been applied in the solution of the charged particle pencil beam transport equation. Pencil beam equations are used to model, e.g., problems of collimated electron and photon particles penetrating piecewise homogeneous regions. The collisions between the beam particles and particles from beams with different directions cause deposit of some part of the energy carried by the beams at the collision sites. To obtain a desired “amounts of energy deposited at certain parts of the target region” (dose) is of crucial interest in radiative cancer therapy.


Monte Carlo Technique Rectangular Domain Pencil Beam Boltzmann Transport Equation Discrete Ordinate 
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  1. [Ag03]
    Agostinelli, S., et al.: Geant4—a simulation toolkit. Nuclear Instruments Methods Phys. Research A, 506, 250–303 (2003).CrossRefGoogle Scholar
  2. [BoLa95]
    Börges, C., Larsen, E.W.: The transversely integrated scalar flux of a narrowly focused particle beam. SIAM J. Appl. Math., 55, 1–22 (1995).CrossRefMathSciNetGoogle Scholar
  3. [BoLa96]
    Börges, C., Larsen, E.W.: On the accuracy of the Fokker–Planck and Fermi pencil beam equations for charged particle transport. Medical Phys., 23, 1749–1759 (1996).CrossRefGoogle Scholar
  4. [Br93]
    Briesmeister, J.F.: MCNP—a general Monte Carlo N-particle transport code. Los Alamos National Laboratory Report LA-12625-M (1993).Google Scholar
  5. [Da92]
    DANTSYS 3.0: One-, Two-, and Three-Dimensional Multigroup, Discrete Ordinates Transport Code System, RSICC Computer Code Collection CCC-547, Oak Ridge National Laboratory (1992).Google Scholar
  6. [DaMa79]
    Davies, B., Martin, B.: Numerical inversion of the Laplace transform: a survey and comparison of methods. J. Comput. Phys., 33, 1–32 (1979).MATHCrossRefMathSciNetGoogle Scholar
  7. [Da57]
    Davison, B.: Neutron Transport Theory, Oxford University Press, London (1957).MATHGoogle Scholar
  8. [D096]
    DOORS 3.1: One-, Two-, and Three-Dimensional Discrete Ordinates Neutron/Photon Transport Code System, RSICC Code Package CCC-650, Oak Ridge National Laboratory (1996).Google Scholar
  9. [Ho07]
    Hoff, G.: Personal communication. Pontifícia Universidade Católica do Rio Grande do Sul, Porto Alegre, RS, Brazil (2007).Google Scholar
  10. [IC89]
    International Commission on Radiation Units and Measurements, Tissue Substitutes in Radiation Dosimetry and Measurement, ICRU 44, Bethesda, Maryland (1989).Google Scholar
  11. [Ka00]
    Kawrakow, I.: Accurate condensed history Monte Carlo simulation of electron transport. I: EGSnrc, the new EGS4 version. Medical Phys., 27, 485–498 (2000).CrossRefGoogle Scholar
  12. [Kr66]
    Kreider, D.L.: An Introduction to Linear Analysis, Addison-Wesley, Reading, MA (1966).MATHGoogle Scholar
  13. [LMFB97]
    Larsen, E.W., Miften, M.M., Fraass, B.A., Brinvis, I.D.: Electron dose calculations using the method of moments. Medical Phys., 24, 111–125 (1997).CrossRefGoogle Scholar
  14. [Po83]
    Pomraning, G.C.: Flux-limited diffusion and Fokker–Planck equations. Nuclear Sci. Engrg., 85, 116–126 (1983).Google Scholar
  15. [Re85]
    Reimer, L.: Scanning Electron Microscopy, Springer, Berlin (1985).Google Scholar
  16. [Ro07]
    Rodriguez, B.D.A.: Methodology for obtaining a solution for the Boltzmann transport equation considering Compton scattering simulated by Klein–Nishina. Doctoral dissertation, Universidade Federal do Rio Grande do Sul (2007).Google Scholar
  17. [RoViVo06]
    Rodriguez, B.D.A., Vilhena, M.T., Borges, V.: The determination of the exposure buildup factor formulation in a slab using the LTSN method. Kerntechnik, 71, 182–184 (2006).Google Scholar
  18. [SeVi94]
    Segatto, C.F., Vilhena, M.T.: Extension of the LTSN formulation for discrete ordinates problems without azimuthal symmetry. Ann. Nuclear Energy, 21, 701–710 (1994).CrossRefGoogle Scholar
  19. [SeViPa00]
    Segatto, C.F., Vilhena, M.T., Pazos, R.P.: On the convergence of the spherical harmonics approximations. Nuclear Sci. Engrg,. 134, 114–119 (2000).Google Scholar
  20. [StSe86]
    Stroud, A.H., Secrest, E.: Gaussian Quadrature Formulas, Prentice-Hall, Englewood Cliffs, NJ (1986).Google Scholar
  21. [Ta67]
    Tannenbaum, B.S.: Plasma Physics, McGraw-Hill, New York (1967).Google Scholar
  22. [ViBa95]
    Vilhena, M.T., Barichello, L.B.: An analytical solution for the multigroup slab geometry discrete ordinates problems. Transport Theory Statistical Phys., 24, 1029–1037 (1995).Google Scholar
  23. [ViSeBa95]
    Vilhena, M.T., Segatto, C.F., Barichello, L.B.: A particular solution for the Sn radiative transfer problems. J. Quant. Spectrosc. Radiat. Transfer, 53, 467–469 (1995).CrossRefGoogle Scholar
  24. [Wr01]
    Wright, D.H.: Physics Reference Manual, (2001). See User Documents at the Geant4 Web page under Documentation.

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© Birkhäuser Boston 2010

Authors and Affiliations

  1. 1.Universidade Federal do Rio GrandeRio GrandeBrazil

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