Generalized Fokker-Planck Theory for Electron and Photon Transport in Biological Tissues: Application to Radiotherapy

In this paper, we study a deterministic method for particle transport in biological tissues. The method is specifically developed for dose calculations in cancer therapy and for radiological imaging. Generalized Fokker-Planck (GFP) theory [Leakeas and Larsen, Nucl. Sci. Eng. 137 (2001), pp. 236-250] has been developed to improve the Fokker-Planck (FP) equation in cases where scattering is forward-peaked and where there is a sufficient amount of large-angle scattering. We compare grid-based numerical solutions to FP and GFP in realistic medical applications. First, electron dose calculations in heterogeneous parts of the human body are performed. Therefore, accurate electron scattering cross sections are included and their incorporation into our model is extensively described. Second, we solve GFP approximations of the radiative transport equation to investigate reflectance and transmittance of light in biological tissues. All results are compared with either Monte Carlo or discrete-ordinates transport solutions.

[1]  S. Incerti,et al.  Geant4 developments and applications , 2006, IEEE Transactions on Nuclear Science.

[2]  H. W. Lewis.  Multiple Scattering in an Infinite Medium , 1950 .

[3]  H Huizenga,et al.  Numerical calculation of energy deposition by broad high-energy electron beams: II. Multi-layered geometry , 1992 .

[4]  J. C. Garth,et al.  ELECTRON/PHOTON TRANSPORT AND ITS APPLICATIONS , 2005 .

[5]  Piero Ravetto,et al.  Quasi-singular angular finite element methods in neutron transport problems , 1995 .

[6]  Michael Herty,et al.  Optimal treatment planning in radiotherapy based on Boltzmann transport calculations , 2007 .

[7]  N. Mason,et al.  Cross Sections for Electron Collisions with Water Molecules , 2005 .

[8]  Datta,et al.  Computational model for coupled electron-photon transport in two dimensions. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[9]  A. Welch,et al.  A review of the optical properties of biological tissues , 1990 .

[10]  Vladimir T. Tikhonchuk,et al.  Reduced multi-scale kinetic models for the relativistic electron transport in solid targets: Effects related to secondary electrons , 2010 .

[11]  H Palmans,et al.  Underdosage of the upper-airway mucosa for small fields as used in intensity-modulated radiation therapy: a comparison between radiochromic film measurements, Monte Carlo simulations, and collapsed cone convolution calculations. , 2002, Medical physics.

[12]  Pedro González-Rodríguez,et al.  Light propagation in tissues with forward-peaked and large-angle scattering. , 2008, Applied optics.

[13]  A. Boyer,et al.  Tissue Inhomogeneity Corrections for Megavoltage Photon Beams , 2004 .

[14]  E W Korevaar,et al.  Accuracy of the phase space evolution dose calculation model for clinical 25 MeV electron beams , 2000, Physics in medicine and biology.

[15]  J. E. Morel.  An improved Fokker-Planck angular differencing scheme , 1985 .

[16]  M Vauhkonen,et al.  Modelling the transport of ionizing radiation using the finite element method , 2005, Physics in medicine and biology.

[17]  A. Ahnesjö,et al.  A pencil beam model for photon dose calculation. , 1992, Medical physics.

[18]  P R Storchi,et al.  Numerical calculation of energy deposition by broad high-energy electron beams. , 1989, Physics in medicine and biology.

[19]  Elastic scattering of electrons and positrons. , 2007, Journal of the ICRU.

[20]  J. Cygler,et al.  Clinical use of a commercial Monte Carlo treatment planning system for electron beams , 2005, Physics in medicine and biology.

[21]  J. Sempau,et al.  PENELOPE-2006: A Code System for Monte Carlo Simulation of Electron and Photon Transport , 2009 .

[22]  B. Rossi,et al.  Cosmic-Ray Theory , 1941 .

[23]  Per Edström,et al.  A Fast and Stable Solution Method for the Radiative Transfer Problem , 2005, SIAM Rev..

[24]  B. Faddegon,et al.  Description and dosimetric verification of the PEREGRINE Monte Carlo dose calculation system for photon beams incident on a water phantom. , 2001, Medical physics.

[25]  Axel Klar,et al.  A Fast and Accurate Moment Method for the Fokker-Planck Equation and Applications to Electron Radiotherapy , 2007, SIAM J. Appl. Math..

[26]  C. Börgers.  Complexity of Monte Carlo and deterministic dose-calculation methods. , 1998, Physics in medicine and biology.

[27]  Firas Mourtada,et al.  Comparison of a finite-element multigroup discrete-ordinates code with Monte Carlo for radiotherapy calculations , 2006, Physics in medicine and biology.

[28]  S Walker,et al.  Electron dose calculation using multiple-scattering theory: evaluation of a new model for inhomogeneities. , 1992, Medical physics.

[29]  G. C. Pomraning.  THE FOKKER-PLANCK OPERATOR AS AN ASYMPTOTIC LIMIT , 1992 .

[30]  E W Korevaar,et al.  Numerical calculation of energy deposition by high-energy electron beams: III-B. Improvements to the 6D phase space evolution model. , 1997, Physics in medicine and biology.

[31]  K.-U. Amthor,et al.  Laser-plasma acceleration of quasi-monoenergetic protons from microstructured targets , 2006, Nature.

[32]  B Mijnheer,et al.  Tolerances for the accuracy of photon beam dose calculations of treatment planning systems. , 2001, Radiotherapy and oncology : journal of the European Society for Therapeutic Radiology and Oncology.

[33]  P. Almond,et al.  Electron beam dose calculations , 1981, Physics in medicine and biology.

[34]  D Jette,et al.  Electron dose calculation using multiple-scattering theory. A. Gaussian multiple-scattering theory. , 1988, Medical physics.

[35]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[36]  B A Fraass,et al.  Electron dose calculations using the Method of Moments. , 1997, Medical physics.

[37]  Emiliano Spezi,et al.  An overview of Monte Carlo treatment planning for radiotherapy. , 2008, Radiation protection dosimetry.

[38]  Dinshaw S. Balsara,et al.  Fast and accurate discrete ordinates methods for multidimensional radiative transfer. Part I, basic methods , 2001 .

[39]  Leonard Eyges,et al.  Multiple Scattering with Energy Loss , 1948 .

[40]  Thomas A. Brunner,et al.  Forms of Approximate Radiation Transport , 2002 .

[41]  Hartmut Hensel,et al.  Deterministic model for dose calculation in photon radiotherapy , 2006, Physics in medicine and biology.

[42]  K R Hogstrom,et al.  Pencil-beam redefinition algorithm for electron dose distributions. , 1991, Medical physics.

[43]  Ines Krajcar Bronić,et al.  Elastic scattering of electrons and positrons (ICRU Report 77) , 2007 .

[44]  Enrico Fermi,et al.  The Ionization Loss of Energy in Gases and in Condensed Materials , 1940 .

[45]  F. Mourtada,et al.  Feasibility of a multigroup deterministic solution method for three-dimensional radiotherapy dose calculations. , 2008, International journal of radiation oncology, biology, physics.

[46]  A Bielajew,et al.  Electron dose calculation using multiple-scattering theory: second-order multiple-scattering theory. , 1989, Medical physics.

[47]  H. Krieger,et al.  Grundlagen der Strahlungsphysik und des Strahlenschutzes , 2004 .

[48]  Michael Herty,et al.  Optimal treatment planning governed by kinetic equations , 2009, 0904.0740.

[49]  P R Storchi,et al.  Numerical calculation of energy deposition by high-energy electron beams: III. Three-dimensional heterogeneous media. , 1994, Physics in medicine and biology.

[50]  Jari P. Kaipio,et al.  A finite-element model of electron transport in radiation therapy and a related inverse problem , 1999 .

[51]  Edward W. Larsen,et al.  Generalized Fokker-Planck Approximations of Particle Transport with Highly Forward-Peaked Scattering , 2001 .

[52]  L. C. Henyey,et al.  Diffuse radiation in the Galaxy , 1940 .

[53]  P. Andreo.  Monte Carlo techniques in medical radiation physics. , 1991, Physics in medicine and biology.

[54]  Simon M. Pimblott,et al.  Effect of Elastic Collisions on Energy Deposition by Electrons in Water , 1997 .

[55]  Michael Herty,et al.  Optimal Treatment Planning in Radiotherapy Based on Boltzmann Transport Equations , 2008 .