Validation
of the
GEANT4 Monte Carlo code for
radiotherapy applications
by
Emily S. Poon
Medical Physics Unit McGill University, Montreal
August 2004
A thesis submitted ta McGill University in partial fulfillment of the requirements of the degree of Master in Science
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Conformément
à
la loi canadienneAbstract
GEANT4 is a Monte Carlo code developed in an object-oriented environment. It
Résumé
Acknowledgements
1 would like to express my gratitude to my supervisor, Dr. Frank Verhaegen. 1 appreciate his continuous guidance and encouragement, and his helpful advice on many aspects of this project. Also, his financial support is gratefully acknowledged.
1 wish to thank Dr. Ervin Podgorsak for being a dedicated prof essor, and for providing me an opportunity to study medical physics at McGill University.
1 would also like to acknowledge Iwan Kawrakow for his help in my thesis. The assistance of the GEANT4 team from the Standard Linear Accelerator Center is also greatly appreciated.
Many thanks to Emily Heath and Wamied Abdel Rahman for their help on computer-related problems. 1 appreciate the friendships of Vicky Huang and Jinxian Dai, and their advice on the EGSnrc simulations. 1 would also like to thank Matthieu Lemire for kindly translating my abstract into French.
Table of Contents
Abstract ... i Résumé ... ii Acknowledgements ... ... ... iii Table of Contents ... ... ... ... ... ... iv L · Ist 0 fF' 19ures... Vll1 ... List of Tables . ... ... ... ... ... ... ... xviiChapter 1. Introduction ... 1
1.1. MONTE CARLO TECHNIQUE IN RADIOTHERAPY ... 1
1.2. THESIS OBJECTIVES ... 2
1.3. THESIS ORGANIZATION ... 3
1.4. REFERENCES ... 3
Chapter 2. Monte Carlo Techniques in Radiation Transport ... 4
2.1. INTRODUCTION ... 4
2.2. PHOTON TRANSPORT ... 5
2.2.1. Photon interaction processes ... 5
2.2.2. Photon cross sections ... 7
2.2.3. Photon transport algorithm ... 10
2.3. ELECTRON TRANSPORT ... 12
2.3.1. Electron interaction processes ... 12
2.3.2. Stopping powers ... 14
2.3.3. Multiple scattering theories ... 17
2.3.4. Electron step algorithm ... 17
2.3.5. Condensed history technique ... 19
2.3.6. Electron transport algorithm ... 20
2.4. SCORING AND ANAL YSIS TECHNIQUES ... 22
2.5. VARIANCE REDUCTION TECHNIQUES ... 23
Chapter 3. Fundamentals of the GEANT4 Monte Carlo Code ... 27
3.1. MONTE CARLO CODES FORRADIOTHERAPY APPLICATIONS ... 27
3.2. GLOBAL STRUCTURE ... 29
3.2.1. Mandatory user classes ... 29
3.3.2. Optional user classes ... 30
3.3. GEOMETRY CONSTRUCTION ... 30 3.3.1. Volume definitions ... 31 3.3.2. Solid types ... 31 3.3.3. Repeated volumes ... 33 3.3.4. Time-dependent geometries ... 34 3.4. PARTICLE DESCRIPTIONS ... 34 3.5. PHYSICS PROCESSES ... 35
3.6. TRAC KING AND PARTICLE TRANSPORTATION ... 36
3.6.1. Determination of step size and particle interaction type ... 36
3.6.2. Particle tracking algorithm ... 37
3.7. STACK MANAGEMENT ... 37
3.8. VARIANCE REDUCTION TECHNIQUES ... 38
3.9. SCORING TECHNIQUES ... 39
3.10. USER-INTERFACE ... 39
3.11. REFERENCES ... 40
Chapter 4. Photon and Electron Interactions in GEANT4 ... 43
4.1. INTRODUCTION ... 43
4.1.1. Electromagnetic physics processes in GEANT4 ... 43
4.1.2. Physics in EGSnrc ... 45
4.1.3. Sources of cross section data ... 45
4.1.4. Extraction of cross section data ... 46
4.2. PHOTON INTERACTIONS ... 47
4.2.1. Photoelectric effect ... 47
4.3. ELECTRON INTERACTIONS ... 59
4.3.1. Production thresholds ... 59
4.3.2. Ionization ... 59
4.3.2.1. Continuous energy loss ... 59
4.3.2.2. Simulation of delta-rays ... 66
4.3.3. Bremsstrahlung ... 68
4.3.3.1. Continuous energy loss ... 68
4.3.3.2. Simulation ofbremsstrahlung photons ... 69
4.3.4. Multiple scattering ... 72
4.3.5. Positron annihilation ... 72
4.4. ELECTRON STEP LIMITATIONS ... 72
4.5. ENERGY LOSS EVALUATION ... 73
4.6. BOUNDARY CROSSING ALGORITHM ... 73
4.7. REFERENCES ... 73
Chapter 5. Development of New Functionalities ... 79
5.1. INTRODUCTION ... 79
5.2. PARALLEL PROCESSING ... : ... 80
5.3. LINAC SIMULATIONS ... 81
5.3.1. Material definitions and geometry construction ... 82
5.3.2. Generation ofphase space files ... 82
5.3.3. Physics list ... 82
5.3.4. Input and output files ... 83
5.4. VARIANCE REDUCTION TECHNIQUES ... 83
5.4.1. Bremsstrahlung splitting ... 83
5.4.2. Russian roulette ... 86
5.4.3. Range rejection ... 86
5.5. DOSE CALCULATIONS ... 88
5.5.1. Construction ofvoxels ... 88
5.5.2. Generation of source partic1es and phase space files ... 90
5.7. STATISTICAL UNCERTAINTIES ... 91
5.8. REFERENCES ... 92
Chapter 6. Modeling of Radiotherapy Applications ... 93
6.1. INTRODUCTION ... 93
6.2. MONOENERGETIC INCIDENT BEAMS ... 94
6.2.1. Dose distributions in a water phantom ... 94
6.2.1.1. Incident photon beams ... 94
6.2.1.2. Incident electron beams ... 97
6.2.2. Electron transport through a thin aluminum slab ... 105
6.3. INTERFACE PERTURBATION STUDY ... 107
6.3.1. Lead interface in water ... 107
6.3.1. Air interface in water ... 113
6.4. FANO CAVITY STUDY ... 115
6.4.1. Geometrical setup ... 115
6.4.2. Mass absorption coefficient in water ... 117
6.4.3. Simulation results ... 118
6.4.4. Discussion ... 122
6.5. CLINICAL PHOTON BEAMS ... 123
6.5.1. Simulation setup ... 123
6.5.2. Analysis of photon beam data ... 124
6.5.3. Dose distributions in a water phantom ... 130
6.5.4. Comparisons of relative dose distributions with measurements ... 132
6.6. REFERENCES ... 134
Chapter 7. Summary and Conclusions ... 135
7.1. SUMMARY ... 135
7.2. CONCLUSIONS ... 137
List of Figures
Chapter 1Figure 1.1 . ... 2
Rendering of the ATLAS detector in GEANT4.
Chapter 2
Figure 2.1. . ... 5 Relative importance of the three major types of photon interactions with
the photon energy and the atomic number of the attenuating material as parameters. Figure 2.2. . ... 8 Contribution of each photon interaction type to the total cross section for (a) carbon and (b) tungsten.
Figure 2.3 . ... 10 A simplified photon transport algorithm.
Figure 2.4. ... 12
Schematic diagram showing the impact parameter b of an electron that comes close to an atom with radius a.
Figure 2.5. ... 16 Mass collision and radiative stopping powers for Uquid water and lead.
Figure 2.6. ... 16 Mass collision stopping power for water. The dashed Une is the restricted mass
collision stopping power with a production cutoff of 189 ke V.
Figure 2.7 . ... 18 A typical electron transport step showing the total path t and the scattering angle O. The step has an initial direction üj and a final direction Ü f .
Figure 2.8. ... 19 The (a) path length correction and the (b) ratio (p)/(l) in water versus kinetic
energy for various step sizes as quantified by ESTEPE.
Figure 2.9. . ... 21
Chapter 3
Figure 3.1 . ... 29 Major class categories ofGEANT4.
Figure 3.2. . ... 32 GEANT4 solids. (a) A simple CSG solid of the G4Para class. (b) A G4Polycone solid. (c) A G4BREPSolidCone object. (d) The BaBar detector at SLAC.
Figure 3.3 . ... 33 Repeated structures. (a) A replica volume with two slabs ofmaterials repeated
along an axis. (b) A parameterized volume in which ail components differ in size. Figure 3.4. . ... 40 Graphical user interfaces of the (a) HepRep programfor visualization, and (b)
JAS3 analysis tool.
Chapter4
Figure 4.1 . ... 49 (a) Photoelectric mass attenuation coefficients for water between
la
keV and 1 MeV. (b) Percentage differences compared with the XCOM cross sections.Figure 4.2. ... ... ... .... ... ... ... ... ... ... ... ... ... 49 (a) Photoelectric mass attenuation coefficients for tungsten between
la
keV and 1 MeV. (b) Percentage differences compared with the XCOM cross sections.Figure 4.3. ... ... ... ... ... ... ... 50 Photoelectric mass attenuation coefficients of tungsten for the Standard process in GEANT4, and the XCOM datafrom PEGS4 and NIST.
Figure 4.4. .. ... ... ... ... .... ... ... .... ... .... ... ... ... ... ... ... .... ... 52 (a) Compton mass attenuation coefficients for water between
la
keV and 100 MeV. (b) Percentage differences compared with the XCOM cross sections.Figure 4.5. ... ... ... ... ... ... 52 (a) Compton mass attenuation coefficientsfor tungsten between
la
keV and 100Figure 4.6 . ... 55 (a) Pair production mass attenuation coefficientsfor water between 1.5 MeV and 100 Me V. (b) Percentage differences compared with the XCOM cross sections.
Figure 4.7 . ... 55 (a) Pair production mass attenuation coefficients for tungsten between 1.5 MeV and 100 Me V. (b) Percentage differences compared with the XCOM cross sections.
Figure 4.8. . ... '" '" ... .... ... ... ... .... ... ... ... ... ... ... ... ... ... ... ... ... ... 56 (a) Rayleigh mass attenuation coefficientsfor water between 1 keVand 1 MeV. (b) Percentage differences compared with the XCOM cross sections.
Figure 4.9 . ... 57 (a) Rayleigh mass attenuation coefficients for tungsten between 1 keVand 1 MeV. (b)Percentage differences compared with the XCOM cross sections.
Figure 4.10 . ... 58 Rayleigh cross sections for tungsten as obtainedfrom the EPDL97library and the Low-energy process.
Figure 4.11 . ... 60 Restricted electron collision stopping powers for water with a 10 ke V production cut. Figure 4.12 . ... 62 (a) Restricted electron mass collision stopping powers for water with a 10 ke V
electron production cut. (b) Percentage differences compared with PEGS4.
Figure 4.13. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 62 (a) Restricted electron mass collision stopping powers for tungsten with a 10 ke V electron production cut. (b) Percentage differences compared with PEGS4.
Figure 4.14 . ... 64 Percentage differences in restricted collision stopping power in water for the
Standard model with two different I-values compared with PEGS4. The electron production cut is 10 ke V.
Figure 4.15. ... ... ... ... ... .... ... ... ... ... ... ... ... ... ... 65 Density effect corrections for (a) water and (b) tungsten as obtainedfrom the
Figure 4.16 . ... 65 Percentage difJerences in restricted electron collision stopping power with a 10 ke V cutoffin (a) water and (b) tungstenfor the Standard ionization model with the default and ICRU Report 37 ô-values.
Figure 4.17 . ... 67 (a) Meanfree paths ofMoller scattering in water with a 10 keV electron production cut. (b) Percentage difJerences compared with PEGS4.
Figure 4.18 . ... 67 (a) Mean free paths of Moller scattering in tungsten with a 10 ke V electron
production cut. (b) Percentage difJerences compared with PEGS4.
Figure 4.19 . ... 70
(a) Bremsstrahlung mean free paths for tungsten with a 1 ke V photon production cut. (b) Percentage difJerences compared with PEGS4.
Figure 4.20. ... ... .... .... ... ... .... ... .... ... ... .... ... ... ... ... ... ... 70
(a) Bremsstrahlung mean free paths for tungsten with a 10 ke V photon production cut. (b) Percentage difJerences compared with PEGS4.
Chapter 5
Figure 5.1 . ... 80 Basic structure of the user code for clinical beam simulations.
Figure 5.2. ... 81 Treatment heads ofClinac21EXfor (a) 6 MV photon beams and (b) 6 MeV electron beams.
Figure 5.3 . ... 84 Trajectories of bremsstrahlung photons produced from one incident electron hitting a target made oftungsten and copper (a) without turning on bremsstrahlung
splitting, and (b) with the splitting number set to 20.
Figure 5.4. ... ... ... ... ... ... ... ... ... ... ... ... 85 Angular distribution and energy fluence of photons at the end of a W/Cu target with
Figure 5.5. ... ... ... ... .... .... .... .... .... .... ... .... ... .... ... ... ... ... ... 87
Comparisons of photon energy spectra and angular distributions at 100 cm from the source with and without turning on the range rejection option.
Figure 5.6. . ... ... ... ... ... ... .... ... ... ... .... ... ... ... ... 88
Voxels inside a phantom along (a) the central axis and (b) the lateral direction. A
phsp file generated from clinac21 is used to produce the source particles.
Figure 5.7. ... 90
PDD curves of 6 Me V incident electrons hitting a water phantom with the electron
production range cuts set to (a) 1 mm, (b) 100 pm, (c) 10 pm, and (d) 1 pm.
Chapter 6
Figure 6.1 . ... 95
(a) Depth dose distribution in a water phan tom and (b) percentage difJerences
compared to BEAMnrc for 2 Me V monoenergetic incident photon beams.
Figure 6.2. ... ... ... .... ... ... .... ... .... ... ... ... ... ... ... .... 95
Percentage difJerences in depth dose distributions for 6 Me V incident photon
beams with maximum step sizes of250 pm and 30 cm.
Figure 6.3. . ... ,. ... .... ... ... ... ... ... 96
(a) Depth dose distributions in a water phantom and (b) percentage difJerences
compared to BEAMnrc for 6 Me V monoenergetic incident photon beams.
Figure 6.4. ... 97
Percentage difJerences in depth dose distributions of 1 0 Me V electron beams for
Ir
set to 0.001, 0.01, and 0.2.
Figure 6.5 . ... 98
Percentage difJerences in depth dose distributions for 1 MeV electron beams with photon and electron production cuts of 990 e V and 10 ke V compared to BEAMnrc.
Figure 6.6. ... 99
Figure 6.7 . ... 100
(a) Depth dose distributions in a water phantom and (b) percentage differences
compared to BEAMnrc for 1 Me V incident electron beams.
Figure 6.8. ... 101
Electron energy spectra of a 1 Me V electron beam in a water phantom at depths
of(a) 1 mm, (b) 1.7 mm, and (c) 3 mm.
Figure 6.9 . ... 101
Electron angular distributions of a 1 Me V electron beam in a water phantom at
depths of(a) 1 mm, (b) 1.7 mm, and (c) 3 mm.
Figure 6.10 . ... 102
Electron mean energies of 1 Me V electron beam in a water phantom at depths of
(a) 1 mm, (b) 1.7 mm, and (c) 3 mm.
Figure 6.11. ... ... ... ... ... 103
Electron planar fluences vs position of a 1 Me V electron beam in a water phantom
at depths of (a) 1 mm, (b) 1.7 mm, and (c) 3 mm.
Figure 6.12. . ... ... ... ... ... ... ... ... ... ... 104
(a) Depth dose distributions in a water phantom and (b) percentage differences
compared to BEAMnrc for 10 Me V incident electron beams.
Figure 6.13. ... ... ... .... ... .... ... ... ... 105
Electron energy spectra of a 10 MeV electron beam incident on a water phantom at depths of(a) 1 cm, (b) 2.75 cm, and (c) 4 cm.
Figure 6.14 . ... 106
(a) Electron energy spectra and (b) angular distributions at the end of a thin
aluminum slab with 1 Me V incident electrons for various maximum electron steps.
Figure 6.15 . ... 106
Energy fluence vs position at the end of a thin aluminum slab with 1 Me V incident
electrons.
Figure 6.16 . ... 107
(a) Electron energy spectra and (b) angular distributions at the end of a thin
Figure 6.17 . ... 108 Geometrical setup of a lead interface in water (not drawn to scale). A lead sheet is placed in a water phantom separated into layers of variable thicknesses.
Figure 6.18. . ... 109 (a) Dose perturbation factors and (b) the ratios ofDPF for various maximum
electron step sizes for the ten layers of water in the upstream region. The Standard EM pro cesses were used with 100 kV incident photons.
Figure 6.19 . ... 109 (a) Dose perturbation factors and (b) the ratios ofDPF for various maximum
electron step sizes for the ten layers of water in the upstream region. The Standard EM processes were used with 1.25 MeV incident photons.
Figure 6.20. ... 110 (a) Dose perturbation factors and (b) the ratios ofDPF for variousf,. values in the upstream region using the Standard model with 100 kV incident photons.
Figure 6.21 . ... 111 (a) Dose perturbation factors and (b) the ratios of DPF for the ten layers ofwater in the upstream region. The Standard, Low-energy and Penelope models were used with 100 k V incident photon beams.
Figure 6.22. ... 112 (a) Dose perturbation factors and (b) the ratios of DPF for the ten layers of water in the upstream region. The Standard, Low-energy and Penelope models were used with 1.25 MeV incident photon beams.
Figure 6.23 . ... 113 Geometrical setup of an air interface in water (not drawn to scale). A slab of air is placed in a water phantom separated into layers of variable thicknesses.
Figure 6.25. ... 115
(a) Dimensions ofion chamber. (b) A 1.25 MeV photon broad beam incident on
the jlat end of the pan cake chamber.
Figure 6.26. ... 119
Ratios of the simulated mass absorption coefficient to the actual value as a function of maximum electron step size.
Figure 6.27 ... 120
Ratios of the simulated mass absorption coefficient to the actual value as a function of
Ir.
Figure 6.28. ... 121
Ratios of the simulated mass absorption coefficient to the actual value as a function of electron production threshold.
Figure 6.29 •... 122
Ratios of the simulated mass absorption coefficient to the actual value as a function of maximum electron step size for two cavity sizes.
Figure 6.30 •... 125
Plots of (a) photon energy jluence vs position and (b) angular distributions below
the W/Cu target.
Figure 6.31 •... 126
Photon energy spectra below the W/Cu target for a photon production cut of (a)
10 ke V for al! physics models, and (b) 990 eV for the Standard model.
Figure 6.32 ... 126
Plots of (a) photon energy jluence vs position and (b) angular distributions above
the upper jaws.
Figure 6.33 ... 127
Photon angular distributions above the upper jaws generated from the Low-energy processes with the Tsai and 2BN bremsstrahlung angular generators.
Figure 6.34. ... 128
Plots of (a) photon energy jluence vs position and (b) angular distributions above
Figure 6.35 . ... 128 Plots of (a) photon spectral distribution and (b) energy fluence above the upper
jaws for a combination of physics processes.
Figure 6.36. ... 129 Plots of(a) photon energy fluence vs position and (b) angular distributions at a
distance of 1 00 cm from the particle source.
Figure 6.37. ... ... ... ... .... ... .... ... .... ... ... .... ... ... ... ... ... ... .... ... ... 130 Plots of (a) electron energy spectra and (b) energy fluence vs positon at a distance of 1 00 cm from the particle source.
Figure 6.38. ... 131 (a) Central axis depth dose distributions and (b) percentage differences compared
to DOSXYZnrc for 6 MV c/inical photon beams.
Figure 6.39 . ... 131 (a) Lateral profile at a depth of 1.5 cm and (b) percentage differences compared to DOSXYZnrc for 6 MV clinical photon beams. The field size is 1 Ox1 0 cm2 at 100
cmSSD.
Figure 6.40 . ... 132 (a) Lateral profile at a depth of 1 0 cm and (b) percentage differences compared to DOSXYZnrc for 6 MV clinical photon beams. The field size is 1 Ox1 0 cm2 at 100
cmSSD.
Figure 6.41 . ... 133 Percent depth dose curves of a 6 MV photon beam for a field size of 1 Ox1 0 cm2 at
100 cm SSD.
List of Tables
Chapter 2Table 2.1 . ... 9 Summary of the dependence of the photon interaction cross sections on atomic
number Z and photon energy hv.
Chapter 3
Table 3.1 . ... 27 General-purpose Monte Carlo codes for radiotherapy applications.
Table 3.2. ... ... ... ... .... ... .... ... ... ... .... ... ... ... ... ... ... ... ... ... 30 User action base classes and times of invocation of their associated class methods.
Chapter4
Table 4.1 . ... 46 Photon cross section libraries used by the major MC codes.
Table 4.2. ... ... ... ... ... ... ... .... ... ... ... ... ... ... ... ... .... 48 Uncertainties ofphotoelectric cross sections in the EPDL97library.
Table 4.3. ... ... .... ... .... ... ... ... .... ... .... ... .... ... ... ... ... 63 Mean excitation energies usedfor atomic constituents of compounds.
Table 4.4. ... 63 Comparisons ofmean excitation energiesfor seven materials used in ICRU Report 37 and the Standard ionization model.
Chapter 5
Table 5.1. ... 84 The numbers of bremsstrahlung photons per incident electron, the simulation times, and the rates of photons reaching the scoring region below a W/Cu target. Results are compared with the splitting numbers set to 1 and 20.
Chapter 6
Table 6.1 . ... 95 CPU times per incident partide for 2 Me V photon beam simulations.
Table 6.2. ... 96 CPU times per incident partide for 6 Me V photon beam simulations.
Table 6.3. ... ... ... ... ... ... 99 CPU times per incident partide for 1 Me V electron beam simulations.
Table 6.4. ... ... ... ... ... ... 103 Numbers ofphotons per incident partide at three difJerent depthsfor a 1 MeV
incident electron beam.
Table 6.5. .. ... ... ... ... ... ... ... 105 CPU times per incident partidefor 10 MeV electron beam simulations.
Table 6.6. ... ... ... ... ... ... ... 110 CPU times (in ms) per incident partidefor interface perturbation study ofvarious maximum electron step sizes using the Standard processes.
Table 6.7 . ... 111 CPU times (in ms) per incident partide for the Standard, Low-energy and
Penelope processes used in the lead interface study.
Table 6.8. ... ... ... ... ... ... ... ... ... 113 CPU times (in ms) per incident partidefor the Standard, Low-energy and
Penelope processes used in the air interface study.
Table 6.9 . ... 119 Mass absorption coefficients ofwater and CPU times per incident partidefor
various maximum electron step sizes.
Table 6.10 . ... 120 Mass absorption coefficients ofwater and CPU times per incident partidefor
various
Ir
values.Table 6.11 . ... 121 Mass absorption coefficients ofwater and CPU times per incident partidefor
Table 6.12 . ... 122 Mass absorption coefficients ofwater and CPU times per incident particlefor
various maximum electron step sizes. The cavity was 2 mm in diameter.
Table 6.13. ... 129 CPU times per incident particle for scoring phase space data at 100 cm SSD for 6 MV clinical photon beams.
Chapter One
Introduction
1.1. MONTE CARLO TECHNIQUE IN RADIOTHERAPY
Since the discovery of x-rays by Roentgen in 1895, radiation has been used for the treatment of cancer. Today, the major cancer treatment modalities include radiotherapy, chemotherapy, and surgery. Of the estimated 145,500 new cancer patients in Canada this year1, half will receive radiotherapy treatment.
The goal of radiotherapy is to destroy the malignant tumor through ionizing radiation while minimizing damages to the surrounding healthy tissues. It is of great importance to deliver dose to the patient accurately, since a 5% change in dose can cause the tumor response to change by 10 to 20%2. ICRU Report 24 recommends an overall accuracy of 5% in dose delivery3. In dose ca1culations, an accuracy of 2% should be aimed for.
Many treatment planning systems in clinical use today perform dose calculations by analytical means, based on many simplifying assumptions. It is often difficult to account for inhomogeneities inside the patient body. Altematively, the Monte Carlo (MC) technique calculates dose distributions by simulating the trajectories of the particles through the geometry of interest. Given that the physics and the geometrical setup are modeled adequately, the MC method can produce highly accurate results.
Chapter 1 Introduction
There are numerous general-purpose MC codes available, most of which are written in FORTRAN. Although not a recent development, object-oriented programming has gained widespread popularity in the past decade due to its better capacity to model complex objects. GEANT46 is an object-oriented MC code developed for high energy detector simulations. For example, it is used to simulate the collisions of high energy protons in the ATLAS detector. The rendering of the detector using a GEANT4 visualization package is shown in Figure 1.1. It clearly illustrates the powerful geometry construction capability of GEANT4. With its great flexibility, the code shows good promise for radiotherapy applications. Due to the high level of accuracy required for dose calculations, a comprehensive validation of the code is needed.
Figure 1.1. Rendering of the ATLAS detector in GEANT47.
1.2. THE SIS OBJECTIVES
Chapter 1 Introduction
1.3. THE SIS ORGANIZATION
This thesis consists of seven chapters. Chapter 2 introduces the MC techniques for the simulations of photon and electron interactions. Chapter 3 gives a general overview of the GEANT4 structure and capabilities, with focus on its radiotherapy applications. Chapter 4 discusses the details of the physics implementations, including the cross section data and the final state generations of the major interaction types. The development of a user code for extemal beam radiotherapy simulations will be introduced in Chapter 5. Chapter 6 summarizes the results of various simulation setups. Lastly, Chapter 7 gives a brief summary and conclusions.
1.4. REFERENCES
National Cancer Institute of Canada. 2004, Canadian Cancer Statistics 2004, Toronto, Canada.
2 Van Dyk, 1. 1999, 'Radiation oncology overview', in The Modern Technology of Radiation
Oncology, ed. Van Dyk, Medical Physics Publishing, Madison, pp. 1-17.
ICRU Report 24. 1976, Determination of Absorbed Dose in a Patient /rradiated by Beams of
X or Gamma Rays in Radiotherapy Procedures, International Commission on Radiation Units and
Measurements, Washington.
4 Kawrakow, 1. 2000, 'VMC++, Electron and Photon Monte Carlo Calculations Optimized for
Radiation Treatment Planning', in Advanced Monte Carlo for Radiation Physics, Particle Transport Simulations and Applications: Proceedings of the Monte Carlo 2000 Conference, eds. A. Kling, F. Barao, M. Nakagawa, L. Travora & P. Vas, Springer, Berlin, pp. 229-236.
5
Hartmann Siantar, C.L., Bergstrom, P.M., Chadler, W.P., Chase, L., Cox, L.J., Daly, T.P., Garette, D., Hornstein, S.M., House, R.K., Moses, E.I., Patterson, R.W., Rathkopf, J.A. & Schach von Wittenau, A. 1997, 'Lawrence Livermore National Laboratory's PEREGRINE project', in LLNL Report No. UCRL-JC-126732.
6 Agostinelli, S. et al. 2003, 'GEANT4 - a simulation toolkit', Nucl. /nstrum. Methods Phys.
Res. A, vol. 506, pp. 250-303. 7
GEANT4 Collaboration. 1998. GEANT4 - Gallery ofResults, [Online], Available from: <http://wwwasd.web.cern.ch/wwwasd/geant4/reports/gallery/Geometry/Atlas-withHits.gif> [July 7,2004].
8 Kawrakow, 1. 2000, 'Accurate condensed history Monte Carlo simulation of electron
Chapter Two
Monte Carlo Techniques in Radiation Transport
2.1. INTRODUCTION
The Monte Carlo method is a statistical approach of deriving a macroscopic solution to a problem by the use of random numbers. It involves the random sampling of probability distribution functions that de scribe the problem of interest. Provided that the algorithm is accurate and the physical system is well-modeled, repeated sampling of the distributions will converge to the correct solution.
In the Monte Carlo simulation of radiation transport, particles travel in discrete steps and they undergo various types of interactions along the way. The stochastic nature of particle interactions is simulated using a pseudo-random number generator. The step length and the type of interaction are sampled from the cross section data, with geometrical constraints taken into account. Sampling of the appropriate differential cross sections determines the energy and direction of the resultant particles. In external beam radiotherapy, the geometries typically include the treatment head of a medical linear accelerator (linac) as specified by vendor specifications, and the patient composition as
obtained from computed tomography (CT) scans. The simulation outcomes are
macroscopic quantities such as dose depositions in a patient.
Chapter 2 Monte Carlo Techniques in Radiation Transport
This chapter describes the major components of a Monte Carlo code for the transport of photons and electrons· in matter. The integral parts of the code include2: (1)
the cross-section data of the processes to be simulated, (2) the particle transport algorithms, (3) the specifications of the geometries and quantities to be scored, as well as (4) the analysis of the simulation. Special focus is paid to the first two components because they can greatly affect the accuracy of the simulation results.
2.2. PHOTON TRANSPORT
2.2.1. Photon interaction processes
In radiation physics, the major photon interactions with matter include the photoelectric effect, Compton scattering, pair-production, Rayleigh scattering, and photonuclear interactions3• Among these interactions, the first three are the most
important because they lead to energy transfer to secondary electrons. Figure 2.1 illustrates the regions of their relative predominance as delineated by the lines where the atomic cross sections of the interactions are equal.
~ CI) .a E :::J c CJ
·e
0:i
100 75 50 25 o 0.01Photoeffect Pair production
dominant dominant 6° I,t~ ~ Compton effect ~ dominant 0.1 1 10 100
Photon energy (MeV)
Figure 2.1. Relative importance of the three major types of photon interactions with the photon energy and the atomie number of the attenuating material as parameters.4
Chapter 2 Monte Carlo Techniques in Radiation Transport
Photoelectric efJect
In a photoelectric interaction, the incident photon transfers aIl of its energy to an atomic electron. The photon gets absorbed and the electron is ejected with a vacancy left behind. The electron is mostly emitted from the K-sheIl, and its binding energy must be less than the incident photon energy. The vacancy will lead to a cascade of electron transitions from the outer shells to the inner shells. These transitions will in tum give rise to characteristic x-rays, Auger electrons, or Coster-Kronig electrons. The direction of the photoelectron can be sampled from the Sauter distributions.
Compton scattering
The Compton interaction is also known as incoherent scattering. It is an inelastic process in which the photon interacts with an electron, usually in the valence shell. The electron is set in motion, leaving a vacancy behind while the photon is scattered with reduced energy. The Klein-Nishina6 treatment of the Compton differential cross-section is based on the assumption that the electron is free and stationary. This assumption is invalid for energies below 100 ke V, where binding effects and Doppler broadening become significant.
Pair production
Pair production occurs when a photon is absorbed in the field of the nucleus and results in an electron-positron pair. The threshold energy for this interaction is 1.022 MeV. In the infrequent case that the photon interacts in the field of the atomic electrons, the electron is ejected along with the creation of an electron-positron pair. This process is known as triplet production, and the threshold energy is 4mee2, or 2.044 MeV.
Rayleigh scattering
Chapter 2 Monte Carlo Techniques in Radiation Transport
Photonuc/ear interactions
In a photonuclear interaction, a nucleus is excited by an energetic photon and becomes radioactive. The process results in the emission of a proton in a ()',p) reaction or a neutron in a ()',n) reaction. The ()',p) reaction has little contribution to dose in radiotherapyapplications3. For linacs that can accelerate electrons above 10 MeV, the
()',n) reaction in high-Z materials such as lead or tungsten is of significance in radiation protection. The neutrons produced may lead to induced radioactivity.
2.2.2. Photon cross sections
Cross sections refer to the various kinds of attenuation coefficients. They play an essential role in particle transport simulations. Prior to each interaction, cross section data are sampled to determine how far a particle travels. The probability of occurrence of a certain kind of interaction depends on its relative contribution to the total cross section.
Linear attenuation coefficient
The probability of photon interaction can be expressed in terms of the linear attenuation coefficient Jl, which indicates the fraction of incident photons that will interact per unit thickness of the attenuating medium. It is a total of the linear attenuation coefficient of each photon interaction:
(2.l) where 7, (Je, K, and (JR are the linear attenuation coefficients for photoelectric effect,
Compton scattering, pair production, and Rayleigh scattering respectively. The contribution from photonuclear interactions is often excluded7• The inverse of Il is the mean free path À, in units of length:
.-l=!.
fi
(2.2)
It indicates the average distance traveled by a particle before undergoing an interaction. The exponential nature of photon attenuation can be represented bl:
(2.3) where N is the number of photons transmitted through a medium of thickness x with No
Chapter 2 Monte Carlo Techniques in Radiation Transport
Mass, atomic and electronic attenuation coefficients
Since Il depends on the material density p, it is often scaled by the mass, atomic, or electron density. Typically, the mass attenuation coefficient /lm is in units of cm2/g, the atomic attenuation coefficient ail is in units of cm2/atom, and the electronic attenuation coefficient ell is in units of cm2/electron. These quantities are related as follows4:
pNA pNAZ
Il
=
Pllm=A
ail A ell, (2.4) where NA is the Avogadro's number, Z is the atomic number, and A is the mass number of the attenuator.Cross section dependence on atomic number and energy
Photon interaction cross sections depend on the photon energy and the atomic number of the medium. As an illustrative example, Figure 2.2 shows the fraction of the total cross section for each interaction for carbon and tungsten. The data is obtained from the NIST XCOM database9• Carbon represents low-Z materials whereas tungsten represents high-Z materials. The dependency is summarized in Table 2.1.
(a) c ~ =: 0.1 :=
e
u Bs
...
~ 0.01 o..
u ~ photoelectrio. 1 ~:
\ \ Compton 0.001 u....~ ... _~~ ... .w..L...I..~~~....J 0.001 0.01 0.1 10 100 photon energy (MeV)(b) photoelectric
!
:Il 0.10.01 0.1 1 10 100 photon energy (MeV)
Chapter 2 Monte Carlo Techniques in Radiation Transport
Interaction Photoelectric Compton Pair production Rayleigh
effect scattering scattering
Atomic number -oc r ~ Z3 -oc CYc ~ ZO -ocZ K -oc CYR Z
dependence p p p p
Energy 1 decreases with increases with 1
energy (above
dependence (hvY energy
1.022 MeV) (hvY Table 2.1. Summary of the dependence of the photon interaction cross sections on atomie number Z and photon energy hv.
Photoelectric effect is strongly favored for high-Z materials and it is the dominant interaction at low energies. The cross section increases rapidly with decreasing energy, except around the absorption edges. The predominance of photoelectric effect at low energies justifies the neglect of binding effects for Compton scattering in sorne photon transport algorithms.
For Compton scattering, the electronic attenuation coefficient based on the Klein-Nishina formula is independent of the atomic number. The mass attenuation coefficient is slightly higher for hydrogenous or low-Z materials due to higher electron densities. Compton scattering dominates over a smaller energy range for high-Z materials because of the higher dominance of photoelectric effect and pair production with increasing Z.
The atomic cross section is proportional to Z2 for pair production and Z for triplet production2. Rence, the contribution of triplet production is higher for low-Z materials. Since triplet production accounts for less than 10% of the total cross section for low-Z materials and less than 1 % for high-Z materials, most Monte Carlo codes do not model this interaction explicitly. Instead, it is simulated as a pair production event by including the contribution of triplet production in the pair production cross section.
Chapter 2 Monte Carlo Techniques in Radiation Transport
Photonuc1ear interactions can contribute up to 10% of the total cross section in the region of the giant resonance peak that lies between 5 and 40 Me V 10. !ts cross section
(jph,n has an irregular dependency on the atomic number and photon energy, and it is
sensitive to the isotopic abundances of the elements9• As there are no adequate theoretical models available for (j ph,n , it is often omitted from the total cross section JO.
2.2.3. Photon transport algorithm
In Monte Carlo simulations, partic1es are transported on a history by history basis. A history entails the complete track of a partic1e and its descendants in the geometry of interest until the energy of each partic1e falls below a cutoff value. A general algorithm of photon transport is best described by Rogers and Bielajew2, and the logic flow is
reproduced in Figure 2.3. This is a simplified algorithm that involves only one medium, and it does not account for the tracking of secondary electrons.
Place initial photon's parameters on stack 1 + - - - ,
Pick up energy, position, direction, geometry of current parti cie from top of stack
y
Determine distance to next interaction Transport photon taking geometry into account
Determine type of i ntera ction - Photoelectric
- Compton - Pair production - Rayleigh
y
Chapter 2 Monte Carlo Techniques in Radiation Transport
At the start of a new history, the parameters of a photon, such as its energy, position and direction, are sampled and pushed onto a stack. Its energy is first compared with a user-defined cutoff value. This procedure increases efficiency by depositing the energy of the photon locally when it is too small to be of significance, rather than tracking its full trajectories until it gets absorbed.
In general, the distance to the next interaction, x, can be sampled from the total cross section Il, in units of cm-1:
1
x
=
--lnRp (2.5)f.J.
where RI is a random number uniformly distributed between 0 and 1. The actual step length may be limited by boundary constraints because all partic1es have to stop at the boundaries between geometrical volumes.
If the photon remams m the volume of interest after it is transporte d, the branching ratios of the photon interactions will be sampled to determine which interaction will follow. For example, ifphotoelectric effect, Compton scattering, and pair production are considered in the simulation, then:
(2.6) The sampling requires a random number, R2, uniformly distributed between 0 and 1. Since the contribution of each interaction reflects its probability of occurrence, it can be determined that photoelectric effect will happen if R2 ~
ri
f.J., Compton scattering will happen ifri
f.J. < R2 ~(r+
(J'c)1
f.J. , and pair production otherwise.Chapter 2 Monte Carlo Techniques in Radiation Transport
2.3. ELECTRON TRANSPORT
2.3.1. Electron interaction processes
Electron transport is a lot more complicated than photon transport because electrons typically experience millions of interactions before losing all of their kinetic energies through Coulomb interactions. The kind of interaction that an electron will undergo depends on the relationship between the impact parameter b and the atomic radius a. The impact parameter is the perpendicular distance between the undisturbed electron path and the atomic nucleus before an interaction. A schematic diagram is shown in Figure 2.412•
undisturbed electron path
---~----~----+e-b
Figure 2.4. Schematic diagram showing the impact parameter b of an electron that comes close to an atom with radius a.
Soft collisions (b » a)
Chapter 2 Monte Carlo Techniques in Radiation Transport
Hard collisions (b ~)
When the impact parameter is comparable to the atomic size, an electron may lose a large amount of kinetic energy to an orbital electron through hard collision. The orbital electron gets ejected and forms a delta ray. As the two electrons are indistinguishable, the incident electron is assumed by convention to be more energetic after the collision. The maximum energy transfer to the orbital electron is thus half of the kinetic energy of the incident electron. For free electrons, the process is governed by the Meller cross section 13. In the case of positron-electron scattering, where the positron can transfer aIl of its kinetic energy to the electron, the Bhabha cross section 14 is used. Hard collisions also contribute to nearly half of the energy transfer of the incident electron.
Interactions with atomic nuc/ei (h « a)
In the event that an electron cornes close to the atomic nucleus, it will most likely undergo elastic scattering. Its trajectory gets deflected through Rutherford scattering, nuclear reactions, or nuclear scattering12, but the energy loss tends to be negligible. Multiple scattering describes the cumulative effect of such interactions. As the cross section per atom is proportional to Z2, the amount of scattering is greater for high-Z materials. The angular distribution is Gaussian at small angles and it becomes like Rutherford scattering at larger angles 15 • A number of multiple scattering theories have
been developed and they will be described in Section 2.3.3.
Chapter 2 Monte Carlo Techniques in Radiation Transport
Positron annihilations
A positron can annihilate with an electron to produce a pair of photons. In the case that the positron is at rest, the photons will be directed oppositely, each of energy 511 keY. In-flight annihilation happens when the annihilating photon is still in motion. The kinetic energy of the positron will be distributed among the two photons.
2.3.2. Stopping powers
The expectation value of the kinetic energy loss T of a charged partic1e per unit pathlength x is known as the stopping power. In a medium of density p, the mass stoppingpower
si
p is defined, in units of MeV·cm2/g, as:~
=
~(:}
(2.7)The component of the stopping power due to soft and hard collisions is called collision stopping power Scol' and the component due to bremsstrahlung productiont is known as radiative stopping power S,ad:
S Scol S,ad
- = - + - - . (2.8)
p p p
The range R of an electron is the mean value of its pathlength:
(2.9)
where Ti is the initial electron kinetic energy.
Collision stopping power
Berger and Seltzerl6 derived Scol
1
p for electrons from the Bethe theoryl7 for soft collisions and the Meller differential cross section for hard collisions. For positrons, the Bhabha differential cross section is used in place ofMeller. It can be expressed aslS:Chapter 2 Monte Carlo Techniques in Radiation Transport
where 2w/meC2
lu
=
0.151536MeV ·cm 2 / g,p=v/c,
F-Cr)
=
(1-p2) ·[1 + r2 /8-(2r + 1)·ln2] for electrons,F+(r)
=
2ln2-(p2 /12).[23+ 14/(r + 2)+ lO/(r + 2)2 +4/(r + 2)3] for positrons,1 is the mean excitation energy, and ô is the density effect correction.
The mean excitation energy is the average value of all ionization and excitation potentials of an atom in the stopping medium. Since it is independent of the imparting particle type, values of 1 are often determined from measurements with heavy charged particles. The density effect correction accounts for the decrease in collision stopping power as a result of the polarization of the medium by the passing particle. It is more important for dense media at high energies. The 1 and ô values provided in ICRU Report
3i
8are recommended for use in precise dosimetry applications.
Radiative stopping power
The mass radiative stopping power is derived from the theory of Bethe and Heitlerl9. It can be expressed as3:
(2.11 )
where CJ'o is a constant that is equal to
_1_(~J2,
or 5.80xlO·28 cm2/atom, and Br is a137 mec
function that varies slowly between 16/3 and 15 for energies up to 100 MeV.
Figure 2.5 shows the mass collision and radiative stopping powers for water and lead2o• As the collision stopping power is proportional to the electron density due to the leading term Z / A, it is higher for low-Z or hydrogenous materials such as water. The radiative stopping power is higher for lead because of the leading term NAZ2 / A. Above
Chapter 2
CI
c
ci e.
Monte Carlo Techniques in Radiation Transport
10 ,,---'-....--.:-.,'-•• , - , -... -"' ... ;"""-'_ ._3J!!.!_IIfater _ _ _ ~.--..._---~ Sco/p,lead ~ 0.01 := Sradlp, water E 0.001 ~~~~-'--~~~"'-'--~~~-'-'-~~~....; 0.01 0.1 1 10 100
klnetlc energy (MeV)
Figure 2.5. Mass collision and radiative stopping powers for Uquid water and lead.
Restricted stopping power
The restricted collision stopping power is the fraction of the collision stopping power that represents the energy transfer to secondary partic1es below an energy cutoff L1.
Similarly, the restricted radiative stopping power is the portion of radiative stopping power that accounts for the creation of bremsstrahlung photons below a cutoff energy. Figure 2.6 compares the restricted and unrestricted electron collision stopping power for water. Since the primary electron can lose at most half of its kinetic energy, the two curves are identical below 2~. Restricted stopping powers are used in Class II condensed history algorithm for electron transport. Details will be discussed in Section 2.3.4.
100 , - - - ,
1
CI C'c.
a..-Ci ~ NE UI u 10 c o > CI) iii ~= ...
'8
UI~
0.01 - - unrestricted- - - . restricted, 189 keV cutoff
...
_---0.1 1 10
energy (MeV) 100
Chapter 2 Monte Carlo Techniques in Radiation Transport
2.3.3. Multiple scattering theories
The angular deflections of electrons due to a large number of elastic Coulomb interactions can be modeled as a multiple scattering process. A number of multiple scattering theories have been developed, all based on the assumption that the particles are in an infinite, homogeneous medium.
Molière theory
The Molière theory21,22 was originally developed as a small-angle theory. It has a simple analytical form, and requires little pre-ca1culated data23. Renee, it is efficient at selecting scattering angles for randomly-sampled step lengths. As the theory is only valid for scattering angles less than 20°, the electron step size has to be limited to satisfy this requirement. On the other hand, the step length cannot be too short because the theory will break down if too few elastic collisions are involved2\ which makes it inadequate for sorne interface studies. Bielajew has developed an exact numerical solution25 to address this problem. Since the theory does not account for spin and relativistic effects, it underestimates the scattering angles for low-energy electrons in high-Z materials26.
Goudsmit-Saunderson theory
Goudsmit-Saunderson theory27 can be used for all scattering angles, and it is applicable for a wider range of step sizes. This theory is more accurate, but sampling from the multiple-scattering distributions is complicated. The implementation can be made efficient by storing the pre-computed distributions for a selected set of step lengths25. Kawrakow and Bielajew23 have derived a scheme to apply the formalism for the case where electron step lengths are stochastic. This theory can be used with elastic-scattering cross sections that incorporate spin and relativistic effects.
2.3.4. Electron step algorithm
Chapter 2 Monte Carlo Techniques in Radiation Transport
the geometrical path, which is the closest distance between the initial position
x
j and thefinal position
x
f of the step.x.
lu.
lFigure 2.7. A typical electron transport step showing the total path t and the scattering angle 8. The step has an initial direction üj and a final direction Ü f .
Path length correction
The projection of the geometrical path along the initial step direction is denoted as 1 in the diagram. The curvature of the transport step can be accounted for using the path length correction PLè':
t-(/)
PLC=--.
1 (2.12)
Figure 2.8a shows the PLC in water as a function of the electron kinetic energy for various step lengths as quantified by the fractional energy loss per step, ESTEPE. The effect is significant except when the kinetic energy is high or when a small ESTEPE value is chosen. PLC should not be neglected unless very small step lengths are used.
Lateral displacement
The lateral displacement of the transport step is denoted as p in Figure 2.7. The impact of ignoring p in Monte Carlo simulations is not as significant, especially when a small step length is used. Figure 2.8b shows the ratio of p to 1 as a function of electron kinetic energy for various step sizes. The same trends are observed as for the PLC.
Chapter 2 Monte Carlo Techniques in Radiation Transport
(a ) 0.3 (b) 0.6
25% energy losslstep 25% energy loss/step
€ c 20% ·B 0.2
~
15% 0 u~
10% .S! 0.1~
5%S
20%i
0.4 ~ 15% :il ~ 10% '6 ~ 0.2 5% .1! CIl-r
0.0 L-'-'-L..LJ..LLll-....L.-L....L..L..L.w.L--'-.J...L..L..L..IJJ.l.---L-L..L.LJ.J.JJ.J 0.1 1 10 100 0.01 0.1 1 10 100kinetic energy (MeV) kinetic energy (MeV)
Figure 2.8. The (a) path length correction and (b) the ratio (p)/(l) in water versus kinetic energy for various step sizes as quantified by ESTEPE9•
Boundary Crossing
As the multiple scattering theories are only applicable for infinite, homogeneous geometries, care must be taken in the transport of partic1es in the vicinity of boundaries. A proper boundary crossing scheme also ensures that the energy deposition along a step is assigned to the appropriate regions. Boundary crossing algorithms are largely code-dependent. The partic1e may stop in the boundary, take smaller step as it approaches the boundary, or tum into single scattering mode30•
2.3.5. Condensed history technique
The condensed history (CH) technique III electron transport is developed by Berge~l to simulate a large number of electron Coulomb interactions as a single step.
The implementations of the CH algorithm can be categorized as Class l or Class II.
Class 1 CH scheme
In the Class l scheme, the overall effects of all electron interactions are accounted for using the CSDA model. Electrons traverse in a series of steps, with the step length chosen to conform to a pre-defined energy-loss grid32• The amount of energy loss is
Chapter 2 Monte Carlo Techniques in Radiation Transport
incorporate the generation of delta electrons, but there is no direct correlation between the energy and direction of the primary electron and the creation of delta electrons.
Class II CH scheme
In the Class II scheme, the user needs to define production thresholds for the secondary photons and electrons. Particles generated with energies above the thresholds are simulated as discrete events. The energy loss of the primary electron corresponds to the energy of the secondary particle. Simulations of sub-threshold bremsstrahlung photons and delta rays are grouped as a step. The step size is randomly sampled from the interaction cross sections and is limited by boundaries. Restricted stopping powers account for the energy loss. There is a double-counting of the angular deflections of the delta electrons because the multiple scattering process also takes into account scattering from orbital electrons2, but the effect is usually trivial. The Class II implementation is more accurate in principle, but the differences in simulation results compared with those of the Class 1 scheme are small except for sorne specialized applications23•
2.3.6. Electron transport algorithm
Figure 2.9 illustrates a simplified electron transport algorithm for both the Class 1 and Class II CH implementations. The logic flow is similar to photon transport, except that the CSDA model is also incorporated. Most of the calculation time is spent simulating the multiple scattering steps and accounting for the energy loss in between discrete interactions.
Chapter 2 Monte Carlo Techniques in Radiation Transport
Place initial electron parameters on stack
y
+
Pick up energy, position,
[ Stack empty?
1
N direction, geometry ct currentparticle from top of stack
...
1 T erminate history N Bectron energy > cutoff
and electron in geometry?
+v
N [ Class" calculation? 1 y
~
l
Select multiple scatter
of-- Sam pie distance to _1
step size and transport discrete interaction
+
•
Sample deflection angle Select multiple scatter
and change direction step size and transport
1+-+
•
l
Sample energy loss Sample deflection angleE= E- Eloss and change direction
•
•
Is a secondary created y Calculate energ,r loss
during the step? E = E - Eloss(restricted s.p.)
+N
•
~
Has electron left geometry? J yt
Has electron left geometry?Y-f'J +N
~
Bectron energy < cutoff?}M-
y [ Bectron energy < cutoff?+N
y
r
Reached point of}--a
l
discrete interaction? Sample discrete interaction- knock-on - bremsstrahlung
•
Sample energy and direction of secondary, store parameters on stack
'IF
N [ Class" calculation?
+y
Change energy and direction of primary as a
result of interaction
Chapter 2 Monte Carlo Techniques in Radiation Transport
2.4. SCORING AND ANAL YSIS TECHNIQUES
The pUl-pose of running Monte Carlo simulations is to score sorne physical quantities of interest. This involves specifying certain sections of the geometry as the
scoring regions. It is often preferable to set the scoring regions to coincide with the geometric regions2• This will eliminate ambiguities in the allocation of energy
depositions to the appropriate regions, as particles must stop or take very small steps when they approach a geometrical boundary. If the geometries of interest are too complicated, it may be necessary to use scoring regions that are different from the geometric regions. Short electron step sizes should then be selected for better accuracy.
Dose de position calculations
In calculating dose distributions in a phantom, the number of scoring voxels can greatly affect the simulation time26. In sorne cases, the reciprocity theorem34 can be
applied. According to this theorem, the dose deposited in a cylindrical detector of radius
rd for an incident circular beam of radius rb is equivalent to the dose deposited in a cylindrical detector of radius rb for an incident circular beam of radius rd. Much efficiency can be gained by using a narrow beam with a large detector.
Scoring of phase space parameters
The information of particles reaching a scoring plane can be stored in a phase space file. Each entry in the file contains data such as the charge, energy, position, direction, and weight associated with a particle. The phase space data can be used to generate new particle histories, or to obtain quantities such as the planar fluence, angular distribution and mean energy of the particles crossing the scoring plane.
Tagging the histories of partie/es
Chapter 2 Monte Carlo Techniques in Radiation Transport
Statistical uncertainties
A direct approach of estimating uncertainties in the precision of Monte Carlo simulations is to divide the simulation of N histories into n equal batches, and calculate the standard error on the mean s; for the scored quantity X:11
where X is the mean value of x:
s-x
=
I
n ( x.-x-)2
i=l 1 n(n-1)
1 nx=-
I xj • n j=l (2.13) (2.14)The final result can be reported as x
=
x
± s;. The number of batches should be large enough to avoid fluctuations in the uncertainties35•The history by history method35 is a more efficient and accurate way of estimating uncertainties. Instead of using statistical batches, the scored quantities are grouped by primary histories. This approach takes into account correlations between particles belonging to the same primary history. However, there is no direct way to account for particle correlations when phase space data are reused.
2.5. VARIANCE REDUCTION TECHNIQUES
Variance reduction techniques are often employed in Monte Carlo simulations to increase efficiency E, which can be defined as2:
1
8 = - - 2 '
T·s (2.15)
Chapter 2 Monte Carlo Techniques in Radiation Transport
Variance reduction algorithms can be c1assified into four major categories33• In
the truncation methods, sorne aspects of the simulation that contribute little to the outcomes are truncated. Examples include not modeling the irrelevant parts of the geometry, and imposing an energy cutoff below which particles are terminated. The population control methods controls the quantities of the particles to be sampled based on their levels of importance. Methods such as particle splitting and Russian roulette belong to this category. The modified sampling methods modify the sampling distributions to improve the statistics, which may involve biasing the particle source or forcing an interaction to occur. Lastly, the partially deterministic methods incorporate sorne deterministic features in the sampling, which may alter the random number sequence.
2.6. REFERENCES
Corander, J & Sillanpaa, M.J. 2002, 'A unified approach to joint modeling of multiple
quantitative and qualitative traits in gene mapping', Journal of Theoretical Biology, vol 218, no.4, pp. 435-446.
2
Rogers, D.W.O. & Bielajew, AF. 1990, 'Monte Carlo techniques of electron and photon transport for radiation dosimetry', in The Dosimetry of Ionizing Radiation Vol III, eds K.R. Kase, B.E. Bjamgard & F.R. Attix, Academic Press, New York, pp. 427 - 533.
3 Attix, F.H. 1986, Introduction to Radiological Physics and Radiation Dosimetry, John Wiley
& Sons, New York.
4 Podgorsak, E.B. 2003, 'Basic radiation physics', in Review of Radiation Oncology Physics: A
Handbookfor Teachers and Students, ed E. B. Podgorsak, IAEA, Vienna, pp.l-36.
5 Sauter, S. 1931, 'Über den atomaren Photoeffekt in der K-Schale nach der relativistischen
Wellenmechanik Diracs' ,Ann. Physik, vol.ll, pp. 454-488.
6
Klein, O. & Nishina, Y. 1929, 'Über die Streuung von Strahlung durch freie Elektronen nach der neuen relativistischen Quantendynamik von Dirac' , Z. for Physik, vol. 52, pp. 853-863.
7 Hubbell, J.H. 1998, 'Review of photon interaction cross section data in the medical and
biological context', Phys. Med. Biol., vol.44, pp. RI-R22.
8 Johns, H.E. & Cunningham, J.R. 1983, The Physics ofRadiology, 4th edn, Charles C Thomas,
Springfield.
9
Berger, M.J., Hubbell, J.H., Seltzer, S.M., Coursey, J.S. & Zucker, O.S. 1999, NIST XCOM:
Photon Cross Sections Database, [Online
J,
A vailable from:<http://physics.nist.govlPhysRefData/XcomlTextlXCOM.html> [12 May 2004
J
10 Hirayama, H. 2000. 'Lecture notes on photon interactions and cross sections', KEK Internai
