September  2014, 7(3): 433-461. doi: 10.3934/krm.2014.7.433

New insights into the numerical solution of the Boltzmann transport equation for photons

1. 

Department of Arts and Sciences, Ahsanullah University of Science and Technology, 141-142 Love Road, Tejgaon Industrial Area, Dhaka-1208, Dhaka, Bangladesh

2. 

Department of Applied Mathematics, Faculty of Mathematics, University of Santiago de Compostela, Campus Vida, 15782 Santiago de Compostela, Spain

Received  August 2013 Revised  April 2014 Published  July 2014

This paper has been thought to describe a numerical algorithm, based on the expansion in orders of scattering, for solving the steady Boltzmann transport equation for photons, and to show some subsequent numerical results which have been obtained by developing an own Matlab® code. The spatial domain is assumed to be a rectangular-shaped container with air and a water phantom inside, albeit the method can be extended to arbitrarily shaped convex domains filled with some other heterogeneous material. High energy x-rays enter the domain through a small rectangle located on its upper face.
Citation: Taposh Kumar Das, Óscar López Pouso. New insights into the numerical solution of the Boltzmann transport equation for photons. Kinetic & Related Models, 2014, 7 (3) : 433-461. doi: 10.3934/krm.2014.7.433
References:
[1]

K. B. Bekar and Y. Y. Azmy, Revisiting the TORT solutions to the NEA suite of benchmarks for 3D transport methods and codes over a range in parameter space,, in Proceedings of the International Conference on Mathematics, (2009), 3.   Google Scholar

[2]

C. Börgers, Complexity of Monte Carlo and deterministic dose-calculation methods,, Phys. Med. Biol., 43 (1998), 517.  doi: 10.1088/0031-9155/43/3/004.  Google Scholar

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T. K. Das, Numerical Solution of the Boltzmann Transport Equation for Photons and of some Equations Derived from the Fokker-Planck Approximation for Electrons. Application to Radiotherapy,, Ph.D thesis, (2012).   Google Scholar

[4]

C. M. Davisson and R. D. Evans, Gamma-ray absorption coefficients,, Rev. Mod. Phys., 24 (1952), 79.  doi: 10.1103/RevModPhys.24.79.  Google Scholar

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R. Duclous, B. Dubroca and M. Frank, A deterministic partial differential equation model for dose calculation in electron radiotherapy,, Phys. Med. Biol., 55 (2010), 3843.  doi: 10.1088/0031-9155/55/13/018.  Google Scholar

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U. Fano, L. V. Spencer and M. J. Berger, Penetration and diffusion of X rays,, in Neutrons and Related Gamma Ray Problems (ed. S. Flügge), (1959), 660.  doi: 10.1007/978-3-642-45920-7_2.  Google Scholar

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M. Frank, H. Hensel and A. Klar, A fast and accurate moment method for the Fokker-Planck equation and applications to electron radiotherapy,, SIAM J. Appl. Math., 67 (): 582.  doi: 10.1137/06065547X.  Google Scholar

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I. M. Gel'fand and G. E. Shilov, Generalized Functions. Vol. 1. Properties and Operations,, Translated from the Russian by Eugene Saletan, (1966).   Google Scholar

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K. A. Gifford, J. L. Horton Jr., T. A. Wareing, G. Failla and F. Mourtada, Comparison of a finite-element multigroup discrete-ordinates code with Monte Carlo for radiotherapy calculations,, Phys. Med. Biol., 51 (2006), 2253.  doi: 10.1088/0031-9155/51/9/010.  Google Scholar

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K. A. Gifford, M. J. Price, J. L. Horton Jr., T. A. Wareing and F. Mourtada, Optimization of deterministic transport parameters for the calculation of the dose distribution around a high dose-rate 192Ir brachytherapy source,, Med. Phys., 35 (2008), 2279.  doi: 10.1118/1.2919074.  Google Scholar

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K. A. Gifford, T. A. Wareing, G. A. Failla, J. L. Horton Jr., P. J. Eifel and F. Mourtada, Comparison of a 3D multi-group $S_N$ particle transport code with Monte Carlo for intracavitary brachytherapy of the cervix uteri,, J. Appl. Clin. Med. Phys., 11 (2010), 2.   Google Scholar

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P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Pitman, (1985).  doi: 10.1137/1.9781611972030.  Google Scholar

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L. L. Gunderson and J. E. Tepper, eds., Clinical Radiation Oncology,, Third edition, (2012).   Google Scholar

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H. Hensel, R. Iza-Teran and N. Siedow, Deterministic model for dose calculation in photon radiotherapy,, Phys. Med. Biol., 51 (2006), 675.  doi: 10.1088/0031-9155/51/3/013.  Google Scholar

[15]

H. E. Johns and J. R. Cunningham, The Physics of Radiology,, Fourth edition, (1983).   Google Scholar

[16]

U. Karimov, A Deterministic Model for Computing the Radiation Dose in Cancer Treatment,, Proyecto Fin de Máster, (2010).   Google Scholar

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J. A. López, Aspectos Médico-Biológicos y Físicos de la Radioterapia como Tratamiento del Cáncer. Modelo Determinista para el Cálculo de la Dosis Absorbida,, Proyecto Fin de Máster, (2008).   Google Scholar

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L. J. Lorence Jr., J. E. Morel and G. D. Valdez, Physics Guide to CEPXS: A Multigroup Coupled Electron-Photon Cross-Section Generating Code,, SANDIA Report SAND89-1685, (1989), 89.   Google Scholar

[19]

L. J. Lorence, Jr., W. E. Nelson and J. E. Morel, Coupled electron-photon transport calculations using the method of discrete ordinates,, IEEE Trans. Nucl. Sci., 32 (1985), 4416.  doi: 10.1109/TNS.1985.4334134.  Google Scholar

[20]

M. F. Modest, Radiative Heat Transfer,, Third edition, (2013).   Google Scholar

[21]

N. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions,, Third edition, (1965).   Google Scholar

[22]

E. Olbrant and M. Frank, Generalized Fokker-Planck theory for electron and photon transport in biological tissues: Application to radiotherapy,, Comput. Math. Methods Med., 11 (2010), 313.  doi: 10.1080/1748670X.2010.491828.  Google Scholar

[23]

G. H. Peebles, Gamma-Ray Transmission through Finite Slabs,, Rand Corporation Report R-240, (1952).   Google Scholar

[24]

G. H. Peebles and M. S. Plesset, Transmission of gamma-rays through large thicknesses of heavy materials,, Phys. Rev., 81 (1951), 430.  doi: 10.1103/PhysRev.81.430.  Google Scholar

[25]

O. N. Vassiliev, T. A. Wareing, J. M. McGhee, G. A. Failla, M. R. Salehpour and F. Mourtada, Validation of a new grid-based Boltzmann equation solver for dose calculation in radiotherapy with photon beams,, Phys. Med. Biol., 55 (2010), 581.  doi: 10.1088/0031-9155/55/3/002.  Google Scholar

[26]

M. L. Williams, D. Ilas, E. Sajo, D. B. Jones and K. E. Watkins, Deterministic photon transport calculations in general geometry for external beam radiation therapy,, Med. Phys., 30 (2003), 3183.  doi: 10.1118/1.1621135.  Google Scholar

[27]

J. Yuan, D. Jette and W. Chen, Deterministic photon kerma distribution based on the Boltzmann equation for external beam radiation therapy,, Med. Phys., 35 (2008), 2839.  doi: 10.1118/1.2962248.  Google Scholar

show all references

References:
[1]

K. B. Bekar and Y. Y. Azmy, Revisiting the TORT solutions to the NEA suite of benchmarks for 3D transport methods and codes over a range in parameter space,, in Proceedings of the International Conference on Mathematics, (2009), 3.   Google Scholar

[2]

C. Börgers, Complexity of Monte Carlo and deterministic dose-calculation methods,, Phys. Med. Biol., 43 (1998), 517.  doi: 10.1088/0031-9155/43/3/004.  Google Scholar

[3]

T. K. Das, Numerical Solution of the Boltzmann Transport Equation for Photons and of some Equations Derived from the Fokker-Planck Approximation for Electrons. Application to Radiotherapy,, Ph.D thesis, (2012).   Google Scholar

[4]

C. M. Davisson and R. D. Evans, Gamma-ray absorption coefficients,, Rev. Mod. Phys., 24 (1952), 79.  doi: 10.1103/RevModPhys.24.79.  Google Scholar

[5]

R. Duclous, B. Dubroca and M. Frank, A deterministic partial differential equation model for dose calculation in electron radiotherapy,, Phys. Med. Biol., 55 (2010), 3843.  doi: 10.1088/0031-9155/55/13/018.  Google Scholar

[6]

U. Fano, L. V. Spencer and M. J. Berger, Penetration and diffusion of X rays,, in Neutrons and Related Gamma Ray Problems (ed. S. Flügge), (1959), 660.  doi: 10.1007/978-3-642-45920-7_2.  Google Scholar

[7]

M. Frank, H. Hensel and A. Klar, A fast and accurate moment method for the Fokker-Planck equation and applications to electron radiotherapy,, SIAM J. Appl. Math., 67 (): 582.  doi: 10.1137/06065547X.  Google Scholar

[8]

I. M. Gel'fand and G. E. Shilov, Generalized Functions. Vol. 1. Properties and Operations,, Translated from the Russian by Eugene Saletan, (1966).   Google Scholar

[9]

K. A. Gifford, J. L. Horton Jr., T. A. Wareing, G. Failla and F. Mourtada, Comparison of a finite-element multigroup discrete-ordinates code with Monte Carlo for radiotherapy calculations,, Phys. Med. Biol., 51 (2006), 2253.  doi: 10.1088/0031-9155/51/9/010.  Google Scholar

[10]

K. A. Gifford, M. J. Price, J. L. Horton Jr., T. A. Wareing and F. Mourtada, Optimization of deterministic transport parameters for the calculation of the dose distribution around a high dose-rate 192Ir brachytherapy source,, Med. Phys., 35 (2008), 2279.  doi: 10.1118/1.2919074.  Google Scholar

[11]

K. A. Gifford, T. A. Wareing, G. A. Failla, J. L. Horton Jr., P. J. Eifel and F. Mourtada, Comparison of a 3D multi-group $S_N$ particle transport code with Monte Carlo for intracavitary brachytherapy of the cervix uteri,, J. Appl. Clin. Med. Phys., 11 (2010), 2.   Google Scholar

[12]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Pitman, (1985).  doi: 10.1137/1.9781611972030.  Google Scholar

[13]

L. L. Gunderson and J. E. Tepper, eds., Clinical Radiation Oncology,, Third edition, (2012).   Google Scholar

[14]

H. Hensel, R. Iza-Teran and N. Siedow, Deterministic model for dose calculation in photon radiotherapy,, Phys. Med. Biol., 51 (2006), 675.  doi: 10.1088/0031-9155/51/3/013.  Google Scholar

[15]

H. E. Johns and J. R. Cunningham, The Physics of Radiology,, Fourth edition, (1983).   Google Scholar

[16]

U. Karimov, A Deterministic Model for Computing the Radiation Dose in Cancer Treatment,, Proyecto Fin de Máster, (2010).   Google Scholar

[17]

J. A. López, Aspectos Médico-Biológicos y Físicos de la Radioterapia como Tratamiento del Cáncer. Modelo Determinista para el Cálculo de la Dosis Absorbida,, Proyecto Fin de Máster, (2008).   Google Scholar

[18]

L. J. Lorence Jr., J. E. Morel and G. D. Valdez, Physics Guide to CEPXS: A Multigroup Coupled Electron-Photon Cross-Section Generating Code,, SANDIA Report SAND89-1685, (1989), 89.   Google Scholar

[19]

L. J. Lorence, Jr., W. E. Nelson and J. E. Morel, Coupled electron-photon transport calculations using the method of discrete ordinates,, IEEE Trans. Nucl. Sci., 32 (1985), 4416.  doi: 10.1109/TNS.1985.4334134.  Google Scholar

[20]

M. F. Modest, Radiative Heat Transfer,, Third edition, (2013).   Google Scholar

[21]

N. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions,, Third edition, (1965).   Google Scholar

[22]

E. Olbrant and M. Frank, Generalized Fokker-Planck theory for electron and photon transport in biological tissues: Application to radiotherapy,, Comput. Math. Methods Med., 11 (2010), 313.  doi: 10.1080/1748670X.2010.491828.  Google Scholar

[23]

G. H. Peebles, Gamma-Ray Transmission through Finite Slabs,, Rand Corporation Report R-240, (1952).   Google Scholar

[24]

G. H. Peebles and M. S. Plesset, Transmission of gamma-rays through large thicknesses of heavy materials,, Phys. Rev., 81 (1951), 430.  doi: 10.1103/PhysRev.81.430.  Google Scholar

[25]

O. N. Vassiliev, T. A. Wareing, J. M. McGhee, G. A. Failla, M. R. Salehpour and F. Mourtada, Validation of a new grid-based Boltzmann equation solver for dose calculation in radiotherapy with photon beams,, Phys. Med. Biol., 55 (2010), 581.  doi: 10.1088/0031-9155/55/3/002.  Google Scholar

[26]

M. L. Williams, D. Ilas, E. Sajo, D. B. Jones and K. E. Watkins, Deterministic photon transport calculations in general geometry for external beam radiation therapy,, Med. Phys., 30 (2003), 3183.  doi: 10.1118/1.1621135.  Google Scholar

[27]

J. Yuan, D. Jette and W. Chen, Deterministic photon kerma distribution based on the Boltzmann equation for external beam radiation therapy,, Med. Phys., 35 (2008), 2839.  doi: 10.1118/1.2962248.  Google Scholar

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