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 and 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, Computational Methods & Reactor Physics (M&C 2009), Saratoga Springs, New York, May 3-7, 2009, on CD-ROM, American Nuclear Society, LaGrange Park, IL, 2009.

[2]

C. Börgers, Complexity of Monte Carlo and deterministic dose-calculation methods, Phys. Med. Biol., 43 (1998), 517-528. doi: 10.1088/0031-9155/43/3/004.

[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, Universidad de Santiago de Compostela, 2012.

[4]

C. M. Davisson and R. D. Evans, Gamma-ray absorption coefficients, Rev. Mod. Phys., 24 (1952), 79-107. doi: 10.1103/RevModPhys.24.79.

[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), 3843-3857. doi: 10.1088/0031-9155/55/13/018.

[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), Encyclopedia of Physics, Vol. 38, Part 2, Springer, Berlin Heidelberg, 1959, 660-817. doi: 10.1007/978-3-642-45920-7_2.

[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.

[8]

I. M. Gel'fand and G. E. Shilov, Generalized Functions. Vol. 1. Properties and Operations, Translated from the Russian by Eugene Saletan, second printing, Academic Press, New York-London, 1966.

[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), 2253-2265. doi: 10.1088/0031-9155/51/9/010.

[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), 2279-2285. doi: 10.1118/1.2919074.

[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), 2-9.

[12]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, MA, 1985. doi: 10.1137/1.9781611972030.

[13]

L. L. Gunderson and J. E. Tepper, eds., Clinical Radiation Oncology, Third edition, Saunders, an imprint of Elsevier Inc., Philadelphia, PA, 2012.

[14]

H. Hensel, R. Iza-Teran and N. Siedow, Deterministic model for dose calculation in photon radiotherapy, Phys. Med. Biol., 51 (2006), 675-693. doi: 10.1088/0031-9155/51/3/013.

[15]

H. E. Johns and J. R. Cunningham, The Physics of Radiology, Fourth edition, Charles C. Thomas Publisher, Springfield, IL, 1983.

[16]

U. Karimov, A Deterministic Model for Computing the Radiation Dose in Cancer Treatment, Proyecto Fin de Máster, Máster en Ingeniería Matemática, Universidad de Santiago de Compostela, 2010.

[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, Máster en Ingeniería Matemática, Universidad de Santiago de Compostela, 2008.

[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.

[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), 4416-4420. doi: 10.1109/TNS.1985.4334134.

[20]

M. F. Modest, Radiative Heat Transfer, Third edition, Academic Press (an imprint of Elsevier Inc.), Oxford, UK, 2013.

[21]

N. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions, Third edition, Oxford University Press, London, 1965.

[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), 313-339. doi: 10.1080/1748670X.2010.491828.

[23]

G. H. Peebles, Gamma-Ray Transmission through Finite Slabs, Rand Corporation Report R-240, Dec. 1, 1952.

[24]

G. H. Peebles and M. S. Plesset, Transmission of gamma-rays through large thicknesses of heavy materials, Phys. Rev., 81 (1951), 430-439. doi: 10.1103/PhysRev.81.430.

[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), 581-598. doi: 10.1088/0031-9155/55/3/002.

[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), 3183-3195. doi: 10.1118/1.1621135.

[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), 2839-2846. doi: 10.1118/1.2962248.

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, Computational Methods & Reactor Physics (M&C 2009), Saratoga Springs, New York, May 3-7, 2009, on CD-ROM, American Nuclear Society, LaGrange Park, IL, 2009.

[2]

C. Börgers, Complexity of Monte Carlo and deterministic dose-calculation methods, Phys. Med. Biol., 43 (1998), 517-528. doi: 10.1088/0031-9155/43/3/004.

[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, Universidad de Santiago de Compostela, 2012.

[4]

C. M. Davisson and R. D. Evans, Gamma-ray absorption coefficients, Rev. Mod. Phys., 24 (1952), 79-107. doi: 10.1103/RevModPhys.24.79.

[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), 3843-3857. doi: 10.1088/0031-9155/55/13/018.

[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), Encyclopedia of Physics, Vol. 38, Part 2, Springer, Berlin Heidelberg, 1959, 660-817. doi: 10.1007/978-3-642-45920-7_2.

[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.

[8]

I. M. Gel'fand and G. E. Shilov, Generalized Functions. Vol. 1. Properties and Operations, Translated from the Russian by Eugene Saletan, second printing, Academic Press, New York-London, 1966.

[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), 2253-2265. doi: 10.1088/0031-9155/51/9/010.

[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), 2279-2285. doi: 10.1118/1.2919074.

[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), 2-9.

[12]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, MA, 1985. doi: 10.1137/1.9781611972030.

[13]

L. L. Gunderson and J. E. Tepper, eds., Clinical Radiation Oncology, Third edition, Saunders, an imprint of Elsevier Inc., Philadelphia, PA, 2012.

[14]

H. Hensel, R. Iza-Teran and N. Siedow, Deterministic model for dose calculation in photon radiotherapy, Phys. Med. Biol., 51 (2006), 675-693. doi: 10.1088/0031-9155/51/3/013.

[15]

H. E. Johns and J. R. Cunningham, The Physics of Radiology, Fourth edition, Charles C. Thomas Publisher, Springfield, IL, 1983.

[16]

U. Karimov, A Deterministic Model for Computing the Radiation Dose in Cancer Treatment, Proyecto Fin de Máster, Máster en Ingeniería Matemática, Universidad de Santiago de Compostela, 2010.

[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, Máster en Ingeniería Matemática, Universidad de Santiago de Compostela, 2008.

[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.

[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), 4416-4420. doi: 10.1109/TNS.1985.4334134.

[20]

M. F. Modest, Radiative Heat Transfer, Third edition, Academic Press (an imprint of Elsevier Inc.), Oxford, UK, 2013.

[21]

N. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions, Third edition, Oxford University Press, London, 1965.

[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), 313-339. doi: 10.1080/1748670X.2010.491828.

[23]

G. H. Peebles, Gamma-Ray Transmission through Finite Slabs, Rand Corporation Report R-240, Dec. 1, 1952.

[24]

G. H. Peebles and M. S. Plesset, Transmission of gamma-rays through large thicknesses of heavy materials, Phys. Rev., 81 (1951), 430-439. doi: 10.1103/PhysRev.81.430.

[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), 581-598. doi: 10.1088/0031-9155/55/3/002.

[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), 3183-3195. doi: 10.1118/1.1621135.

[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), 2839-2846. doi: 10.1118/1.2962248.

[1]

Yan Guo, Juhi Jang, Ning Jiang. Local Hilbert expansion for the Boltzmann equation. Kinetic and Related Models, 2009, 2 (1) : 205-214. doi: 10.3934/krm.2009.2.205

[2]

Alexander Bobylev, Raffaele Esposito. Transport coefficients in the $2$-dimensional Boltzmann equation. Kinetic and Related Models, 2013, 6 (4) : 789-800. doi: 10.3934/krm.2013.6.789

[3]

Stefan Possanner, Claudia Negulescu. Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport. Kinetic and Related Models, 2011, 4 (4) : 1159-1191. doi: 10.3934/krm.2011.4.1159

[4]

Martin Frank, Thierry Goudon. On a generalized Boltzmann equation for non-classical particle transport. Kinetic and Related Models, 2010, 3 (3) : 395-407. doi: 10.3934/krm.2010.3.395

[5]

John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems and Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181

[6]

Bertrand Lods, Clément Mouhot, Giuseppe Toscani. Relaxation rate, diffusion approximation and Fick's law for inelastic scattering Boltzmann models. Kinetic and Related Models, 2008, 1 (2) : 223-248. doi: 10.3934/krm.2008.1.223

[7]

Enrique Fernández-Cara, Juan Límaco, Laurent Prouvée. Optimal control of a two-equation model of radiotherapy. Mathematical Control and Related Fields, 2018, 8 (1) : 117-133. doi: 10.3934/mcrf.2018005

[8]

Nicolas Fournier. A recursive algorithm and a series expansion related to the homogeneous Boltzmann equation for hard potentials with angular cutoff. Kinetic and Related Models, 2019, 12 (3) : 483-505. doi: 10.3934/krm.2019020

[9]

Lingbing He, Yulong Zhou. High order approximation for the Boltzmann equation without angular cutoff. Kinetic and Related Models, 2018, 11 (3) : 547-596. doi: 10.3934/krm.2018024

[10]

Yohan Penel. An explicit stable numerical scheme for the $1D$ transport equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 641-656. doi: 10.3934/dcdss.2012.5.641

[11]

Armando Majorana. A numerical model of the Boltzmann equation related to the discontinuous Galerkin method. Kinetic and Related Models, 2011, 4 (1) : 139-151. doi: 10.3934/krm.2011.4.139

[12]

Yves Achdou, Fabio Camilli, Lucilla Corrias. On numerical approximation of the Hamilton-Jacobi-transport system arising in high frequency approximations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 629-650. doi: 10.3934/dcdsb.2014.19.629

[13]

Victor Ginting. An adjoint-based a posteriori analysis of numerical approximation of Richards equation. Electronic Research Archive, 2021, 29 (5) : 3405-3427. doi: 10.3934/era.2021045

[14]

Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080

[15]

Marcel Braukhoff. Global analytic solutions of the semiconductor Boltzmann-Dirac-Benney equation with relaxation time approximation. Kinetic and Related Models, 2020, 13 (1) : 187-210. doi: 10.3934/krm.2020007

[16]

Zheng-an Yao, Yu-Long Zhou. High order approximation for the Boltzmann equation without angular cutoff under moderately soft potentials. Kinetic and Related Models, 2020, 13 (3) : 435-478. doi: 10.3934/krm.2020015

[17]

Esther S. Daus, Shi Jin, Liu Liu. Spectral convergence of the stochastic galerkin approximation to the boltzmann equation with multiple scales and large random perturbation in the collision kernel. Kinetic and Related Models, 2019, 12 (4) : 909-922. doi: 10.3934/krm.2019034

[18]

Xu Yang, François Golse, Zhongyi Huang, Shi Jin. Numerical study of a domain decomposition method for a two-scale linear transport equation. Networks and Heterogeneous Media, 2006, 1 (1) : 143-166. doi: 10.3934/nhm.2006.1.143

[19]

Florian De Vuyst, Francesco Salvarani. Numerical simulations of degenerate transport problems. Kinetic and Related Models, 2014, 7 (3) : 463-476. doi: 10.3934/krm.2014.7.463

[20]

Nicolas Lecoq, Helena Zapolsky, P.K. Galenko. Numerical approximation of the Chan-Hillard equation with memory effects in the dynamics of phase separation. Conference Publications, 2011, 2011 (Special) : 953-962. doi: 10.3934/proc.2011.2011.953

2020 Impact Factor: 1.432

Metrics

  • PDF downloads (110)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]