The M₂ Model for Dose Simulation in Radiation Therapy

References

• Alldredge, G. W., Hauck, C. D., O’Leary, D. P., Tits, A. L. (2014). Adaptive change of basis in entropy-based moment closures for linear kinetic equations. J. Comp. Phys. 74(4):489--508.
   Google Scholar
• Alldredge, G. W., Hauck, C. D., Tits, A. L. (2012). High-order entropy-based closures for linear transport in slab geometry II: A computational study of the optimization problem. SIAM J. Sci. Comput. 34(4):361–391.
   Web of Science ®Google Scholar
• Anile, A. M., Pennisi, S. (1992). Thermodynamic derivation of the hydrodynamical model for charge transport in semiconductors. Phys. Rev. B 100:46:13186–13193.
   Google Scholar
• Barò, J., Sempau, J., Fernández-Varea, J. M., Salvat, F. (1995). Penelope: An algorithm for Monte Carlo simulation of the penetration and energy loss of electrons in matter. Nucl. Instrum. Methods 100:31–46.
   Web of Science ®Google Scholar
• Barò, J., Sempau, J., Fernández-Varea, J. M., Salvat, F. (1996). Penelope: An algorithm and computer code for Monte Carlo simulation of electron-photon shower. CIEMAT Technical Report 799. CIEMAT: Madrid. Accessed 15 March 2016. http://www.iaea.org/inis/collection/NCLCollectionStore/_Public/38/094/38094326.pdf
   Google Scholar
• Berthon, C., Charrier, P., Dubroca, B. (2007). An HLLC scheme to solve the M1 model of radiative transfer in two space dimensions. J. Sci. Comp. 31(3):347–389.
   Web of Science ®Google Scholar
• Borwein, J., Lewis, A. (1991). Duality relationships for entropy-like minimization problems. SIAM J. Control Optim. 29(2):325–338.
   Web of Science ®Google Scholar
• Borwein, J., Lewis, A. (1992a). Partially finite convex programming, part I: Quasi relative interiors and duality theory. Math. Program. 57:15–48.
   Web of Science ®Google Scholar
• Borwein, J., Lewis, A. (1992b). Partially finite convex programming: Part II. Math. Program. 57:49–83.
   Web of Science ®Google Scholar
• Caron, J., Feugeas, J. -L., Dubroca, B., Kantor, G., Dejean, C., Birindelli, G., Pichard, T., Nicolaï, Ph, d’Humières, E., Frank, M., Tikhonchuk, V. (2015). Deterministic model for the transport of energetic particles. Application in the electron radiotherapy. Eur. J. Med. Phys. 31:912–921.
   Google Scholar
• Dubroca, B., Feugeas, J. L. (1999). Hiérarchie des modèles aux moments pour le transfert radiatif. C.R. Acad. Sci. Paris 329:915–920.
   Web of Science ®Google Scholar
• Duclous, R., Dubroca, B., Frank, M. (2010). A deterministic partial differential equation model for dose calculation in electron radiotherapy. Phys. Med. Biol. 55:3843–3857.
   PubMed Web of Science ®Google Scholar
• Grad, H. (1949). On the kinetic theory of rarefied gases. Comm. Pure App. Math. 2(4):331–407.
   Web of Science ®Google Scholar
• Guisset, S., Brull, S., Dubroca, B., Humières, E., Karpov, S., Potapenko, I. (2016). Asymptotic-preserving scheme for the Fokker-Planck-Landau-Maxwell system in the quasi-neutral regime. Comm. Comp. Phys. 19(2):301--328.
   Web of Science ®Google Scholar
• Guisset, S., Moreau, J. G., Nuter, R., Brull, S., Humieres, E., Dubroca, B., Tikhonchuk, V. T. (2015). Limits of the M1 and M2 angular moments models for kinetic plasma physics studies. J. Phys. A: Mathematical and Theoretical. 48(33):335501
   Web of Science ®Google Scholar
• Hauck, C., McClarren, R. (2010). Positive PN closures. SIAM J. Sci. Comput. 32(5):2603–2626.
   Web of Science ®Google Scholar
• Hauck, C. D. (2011). High-order entropy-based closures for linear transport in slab geometry. Commun. Math. Sci. 9(1):187–205.
   Web of Science ®Google Scholar
• Hensel, H., Iza-Teran, R., Siedow, N. (2006). Deterministic model for dose calculation in photon radiotherapy. Phys. Med. Biol. 51:675–693.
   PubMed Web of Science ®Google Scholar
• Rosa, S., Mascali, G., Romano, V. (2009). Exact maximum entropy closure of the hydrodynamical model for SI semiconductors: The 8-moment case. SIAM J. Appl. Math. 70(3):710–734.
   Web of Science ®Google Scholar
• Leveremore, C. D. (1984). Relating Eddington factors to flux limiters. J. Quant. Spectros. Radiat. Transf. 31:149–160.
   Web of Science ®Google Scholar
• Levermore, C. D. (1996). Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83(5–6):1021–1065.
   Web of Science ®Google Scholar
• Lewis, E. E., Miller, W. F. (1993). Computational Methods of Neutron Transport. La Grange Park, IL: American Nuclear Society.
   Google Scholar
• Mallet, J., Brull, S., Dubroca, B. (2014). An entropic scheme for an angular moment model for the classical Fokker-Planck-Landau equation of electrons. Comm. Comp. Phys. 15(2):422–450.
   Web of Science ®Google Scholar
• Mallet, J., Brull, S., Dubroca, B. (2015). General moment system for plasma physics based on minimum entropy principle. Kinet. Relat. Models. 8(3):533–558.
   Web of Science ®Google Scholar
• McClarren, R. G., Hauck, C. D. (2010). Robust and accurate filtered spherical harmonics expansion for radiative transfer. J. Comp. Phys. 229(16):5597–5614.
   Web of Science ®Google Scholar
• Olbrant, E., Frank, M. (2010). Generalized Fokker-Planck theory for electron and photon transport in biological tissues: Application to radiotherapy. Comput. Math. Methods Med. 11(4):313–339.
   PubMed Web of Science ®Google Scholar
• Pichard, T., Alldredge, G. W., Brull, S., Dubroca, B., Frank, M. (in press) An approximation of the M2 closure: Application to radiotherapy dose simulation. Submitted for publication in J. Sci. Comput.
   Google Scholar
• Pichard, T., Aregba-Driollet, D., Brull, S., Dubroca, B., Frank, M. (2016). Relaxation schemes for the M1 model with space-dependent flux: Application to radiotherapy dose calculation. Commun. Comp. Phys. 19(1):168--191.
   Web of Science ®Google Scholar
• Pomraning, G. C. (1991). The Fokker-Planck operator as an asymptotic limit. Math. Mod. Meth. Appl. Sci. 2(1):21–36.
   Google Scholar