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An approximation of the \(M_2\) closure: application to radiotherapy dose simulation. (English) Zbl 1386.82064

Summary: Particle transport in radiation therapy can be modelled by a kinetic equation which must be solved numerically. Unfortunately, the numerical solution of such equations is generally too expensive for applications in medical centers. Moment methods provide a hierarchy of models used to reduce the numerical cost of these simulations while preserving basic properties of the solutions. Moment models require a closure because they have more unknowns than equations. The entropy-based closure is based on the physical description of the particle interactions and provides desirable properties. However, computing this closure is expensive. We propose an approximation of the closure for the first two models in the hierarchy, the \(M_1\) and \(M_2\) models valid in one, two or three dimensions of space. Compared to other approximate closures, our method works in multiple dimensions. We obtain the approximation by a careful study of the domain of realizability and by invariance properties of the entropy minimizer. The \(M_2\) model is shown to provide significantly better accuracy than the \(M_1\) model for the numerical simulation of a dose computation in radiotherapy. We propose a numerical solver using those approximated closures. Numerical experiments in dose computation test cases show that the new method is more efficient compared to numerical solution of the minimum entropy problem using standard software tools.

MSC:

82C70 Transport processes in time-dependent statistical mechanics
82D99 Applications of statistical mechanics to specific types of physical systems
94A17 Measures of information, entropy
92C50 Medical applications (general)
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
92-08 Computational methods for problems pertaining to biology
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[1] Alldredge, G.W., Hauck, C.D., O’Leary, D.P., Tits, A.L.: Adaptive change of basis in entropy-based moment closures for linear kinetic equations. J. Comput. Phys. 258, 489-508 (2014) · Zbl 1349.82066 · doi:10.1016/j.jcp.2013.10.049
[2] Alldredge, G.W., Hauck, C.D., Tits, A.L.: High-order entropy-based closures for linear transport in slab geometry II: a comutational study of the optimization problem. SIAM J. Sci. Comput. 34(4), 361-391 (2012) · Zbl 1297.82032 · doi:10.1137/11084772X
[3] Alldredge, G.W., Li, R., Li, W.: Approximating the \[{M_2}\] M2 method by the extended quadrature method of moments for radiative transfer in slab geometry. Kin. Rel. Mod. 9(2), 237-249 (2016) · Zbl 1333.78018
[4] Anile, A.M., Pennisi, S.: Thermodynamic derivation of the hydrodynamical model for charge transport in semiconductors. Phys. Rev. B 46, 13186-13193 (1992) · Zbl 1174.82324 · doi:10.1103/PhysRevB.46.13186
[5] Berntsen, J., Espelid, T.O., Genz, A.: Algorithm 698: Dcuhre: an adaptive multidemensional integration routine for a vector of integrals. ACM Trans. Math. Softw. 17(4), 452-456 (1991). http://netlib.org/toms/ · Zbl 0900.65053
[6] Berthon, C., Charrier, P., Dubroca, B.: An HLLC scheme to solve the \[{M_1}\] M1 model of radiative transfer in two space dimensions. J. Sci. Comput. 31(3), 347-389 (2007) · Zbl 1133.85003 · doi:10.1007/s10915-006-9108-6
[7] Berthon, C., Frank, M., Sarazin, C., Turpault, R.: Numerical methods for balance laws with space dependent flux: application to radiotherapy dose calculation. Commun. Comput. Phys. 10(5), 1184-1210 (2011) · Zbl 1373.76116 · doi:10.4208/cicp.020810.171210a
[8] Borwein, J., Lewis, A.: Duality relationships for entropy-like minimization problems. SIAM J. Control Optim. 29(2), 325-338 (1991) · Zbl 0797.49030 · doi:10.1137/0329017
[9] Borwein, J., Lewis, A.: Partially finite convex programming, part I: quasi relative interiors and duality theory. Math. Program. 57, 15-48 (1992) · Zbl 0778.90049 · doi:10.1007/BF01581072
[10] Borwein, J., Lewis, A.: Partially finite convex programming: part II. Math. Program. 57, 49-83 (1992) · Zbl 0778.90050 · doi:10.1007/BF01581073
[11] Brunner, T.A., Holloway, J.P.: One-dimensional riemann solvers and the maximum entropy closure. J. Quant. Spectrosc. Radiat. Transf. 69(5), 543-566 (2001) · doi:10.1016/S0022-4073(00)00099-6
[12] Caron, J., Feugeas, J.-L., Dubroca, B., Kantor, G., Dejean, C., Birindelli, G., Pichard, T., Nicolaï, Ph, d’Humières, E., Frank, M., Tikhonchuk, V.: Deterministic model for the transport of energetic particles. Application in the electron radiotherapy. Phys. Med. 31, 912-921 (2015) · doi:10.1016/j.ejmp.2015.07.148
[13] Chandrasekhar, S.: Radiative Transfer. Dover, New York (1950) · Zbl 0037.43201
[14] Chetty, I.J., Curran, B., Cygler, J.E., DeMarco, J.J., Ezzell, G., Faddegon, B.A., Kawrakow, I., Keall, P.J., Liu, H., Ma, C.-M.C., Rogers, D.W.O., Seuntjens, J., Sheikh-Bagheri, D., Siebers, J.V.: Report of the AAPM task group no. 105: issues associated with clinical implementation of Monte Carlo-based photon and electron external beam treatment planning. Med. Phys. 34(12), 4818-4853 (2007) · doi:10.1118/1.2795842
[15] Curto, R., Fialkow, L.A.: Recusiveness, positivity, and truncated moment problems. Houst. J. Math. 17(4), 603-634 (1991) · Zbl 0757.44006
[16] Curto, R., Fialkow, L.A.: A duality prood to Tchakaloff’s theorem. J. Math. Anal. Appl. 269, 519-536 (2002) · Zbl 1001.41014 · doi:10.1016/S0022-247X(02)00034-3
[17] Dubroca, B., Feugeas, J.L.: Hirarchie des modles aux moments pour le transfert radiatif. C.R. Acad. Sci. Paris 329, 915-920 (1999) · Zbl 0940.65157 · doi:10.1016/S0764-4442(00)87499-6
[18] Dubroca, B., Frank, M.: An iterative method for transport equations in radiotherapy. Prog. Ind. Math. ECMI 2008, 407-412 (2010) · Zbl 1308.92048
[19] Duclous, R., Dubroca, B., Frank, M.: A deterministic partial differential equation model for dose calculation in electron radiotherapy. Phys. Med. Biol. 55, 3843-3857 (2010) · doi:10.1088/0031-9155/55/13/018
[20] Friedriechs, K.O., Lax, P.D.: Systems of conservation equations with a convex extension. Proc. Nat. Acad. Sci. USA 68(8), 1686-1688 (1971) · Zbl 0229.35061 · doi:10.1073/pnas.68.8.1686
[21] Grad, H.: On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2(4), 331-407 (1949) · Zbl 0037.13104 · doi:10.1002/cpa.3160020403
[22] Guisset, S., Brull, S., d’Humières, E., Dubroca, B., Karpov, S., Potapenko, I.: Asymptotic-preserving scheme for the M1-Maxwell system in the quasi-neutral regime. Commun. Comput. Phys. 19(2), 301-328 (2016) · Zbl 1373.76169
[23] Guisset, S., Moreau, J.G., Nuter, R., Brull, S., d’Humieres, E., Dubroca, B., Tikhonchuk, V.T.: Limits of the \[{M_1}\] M1 and \[{M_2}\] M2 angular moments models for kinetic plasma physics studies. J. Phys. A Math. Theor. 48(33), 335501 (2015) · Zbl 1329.82124 · doi:10.1088/1751-8113/48/33/335501
[24] Harten, A., Lax, P., Van Leer, B.: On upstream differencing and Gudonov-type schemes for hyperbolic conservation laws. SIAM Rev. 25(1), 35-61 (1983) · Zbl 0565.65051 · doi:10.1137/1025002
[25] Hauck, C., McClarren, R.: Positive \[P_N\] PN closures. SIAM J. Sci. Comput. 32(5), 2603-2626 (2010) · Zbl 1385.70034 · doi:10.1137/090764918
[26] Hauck, C.D.: High-order entropy-based closures for linear transport in slab geometry. Commun. Math. Sci 9(1), 187-205 (2011) · Zbl 1284.82050 · doi:10.4310/CMS.2011.v9.n1.a9
[27] Hauck, C.D., Levermore, C.D., Tits, A.L.: Convex duality and entropy-based moment closures: characterizing degenerate densities. SIAM J. Control Optim. 47, 1977-2015 (2007) · Zbl 1167.49033 · doi:10.1137/070691139
[28] Hensel, H., Iza-Teran, R., Siedow, N.: Deterministic model for dose calculation in photon radiotherapy. Phys. Med. Biol. 51, 675-693 (2006) · doi:10.1088/0031-9155/51/3/013
[29] Junk, M.: Maximum entropy for reduced moment problems. Math. Mod. Methods Appl. Sci. 10(1001-1028), 2000 (1998)
[30] Kershaw, D.: Flux Limiting Nature’s Own Way. Technical Report, Lawrence Livermore Laboratory (1976) · Zbl 1388.65043
[31] Larsen, E.W., Miften, M.M., Fraass, B.A., Bruinvis, I.A.: Electron dose calculations using the method of moments. Med. Phys. 24(1), 111-125 (1997) · doi:10.1118/1.597920
[32] La Rosa, S., Mascali, G., Romano, V.: Exact maximum entropy closure of the hydrodynamical model for Si semiconductors: The 8-moment case. SIAM J. Appl. Math. 70(3), 710-734 (2009) · Zbl 1197.82118 · doi:10.1137/080714282
[33] Levermore, C.D.: Relating Eddington factors to flux limiters. J. Quant. Spectrosc. Radiat. Transf. 31, 149-160 (1984) · doi:10.1016/0022-4073(84)90112-2
[34] Levermore, C.D.: Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83(5-6), 1021-1065 (1996) · Zbl 1081.82619 · doi:10.1007/BF02179552
[35] Lewis, E.E., Miller, W.F.: Computational Methods of Neutron Transport. American Nuclear Society (1993) · Zbl 0594.65096
[36] Mallet, J., Brull, S., Dubroca, B.: An entropic scheme for an angular moment model for the classical Fokker-Planck-Landau equation of electrons. Commun. Comput. Phys. 15(2), 422-450 (2014) · Zbl 1388.65043 · doi:10.4208/cicp.050612.030513a
[37] Mallet, J., Brull, S., Dubroca, B.: General moment system for plasma physics based on minimum entropy principle. Kin. Rel. Mod. 8(3), 533-558 (2015) · Zbl 1328.35244 · doi:10.3934/krm.2015.8.533
[38] Maple™. Technical report, Maplesoft, a division of Waterloo Maple Inc. (2016)
[39] Mayles, P., Nahum, A., Rosenwald, J.C. (eds.): Handbook of Radiotherapy Physics: Theory and Practice. Taylor & Francis, London (2007)
[40] McClarren, R.G., Hauck, C.D.: Robust and accurate filtered spherical harmonics expansion for radiative transfer. J. Comput. Phys. 229, 5597-5614 (2010) · Zbl 1193.82043 · doi:10.1016/j.jcp.2010.03.043
[41] McDonald, J., Torrilhon, M.: Affordable robust moment closures for cfd based on the maximum-entropy hierarchy. J. Comput. Phys. 251, 500-523 (2013) · Zbl 1349.82057 · doi:10.1016/j.jcp.2013.05.046
[42] Mead, L.R., Papanicolaou, N.: Maximum entropy in the problem of moments. J. Math. Phys. 25(8), 2404-2417 (1984) · Zbl 1308.35157 · doi:10.1063/1.526446
[43] Moré, J.J., Garbow, B.S., Hillstrom, K.E.: User Guide for MINPACK-1 (1980). http://www.netlib.org/minpack/
[44] Olbrant, E., Frank, M.: Generalized Fokker-Planck theory for electron and photon transport in biological tissues: application to radiotherapy. Comput. Math. Methods Med. 11(4), 313-339 (2010) · Zbl 1202.92045 · doi:10.1080/1748670X.2010.491828
[45] Pichard, T., Alldredge, G.W., Brull, S., Dubroca, B., Frank, M.: The \[{M_2}\] M2 model for dose simulation in radiation therapy. J. Comput. Theor. Transport. 45(3), 174-183 (2016) · Zbl 07503197
[46] Pichard, T., Aregba-Driollet, D., Brull, S., Dubroca, B., Frank, M.: Relaxation schemes for the \[{M_1}\] M1 model with space-dependent flux: application to radiotherapy dose calculation. Commun. Comput. Phys. (2014, to appear) · Zbl 1388.65089
[47] Piessens, R., De Doncker-Kapenga, E., Überhuber, C.W.: QUADPACK: A Subroutine Package for Automatic Integration. Springer edition (1983). http://www.netlib.org/quadpack/ · Zbl 0508.65005
[48] Pomraning, G.C.: The Fokker-Planck operator as an asymptotic limit. Math. Mod. Methods Appl. Sci. 2(1), 21-36 (1991) · Zbl 0796.45013 · doi:10.1142/S021820259200003X
[49] Schneider, F.: Kershaw closures for linear transport equations in slab geometry I: model derivation. J. Comput. Phys. 322(C), 905-919 (2016) · Zbl 1351.82086
[50] Schneider, J.: Entropic approximation in kinetic theory. ESAIM Math. Model. Numer. 38(3), 541-561 (2004) · Zbl 1084.82010 · doi:10.1051/m2an:2004025
[51] Sempau, J., Salvat, F., Fernández-Varea, J. M.: PENELOPE-2011: A Code System for Monte Carlo Simulation of Electron and Photon Transport (2011) · Zbl 1084.82010
[52] Spezi, E., Lewis, G.: An overview of Monte Carlo treatment planning for radiotherapy. Radiat. Prot. Dos. 131(1), 123-129 (2008) · doi:10.1093/rpd/ncn277
[53] Vikas, V., Hauck, C.D., Wang, Z.J., Fox, R.O.: Radiation transport modeling using extended quadrature method of moments. J. Comput. Phys. 246, 221-241 (2013) · Zbl 1349.78080 · doi:10.1016/j.jcp.2013.03.028
[54] Wareing, T.A., McGhee, J.M., Archambault, Y., Thompson, S.: Acuros XB advanced dose calculation for the Eclipse™ treatment planning system. Clinical Perspectives (2010)
[55] Zankowski, C., Laitinen, M., Neuenschwander, H.: Fast electron Monte Carlo for eclipse™. Technical Report, Varian Medical System · Zbl 0778.90050
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