September  2012, 5(3): 485-503. doi: 10.3934/krm.2012.5.485

Optimization of a model Fokker-Planck equation

1. 

RWTH Aachen University, Templergraben 55, D-52056 Aachen, Germany, Germany, Germany

Received  October 2011 Revised  March 2012 Published  August 2012

We discuss optimal control problems for the Fokker--Planck equation arising in radiotherapy treatment planning. We prove existence and uniqueness of an optimal boundary control for a general tracking--type cost functional in three spatial dimensions. Under additional regularity assumptions we prove existence of a continuous necessary first--order optimality system. In the one--dimensional case we analyse a numerical discretization of the Fokker--Planck equation. We prove that the resulting discrete optimality system is a suitable discretization of the continuous first--order system.
Citation: Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic & Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485
References:
[1]

R. Barnard, M. Frank and M. Herty, Optimal radiotherapy treatment planning using minimum entropy models, preprint, 2011. Google Scholar

[2]

N. Bellomo and P. K. Maini, Preface (Special issue on cancer modelling), Math. Mod. Math. Appl. Sci., 15 (2005), iii-viii.  Google Scholar

[3]

N. Bellomo and P. K. Maini, Preface (Special issue on cancer modelling II), Math. Mod. Math. Appl. Sci., 16 (2006), iii-vii.  Google Scholar

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N. Bellomo and P. K. Maini, Preface (Special issue on cancer modelling), Math. Mod. Math. Appl. Sci., 17 (2007), iii-vii.  Google Scholar

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K. K. Bucci, A. Bevan and M. Roach III, Advances in radiation therapy: Conventional to 3d, to IMRT, to 4d, and beyond}, CA Cancer J. Clin., 55 (2005), 117-134. Google Scholar

[6]

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.  Google Scholar

[7]

C. Börgers, The radiation therapy planning problem, in "Computational Radiology and Imaging" (Minneapolis, MN, 1997), IMA Volumes in Mathematics and its Applications, 110, Springer, New York, (1999), 1-16.  Google Scholar

[8]

T. Brunner, "Forms of Approximate Radiation Transport," Sandia Report, 2002. Google Scholar

[9]

R. G. Dale, The application of the linear-quadratic dose-effect equation to fractionated and protracted radiotherapy, Br. J. Radiol., 58 (1985), 515-528. doi: 10.1259/0007-1285-58-690-515.  Google Scholar

[10]

P. Degond and S. Mas-Gallic, Existence of solutions and diffusion approximation for a model Fokker-Planck equation, Transport Theory and Statistical Physics, 16 (1987), 589-636. doi: 10.1080/00411458708204307.  Google Scholar

[11]

B. Dubroca and J.-L. Feugeas, Étude théorique et numérique d'une hiérarchie de modèles aux moments pout le transfert radiatif, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), 915-920.  Google Scholar

[12]

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

[13]

M. Frank, Approximate Models for Radiative Transfer, Bulletin of the Institute of Mathematics Academia Sinica (New Series), 2 (2007), 409-432.  Google Scholar

[14]

M. Frank, B. Dubroca and A. Klar, Partial moment entropy approximation to radiative heat transfer, Journal of Computational Physics, 218 (2006), 1-18. doi: 10.1016/j.jcp.2006.01.038.  Google Scholar

[15]

M. Frank, M. Herty and A. N. Sandjo, Optimal radiotherapy treatment plannig governed by kinetic equations, Mathematical Models and Methods in Applied Sciences, 20 (2010), 661-678. doi: 10.1142/S0218202510004386.  Google Scholar

[16]

M. Frank, M. Herty and M. Schäfer, Optimal treatment plannig in radiotherapy based on Boltzmann transport calculation, Mathematical Models and Methods in Applied Sciences, 18 (2008), 573-592. doi: 10.1142/S0218202508002784.  Google Scholar

[17]

K. A. Gifford, J. L. Horton Jr., T. A. Wareing, G. Failla and F. Mourtada, Comparioson of a finite-element multigroup discrete-ordinates code with Monte Carlo for radiotherapy calculations, Phys. Med. Biol., 51 (2006), 2253-2265. Google Scholar

[18]

H. Hensel, R. Iza-Teran and Norbert Siedow, Deterministic model for dose calculation in photon radiotherapy, Physics in Medicine and Biology, 51 (2006), 675-693. doi: 10.1088/0031-9155/51/3/013.  Google Scholar

[19]

M. Herty and A. N. Sandjo, On Optimal treatment plannig in radiotherapy governed by transport equations, Mathematical Models and Methods in Applied Sciences, 21 (2011), 345-359. doi: 10.1142/S0218202511005076.  Google Scholar

[20]

M. Herty, R. Pinnau and M. Seaid, Optimal control in radiative transfer, Optimization Methods and Software, 22 (2007), 917-936.  Google Scholar

[21]

E. W. Larsen, M. M. Miften, B. A. Fraass and I. A. D. Bruinvis, Electron dose calculations using the method of moments, Med. Phys., 24 (1997), 111-125. doi: 10.1118/1.597920.  Google Scholar

[22]

E. W. Larsen, Tutorial: The nature of transport calculations used in radiation oncology, Transp. Theory Stat. Phys., 26 (1997). Google Scholar

[23]

J. L. Lions, "Équations Differentielles Operationnelles et Problèmes aux Limites," Die Grundlehren der mathematischen Wissenschaften, Band 111, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.  Google Scholar

[24]

D. Jackson, "Fourier Series and Orthogonal Functions," Carus Mathematical Monograph Series, No. 6, Mathematical Assoc. of America, 1941.  Google Scholar

[25]

K.-H. Küfer, M. Monz, A. Scherrer, P. Süss, F. Alonso, A. S. A. Sultan, T. Bortfeld and C. Thieke, Multicriteria optimizaton in intensity modulated radiotherapy planning, in "Handbook of Optimization in Medicine," Springer Optim. Appl., 26, Springer, New York, (2009), 123-167.  Google Scholar

[26]

J. C. Mark, "The Spherical Harmonics Method. I. (General Development of the Theory)," Document no. CRT-340 (N.R.C. 1588), National Research Council of Canada, Division of Atomic Energy, 1945.  Google Scholar

[27]

J. C. Mark, "The Spherical Harmonics Method. II. (Application to Problems with Plane and Spherical Symmetry," Document no. CRT-338 (N.R.C. 1589), National Research Council of Canada, Division of Atomic Energy, 1945.  Google Scholar

[28]

R. N. Slaybaugh, M. L. Williams, D. Ilas, D. E. Peplow, B. L. Kirk, T. L. Nichols, Y. Y. Azmy and M. P. Langer, Radiation treatment planning using discrete ordinates codes, Transactions of the American Nuclear Society, 96 (2007), 343-345. Google Scholar

[29]

D. M. Shepard, M. C. Ferris, G. H. Olivera and T. R. Mackie, Optimizing the delivery of radiation therapy to cancer patients, SIAM Rev., 41 (1999), 721-744. Google Scholar

[30]

J. Tervo and P. Kolmonen, Inverse radiotherapy treatment planning model applying Boltzmann-transport equation, Math. Models. Methods. Appl. Sci., 12 (2002), 109-141. doi: 10.1142/S021820250200157X.  Google Scholar

[31]

G. G. Steel, J. M. Deacon, G. M. Duchesne, A. Horwich, L. R. Kelland and J. H. Peacock, The dose-rate effect in human tumour cells, Radiotherapy and Oncology, 9 (1987), 299-310. Google Scholar

[32]

G. G. Steel, J. D. Down, J. H. Peacock and T. C. Stephens, Dose-rate effects and the repair of radiation damage, Radiotherapy and Oncology, 5 (1986), 321-331. Google Scholar

[33]

H. Struchtrup, On the number of moments in radiative transfer problems, Annals of Physics, 266 (1998), 1-26. doi: 10.1006/aphy.1998.5791.  Google Scholar

[34]

J. Tervo, On coupled Boltzmann transport equation related to radiation therapy, J. Math. Anal. Appl., 335 (2007), 819-840. doi: 10.1016/j.jmaa.2007.01.092.  Google Scholar

[35]

J. Tervo, M. Vauhkonen and E. Boman, Optimal control model for radiation therapy inverse planning applying the Boltzmann transport equation, Linear Algebra and its Applications, 428 (2008), 1230-1249. doi: 10.1016/j.laa.2007.03.003.  Google Scholar

[36]

J. Tervo, P. Kolmonen, M. Vauhkonen, L. M. Heikkinen and J. P. Kaipio, A finite-element model of electron transport in radiation therapy and related inverse problem, Inv. Probl., 15 (1999), 1345-1361. doi: 10.1088/0266-5611/15/5/316.  Google Scholar

[37]

F. Tröltzsch, "Optimal Control of Partial Differential Equations. Theory, Methods and Applications," Graduate Studies in Mathematics, 112, AMS, Providence, RI, 2010.  Google Scholar

show all references

References:
[1]

R. Barnard, M. Frank and M. Herty, Optimal radiotherapy treatment planning using minimum entropy models, preprint, 2011. Google Scholar

[2]

N. Bellomo and P. K. Maini, Preface (Special issue on cancer modelling), Math. Mod. Math. Appl. Sci., 15 (2005), iii-viii.  Google Scholar

[3]

N. Bellomo and P. K. Maini, Preface (Special issue on cancer modelling II), Math. Mod. Math. Appl. Sci., 16 (2006), iii-vii.  Google Scholar

[4]

N. Bellomo and P. K. Maini, Preface (Special issue on cancer modelling), Math. Mod. Math. Appl. Sci., 17 (2007), iii-vii.  Google Scholar

[5]

K. K. Bucci, A. Bevan and M. Roach III, Advances in radiation therapy: Conventional to 3d, to IMRT, to 4d, and beyond}, CA Cancer J. Clin., 55 (2005), 117-134. Google Scholar

[6]

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.  Google Scholar

[7]

C. Börgers, The radiation therapy planning problem, in "Computational Radiology and Imaging" (Minneapolis, MN, 1997), IMA Volumes in Mathematics and its Applications, 110, Springer, New York, (1999), 1-16.  Google Scholar

[8]

T. Brunner, "Forms of Approximate Radiation Transport," Sandia Report, 2002. Google Scholar

[9]

R. G. Dale, The application of the linear-quadratic dose-effect equation to fractionated and protracted radiotherapy, Br. J. Radiol., 58 (1985), 515-528. doi: 10.1259/0007-1285-58-690-515.  Google Scholar

[10]

P. Degond and S. Mas-Gallic, Existence of solutions and diffusion approximation for a model Fokker-Planck equation, Transport Theory and Statistical Physics, 16 (1987), 589-636. doi: 10.1080/00411458708204307.  Google Scholar

[11]

B. Dubroca and J.-L. Feugeas, Étude théorique et numérique d'une hiérarchie de modèles aux moments pout le transfert radiatif, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), 915-920.  Google Scholar

[12]

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

[13]

M. Frank, Approximate Models for Radiative Transfer, Bulletin of the Institute of Mathematics Academia Sinica (New Series), 2 (2007), 409-432.  Google Scholar

[14]

M. Frank, B. Dubroca and A. Klar, Partial moment entropy approximation to radiative heat transfer, Journal of Computational Physics, 218 (2006), 1-18. doi: 10.1016/j.jcp.2006.01.038.  Google Scholar

[15]

M. Frank, M. Herty and A. N. Sandjo, Optimal radiotherapy treatment plannig governed by kinetic equations, Mathematical Models and Methods in Applied Sciences, 20 (2010), 661-678. doi: 10.1142/S0218202510004386.  Google Scholar

[16]

M. Frank, M. Herty and M. Schäfer, Optimal treatment plannig in radiotherapy based on Boltzmann transport calculation, Mathematical Models and Methods in Applied Sciences, 18 (2008), 573-592. doi: 10.1142/S0218202508002784.  Google Scholar

[17]

K. A. Gifford, J. L. Horton Jr., T. A. Wareing, G. Failla and F. Mourtada, Comparioson of a finite-element multigroup discrete-ordinates code with Monte Carlo for radiotherapy calculations, Phys. Med. Biol., 51 (2006), 2253-2265. Google Scholar

[18]

H. Hensel, R. Iza-Teran and Norbert Siedow, Deterministic model for dose calculation in photon radiotherapy, Physics in Medicine and Biology, 51 (2006), 675-693. doi: 10.1088/0031-9155/51/3/013.  Google Scholar

[19]

M. Herty and A. N. Sandjo, On Optimal treatment plannig in radiotherapy governed by transport equations, Mathematical Models and Methods in Applied Sciences, 21 (2011), 345-359. doi: 10.1142/S0218202511005076.  Google Scholar

[20]

M. Herty, R. Pinnau and M. Seaid, Optimal control in radiative transfer, Optimization Methods and Software, 22 (2007), 917-936.  Google Scholar

[21]

E. W. Larsen, M. M. Miften, B. A. Fraass and I. A. D. Bruinvis, Electron dose calculations using the method of moments, Med. Phys., 24 (1997), 111-125. doi: 10.1118/1.597920.  Google Scholar

[22]

E. W. Larsen, Tutorial: The nature of transport calculations used in radiation oncology, Transp. Theory Stat. Phys., 26 (1997). Google Scholar

[23]

J. L. Lions, "Équations Differentielles Operationnelles et Problèmes aux Limites," Die Grundlehren der mathematischen Wissenschaften, Band 111, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961.  Google Scholar

[24]

D. Jackson, "Fourier Series and Orthogonal Functions," Carus Mathematical Monograph Series, No. 6, Mathematical Assoc. of America, 1941.  Google Scholar

[25]

K.-H. Küfer, M. Monz, A. Scherrer, P. Süss, F. Alonso, A. S. A. Sultan, T. Bortfeld and C. Thieke, Multicriteria optimizaton in intensity modulated radiotherapy planning, in "Handbook of Optimization in Medicine," Springer Optim. Appl., 26, Springer, New York, (2009), 123-167.  Google Scholar

[26]

J. C. Mark, "The Spherical Harmonics Method. I. (General Development of the Theory)," Document no. CRT-340 (N.R.C. 1588), National Research Council of Canada, Division of Atomic Energy, 1945.  Google Scholar

[27]

J. C. Mark, "The Spherical Harmonics Method. II. (Application to Problems with Plane and Spherical Symmetry," Document no. CRT-338 (N.R.C. 1589), National Research Council of Canada, Division of Atomic Energy, 1945.  Google Scholar

[28]

R. N. Slaybaugh, M. L. Williams, D. Ilas, D. E. Peplow, B. L. Kirk, T. L. Nichols, Y. Y. Azmy and M. P. Langer, Radiation treatment planning using discrete ordinates codes, Transactions of the American Nuclear Society, 96 (2007), 343-345. Google Scholar

[29]

D. M. Shepard, M. C. Ferris, G. H. Olivera and T. R. Mackie, Optimizing the delivery of radiation therapy to cancer patients, SIAM Rev., 41 (1999), 721-744. Google Scholar

[30]

J. Tervo and P. Kolmonen, Inverse radiotherapy treatment planning model applying Boltzmann-transport equation, Math. Models. Methods. Appl. Sci., 12 (2002), 109-141. doi: 10.1142/S021820250200157X.  Google Scholar

[31]

G. G. Steel, J. M. Deacon, G. M. Duchesne, A. Horwich, L. R. Kelland and J. H. Peacock, The dose-rate effect in human tumour cells, Radiotherapy and Oncology, 9 (1987), 299-310. Google Scholar

[32]

G. G. Steel, J. D. Down, J. H. Peacock and T. C. Stephens, Dose-rate effects and the repair of radiation damage, Radiotherapy and Oncology, 5 (1986), 321-331. Google Scholar

[33]

H. Struchtrup, On the number of moments in radiative transfer problems, Annals of Physics, 266 (1998), 1-26. doi: 10.1006/aphy.1998.5791.  Google Scholar

[34]

J. Tervo, On coupled Boltzmann transport equation related to radiation therapy, J. Math. Anal. Appl., 335 (2007), 819-840. doi: 10.1016/j.jmaa.2007.01.092.  Google Scholar

[35]

J. Tervo, M. Vauhkonen and E. Boman, Optimal control model for radiation therapy inverse planning applying the Boltzmann transport equation, Linear Algebra and its Applications, 428 (2008), 1230-1249. doi: 10.1016/j.laa.2007.03.003.  Google Scholar

[36]

J. Tervo, P. Kolmonen, M. Vauhkonen, L. M. Heikkinen and J. P. Kaipio, A finite-element model of electron transport in radiation therapy and related inverse problem, Inv. Probl., 15 (1999), 1345-1361. doi: 10.1088/0266-5611/15/5/316.  Google Scholar

[37]

F. Tröltzsch, "Optimal Control of Partial Differential Equations. Theory, Methods and Applications," Graduate Studies in Mathematics, 112, AMS, Providence, RI, 2010.  Google Scholar

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