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ON OPTIMAL TREATMENT PLANNING IN RADIOTHERAPY GOVERNED BY TRANSPORT EQUATIONS

MICHAEL HERTY

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

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and

ALBERT N. SANDJO

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

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https://doi.org/10.1142/S0218202511005076Cited by:5 (Source: Crossref)

Abstract

This paper is devoted to the study of optimal control problems arising in radiotherapy planning problems. The distribution of the radiative intensity in the patient's body is described by a Boltzmann-integro differential equation with position, angular and energy-dependent scattering and absorption coefficients and an energy loss term. The presented discussion is the last in the series of Refs. 13 and 14 discussing radiotherapy problems using the Boltzmann transport equation. We show the existence, uniqueness and regularity of an optimal control using evolution group theory. We extend existing results in order to treat the important case of energy-dependent scattering coefficients.

Keywords:

AMSC: 49J20, 35R09

References

N. Bellomo, N. K. Li and P. K. Maini, Math. Models Methods Appl. Sci. 18, 593 (2008). Link, Web of Science, Google Scholar
N. Bellomo and P. Maini, Math. Models Methods Appl. Sci. 15, iii (2005). Web of Science, Google Scholar
N. Bellomo and P. K. Maini, Math. Models Methods Appl. Sci. 16, iii (2006). Web of Science, Google Scholar
N. Bellomo and P. K. Maini, Math. Models Methods Appl. Sci. 17, 1641 (2007). Link, Web of Science, Google Scholar
C. Börgers, Phys. Med. Biol. 43, 517 (1998). Crossref, Web of Science, Google Scholar
C. Börgers , The Radiation Therapy Planning Problem , IMA Volumes in Mathematics and Its Applications 110 ( Springer , 1999 ) . Crossref, Google Scholar
T. Bortfeld, Dosiskonformation in der tumortherapie mit externer ionisierender strahlung: Physikalische möglichkeiten und grenzen, Habilitationsschrift der Ruprechts-Karls-Universität Heidelberg . Google Scholar
K. K. Bucci, A. Bevan and M. Roach III, CA Cancer J. Clin. 55, 117 (2005), DOI: 10.3322/canjclin.55.2.117. Crossref, Web of Science, Google Scholar
R. F. Curtain and A. J. Pritchard , Infinite-Dimensional Linear Systems Theory , Lecture Notes in Control and Information Sciences 8 ( Springer , 1978 ) . Crossref, Google Scholar
R. F. Curtain and H. Zwart , An Introduction to Infinite-Dimensional Linear Systems Theory , Texts in Applied Mathematics 21 ( Springer , 1995 ) . Crossref, Google Scholar
R. Dautray and J. L. Lions , Mathematical Analysis and Numerical Methods for Science and Technology ( Springer , 1993 ) . Crossref, Google Scholar
M. Frank, H. Hensel and A. Klar, SIAM J. Appl. Math. 67, 582 (2007), DOI: 10.1137/06065547X. Crossref, Web of Science, Google Scholar
M. Frank, M. Herty and A. N. Sandjo, Math. Models Methods Appl. Sci. 20, 661 (2010). Link, Web of Science, Google Scholar
M. Frank, M. Herty and M. Schäfer, Math. Models Methods Appl. Sci. 18, 573 (2008). Link, Web of Science, Google Scholar
N. A. Gentile, J. Comput. Phys. 172, 543 (2001), DOI: 10.1006/jcph.2001.6836. Crossref, Web of Science, Google Scholar
K. A. Giffordet al., Phys. Med. Biol. 51, 2253 (2006), DOI: 10.1088/0031-9155/51/9/010. Crossref, Web of Science, Google Scholar
A. Gustafsson, B. K. Lind and A. Brahme, Med. Phys. 21, 343 (1994), DOI: 10.1118/1.597302. Crossref, Web of Science, Google Scholar
H. Hensel, R. Iza-Teran and N. Siedow, Phys. Med. Biol. 51, 675 (2006), DOI: 10.1088/0031-9155/51/3/013. Crossref, Web of Science, Google Scholar
M. Herty, R. Pinnau and M. Seaïd, Optim. Methods Softw. 22, 917 (2007), DOI: 10.1080/10556780701405783. Crossref, Web of Science, Google Scholar
M. Herty, R. Pinnau and G. Thömmes, ZAMM Z. Angew. Math. Mech. 87, 333 (2007), DOI: 10.1002/zamm.200610316. Crossref, Web of Science, Google Scholar
T. Krieger and O. Sauer, Phys. Med. Biol. 50, 859 (2005), DOI: 10.1088/0031-9155/50/5/010. Crossref, Web of Science, Google Scholar
E. W. Larsen, Transp. Theory Stat. Phys. 26, 739 (1997), DOI: 10.1080/00411459708224421. Crossref, Google Scholar
E. W. Larsenet al., Med. Phys. 24, 111 (1997), DOI: 10.1118/1.597920. Crossref, Web of Science, Google Scholar
B. Mercier, SIAM J. Math. Anal. 18, 393 (1987), DOI: 10.1137/0518030. Crossref, Web of Science, Google Scholar
G. Nickel, Math. Nachr. 212, 101 (2000). Crossref, Web of Science, Google Scholar
A. Pazy , Semigroups of Linear Operators and Applications to Partial Differential Equations , Applied Mathematical Sciences 44 ( Springer , 1983 ) . Crossref, Google Scholar
D. M. Shepardet al., SIAM Rev. 41, 721 (1999), DOI: 10.1137/S0036144598342032. Crossref, Web of Science, Google Scholar
J. Tervo, Math. Methods Appl. Sci. 32, 899 (2009), DOI: 10.1002/mma.1072. Crossref, Web of Science, Google Scholar
J. Tervo and P. Kolmonen, Math. Models Methods Appl. Sci. 12, 109 (2002). Link, Web of Science, Google Scholar
J. Tervoet al., Inv. Probl. 15, 1345 (1999), DOI: 10.1088/0266-5611/15/5/316. Crossref, Web of Science, Google Scholar
J. Tervo, M. Vauhkonen and E. Boman, Lin. Alg. Appl. 428, 1230 (2008). Crossref, Web of Science, Google Scholar