On optimal treatment planning in radiotherapy governed by transport equations. (English) Zbl 1227.49006
Summary: 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 [M. Frank, M. Herty and A. N. Sandjo, Math. Models Methods Appl. Sci. 20, No. 4, 661–678 (2010; Zbl 1255.49007); M. Frank, M. Herty and M. Schäfer, Math. Models Methods Appl. Sci. 18, No. 4, 573–592 (2008; Zbl 1143.76058)] 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.
MSC:
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
| 35R09 | Integro-partial differential equations |
| 35Q20 | Boltzmann equations |
| 49N60 | Regularity of solutions in optimal control |
| 92C50 | Medical applications (general) |
References:
| [1] | Bellomo, N.Li, N. K.Maini, P. K., Math. Models Methods Appl. Sci.18, 593 (2008). · Zbl 1151.92014 |
| [2] | Bellomo, N.Maini, P., Math. Models Methods Appl. Sci.15, iii (2005). |
| [3] | Bellomo, N.Maini, P. K., Math. Models Methods Appl. Sci.16, iii (2006). |
| [4] | Bellomo, N.Maini, P. K., Math. Models Methods Appl. Sci.17, 1641 (2007). · Zbl 1210.00049 |
| [5] | Börgers, C., Phys. Med. Biol.43, 517 (1998). |
| [6] | Börgers, C., The Radiation Therapy Planning Problem, 110, 1999, Springer · Zbl 0972.92015 |
| [7] | T. Bortfeld, Dosiskonformation in der tumortherapie mit externer ionisierender strahlung: Physikalische möglichkeiten und grenzen, Habilitationsschrift der Ruprechts-Karls-Universität Heidelberg. |
| [8] | Bucci, K. K.Bevan, A.Roach, M., CA Cancer J. Clin.55, 117 (2005), DOI: 10.3322/canjclin.55.2.117. |
| [9] | Curtain, R. F.; Pritchard, A. J., Infinite-Dimensional Linear Systems Theory, 8, 1978, Springer · Zbl 0389.93001 |
| [10] | Curtain, R. F.; Zwart, H., An Introduction to Infinite-Dimensional Linear Systems Theory, 21, 1995, Springer · Zbl 0839.93001 |
| [11] | Dautray, R.; Lions, J. L., Mathematical Analysis and Numerical Methods for Science and Technology, 1993, Springer |
| [12] | Frank, M.Hensel, H.Klar, A., SIAM J. Appl. Math.67, 582 (2007), DOI: 10.1137/06065547X. |
| [13] | Frank, M.Herty, M.Sandjo, A. N., Math. Models Methods Appl. Sci.20, 661 (2010). · Zbl 1255.49007 |
| [14] | Frank, M.Herty, M.Schäfer, M., Math. Models Methods Appl. Sci.18, 573 (2008). · Zbl 1143.76058 |
| [15] | Gentile, N. A., J. Comput. Phys.172, 543 (2001), DOI: 10.1006/jcph.2001.6836. |
| [16] | Gifford, K. A.et al., Phys. Med. Biol.51, 2253 (2006), DOI: 10.1088/0031-9155/51/9/010. |
| [17] | Gustafsson, A.Lind, B. K.Brahme, A., Med. Phys.21, 343 (1994), DOI: 10.1118/1.597302. |
| [18] | Hensel, H.Iza-Teran, R.Siedow, N., Phys. Med. Biol.51, 675 (2006), DOI: 10.1088/0031-9155/51/3/013. |
| [19] | Herty, M.Pinnau, R.Seaïd, M., Optim. Methods Softw.22, 917 (2007), DOI: 10.1080/10556780701405783. |
| [20] | Herty, M.Pinnau, R.Thömmes, G., ZAMM Z. Angew. Math. Mech.87, 333 (2007), DOI: 10.1002/zamm.200610316. |
| [21] | Krieger, T.Sauer, O., Phys. Med. Biol.50, 859 (2005), DOI: 10.1088/0031-9155/50/5/010. |
| [22] | Larsen, E. W., Transp. Theory Stat. Phys.26, 739 (1997), DOI: 10.1080/00411459708224421. |
| [23] | Larsen, E. W.et al., Med. Phys.24, 111 (1997), DOI: 10.1118/1.597920. |
| [24] | Mercier, B., SIAM J. Math. Anal.18, 393 (1987), DOI: 10.1137/0518030. |
| [25] | Nickel, G., Math. Nachr.212, 101 (2000). · Zbl 1016.47031 |
| [26] | Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, 44, 1983, Springer · Zbl 0516.47023 |
| [27] | Shepard, D. M.et al., SIAM Rev.41, 721 (1999), DOI: 10.1137/S0036144598342032. |
| [28] | Tervo, J., Math. Methods Appl. Sci.32, 899 (2009), DOI: 10.1002/mma.1072. |
| [29] | Tervo, J.Kolmonen, P., Math. Models Methods Appl. Sci.12, 109 (2002). · Zbl 1016.92017 |
| [30] | Tervo, J.et al., Inv. Probl.15, 1345 (1999), DOI: 10.1088/0266-5611/15/5/316. |
| [31] | Tervo, J.Vauhkonen, M.Boman, E., Lin. Alg. Appl.428, 1230 (2008). · Zbl 1195.92042 |
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