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Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

A class of optimization problems in radiotherapy dosimetry planning

Pages: 1651 - 1672, Volume 17, Issue 6, September 2012      doi:10.3934/dcdsb.2012.17.1651

 
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Juan Carlos López Alfonso - Instituto de Matemática Interdisciplinar (IMI) and Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Plaza de las Ciencias 3, Madrid 28040, Spain (email)
Giuseppe Buttazzo - Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy (email)
Bosco García-Archilla - Departamento de Matemática Aplicada, Escuela Superior de Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos, Sevilla 41092, Spain (email)
Miguel A. Herrero - Instituto de Matemática Interdisciplinar (IMI) and Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Plaza de las Ciencias 3, Madrid 28040, Spain (email)
Luis Núñez - Servicio de Radiofsica y Protección Radiológica, Hospital Universitario Puerta de Hierro Majadahonda, Manuel de Falla 1, 28222 Majadahonda, Madrid, Spain (email)

Abstract: Radiotherapy is an important clinical tool to fight malignancies. To do so, a key point consists in selecting a suitable radiation dose that could achieve tumour control without inducing significant damage to surrounding healthy tissues. In spite of recent significant advances, any radiotherapy planning in use relies principally on experience-based decisions made by clinicians among several possible choices.
    In this work we consider a mathematical problem related to that decision-making process. More precisely, we assume that a well-defined target region, called planning target volume (PTV), is given. We then consider the question of determining which radiation distribution is able to achieve a maximum impact on tumour cells and a minimum one in healthy ones. Such dose distribution is defined as the solution of a multi-parameter minimization problem over the PTV and healthy tissues, subject to a number of constraints arising from clinical and technical requirements. For any choice of parameters, sufficient conditions for the existence of a unique solution of that problem are derived. Such solution is then approximated by means of a suitable numerical algorithm. Finally, some examples are considered, on which the dependence on model parameters of different clinical efficiency indexes is discussed.

Keywords:  Radiotherapy dosimetry planning, linear quadratic model, optimization, numerical simulation, variational problems.
Mathematics Subject Classification:  Primary: 65K10, 92C50; Secondary: 81T80.

Received: September 2011;      Revised: January 2012;      Available Online: May 2012.

 References