*Published Paper*

**Inserted:** 9 oct 2017

**Journal:** Discrete Contin. Dyn. Syst. Ser. B

**Volume:** 17

**Number:** 6

**Pages:** 1651-1672

**Year:** 2012

**Doi:** 10.3934/dcdsb.2012.17.1651

**Links:**
paper at the journal web site

**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:**
Variational problems, optimization, Radiotherapy dosimetry planning, Linear quadratic model, numerical simulation