The ill- conditionning in stereotaxic irradiation: Optimization of isodoses arrays distribution using the singular values decomposition

Abstract

Inverse problems are encountered in physical problems over a large field of applications. In medicine many applications treat the inverse problem. For example the problems of X-rays or gamma-rays Computed Tomography, radiography, positron imaging are of this kind. The inverse problem was encountered in classical radiotherapy, in a partial form, searching the ‘best possible’ radiation dose distribution for a given treatment situation. The last years the stereotaxic radiosurgery technique was used for the treatment of small brain malformations. The great precision of this technique and the high level of the dose delivered during unique irradiation, necessitate the use of one optimisation treatment method. We propose an optimisation method which has the following caracteristics. In opposition to the classical radiotherapy methods, the dose distribution formation procedure is considered in its globality, taking into account all the pixels of the dose matrix. We realize a deep analysis of the radiosurgery problem, examining the conditionning of the problem. This analysis is based on the singular values decomposition. The goal of the method is to find physically acceptable solutions of the weights vector for a given irradiation configuration, for obtain a predifined dose distribution.