On the Use of Second Derivatives in Optimization of Radiation Therapy

Fredrik Carlsson
Royal Institute of Technology, Sweden

Anders Forsgren
Royal Institute of Technology, Sweden

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Ingår i: Nordic MPS 2004. The Ninth Meeting of the Nordic Section of the Mathematical Programming Society

Linköping Electronic Conference Proceedings 14:9, s.

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Publicerad: 2004-12-28


ISSN: 1650-3686 (tryckt), 1650-3740 (online)


The goal of external-beam radiation therapy of cancer is to obtain an acceptable balance between tumor control and complications to the normal tissue surrounding the tumor. During the last decade; the field has experienced a rapid progress. New technology has improved the accuracy of the beam delivery significantly. Together with the development of faster computers; this has led the way for so called ’intensity modulated radiation therapy’ (IMRT).

In IMRT; the clinician specifies certain characteristics of the desired dose distribution by introducing objective functions for the tumor and for the critical organs close to the tumor. A discretization of the incident beams and of the treatment volume of the patient is performed and an optimization problem is formulated. In general; the IMRT problem is large-scale and has a non-convex nature; often with linear and non-linear constraints. In this study we investigate how the Hessian affects the optimization performance for a quasi-Newton algorithm used in a commercial treatment planning system. Currently; the initial Hessian fed into the algorithm is diagonal. The influence of including more accurate curvature information; represented as off-diagonal elements; is explored for three patient cases.

A more accurate initial Hessian results in a much faster progress of optimization than when using a diagonal initial Hessian. Furthermore; the optimal beam profiles differ significantly; with an accurate Hessian they are very jagged compared to the smooth profiles obtained with a diagonal Hessian. Jagged profiles are; in general; not desirable since they are harder to deliver; but for a certain class of IMRT problems they are preferable. The results also indicate that the IMRT problem is an ill-posed inverse problem in the sense that very different fluence profiles can produce almost identical dose distributions.


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