Fachbereich Mathematik - GSI

Abstract
In the GSI therapy pilot project from 1997 until 2008 about 450 cancer patients
were successfully treated with carbon ions. Because of the promising healing rates
the clinical radiotherapy facility HIT was opened in 2009. However, research for the
heavy-ion therapy is still in progress at GSI.
For the treatment planning the software TRiP is used. An essential part of the
treatment planning ist the dose optimization. The aim of the dose optimization is
to achieve a homogeneous target dose distribution as close as possible to the prescribed
dose distribution by an appropriate sparing of healthy tissue and critical
structures like the brainstem. These requirements can be mathematically expressed
by an optimization problem, where the free optimization parameters are the particle
numbers for the rasterspots. If biological effects are taken into account, the optimization
problem leads to a nonlinear, finite dimensional and restricted minimization
problem. A theoretical examination and a solution of the optimization problem is
the core area of this master-thesis.
By utilizing a special variant of the Weierstraß extreme value theorem, which
requires the lower semi-continuity and radial unboundness of the objective function,
the existence of a global minimum of the optimization problem can be proofed. Due
to the nonconvexity of the objective funtion the uniqueness of a global minimum
can not be proofed straightforward because standard techniques for such proofs can
not be applied. A numerical approach, which indicates, that the minimum of the
objective function is unique, is discussed. Furthermore, the objective function is
approximated with an appropriate smooth function. This is necessary to derive the
Karush-Kuhn-Tucker-conditions for the optimization problem.
Due to the nonlinearity of the objective function the optimization problem can
only be solved with numerical methods. For this purpose the Linesearch-Techniques
method of steepest descent, conjugated gradients and the inverse BFGS-method were
implemented and examined. Several variants of these numerical solvers are presented
in this work. The convergence results show, that the Fletcher-Reeves variant of
the conjugated gradient method has the best numerical properties. The results of
the inverse BFGS-method are similar concerning the iteration steps but require by
a factor of 1.5 more computation time. The method of steepest descent is slow
concerning the iteration steps and computation time.
The convergence results show, that currently the Fletcher-Reeves variant of the
conjugated gradient method is the best algorithm for the optimization step in the
treatment planning procedure. With this method complete patient plans can be
optimized in an acceptable computation time. Furthermore, this method doesn’t
require much memory space and is robust.
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