Multiple Objective Linear Programming in Radiotherapy Treatment Planning

by Shao, Lizhen

Abstract (Summary)
The aim of intensity modulated radiation therapy (IMRT) is to kill tumor cells while at the same time protecting the surrounding tissue and organs from the damaging effect of radiation. To achieve these goals computerized inverse planning systems are used. Given the number of beams and beam directions, beam intensity profiles that yield the best dose distribution under consideration of clinical and physical constraints are calculated. This is called beam intensity optimization problem. In this thesis, we first review existing mathematical models and computation methods for the beam intensity optimization problem. Next, we formulate the beam intensity optimization problem as a multiobjective linear programme (MOLP) with three objectives. For clinical cases this optimization problem involves thousands of variables and tens of thousands of constraints and existing methods such as multiobjective simplex methods can not handle it. The rest of the thesis is dedicated to developing methods to solve this large MOLP efficiently and to the application in the beam intensity optimization problem. Benson (1998c) argues that solving an MOLP in objective space needs less computation time than solving it in decision space if the number of objectives of the MOLP is much smaller than the number of variables. Moreover, the constraint matrix of the problem relies on the calculation of dose deposited in tissue. Since this calculation is always imprecise solving the MOLP exactly is not necessary in practice. This motivates us to develop algorithms for solving an MOLP in objective space approximately. We summarize Benson’s outer approximation algorithm for solving MOLPs in objective space and propose some small changes to improve computational performance. Then in order to approximate the true nondominated set we propose a modification of Benson’s algorithm which is called an approximation version of Benson’s algorithm. Our approximation algorithm computes an inner and an outer approximation of the nondominated set. We prove that the inner approximation provides a set of ?-nondominated points. The geometric duality theory of Heyde and L¨ohne (2006) defines a dual to an MOLP and it assures us to be able to find the nondominated set of the primal MOLP by solving its dual MOLP. Based on this we develop a dual variant of Benson’s outer approximation algorithm to solve the dual MOLP in objective space. We prove that solving the dual provides a weight set decomposition. We compare the primal algorithm and the dual algorithm on small illustrative and on radiotherapy examples. Furthermore, we propose an algorithm to solve the dual MOLP approximately but within specified tolerance. This approximate solution set can be used to calculate an approximation of the nondominated set of the primal MOLP.We show that this set is an ?-nondominated set of the original primal MOLP and provide numerical evidence that this approach can be faster than solving the primal MOLP approximately. Considering that the set of nondominated points is infinite, it is not very useful from the planners’ point of view. We address the problem of finding well distributed nondominated points for an MOLP.We propose a method which combines the global shooting and normal boundary intersection methods. By doing so, we overcome the limitation of normal boundary intersection method that parts of the nondominated set may be missed. Discrepancy analysis of the nondominated points from a geometry point of view shows that this method produces evenly distributed nondominated points. Moreover, the coverage error and the uniformity level can be measured. Finally, we apply the algorithms developed to the beam intensity optimization problem of 3D clinical cases with voxel size of 5mm and 3mm. A technique of reducing the resolution in normal tissue has been used to reduce the computation time. The results clearly illustrate the advantages of our methods.
Bibliographical Information:

Advisor:Assoc. Prof. Matthias Ehrgott; Prof. David Ryan

School:The University of Auckland / Te Whare Wananga o Tamaki Makaurau

School Location:New Zealand

Source Type:Master's Thesis

Keywords:fields of research 290000 engineering and technology


Date of Publication:01/01/2008

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