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  • Optimization of Beam

    Orientations in Intensity

    Modulated Radiation Therapy

    Planning

    Vom Fachbereich Mathematikder Technischen Universitat Kaiserslauternzur Erlangung des akademischen Grades

    Doktor der Naturwissenschaften(Doctor rerum naturalium, Dr. rer. nat.)

    genehmigte

    Dissertationvon

    Ahmad Saher Azizi Sultan

    Erstgutachter:

    PD Dr. habil. Karl-Heinz Kufer

    Zweitgutachter:

    Prof. Dr. Uwe Oelfke

    Datum der Disputation: 23. Oktober 2006

    D 386

  • Dedicated to the memory of my first teachers;

    Prof. Mohammed Ibn Abd-Allah

    and

    my parents

  • Acknowledgments

    This work was supported by the department of optimization at the Fraun-hofer Institute for Industrial Mathematics (ITWM).

    I am deeply indebted to my supervisor PD Dr. habil. Karl-Heinz Kufer forintroducing me to this challenging mathematical topic, his extensive sup-port and guidance. Special thanks are addressed to him and my colleaguesDr. Alexander Lavrov, Dr. Thomas Hanne, Dr. Alexander Scherrer, FernandoAlonso , Michael Monz and Philipp Suess for their supports and fruitful dis-cussions through a long lasting teamwork.

    I also would like to thank Prof. Dr. Thomas Bortfeld from the MassachusettsGeneral Hospital (MGH) and Harvard Medical School in Boston for his criti-cal remarks, and Prof. Dr. Sven Krumke from the department of optimizationat the university of Kaiserslautern for helpful discussions.

    Further thanks go to our scientific partners from the German Cancer Re-search Center (DKFZ) in Heidelberg, in particular Dr. Dr. Christian Thiekefrom the Clinical Cooperation Unit Radiation Oncology and Prof. Dr. UweOelfke from the Department of Medical Physics in Radiation Therapy for asuccessful cooperation.

    Finally, I thank my family for their love and encouragement during my stud-ies.

    Kaiserslautern, June 2006

    Ahmad Saher Azizi Sultan

  • 2

    Abstract

    For the last decade, optimization of beam orientations in intensity-modulatedradiation therapy (IMRT) has been shown to be successful in improving thetreatment plan. Unfortunately, the quality of a set of beam orientationsdepends heavily on its corresponding beam intensity profiles. Usually, astochastic selector is used for optimizing beam orientation, and then a sin-gle objective inverse treatment planning algorithm is used for the optimiza-tion of beam intensity profiles. The overall time needed to solve the inverseplanning for every random selection of beam orientations becomes excessive.Recently, considerable improvement has been made in optimizing beam in-tensity profiles by using multiple objective inverse treatment planning. Suchan approach results in a variety of beam intensity profiles for every selectionof beam orientations, making the dependence between beam orientations andits intensity profiles less important. This thesis takes advantage of this prop-erty to accelerate the optimization process through an approximation of theintensity profiles that are used for multiple selections of beam orientations,saving a considerable amount of calculation time. A dynamic algorithm (DA)and evolutionary algorithm (EA), for beam orientations in IMRT planningwill be presented. The DA mimics, automatically, the methods of beamseye view and observers view which are recognized in conventional conformalradiation therapy. The EA is based on a dose-volume histogram evaluationfunction introduced as an attempt to minimize the deviation between themathematical and clinical optima. To illustrate the efficiency of the algo-rithms they have been applied to different clinical examples. In comparisonto the standard equally spaced beams plans, improvements are reported forboth algorithms in all the clinical examples even when, for some cases, fewerbeams are used. A smaller number of beams is always desirable without com-promising the quality of the treatment plan. It results in a shorter treatmentdelivery time, which reduces potential errors in terms of patient movementsand decreases discomfort.

  • Contents

    1 Introduction 5

    2 The multicriteria environment in IMRT 102.1 From IMRT to multicriteria optimization . . . . . . . . . . . . 10

    2.1.1 Treatment plan evaluations . . . . . . . . . . . . . . . 112.1.2 Pareto solutions and the planning domain . . . . . . . 132.1.3 Solution strategies . . . . . . . . . . . . . . . . . . . . 15

    2.2 Database navigation . . . . . . . . . . . . . . . . . . . . . . . 20

    3 Beam orientations in focus 223.1 Complexity of beam orientations . . . . . . . . . . . . . . . . 22

    3.1.1 Beam orientation is not convex . . . . . . . . . . . . . 223.2 Hardness of beam orientations . . . . . . . . . . . . . . . . . 25

    3.2.1 An introduction to complexity theory . . . . . . . . . 253.2.2 Example construction and reduction . . . . . . . . . . 29

    3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

    4 A dynamic algorithm 364.1 A mathematical model . . . . . . . . . . . . . . . . . . . . . . 36

    4.1.1 Automated beams eye view . . . . . . . . . . . . . . . 374.1.2 Automated observers view . . . . . . . . . . . . . . . . 384.1.3 Insight . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

    4.2 A dynamic algorithm . . . . . . . . . . . . . . . . . . . . . . . 434.2.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 444.2.2 Iterative analysis . . . . . . . . . . . . . . . . . . . . . 46

    4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3.1 An artificial coplanar case study . . . . . . . . . . . . . 504.3.2 Prostate carcinoma . . . . . . . . . . . . . . . . . . . . 524.3.3 Head and neck tumor . . . . . . . . . . . . . . . . . . . 544.3.4 Paraspinal case . . . . . . . . . . . . . . . . . . . . . . 55

    3

  • CONTENTS 4

    5 A DVH-Evaluation scheme and an EA 585.1 Introduction and motivation . . . . . . . . . . . . . . . . . . . 585.2 A Mathematical model . . . . . . . . . . . . . . . . . . . . . 59

    5.2.1 The DVH concept as a function . . . . . . . . . . . . . 595.2.2 DVH evaluation scheme . . . . . . . . . . . . . . . . . 605.2.3 Treatment plan evaluation . . . . . . . . . . . . . . . . 615.2.4 Intensity maps approximation . . . . . . . . . . . . . . 615.2.5 Optimization of beam orientations . . . . . . . . . . . 64

    5.3 An evolutionary Algorithm . . . . . . . . . . . . . . . . . . . . 655.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 655.3.2 Implementation of evolutionary algorithm . . . . . . . 66

    5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.4.1 Parameters values . . . . . . . . . . . . . . . . . . . . . 705.4.2 Numerical results of the EA . . . . . . . . . . . . . . . 715.4.3 Beam orientations results . . . . . . . . . . . . . . . . 74

    6 Comparison between the DA and EA 796.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

    6.1.1 Prostate carcinoma . . . . . . . . . . . . . . . . . . . . 796.1.2 Paraspinal case . . . . . . . . . . . . . . . . . . . . . . 816.1.3 Head and neck tumor . . . . . . . . . . . . . . . . . . . 83

    6.2 Discusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

    7 Summary 86

    8 Further research 888.1 Improving the DA . . . . . . . . . . . . . . . . . . . . . . . . . 888.2 A two phases algorithm . . . . . . . . . . . . . . . . . . . . . . 888.3 Biological parameters . . . . . . . . . . . . . . . . . . . . . . . 89

    A Complementary definitions 90A.1 Multicriteria combinatorial optimization problems . . . . . . . 90A.2 Binary relation and some properties . . . . . . . . . . . . . . . 90

    B Abbreviations 91

  • Chapter 1

    Introduction

    Almost immediately after the discovery of X-rays, the biological effects ofemitting radiation to human cells were recognized, therefore, being addressedand studied. The result was fortunate. It was found that, although high dosesof radiation can kill both cancerous and normal cells, the cancerous cells aremore sensitive to radiation than the normal cells. This is due to the fact thatthe repair mechanism of cancerous cells is less efficient than that of normalones. Thus normal cells are more likely to fully recover from the effects ofradiation [NCI97]. This natural fact, together with a good radiation treat-ment plan, has proved radiation therapy to be a successful tool in cancertreatments. Consequently, radiation therapy becomes one of the most-usedtreatment tools during the fight against cancer. Approximately, fifty percentof the people who are diagnosed with life threatening forms of cancer aretreated with radiation, either exclusively or in combination with surgery andchemotherapy.

    The majority of radiation therapy treatments are performed using highenergetic photon beams generated by a linear accelerator and delivered tothe patient from different directions (see figure 1.1). The aperture of eachindividual beam is created so that it adheres to the shape of the tumor. Thiscan be done by using an automatic device called a multi-leave collimator(MLC) in which a number of parallel leaves are projected into the primarybeam to create the required tumorodical shape [VRA]. For depiction seefigure (1.2).

    In conventional conformal radiation therapy, the adjustment of the MLCaperture can not be altered while the linear accelerator is turned on. Thusthe resulting radiation will have uniform intensity throughout the beam aper-ture. In many cases, especially for irregularly shaped non-convex targets, this

    5

  • CHAPTER 1. INTRODUCTION 6

    Figure 1.1: The gantry can move around the couch on which the patient lies and thecouch itself may be moved to alter the directions

    limits the possibilities of achieving dose distribution with a good

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