dose plan optimization in hdr brachytherapy using...

Linköping Studies in Science and Technology, ThesisNo. 1486

Dose Plan Optimization inHDR Brachytherapy using Penalties

Properties and Extensions

Åsa Holm

Department of MathematicsLinköping University

SE–581 83 Linköping, SwedenLinköping 2011

Linköping Studies in Science and Technology, ThesisNo. 1486

Dose Plan Optimization in HDR Brachytherapy using PenaltiesProperties and Extensions

[email protected]

Division of OptimizationDepartment of Mathematics

Linköping UniversitySE–581 83 Linköping

Sweden

ISBN 978-91-7393-162-5 ISSN 0280-7971

Printed by LiU-Tryck, Linköping, Sweden 2011

Abstract

High dose-rate (HDR) brachytherapy is a specific type of radiotherapy used to treat tu-mours of for example the cervix, prostate, and breasts. In HDR brachytherapy applicatorsare implanted into or close to the tumour volume. A radioactive source is moved throughthese applicators and stops at certain positions, known as dwell points. For each patientan anatomy-based dose plan is created that decides for example where to place the appli-cators, which dwell points to use, and for how long. The aim when creating a dose planis to deliver an as high dose as possible to the tumour while simultaneously keeping thedose to the surrounding healthy organs as low as possible.

In order to improve the quality of dose plans mathematical optimization methods aretoday used in clinical practice. Usually one solves a linear penalty model that minimizesa weighted deviation from dose intervals provided by a physician. In this thesis we studycertain properties and alterations of this model.

One interesting property of the model that we study is the distribution of the basicvariables. We show that due to the distribution of these variables only a limited numberof dwell positions can be used. Since relatively few dwell positions are used some ofthe corresponding dwell times have to be long in order for the desired overall dose levelto be reached. These long dwell times have been observed in clinical practice and areconsidered to be a problem.

Another property that we study is the correlation between the objective value of thelinear penalty model and dose-volume parameters used for evaluation of dose plans. Weshow that the correlation is weak, which implies that optimizing the linear penalty modeldoes not give a solution to the correct problem.

Some alternative models are also considered. One that includes into the optimiza-tion the decision of where to place the applicators, when HDR brachytherapy is appliedfor prostate cancer, and one that reduces the long dwell times by using piecewise linearpenalties. The solutions to both models show significant improvements.

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Populärvetenskaplig sammanfattning

Denna avhandling handlar om hur man med hjälp av matematisk optimering kan skapabättre dosplaner för HDR brachyterapi, så att behandlingen blir mer framgångsrik ochbiverkningarna minskar.

Varje år avlider cirka 22 000 svenskar på grund av cancer. Det finns ett flertal olikasätt att behandla cancer och strålbehandling är en sådan metod. Av de svenskar som di-agnostiseras kommer ungefär hälften att någon gång genomgå strålbehandling. Det finnsmånga olika typer av strålbehandling. En av dessa är HDR brachyterapi, där radioaktivakällor placeras i eller mycket nära tumören som skall behandlas. Den strålning som avgesfrån källorna tas upp av kringliggande vävnad, varför både tumören och frisk vävnad fårdoser av radioaktiv strålning.

Målet med behandlingen är att ge en tillräckligt hög dos till tumören samtidigt somfrisk vävnad skadas så lite som möjligt. Det som påverkar hur hög dosen blir till oli-ka delar av behandlingsområdet är var de radioaktiva källorna placeras och hur länge destannar i dessa positioner. Att planera var och hur länge källorna placeras kallas att skapaen dosplan. Förr skapades alla dosplaner manuellt, vilket är mycket tidskrävande. Underde senaste 10-15 åren har dock matematisk optimering mer och mer tillämpats för att hjäl-pa processen. Idag används oftast dosplaneringssystem som automatiskt skapar dosplanersom sedan utvärderas av ansvarig läkare eller radiofysiker. De flesta dosplaneringssystemanvänder sig av optimering för att skapa dosplaner, vanligen genom att lösa en linjär mo-dell. Den linjära modellen utgår från dosintervall, som användaren anger, för de vävnadersom kommer att påverkas av strålningen. Avvikelser från dessa intervall straffas linjärt,och målet är att minimera det totala straffet för en plan.

I denna avhandlingen har vi studerat egenskaper hos denna linjära modell. Vi harbland annat visat att modellen har en egenskap som gör att få strålpositioner kan väljasoch därför blir några av bestrålningstiderna mycket långa. Dessa långa tider har ocksånoterats i praktiken och anses vara ett problem eftersom de kan orsaka nekros. Vi hardärför förändrat modellen för att minska effekten av denna egenskap. En annan egenskapsom vi studerat är korrelationen mellan målfunktionen i modellen och de kliniska måttsom används för utvärdering av dosplaner. Våra studier visar att korrelationen är svag, detvill säga att det minsta minsta straffet inte ger de bästa planerna, vilket tyder på att manegentligen löser fel problem.

När HDR brachyterapi används för prostatacancer förs först ett antal ihåliga nålar ini prostatan. Den radioaktiva källan kan sedan enbart positioneras i dessa nålar, vilket göratt placeringen av nålarna är mycket viktig för att kunna skapa en bra dosplan. Valet avplacering av nålarna sker idag helt manuellt. Vi har i denna avhandlingen utökat optime-ringsmodellen för källtiderna till att även inkludera var nålarna placeras. Genom att göradetta är det möjligt att med hjälp av optimering finna ännu bättre planer.

vii

Acknowledgments

First of all I would like to start by thanking my supervisors. Torbjörn Larsson for all ourinteresting discussions about how to twist and tune optimization to suit the problem athand, and for helping me to take the next step. Åsa Carlsson Tedgren for introducing meto the field of brachytherapy and putting up with all of my questions about small details.I also like to thank the research school in interdisciplinary mathematics for giving methe opportunity to work in my favorite field; how to apply optimization to real worldproblems.

The doctors and radiophysicists, especially Håkan Hedtjärn, at the University Hospitalin Linköping deserve a big thank you for providing me with data for my tests, answeringquestions about the treatment and showing me around.

I would also like to thank my present and former colleagues at the Department ofMathematics, I always enjoy having discussions with you. A special thanks to all thePhD-student, former and present, for sharing your experiences and knowledge, it is alwayseasier when you know your not alone.

The process of writing a licentiate thesis, or even undergoing PhD studies is veryfocused on the person doing it and the subject of research. For me however it would nothave been possible without the support of people around me. There are many personswho without knowing it have made it easier, I would like to mention those who have beenmost important to me.

Mikael Call, my office roommate for the last two and a half years have been invalu-able. Helping me with small and big things, making sure I take the breaks I need, cheeringme up and putting up with me winning from time to time. Svante Vikingson for all of ourdiscussions about what it means to be a PhD-student and for his encouragement to aspireto become better. All my other friends for making sure that even when work is though Ihave something to look forward to, playing games, watching movies or just talking.

But none have been as important as my family, always believing in me and encourag-ing me. Thanks Freya for always being happy when I come home, nothing can help raisemy spirit as much after a hard days work as being greeted in the door by you, jumpingaround and trying to kiss me and forcing me to play. Thank you mum and dad for alwayssupporting me no matter what! Lastly Rolle, there are not words enough to describe howimportant you are to me, I love you.

My sincerest thanks

Åsa HolmLinköping, April 18, 2011

ix

Contents

1 Introduction 1

I Background 3

2 Cancer and Radiotherapy 52.1 The human cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Cancer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Particle radiation . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.2 Electromagnetic radiation . . . . . . . . . . . . . . . . . . . . . 102.3.3 Sources of radiation . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Biological impact of radiation . . . . . . . . . . . . . . . . . . . . . . . 112.4.1 How radiation damages the cell . . . . . . . . . . . . . . . . . . 112.4.2 Measuring biological effect of radiation . . . . . . . . . . . . . . 122.4.3 Biological impact of radiation on tissue . . . . . . . . . . . . . . 13

2.5 Radiotherapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.5.1 Fractionated radiotherapy . . . . . . . . . . . . . . . . . . . . . 152.5.2 IMRT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.5.3 Brachytherapy . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.5.4 HDR brachytherapy for prostate cancer . . . . . . . . . . . . . . 172.5.5 Treatment plans . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5.6 Evaluation of dose plans . . . . . . . . . . . . . . . . . . . . . . 21

3 Optimization of Radiotherapy 253.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

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xii Contents

3.2.1 Dose point generation . . . . . . . . . . . . . . . . . . . . . . . 263.2.2 Dose calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Earlier models and research . . . . . . . . . . . . . . . . . . . . . . . . . 283.3.1 HDR brachytherapy . . . . . . . . . . . . . . . . . . . . . . . . 283.3.2 IMRT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Main Optimization Contributions of Our Research 354.1 Properties of the linear penalty model . . . . . . . . . . . . . . . . . . . 354.2 Optimizing catheter positions . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Bibliography 43

II Papers 47

A Impact of Using Linear Optimization Models in Dose Planning for HDRBrachytherapy 491 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522 Mathematical model and theory . . . . . . . . . . . . . . . . . . . . . . 53

2.1 Analysis of the standard model . . . . . . . . . . . . . . . . . . . 542.2 Alternative penalty . . . . . . . . . . . . . . . . . . . . . . . . . 57

3 Materials and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 594 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 63References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

B Integrated Optimization of Catheter Positioning and Dwell Time Distribu-tion in Prostate HDR Brachytherapy 651 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693 Mathematical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.1 Original model for the DTDOP . . . . . . . . . . . . . . . . . . 713.2 Alternative model for the DTDOP . . . . . . . . . . . . . . . . . 723.3 Integrating catheter placement with DTDOP . . . . . . . . . . . 73

4 Heuristics for CLP and CPLP . . . . . . . . . . . . . . . . . . . . . . . . 754.1 The neighbourhood and its characteristics . . . . . . . . . . . . . 754.2 Tabu search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.3 Variable neighbourhood search . . . . . . . . . . . . . . . . . . . 794.4 Genetic algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5 Computational studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.1 Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . 82

6 Conclusion and future Research . . . . . . . . . . . . . . . . . . . . . . 87References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

xiii

C On the Correlation Between DVH Parameters and Linear Penalties in Opti-mization of HDR Prostate Brachytherapy Dose Plans 911 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 942 Methods and material . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

xiv Contents

1Introduction

This thesis is concerned with the problem of how to generate high quality dose plans forhigh dose-rate (HDR) brachytherapy. HDR brachytherapy is one type of radiotherapy,and is used to treat several sorts of cancer such as tumours of the cervix, oesophagus,lungs, breasts, skin and prostate. As with all types of treatment, planning is needed beforetreatment can actually commence. One of the planning steps is to create a dose plan thatfor example decides how the radiation should be delivered.

During the last decade, the use of optimization as an aid when creating dose plansfor HDR brachytherapy has increased. Our research has been focused on characteristicsof the optimization model commonly used in clinical practice, and how this model couldbe extended to include more decisions in the planning process. Parts of our researchare general and should be applicable for all tumours treated with HDR brachytherapy.In some parts it has been necessary to restrict ourselves to one specific treatment area,prostate cancer.

Applying optimization to real world problems often require a thorough understandingof the application under consideration. Chapter 2 therefore introduces important knowl-edge related to HDR brachytherapy, without assuming previous knowledge about cancer,radiation or radiotherapy. Readers that already have this knowledge can skip Chapter 2.Readers that are not interested in understanding more than is absolutely necessary aboutthe treatment can skip most of Chapter 2, except Sections 2.5.3, 2.5.4 and 2.5.6. For thosethat would like to learn more we recommend the books ’Strålbehandling’ by Degerfältet al. 14 and ’Medicinsk fysik’ by Berglund and Jönsson8 for a general introduction toradiotherapy. For those interested in radiobiology, the book ’Basic clinical radiobiology’by Joiner and van der Kogel20 provides an in-depth description.

The thesis is organized as follows. It consist of two parts, Part I that introduces thefield of optimization of dose plans for high dose-rate brachytherapy and summarizes theimportant contribution from our research, and Part II that include three research papers.Part I starts with Chapter 2 that, as mentioned above, introduces the essential backgroundto HDR brachytherapy. The chapter explains why and how the treatment works, the plan-

1

2 1 Introduction

ning that is needed, and how to evaluate dose plans. The actual dose plan optimizationproblem is introduced in Chapter 3. The framework needed to use optimization for gen-erating dose plans and some earlier research in the area, are also presented in Chapter 3.The important optimization contributions of our research papers are presented in Chapter4, where we also present some other minor contributions.

The main contributions of the research papers included in Part II are:

Paper A - Impact of Using Linear Optimization Models in Dose Planning for HDRBrachytherapyShows that certain properties of the optimization model mostly used in clinicalpractice are the cause of the inhomogeneities in the plans that physicians find trou-blesome.

Paper B - Integrated Optimization of Catheter Positioning and Dwell Time Distri-bution in Prostate HDR BrachytherapyExtends, in the case of prostate cancer, the optimization model mostly used in clin-ical practice to also include one part of the dose planning process, catheter posi-tioning, which today is performed manually.

Paper C - On the Correlation Between DVH Parameters and Linear Penalties inOptimization of HDR Prostate Brachytherapy Dose PlansShows that the correlation, between the objective function of the optimization modelmostly used in clinical practice and parameters used for evaluating plans, is weakwhen applied to the case of prostate cancer.

Part I

Background

3

2Cancer and Radiotherapy

Radiotherapy is one kind of treatment used for cancer. To understand how optimiza-tion can be used to improve the treatment an understanding of both the disease and thetreatment is needed. This chapter will provide a comprehensive introduction into thisfield. The organisation of the chapter is as follows: it starts by describing the humancell and cancer, thereafter the concept of radiation is introduced, continuing with the ef-fect radiation has on cells and tissue, and ending with a description of different types ofradiotherapy.

2.1 The human cell

Humans are made up of tissue, which in turn consists of eukaryotic cells (we will fromnow on refer to eukaryotic cells as only cells), and in each human there are approximately1014 cells9. The cells are responsible for all processes in the human body. The cellconsists of many different components and one of these is the nucleus, where all DNA(deoxyribonucleic acid) is located. The DNA contains the genetic instructions used forthe development and functioning of the cell and in human cells it is divided into severallinear bundles called chromosomes.

DNA consists of two polynucleotide strands wrapped around each other to form adouble helix. Each strand includes nucleobases. These bases are the genetic instructionsand they are paired with bases on the other strand by hydrogen bonds. The base pairing iscomplementary, which in essence means that if you know one of the bases in the pair youautomatically know the other as well. An illustration of DNA is shown in Figure 2.1.11

During a cell’s life it goes through a number of different phases, and these phasesmake up the cell cycle. The phases are:

G0 (Gap 0) This is a resting phase were the cell has, temporarily or permanently, left thecell cycle. No cell division occurs and the cell tends to its normal duties.

5

6 2 Cancer and Radiotherapy

Figure 2.1: Illustration of the DNA, which is shaped as a double helix.

G1 (Gap 1) The cell grows and prepares for synthesis. At the end of this phase, justprior to the S phase, there is a checkpoint. At this checkpoint the cell ensures thatit is ready for synthesis, especially the DNA is checked for errors. If any dam-ages are detected the cell tries to either repair the damages or undergoes apoptosis(programmed cell death).

S (Synthesis) During this phase the DNA is duplicated, this is called replication.

G2 (Gap 2) The cell continue to grow. At the end of this phase just before mitosis startsthere is another DNA error checkpoint.

M (Mitosis) This is the phase where the cell is divided into two new cells. Mitosis is avery complicated process consisting of several subphases as well as a checkpoint.

In Figure 2.2 an illustration of the stages of the cell cycle is given.9

2.2 Cancer

Cancer is apart from heart diseases the most common cause of death throughout the world.For the year of 2007 there were estimates of 12 million new cancer cases and around 7.6million deaths related to cancer worldwide. This equals that that 1 out of 8 deaths iscaused by cancer.17

Cancer is not one disease but rather a group of diseases comprising over 200 differ-ent types, all characterised by uncontrolled growth and spread of abnormal cells17. Thedifferent types are usually named by the type of cell it starts in. Cancer diseases arecaused by multiple changes (called mutations) in the DNA of a cell, changing the prop-erties of that cell. Mutations can be caused by ionising radiation, tobacco etcetera, but

2.2 Cancer 7

��

��

M

I

G G1 0

M

G2

Cell

S

Figure 2.2: Illustration of stages of the cell cycle.

most of them are spontaneous and occur frequently during the cell division in all typesof cells. The difference with the mutations that cause cancer is that they give the cancercells a competitive advantage over their neighbouring cells, usually by changing the DNAencoding for proteins responsible for controlling the cell cycle. The properties the muta-tions need to cause in the cells are different for different types of cancers, but a numberof key features can be distinguished:

• Cancer cells can reproduce without receiving growth-chemical signals that normalcells require.

• Cancer cells can continue to grow even though they receive “stop-growth“ signals.

• Cancer cells are less likely to kill themselves when their DNA has been damaged.

• Cancer cells can influence the body to grow new blood vessels to supply the cancercell with nutrients.

• Cancer cells can proliferate indefinitely unlike normal cells that can only divide alimited number of times.

• Cancer cells can break loose and travel to other parts of the body, this is calledmetastasis.

It is the last feature that makes cancer so lethal, out of 10 deaths in cancer, 9 are due tometastasis.1, 11

The choice of treatment type used for a patient with cancer does not only dependon the type of cancer the patient has but also the extent of its spread, its sensitivity totreatment and factors related to the patients physical, psychological and social needs16.The most common treatment types are:

Surgery If the tumour is still localised it could be surgically removed. The goal is toremove all cancerous cells, this often require removing normal surrounding tissueand usually surrounding lymph nodes as well since cancer often metastasis throughthese.


Chemotherapy One property of cancerous cells is that they divide rapidly. In chemother-apy this is used by the delivery of drugs that target dividing cells, either causingthem to undergo apoptosis or impairing mitosis. Since cancer cells are not the onlycells dividing in the body side-effects are common, especially to cells that dividerapidly under normal circumstances such as bone marrow, digestive tract and hairfollicles cells.

Radiotherapy Normal cells are better than cancerous cells at repairing damages to theirDNA, and DNA damages in cells often lead to cell death or at least reduced cellreproduction. Radiotherapy uses these facts by trying to damage the DNA in thecancerous cells with ionising radiation.

These treatment types as well as the more uncommon ones can be used alone or togetherin different combinations.

2.3 Radiation

Radiation is a process where energy is transported through space either as electromagneticwaves (referred to as electromagnetic radiation) or as particles. One feature of radiationis that the energy radiates from its source, that is, the energy travels outwards in straightlines in all directions. Radiation is usually classified in one of two ways, either by how theenergy is transported, as particle radiation or as electromagnetic radiation, or by the effectthe radiation has on the irradiated medium. Below different types of radiation are de-scribed according to the first classification but first the reason for the second classificationis presented.

The energy that is transported can be deposited into some irradiated medium throughinteraction between the radiation and the medium. When the energy is transferred to theatoms of the irradiated medium, the electrons in the atoms may enter an excited state. Thismeans that the electrons have gained energy and thereby moved to a higher orbit. If theenergy is higher the result is more drastic and electrons may leave the atom, an illustrationof this can be found in Figure 2.3. The process that causes electrons to leave the atoms iscalled ionisation, since the atom has lost an electron it has changed from being neutrallycharged to being positively charged, which significantly alters the characteristics of theatom. Depending on the energy of the radiation the irradiated medium could becomeionised, this is what is used for the second classification of radiation. Types of radiationthat have high enough energy to cause ionisation is called ionising and if the energy islower it is called non-ionising. Examples of ionising radiation are α-particles, β-particles,X-rays and γ-rays (all of these are of interest when considering radiotherapy and aretherefore presented in the following subsections), and examples of non-ionising radiationis radio waves and visible light.

2.3.1 Particle radiation

In particle radiation the energy is transported by subatomic particles, and the energy con-sists of the kinetic energy of the particle. Some common particles in this context are

2.3 Radiation 9

Radiation interacts with the atom The electron enters an excited state

The electron leaves the atom, sothe atom becomes ionized

Figure 2.3: Transfer of energy to an electron by radiation can cause the electron toenter an excited state or if the energy is high enough to leave the atom.

electrons, positrons, α-particles, neutrons and protons. Below two of the common parti-cles are described more, especially how they interact with the irradiated medium.

An α-particle consists of two protons and two neutrons but no electrons, that is, α-particles are helium nucleuses. They deposit their energy to a medium through collisionswith the atoms of the medium, or rather mainly the electrons of the atoms. Through thecollision energy is transported from the α-particle to the electron, but since the α-particleis much heavier than electrons they only loss a small part of their energy and the collisiondoes not change their direction. This means that to deposit all of its energy it has tocollide with many electrons, however since the α-particle is quite big it will collide withall electrons that it passes and hence lose its energy quite quickly. As a result of this theα-particles cause great damage along their path but also have a very short range, only lessthan a millimetre in tissue and they can be stopped by a single sheet of paper.

A β-particle is either an electron or a positron (that is, the electron’s anti-particle). Aswith α-particles they deposit their energy through collisions, the difference being that theβ-particle is much lighter. This implies that their directions may change as a consequenceof the collisions, and β-radiation does therefore have curved tracks. The range of β-particles is short, however much longer than α-particles. In tissue the range is measuredin millimetres and β-particles can be stopped be an aluminium plate.


2.3.2 Electromagnetic radiation

Electromagnetic radiation is a special form of energy that consists of electric and mag-netic oscillations; it has no mass or kinetic energy in the strict sense. Two properties areimportant when considering electromagnetic radiation, frequency and wavelength. Wave-length is the spatial period of the wave, that is the length between two crests (see Figure2.4). Frequency is the number of waves per time unit. There is a simple relationshipbetween frequency (f ) and wavelength (λ), namely v = fλ where v is the velocity ofthe radiation. Electromagnetic radiation also exhibits particle properties, especially whenconsidering short times and distances, and is then referred to as photons. The photonsare associated with waves with frequency proportional to the energy they carry, and theenergy per photon is E = hf , where h is Plank’s constant.

Wavelength

Figure 2.4: Illustration of wavelength.

Electromagnetic radiation is classified into different types depending on the frequencyof the radiation (and thereby also the energy), examples of groups are radio waves, infra-red radiation, visible light, X-rays and γ-rays.

X-rays and γ-rays are physically identical, the difference being only the origin. X-rays are emitted by electrons outside the nucleus while γ-rays are emitted by the nucleus.The range of X-rays and γ-rays is long; it can reach several meters in tissue and hundredsof meters in air. To stop γ-rays or X-rays meter-thick concrete layers or decimetre-thicklayers of lead are needed. The energy of X-rays and γ-rays can be from 12eV and up.

2.3.3 Sources of radiation

In radiotherapy there are mainly two types of sources for radiation that are used: radionu-clides and accelerators. Below these two are described shortly.

Radionuclides

Isotopes are atoms that contain the same numbers of protons but a different number ofneutrons. Unstable isotopes that undergo radioactive decay are called radionuclides. Ra-dioactive decay means that the unstable isotopes send out one or more particles and/orelectromagnetic radiation to get rid of excess energy (in rare cases it can also split intotwo approximately equal parts). There exist a number of different decay processes andthe type and level of the energy emitted varies between radionuclides. The usual types areα-particles, β-particles and γ-rays.

The decay processes follow a probabilistic behaviour, and this is the reason why notall atoms decay immediately and simultaneously. The probability of decay varies betweenradionuclides, causing different nuclides to decay at different rates. The period of time

2.4 Biological impact of radiation 11

required for half of the radionuclides to decay is called half-life. The half-life is closelyconnected to the activity of a radionuclide, which is number of decays per second.

Accelerators

A particle accelerator is a device that increases the speed of charged particles by usingelectromagnetic fields. In medicine accelerators are often used to create X-rays. This isdone by accelerating electrons, produced by thermionic emission in the cathode, towardsthe anode, consisting of a metal with high melting temperature, by electric forces. Uponimpact with the anode the electrons are rapidly decelerated, and as a result the kineticenergy is transform into other kinds of energy, mostly heat, but around one percent isemitted as X-rays. The energy of the X-rays depends on the kind of metal in the anodeand the velocity and kinetic energy of the electrons. Depending of the intended use ofthe X-rays the mean photon energy is different, when used for diagnostic purposes suchas mammography, dentistry and computed tomography it is in the range 20-60keV, whilethe mean photon energy is in the range 3-10MeV when used for radiotherapy.

Accelerators can also be used in radiotherapy to deliverer electron radiation, the en-ergy of the electrons is then somewhere in the range 2-42MeV.

2.4 Biological impact of radiation

2.4.1 How radiation damages the cell

As described in Section 2.3 radiation deposits energy to the matter with which it interact,which might cause ionisation. This might break or change the structure of the matter, allmolecules in a cell can therefore be damaged by radiation. It has however been shownthat the ”target“ in the cell most sensitive to ionising radiation is the DNA-helix.20 Adisturbance in the combinations of the DNA string can yield devastating effects on thefunction of the cell. The damage imposed on the DNA by the radiation arises from twotype of effects: the direct and the indirect effect. The most prominent effect depends onthe type of radiation. For heavy particles, such as α-particles, the direct effect is the mostsignificant. For photon-radiation and β-particles on the other hand the indirect effect isthe most significant.9

Damage due to the direct effects means that the radiation causes ionisation directlyin the atoms of the DNA. The electrons that leave the atoms cause bonds to be broken,which can break the DNA strand.9

How damage due to the indirect effect occurs is more complicated. The radiationthen interacts with the water in the cell, causing the water molecules to break and createfree radicals. Radicals are highly reactive molecules, with unpaired electrons. Some ofthe radicals created, or products of the radicals, can travel through the cell to the nucleuswhere they act as a toxic for the DNA causing strands of the DNA to break.9

The breaks of the DNA strand can be of two types: single-strand break and double-strand break. Here single-strand break means that only one of the strands of the DNA-helix is broken, and double-strand break means that both strands are broken (with less than5 base pairs between the breaks). Damages such as breaks of the DNA can be repaired;however single-strand breaks are easier to repair since the other strand can be used as a


template. Not all damages can be repaired and sometimes misrepair occurs; this can causethe cell to die. There are many factors that affect the cells’ ability to repair DNA-damage,such as in which phase of the cell cycle the cell is when the damage occurs, the numberof breaks, and the type of cell. The ability to repair damage is lower for tumour cells thanfor healthy tissue.9

2.4.2 Measuring biological effect of radiation

The impact of the radiation is hence not due to high energy levels but rather that the wrongmolecules are ionised8. There is however a connection between the energy absorbedand the probability of producing harmful effects. In radiotherapy a basic quantity usedfor measuring the absorbed energy is the absorbed dose. This is defined as the amountof absorbed energy per unit mass of absorbent material and is measured in Gray (Gy).It is also common to talk about the equivalent dose, that in addition to absorbed dosealso consider that different types of radiation affect the tissue differently. This is doneby multiplying the absorbed dose by a radiation weighting factor for the radiation type.Below we will mean absorbed dose when we say dose. Another important concept in thiscontext is dose-rate which is absorbed dose per time unit.14

To measure the connection between the energy absorbed and the probability of pro-ducing harmful effects, a common method is to observe cell survival after radiation withvarious doses. Curves showing survival against dose are called cell survival curves. If thedose is plotted on the x-axis and survival on the y-axis a sigmoid curve is obtained (seeFigure 2.5a), however it is more common to plot the logarithm of the survival curve onthe y-axis and then a semilogaritmic curve is obtained (see Figure 2.5b). There are manymodels trying to describe the observed curve, one that is widespread in both experimentaland clinical radiobiology is the linear-quadratic (LQ) model. In the LQ-model the for-mula describing cell survival probability S (which is the same as surviving fraction) isS = e−αD−βD

2, where D is the dose and α and β are parameters.20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Dose

Sur

viva

l fac

tor

(a) Sigmoid curve

10−3

10−2

10−1

100

Dose

Sur

viva

l fac

tor

(b) Semilogaritmic curve

Figure 2.5: Cell survival curves.

There are many factors that affect the cell survival curve. Some of these are:

Cell cycle The cells’ sensitivity to radiation varies throughout the cell cycle. It is mostsensitive while in mitosis or late G2, and most resistant in the S phase and G0phase. This is one reason for the success of radiotherapy as treatment for cancer,

2.4 Biological impact of radiation 13

since tumour cells divide more rapidly the probability of cells being in mitosis ishigher for tumour cells than for healthy cells.

Oxygen Lack of oxygen makes cells more resistant to radiation. The reason for thisis that oxygen increases the toxicity of free radicals. Healthy cells are always welloxygenated while there are often tumour cells that have poor access to oxygen (cellsthat have poor access to oxygen are called hypoxic).

Dose-rate Using a lower dose-rate yields higher survival fraction. The reason for this isthat all cells get more time to repair damages and proliferate.

Radiation type Radiosensitivity is higher for heavy particles such as α-particles than forβ-particles and photons, due to higher probability for double strand breaks.9

2.4.3 Biological impact of radiation on tissue

When using radiation as a treatment the interesting result is not the effect on each cellbut rather how the tumour and surrounding healthy tissue respond. Dose-response curvesshow how the incidence of a radiation effect depends on dose. Examples of radiationeffects could be cancer cure, or side-effects of different types. All dose-response curveshave a sigmoid shape. The dose-response curve related to tumour cure (control) is oftencalled tumour control (cure) probability (TCP). TCP can be mathematically modelled andseveral models exist, and one of the more common defines TCP as:

TCP =n∏i=1

TCP (di, Ni),

TCP (di, Ni) = eNi∗Si = eNie−αdi−βd

2i .

Here the tumour volume is divided into n volumes with Ni tumour cells in volume i, anddi is the dose to volume i. As can be seen the surviving fraction of cells is used in thisfunction and modelled by the LQ-model.9, 20

Creating models for healthy tissue response is harder since many different side-effectscan occur and there are different levels of severity of each side-effect. For this thesis wesettle with noting that there exists models also for normal tissue response and that theseyield a measure called normal tissue complication probability (NTCP). TCP and NTCPcan be used to estimate the success of treatment regarding both cure and side-effects.9

The farther apart the curves for TCP and NTCP with respect to dose are, the better thechances for a good treatment result. In this context a concept called therapeutic ratio (TR)is often introduced. Therapeutic ratio is the relationship between tumour control doseand the tolerance dose for normal tissue, and should be as high as possible. The formulais TR=DNTCP/DTCP where DTCP is dose relative the likelihood of cure and DNTCPis dose relative the likelihood of side-effects. The concept of TR is illustrated in Figure2.6.14


TCP

NTCP

Pro

babi

lity

p

Dosea b

Figure 2.6: Illustration of therapeutic ratio (TR). For the probability p for tumourcontrol and impact on normal tissue the dose required is a and b, respectively. HenceTR=a/b.

2.5 Radiotherapy

Radiotherapy is the use of ionising radiation as treatment for cancer and a few otherdiseases. As described in Section 2.4 radiation damages cells in predicable ways, andcancer cells are for several reasons more sensitive to radiation than healthy cells. This isa fundamental reason that makes radiotherapy a viable choice.

Radiotherapy can be divided into three subgroups:

External beam radiotherapy During external beam radiotherapy the radioactive sourceis localised outside the body. The source is most commonly a linear accelerator thatgenerates electron or photon radiation, but protons and heavier ions are also used.

Brachytherapy During brachytherapy the radioactive source is localised inside or nextto the area requiring treatment. Brachy is a Greek word meaning ”short-distance“.

Systemic radioisotope therapy During systemic radioisotope therapy radioisotopes thattarget specific cells are delivered through infusion (into the bloodstream) or inges-tion. Targeting can be due to chemical properties of the isotope or by attaching theradioisotope to another molecule or antibody that guides it to the target tissue.

Of the three subgroups external beam therapy is the most common one. Some specifictreatment types are described in more detail in Sections 2.5.2-4.

The portion of patients with cancer in Sweden that were given radiotherapy at somepoint during their treatment was 47% in 2001. Approximately half of the radiotherapytreatments were given as a curative treatment, the rest as palliative treatment (where cureis not possible and the aim is pain relief or local disease control).34

Radiotherapy is in itself painless, however due to the damage of healthy cells side-effects may occur. Most of the side-effects that occur are predictable and expected, and

2.5 Radiotherapy 15

they are usually limited to the organs that receive radiation. The nature, severity, andlongevity of side-effects depend on, among others, the treatment area, the type of radia-tion, the dose, and the patient. Organs that might suffer from side-effects due to radiationare referred to as organs-at-risk.

2.5.1 Fractionated radiotherapy

As is well known radiotherapy is usually delivered in several fractions, that is, the doseis not delivered all at once but divided into fractions delivered with some time inbetween.There are four main reasons for this:

Repopulation Both normal cells and tumour cells proliferate even after radiation expo-sure. The increase in tumour cells does of course work against the treatment butthe increase in normal cells instead reduce the risk of side-effects and is thereforein favour of the treatment.

Repair Since normal cells are better at repairing damage due to radiation than tumourcells, allowing enough time between fractions for normal cells to repair is in favourof the treatment.

Redistribution Since radiosensitivity varies throughout the cell cycle fractionated radio-therapy increases the probability of tumour cells being exposed to radiation duringa sensitive phase. Since tumour cells divide more frequently than normal cells thiswill cause more damage to tumour cells than to normal cells, since mitosis is oneof the most sensitive phases.

Reoxygenation As noted above hypoxic cells are more resistant to radiation than welloxygenated cells. When well oxygenated tumour cells die the hypoxic cells willbecome increasingly oxygenated thereby increasing their sensitivity to radiation.9

2.5.2 IMRT

Intensity-modulated radiation therapy (IMRT) is one type of external radiotherapy. Radi-ation beams, of either photons or electrons, are formed by a linear accelerator and travelthrough a gantry that can rotate around the patient, see Figure 2.7a. Since the gantry canrotate radiation can be directed at the patient from any angle. The head of the gantryaccommodates a focusing apparatus, most modern facilities use a multileaf collimator,in Figure 2.7b an illustration of a multileaf collimator is found. When the beam passesthrough the multileaf collimator parts of the beam can be blocked, and by changing whichparts that are blocked the intensity of different parts of the beam, called beamlets, can bedifferentiated. This allows the dose to conform to the 3-D shape of the tumour.15

2.5.3 Brachytherapy

As mentioned above brachytherapy is a form of radiotherapy where the radioactive sourceis placed inside or next to the area requiring treatment. It is used to treat tumours of thecervix, oesophagus, lungs, breasts, skin, and prostate. A key feature of brachytherapyis that the radiation affects only a small area around the source, and since the source is


(a) Standard equipment for IMRT. Thegantry rotates around the patient.

Tumour

Leaves(b) Illustration of a multileaf collima-tor.

Figure 2.7: Equipment used for IMRT.

placed directly at the site of the tumour, the healthy tissue further away from the source isexposed to less radiation than with other techniques. Another advantage of brachytherapyis that errors due to patient movement, or movement of the tumour within the body, arereduced since the radiation sources retain their position in relation to the tumour.

Most radioactive sources used for brachytherapy are radionuclides enclosed withina non-radioactive capsule. Different types of radionuclides are used and examples are:Iridium-192 (192Ir), Iodine-125 (125I), and Ruthenium-106 (106Ru).19

The radioactive source could be delivered manually, but due to radiation exposureto clinical staff they are usually delivered using a technique known as afterloading. Inafterloading, applicators, that are non-radioactive, are positioned in the treatment area andthe radioactive source is then subsequently inserted through the applicators. The insertionof the radioactive source could be done by manual afterloading where clinical staff usesappropriate handling tools, or by remote afterloading. When using remote afterloadingapplicators are after positioning connected to an ’afterloader’ machine through connectingguiding tubes. When the clinical staff has left the treatment room the machine applies thesource which has until then been inside a radiation shielded safe.19

Different types of brachytherapy are classified according to three characteristics:

• Source placement:

Intracavitary Therapy The applicators and radioactive sources are inserted intoan existing body cavity such as the vagina.

Interstitial Therapy The applicators and radioactive sources are inserted directlyinto tissue using for example needles or wires. This kind of treatment is usedfor treatment areas such as prostate and breast.

Intralumenal therapy The applicators and radioactive sources are inserted into alumen, such as the oesophagus.

Intravascular The applicators and radioactive sources are inserted into an artery.

2.5 Radiotherapy 17

• Duration:

Temporary The radioactive source is removed after treatment, where the treatmentduration is usually between a few minutes and a day.

Permanent Small low dose-rate radioactive seeds are placed into the treatment siteand left there to decay. After some time the radiation emitted decays to almostzero, and they can hence remain with no lasting effect.

• Dose-rate No universally accepted definitions exists, however most accept the fol-lowing:

LDR Low dose-rate corresponds to around 0.5-1 Gy/h.

MDR Medium dose-rate corresponds to around 1-12 Gy/h.

HDR High dose-rate corresponds to above 12 Gy/h, however typically the dose-rate is around 2 Gy/min which is around 10 times as much.

PDR A specific technique where HDR ’pulses’ (typically 5-10 minutes long) arerepeated at short intervals (typically once per hour).19

2.5.4 HDR brachytherapy for prostate cancer

Since the focus of our research has been how to optimize dose plans for HDR brachyther-apy and especially HDR brachytherapy for prostate cancer, this section will more thor-oughly cover this treatment.

The prostate is a male gland located at the top of the urethra, see Figure 2.8. It isa part of the male reproductive system and contributes to the production and storage ofseminal fluid. The prostate is normally about three inches long and weighs 20 grams foradult males.

Prostate cancer is one of the most common types of cancers for men and in 2002 esti-mations showed that 700 000 new cases occurred each year10. It mainly affects older men,few are diagnosed before they are fifty, and half are not diagnosed before they are sev-enty12. Even though many cases of prostate cancer never develop symptoms or undergotherapy and the patients eventually die of other causes, 8740010 deaths were recorded inEurope during 2006.

Possible treatment options for prostate cancer include among other watchful waiting,external beam radiotherapy, high dose-rate (HDR) brachytherapy, low dose-rate (LDR)brachytherapy, and prostatectomy. Which treatment or combination of treatments that ischosen depends on several factors such as the stage of the cancer, age and general healthof the patient, patient preferences, and quality of life aspects30.

Brachytherapy for prostate cancer was used as a treatment as early as in the 1920’s28,however the use of remote afterloading with high dose-rate 192Ir was not introduced untilthe late 1980’s29. When using HDR brachytherapy for prostate cancer hollow needles, inthe following called catheters, are inserted into the treatment area through the perineum,hence HDR brachytherapy is an interstitial therapy. Catheters, see Figure 2.9b for anexample, are usually placed using a fixed template29, in Figure 2.9a an example of atemplate is shown, and by using transrectal ultrasound for guidance. For an illustrationof HDR brachytherapy for prostate cancer see Figure 2.9c. In Figure 2.9d an equipment


Rectum

Bladder

ProstataUrethra

Figure 2.8: Illustration of the prostate and nearby organs.

setup is shown. The afterloader moves the 192Ir source through the catheters in specifiedsteps, stopping at certain positions called dwell points. The length of a stop is calleddwell time. The dwell points are evenly distributed with a distance of 2.5 mm to 10.0 mmbetween points.

When constructing treatment plans for HDR brachytherapy the entire prostate is ingeneral considered to be the target. The main organs-at-risk are rectum and urethra, andoften also the bladder.

2.5.5 Treatment plans

Before treatment with radiation commence planning is needed, this is often called treat-ment planning. One of the first steps is to obtain images of the treatment area using forexample CT (computed tomography, which is a kind of X-ray), MRI (magnetic resonanceimaging) or ultrasound. This yields a number of cross sections of the treatment area thattogether create a 3-D-visualisation (or 3-D-representation) of the treatment volume; inFigure 2.10 an example of such a cross section obtained by ultrasound is found. On theseimages the target volume and organs-at-risk close to the target volume are contoured. Thetarget volume is usually contoured in different levels:

GTV: Gross tumour volume, that is the part of the volume with known tumour growth.

CTV: Clinical Target Volume includes, in addition to volume included in GTV, also vol-umes where tumour growth are suspected due to closeness to the tumour or lymphnodes with high probability of spread.

PTV: Planning target volume is the volume that is intended for treatment. It includesCTV but also a margin to include possible movement etcetera during treatment.

2.5 Radiotherapy 19

(a) Example of a template for insertingcatheters.

(b) Example of a catheter.

(c) Illustration of the treatment. (d) An example of the setup used for the treatment.

Figure 2.9: Equipment used in HDR brachytherapy for prostate cancer.

For each target volume a dose is prescribed, and tolerated dose levels are specified for eachorgan-at-risk. When defining target volumes and radiation doses considerations taken arefor example:

• The goal of the treatment, that is whether it is intended to be curative (cure thecancer) or palliative (reduce or prevent symptoms caused by cancer).

• Radiosensitivity and growth pattern of the tumour.

• Risk for side-effects.

• Variations in setup between treatment sessions.


Figure 2.10: Ultrasound image of the treatment area for a prostate cancer patient.

• If and how the treatment is divided into fractions.14

Knowing what doses that are desired for each volume, the next step is to decide how todeliver radiation so that the dose conforms as well as possible to the prescription for targetvolumes and tolerated levels for organs-at-risk. In IMRT this corresponds to choosing di-rections and sizes for the treatment fields, in permanent LDR brachytherapy it correspondsto choosing positions and strengths of the sources to implant, and for HDR brachytherapyit corresponds to choosing where and for how long to stop the radioactive source. Thisstep is often referred to as creating a dose plan and the goals are:

• To reach the prescribed dose to the target.

• A homogenous dose distribution within the target.

• As low dose as possible (and below the tolerated dose level) to organs-at-risk.

• As low total dose as possible.

• A plan that is realisable and can be repeated with high precision.

It is generally impossible to fulfil all the goals and trade-offs are inevitable.14

In most cases the dose plan is created for the entire treatment volume simultaneously,referred to as 3-D planning. Using the 3-D-representation an exact calculation of the doseto each point in the treated volume can be made (given the shape and density of the patientas well as how the radiation is delivered). There are different ways to generate the doseplan. Some generate dose plans manually by iteratively changing how the radiation isdelivered and evaluating the generated dose distribution (the dose to each point). Othersuse software that generates plans by using different kinds of optimization techniques, inSection 3.3 some of these optimization techniques are presented.

There are other steps included in treatment planning as well, such as plans for fixationand dose simulations; however these will not be covered since they do not affect theoptimization process.

2.5 Radiotherapy 21

2.5.6 Evaluation of dose plans

Before applying a dose plan to a patient, evaluation is needed. One common method forevaluation is to graphically illustrate the dose distribution and visually inspect it. Visualevaluation provides information about if and where hot spots (volumes receiving a veryhigh dose) and cold spots (volumes receiving a low dose) are located and the size of suchvolumes. It is also quite easy to see how the dose conforms to the target. In Figure 2.11an example of how the dose can be illustrated to enable visual inspection is found.

Figure 2.11: Graphical illustration of dose distribution in a cross section. Darkcolours represent high doses and light colours represent low doses.

Another common method for evaluation of dose plans is dose-volume histograms(DVH) which describe the dose distribution for a structure. There are two kinds of DVH:s,cumulative and differential. A differential DVH illustrates for each possible dose howlarge part of the volume that receives exactly that dose, see an illustration in Figure 2.12b.A cumulative DVH on the other hand illustrates how much of the volume that receives acertain dose or more (for each possible dose), see an example in Figure 2.12a. CumulativeDVH:s are more common and hence when we write only DVH we refer to the cumulativeDVH. Ideally the entire target volume receives the prescription dose, corresponding tothe DVH in Figure 2.13a, while organs-at-risk receive no dose at all, corresponding to theDVH in Figure 2.13b.

From the DVH certain measures (often called dose-volume parameters, dosimetricindices, or DVH-based parameters) can be extracted. These measures are of two differenttypes: D-measures and V-measures. A V-measure is the percentage of the volume thatreceives a certain dose, and such measures can be easily extracted from the DVH asillustrated in Figure 2.14b. An example of a V-measure is V100, which measures thepercentage of the volume that receives 100% of the target prescription dose or more. D-measures are the reversals of V-measures, they measure the dose that a certain percentageof volume receives. An illustration of how D-measures can be extracted from the DVHis found in Figure 2.14a. An example of a D-measure is D90, which is the lowest dose


Dose

Percent of volume

Prescription dose

100% ofvolume

(a) Cumulative DVH.Dose

Percent of volume

Prescription dose

(b) Differential DVH.

Figure 2.12: The two types of dose-volume histograms (DVH).

Dose

Percent of volume

Prescription dose

100% ofvolume

(a) Ideal cumulative DVH for target.Dose

Percent of volume

Prescription dose

100% ofvolume

(b) Ideal cumulative DVH for organ-at-risk.

Figure 2.13: Ideal dose-volume histograms.

2.5 Radiotherapy 23

Dose

Percent of volume

Prescription dose

100% ofvolume

x

Dx

(a) Illustration of how Dx is calcu-lated.

Dose

Percent of volume

Prescription dose

100% ofvolume

x

Vx

(b) Illustration of how Vx is calcu-lated.

Figure 2.14: Illustration of how dosimetric indices can be obtained from the dose-volume histograms.

received by the 90% of the volume that receive the highest dose.There are a lot of other measures used as well, such as gEUD6, COIN5, HI22 etcetera,

however since we have not used these at all they will not be covered here.

3Optimization of Radiotherapy

This chapter will introduce the reader to optimization of dose plans in radiotherapy, start-ing with a general problem formulation and then presenting the framework needed forperforming the optimization. The chapter ends with presenting earlier research related toour research.

3.1 Problem formulation

Optimization within the field of radiotherapy is mostly focused on creating good (’near-optimal’) treatment plans. A quite general problem formulation is:

Given the patient’s anatomy what is the best way to deliver a tumoricidaldose of radiation to the cancerous region while limiting the dose of radiationto organs-at-risk surrounding the cancer so they can survive the treatment.

What needs to be determined is thus how to deliver the dose, that is, creating a dose plan.Translating the general problem formulation into an optimization model is difficult.

One reason for this is of course that the goal of tumoricidal dose of radiation to the can-cerous region conflicts with limiting the dose of radiation to organs-at-risk. Another dif-ficulty is that if any region of the anatomy receives an unreasonably high dose, all cellswithin this region will die, and if the region is large enough this will cause an unwantedcondition called necrosis. Yet another difficulty is that different organs react to radiationin different ways.18

The treatment goal may also vary between patients; in many cases it is of course todeliver a tumoricidal dose of radiation to the cancerous region while keeping the dose toorgans-at-risk under control. For a terminally ill patient however the goal is not aboutcuring the cancer but rather to increase quality of life, and this might for example meanthat minimizing dose to certain organs-at-risk is more important than a tumoricidal dose.

25

26 3 Optimization of Radiotherapy

In cases when the likelihood of success is very low when sparing organs-at-risk the ques-tion of to what degree the patient is willing to risk damage to the organs-at-risk arises.Taking all these cases in to account it is clear that optimization models need to be flexibleto accommodate different situations.18

Given the difficulties and variations in goal it is not surprising that there have beenmany different models suggested and in Section 3.3 some of these are presented. Allthese models do however try to solve the same problem, namely the general problemformulation above.

3.2 Framework

Given the problem formulation there is a need to be able to describe how the dose dis-tribution depends on the variables chosen. A natural question that arises in this contextis where the dose should be calculated. It is clear that it is not possible to calculate itfor every single cell, so some kind of discretization is needed. In Section 3.2.1 differentchoices of discretizations are described, all yielding a number of points that represent thetotal volume, the generated points are called dose-calculation points. For each of thesepoints a description of how the dose depends on the variables is needed, and in Section3.2.2 this description is given for the case of HDR brachytherapy.

3.2.1 Dose point generation

There are a number of different suggestions for how to discretize the treatment area andthe choice depends on for example what the points should be used for, and the kind oftreatment considered. When evaluating doses it is common to use a very fine squaregrid, where each point in the grid represents a small part of the treatment volume. Wehave when evaluating our generated dose plans used such a square grid where each pointrepresents 0.25 mm3 of the treatment volume. However if such a fine grid were to beused during the optimization phase the problem size would be very large, since with aresolution of 0.25 mm3 per point around 2 million points are needed to cover the treatmentvolume for prostate HDR brachytherapy. For this reason it is common to use a sparserresolution in the optimization phase.

When considering IMRT it is common to use a square grid also for the optimiza-tion phase, but letting each point represent a larger volume. In IMRT this works wellsince the dose distributions are quite uniform and hence a high level of accuracy canbe achieved using quite few points. Dose distributions in brachytherapy on the otherhand are significantly more nonuniform and more care is needed when choosing points.The common technique to generate dose-calculation points in brachytherapy seems to bethe one presented by Lahanas et al. 23 and this is the technique we have used. Here thedose-calculation points are generated both within the volume of each structure and on thesurface of the volume of each structure. The points within each structure are generatedby using triplets of Sobol sequences (a type of quasi-random low-discrepancy sequences)inside a bounding box of the structure, and each point is then investigated to determine ifit is inside the structure or not. To generate surface points a triangulation of the surfaceof the structure is needed. Given the triangulation, the number of points to generate in

3.2 Framework 27

each triangle is chosen using the Stochastic Universal Sampling Algorithm. Then for eachtriangle the chosen number of points are randomly distributed by the random generationof barycentric coordinates.23

3.2.2 Dose calculations

When calculating doses the first step is usually to calculate the dose-rate contribution fromone dwell point (seed or beamlet) to one dose point per time unit. Given these dose-ratesthe total dose in a dose-calculation point can be easily calculated by multiplying the dose-rate with the dwell time for each dwell point and then summing this up. This means thatif dij is the dose-rate contribution from dwell point j to dose-calculation point i, then thetotal dose in dose-calculation point i amounts to

∑j dijtj .

As mentioned in Section 2.5.4 the usual radioactive source in HDR brachytherapy is192Ir. Iridium-192 decays by beta and photon emission19. Regardless of manufacturerall 192Ir sources for HDR brachytherapy are very similar. They consist of a cylinder of192Ir enclosed in stainless steel. The steel enclosure is thick enough to stop all electronsemitted by β-decay. The photons with high energy are however not affected by the steeland are used for the radiotherapy.

By Monte Carlo simulations of photons in a geometry consisting of the source placedin a large volume of water one obtains a good approximation of the dose-rate from an 192Irsource to the prostate. The results obtained by these simulations can be parameterised indifferent ways; one of these is described below.

��

��

��

��

��

��

��

Dose point

vr

Source

Figure 3.1: Illustration of how to measure the angle (v) and radius (r) for calculatingdoses when using an 192Ir source in HDR brachytherapy of the prostate.

The dose-rate at a point, DRtot, as a function of of the distance to the centre of thesource (in polar coordinates r and v, see Figure 3.1) is calculated as:

[DRtot(r, v)]MC = [DRprim(r, v)]MC + [DRscat(r, v)]MC

[DRprim(r, v)]MC =1

4πr2a1(v)e−(a2(v)−a3(v)r)r


[DRscat(r, v)]MC =1

4πr2

(b1(v)

[(1 + b2(v))− e−b3(v)r

]e−b4(v)r

).

Here, a1, a2, a3, b1, b2, b3, and b4 are parameters that depend on v. Note that this isnot the actual dose-rate obtained at the hospital, thereof the index MC. At the hospitalone has to adjust the dose-rate to the strength of the source that is actually used. Thereason for this is that the strength depends on how old the source is, since the activitydecreases due to the radioactive decay. The strength of the source is measured in thephysical quantity reference air kerma rate (RAKR), and the actual dose-rate is

[DRtot(r, θ)]hospital =[RAKR]hospital

[RAKR]MC

[DRtot(r, θ)]MC ,

where [RAKR]MC is a known constant.

3.3 Earlier models and research

The first model for optimizing dose plans was proposed already in 1968 by Bahr et al. 4

and was a linear model. Since then many different models and methods have been pro-posed. Good comprehensive reviews of external radiotherapy optimization literature havebeen presented by Shepard et al. 33 and Ehrgott et al. 15. Below we focus on the modelsand methods of interest for our research.

3.3.1 HDR brachytherapy

When creating dose plans for HDR brachytherapy there are mainly two decisions that canbe optimized, namely, where to place the catheters, called catheter placement or catheterpositioning, and for how long the source should dwell at each possible position, calleddwell time distribution. The focus of research has so far been on the dwell time distribu-tion within already implanted applicators, not catheter positioning.

Optimization of dwell time distribution

Several different models or methods for the dwell time distribution optimization prob-lem (DTDOP) have been proposed during the last decade. Among the proposed modelsmany seem to utilise a common concept; it is assumed that physicians provide upperand lower limits on the dose for each structure in the treatment area (tumours, organs-at-risk and other healthy tissue) and the objective function of these models then penalisevalues above or below these limits. The difference between models is how the devia-tion is penalised, and if the penalty for each structure is treated as separate objectives(multi-objective problems) or if a weighted sum of the penalties is used (single objectiveproblem). For example Milickovic et al. 27, Lahanas et al. 24, and Lessard and Pouliot25

have proposed such models. A general model of this type, formulated in a way that makesit easy to understand, but not necessarily to solve, is presented in Table 3.1.

The model that seems to be the one most commonly used in clinical practice is thegeneral model with α = 1 and where Θ(x) is the Heaviside function. It was first intro-duced by Lessard and Pouliot25, who solve it using fast-simulated annealing. They refer

3.3 Earlier models and research 29

ParametersS = Number of structures.Ns = Number of dose-calculation points in structure s.n = Number of dwell points.Dmins = Lower dose limit for structure s.

Dmaxs = Upper dose limit for structure s.

dsij = Dose-rate contribution from dwell point j to dose-calculation point i in structure s.

$mins = Objective weight for underdosage of structure s.

$maxs = Objective weight for overdosage of structure s.

α = Model dependent parameter.Θ(x) = Heaviside function or an approximation of the Heaviside function.VariablesDosesi = Dose to dose-calculation point i in structure s.Usi = Underdosage of dose-calculation point i in structure s.Osi = Overdosage of dose-calculation point i in structure s.tj = Dwell time at point j.

Objective function in single objective case

minS∑s=1

Ns∑i=1

($mins Θ(Usi )(Usi )α +$max

s Θ(Osi )(Osi )α)

Objective function in multi-objective case

min f(U,O) = (fmin1 (U), . . . , fminS (U), fmax1 (O), . . . , fmaxS (O))

fsmin(U) =Ns∑i=1

(Θ(Usi )(Usi )α), fsmax(U) = (Θ(Osi )(Osi )α) ∀s

Subject to

Dosesi =n∑j=1

dsijtj ∀i, s

Usi = Dmins −Dosesi ∀i, s

Osi = Dosesi −Dmaxs ∀i, s

tj ≥ 0 ∀j

Table 3.1: A general description of many of the proposed models for HDRbrachytherapy.


to their method as IPSA (inverse planning simulated annealing). In 2006 Alterovitz et al. 2

showed that the model can be formulated as a linear problem, and hence can be solved eas-ily to global optimality using for example the simplex method. This linear model, whichwe in the following will refer to as the linear penalty model, is presented in Table 3.2.

For plans generated by solving the linear penalty model it is common with a few dom-inating positions with dwell times large compared to the rest6, 13. Long dwell times dohowever cause hot spots around the corresponding positions. Concern has been expressedregarding the unknown effect of such hot spots and therefore more homogeneous solu-tions are preferred13. A few attempts to deal with the problem of long dwell times havebeen introduced. Chanjon et al. 13 suggest introducing artificial normal tissue around thecatheters or explicit ceilings on maximum dwell time. Baltas et al. 6 introduce dwell timegradients and their results using these are promising26.

In addition to the different versions of the general model only a few researchers havepresented models for DTDOP. Ruotsalainen et al. 32 propose an interactive multi-objectiveapproach where different measures describing the dose distribution, such as V PTV100 andmaximum dose deviation in PTV from prescription dose, are used as objective functions.Beliën et al. 7 suggest a model similar to the linear penalty model but that also includesdose volume constraints. Adding such constraints requires introducing a binary variablefor each dose-calculation point within the volume of interest. To solve this model theyuse a hybrid simulated annealing linear programming approach.

Optimization of catheter positioning

As mentioned above few researchers have used optimization to choose catheter positions.The attempts we know of are one by Ayotte et al. 3 and one by Karabis et al. 21.

Ayotte et al. 3 use an iterative approach, starting with a high number of catheters andthen gradually removing catheters based on the fraction of total dwell time attributable toeach catheter. To find the optimal dwell times in each iteration they use the linear penaltymodel. Their results show that the objective was improved, compared to using a prioridetermined catheter positions, and that dose to healthy tissue is reduced.

Karabis et al.21 use an approach where the problem is modelled as a mixed integernonlinear problem or a mixed integer linear problem, by first introducing a set of feasiblecatheter positions and then assigning to each a binary variable that describes if the positionis used or not. They use two approaches when trying to solve the model: the optimiza-tion software CPLEX and a heuristic method. CPLEX has difficulties solving the modeland succeeds only when relatively few possible positions are introduced. Their heuristicon the other hand, not described in detail but consisting of a combination of simulatedannealing and a scoring method for the binary part and a quasi-Newton optimization forthe continuous part, finds provably good solutions for many instances (close to the lowerbound of CPLEX).

3.3.2 IMRT

Conventional external radiotherapy and IMRT are the types of radiotherapy that have re-ceived the most attention by optimization experts. The creation of dose plans for IMRTis typically divided into three steps:


ParametersS = Number of structures.Ns = Number of dose-calculation points in structure s.n = Number of dwell points.Dmins = Lower dose limit for structure s.


Mmins = Penalty for underdosage of structure s.

Mmaxs = Penalty for overdosage of structure s.

dsij = Dose-rate contribution from dwell point j to dose-calculation point i in structure s.Variableswsi = Penalty for dose-calculation point i in structure s.tj = Dwell time at point j.

Objective function

minS∑s=1

Ns∑i=1

wsiNs

Subject to

wsi +n∑j=1

Mmins dsijtj ≥Mmin

s Dmins ∀i, s

wsi −n∑j=1

Mmaxs dsijtj ≥ −Mmax

s Dmaxs ∀i, s

wsi ≥ 0 ∀i, stj ≥ 0 ∀j

Table 3.2: The linear penalty model which is the model usually used in clinicalpractice when optimizing dose plans. It was originally presented by Alterovitz etal.2.


1. The beam angle optimization problem: The selection of the number of beams andthe directions from which the gantry delivers radiation. Also called the geometryproblem.

2. The fluence map optimization problem: Choosing the intensity pattern for each beam(that is the intensity of each beamlet). Also referred to as the intensity problem.

3. The segmentation problem: Deciding how to operate the multileaf collimator to ef-ficiently administer the treatment. Also called the realization problem.

The segmentation problem does not correspond to any problem in brachytherapy andhence we will not go into any research in this area. Beam angle optimization is similar tocatheter placement, and it seems that the general approach is the same as the approach byKarabis et al. 21: generate a number of possible angles (positions) and choose the subsetthat yields the best fluence map (dwell time distribution). There are some different ap-proaches to do this, but none will be presented here since we have not used any of these asinspiration for our research. The interested reader is referred to the survey by Ehrgott15

for further reading and references.Fluence map optimization is very similar to dwell time distribution, and hence models

are very similar. There are however fundamental differences in which dose distributionsthat are possible to achieve by changes in intensity (time), and this requires special con-siderations. In HDR brachytherapy dose peaks are inevitable, in IMRT on the other handsuch peaks do not, or should at least not, occur. An example of how this difference affectsmodels is that hard dose constraints (constraints limiting the dose in a dose-calculationpoint to a certain value) are less applicable in brachytherapy.

The most prevalent formulation in IMRT seems to be the weighted least square model,which minimizes the weighted sum of averaged squared deviations from the prescribeddose for each structure15. The model of most interest to us is however the one presented byRomeijn et al. 31. They use a piecewise linear convex function to calculate the penalty fordeviation from a prescribed dose (or dose interval). This yields a simple but very flexiblemodel, and if using the same notation as for the linear penalty function the model is:Objective function

minS∑s=1

Ns∑i=1

wsiNs

Subject to

cks +n∑j=1

Mks d

sijtj ≤ wis ∀i, s, k

n∑j=1

dsijtj ≤ Dmaxs ∀i, s

n∑j=1

dsijtj ≥ Dmins ∀i, s

wsi ≥ 0 ∀i, stj ≥ 0 ∀j.


Here Mks is the slope and cks the y-intercept for segment k of the piecewise linear penalty

function for structure s. Romeijn et al. also introduce a way to approximate dose-volumeconstraints by a set of linear constraints. Both versions of their model, with or withoutthe approximate dose-volume constraints, yield excellent results.

4Main Optimization Contributions of

Our Research

Earlier chapters have given an introduction to optimization in radiotherapy as well aspresented some different models and methods. This chapter will outline our contributionto the field by giving brief overviews of the papers presented in part II and also presentingsome other minor contributions. The common theme of our research has been the linearpenalty model described in Table 3.2, mainly because this model is the one commonlyused in clinical practice when optimizing dose plans.

4.1 Properties of the linear penalty model

Since the linear penalty model is the model most commonly used in practice, it is ofcourse of interest to understand the properties of the dose plans generated by this model,and this is the main goal of Paper A. The property of interest in this paper is mainly howmany variables of different types that can be basic variables. Since the model is a linearoptimization model an extreme point of the feasible set is optimal (alternative optimacould exist but tests show that is very unlikely). As is well-known the number of basicvariables at an extreme point equals to the number of constraints. In the linear penaltymodel there are three types of variables: times (tj), penalties (wsi ), and slacks (denotedby smin/maxis ), and 2

∑Ss=1Ns constraints. We investigate which kind of variables that

become basic in different cases.Consider a specific dose-calculation point, ı, in structure s and assume that the total

dose to this point is Dosesı =∑nj=1 d

sıjtj . Writing on standard form the constraints

associated with the dose-calculation point are:

wsı − sminıs = Mmins (Dmin

s −Dosesı ) (4.1)

wsı − smaxıs = Mmaxs (Dosesı −Dmax

s ) (4.2)

wsı ≥ 0. (4.3)

35

36 4 Main Optimization Contributions of Our Research

Depending on the value of Dosesı it is possible to deduct which variables that have to bebasic, in Table 4.1 the variables that have to be basic in each case are displayed. It is pos-sible to illustrate this result as in Figure 4.1, where segments (1), (2) and (3) correspondto conditions 4.1, 4.2, and 4.3 respectively.

Dosesı ∈ Basic variables No. basic variables[0, Dmin

s ) wsi, smaxis

2[Dmin

s ] smaxis

1(Dmin

s , Dmaxs ) smin

is, smaxis

2[Dmax

s ] sminis

1(Dmax

s , ∞) wsi, sminis

2

Table 4.1: Number of forced basic variables from Eq. (4.1) and Eq. (4.2).

Now let nd be the number of dose-calculation points with Dosesı equal to Dmins

or Dmaxs . Then the number of basic variables that are slacks or penalties is at least

2∑sNs − nd, and hence the maximum number of basic time variables is nd. It is there-

fore possible to conclude that the total number of positive dwell times is limited by thenumber of dose-calculation points on their upper or lower dose limits.

As mentioned in Section 3.3.1 plans generated by the linear penalty model tend to bedominated by a few dwell positions with times large compared to the rest13. Given theresult above this is not surprising since in practice relatively few values of Dosesı willequal Dmin

s or Dmaxs , hence only a few dwell time variables become non-zero (basic),

and some of these times then have to take large values, in order to reach the desired overalldose level. The long dwell times observed in clinical practice can thus be explained bythe distribution of basic variables, and this is the primary contribution of Paper A.

Paper A also presents an alternative model that uses piecewise linear penalties insteadof the linear ones. Comparing the dose plans obtained by the linear penalty model tothose from the alternative model yields quite similar dose-volumes parameters, and thisindicates that the solutions are almost clinically equivalent. When comparing dwell timesthere is however a large difference, since maximum dwell times are greatly reduced (aver-age 30%) and the number of active dwell positions is increased (average 20%), implyinga decrease in the undesired hot spots.

Paper C also considers a property of the linear penalty model, namely the relationshipbetween dose-volume parameters and the objective function. Since physicians use dose-volume parameters to evaluate dose plans it is of interest to see if there is a correlationbetween these parameters and the linear penalty function. In Paper C experiments areused to investigate the correlation between dose-volume parameters and the objectivevalue. The results in the paper indicate that the correlation is poor. It seems that theobjective function can make a rough division of plans into better and worse, however itcan not distinguish the best solution with respect to dose-volume parameters. If there isno correlation then it is unlikely that solutions found by solving the linear penalty modelare close to being optimal in the sense of the dose-volume parameters. Given the resultsfrom Paper C it hence seems that the linear penalty problem does not find an optimalsolution with respect to the clinical parameters used for evaluating plans.

4.2 Optimizing catheter positions 37

Figure 4.1: Illustration of the variables that have to be basic variables given the doseat dose-calculation point ı, in structure s.

4.2 Optimizing catheter positions

As mentioned in Section 3.3.1 there have been a few attempts to include the positioningof the catheters into the optimization of dose plans. These attempts are however eithervery simple or only scantily described. In Paper B we develop an optimization modelthat includes catheter placement by introducing a set of feasible catheter positions andassigning to each position a binary variable that describes if it is used or not; the modelcan be found in Table 4.2. The suggested model is however computationally demandingto solve and we have therefore also developed three heuristics (a tabu search, a variableneighbourhood search, and a genetic algorithm) to solve the new model.

Of the developed heuristics, variable neighbourhood search is clearly the best, outper-


ParametersS = Set of structures.Ns = Number of dose-calculation points in structure s.K = Set of possible catheter positions.nk = Number of dwell points in catheter k.Dmins = Lower dose limit for structure s.


Mmins = Penalty for underdosage of structure s.

Mmaxs = Penalty for overdosage of structure s.

dsij = Dose-rate contribution from dwell point j to dose-calculation point i in structure s.

B1 = Large number.B2 = Large number.Variableswsi = Penalty for dose-calculation point i in structure s.tj = Dwell time in point j.yk = Binary variable for catheter k.

Objective function

min∑s∈S

Ns∑i=1

wsiNs

Subject to

wsi +∑k∈K

nk∑j=1

Mmins dsikjt

kj ≥Mmin

s Dmins ∀i = 1 . . . Ns, s ∈ S

wsi −∑k∈K

nk∑j=1

Mmaxs dsikjt

kj ≥ −Mmax

s Dmaxs ∀i = 1 . . . Ns, s ∈ S∑

k∈K

yk = C

tkj ≤ B1yk ∀j = 1 . . . nk, k ∈ Knk∑j=1

tkj ≤ B2yk ∀k ∈ K

wsi ≥ 0 ∀i = 1 . . . Ns, s ∈ Stkj ≥ 0 ∀j = 1 . . . nk, k ∈ Kyk ∈ {0, 1} ∀k ∈ K.

Table 4.2: The model presented in Paper B to optimize catheter positioning anddwell time distribution simultaneously.

4.2 Optimizing catheter positions 39

forming the other heuristics in all test cases and a state-of-the-art optimization software(CPLEX) in most cases. The variable neighbourhood search heuristic utilise three kindsof neighbourhoods:

Neighbourhood 1: Move one catheter within a subset R of K, or, mathematically, K2 ∈N(K1) if (|K1∩K2| = C−1)∧(K14K2 ⊆ R). The set R is generated uniformlyrandomly in each iteration as to include both used and unused catheters.

Neighbourhood 2: Move one catheter, or, mathematically,K2 ∈ N(K1) if (|K1∩K2| =C − 1) ∧ (K1 4K2 ⊆ K).

Neighbourhood 3: As neighbourhood 1, but 2 catheters are to be moved, that is, mathe-matically, K2 ∈ N(K1) if (|K1 ∩K2| = C − 2) ∧ (K1 4K2 ⊆ R).

Here K1 and K2 are two solutions, N(K1) are the neighbours of K1, and4 is the sym-metric difference. A pseudo-code of our variable neighbourhood search can be found inTable 4.3. The best neighbour is found by solving an integer problem, since an explicitenumeration of all neighbours is time consuming. The integer problem that is solved issimilar to the one in Table 4.2 but with additional constraints. With these constraints theproblem can be solved to optimality fairly quickly.

Solution=generate a random solutionCurrentNeighbourhood=1Counter=0while not fulfilled stopping criteria do

NewSolution= best neighbour of Solution in CurrentNeighbourhoodif NewSolution is better than Solution then

CurrentNeighbourhood=1Solution=NewSolutionCounter=0

elseCounter=Counter+1;if Counter>CounterLimit(CurrentNeighbourhood) then

CurrentNeighbour=CurrentNeighbour+1Counter=0if CurrentNeighbour>3 then

Perturb Solution randomlyCurrentNeighbour=1

end ifend if

end ifend while

Table 4.3: Pseudo-code for our variable neighbourhood search.

The tabu search performs quite evenly with CPLEX given equal time, and the geneticalgorithm performs quite poorly, around 50% worse than CPLEX. The most interestingresult from Paper B is however probably the improvement in objective value achieved


Settings Lower bound Upper boundStandard 133 228No cuts 134 214

No cuts+subcuts 194 203Table 4.4: Runs of 15 hour with CPLEX (10.0) using different settings (cuts or not)and our restricting inequalities subcuts for one patient.

when integrating catheter positioning in the optimization, compared to being restricted tothe positions used in clinical practice; the mean improvement was 53%.

Before starting to develop the heuristics presented in Paper B we investigated if tuningCPLEX or adding additional constraints could improve the solutions found when tryingto solve the model in Table 4.2. What we noticed was that when turning off the generationof cuts in CPLEX, the solutions found improved (see Table 4.4), which implies that thestandard integer cuts does not work well for the given formulation. It might therefore beinteresting to investigate if it is possible to find specialised cuts, if one intends to use thismodel.

We also considered adding some restricting inequalities that in practice are likelyto hold for good solutions. Our in-equalities limited the number of catheters used insubvolumes. They are reasonable because if many catheters are used in one subvolumethere will not be enough catheters left to cover the rest of the treatment volume. Sincethe possible catheters positions we used in our tests correspond to the possible positionsof the template they formed a square grid. The inequalities added (referred to as subcuts)limited the number of catheters used in each square to two. The subcuts are illustrated inFigure 4.2 and they are mathematically described by creating a set Sqi for each square i,containing the catheters in the square and then restricting the y-sum for each square:∑

k∈Sqi

yk ≤ 2 ∀i. (4.4)

Tests show that these subcuts support finding better solutions (see Table 4.4) even thoughthey restrict the feasible space. In the paper we choose not to include the subcuts in

Figure 4.2: The subcuts limit the number of used catheters of those within the boxto at most two.

4.3 Future research 41

the model since we wanted to avoid unnecessary restrictions, however we turned off thegeneration of cuts in CPLEX since this improve the solutions found as shown above.

4.3 Future research

We have in Paper A and Paper C shown that the linear penalty model has some undesiredproperties. Future research should hence aim to develop new models that do not havethese properties and that better include what physicians look for in good dose plans. Ourbelief is that one way to achieve this is to more explicitly include dose-volume parame-ters, or other measures used for evaluation, in the optimization process. This is howeverdifficult since such parameters often are computationally demanding and non-convex.

Since Paper B showed that including catheter positioning in the optimization processimproved the quality of the plans, this decision should also be included in the new models.Such models including both catheter positioning and dose-volume parameters will prob-ably be very hard to solve, therefore tailored solution techniques are probably needed.

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