AUTOMATED RADIATION THERAPY TREATMENT PLANNING BY ...1258306/FULLTEXT01.pdf · Abstract Every...

AUTOMATED RADIATION THERAPY TREATMENT PLANNINGBY INCREASED ACCURACY OF OPTIMIZATION TOOLS

Automated radiationtherapy treatment planningby increased accuracy ofoptimization tools

LOVISA ENGBERG

Doctoral thesis in applied and computational mathematicsStockholm, Sweden 2018

KTH Royal Institute of TechnologySchool of Engineering Sciences

TRITA-SCI-FOU 2018:43ISBN 978-91-7729-943-1

Optimeringslära och systemteoriMatematiska institutionen

KTH100 44 Stockholm

Akademisk avhandling som med tillstånd av KTH i Stockholm framlägges till offentliggranskning för avläggande av teknologie doktorsexamen fredagen den 23 november2018 kl 10:00 i sal F3, KTH, Lindstedtsvägen 26, Stockholm.

© Lovisa Engberg, oktober 2018

Tryck: Universitetsservice US-AB, Stockholm

Abstract

Every radiation therapy treatment is preceded by a treatment planning phase. In thisphase, a treatment plan that specifies exactly how to irradiate the patient is designed bythe treatment planner. Since the introduction of intensity-modulated radiation therapyinto clinical practice in the 1990’s, treatment planning involves, and requires, the use ofadvanced optimization tools due to the largely increased degrees of freedom in treatmentspecifications compared to earlier radiation therapy techniques.

The aim of treatment planning is to create a plan that results in the, in some sense, besttreatment—a treatment that at the same time reflects the patient-specific clinical goals,achieves the best possible quality, and adheres to other possible preferences of the on-cologist or of the clinic. Despite dedicated treatment planning systems available withadvanced optimization tools, treatment planning is often referred to as a complicated pro-cess involving many iterations with successively adjusted parameters. Over the years,a request has emerged from the clinical and treatment planners’ side to make treatmentplanning less time-consuming and more straightforward, and the methods subsequentlydeveloped as a response have come to be referred to as methods for automated treatmentplanning.

In this thesis, a framework for automated treatment planning is proposed and its po-tential and flexibility investigated. The focus is placed on increasing the accuracy of theoptimization tools, aiming at achieving a less complicated treatment planning process thatis driven by intuition rather than, as currently, trial and error. The suggested framework iscontrasted to a class of methods dominating in the literature, which applies a more clas-sical view of automation to treatment planning and strives towards reducing any type ofhuman interaction. To increase the accuracy of the optimization tools, the underlying so-called objective functions are reformulated to better correlate with measures of treatmentplan quality while possessing mathematical properties favorable for optimization. An im-portant step is to show that the suggested framework not only is theoretically desirable,but also useful in practice. An interior-point method is therefore tailored to the specificstructure of the novel optimization formulation, and is applied throughout the thesis, todemonstrate tractability. Numerical studies support the idea of the suggested frameworkequipping the treatment planner with more accurate and thereby less complicated tools tomore straightforwardly handle the intrinsically complex process that constitutes treatmentplanning.

Keywords: Optimization, intensity-modulated radiation therapy, radiation therapytreatment planning, automated radiation therapy treatment planning, interior-pointmethods

vii

Sammanfattning

Varje strålbehandling föregås av en dosplaneringsfas. Under dosplaneringsfasen ska-pas den strålbehandlingsplan som exakt beskriver hur strålbehandlingen ska genomföras.Sedan 1990-talet och den så kallade intensitetsmodulerade strålbehandlingens inträdei klinisk praxis har dosplanering kommit att betyda och rent av kräva användande avavancerade optimeringsverktyg – en konsekvens av den kraftigt ökade mängden frihets-grader jämfört med tidigare strålbehandlingstekniker.

Det övergripande målet med dosplanering är att skapa en plan som i någon meningger den bästa strålbehandlingen. En sådan behandling ska i synnerhet spegla de kliniskamål som satts upp för den enskilda patienten, i allmänhet uppnå bästa möjliga kvalitetsamt förhålla sig till eventuella övriga önskemål från onkologen eller kliniken. Utbudetav dosplaneringssystem med avancerade optimeringsverktyg är stort och användandet ut-brett, men trots detta beskrivs ofta dosplanering som en komplicerad process där finjuster-ing av parametrar utgör en väsentlig del. Därför har efterfrågan på hjälpmedel för mindretidskrävande och mer rättfram dosplanering under det senaste årtiondet vuxit fram. Demetoder som utvecklats som svar benämns som metoder för automatiserad dosplanering.

I det här arbetet föreslås och utvärderas ett ramverk för automatiserad dosplanering.Fokus har lagts på optimeringsverktygen och att förbättra noggrannheten i dessa, föratt därigenom skapa förutsättningar för mindre komplicerad dosplanering där intuitionsnarare än ett tidskrävande experimenterande driver processen framåt. Ramverket somhär föreslås ställs i kontrast till en annan, dominerande klass av föreslagna metoder förautomatiserad dosplanering som bygger på en mer klassisk syn på automatisering, det villsäga, som strävar efter att minska människa-datorinteraktion i allmänhet. Förbättring avoptimeringsverktygens noggrannhet uppnås genom att omformulera de bakomliggande såkallade målfunktionerna till alternativ som bättre korrelerar med givna kvalitetsmått ochsom samtidigt har matematiska egenskaper som är önskvärda vid optimering. Ett viktigtsteg är dock att visa att det föreslagna ramverket inte bara är teoretiskt lämpligt, utanatt det också är praktiskt hanterbart ur beräkningssynpunkt. En inrepunktsmetod anpas-sas till den specifika strukturen på det nya, storskaliga optimeringsproblemet för att visajust detta. Fallstudier stödjer idén om att det föreslagna ramverket ger mer noggrannaoch därmed lätthanterliga optimeringsverktyg, med vilka dosplaneringens ofrånkomligakomplexitet kan hanteras på ett mer effektivt sätt.

Nyckelord: Optimering, intensitetsmodulerad strålbehandling, dosplanering,automatiserad dosplanering, inrepunktsmetoder

viii

Preface

Pursuing your doctoral studies is, thank goodness, a once-in-a-lifetime experi-ence. The following words are to show why my decision in 2013 to take on thischallenge is not one that I regret.

I am grateful

to Johan Löf, the CEO of RaySearch Laboratories, for not just funding a five-year research project, but also for thereby creating an opportunity for, inthis case, me to pursue my doctoral studies;

to Anders Forsgren, my main supervisor, for his support and guidance on a bothacademic and personal level; Anders’ to me unconditional encouragementhas been a key in making this journey enjoyable; and

to Kjell Eriksson and Björn Hårdemark, my industrial supervisors, for takingactive parts in this semi-academic project in an era of intensive expansionof RaySearch Laboratories; for ideas, understanding, and support.

There are many similarities of this thesis with the first movement of the MoonlightSonata by Beethoven—another piece (among very few) which I know by heart.In particular, perhaps more important than using technical skill, “this whole piecemust be played very delicately”. I am indeed grateful to all of my supervisors forrespecting my way of facing research ideas: let’s say, very delicately.

Superceding “Albin and Rasmus” as a third-generation industrial graduate stu-dent at RaySearch Laboratories has been a challenge for my self-confidence; itwould have been, I believe, for anyone’s. But it has been the more rewarding inmany other ways. I am certain that seeing the strong competences of the entireResearch Department has pushed me towards better achievements. The same canbe said about the Division of Optimization and Systems Theory at KTH, and I amparticularly grateful for the inclusive atmosphere provided by this group.

Slutligen, tack till mamma, pappa, Fredrika och Emil, som alltid står bakom mig.Och tack till dig, Per, för att du alltid står bredvid mig.

Lidingö, October 2018Lovisa Engberg

ix

Contents

Introduction 11 Radiation therapy . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Intensity-modulated radiation therapy (IMRT) . . . . . . . 11.2 Dose calculation . . . . . . . . . . . . . . . . . . . . . . 21.3 IMRT delivery techniques . . . . . . . . . . . . . . . . . 3

2 Treatment planning . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 From forward to inverse treatment planning . . . . . . . . 42.2 Automated treatment planning . . . . . . . . . . . . . . . 5

3 Treatment plan optimization . . . . . . . . . . . . . . . . . . . . 63.1 Measures of plan quality . . . . . . . . . . . . . . . . . . 83.2 Objective functions . . . . . . . . . . . . . . . . . . . . . 9

4 Methods for treatment plan optimization . . . . . . . . . . . . . . 124.1 Optimization method applied in this thesis . . . . . . . . . 124.2 Fluence map or machine parameter optimization? . . . . . 13

5 Thesis summary and contributions . . . . . . . . . . . . . . . . . 155.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 155.2 Summary of appended papers . . . . . . . . . . . . . . . 155.3 Main contributions . . . . . . . . . . . . . . . . . . . . . 17

6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

A Explicit optimization of plan quality measures in IMRTtreatment planning 29A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30A.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

A.2.1 Conventional planning objectives . . . . . . . . . . . . . 32A.2.2 Proposed planning objectives . . . . . . . . . . . . . . . . 33A.2.3 A note on planning constraints . . . . . . . . . . . . . . . 36A.2.4 A note on maximum and minimum dose . . . . . . . . . . 36

xi

A.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37A.3.1 Patient cases . . . . . . . . . . . . . . . . . . . . . . . . 37A.3.2 DVH statistics of treatment plan cohorts . . . . . . . . . . 38

A.4 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . 40A.4.1 Interior-point method for specific problem structure . . . . 41A.4.2 Performance of method implementation . . . . . . . . . . 43

A.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43A.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

B Increased accuracy of planning tools for optimization ofDMLC delivery of radiotherapy through reformulatedobjective functions 53B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54B.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

B.2.1 Formulation of MCO objective functions . . . . . . . . . 56B.2.2 Modelling of DMLC deliverability . . . . . . . . . . . . . 57B.2.3 On solving the proposed MCO formulation . . . . . . . . 60

B.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62B.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65B.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69B.A Full proposed formulation . . . . . . . . . . . . . . . . . . . . . 69B.B List of PTV and OAR requirements . . . . . . . . . . . . . . . . 70B.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

C On tradeoffs between treatment time and plan quality ofVMAT with sliding-window delivery 79C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80C.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

C.2.1 Optimization formulation . . . . . . . . . . . . . . . . . 82C.2.2 Accurate dose computation for sliding-window VMAT . . 83C.2.3 Heuristic methods . . . . . . . . . . . . . . . . . . . . . 85

C.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87C.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91C.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92C.A A MILP formulation of dose constraints . . . . . . . . . . . . . . 93C.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

xii

Introduction

This thesis consists of an introduction and three appended papers. The purposeof the introduction is to give the specific radiation therapy treatment planningand optimization contexts within which the research has been conducted. Themain contributions of the thesis to the field of automated treatment planning aresummarized in Section 5.

1 Radiation therapy

Radiation therapy has been used as a treatment of cancer since the end of the19th century. What started as a treatment technique of experimental nature, tosome extent driven by hypotheses and expectations, has now become an art ofgreat precision and accuracy in medical, physical, and mathematical aspects. To-day, the field of radiation therapy relies on a multitude of technologies, such asmedical physics, radiobiology, medical imaging, image processing, mathematicaloptimization, and software and hardware development.

1.1 Intensity-modulated radiation therapy (IMRT)

The overall goal of radiation therapy is to deliver a high, uniform radiation doseto the tumor while sparing surrounding healthy tissue to avoid complications. Inexternal radiation therapy, this goal is achieved by irradiating the patient withmultiple beams from different directions, so as to have an intersecting irradiatedvolume that as accurately as possible conforms to the shape of the tumor.

Since the 1970’s, radiation therapy has evolved from delivering simple rectan-gular (cross-sectionally) beams, via three-dimensional conformal radiation ther-apy (3D-CRT) with tumor projection-shaped beams and a better ability to controlthe shape of the intersecting irradiated volume, to intensity-modulated radiationtherapy (IMRT) with a potential to offer highly conformal treatments and a sig-nificantly reduced risk of complications. IMRT became practically possible bythe introduction of the multileaf collimator (MLC) (Figure 1) that has been com-mercially available since the 1990’s. The MLC is a device mounted at the head ofthe treatment machine to control the shape of the beam. Its radiation-absorbingleaves, arranged on opposing sides, can be moved against or away from each other

1

2 INTRODUCTION

Figure 1. A schematic illustration of a multileaf collimator (MLC). The opposingradiation-absorbing leaves can be moved into different positions to sculpt the beamas it passes through the aperture.

into different positions to form almost any aperture through which the beam canpass. The increased conformity and sparing of healthy tissue obtained with IMRTis due to the modulation of not only the shape of the beam as in 3D-CRT, but alsoits intensity: different parts of the beam can be intensified or toned down depend-ing on what will be encountered along its path through the body. How to obtain acertain intensity pattern using the MLC depends on the choice of IMRT deliverytechnique and is further described in Section 1.3.

Comprehensive reviews of the history of IMRT are given by Webb [55] andBortfeld [9].

1.2 Dose calculation

The beams consist of megavoltage X-rays that are produced inside the treatmentmachine, the linear accelerator. On their way through a medium such as thehuman body, the X-rays deposit parts of their energy and create what is referredto as dose (energy per unit mass [Gy]).

From a mathematical perspective (here, a simplification of the physical per-spective), calculating the resulting dose inside the body given a set of IMRTbeams starts by discretizing the three-dimensional body into m voxels (volumepixels) and the two-dimensional planes orthogonal to the incident beam direc-tions into n bixels (beam pixels). An n-dimensional vector x, the fluence map, isintroduced to represent the modulated intensity of the IMRT beam at each bixel,and an m-dimensional vector d, the dose distribution, to represent the resulting

AUTOMATED PLANNING BY INCREASED ACCURACY OF OPTIMIZATION TOOLS 3

dose. Given a fluence map, going to the dose distribution is a straightforward lin-ear operation if also assuming an m × n-dimensional dose deposition matrix Pavailable:

d = Px. (1)

However, calculating the dose deposition matrix P given the body geometry andbeam configuration requires advanced dose calculation algorithms. In commercialtreatment planning systems, the dose deposition matrix is seldom generated ex-plicitly for memory-saving reasons and due to it only being valid for a given beamconfiguration—it is often computationally more efficient to apply a dose calcula-tion algorithm to generate the distribution d directly whenever needed [59]. Al-gorithms of different computational complexity can be used depending on the de-sired accuracy. For instance, a more efficient algorithm is commonly used duringtreatment planning, but a clinically accurate dose distribution is always calculatedand quality-checked before going forward with the radiation therapy treatment.See, e.g., [39] for an introduction from the physical perspective to dose calcula-tion algorithms used in radiation therapy.

It should be mentioned that besides photons (X-rays), which is the most widelyused treatment modality, also protons and even heavier particles are used in radi-ation therapy. As only photon-based radiation therapy is considered in this thesis,an introduction to proton and particle therapy is not included. See, e.g., [53] fora review on proton-based radiation therapy, and [48] for an introduction to protontherapy treatment planning.

1.3 IMRT delivery techniques

A modulated beam intensity is the result of using multiple, consecutive MLCapertures. The aperture can be varied both statically, with the treatment machineidling while the MLC leaves are moved, or dynamically, with a continuouslychanging aperture. Static and dynamic MLC motions define two different IMRTdelivery techniques referred to as SMLC (or, “step-and-shoot”) and DMLC, re-spectively. From the mathematical point of view, different delivery techniquescould require different approaches concerning both the formulation and the solv-ing of the associated optimization problem. However, choosing one above theother is always a clinical decision: it depends, e.g., on treatment time limitations,SMLC being in general slightly slower due to the idling phases; treatment ma-chine capabilities or the level of “wear and tear” acceptance, DMLC being ingeneral more demanding in that sense; and institutional or the oncologist’s ownpreferences [1, 17].

4 INTRODUCTION

By allowing continuous rotation of the treatment machine head around thepatient during irradiation instead of restricting to discrete beam directions as as-sumed above, the concepts of a technically more complex IMRT delivery tech-nique are outlined. This delivery technique most often goes by the name volumetric-modulated arc therapy (VMAT). In its currently most widely used form, VMATwas first formulated in mathematical terms by Otto [43]. The primary advantageis the opportunity to obtain highly time-efficient treatments, but as with SMLCand DMLC, treatment machine and quality control considerations also play a rolein the clinical decision on whether to treat using VMAT. In addition, VMAT isassociated with even greater mathematical challenges and places larger demandson both human and software resources.

2 Treatment planning

Every IMRT treatment is preceded by a treatment planning phase. In this phase,the treatment plan that specifies exactly how to irradiate the patient—from whatdirections, using which beam shapes, et c.—is generated by the treatment plan-ner. The aim of treatment planning is to find a plan that results in the treatmentfulfilling the clinical goals—a patient-specific adaptation of the overall radiationtherapy goal defined in Section 1.1—defined by the oncologist.

2.1 From forward to inverse treatment planning

Historically, there have been two ways of proceeding with treatment planning: ina forward or in an inverse manner. Forward planning refers to the procedure ofgenerating treatment plans by hand, i.e., by manually defining all treatment planspecifications and then calculating the resulting dose distribution using dedicatedsoftware. If the dose distribution cannot be deemed good enough with respectto the clinical goals, the manually set treatment plan specifications have to bemanually revised. It is easy to understand that forward planning is essentiallyonly applicable to “at most” 3D-CRT, since for IMRT, there are too many planspecifications to consider. The idea behind inverse planning is to formulate amathematical optimization problem that takes information about the clinical goalsas a parameter input. By then applying methods for optimization, the ambitionand expectations are to directly obtain the treatment plan that in some sense bestfulfills the clinical goals.

The concepts of inverse planning were first described in the early 1980’s byBrahme et al. [12]. In two later publications, Webb [54] and Bortfeld et al. [11]


contributed to establishing the optimization framework. But the optimized treat-ment plans were expressed in terms of heterogeneous fluence maps that requireIMRT delivery, and it was therefore not until the MLC was in place in the 1990’sthat the benefit of inverse planning could be implemented into practice—inverseplanning and MLCs together formed IMRT. In the early 2000’s, inverse planningand IMRT were widely commercialized and became clinical routine within a fewyears [38]. Since then, forward planning is seldom being considered. The devel-opment of software for inverse planning is now a high-technology industry, andthe treatment planning systems of today can offer a much broader spectrum oftools than during the forward planning era, when essentially computerized dosecalculation was sufficient.

Understanding the current inverse planning process is of utmost importancefor this thesis. Despite the use of advanced optimization methods, the process iscomplicated and requires both experience and skills from the treatment planner—a comprehensive study by Nelms et al. [42] reports that the level of experienceof the treatment planner, among other factors, affects the quality of the treatmentplan. For each patient, the treatment planner takes the individually set clinicalgoals into consideration when specifying the objectives of the treatment in thetreatment planning system. The objectives provided are translated by the soft-ware into objective functions of an optimization problem (see Section 3). Afteroptimization, the treatment planner examines the resulting dose distribution andverifies that the clinical goals, at least to a satisfiable degree, are met. If not, theobjectives are updated, and the dose distribution is re-optimized; this procedure isrepeated until the treatment plan is approved by the oncologist.

The requirements on experience and skills for successful inverse planning aredue to the conflicting nature of the clinical goals (indeed, to irradiate the tumor isin conflict with sparing the surrounding tissue), and to the difficulty in assessingthis conflict. The skilled treatment planner need to be aware of how an objectivefunction with conflicting constituents is handled during optimization, and to beable to roughly predict how the resulting dose distribution is affected by a certainchoice of objectives; thus knows how to adapt the objectives in order to approachthe clinical goals.

2.2 Automated treatment planning

In the past decade, a request has emerged from the clinical and treatment planners’side to make inverse treatment planning less time-consuming and more straight-forward. It is not uncommon that several person-hours are spent on treatment

6 INTRODUCTION

planning for each new patient case. The response from the research communityand from the industry has been a new treatment planning paradigm, automatedtreatment planning. There is no strict definition of automated planning, and thelevel of user interaction at which the treatment planning process is considered au-tomated varies—as does the level claimed necessary to leave the treatment plannerwith sufficient control over the planning process.

The most prominent trend in automated planning is the use of machine learn-ing techniques and the utilization of the fact that many cases share patient-geo-metrical and clinical-goal similarities. Given the recent explosion of literatureinvolving machine learning, treatment planning can be concluded having enteredyet another field of high-technological research and development: computer sci-ence. Examples of machine learning methods for treatment planning can be foundin [4,36,37,51,52,58] (in fact, a very early publication discussing machine learn-ing in treatment planning dates back to 1992 [6]). Similar to other machine learn-ing applications, the methods rely on the existence of a database consisting of,in this case, patient geometry information and associated treatment plans. Thepredicted data for a new patient is either used as decision support during the treat-ment planning process or is already on the form of a treatment plan ready to be re-viewed. Instead of using machine learning techniques, some have adopted a moremechanical view of automation. For instance, algorithms or scripts to mimic thetreatment planner’s successive adaptation of objectives to eventually reach clin-ical goal fulfillment have been developed, usually based on a specific treatmentplanning system. Examples can be found in [25, 29, 61].

Common for the abovementioned methods is that they aim at handling symp-toms instead of underlying causes of a cumbersome treatment planning process.The contributions of this thesis to the field of automated treatment planning, assummarized in Section 5, involve attempts to overcome the causes.

3 Treatment plan optimization

Treatment plan optimization refers to the matter of formulating and solving themathematical problems arising in inverse treatment planning. The formulation ofa general treatment plan optimization problem is given by

minimizex∈X , d∈Rm

f(d)

subject to c(d) ≤ 0,

d = Px,

(2)


where X denotes the set of fluence maps, f(d) is the objective function to beminimized, and c(d) ≤ 0 (and d = Px) the constraints. Depending on themathematical characteristics of the functions f and c and the set X , (2) can bemore or less difficult to solve. Slightly simplified, (2) can be solved to globaloptimality if f and c are convex functions and X a convex set; otherwise, (2) isnonconvex, in which case only local optima can in general be found.

The fact that several conflicting clinical goals are to be handled makes inverseplanning a multicriteria optimization (MCO) problem. The multicriteria objectivefunction is given by the vector of scalar-valued constituent objective functions,[

f1(d), · · · , fK(d)]T.

Traditionally, the multicriterial nature of inverse planning has been handled byweighted-sum scalarization, i.e., accumulation of the constituent objective func-tions using positive weighting factors wk, k = 1, . . . ,K, into the singlecriteriaobjective function

f(d) =K∑k=1

wkfk(d). (3)

The weighting factors are specified by the treatment planner along with the ob-jectives of the treatment and are interpreted as priorities: the larger the weight-ing factor, the more emphasis should be given to fulfill the associated objective.In the last decade, dedicated methods to handle the MCO problem have beendeveloped by which the manual specification of weighting factors is eliminatedthrough incorporation into an outer formalism. A welcome consequence is a re-duction in workload of the treatment planner [20]. The most studied class of MCOmethods for inverse planning is a posteriori methods. These methods generatea well-distributed (in constituent objective function values) set of a fixed num-ber of treatment plans, each optimal to a weighted-sum instance, between whichthe treatment planner can then interactively “navigate” to explore treatment plansapproximately optimal to any weighted-sum instance. Examples of a posteriorimethods can be found in [7, 19, 21, 41].

It is in its place to recall three already introduced key concepts of inverseplanning, and to define a fourth one:

• the initial patient-specific clinical goals specified by the oncologist,

• the objectives specified by the treatment planner based on the clinical goals;objectives include extensions of the clinical goals with artificial “help ob-

8 INTRODUCTION

Figure 2. A (cumulative) dose-volume histogram (DVH) example illustrated for atumor (red) and a critical structure (blue). The interpretation of the point (d̂, v̂) isthat a volume fraction v̂ receives a dose of at least d̂ Gy.

jectives” and modifications of overly optimistic or too loose clinical goals—again, the skilled treatment planner knows how to adapt the objectives inorder to approach (or exceed) the clinical goals,

• the constituent objective functions into which the objectives are mathemat-ically translated by the treatment planning system, and

• the measures of plan quality used to evaluate the quantifiable quality of agiven treatment plan.

These four concepts are strongly interconnected—e.g., clinical goals and objec-tives are usually formulated in terms of explicit thresholds of measures of planquality—but are kept distinguished from each other here to give proper under-standing of the current treatment planning process.

3.1 Measures of plan quality

Unfortunately from the mathematician’s perspective, there is no general agree-ment on what exactly makes a high-quality treatment plan and dose distribution,and certainly not an optimal—the clinical goals are not self-contained in thatsense. There is, however, some quantifiable measures that dominate in qualityassessment of treatment plans.

An essential tool in quality assessment of treatment plans is the (cumulative)dose-volume histogram (DVH) (Figure 2). The DVH is a way of visualizing thethree-dimensional dose distribution in any tumor, organ, or other structure in atwo-dimensional graph. The interpretation of a point (d̂, v̂) on the curve is that a


volume fraction v̂ of the structure receives a dose of at least d̂ Gy. Two importantmeasures of plan quality are given by the parameterizations of the DVH curve byd̂ and v̂ as functions of the dose distribution, referred to as and denoted by

• the dose-at-volume [Gy] measure, d̂ = D(d; v̂), and

• the volume-at-dose [%] measure, v̂ = V (d; d̂).

The minimum, maximum, and average dose [Gy] measures are also being fre-quently evaluated and reported. A majority of the oncologist’s clinical goals andthe treatment planner’s objectives are expressed in terms of lower (for tumors) orupper bounds on these five measures (as recommended by the Radiation TherapyOncology Group (RTOG), Philadelphia, PA, USA). Commonly reported valuesfor tumors are D(d; vref) for vref in the ranges 1–5 % and 95–99 %, and V (d; dref)for dref in the range 90–110 % of the prescribed tumor dose; for organs and healthytissue, a greater variation is seen due to the different sensitivity to radiation [30].Specific combinations of doses-at-volume or volumes-at-dose are sometimes con-sidered, in particular

• a homogeneity index, often (D(d; 2)−D(d; 98))/D(d; 50) [30], and

• a conformity index, often the ratio between the body volume receiving asignificant dose and the tumor volume [32].

Also relating to the DVH is the mean-tail-dose [Gy] measure (Figure 3). Themean-tail-dose was introduced by Romeijn et al. [47] as an alternative to dose-at-volume and volume-at-dose for its favorable mathematical properties, but themeasure has not yet been used in clinical quality assessment. The upper and lowermean-tail-dose is defined as the average dose of the upper and lower DVH “tail”,i.e., the average dose of the part of the curve at right or left of a point (d̂, v̂) on thecurve.

Measures with biological connections are sometimes evaluated and reported,such as the tumor-control probability (TCP) and the normal-tissue complicationprobability (NTCP); but their use in quality assessment is as of today not recom-mended, or at least limited, due to the uncertainty associated with, and continualrefinement of, biological models [30].

3.2 Objective functions

As described in Section 2.1 and recalled in the beginning of this chapter, thetreatment planner specifies the objectives of the treatment based on the clinical

10 INTRODUCTION

Figure 3. An illustration of mean-tail-dose. The upper and lower mean-tail-dosesare defined as the average dose of the upper (blue) and lower (red) DVH tail givenby a volume fraction v̂.

goals, whereafter the treatment planning system translates each of these into acorresponding constituent objective function of a treatment plan MCO problem.Consequently, the objective functions constitute the “optimization tools” avail-able to the treatment planner and are central to this thesis. The importance andchallenges of finding accurate formulations of objective functions with close con-nections to the clinical goals and associated measures of plan quality were broughtup already in 1994 by Mohan et al. [40] and were still discussed on a fundamental(and practical) level in 2005 in a comprehensive paper by Kessler et al. [31].

By convention, and already in the first publications on treatment plan opti-mization [11,54], a penalty-function based paradigm inherited from the analogousproblem of image reconstruction has been used when formulating objective func-tions for inverse planning. In the clinical treatment planning systems of today,the objective functions commonly found are formulated as quadratic penaltiesassociated with violation of the input objectives. A conventional objective func-tion aimed at controlling the dose-at-volume objective D(d; vref) ≤ dref or thevolume-at-dose objective V (d; dref) ≤ vref imposed on a structure with voxels S,S ⊂ {1, . . . ,m}, is given by

f(d) =∑i∈S:

dref≤ di≤D(d;vref)

∆Si

(di − dref)2 , (4)

where di is the dose in voxel i and ∆Si denotes the relative volume of the structure

in voxel i [8]. Figure 4 gives a helpful visual interpretation in the DVH graph tomake sense of which voxels are considered in the summation. The expressions


Figure 4. A DVH illustration of the conventional objective functions aimed atcontrolling the dose-at-volume or volume-at-dose objectives D(d; vref) ≤ dref orV (d; dref) ≤ vref (left), and D(d; vref) ≥ dref or V (dref) ≥ vref (right). The high-lighted areas point out the violations which are quadratically (in dose) penalized.

for the reversed objectives, D(d; vref) ≥ dref and V (d; dref) ≥ vref, are analogous,

f(d) =∑i∈S:

D(d;vref)≤ di≤ dref

∆Si

(dref − di

)2. (5)

Under minimization, (4) and (5) push the DVH curve towards the point (dref, vref).

The conventional objective functions are nonconvex and nondifferentiable dueto the dependence of the summation index on the dose-at-volume measure, whichrequires the use of integer variables for exact handling. Exact formulations usinginteger variables have been used in [27,33–35]. Nonconvex and nondifferentiableoptimization problems are in general difficult to handle, yet computational ex-perience in treatment plan optimization indicates that the optimization methodscommonly applied are able to overlook these theoretical drawbacks of (4) and(5) [60]. However, a known—although often disregarded—issue is the fact thatthe gradient of these quadratic penalties vanishes as the function approaches itsminimum, and in case of a conventional constraint (0 ≥ c(d) := f(d)), vanishesinside the feasible region [23]. Consequently, when applying gradient-based op-timization methods, strict fulfillment of constraints is difficult to reach.

In this thesis, a framework with novel objective functions is suggested. Onepurpose is to overcome treatment planning difficulties associated with noncon-vexities, nondifferentiabilities, and vanishing gradients of the objective functions;another is to improve correlation with measures of plan quality.

12 INTRODUCTION

4 Methods for treatment plan optimization

Optimization methods are applied to solve treatment plan optimization problems.Different approaches are used depending on the mathematical characteristics ofthe formulation, such as whether it is convex, linear, or nonlinear, or whether itsobjective functions and constraints are differentiable.

Given the conventional objective functions, the weighted-sum instances of (2)are constrained nonlinear programming problems, or nonlinear programs. Theoptimization method primarily used in treatment planning systems to solve theseproblems—as well as used in the early publications of inverse planning—is bysequential quadratic programming (SQP) [11, 28]. SQP amounts to solving a se-quence of positive definite quadratic programs, for which there in turn exist meth-ods to find the global optimum. A comprehensive introduction to SQP is givenby Gill et al. [24], and its successful application to treatment plan optimization,where typically only a few iterations are required, has been studied by Carlsson etal. [15] and Carlsson and Forsgren [14]. Other optimization methods seen in theliterature to solve the conventional nonlinear program include, e.g., interior-pointmethods; examples can be found in [2, 13].

4.1 Optimization method applied in this thesis

The objective functions and deliverability constraints considered in this thesis re-sult in the weighted-sum instances of (2) being large-scale linear programmingproblems, or linear programs. Linear programs constitute a well-studied segmentof optimization problems and a spectra of general-purpose optimization methodscan be applied, such as simplex, active-set, and interior-point methods [26]. Themethod applied in Papers A–C is of the latter class, of which a thorough intro-duction can be found in Wright [57]. By exploiting the structure imposed bythe specific optimization problem on arising systems of linear equations, interior-point methods can be adapted to improve efficiency in solving large-scale linearprograms. Examples of such structure utilization in engineering applications arefound in [16, 56]. Below, in demonstrating where the structure-exploiting op-portunity occurs, some fundamental linear programming concepts are referenced,such as primal, dual, and optimality conditions. The interested reader may againrefer to Wright [57] but understanding of these concepts is not necessary in orderto follow the reasoning in the following paragraph.


Given a standard-form linear program and its corresponding dual problem,

minimizez∈Rn

cT z maximizey∈Rm, s∈Rn

bT y

subject to Az = b, subject to AT y + s = c,

z ≥ 0, s ≥ 0,

(6)

a general interior-point method approaches the optimal solution (z∗, y∗, s∗) bytaking steps of length αk > 0 from the current iterate (zk, yk, sk) along the direc-tion given by the system of linearized perturbed optimality conditions, 0 AT I

A 0 0Sk 0 Zk

∆zk∆yk∆sk

= −

AT yk + sk − cAzk − b

ZkSke− µe

, (7)

where Sk = diag(sk) and Zk = diag(zk), e is the vector of ones, and µ is a posi-tive method parameter that is successively decreased to phase out the perturbation.The steplength αk must ensure nonnegativity of the iterates, thus is chosen suchthat

(zk+1, sk+1) = (zk, sk) + αk(∆zk,∆sk) > 0. (8)

Instead of the system of linear equations given in (7), smaller systems where ∆skor (∆zk,∆sk) have been eliminated are sometimes considered:(

−Z−1k Sk AT

A 0

)(∆zk∆yk

)= −

(AT yk + µZ−1k e− c

Azk − b

), (9)

or(AS−1k ZkA

T)

∆yk = −(AS−1k Zk

(AT yk + µZ−1k e− c

)− (Azk − b)

). (10)

Now, depending on a known structure of the coefficient matrix A, solving (7),(9), or (10) may be done computationally more efficiently than the direct ap-plication of a standard factorization algorithm, namely, by applying a specificblock-eliminating approach and a so-called Schur complement technique. In fact,existence of such an exploitable structure in A could be critical for the tractabilityof a problem in a given application, as seen in Paper A.

4.2 Fluence map or machine parameter optimization?

A common categorization of formulations of treatment plan optimization is bywhether the set of fluence maps X considered in (2) contains all positive fluence

14 INTRODUCTION

maps, i.e., if X = {x ∈ Rn : x ≥ 0}, or if only the fluence maps resulting frompractically realizable MLC apertures are included. The two categories have beennamed fluence map optimization (FMO) and direct machine parameter optimiza-tion (DMPO), and the latter is said to take the deliverability of the treatment planinto account.

As one might expect, there are pros and cons associated with both FMO andDMPO. FMO problems are generally considered easier to solve due to the relax-ation of deliverability constraints. On the other hand, the optimal fluence mapobtained using FMO must be post-processed and converted into machine param-eters with respect to the chosen IMRT delivery technique. An obvious risk isdegradation of the solution after conversion. Early examples of post-processingalgorithms include [10, 18], and more recent suggestions for the mathematicallycomplex VMAT delivery can be found in [5, 49]. As to DMPO problems, de-pending on the IMRT delivery technique, these can be almost arbitrarily difficultto both formulate and solve. Yet, on the other hand, studies have reported thatDMPO results in superior treatment plans compared to FMO with subsequent con-version [46,50]. Formulations of DMPO have been developed for most clinicallyused delivery techniques; to give a few examples, see [22,50] for SMLC, [44] forDMLC, and [45] for VMAT delivery.

Comprehensive formulations of DMPO are more seldom considered with in-creasing mathematical complexity of the IMRT delivery technique. While FMOwith conversion is one way forward, another is to introduce limitations of or makeassumptions regarding the delivery to simplify the DMPO problem and to im-prove its tractability. For instance, Papp and Unkelbach [44] have shown thatassuming DMLC delivery by unidirectional MLC leaf motions, “sweeps”, the setof physically realizable fluence maps can be described using linear, hence convexinequalities. A drawback with making such assumptions is that the resulting sim-plified DMPO problem may omit high-quality treatment plans. On the other hand,solving a simplified optimization problem to global optimum may give better re-sults than tackling a more comprehensive formulation, and may be the approachless sensitive to parameter perturbations—a desirable property in the design ofaccurate optimization tools in this thesis.

FMO is considered in Paper A, while DMPO and specifically the formulationby Papp and Unkelbach is considered in Papers B and C.


5 Thesis summary and contributions

Contributions of this thesis are within the field of automated treatment planning.The main contributions are summarized below, preceded by a motivation for thepath taken and a summary of each of the appended papers.

5.1 Motivation

During the commercialization of inverse treatment planning, forward planningwas often contrasted as a trial-and-error process due to the required iterative man-ual revision of treatment plan specifications. Ironically, the same epithet haslately come to be applied to inverse planning for reasons described in Sections 2.1and 3.2. Central to the direction taken by this thesis are the questions: How couldthat be, and how can it be avoided?

A view similar to that presented by Andersson et al. [3] is adopted: the treat-ment planning process is recognized as an inherently complex process, but a dis-tinction is made to complicated optimization tools to deal with the process. Itis assumed desirable that objective functions of the treatment plan optimizationproblem (i.e., the optimization tools) should strive towards clinical goal fulfill-ment, and in case fulfillment is impossible, should strive towards the best valuespossible in the associated measures of plan quality. It is hypothesized that suchbehavior of the optimization tools would require less trial and error, thus offer astreamlined treatment planning process.

5.2 Summary of appended papers

The co-authors of the three appended papers have acted as academic and industrialadvisors, suggesting directions of the research and supervising the work.

Paper A: Explicit optimization of plan quality measures in intensity-modulated radiation therapy treatment planning

Paper A is co-authored with Anders Forsgren, Kjell Eriksson, and BjörnHårdemark, and has been published in Medical Physics, Vol. 44, No. 6,pp. 2045–2053, 2017.

In this paper, a novel formulation of objective functions for IMRT treatment planoptimization is presented. The purpose of the novel formulation is to overcomethe known issues of the conventional penalty-based objective functions causing

16 INTRODUCTION

the trial-and-error behavior of inverse treatment planning. Two important aspectsof the proposed objective functions help in achieving this purpose: (1) the conven-tional penalty-based paradigm is abandoned in favor of a more explicit approach,and (2) convex approximations are used whenever needed in order to obtain a con-vex optimization problem. Fluence map optimization with the proposed objectivefunctions requires the introduction of a large number—of the order of the numberof voxels—of variables and constraints. Practical handling of this optimizationproblem is demonstrated through the adaptation of an interior-point method forexploitation of the problem structure, so as to obtain a reduction in problem sizeby several orders of magnitude. Numerical experiments confirm that the methodefficiently solves the given optimization problem, and comparison to a commer-cially available generic solver indicates that the problem-structure exploitation isthe key in achieving this result.

Numerical results from two patient cases indicate that fluence map optimiza-tion with the proposed objective functions results in improved plan quality metricsand higher feasibility to constraints in comparison to the conventional penalty-based functions. It is concluded that the novel formulation appears to better cor-relate with plan quality metrics, but that evaluation in a more clinically realisticsetting is needed.

Paper B: Increased accuracy of planning tools for optimization of dynamicmultileaf collimator delivery of radiotherapy through reformulatedobjective functions

Paper B is co-authored with Kjell Eriksson and Anders Forsgren, and has beenpublished in Physics in Medicine & Biology, Vol. 63, No. 12, p. 125012, 2018.

In this paper, the explicit approach to treatment plan optimization presented inPaper A is extended with a DMLC deliverability model from the literature. Thepurpose is to stress-test the proposed objective functions in a more clinically re-alistic setting, including comparison in the domain of deliverable treatment plansand final clinically accurate dose calculation. It is demonstrated that the problemstructure needed to perform the algebraic manipulations of the tailored interior-point method developed in Paper A can be preserved despite the additional de-liverability constraints, and that the given optimization problems can be solvedefficiently.

Numerical results from three patient cases are in line with the outcome ofPaper A: direct machine parameter optimization for DMLC delivery with the pro-


posed objective functions results in improved plan quality metrics in compari-son to the conventional penalty-based functions, in this study represented by theMCO module in the treatment planning system RayStation (RaySearch Laborato-ries, Stockholm, Sweden). In addition, the generated treatment plans show betterfeasibility to constraints subsequent to accurate dose computation.

Paper C: On tradeoffs between treatment time and plan quality ofvolumetric-modulated arc therapy with sliding-window delivery

Paper C is co-authored with Anders Forsgren, and has been submitted to Physicsin Medicine & Biology.

In this paper, an accurate formulation of direct machine parameter optimizationfor so-called sliding-window VMAT delivery is presented. The formulation isbased on an algorithm for VMAT optimization from the literature. The purposeis to, in light of the accurate formulation, investigate the effects on plan qualityand treatment efficiency when decreasing the number of sliding-window sweepsdelivered as the machine head rotates around the patient. While it is generally truethat many sweeps lead to better plan quality given a generous treatment time, it ishypothesized advantageous to decrease the number of sweeps if a highly efficienttreatment is required. The algorithm from the literature is generalized, and analgorithmic modification is suggested for better handling of the suggested settingwith fewer sliding-window sweeps.

Numerical results from two patient cases indicate that with tighter treatmenttime restrictions, at a certain point, it is beneficial to decrease the number ofsliding-window sweeps to maintain high plan quality. It is observed that the sug-gested modified algorithm performs better than the original algorithm in terms ofobjective function value.

5.3 Main contributions

The main contribution of this thesis is the framework for—and novel perspec-tive on—automated treatment planning, within which it is demonstrated how in-creased accuracy of the optimization tools provided to the treatment planner couldopen up for streamlining of the treatment planning process. The core of the sug-gested framework is mathematical tractability and strong correlation to measuresof plan quality.

In Paper A, methodological drawbacks of the conventional penalty-based ob-jective functions given in (4) and (5) are identified as one factor that makes the

18 INTRODUCTION

current treatment planning process a tedious task. In addition to the noncon-vexity, nondifferentiability, and vanishing gradients discussed in Section 3.2, thedrawbacks include an implicit and not so clear relationship to the plan qualitymeasures. The suggested framework for automated treatment planning is the re-sult of a novel formulation of objective functions with which the penalty-basedparadigm is abandoned in favor of a more explicit relationship to, thus bettercorrelation with the widely used dose-at-volume measure. Paper B strengthensthe conclusions made in Paper A regarding improved correlation by presenting anumerical study where cohorts of DMLC deliverable treatment plans generatedwithin the conventional and the suggested frameworks are compared.

It is in general expected that a framework for automated treatment planningshould be compatible with most IMRT delivery techniques. For the suggestedframework, compatibility includes maintained mathematical tractability and cor-relation with measures of plan quality. Paper C approaches such a situation for themathematically more complex VMAT delivery technique by giving a deliverabil-ity model for a direct machine parameter optimization problem, and by suggestingimprovements of an existing heuristic to handle the resulting optimization prob-lem.

A possible drawback with the proposed objective functions is largely increaseddimensions of the treatment plan optimization problem. In Paper A, tractabilityof the resulting optimization problem is demonstrated through the exploitation ofits structure in an interior-point method along the lines described in Section 4.1;and in Papers B and C, compatibility of the tailored interior-point method withrespectively a DMLC and VMAT delivery model is demonstrated.

6 Bibliography

[1] P. Alaei, P. D. Higgins, R. Weaver, and N. Nguyen. Comparison of dynamic andstep-and-shoot intensity-modulated radiation therapy planning and delivery. Med.Dosim., 29(1):1–6, 2004.

[2] D. M. Aleman, D. Glaser, H. E. Romeijn, and J. F. Dempsey. Interior point algo-rithms: guaranteed optimality for fluence map optimization in IMRT. Phys. Med.Biol., 55(18):5467–5482, 2010.

[3] A. W. Andersson, A. Jansson, B. Sandblad, and S. Tschirner. Recognizing complex-ity: Visualization for skilled professionals in complex work situations. In Build-ing bridges: HCI, visualization, and non-formal modeling, pages 47–66. SpringerBerlin Heidelberg, 2014.


[4] L. M. Appenzoller, J. M. Michalski, W. L. Thorstad, S. Mutic, and K. L. Moore. Pre-dicting dose-volume histograms for organs-at-risk in IMRT planning. Med. Phys.,39(12):7446–61, 2012.

[5] M. Balvert and D. L. Craft. Fast approximate delivery of fluence maps for IMRTand VMAT. Phys. Med. Biol., 62(4):1225–1247, 2017.

[6] J. Berger. ROENTGEN: Case-based reasoning and radiation therapy planning. Proc.Annu. Symp. Comput. Appl. Med. Care, pages 210–214, 1992.

[7] R. Bokrantz. Multicriteria optimization for managing tradeoffs in radiation therapytreatment planning. PhD thesis, KTH Royal Institute of Technology, 2013.

[8] T. R. Bortfeld. Optimized planning using physical objectives and constraints. Semin.Radiat. Oncol., 9(1):20–34, 1999.

[9] T. R. Bortfeld. IMRT: A review and preview. Phys. Med. Biol., 51(13):363–379,2006.

[10] T. R. Bortfeld, D. L. Kahler, T. J. Waldron, and A. L. Boyer. X-ray field compensa-tion with multileaf collimators. Int. J. Radiat. Oncol., 28(3):723–730, 1994.

[11] T. R. Bortfeld, J. Unkelbach, R. Boesecke, and W. Schlegel. Methods of imagereconstruction from projections applied to conformation radiotherapy. Phys. Med.Biol., 35(10):1423–1434, 1990.

[12] A. Brahme, J.-E. Roos, and I. Lax. Solution of an integral equation encountered inradiation therapy. Phys. Med. Biol., 27(10):1221, 1982.

[13] S. Breedveld, B. van den Berg, and B. Heijmen. An interior-point implementa-tion developed and tuned for radiation therapy treatment planning. Comput. Optim.Appl., 68(2):209–242, 2017.

[14] F. Carlsson and A. Forsgren. Iterative regularization in intensity-modulated radia-tion therapy optimization. Med. Phys., 33(1):225–234, 2006.

[15] F. Carlsson, A. Forsgren, H. Rehbinder, and K. Eriksson. Using eigenstructure ofthe Hessian to reduce the dimension of the intensity modulated radiation therapyoptimization problem. Ann. Oper. Res., 148(1):81–94, 2006.

[16] J. Castro. A specialized interior-point algorithm for multicommodity network flows.SIAM J. Optimiz., 10(3):852–877, 2000.

[17] C. S. Chui, M. F. Chan, E. Yorke, S. Spirou, and C. C. Ling. Delivery of intensity-modulated radiation therapy with a conventional multileaf collimator: Comparisonof dynamic and segmental methods. Med. Phys., 28(12):2441–2449, 2001.

20 INTRODUCTION

[18] D. J. Convery and M. E. Rosenbloom. The generation of intensity-modulated fieldsfor conformal radiotherapy by dynamic collimation. Phys. Med. Biol., 37(6):1359–1374, 1992.

[19] D. L. Craft, T. F. Halabi, H. A. Shih, and T. R. Bortfeld. Approximating convexPareto surfaces in multiobjective radiotherapy planning. Med. Phys., 33(9):3399–3407, 2006.

[20] D. L. Craft, T. S. Hong, H. A. Shih, and T. R. Bortfeld. Improved planning timeand plan quality through multicriteria optimization for intensity-modulated radio-therapy. Int. J. Radiat. Oncol., 82(1):83–90, 2012.

[21] D. L. Craft and C. Richter. Deliverable navigation for multicriteria step and shootIMRT treatment planning. Phys. Med. Biol., 58(1):87–103, 2013.

[22] W. De Gersem, F. Claus, C. De Wagter, B. Van Duyse, and W. De Neve. Leafposition optimization for step-and-shoot IMRT. Int. J. Radiat. Oncol., 51(5):1371–1388, 2001.

[23] A. Fredriksson. Automated improvement of radiation therapy treatment plans byoptimization under reference dose constraints. Phys. Med. Biol., 57:7799–7811,2012.

[24] P. E. Gill, W. Murray, and M. A. Saunders. SNOPT: An SQP algorithm for large-scale constrained optimization. SIAM J. Optimiz., 12(4):979–1006, 2002.

[25] D. Gintz, K. Latifi, J. Caudell, B. E. Nelms, G. Zhang, E. Moros, and V. Feygelman.Initial evaluation of automated treatment planning software. J. Appl. Clin. Med.Phys., 17(3):331–346, 2016.

[26] I. Griva, S. Nash, and A. Sofer. Linear and nonlinear optimization. Society forIndustrial and Applied Mathematics, 2009.

[27] T. Halabi, D. L. Craft, and T. R. Bortfeld. Dose-volume objectives in multi-criteriaoptimization. Phys. Med. Biol., 51(15):3809–3818, 2006.

[28] B. Hårdemark, A. Liander, H. Rehbinder, and J. Löf. Direct machine parameteroptimization with RayMachine® in Pinnacle3® (White Paper). RaySearch Labo-ratories, Stockholm, Sweden, 2003.

[29] I. Hazell, K. Bzdusek, P. Kumar, C. R. Hansen, A. Bertelsen, J. G. Eriksen, andJ. Johansen. Automatic planning of head and neck treatment plans. J. Appl. Clin.Med. Phys., 17(1):272–282, 2016.

[30] International Commission on Radiation Units and Measurements. ICRU report 83:Prescribing, recording, and reporting photon-beam intensity-modulated radiationtherapy (IMRT). J. ICRU, 10(1):1, 2010.


[31] M. L. Kessler, D. L. McShan, M. A. Epelman, K. A. Vineberg, A. Eisbruch, T. S.Lawrence, and B. A. Fraass. Costlets: A generalized approach to cost functions forautomated optimisation of IMRT plans. Optim. Eng., 6:421–448, 2005.

[32] T. Knöös, I. Kristensen, and P. Nilsson. Volumetric and dosimetric evaluationof radiation treatment plans: Radiation conformity index. Int. J. Radiat. Oncol.,42(5):1169–1176, 1998.

[33] M. Langer, R. Brown, M. Urie, J. Leong, M. Stracher, and J. Shapiro. Large scaleoptimization of beam weights under dose-volume restrictions. Int. J. Radiat. Oncol.,18:887–893, 1990.

[34] E. K. Lee, T. Fox, and I. Crocker. Optimization of radiosurgery treatment planningvia mixed integer programming. Med. Phys., 27(5):995–1004, 2000.

[35] E. K. Lee, T. Fox, and I. Crocker. Integer programming applied to intensity-modulated radiation therapy treatment planning. Ann. Oper. Res., 119(1–4):165–181, 2003.

[36] C. McIntosh and T. G. Purdie. Voxel-based dose prediction with multi-patientatlas selection for automated radiotherapy treatment planning. Phys. Med. Biol.,62(2):415–431, 2017.

[37] C. McIntosh, M. Welch, A. McNiven, D. A. Jaffray, and T. G. Purdie. Fully auto-mated treatment planning for head and neck radiotherapy using a voxel-based doseprediction and dose mimicking method. Phys. Med. Biol., 62(15):5926–5944, 2017.

[38] L. K. Mell, A. K. Mehrotra, and A. J. Mundt. Intensity-modulated radiation therapyuse in the U.S., 2004. Cancer, 104(6):1296–1303, 2005.

[39] P. Metcalfe, T. Kron, and P. Hoban. The physics of radiotherapy X-rays and elec-trons. Medical Physics Pub., 2007.

[40] R. Mohan, X. Wang, A. Jackson, T. R. Bortfeld, A. L. Boyer, G. J. Kutcher, S. A.Leibel, Z. Fuks, and C. Clifton Ling. The potential and limitations of the inverseradiotherapy technique. Radiother. Oncol., 32(3):232–248, 1994.

[41] M. Monz, K.-H. Küfer, T. R. Bortfeld, and C. Thieke. Pareto navigation—Algorithmic foundation of interactive multi-criteria IMRT planning. Phys. Med.Biol., 53(4):985–998, 2008.

[42] B. E. Nelms, G. Robinson, J. Markham, K. Velasco, S. Boyd, S. Narayan,J. Wheeler, and M. L. Sobczak. Variation in external beam treatment plan qual-ity: An inter-institutional study of planners and planning systems. Pract. Radiat.Oncol., 2(4):296–305, 2012.

[43] K. Otto. Volumetric modulated arc therapy: IMRT in a single gantry arc. Med.Phys., 35(1):310–317, 2008.

22 INTRODUCTION

[44] D. Papp and J. Unkelbach. Direct leaf trajectory optimization for volumetric modu-lated arc therapy planning with sliding window delivery. Med. Phys., 41(1):011701,2014.

[45] F. Peng, X. Jia, X. Gu, M. A. Epelman, H. E. Romeijn, and S. B. Jiang. A newcolumn-generation-based algorithm for VMAT treatment plan optimization. Phys.Med. Biol., 57(14):4569–4588, 2012.

[46] M. Rao, D. Cao, F. Chen, J. Ye, V. Mehta, T. Wong, and D. Shepard. Comparisonof anatomy-based, fluence-based and aperture-based treatment planning approachesfor VMAT. Phys. Med. Biol., 55(21):6475–6490, 2010.

[47] H. E. Romeijn, R. K. Ahuja, J. F. Dempsey, and A. Kumar. A new linear pro-gramming approach to radiation therapy treatment planning problems. Oper. Res.,54(2):201–216, 2006.

[48] M. Schwarz. Treatment planning in proton therapy. Eur. Phys. J. PLUS, 126(7):67,2011.

[49] D. M. Shepard, D. Cao, M. K. N. Afghan, and M. A. Earl. An arc-sequencingalgorithm for intensity modulated arc therapy. Med. Phys., 34(2):464–470, 2007.

[50] D. M. Shepard, M. A. Earl, X. A. Li, S. Naqvi, and C. X. Yu. Direct aperture op-timization: A turnkey solution for step-and-shoot IMRT. Med. Phys., 29(6):1007–1018, 2002.

[51] S. Shiraishi and K. L. Moore. Knowledge-based prediction of three-dimensionaldose distributions for external beam radiotherapy. Med. Phys., 43(1):378–387, 2016.

[52] J. Skarpman Munter and J. Sjölund. Dose-volume histogram prediction using den-sity estimation. Phys. Med. Biol., 60(17):6923–36, 2015.

[53] A. R. Smith. Proton therapy. Phys. Med. Biol., 51(13):491–504, 2006.

[54] S. Webb. Optimisation of conformal radiotherapy dose distributions by simulatedannealing. Phys. Med. Biol., 34(10):1349–1370, 1989.

[55] S. Webb. The physical basis of IMRT and inverse planning. Brit. J. Radiol.,76(910):678–689, 2003.

[56] A. G. Weldeyesus and J. Gondzio. A specialized primal-dual interior point methodfor the plastic truss layout optimization. Comput. Optim. Appl. (to appear), 2018.

[57] S. Wright. Primal-dual interior-point methods. Society for Industrial and AppliedMathematics, 1997.


[58] B. Wu, F. Ricchetti, G. Sanguineti, M. Kazhdan, P. Simari, R. Jacques, R. Taylor,and T. R. McNutt. Data-driven approach to generating achievable dose-volumehistogram objectives in intensity-modulated radiotherapy planning. Int. J. Radiat.Oncol., 79(4):1241–1247, 2011.

[59] Q. Wu and R. Mohan. Algorithms and functionality of an intensity modulated ra-diotherapy optimization system. Med. Phys., 27(4):701–711, 2000.

[60] Q. Wu and R. Mohan. Multiple local minima in IMRT optimization based on dose-volume criteria. Med. Phys., 29(7):1514–1527, 2002.

[61] I. Xhaferllari, E. Wong, K. Bzdusek, M. Lock, and J. Chen. Automated IMRTplanning with regional optimization using planning scripts. J. Appl. Clin. Med.Phys., 14(1):4052, 2013.

AUTOMATED RADIATION THERAPY TREATMENT PLANNING BY ...1258306/FULLTEXT01.pdf · Abstract Every...

Documents

Transcript of AUTOMATED RADIATION THERAPY TREATMENT PLANNING BY ...1258306/FULLTEXT01.pdf · Abstract Every...