Automated radiation therapy treatment planning by increased accuracy of optimization tools

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AUTOMATED RADIATION THERAPY TREATMENT PLANNING BY INCREASED ACCURACY OF OPTIMIZATION TOOLS

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Automated radiation

therapy treatment planning by increased accuracy of optimization tools

LOVISA ENGBERG

Doctoral thesis in applied and computational mathematics Stockholm, Sweden 2018

KTH Royal Institute of Technology School of Engineering Sciences

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

Optimeringslära och systemteori Matematiska institutionen KTH 100 44 Stockholm

Akademisk avhandling som med tillstånd av KTH i Stockholm framlägges till offentlig granskning för avläggande av teknologie doktorsexamen fredagen den 23 november 2018 kl 10:00 i sal F3, KTH, Lindstedtsvägen 26, Stockholm.

Tryck: Universitetsservice US-AB, Stockholm

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Abstract

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

The aim of treatment planning is to create a plan that results in the, in some sense, best treatment—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 oncologist or of the clinic. Despite dedicated treatment planning systems available with advanced optimization tools, treatment planning is often referred to as a complicated process involving many iterations with successively adjusted parameters. Over the years, a request has emerged from the clinical and treatment planners’ side to make treatment planning less time-consuming and more straightforward, and the methods subsequently developed as a response have come to be referred to as methods for automated treatment planning.

In this thesis, a framework for automated treatment planning is proposed and its potential and flexibility investigated. The focus is placed on increasing the accuracy of the optimization tools, aiming at achieving a less complicated treatment planning process that is driven by intuition rather than, as currently, trial and error. The suggested framework is contrasted 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 of human interaction. To increase the accuracy of the optimization tools, the underlying so- called objective functions are reformulated to better correlate with measures of treatment plan quality while possessing mathematical properties favorable for optimization. An important 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 specific structure of the novel optimization formulation, and is applied throughout the thesis, to demonstrate tractability. Numerical studies support the idea of the suggested framework equipping the treatment planner with more accurate and thereby less complicated tools to more straightforwardly handle the intrinsically complex process that constitutes treatment planning.

Keywords: Optimization, intensity-modulated radiation therapy, radiation therapy treatment planning, automated radiation therapy treatment planning, interior-point methods

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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 inträde i klinisk praxis har dosplanering kommit att betyda och rent av kräva användande av avancerade 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 mening ger den bästa strålbehandlingen. En sådan behandling ska i synnerhet spegla de kliniska mål som satts upp för den enskilda patienten, i allmänhet uppnå bästa möjliga kvalitet samt förhålla sig till eventuella övriga önskemål från onkologen eller kliniken. Utbudet av 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 mindre tidskrävande och mer rättfram dosplanering under det senaste årtiondet vuxit fram. De metoder 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, för att därigenom skapa förutsättningar för mindre komplicerad dosplanering där intuition snarare än ett tidskrävande experimenterande driver processen framåt. Ramverket som här föreslås ställs i kontrast till en annan, dominerande klass av föreslagna metoder för automatiserad dosplanering som bygger på en mer klassisk syn på automatisering, det vill säga, som strävar efter att minska människa-datorinteraktion i allmänhet. Förbättring av optimeringsverktygens noggrannhet uppnås genom att omformulera de bakomliggande så kallade målfunktionerna till alternativ som bättre korrelerar med givna kvalitetsmått och som samtidigt har matematiska egenskaper som är önskvärda vid optimering. Ett viktigt steg är dock att visa att det föreslagna ramverket inte bara är teoretiskt lämpligt, utan att det också är praktiskt hanterbart ur beräkningssynpunkt. En inrepunktsmetod anpas- sas till den specifika strukturen på det nya, storskaliga optimeringsproblemet för att visa just detta. Fallstudier stödjer idén om att det föreslagna ramverket ger mer noggranna och därmed lätthanterliga optimeringsverktyg, med vilka dosplaneringens ofrånkomliga komplexitet kan hanteras på ett mer effektivt sätt.

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

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Preface

Pursuing your doctoral studies is, thank goodness, a once-in-a-lifetime experience. The following words are to show why my decision in 2013 to take on this challenge 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, in this case, me to pursue my doctoral studies;

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

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

There are many similarities of this thesis with the first movement of the Moonlight Sonata by Beethoven—another piece (among very few) which I know by heart.

In particular, perhaps more important than using technical skill, “this whole piece must be played very delicately”. I am indeed grateful to all of my supervisors for respecting 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; it would have been, I believe, for anyone’s. But it has been the more rewarding in many other ways. I am certain that seeing the strong competences of the entire Research Department has pushed me towards better achievements. The same can be said about the Division of Optimization and Systems Theory at KTH, and I am particularly 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 2018 Lovisa Engberg

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Introduction

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

1 Radiation therapy

Radiation therapy has been used as a treatment of cancer since the end of the 19th century. What started as a treatment technique of experimental nature, to some extent driven by hypotheses and expectations, has now become an art of great precision and accuracy in medical, physical, and mathematical aspects. To- day, the field of radiation therapy relies on a multitude of technologies, such as medical physics, radiobiology, medical imaging, image processing, mathematical optimization, 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 dose to the tumor while sparing surrounding healthy tissue to avoid complications. In external radiation therapy, this goal is achieved by irradiating the patient with multiple beams from different directions, so as to have an intersecting irradiated volume 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 therapy(3D-CRT) with tumor projection-shaped beams and a better ability to control the shape of the intersecting irradiated volume, to intensity-modulated radiation therapy(IMRT) with a potential to offer highly conformal treatments and a sig- nificantly reduced risk of complications. IMRT became practically possible by the 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 of the treatment machine to control the shape of the beam. Its radiation-absorbing leaves, arranged on opposing sides, can be moved against or away from each other

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2 INTRODUCTION

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

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

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

1.2 Dose calculation

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

From a mathematical perspective (here, a simplification of the physical perspective), calculating the resulting dose inside the body given a set of IMRT beams starts by discretizing the three-dimensional body into m voxels (volume pixels) and the two-dimensional planes orthogonal to the incident beam directions into n bixels (beam pixels). An n-dimensional vector x, the fluence map, is introduced 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

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AUTOMATED PLANNING BY INCREASED ACCURACY OF OPTIMIZATION TOOLS 3 dose. Given a fluence map, going to the dose distribution is a straightforward linear operation if also assuming an m × n-dimensional dose deposition matrix P available:

d = P x. (1)

However, calculating the dose deposition matrix P given the body geometry and beam configuration requires advanced dose calculation algorithms. In commercial treatment 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 beam configuration—it is often computationally more efficient to apply a dose calculation 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 during treatment planning, but a clinically accurate dose distribution is always calculated and quality-checked before going forward with the radiation therapy treatment.

See, e.g., [39] for an introduction from the physical perspective to dose calculation algorithms used in radiation therapy.

It should be mentioned that besides photons (X-rays), which is the most widely used treatment modality, also protons and even heavier particles are used in radiation 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] for a review on proton-based radiation therapy, and [48] for an introduction to proton therapy treatment planning.

1.3 IMRT delivery techniques

A modulated beam intensity is the result of using multiple, consecutive MLC apertures. The aperture can be varied both statically, with the treatment machine idling while the MLC leaves are moved, or dynamically, with a continuously changing aperture. Static and dynamic MLC motions define two different IMRT delivery techniques referred to as SMLC (or, “step-and-shoot”) and DMLC, respectively. From the mathematical point of view, different delivery techniques could require different approaches concerning both the formulation and the solving of the associated optimization problem. However, choosing one above the other 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 machine capabilities or the level of “wear and tear” acceptance, DMLC being in general more demanding in that sense; and institutional or the oncologist’s own preferences [1, 17].

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4 INTRODUCTION

By allowing continuous rotation of the treatment machine head around the patient during irradiation instead of restricting to discrete beam directions as assumed above, the concepts of a technically more complex IMRT delivery technique are outlined. This delivery technique most often goes by the name volumetric- modulated arc therapy(VMAT). In its currently most widely used form, VMAT was first formulated in mathematical terms by Otto [43]. The primary advantage is the opportunity to obtain highly time-efficient treatments, but as with SMLC and DMLC, treatment machine and quality control considerations also play a role in the clinical decision on whether to treat using VMAT. In addition, VMAT is associated with even greater mathematical challenges and places larger demands on 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 what directions, using which beam shapes, et c.—is generated by the treatment planner. The aim of treatment planning is to find a plan that results in the treatment fulfilling the clinical goals—a patient-specific adaptation of the overall radiation therapy 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: in a forward or in an inverse manner. Forward planning refers to the procedure of generating treatment plans by hand, i.e., by manually defining all treatment plan specifications and then calculating the resulting dose distribution using dedicated software. If the dose distribution cannot be deemed good enough with respect to the clinical goals, the manually set treatment plan specifications have to be manually revised. It is easy to understand that forward planning is essentially only applicable to “at most” 3D-CRT, since for IMRT, there are too many plan specifications to consider. The idea behind inverse planning is to formulate a mathematical optimization problem that takes information about the clinical goals as a parameter input. By then applying methods for optimization, the ambition and expectations are to directly obtain the treatment plan that in some sense best fulfills the clinical goals.

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

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AUTOMATED PLANNING BY INCREASED ACCURACY OF OPTIMIZATION TOOLS 5 contributed to establishing the optimization framework. But the optimized treatment plans were expressed in terms of heterogeneous fluence maps that require IMRT delivery, and it was therefore not until the MLC was in place in the 1990’s that the benefit of inverse planning could be implemented into practice—inverse planning and MLCs together formed IMRT. In the early 2000’s, inverse planning and IMRT were widely commercialized and became clinical routine within a few years [38]. Since then, forward planning is seldom being considered. The development of software for inverse planning is now a high-technology industry, and the treatment planning systems of today can offer a much broader spectrum of tools than during the forward planning era, when essentially computerized dose calculation was sufficient.

Understanding the current inverse planning process is of utmost importance for this thesis. Despite the use of advanced optimization methods, the process is complicated and requires both experience and skills from the treatment planner—

a comprehensive study by Nelms et al. [42] reports that the level of experience of the treatment planner, among other factors, affects the quality of the treatment plan. For each patient, the treatment planner takes the individually set clinical goals into consideration when specifying the objectives of the treatment in the treatment planning system. The objectives provided are translated by the software into objective functions of an optimization problem (see Section 3). After optimization, the treatment planner examines the resulting dose distribution and verifies that the clinical goals, at least to a satisfiable degree, are met. If not, the objectives are updated, and the dose distribution is re-optimized; this procedure is repeated until the treatment plan is approved by the oncologist.

The requirements on experience and skills for successful inverse planning are due to the conflicting nature of the clinical goals (indeed, to irradiate the tumor is in conflict with sparing the surrounding tissue), and to the difficulty in assessing this conflict. The skilled treatment planner need to be aware of how an objective function with conflicting constituents is handled during optimization, and to be able to roughly predict how the resulting dose distribution is affected by a certain choice of objectives; thus knows how to adapt the objectives in order to approach the 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 straightforward. It is not uncommon that several person-hours are spent on treatment

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6 INTRODUCTION

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

The most prominent trend in automated planning is the use of machine learning techniques and the utilization of the fact that many cases share patient-geo- metrical and clinical-goal similarities. Given the recent explosion of literature involving machine learning, treatment planning can be concluded having entered yet another field of high-technological research and development: computer sci- ence. Examples of machine learning methods for treatment planning can be found in [4, 36, 37, 51, 52, 58] (in fact, a very early publication discussing machine learning in treatment planning dates back to 1992 [6]). Similar to other machine learning applications, the methods rely on the existence of a database consisting of, in this case, patient geometry information and associated treatment plans. The predicted data for a new patient is either used as decision support during the treatment 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 more mechanical view of automation. For instance, algorithms or scripts to mimic the treatment planner’s successive adaptation of objectives to eventually reach clinical goal fulfillment have been developed, usually based on a specific treatment planning system. Examples can be found in [25, 29, 61].

Common for the abovementioned methods is that they aim at handling symp- tomsinstead of underlying causes of a cumbersome treatment planning process.

The contributions of this thesis to the field of automated treatment planning, as summarized in Section 5, involve attempts to overcome the causes.

3 Treatment plan optimization

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

minimize

x∈X , d∈R^m f (d) subject to c(d) ≤ 0,

d = P x,

(2)

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AUTOMATED PLANNING BY INCREASED ACCURACY OF OPTIMIZATION TOOLS 7 where X denotes the set of fluence maps, f (d) is the objective function to be minimized, and c(d) ≤ 0 (and d = P x) the constraints. Depending on the mathematical characteristics of the functions f and c and the set X , (2) can be more or less difficult to solve. Slightly simplified, (2) can be solved to global optimality if f and c are convex functions and X a convex set; otherwise, (2) is nonconvex, in which case only local optima can in general be found.

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

f₁(d), · · · , f_K(d)T

.

Traditionally, the multicriterial nature of inverse planning has been handled by weighted-sum scalarization, i.e., accumulation of the constituent objective functions using positive weighting factors wk, k = 1, . . . , K, into the singlecriteria objective function

f (d) =

K

X

k=1

w_kf_k(d). (3)

The weighting factors are specified by the treatment planner along with the objectives of the treatment and are interpreted as priorities: the larger the weighting factor, the more emphasis should be given to fulfill the associated objective.

In the last decade, dedicated methods to handle the MCO problem have been developed by which the manual specification of weighting factors is eliminated through incorporation into an outer formalism. A welcome consequence is a reduction in workload of the treatment planner [20]. The most studied class of MCO methods for inverse planning is a posteriori methods. These methods generate a well-distributed (in constituent objective function values) set of a fixed number of treatment plans, each optimal to a weighted-sum instance, between which the treatment planner can then interactively “navigate” to explore treatment plans approximately optimal to any weighted-sum instance. Examples of a posteriori methods can be found in [7, 19, 21, 41].

It is in its place to recall three already introduced key concepts of inverse planning, 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-

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8 INTRODUCTION

Figure 2.A (cumulative) dose-volume histogram (DVH) example illustrated for a tumor (red) and a critical structure (blue). The interpretation of the point ( ˆd, ˆv) is that a volume fraction ˆv receives a dose of at least ˆd Gy.

jectives” and modifications of overly optimistic or too loose clinical goals—

again, the skilled treatment planner knows how to adapt the objectives in order to approach (or exceed) the clinical goals,

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

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

These four concepts are strongly interconnected—e.g., clinical goals and objectives are usually formulated in terms of explicit thresholds of measures of plan quality—but are kept distinguished from each other here to give proper understanding 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 that sense. There is, however, some quantifiable measures that dominate in quality assessment 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 the three-dimensional dose distribution in any tumor, organ, or other structure in a two-dimensional graph. The interpretation of a point ( ˆd, ˆv) on the curve is that a

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AUTOMATED PLANNING BY INCREASED ACCURACY OF OPTIMIZATION TOOLS 9 volume fraction ˆv of the structure receives a dose of at least ˆd Gy. Two important measures of plan quality are given by the parameterizations of the DVH curve by d and ˆˆ v as functions of the dose distribution, referred to as and denoted by

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

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

The minimum, maximum, and average dose [Gy] measures are also being fre- quently evaluated and reported. A majority of the oncologist’s clinical goals and the treatment planner’s objectives are expressed in terms of lower (for tumors) or upper bounds on these five measures (as recommended by the Radiation Therapy Oncology Group (RTOG), Philadelphia, PA, USA). Commonly reported values for tumors are D(d; v^ref) for v^refin the ranges 1–5 % and 95–99 %, and V (d; d^ref) for d^refin the range 90–110 % of the prescribed tumor dose; for organs and healthy tissue, 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 considered, 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 a significant dose and the tumor volume [32].

Also relating to the DVH is the mean-tail-dose [Gy] measure (Figure 3). The mean-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 the measure has not yet been used in clinical quality assessment. The upper and lower mean-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 ( ˆd, ˆv) on the curve.

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

3.2 Objective functions

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

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10 INTRODUCTION

Figure 3.An illustration of mean-tail-dose. The upper and lower mean-tail-doses are defined as the average dose of the upper (blue) and lower (red) DVH tail given by a volume fraction ˆv.

goals, whereafter the treatment planning system translates each of these into a corresponding constituent objective function of a treatment plan MCO problem.

Consequently, the objective functions constitute the “optimization tools” available to the treatment planner and are central to this thesis. The importance and challenges of finding accurate formulations of objective functions with close connections to the clinical goals and associated measures of plan quality were brought up 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 optimization [11,54], a penalty-function based paradigm inherited from the analogous problem of image reconstruction has been used when formulating objective functions for inverse planning. In the clinical treatment planning systems of today, the objective functions commonly found are formulated as quadratic penalties associated with violation of the input objectives. A conventional objective function aimed at controlling the dose-at-volume objective D(d; v^ref) ≤ d^ref orthe volume-at-dose objective V (d; d^ref) ≤ v^refimposed on a structure with voxels S, S ⊂ {1, . . . , m}, is given by

f (d) = X

i∈S:

d^ref≤ di≤ D(d;v^ref)

∆^S_i di− d^ref2

, (4)

where diis the dose in voxel i and ∆^S_i denotes the relative volume of the structure in voxel i [8]. Figure 4 gives a helpful visual interpretation in the DVH graph to make sense of which voxels are considered in the summation. The expressions

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AUTOMATED PLANNING BY INCREASED ACCURACY OF OPTIMIZATION TOOLS 11

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

for the reversed objectives, D(d; v^ref) ≥ d^refand V (d; d^ref) ≥ v^ref, are analogous, f (d) =X

i∈S:

D(d;v^ref) ≤ di≤ d^ref

∆^S_i d^ref− d_i2

. (5)

Under minimization, (4) and (5) push the DVH curve towards the point (d^ref, v^ref).

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

In this thesis, a framework with novel objective functions is suggested. One purpose 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.

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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 of the formulation, such as whether it is convex, linear, or nonlinear, or whether its objective functions and constraints are differentiable.

Given the conventional objective functions, the weighted-sum instances of (2) are constrained nonlinear programming problems, or nonlinear programs. The optimization method primarily used in treatment planning systems to solve these problems—as well as used in the early publications of inverse planning—is by sequential quadratic programming(SQP) [11, 28]. SQP amounts to solving a se- quence of positive definite quadratic programs, for which there in turn exist methods to find the global optimum. A comprehensive introduction to SQP is given by 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 et al. [15] and Carlsson and Forsgren [14]. Other optimization methods seen in the literature to solve the conventional nonlinear program include, e.g., interior-point methods; 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 result in the weighted-sum instances of (2) being large-scale linear programming problems, or linear programs. Linear programs constitute a well-studied segment of optimization problems and a spectra of general-purpose optimization methods can be applied, such as simplex, active-set, and interior-point methods [26]. The method applied in Papers A–C is of the latter class, of which a thorough introduction can be found in Wright [57]. By exploiting the structure imposed by the specific optimization problem on arising systems of linear equations, interior- point methods can be adapted to improve efficiency in solving large-scale linear programs. Examples of such structure utilization in engineering applications are found in [16, 56]. Below, in demonstrating where the structure-exploiting opportunity occurs, some fundamental linear programming concepts are referenced, such as primal, dual, and optimality conditions. The interested reader may again refer to Wright [57] but understanding of these concepts is not necessary in order to follow the reasoning in the following paragraph.

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AUTOMATED PLANNING BY INCREASED ACCURACY OF OPTIMIZATION TOOLS 13 Given a standard-form linear program and its corresponding dual problem,

minimize

z∈Rⁿ c^Tz maximize

y∈R^m, s∈Rⁿ b^Ty

subject to Az = b, subject to A^Ty + s = c,

z ≥ 0, s ≥ 0,

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





0 A^T I

A 0 0

Sk 0 Zk









∆z_k

∆y_k

∆sk



= −





A^Ty_k+ s_k− c Az_k− b ZkSke − µe



, (7)

where S_k= diag(s_k) and Z_k= diag(z_k), e is the vector of ones, and µ is a positive method parameter that is successively decreased to phase out the perturbation.

The steplength αk must ensure nonnegativity of the iterates, thus is chosen such that

(z_k+1, s_k+1) = (z_k, s_k) + α_k(∆z_k, ∆s_k) > 0. (8) Instead of the system of linear equations given in (7), smaller systems where ∆s_k or (∆zk, ∆s_k) have been eliminated are sometimes considered:

−Z_k⁻¹S_k A^T

A 0

∆z_k

∆y_k

= −A^Ty_k+ µZ_k⁻¹e − c Az_k− b

, (9)

or

AS_k⁻¹Z_kA^T ∆y_k= − AS_k⁻¹Z_k A^Ty_k+ µZ_k⁻¹e − c − (Az_k− 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 application of a standard factorization algorithm, namely, by applying a specific block-eliminating approach and a so-called Schur complement technique. In fact, existence of such an exploitable structure in A could be critical for the tractability of 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 by whether the set of fluence maps X considered in (2) contains all positive fluence

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14 INTRODUCTION

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

As one might expect, there are pros and cons associated with both FMO and DMPO. FMO problems are generally considered easier to solve due to the relax- ation of deliverability constraints. On the other hand, the optimal fluence map obtained using FMO must be post-processed and converted into machine parameters with respect to the chosen IMRT delivery technique. An obvious risk is degradation of the solution after conversion. Early examples of post-processing algorithms include [10, 18], and more recent suggestions for the mathematically complex VMAT delivery can be found in [5, 49]. As to DMPO problems, depending on the IMRT delivery technique, these can be almost arbitrarily difficult to both formulate and solve. Yet, on the other hand, studies have reported that DMPO results in superior treatment plans compared to FMO with subsequent conversion [46, 50]. Formulations of DMPO have been developed for most clinically used delivery techniques; to give a few examples, see [22, 50] for SMLC, [44] for DMLC, and [45] for VMAT delivery.

Comprehensive formulations of DMPO are more seldom considered with increasing mathematical complexity of the IMRT delivery technique. While FMO with conversion is one way forward, another is to introduce limitations of or make assumptions regarding the delivery to simplify the DMPO problem and to improve its tractability. For instance, Papp and Unkelbach [44] have shown that assuming DMLC delivery by unidirectional MLC leaf motions, “sweeps”, the set of physically realizable fluence maps can be described using linear, hence convex inequalities. A drawback with making such assumptions is that the resulting simplified DMPO problem may omit high-quality treatment plans. On the other hand, solving a simplified optimization problem to global optimum may give better results than tackling a more comprehensive formulation, and may be the approach less sensitive to parameter perturbations—a desirable property in the design of accurate optimization tools in this thesis.

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

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AUTOMATED PLANNING BY INCREASED ACCURACY OF OPTIMIZATION TOOLS 15 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 the path taken and a summary of each of the appended papers.

5.1 Motivation

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

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

5.2 Summary of appended papers

The co-authors of the three appended papers have acted as academic and industrial advisors, 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 Björn Hårdemark, 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 plan optimization is presented. The purpose of the novel formulation is to overcome the known issues of the conventional penalty-based objective functions causing

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16 INTRODUCTION

the trial-and-error behavior of inverse treatment planning. Two important aspects of the proposed objective functions help in achieving this purpose: (1) the conventional 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 convex optimization problem. Fluence map optimization with the proposed objective functions requires the introduction of a large number—of the order of the number of voxels—of variables and constraints. Practical handling of this optimization problem is demonstrated through the adaptation of an interior-point method for exploitation of the problem structure, so as to obtain a reduction in problem size by several orders of magnitude. Numerical experiments confirm that the method efficiently solves the given optimization problem, and comparison to a commer- cially available generic solver indicates that the problem-structure exploitation is the key in achieving this result.

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

Paper B: Increased accuracy of planning tools for optimization of dynamic multileaf collimator delivery of radiotherapy through reformulated

objective functions

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

In this paper, the explicit approach to treatment plan optimization presented in Paper A is extended with a DMLC deliverability model from the literature. The purpose is to stress-test the proposed objective functions in a more clinically realistic setting, including comparison in the domain of deliverable treatment plans and final clinically accurate dose calculation. It is demonstrated that the problem structure needed to perform the algebraic manipulations of the tailored interior- point method developed in Paper A can be preserved despite the additional deliverability constraints, and that the given optimization problems can be solved efficiently.

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

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AUTOMATED PLANNING BY INCREASED ACCURACY OF OPTIMIZATION TOOLS 17 posed objective functions results in improved plan quality metrics in comparison to the conventional penalty-based functions, in this study represented by the MCO module in the treatment planning system RayStation (RaySearch Laborato- ries, Stockholm, Sweden). In addition, the generated treatment plans show better feasibility to constraints subsequent to accurate dose computation.

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

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

In this paper, an accurate formulation of direct machine parameter optimization for so-called sliding-window VMAT delivery is presented. The formulation is based on an algorithm for VMAT optimization from the literature. The purpose is to, in light of the accurate formulation, investigate the effects on plan quality and treatment efficiency when decreasing the number of sliding-window sweeps delivered as the machine head rotates around the patient. While it is generally true that many sweeps lead to better plan quality given a generous treatment time, it is hypothesized advantageous to decrease the number of sweeps if a highly efficient treatment is required. The algorithm from the literature is generalized, and an algorithmic modification is suggested for better handling of the suggested setting with fewer sliding-window sweeps.

Numerical results from two patient cases indicate that with tighter treatment time restrictions, at a certain point, it is beneficial to decrease the number of sliding-window sweeps to maintain high plan quality. It is observed that the suggested modified algorithm performs better than the original algorithm in terms of objective function value.

5.3 Main contributions

The main contribution of this thesis is the framework for—and novel perspective on—automated treatment planning, within which it is demonstrated how increased accuracy of the optimization tools provided to the treatment planner could open up for streamlining of the treatment planning process. The core of the suggested framework is mathematical tractability and strong correlation to measures of plan quality.

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

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18 INTRODUCTION

current treatment planning process a tedious task. In addition to the noncon- vexity, nondifferentiability, and vanishing gradients discussed in Section 3.2, the drawbacks include an implicit and not so clear relationship to the plan quality measures. The suggested framework for automated treatment planning is the result of a novel formulation of objective functions with which the penalty-based paradigm is abandoned in favor of a more explicit relationship to, thus better correlation with the widely used dose-at-volume measure. Paper B strengthens the conclusions made in Paper A regarding improved correlation by presenting a numerical study where cohorts of DMLC deliverable treatment plans generated within the conventional and the suggested frameworks are compared.

It is in general expected that a framework for automated treatment planning should be compatible with most IMRT delivery techniques. For the suggested framework, compatibility includes maintained mathematical tractability and correlation with measures of plan quality. Paper C approaches such a situation for the mathematically more complex VMAT delivery technique by giving a deliverability model for a direct machine parameter optimization problem, and by suggesting improvements of an existing heuristic to handle the resulting optimization problem.

A possible drawback with the proposed objective functions is largely increased dimensions of the treatment plan optimization problem. In Paper A, tractability of the resulting optimization problem is demonstrated through the exploitation of its 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 with respectively a DMLC and VMAT delivery model is demonstrated.

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Automated radiation therapy treatment planning by increased accuracy of optimization tools

Automated radiation

therapy treatment planning by increased accuracy of optimization tools

Preface

Contents

Introduction