ACTAUNIVERSITATIS

UPSALIENSISUPPSALA

2016

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Medicine 1264

Probabilistic treatment planningbased on dose coverage

How to quantify and minimize the effects ofgeometric uncertainties in radiotherapy

DAVID TILLY

ISSN 1651-6206ISBN 978-91-554-9720-0urn:nbn:se:uu:diva-304180

Dissertation presented at Uppsala University to be publicly examined in Skoogsalen, Ing.78-79, Akademiska Sjukhuset, Uppsala, Friday, 25 November 2016 at 13:00 for the degree ofDoctor of Philosophy (Faculty of Medicine). The examination will be conducted in English.Faculty examiner: Professor Uwe Oelfke (The Institute of Cancer Research, London).

AbstractTilly, D. 2016. Probabilistic treatment planning based on dose coverage. How to quantifyand minimize the effects of geometric uncertainties in radiotherapy. Digital ComprehensiveSummaries of Uppsala Dissertations from the Faculty of Medicine 1264. 51 pp. Uppsala: ActaUniversitatis Upsaliensis. ISBN 978-91-554-9720-0.

Traditionally, uncertainties are handled by expanding the irradiated volume to ensure target dosecoverage to a certain probability. The uncertainties arise from e.g. the uncertainty in positioningof the patient at every fraction, organ motion and in defining the region of interests on theacquired images. The applied margins are inherently population based and do not exploit thegeometry of the individual patient. Probabilistic planning on the other hand incorporates theuncertainties directly into the treatment optimization and therefore has more degrees of freedomto tailor the dose distribution to the individual patient. The aim of this thesis is to create aframework for probabilistic evaluation and optimization based on the concept of dose coverageprobabilities. Several computational challenges for this purpose are addressed in this thesis.

The accuracy of the fraction by fraction accumulated dose depends directly on the accuracyof the deformable image registration (DIR). Using the simulation framework, we could quantifythe requirements on the DIR to 2 mm or less for a 3% uncertainty in the target dose coverage.

Probabilistic planning is computationally intensive since many hundred treatments mustbe simulated for sufficient statistical accuracy in the calculated treatment outcome. A fastdose calculation algorithm was developed based on the perturbation of a pre-calculated dosedistribution with the local ratio of the simulated treatment’s fluence and the fluence of the pre-calculated dose. A speedup factor of ~1000 compared to full dose calculation was achieved withnear identical dose coverage probabilities for a prostate treatment.

For some body sites, such as the cervix dataset in this work, organ motion must be included forrealistic treatment simulation. A statistical shape model (SSM) based on principal componentanalysis (PCA) provided the samples of deformation. Seven eigenmodes from the PCA wassufficient to model the dosimetric impact of the interfraction deformation.

A probabilistic optimization method was developed using constructs from risk managementof stock portfolios that enabled the dose planner to request a target dose coverage probability.Probabilistic optimization was for the first time applied to dataset from cervical cancer patientswhere the SSM provided samples of deformation. The average dose coverage probability of allpatients in the dataset was within 1% of the requested.

Keywords: Radiotherapy, treatment simulation, probabilistic planning, dose calculation,probabilistic optimization, statistical shape model

David Tilly, Department of Immunology, Genetics and Pathology, Medical Radiation Science,Rudbecklaboratoriet, Uppsala University, SE-751 85 Uppsala, Sweden.

© David Tilly 2016

ISSN 1651-6206ISBN 978-91-554-9720-0urn:nbn:se:uu:diva-304180 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-304180)

To my family

List of Papers

This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

I Tilly, D., Tilly, N. and Ahnesjö, A. (2013) Dose mapping sensi-

tivity to deformable registration uncertainties in fractionated ra-diotherapy – applied to prostate proton treatments. BMC Medical Physics, 13(1):2

II Tilly, D. and Ahnesjö, A. (2015) Fast dose algorithm for genera-tion of dose coverage probability for robustness analysis of frac-tionated radiotherapy. Physics in Medicine and Biology, 60(14):5439-5454

III Tilly, D., van de Schoot A.J., Grusell E., Bel A. and Ahnesjö, A. (2016) Dose coverage calculation using a statistical shape model – applied to cervical cancer radiotherapy. Accepted for publica-tion in Physics in Medicine and Biology

IV Tilly, D., Holm, Å., Grusell E. and Ahnesjö, A. (2016) Probabil-istic optimization of the dose coverage – applied to radiotherapy treatment planning of cervical cancer. Manuscript

Reprints were made with permission from the respective publishers.

Contents

1. Introduction ................................................................................................. 9 1.1 The probability of delivering the prescribed dose .............................. 10

1.1.1 Margin based planning ............................................................... 10 1.1.2 Limitations of the margin concept .............................................. 11 1.2.3 CTV based probabilistic planning without margins ................... 11

1.2 Aim of the thesis ................................................................................ 12

2. Treatment simulation framework for probabilistic planning .................... 13 2.1 Calculating treatment scenario dose ................................................... 13

2.1.1 Uncertainty model ...................................................................... 14 2.1.2 Accumulating dose in a deforming patient ................................. 15 2.1.3 Dose calculation models ............................................................. 15

2.2 Probabilistic plan evaluation .............................................................. 17 2.2.1 Treatment planning criteria ......................................................... 17 2.2.2 Mathematical formulation of dose coverage probability concepts ............................................................................................... 18

3. Challenges of simulating treatment scenarios ........................................... 21 3.1 Assessing the accuracy of the accumulated dose ............................... 21 3.2 Reducing the dose calculation burden ................................................ 23 3.3 Patient geometry sampling using a statistical shape model ................ 25

4. Optimization towards a requested dose coverage probability ................... 33 4.1 Expected percentile underdosage ....................................................... 33 4.2 Defining the optimization problem .................................................... 34 4.3 The target coverage constraint tolerance ............................................ 37 4.4. Convergence to the requested dose coverage probability ................. 38

5. Summary and directions for future research ............................................. 42

6. Summary in Swedish ................................................................................ 44

7. Acknowledgements ................................................................................... 46

Abbreviations and list of symbols

CTV Clinical target volume CVaR Conditional Value at Risk DIR Deformable image registration DVCM Dose volume coverage map DVF Deformation vector field DVH Cumulative dose volume histogram EPUD ⟨ ⟩ Expected Percentile UnderDosage GPU Graphics processor unit IM Internal margin ITV Internal target volume OAR Organ at risk PCA Principal component analysis PTV Planning target volume PUD Percentile UnderDosage (dose coverage probability

for targets) ROI Region of interest SM Setup margin SSM Statistical shape model VaR Value at Risk Dose coverage Probability density function of , Dose coverage probability Probability , The prescribed PUD

DVH Iso-probability line in the DVCM Number of treatment scenarios

Number of fractions Number of volume elements (voxels) in the patient Number of beamlets in the treatment plan Fixed reference geometry Geometry of the :th fraction

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1. Introduction

Cancer tumours have been treated with ionizing radiation since shortly after the discovery of x-rays in the late 1800’s. Approximately 30% of all cancer cure is attributed to radiotherapy, and half of all cancer patients receive radia-tion at some point during their treatment (Cancerfonden 2015). Modern radi-ation oncology is a highly evolved discipline employing sophisticated meth-ods and apparatus in multiple areas of medical physics. The most common radiation treatment is external photon beam radiotherapy where electrons from linear accelerators are used to produce high energy Bremsstrahlung photons (0.2-10 MeV incident on the patient compared to 10-120 keV in diagnostic x-ray imaging).

Ionizing radiation cause damages to the DNA that can make the cells una-ble to reproduce if the damage is not repaired correctly. Localized tumour vol-umes are irradiated from multiple directions with cross-fire techniques to con-centrate the dose to the tumour cells and minimize side effects from dose bur-dens to normal tissues.

The common sequence of procedures in radiotherapy is to image the patient using computerized tomography or magnetic resonance imaging prior to the treatment, then construct a treatment plan based on the images and finally de-liver the treatment according to the plan. Modern radiotherapy may also con-tain treatment plan adaptation to accommodate to changes occurring during the course of treatment.

Radiobiology research has proven that dividing the treatment into fractions can increase the therapeutic window (Withers 1975), i.e. we are able to in-crease the target dose and thus control the tumour without increasing the risk of normal tissue complications. The patients receive typically one fraction per day of the total dose, with the number of fractions varying depending on can-cer type and treatment protocol from a few up to 40 yielding a total tumour dose in the range of 35 to 100 Gy.

At every fraction there are uncertainties arising from e.g. treatment deliv-ery, patient positioning and patient organ motion. A benefit of fractionation is the averaging effect, i.e. an erroneous dose at a single fraction is of less im-portance for a treatment consisting of many fractions than of one with only a few fractions. It is possible to estimate the effects of various uncertainties by repeated simulation of the entire treatment with the relevant uncertainties sam-pled for each fraction. The aim of this thesis is to find efficient algorithms for quantifying, analysing and minimizing the effect of these uncertainties.

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1.1 The probability of delivering the prescribed dose 1.1.1 Margin based planning Uncertainties in radiotherapy are traditionally dealt with by expanding the ir-radiated volume and treat with margins as schematically shown in figure 1. The gross tumour volume (GTV) defines the visible (or demonstrated) tumour volume and the clinical target volume (CTV) include a margin around the GTV for microscopically spread disease (ICRU 1993). An optional internal margin (IM) is added to the CTV to form the internal target volume (ITV) accounting for the variation in anatomical location and shape of the CTV (ICRU 1999). Finally, a setup margin (SM) accounting for the variation in beam delivery is combined with the ITV to construct the planning target vol-ume (PTV). The intention of the CTV to PTV margin is to simplify the treat-ment planning process by prescribing to the PTV and thus ensure sufficient dose to the tumour.

The uncertainties are commonly divided into systematic (for all fractions planned on a common image) and random errors (specific to each fraction) (van Herk et al 2000). Example of systematic errors are setup-errors at the imaging session prior to treatment, uncertainty in defining the organs in the acquired images and dose calculation inaccuracies. Random errors include daily setup-errors and organ motion not else accounted for in the treatment planning.

Van Herk et al (2000) developed a commonly used recipe for determination of margin sizes based on dose coverage probabilities. The recipe is constructed such that the combination of the IM and SM margins ensures that the planned dose to the PTV edge (often 95% of the intended dose) in reality will be de-livered to the entire CTV in 90% of the treatments. Note that the IM and SM should not be combined linearly as they contain uncorrelated uncertainties that should be added quadratically (ICRU 1999). The result is a probabilistic pre-scription that the entire CTV in 90% of the treatments receives at least 95% of the dose prescribed to the interior of the CTV. The probability of underdos-age is however implicit and rarely reported.

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Figure 1. Illustration of how to perform margin based treatment planning using the standard volumes recommended by ICRU. The ITV takes the motion of the CTV into account. The SM relates to the uncertainty in beam delivery and is combined with the IM to form the PTV such that the planned dose to the PTV edge (black curve), often 95% of intended target dose, is delivered to the entire CTV in 90% of the treatments (van Herk et al 2000).

1.1.2 Limitations of the margin concept The van Herk margin recipe is population based and does not take into account the specific geometry of the individual patient. Moreover, it assumes normal distributed errors and utilizes a simplified dose calculation model where ran-dom errors are accounted for by convolution and thereby assuming an infinite number of fractions. Van Herk et al (2003) showed improvement of the con-volution model, for a 36 fraction treatment, using effective values of the error standard deviations. The margin needs to be larger for a finite number of frac-tions to fulfil the same probability of dose coverage because the consequences of a “miss” become larger. Gordon and Siebers (2007) determined the condi-tions for when the van Herk recipe is accurate and also developed an alterna-tive margin recipe that can ensure target coverage down to five fractions.

1.2.3 CTV based probabilistic planning without margins Probabilistic planning where the quantified uncertainties are incorporated di-rectly into the treatment optimization to produce a plan that is robust with respect to the involved has been proposed as an alternative to margin based planning (e.g. Baum et al 2006, Birkner et al 2003, Chan et al 2006, Li and Xing 2000, McShan et al 2006, Moore et al 2009, Rehbinder et al 2004, Sobotta et al 2010, Stroom et al 1999, Trofimov et al 2005, Witte et al 2007, Fredriksson et al 2011). The proposals have in common that they do not use

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the PTV concept but rather optimize the treatment plan with respect to the CTV directly. The uncertainties that affect the resulting dose distribution are sampled from their respective probability density functions in a process di-rectly included into the treatment plan optimization. The produced plans have been shown to be robust with respect to the uncertainties studied but the opti-mization strategies are not designed to ensure a specific dose coverage proba-bility goal. This presents a problem in the clinic if a probabilistic planning paradigm is to be implemented since dose coverage probability has long been a corner stone in the margin based planning commonly used.

Inclusion of dose coverage probability into a probabilistic framework would enable clinicians to use similar prescriptions as in margin based plan-ning but allow for patient specific improvements (lower risk for side effects or improved probability for cure) by direct utilization of patient specific cir-cumstances (geometry etc.). Gordon et al (2010) suggested a probabilistic framework where the optimization problem was formulated to fulfil a dose coverage probability. Their probabilistic plans demonstrated better target dose coverage compared to margin based plans without compromising organ at risk doses for prostate patients. Bohoslavsky et al. (2013) used dose coverage probabilities to evaluate probabilistic plans versus margin based plans and showed that they could reduce the dose to the rectum wall while maintaining adequate target dose coverage.

1.2 Aim of the thesis The aim of this thesis was to develop calculation models tailored for probabil-istic planning of radiotherapy as to enable evaluation and optimization of treatment plans based on dose coverage probability. In particular these aspects have been addressed:

• Sensitivity of the fraction by fraction accumulated dose to the un-certainty in the tracking of tissue from fraction to fraction (Paper I)

• Efficiency of the repeated dose calculation needed for the simula-tion of many treatments (Paper II)

• Sampling of organ deformation for the generation of scenarios in-cluded in the probabilistic planning process (Paper III)

• Probabilistic optimization where the dose planner can explicitly re-quest a dose coverage probability (Paper IV)

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2. Treatment simulation framework for probabilistic planning

2.1 Calculating treatment scenario dose The effect of uncertainties can in principle be calculated analytically if the dose is calculated using pencil kernel methods and the uncertainties modelled by normal distributions (Bangert et al 2013). The use of pencil kernel methods may not be accurate enough in case of heterogeneous patient anatomy, e.g. in lung patients, and uncertainties may not always be assumed to be normally distributed. Thus, as in this work, probabilistic planning relies on repeated simulations of a treatment in order to assess the effect of the involved uncer-tainties. We define a treatment scenario as one realization of a complete treat-ment where, for each fraction, the uncertainties have been sampled from their respective probability distributions. Each scenario will result in a dose distri-bution and yield scenario specific dose volume histogram (DVH) which can be used as input to a probabilistic analysis. The details of simulating a scenario where the position and shape of the patient geometry are sampled at each frac-tion, see figure 2, will be explained in the following subsections.

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Figure 2. The flow of calculations in the simulation of a single treatment scenario with fractions. A patient geometry sampler can provide the fraction specific pa-tient geometry by resampling a fixed reference image using a geometric defor-mation ( ) sampled from a shape model. The dose is calculated in and accumu-lated in by mapping back the dose through . Dose coverage probabilities based on all scenario DVHs can then be evaluated with treatment criteria defined for prob-abilistic planning.

2.1.1 Uncertainty model The uncertainties are commonly divided into systematic uncertainties, com-mon to all fractions, and random uncertainties, specific to each fraction (van Herk et al 2000). The image used for treatment planning is simply a snapshot of the patient which introduce a systematic error common to all fractions in the treatment based on that image. The setup-error at each fraction and the inter-fraction motion is considered part of the random error. The sampled er-rors during the simulations must adhere to this convention, i.e. systematic er-rors sampled once per scenario and random errors sampled at every fraction.

Patient geometry sampler

Fraction

⋯ ( ( ))

F Shape model

Fraction 1

Scenario DVH

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In this thesis we have considered setup-errors and organ motion but other un-certainties such as the uncertainty in the definition the ROIs can also be con-sidered. The distribution of setup-errors can be extracted from collected clin-ical setup-error data if available, or be taken from the literature as done in this work.

2.1.2 Accumulating dose in a deforming patient The total dose from fractionated treatments can be accumulated into a refer-ence geometry for analysis. This requires the relation between the geometries of the different fractions to be known. The image used for treatment planning is often chosen to be the fixed frame of reference . However, sometimes it can be more relevant to accumulate the dose to an intermediate or the latest acquired image of a treatment. Deformable image registration (DIR), see e.g. (Hill et al 2001), can be used to calculate the transform between the image acquired during a treatment and and thereby enable the estimation of the total delivered dose (Yan et al 1999). The quality of the DIR will directly affect the accuracy of the calculated dose which is the topic of Paper I.

If the location of each tissue element (voxel) in fraction specific images , = 1, … , can be related to by the geometrical transforms ( ) of po-sition , then the total scenario dose ( ) is calculated by summing the cor-responding fraction doses ( ) calculated in ( ) = ( ) .

(1)

Given the task to simulate treatment scenarios, a patient geometry sampler can be used to provide the deformations ( ), and the images needed to per-form the calculations. For Paper I where the sensitivity of the accumulated dose to uncertainties in the DIR was investigated, a schematic prostate organ model with an assumed deformation decreasing exponentially from the pros-tate edge was chosen. A more sophisticated model based on principal compo-nent analysis (PCA) of deformations extracted using DIR was chosen for Pa-pers III and IV where probabilistic evaluation and optimization tools were de-veloped and tested on dataset from cervix patients. The scenario DVHs calcu-lated from the scenario doses forms the basis of a statistical analysis assessing the effect of the uncertainties (described in section 2.2).

2.1.3 Dose calculation models The dose calculation burden is significantly increased for probabilistic plan-ning compared to standard treatment planning since many treatment scenarios must be simulated each consisting of a number of fractions. Critical for the overall calculation time is to determine how many scenarios is needed to get

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sufficient statistical accuracy. We can estimate the number of scenarios needed if we assume the dose coverage probability to be a binomial distribu-tion that is either passing or failing a dose-volume prescription. To achieve a ±3% (95% CI) uncertainty for a 90% probability level approximately 400 treatment scenarios are needed.

A brute force approach with explicit calculation for every fraction of the dose using a full-fledged dose engine is not feasible in clinical routine, despite the increasing performance of modern computers. Various approximations can be made to speed up the dose calculation. The “static dose cloud” approx-imation assumes that the dose as a function of location does not change due to deformations such that ( ) ≈ ( ) ,

(2)

where ( ) is the fraction dose calculated in the reference geometry . The static dose cloud approximation has been shown to be a valid approximation for prostate patients (Craig et al 2003) and in Paper III the same assumption is made for cervix patients. The dose needs then to be calculated only once with a full-fledged dose engine. Further speedup can be achieved by imple-menting the simulations on massively parallel hardware such as a graphics processor unit (GPU).

During the treatment optimization it is necessary to calculate the dose and gradient with respect to the optimization variables. Beamlets are often used for this purpose where a beamlet is defined as the dose per unit incident flu-ence from a pixel in the plane where the fluence is defined (bixel). The dose calculated in is calculated by ( ) = , ∙

(3)

where , is the dose from beamlet to voxel with location , and ∈ℝ are the beamlet weights. Equation (3) is a matrix-vector multiplication which can be performed efficiently using highly optimized linear algebra soft-ware packages. It is common to use the beamlet weights, at least at some stage, as optimization variables in the treatment optimization, as was done for the probabilistic optimization in Paper IV. The beamlet weights can subsequently be converted into machine configurations. The use of beamlets and the static dose cloud approximation is combined in Paper IV. Since the dose as a func-tion of location is assumed to not change due to organ motion, c.f. equation (2), we can perform the optimization with only a single set of beamlets. In general, a different set of beamlets is needed per fraction and scenario which will result in extremely long calculation times.

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In Paper II, an efficient dose calculation algorithm was developed specifi-cally for the purpose of the repeated dose calculation involved in simulating many treatment scenarios. The efficiency of the algorithm arise from being based on perturbation of a pre-calculated dose distribution.

2.2 Probabilistic plan evaluation

2.2.1 Treatment planning criteria Prescriptions in conventional radiotherapy following recommendations in ICRU Report 83 (ICRU 2010) are specified using (a number of) criteria based on dose coverage and which are extracted from DVHs for the ROIs. As an example, the PTV % is the dose that 98% of the PTV volume at least receives and can be used to describe a minimum dose coverage criterion. Cen-tral to this thesis is to enable prescriptions based on probabilistic terms suita-ble for both treatment planning and reporting. Similar to Gordon et al (2009) we specify the target minimum dose of a prescription by requiring that , , defined as the dose coverage assured with probability , must be at least some threshold dose , i.e. , ≥ , . (4) We term , as the Percentile UnderDosage (PUD) in the context of a target minimum dose criterion. Further, the expectation value of the target for all scenarios with ≤ , is termed the Expected Percentile UnderDosage (EPUD) and is denoted ⟨ ⟩ . EPUD is a central concept in the probabilistic optimization in Paper IV. Figure 3 illustrates the used statistical concepts.

The criterion for the CTV equivalent to the margin recipe, as described in 1.1.1, is in this notation written as %, % ≥ %, % (5) i.e. the entire CTV should receive a prescribed dose %, % in 90% of all treatments. As % is a numerically unstable measure, the near minimum % is recommended for explicit calculations (ICRU 2010). Similar criteria can be defined for organ at risks where volume based requirements are com-mon to limit the upper dose coverage. Along these lines we define , as the fractional volume receiving at most dose to the probability .

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Figure 3. (left) The scenario DVHs from many treatment simulations. The probabil-ity distribution % of % (right) is estimated by extracting % from the simu-lated DVHs as indicated by the dashed arrows. The PUD and EPUD is used to char-acterize the lower part of %.

2.2.2 Mathematical formulation of dose coverage probability concepts A treatment plan is defined by machine configurations consisting of MLC leaf positions, monitor units etc. For our purpose of developing an alternative treat-ment optimization method in Paper IV, we neglect for simplicity machine spe-cific data and base all dose calculations on beamlets. We define as the beam-let weights which will then be the optimization variables.

The uncertainties included in the simulations, setup-errors and organ mo-tion divided into systematic and random errors, are combined in a multivariate random vector where the individual variables describe different types of un-certainties. In particular, we let the sample space of the patient anatomy be defined by whose elements are the parameters of the patient geometry sam-pler that provides the displacement transforms ( , ). The total scenario dose ( , , ) given by is a stochastic variable in as it depends on ( , ).

A treatment plan is based (and evaluated) on ROIs defined by contours drawn on images acquired before and in some cases during the treatment. For calculation purposes we divide the patient geometry into voxels with po-sitions { , = 1,… , }. The ROIs are defined in and represented by the binary maps ( ) which states if is inside or outside of the ROI. The volume in units of number of voxels can thus be calculated as

98%

%,%

⟨ %⟩%

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= ( ). (6)

A relative cumulative dose volume histogram is defined as the fractional vol-ume receiving at least a dose , DVH( , , ) = 1 − ( , , ) ∙ ( )

(7)

where is the Heaviside function. The dose coverage , i.e. the dose that a fractional volume at least receives, depends on as well as the geometrical uncertainties , ( , ) = min{ ∶ DVH( , , ) ≥ }, (8) and is also a stochastic variable. Given , the probability density function of can be approximated by extracting from a set of simulated scenario doses, see figure 3. The probability that will exceed some level ∈ ℝ,

, can be determined from through ( , , ) = ( ( , ) ≥ ) = ( , )d .

(9)

The PUD can now be written as the dose coverage that a treatment outcome has the probability to meet or exceed , ( ) = max ∈ ℝ ∶ ( , , ) ≥ . (10)

A dose volume coverage map (DVCM) (Gordon and Siebers 2009) is a 2D map of ( , , ) as a function of dose level and volume for a given , see illustration in figure 4 on how it may be constructed. The DVCM is a generalization of the population dose histograms (DPH) used in the derivation of the margin size (van Herk et al 2000). The DPH is the probability that the entire CTV ( %) receives at least and represents a horizontal line in the DVCM for =100%. By extracting an iso-probability line in the DVCM a pseudo-DVH is obtained that can used for evaluating versus the treatment cri-teria. We denote the 90% iso-probability line in the DVCM as DVH %. For OARs, we choose to calculate the probability that receives at most some dose , which mean that DVH % indicate the dose-volume combination that the OAR to 90% probability at most receives.

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Figure 4. (left) Construction of a DVCM based on 4 DVHs. The for an arbitrary position ( , ) in the dose-volume map is approximated by counting the number of DVHs that are above (or right) of ( , ). (right) Example of a CTV DVCM based on a 100 scenarios.

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3. Challenges of simulating treatment scenarios

3.1 Assessing the accuracy of the accumulated dose

The accumulated dose in adaptive radiotherapy relies on deformable image registration (DIR) (c.f. equation (1)). DIR enable tracking of tissue element locations through several fractions. DIR is associated with uncertainties and is still an active field of research (Hill et al 2001, Oliveira and Tavares 2014). There exist no ground truth when assessing the accuracy of DIR. It is neces-sary to identify landmarks in the images that are registered. Multi-institutional studies show that there exist CT-CT registration algorithms with an average mean absolute error of ~1 mm (Brock 2009, Kashani et al 2008, Murphy et al 2011). Therefore a way to quantify the required accuracy in the DIR to achieve accurate fraction by fraction dose accumulation was developed in Paper I.

An analytical patient geometry sampler for a prostate patient anatomy was devloped in Paper I to generate the for the treatment simulations. The model provided the ground truth needed for dose accumulation without errors. By adding a known error to we can perform numerical experiments and study the effect of the error on calculated treatment outcome. Treatment simulations were performed according to figure 2 where the analytical model including the added error served as the shape model.

The volumes of the prostate, rectum and bladder in the analytical model were averages of volumes from 15 prostate patients treated at Akademiska Sjukhuset, Uppsala, Sweden. The prostate was assumed to move as an incom-pressible sphere and deforming the surrounding tissue. The tissue closest to the prostate edge was assumed to move in the same direction of the prostate motion with the magnitude of the deformation decreasing exponentially from the prostate edge. The rate of decrease was chosen to give reasonable change in rectum and bladder volumes. The inter-fraction motion of the prostate was sampled from a 3D normal distribution (no cross correlation) with standard deviations 4, 1 and 4 mm in the anterior-posterior, lateral and superior-inferior directions, respectively, consistent with the literature (Byrne 2005). The vir-tual patient in Paper I was planned with hypofractionated (6.7Gy × 5fx) spot scanned proton therapy whose sharp distal penumbra is sensitive to errors in the dose mapping.

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A sampled error , ( ) represented by a varying B-Spline vector field was added to the analytical deformation , ( ) such that the sampled deformation of the :th fraction became ( ) = , ( ) + , ( ). (11) The B-spline control points were sampled from 3D normal distribution (zero correlation) with zero mean. The magnitude of , was increased by increas-ing the standard deviation in the generation of the B-spline field. The mean absolute error ⟨ , ( ) ⟩ was used to specify the level of registration error in the results.

The dose of 400 scenarios was calculated for each magnitude of the error , . By comparing the calculated treatment outcome (TCP, NTCP, target % and OAR %) with the nominal scenario, , = , it was possible to assess the effect of the introduced deformable image registration uncertainty.

The simulations showed that the target % was the most sensitive meas-ure to the registration error. A requirement that the calculated error in % is less than 3% translated into an accuracy requirement of approximately 2 mm for the registrations. Figure 5 shows that the target coverage is more sensitive to the registration error than the bladder and rectum coverage in terms of DVH %.

Figure 5. Dose coverage probability, DVH %, for the nominal (without error) and 4 mm ⟨ ,e( ) ⟩ error. The dose is relative a target dose of 6.7Gy × 5fx = 33.5Gy.

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3.2 Reducing the dose calculation burden Advances in algorithms and hardware, e.g. employing massive parallel hard-ware such as GPUs, have reduced the calculation time of a single treatment plan down to ~1s (Xun et al 2014). At this speed the calculation time for the simulation of 400 scenarios of a 5fx hypofractionated treatment will be > 30 min. If the dose coverage probability is incorporated in the iterations of the treatment optimization (possibly evaluated at every iteration) there will be a need for algorithms that are several orders of magnitude faster.

The common static dose cloud approximation assumes that the dose distri-bution as a function of does not change due to anatomical deformations. Since only a single explicit dose calculation is made the calculation burden becomes very small. For the simulation of male pelvic treatments one can as-sume that the static dose approximation gives sufficiently small errors (Craig et al 2003) compared to full calculations. The same approximation is utilized in Papers III and IV for the cervix patients. Body sites with more pronounced heterogeneities, such as the lung will require a more accurate dose calculation (Craig et al 2003). Even with the static dose cloud approximation the dose distribution will be time consuming for the many hundred scenarios needed. This computational burden was the motivation for the fast algorithm devel-oped in Paper II for the repetitive calculation of treatment scenario doses.

The main idea behind the algorithm is to approximate dose perturbations as proportional to perturbations in the incident energy fluence, as seen in the coordinate system fixed in the patient. A correction factor based algorithm can then be constructed provided there exist a similar pre-calculated (non-zero) dose distribution for all points of interest for a known (non-zero) en-ergy fluence to use as a basis for corrections. The fluence of the nominal (i.e. no error) beam configuration is effectively zero outside the beam and thus not suited as basis for factorized correction factors. The average fluence (as seen by ) for an infinite number of fractions subject to positional error can be calculated by convolving the nominal fluence with the error distribu-tion and is non-zero outside the nominal beam. The corresponding dose dis-tribution ( ) is calculated using a dose engine that can take a fluence dis-tribution as input. is wider, i.e. non-zero for all points of interest, and thus suitable for a correction factor based algorithm.

The fluence per fraction for a single beam direction of a scenario as seen by can be expressed as ( , ) = 1 − , −

(12)

where and are the projection of on the plane where the fluence is defined, and and are the sampled positional errors of the beam lateral coordinates

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at the :th fraction. The correction factor applied to could as a first approx-imation be the local ratio of and . The ratio can be made more robust by smoothing with convolution by a pencil kernel for a representative depth and thus effectively performing the correction in the smoother dose domain. As the primary and scatter dose components have different characteristics (Ahnesjö et al 1992) the smoothing is done per component using separate ker-nels. This requires that the primary and scatter dose components can be calcu-lated separately in (i.e. and ). The perturbation based primary dose of the scenario is thus calculated as ( ) ≈ ( , ) ⊗( , ) ⊗ ∙ ( )

(13)

where is the primary dose kernel. The scatter dose component ( ) is

calculated analogously using a scatter dose kernel and . The total sce-nario dose is then the sum of and .

The speed-up of the algorithm is attributed to the fraction by fraction ac-cumulation is made using the 2D fluence distributions rather than the 3D dose distributions. The speedup will be greater the more fractions per sce-nario there are. The dose calculation for a single point, after the convolution ( , ) ⊗ has been made, requires only two scaling and one summation

since , and the denominator in equation (13) can all be pre-calcu-lated. Since all points are independent it is straightforward to implement the algorithm using massively parallel hardware such as a GPU.

The effect of organ motion can to some degree be incorporated by weighting the fluence per fraction in equation (12) with the change in depth dose and inverse square law, as shown in Paper II. This is only valid for tissues that moves as a rigid body compared to the incident fluence. Setup-error and organs moving as a rigid body are therefore modelled well even in heteroge-neous media such as a lung-like virtual phantom as shown in Paper II. In fact, the 2%/2mm gamma (Low et al 1998) pass rate for points belonging to a mov-ing tumour inside the virtual lung was practically 100%.

Regions with a deforming geometry must be divided into sub-regions with motion that can be approximated as a rigid body. The accumulation of fluence would then be performed separately per region. The increased dose burden would be limited as the pre-calculated properties can be reused between sub-regions.

Figure 6 shows the excellent agreement in DVH % between the full dose calculation (explicit dose calculate for every fraction using a pencil kernel method (Ahnesjö et al 1992)) and the perturbation algorithm based on the simulation of setup-errors and prostate motion for a hypofractionated

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(5Gy × 7fx = 35Gy) VMAT treatment. A speedup of a factor of ~1000, using only the CPU, over the full dose calculation was achieved.

Figure 6. DVH %based on dose coverage calculations for a 5fx VMAT treatment using the perturbation algorithm (solid) and full dose calculation (dots). The dose is relative 5Gy × 7fx = 35Gy.

3.3 Patient geometry sampling using a statistical shape model Cervical cancer patients exhibit substantial inter-fraction deformation of or-gans. The deformation of the uterus, that can be several centimetres, is mainly attributed to the varying filling of the bladder (Jadon et al 2014). It is of inter-est to investigate probabilistic planning methods including a realistic defor-mation model due to the large treated volumes. A probabilistic evaluation tool was therefore developed in Paper III where a patient geometry sampler pro-vided the patient geometry deformations for the simulations. The treatment scenario calculations were implemented on GPU hardware for efficiency. The clinical plans of five cervical cancer patients treated at Academic Medical Center, Amsterdam, The Netherlands were retrospectively evaluated using the probabilistic tool. To our knowledge, this work was the first time probabilistic

26

evaluation of treatment plans was shown for cervix patients where the defor-mation was included.

The patients were treated with IMRT or single-arc therapy (2Gy × 23fx, 6MV photons). The primary CTV encompassed the GTV, cervix, corpus-uterus, and the upper part of the vagina. An ITV was created by expanding the primary CTV using a 10 mm margin. The left and right lymph nodes were included in the total CTV. A PTV was created using an 8 mm margin around the union of the ITV and lymph nodes.

Statistical shape modelling (SSM) (Cootes et al 1995) have been used ex-tensively for segmentation purposes (Heimann and Meinzer 2009) and for modelling of patient specific lung respiratory motion (Zhang et al 2007). SSM based on PCA have been used to model deformations in prostate radiotherapy patients using deformations extracted by point based DIR (Söhn et al 2005, Budiarto et al 2011) and DIR using both image and contour information (Xu et al 2014). Söhn et al and Xu et al modelled intra-patient deformation whereas Budiarto et al collected deformations from several prostate patients.

It was investigated in Paper III if a SSM based on PCA could provide a compact and efficient means of sampling realistic deformations for cervical cancer patients. The model was based on the five cervical cancer patients ( = 5) each with 4-6 segmented follow-up CT. The deformations were extracted using intra-patient deformable image registrations (ADMIRE, v1.11, Elekta AB) (Elekta 2015) where a normalized sum of squared differ-ence similarity measure was used together with the contours to drive the reg-istrations. The inclusion of contours in the DIR was crucial to the result, and thus it was imperative that the segmentation was done consistently for all pa-tients. All segmentations were performed by experienced physicians and thereafter carefully reviewed to ensure consistency. The bladder, rectum, fe-murs and the union of corpus, cervix and vagina was included in the DIR. The Dice (Dice 1945) scores when comparing the DIR were generally above 0.9 and the mean distance to surface ~1 mm for all structures in the DIR, indicat-ing that the DIR provided a sound basis for the SSM. The intra-patient defor-mations were mapped using DIR to an average patient (Ehrhardt et al 2011) serving as a common frame of reference where the deformations were col-lected.

A shape deformation vector, defined in , for the :th series of the :th

patient was constructed by discretizing the deformation ( ) collected in in voxels = ( ), ( ), … , ( ) ∈ ℝ . (14) The average shape deformation = 1∙ , (15)

27

where is the number of repeat images per patient, was used to centre the shape deformations. With PCA one can decompose the correlated centred de-

formations − into uncorrelated eigenmodes by finding the eigensystem of the covariance matrix of all centred deformations. Furthermore, the PCA ranks the eigenmodes according to what degree each eigenmode explain the variance in the data. The highest eigenvalue means that the corresponding eigenmode explain the variance in the data to the highest degree and is said to be a dominating eigenmode. The PCA of the patient data in Paper III show that 90% of the variance in the deformation can be explained by the 15 domi-nating eigenmodes.

A deformation vector field can be sampled using the first eigenmodes, ≈ + ∙ | | = 1,

(16)

where the coefficients are sampled from probability distributions with zero mean and variance equal to their respective eigenvalues. The assumption was made that normal distributed can reconstruct the original data. The defor-mations ( ) used in the treatment simulations could be created very effi-ciently by sampling random numbers from respective normal distribution and summing the weighted eigenvectors.

The residual error of truncating the sum of eigenmodes can be quantified.

The optimal coefficients for the reconstruction of − using is its projec-tion onto the eigenvectors ( ) = − ∙ . (17) The truncation error is calculated as ( ) = − − ( ) ∙

(18)

which is a ℝ vector that can be transformed to represent the geometrical error as a function of location. The distribution of the truncation error for the

different ROIs can be calculated by combining from all patients and series. Figure 7 shows the median truncation error as well as the probability density function as more eigenmodes are added.

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Figure 7. The median residual error per ROI when deformations are approximated using varying numbers of the dominating eigenmodes. The grayscale map indicate the probability density function of the truncation error for a chosen number of eigenmodes (all three ROIs combined). The median errors are higher than the maxi-mum value of the probability density function due to the asymmetrically shaped probability density function with a tail towards larger errors. The grayscale is nor-malized such that for each eigenmode added, the error level with highest probability is black. The union of corpus/cervix/vagina is denoted CCV.

Figure 7 shows a larger truncation error for the same compared to the results of Budiarto et al for prostate patients which reached 1 mm average truncation error using only seven eigenmodes (2011). The difference may be due to dif-ferences in deformation patterns between cervix patients and prostate patients. Budiarto et al modelled the prostate surface whereas we modelled the entire patient which also might contribute to the difference.

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Figure 8. Evaluation of the planned dose, based on 1000 simulated scenarios using 15 eigenmodes, versus a PUD criteria of %, % = 95%of46Gy = 43.7Gy, and the bladder , % and rectum , %. The criterion for CTV overdosage was 105%of46Gy = 48.3Gy. The figure is adapted from Paper III.

Figure 8 shows the result of the probabilistic evaluation of the clinical plan. The planned dose was evaluated versus the probabilistic evaluation criteria using the RTOG recommendations to limit the bladder to 35% and the rectum to 60% (Jhingran et al 2012). We chose the bladder , % and rectum , % as the corresponding probabilistic measures. The CTV

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target minimum coverage was evaluated versus 98%,90%presc = 95%of46Gy. The PUD was equal or above 98%,90%presc for patients #1, #2 and #5 whereas patients #3 and #4 was showing some slight loss of coverage. At the same time the bladder , % was lower than the planned for all patients so the probabilistic evaluation show that there might be room to increase the target coverage. An alternative could be to allow the bladder dose to increase and thereby possible lower the rectum dose that is clearly above the RTOG rec-ommended limit.

The probability density functions based on the simulations describe the spread of the calculated dosimetric outcome of the treatment, as seen for pa-tient #4 in figure 9. Again the PUD is lower than %, %, but is can also be seen that there is no probability of overdosage as the narrow %is entirely below the overdosage criterion 105% of the intended dose. The %is slightly below the intended target dose of 46 Gy. The probability density func-tions for the bladder and rectum confirm the results in figure 8.

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Figure 9. Probability density functions based on the clinical plans for patient #4 (CTV %, %, %; Bladder ; Rectum ). The PUD is evaluated ver-sus an prescribed %, % equal to 43.7 Gy.

The calculation speed of the simulations depends highly on the number of eigenmodes used. The calculation time, using the GPU, for simulating 1000 scenarios using 15 eigenmodes were ~13 min. It is therefore interesting to investigate how the number of eigenmodes used in the simulations affects the calculated , . Figure 10 shows that the difference in , is within 0.5 Gy when using seven eigenmodes compared to using fifteen eigenmodes. The rectum , % was most sensitive to using few eigenmodes. A conclusion from figures 7 and 10 is that more eigenmodes needs to be included for pure geometrical investigations compared to dosimetric investigations since the dose boarders are relatively diffuse.

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Figure 10. The difference in dose of the calculated dose coverage probabilities when basing the SSM on different number of eigenmodes. The result is based on 10000 scenarios. Figure from Paper III.

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4. Optimization towards a requested dose coverage probability

4.1 Expected percentile underdosage Optimization methods have been developed in finance to handle a volatile stock market. A common risk measure is to specify the percentile loss of a stock portfolio. This concept is very similar to the problem in radiotherapy where we require a minimum dose coverage to a specified probability. In Pa-per IV, we borrowed modern optimization methods developed for stock port-folio management and applied them to radiotherapy. The aim was to develop a probabilistic optimization where the dose planner requests a dose coverage to a specified probability. The chosen planning paradigm was to require a min-imum target dose coverage probability, expressed as a PUD criteria, or pre-scription, by , ≥ , . The optimization problem was set up to satisfy the PUD criterion as a constraint while limiting the max dose to the organ at risks and the CTV overdosage.

The mathematically equivalent measure to PUD from risk management of stock portfolios is called Value at Risk (VaR) (Jorion 2006). VaR, and thus also PUD, has been shown to be unsuitable for direct optimization due to non-convexity (Artzner et al 1999). In this thesis we explore a related risk measure, Conditional Value at Risk (CVaR) which is proven to be convex and thus can be optimized with standard optimization techniques (Rockafellar and Uryasev 2000). For use in the context of radiation treatment optimization, we make use of the Expected Percentile UnderDosage (EPUD) introduced in section 2.2.1 which is mathematically identical to CVaR and therefore also convex. It is defined as the average for all ≤ , , ⟨ ⟩ ( ) = E ( , ) ≤ , ( ) = 11 − ∙ ( , )d, ( ) .

(19)

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4.2 Defining the optimization problem The ( , ) is not a convex measure in (Deasy 1997) and therefore we make use of the convex quadratic penalty functions to be minimized ( , , , )= 1 max 0, − ( , , ) ( )

(20)

for a target under dosage criterion with threshold dose , and ( , , , )= 1 max 0, ( , , ) − ( ),

(21)

for a maximum dose criterion with the threshold dose . The can be strongly correlated with the target % if is suitably chosen, as exempli-fied for one of the cervix patients, see figure 11. If is chosen to be exactly , then for most scenarios will be zero or very close to zero which

makes for a weak correlation. However, if is chosen slightly above , a stronger correlation is achieved. It was found that setting 5% above , yielded a good correlation, see figure 11. The correlation becomes stronger as the dose coverage increases as exemplified for 1fx and 23fx. This correlation suggests that can be used as a convex surrogate for % during optimization.

For a given , ( , ) is a random variable with probability density func-tion ( , ) on the same sample space as . The parameter ∈ ℝ spans all values of . Through the simulation of treatment scenarios ( , ) can be estimated. We can define , ( ) and ⟨ ⟩ ( ) as measures of the PUD and EPUD, analogous to , and ⟨ ⟩ , , ( ) = max ∈ ℝ ∶ ( , )d ≥ .

(22)

and ⟨ ⟩ ( ) = E ( , ) ≤ , ( ) , (23) where the latter is convex and can be used in the optimization.

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Figure 11. Correlation between calculated values of and % for 100 scenarios (one dot per scenario) for patient #4 in the cervix dataset based on 1fx and 23 fx us-ing the same treatment plan. The threshold dose was set to 5% above the %, %. The linear fits can be used to determine the value of that correspond

to the PUD requirement %, %.

The optimization procedure relies on the correlation between the % and as evaluated by dose distributions based on a given x. The effect of fractiona-tion was for simplicity not included in Paper IV. Figure 11 illustrates that the correlation is stronger for plans with higher %. The averaging effect of fractionation will push % upwards compared to single fraction. In this sense, Paper IV is a proof of principle in that it uses 1fx with an initial weak correlation. It is reasonable then that the method will work also when fraction-ation is included.

Rockafellar and Uryasev (2000) defined a help function that can be used for optimizing functions based on CVaR such as ⟨ ⟩ ( ) with respect to , ( , ) = + 11 − ( − ) ∙ ( , )d .

(24)

( , ) is convex with respect to , and moreover with respect to under the assumption that ( , ) is convex in . Rockafellar and Uryasev showed that , ( ) and ⟨ ⟩ ( ) can be calculated using ( , ) as

36

, ( ) = argmin ( , ) (25) and ⟨ ⟩ ( ) = min ( , ). (26) It follows that if ( ∗, ∗) is a minimizer to ( , ) ( ∗, ∗) = argmin , ( , ) (27)

then ∗ is also a minimizer to ⟨ ⟩ ( ) ∗ = argmin⟨ ⟩ ( ). (28) Moreover, Rockafellar and Uryasev showed that the minimization of ⟨ ⟩ ( ) with respect to can then be done simultaneously as with respect to ( , ). As stated in 4.1, the chosen treatment planning paradigm is that the PUD cri-terion is set as a constraint. By constraining ( , ), the upper tail of the is controlled and due to the correlation between % and also the lower tail of % . Lets define the upper bound of ( , ) as which will be used as a parameter to control in the optimization.

can be approximated using sampled scenarios by ( , ) = + 11 − 1 max(0, ( , , , ) − )

(29)

where { , = 1,… , } are the samples of per scenario and is the binary map of the CTV.

The dose to the organ at risks and normal tissue is minimized by penalty functions of the type , see equation (21). The optimization problem when including all ROIs becomes, min, ∙ 1 , , , ,subject to ( , ) ≤> ,

(30)

where is an importance weight per ROI, , is the max tolerance dose, and > in this work denotes the positivity of the beamlet weights. Note that the objective function minimizes the dose to normal tissue as well as limiting the CTV max dose since is part of { }. The optimization problem (30) is constrained by to produce a treatment plan that satisfies a target coverage probability, while limiting the max dose to the CTV as well as the dose to the organ at risks. However, the particular value ∗ for giving the requested

37

PUD is a priori unknown. Next we will detail an iterative process of determin-ing ∗. 4.3 The target coverage constraint tolerance The aim of the algorithm is to fulfill a PUD constraint, i.e. , ≥ ,

which will be achieved by determining ∗ and require that ≤ ∗. It is pos-sible to determine determine ∗ by trial and error but this may require many iterations. An iterative method has therefore been designed to gradually im-prove an estimate of ∗. This method is based on linear fits of the correlation found between the outcome { ( , )} and { ( , )} based on simu-lated scenarios for a specific treatment setup x. As stated in 4.2, the fits will be badly conditioned if is chosen too close to , .

We start by first defining ∗ as the value of , when = ∗ (i.e. ful-filling the , constraint). Given a solution to problem (30) and its corre-lation function we can estimate ∗ by ( , ). Then we recognize that ∗ can be written as ∗ plus the difference between the two. We then estimate ( ∗ − ∗) by the result of the current solution ⟨ ⟩ − , , which

can be calculated directly from ( , ) or by solving problems (25) and (26). By combining the estimates of ∗ and ( ∗ − ∗) we arrive at a new es-timate of ∗. This procedure can be repeated as an iterative algorithm until , is within some tolerance of , , see pseudo code in algorithm 1.

Algorithm 1. Iterative algorithm for determining ∗ ← High value

←Solve current problem (30) using while , ( ) − , < do

← correlation { ( , , , ), ( , )} ← ( , ) , ( ) and ⟨ ⟩ ( ) ← Solution to (25) and (26) ← + ⟨ ⟩ ( ) − , ( ) ← max( , 0.5 ) ← Solve current problem (30) using ← i + 1

A high value for should be chosen such that > ∗ and the solution to problem (30) results in a low , . Algorithm 1 will iteratively update until PUD sufficiently close to the requested PUD (i.e. sufficiently close to ∗).

38

This iterative algorithm will converge to ∗ if the estimate of ∗ through becomes accurate and if is not widened with increasing iterations (i.e. − non-increasing). It is reasonable to assume that is not widened

since will produce high values of and move to lower values of with more iterations. This will lead to smaller ( − ) provided that the shape of does not change too much. In this work we limit the decrease in

to 50% as a conservative constraint on the updates. The convergence of the PUD to the requested , using algorithm 1 as well as the approximation of only using 100 scenarios was investigated using the cervical cancer pa-tients.

4.4. Convergence to the requested dose coverage probability The method of optimizing towards a requested PUD was applied to four cervix patients (#2 - #5) from the dataset described in 3.3. The probabilistic prescrip-tion chosen in this work, and implemented as part of problem (30), was 98%,90%presc = 95% of 46 Gy = 43.7 Gy. The max dose, implemented as in equation (21), to both the bladder and rectum was 90% of 46 Gy, 105% of 46 Gy for the CTV and to the rest of the normal tissue 75% of 46 Gy. The number of scenarios was chosen to 100 as a reasonable trade-off between PUD accu-racy and calculation time. The optimization problem (30) was implemented in C++ using the nonlinear optimization library Ipopt (Wachter and Biegler 2006).

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Figure 12. The convergence of %, % towards the requested %, % as the con-straint tolerance is tightened per iteration.

Figure 12 shows the convergence within 10 iterations of %, % to the re-quested %, %. The conservative 50% update restriction in the decrease of

was active for patients #2, #3 and #5 during the first iterations when the correlation was weaker. Patient #4 exhibit the weakest correlation, espe-cially for the first few iterations, but the algorithm still decreased until the correlation became stronger and in the end converged.

The number of scenarios during the first 10 iterations of the optimization, was limited to 100 for increased calculation speed. This approximation caused a difference in %, % up to 3% (not shown here) compared to the verifica-tion calculation using 1000 scenarios. An 11th iteration using 400 scenarios was therefore performed using the same as the 10th iteration. The calculation time of the optimization per iteration (i.e. solving problem (30)) increased a factor 10 with the increase in thus motivating that only the final 11th itera-tion is performed using = 400. The resulting PUD of the 11th iteration was then within 0.3 Gy of %, %, see table 1. The %, % after the verifica-tion calculations for all four patients were within 1.3% (0.6 Gy) of the %, % with an average of 99.3%. The differences in %, % between the 11th iteration and the verification calculations are small.

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The results strongly indicate that the algorithm is capable of fulfilling the probabilistic treatment criterion provided the scenarios cover enough of the sample space. Table 1. Treatment planning criteria %, % [Gy] after optimization and verifica-tion calculations. The requested %, %= 43.7 Gy.

Patient # Optimization (10th iteration) = 100

Optimization (11th iteration) = 400

Verification =1000

1 43.7 43.4 43.2 2 43.7 43.8 43.8 3 43.7 43.7 43.4 4 43.6 43.4 43.1

Probability density functions for the CTV, based on the verification calcula-tions are shown in figure 13. The % in relation to 105% of 46 Gy = 48.3 Gy show that the CTV overdosage was limited.

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Figure 13. Probability density functions for CTV % (red), % (blue) and % (green) extracted from the 1000 verification scenarios. The resulting PUD (solid black) are close to the prescribed %, % = 43.7Gy (dashed black) for all 4 patients. Figure from Paper IV.

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5. Summary and directions for future research

Radiotherapy is an evolving and improving therapy that is an important tool for the treatment of many cancer diseases. A challenge when delivering radi-otherapy is to manage the associated uncertainties in practically all steps from segmentation uncertainty to organ motion. Traditionally this is handled by ex-panding the irradiated volume by applying margins around the targeted tu-mour. The limitation of margins is that they are population based and more or less isotropic, and thus do not exploit the particular geometrical circumstances per individual patient. Probabilistic planning has emerged as an alternative to margin based planning to explore the possibilities to improve the treatments. The aim of this thesis was to create a comprehensive simulation framework for probabilistic treatment plan optimization and evaluation based on the prin-ciple of dose coverage probability.

A necessary tool for simulating treatment scenarios is the ability to effi-ciently and accurately accumulate the total dose of the fractionated treatment. The accuracy of the accumulated dose relies on deformable image registration to track the location of the tissue through the fractions. We concluded that an accuracy of 3% in the target dose coverage % required a 2 mm registration uncertainty (or less).

A fast dose calculation based on perturbation was developed to relieve some of the dose calculation burden when calculating the dose coverage prob-ability where several hundred scenarios are needed for sufficient statistical accuracy. The main idea is that a perturbation in incident energy fluence is proportional to the perturbation in dose. The perturbation is done separately for primary and scatter dose, and regularized using pencil kernel convolution for increased accuracy. The fast dose calculation demonstrated a speedup of ~1000 compared to full dose calculations and an accuracy of 2%/2mm for a highly modulated IMRT beam as well as for a target moving in a virtual lung phantom. The calculated dose coverage probabilities were indistinguishable from full dose calculation for a hypofractionated prostate patient. Future ap-plications of the fast dose algorithm could include gradient calculation for use in optimization and possibly also real-time dose calculation while the patient is being irradiated for use in contexts such as MR-Linac (Raaymakers et al 2009).

The ability to acquire multiple images per patient, even daily, and the in-crease in computing power in modern computers makes it possible to simulate complex organ deformation patterns, such as the substantial deformation of

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the cervix patients explored in this thesis, and directly incorporate them during the treatment plan design. In this work, a statistical shape model (SSM) pro-vided compact and efficient sampling of patient geometries where as few as seven eigenmodes were found to be enough to model the dosimetric impact of the interfraction deformation. Future work include a clinical validation of the SSM to enable the use of the probabilistic planning tool in clinical routine.

The idea behind the probabilistic optimization algorithm in this work is to generalize dose coverage prescriptions to include also the probability which now is implicit for margin based prescriptions. The algorithm yields plans with the requested dose coverage probability while minimizing the dose to the normal tissue. For this purpose, modern optimization methods from financial risk management, in particular the Conditional Value at Risk measure, was translated into radiotherapy concepts. Thus, the concepts of the Percentile Un-derDosage (PUD) and Expected Percentile UnderDosage (EPUD) were in-troduced as central tools in the optimization. PUD is the percentile dose cov-erage based on all simulated scenarios and EPUD is the expected dose cover-age of all scenarios with lower dose coverage than PUD.

Radiation treatment of cervical cancer patients, such as the ones in the da-taset used in this study, require large margins to ensure target coverage and thus causing extensive dose burden to the normal tissue. Probabilistic optimi-zation including a model for the organ motion has to our knowledge never been evaluated as an alternative planning method for this patient group. It is therefore of high interest to compare the potential benefit of PUD based plans over traditional margin based plans.

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6. Summary in Swedish

Tumörer har behandlats med joniserande strålning sedan slutet av 1800-talet. Ungefär varannan cancerpatient får strålning som del av behandlingen, vilket innebär att var 6:e invånare i utvecklade länder förväntas någon gång få strål-behandling. Enligt Cancerfonden står strålbehandling för 30% av all botad cancer. Den vanligaste formen är extern strålbehandling där linjäraccelerato-rer genererar högenergetiska fotoner. Strålningen skadar cellernas DNA och därmed deras förmåga att dela sig. Målet är att fokusera strålningen på ett så-dant sätt att maximal skada åsamkas tumören utan att skada omkringliggande vävnad i för hög grad. Detta åstadkoms vanligen genom att stråla från många riktningar och därmed maximalt koncentrera stråldosen till tumörområdet.

Det är ofta en fördel att dela upp strålbehandling i delbehandlingar, frakt-ioner, vanligtvis en per dag, där allt från några få upp till ca 40 fraktioner förekommer. Fraktioneringen motiveras av att friska celler återhämtar sig bättre än tumörceller och skillnaden i överlevnadsgrad kan därför accentueras med fler fraktioner.

Modern radioterapi är mycket tekniskt avancerad där tredimensionell bild-tagning med hjälp av t.ex. datortomograf och magnetkamera sker till grund för planering av behandlingen. Bilder tas även i samband med delbehandling-arna för att öka träffsäkerheten. Dosplanering sker genom att simulera strål-ningen i patienten med datorns hjälp och ofta via matematisk optimering hitta dom maskinställningar som ger optimal avvägning mellan dos till tumör och dos till frisk vävnad. Svårigheten att entydigt definiera tumör och riskorgan på bilder, positionera tumören exakt likadant vid varje behandlingstillfälle, samt rörelse och deformation av olika organ är några av dom största osäker-heterna inom radioterapi. Traditionellt hanteras osäkerheterna genom att ut-vidga området som strålas för att därigenom säkerställa att tumören får den önskade dosen. Storleken på marginalen sätts utifrån det statistiska måttet att minst 90% av alla behandlingar ska resultera i att hela tumören får minst den avsedda dosen. Dosplanering med marginaler har begränsningar då margina-lernas storlek baserar sig på hela patientgrupper och inte tar hänsyn till den individuella patientens geometri.

Den upprepade bildtagningen samt utvecklingen av datorkraft möjliggör så kallad probabilistisk planering som ett alternativ till marginalbaserad dospla-nering. Probabilistisk planering bygger på att ett stort antal, hundratals eller till och med tusentals, datorsimuleringar av slumpade behandlingar för den

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individuella patienten utförs och används som underlag för en statistisk ana-lys. Sjukvårdspersonalen kan då göra en mer välinformerad avvägning mellan dos till tumören kontra risk för biverkningar. Målet med denna avhandling var att skapa ett ramverk för probabilistisk planering där man kan ta hänsyn till organrörelse på ett effektivt sätt.

Probabilistisk planering är extremt beräkningsintensivt där de hundra-tals/tusentals simuleringarna inte kan beräknas på vanligt sätt på grund av långa beräkningstider. En snabb dosberäkningsalgoritm utvecklades speciellt för detta ändamål där en förberäknad dosfördelning korrigeras baserat på va-riationer i det resulterande strålfältet. En simulering av en prostata-behandling med positioneringsfel snabbades upp ungefär 1000 gånger jämfört med en fullständig dosberäkning utan att försämra noggrannheten i de statistiska mått som behandlingsplanen utvärderades mot.

Ett specifikt mål var att med hjälp av moderna matematiska optimerings-metoder, lånade från riskbedömningar vid förvaltning av aktieportföljer, möj-liggöra att direkt specificera till vilken sannolikhet behandlingskriteriet ska uppfyllas för därigenom maximalt kunna utnyttja möjligheter att sänka do-serna till de omgivande organen utan att äventyra chansen för bot av tumören. Övergången från marginalbaserad, där sannolikheten är underförstådd, till den mer avancerade probabilistiska planeringen kan då också ske utan att förändra grunden för för hur behandlingsmålen är satta.

På grund av den omfattande organrörelsen hos patienter med livmoder-halscancer krävs stora marginaler vilket medför biverkningar på framförallt tarmar, blåsa och rektum. Det är därför viktigt att testa alternativa planerings-metoder för att eventuellt kunna lindra dessa biverkningar. För första gången kunde probabilistisk planering appliceras, retrospektivt, på patienter med liv-moderhalscancer där rörelsen inkorporerades direkt i optimeringsprocessen. I uppföljningsstudier kan de eventuella fördelarna med den utvecklade proba-bilistiska planeringen utvärderas mot traditionell marginalbaserad planering.

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7. Acknowledgements

My main supervisor Anders Ahnesjö for teaching me how to think like a sci-entist. Sometimes it seems like you have a magic wand for algorithms, which you can wave to help me get out (usually by a convolution or two!) of what-ever algorithmic corner I have been stuck in. There is very little in radiation physics and radiotherapy in general that I haven't learned from you! Erik Grusell for taking on the role of a supervisor in the middle of this project, and for lending me your very sharp and mathematically stringent intellect. There is no doubt that the last papers have benefitted from your participation. My co-authors Åsa Holm, Nina Tilly, Stijn van de Schoot and Arjan Bel for fruitful collaboration. Thanks also to you anonymous patients for letting me perform my numerical experiments on you! My PhD colleagues and fellow researchers at Ackis, Karin, Carl, Gloria, Fer-nanda, Kenneth, Eric, Erik and David for interesting coffee and beer conver-sations about being a PhD or whatever else that came to mind. Everybody at Sjukhusfysik for your warm welcome and providing a second great place to work.

My unofficial supervisor for many years at Elekta, Mikael Saxner for teaching me all the ins and outs of implementing dose calculation algorithms. Noone could be easier to work with. I especially admire your fantastic ASCII graphics in the code explaining the complex algorithms! Everybody at Klostergatan, thanks for truly being like family over the years. It has been very reassuring to know that you have always been there. See you at after-work!

This work has been financed by Nucletron and later on Elekta. Thanks to Per Ekström who helped me get started, Dan Fristedt for always having my back and letting me continue my research, and to Jonas Gårding and Virgil Willcut for allowing me to finish.

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Mina Skellefte-kompisar – ni vet vilka ni är – för ölkvällar på Monk’s skidre-sor, midsommarfester och allt mest annat vi hittat på. Jag ser fram emot mer sånt! Tack till alla mina vänner för att ni gör mitt liv mer intressant. Mina föräldrar Jan och Karin, samt mina svärföräldrar Sven-Åke and Ella-Greta för all hjälp under den sista tiden - ett enormt stort tack. Utan er hade den här boken aldrig blivit klar! Tack också till resten av min familj, Simon och Jenny m familj, Sofia och Gustav m familj, och Markus. Nina, jag kan inte tänka mig en bättre förste medförfattare till vår familjs liv. Tack också för att du (av någon outgrundlig anledning) alltid, alltid trott på mig och min förmåga att klara detta. Enya, tack för att du finns till. Pappa har skrivit klart boken nu!! Jag älskar er!

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Acta Universitatis UpsaliensisDigital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Medicine 1264

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A doctoral dissertation from the Faculty of Medicine, UppsalaUniversity, is usually a summary of a number of papers. A fewcopies of the complete dissertation are kept at major Swedishresearch libraries, while the summary alone is distributedinternationally through the series Digital ComprehensiveSummaries of Uppsala Dissertations from the Faculty ofMedicine. (Prior to January, 2005, the series was publishedunder the title “Comprehensive Summaries of UppsalaDissertations from the Faculty of Medicine”.)

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