The introduction of Magnetic Resonance (MR) Imaging in treatment planning for radiotherapy of prostate cancer give rise to new challenges in defining the treatment volume, or the target. Imaging with MR have several advantages, especially better soft-tissue contrast, compared with the standard image modality, Computed Tomography (CT). The purpose of this project was to determine how the target definition varies with the choice of image modality and the systematic differences between them. The purpose was also to determine how the inter- and intra physician variability influences the delineation, depending on which image modality that is used. In the project, five physicians delineated the prostate gland on CT and MR images for nine patients. The physicians had no information of which image series that was connected, and were thus delineating independent. After the delineation, the CT and MR image series was set in the same geometrical coordinate system in the treatment planning system Oncentra. The target delineations were analysed by comparing the radial distances from the centre of mass in different directions, such as anterior, posterior, etc. The radial distances were later used to evaluate the variability of the delineations and to determine the inter and intra physician variability in different directions of the targets. ANOVA was also preformed to determine if there is a significant difference between the parameters, as the image modality and the image modalities influence on the physicians delineations. A model was made to investigate how the MR delineations differ from a CT defined volume that 95% of the delineations cover, called the ideal CT delineation. From this, the median deviation in different directions  was analysed and it was found that the median value of the MR delineations in different directions was between -0.10-2.27 mm larger than the ideal CT delineation. The fluctuations between the delineations was, however, large. The target volume was larger for CT defined volumes in 87% of the cases, compared with MR targets. The inter physician variability was found to be between 0.54-2.17 mm for the CT based delineations and 0.68-2.08 mm for the MR based target delineations. The intra physician variability was larger than the inter physician variability, between 0.74-2.51 mm for CT based delineations and 0.85-1.45 mm for MR based delineations. The median variability of the delineations were not uniform around the target volume but were larger for example in the superior and inferior directions and had its minimum in the posterior direction. The ANOVA tests showed a significant difference between MR and CT based target delineations, it also shows a relation between the delineating physician and the image modality, meaning that the physicians are delineating different on CT and MR images. Target volumes defined on MR images are in general smaller than the CT defined targets. The soft tissue contrast in MR images makes the delineating process easier, however, the analysis of the variability in this project indicates that the variations of MR based target delineations are larger than the CT based. The large variation of the delineations implies that clinical tests should be made to ensure a proper dose coverage before MR could be used as the only image modality in radiotherapy treatment planning of prostate cancer.

Cancer is one of the deadliest diseases today, and with increasingly larger and older populations, cancer constitutes an enormous contemporary and future challenge. Luckily, advances in technology and medicine are continuously contributing to a decrease in cancer mortality, and to the reduction of treatment side effects. The aim of this Master's thesis is to be a part of these advances, thereby increasing the survival chances and well-being of future cancer patients. The thesis regards specifically the improvement of radiation therapy, a form of treatment utilized in both curative and palliative cancer care. In radiation therapy, ionizing radiation is directed at cancerous cells in the body. The radiation prevents the further proliferation of malignant cells by damaging their DNA. However, the radiation is also harmful to healthy cells. It is therefore of utmost importance that the irradiation of the patient is done in such a way to spare the critical organs in the vicinity of the tumor. To obtain the best possible treatment, mathematical optimization algorithms are utilized. Using physical models of how radiation travels in the body, it is possible to calculate what effect the irradiation of the patient will have. To quantify the quality of the treatment, mathematical functions are used, which evaluate the radiation dose under certain criteria. Once these functions are defined, algorithms can be applied that find the optimal treatment with regard to the given criteria. The formulation of these functions and their properties is the main focus of this thesis. Using clinical evaluation criteria previously used to assess treatments, a framework for optimizing functions that directly correlate to the clinical goals is constructed. The framework is examined and used to generate radiation therapy plans for three cancer patients. In each of the cases, the constructed treatment plans demonstrate high quality, often better than or comparable to the plans created by experienced dose planners using existing tools. A particularly interesting application of the developed framework is the automatic generation of treatments. This relies on the clinician giving the clinical goals as input to the algorithm. A plan is then generated with maximal goal fulfillment. This eliminates the tedious and time consuming process of parameter tuning to achieve a satisfactory plan. Several studies have demonstrated the ability of automatic planning to retain the plan quality while substantially improving planning efficiency.

<p>Silicon diodes compared to other types of dosimeters have several attractive properties, such as an excellent spatial resolution, a high sensitivity, and clinically practical to use. These properties make silicon diodes a preferred dosimeter for relative dosimetry for several types of measurements in small field dosimetry, e.g., stereotactic treatments and intensity modulated radiotherapy (IMRT). Silicon diodes are, however, limited by an energy dependent response variation in photon beams, resulting in that the diode readout per dose to the phantom medium varies with photon spectral changes, thereby introducing a significant uncertainty in the measured data. The traditional solution for the energy dependent over-response caused by low-energy photons is to use diodes with a shielding filter of high atomic number. These shielded diodes, however, show an incorrect readout for small fields due to electrons scattered from the shielding (Griessbach <em>et al</em>. 2005). In regions with degraded lateral electron equilibrium (LEE) shielded diodes over-respond due to an increased degree of LEE, as a consequence of the high density shielding (Lee <em>et al</em>. 2002).</p><p>In this work a prototype software that corrects for the energy dependent response of a silicon diode is developed and validated for small field sizes. The developed software is based on the novel concept of Monte Carlo (MC) simulated fluence pencil beam kernels to calculate spectra (Eklund and Ahnesjö 2008), and the spectra based silicon diode response model proposed by Eklund and Ahnesjö (2009). The software was also extended to include correction of ionization chambers, for the energy dependent Spencer-Attix water/air stopping power ratio (<em>s</em>_w,air). The calculated <em>s</em>_w,air are shown to be in excellent agreement with published values to better than 0.1% for most values, the maximum deviation being 0.3%.</p><p>Measured relative depth doses, relative profiles, and output factors in water, for small square field sizes, for 6 MV and 15 MV clinical photon beams are presented in this work. The results show that the unshielded Scanditronix-Wellhöfer EFD^3G silicon diode response, corrected by the developed software, is in excellent agreement with reference ionization chamber measurements (corrected for change in <em>s</em>_w,air), the maximum deviation being 0.4%.</p><p>Measurements with two types of shielded diodes, namely Scanditronix-Wellhöfer PFD silicon diodes (FP1990 and FP2730), are also included in this work. The shielded diodes are shown to have an over-response as large as 2-3.5% for field sizes smaller than 5 cm x 5 cm. The presented results also suggest a difference in accuracy as large as 0.5-1% between the two types of shielded diodes, where the spectral composition at the measurement position dictates which type of diode is more accurate.</p><p>The fast correction of silicon diodes provided by the developed software is more accurate than shielded diodes for small field sizes, and can in radiotherapeutic clinical practice increase the dosimetric accuracy of silicon diodes.</p>

Silicon diodes compared to other types of dosimeters have several attractive properties, such as an excellent spatial resolution, a high sensitivity, and clinically practical to use. These properties make silicon diodes a preferred dosimeter for relative dosimetry for several types of measurements in small field dosimetry, e.g., stereotactic treatments and intensity modulated radiotherapy (IMRT). Silicon diodes are, however, limited by an energy dependent response variation in photon beams, resulting in that the diode readout per dose to the phantom medium varies with photon spectral changes, thereby introducing a significant uncertainty in the measured data. The traditional solution for the energy dependent over-response caused by low-energy photons is to use diodes with a shielding filter of high atomic number. These shielded diodes, however, show an incorrect readout for small fields due to electrons scattered from the shielding (Griessbach et al. 2005). In regions with degraded lateral electron equilibrium (LEE) shielded diodes over-respond due to an increased degree of LEE, as a consequence of the high density shielding (Lee et al. 2002). In this work a prototype software that corrects for the energy dependent response of a silicon diode is developed and validated for small field sizes. The developed software is based on the novel concept of Monte Carlo (MC) simulated fluence pencil beam kernels to calculate spectra (Eklund and Ahnesjö 2008), and the spectra based silicon diode response model proposed by Eklund and Ahnesjö (2009). The software was also extended to include correction of ionization chambers, for the energy dependent Spencer-Attix water/air stopping power ratio (sw,air). The calculated sw,air are shown to be in excellent agreement with published values to better than 0.1% for most values, the maximum deviation being 0.3%. Measured relative depth doses, relative profiles, and output factors in water, for small square field sizes, for 6 MV and 15 MV clinical photon beams are presented in this work. The results show that the unshielded Scanditronix-Wellhöfer EFD3G silicon diode response, corrected by the developed software, is in excellent agreement with reference ionization chamber measurements (corrected for change in sw,air), the maximum deviation being 0.4%. Measurements with two types of shielded diodes, namely Scanditronix-Wellhöfer PFD silicon diodes (FP1990 and FP2730), are also included in this work. The shielded diodes are shown to have an over-response as large as 2-3.5% for field sizes smaller than 5 cm x 5 cm. The presented results also suggest a difference in accuracy as large as 0.5-1% between the two types of shielded diodes, where the spectral composition at the measurement position dictates which type of diode is more accurate. The fast correction of silicon diodes provided by the developed software is more accurate than shielded diodes for small field sizes, and can in radiotherapeutic clinical practice increase the dosimetric accuracy of silicon diodes.

Treating cancer with radiotherapy requires precise planning. Several planning pipelines rely on reference dose mimicking, where one tries to find machine parameters best mimicking a given reference dose. Dose mimicking relies on having a function that quantifies dose similarity well, necessitating methods for feature extraction of dose images. In this thesis we investigate ways of extracting features from clinical doseimages, and propose a few proof-of-concept dose mimicking functions using the extracted features. We extend current techniques and lay the foundation for new techniques for feature extraction, using mathematical frameworks developed in entirely different areas. In particular we give an introduction to wavelet theory, which provides signal decomposition techniques suitable for analysing local structure, and propose two different dose mimicking functions using wavelets. Furthermore, we extend ROI-based mimicking functions to use artificial ROIs, and we investigate variational autoencoders and their application to the clinical dose feature extraction problem. We conclude that the proposed functions have the potential to address certain shortcomings of current dose mimicking functions. The four methods all seem to approximately capture some notion of dose similarity. Used in combination with the current framework they have the potential of improving dose mimickingresults. However, the numerical tests supporting this are brief, and more thorough numerical investigations are necessary to properly evaluate the usefulness of the new dose mimicking functions. / Behandling av cancer med strålterapi kräver precis planering. Flera olika planeringsramverk bygger på doshärmning, som innebär att hitta de maskinparametrar som bäst härmar en given referensdos. För doshärmning behövs en funktion som kvantifierar likheten mellan två doser, vilket kräver ett sätt att extrahera utmärkande egenskaper – så kallade features – från dosbilder. I det här examensarbetet undersöker vi olika matematiska metoder för att extrahera features från kliniska dosbilder, och presenterar några olika förslag på prototyper till doshärmningsfunktioner, konstruerade utifrån extraherade features. Vi utvidgar nuvarande tekniker och lägger grunden för nya tekniker genom att använda matematiska ramverk utvecklade för helt andra syften. Speciellt så ger vi en introduktion till wavelet-teori, som ger matematiska verktyg för att analysera lokala beteenden hos signaler, exempelvis bilder. Vi föreslår två olika doshärmningsfunktioner som utnyttjar wavelets, och utvidgar ROI-baseraddoshärmning genom att introducera artificiella ROIar. Vidare så undersökervi så kallade variational autoencoders  och möjligheten att använda dessa för extrahering av features från dosbilder. Vi kommer fram till att de föreslagna funktionerna har potential att åtgärda vissa begränsningar som finns hos de doshärmningsfunktioner som används idag. De fyra metoderna verkar alla approximativt kvantifiera begreppet doslikhet. Användning av dessa nya metoder i kombination med nuvarande ramverk för doshärmning har potential att förbättra resultaten från doshärmning. De numeriska undersökningar som underbygger dessa slutsatser är dock inte särskilt ingående, så mer noggranna numeriska tester krävs för att kunna ge några definitiva svar angående de presenterade doshärmningsfunktionernas användbarhet ipraktiken.

Cancer is the second largest mortal disease in Sweden, and high efforts are made to develop the treatment of cancer. One of the main treatment methods is radiotherapy, which uses ionizing radiation to damage the cancerous cells. This has the chance of stopping the cell reproduction, and the goal is to reduce the tumor and stop the tumor growth. The most common forms of radiotherapy uses external beams to irradiate the tumor. In intensity modulated radiotherapy, IMRT, the beam fluences are optimized to give a highly conformal dose, i.e. a dose distribution which is restricted to the tumor and has low dose values outside of the tumor. A conformal dose is necessary to spare healthy tissue and sensitive organs, and thus keep the side-effects of the treatment at an acceptable level. The optimized beam shapes are created using a multileaf collimator, MLC. Finding the leaf positions and dose levels is formulated as a problem in the framework of mathematical optimization. Currently, one of the limitations in delivering conformal dose is due to patient movement during the treatment. In IMRT, the beams are delivered by consecutive segments, and the exact pairing of the segments with the patient position will have an impact on the delivered dose. This is called the interplay effect, and can cause both underdosage of the tumor and overdosage of the surrounding tissue. There are methods of mitigating the interplay effect. For example, the beam could be restricted to a single phase of the motion by repeatedly turning it on and off. This is known as gating. However, gating and many other interplay mitigation techniques lead to prolonged treatment times, which decreases the clinical throughput, causes higher patient discomfort and gives higher uncertainties in the delivered dose. This makes it desirable to find methods which avoid prolonged treatment times, while still giving highly conformal doses. Ideally, the best method would be to have a beam which follows any target movement. This idea is known as target tracking. In this thesis, an optimization method is suggested which includes the interplay effect in the treatment optimization. Two main treatment strategies are proposed. The method which is simplest to implement clinically is to create plans which are robust against uncertainties in the times for the patient motion. The resulting doses are found to give acceptable target covering where similar, conventional plans give a significant target underdose. To further increase the conformality of the doses, a non-robust method paired with gating technology is suggested. This method can effectively be seen as a target tracking method, and has the possibility to give highly conformal doses under acceptable treatment times.

Cancer is a group of diseases that are characterized by abnormal cell growth and is considered a leading cause of death globally. There are a number of different cancer treatment modalities, one of which is radiation therapy. In radiation therapy treatment planning, it is important to make sure that enough radiation is delivered to the tumor and that healthy organs are spared, while also making sure to account for uncertainties such as misalignment of the patient during treatment. To reduce the workload on clinics, data-driven automated treatment planning can be used to generate treatment plans for new patients based on previously delivered plans. In this thesis, we propose a novel method for robust automated treatment planning where a deep learning model is trained to deform a dose in accordance with a set of potential scenarios that account for the different uncertainties while maintaining certain statistical properties of the input dose. The predicted scenario doses are then used in a robust optimization problem with the goal of finding a treatment plan that is robust to these uncertainties. The results show that the proposed method for deforming doses yields realistic doses of high quality and that the proposed pipeline can potentially generate doses that conform better to the target than the current state of the art but at the cost of dose homogeneity. / Cancer är ett samlingsnamn för sjukdomar som karaktäriseras av onormal celltillväxt och betraktas som en ledande dödsorsak globalt. Det finns olika typer av cancerbehandling, varav en är strålterapi. Inom strålterapiplanering är det viktigt att säkerställa att tillräckligt med strålning ges till tumören, att friska organ skonas, och att osäkerheter som felplacering av patienten under behandlingen räknas med. För att minska arbetsbelastningen på kliniker används data-driven automatisk strålterapiplanering för att generera behandlingsplaner till nya patienter baserat på tidigare levererade behandlingar. I denna uppsats föreslår vi en ny metod för robust automatisk strålterapiplanering där en djupinlärningsmodell tränas till att deformera en dos i enlighet med en mängd potentiella scenarion som motsvarar de olika osäkerheterna medan vissa statistiska egenskaper bibehålls från originaldosen. De predicerade scenariodoserna används sedan i ett robust optimeringsproblem där målet är att hitta en behandlingsplan som är robust mot dessa osäkerheter. Resultaten visar att den föreslagna metoden för dosdeformation ger realistiska doser av hög kvalitet, vilket i sin tur kan leda till robusta doser med högre doskonformitet än tidigare metoder men på bekostnad av doshomogenitet.

Treatment planning for radiation therapy inherently involves tradeoffs, such as between tumor control and normal tissue sparing, between time-efficiency and dose quality, and between nominal plan quality and robustness. The purpose of this thesis is to develop methods that can facilitate decision making related to such tradeoffs. The main focus of the thesis is on multicriteria optimization methods where a representative set of treatment plans are first calculated and the most appropriate plan contained in this representation then selected by the treatment planner through continuous interpolation between the precalculated alternatives. These alternatives constitute a subset of the set of Pareto optimal plans, meaning plans such that no criterion can be improved without a sacrifice in another. Approximation of Pareto optimal sets is first studied with respect to fluence map optimization for intensity-modulated radiation therapy. The approximation error of a discrete representation is minimized by calculation of points one at the time at the location where the distance between an inner and outer approximation of the Pareto set currently attains its maximum. A technique for calculating this distance that is orders of magnitude more efficient than the best previous method is presented. A generalization to distributed computational environments is also proposed. Approximation of Pareto optimal sets is also considered with respect to direct machine parameter optimization. Optimization of this form is used to calculate representations where any interpolated treatment plan is directly deliverable. The fact that finite representations of Pareto optimal sets have approximation errors with respect to Pareto optimality is addressed by a technique that removes these errors by a projection onto the exact Pareto set. Projections are also studied subject to constraints that prevent the dose-volume histogram from deteriorating. Multicriteria optimization is extended to treatment planning for volumetric-modulated arc therapy and intensity-modulated proton therapy. Proton therapy plans that are robust against geometric errors are calculated by optimization of the worst case outcome. The theory for multicriteria optimization is extended to accommodate this formulation. Worst case optimization is shown to be preferable to a previous more conservative method that also protects against uncertainties which cannot be realized in practice. / En viktig aspekt av planering av strålterapibehandlingar är avvägningar mellan behandlingsmål vilka står i konflikt med varandra. Exempel på sådana avvägningar är mellan tumörkontroll och dos till omkringliggande frisk vävnad, mellan behandlingstid och doskvalitet, och mellan nominell plankvalitet och robusthet med avseende på geometriska fel. Denna avhandling syftar till att utveckla metoder som kan underlätta beslutsfattande kring motstridiga behandlingsmål. Primärt studeras en metod för flermålsoptimering där behandlingsplanen väljs genom kontinuerlig interpolation över ett representativt urval av förberäknade alternativ. De förberäknade behandlingsplanerna utgör en delmängd av de Paretooptimala planerna, det vill säga de planer sådana att en förbättring enligt ett kriterium inte kan ske annat än genom en försämring enligt ett annat. Beräkning av en approximativ representation av mängden av Paretooptimala planer studeras först med avseende på fluensoptimering för intensitetsmodulerad strålterapi. Felet för den approximativa representationen minimeras genom att innesluta mängden av Paretooptimala planer mellan inre och yttre approximationer. Dessa approximationer förfinas iterativt genom att varje ny plan genereras där avståndet mellan approximationerna för tillfället är som störst. En teknik för att beräkna det maximala avståndet mellan approximationerna föreslås vilken är flera storleksordningar snabbare än den bästa tidigare kända metoden. En generalisering till distribuerade beräkningsmiljöer föreslås även. Approximation av mängden av Paretooptimala planer studeras även för direkt maskinparameteroptimering, som används för att beräkna representationer där varje interpolerad behandlingsplan är direkt levererbar. Det faktum att en ändlig representation av mängden av Paretooptimala lösningar har ett approximationsfel till Paretooptimalitet hanteras via en metod där en interpolerad behandlingsplan projiceras på Paretomängden. Projektioner studeras även under bivillkor som förhindrar att den interpolerade planens dos-volym histogram kan försämras. Flermålsoptimering utökas till planering av rotationsterapi och intensitetsmodulerad protonterapi. Protonplaner som är robusta mot geometriska fel beräknas genom optimering med avseende på det värsta möjliga utfallet av de föreliggande osäkerheterna. Flermålsoptimering utökas även teoretiskt till att innefatta denna formulering. Nyttan av värsta fallet-optimering jämfört med tidigare mer konservativa metoder som även skyddar mot osäkerheter som inte kan realiseras i praktiken demonstreras experimentellt. / <p>QC 20130527</p>

In radiation therapy treatment planning, recent works have used machine learning based on historically delivered plans to automate the process of producing clinically acceptable plans. Compared to traditional approaches such as repeated weighted-sum optimization or multicriteria optimization (MCO), automated planning methods have, in general, the benefits of low computational times and minimal user interaction, but on the other hand lack the flexibility associated with general-purpose frameworks such as MCO. Machine learning approaches can be especially sensitive to deviations in their dose prediction due to certain properties of the optimization functions usually used for dose mimicking and, moreover, suffer from the fact that there exists no general causality between prediction accuracy and optimized plan quality.In this thesis, we present a means of unifying ideas from machine learning planning methods with the well-established MCO framework. More precisely, given prior knowledge in the form of either a previously optimized plan or a set of historically delivered clinical plans, we are able to automatically generate Pareto optimal plans spanning a dose region corresponding to plans which are achievable as well as clinically acceptable. For the former case, this is achieved by introducing dose--volume constraints; for the latter case, this is achieved by fitting a weighted-data Gaussian mixture model on pre-defined dose statistics using the expectation--maximization algorithm, modifying it with exponential tilting and using specially developed optimization functions to take into account prediction uncertainties.Numerical results for conceptual demonstration are obtained for a prostate cancer case with treatment delivered by a volumetric-modulated arc therapy technique, where it is shown that the methods developed in the thesis are successful in automatically generating Pareto optimal plans of satisfactory quality and diversity, while excluding clinically irrelevant dose regions. For the case of using historical plans as prior knowledge, the computational times are significantly shorter than those typical of conventional MCO. / Inom strålterapiplanering har den senaste forskningen använt maskininlärning baserat på historiskt levererade planer för att automatisera den process i vilken kliniskt acceptabla planer produceras. Jämfört med traditionella angreppssätt, såsom upprepad optimering av en viktad målfunktion eller flermålsoptimering (MCO), har automatiska planeringsmetoder generellt sett fördelarna av lägre beräkningstider och minimal användarinteraktion, men saknar däremot flexibiliteten hos allmänna ramverk som exempelvis MCO. Maskininlärningsmetoder kan vara speciellt känsliga för avvikelser i dosprediktionssteget på grund av särskilda egenskaper hos de optimeringsfunktioner som vanligtvis används för att återskapa dosfördelningar, och lider dessutom av problemet att det inte finns något allmängiltigt orsakssamband mellan prediktionsnoggrannhet och kvalitet hos optimerad plan. I detta arbete presenterar vi ett sätt att förena idéer från maskininlärningsbaserade planeringsmetoder med det väletablerade MCO-ramverket. Mer precist kan vi, givet förkunskaper i form av antingen en tidigare optimerad plan eller en uppsättning av historiskt levererade kliniska planer, automatiskt generera Paretooptimala planer som täcker en dosregion motsvarande uppnåeliga såväl som kliniskt acceptabla planer. I det förra fallet görs detta genom att introducera dos--volym-bivillkor; i det senare fallet görs detta genom att anpassa en gaussisk blandningsmodell med viktade data med förväntning--maximering-algoritmen, modifiera den med exponentiell lutning och sedan använda speciellt utvecklade optimeringsfunktioner för att ta hänsyn till prediktionsosäkerheter.Numeriska resultat för konceptuell demonstration erhålls för ett fall av prostatacancer varvid behandlingen levererades med volymetriskt modulerad bågterapi, där det visas att metoderna utvecklade i detta arbete är framgångsrika i att automatiskt generera Paretooptimala planer med tillfredsställande kvalitet och variation medan kliniskt irrelevanta dosregioner utesluts. I fallet då historiska planer används som förkunskap är beräkningstiderna markant kortare än för konventionell MCO.

In optimization of intensity-modulated radiation therapy treatment plans, dose-volumehistogram (DVH) functions are often used as objective functions to minimize the violationof dose-volume criteria. Neither DVH functions nor dose-volume criteria, however,are ideal for gradient-based optimization as the former are not continuously differentiableand the latter are discontinuous functions of dose, apart from both beingnonconvex. In particular, DVH functions often work poorly when used in constraintsdue to their being identically zero when feasible and having vanishing gradients on theboundary of feasibility.In this work, we present a general mathematical framework allowing for direct optimizationon all DVH-based metrics. By regarding voxel doses as sample realizations ofan auxiliary random variable and using kernel density estimation to obtain explicit formulas,one arrives at formulations of volume-at-dose and dose-at-volume which are infinitelydifferentiable functions of dose. This is extended to DVH functions and so calledvolume-based DVH functions, as well as to min/max-dose functions and mean-tail-dosefunctions. Explicit expressions for evaluation of function values and corresponding gradientsare presented. The proposed framework has the advantages of depending on onlyone smoothness parameter, of approximation errors to conventional counterparts beingnegligible for practical purposes, and of a general consistency between derived functions.Numerical tests, which were performed for illustrative purposes, show that smoothdose-at-volume works better than quadratic penalties when used in constraints and thatsmooth DVH functions in certain cases have significant advantage over conventionalsuch. The results of this work have been successfully applied to lexicographic optimizationin a fluence map optimization setting. / Vid optimering av behandlingsplaner i intensitetsmodulerad strålterapi används dosvolym- histogram-funktioner (DVH-funktioner) ofta som målfunktioner för att minimera avståndet till dos-volymkriterier. Varken DVH-funktioner eller dos-volymkriterier är emellertid idealiska för gradientbaserad optimering då de förstnämnda inte är kontinuerligt deriverbara och de sistnämnda är diskontinuerliga funktioner av dos, samtidigt som båda också är ickekonvexa. Speciellt fungerar DVH-funktioner ofta dåligt i bivillkor då de är identiskt noll i tillåtna områden och har försvinnande gradienter på randen till tillåtenhet. I detta arbete presenteras ett generellt matematiskt ramverk som möjliggör direkt optimering på samtliga DVH-baserade mått. Genom att betrakta voxeldoser som stickprovsutfall från en stokastisk hjälpvariabel och använda ickeparametrisk densitetsskattning för att få explicita formler, kan måtten volume-at-dose och dose-at-volume formuleras som oändligt deriverbara funktioner av dos. Detta utökas till DVH-funktioner och så kallade volymbaserade DVH-funktioner, såväl som till mindos- och maxdosfunktioner och medelsvansdos-funktioner. Explicita uttryck för evaluering av funktionsvärden och tillhörande gradienter presenteras. Det föreslagna ramverket har fördelarna av att bero på endast en mjukhetsparameter, av att approximationsfelen till konventionella motsvarigheter är försumbara i praktiska sammanhang, och av en allmän konsistens mellan härledda funktioner. Numeriska tester genomförda i illustrativt syfte visar att slät dose-at-volume fungerar bättre än kvadratiska straff i bivillkor och att släta DVH-funktioner i vissa fall har betydlig fördel över konventionella sådana. Resultaten av detta arbete har med framgång applicerats på lexikografisk optimering inom fluensoptimering.