Cancer is a widespread type of diseases that each year affects millions of people. It is mainly treated by chemotherapy, surgery or radiation therapy, or a combination of them. One modality of radiation therapy is high dose-rate brachytherapy, used in treatment of for example prostate cancer and gynecologic cancer. Brachytherapy is an invasive treatment in which catheters (hollow needles) or applicators are used to place the highly active radiation source close to or within a tumour. The treatment planning problem, which can be modelled as a mathematical optimization problem, is the topic of this thesis. The treatment planning includes decisions on how many catheters to use and where to place them as well as the dwell times for the radiation source. There are multiple aims with the treatment and these are primarily to give the tumour a radiation dose that is sufficiently high and to give the surrounding healthy tissue and organs (organs at risk) a dose that is sufficiently low. Because these aims are in conflict, modelling the treatment planning gives optimization problems which essentially are multiobjective. To evaluate treatment plans, a concept called dosimetric indices is commonly used and they constitute an essential part of the clinical treatment guidelines. For the tumour, the portion of the volume that receives at least a specified dose is of interest while for an organ at risk it is rather the portion of the volume that receives at most a specified dose. The dosimetric indices are derived from the dose-volume histogram, which for each dose level shows the corresponding dosimetric index. Dose-volume histograms are commonly used to visualise the three-dimensional dose distribution. The research focus of this thesis is mathematical modelling of the treatment planning and properties of optimization models explicitly including dosimetric indices, which the clinical treatment guidelines are based on. Modelling dosimetric indices explicitly yields mixedinteger programs which are computationally demanding to solve. The computing time of the treatment planning is of clinical relevance as the planning is typically conducted while the patient is under anaesthesia. Research topics in this thesis include both studying properties of models, extending and improving models, and developing new optimization models to be able to take more aspects into account in the treatment planning. There are several advantages of using mathematical optimization for treatment planning in comparison to manual planning. First, the treatment planning phase can be shortened compared to the time consuming manual planning. Secondly, also the quality of treatment plans can be improved by using optimization models and algorithms, for example by considering more of the clinically relevant aspects. Finally, with the use of optimization algorithms the requirements of experience and skill level for the planners are lower. This thesis summary contains a literature review over optimization models for treatment planning, including the catheter placement problem. How optimization models consider the multiobjective nature of the treatment planning problem is also discussed.

Cette thèse développe les statistiques géométriques pour l'analyse de lavariabilité normale et pathologique des formes d'organe en anatomienumérique. Les statistiques géométriques s’intéressent aux données issues devariétés avec structures géométriques additionnelles. En anatomie numérique,les formes d'un organe peuvent être vues comme des déformations d'un organede référence - i.e. comme éléments d'un groupe de Lie, une variété avec unestructure de groupe - ou comme les classes d'équivalence de leur configuration3D sous l'action de transformations - i.e. comme éléments d'un quotient, unevariété avec une stratification. Les images médicales peuvent êtrereprésentées par des variétés avec une distribution horizontale. Lacontribution de cette thèse est d'étendre les statistiques géométriques au delàdes géométries riemanniennes ou métriques maintenant classiques pourprendre en compte des structures additionnelles. Premièrement, nousdéfinissons les statistiques géométriques sur les groupes de Lie. Nousproposons une construction algorithmique de (pseudo-)métriqueRiemannienne, compatible avec la structure de groupe, lorsqu'elle existe. Noustrouvons que certains groupes n'admettent pas de telle (pseudo-)métrique etdéfendons l'idée de statistiques non-métriques sur les groupes de Lie. Ensuite,nous utilisons les statistiques géométriques pour analyser l'algorithme decalcul d'organe de référence, reformulé avec des espaces quotient. Nousmontrons son biais et suggérons un algorithme amélioré. Enfin, nousappliquons les statistiques géométriques au traitement d'images, engénéralisant les structures sous-Riemanniennes, utilisées en 2D, au 3D / This thesis develops Geometric Statistics to analyze the normal andpathological variability of organ shapes in Computational Anatomy. Geometricstatistics consider data that belong to manifolds with additional geometricstructures. In Computational Anatomy, organ shapes may be modeled asdeformations of a template - i.e. as elements of a Lie group, a manifold with agroup structure - or as the equivalence classes of their 3D configurations underthe action of transformations - i.e. as elements of a quotient space, a manifoldwith a stratification. Medical images can be modeled as manifolds with ahorizontal distribution. The contribution of this thesis is to extend GeometricStatistics beyond the now classical Riemannian and metric geometries in orderto account for these additional structures. First, we tackle the definition ofGeometric Statistics on Lie groups. We provide an algorithm that constructs a(pseudo-)Riemannian metric compatible with the group structure when itexists. We find that some groups do not admit such a (pseudo-)metric andadvocate for non-metric statistics on Lie groups. Second, we use GeometricStatistics to analyze the algorithm of organ template computation. We show itsasymptotic bias by considering the geometry of quotient spaces. We illustratethe bias on brain templates and suggest an improved algorithm. We then showthat registering organ shapes induces a bias in their statistical analysis, whichwe offer to correct. Third, we apply Geometric Statistics to medical imageprocessing, providing the mathematics to extend sub-Riemannian structures,already used in 2D, to our 3D images

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The goal of elastography is to map the mechanical properties of soft tissues associated with health and disease. The mechanical property of interest in this work is the complex shear modulus, composed of a real part, the storage modulus, which is a measure of elasticity, and an imaginary part, the loss modulus, which is a measure of viscosity. Together, they determine the speed and attenuation of shear waves in the medium. Elastography techniques based on either ultrasound imaging or MRI can image shear wave propagation and thus are capable of measuring shear wave speed and attenuation. Dispersion, or the frequency-dependence of material parameters, is a primary confounding factor when comparing measurements between different shear wave elastography implementations. Prior attempts at quantifying this frequency-dependence suffered from inaccurate modeling assumptions and low signal-to-noise ratios (SNR). To overcome these limitations, a high-fidelity forward model of shear wave propagation in homogeneous media was developed. The model is an exact semi-analytical solution of Navier's equation and is well-suited for acoustic radiation force impulse shear wave elastography (ARFI-SWE) because it does not require precise knowledge of the strength of the source, nor its spatial or temporal distribution. Unlike models used in ARFI-SWE heretofore, it accounts for the vector polarization of shear waves and exactly represents geometric spreading of the shear wavefield, whether spherical, cylindrical, or neither. Furthermore, it is material-model independent, i.e. it makes no assumption about the frequency-dependence of material parameters. It overcomes the problem of low SNR through spatial averaging and enables estimation of the frequency-dependent complex shear modulus over a wider frequency range than has hitherto been possible. This improved ARFI-SWE was named Shear Wave Rheometry (SWR). By combining SWR with a novel torsional vibration rheometry, dispersion in tissue-mimicking gels was quantified from 1--1800 Hz. The measurements show sizable frequency-dependent variation in the shear modulus of gelatin, a material often assumed to be non-dispersive based on narrow-band measurements. SWR measurements in ex vivo bovine liver tissue yielded complex shear modulus estimates from 25--250 Hz and showed that liver tissue exhibits significant dispersion in this frequency range: a factor of 4 increase in the storage modulus and a factor of 10 increase in the loss modulus. Quality metrics showed that liver tissue can be reasonably approximated as homogeneous and isotropic for ARFI-SWE measurements in this frequency range. Results demonstrate that accounting for dispersion is essential for meaningful comparisons of measurements between systems. Moreover, improved tissue characterization enabled by SWR may have clinical relevance, for example, in the diagnosis and monitoring of chronic liver disease.

abstract: This dissertation develops a second order accurate approximation to the magnetic resonance (MR) signal model used in the PARSE (Parameter Assessment by Retrieval from Single Encoding) method to recover information about the reciprocal of the spin-spin relaxation time function (R2*) and frequency offset function (w) in addition to the typical steady-state transverse magnetization (M) from single-shot magnetic resonance imaging (MRI) scans. Sparse regularization on an approximation to the edge map is used to solve the associated inverse problem. Several studies are carried out for both one- and two-dimensional test problems, including comparisons to the first order approximation method, as well as the first order approximation method with joint sparsity across multiple time windows enforced. The second order accurate model provides increased accuracy while reducing the amount of data required to reconstruct an image when compared to piecewise constant in time models. A key component of the proposed technique is the use of fast transforms for the forward evaluation. It is determined that the second order model is capable of providing accurate single-shot MRI reconstructions, but requires an adequate coverage of k-space to do so. Alternative data sampling schemes are investigated in an attempt to improve reconstruction with single-shot data, as current trajectories do not provide ideal k-space coverage for the proposed method. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2019

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As a non-robotic minimally invasive surgery, endoscopic surgery is one of the widely used surgeries for the medical domain to reduce the risk of infection, incisions, and the discomfort of the patient. The endoscopic surgery procedure, also named surgical workflow in this work, can be divided into different sub-phases. During the procedure, the surgeon inserts a thin, flexible tube with a video camera through a small incision or a natural orifice like the mouth or nostrils. The surgeon can utilize tiny surgical instruments while viewing organs on the computer monitor through these tubes. The surgery only allows a limited number of instruments simultaneously appearing in the body, requiring a sufficient instrument preparation method. Therefore, surgical workflow anticipation, including surgical instrument and phase anticipation, is essential for an intra-operative decision-support system. It deciphers the surgeon's behaviors and the patient's status to forecast surgical instrument and phase occurrence before they appear, supporting instrument preparation and computer-assisted intervention (CAI) systems. In this work, we investigate an unexplored surgical workflow anticipation problem by proposing an Instrument Interaction Aware Anticipation Network (IIA-Net). Spatially, it utilizes rich visual features about the context information around the instrument, i.e., instrument interaction with their surroundings. Temporally, it allows for a large receptive field to capture the long-term dependency in the long and untrimmed surgical videos through a causal dilated multi-stage temporal convolutional network. Our model enforces an online inference with reliable predictions even with severe noise and artifacts in the recorded videos. Extensive experiments on Cholec80 dataset demonstrate the performance of our proposed method exceeds the state-of-the-art method by a large margin (1.40 v.s. 1.75 for inMAE and 2.14 v.s. 2.68 for eMAE).

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