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Finished Master Theses

  Modeling and Solution of the Inverse Problem of Nanoparticle Magnetorelaxometry
Janic Föcke, 2015

Novel approaches for medical methods use Magnetic Nanoparticles (MNP) for the treatment process. These nanoparticles with a diameter of about 100nm can also be used for imaging purposes. In Magnetorelaxometry Imaging (MRXI), magnetic nanoparticles are exposed to an external magnetic field induced by coils outside the region of interest. This causes a torque on the magnetization leading to an alignment of the MNPs. The overlay of the magnetic field induced by each MNP itself can be measured using Superconducting Quantum Interference Devices (SQUIDs). The corresponding mathematical operator that models the MRXI experiment is severely ill-posed and the corresponding inverse problem of determining the particle distribution is difficult to solve directly. In the recent literature only basic regularization techniques like Tikhonov regularization and Truncated Singular Value Decomposition were used to derive a reasonable solution. Using first-order regularization techniques like the Alternating Direction Method of Multipliers (ADMM), it is possible to include a-priori knowledge about the anticipated solution. In this work we derive a sophisticated mathematical model of the MRXI operator and compare the impact of different regularization techniques on the solution, in particular in the case of noisy data, for various simulated settings.  [pdf-file]

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  Flocking-Modelle mit metrischen und topologischen Interaktionen
Miriam Riedel, 2015

In this thesis, flocking-models of birds with two different kinds of interactions are presented. For that a distinction between metrical and topological interactions is made. First discrete models in two dimensions and the amount of a flock's alignment are introduced. Then both models are implemented in MATLAB and the resulting simulations are discussed. Subsequently, the theory of continuous models in one dimension is considered. The model equations regarding the type of interaction are deduced. Finally further considerations and a brief outlook are given.  [pdf-file]

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  PET-MRI Joint Reconstruction via Bregman-TV and Infimal Convolution
Julian Rasch, 2015

Technological advances recently allowed for the combination of Positron Emission To- mography (PET) and Magnetic Resonance Imaging (MRI). Since the obtained images originate from the same object, they are not independent of each other. In this thesis we exploit the inherent structural similarity of both images and present an approach for a joint reconstruction of PET and MR images. We use the difference between edge and level sets of both images as a measure of similarity within a variational framework. The necessary structural information for the comparison of the images is introduced to the problem in form of subgradients of their total variation. We therefore use a weighted sum of Bregman distances with regard to these subgradients as a regularizer and thereby favor piecewise constant images with sharp joint edges. We will prove the well-definedness of the method and the existence of solutions. In order to exclude the sign of jumps across the edges from our considerations, we employ the infimal convolution of Bregman distances. We will derive a suitable primal-dual scheme for the numerical implementation of both ap- proaches. At last, we will evaluate the method numerically on artificial data and illustrate the benefits and advantages over individual reconstructions  [pdf-file]

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  Sparse Dynamic SPECT Modelling and Reconstruction
Carolin Rossmanith, 2014 (Supervision with: Xiaoqun Zhang, Shanghai Jiaotong University)

This work deals with the reconstruction of dynamic Single Photon Emission Computed Tomography (SPECT). We make use of a basis approach for the unknown tracer concentration by assuming that the region of interest can be divided into subregions with spacially constant concentration curves. Applying a regularized variational framework we simultaneously reconstruct both the indicator functions of the subregions as well as the subconcentrations within each region. We are going to present some analysis of the variational model and derive a necessary optimality condition as well as a source condition which allows some basic convergence estimates. Furthermore, two different algorithms are tested on exact and noisy artificial data. In order to verify the reasonability of the reconstruction approach and the chosen variational model we finally analyze and compare the results and present some error measures.  [pdf-file]

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  An SDE Approach to Cancer Therapy including Stem Cells
Julia Kroos, 2014 (Supervision with: Christina Surulescu, Christian Stinner (Kaiserslautern)

Discrete mathematical models for cancer propagation only describe the behavior of cancer cells to a certain extend. In this thesis we present a stochastic model for cancer cells that incorporates intrinsic and extrinsic uncertainties as well as the rather new concept of cancer stem cells. Here, we derive a model for dierentiated cancer cells, cancer stem cells and normal tissue cells, prove the existence of solutions and analyze the behavior of the dierent populations with regards to their persistence times. Subsequently dierent treatment strategies including chemotherapy and radiation therapy and combinations of these therapies are introduced and incorporated in the SDE-model. In order to quantify the treatment strategies we introduced two quality measures: the tumor control probability (TCP) and the normal tissue complication probability (NTCP). For both measures discrete formulas as well as stochastic approaches were derived and compared.  [pdf-file]

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  MathematicalModels for Pancreatic Cancer and Stellate Cells
Daniel Auverkamp, 2014

Pancreatic cancer, which is characterised by its late detection because of no symptoms in early stages, aggressive growth, intense inltration into adjacent tissue, early metastasis, resistance to chemo- and radiotherapy, has a very poor prognosis with a ve-year survival rate of less than 10%. This thesis deals with mathematical modelling of interaction of pancreatic carcinoma cells with stellate cells. A new model, that takes the lling of volume by cells into account, describes the invasion of cancer cells into the surrounding extracellular matrix and interaction between the carcinoma cells and two phenotypes of stellate cells, an inactive and an active, highly motile, chemical expressing cell type, is formulated. Non-dimensionalisation and scaling enable feasible estimation of involved parameter, which are used in the following stability analysis. The work concludes with numerical simulation of the model and variation of it, realised via a combined Crank-Nicholson and Scharfette-Gummel scheme.  [pdf-file]

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  Traction Forces from Phase-Contrast Microscopy
Eva-Maria Brinkmann, 2014

Understanding the cellular ability to migrate is the key to a better comprehension of many biological processes including wound healing and the formation of tumor metastases. However, cellular motility results from a highly complex system of molecular mechanisms among which forces generated by contractions of the cytoskeleton are of vital importance. To study these forces, cells are cultured on an elastic substrate providing an in vitro model of the migration process. As cellular tractions cause the underlying substrate to deform, they can be reconstructed by means of an inverse problem, which however requires an estimate of the corresponding displacement field: The most prominent way of providing this estimate is the elastic substratum method where beads are firmly embedded into the substrate and tracked over time. This is the basis of so-called Traction Force Microscopy. In this thesis we overcome the need for these beads by introducing a novel approach for the reconstruction of cellular traction forces from usual phase-contrast microscopy images of sufficient resolution. Based on a sound mathematical model of the experimental setting, we obtain the displacement field and thus the cellular traction forces by combining linear elasticity theory and a variational framework of image registration. This leads to two alternative minimization problems for which we also show the existence of a solution. Finally, we provide an implementation of the new approach and present first numerical results.  [pdf-file]

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  Nonlinear Reconstruction Methods for Transmission Electron Microscopy
Leonie Zeune, 2014 (Supervision with: Christoph Brune, Twente University, and Ozan Öktem, KTH Stockholm)

Electron tomography (ET) is a technique to recover the three-dimensional structure of an object on a molecular level from a set of two-dimensional transmission electron microscope (TEM) images recorded from different perspectives. These images are corrupted by Poisson as well as Gaussian noise. The resulting inverse problem is severely ill-posed due to a combination of a very low signal-to-noise ratio and an incomplete data problem. In this thesis we present an approach to solve this inverse problem with variational methods. It is based on a statistical modeling of the inverse problem in terms of maximum a posteriori (MAP) likelihood estimations. In contrast to the majority of reconstruction methods in the field of ET, we focus on modeling data corrupted by Poisson noise. Thus, we want to minimize a nonlinear energy functional with the Kullback-Leibler divergence as the data discrepancy term combined with a total variation regularization. In order to solve this optimization problem, we propose an alternating two-step iteration consisting of an expectation-maximization (EM) step and the solution of a weighted Rudin-Osher-Fatemi (ROF) model. The algorithm is adapted to the ane form of the forward model in ET. In order to overcome contrast loss typical for TV-based regularization, we extend the algorithm by iterative regularization based on Bregman distances. Finally, we illustrate the performance of our techniques on synthetic and experimental biological data.  [pdf-file]

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  Methods for automatic mitosis detection and tracking in phase contrast images
Joana Grah, 2014 (Supervision with: Carola Schönlieb, Stefanie Reichelt, Cambridge University, and Rachel Hegemann)

Nowadays, biomedical sciences and more specifically mitotic index analysis in cancer research strongly depend on evaluation and processing of digital microscopy images. This thesis deals with mathematical analysis of phase contrast microscopy images and presents methods for automatic detection and tracking of mitotic cells. We briefly explain the biological background and provide an introduction to mathematical methods of image processing. The main part comprises illustration of established tracking methods as well as a new approach to combine methods for our specific setting within the framework of a variational problem. After discussing some results, we give a summary and an outlook. Finally, we present a Graphical User Interface which facilitates applying the presented methods in MATLAB.  [pdf-file]

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  Quality Measures for Radiation Treatment and their Optimization
Theresa Stocks, 2014 (Supervision with: Thomas Hillen, University of Waterloo)

Abstract A successful radiation treatment destroys the tumor and leaves the damage of surrounding healthy tissue below an admissible level. In this thesis we present a novel modelling for the optimization of cancer treatment which is constrained by early and late complications of healthy cells. Here, an improved LQ-model is used to describe the cell survival and existence of the solution is proved. In this sense an optimal dose and duration for standard as well as hyperfractionated treatment were found. Additionally, we derive and analyze existing models of increasing complexity for TCP and base modelling of NTCP thereof. We are able to improve the one-compartment NTCP model by Gong by making it independent of the carrying capacity. Finally, we propose the derivation and analysis of a new, more general NTCP two-compartment model that includes stem cells as well as differentiated cells.  [pdf-file]

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  Mathematische Modellierung der Fluoreszenztomographie
Lisa Thüß, 2014

In this thesis the fluorescence tomography is presented, an optical imaging technique that uses light to excite an inner fluorescence marker to make certain structures within a biological probe visible. To model the light transport in tissue the radiative transfer equation is introduced and then the diffusion approximation leads us to the diffusion equation. The forward model of the fluorescence tomography is set up and its unique solvability is proved. Finally, we give an example for the numerical implementation of the forward model.  [pdf-file]

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  Sparse Reconstruction and Realistic Head Modeling in EEG/MEG
Sina Tellen, 2013 (Supervision with: Felix Lucka, Carsten Wolters)

The brain is one of the most interesting organs of the human body. Especially the understanding of the human brain functions has aroused researchers interest even from various science fields. The magneto- and the electroencephalography are imaging techniques that record external electromagnetic fields resulting from neural activity. A widely discussed problem in the mathematical area is the ill-posed inverse problem of reconstructing the neuronal sources that produce these measurable fields. One topic in this investigation is the modeling of the human brain as a conductor. A mass of headmodels with varying number of compartments, different structures and conductivities have been investigated from several points of view. In this thesis we examine six headmodels with different complexity with regard to their interaction between concepts of the compressed sensing (CS) and the regularization theory in order to establish significant differences. We show that the conditions or properties provided by the CStheory are only satisfied under strict constraints on the sparsity level. On the other hand, the (Strong) Source Condition, a concept of the regularization theory, leads to more significant trends towards more complex models. This is expressed in terms of a higher percentage of randomly generated samples satisfying a Source Condition in case of more realistic conductor models.  [pdf-file]

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  Verbesserte Partialvolumenkorrektur-Methoden in der PET-Rekonstruktion
Andre Koert, 2013 (Supervision with: Jahn Müller)

This master thesis describes several methods for the correction of the partial volume effect during the image reconstruction process of the positron-emission-tomography. Modified EM-algorithms are defined and optimized by standard regularization methods. In the end of this thesis, the modified EM-methods are compared to the classical reconstruction methods.  [pdf-file]

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  Vergleich von aktuellen Bildrekonstruktionsalgorithmen bei Poisson-Rauschen
Matthias Redecker, 2013 (Supervision with Alex Sawatzky, Daniel Tenbrinck)

Many different state of the art reconstruction algorithms for Poisson problems were presented in the last years. To check if the algorithms hold what the authors promised and to compare them, this thesis adopted some of these algorithms to test them on different synthetic and real data sets. For comparison we will discuss shortly the algorithms, define quality criteria for the reconstruction results and take a look on the convergence speed of the algorithms. Finally we want to evaluate which have performed best.  [pdf-file]

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  Zugexperimente an Biologischen Moleklen - Modelluntersuchung und Simulation
Katharina Wenzel, 2013 (Collaboration with: Andreas Heuer, WWU)

The implementation of AFM-pulling routines by Steered Molecular Dynamics has created a powerful tool to provide information of the unfolding and refolding process of biological molecules. Here, a molecule is stretched under pulling with the help of an umbrella potential where the choice of pull speed and cantilever stiffness can play a critical role in the unfolding pathway. To investigate the influence of these parameters a basic setup for pulling experiments with one particle under brownian motion and external pull force is implemented and the properties of these parameters examined. Additionally several pull experiments under the same circumstances are simulated with the Trp-cage Miniprotein TC5b in vacuo. It can be shown that simulations can lead to very unstable behaviour, especially for higher force speed and stiff cantilever.  [pdf-file]

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  Mathematical Models for Transport in Axons
Lena Frerking, 2012 (Collaboration with: Andreas Püschel, WWU)

This thesis discusses different mathematical models for transport in axons. We regard existing models that deal with transport, which is conducted by motor proteins of the kinesin and dynein superfamilies. We take a closer look to interactions between these motor proteins. In this context, the protein dynactin plays an important role. Besides improving the processivity of dynein motors, it regulates their attachment and detachment behavior. This way, it prevents a dynein motor from pulling a cargo, if this already happens by a kinesin motor. In this thesis, we develop a new mathematical model for axonal transport that considers this recent finding.  [pdf-file]

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  Bewegungskorrigierte PET-Rekonstruktion
Sebastian Suhr, 2012

This thesis discusses motion corrected PET Reconstruction. In the first part we present the theoretical basics of PET Reconstruction and Modelling of the tracer activity. We inquire about Eulerian and Lagrangian Modelling of this activity. For both approaches we discuss continuous derivations as well as discretized minimization technics. Furthermore we present, if some conditions hold, derivatives for regularization functionals for the density image and for the transformation. In the second part we focus on more practical aspects: We introduce a rough estimation for the minimization of the beforehand obtained functional. We test this implementation with software and hardware phantom data.  [pdf-file]

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  A Novel Class of Priors for Edge-Preserving Methods in Bayesian Inversion
Silvia Comelli, 2011 (at University of Milano, collaboration with Vincenzo Capasso)

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