##
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]

##
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]

##
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]

##
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]

##
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]

##
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]

##
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]

##
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]

##
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]

##
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]

##
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]

##
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]

##
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]

##
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]

##
Zugexperimente an Biologischen Molekülen - 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]

##
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]

##
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]

##
A Novel Class of Priors for Edge-Preserving Methods in Bayesian Inversion

Silvia Comelli, 2011 (at University of Milano, collaboration with Vincenzo Capasso)

....
[pdf-file]