Mathematics in Clinical Research and Health Care
The health care industry has experienced a proliferation of innovations aimed at enhancing life expectancy, quality of life, diagnostic and treatment options, as well as the efficiency and cost effectiveness of the health care system. The general trends into patient-specific treatment and molecular medicine require continuous innovations in our ability to understand bio-medical processes in detail. Predictive models for complex bio-medical processes and information-based medicine more and more complement the traditional lab-based approaches to answer these needs. On the contrary, progress in medical technologies in combination with the aging society induce a cost explosion: Health costs here have been rising significantly faster than the overall economy or personal incomes for more than 40 years, a trend that cannot continue forever. Cost-effective therapies, especially for chronic diseases, prevention and hospital logistics seem essential for the future of our health care system.
In the framework of Matheon's application area Life Sciences mathematical research has focused on three domains of expertise: (1) In computational surgery planning the priority lies on implant design. (2) Molecular processes research concentrates on in-silico design and optimization of molecular sensors, ligands and enzymes accompanied by further research on quantum processes. (3) In mathematical systems biology the aim is to bridge the gap between mathematical basic research and application in pharmaceutical industry. These domains of expertise are complemented by activities in medical imaging in Matheon's application area Visualization. The majority of cooperation partners in these fields of research are from the natural and bio-sciences or from (pharmaceutical) industry.
Dr. Clemens Guhlke
DescriptionSensing with nanopores is a promising new technology to analyze macromolecules like DNA strands by low cost/high speed measurements. The sensing device is constructed based on a nanopore embedded into a membrane which separates two electrodes. The system is filled with an electrolyte containing macromolecules to be analyzed. An electric potential is applied to the electrodes and induces an ionic current through the pore. Sensing is based on the observation that this ionic current is influenced by the geometrical configurations of the pore and of the macromolecules positioned within the pore. Under controlled movement of the macromolecule through the pore a characteristic time dependent current signal is generated, which is correlated to the structure of the pore and the macromolecule. Therefore nanopores can be used to count and even to characterize macromolecules in an electrolytic solution. n order to achieve a better understanding of the of phenomena that control the passing time of the analytes (macromolecules) through the nanopore, and to derive a relation between characteristic properties of the macromolecule and the generated current, the project will focus on three groups of tasks: Development of an appropriate nanopore model in the context of non-equilibrium thermodynamics, which accounts for the geometrical properties of pore and analyte, the charged boundary layers, ion diffusion and fluid flow. Combination and analysis of novel numerical discretization schemes, like pressure robust methods for fluid flow and novel finite volume discretization approaches for the PNP system in order to provide physically meaningful numerical models of the double layer structure and its impact on the fluid flow. Use of asymptotic analysis to derive reduced models, which include the relevant features of the complete thermodynamic model in different regimes. X Close project
DescriptionVery recently, magnetic resonance fingerprinting (MRF) has been introduced as a highly promising MRI acquisition scheme which allows for the simultaneous quantification of the tissue parameters (e.g. T1, T2 and others) using a single acquisition process. In MRF, the tissue of interest is excited through a random sequence of pulses without needing to wait for the system to return to equilibrium between pulses. After each pulse, a subset of the signal's Fourier coefficients is collected, as in classical MRI, and a reconstruction of the net magnetization image is performed. These reconstructions suffer from the presence of artifacts since the Fourier coefficients are not fully sampled. The formed sequence of image elements is then compared to a family of predicted sequences (dictionary of fingerprints) each of which corresponds to a specific combination of values of the tissue parameters. This dictionary is computed beforehand by solving the Bloch equations. The idea is that, provided the dictionary is rich enough, every material element (voxel) can be then mapped to its parameter values. While first very promising results have been obtained in biomedical engineering, many aspects of MRF remain widely open and require a proper mathematization for optimizing and robustifying the procedure. The aim of this project is, thus, to provide a quantitative mathematical model for the MRF process, leading to a variational image reconstruction problem subject to dynamical constraints describing magnetization and an embedded reconstruction scheme. This model will be subject to a detailed mathematical analysis and its efficient numerical solution. X Close project
Prof. Dr. Susanna Röblitz
DescriptionOne of the main goals of mathematical modeling related to medical applications is to obtain patient-specific parametrizations and model predictions. In clinical practice, however, the number of available measurements for single patients is usually limited due to time and cost restrictions. This hampers the process of making patient-specific predictions about the outcome of a treatment. On the other hand, data are often available for many patients, in particular if extensive clinical studies have been performed. Empirical Bayes methods can provide a solution to this controversy. Instead of applying Bayes’ rule to each measurement separately, these methods usually boil down to combining all measurements in order to construct an informative prior as a first step and then using this prior for the Bayesian inference of the individual parametrizations in a second step. We want to demonstrate the applicability and benefit of this approach on a high-dimensional model system for predicting patient-specific treatment success rates related to in vitro fertilization in reproductive medicine. X Close project
Prof. Dr. Gitta Kutyniok
Prof. Dr. Christof Schütte
DescriptionIn living organisms, biological cells transition from one state to another. This happens during normal cell development (e.g. aging) or is triggered by events, such as diseases. The time-ordered set of state changes is called a trajectory. Identifying these cell trajectories is a crucial part in bio-medical research to understand changes on a gene and molecular level. It allows to derive biological insights such as disease mechanisms and can lead to new biomedical discoveries and to advances in health-care. With the advent of single cell experiments such as Drop-Seq or inDrop, individual gene expression profiles of thousands of cells can be measured in a single experiment. These large data-sets allow to determine a cell's state based on its gene activity (cell expression profiles, CEPs), which can be expressed as a large feature vector representing its location in some large state space. The main problem with these experiments is that the actual time-information is lost, and needs to be recovered. The state-of-the art solution is to introduce the concept of pseudo-time in which the cells are ordered by CEP similarity. To find robust and biological meaningful trajectories based on CEPs, two main tasks have to be performed: (1) A CEP-based metric has to be learned to define pair-wise distances between CEPs. (2) Given this metric, similar CEP groups and transition paths between those groups should be identified and analysed. X Close project
Prof. Tim Sullivan
Dr. Christoph von Tycowicz
DescriptionThe reconstruction of discretized geometric shapes from empirical data, especially from image data, is important for many applications in medicine, biology, materials science, and other fields. During the last years, a number of techniques for performing such geometrical reconstructions and for conducting shape analysis have been developed. An important mathematical concept in this context are shape spaces. These are high-dimensional quotient manifolds with Riemannian structure, whose points represent geometrical shapes. Using suitable metrics and probability density functions on such manifolds, distances between shapes or statistical shape priors (for utilization in reconstruction tasks) can be defined. A frequently encountered situation is that instead of a set of discrete shapes a series of shapes is given, varying with some parameter (e.g. time). The corresponding mathematical object is a trajectory in shape space. For many analysis questions it is helpful to consider the shape trajectories as such (instead of individual shapes) - often together with co-varying parameters. The focus of this project is to develop new mathematical methods for the analysis, processing and reconstruction of empirically defined shape trajectories. By treating the trajectories as curves in shape space, we plan to exploit the rich geometric structure inherent to these spaces. In consequence, we expect the derived schemes to benefit from a compact encoding of constraints and a superior consistency as compared to their Euclidean counterparts. To develop new mathematical methods for the analysis, processing and reconstruction of empirically defined shape trajectories exploiting the rich geometric structure of shape space. X Close project
Dr.-Ing. Stefan Zachow
DescriptionThis project aims at estimating the unknown parameters of a physics-based joint model together with a systematic sensitivity analysis, to ensure the reliability of a computer-assisted surgery planning tool developed in previous Matheon projects. The estimation shall be carried out using a Bayesian approach in combination with reduced basis methods to achieve feasible computing times. This calibration of the model will be accompanied by a clinical validation based on real patient data, in cooperation with the Orthopaedic Research Center of the university hospital Stavanger. X Close project
Prof. Dr. Christof Schütte
DescriptionWhile simulations of detailed molecular structure, e.g. using atomistic or coarse- grained MD simulation is able to describe the evolution of molecular systems at length/timescales of nanometers/milliseconds, we require a way to bridge from the molecular scale to large-scale/long-time evolutions of molecular superstructures such as actin networks on the scale of micrometers/hours. Such time- and lengthscales while still maintaining some structural, and importantly single-molecule resolution, can be covered by particle-based reaction-diffusion simulations. Molecular kinetic models of small parts of the overall machinery (single molecules and small complexes) can be parametrized with high-throughput MD simulations, enhanced sampling simu- lations, possibly by incorporating constraints from experimental data. In order to ex- plore the long-range and long-time behavior of mixtures and superstructures of many molecules, we set out ot develop a rigorous and computationally efficient coupling be- tween molecular kinetics models and particle-based reaction-diffusion dynamics (Fig. 1). X Close project
DescriptionBased on novel results for smooth and discrete Hodge-type decompositions on manifolds with boundary, this project aims to incorporate discrete boundary-sensitive Hodge decompositions as a central tool for the analysis of blood flow and parameterization of blood vessels. These decompositions provide the following two substantial improvements over existing methods: first, they are able to distinguish harmonic blood flow arising from boundary in- and out ow from harmonic circulations induced by the interior topology of the geometry. Second, they guarantee a theoretically-sound linkage of certain fields with controlled boundary behaviour to cohomological quantities of the geometry, which is the essential and still missing ingredient for the creation of periods to ensure global matching of parameter lines in modern parameterization techniques. X Close project
Prof. Dr. Frank Noé
Prof. Dr. Reinhold Schneider
Dr. Hao Wu
DescriptionThe dynamics of a molecular system can be described by the propagation of probabilities. The project aims at estimating coarse grained models of probability densities for molecular dynamics (MD) by nonlinear projections from a high dimensional space onto a low dimensional space. Molecular processes such as protein kinetics from all-atom simulations and the like suffer from the high dimensionality of the underlying space. To overcome this, projections from the high dimensional space onto a low dimensional space have been introduced, such that the system can be described on a coarser scale by using less degrees of freedom. In the present project we apply low rank tensor approximations, to tackle the curse of dimensions. We will use Observable Operator models (OOM) to estimate the dynamics using data from short time simulation. X Close project
Dr. Martin Weiser
DescriptionDuring brain development, synaptic connection patterns are formed in an extremely robust manner. As the interconnection patterns are much too complex to be encoded directly in the genome, they must emerge from simpler rules. In this project we investigate mechanistic stochastic models of axon growth and filopodial dynamics, checking whether their simulation leads to connection patterns and dynamics as observed in vivo, and with the same robustness. X Close project
Dr. Stefan Klus
Prof. Dr. Christof Schütte
DescriptionCellular processes are governed by diffusion, transport, and interactions of its constituents. For many processes the spatial inhomogeneity of cells is of secondary importance; modelling such processes means finding appropriate kinetic models of the underlying cellular reaction networks (CRNs). The availability of such models is key to many areas of the life sciences ranging from computational biology to system medicine and is essential for understanding the fundamentals of cellular behavior, its malfunction under external stress and its restoration by regenerative interventions. X Close project
Dr.-Ing. Stefan Zachow
DescriptionThis project aims at the development, analysis and implementation of algorithms for computer-assisted planning in hip surgery and hip joint replacement by fast virtual test. Fast forward simulations of patient-specific motion of hip joints and implants in 3D shall be enabled by exploiting suitable a priori information. To this end, we will derive, analyze, and implement reduced basis methods for heterogeneous joint models (reduced approximation). X Close project
Prof. Dr. Gitta Kutyniok
Prof. Dr. Christof Schütte
DescriptionTumor diseases rank among the most frequent causes of death in Western countries coinciding with an incomplete understanding of the underlying pathogenic mechanisms and a lack of individual treatment options. Hence, early diagnosis of the disease and early relapse monitoring are currently the best available options to improve patient survival. In this project, we aim for the identification of disease specific sets of biological signals that reliably indicate a disease outbreak (or status) in an individual. Such biological signals (e.g. proteomics or genomics data) are typically very large (millions of dimensions), which significantly increases the complexity of algorithms for analyzing the parameter space or makes them even infeasible. However, these types of data usually exhibit a very particular structure, and at the same time, the set of disease specific features is very small compared to the ambient dimension. Such a high-dimensional setting naturally calls for the application of the concept of sparse classifiers, which has been extensively studied in the fields of compressed sensing and statistical learning during the last decade. Our research focuses on both algorithmic improvements of available methods as well as theoretical results such as recovery guarantees for general data models. X Close project
DescriptionA fundamental problem in machine vision asks to generate geometric information about a scene in 3-space from several camera images. This is relevant, e.g., in the context of augmented reality frameworks for eye surgery simulation. It is the goal of this project to apply techniques from geometric combinatorics and algebraic geometry for analyzing the picture space to allow for a profound computational preprocessing. X Close project
PD Dr. Marcus Weber
DescriptionDevelopment and spread of drug resistant microorganisms is a major health issue which, accompanied by an attrition in drug development, is expected to worsen in the near future. The source of drug resistance development is the inadequate use of antimicrobials: Inadequate therapies insufficiently suppress susceptible strains, which may give rise to a drug resistant type. At the same time, inadequate therapy exerts enough selective pressure to provide the newly emerged resistant strain with a selective advantage that allows it to become fixed in the population. In recent years, we have elaborated the idea, that an optimal switching between existing antimicrobial drugs may mitigate drug resistance development in the individual. Drug resistance development is an intrinsically stochastic process. This process can be accurately described by the chemical master equation (CME). A major mathematical drawback is the fact that the CME cannot be solved directly due to its numerical complexity. Therefore, computation of an optimal control/therapy based on a direct numerical solution of the CME is usually not feasible. The aim of the proposed project is to mathematically characterize and develop optimal control policies derived from approximations of the CME, and to use the developed methods to suggest drug mitigating therapies to clinical partners in the field of HIV-1 and antibiotic resistance. X Close project
Prof. Dr. Susanna Röblitz
Prof. Dr. Heike Siebert
DescriptionMathematical modelling in biological and medical applications is almost always faced with the problem of incomplete and noisy data. Rather than adding unsupported assumptions to obtain a unique model, a different approach generates a pool of models in agreement with all available observations. Analysis and classification of such models allow linking the constraints imposed by the data to essential model characteristics and showcase different implementations of key mechanisms. Within the project, we aim at combining the advantages of logical and continuous modeling to arrive at a comprehensive system analysis under data uncertainty. Model classification will integrate qualitative aspects such as characteristics of the network topology with more quantitative information extracted from clustering of joint parameter distributions derived from Bayesian approaches. The theory development is accompanied by and tested in application to oncogenic signaling networks. X Close project
Prof. Dr. Christof Schütte
DescriptionIn biotechnology, systems biology, or reaction engineering one is faced with large systems of ordinary differential equations (ODE) that are used to describe the kinetics of the reaction network of interest. These ODE models contain a large number of mostly unknown kinetic parameters that one needs to infer from usually sparse and noisy experimental data. Typically, inverse problems like classical parameter identification are associated with ill-posed behaviour. However, Bayesian approaches can be used to recover joint parameter distributions and allow for the quantification of uncertainty and risk in a way demanded by the applications. In this project, we want to overcome the computational limitations of classical Markov-chain Monte-Carlo methods by developing new algorithmic approaches to Bayesian inverse problems using, e.g., sparse approximation results or empirical Bayes methods. The methods will directly be applied to large-scale networks in systems biology. X Close project
Prof. Dr. Christof Schütte
Project BackgroundPancreatic cancer is the fifth leading cause of cancer death in Germany (see DKFZ Report, 2010). It is estimated that in 2030 it will be the second leading cause of cancer death incurring a cost of about 15,8 Billion US-Dollar worldwide to the public health systems.
Cancer is a systems disease"Cancer is no more a disease of cells than a traffic jam is a disease of cars. A lifetime of study of the internal-combustion engine would not help anyone to understand our traffic problems.'" (Smithers1962). It is accepted that gene mutations are part of the process of cancer, but mutations alone are not enough. Cancer involves an interaction between neoplastic cells and surrounding tissue on many different levels, e.g. interaction of RNA molecules, proteins, and metabolites. But most available models are limited to only one or very few levels of interactions and describe a rather static view.
From single to multi source: data integration on a systems levelCurrent high-throughput -omics technologies have dramatically eased the production of part lists for a variety of organisms. What is still missing are the dynamic interactions among an organism's molecular parts, and the interactions between different biological levels, such as transcriptomics and proteomics. This is pivotal to better understanding of an organism's biology, and - in our case - to understand pancreas cancer.
Therefore, the aim of this project is two-fold: (1) use data acquired in our earlier projects to create a holistic integration of the aforementioned sources and levels for modeling pancreas cancer, which we call Network-of-Networks or short: NoN (in our context networks of different -omics levels, such as genomics, transcriptomics, proteomics and metabolomics. (2) A NoN is a very large and complex object and its structure differs significantly from other biological networks. Thus, new methods for complexity reduction and analyzing NoNs will be developed in this project.
The goalIn this project we aim to develop a new method that can be used to solve this task: the identification of minimal, yet robust fingerprints from very high-dimensional, noisy -omics data. Our method will be based on ideas from the areas of compressed sensing and machine learning. X Close project
Dr. Martin Weiser
Dr.-Ing. Stefan Zachow
DescriptionMedical imaging is essential in diagnostics and surgery planning. For representation of bony structures different imaging modalities are used; the leading methods are X-ray projection (projectional radiography) and CT. Disadvantage of these imaging techniques is the ionization caused by X-rays, particularly in CT, where the dose is 250-500 times higher than in classic X-ray projection. From the clinical perspective therefore one would like to replace CT acquisitions by a few possible X-ray projections. The project deals with the ill-posed inverse problem of 3D reconstruction of bony structures from 2D radiographs. Virtual radiographs are generated from virtual bone structure models; these are compared with clinical patient images and incrementally changed until a sufficiently accurate bone model is found whose virtual projections fit to the measured data. By using a statistical shape model as prior knowledge it is possible to formulate a well-posed optimization problem in a Bayesian setting. Using gradient methods and multilevel/multiresolution methods for both the reconstruction parameters and image data, good computational performance is achieved. Uncertainty quantification techniques can be applied to describe the spatially varying accuracy of the reconstructed model. Finding best X-ray projections (recording directions) minimizing both uncertainty and radiation exposure leads to a design of experiments problem. Two flavors of this design optimization are considered: An all-at-once approach finding the best image acquisition setup before any X-ray projections are performed, and a sequential approach determining the best next projection direction based on the accumulated knowledge gained from the previously taken images. X Close project
Dr.-Ing. Stefan Zachow
DescriptionThis project aims at a software environment supporting computer-assisted planning for total hip joint replacement by suggesting implant positions optimized for longevity of bone implants. The aim is to pre-operatively assess stress distribution in bone and to determine an optimal implant position with respect to natural function and stress distribution to prevent loosening, early migration, stress shielding, undesired bone remodeling, and fracture. Increasing the longevity of implants will help to enhance quality of life and reduce the cost of health care in aging societies. Focus of the research is the development of efficient optimization algorithms by adaptive quadrature of the high-dimensional space of daily motions and appropriate choice of tolerances for the underlying dynamic contact solver. X Close project
DescriptionThe chemical master equation is a fundamental equation in chemical kinetics. It underlies the classical reaction-rate equations and takes the stochastic effects into account that cannot be neglected in the case of small population numbers.
There is an ongoing effort to tackle the chemical master equation numerically. The major challenge is its high dimensionality: for a system of d interacting species the chemical master equation is a differential equation with state space N_0^d, N_0 the set of nonnegative integers.
The main goal of project A-CH10 is build a sound mathematical basis for the numerical approximation of the chemical master equation and to put numerical methods for this equation on a firm mathematical ground. X Close project