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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.

Projects

Project heads
Staff
CH1
Project heads Prof. Dr. Ralf Kornhuber
Dr.-Ing. Stefan Zachow
Staff Dr. Jonathan Youett
Duration: - Status: running Located at: Freie Universität Berlin (FU Berlin)

Description

This 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).

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CH2
Project heads Prof. Dr. Tim Conrad
Prof. Dr. Gitta Kutyniok
Prof. Dr. Christof Schütte
Staff Nada Cvetkovic
Martin Genzel
Duration: 01.06.2014 - 31.05.2017 Status: running Located at: Freie Universität Berlin (FU Berlin) , Technische Universität Berlin (TU Berlin)

Description

Tumor 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.

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CH3
Project heads Prof. Dr. Michael Joswig
Staff André Wagner
Duration: 01.06.2014 - 31.05.2017 Status: running Located at: Technische Universität Berlin (TU Berlin)

Description

A 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.

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CH4
Project heads Prof. Dr. Carsten Hartmann
Dr. Max von Kleist
PD Dr. Marcus Weber
Staff Wei Zhang
Duration: 01.06.2014 - 31.05.2017 Status: running Located at: Freie Universität Berlin (FU Berlin)

Description

Development 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.

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CH5
Project heads Prof. Dr. Alexander Bockmayr
Prof. Dr. Susanna Röblitz
Prof. Dr. Heike Siebert
Staff Stefanie Kasielke
Adam Streck
Duration: 01.06.2014 - 31.05.2017 Status: running Located at: Freie Universität Berlin (FU Berlin) , Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)

Description

Mathematical 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.

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CH6
Project heads Prof. Dr. Susanna Röblitz
Prof. Dr. Christof Schütte
Staff Dr Ilja Klebanov
Duration: 01.06.2014 - 31.05.2017 Status: running Located at: Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)

Description

In 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.

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CH7
Project heads Prof. Dr. Tim Conrad
Prof. Dr. Christof Schütte
Staff
Duration: 01.06.2014 - 31.05.2017 Status: running Located at: Freie Universität Berlin (FU Berlin)

Description

Project Background

Pancreatic 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 level

Current 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.

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CH8
Project heads Hon Prof. Hans-Christian Hege
Dr. Martin Weiser
Dr.-Ing. Stefan Zachow
Staff Dennis Jentsch
Duration: 01.06.2014 - 31.05.2017 Status: running Located at: Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)

Description

Medical 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.

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CH9
Project heads Dr. Martin Weiser
Dr.-Ing. Stefan Zachow
Staff Marian Moldenhauer
Duration: 01.06.2014 - 31.05.2017 Status: running Located at: Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB)

Description

This 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.

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CH10
Project heads Prof. Dr. Harry Yserentant
Staff Janina Oertel
Duration: 01.06.2014 - 31.05.2017 Status: running Located at: Technische Universität Berlin (TU Berlin)

Description

The 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.

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