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  • 1
    Language: English
    In: Journal of Mathematical Biology, 2015, Vol.71(3), pp.551-582
    Description: Gliomas are a class of rarely curable tumors arising from abnormal glia cells in the human brain. The understanding of glioma spreading patterns is essential for both radiological therapy as well as surgical treatment. Diffusion tensor imaging (DTI) allows to infer the white matter fibre structure of the brain in a noninvasive way. Painter and Hillen (J Theor Biol 323:25–39, 2013) used a kinetic partial differential equation to include DTI data into a class of anisotropic diffusion models for glioma spread. Here we extend this model to explicitly include adhesion mechanisms between glioma cells and the extracellular matrix components which are associated to white matter tracts. The mathematical modelling follows the multiscale approach proposed by Kelkel and Surulescu (Math Models Methods Appl Sci 23(3), 2012). We use scaling arguments to deduce a macroscopic advection-diffusion model for this process. The tumor diffusion tensor and the tumor drift velocity depend on both, the directions of the white matter tracts as well as the binding dynamics of the adhesion molecules. The advanced computational platform DUNE enables us to accurately solve our macroscopic model. It turns out that the inclusion of cell binding dynamics on the microlevel is an important factor to explain finger-like spread of glioma.
    Keywords: Glioma modelling ; Multiscale model ; DTI data ; Kinetic model
    ISSN: 0303-6812
    E-ISSN: 1432-1416
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  • 2
    Language: English
    In: Mathematical Models and Methods in Applied Sciences, 30 June 2017, Vol.27(7), pp.1355-1390
    Description: We propose a multiscale model for tumor cell migration in a tissue network. The system of equations involves a structured population model for the tumor cell density, which besides time and position depends on a further variable characterizing the cellular state with respect to the amount of receptors bound to soluble and insoluble ligands. Moreover, this equation features pH-taxis and adhesion, along with an integral term describing proliferation conditioned by receptor binding. The interaction of tumor cells with their surroundings calls for two more equations for the evolution of tissue fibers and acidity (expressed via concentration of extracellular protons), respectively. The resulting ODE-PDE system is highly nonlinear. We prove the global existence of a solution and perform numerical simulations to illustrate its behavior, paying particular attention to the influence of the supplementary structure and of the adhesion.
    Keywords: Cancer Cell Migration through Tissue ; Multiscale Model ; Structured Population Model ; Ph-Taxis ; Cell–Cell and Cell–Tissue Adhesion ; Receptor Binding ; Nonlocal Pde-Ode System ; Extracellular Proton Dynamics ; Nonlinear Diffusion ; Global Existence ; Integro-Differential Equations ; Engineering ; Mathematics
    ISSN: 0218-2025
    E-ISSN: 1793-6314
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  • 3
    Language: English
    In: Mathematics and Computers in Simulation, July 2017, Vol.137, pp.266-285
    Description: Subsurface fractures play an important role in many modern energy technologies (e.g. geothermal energy, fracking, nuclear waste management). Real world experiments concerning fracture propagation are usually expensive and time consuming, therefore numerical simulations become more and more important in this area. The main challenge for numerical methods is the evolving domain. Standard finite element (FE) methods require remeshing to resolve the crack surface once a fracture starts propagating. To overcome this problem we use a phase field approach to regularize the crack surface. Thereby we consider quasi static evolution in fluid filled media. For the one-dimensional case -convergence of the approximating functional to the potential energy of the system is shown. Based on this model we propose a discontinuous Galerkin (DG) formulation for the displacement. This takes into account displacement jumps at the crack surface. Numerical experiments compare our method with a standard FE approach.
    Keywords: Fracture Propagation ; Phase Field ; Gamma Convergence ; Discontinuous Galerkin ; Computer Science
    ISSN: 0378-4754
    E-ISSN: 1872-7166
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  • 4
    Language: English
    In: ACM Transactions on Mathematical Software (TOMS), 10 October 2017, Vol.44(2), pp.1-20
    Description: Implicitly described domains are a well-established tool in the simulation of time-dependent problems, for example, using level-set methods. To solve partial differential equations on such domains, a range of numerical methods was developed, for example, the Immersed Boundary method, the Unfitted Finite Element or Unfitted Discontinuous Galerkin methods, and the eXtended or Generalised Finite Element methods, just to name a few. Many of these methods involve integration over cut-cells or their boundaries, as they are described by sub-domains of the original level-set mesh. We present a new algorithm to geometrically evaluate the integrals over domains described by a first-order, conforming level-set function. The integration is based on a polyhedral reconstruction of the implicit geometry, following the concepts of the marching cubes algorithm. The algorithm preserves various topological properties of the implicit geometry in its polyhedral reconstruction, making it suitable for Finite Element computations. Numerical experiments show second-order accuracy of the integration. An implementation of the algorithm is available as free software, which allows for an easy incorporation into other projects. The software is in productive use within the DUNE framework (Bastian et al. 2008a).
    Keywords: Level-Sets ; Cut-Cell Methods ; Geometry Reconstruction ; Implicit Domains ; Marching Cubes ; Numerical Quadrature ; Surface Reconstruction ; Sciences (General) ; Computer Science
    ISSN: 00983500
    E-ISSN: 1557-7295
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  • 5
    Language: English
    In: Optics letters, 15 January 2017, Vol.42(2), pp.227-230
    Description: We present a simple and fast phase aberration compensation method in digital holographic microscopy (DHM) for quantitative phase imaging of living cells. By analyzing the frequency spectrum of an off-axis hologram, phase aberrations can be compensated for automatically without fitting or pre-knowledge of the setup and/or the object. Simple and effective computation makes the method suitable for quantitative online monitoring with highly variable DHM systems. Results from automated quantitative phase imaging of living NIH-3T3 mouse fibroblasts demonstrate the effectiveness and the feasibility of the method.
    Keywords: Algorithms ; Holography -- Methods ; Microscopy -- Methods
    ISSN: 01469592
    E-ISSN: 1539-4794
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  • 6
    Description: Implicitly described domains are a well established tool in the simulation of time dependent problems, e.g. using level-set methods. In order to solve partial differential equations on such domains, a range of numerical methods was developed, e.g. the Immersed Boundary method, Unfitted Finite Element or Unfitted discontinuous Galerkin methods, eXtended or Generalised Finite Element methods, just to name a few. Many of these methods involve integration over cut-cells or their boundaries, as they are described by sub-domains of the original level-set mesh. We present a new algorithm to geometrically evaluate the integrals over domains described by a first-order, conforming level-set function. The integration is based on a polyhedral reconstruction of the implicit geometry, following the concepts of the Marching Cubes algorithm. The algorithm preserves various topological properties of the implicit geometry in its polyhedral reconstruction, making it suitable for Finite Element computations. Numerical experiments show second order accuracy of the integration. An implementation of the algorithm is available as free software, which allows for an easy incorporation into other projects. The software is in productive use within the DUNE framework.
    Keywords: Computer Science - Numerical Analysis ; Mathematics - Numerical Analysis ; G.1.2 ; G.1.4 ; I.3.5
    Source: Cornell University
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  • 7
    Description: Motivated by considering partial differential equations arising from conservation laws posed on evolving surfaces, a new numerical method for an advection problem is developed and simple numerical tests are performed. The method is based on an unfitted discontinuous Galerkin approach where the surface is not explicitly tracked by the mesh which means the method is extremely flexible with respect to geometry. Furthermore, the discontinuous Galerkin approach is well-suited to capture the advection driven by the evolution of the surface without the need for a space-time formulation, back-tracking trajectories or streamline diffusion. The method is illustrated by a one-dimensional example and numerical results are presented that show good convergence properties for a simple test problem. Comment: In proceedings of ALGORITMY 2016
    Keywords: Mathematics - Numerical Analysis ; 35l02, 35l65, 35q90, 35r01, 65n12, 65n30
    Source: Cornell University
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  • 8
    Language: English
    In: Mathematical biosciences and engineering : MBE, 01 April 2016, Vol.13(2), pp.443-60
    Description: Glioma is a broad class of brain and spinal cord tumors arising from glia cells, which are the main brain cells that can develop into neoplasms. They are highly invasive and lead to irregular tumor margins which are not precisely identifiable by medical imaging, thus rendering a precise enough resection very difficult. The understanding of glioma spread patterns is hence essential for both radiological therapy as well as surgical treatment. In this paper we propose a multiscale model for glioma growth including interactions of the cells with the underlying tissue network, along with proliferative effects. Our current accounting for two subpopulations of cells to accomodate proliferation according to the go-or-grow dichtomoty is an extension of the setting in [16]. As in that paper, we assume that cancer cells use neuronal fiber tracts as invasive pathways. Hence, the individual structure of brain tissue seems to be decisive for the tumor spread. Diffusion tensor imaging (DTI) is able to provide such information, thus opening the way for patient specific modeling of glioma invasion. Starting from a multiscale model involving subcellular (microscopic) and individual (mesoscale) cell dynamics, we perform a parabolic scaling to obtain an approximating reaction-diffusion-transport equation on the macroscale of the tumor cell population. Numerical simulations based on DTI data are carried out in order to assess the performance of our modeling approach.
    Keywords: Models, Biological ; Neoplasm Invasiveness ; Glioma -- Pathology
    E-ISSN: 1551-0018
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  • 9
    Language: English
    In: Journal of Computational Physics, 15 September 2014, Vol.273, pp.227-242
    Description: Optimal control for cardiac electrophysiology based on the bidomain equations in conjunction with the Fenton–Karma ionic model is considered. This generic ventricular model approximates well the restitution properties and spiral wave behavior of more complex ionic models of cardiac action potentials. However, it is challenging due to the appearance of state-dependent discontinuities in the source terms. A computational framework for the numerical realization of optimal control problems is presented. Essential ingredients are a shape calculus based treatment of the sensitivities of the discontinuous source terms and a marching cubes algorithm to track iso-surface of excitation wavefronts. Numerical results exhibit successful defibrillation by applying an optimally controlled extracellular stimulus.
    Keywords: Bidomain Model ; Fenton–Karma Ionic Model ; Defibrillation ; Neumann Boundary Stimulation ; Optimal Control ; State-Dependent Discontinuities ; Applied Sciences
    ISSN: 0021-9991
    E-ISSN: 1090-2716
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  • 10
    Language: English
    In: SIAM Journal on Scientific Computing, 2017, Vol.39(4), pp.1435-1465
    Description: Engineers manually optimizing a structure using finite element based simulation software often employ an iterative approach where in each iteration they change the structure slightly and resimulate. Standard finite element based simulation software is usually not well suited for this workflow, as it restarts in each iteration, even for tiny changes. In settings with complex local microstructure, where a fine mesh is required to capture the geometric detail, localized model reduction can improve this workflow. To this end, we introduce ArbiLoMod, a method which allows fast recomputation after arbitrary local modifications. It employs a domain decomposition and a localized form of the reduced basis method for model order reduction. It assumes that the reduced basis on many of the unchanged domains can be reused after a localized change. The reduced model is adapted when necessary, steered by a localized error indicator. The global error introduced by the model order reduction is controlled by a robust and efficient localized a posteriori error estimator, certifying the quality of the result. We demonstrate ArbiLoMod for a coercive, parameterized example with changing structure.
    Keywords: Model Order Reduction ; Reduced Basis Method ; Domain Decomposition ; A Posteriori Error Estimation ; 65n55 ; 65n30
    ISSN: 1064-8275
    E-ISSN: 1095-7197
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