28 research outputs found
Robust preconditioners for PDE-constrained optimization with limited observations
Regularization robust preconditioners for PDE-constrained optimization
problems have been successfully developed. These methods, however, typically
assume that observation data is available throughout the entire domain of the
state equation. For many inverse problems, this is an unrealistic assumption.
In this paper we propose and analyze preconditioners for PDE-constrained
optimization problems with limited observation data, e.g. observations are only
available at the boundary of the solution domain. Our methods are robust with
respect to both the regularization parameter and the mesh size. That is, the
condition number of the preconditioned optimality system is uniformly bounded,
independently of the size of these two parameters. We first consider a
prototypical elliptic control problem and thereafter more general
PDE-constrained optimization problems. Our theoretical findings are illuminated
by several numerical results
Mathematical Modeling of the Human Brain
This open access book bridges common tools in medical imaging and neuroscience with the numerical solution of brain modelling PDEs. The connection between these areas is established through the use of two existing tools, FreeSurfer and FEniCS, and one novel tool, the SVM-Tk, developed for this book. The reader will learn the basics of magnetic resonance imaging and quickly proceed to generating their first FEniCS brain meshes from T1-weighted images. The book's presentation concludes with the reader solving a simplified PDE model of gadobutrol diffusion in the brain that incorporates diffusion tensor images, of various resolution, and complex, multi-domain, variable-resolution FEniCS meshes with detailed markings of anatomical brain regions. After completing this book, the reader will have a solid foundation for performing patient-specific finite element simulations of biomechanical models of the human brain
Mathematical Modeling of the Human Brain
This open access book bridges common tools in medical imaging and neuroscience with the numerical solution of brain modelling PDEs. The connection between these areas is established through the use of two existing tools, FreeSurfer and FEniCS, and one novel tool, the SVM-Tk, developed for this book. The reader will learn the basics of magnetic resonance imaging and quickly proceed to generating their first FEniCS brain meshes from T1-weighted images. The book's presentation concludes with the reader solving a simplified PDE model of gadobutrol diffusion in the brain that incorporates diffusion tensor images, of various resolution, and complex, multi-domain, variable-resolution FEniCS meshes with detailed markings of anatomical brain regions. After completing this book, the reader will have a solid foundation for performing patient-specific finite element simulations of biomechanical models of the human brain
Estimates of the permeability of extra-cellular pathways through the astrocyte endfoot sheath
Abstract Background Astrocyte endfoot processes are believed to cover all micro-vessels in the brain cortex and may play a significant role in fluid and substance transport into and out of the brain parenchyma. Detailed fluid mechanical models of diffusive and advective transport in the brain are promising tools to investigate theories of transport. Methods We derive theoretical estimates of astrocyte endfoot sheath permeability for advective and diffusive transport and its variation in microvascular networks from mouse brain cortex. The networks are based on recently published experimental data and generated endfoot patterns are based on Voronoi tessellations of the perivascular surface. We estimate corrections for projection errors in previously published data. Results We provide structural-functional relationships between vessel radius and resistance that can be directly used in flow and transport simulations. We estimate endfoot sheath filtration coefficients in the range L p = 2 × 10 - 11 m Pa - 1 s - 1 to 3 × 10 - 10 m Pa - 1 s - 1 , diffusion membrane coefficients for small solutes in the range C M = 5 × 10 2 m - 1 to 6 × 10 3 m - 1 , and gap area fractions in the range 0.2–0.6%, based on a inter-endfoot gap width of 20 nm. Conclusions The astrocyte endfoot sheath surrounding microvessels forms a secondary barrier to extra-cellular transport, separating the extra-cellular space of the parenchyma and the perivascular space outside the endothelial layer. The filtration and membrane diffusion coefficients of the endfoot sheath are estimated to be an order of magnitude lower than those of the extra-cellular matrix while being two orders of magnitude higher than those of the vessel wall
Simulation of water and tracer transport in brain tissue with explicit resolution of microvessels
Poster presented at the BrainH2O symposium, Kopenhagen (16-18 August 2021).
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Dynamics of a neuron-glia system: the occurrence of seizures and the influence of electroconvulsive stimuli
In this paper, we investigate the dynamics of a neuron–glia cell system and the underlying mechanism for the occurrence of seizures. For our mathematical and numerical investigation of the cell model we will use bifurcation analysis and some computational methods. It turns out that an increase of the potassium concentration in the reservoir is one trigger for seizures and is related to a torus bifurcation. In addition, we will study potassium dynamics of the model by considering a reduced version and we will show how both mechanisms are linked to each other. Moreover, the reduction of the potassium leak current will also induce seizures. Our study will show that an enhancement of the extracellular potassium concentration, which influences the Nernst potential of the potassium current, may lead to seizures. Furthermore, we will show that an external forcing term (e.g. electroshocks as unidirectional rectangular pulses also known as electroconvulsive therapy) will establish seizures similar to the unforced system with the increased extracellular potassium concentration. To this end, we describe the unidirectional rectangular pulses as an autonomous system of ordinary differential equations. These approaches will explain the appearance of seizures in the cellular model. Moreover, seizures, as they are measured by electroencephalography (EEG), spread on the macro–scale (cm). Therefore, we extend the cell model with a suitable homogenised monodomain model, propose a set of (numerical) experiment to complement the bifurcation analysis performed on the single–cell model. Based on these experiments, we introduce a bidomain model for a more realistic modelling of white and grey matter of the brain. Performing similar (numerical) experiment as for the monodomain model leads to a suitable comparison of both models. The individual cell model, with its seizures explained in terms of a torus bifurcation, extends directly to corresponding results in both the monodomain and bidomain models where the neural firing spreads almost synchronous through the domain as fast traveling waves, for physiologically relevant paramenters
Transitional hemodynamics in intracranial aneurysms - Comparative velocity investigations with high resolution lattice Boltzmann simulations, normal resolution ANSYS simulations, and MR imaging
Purpose: Blood flow in intracranial aneurysms has, until recently, been considered to be disturbed but still laminar. Recent high resolution computational studies have demonstrated, in some situations, however, that the flow may exhibit high frequency fluctuations that resemble weakly turbulent or transitional flow. Due to numerous assumptions required for simplification in computational fluid dynamics (CFD) studies, the occurrence of these events, in vivo, remains unsettled. The detection of these fluctuations in aneurysmal blood flow, i.e., hemodynamics by CFD, poses additional challenges as such phenomena cannot be captured in clinical data acquisition with magnetic resonance (MR) due to inadequate temporal and spatial resolutions. The authors purpose was to address this issue by comparing results from highly resolved simulations, conventional resolution laminar simulations, and MR measurements, identify the differences, and identify their causes. Methods: Two aneurysms in the basilar artery, one with disturbed yet laminar flow and the other with transitional flow, were chosen. One set of highly resolved direct numerical simulations using the lattice Boltzmann method (LBM) and another with adequate resolutions under laminar flow assumption were conducted using a commercially available ANSYS Fluent solver. The velocity fields obtained from simulation results were qualitatively and statistically compared against each other and with MR acquisition. Results: Results from LBM, ANSYS Fluent, and MR agree well qualitatively and quantitatively for one of the aneurysms with laminar flow in which fluctuations were \u3c 80 Hz. The comparisons for the second aneurysm with high fluctuations of \u3e ∼600 Hz showed vivid differences between LBM, ANSYS Fluent, and magnetic resonance imaging. After ensemble averaging and down-sampling to coarser space and time scales, these differences became minimal. Conclusions: A combination of MR derived data and CFD can be helpful in estimating the hemodynamic environment of intracranial aneurysms. Adequately resolved CFD would suffice gross assessment of hemodynamics, potentially in a clinical setting, and highly resolved CFD could be helpful in a detailed and retrospective understanding of the physiological mechanisms. C 2016 American Association of Physicists in Medicine. [http://dx.doi.org/10.1118/1.4964793]