68 research outputs found
Derivation of a bidomain model for bundles of myelinated axons
The work concerns the multiscale modeling of a nerve fascicle of myelinated axons. We present a rigorous derivation of a macroscopic bidomain model describing the behavior of the electric potential in the fascicle based on the FitzHugh–Nagumo membrane dynamics. The approach is based on the two-scale convergence machinery combined with the method of monotone operators
On uniqueness theorems for the inverse problem of Electrocardiography in the Sobolev spaces
We consider a mathematical model related to reconstruction of cardiac
electrical activity from ECG measurements on the body surface. An application
of recent developments in solving boundary value problems for elliptic and
parabolic equations in Sobolev type spaces allows us to obtain uniqueness
theorems for the model. The obtained results can be used as a sound basis for
creating numerical methods for non-invasive mapping of the heart.Comment: arXiv admin note: substantial text overlap with arXiv:2106.0412
Efficient time splitting schemes for the monodomain equation in cardiac electrophysiology
Approximating the fast dynamics of depolarization waves in the human heart described by the monodomain model is numerically challenging. Splitting methods for the PDE-ODE coupling enable the computation with very fine space and time discretizations. Here, we compare different splitting approaches regarding convergence, accuracy and efficiency. Simulations were performed for a benchmark configuration with the Beeler–Reuter cell model on a truncated ellipsoid approximating the left ventricle including a localized stimulation. For this benchmark configuration, we provide a reference solution for the transmembrane potential. We found a semi-implicit approach with state variable interpolation to be the most efficient scheme. The results are transferred to a more physiological setup using a bi-ventricular domain with a complex external stimulation pattern to evaluate the accuracy of the activation time for different resolutions in space and time
Proyecto Docente e Investigador, Trabajo Original de Investigación y Presentación de la Defensa, preparado por Germán Moltó para concursar a la plaza de Catedrático de Universidad, concurso 082/22, plaza 6708, área de Ciencia de la Computación e Inteligencia Artificial
Este documento contiene el proyecto docente e investigador del candidato Germán Moltó MartÃnez presentado como requisito para el concurso de acceso a plazas de Cuerpos Docentes Universitarios. Concretamente, el documento se centra en el concurso para la plaza 6708 de Catedrático de Universidad en el área de Ciencia de la Computación en el Departamento de Sistemas Informáticos y Computación de la Universitat Politécnica de València. La plaza está adscrita a la Escola Técnica Superior d'Enginyeria Informà tica y tiene como perfil las asignaturas "Infraestructuras de Cloud Público" y "Estructuras de Datos y Algoritmos".También se incluye el Historial Académico, Docente e Investigador, asà como la presentación usada durante la defensa.Germán Moltó MartÃnez (2022). Proyecto Docente e Investigador, Trabajo Original de Investigación y Presentación de la Defensa, preparado por Germán Moltó para concursar a la plaza de Catedrático de Universidad, concurso 082/22, plaza 6708, área de Ciencia de la Computación e Inteligencia Artificial. http://hdl.handle.net/10251/18903
Well-posedness for a modified bidomain model describing bioelectric activity in damaged heart tissues
We prove the existence and the uniqueness of a solution for a modified bidomain model, describing the electrical behaviour of the cardiac tissue in pathological situations. The main idea is to reduce the problem to an abstract parabolic setting, which requires to introduce several auxiliary differential systems and a non-standard bilinear form.
The main difficulties are due to the degeneracy of the bidomain system and to its non-standard coupling with the diffusion equation
Modeling Excitable Tissue
This open access volume presents a novel computational framework for understanding how collections of excitable cells work. The key approach in the text is to model excitable tissue by representing the individual cells constituting the tissue. This is in stark contrast to the common approach where homogenization is used to develop models where the cells are not explicitly present. The approach allows for very detailed analysis of small collections of excitable cells, but computational challenges limit the applicability in the presence of large collections of cells
A positive cell vertex godunov scheme for a beeler-reuter based model of cardiac electrical activity
International audienceThe monodomain model is a widely used model in electrocardiology to simulate the propagation of electrical potential in the myocardium. In this paper, we investigate a positive nonlinear control volume finite element (CVFE) scheme, based on Godunov's flux approximation of the diffusion term, for the monodomain model coupled to a physiological ionic model (the Beeler-Reuter model) and using an anisotropic diffusion tensor. In this scheme, degrees of freedom are assigned to vertices of a primal triangular mesh, as in conforming finite element methods. The diffusion term which involves an anisotropic tensor is discretized on a dual mesh using the diffusion fluxes provided by the conforming finite element reconstruction on the primal mesh and the other terms are discretized by means of an upwind finite volume method on the dual mesh. The scheme ensures the validity of the discrete maximum principle without any restriction on the transmissibility coefficients. By using a compactness argument, we obtain the convergence of the discrete solution and as a consequence, we get the existence of a weak solution of the original model. Finally, we illustrate the efficiency of the proposed scheme by exhibiting some numerical results
Parallel Newton-Krylov-BDDC and FETI-DP deluxe solvers for implicit time discretizations of the cardiac Bidomain equations
Two novel parallel Newton-Krylov Balancing Domain Decomposition by
Constraints (BDDC) and Dual-Primal Finite Element Tearing and Interconnecting
(FETI-DP) solvers are here constructed, analyzed and tested numerically for
implicit time discretizations of the three-dimensional Bidomain system of
equations.
This model represents the most advanced mathematical description of the
cardiac bioelectrical activity and it consists of a degenerate system of two
non-linear reaction-diffusion partial differential equations (PDEs), coupled
with a stiff system of ordinary differential equations (ODEs).
A finite element discretization in space and a segregated implicit
discretization in time, based on decoupling the PDEs from the ODEs, yields at
each time step the solution of a non-linear algebraic system.
The Jacobian linear system at each Newton iteration is solved by a Krylov
method, accelerated by BDDC or FETI-DP preconditioners, both augmented with the
recently introduced {\em deluxe} scaling of the dual variables.
A polylogarithmic convergence rate bound is proven for the resulting parallel
Bidomain solvers.
Extensive numerical experiments on linux clusters up to two thousands
processors confirm the theoretical estimates, showing that the proposed
parallel solvers are scalable and quasi-optimal
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