138 research outputs found
Incorporating Inductances in Tissue-Scale Models of Cardiac Electrophysiology
In standard models of cardiac electrophysiology, including the bidomain and
monodomain models, local perturbations can propagate at infinite speed. We
address this unrealistic property by developing a hyperbolic bidomain model
that is based on a generalization of Ohm's law with a Cattaneo-type model for
the fluxes. Further, we obtain a hyperbolic monodomain model in the case that
the intracellular and extracellular conductivity tensors have the same
anisotropy ratio. In one spatial dimension, the hyperbolic monodomain model is
equivalent to a cable model that includes axial inductances, and the relaxation
times of the Cattaneo fluxes are strictly related to these inductances. A
purely linear analysis shows that the inductances are negligible, but models of
cardiac electrophysiology are highly nonlinear, and linear predictions may not
capture the fully nonlinear dynamics. In fact, contrary to the linear analysis,
we show that for simple nonlinear ionic models, an increase in conduction
velocity is obtained for small and moderate values of the relaxation time. A
similar behavior is also demonstrated with biophysically detailed ionic models.
Using the Fenton-Karma model along with a low-order finite element spatial
discretization, we numerically analyze differences between the standard
monodomain model and the hyperbolic monodomain model. In a simple benchmark
test, we show that the propagation of the action potential is strongly
influenced by the alignment of the fibers with respect to the mesh in both the
parabolic and hyperbolic models when using relatively coarse spatial
discretizations. Accurate predictions of the conduction velocity require
computational mesh spacings on the order of a single cardiac cell. We also
compare the two formulations in the case of spiral break up and atrial
fibrillation in an anatomically detailed model of the left atrium, and [...].Comment: 20 pages, 12 figure
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
BPX preconditioners for the Bidomain model of electrocardiology
The aim of this work is to develop a BPX preconditioner for the Bidomain model of electrocardiology. This model describes the bioelectrical activity of the cardiac tissue and consists of a system of a non-linear parabolic reaction\u2013diffusion partial differential equation (PDE) and an elliptic linear PDE, modeling at macroscopic level the evolution of the transmembrane and extracellular electric potentials of the anisotropic cardiac tissue. The evolution equation is coupled through the non-linear reaction term with a stiff system of ordinary differential equations, the so-called membrane model, describing the ionic currents through the cellular membrane. The discretization of the coupled system by finite elements in space and semi-implicit finite differences in time yields at each time step the solution of an ill-conditioned linear system. The goal of the present study is to construct, analyze and numerically test a BPX preconditioner for the linear system arising from the discretization of the Bidomain model. Optimal convergence rate estimates are established and verified by two- and three-dimensional numerical tests on both structured and unstructured meshes. Moreover, in a full heartbeat simulation on a three-dimensional wedge of ventricular tissue, the BPX preconditioner is about 35% faster in terms of CPU times than ILU(0) and an Algebraic Multigrid preconditioner
A staggered-in-time and non-conforming-in-space numerical framework for realistic cardiac electrophysiology outputs
Computer-based simulations of non-invasive cardiac electrical outputs, such
as electrocardiograms and body surface potential maps, usually entail severe
computational costs due to the need of capturing fine-scale processes and to
the complexity of the heart-torso morphology. In this work, we model cardiac
electrical outputs by employing a coupled model consisting of a
reaction-diffusion model - either the bidomain model or the most efficient
pseudo-bidomain model - on the heart, and an elliptic model in the torso. We
then solve the coupled problem with a segregated and staggered in-time
numerical scheme, that allows for independent and infrequent solution in the
torso region. To further reduce the computational load, main novelty of this
work is in introduction of an interpolation method at the interface between the
heart and torso domains, enabling the use of non-conforming meshes, and the
numerical framework application to realistic cardiac and torso geometries. The
reliability and efficiency of the proposed scheme is tested against the
corresponding state-of-the-art bidomain-torso model. Furthermore, we explore
the impact of torso spatial discretization and geometrical non-conformity on
the model solution and the corresponding clinical outputs. The investigation of
the interface interpolation method provides insights into the influence of
torso spatial discretization and of the geometrical non-conformity on the
simulation results and their clinical relevance.Comment: 26 pages,11 figures, 3 table
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
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 problem with the Beeler–Reuter cell model on a truncated ellipsoid approximating the left ventricle including a localized stimulation. For this 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
An introduction to mathematical and numerical modeling in heart electrophysiology
The electrical activation of the heart is the biological process that regulates the contraction of the cardiac muscle, allowing it to pump blood to the whole body. In physiological conditions, the pacemaker cells of the sinoatrial node generate an action potential (a sudden variation of the cell transmembrane potential) which, following preferential conduction pathways, propagates throughout the heart walls and triggers the contraction of the heart chambers. The action potential propagation can be mathematically described by coupling a model for the ionic currents, flowing through the membrane of a single cell, with a macroscopical model that describes the propagation of the electrical signal in the cardiac tissue. The most accurate model available in the literature for the description of the macroscopic propagation in the muscle is the Bidomain model, a degenerate parabolic system composed of two non-linear partial differential equations for the intracellular and extracellular potential. In this paper, we present an introduction to the fundamental aspects of mathematical modeling and numerical simulation in cardiac electrophysiology
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