39 research outputs found
Derivation of the bidomain equations for a beating heart with a general microstructure
A novel multiple scales method is formulated that can be applied to problems which have an almost\ud
periodic microstructure not in Cartesian coordinates but in a general curvilinear coordinate system.\ud
The method is applied to a model of the electrical activity of cardiac myocytes and used to derive a\ud
version of the bidomain equations describing the macroscopic electrical activity of cardiac tissue. The\ud
treatment systematically accounts for the non-uniform orientation of the cells within the tissue and for\ud
deformations of the tissue occurring as a result of the heart beat
Modelling the effect of gap junctions on tissue-level cardiac electrophysiology
When modelling tissue-level cardiac electrophysiology, continuum
approximations to the discrete cell-level equations are used to maintain
computational tractability. One of the most commonly used models is represented
by the bidomain equations, the derivation of which relies on a homogenisation
technique to construct a suitable approximation to the discrete model. This
derivation does not explicitly account for the presence of gap junctions
connecting one cell to another. It has been seen experimentally [Rohr,
Cardiovasc. Res. 2004] that these gap junctions have a marked effect on the
propagation of the action potential, specifically as the upstroke of the wave
passes through the gap junction.
In this paper we explicitly include gap junctions in a both a 2D discrete
model of cardiac electrophysiology, and the corresponding continuum model, on a
simplified cell geometry. Using these models we compare the results of
simulations using both continuum and discrete systems. We see that the form of
the action potential as it passes through gap junctions cannot be replicated
using a continuum model, and that the underlying propagation speed of the
action potential ceases to match up between models when gap junctions are
introduced. In addition, the results of the discrete simulations match the
characteristics of those shown in Rohr 2004. From this, we suggest that a
hybrid model -- a discrete system following the upstroke of the action
potential, and a continuum system elsewhere -- may give a more accurate
description of cardiac electrophysiology.Comment: In Proceedings HSB 2012, arXiv:1208.315
The cardiac bidomain model and homogenization
We provide a rather simple proof of a homogenization result for the bidomain
model of cardiac electrophysiology. Departing from a microscopic cellular
model, we apply the theory of two-scale convergence to derive the bidomain
model. To allow for some relevant nonlinear membrane models, we make essential
use of the boundary unfolding operator. There are several complications
preventing the application of standard homogenization results, including the
degenerate temporal structure of the bidomain equations and a nonlinear dynamic
boundary condition on an oscillating surface.Comment: To appear in Networks and Heterogeneous Media, Special Issue on
Mathematical Methods for Systems Biolog
GEMS: A Fully Integrated PETSc-Based Solver for Coupled Cardiac Electromechanics and Bidomain Simulations
Cardiac contraction is coordinated by a wave of electrical excitation which propagates through the heart. Combined modeling of electrical and mechanical function of the heart provides the most comprehensive description of cardiac function and is one of the latest trends in cardiac research. The effective numerical modeling of cardiac electromechanics remains a challenge, due to the stiffness of the electrical equations and the global coupling in the mechanical problem. Here we present a short review of the inherent assumptions made when deriving the electromechanical equations, including a general representation for deformation-dependent conduction tensors obeying orthotropic symmetry, and then present an implicit-explicit time-stepping approach that is tailored to solving the cardiac mono- or bidomain equations coupled to electromechanics of the cardiac wall. Our approach allows to find numerical solutions of the electromechanics equations using stable and higher order time integration. Our methods are implemented in a monolithic finite element code GEMS (Ghent Electromechanics Solver) using the PETSc library that is inherently parallelized for use on high-performance computing infrastructure. We tested GEMS on standard benchmark computations and discuss further development of our software
Nonlinear physics of electrical wave propagation in the heart: a review
The beating of the heart is a synchronized contraction of muscle cells
(myocytes) that are triggered by a periodic sequence of electrical waves (action
potentials) originating in the sino-atrial node and propagating over the atria and
the ventricles. Cardiac arrhythmias like atrial and ventricular fibrillation (AF,VF)
or ventricular tachycardia (VT) are caused by disruptions and instabilities of these
electrical excitations, that lead to the emergence of rotating waves (VT) and turbulent
wave patterns (AF,VF). Numerous simulation and experimental studies during the
last 20 years have addressed these topics. In this review we focus on the nonlinear
dynamics of wave propagation in the heart with an emphasis on the theory of pulses,
spirals and scroll waves and their instabilities in excitable media and their application
to cardiac modeling. After an introduction into electrophysiological models for action
potential propagation, the modeling and analysis of spatiotemporal alternans, spiral
and scroll meandering, spiral breakup and scroll wave instabilities like negative line
tension and sproing are reviewed in depth and discussed with emphasis on their impact
in cardiac arrhythmias.Peer ReviewedPreprin
A mathematical model for mechanically-induced deterioration of the binder in lithium-ion electrodes
This study is concerned with modeling detrimental deformations of the binder
phase within lithium-ion batteries that occur during cell assembly and usage. A
two-dimensional poroviscoelastic model for the mechanical behavior of porous
electrodes is formulated and posed on a geometry corresponding to a thin
rectangular electrode, with a regular square array of microscopic circular
electrode particles, stuck to a rigid base formed by the current collector.
Deformation is forced both by (i) electrolyte absorption driven binder
swelling, and; (ii) cyclic growth and shrinkage of electrode particles as the
battery is charged and discharged. The governing equations are upscaled in
order to obtain macroscopic effective-medium equations. A solution to these
equations is obtained, in the asymptotic limit that the height of the
rectangular electrode is much smaller than its width, that shows the
macroscopic deformation is one-dimensional. The confinement of macroscopic
deformations to one dimension is used to obtain boundary conditions on the
microscopic problem for the deformations in a 'unit cell' centered on a single
electrode particle. The resulting microscale problem is solved using numerical
(finite element) techniques. The two different forcing mechanisms are found to
cause distinctly different patterns of deformation within the microstructure.
Swelling of the binder induces stresses that tend to lead to binder
delamination from the electrode particle surfaces in a direction parallel to
the current collector, whilst cycling causes stresses that tend to lead to
delamination orthogonal to that caused by swelling. The differences between the
cycling-induced damage in both: (i) anodes and cathodes, and; (ii) fast and
slow cycling are discussed. Finally, the model predictions are compared to
microscopy images of nickel manganese cobalt oxide cathodes and a qualitative
agreement is found.Comment: 25 pages, 11 figure