1,845 research outputs found
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
A multiresolution space-time adaptive scheme for the bidomain model in electrocardiology
This work deals with the numerical solution of the monodomain and bidomain
models of electrical activity of myocardial tissue. The bidomain model is a
system consisting of a possibly degenerate parabolic PDE coupled with an
elliptic PDE for the transmembrane and extracellular potentials, respectively.
This system of two scalar PDEs is supplemented by a time-dependent ODE modeling
the evolution of the so-called gating variable. In the simpler sub-case of the
monodomain model, the elliptic PDE reduces to an algebraic equation. Two simple
models for the membrane and ionic currents are considered, the
Mitchell-Schaeffer model and the simpler FitzHugh-Nagumo model. Since typical
solutions of the bidomain and monodomain models exhibit wavefronts with steep
gradients, we propose a finite volume scheme enriched by a fully adaptive
multiresolution method, whose basic purpose is to concentrate computational
effort on zones of strong variation of the solution. Time adaptivity is
achieved by two alternative devices, namely locally varying time stepping and a
Runge-Kutta-Fehlberg-type adaptive time integration. A series of numerical
examples demonstrates thatthese methods are efficient and sufficiently accurate
to simulate the electrical activity in myocardial tissue with affordable
effort. In addition, an optimalthreshold for discarding non-significant
information in the multiresolution representation of the solution is derived,
and the numerical efficiency and accuracy of the method is measured in terms of
CPU time speed-up, memory compression, and errors in different norms.Comment: 25 pages, 41 figure
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
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
Convergence of discrete duality finite volume schemes for the cardiac bidomain model
We prove convergence of discrete duality finite volume (DDFV) schemes on
distorted meshes for a class of simplified macroscopic bidomain models of the
electrical activity in the heart. Both time-implicit and linearised
time-implicit schemes are treated. A short description is given of the 3D DDFV
meshes and of some of the associated discrete calculus tools. Several numerical
tests are presented
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