10 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
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
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
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 periodic microstructure not in Cartesian coordinates but in a general curvilinear coordinate system. The method is applied to a model of the electrical activity of cardiac myocytes and used to derive a version of the bidomain equations describing the macroscopic electrical activity of cardiac tissue. The treatment systematically accounts for the nonuniform orientation of the cells within the tissue and for deformations of the tissue occurring as a result of the heart bea
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 periodic microstructure not in Cartesian coordinates but in a general curvilinear coordinate system. The method is applied to a model of the electrical activity of cardiac myocytes and used to derive a version of the bidomain equations describing the macroscopic electrical activity of cardiac tissue. The treatment systematically accounts for the non-uniform orientation of the cells within the tissue and for deformations of the tissue occurring as a result of the heart beat