The Defibrillation Beeler-Reuter Model
Catherine
Lloyd
Bioengineering Institute, University of Auckland
Model Structure
ABSTRACT: The goal of this simulation study is to examine, in a sheet of myocardium, the contribution of anode and cathode break phenomena in terminating a spiral wave reentry by the defibrillation shock. The tissue is represented as a homogeneous bidomain with unequal anisotropy ratios. Two case studies are presented in this article: tissue that can electroporate at high levels of transmembrane potential, and model tissue that does not support electroporation. In both cases, the spiral wave is initiated via cross-field stimulation of the bidomain sheet. The extracellular defibrillation shock is delivered via two small electrodes located at opposite tissue boundaries. Modifications in the active membrane kinetics enable the delivery of high-strength defibrillation shocks. Numerical solutions are obtained using an efficient semi-implicit predictor-corrector scheme that allows one to execute the simulations within reasonable time. The simulation results demonstrate that anode and/or cathode break excitations contribute significantly to the activity during and after the shock. For a successful defibrillation shock, the virtual electrodes and the break excitations restrict the spiral wave and render the tissue refractory so it cannot further maintain the reentry. The results also indicate that electroporation alters the anode/cathode break phenomena, the major impact being on the timing of the cathode-break excitations. Thus, electroporation results in different patterns of transmembrane potential distribution after the shock. This difference in patterns may or may not result in change of the outcome of the shock.
The original paper reference is cited below:
Anode/cathode make and break phenomena during defibrillation: Does electroporation make a difference?, Skouibine, K., Trayanova, N., Moore, P. 1999, IEEE Transactions on Biomedical Engineering, 46, 769-777. PubMed ID: 10396895
model diagram
Schematic diagram of the cell model.
$\frac{d V}{d \mathrm{time}}=\frac{\mathrm{I\_stim}-\mathrm{i\_Na}+\mathrm{i\_s}+\mathrm{i\_x1}+\mathrm{i\_K1}}{C}$
$\mathrm{i\_Na}=\mathrm{g\_Na}m^{3.0}h(V-\mathrm{E\_Na})$
$\mathrm{alpha\_m}=\begin{cases}890.9437890\frac{e^{0.0486479V-4.8647916}}{1.0+5.93962526e^{0.0486479V-4.8647916}} & \text{if $V> 100.0$}\\ 0.9\frac{V+42.65}{1.0-e^{-0.22V-9.3830}} & \text{otherwise}\end{cases}$
$\mathrm{beta\_m}=\begin{cases}1.437e^{-0.085V-3.37875} & \text{if $V> -85.0$}\\ \frac{100.0}{1.0+0.486479e^{0.2597504V+22.0787804}} & \text{otherwise}\end{cases}$
$\frac{d m}{d \mathrm{time}}=\mathrm{alpha\_m}(1.0-m)-\mathrm{beta\_m}m$
$\mathrm{alpha\_h}=\begin{cases}0.1e^{-0.193V-15.37245} & \text{if $V> -90.0$}\\ -12.0662845-0.1422598V & \text{otherwise}\end{cases}$
$\mathrm{beta\_h}=\frac{1.7}{1.0+e^{-0.095V-1.9475}}$
$\frac{d h}{d \mathrm{time}}=\mathrm{alpha\_h}(1.0-h)-\mathrm{beta\_h}h$
$\mathrm{i\_K1}=0.35e-2(4.0\frac{e^{0.04(V+85.0)}-1.0}{e^{0.08(V+53)}+e^{0.04(V+53.0)}}+0.2\frac{V+23.0}{1.0-e^{-0.04(V+23.0)}})$
$\mathrm{i\_x1}=\mathrm{x1}\times 0.8e-2\frac{e^{0.04(V+77.0)}-1.0}{e^{0.04(V+35.0)}}$
$\mathrm{alpha\_x1}=\begin{cases}151.7994692\frac{e^{0.0654679V-26.1871448}}{1.0+1.5179947e^{0.0654679V-26.1871448}} & \text{if $V> 400.0$}\\ 0.0005\frac{e^{V\times 0.083+4.150}}{1.0+e^{0.057V+2.850}} & \text{otherwise}\end{cases}$
$\mathrm{beta\_x1}=0.0013\frac{e^{V\times -0.06-1.2}}{1.0+e^{-0.04V-0.8}}$
$\frac{d \mathrm{x1}}{d \mathrm{time}}=\mathrm{alpha\_x1}(1.0-\mathrm{x1})-\mathrm{beta\_x1}\mathrm{x1}$
$\mathrm{E\_s}=-82.3-13.0287\ln (\mathrm{Cai}\times 0.001)$
$\mathrm{i\_s}=\mathrm{g\_s}df(V-\mathrm{E\_s})$
$\frac{d \mathrm{Cai}}{d \mathrm{time}}=\begin{cases}0.0 & \text{if $V> 200.0$}\\ \mathrm{i\_s}\times -0.01+0.07(0.0001-\mathrm{Cai}) & \text{otherwise}\end{cases}$
$\mathrm{alpha\_d}=\frac{0.095e^{-\left(\frac{V-5.0}{100.0}\right)}}{1.0+e^{-\left(\frac{V-5.0}{13.89}\right)}}$
$\mathrm{beta\_d}=\frac{0.07e^{-\left(\frac{V+44.0}{59.0}\right)}}{1.0+e^{\frac{V+44.0}{20.0}}}$
$\frac{d d}{d \mathrm{time}}=\frac{\mathrm{alpha\_d}}{\mathrm{R\_d}}(1.0-d)-\mathrm{beta\_d}d$
$\mathrm{alpha\_f}=\frac{0.012e^{-\left(\frac{V+28.0}{125.0}\right)}}{1.0+e^{\frac{V+28.0}{6.67}}}$
$\mathrm{beta\_f}=\frac{0.0065e^{-\left(\frac{V+30.0}{50.0}\right)}}{1.0+e^{-\left(\frac{V+30.0}{5.0}\right)}}$
$\frac{d f}{d \mathrm{time}}=\frac{\mathrm{alpha\_f}}{\mathrm{R\_f}}(1.0-f)-\mathrm{beta\_f}f$
Added more metadata.
The University of Auckland
The Bioengineering Research Group
The kinetics of the f gate.
Autumn
Cuellar
A
Changed some units and added a stimulus current and touched up some
of the equations.
The opening rate for the h gate.
The main component for the model, defining the transmembrane
potential.
The voltage-dependent activation gate for the slow inward current -
the d gate.
The main differential equation for the model, specifing membrane
potential as the summation of all ionic currents and an applied
stimulus current.
2001-12-28
While the governing equations for the time dependent outward
potassium current are unchanged, the gating variables (alpha_x1 and
beta_x1) are modified.
2003-04-01
The voltage-dependent inactivation gate for the fast sodium channel
- the h gate.
1999-07-01
The kinetics of the m gate.
The time rate of change of intracellular calcium concentration.
The opening rate of the d gate.
The opening rate of the f gate.
10396895
The closing rate for the h gate.
The closing rate of the d gate.
The gating variable for the time-dependent outward potassium
current - the x1 gate.
The fast sodium current, primarily responsible for the upstroke of
the action potential.
The kinetics of the x1 gate.
The closing rate of the f gate.
Changed the model name so the model loads in the database easier.
The closing rate of the x1 gate.
Updated metadata to conform to the 16/1/02 CellML Metadata 1.0
Specification.
David
Nickerson
P
Bioengineering Institute, The University of Auckland
c.lloyd@auckland.ac.nz
Catherine
Lloyd
May
2002-01-21
The opening rate for the m gate.
The defibrillation Beeler-Reuter model
keyword
excitation-contraction coupling
ventricular myocyte
electrophysiology
defibrillation
cardiac
IEEE Trans Biomed Eng.
2003-07-28
Anode/cathode make and break phenomena in a model of defibrillation
46
769
777
P
Moore
K
2003-04-05
Autumn
Cuellar
A.
This is the CellML description of the defibrillation Beeler-Reuter model. The original Beeler-Reuter model was modified by Drouhard and Roberge (1987) to improve the fast sodium kinetics. This model was then further modified by Skouibine et al (1999) to handle potentials outside the range of normal physiological activity, allowing the model to be used in defibrillation studies.
The closing rate for the m gate.
Calculation of the slow inward current.
K
Skouibine
B
Calculation of the fast sodium current.
The voltage-dependent activation gate for the fast sodium channel -
the m gate.
Autumn
Cuellar
A
The formula for the time independent outward potassium current of
the defibrillation Beeler-Reuter model is the same as for the
original Beeler-Reuter (1977) model.
The opening rate of the x1 gate.
A minor change is made to the intracellular calcium ion tracking to
limit the movement of calcium ions at large potentials. In addition
to these changes, a scale factor can be added to the time dependent
d and f gates to allow the scaling of the action potential duration.
This enables a better representation of an action potential from
ischemic tissue.
The kinetics of the d gate.
Calculation of the reversal potential for the slow inward current.
N
Trayanova
A
The kinetics of the h gate.
The voltage-dependent inactivation gate for the slow inward current
- the f gate.
Catherine Lloyd