Early afterdepolarisations and ventricular arrhythmias in cardiac tissue: a computational study

Abstract

Afterdepolarisations are associated with arrhythmias in the heart, but are difficult to study experimentally. In this study we used a simplified computational model of 1D and 2D cardiac ventricular tissue, where we could control the size of the region generating afterdepolarisations, as well as the properties of the afterdepolarisation waveform. Provided the size of the afterdepolarisation region was greater than around 1 mm, propagating extrasystoles were produced in both 1D and 2D. The number of extrasystoles produced depended on the amplitude, period, and duration of the oscillatory EAD waveform. In 2D, re-entry was also initiated for specific combinations of EAD amplitude, period, and duration, with the afterdepolarisation region acting as a common pathway. The main finding from this modelling study is therefore that afterdepolarisations can act as potent sources of propagating extrasystoles, as well as a source of re-entrant activation.

Appendix

The equations for the 4VSIM model are based on four state variables; u (equivalent to membrane voltage in an ionic model), v, w, and s, which are given by the following differential equations. In each case H denotes the Heaviside step function, so H(x) has the value 1 if x > 0, and 0 otherwise;

$$ \frac{{{\text{d}}u}}{{{\text{d}}t}} = - (J_{\text{fi}} + J_{\text{si}} + J_{\text{so}} ) $$

$$ \frac{{{\text{d}}v}}{{{\text{d}}t}} = (1 - H(u - u_{c} ))\frac{(1 - v)}{{\tau_{v}^{ - } }} - H(u - u_{c} )\frac{v}{{\tau_{v}^{ + } }} $$

$$ \frac{{{\text{d}}w}}{{{\text{d}}t}} = (1 - H(u - u_{c} ))\frac{(1 - w)}{{\tau_{w}^{ - } }} - H(u - u_{c} )\frac{w}{{\tau_{w}^{ + } }} $$

$$ \frac{{{\text{d}}s}}{{{\text{d}}t}} = \left( {0.5\left( {1 + \tanh ((u - u_{c,{\rm{si}}} )/x_{k} )} \right) - s} \right)r_{s} $$

The following equations are used to calculate the parameters that depend on u, v, w and s.

$$ \tau_{v}^{ - } = \tau_{v2}^{ - } H(u - u_{v} ) + \tau_{v1}^{ - } (1 - H(u - u_{v} )) $$

$$ r_{s} = r_{s}^{ + } H(u - u_{c} ) + r_{s}^{ - } (1 - H(u - u_{c} )) $$

$$ J_{\text{fi}} = \frac{{ - vH(u - u_{c} )(u - u_{c} )(1 - u)}}{{\tau_{d} }} $$

$$ J_{\text{si}}=\frac{ - ws}{{\tau_{\text{si}} }} $$

$$ J_{\text{so}} = \frac{{u(1 - H(u - u_{c} ))}}{{\tau_{o} }} + H(u - u_{c} )\tau_{\text{so}} $$

$$ \tau_{\text{so}} = \tau_{{\text{so}}1} + \frac{{\tau_{{\text{so}}2} }}{2}(1 + \tanh ((u - u_{\text{so}} )/x_{{\text{so}}k} )) $$

The fixed parameters are given in the following table:

Parameter	Value
u c	0.23
τ + v	3.33
u v	0.055
τ − v1	19.6
τ − v2	1250.0
τ + w	100.0
τ − w	150.0
u c,si	0.65
x k	0.23
r + s	0.02
r − s	0.08
τ d	0.118
τ si	5.0922
τ o	15.0
τ so1	0.009
τ so2	0.078
x sok	0.25
u so	0.8