- Open Access
Dispersion of cardiac action potential duration and the initiation of re-entry: A computational study
© Clayton and Holden; licensee BioMed Central Ltd. 2005
Received: 11 November 2004
Accepted: 18 February 2005
Published: 18 February 2005
The initiation of re-entrant cardiac arrhythmias is associated with increased dispersion of repolarisation, but the details are difficult to investigate either experimentally or clinically. We used a computational model of cardiac tissue to study systematically the association between action potential duration (APD) dispersion and susceptibility to re-entry.
We simulated a 60 × 60 mm 2 D sheet of cardiac ventricular tissue using the Luo-Rudy phase 1 model, with maximal conductance of the K+ channel gKmax set to 0.004 mS mm-2. Within the central 40 × 40 mm region we introduced square regions with prolonged APD by reducing gKmax to between 0.001 and 0.003 mS mm-2. We varied (i) the spatial scale of these regions, (ii) the magnitude of gKmax in these regions, and (iii) cell-to-cell coupling.
Changing spatial scale from 5 to 20 mm increased APD dispersion from 49 to 102 ms, and the susceptible window from 31 to 86 ms. Decreasing gKmax in regions with prolonged APD from 0.003 to 0.001 mS mm-2 increased APD dispersion from 22 to 70 ms, and the susceptible window from <1 to 56 ms. Decreasing cell-to-cell coupling by changing the diffusion coefficient from 0.2 to 0.05 mm2 ms-1 increased APD dispersion from 57 to 88 ms, and increased the susceptible window from 41 to 74 ms.
We found a close association between increased APD dispersion and susceptibility to re-entrant arrhythmias, when APD dispersion is increased by larger spatial scale of heterogeneity, greater electrophysiological heterogeneity, and weaker cell-to-cell coupling.
Cardiac disease remains an important cause of sudden death in the industrialised world, and in many cases the lethal events are the cardiac arrhythmias called ventricular tachycardia (VT) and ventricular fibrillation (VF). Spontaneous episodes of VT and VF occur in patients where cardiac disease or congenital abnormality has remodelled either the structure or function of cardiac cells and tissue. There is abundant experimental evidence to support the idea that VT and VF are sustained by re-entry [1, 2], but the initiation of re-entry in a particular individual is not well understood, and so is difficult to either predict or prevent.
Slow conduction and unidirectional block have long been known to facilitate re-entry , and experimental studies have established a link between regional differences in repolarisation, and an increased vulnerability to re-entrant arrhythmias following one or more premature stimuli [4–7]. One of the earliest computer models of activation in cardiac tissue was used to demonstrate that regional differences in repolarisation can allow fibrillation to develop following a premature stimulus . Further experimental studies have found that steep gradients in repolarisation correlate with arcs of conduction block around which re-entry circulates [9–12], and have suggested that regions with longer refractory period must be of a critical size for sustained re-entry to occur . Computational and theoretical studies [14–16] have also shown how a region with prolonged repolarisation can block a premature excitation resulting in initiation of re-entry, and that the size of the inhomogeneity determines the characteristics and persistence of re-entry.
Regional differences in repolarisation are often described as action potential duration (APD) dispersion. The difference between the longest and shortest observed APD is a conceptually simple and easily obtained quantity and has widely been used to measure APD dispersion, although other indices have been proposed . Some experimental studies have established critical values of APD dispersion above which re-entry is initiated consistently . In others the gradient of APD has been measured, and spatial gradients of between 2 and 12.5 ms mm-1 were associated with block and re-entry [9, 19, 20]. Spatial APD gradients arise from regional differences in ion channel function, but their magnitude depends on electrotonic current flow during repolarisation. APD dispersion can be produced by the spatial scale of regional differences and the magnitude of functional heterogeneity, and is modulated by electrotonic current flow which depends on the strength of cell-to-cell coupling [16, 21, 22]. The relative effect of these three quantities on APD dispersion and vulnerability to re-entry is important because both disease and congenital abnormalities can result in changes to one or more of them. However, it is difficult to control these tissue properties independently in experiments.
Computational models offer a powerful research tool for addressing these questions, because the properties of a virtual tissue can be controlled precisely and independently in a way that would be extremely difficult to achieve experimentally. The purpose of this study was therefore to investigate systematically how measured APD dispersion and vulnerability to re-entry in a computational model of ventricular tissue are related to: (i) the spatial scale of heterogeneity, (ii) the magnitude of differences in K+ channel conductance between regions with short and long APD, and (iii) strength of cell-to-cell coupling.
2.1 Computational model of electrical activation
We simulated electrical activation in a 2 D isotropic monodomain virtual tissue 
where V m is membrane voltage, C m specific membrane capacitance, D a diffusion coefficient and I ion current flow through the cell membrane per unit area. We used the Luo-Rudy phase 1 (LR1) model  to give I ion ,
where I Na , I Ca (described as I si in the original model) and I K are time and voltage dependent currents flowing through Na+, Ca2+, and K+ channels, IK 1a time-independent K+ current, I Kp a plateau K+ current, and I b a background current. We changed two parameters from the original Luo and Rudy paper . We reduced maximum Na+ conductance from 0.23 mS mm-2 to 0.16 mS mm-2 as in the later version of the model , and we reduced the maximum conductance of the slow inward current from 0.0009 mS mm-2 to 0.0005 mS mm-2 to produce an APD comparable to that in the canine ventricle. We controlled repolarisation by varying maximum K+ conductance (gKmax) from the default value of 0.00282 mS mm-2 to a value between 0.001 mS mm-2 and 0.004 mS mm-2 (see below).
2.2 Numerical methods
We solved equation 1 and the LR1 equations using an explicit Euler method, with both a lookup table of the voltage dependent parameters in the LR1 model, and an adaptive operator splitting technique . We applied no-flux boundary conditions, set C m to 0.001 μF mm-2, and set D to between 0.05 and 0.2 mm2 ms-1. We used an adaptive timestep of either 0.02 or 0.1 ms depending on the magnitude of dV m /dt at each grid point . With a space step of 0.2 mm, and D of 0.1 mm2 ms-1 we obtained a conduction velocity (CV) for a stable plane wave of 0.56 m s-1, with a speedup of about two times and an error in CV of 2.5 % compared to computations with a fixed timestep of 0.01 ms. Simulations with smaller fixed and adaptive timesteps yielded plane waves with a comparable CV. Changing the space step to 0.25 mm and 0.15 mm resulted in a change of CV for a plane wave of <5 % compared to the CV computed with a space step of 0.2 mm. These findings indicated the stability of our numerical method.
2.3 Virtual tissue and heterogeneity
In our reference virtual tissue, the 40 × 40 mm heterogeneous region was divided into 16 squares giving heterogeneity with a spatial scale of 10 mm. In half of the 16 squares gKmax was set to 0.001 mS mm-2, giving a functional heterogeneity with a difference in gKmax (Δgkmax) of 0.003 mS mm-2. The diffusion coefficient in the whole virtual tissue was set to 0.1 mm2 ms-1.
We varied the spatial scale of heterogeneity by changing the size of heterogeneity in the reference virtual tissue from 10 mm to 20 mm and 5 mm (Figure 2a), varied Δgkmax from 0.003 to 0.002 and 0.001 mS mm-2 by changing gKmax in the alternate squares from 0.001 to 0.002 and 0.003 mS mm-2 respectively (Figure 2b), and varied the strength of cell-to-cell coupling by changing the diffusion coefficient from 0.1 to 0.05 and 0.2 mm2 ms-1 (Figure 2c).
2.4 APD dispersion
Action potentials were initiated in each virtual tissue by holding the membrane voltage along one edge at 0 mV for 2 ms. We measured APD to 90% recovery (APD90) at every grid point. We estimated APD dispersion during steady pacing with three measured that have been used in experimental studies . First we measured APD across the whole virtual tissue, and determined the difference between the maximum and minimum APD (APDdiff). Second we measured the standard deviation of APD (APDSD) across the whole virtual tissue. Finally we measured the APD difference between each grid point and its neighbours 1 mm above, below, left and right, and determined the maximum value of measurements throughout the whole virtual tissue (maxLD) .
2.5 Vulnerability to re-entry
A spatially homogenous control virtual tissue with uniform gKmax supported propagating plane waves following S1 stimulation along one edge. These propagating plane waves had a depolarising wavefront and a repolarising wave back both aligned parallel to the edge that was stimulated. A premature S2 stimulus delivered to the same edge as the S1 stimulus therefore resulted either in block or in a propagating plane wave.
In the heterogeneous virtual tissues the wave back was not a plane wave because some regions repolarised more quickly than others. A premature S2 stimulus could produce a wavefront that would encounter a mixture of recovered and refractory tissue, and hence elicit wavebreak and re-entry. We therefore assessed vulnerability to re-entry in the heterogeneous virtual tissues by delivering two S1 stimuli to one edge at 500 ms intervals, and then a premature S2 stimulus to the same edge. We varied the timing of the S2 stimulus in steps of 1 ms. The virtual tissue response was characterised as either block if the S2 stimulus failed to propagate, re-entry if the S2 stimulus elicited re-entry that completed more than one cycle, wavebreak if the S2 stimulus elicited a wave that broke but did not re-enter, or propagation if the S2 stimulus elicited a wave that propagated without wavebreak.
Vulnerability to re-entry is typically estimated by applying a local premature stimulus that interacts with repolarising tissue, and the vulnerable window is the range of stimulus strength and timing that elicits re-entry. In this study, we investigated the initiation of re-entry by S1 S2 stimulation from the same stimulus site, and we estimated the vulnerability of each virtual tissue to re-entry from the range of S2 intervals that resulted in either wavebreak or re-entry. To avoid confusion we refer to our estimate of vulnerability as susceptibility to re-entry, and to the range of S2 intervals as the width of the susceptible window.
2.6 Potential antiarrhythmic strategies
We made a preliminary investigation into two candidate mechanisms for reducing susceptibility, inactivation of the Na+ channel and conductance of the time independent K+ channel gK 1.
Block of an action potential occurs if there is insufficient Na+ current to support a propagating wavefront ; when Na+ channels have not recovered from inactivation, then a propagating wave blocks and dissipates . Na+ channel inactivation is controlled by the j-gate in the LR1 model . We prolonged recovery of Na+ channels from inactivation throughout the virtual tissue by multiplying the time constant of Na+ channel inactivation τj by 10 .
The time independent K+ current iK 1is a voltage dependent current that holds the membrane at its resting potential. It is activated during repolarisation and at rest, and is also activated close to the core of re-entrant waves [30, 31]. We investigated the effect of doubling the conductance of iK 1throughout the virtual tissue.
3.1 Propagation and APD dispersion during pacing
When re-entry was initiated, we observed break-up into multiple re-entrant wavelets with up to 18 phase singularities. The mechanism of instability was likely to be a combination of the spatial heterogeneity leading to localised conduction block combined with dynamical instability resulting from steep APD restitution  (Figure 1), but was not investigated explicitly. In some simulations the re-entrant waves coalesced and re-entry spontaneously terminated. However, there was no clear association between this observation and S2 timing, spatial scale, functional heterogeneity, or coupling.
3.2 Susceptibility to re-entry
3.3 Potential antiarrhythmic strategies
Although prolonging τj by a factor of 10 had only a small (< 2 ms) effect on maximum and minimum APD, the width of the susceptible window was decreased from 56 ms to 39 ms. The lower bound of the susceptible window moved from 198 to 216 ms, reflecting an increase in the refractory period of the virtual tissue as well as more prominent conduction velocity restitution.
Doubling gK 1increased current flow across the membrane during repolarisation, shortening APD by about 10% and decreasing APDdiff from 70 ms to 49 ms. The width of the susceptible window was also reduced from 56 ms to 42 ms when gK 1was doubled. The lower bound of the susceptible window moved from 198 ms to 175 ms, as a result of the shorter APD.
In this study we have used a computational model of cardiac tissue to dissect out the effects of spatial scale, Δgkmax, and strength of cell-to-cell coupling on APD dispersion and susceptibility to re-entry. Wavebreaks and re-entry can be created in cardiac tissue when an activation wavefront encounters a gradient of recovery. Experimental studies have therefore found an association between increased APD dispersion and greater susceptibility to re-entry because a premature stimulus is more likely to be blocked in tissue with regions of prolonged APD. In this study we found that large spatial scale heterogeneity, large Δgkmax, and reduced strength of cell-to-cell coupling all increased both APD dispersion and susceptibility to re-entry. Tissue heterogeneities produce APD dispersion, and APD dispersion is modulated by electrotonic current flow . In this study, spatial scale and Δgkmax affected APD dispersion directly, whereas changing the strength of cell-to-cell coupling affected electrotonic current flow. This study indicates that each of these factors could be an important component in arrhythmogenesis, and that the susceptibility of a heterogenous tissue to re-entry can be estimated from simple measures of APD dispersion.
4.1 Relation to other work
Although the spatial scale of heterogeneity has been identified as potentially important in other studies , there is little information in the experimental literature to indicate how spatial scale affects susceptibility to re-entry. In one experimental study a ~1 cm2 region of thin layer of rabbit ventricular epicardium was cooled to produce a small region with prolonged APD, producing a dispersion of refractory periods ranging from 27 and 45 ms . Re-entry could be initiated in this preparation using 4 increasingly premature stimuli. The spatial scale of heterogeneity in this experimental study was comparable to the reference virtual tissue used in our present study, but the effects of Δgkmax and strength of cell-to-cell coupling in the experimental study are difficult to establish. Nevertheless the initiation of re-entry by block and retrograde activation in the region with prolonged APD followed a broadly similar pattern to our simulations, although the activation pathways in the experimental study were more complex, presumably due to anisotropic conduction in the rabbit ventricle. A decrease in strength of cell-to-cell coupling results in slowed conduction, and this is a common finding in tissue damaged by ischaemia, infarction , and other pathology . Several studies have shown that decreasing cell-to-cell coupling can expose ionic heterogeneities [22, 35–37].
Recent computational studies have addressed the influence of heterogeneous acetylcholine distribution on the vulnerability to and stability of re-entry in the atria [15, 32]. The findings of these studies are broadly similar to the present study, although the effects of cell-to-cell coupling were not explicitly addressed, and initiation of re-entry was by either crossfield stimulation  or by S1 and S2 stimulation at different sites . Both of these protocols would be expected to induce re-entry even in uniform tissue. In the present study both S1 and S2 stimuli were delivered from the same location, which does not initiate re-entry in uniform tissue, allowing us to examine the effect of heterogeneity on the initiation of re-entry in isolation.
The dynamical behaviour of APD is recognised as important not only for the stability of re-entry  but also in the development of alternans . Recent experimental  and computational [38, 40] studies have shown that APD dispersion can arise dynamically leading to discordant alternans, wavebreak, and re-entry in tissue that is either homogeneous or in which the ionic properties vary smoothly[38, 41, 42]. In the present study we measured APD dispersion at a fixed cycle length of 500 ms. The APD restitution curves given in Figure 1 indicate that APD dispersion could have been affected by pacing at shorter cycle lengths. This observation raises the possibility that heterogenous APD restitution could act to amplify APD dispersion.
The effect of heterogeneity on the stability of spiral waves has been investigated by Xie et al . This study found that the amount of heterogeneity required to destabilise re-entry decreased as the degree of dynamical instability resulting from a steep APD restitution curve increased. In the present study we were interested in the initiation of re-entry rather than the stability of re-entry once initiated.
4.2 APD dispersion and susceptibility to arrhythmias
Normal ventricular tissue is remarkably resistant to the initiation of re-entry, but this robustness is greatly reduced by actions that increase the spatial dispersion of refractoriness. In this computational study we have shown that regional differences in repolarisation have an interlinked effect not only on the initiation of re-entry but also on measures of APD dispersion. Measures of APD dispersion are valuable in clinical practice because they could provide an estimate of arrhythmia risk, and various indices have been developed in experimental studies . In our present study we have found that relatively simple measures of APD dispersion obtained from the tissue were related to the width of the susceptible period for re-entry.
4.3 Potential antiarrhythmic strategies
These preliminary investigations suggest that, in our model, susceptibility to re-entry could be reduced if recovery of Na+ channels from inactivation can be prolonged, or if the conductance of the iK 1channel can be increased. The effect of this kind of intervention in the intact heart may however be more complex. Other computational studies have shown that modifying the kinetics of the Na+ channel can have a pro-arrhythmic effect. Delaying recovery of Na+ channels from inactivation can increase the slope of the APD restitution curve and hence the likelihood of alternans and re-entry , and reducing Na+ channel conductance increases the vulnerable window . Differences in the spatio-temporal complexity of VF between left and right ventricles have been attributed to differences in the current density of the iK 1channel in experimental studies . Although this experimental finding is not directly connected to the effects of the iK 1channel conductance on susceptibility to re-entry investigated in the present study, it does highlight the potential importance of this channel for the mechanisms of re-entry.
The influence of individual ion channel currents on the initiation and subsequent behaviour of re-entry is an important direction for future research, but will require more biophysically detailed cell models than the LR1 model used in this study.
4.4 Limitations of the study
The electrical behaviour of cardiac tissue is complex, and depends on processes that act at tissue, cell, sub-cellular, and molecular levels. Computational models of electrical activation and conduction in the heart aim to simulate processes that are relevant to the research question, and simplifications are made accordingly. This study involved a large number of computations to establish susceptibility to re-entry, and so we chose to use a model that was a compromise between fidelity to real cardiac tissue and computational requirements.
More detailed versions of the LR model and others incorporating a fuller description of ion channels, pumps, exchangers, as well as Ca2+ storage and release have been developed [22, 25, 45, 46]. In tissue with regional ischaemia, Ca2+ handling may become heterogeneous in addition to APD, and so it is possible that susceptibility to re-entry could also be modified if this additional feature is taken into account.
In the present study we chose to use an idealised geometrical heterogeneity based on square regions because this approach allowed us to assess the initiation of re-entry under well controlled conditions. In real cardiac tissue we would expect the heterogeneities to be much more irregular in shape and gradient, and the conditions that favour re-entry to be dependent on the relative location of the heterogenous region and the stimulus site.
The behaviour of re-entry in 3 D tissue is more complex than in 2 D, especially when the effects of rotational anisotropy and transmural differences in action potential shape and duration are taken into account . Studies relating APD dispersion and susceptibility to re-entry in anatomically detailed 3 D tissue are another important project for the future. Recent computational studies indicate that the mechanical properties can not only modify the behaviour of re-entrant waves , but also that stretch activated channels in the cell membrane can contribute to susceptibility to re-entry if the tissue is stretched during repolarisation [49, 50]. Since electrical repolarisation occurs at the same time as force generation in cardiac cells, the effect of cardiac mechanics on susceptibility to re-entry remains an important research question.
This work was supported by the British Heart Foundation through the award of Basic Science Lectureship BS98001, and project grant PG/03/102/15852 to RHC. We are also grateful to the United Kingdom Medical Research Council and Engineering and Physical Sciences Research Council for additional financial support.
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