Dependence of phase-2 reentry and repolarization dispersion on epicardial and transmural ionic heterogeneity: a simulation study

Abstract

Aims

Phase-2 reentry (P2R) is a local arrhythmogenic phenomenon where electrotonic current propagates from a spike-and-dome action potential region to re-excite a loss-of-dome action potential region. While ionic heterogeneity has been shown to underlie P2R within the epicardium and has been hypothesized to occur transmurally, we are unaware of any study that has investigated the effects of combining these heterogeneities as they occur in the heart. Thus, we tested the hypothesis that P2R can result by either epicardial or transmural heterogeneity and that the realistic combination of the two would increase the likelihood of P2R.

Methods and results

We used computational ionic models of cardiac myocyte dynamics to investigate initiation and development of P2R in simulated tissues with different ionic heterogeneities. In one-dimensional transmural cable simulations, P2R occurred when the conductance of the transient outward current in the epicardial region was near the range for which epicardial action potentials switched intermittently between spike-and-dome and loss-of-dome morphologies. Phase-2 reentry was more likely in two-dimensional tissue simulations by both epicardial and transmural heterogeneity and could expand beyond its local initiation site to create a macroscopic reentry.

Conclusion

The characteristics and stability of action potential morphology in the epicardium are important determinants of the occurrence of both transmural and epicardial P2R and its associated arrhythmogenesis.

What's new?

• Transmural heterogeneity can act as a trigger of phase-2 reentry (P2R) when the transient outward current in the epicardium is in a critical range, in which the epicardial action potential switches intermittently between a spike-and-dome morphology and a loss-of-dome morphology.

• However, transmural heterogeneity poses a smaller risk than epicardial heterogeneity in triggering P2R in simulations.

• Phase-2 reentry is facilitated in two-dimensional tissue simulations with combined transmural and epicardial heterogeneity. In these cases, P2R is initiated in the epicardium, due to electrotonic current contributions from both the transmural and the epicardial direction.

• In two-dimensional tissue simulations, P2R may trigger reentrant spiral waves that could serve as initiation of ventricular fibrillation in the intact heart.

Introduction

Ventricular action potentials (APs) exhibit considerable morphological variation. In many species, including humans and dogs, cardiomyocytes isolated from the epicardium exhibit spike-and-dome APs with a characteristic notch due to rapid partial repolarization being followed by secondary depolarization. In contrast, endocardial myocytes have no such notch.^1,2 Underlying this distinction is a difference in the levels of the transient outward current (I_to), with endocardial cells expressing little or no I_to and epicardial myocytes having a prominent I_to.^3,4 Drug-induced augmentation or artificial injection of I_to in myocytes, where it is intrinsically low or absent, can indeed cause a salient notch.^4–6 Furthermore, excessive I_to injection/augmentation in such cells causes a second phenotype transition to a loss-of-dome AP characterized by rapid repolarization and short duration.^4–6

The intrinsic heterogeneity in AP morphology may be amplified in disease states such as Brugada syndrome and myocardial ischaemia.^7,8 Brugada syndrome is associated mainly with loss-of-function mutations in the SCN5A gene that encodes the α-subunit of the cardiac sodium channel, causing decreased sodium current (I_Na).^7,9,10 Because I_Na and I_to counteract each other, a reduced I_Na can make cells more susceptible to the loss-of-dome transition.^5,11 During ischaemia, depletion of ATP activates the outward I_K(ATP) current. Thus, in Brugada syndrome and ischaemia, regional excess of outward current relative to inward current may result in a phenotype transition to a loss-of-dome AP. Myocytes with larger endogenous I_to, such as epicardial cells, are more vulnerable to such perturbation and more likely to undergo the bifurcation. Thus, one of the ways in which arrhythmogenesis may occur in Brugada syndrome and ischaemia is through the formation of a heterogeneous substrate, in which some region(s) exhibits APs of much shorter duration than others. Such a substrate sets the stage for formation of phase-2 reentry (P2R), in which conduction of electrotonic current from phase 2 of spike-and-dome AP sites to already recovered loss-of-dome sites causes abnormal re-excitation in the latter.¹² If sufficiently large and ill-timed, this abnormal re-excitation may trigger ventricular fibrillation.

In previous work, we investigated P2R mechanisms in simulated epicardial tissue of the right ventricle. The epicardium of the right ventricle has a relatively large and heterogeneous expression of I_to channels and has been associated with P2R formation in both canine models and humans.^13–17 Yet, transmural heterogeneity, with endocardial, mid-myocardial, and epicardial regions, has also been hypothesized to facilitate P2R.^4,18–20 Therefore, this present study focuses on P2R development in simulated transmural tissue. We found that the intrinsic heterogeneity within the ventricular wall, in combination with dome/loss-of-dome switching behaviour in the epicardium, could produce P2R in the transmural direction. Using two-dimensional (2D) tissue simulations, we also found that P2R could degenerate into macroscopic reentry, creating spiral waves and prolonging the time of tissue excitation.

Methods

Single-cell computational models and methods

As in our previous work,¹³ we used the Luo–Rudy dynamic model,²¹ modified with an updated L-type calcium current formalism²² and incorporation of I_to.²³ To simulate endo-, mid-, and epicardial cells, we modified the conductances of I_to (G_to) and I_Ks (G_Ks). We set the nominal epicardial G_to value to 1.1 mS/µF,¹³ and investigated a range of conductances around this baseline as the exact epicardial G_to gradient remains unknown. Mid- and endocardial G_to values were set to 0.935 and 0.0 mS/µF, respectively, based on experimental^3,23,24 and computational work.²⁵ To account for the prolonged AP duration (APD) and decreased I_Ks in mid-myocardial cells,²⁶ we scaled G_Ks in mid-myocardial cells by a factor of 0.24. This adjustment produced a mid-myocardial APD 37% longer than the epicardial APD, consistent with experimental data.^2,26–28

We used the forward Euler method with a time step of Δt= 0.005 ms to integrate transmembrane potential and ionic concentrations, while the gating variables were computed from their analytical expressions. Square-wave stimuli of 200 mA and 0.5 ms duration were applied to excite the cells at a pacing cycle length of 1000 ms. To minimize transients, we ran single-cell simulations for each cell type for 500 excitations, saved all system variables, and used those as initial conditions for the corresponding 1D cable or 2D sheet simulations.¹³

One-dimensional cable simulations

To simulate a transmural cable, we used the setup by Gima and Rudy,²⁵ composed of 165 connected computational cells representing a 1.65 cm long fibre. The cable consisted of endocardial (cells 1–60), mid-myocardial (cells 61–105), and epicardial (cells 106–165) cells, where each cell type was modelled incorporating the heterogeneities of ion-channel densities described above. Gap junction connectivity was set to 1.73 µS for most cells, except for a five-fold decrease at the mid-to-epicardium transition region (cells 104–107) mimicking the experimentally described local increase in resistivity.²⁵ Stimuli were applied to the five end cells, either endocardially to simulate normal propagation or epicardially to simulate ectopic activity. Each cable simulation ran for 30 beats, with a pacing cycle length of 1000 ms. Although simulations were started from initial conditions corresponding to their particular parameter values, there may still be transient behaviours owning to the difference in dynamics between tissue-coupled and single cells. Hence, we distinguished between transient and stable P2R, with transient P2R defined as P2R occurring in the first beats only, whereas stable P2R was defined as later occurrences of P2R.

Two-dimensional sheet simulations

Two-dimensional sheets were composed of 300 × 165 grid points representing a tissue slice 3 cm long and 1.65 cm wide in the transmural direction. Transmurally, sheets were heterogeneous in the same manner as the 1D transmural cables. To mimic right ventricular ionic heterogeneity, a range of continuous G_to gradients were simulated along the epicardium.

Coupling in the transmural direction was the same as for the 1D cables, while the transverse direction (along the different layers) was coupled with a diffusion coefficient of D=0.0007 cm²/ms,¹³ yielding a conduction velocity of ∼50 cm/s. Stimuli were applied to the first five rows of the endocardium. Simulation duration was 10 s with a pacing cycle length of 1000 ms.

Results

Switching behaviour in transmural cable simulations

Epicardial levels of G_to may vary considerably throughout the right ventricle.^8,29,30 Therefore, to determine a possible role for transmural heterogeneity in P2R, we first investigated P2R occurrence in 1D transmural cable simulations when scanning through a range of plausible epicardial G_to (G_to,epi) values. Excitation was induced at the endocardial end and spread towards the mid- and epicardium. While endocardial and mid-myocardial APs always exhibited no and moderate notches, respectively, due to their smaller G_to values, the AP morphology in the epicardium varied (Figure 1). As expected, the baseline G_to,epi (1.1 mS/µF) produced the typical spike-and-dome morphology in the epicardium (Figure 1A), while substantially increased G_to,epi generated abbreviated APs with loss-of-dome morphology (Figure 1C), because the outward current (I_to) overwhelmed the inward currents. However, for intermediate G_to,epi values (between 1.3 and 1.6 mS/µF), the epicardium displayed both the loss-of-dome and spike-and-dome morphologies, with APs switching intermittently between the two (Figure 1B shows an example with G_to=1.5 mS/µF). These results correlate with the switching behaviour defined for single cells in our previous work (range 1.25–1.65 mS/µF¹³). The ratio of the different morphologies in a given simulation varied depending on G_to,epi value, with more spike-and-dome APs for G_to,epi closer to 1.3 mS/µF and more loss-of-dome APs for G_to,epi closer to 1.6 mS/µF.

Figure 1.

Transmural cable simulations with endocardial stimulation. Cables are composed of endocardium (0≤x≤6 mm), mid-myocardium (6<x≤10.5 mm), and epicardium (10.5<x≤16.5 mm). The first column displays a few beats for each simulation, which lasted 30 s. The second column displays a magnified view of one of the beats from the first column. Action potential duration (APD) along the cable is presented in the third column; dashed lines represent the borders between the different layers. (A) With G_to,epi=1.1 mS/µF there is little transmural repolarization dispersion. No loss-of-dome APs or P2R instances occurred. (B) G_to,epi=1.5 mS/µF caused dynamical changes in epicardial AP morphology. An example of P2R occurrence is seen on the 10th beat (stimulated at 10.0 s) of the simulation, following an epicardial loss-of-dome AP. The P2R-induced AP has a spike-and-dome morphology [red points in B(3) give its duration]. (C) G_to,epi=2.0 mS/µF, epicardial APs are always loss-of-dome; no P2R occurred.

Phase-2 reentry occurrence in transmural cable simulations

Scanning a range of G_to,epi values revealed a reasonable correlation between P2R occurrence and switching dynamics (Table 1). For G_to,epi<1.3 mS/µF, P2R did not occur because the epicardium always manifested the spike-and-dome morphology. Therefore, although the epicardium and the mid-myocardium produced slightly different APs (deeper notch in the epicardium, longer APD in mid-myocardium), the two neighbouring regions always exhibited comparable AP variants (Figure 1A).

Table 1.

Simulation results for transmural cables for which epicardial G_to was varied systematically between 0.0 and 2.9 mS/µF; G_to values <1.4 never resulted in P2R (not shown)

Gto,epi	Endocardial stimulation (%P2R)	Epicardial stimulation (%P2R)
1.4	0%	0%
1.5	10%	3%
1.6	7%	7%
1.7	7%	10%
1.8	3%	7%
1.9	3%	(7%)
2.0	(7%)	(7%)
2.1	(7%)	(10%)
2.2	(3%)	(7%)
2.3	(3%)	(7%)
2.4	(3%)	(7%)
2.5	(3%)	(3%)
2.6	(3%)	(3%)
2.7	(3%)	(3%)
2.8	0%	(3%)
2.9	0%	3%

Stimulation was applied at either the endocardial or the epicardial end of the cable as indicated. Values give the percentage of beats in a given simulation for which P2R occurred. Values in parentheses indicate transient P2R, i.e. P2R like owning to initial conditions being slightly off. Stable P2R was facilitated by G_to,epi values within the 1.5–1.9 mS/µF range.

For cables with G_to,epi>1.9 mS/µF, P2R did not occur or occurred only transiently (Table 1) even though there was a steep dispersion of repolarization between the epi- and the mid-myocardium as the epicardium displayed only the loss-of-dome morphology [Figure 1C (1) shows a representative simulation]. The APs in the mid-myocardium–epicardium transition zone changed gradually between the mid-myocardial spike-and-dome APs and the epicardial loss-of-dome APs [Figure 1C (2 and 3)].

Stable P2R was triggered for intermediate G_to,epi values (Table 1). In these simulations the epicardium manifested the spike-and-dome morphology in some beats and the loss-of-dome morphology in others [see example in Figure 1B (1)]. For some of those beats, P2R occurred, with the inner epicardium (where it borders the mid-myocardium) manifesting a prolonged spike-and-dome morphology [compare Figure 1B (3) with Figure 1C (3)] and the outer epicardium having the loss-of-dome morphology. The P2R-induced additional excitation had a shortened spike-and-dome morphology and intermediate duration [red curve in Figure 1B (3)]. Phase-2 reentry occurred in 3 out of the 30 beats in a similar manner during this simulation.

Epicardial excitation of transmural cables

We hypothesized that epicardial stimulation mimicking ectopic activation would increase the likelihood of P2R initiation as it would delay repolarization of the mid-myocardium relative to the epicardium. However, changing the stimulus site to the epicardium caused only very slight changes in P2R occurrence (Table 1). One noticeable change was the occurrence of P2R in a single beat in a simulation with a very high value of G_to,epi. This value is far from the switching behaviour range, and indeed the P2R-induced reactivation of the epicardium was of the loss-of-dome morphology (not shown), whereas for simulations with G_to,epi values inside the switching behaviour range the abnormal excitation was always of the spike-and-dome morphology.

Phase-2 reentry occurrence with epicardial and transmural heterogeneity

To investigate P2R formation in a more realistic setting, we simulated activation in 2D sheets with both transmural and epicardial ionic heterogeneity. Transmurally, the sheet was composed as the cables, while different types of G_to gradients were applied along the epicardium. An example is illustrated in Figure 2A (linear epicardial G_to gradient from 2 to 1.5 mS/µF), with one AP wave through the tissue (Figure 2B). The three transmural regions have clearly different repolarization dynamics, with early repolarization in the epicardium (due to loss-of-dome APs; see Supplementary material online,Figure S1), later repolarization in the endocardium, and final repolarization in the mid-myocardium. The epicardial G_to heterogeneity caused the left side of the epicardium to repolarize sooner than the right side. This symmetry break had the secondary effect of delaying mid-myocardial repolarization in the right compared with the left side. Complete repolarization occurred 162 ms after stimulus application.

Figure 2.

(A) Schematic representation of the 2D tissue. Linear epicardial G_to gradient between 1.5 and 2.0 mS/µF is shown. (B) Membrane potential snapshots from the fifth beat of the simulation. Horizontal dotted lines delineate transmural layers: ENDO (endocardium), MID (mid-myocardium), and EPI (epicardium). A stimulus was applied along the endocardial edge at t=5.0 s. The activation wavefront reached the epicardial tissue edge at t=5.028 s. At t=5.046 s, the left side of the epicardium which has the largest G_to value, had nearly repolarized. The mid-myocardium repolarized last. Although the mid-myocardial spike-and-dome AP morphology bordered an epicardial loss-of-dome morphology (t=5.104 s), there were no P2R occurrences in this beat.

As was the case in the transmural cable simulations, the dynamics in the 2D sheet could change from beat to beat. Indeed, following the next stimulus in this simulation, P2R did occur (Figure 3). On this beat, rather than losing the AP dome, the epicardial tissue with the lower G_to values developed a deep notch and a dome (Figure 3; see Supplementary material online, Figure S2D). Thus, dome vs. loss-of-dome dispersion arose in both the transmural and the epicardial directions. Phase-2 reentry initiated at the junction of the borders separating the two morphologies (at t=6.096 s in Figure 3), and triggered an additional wave of excitation propagating along the recovered epicardium (t=6.136 s) until reaching the left edge of the tissue and extinguishing (t=6.156 s). The APs triggered by this epicardial wave also had divergent morphologies, with spike-and-dome in the middle of the tissue, and loss-of-dome to the left (where G_to is larger; Figure 3 and see Supplementary material online, Figure S2). This repolarization variance initiated a second P2R occurrence, triggering a pair of figure-of-eight reentrant waves into the recovered epi- and mid-myocardium (t=6.196–6.212 s in Figure 3). The simulated tissue was of insufficient size to sustain the reentrant waves for more than one rotation and the tissue was completely repolarized 362 ms after endocardial stimulation.

Figure 3.

Membrane potential snapshots from the sixth beat of the 2D tissue simulation with linear epicardial G_to gradient from 1.5 to 2.0 mS/µF. The stimulus was applied along the endocardial edge at t=6.0 s and resulted in two P2R occurrences: one at t=6.096 s, which triggered a second activation in the epicardium, and a second at t=6.196 s, which initiated reactivation of the mid- and endocardium. Phase-2 reentry also occurred at beats 1, 2, and 10, while the reminder of the beats showed normal propagation.

Mechanisms of phase-2 reentry in two-dimensional sheets with epicardial and transmural G_to gradients

We employed a range of different epicardial G_to gradients in the 2D sheet, as summarized in Table 2. In these 2D sheets with both transmural and epicardial heterogeneity, P2R occurred much more frequently than in 1D transmural (Table 1) or 1D epicardial cables (range 0–48%¹³). In particular, with a linear gradient from 2.0 mS/µF to either 0.0, 1.0, or 1.5 mS/µF, P2R occurred in 10 out of the 10 beats (Table 2). The example with the linear gradient from 2.0 to 1.5 mS/µF (Figures 2 and 3) resulted in P2R in 40% of the beats. In general, when using a linear epicardial G_to gradient, P2R always occurred in beats where epicardial APs had divergent morphologies with a loss-of-dome and a spike-and-dome region. This situation arose for all beats in simulations with the steeper linear gradients (2.0 to 1.0, 0.5, and 0.0 mS/µF) and in some beats with the flatter linear gradient (2.0 to 1.5 mS/µF) as shown in Figure 3. In the remaining beats of the simulation with the linear 2.0 to 1.5 mS/µF gradient, the epicardial APs had a loss-of-dome everywhere (as in the example beat in Figure 2).

Table 2.

Summary of simulation results in 2D tissue with both transmural and epicardial ionic heterogeneity

	2.0→0.0	2.0→0.5	2.0→1.0	2.0→1.5
Linear	100%	100%	100%	40%
Boltzmann	40%	40%	60%	40%

Gradients (top row) indicate G_to values in mS/µF going left→right along the epicardium. Percentage values give P2R occurrence in 10 beats (all stable). The steeper linear gradients caused P2R in all beats, while the Boltzmann function gradients gave rise to P2R in about half the beats.

Using a Boltzmann function to describe the epicardial G_to gradient (resulting in a steep, but smooth transition between two different values¹³) also resulted in many P2R occurrences (Table 2). The appearance of P2R is very similar (40% of beats) when using the Boltzmann gradient or the linear gradient from 2.0 to 1.5 mS/µF. One would indeed expect the gradient type to be of less significance when gradients are relatively flat. For the more disparate Boltzmann function gradients (2.0 to 1.0, 0.5, and 0.0 mS/µF), P2R did not always occur when there was a spatial AP morphological discrepancy, with a loss-of-dome epicardial region neighbouring spike-and-dome epicardium. These results are consistent with our previous findings that a steep change in ionic parameters is not necessary for P2R generation and that the existence of loss-of-dome and spike-and-dome neighbouring locations is not sufficient to trigger P2R, in particular, when G_to values are outside the switching behaviour range.¹³

Discussion

Transmural and epicardial phase-2 reentry

In the normal heart, transmural and apico/basal gradients in I_to and other ionic currents may help synchronize repolarization. However, such gradients may become pathologically proarrhythmic, e.g. by increasing heterogeneity in the area of the right ventricular outflow tract.

In this modelling study, we showed that transmural heterogeneity can indeed induce P2R when I_to is elevated. Furthermore, the mechanism of P2R formation builds on the same AP instability mechanism as for simulated epicardial tissue, in which G_to sensitivity causes cells or regions to switch intermittently between spike-and-dome and loss-of-dome morphologies.¹³ Accordingly, our findings suggest that such switching behaviour may be one of the bases for the development of P2R in cardiac tissue. Alternating loss of the epicardial AP dome also produced P2R in a canine Brugada model.¹⁴ Furthermore, epicardial dispersion of repolarization has previously been recognized as a primary factor for P2R formation in canine and computational Brugada models.^11,15,31

In general, the transmural cable simulations presented here had less frequent P2R occurrence than the epicardial cables in our previous work,¹³ even though the transmural cables had added I_Ks heterogeneity. This suggests that the transmural I_Ks heterogeneity did not play a major role in P2R development. The difference in P2R prevalence may be due to variations in AP morphology in the different regions, and/or because of the longer length of the epicardial cables compared with the transmural ones.

The P2R incidence was also increased in the 2D sheets with both transmural and epicardial ionic heterogeneity compared with either epicardial or transmural 1D cables. This increase is notable because one might expect a decrease in P2R occurrence in 2D simulations due to a suppression of local instability when residing in a bigger tissue with a larger current sink. Several factors may contribute to this increase in P2R occurrence. In particular, the epicardial heterogeneity essentially sweeps a range of G_to values and hence scans a variety of dome and loss-of-dome dynamics in each beat, thus increasing the possibility of P2R. This is particularly apparent when the gradient is linear and P2R did in fact occur more frequently for the linear gradients (Table 2). In addition, in the 2D simulations, the electrotonic current causing propagation of the dome has both transmural and epicardial components, thus facilitating the generation of sufficient current to trigger re-excitation. Indeed, P2R almost always initiated at the junction of the borders separating the mid-myocardial spike-and-dome region and the epicardial spike-and-dome region from the epicardial loss-of-dome region (as in Figure 3, t=6.096 s).

Because most P2R incidences initiated and propagated in the epicardium, we examined whether P2R might occur with complete block of transverse epicardial propagation. We hypothesized that in case of such a block, P2R would arise in the epicardium transmurally. In a simulation of 2D transmural tissue lacking transverse epicardial diffusion, P2R did indeed take place frequently (9 out of the 10 beats) along the transmural direction. Thus, the epicardium plays a key role in P2R initiation, where P2R may develop either along the epicardial axis or along the transmural direction within the epicardium.

Phase-2 reentry and Brugada syndrome

Many loss-of-function mutations in the SCN5A gene have been reported in Brugada syndrome patients, but these account for only about 20% of Brugada cases.^7,9,10 The resulting reduced I_Na may lead to the formation of an arrhythmogenic substrate due to conduction slowing and delayed depolarization, but may also result in a propensity for P2R-induced extrasystoles by decreasing the inward current available to counteract I_to. Failure of trafficking of sodium channels to the membrane associated with genetic variations in the MOG1 and GPD1L genes^32,33 may have similar effects. Recently, some patients with Brugada syndrome have been found to harbour mutations in the KCND3 gene,³⁴ which encodes the Kv4.3 I_to channel, or in the KCNE3³⁵ and KCNE5 genes,³⁶ which encode I_to-modulating proteins. These mutations lead to augmented I_to, and in the cases where the amount of I_to increase was incorporated into a mathematical model, it was found to be sufficient to cause loss-of-dome APs.^34,36

Evidently, P2R is not the sole arrhythmogenic pathway in Brugada syndrome and/or I_to-induced AP divergence. The dispersion in repolarization due to I_to heterogeneity in itself forms an arrhythmogenic substrate with an increased propensity for unidirectional conduction block by a closely coupled extrasystole (formed by any cause, i.e. not necessarily P2R-induced). Indeed, in our transmural simulations, repolarization dispersion increased from around 70 ms/cm to more than 400 ms/cm when a spike-and-dome/loss-of-dome boundary formed. The latter value far exceeds the wide range of unidirectional block thresholds found in ex vivo studies under a variety of experimental conditions (10–120 ms/cm; see³⁷).

I_to and action potential instability

In contrast to the increase in I_to due to some Brugada syndrome-associated mutations, I_to is negatively impacted by remodelling processes in several prominent pathologies, including heart failure, hypertrophy, and chronic atrial fibrillation.³⁸ Recent studies have suggested a role for I_to in APD prolongation and early afterdepolarization (EAD) formation.^39,40 Potentially, due to their shared dependence on I_to, the AP dome vs. loss-of-dome instability associated with I_to in this study and elsewhere^13,36 shares several similarities with the EAD-forming AP instability. In particular, EADs can exhibit a switching behaviour, occurring on only a subset of APs^41,42 and may be facilitated by AP heterogeneity and repolarization dispersion.⁴³ Furthermore, the source–sink conditions under which an EAD is capable of propagating and triggering reentry may be similar to those determining P2R generation and propagation. In line with previous modelling studies, we found here that P2R initiation is due to a combination of electrotonic current from neighbouring regions causing local reactivation of I_Ca, followed by normal I_Na-generated propagation.^13,44 Likewise, EAD formation is typically due to I_Ca reactivation. Although not found in our simulations, other modelling studies have found that under some circumstances both P2R and EADs may result in I_Ca wave propagation.^44,45

Study limitations

First, the division of the transmural dimension into three sharply bordered regions with distinct ionic properties is an oversimplification. However, the precise distribution and properties of myocytes through the ventricular wall and the extent to which this causes functional heterogeneity in the intact heart remains controversial.⁴⁶ Secondly, because the exact range of epicardial G_to in the right ventricle is unknown, a broad range of G_to values was used. This range was based primarily on canine data and extrapolation to the human heart must be done cautiously. Thirdly, because of the linkage of P2R initiation to the epicardium, we did not explore the possibility of mid-myocardial increases in G_to. Fourthly, we investigated only one ventricular model and one basic cycle length (1000 ms; chosen because most Brugada-related arrhythmia occur during rest). Further studies are needed to address rate and model dependence. Fifthly, sustained reentry was not demonstrated in this study, perhaps due to small tissue dimensions (3 cm × 1.65 cm); larger tissue simulations may resolve this. Finally, it should be noted while we have focused on ionic heterogeneity as a P2R substrate, P2R may also be facilitated by structural heterogeneity⁴⁷ and that I_to may also regulate excitation failure in such setting.⁴⁸

Supplementary material

Supplementary material is available at Europace online.

Conflict of interest: none declared.

Funding

This work was supported by National Institutes of Health grants R01HL101190 and R01HL094620 and the Tri-Institutional Training Program in Computational Biology and Medicine.