AP: Action potential.

APA: Action potential amplitude.

APD: Action potential duration.

BCL: Basic cycle length.

CV: Conduction velocity.

CTL: Control condition.

DT-MRI: Diffusion tensor magnetic resonance imaging.

EADs: Early afterdepolarizations.

ERP: Effective refractory period.

HF: Heart failure.

MUV: Maximum upstroke velocity.

1D: One-dimensional (1D).

OS: Overshoot (OS) of action potential.

PP: Plateau potential.

PF: Purkinje fiber.

PVJ: Purkinje fiber-ventricle junction.

RMP: Resting membrane potential.

SF: Safety factor (SF).

3D: Three-dimensional.

VW: Vulnerable window.

Two families of canine ventricular and PF cell models were developed under both CTL and HF conditions. One is based on updating and modification of the cellular models for canine ventricular cells developed by Benson et al. (2008) (see Supplementary Text S1.1 for CTL and Supplementary Text S1.2 for HF ventricular cell model development), and the other is based on modification of the canine PF cell model developed by Aslanidi et al. (2009) (see Supplementary Text S1.3 for CTL and Supplementary Text S1.4 for HF PF cell model development). The modifications to the basal models of ventricular and PF cells are summarised in Tables 1–4, which summarizes all changes made in simulating the electrophysiological properties of the ventricular and PF cells based on comprehensive reviews of extant experimental data. Further details about the HF-induced changes in the electrophysiological properties of the cell models can be found in the Supplementary Tables S1–S8. Justification and comparison of the simulation results for HF-induced change in ionic properties to experimental data are shown in Supplementary Figures S1–S9. Detailed equations, parameters and initial conditions for the ventricular and PF cell models are presented in Supplementary Text S1 respectively.

Action potentials in each cell model were evoked by an external stimulus with a magnitude of -45.0 pA/pF and duration of 1 ms. All simulations were performed at a BCL of 1000 ms and for a duration of 100 s. For each of the simulations, the duration of the AP was measured after numerical solution of the model reached its steady state. All APDs mentioned in this study refer to APD₉₀.

Simulations have been done in Endo, M and Epi cells for characterising the functional impact of each individual HF-remodelled ionic current in contribution to APD prolongation. Results obtained from the three cell type models are qualitatively similar. As for illustration, results from the M cell model are shown here for the relative role of each individual HF-remodelled ionic current in contribution to APD prolongation in the ventricular action potentials.

Figures 3A–I show results when each of the HF-remodelled ionic currents alone was considered. Computed APs were superimposed on those in the CTL condition. The changes in APD and other AP morphological characteristics were summarized in Table 6. A bar chart plot of the APD comparison was also provided in Supplementary Figure S12. Simulation data suggested that each of the HF-remodelled I _NaL , I _K1 , I _Ks , I _CaL and I _NaK contributed individually to increased APD (the increase was larger than 10 ms), but the HF-remodelled I _Na or the NCX produced a modest APD prolongation (i.e., <10 ms). In contrast, HF-induced remodelling of I _to1 or [Ca²⁺]_i, abbreviated the APD. The HF-remodelled I _Na , I _to1 and [Ca²⁺]_i also altered other AP characteristics, such as the upstroke velocity and RMP.

Simulations were also performed with the HF-induced change in each individual ionic current was in turn excluded (i.e., current properties were adjusted back to the CTL condition) to investigate consequent effects on AP. Results are shown in Figures 3J–R. Computed APs were superimposed on those in the HF condition when all HF-induced changes were considered. Table 7 lists the effect of excluding remodelling of each current on the APD. Corresponding bar charts are shown in Supplementary Figure S12. Simulation data showed that the adjustment of I _Na , I _NaL , I _K1 , I _Ks , I _CaL, the NCX, [Ca²⁺]_i and I _NaK back to the CTL condition individually resulted in a dramatic reduction of the APD by 20–89 ms. However, the adjustment of I _to1 back to the CTL condition resulted in a 36 ms increase of the APD.

Figures 4A–D show simulated AP propagation along two sets of 1D strands in both CTL and HF conditions. Corresponding APD distribution are plotted in Figures 4E,F. Due to electrical coupling, the distribution of ventricular APDs was smoothed compared to the APD differences among isolated cells. The APDs of ENDO and EPI region were increased whereas the midmyocardial APD was abbreviated.

In the 1D model with PF present, the measured APD of PF cells gradually decreased towards ventricles in the CTL condition, whereas it gradually increased towards ventricles in the HF condition. This was due to the marked increase in ventricular APDs in the HF condition. As a result, in contrast to observations with single cell model, the APD of PF in the intact model was actually increased in the HF condition compared to that in CTL condition.

Supplementary Figure S13 examines the AP morphology of the Endo, M, Epi and PF cell models in the 1D model. APs were recorded from the middle of each cell region; the recorded APs were time-shifted such that the APs were superimposed on each other for a direct comparison. In the HF condition, simulated APs from the Endo, M and Epi cell regions of the 1D tissue model showed similar changes to those seen in uncoupled single cells, which included a reduced OS and APA, and a prolonged APD. However, the elevation to the RMP was negligible compared to the CTL condition. On the other hand, the APs of PF cells in the 1D model showed a reduced OS, an increased PP and a slightly elevated RMP, all of which were similar to those of the uncoupled PF cell in the HF condition, whereas the APD was slightly prolonged.

Figure 5A shows computed CV along the 1D PVJ strand. The resultant CV was decreased by 30% in PF cell regions and 41% in ventricular cell regions in the HF condition. Our simulations suggested a relatively continuous conduction within each of the PF and ventricular cell regions. However, there was a sudden CV drop at the PVJ in both conditions. This CV drop suggested a discontinuous AP conduction at the PVJ (Wiedmann et al., 1996), which may be pro-arrhythmic and likely to cause conduction delay or conduction failure (Gilmour and Watanabe, 1994; Li et al., 1994; Wiedmann et al., 1996; Morley et al., 2005). Note that a CV drop and then recovery was also observed in the ventricle strand at the vicinity of the M-EPI border, which was attributable to a five-fold decrease of the intercellular coupling at the M-EPI border (Gima and Rudy, 2002; Zhang et al., 2008).

To characterise possible pro-arrhythmic effects arising from the conduction discontinuity at the PVJ, we also investigated the conduction SF and VW of the cardiac tissue to unidirectional conduction block.

Figure 5B showed computed SF along the strand with PF in both CTL and HF conditions. Simulations in the CTL condition showed a slight decrease in SF near the boundary between PF and ventricular tissue, followed by a sharp increase of SF at the PVJ.

The reduced CV before the PVJ led to longer time delay, which might enhance the imbalance between I _out and I _in (see the Method section), leading to the slight decrease of SF (Aslanidi et al., 2009). At the PVJ, a decrease of the intercellular coupling (i.e., D _PF > D _ventricle ) reduced electrical loading. As a result, less current leaked into the ventricular tissue at any given moment, and a larger fraction of the charge provided by the ionic current (I _ion ) was stored by the cell at the junction, which increased SF (Aslanidi et al., 2009).

Similarly, in the HF condition, decreased coupling at PVJ increased SF. However, there was a sharp dip of SF at the PVJ just after the peak site, indicating a likely failure in electrical conduction from the PF cell region to the ventricular cell region. This might be due to the reduced excitability of downstream neighbourhood of cells (Shaw and Rudy, 1997).

Figures 5C–E show computed VW in the 1D strand in both CTL and HF conditions. In the CTL condition, there was a small VW across the 1D strand with no significant differences in the width of the VW between PF and ventricular cells. This suggests an unlikely occurrence of unidirectional block, consequently making it difficult to initiate re-entry. However, in the HF condition, the VW was dramatically increased, especially in the regions of the PF and the PVJ. The difference of the width of the VW between the PF cell region and the ventricular wall was also increased. At the PVJ, the width of the VW was significantly (almost 25 times) greater in the HF condition than that in the CTL condition. These results indicated that there was an increased probability of the occurrence of unidirectional block around the region of the PVJ.

It is well established that HF can prolong the APD of ventricular cells in different species, including canine (Kaab et al., 1996; Undrovinas et al., 1999), human (Beuckelmann et al., 1992) and rabbit (Rose et al., 2005). Experiments have also shown that HF abbreviated the APD of PF cells (Li et al., 1993; Han et al., 2001; Maguy et al., 2009). Our simulation data at the single cell level also showed prolongation and abbreviation of ventricular and PF APDs respectively. This is in agreement with prior experimental findings, implying that the experimentally observed ion current and Ca²⁺-handling remodelling were sufficient to account for the APD changes in ventricle and PF cells as observed experimentally.

The simulated ventricular APD prolongation can be accounted for by the combined action of all ion current and Ca²⁺-handling remodelling. However, the relative roles of each of the remodelled ionic currents in contributing to APD prolongation varied. Our simulation results suggested that for HF-remodelled I _NaL , I _K1 , I _Ks , I _CaL and I _NaK , each of them contributed a marked increase in the APD prolongation. For I _NaL , HF increased its current density but slowed down its inactivation time course, which led to an increased current amplitude during the time course of AP, thereby contributing to plateau prolongation. HF decreased the current density of I _K1, I _Ks and I _NaK , which led to a slowing of repolarization, thereby also contributing to AP prolongation. For I _CaL , HF shifted its steady state activation curve leftward that resulted in an increased window current (Beetz et al., 2009), leading to an elevated membrane potential in phase 3 and a prolonged APD. However, the effect of the HF-induced decrease in I _to1 contributed to a reduced APD, which initially seemed counter-intuitive, given that a reduced outward K⁺ channel current would normally increase the APD. We further analysed our simulation data and found that this was due to an interplay between ionic currents. A reduction in I _to increased the plateau membrane potential, leading to an increased activation of the delayed rectifier potassium channel currents that promoted repolarization, which is consistent with a previous observation (Grant, 2009). However, the integrated effects of HF on the different ionic currents outweigh such an effect, with the net consequence of APD prolongation.

It has been noted that HF had different effects in modulating APs in ventricle and PF cells, which might be attributable to the differences in the HF-induced remodelling on ionic properties between the two cell types (Li et al., 1993; Han et al., 2001; Maguy et al., 2009). Our simulation results supported this notion. In the HF condition, I _Na , I _K1 , I _to1 , I _Ks , I _CaL and [Ca²⁺]_i were reduced in both PF and ventricular cells, whilst I _NaL and the NCX were increased and I _NaK was reduced in ventricular cells; none of them was changed in the PF cell. On the other hand, I _Kr was reduced in the PF cell but not in ventricular cells (Li et al., 2002); additionally, in PF there was a reduction of I _CaT , which was not present in canine ventricular cell models. Those differences in the HF-induced remodelling on K⁺ currents (outward current) and Ca²⁺ currents (inward current) along with an increased I _NaL (inward current) in ventricular cells can explain the different modulation of HF on cellular APs between the PF (shortened AP) and ventricular (prolonged AP) cells.

Experimentally it has been shown that HF is associated with abnormal electrical wave conduction in the intact PF-ventricular tissue, which provides a pro-arrhythmic substrate that favours increased vulnerability to and sustainability of arrhythmias in the HF condition (Peters et al., 1997; van Rijen et al., 2004; Cabo et al., 2006). Abnormal conduction manifested as a reduced electrical wave conduction velocity in the PF and ventricular tissues, but an increased conduction time delay at the PVJ. In ventricular tissue, about ∼40% reduction in the conduction velocity has been observed (Akar et al., 2007; Maguy et al., 2009), which may be attributable to reduced Cx43 proteins (Peters et al., 1997; van Rijen et al., 2004; Cabo et al., 2006). Our simulation data provide evidence to support this proposition. It has been shown that both HF-induced remodelling in ionic properties and the intercellular coupling contributed to the CV reduction, which was by about 40% in the ventricular and 30% in the PF tissue models. Such CV reduction was attributable to the integral action of reduced maximal upstroke velocity of AP arising from the ionic remodelling and the intercellular coupling arising from the reduction of Cx43. The simulated reductions in CV by HF in the tissue models are numerically in agreement with experimental observation (Akar et al., 2007; Maguy et al., 2009). Due to the reduced CV, the excitation wavelength was reduced, which facilitated the formation and maintenance of re-entrant excitation waves in a limited, pro-arrhythmic cardiac tissue substrate.

A conduction time delay at the PVJ arising from the source-sink mismatch provides a potential cause for the genesis of unsafe conduction and/or even conduction failure that leads to formation of re-entrant arrhythmia (Gilmour and Watanabe, 1994; Li et al., 1994; Wiedmann et al., 1996; Morley et al., 2005). In this study, we have shown that HF increased markedly such a conduction delay and consequently vulnerability to the genesis of conduction block at the PVJ was augmented. Due to the integral action of a reduced maximal upstroke velocity and intercellular electrical coupling in the HF condition, the driving force for the PF to excite the ventricular tissue was reduced, leading to a sharp decrease of conduction SF at PVJ, although the electronic load of the ventricular tissue to the PF was reduced due to the reduced intercellular coupling. Therefore, in the HF condition, there was less current flowing into downstream neighbour cells than the current received from upstream PF cells, leading to an increased likelihood of occurrence of conduction failure, which would require an increased thickness of PF strands to overcome the source-sink mismatch in ensuring the PF to successfully drive the ventricle as observed by Aslanidi et al. (2009). At the PVJs where the PF strands are thin, the reduced SF may predispose to conduction failure that leads to formation of re-entry.

Reduced transmural dispersion of repolarisation in ventricles in the HF condition has been observed experimentally and has been believed to underlie the electrophysiological mechanism for the genesis of unidirectional block, formation of re-entry and arrhythmogenicity (Li et al., 2002; Bai et al., 2006; Wang and Hill, 2010). In this study, such reduced ventricular transmural dispersion was also seen in our simulations, which is consistent with experimental observations (Zhu et al., 2009; Osadchii, 2017). However, it is difficult to causally link the reduced ventricular repolarisation dispersion to the increased pro-arrhythmogenesis in the HF condition as the reduced dispersion might be expected to be anti-arrhythmic. However, the paradoxical effects of the reduced dispersion of repolarisation can be explained by the altered repolarisation dispersion sequence between PF and the ventricle junctions, which, to our best knowledge, has not been investigated previously. In the control condition, the repolarisation of PF strand was later than that of the adjacent ventricle tissue, preventing the re-entry of excitation waves from ventricle tissue to the PF at the junction, even at points where the PF failed to drive the ventricular tissue. However, in the HF condition, the repolarisation sequence between PF and ventricles was reversed, causing the repolarisation of PF to be earlier than that of the adjacent ventricular tissue due to the different modulatory role of HF on the APDs of the ventricle and PF cells (Figure 2). At the PVJs where PF are thin, PFs failed to excite the adjacent ventricular tissue due to the augmented source-sink mismatch and the reduced excitability of ventricular cells in the HF condition. However, at the PVJs where PF are thick enough, the conduction wave propagated into ventricle with a slow conduction velocity and long APD. When the excitation wave reached the PVJ sites where the conduction from PF to ventricle failed at the first instance, the ventricular APs with a longer ventricular APD or later completion of the repolarisation process provided a stimulus trigger to evoke a second excitation in the PF, opening a “gate” allowing the conduction wave to re-enter from the ventricle back to the thinner PF. Consequently the ventricular action potentials re-entered the thinner PF bundle from the ventricle and triggered the genesis of re-entrant wave at the PVJ as seen in Figure 8.

Altered repolarisation dispersion at the PVJ by pathological conditions has been observed in sheep and ovine models (Walton et al., 2014; Martinez et al., 2018), as well as in experimental and computational murine models of CPVT (Blackwell et al., 2022). It is clear from the studies of repolarization heterogeneity that the Purkinje fiber ventricular junction influences dispersion of repolarization. Moreover, in the study of Blackwell et al. (2022), which was a combined experimental and simulation investigation, the Purkinje–myocardial junction was seen to be involved as the anatomic origin of ventricular arrhythmia in the murine CPVT model. The ablation of subendocardial cells adjacent to PF was sufficient to protect against catecholamine induced arrhythmias and, in simulations, DADs in ventricular myocardium could trigger action potentials in PFs, but not vice versa.

All together this suggests that the impact of HF on the conduction discontinuity and dispersion of repolarisation at the PVJ plays an important role for the initiation of re-entry, adding new understanding of the pro-arrhythmic mechanisms of HF.

Several other studies have developed models to simulate the effects of HF-remodelling on modulating ventricular APDs (e.g. Priebe and Beuckelmann, 1998; Flaim et al., 2006; Trenor et al., 2012), most of which concentrated on a few HF-remodelled ion currents. The ventricular and PF models employed in the present study took into consideration comprehensive HF-induced ion channel, Ca²⁺ handling and intercellular electrical coupling remodelling, based on available information, allowing the reproduction of major changes in electrophysiology observed experimentally.

In this study, we have shown that HF augmented discontinuous conduction at the PVJ due to the source-sink mismatch. This may cause conduction failure at some PVJ sites where the PFs were not able to drive the ventricle. As such, a “gate” is produced allowing a previous excitation wave to retrogradely re-enter the PF network, forming a re-entrant pathway. Our observations support the “gate hypothesis” proposed by Myerberg et al. (Myerburg et al., 1970; Myerburg et al., 1971), which suggested that the area with the maximum APD in PF determines the minimal AP coupling intervals for successful conduction at the PVJ, thus preventing premature action potentials to propagate into the ventricle (Myerburg et al., 1971). Based on this hypothesis, Lazzara et al. (Lazzara et al., 1976) subsequently proposed a theory to explain the formation of PF-ventricle re-entry, that in some conditions in which conduction to ventricle succeeded in one PF branch but failed in another, ventricular conduction might retrogradely propagate through the gate previously in refractory period and form a re-entrant wave (Lazzara et al., 1976). Experimental observations conducted by Logantha et al. (Logantha et al., 2021) provided data to support this theory.

This study provides insights into the mechanisms underlying the “gate hypothesis.”

We have shown that the formation of PF-ventricle re-entry in the HF condition is associated with two major factors: one is the impaired driving power of the PF to the ventricle and the other is the reduced APD dispersion at the PVJ, rather than a large APD dispersion as foreseen by the “gate hypothesis.” Due to the weakened driving power, conduction might fail at PVJs with the width of PF below a critical value. Due to the reduced APD dispersion at the PVJs by HF, PF repolarised earlier than the ventricle, allowing ventricular excitation retrogradely propagating into the PF. In addition, slow conduction in the HF condition also reduced the excitation wavelength, facilitating the initiation and maintenance of re-entrant waves.

We incorporated investigation of the effects of HF-induced electrical and gap junctional coupling remodelling on the discontinuous conduction at the PVJ. Our results have shown that in the HF condition, the critical width of PF to drive the ventricle was significantly increased, attributable to the reduced intercellular coupling and the amplitude of cellular action potentials. Our results are consistent with a previous computational study of Behradfar et al. (Behradfar et al., 2014), who investigated the impact of electrical resistance (equivalent to the diffusion coefficient in this study) and size of PVJs on arrythmogenesis at the PVJ. Their data showed that when the resistance was high (i.e., low D in our study), there wasn’t enough charge passed from PF to downstream neighbouring ventricle to fire an action potential. When the resistance was too low, it suppressed prejunctional action potentials, also leading to conduction failure. Our results of the conduction failure at the PVJ in the HF condition when the diffusion coefficient was reduced match the findings of Behradfar et al. (Behradfar et al., 2014).

In this study, from the PF strand to the PVJ and Epi end of the 1D PF-ventricle model, a gradual decrease (i.e. downward gradient) in APD was shown in CTL condition, but it was changed to an upward gradient in HF due to a dramatic APD increase in the ventricle (see Figure 4). These simulation data differ from those shown in (Li et al., 2015), for which rabbit PF and ventricle cell models were used: in that study a downward gradient in APD the PF to the ventricular muscle in the CTL condition was enhanced in HF condition. This difference may be due to the species difference between rabbit and canine, where a significant APD increase in the PF cell was observed in the HF condition; however, there is little APD change in the HF condition as compared to CTL seen in simulations and experimentations (Li et al., 2015).

HF is associated with high risk of cardiac arrythmias causing morbidity and mortality (Dean and Lab, 1989; Aistrup et al., 2011; Masarone et al., 2017). However, incomplete understanding of the mechanisms underlying the pro-arrhythmic effects of HF impedes effective prevention and treatment of cardiac arrhythmias in this condition. In this study, we adopted a computational modelling approach to analyse the integrated and individual roles of HF-induced changes in the electrical properties and kinetics of the ion channels and transporters in modulating cellular action potentials of PF and ventricle cells, the regional repolarization dispersion and discontinues electrical conduction at the PVJ. The identified major role of HF-remodelled ion channels (including the augmented I_NaL and decreased I_K1) in this may provide possible novel target(s) in antiarrhythmic therapy. In addition, we have found that the discontinuous conduction at the PVJ forms a pro-arrhythmic substrate in the HF condition, due to the altered dispersion of repolarisation and increased source-sink mismatch. Further experimental and clinical studies are warranted to analyse possible effects of targeting these features on generation of re-entry at the PVJ, in order to evaluate whether they constitute valuable approaches for anti-arrhythmia strategy in HF conditions.