Curvature-mediated source and sink effects on the genesis of premature ventricular complexes in long QT syndrome

Abstract

Premature ventricular complexes (PVCs) are spontaneous excitations occurring in the ventricles of the heart that are associated with ventricular arrhythmias and sudden cardiac death. Under long QT conditions, PVCs can be mediated by repolarization gradient (RG) and early afterdepolarizations (EADs), yet the effects of heterogeneities or geometry of the RG or EAD regions on PVC genesis remain incompletely understood. In this study, we use computer simulation to systematically investigate the effects of the curvature of the RG border region on PVC genesis under long QT conditions. We show that PVCs can be either promoted or suppressed by negative or positive RG border curvature depending on the source and sink conditions. When the origin of oscillation is in the source region and the source is too strong, a positive RG border curvature can promote PVCs by causing the source area to oscillate. When the origin of oscillation is in the sink region, a negative RG border curvature can promote PVCs by causing the sink area to oscillate. Furthermore, EAD-mediated PVCs are also promoted by negative border curvature. We also investigate the effects of wavefront curvature and show that PVCs are promoted by convex but suppressed by concave wavefronts; however, the effect of wavefront curvature is much smaller than that of RG border curvature. In conclusion, besides the increase of RG and occurrence of EADs caused by QT prolongation, the geometry of the RG border plays important roles in PVC genesis, which can greatly increase the risk of arrhythmias in cardiac diseases.

NEW & NOTEWORTHY The effects of the curvature or geometry of the repolarization gradient region and wavefront curvature on the genesis of premature ventricular complexes are systematically investigated using computer modeling and simulation. Premature ventricular complexes can be promoted by either positive or negative curvature of the gradient region depending on the source and sink conditions. The underlying mechanisms of the curvature effects are revealed, which provides mechanistic insights into arrhythmogenesis in cardiac diseases.

INTRODUCTION

Premature ventricular complexes (PVCs) are extra spontaneous excitations occurring in the ventricles of the heart. PVCs are associated with ventricular arrhythmias and sudden cardiac death (1–4), as well as the development of cardiomyopathy (5–7). As summarized in our recent review article (8), PVCs can serve as triggers to initiate reentrant arrhythmias via the traditional trigger and substrate interactions or degenerate directly into reentrant arrhythmias. The causes of PVCs can be automaticity (9–12), early afterdepolarization (EAD), or delayed afterdepolarization (DAD)-induced triggered activity (13–17). In these mechanisms, the automaticity or triggered activity is the source that propagates out of the source area once the source-sink condition is satisfied. Under long QT syndrome (LQTS) conditions, the action potential (AP) is prolonged and PVCs can be induced by repolarization gradient (RG) (18–23), which occur when RG is large and L-type calcium current (I_Ca,L) is enhanced, regardless of the presence of EADs. PVCs can also be induced by phase-2 EADs when RG is small (19, 20). For both RG- and EAD-mediated PVCs, they are not a result of propagation of EAD-induced triggered activity as in the traditional mechanism but caused by tissue-scale dynamical instabilities (20, 24, 25). The RG-mediated PVCs are supported by the PVC properties observed in experiments (18, 26–30) that the earliest sites of PVCs are in the RG region.

In all the mechanisms of PVCs, whether they are caused by propagation of automaticity and triggered activity or by the tissue-scale dynamical instabilities, the well-known source and sink effects still apply. In other words, a strong enough source (such as a sufficient number of source cells) is needed to overcome the sink effect of the surrounding cells to initiate a PVC (17). Besides the number of cells, the geometry of the source area is also important. For example, the border between the source area and surrounding sink area can be curved and the curvature of the border region affects the PVC genesis. For the RG-mediated PVCs, in a location with negative RG border curvature, it has a more favorable source condition than other locations, and therefore, one would anticipate that PVCs will tend to occur at the location with negative border curvature. This is indeed the case as shown in computer simulations (31–33). However, in a recent study by Teplenin et al. (24), the authors demonstrated in both experiments and computer simulations that PVCs were promoted by positive RG border curvature, that is, PVCs originated from the locations with the largest positive RG border curvature. Therefore, a question arises: how to reconcile the seemingly contradictory results that both negative and positive border curvature can promote PVCs and what are the underlying mechanisms?

Another widely known source-sink effect is the wavefront curvature on conduction, known as the eikonal relation (34–36), that is,

$$\text{CV\hspace{0.17em}}={\text{CV}}_0-D\kappa ,$$ (1)

where CV is the conduction velocity of the curved wavefront and CV₀ is that of the planar wave. D is the diffusion constant of the tissue and κ is the curvature of the wavefront. κ = 0 for a planar wave, κ < 0 for a concave wave, and κ > 0 for a convex wave. Because of the source-sink effect, a convex wave is slower and a concave wave is faster than a planar wave. If the curvature of a convex wave is too large, conduction may fail because of a too big sink effect from the surrounding cells. However, the wavefront curvature affects not only conduction but also repolarization. We have shown previously (36) that the wavefront curvature results in a waveback (or repolarization front) curvature that affects the AP duration (APD), similar to the eikonal relation, that is,

$$\text{APD\hspace{0.17em}}={\text{APD}}_0+\alpha D\kappa ,$$ (2)

where APD₀ is the APD of the planar wave and α is a coefficient. Therefore, APD is shorter for a concave wave (κ < 0) but longer for a convex wave (κ > 0). Because of the effect of wavefront curvature on APD, one would expect that the wavefront curvature may affect the genesis of PVCs.

In this study, we investigated the curvature effects on PVC genesis under the condition of APD or QT prolongation, a condition occurring in LQTS. We carried out one-dimensional (1-D) cable and two-dimensional (2-D) tissue simulations using a total of nine AP models. We used the 1991 Luo and Rudy (LR1) model (37) to perform a systematic investigation of the effects of RG border and wavefront curvature on PVC genesis and examine the results in eight physiologically more detailed AP models. We revealed the conditions and mechanisms by which curvature promotes or suppresses PVCs via the source and sink effects. The results of the LR1 model were examined using eight different AP models that have more physiological details of ionic currents and calcium cycling.

METHODS

Tissue Model

We carried out 2-D tissue simulations using the following partial differential equation for voltage (V):

$$\frac{\partial V}{\partial t}=-\frac{I_{\text{ion}}+I_{\text{sti}}}{C_{\text{m}}}+D(\frac{{\partial }^{2}V}{\partial x^2}+\frac{{\partial }^{2}V}{\partial y^2}),$$ (3)

where C_m = 1 μF/cm² is the membrane capacitance, I_ion is the total ion current density, and I_sti is the stimulation pulse. D is the diffusion constant, and we used D = 0.001 cm²/ms for the LR1 model as used in Zhang and Qu (20) and D = 0.0005 cm²/ms for other AP models as used in Zhang et al. (19). RG heterogeneity and geometry are modeled by heterogeneous distribution of the time-dependent potassium currents. The specific changes are clarified for each case in each AP model in results.

Reduction of a 2-D Tissue Model into a 1-D Cable Model

Under certain special conditions, one can mathematically reduce the 2-D tissue model into a 1-D cable model. In other words, one only needs to simulate a 1-D cable to obtain the dynamics of the 2-D tissue. For example, for a 2-D tissue with a circular heterogeneous region, it can be reduced to a 1-D cable in the polar coordinate system, that is, one can transform Eq. 3 from the Cartesian coordinate system to a polar coordinate system as:

$$\frac{\partial V}{\partial t}=-\frac{I_{\text{ion}}+I_{\text{sti}}}{C_{\text{m}}}+D\frac{1}{\text{r}}\frac{\partial V}{\partial \text{r}}+D\frac{{\partial }^{2}V}{\partial {\text{r}}^2}.$$ (4)

The origin of the polar coordinate system is the center of the circular heterogeneous region. Equation 4 is equivalent to Eq. 3 for certain pacing protocols. First, pacing from the center (r = 0) of the circular region results in an outward radial (convex) wave. The front curvature $\kappa =\hspace{0.17em}\frac{1}{\text{r}}$ changes along the direction of propagation, and at the RG border, the border curvature is the same as the wavefront curvature. Second, pacing the whole tissue simultaneously to avoid wave propagation. Under this condition, only the border curvature effect is present.

However, in real systems, the wavefront curvature and the RG border curvature are different and both can be heterogeneous in space. An exact dimension reduction as for the circular RG heterogeneity is not feasible. To investigate the curvature effect in general, we generalize Eq. 4 back into a Cartesian coordinate by setting the curvature as a constant, that is, we replace $\frac{1}{\text{r}}$ in Eq. 4 with κ and change r to x, then Eq. 4 becomes:

$$\frac{\partial V}{\partial t}=-\frac{I_{\text{ion}}+I_{\text{sti}}}{C_{\text{m}}}+D\kappa \frac{\partial V}{\partial x}+D\frac{{\partial }^{2}V}{\partial x^2}.$$ (5)

We have used Eq. 5 previously (36) to investigate the dependence of CV and APD on wavefront curvature (see Eqs. 1 and 2) in homogenous tissue. Unlike in Eq. 4 where the curvature is determined by the position of the wavefront, in Eq. 5, κ is a parameter independent of position and one can change it independently to systematically explore the curvature effects. Although Eq. 5 cannot exactly simulate the 2-D tissue, as we will show later in this article, it can be a very good approximation for investigating the curvature effects, in particular when the whole tissue is paced simultaneously. Equation 5 will be used for all the 1-D simulations in this paper.

Action Potential Models

The majority of the simulations were done using the 1991 Luo and Rudy (LR1) guinea pig ventricular AP model (37). To examine the generality of the LR1 model results, we also simulated eight other AP models that were used in our previous studies (18, 19) for the investigation of PVC genesis under LQTS conditions. The models are as follows: 1) the 1994 Luo and Rudy (LRd) guinea pig ventricular AP model (38, 39); 2) the 2004 Shannon et al. (SB) rabbit ventricular AP model (40); 3) the 2004 Hund and Rudy (HRd) canine ventricular AP model (41); 4) the 2004 ten Tusscher et al. (TP04) human epicardial ventricular AP model (42); 5) the 2010 Grandi et al. (GB) human epicardial ventricular AP model (43); 6) the 2011 O’Hara et al. (ORd) human endocardial ventricular AP model (44); 7) the 2019 Tomek et al. (ToRORd) human endocardial ventricular AP model (45), which is modified from the ORd model; and 8) the modified UCLA rabbit ventricular AP model (labeled as UCLA_H) by Huang et al. (18). The original codes for the AP models were downloaded from the websites of the original authors and incorporated into our own C++ and CUDA codes for simulation.

Parameters were changed to generate EADs in single cells and PVCs in tissue. For the LR1 model [the parameter setting was the same as in Fig. 2 in Zhang and Qu (20)], we sped up the time constants of the slow inward current (I_si) as follows: τ_d (V) →0.3τ_d (V) and τ_f (V) → 0.5 τ_f (V). We slowed the time constant of the time-dependent potassium current (I_K) as follows: τ_X (V) →6τ_X (V). We changed the maximum conductance of the inwardly rectifying potassium current (I_K1) to G_K1 = 0.4 × 0.6047 mS/cm². RG heterogeneity was simulated by heterogeneous distribution of the maximum conductance (G_K) of I_K.

For all other AP models, the control parameters were the same as in the original models. LQTS type 2 condition was simulated, in which the rapid component of the delayed rectifier potassium current (I_Kr) was removed (G_Kr = 0). The slow component of this current (I_Ks) was altered in space to simulate RG heterogeneity (see Table 1). For the detailed AP models, the control parameter settings for PVC genesis were the same as in Fig. 4 of Huang et al. (18) for the UCLA_H model, and in the cases of LQTS type 2 in Fig. 2 of Zhang et al. (19) for all other models. RG was modeled by heterogeneous distribution of slow delayed rectifier potassium current conductance (G_Ks), To generate PVCs, we had to alter G_Ks in the short APD area differently for different AP models, which are listed in Table 1. We labeled fold change (α) of G_Ks from the control value as α(G_Ks)_long in the long APD area and α(G_Ks)_short in the short APD area. The parameter σ descripting RG for each model was defined as σ = G_Ks-long/G_Ks-short.

Table 1.

Fold changes of G_Ks in the short APD area for each model

Model	LRd	SB	HRd	TP04	GB	ORd	ToRORd	UCLAh
α(GKs)short	2	10	20	2	30	10	20	1

α(G_Ks)_short, fold change of G_Ks from the control value in the short APD area; APD, action potential (AP) duration; GB, 2010 Grandi et al. (43) human epicardial ventricular AP model; G_Ks, slow delayed rectifier potassium current conductance; HRd, 2004 Hund and Rudy (41) canine ventricular AP model; LRd, 1994 Luo and Rudy (38, 39) Guinea pig ventricular AP model; ORd, 2011 O’Hara et al. (44) human endocardial ventricular AP model; SB, 2004 Shannon et al. (40) rabbit ventricular AP model; ToRORd, 2019 Tomek et al. (45) human endocardial ventricular AP model; TP04, 2004 ten Tusscher et al. (42) human epicardial ventricular AP model; UCLA_H, Huang et al. (18) modified University of California, Los Angeles (UCLA), rabbit ventricular AP model.

Figure 4.

Curvature-dependent premature ventricular complex (PVC) firing time. Simulations are done using a one-dimensional (1-D) cable in the same way as in Figs. 2 and 3. Showing are PVC firing time (T) vs. κ. T is recorded from the last cell in the short action potential duration (APD) end. For a given κ, the data points for T are colored in the following order: black for the first PVC, red for the second PVC, and blue for the third PVC, and so on, as indicated by the numbers in A. Because of the complex PVC dynamics, in some regimes of κ, the data points are well organized and ordered (such as the PVC regions in the positive κ regimes) but can look very irregular in other regimes of κ (such as the negative κ regime in C). The κ regimes in which no PVCs occur are labeled as “no PVC zone” with boundaries marked by the vertical cyan dashed lines. A–C: parameters are the same as for Fig. 2B. except G_K-short = 1.25 mS/cm² (A), G_K-short = 0.5 mS/cm² (B), and G_K-short = 0.3 mS/cm² (C). D: parameters are the same for Fig. 1C. Horizontal red dashed lines are reference lines for the minimum T for the same PVC beat for beats 1 (black) and 5 (olive). Dashed arrow indicates shift of minimum T in positive κ range. Note that a comparable minimum T for beat 1 can also occur in the negative curvature region (κ ≈ –2.5 cm⁻¹).

Figure 2.

Phase diagrams of premature ventricular complex (PVC) dynamics. Simulations are done using a one-dimensional (1-D) cable (Eq. 5) with κ as a parameter. The 1991 Luo and Rudy model (37) (LR1) model is used. The length of the cable is 3 cm (200 computational cells) with G_K in one half [G_K-long, long action potential duration (APD) area] being different from the other half (G_K-short, short APD area). A stimulus is applied to all cells in the cable at t = 0 (A, inset). Colored or gray regions in the plots are the parameter combinations giving rise to PVCs and no PVCs occur in the white region. We call this type of plots as phase diagrams. A: phase diagram in κ-G_K-short plane for G_si = 0.075 mS/cm² and G_K-long = 0 mS/cm² showing two PVC regions that occur only in the negative curvature regime. Region I is transient oscillations occurring in the short APD area, and region II is sustained oscillations in the short APD area. B: phase diagram in κ-G_K-short plane for a high G_si, that is, G_si = 0.1 mS/cm² and G_K-long = 0 mS/cm². High G_si causes a new PVC region (region III) in the phase diagram, which occurs in the positive curvature regime. C: phase diagram in κ-G_K-short plane for G_si and G_K-long being higher, that is, G_si = 0.1 mS/cm² and G_K-long = 0.1 mS/cm². All three PVC regions become larger and regions II and III are linked together. D: phase diagram in κ-G_K-long plane for G_si = 0.075 mS/cm² and G_K-short = 1 mS/cm². E: phase diagram in κ-G_K-long plane for G_si = 0.1 mS/cm² and G_K-short = 1 mS/cm². F: phase diagram in κ-G_K-short plane for G_si = 0.1 mS/cm² and G_K-long = 0 mS/cm² with V clamped at the plateau phase in the long APD area. Region III disappears from the diagram. For A, B, and F, we can color different regions based on transient and persistent oscillations. For C (also for E), we cannot distinguish regions II and III and thus we did not color the plot. For D and E, unlike in A and B where repolarization fails in the long APD area, we can distinguish transient oscillation from persistent oscillation, but in D and E, long APD area can repolarize (see Fig. 3D), making the coloring impossible. Symbols in B and E mark the locations where examples are taken for Fig. 3.

Pacing Protocols and Initial Conditions

The initial conditions were the same as the ones in the original codes of the AP models downloaded from the repository sites of the original authors. The durations and strengths of the stimulation pulses were also the same as in the original code of each model. For the 1-D and 2-D simulations investigating the RG border curvature on PVC genesis, all cells were given the same initial condition, and a single pacing stimulus was applied simultaneously to all cells (at t = 0 for the LR1 model and t = 100 ms for other models) to avoid the confluent effects of conduction. We compared pacing from one end with simultaneous pacing, and the results only differ slightly (see Supplemental Fig. S1; note: all Supplemental Figures are available at https://doi.org/10.6084/m9.figshare.25395589.v1). To investigate the wavefront curvature effects on PVC genesis, we carried out simulations in homogeneous tissue and paced the 1-D cable from one end or the 2-D tissue from one side using the S1S2 pacing protocol. The multiple S1 beats avoid the small heterogeneities caused by the time delay during conduction of the first pacing beat. The S1S1 and S1S2 intervals are clarified in each case.

Numerical Methods

Computer simulations were carried out using GPU computation with C++ and CUDA programing. Equations 3 and 5 are discretized using the central difference scheme and integrated using a forward Euler method with Δx = Δy = 0.015 cm and Δt = 0.01 ms. No-flux boundary conditions were used.

RESULTS

RG Border Curvature on PVC Genesis

To investigate the effects of RG border curvature on PVC genesis, we used a pacing protocol in which all cells in the whole tissue were paced simultaneously so that there was no wave conduction in the tissue. By pacing the tissue with this protocol, we excluded the confluent effects of wave conduction and wavefront curvature on PVC genesis.

RG border curvature-dependent PVC genesis.

We first carried out simulations of 2-D heterogeneous tissue to show curvature-dependent PVC genesis. In these 2-D tissue simulations, we gave a single stimulus to all cells in the whole tissue at t = 0. Figure 1A shows a case in which PVCs emerge from the negatively curved RG borders. This agrees with the previous observation (31–33) that negative border curvature promotes PVCs. Figure 1B shows a case in which PVCs emerge from the corners of the RG border at which the curvatures are positive and the largest. This pattern remains the same for many PVC beats, and thus it is stable. This agrees with a previous study (24) that positive RG border curvature promotes PVCs. Figure 1C shows a more complex case in which PVCs emerge from different locations of the RG border in different PVC beats. In the first beat, the PVC emerges at the corners, which have the largest curvature. In the second beat, the PVC emerges slightly away from the corners, which have a smaller curvature than the corners. Many PVC beats later, the PVCs emerge from the sides whose curvature is zero.

Figure 1.

Repolarization gradient (RG) border curvature and premature ventricular complex (PVC) genesis in two-dimensional (2-D) tissue. Simulations are done in a 15.36 × 15.36 cm² tissue using the 1991 Luo and Rudy model (37) (LR1) model. A single stimulus is applied at t = 0 to all cells in the tissue. A: voltage snapshots showing PVCs emerging at the sites with the largest negative RG border curvature. G_si = 0.18 mS/cm², G_K = 0.92 mS/cm² in the long action potential duration (APD) area (labeled as G_K-long), and G_K = 1 mS/cm² in the short APD area (labeled as G_K-short). The long APD area is composed of two circles of radius 3.84 cm with their centers 3 cm apart. B: voltage snapshots showing PVCs emerging from locations with largest positive RG border curvature. G_si = 0.1 mS/cm², G_K-long = 0 mS/cm², and G_K-short = 1.25 mS/cm². The long APD area is a square of side length 7.68 cm. C: voltage snapshots showing PVCs emerging from different locations at different PVC beats. G_si = 0.35 mS/cm², G_K-long = 0.4 mS/cm², and G_K-short = 1.05 mS/cm². The long APD area is a square of side length 7.68 cm.

Curvature-Mediated Source and Sink Effects on PVC Genesis

To understand the complex PVC behaviors shown in Fig. 1, we carried out simulations using the generalized 1-D cable equation (Eq. 5) and varied curvature (κ) to systematically explore the curvature effects (see methods for the rationale for using this equation) on PVC genesis. We simulated a 1-D cable of 200 computational cells in which APD is longer in the first 100 cells (Fig. 2A, inset). A single pacing beat was given at t = 0 to all cells in the cable. Figure 2 shows phase diagrams plotting the PVC regions. In these diagrams, the x-axis is curvature (κ) and the y-axis is G_K of the short APD region (Fig. 2, A–C and F, labeled as G_K-short) or the long APD (Fig. 2, D and E, labeled as G_K-long). Figure 2A is the phase diagram when G_si (the maximum conductance of I_si in the LR1 model, corresponding to I_Ca,L in other models) is relatively small, in which PVCs occur only when κ < 0 and in two regions (I and II) with a small overlap. When G_si is increased, a new PVC region (III) appears, which only occurs for κ > 0 and is separated from regions I and II. If one increases the G_K (G_K-long) of the long APD area, region III moves toward the negative curvature direction, and regions II and III join together (Fig. 2C). Note that region III is insensitive to the change in G_K (G_K-short) of the short APD area unless G_K-short becomes small. We also scanned the G_K (G_K-long) of the long APD region, which is shown in Fig. 2, D and E.

In Fig. 2, the different PVC regions correspond to different PVC mechanisms. In region I, PVCs are caused by transient oscillations originating from the short APD area (Fig. 3A). This was further demonstrated by the following simulation: when V of the long APD area was held at a constant in the plateau phase, the transient oscillation remained (Fig. 3A, bottom). In region II, PVCs are persistent oscillations originating from the short APD area (Fig. 3B). Similar to region I, clamping V of the long APD area at a constant in the plateau phase does not affect the PVCs (Fig. 3B, bottom). Therefore, in both regions I and II, the long APD area only provides a source for the short APD area to become oscillatory, resulting in PVCs. We call this PVC mechanism as the lift-up mechanism, which has been demonstrated in our previous study (18). The lift-up mechanism tends to occur when the sink is relatively weak (small G_K-short) while the source is strong. The role of the RG border curvature is as follows: negative border curvature enhances the source to potentiate PVCs, whereas positive curvature weakens the source to suppress PVCs, and thus regions I and II are on the negative border curvature side in the phase diagrams. The example shown in Fig. 1A belongs to this case.

Figure 3.

Mechanisms of premature ventricular complex (PVC) genesis. Shown are space-time plots of V from the same one-dimensional (1-D) cable simulations as in Fig. 2. The parameters for each panel are indicated by the same symbols in Fig. 2, B and E, as the ones marked on each panel. *PVCs or oscillations. A: transient oscillation in the short action potential duration (APD) area. G_si = 0.1 mS/cm², G_K-long = 0 mS/cm², G_K-short = 0.6 mS/cm², and κ = –18 cm⁻¹. B: lift-up oscillation. G_si = 0.1 mS/cm², G_K-long = 0 mS/cm², G_K-short = 0.5 mS/cm², and κ = –10 cm⁻¹. C: pulldown oscillation. G_si = 0.1 mS/cm², G_K-long = 0 mS/cm², G_K-short = 1.25 mS/cm², and κ = 1.5 cm⁻¹. D: early afterdepolarization (EAD)-mediated PVCs. Pentagon, G_si = 0.1 mS/cm², G_K-long = 0.425 mS/cm², G_K-short = 1 mS/cm², and κ = 2.5 cm⁻¹; diamond, G_si = 0.1 mS/cm², G_K-long = 0.15 mS/cm², G_K-short = 1 mS/cm², and κ = –7 cm⁻¹; circle, G_si = 0.1 mS/cm², G_K-long = 0 mS/cm², G_K-short = 0.443 mS/cm², and κ = –3 cm⁻¹. A–C (bottom): voltage in the long APD area (magenta) is clamped at a constant in the plateau (clamped time intervals are indicated by double arrows).

In region III, PVCs are oscillations originating from the long APD area (Fig. 3C) since the oscillations and the PVCs disappear when V of the long APD area is held at a constant in the plateau phase (Fig. 3C, bottom). We also redid the phase diagram of Fig. 2B by clamping V of the long APD area, which is shown in Fig. 2F. Region III was completely eliminated, whereas regions I and II shrank somewhat. In this mechanism of PVCs, the sink effect from the short APD area needs to be strong enough to pull down the voltage of the source cells into the oscillatory regime for oscillation, and we call this mechanism of PVCs as the pulldown mechanism. In other words, when the source is too strong, the voltage of the cells in the long APD area is above the voltage range for oscillation. Therefore, a strong enough pulldown force from the cells in the short APD area is needed to bring the cells in long APD area into the voltage regime of oscillation. The pulldown mechanism tends to occur when both the sink and source are strong (large G_si and G_K-short). We have also demonstrated this PVC mechanism in our previous study (18). The role of RG border curvature in this mechanism is as follows: positive RG border curvature plays a role in weakening the source to promote oscillation. The example shown in Fig. 1B belongs to this case. This usually occurs in a relatively narrow range of κ (see region III in Fig. 2, B and C): too small or too large curvature will cause the exit of oscillation. This range of κ is not limited to positive curvature; it can span a range of both negative and positive curvature as shown in Fig. 2C, or even purely negative curvature. The underlying mechanism is that if the source is too weak, a negative (or a less positive) curvature can enhance the source to bring it into the oscillatory regime, and if the source is too strong, a positive (or a less negative) curvature can weaken the source to bring it into the oscillatory regime.

Besides the lift-up and pulldown oscillation generated by PVCs, PVCs can be EAD mediated (Fig. 3D), which has been analyzed in detail in our previous studies (18–20). The EAD-mediated PVCs belong to the lift-up mechanism, which is suppressed by positive RG border curvature and promoted by negative RG border curvature (all examples in Fig. 3D are for κ < 0). The case shown in Fig. 1A and the ones in previous studies (31–33) are EAD-mediated PVCs promoted by negative RG border curvature.

Unlike the case shown in Fig. 1B in which the PVCs always originate from the corners, forming a stable PVC pattern, in the case in Fig. 1C, the PVCs at different PVC beats originate from locations with different curvature. To understand this phenomenon and other more complex curvature-dependent PVC behaviors, we calculated the curvature-dependent timing (T) of PVC genesis in the 1-D cable (Eq. 5). T is defined as the time interval from t = 0 (when the stimulus is given) to the time of the PVC firing in the short APD area. Figure 4A shows T versus κ for the same parameter set as in Fig. 1B (or the same as in Fig. 2B, except that G_K-short = 1.25 mS/cm²). Under this parameter set, PVCs are promoted by positive curvature only as already shown in Fig. 2B. PVCs occur mainly when 0.9 < κ < 1.9 cm⁻¹ and T increases as κ increases. Therefore, for an RG heterogeneity as in Fig. 1B, PVCs cannot occur at the sides (κ = 0) but close to or at the corners where κ is large enough, and thus we observed a stable PVC pattern in Fig. 1B. Figure 4B shows T versus κ for the same parameters as in Fig. 2B and G_K-short = 0.5 mS/cm², PVCs can occur at both negative and positive κ, with first PVC for negative κ begins at a much later time. When G_K-short is reduced further (Fig. 4C), the first PVC in both negative and positive κ occurs at roughly the same time. Finally, we calculated T versus κ for the parameter set as in Fig. 1C. The minimum T (and thus the earliest timing of the PVC) shifts toward smaller curvature (Fig. 4D, arrow) at a later time. This explains that the later PVC beats in Fig. 1C emerge at a smaller κ (from the corners to the sides) because of the curvature-dependent timing. Note that in Fig. 1C, the RG area only exhibits nonnegative border curvature, but if negative border curvature is present, the earliest location of PVCs can also be the location of negative curvature since the T for the negative curvature can be as early as that for the positive curvature (e.g., the minimum T of the first PVC beat for κ < 0 is similar to that for κ > 0, see the horizontal dashed reference line in Fig. 4D).

As indicated in the inset of Fig. 2A, the 1-D cable is paced simultaneously in the simulations. When we paced the 1-D cable from either the long APD end or the short APD end to allow wave conduction, we observed little change in the phase diagrams (compare Supplemental Fig. S1 with Fig. 2B). Moreover, in the simulations above, we switched from G_K-long to G_K-short suddenly. Although the diffusive coupling gives rise to gradual changes in voltage and APD, as shown in our previous studies (18, 20), altering RG can alter the PVC dynamics. Here, we also changed G_K-long to G_K-short gradually and redid the phase diagram of Fig. 2B. The results are shown in Supplemental Fig. S2, which show that reducing the RG gradient enhances the lift-up mechanisms of PVCs, agreeing with our previous observations (18, 20).

Linking 1-D Predictions Quantitatively to PVC Genesis in 2-D Tissue

As discussed earlier, we can use the 1-D cable simulation results to qualitatively explain the 2-D results in Fig. 1, but a quantitative match would further validate the usefulness of the generalized 1-D equation (Eq. 5). To examine if the 1-D cable predictions can be more quantitatively linked to PVC genesis in 2-D tissue, we carried out 2-D tissue simulations with different RG border curvatures. As shown in Fig. 5A, we placed a triangular area of long APD in the tissue. Because of cell-to-cell coupling, the tip region is smoothed to form a smoothly curved border (Fig. 5A, right, blowup view). We then calculated the RG border curvature, which changes along the border (Fig. 5B). In Fig. 5C, we plotted the maximum curvature (κ_max) versus the angle (θ) of the triangle. Note that when θ > 180°, κ becomes negative. The measured κ_max can be well fitted by a tangential function in the following form:

$${\kappa }_{\text{max}}=\alpha \times \text{tan}(\frac{\theta -180}{2}),$$ (6)

where α is a fitting parameter. In the two cases shown in Fig. 5C, α = –4.5 and α = –2 for G_K-short = 1.25 mS/cm² and G_K-short = 0.5 mS/cm², respectively.

Figure 5.

Quantitatively matching one-dimensional (1-D) predictions to two-dimensional (2-D) tissue simulation results. Two-dimensional tissue simulations are done in a 15.36 × 15.36 cm² tissue using the 1991 Luo and Rudy model (37) (LR1) model with the parameters the same as for Fig. 4A. except G_K-short = 1.25 mS/cm². A single stimulus is applied to all cells at t = 0. A (left): voltage snapshot showing that a triangular long action potential duration (APD) area is placed in the tissue. White solid line, border of triangular area; black dashed line, polynomial fit of the repolarization gradient (RG) border using the iso-voltage line (−37 mV) at a time point when the short APD region fully repolarizes. A, right: blowup view of the area marked by the red square. B: calculated border curvature κ vs. position x for the example shown in A. κ is calculated as follows: the iso-voltage line (black dashed curve in A) is fitted with a polynomial function, and the standard formula is then used to calculate κ of the curve. C: maximum κ (κ_max) vs. the angle θ of the triangle for G_K-short = 1.25 mS/cm² (black dots) and G_K-short = 0.5 mS/cm² (red dots). Black line, ${\kappa }_{\text{max}}\hspace{0.17em}=\hspace{0.17em}-4.5*\text{tan}(\frac{\theta -180}{2})$; red line, ${\kappa }_{\text{max}}=-2\text{*tan}(\frac{\text{θ}-180}{2})$. D–G: voltage line scans along the white dashed line in A for different angle θ. *Premature ventricular complex (PVC) beats.

We first examined the predictions of Fig. 4A. Based on Fig. 4A, when κ < 0.9 cm⁻¹, no PVCs occur. When 0.9 < κ < 1.25 cm⁻¹, only one PVC occurs although the long APD area remains depolarized. When 1.4 < κ < 2 cm⁻¹, persistent PVCs occur. When κ > 2 cm⁻¹, no PVCs occur. In the 2-D tissue simulations, for θ = 150°, which corresponds to κ_max = 1.2 cm⁻¹ calculated using Eq. 6 with α = –4.5, no PVCs occur (Fig. 5D). For θ = 145° (κ_max = 1.42 cm⁻¹), one PVC occurs (Fig. 5E). This indicates that the threshold curvature for one PVC in 2-D is between κ_max = 1.2 cm⁻¹ and κ_max = 1.42 cm⁻¹, which is slightly larger than 0.9 < κ < 1.25 cm⁻¹ predicted by the 1-D cable equation. For θ = 125° (κ_max = 2.38 cm⁻¹), repetitive PVCs occur (Fig. 5F). In the 5-s period shown in Fig. 5F, there are 10 PVCs. In the 1-D cable simulations, there are eight PVCs in 5 s for the border curvature around κ = 1.5 cm⁻¹ (Fig. 4A). Although the 1-D cable predictions and 2-D simulation results are not matched exactly, it is still a good quantitative match. For θ = 30° (κ_max = 16.8 cm⁻¹), repetitive PVCs occur (Fig. 5G) but with a longer period than that for θ = 125°. Note that the 1-D cable prediction is that no PVCs occur when κ > 2 cm⁻¹ (Figs. 2B and 4A), but PVCs still occur for any κ_max > 1.42 cm⁻¹ in 2-D tissue (e.g., Fig. 5G). This is because in 2-D tissue, κ decreases from κ_max to zero (see Fig. 5B) and thus there is always curvature satisfying the requirement. Therefore, in 2-D tissue, as long as κ_max is larger than the threshold κ, then PVCs can occur.

We then examined the predictions of Fig. 4B. Based on Fig. 4B, no PVCs occur when –3.6 < κ < 1.25 cm⁻¹ and $\kappa >2$ cm⁻¹. In the 2-D tissue (Fig. 6), we set a V-shaped long APD area, which includes both positive and negative border curvature of the same magnitude. For θ = 120°, which corresponds to κ_max = ±1.15 cm⁻¹ calculated using Eq. 6 with α = ±2, no PVCs occur, agreeing with the 1-D prediction. For θ = 110° (κ_max = ±1.4 cm⁻¹) and θ = 90° (κ_max = ±2 cm⁻¹), repetitive PVCs occur in the positive κ side, agreeing with the 1-D predictions that repetitive PVCs occur in this positive curvature range but not in this negative curvature range (κ < –3.6 cm⁻¹ is required for PVCs to occur in negative curvature). There are seven PVCs in 5 s on the positive κ side, agreeing with the 1-D prediction of six PVCs in 5 s (Fig. 4B). For θ = 80° (κ_max = ±2.4 cm⁻¹), in addition to repetitive PVCs on the positive κ side, PVCs occur in the negative κ side. As also predicted in Fig. 4B, the PVCs occur in a much later time (>8 s) and longer PVC intervals than the positive κ side. These simulations indicate that the 2-D results agree well with the predictions of the 1-D cable equation shown in Fig. 4B.

Figure 6.

Quantitatively matching one-dimensional (1-D) predictions to two-dimensional (2-D) tissue simulation results. A V-shaped long action potential duration (APD) area is placed in the 2-D tissue (leftmost), which exhibits both positive and negative repolarization gradient (RG) border curvature. Parameters are the same as for Fig. 4B. except G_K-short = 0.5 mS/cm². The tissue size is 15.36 × 15.36 cm². A single stimulus is given to all cells at t = 0. A–D: voltage line scans along the white dashed line (leftmost) for different angle θ. k_max and θ relationship for G_K-short = 0.5 mS/cm² is shown in Fig. 5C as red dots. *Premature ventricular complex (PVC) beats. E: voltage snapshots taken from time window marked by dashed box in D, showing PVCs arising from both positive (t = 8,020 ms) and negative (t = 8,355 ms) RG border curvature in the same tissue.

Limit of Curvature Effects

In previous studies (19, 20), we showed that in the absence of RG border curvature (κ = 0), the lowest boundary (or the limit) of the PVC regions in the phase diagram is the onset boundary of EADs in single cells (e.g., the lowest red curve in Fig. 7 as indicated by the arrow). Now we ask the following question: can RG border curvature push the PVC regions beyond this limit? We carried out simulations using Eq. 5 and plotted the phase diagram in the σ-G_si plane for a given curvature κ, as the phase diagrams (κ = 0) in our previous studies (19, 20). σ describes the degree of RG, defined as σ = G_K-long/G_K-short. Figure 7A plots the phase diagram for κ = 2 cm⁻¹ (transparent olive) on that for κ = 0 (gray). This positive curvature eliminates the EAD-mediated PVCs in the diagram and reduces the RG-mediated PVC region. Figure 7B is for κ = −2 cm⁻¹, in which the negative border curvature enlarges the RG-mediated PVC region, as well as the EAD-mediated PVC regions. However, the PVC region cannot go below the single-cell EAD boundary no matter how large the negative κ is. To show this, we plot the PVC region in the κ-G_si plane with a fixed σ (=0.05). When G_si is large, the boundary of the PVC region changes with κ. However, when G_si is reduced to a certain value, the boundary of the PVC region is not dependent on κ anymore (the boundary becomes horizontal). This G_si is exactly the boundary for the onset of single-cell EADs (linked by the dashed horizontal line in Fig. 7, B and C). This holds true for any σ values we use. This same PVC boundary behavior has already been shown in Fig. 2, D and E (the horizontal boundaries in these phase diagrams). Therefore, the negative border curvature promotes PVCs but cannot move beyond the single-cell EAD boundary. Note that PVCs are suppressed in the upper left corner (arrow) of the phase diagram in Fig. 7C, indicating that PVCs promoted by large G_si may be suppressed by negative curvature because of too much source.

Figure 7.

Limit of border curvature effects on premature ventricular complex (PVC) genesis in tissue with the 1991 Luo and Rudy model (37) (LR1) model. Simulations are done in a one-dimensional (1-D) cable (see schematic diagram at A, bottom) in the same way as in Fig. 2. A single stimulus is applied to all cells at t = 0. A: phase diagram showing PVC regions (olive and gray) in the σ-G_si plane. σ = G_K-long/G_K-short with G_K-short = 1 mS/cm² while varying G_K-long to change σ. Red lines, parameter points at which a new early afterdepolarization (EAD) occurs in the action potential (AP), which are determined from a single cell with the same parameters as for the long AP duration (APD) area (i.e., a single cell with the same G_K-long as in the cable). Numbers 1–4 mark the onsets of first 4 EADs in the AP as G_si increases. “EAD boundary” is the onset of the first EAD in the single cell, below which no EADs can occur. Gray, PVC regions for κ = 0; olive, κ = 2 cm⁻¹. Positive repolarization gradient (RG) border curvature suppresses PVCs (compare olive region with gray). B: same as A but substituting κ = 2 cm⁻¹ by κ = –2 cm⁻¹. Negative RG border curvature promotes PVCs. C: PVC regions (gray) in the κ-G_si plane for σ = 0.5. Vertical and horizontal blue dashed lines in B intersect at the EAD boundary for σ = 0.5, with horizontal line being exact the limit of κ effects on PVC genesis in C.

Curvature Effects on PVC Genesis in Different AP Models

We then carried out simulations to investigate the RG border curvature effects on PVC genesis using additional eight different physiologically more detailed AP models to examine the results of the LR1 model. We modeled a condition of LQTS type 2 in which I_Kr was removed in each model. The simulations are done using the same 1-D cable model (Eq. 5) as in Fig. 7. RG is modeled by changing G_Ks suddenly in the cable as for the LR1 model. The phase diagrams for κ = 0 and the single-cell EAD boundaries shown in Fig. 8 are replotted from our previous studies (18, 19). For each model, the phase diagrams are plotted the same way as for the LR1 model shown in Fig. 7, that is, we show the phase diagrams on σ-G_Ca,L plane for κ = 0, κ = 2, and κ = −2 cm⁻¹, and the phase diagram on κ-G_Ca,L plane for a fixed σ. Similar to the LR1 model, σ is defined as σ = G_Ks-long/G_Ks-short. In general, PVCs are suppressed by positive κ but promoted by negative κ for all models, agreeing with those of the LR1 model. However, not all models obey the rule that the limit of the RG border curvature effect is the EAD boundary (see the horizontal dashed red lines in each case). For the LRd, GB, ORd, ToRORd, and UCLA_H models (Fig. 8, A and E–H), the limit of the PVC boundary is exactly the single-cell EAD boundary. However, for the SB, HRd, and TP04 models (Fig. 8, B–D), the border curvature can push the PVC boundary below the EAD boundary.

Figure 8.

Effects of border curvature on premature ventricular complex (PVC) genesis under long QT syndrome (LQTS) type 2 in eight different action potential (AP) models. Simulations are done in a one-dimensional (1-D) cable the same way as in Fig. 7. Shown are phase diagrams plotted in the same way as Fig. 7 for each model, that is, PVC regions (olive and gray) in the σ-G_Ca,L plane for a fixed κ (left and middle) and in the κ-G_Ca,L plane for a fixed σ (right). α(G_Ca,L) is fold of the maximum conductance of L-type calcium current (I_Ca,L). LQTS type 2 is modeled by setting G_Kr = 0. Repolarization gradient (RG) gradient is modeled by changing G_Ks suddenly in the cable similar to Fig. 7 with σ defined as σ = G_Ks-long/G_Ks-short. G_Ks-short is fixed [see Table 1 for G_Ks-short value for each model], whereas G_Ks-long is varied to change σ. Red line, early afterdepolarization (EAD) boundary determined from a single cell with the same parameters as for the cells in the long AP duration (APD area, i.e., the same G_Ks-long). Differing from Fig. 7. only the EAD boundary, that is, the onset point of the first EAD in the AP, is plotted. No EADs can be observed in single cells below the EAD boundary. Gray, PVC regions for κ = 0; olive, κ = 2 cm⁻¹ or κ = –2 cm⁻¹. Vertical and horizontal blue dashed lines intersect at the EAD boundary for the given σ to indicate limit of curvature effect. A: 1994 Luo and Rudy (38, 39) guinea pig ventricular AP model (LRd). B: 2004 Shannon et al. (40) rabbit ventricular AP model (SB). C: 2004 Hund and Rudy (41) canine ventricular AP model (HRd). D: 2004 ten Tusscher et al. (42) human epicardial ventricular AP model (TP04). E: 2010 Grandi et al. (43) human epicardial ventricular AP model (GB). F: 2011 O’Hara et al. (44) human endocardial ventricular AP model (ORd). G: 2019 Tomek et al. (45) human endocardial ventricular AP model (ToRORd). H: Huang et al. (18) modified University of California, Los Angeles (UCLA), rabbit ventricular AP model (UCLA_H).

It is not clear why some models do but others do not obey the same rule as in the LR1 model. One observation is that the ones obeying the rule are much easier to exhibit EADs than those not obeying the rule. Another observation is that for those that do not obey the rule, if we couple two cells together with both cells below the EAD boundary so that they do not exhibit EADs individually, the coupled system can exhibit EADs (Supplemental Fig. S3A). For those obeying the rule, coupling two non-EAD cells cannot produce EADs (Supplemental Fig. S3, B and C). Although the two-cell system is not equivalent to the heterogeneous 1-D cable, we can imply from this observation that for those that do not obey the curvature limit rule, tissue-scale EADs may be induced because of cell coupling, whereas for those obeying the rule, no tissue-scale EADs can be observed unless EADs already occur in the single cells. Note that the limit of the PVC boundary can be satisfied partially in the HRd and TP04 models (indicated by arrows in Fig. 8, C and D). One can observe multiple horizontal PVC boundaries in the ORd and ToRORd models (Fig. 8, F and G) due to multiple EAD boundaries in each model.

The results shown in Fig. 8 indicate that in all models, positive border curvature suppresses PVCs and negative border curvature promotes PVCs. However, as observed in the LR1 model, PVCs are also promoted by positive border curvature in all the AP models when the source in the long APD area and the sink in short APD area are both strong enough, with similar phase diagrams shown in Fig. 2B (see Supplemental Fig. S4 for results of the LRd and TP04 model).

Modulation of PVC Genesis by Wavefront Curvature

In the sections above, we paced the whole tissue simultaneously (and thus no wave conduction) to investigate the effects of RG border curvature to avoid the confluent effects of conduction. Wavefront curvature not only affects conduction velocity but also repolarization (36), and thus it can affect the PVC genesis. One typical case is in a tissue structure where a narrow pathway suddenly expands, which can result in a large wavefront curvature that affects conduction (46). This structure can occur in fibrotic tissue or at the Purkinje-ventricular junctions. Figure 9, A and B, shows a case in which a PVC is promoted at the sudden opening of a narrow pathway. We used the S1S2 pacing protocol with many S1 beats to avoid the effects of the initial condition. However, when the pathway is wide enough, PVCs can no longer occur (Fig. 9C). To show how the wavefront curvature affects the PVC genesis, we used the generalized 1-D cable model (Eq. 5) with a linear function κ (x) = βx to describe the curvature gradient. Figure 9D shows the phase diagram using this function for heterogeneous curvature, which shows that PVCs are promoted by positive wavefront curvature and suppressed by negative wavefront curvature but the effect is small.

Figure 9.

Effects of wavefront curvature on premature ventricular complex (PVC) genesis. Simulations are done in a two-dimensional (2-D) tissue connected with a narrow channel. The 1991 Luo and Rudy model (37) (LR1) model is used. To avoid the influence of the initial condition, a S1S2 protocol is used in which the S1S1 interval is 500 ms and S1S2 interval is 1,500 ms. Nine S1 beats are applied before the S2 beat. The pacing stimuli are applied at the beginning of channel eliciting electrical waves conducting from channel to the tissue. A: PVC promoted in a tissue with 0.3-cm channel. Top: voltage snapshots from different time points of the S2 beat that propagates from left to right. Bottom: voltage snapshots from different time points of the PVC beat. Arrows indicate the initiation and propagation of the PVC that propagates from right to left in B. Voltage recorded from a horizontal line through the channel in the simulation in A. C: same as B but for a wider channel (0.75 cm). No PVC occurs. D: PVC region in the β-G_si plane from a one-dimensional (1-D) cable (Eq. 5) with κ = βx. The same S1S2 protocol is used. For A–C, parameters are G_si = 0.306412 mS/cm² and G_si = 1.1 mS/cm², and the rectangular tissue size is 3.84 × 7.68 cm². For D, G_K = 1 mS/cm² and the cable length is 3 cm.

Effects of Propagation Direction on PVC Genesis

Besides the wavefront curvature, the conduction itself can affect the genesis of PVCs. This is because during conduction, there is a voltage gradient between cells, which generates a diffusive current to affect repolarization and dynamics (47). This current depends on the RG at the boundary and the direction of conduction. To examine how big the effects of conduction on PVC genesis, we first carried out simulations of a 1-D cable without curvature (κ = 0) and a long APD area in the center of the cable (Fig. 10A, inset). The cable was paced from the left end and the paced beat conducted from left to right. We used the S1S2 pacing protocol with many S1 beats to avoid the effects of the initial condition. We observed three modes of PVC conduction: 1) PVCs conduct in both directions, 2) PVCs conduct toward left only, and 3) PVCs conduct toward right only. For the RG-mediated PVCs, we observed two modes of PVC conduction: modes 1 and 2. Figure 10A shows a phase diagram in a small parameter window from Fig. 7A showing that the mode 2 conduction occurs at the PVC boundary region (red) and the majority is the mode 1 conduction (blue). For the EAD-mediated PVCs, we observed all three modes of conduction (Fig. 10B). The different modes of conduction are a result of the difference in diffusive current at the border region (Fig. 10C): the diffusive current is larger at the ramp-up border (olive) than that at the ramp-down border (magenta). In the RG-mediated PVC case, a larger diffusive current at the ramp-up border helps the genesis of a PVC at this border that conducts toward left (mode 2). However, for the EAD-mediated PVCs, this current can either promote or suppress PVCs at the border because of the requirement that the source and the sink have to be properly matched for PVCs to occur (20), and thus we observed all three modes of PVC conduction.

Figure 10.

Effects of propagation direction on premature ventricular complex (PVC) genesis. The 1991 Luo and Rudy model (37) (LR1) model is used for the simulations. To avoid the influence of initial condition in the simulations, we use a S1S2 pacing protocol as in Fig. 9 with a S1S1 interval 400 ms (21 beats) and a S1S2 interval 2,000 ms in both the one-dimensional (1-D) cable and two-dimensional (2-D) tissue. A: phase diagram shows PVC conduction modes (colored in the same way as arrows, indicating conduction directions in inset) in the σ-G_si plane for the repolarization gradient (RG)-mediated PVCs in a heterogeneous 1-D cable (cable setup, inset). Cable length is 4.5 cm (300 computational cells) with the middle segment (red) being the long action potential duration (APD) area. Stimuli are applied at the left end of the cable. Inset: conduction direction of the paced (S1 and S2) beats (black arrows) and the three modes of PVC genesis (colored arrows): 1) PVC occurs in both RG borders (blue arrows), 2) PVC occurs in the left border (red arrow), and 3) PVC occurs in the right border (cyan arrow). Parameters are the same as in Fig. 7A but from a smaller window. B: same as A but for early afterdepolarization (EAD)-mediated PVCs. In this case, all three modes of PVC genesis can be observed. Parameters are the same as in Fig. 7A but from a smaller window. C: diffusive current recorded from the two borders of the cable (inset). Current is slightly larger in the upward RG border (olive) than that of the downward border (magenta). G_si = 0.16 mS/cm², G_K = 0.385 mS/cm² in the long APD area, and G_K = 1 mS/cm² in the short APD area. D–F: voltage snapshots taken from different time points showing PVCs arising from different border locations in a 2-D tissue with a circular long APD area (dashed circle). The times on above the panels are counted from the last S1 beat. Parameters are the same for the three cases, except slightly different G_si values: G_si = 0.211 mS/cm² (D), G_si = 0.2118 mS/cm² (E), and G_si = 0.2123 mS/cm² (F). Size of the 2-D tissue is 15.36 × 15.36 cm², and radius of the circular area is 6 cm. G_K = 0.96 mS/cm² in the long APD area, and G_K = 1 mS/cm² in the short APD area. *Origins of PVCs.

To show this in 2-D tissue, we set a circular area of long APD in the center of the tissue to maintain symmetry. The tissue was paced from the bottom using the same S1S2 pacing protocol to result in planar waves conducting upward (see Fig. 10, D–F). Depending on the parameters, the PVCs occurred at the lower border (Fig. 10D), two sides (Fig. 10E), and upper border (Fig. 10F).

DISCUSSION

In this study, we performed computer simulations to systematically investigate the effects of RG border curvature and wavefront curvature on PVC genesis under the conditions of QT prolongation. Based on the transformed 1-D equations (Eqs. 4 and 5), the curvature generates an additional diffusive current (the curvature terms in the 1-D equations) to affect PVC genesis. Using this transformation, we could systematically investigate the curvature effects using the simplified 1-D equations which allowed us to gain mechanistic insights. We showed that PVCs could be either promoted or suppressed by negative or positive RG curvature depending on the source and sink conditions of the tissue. Our specific findings are as follows: 1) when the origin of the PVC is in the sink area, negative RG border curvature provides a source to potentiate oscillation in the short APD area for PVC genesis; 2) when the origin of the PVC is in the source area, positive RG border curvature provides a sink to potentiate oscillation in the long APD area for PVC genesis; 3) under the condition when the source is strong and the sink is proper, PVCs can be promoted simultaneously by both positive and negative RG border curvature in the same tissue; 4) the negative RG border curvature can substantially promote EAD-mediated and lift-up mechanism of PVCs, and thus can greatly increase the risk of arrhythmogenesis; 5) PVCs are promoted by convex wavefront and suppressed by concave wavefront; however, the effect of wavefront curvature on PVC genesis is much smaller than that of RG border curvature; and 6) PVC genesis can be modulated by conduction direction because of the conduction-dependent difference in diffusive current in the RG border region.

RG Border Curvature-Mediated Source and Sink Effects on PVC Genesis

In the traditional mechanisms of PVC genesis, the formation of a PVC is subjected to the well-known source-sink effect (17), i.e., an enough number of source cells exhibiting automaticity, or EAD- or DAD-induced triggered activity is needed to overcome the sink effect of the surrounding cells to initiate a PVC. However, under the LQTS conditions, the EAD amplitude is relatively small and the takeoff potential is much higher (in phase 2 of the AP) than the resting potential (19, 48–50), and thus EADs cannot propagate. In recent studies (18–20, 25), we have shown that the RG-mediated PVCs are oscillations caused by Hopf bifurcations (25), whereas the EAD-mediated PVCs depend on how far away the EAD is from the homoclinic bifurcation point (20). Both mechanisms are tissue-scale dynamical instabilities originating from the steep RG region, which cause the earliest site of the PVC to occur in the steep RG region. This agrees with the experimental observation that the PVCs originated in the RG region (18, 26–30).

Although the latter mechanisms of PVCs require certain conditions for dynamical instabilities to occur, the source and sink effects are still important and the RG border curvature modulates the source and sink effects to potentiate or suppress PVCs. As shown in our previous study (18), in the RG-mediated PVCs, there are two oscillation modes. One mechanism of oscillation is that the short APD cells pull down the long APD cells to oscillate, and the oscillation stops if the voltage of the long APD cells is clamped at the plateau voltage (e.g., Fig. 3C). In other words, the coupling of short and long APD cells causes the oscillation to occur on the long APD side. The other mechanism of oscillation is that the long APD cells lift up the short APD cells to oscillate but the oscillation continues if the voltage of the long APD cells is clamped at the plateau voltage (e.g., Fig. 3B). In other words, the coupling of the short and long APD cells causes the oscillation to occur on the short APD side. Adding the curvature effects to the mechanisms of oscillation, one can have the following insights:

• 1. For the lift-up mechanism, PVCs are promoted by negative border curvature but suppressed by positive border curvature (Figs. 2, 4, 7, and 8). This is because the negative border curvature provides additional source to the cells in the long APD area, helping to lift the cells in the short APD area to oscillate. However, if the curvature is too negative, the oscillation can also stop because of too much source from the curvature. In other words, for oscillation to occur, the source and sink need to be properly matched.
• 2. For the pulldown mechanism, PVCs are generally promoted by positive RG border curvature but can also be promoted by negative RG border curvature (Fig. 2, B and C). This depends on the strength of the source. If the source is very strong, then a positive RG border curvature helps the sink to pulldown the source for oscillation. If the source is not strong enough, a negative curvature can avoid too much sink to pull down the source cells without oscillation. Again, a too negative or too positive RG border curvature will suppress the oscillation because of the mismatch between the source and sink effects.

In addition to the RG-mediated PVC mechanisms, RG border curvature also modulates the genesis of the EAD-mediated PVCs. In general, EAD-mediated PVCs are suppressed by positive RG border curvature but promoted by negative RG border curvature (see the curvature effects for the LR1, ORd, and ToRORd models shown in Figs. 7 and 8).

As shown in our previous studies (18–20), the RG-mediated PVCs require a stronger I_Ca,L and a larger RG than the EAD-mediated PVCs, and the lowest PVC boundary is the single-cell EAD boundary. The negative RG border curvature can help push the PVC boundary to this limit in most of the AP models but can push beyond this limit in other AP models. As shown in our simulations, the AP models obeying this limit are those prone to EADs, but those violating this limit are less prone to EADs. Moreover, in the AP models less prone to EADs, the occurrence of EADs and PVCs requires a very high I_Ca,L conductance (Fig. 8).

In the presence of RG border curvature, the gap junction coupling may become nontrivial. In our previous study (19), we showed that altering diffusion constant D has a small effect on PVC genesis in the absence of curvature. Theoretically, if one considers the cardiac tissue as a continuum described by the following partial differential equation: $\frac{\partial V}{\partial t}\hspace{0.17em}=\hspace{0.17em}-\frac{I_{\text{ion}}}{C_{\text{m}}}+D\frac{{\partial }^{2}V}{\partial x^2}$, and for a homogeneous system, one can use the transformation: $\xi =\hspace{0.17em}x/\sqrt{D}$, to obtain a new differential equation as: $\frac{\partial V}{\partial t}=-\frac{I_{\text{ion}}}{C_{\text{m}}}+\frac{{\partial }^{2}V}{\partial {\xi }^2}$. D disappears in the equation and thus D does not affect the dynamics except for the rescaling of space. The transformation is still valid when one changes a parameter suddenly in space (i.e., a step function of space as used in our current study) to model the heterogeneity. However, in a discretized system (the cardiac tissue is indeed discretized), this transformation is only approximate. This is why we only observed a small effect of gap junction coupling on PVC genesis in our previous study (19). In the presence of RG border curvature; however, the gap junction coupling may exhibit nontrivial effects. For example, if one uses the same transformation, then Eq. 5 becomes: $\frac{\partial V}{\partial t}\hspace{0.17em}=\hspace{0.17em}-\frac{I_{\text{ion}}}{C_{\text{m}}}+\sqrt{D}\kappa \frac{\partial V}{\partial \xi }+\frac{{\partial }^{2}V}{\partial {\xi }^2}$, indicating that the system cannot be simply rescaled (D still appears in the equation) and the gap junction coupling effect is nontrivial. Moreover, when the parameter change in space is not a step function but gradual, the transformation is no longer valid in either the presence or absence of RG border curvature. To demonstrate these effects, we carried out 1-D cable simulations with the absence and presence of RG border curvature for a sudden change in G_K in space (Supplemental Fig. S5). When κ = 0, the PVC region is independent of D, that is, the boundary of the PVC region does not change as D changes (Supplemental Fig. S5A). When κ > 0, the PVC region increases as D decreases (Supplemental Fig. S5, B and C). Moreover, the boundary of the PVC region obeys $\sqrt{D}\kappa =\text{constant}$ (Supplemental Fig. S5C), which is implied from the transformed 1-D equation above.

Finally, in the examples shown in this paper for the two oscillation mechanisms, the long APD area fails to repolarize, maintaining persistent oscillations. As shown in our previous studies (18, 19), the same mechanisms still apply when the long APD area repolarizes and the number of PVCs depends on APD in the long APD area.

Effects of Wave Propagation and Wavefront Curvature on PVC Genesis

Conduction in the real heart is not planar waves but curved wavefronts. A curved wavefront gives rise to a curved wave back which affects the repolarization and thus the genesis of PVCs. Our simulations show that PVCs are promoted by convex wavefronts or positive wavefront curvature. However, the wavefront curvature effect is small (Fig. 9D), in particular, compared with the effects of the RG border curvature. Furthermore, the genesis of PVCs may also be modulated by the conduction direction relative to the directions of the RG because of the difference in diffusive coupling.

Implications of Arrhythmogenesis in Cardiac Diseases

QT prolongation occurs in many cardiovascular diseases, such as drug induced and inherited LQTS, heart failure, and ischemia. PVCs can serve as triggers for reentry arrhythmias via the classical trigger and interaction mechanisms or can degenerate directly into reentry arrhythmias as reviewed in our recent article (8). The PVCs mediated either by RG or EADs originate from the RG region and propagate unidirectionally. Depending on the size and the time of repolarization of the long APD area, they can form quasitarget waves or figure-of-eight reentry (8). The curvature effects of the RG border region potentiate the localization of the initial firing of the PVCs and can potentiate the formation of reentry. The effects of wavefront curvature due to sudden opening can occur in heart failure and ischemia in which fibrosis exists. It can also occur at the Purkinje-ventricular junctions. Therefore, besides that a sufficient number of source cells is required for PVC genesis in cardiac diseases, the geometry of the RG region is also important. Although positive RG border curvature can promote PVCs, it occurs when both the source and the sink are strong. However, negative RG border curvature can lower the threshold for arrhythmias to the threshold of single-cell EADs or beyond, which can greatly increase the risk of arrhythmias. Our results also imply that in the real hearts, the geometry of the RG area can vary among individuals, and even if the QT interval and other biological parameters are identical for two individuals, their risk of arrhythmias can be greatly different if the RG geometries differ.

Limitations

In most of our simulations, a single stimulus with a given initial condition was used to induce PVCs. This protocol allows us to extensively explore the parameter space and thus pinpoint the mechanisms since it is computationally more efficient than using other pacing protocols. For example, in the S1S2 or S1S1 protocol, one paces the heart with many beats, which not only requires much more computer time but also can engage very complex dynamics (51). This can make it much more complex for obtaining the phase diagrams. In the 1-D and 2-D models simulated in this study, the RG was modeled by changing one parameter (the maximum conductance of a potassium current) suddenly in space, which is the simplest way of modeling RG heterogeneity. In real cardiac tissue, the RG heterogeneity can result from gradual changes in parameters and from contributions of many ionic currents. As we showed previously (18, 20) and in Supplemental Fig. S2 in this study, the steepness of RG can exhibit large impacts on PVC genesis and the impact depends on the mechanisms of PVCs. We did not model nonuniform heterogeneities of repolarization and structural heterogeneities, such as random fibrosis in heart failure. These factors can have additional effects on PVC genesis. Furthermore, conduction in the real heart is in a three-dimensional (3-D) system and thus the curvature effects can be more complex. Following the same approach, one can transform of 3-D tissue with a spherical long APD area into a 1-D cable equation in the polar coordinate system as

$$\frac{\partial V}{\partial t}=-\frac{I_{\text{ion}}+I_{\text{sti}}}{C_{\text{m}}}+D\frac{2}{\text{r}}\frac{\partial V}{\partial \text{r}}+D\frac{{\partial }^{2}V}{\partial {\text{r}}^2}.$$ (7)

Similarly, one can extend the generated 1-D cable equation (Eq. 5) to:

$$\frac{\partial V}{\partial t}=-\frac{I_{\text{ion}}+I_{\text{sti}}}{C_{\text{m}}}+2D\kappa \frac{\partial V}{\partial x}+D\frac{{\partial }^{2}V}{\partial x^2}.$$ (8)

If one sets κ′ = 2κ in Eq. 8, then it becomes Eq. 5, and all results obtained using Eq. 5 in the present paper apply to 3-D tissue. To generate PVCs, we had to change some of the parameters (mainly the maximum conductance of I_Ks and I_Ca,L) many folds in some of the AP models (19), which can be unrealistically large. The results may also depend on the types of LQTS, and we only simulated LQTS type 2 in this study. Nevertheless, the mechanisms of PVC genesis remain valid in different AP models and thus should be applicable in general.

Conclusions

Besides the presence of RG or EADs, the genesis of PVCs depends on wavefront curvature, propagation direction, and the geometry or curvature of the RG region. The curvature or conduction direction modulates the diffusive current to affect the source and sink effects, which can either promote or suppress PVCs depending on the conditions and mechanisms of PVCs. Therefore, under LQTS, the genesis of PVCs requires dynamical instabilities that are determined by not only the RG and I_Ca,L conductance but also the geometry of the RG region via the source and sink effects. The negative RG border curvature can lower the threshold for arrhythmias to the threshold of single-cell EADs or beyond, greatly increasing the risk of arrhythmias.

DATA AVAILABILITY

Data will be made available upon reasonable request.

SUPPLEMENTAL DATA

Supplemental Figs. S1–S5: https://doi.org/10.6084/m9.figshare.25395589.v1.

GRANTS

This work is supported by National Heart, Lung, and Blood Institute Grants R01 HL134709, R01 HL139829, R01 HL157116, and P01 HL164311. Z.Z. is supported by National Natural Science Foundation of China Grants 12375033, 11975131, and 12235007; Zhejiang Provincial Natural Science Foundation of China Grant LY23A050002; and Ningbo University K. C. Wong Magna Fund. Y.Z. is supported by Ningbo University Graduate Student Scientific Research and Innovation Project grants.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

Z.Q. conceived and designed research; Y.Z. and Z.Z. performed experiments; Y.Z., Z.Z., and Z.Q. analyzed data; Z.Z. and Z.Q. interpreted results of experiments; Y.Z., Z.Z., and Z.Q. prepared figures; Z.Q. drafted manuscript; Z.Z. and Z.Q. edited and revised manuscript; Y.Z., Z.Z., and Z.Q. approved final version of manuscript.