A Computational Study of the Factors Influencing the PVC-Triggering Ability of a Cluster of Early Afterdepolarization-Capable Myocytes

Premature ventricular complexes (PVCs), which are abnormal impulse propagations in cardiac tissue, can develop because of various reasons including early afterdepolarizations (EADs). We show how a cluster of EAD-generating cells (EAD clump) can lead to PVCs in a model of cardiac tissue, and also investigate the factors that assist such clumps in triggering PVCs. In particular, we study, through computer simulations, the effects of the following factors on the PVC-triggering ability of an EAD clump: (1) the repolarization reserve (RR) of the EAD cells; (2) the size of the EAD clump; (3) the coupling strength between the EAD cells in the clump; and (4) the presence of fibroblasts in the EAD clump. We find that, although a low value of RR is necessary to generate EADs and hence PVCs, a very low value of RR leads to low-amplitude EAD oscillations that decay with time and do not lead to PVCs. We demonstrate that a certain threshold size of the EAD clump, or a reduction in the coupling strength between the EAD cells, in the clump, is required to trigger PVCs. We illustrate how randomly distributed inexcitable obstacles, which we use to model collagen deposits, affect PVC-triggering by an EAD clump. We show that the gap-junctional coupling of fibroblasts with myocytes can either assist or impede the PVC-triggering ability of an EAD clump, depending on the resting membrane potential of the fibroblasts and the coupling strength between the myocyte and fibroblasts. We also find that the triggering of PVCs by an EAD clump depends sensitively on factors like the pacing cycle length and the distribution pattern of the fibroblasts.


Introduction
Life-threatening cardiac arrhythmias, like ventricular fibrillation (VF), are associated with the abnormal propagation of waves of electrical activation through cardiac tissue [1,2]. The degeneration from a normal heart beat to an irregular heart beat (like in VF) can occur, inter alia, if The remaining part of this paper is organized as follows. The Section entitled Materials and Methods describes the models we use and the numerical methods we employ to study them. The Section entitled Results contains our results, from tissue-level simulations. The Section entitled Discussions is devoted to a discussion of our results in the context of earlier numerical and experimental studies.

Materials and Methods
For our myocyte cell we use the O'Hara-Rudy model (ORd) for a human ventricular cell [28] with the modifications as implemented in Ref. [30], where the fast sodium current (I Na ), of the original ORd model, has been replaced with that of the model due to Ten Tusscher and Panfilov [29] (This was actually suggested by O'Hara and Rudy themselves (see Ref. [31])). This modification is implemented because of the slow conduction velocity of the original ORd model in cardiac-tissue simulations [30]. To simulate fibrosis in our study, the collagen deposits from the fibroblasts are modelled as inexcitable point obstacles by setting the diffusion constant D = 0. The fibroblasts are modelled as passive cells, as is done in other computational studies [32,33]; for these fibroblasts we use the model given by MacCannell, et al. [34]. The membrane capacitance of the fibroblasts is taken to be 6.3 pF, and their membrane conductance is 4 nS. The default gap-junctional conductance between fibroblasts and myocytes is 8 nS, unless mentioned otherwise in the text.
In a unit of a myocyte-fibroblast composite, the membrane potential of the myocyte V m is governed by the ordinary differential equation (ODE) where C m is the myocyte capacitance, which has a value of 185 pF; I ion is the sum of all the ionic currents of the myocyte, and I gap is the gap-juctional current between the fibroblast and myocyte. We give I ion and I gap below: I ion ¼ I Na þ I to þ I CaL þ I CaNa þ I CaK þ I Kr þ I Ks þ I K1 þ I NaCa þ I NaK þ I Nab þ I Cab þ I Kb þ I pCa ; ð2Þ here V f is the membrane potential of the fibroblast, and G gap is the gap-junctional conductance between the fibroblast and myocyte. A glossary of all the ionic currents of the myocyte I ion is given in Table 1.
The membrane potential of the fibroblast is governed by the equation where C f is the membrane capacitance of the fibroblast, and I f is the fibroblast current, here G f is the membrane conductance of the fibroblast, and E f is the resting membrane potential of the fibroblast.
The spatiotemporal evolution of the membrane potential (V m ) of the myocytes in tissue is governed by a reaction-diffusion equation, which is the following partial-differential equation (PDE): where D is the diffusion constant between the myocytes. I gap = 0 if no fibroblast is attached to the myocyte.

Numerical Methods
We use a forward-Euler method to solve the ODEs Eqs (1) and (4) for V m and V f , respectively, and also for the ODEs for the gating variables of the ionic currents of the myocyte. For solving the PDE Eq (5), we use the forward-Euler method for time marching with a fivepoint stencil for the Laplacian. We set D = 0.0012 cm 2 /ms. The temporal and spatial resolutions are set to be δx = 0.02 cm and δt = 0.02 ms, respectively. The conduction velocity in the tissue, with the above set of parameters, is 65 cm/s. In our two-dimensional (2D) tissue simulations, we use a domain size of 448 × 448 grid points, which translates into a physical size of 8.96 × 8.96 cm 2 . All our 2D simulations are carried out for a duration of 15 seconds. For pacing the tissue, we use an asymmetric pacing protocol, in which an external stimulus is applied over a small region (0.14 ×3 cm 2 ) on the lower boundary of the domain as shown in Fig 1. The strength and duration of the stimulus of this asymmetric pulse are -150 μA/μF and 3 ms, respectively.

Results
We now present the results of our studies of the factors that are important in triggering PVCs by an EAD clump. We first elucidate the effects of the repolarization reserve (RR) on PVCs.
We then investigate the effects of the size of EAD clumps on PVCs. We follow this with a study of the effects of the coupling strengths on PVC triggering. We then investigate how the presence of fibroblasts modulate the triggering of PVCs. Finally, we present a few representative results to show how a spiral wave interacts with an EAD clump.

Role of Repolarization reserve in triggering PVCs
The repolarization reserve (RR) is defined as the ability of the cell to repolarize after it is depolarized. and hence the RR in the type-II AP is lower than that of the type-I AP. In the type-I AP (see Fig 2) the EAD oscillations have an amplitude that increases with time until the AP repolarizes to its resting value. By contrast, the type-II AP has decaying EAD oscillations, which relax to a potential higher than the normal resting membrane potential. (Type-II APs, with repolarization failure, are not just a result of computational models, but, they are also seen in experiments [35].) The amplitude of the EAD oscillations in the type-II AP is lower compared to that for the type-I AP.
We now consider two clumps; one of these yields APs of type-I and the other APs of type-II. The clumps are circular and are of the same radius R = 2.4 cm. The circular clumps are embedded in the middle of our simulation domain and we then pace the domain by using asymmetric pulses at a pacing cycle length (PCL) of 1000 ms. Fig 3 shows the sequence of pseudocolor plots of V m that we obtain at different times as we pace the tissue. The white circular contour in Fig 3 marks the periphery of the EAD clump. The clump with a type-I AP (top panel) triggers PVCs, whereas the clump with a type-II AP (bottom panel) does not trigger PVCs (see S1 Video). Even if we increase R, a clump with type-II APs does not lead to PVCs: although type-II APs arise from a lower RR than do type-I APs, the amplitude of EAD oscillations for the former are too weak to excite cells that lie near the EAD-clump boundary.

Effect of the size of EAD clumps on PVC triggering
We explore the dependence of PVC triggering on the radius R of an EAD clump for type-I APs. Henceforth we refer to the type-I EAD cells as EAD cells. We pace the tissue by using asymmetric pulses at a PCL of 1000 ms.  , we plot the number N of PVCs, triggerd within the 15 seconds of our simulation time, versus R for PCL = 1000 ms (blue curve) and PCL = 1400 ms (red curve). These plots show that N increases, roughly, with R; but the details of the plots in Fig 5 depend on PCL, which is because of the rate dependence of EADs (see, e.g., Ref. [9]); and the non-monotonic and "noisy" behavior of N as a function of R or PCL is a manifestation of the sensitive dependence on parameter values because of the underlying spatiotemporal chaos in our extended dynamical system (this sensitive dependence on parameter values has also been shown extensively, in studies of inhomogeneities in mathematical models for cardiac tissue, in Refs [36,37]). Even the threshold size of the EAD clump, after which it triggers PVCs, depends on factors including the PCL and parameter set we use for the EAD cells.

Effect of coupling strength in triggering PVCs
Here we show how the reduction in the coupling strength, which we achieve by reducing the diffusion constant between the EAD cells, assists in triggering PVCs. We take EAD clumps of the same radius R = 2 cm, but with different values of the diffusion constants D inside the clump. In  PVCs. However, if the coupling strength is reduced, because the cells are weakly coupled, more cells in the clump retain their ability to produce EADs and, therefore, can trigger PVCs. If the coupling strength in the clump is extremely reduced, as in the bottom panel (D = 0.2xD o ), the clump supports waves of small wavelengths and hence small-wavelength spirals develop inside the clump; this increases the PVC-triggering rate. The existence of such mini spirals, inside a region of ionic inhomogeneity, with a low value of D, has also been seen in experiments [38]. Fig 5(B) shows the number N of PVCs for different values of D for PCL = 1000 ms and PCL = 1400 ms, which are indicated by blue and red curves, respectively. The diffusion constant on the horizontal axis is normalized by its normal value (D o = 0.0012 cm 2 /s). This plot shows that N increases as D is reduced and almost saturates for low values of D. Note that N depends sensitively on PCL.

Effects of fibroblasts on PVC triggering
We now present the effects of fibroblasts on the triggering of PVCs by an EAD clump.
Fibrosis and PVCs: We show how the interruption of the coupling between EAD myocytes by collagen deposits, simulated here as inexcitable point obstacles, facilitates the triggering of PVCs. We populate the EAD clump of radius R = 2 cm with point obstacles in a diffuse pattern [33,39,40].  point obstacles. Henceforth we refer to P f as the percentage of fibrosis. The clump with 15 percent fibrosis triggers PVCs, whereas the one with 10 percent fibrosis does not (see S4 Video). The number of PVCs increases with P f up to 40%, as shown in Fig 5(C), when we use PCL = 1000 ms (blue curve) and PCL = 1400 ms (red curve). This increase in the number of PVCs is because, as P f increases, the mean number of neighboring myocytes reduces, which in turn lowers the local source-sink mismatch, and thus promotes PVCs. However, after P f =  Premature Ventricular Complexes from Cluster of Early Afterdepolarization-Capable Myocytes 40%, PVC triggering decreases, because there is a trade-off between P f , which reduces the source-sink mismatch, and the number of EAD cells required for the triggering of PVCs. The number of PVCs drops to zero at P f = 55% for both PCL = 1000 ms and PCL = 1400 ms.
Effects of myocyte-fibroblast coupling on PVC triggering: Fibroblasts can form heterocellular couplings with myocytes and influence the EAD-triggering ability of EAD cells, and can, thereby, modulate the triggering of PVCs. To investigate this, we take an EAD clump of radius R = 2.2 cm, which does not trigger PVCs (cf . Fig 4), and attach each of the EAD cells in the clump with a fibroblast. This clump is, therefore, a layer of EAD myocytes with a layer of fibroblasts on top of it as in Ref. [32]. As before, we pace the tissue at PCL = 1000ms. If the EAD clump is attached with fibroblasts that have a resting membrane potential E f = -30 mV, the clump triggers PVCs as shown in Fig 8 (bottom panel). However, if E f is reduced to -35 mV, the clump does not trigger PVCs (top panel) [see S5 Video]. This can be understood by examining the effect of E f on an EAD myocyte-fibroblast composite. Fig 9(A), 9(B) and 9(C) show the V m of an isolated EAD myocyte, an EAD myocyte coupled to a fibroblast with E f = -35 mV, and -30 mV, respectively, with PCL = 1000 ms. The APs of the EAD myocyte, attached to a fibroblast with E f = -30 mV (Fig 9(C)), show an enhancement in the EAD oscillations as compared to the APs of an isolated EAD myocyte in the sense that the last AP in Fig 9(C) shows non-decaying EAD oscillations, unlike the ones in Fig 9(A), which always repolarize. This implies that the coupling of an EAD myocyte to a fibroblast, of E f = -30 mV, enhances the EAD oscillations, and hence increases the PVC-triggering ability of the EAD clump. By contrast, in Fig 9(B), the fibroblast with E f = -35 mV suppresses the EADs of the myocyte and, therefore, subdues the PVC-triggering ability of the EAD clump. Thus, the PVC-triggering ability of an EAD clump, attached to fibroblasts, strongly depends on the E f of the fibroblasts: the ability of the clump to trigger PVCs may either be enhanced (at high values of E f ) or suppressed (at low values of E f ). The other factor that regulates the PVC-triggering ability of an EAD clump is the gap-junctional coupling strength G gap between the myocyte and the fibroblast. In an EAD myocytefibroblast composite, G gap modulates the amount of influence of the fibroblasts on the EAD myocyte. So, if G gap is high, the fibroblast may either assist or impede the EAD-triggering ability of the EAD myocyte depending on the value of E f ; however, if G gap is low, the fibroblast does not influence the electrophysiology of the EAD myocyte significantly, and the EAD myocyte retains its ability to trigger EADs. Therefore, a PVC-triggering clump, which does not trigger PVCs when coupled strongly (high G gap ) to fibroblasts of low E f , may regain its ability to trigger PVC if G gap is reduced. To demonstrate this, we take an EAD clump of R = 2.4 cm, which triggers PVCs without the presence of fibroblasts (cf. Fig 4, bottom panel). We first attach the clump with fibroblasts of E f = -35 mV with G gap = 8 nS, and observe that the PVC triggering ability of the clump is suppressed, as shown in Fig 10, top panel. But if we reduce the G gap to 1 nS, the clump regains its ability to trigger PVCs as shown in Fig 10, bottom panel (see Effects of randomly attaching fibroblasts to an EAD clump: We have, so far, employed an EAD myocyte-fibroblast bilayer clump in which a uniform layer of fibroblasts is placed atop a layer of EAD myocytes. We now explore a similar bilayer clump but with a random array of fibroblasts in the top layer. We study such a model because, in a diseased heart, the distribution of fibroblasts is random [41]; furthermore, the density of fibroblasts depends on the age of a patient [42]. To investigate the effects of inhomogeneously distributed fibroblasts, we carry out an illustrative simulation of an EAD clump with radius R = 2.2 cm, on top of which we have a layer of fibroblasts distributed randomly. The percentage P a of the fibroblasts is 40%, i.e., in the top layer only 40% of the sites are occupied by fibroblasts. We show this random array of fibroblasts in the top panel of Fig 11, in which brown, green, and blue colors indicate, respectively, EAD myocyte-fibroblast composites, EAD myocytes, and normal myocytes. This clump, attached randomly to fibroblasts with E f = -35 mV, triggers PVCs as shown in Fig 11 (bottom panel) when we pace with PCL = 1000 ms. This triggering of PVCs by the clump is counterintuitive, because this value of E f = -35 mV suppresses EADs as shown in Fig 9(C), and hence should prohibit the triggering of PVCs as in Fig 8 (top panel). This result implies that the inhomogeneous distribution of fibroblasts plays a role in triggering PVCs.
To study the dependence of PVC triggering on the spatial distribution of fibroblasts, we take a one-dimensional cable of 280 myocytes and study the four different EAD myocyte-fibroblast distribution patterns, which are shown in the top panels of Fig 12(A), 12(B), 12(C) and 12 (D). The blue, green, and brown colors indicate, respectively, normal myocytes, EAD myocytes, and EAD myocyte-fibroblast composites. Pattern (A) has an EAD segment of 160 EAD myocytes (60 x 220) sandwiched between normal myocytes; pattern (B) has fibroblasts , we see that this plateau is rugged in the sense that it shows many sharp peaks and valleys, which are associated with the nonuniform distribution of fibroblasts (top panel of Fig 12(D)). The presence of such local minima in the V m of the EAD segment favors the formation of EADs and, thereby, PVCs. To illustrate this, we show, in Fig 13, three plots of action potentials (APs) recorded from the central myocyte of the EAD segment (cell number 140) for the patterns of Fig 12(A), 12(B) and 12(C) (blue, red, and black APs, respectively). We see an EAD in the black curve, but no EADs in the other two, which can be explained as follows. The gap-junctional current I gap between the myocyte and fibroblast, in the early phase of the action potential, provides a transient  outward current (resembling the transient outward potassium current I to ) [43], where I gap flows from the myocyte to the fibroblast. This reduces the plateau voltage of the AP; such a lowering of the plateau voltage is known to promote EADs, because the lowering of the plateau potential delays the activation of I ks and enhances the inward calcium current I CaL [44]. This is why we find EADs in the black AP in Fig 13 (fibroblast pattern as in Fig 12(C)) but not in the blue AP (without fibroblasts as in Fig 12(A)). The red AP in Fig 13 (fibroblast pattern as in Fig  12(B)) has a low plateau voltage, like its black counterpart, but it does not show EADs, because it has a steeper repolarizing phase than the black AP. This difference in steepness follows from the qualitative difference in the plots of V m , along the cable in Fig 12(C) and 12(B). In Fig 12  (C), the maxima in V m lead to an influx of current from the regions with EAD myocytes into the region with fibroblasts; such an influx is absent in Fig 12(B), because the plot of V m in the Fig 13. Action-potential plots showing the enhancement in the EAD depending on the EAD myocyte-fibroblast pattern (Fig 12). Plots of V m versus time showing the three action potentials recorded from the middle myocyte of the EAD segment (cell number 140) for three different configurations of the cable: the blue AP is for the case where the EAD segment has no fibroblasts attached to it (Fig 12(A)); the red AP arises when the EAD segment is homogeneously coupled to fibroblasts (Fig 12(B)); and the black AP is obtained when the fibroblasts are attached to the middle of the EAD segment (Fig 12  (C)). EAD segment has a flat plateau. Similarly, the peak-and-valley structure in the plot of V m in the EAD segment in Fig 12(D) leads to EADs, and, thereby, to PVCs. Thus, the triggering of PVCs depends not only on the electrophysiological properties of fibroblasts and myocytes, but also on the spatial pattern of fibroblasts and EAD myocytes. Note that, although E f = -35 mV is low enough to suppress EADs in an isolated EAD myocyte-fibroblast composite (Fig 9(B)), this does not necessarily imply the suppression of EADs (or PVCs) in tissue; this is an example where the single-cell level results cannot be used to predict the results at the level of tissue.
We show in Fig 14, for a 1D cable with randomly attached fibroblasts in its EAD segment, the different numbers of PVC triggerings, indicated by different colors, in the E f -P a plane, for two different realizations of the random configuration of fibroblasts. Fig 14 illustrates the sensitive dependence of PVC triggerings on E f , P a , and distribution pattern of fibroblasts on the EAD clump. There are no PVCs in the region with low values of E f and P a .
Experimental evidence shows that the membrane conductance G f of fibroblasts can depend on the voltage V f [45]. Therefore, we have carried out illustrating studies in which G f has a nonlinear dependence on V f . In particular, we use two different values of G f , like in Refs. [46,47], instead of using a constant value of G f . The value of G f is set to 2 nS if V f is below -20 mV, and 4 nS for V f above -20 mV. Specifically, we find that high E f values trigger PVCs (S1 Fig); reducing the myocyte-fibroblast coupling strength G gap allows the EAD clump to regain its ability to trigger PVCs (otherwise it does not trigger PVCs at high values of G gap ) (S2 Fig); and distributing the fibroblasts randomly assists the triggering of PVCs (S3 Fig).

Spiral-Wave Dynamics
Here we study the spatiotemporal evolution of a spiral wave, which we initiate in the presence of an EAD clump. The size of the simulation domain is 19.2 × 19.2 cm 2 . We initiate the spiral near to the clump using S1-S2 cross-field protocol [48,49]. The spiral is attracted by the clump and it gets anchored to the clump, as we show in the middle and bottom panels of Fig 15. Such an Premature Ventricular Complexes from Cluster of Early Afterdepolarization-Capable Myocytes attraction of spiral waves has been observed in the presence of small ionic inhomogeneities in Ref. [50]. Our study illustrates the attraction of spiral waves also in the case of an EAD clump. In the presence of EAD clumps, the EADs trigger excitations, which can disrupt the motion of spiral tips, as we indicate by the white arrows in the middle and bottom panels of Fig 15 at times 13900 ms and 13960 ms, respectively. The excitations block the progression of the spiral tips and, in turn, generate new spiral tips (see S7 Video). Such excitations can destroy the periodicity of the spiral. If the radius of the clump is small, the periodicity of the spiral is maintained and the frequency of the spiral is reduced; but, if the radius is large, the excitations destroy the periodicity of the spiral. Fig 16(A), 16(B) and 16(C) show the averaged power spectra from four representative points, located near the four corners of the simulation domain, for the freely rotating spiral (Fig 15 top panel), the spiral attached to the clump of radius R = 2 cm (Fig 15 middle  panel), and the one of radius R = 3 cm (Fig 15 bottom panel), respectively. This figure shows that the spiral attached to the clump of radius R = 2 cm still executes a periodic motion, with the frequency reduced to 3.64 Hz as compared to the freely rotating spiral with a frequency of 4.38 Hz. However, the temporal evolution of the spiral, attached to the clump of radius R = 3 cm, is quasiperiodic with two incommensurate frequencies; the peaks in Fig 16(C) can be indexed as n 1 ω 1 + n 2 ω 2 , where n 1 and n 2 are integers and ω 1 ' 2.73 Hz and ω 2 ' 3.64 Hz.

Discussion
Regional heterogeneities occur in the heart because of pathologies like myocardial ischemia [51,52]; such heterogeneities may also exist inherently in ventricles in a normal heart [53,54]. Thus, it is important to investigate how pathological cells in a localized region are arrhythmogenic. Here, we investigate the factors that can assist a cluster of EAD cells to trigger PVCs. First, we show that not all APs with EADs lead to PVCs. If the repolarization reserve of the myocyte is such that the EADs have a low amplitude and decaying oscillations as in type-II APs (Fig 2(B)), the EAD clumps cannot trigger PVCs. Second, a clump of EAD myocytes triggers PVCs only after a particular threshold size, characterized here by radius R; and the number N of PVC triggerings increases, roughly speaking, with R; however, the plot of N versus R depends sensitively on parameters such as the pacing cycle length PCL. Third, we show that a reduction in the coupling strength D, between the EAD cells, also assists in triggering PVCs, because, if D is low, more cells in the clump retain their ability to trigger EADs, and, thereby, facilitate the triggering of PVCs. Fourth, we show that fibrosis aids in the triggering of PVCs. The presence of fibrosis, i.e., the interspersion of collagen deposits from fibroblasts (modelled as inexcitable obstacles), reduces the effective number of neighboring myocytes, and, thereby, lowers the local source-sink mismatch, which thus aids in the triggering of PVCs. We show that N increases with the percentage of fibrosis P f , up to 40%, and then decreases as P f increases, indicating a trade-off between the percentage of fibrosis and the number of EAD myocytes for PVC triggerings. This finding, that the increase of the percentage of fibrosis up to an intermediate value facilitates the triggering of PVCs, agrees with the experimental study conducted on intact rat hearts by Morita, et al. [15].
In vitro studies show that, the formation of gap-junctional coupling between fibroblasts and myocytes can be arrhythmogenic, as the coupling can promote EADs [55] or induce ectopic activity [56]. We show how the gap-junctional coupling between the fibroblasts and the EAD myocytes modulates the PVC-triggering ability of the EAD clumps. We demonstrate that the resting membrane potential E f of the fibroblasts plays an important role in modulating the PVC-triggering ability of the EAD clump. The fibroblasts either assist or impede the triggering of PVCs, depending on the value of E f . In an EAD myocyte-fibroblast composite, low values of E f (< -40 mV) suppress EADs and hence suppress PVCs; high values of E f (E f > -32 mV) enhance EADs and hence promote PVCs. Another important factor is the gap-junctional coupling strength (G gap ) between the fibroblasts and myocytes, the conduit for the fibroblast's influence on the electrophysiology of EAD myocytes.
We also find that the triggering of PVCs by an EAD clump, with fibroblasts attached, depends sensitively on the distribution pattern of the fibroblasts. In an EAD clump, which is attached to fibroblasts distributed inhomogeneously, the presence of local minima and maxima in V m , associated with the inhomogeneous distribution of fibroblasts, promotes PVCs. In such EAD clumps, we observe the triggering of PVCs, even if E f is low enough to suppress EADs in the isolated EAD myocyte-fibroblast composite. This result is an example of a case where a single-cell result cannot be extrapolated naïvely to predict tissue-level results. To conclude the part of our study of EAD clumps with fibroblasts, we have shown that the proliferation of fibroblasts in tissue can influence the PVC-triggering ability of an EAD clump in two ways: 1) By depositing collagens in the tissue, which in turn reduces local source-sink mismatch and promotes PVCs; 2) by forming gap-junctional coupling with the myocytes, which either promotes or impedes PVCs, depending on the electrical properties of the fibroblasts and the distribution pattern of the fibroblasts.
Finally, we have shown, with a few representative simulations, how an EAD clump can interact with a spiral, which is initiated near the clump. We observe that the spiral gets anchored to the clump, and the excitations triggered by the EAD clump annihilate the spiral tip and also, in turn, generate a new spiral tip. We find that, if R is small, the spiral rotates around the clump in a periodic manner, but, if R is large, the spiral rotates quasiperiodically.
A disease like heart failure is associated with the features that we have investigated in this paper, namely, the promotion of EADs [24][25][26][27], a reduced coupling strength because of the reduction in the expression of the gap-junctional protein connexin43 [22,23], and also fibrosis [20,21]. Such failing hearts can thus be susceptible to premature ventricular complexes (PVCs). Therefore, our detailed in silico study of the various factors that promote PVCs in a mathematical model for cardiac tissue help us in understanding and characterizing the risk of fatal cardiac diseases associated with PVCs.
We end our discussion with some limitations of our study. Our tissue simulations are restricted to 2D domain without tissue anisotropy. Such tissue anisotropy can affect the triggering of PVCs. An earlier study by Sato, et al. [10] has investigated the effects of tissue anisotropy on the propagation of EADs in 2D tissue; however, simulations in 3D tissue with fiber architecture, or in anatomically-realistic heart can improve our understanding of the propagation of EADs, manifesting as PVCs, in mammalian hearts. We have used an isotropic monodomain representation of cardiac tissue; our study needs to be extended to other tissue models, such as those that use bidomain representations [57], or novel fractional descriptions accounting for tissue microstructure [58]. The maximum fibroblast-myocyte coupling ratio we use in our study is equal to 1, which means that the maximum number of fibroblasts that is connected to a myocyte is 1; we have not investigated the effects of higher fibroblast-myocyte ratio on the triggering of PVCs. However, an earlier study by Morita et al. [15] has found that PVC triggering is prominent at intermediate values of fibroblast-myocyte ratio. Also, the fibroblasts, in our study, are modelled as passive cells by using the model given by MacCannell, et al. [34]. However, there is evidence that fibroblasts can also behave as active cells [34,59,60]. A detailed study of the effects of such active fibroblasts on EAD myocytes and their influence on the triggering of PVCs lies beyond the scope of the present investigation.
Supporting Information S1 Video. Dependence of PVC triggering on the repolarization reserve. Video showing the triggering of PVCs, by an EAD clump, which yields type-I AP (Fig 2(A)). The EAD clump with type-II AP (Fig 2(B)) does not trigger PVCs. The clumps have the same radii R = 2.4 cm. For this video, we use 10 frames per second with each frame separated from the succeeding frame by 20ms in real time. the presence of EAD clumps of radii R = 2 cm, and R = 3 cm, respectively. The spiral anchored to the small clump (R = 2 cm) rotates in a periodic motion, whereas the spiral anchored to the large clump (R = 3 cm) does not exhibit a periodic motion, but rather rotates in a quasiperiodic manner (cf. Fig 16). The EAD clumps trigger excitations that block the progression of the spiral tip and, in turn, generate a new spiral tip. For this video, we use 10 frames per second with each frame separated from the succeeding frame by 20ms in real time.