Comparison of simulated results with experiments – model validation

We initially compared the simulation results with in vitro findings for AP and CaT characteristics in cAF vs. normal sinus rhythm (nSR). The model reproduces one of the hallmarks of electrophysiological cAF-remodeling, AP shortening (Figure 1B). Measured as a decrease of APD₉₀, in simulations AP shortening (31.9%) corresponds closely with previous in vitro studies (Figure 1C). In addition, the more negative (5.5% increase) resting membrane potential (RMP) observed here in cAF as compared to nSR cells is in line with experimental findings (Figure 1C).

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Figure 1. Illustration of cAF-remodeling processes accounted for in the model and consequent changes in electrophysiological properties and Ca²⁺ dynamics.

(A) Schematic presentation of the cell model. Ionic currents and ion concentrations are referred to with I_X and [X^z]_compartment, respectively. Furthermore, NKA =  sodium potassium ATPase, NCX =  sodium Ca²⁺ exchanger, PMCA =  plasma membrane Ca²⁺ ATPase, SERCA =  sarcoplasmic reticulum Ca²⁺ ATPase, PLB =  phospholamban, SLN =  sarcolipin and CRU =  calcium release unit or ryanodine receptor. Colour coding with red and blue refers to increased and decreased activity and/or expression of cellular components (proteins involved in ion transport), respectively. (B & C) cAF-remodeling shortens the AP and hyperpolarizes the membrane. Simulation results are compared to in vitro findings of Yu et al. [42] and Dobrev et al. [43]. (D & E) cAF-remodeling decreases the amplitude of CaT, diastolic Ca²⁺ concentration and SR Ca²⁺ content, corresponding to in vitro results of Voigt et al. [21] (cyan bar), Voigt et al. [14] (red bar), Wakili et al. [10] (diastolic Ca²⁺) and Grandi et al. [11] (CaT_amp and SR Ca²⁺ content). (F & G) Spatiotemporal view of the CaT along the radial direction of the virtual cell in nSR and cAF (x =  distance from sarcolemma).

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Cellular cAF-remodeling also causes dramatic changes to Ca²⁺ dynamics (Figure 1D). In simulations, both reduced diastolic [Ca²⁺]_i (−29.1% for cAF vs. nSR) and the decreased CaT_amp (−62.3% for cAF vs. nSR) match in vitro findings well (Figure 1E). Additionally, a small reduction in sarcoplasmic reticulum (SR) Ca²⁺ content (−23% for cAF vs. nSR), measured as the integral of I_NCX during a caffeine pulse (Figure S2) is observed, which corresponds well with the 18% decrease reported in vitro in cAF myocytes [11]. Furthermore, CaT peak is delayed in simulations (by 49.8% for cAF vs. nSR), which compares well qualitatively with results obtained from a canine AF model [10]. The CaT decay time constant was also increased by 35.6% in cAF vs. nSR, within the reported range for in vitro results (28% [14] and 80% [11]). The spatiotemporal presentation of CaT in Figure 1 F&G shows there is virtually no rise in [Ca²⁺]_i in the central parts of the cAF-remodeled cell, which also corresponds well to in vitro findings [10].

Simulation results also accurately represent the non-linear nature of cardiac myocyte Ca²⁺ dynamics: although the maximum conductance of I_CaL is decreased by 65% in the cAF model as compared to the nSR model, the total Ca²⁺ influx is decreased only by 39.6% overall, as there is less Ca²⁺ dependent inactivation of I_CaL in cAF. These results are also in line with in vitro findings of 42% and 22% reduction in peak vs. integrated I_CaL, respectively [14].

1D tissue simulations reveal restitution properties that also correspond well to in vivo findings in nSR vs. cAF (Figure 2 A–D). Relative APD₉₀ and conduction velocity (CV) changes lie within the measured standard deviation. The model reproduces relative ERP for the nSR case quite well, although this is rather low for the cAF case. The rotor center in a 2D tissue patch for the nSR and the cAF in silico models are depicted in Figure 2 E and F, respectively. The rotor center trajectory of the nSR variant consumes greater area than that of cAF, representing the stabilization of reentrant waves associated with this electrophysiological remodeling. Furthermore, the rotor is meandering comparatively stable during the simulation time of 8 s for the cAF case, whereas the instable rotor center of the nSR case drifts collides with the geometry boundary and vanishes. Movies showing rotor movement for both cases can be found in Supporting Information (Video S1–S2). Simulated dominant frequencies are 4.15 Hz for the physiological case and 11.23 Hz for the cAF model, which compares favorably to the range of measured values of 11.6±2.9 Hz [15].

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Figure 2. Electrophysiological properties in cAF tissue in silico.

A–D) Restitution properties in a 1D tissue beam, compared to in vivo results of Franz et al. [27] for action potential duration at 90% repolarisation (A), Yu et al. [28] for effective refractory period (B), Feld et al. [29] for conduction velocity (C) and wavelength (D). Simulation results are normalised to BCL =  [0.8 0.7 0.6 1.0] s in (A–D), respectively. E&F) Mapping of rotor center trajectories after initiation in 2D tissue shows that in cAF (F) the meandering trajectory occupies a lot less space compared to nSR (E).

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Analysis of cellular remodeling targets in cAF

To evaluate the contribution of individual variables to cAF remodeling, we next simulated changes in each cellular component separately in the model and calculated three resultant biomarkers APD₉₀, AP_tri and CaT_amp for all cell variants. Each cAF-remodeled component was included one at a time, and the AP and CaT characteristics of all cell variants were compared to the nSR myocyte (Figure 3). In simulations, reduced I_CaL alone decreased APD₉₀ by 17.3%, while increased I_K1 caused an even greater reduction of APD₉₀ (by 52.7%). Increased I_NCX and cell dilation each had the opposite effect: APD₉₀ increased by 21.5% and 7.4%, respectively. AP_tri was substantially decreased in cAF (−35.9%) vs. nSR, which appeared to be primarily related to increased I_K1 and reduced I_CaL, as these singular modifications reduced AP_tri by 58.9% and 16.6%, respectively. While cell dilation had almost no effect on AP_tri (+3.1%), cAF-remodeled NCX increased AP_tri quite dramatically (by 22.9%). On the other hand, CaT_amp was impacted most by reduced I_CaL (−46.0%), cell dilation (−20.0%) and increased NCX activity (−12.4%). These modifications also hampered the propagation of intracellular Ca²⁺ waves from sarcolemma to cell center (right column of Figure 3). The crucial role of increased I_K1 and reduced I_CaL in cAF remodeling was further demonstrated in tachy pacing (BCL = 250 ms) simulations. Results showed that without these two remodeling targets, the virtual cell is unable to recover excitability between stimuli during such a fast pacing regime (Figure S5). Investigation of restitution properties in 1D tissue simulations (Figure S7) also revealed the dominant effects of changes in I_CaL and I_K1 on APD, ERP, CV and WL restitution (as known from measurements and illustrated in Figure 2 A–C). Interestingly, the reduction of I_CaL alone led to alternans in electrical properties at higher rates.

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Figure 3. Effect of individual remodeling targets on CaT and AP characteristics in cAF.

(A) normal sinus rhythm (nSR). (B) chronic atrial fibrillation (cAF: all). (C–F) four remodeled cellular components separately (L-type Ca²⁺ current, I_CaL; inward rectified K⁺ current, I_K1; Na⁺/Ca²⁺ exchanger current, I_NCX; and increased cell volume, dilation), respectively. Columns from left to right: action potential (AP), Ca²⁺ transient (CaT) averaged over cell volume, and spatiotemporal presentation of CaT (x =  distance from sarcolemma). Colour scale for right column: 0–1.5 µM corresponds to dark blue – dark red (similar to Figure 1 F&G). All results are obtained at BCL = 1000 ms.

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Similar analysis using the same three biomarkers was also carried out for the cAF-remodeled I_to, I_Kur, SERCA and RyR; the results are shown in Supporting Information (Figure S3). Surprisingly, the effects of SERCA and RyR remodeling on both CaT_amp and AP morphology were very small as compared to the effects of, for example, cAF-remodeled I_CaL and NCX. However, it has been shown in animal studies that increased RyR sensitivity has only transient effects on CaT_amp, as reduced SR Ca²⁺ content balances the effect of increased sensitivity [16]. Furthermore, increased PLB and decreased SLN expressions have opposing effects on the Ca²⁺ affinity of SERCA, so these modifications partially balance one another in the cAF model.

To explore putative targets among the remodeled cellular components for reversing the cAF phenotype, we performed simulations in which we excluded each one of these components independently, and then compared AP and CaT characteristics to the full cAF model (Figure 4). Neglecting the effect of I_K1 remodeling caused a substantial increase in APD₉₀ (+79.7%), while similarly excluding the effects of I_CaL and NCX remodeling, as well as cell dilation had only relatively minor effects (+2.5%, −8.7% and +1.0% change in APD₉₀, respectively). Interestingly, neglecting the effect of remodeled I_K1 renders the virtual cell bistable: depending on initial conditions either a normal or unresponsive/depolarized steady-state is reached via normal pacing (BCL = 1000 ms; data not shown). The second biomarker, AP_tri, was changed by −12.4%, 112.7%, −33.9% and −17.2% in comparison to cAF, when the remodeling of I_CaL, I_K1, NCX and cell dilation, respectively, were independently reversed. When compared to nSR, AP_tri values were not well restored: −43.8%, +36.4%, −57.6% and −46.9% for the I_CaL, I_K1, NCX and cell dilation, respectively. On the contrary, CaT_amp was almost completely restored when the effects of remodeling (reduced) I_CaL (+94.4%) were reversed, and enhanced to a smaller extent (+38.2%) if the virtual cell was not dilated (Figures 4 B&E, right column). The vital role of increased I_K1 and reduced I_CaL in cAF remodeling was further demonstrated in tachy pacing (BCL = 250 ms) simulations; omitting either of these remodeling targets renders the virtual cell unresponsive to pacing at such a rapid rate (Figure S6). 1D restitution simulations revealed similar results as in single cell simulations (Figure S8). An increase of APD, ERP, CV and WL was only significant in the cases wherein I_CaL or I_K1 remodeling were omitted. Interestingly, reduction of I_K1 led to alternans at higher rates in this case. Similar analysis was performed for cAF-remodeled I_to, I_Kur, SERCA and RyR (results shown in Supporting Information, Figure S4).

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Figure 4. Effect of reversing remodeling of individual targets on CaT and AP characteristics in cAF.

(A) chronic atrial fibrillation (cAF: all). (B–E) four restored cellular components (L-type Ca²⁺ current, I_CaL; inward rectified K⁺ current, I_K1; Na⁺/Ca²⁺ exchanger current, I_NCX; and increased cell volume, dilation), respectively. Columns from left to right: action potential (AP), Ca²⁺ transient (CaT) averaged over cell volume, and spatiotemporal presentation of CaT (x =  distance from sarcolemma). Colour scale for right column: 0–1.5 µM corresponds to dark blue – dark red (similar to Figure 1 F&G). All results are obtained at BCL = 1000 ms.

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The strong link between intracellular Ca²⁺ and AP morphology persists in cAF

In a previous study, we showed that SR Ca²⁺ release is a strong modulator of APD [13]. Here, we used a similar approach to investigate to what extent AP shortening and triangulation in cAF might be reversed if intracellular Ca²⁺ dynamics were restored to match those in nSR. Figure 5A shows the subsarcolemmal CaT (CaT_ss) clamp used in simulations. Restoring CaT_ss had substantial effect on AP shape, increasing APD₉₀ by 18.9% and AP_tri by 16.1%. Figures 5 C and D illustrate the underlying changes in I_NCX and I_CaL responsible for modifying these late and early stages of repolarization, respectively.

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Figure 5. The strong link between intracellular Ca²⁺ and AP shape still exist in the cAF-remodeled virtual cell.

Clamping the CaT (A) in the subsarcolemmal space to be normal (as in nSR) speeds up the initial and slows later repolarisation phases of membrane voltage (B), due to indirect changes in I_NCX (C) and I_CaL (D).

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Na⁺ accumulation during fast pacing

As the typical rate of electrical activation of cells in cAF is dramatically faster than in nSR, we next analyzed accumulation of intracellular Na⁺ and Ca²⁺ during increasingly fast pacing (Figure 6 B and C). Previous studies have already established that intracellular Na⁺ accumulation, which is inherently linked to Ca²⁺ accumulation via NCX, is an important mechanism for AP shortening during fast pacing [11], [13]. Motivated by the finding that restoring the intracellular CaT appears to impart a beneficial effect on AP shape (increased APD₉₀; Figure 5), we analyzed the effect of independently reversing remodeling of cellular components affecting CaT properties the most: I_CaL reduction, increased NCX, and cell dilation. As results in Figure 6 reveal, reversing remodeling of I_CaL affects Na⁺ and Ca²⁺ accumulation most dramatically of the three. Interestingly, in addition to increasing the magnitude of ion accumulation, there is also dramatic shift in the ionic dynamics. Specifically, when I_CaL is restored to a “healthy level” in a cAF-remodeled virtual cell, the regime of Ca²⁺ overload is shifted to larger, more physiologically relevant BCLs (Figure 6A). Similar analysis was performed for all the other cell model variants (results shown in Supporting Information, Figure S10).

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Figure 6. Intracellular Ca²⁺ and Na⁺ accumulation in five virtual cell variants during increasingly fast pacing.

(A&B) Ca²⁺ accumulation and corresponding activation of the reverse mode of NCX in four cell model variants (normalized to nSR). (C&D) Na⁺ accumulation and corresponding activation of the forward mode of NCX (normalized to nSR).

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To show directly that Na⁺ accumulation is still a mechanism responsible for AP shortening in drastically remodeled cells, we clamped Na⁺ concentration to its steady-state value when pacing the model at BCL = 1000 ms, while all other variables represent a steady-state at BCL = 167 ms. The late phase of AP repolarization is slowed substantially (Figure S9E) during Na⁺ clamp, as there is less intracellular [Na⁺] to activate the Na⁺/K⁺ ATPase (NKA) current (Figure S9F) than when Na⁺ is allowed to accumulate normally.

Inducibility of DADs in cAF-remodeled cells

Delayed afterdepolarizations (DADs) have been linked to various arrhythmogenic diseases; however their role in cAF has not yet been elucidated [17]. The main mechanism for induction of cellular DADs in human atrial cells has been shown to be NCX [18]. It was thus of special interest to examine, how remodeled Ca²⁺ handling might affect the inducibility of DADs in these cells. First, we tested whether DADs could be induced via an extra opening of RyRs during diastole (Figure 7A). While it was not possible to induce DADs in the cAF-remodeled virtual cell (Figure 7B), the subsequent activation of NCX (Figure 7B) in the nSR model variant did elicit DADs, as the inward current sufficiently depolarized the virtual cell to elicit an AP. A possible explanation for this surprising finding is the reduced SR Ca²⁺ content in cAF cells. To test this, we employed the same protocol used in cAF versus nSR virtual cells in a cell variant featuring RyR remodeling only (with all other features identical to nSR model). Even maximal opening of the RyR was not enough to activate NCX and induce a DAD in this variant (cAF: RyR in Figure 7), supporting the hypothesis that it was not possible to induce DADs in cAF cells due to reduced SR Ca²⁺ content. In this case, however, membrane potential following RyR opening was closer to the AP initiation threshold than in the original cAF cell variant (Figure 7B).

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Figure 7. Inducibility of DADs is decreased in cAF due to the stabilizing effect of remodeled I_K1.

DADs were induced with either an extra Ca²⁺ release from the junctional sarcoplasmic reticulum (A–C) or an extra current stimulus (D&E) at the time point of 1 second. Same protocol was used to study four model variants: normal sinus rhythm (nSR), chronic atrial fibrillation with all modifications (cAF: all), only modified inward rectified K⁺ current (cAF: IK1), and only ryanodine receptor Ca²⁺ sensitivity (cAF: RyR).

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An additional mechanism to explain the lack of DADs in the cAF-remodeled cell could be increased I_K1, which might stabilize the membrane potential such that pathological opening of RyRs during diastole would not induce a DAD. To investigate this possibility, we performed simulations in which DADs were induced by current injection during diastole (Figure 7D). A 65% greater current amplitude was needed to induce a DAD in cAF as compared to nSR, suggesting that the cAF model membrane was indeed more stable with respect to depolarization. To further isolate the role of increased I_K1, we ran corresponding simulations in a model variant that included only the remodeling of I_K1 (cAF: IK1 in Figure 7). Compared to nSR, a 69% greater current amplitude was needed to induce a DAD in this model variant (Figure 7D).

These results implicate two mechanisms that dramatically reduce the inducibility of DADs in cAF-remodeled virtual cells. First, when SR Ca²⁺ content is reduced, it is not possible to release a sufficient amount of Ca²⁺ from the SR to activate NCX to the extent that would elicit a DAD. Second, increased I_K1 decreases cell excitability and stabilizes the membrane potential against DADs in cAF, as with a RMP hyperpolarization of −3.7 mV, a larger depolarizing current is needed to reach the AP threshold. Thus, reduced SR Ca²⁺ load together with increased I_K1 actually overcompensates for the combined, contradictory effects of increased RyR sensitivity and increased expression of NCX, such that DAD inducibility is reduced rather than enhanced in our in silico model of cAF.

Central targets in cAF remodeling

Simulation results are summarized in Figure 8 in a heat map-like presentation, where each individual modification is rated as based on its impact in remodeling (from nSR to cAF) and in reverse remodeling (from cAF closer to nSR) on selected AF biomarkers. It is apparent that individual modifications have differential impact on cell function depending whether each is involved in remodeling or reverse remodeling (i.e. the isolated modification is aimed at improving function following cAF remodeling). For example, while cell dilation affects several functional variables (CaT, Na⁺ accumulation, DAD inducibility and cell excitability) during the remodeling process, when excluded from cAF-remodeled cells, cell dilatation affects only cell excitability. It is also clear that the cAF phenotype is more resistant to modifications than the nSR phenotype (see larger gray area in Figure 8). In evaluating the impact of isolated features of remodeling, results also show that the increase in I_K1 is the most central component in cAF remodeling, since it affects all functional markers.

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Figure 8. Summary of cellular remodeling affecting electrophysiological properties in cAF.

Analysis was done for both individual inclusion (Remodeling) and exclusion (Restoring) of cellular components to elucidate their contribution action potential duration at 90% repolarisation (APD₉₀), Ca²⁺ transient amplitude (CaT_amp), action potential triangulation (AP_tri), Na⁺ accumulation, inducibility of delayed afterdepolarisations (DAD induc.) and electrical excitability of the cell (cell exc.).

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Ca²⁺ signaling is altered in cAF

The importance of abnormal intracellular Ca²⁺ handling in the pathophysiology of AF is becoming clear [19], [20]. According to our simulations, Ca²⁺ signals are severely blunted in cAF (smaller CaT_amp), which corresponds well with experimental data [10], [11], [14], [21]. These changes are primarily due to decreased I_CaL, and secondarily to the increased activity of NCX. Changes in other remodeling targets involved in Ca²⁺ handling, such as RyR and SERCA, had only minor impact on Ca²⁺ signals (Figures 3 and 4 and Figures S2 and S3). Reduction in I_CaL exerts its effect not only by limiting the Ca²⁺ influx and thus the immediate trigger for Ca²⁺-induced Ca²⁺ release during the AP, but also reduces SR Ca²⁺ content over time, thereby further reducing the strength of SR Ca²⁺ release.

In atrial myocytes, Ca²⁺-induced Ca²⁺ release involves two separate phases: the Ca²⁺ influx first activates RyR clusters in the vicinity of the sarcolemma and Ca²⁺ release from these junctional release sites triggers a propagating Ca²⁺ wave activating adjacent RyRs located deeper within the cell [10], [13], [22]. However, recent data suggest that larger mammals, including humans, might actually have a more developed network of t-tubules in their atrial cells than previously thought [23]. In fact, t-tubules are present in the ovine atrial myocytes at low density and strongly reduced in AF, leading to gradual loss of synchronization of Ca²⁺ signals [24]. This feature of cAF-remodeling is a very interesting topic for future study, particularly as increased spatial heterogeneity in Ca²⁺ diffusion within the cell has been shown to promote the genesis of Ca²⁺ alternans [25].

Our simulations and in vitro data [10] have shown the vulnerability of the fire-diffuse-fire mechanism to disruption in cAF; suppression of Ca²⁺ influx during AF remodeling leads to severely compromised transverse propagation of Ca²⁺ inside the cell. Ca²⁺ ions are thus only circulated within the volume just beneath the sarcolemma in this case; a defect aggravated by cAF-induced cell dilatation, which increases the volume of the junctional space and further dilutes [Ca²⁺] in the subsarcolemmal space. This redistribution of Ca²⁺ is likely to contribute to the suppression of contraction during AF, as with impaired propagation, Ca²⁺ signals do not reach the contractile elements located centrally within the cell. Furthermore, this alteration likely has a profound indirect effect on energy expenditure of the cAF-remodeled cells, since contractile elements are not activated to same extent and thus consume less energy than in healthy cardiomyocytes [26].

According to experimental data from patients suffering from chronic AF, contractile force of atrial tissue can be restored with increased extracellular [Ca²⁺] [7]. This implies that remodeling of the cellular contractile elements involved in cAF has a lesser role in depressing contraction in cAF as compared to the impact of altered Ca²⁺ signaling in the disease. Furthermore, the data also suggest that Ca²⁺ influx is the single most influential variable when considering cAF-induced contractile dysfunction in the light of electrical remodeling. This view is supported by our simulations, wherein cAF-induced I_CaL downregulation alone reduces CaT_amp by 46% and induces defects in Ca²⁺ signal propagation (Figure 3).

AP morphology changes in cAF

A number of remodeling targets in AF have been proposed to contribute to changes in the AP waveform [5], [17]. Our simulations support the conclusion that the two most important elements leading to AP shortening in AF are increased I_K1 and decreased I_CaL. Increased I_K1 alone reduces APD by 52.7% and is also the single most influential factor contributing to AP triangulation (Figure 3). Furthermore, the membrane potential is hyperpolarized in diastole by the cAF-related remodeling of I_K1 (which is a repolarizing current and one of the main contributors to the maintenance of the RMP in cardiac myocytes). Interestingly, some of the remodeling modifications also act to lengthen the AP, e.g. increases in I_NCX and cell volume. Increase in the NCX current promotes augmented inward current during AP repolarization when it exchanges cytosolic Ca²⁺ ions for extracellular Na⁺ ions at a ratio of 1∶3. The effect of the cell volume on AP is a bit more indirect; the volume increase appears to delay the Ca²⁺ removal from the cytosol, which in turn increases I_NCX during late repolarization (Figure 3).

Intracellular Ca²⁺ signals and the AP are tightly coupled in human atrial myocytes inherently, and this coupling seems to be an essential part of AF remodeling. Hence, as compared to changes in e.g. K⁺ current densities which have more straightforward effects on AP, changes in variables involved in Ca²⁺ signaling, like I_CaL and I_NCX, modulate not only the AP directly, but have more adverse consequences through their effects on intracellular Ca²⁺ signals. In our simulations, remodeling of I_CaL alone reduced APD₉₀ by 17.3% (compared to AP shortening by 31.9% in cAF, Figure 3), while normalization of CaT in cAF cells lengthened the AP by 18.7% (Figure 5). This suggests that effects of I_CaL on AP are mediated only partly by direct impact of the current on V_m and that major effects come via the secondary suppression of CaT.

Tissue electrophysiology is influenced by cAF

Changes in AP morphology also impact tissue electrophysiology (see Figure 2). The simulated tissue APD₉₀ is reduced by around 30%, which is in agreement with the available in vivo data [27]. Similarly, the simulated ERP is reduced by about 20%. In this case, the measured data from Yu et al. [28] revealed reduction by a lesser extent (around 10%) as compared to simulation data. These differences might be due to different stages of remodeling. CV is not influenced to a great degree in our simulations, as we did not include gap junction remodeling. Feld et al. [29] measured a reduced CV in cAF, suggesting that there might be changes in conductive tissue properties during cAF. The reduced wavelength (the product of ERP and CV) in the simulated cAF case suggests the higher chance of the maintenance of AF following rotor initiation. The simulated rotor center trajectories (Figure 2) show that these anchor more easily in cAF, evincing greater stability, whereas in nSR, the rotor core tends to meander and subsequently might be eliminated at a boundary or an anatomical obstacle. The simulated dominant frequencies also demonstrate the higher chance of a permanent fibrillation in the cAF case.

Sodium accumulation is attenuated in cAF

Intracellular Na⁺ accumulation has been established as an important mechanism for AP shortening during fast pacing in previous studies [11], [13]. In cardiac myocytes, [Na⁺]_i is mainly dictated by the balance between Na⁺ influx during an AP (upon activation of I_Na and I_NCX) and Na⁺ efflux (through NKA and NCX). Therefore, Na⁺ fluxes are tightly coupled with both [Ca²⁺]_i and activation frequency, both of which are drastically altered in AF. According to our simulations, high frequency activity induces substantial Na⁺ accumulation in cAF cells, and this accumulation acts to shorten the AP upon activation of NKA, although this mechanism is less prominent than in nSR cells. In all types of cardiac myocytes, Na⁺ accumulation can result indirectly via Ca²⁺ overload which itself automatically results from high frequency pacing. In cAF cells, Ca²⁺ overload is limited by remodeling (reduced I_CaL), which drastically suppresses the AP-evoked CaT. Thus, there is less Ca²⁺ to be extruded by NCX and consequently a lesser degree of Na⁺ accumulation.

To demonstrate the link between [Ca²⁺]_i and [Na⁺]_i, we normalized the I_CaL in cAF model and noticed that pacing-induced Ca²⁺ and Na⁺ accumulation were both augmented (Figure 6). It could be hypothesized that altered Ca²⁺ and Na⁺ balances are actually among the features of cAF cells that enable sustained high frequency activity. When cardiomyocytes act to restore normal levels of [Ca²⁺]_i and [Na⁺]_i, vast amounts of ATP are consumed by SERCA (to pump Ca²⁺ to the SR) and by NKA (to pump Na⁺ to the extracellular space). Thus, when ion gradients are smaller, cAF cells can maintain high frequency of activation at lower energy costs [26].

Inducibility for DADs is not increased in cAF-remodeled cells

The main mechanism for induction of cellular DADs in human atrial cells has been shown to be activation of NCX [18]. As NCX is overexpressed in cAF, we expected to see a lower threshold for DADs in simulations; however, results revealed the opposite finding. In fact, an extra Ca²⁺ release from the SR was not enough to trigger DADs in the cAF-remodeled virtual cell. This result appears to contradict the recent in vitro findings that showed increased spontaneous Ca²⁺ waves in cells of AF patients [30], [31]. Possible explanations for this discrepancy include the measurement conditions (experiments carried out at room temperature) and pooled patient population (no separation for paroxysmal, persistent and chronic AF). Indeed, further analysis of our simulation results showed that the mechanism explaining this surprising finding was the reduced cellular excitability due to increase of inward rectifying K⁺ currents.

Our results suggesting reduced DAD inducibility in cAF contradict the recent finding that enhanced SR Ca²⁺ leak and NCX function underlie DADs in patients with cAF [14]. In another study, however, DADs were not observed in either nSR nor cAF patient tissue despite the fact that the measurement conditions were in favor of such events, as I_Kur was blocked with AVE0118 compound [32]. These controversial results suggest that increased propensity for DADs in cAF might depend, for example, on underlying etiologies in the patient population.

To summarize, DAD inducibility depends on four factors mechanistically: 1) the strength of the input (SR Ca²⁺ load), 2) how this input is transformed into a trigger (sensitivity of RyR), 3) how much depolarizing current this trigger induces (NCX vs. SERCA balance in Ca²⁺ removal from the cytosol), and, ultimately, 4) if the depolarizing current is large enough to depolarize the membrane voltage above the threshold for I_Na activation (which depends on the dynamic balance of depolarizing and repolarizing membrane currents). All four factors are altered in the context of cAF. Because factors #2 and #3 are greater (increase RyR sensitivity and greater net depolarizing current, respectively) one might intuitively infer that DAD inducibility would be increased in cAF-remodeled cells. In our in silico cAF model, however, the reduced SR Ca²⁺ load together with increased I_K1, which reduce the trigger and stabilize the resting membrane potential, respectively, overcompensate for the combined depolarizing effect of increased RyR sensitivity and increased expression of NCX, such that DAD inducibility is actually reduced.

Future computational studies, possibly employing stochastic methods and finer spatial resolution, should address factors #2 and #3 in more detail, when in vitro data on the co-localization of RyR and NCX in cAF vs. nSR human atrial cells becomes available.

What are the key cellular components contributing to remodeling in cAF?

Both AP shortening and loss of contractility are hallmarks of cAF. Our analysis indicates that, at the cellular level, these changes are strongly coupled to the increased I_K1 and decreased I_CaL conductances, respectively. In fact, without the I_K1 modification, the cAF-remodeled cell becomes unresponsive during more rapid pacing due to sustained depolarization of the membrane voltage (which inactivates fast sodium channels). Decreased I_CaL conductance, on the other hand, has a more diverse effect. While also contributing to AP shortening, reduced I_CaL is the main mechanism for the diminished intracellular CaT_amp in cAF. The large impact of the remodeling of I_CaL is related to its dual role, since it not only acts as a trigger for Ca²⁺ release from the SR, but also affects Ca²⁺ loading of the SR.

Remodeled I_CaL and I_NCX work in synergy to adapt the cell to abnormally fast reoccurring activation in cAF. While reduced I_CaL and increased I_NCX both reduce Ca²⁺ overload during fast pacing, they also shift Ca²⁺ dynamics from the normal “whole-cell state” to a “subsarcolemmal state”, where Ca²⁺ cycling is limited primarily to the vicinity of the cell membrane. Myocyte hypertrophy exacerbates the effect of remodeled Ca²⁺ handling, in that it further reduces CaT_amp in cAF in addition to the effects of I_CaL and NCX remodeling. The dilation of the cell also increases the delay between the peaks of the AP and the CaT, which may have arrhythmogenic effects in tissue. In fact, Schotten et al. [12] found that myocyte hypertrophy can cause conduction disturbances in the absence of increased fibrosis in a goat model of chronic atrio-ventricular block. As changes in intracellular Ca²⁺ signaling are centrally involved in normal and pathological regulation of myocyte growth, apoptosis and necrosis [33], cell dilation warrants further research to elucidate its role in cAF.

What could be the key targets for restoring cell function in cAF?

Anti-arrhythmic drug therapy to counter AF has long concentrated on agents that may delay atrial repolarization. Drug targets have included, for example, I_Kr and I_Ks; however, more recently agents blocking I_Kur have been studied extensively, because of the current's atria-specificity in human myocardium. More recently, intracellular Ca²⁺ handling has been established as a potential drug target in cAF [19]. As our results showed (Figure 5), restoration of intracellular CaT could, hypothetically, be used to improve AP shape (increase APD₉₀) in cAF to, for example, lengthen the effective refractory period.

The most effective targets for restoring healthy cell properties following cAF-induced electrical remodeling are likely to be those that most impact the cAF phenotype. Our simulations suggest that changes in I_K1 and I_CaL in isolation induce most of the characteristic features of cAF (Figure 8). Therefore, restoring either the K⁺ or Ca²⁺ conductance could, in theory, be effective in limiting the effects of electrical remodeling in the cAF substrate. However, complete reversal of any single cAF-induced change via pharmacological means is not likely to be feasible. Instead, it might be useful to consider therapies that aim at partial restoration of combinations of targets. In such efforts, however, understanding the full implications of altered cellular electrophysiology on tissue and organ dynamics is absolutely essential. To illustrate, consider the partial inhibition of K⁺ currents (for increasing APD₉₀ and thus ERP), in combination with drugs aimed at increasing CaT_amp (for restoring contractility). Partial block of NKA with digoxin, combined with reduced RyR Ca²⁺ leak using a calmodulin kinase II inhibitor, appeared to be beneficial in single cell simulations (Figure S11) and may actually become feasible in the near future, as novel specific blockers of I_K1 are being developed [34]. The 1D restitution results (Figure S12) also illustrated increases in APD, ERP, CV and WL, which may be desirable in terms of protecting against arrhythmia. However, this model variant developed alternans at higher pacing rates. In 2D simulations, these alternans also led to a break-up of a single rotor into two rotors (not captured within the geometry, so excitation vanishes; Figure S12E and Video S3). In a realistic geometry, such wavebreak could lead to stable fibrillatory activity. This finding highlights the need to carry out in silico analysis of potential drug targets at different scales (cell, tissue, organ) to achieve a more realistic understanding of pharmacological effects.

Limitations of the study

Although the human atrial myocyte model employed here has been shown to be the most internally consistent and physiologically accurate to date, particularly regarding intracellular Ca²⁺ handling, in a recent comparison, the model has its limitations [35]. Furthermore, a holistic analysis of cAF as effected in this study is inevitably biased to some extent by the fact that the pathophysiology clinically involves multiple etiologies. Some studies group available data based on, for example, whether patients have a valvular disease or not, while other studies pool the data among AF subtypes and etiologies. Finally, our model of the cAF-remodeled cell is by no means exhaustive, as novel mechanisms of electrical remodeling are reported continuously. Instead, we have included those remodeling targets that have been established in more than one study of human atrial electrophysiology. When novel experimental data on these disease mechanisms accumulate, the model should be updated accordingly.

Conclusions

The results indicate that, at the cellular level, reduced I_CaL and increased I_NCX contribute synergistically to adapt the cell to fast activation rates of cAF by reducing Ca²⁺ overload, which additionally causes a drastic decrease in CaT_amp at normal heart rates. Furthermore, our findings suggest that an increase of I_K1 in cAF is the dominant mechanism responsible for AP shortening in cAF, while the effect of reduced I_CaL is less prominent and the role of remodeled I_to and I_Kur are rather insignificant. Increased I_K1, in synergy with reduced intracellular Ca²⁺ stores, also stabilizing the cAF-remodeled cell against DADs. The results also show that, in addition to remodeling of ion currents and Ca²⁺ handling, cellular hypertrophy is an important mechanism contributing to changes in atrial refractoriness, contractility and arrhythmogenicity. Finally, the intrinsic complexity and interdependency of electrophysiological mechanisms are highlighted by our analysis. The presented results thus suggest that instead of targeting a single cellular component a more holistic approach is worth considering when looking for novel therapeutic approaches for chronic AF.