Simulation Study of Cellular Electric Properties in Heart Failure
Abstract—Patients with severe heart failure are at high risk of sudden cardiac death. In the majority of these patients, sudden cardiac death is thought to be due to ventricular tachyarrhythmias. Alterations of the electric properties of single myocytes in heart failure may favor the occurrence of ventricular arrhythmias in these patients by inducing early or delayed afterdepolarizations. Mathematical models of the cellular action potential and its underlying ionic currents could help to elucidate possible arrhythmogenic mechanisms on a cellular level. In the present study, selected ionic currents based on human data are incorporated into a model of the ventricular action potential for the purpose of studying the cellular electrophysiological consequences of heart failure. Ionic currents that are not yet sufficiently characterized in human ventricular myocytes are adopted from the action potential model developed by Luo and Rudy (LR model). The main results obtained from this model are as follows: The action potential in ventricular myocytes from failing hearts is longer than in nonfailing control hearts. The major underlying mechanisms for this prolongation are the enhanced activity of the Na+-Ca2+ exchanger, the slowed diastolic decay of the [Ca2+]i transient, and the reduction of the inwardly rectifying K+ current and the Na+-K+ pump current in myocytes of failing hearts. Furthermore, the fast and slow components of the delayed rectifier K+ current (IKr and IKs, respectively) are of utmost importance in determining repolarization of the human ventricular action potential. In contrast, the influence of the transient outward K+ current on APD is only small in both cell groups. Inhibition of IKr promotes the development of early afterdepolarizations in failing, but not nonfailing, myocytes. Furthermore, spontaneous Ca2+ release from the sarcoplasmic reticulum triggers a premature action potential only in failing myocytes. This model of the ventricular action potential and its alterations in heart failure is intended to serve as a tool for investigating the effects of therapeutic interventions on the electric excitability of the human ventricular myocardium.
Patients with severe heart failure are at high risk of sudden cardiac death. More than 50% of these patients die suddenly.1 In the majority of these patients, sudden cardiac death is thought to be due to ventricular tachyarrhythmias. The mechanisms underlying these lethal arrhythmias are largely unknown. However, in animal models of heart failure and in humans, there is evidence that reentry as well as nonreentrant mechanisms may play a role.2–4 Triggered activity arising from EADs or DADs and abnormal automaticity are possible cellular mechanisms underlying nonreentrant arrhythmias.5–8 In animal studies, Ca2+ influx through voltage-gated ICa was demonstrated to cause EADs.9,10 Cation influx through Ins(Ca) or Na+ influx via the electrogenic INaCa was found to underlie DADs.11,12 However, it is unknown what influence these currents may have on triggered activity in human ventricular myocardium. In recent years, quantitative studies of ionic currents in human ventricular cells have greatly enlarged our knowledge about the characteristics of the human AP in patients with and without heart failure.13–25 In some of these studies, great variations in shape and duration of the APs have been found. Nevertheless, an unequivocal result was that the AP is prolonged in human ventricular myocytes isolated from patients with heart failure compared with control subjects. Various ionic currents have been shown to be altered in heart failure. IK113,22 and Ito13,19 have been found to be reduced by some groups, although this finding was not undisputed. Wettwer et al14 have found no significant alterations in current densities and kinetics of Ito in failing compared with nonfailing myocytes. Current densities and kinetics of ICa, however, have been shown to be unaltered by most groups.15,16,18
A prominent feature of the single myocyte of the failing heart is an alteration of [Ca2+]i handling26,27 and an enhanced activity of the Na+-Ca2+ exchanger.28,29 It has been postulated that these abnormalities may give rise to arrhythmias in heart failure, but proof for this hypothesis remains lacking. Mathematical models of the cellular AP and its underlying ionic currents may help to elucidate possible arrhythmogenic mechanisms on a cellular level. For this purpose, a model of the ventricular AP based on Hodgkin-Huxley formalisms30 was developed. Selected depolarizing and repolarizing ionic currents and the [Ca2+]i handling incorporated into this model were based on quantitative measurements in single ventricular myocytes isolated from nonfailing and terminally failing human hearts.
Using this model, we evaluated which ionic currents may affect the AP in human myocardium and which cellular abnormalities in human ventricular myocytes from failing hearts may contribute to arrhythmogenesis in heart failure.
Materials and Methods
A variety of ionic currents and electrogenic ion pumps and exchangers that have been described in animals have not been sufficiently characterized in human ventricular myocytes. Therefore, these currents have to be calculated by equations used in an AP model developed for guinea pig ventricular myocytes by Luo and Rudy31 (LR model). However, the major ionic currents, ICa, Ito, IK with its two components (IKr and IKs), and IK1, are based on human data. In addition, the [Ca2+]i transients in human myocytes and their alterations in failing myocytes observed experimentally27 can be simulated by this model. The other currents included into the model, INa, INaCa, and INaK, have to be adopted from the LR model and modulated in such a way that simulations are widely consistent with available human data.
Voltage-clamp data for ICa, Ito, and IK1 and Ca2+ measurements are as described in our previous studies.13,16,19,20,27 Experiments were carried out at 37°C. (See References 13, 16, 19, 20, and 2713 16 19 20 27 for a detailed description of the experimental conditions.)
Under space-clamp conditions, the differential equation describing the time-dependent changes in membrane potential (V) is as follows:
where Cm is the membrane capacitance and Ist is an externally applied stimulus current. The ionic currents IX are calculated by ionic gates using Hodgkin-Huxley–type formalisms.30 All ionic currents are computed for 1 pF of cell membrane capacitance.
The complete set of equations of all ionic currents, ionic exchanger currents, and [Ca2+]i handling is provided in the Appendix.
Fast Na+ Current: INa
INa is calculated using equations of the LR model.31 Sakakibara et al25 have demonstrated that the characteristics of INa in isolated human ventricular myocytes are similar to those of other mammalian species and that INa kinetics are identical in several different disease states. Therefore, we use the same equations for INa in both cell groups.
L-Type Ca2+ Current: ICa
The kinetics of ICa have been shown to be unaltered in myocytes from failing hearts.15,16,18 Therefore, for calculating ICa the same gating parameters are used in both groups. In animal and human studies, it has been clearly demonstrated that the inactivation of ICa is voltage dependent. In addition, there is experimental evidence indicating that inactivation of ICa is also Ca2+ dependent.32,33 This type of regulation of ICa seems also to exist in human ventricular myocytes.17 Consequently, we integrate a proportional factor, fCa, in the equation of ICa that is formulated as follows: fCa=[1+([Ca2+]i/600 nmol/L)]−1. Fitting of experimental ICa is performed with simulated [Ca2+]i transients formulated as follows: A · [exp(−t/τ1)−exp(−t/τ2)]+R. A is a proportional factor, τ1 and τ2 are time constants, and R is the basal Ca2+ level. By this approach, the important Ca2+-dependent inactivation of ICa may be sufficiently incorporated into this model as the simulations of ICa indicate.
Transient Outward K+ Current: Ito
No significant alterations of the kinetics of Ito have been found in failing compared with control myocytes.14 Therefore, the same gating parameters for simulating Ito are used in both groups. On the basis of experimental data,20 the current density of Ito is assumed to be 64% of the value measured in nonfailing myocytes. The steady-state activation and inactivation curves of Ito obtained from fitting the experimental voltage clamp traces are shown in Figure 1⇓.
Data are fitted by a Boltzmann distribution. The parameters V0.5 and k of the Boltzmann equation in the model are compared with those from experimental voltage-clamp studies (Table 1⇓). The differences are negligible and can be explained by the different solutions used in the experiments to block interfering currents.
Delayed Rectifier K+ Current: IK
The existence of two components of the delayed rectifier, a rapidly activating component (IKr) and a slowly activating component (IKs), has been documented by Li et al.21 On the basis of their data, both currents are incorporated into the model. The method for simulating IKr in human ventricular myocytes is the same as that used by Sanguinetti and Jurkiewicz34 in guinea pig myocytes. For simplification, the slow inactivation of IKr during depolarization at +50 mV observed experimentally (Li et al21) is not considered. Quantitative values of IKs are calculated by fitting the experimental voltage-clamp traces recorded in the study of Li et al to a single exponential function. On depolarization, the activation of IKs in human ventricular myocytes follows a sigmoidal time course, as has been reported in guinea pig ventricular myocytes.34 This strong sigmoidal activation has also been found in wild-type IKs.35 Therefore, the second power of activation in the Hodgkin-Huxley formalism of IKs is used to obtain an adequate fit to the measured traces. In animal studies, IKs has been shown to be sensitive to intracellular Ca2+ 36,37, and IKr has been shown to be sensitive to extracellular K+.37 In the only study investigating IKs and IKr in human ventricular myocytes (ie, that of Li et al21), experiments were performed only with 5.0 mmol/L EGTA in the pipette solution and with one extracellular K+ concentration. Thus, it is unclear whether this type of regulation found in animal ventricular myocytes also exists in human ventricular myocytes. Consequently, we do not consider such a regulation of IKs and IKr in our model. At the present time, the properties of IKr and IKs in heart failure are unknown. Therefore, we assume that IKs and IKr are unchanged in heart failure.
Inward Rectifier K+ Current: IK1
The simulated current density of IK1 is assumed to be reduced by 25% at −70 mV in the failing myocyte compared with control myocytes on the basis of results of experimental studies.13,22 Since the time-dependent inactivation of IK1 can be observed only at voltages negative to −110 mV,22 IK1 is assumed to be time independent. As in animal ventricular myocytes, IK1 is also almost solely carried by K+ ions in human ventricular myocytes.22 Therefore, the reversal potential of IK1 is calculated by Nernst’s equation for K+.
Na+-Ca2+ Exchanger Current: INaCa
INaCa is integrated into the model using values from the LR model because data in human ventricular myocytes are not available at present. To compute INaCa in a nonfailing myocyte, only kNaCa is changed to 50% of the value used in the LR model, taking into account the smaller activity of INaCa in human myocytes compared with different animal species.38 With such a value of kNaCa, INaCa simulated in a nonfailing myocyte with a protocol similar to that in experiments by Sham et al38 is in the range of the experimental data (model, 0.50 pA/pF; experiment, 0.54±0.1 pA/pF). In a failing myocyte, we assume 65% greater INaCa than in a nonfailing myocyte. This assumption is based on the observation of an increase of Na+-Ca2+ exchanger activity in myocardium from patients with heart failure.28
Na+-K+ Pump Current: INaK
For simulation of INaK, we use the equation of the LR model. The magnitude of INaK has been chosen in a way such that APD in a nonfailing myocyte at a stimulation frequency of 1 Hz is in the range measured experimentally in single human myocytes from nonfailing heart by our group (unpublished data) and by Peeters et al.39 There is a report suggesting that the concentration of the Na+,K+-ATPase is decreased by 42% in failing hearts.40 This alteration is assumed to represent a proportional decrease in INaK. Therefore, a 42% reduction in INaK of a failing myocyte is incorporated into the model.
To simulate the [Ca2+]i transients in both groups, the approach of the LR model has been chosen.31 In some equations for calculating the Ca2+ homeostasis, the parameters are changed in a way such that simulated [Ca2+]i transients closely resemble those measured in nonfailing and failing human ventricular myocytes.27 The differences in simulations of the intracellular Ca2+ fluxes to the LR model are described below. For more details, see Reference 3131 .
CICR by the SR
The threshold for CICR from the cardiac SR is reduced from 0.18 to 0.005 μmol/L because of the smaller size of the peak ICa in human compared with animal myocytes. The time constants for the activation and deactivation of the release process is set to 4 ms. Experimental studies have revealed that the function and number of the ryanodine channels are widely unaltered in heart failure.41,42 Therefore, the CICR mechanism is assumed to be equal in nonfailing and failing myocytes.
Ca2+ Buffers in the Myoplasm and the SR
There are reports that the affinity of troponin C to Ca2+ is unaltered in heart failure.43 Consequently, because of the great contribution of troponin C to the total myoplasmic Ca2+ buffer capacity, we have used equal myoplasmic buffer concentrations in nonfailing and failing myocytes. For simulating the Ca2+ buffering in the JSR (calsequestrin), we adopted the values of the LR model for our model. There is no evidence of differences in the level of calsequestrin in heart failure.44 Therefore, equal concentrations of calsequestrin have been used in both cell groups. With the approach of Hilgemann and Noble,45 we compute the steady-state buffering process numerically by using Newton’s iterative method.
Ca2+ Uptake and Leakage by the NSR
Reduction of the activity of Ca2+-ATPase of the SR in heart failure, as shown in experimental studies,46,47 is integrated into the model. To obtain the characteristic Ca2+ transients in both cell groups, the scaling factor for Ca2+ uptake, Īup, is set to 0.0045 mmol/(L · ms) in a nonfailing and 0.0015 mmol/(L · ms) in failing myocytes. The Kleak value in both cell groups was chosen in a way such that Ca2+ leakage out of the NSR is equal to the Ca2+ uptake in the NSR at basal [Ca2+]i (nonfailing, Kleak=0.00026 ms−1; failing, Kleak=0.00017 ms−1).
Sarcolemmal Ca2+ Pump
The contribution of the sarcolemmal Ca2+ pump to the extrusion of Ca2+ out of the cell has been shown to be very small.48 Therefore, we do not consider this pump in our model.
A linear Ca2+ background current, ICa,b, is incorporated into the model for balancing the Ca2+ extrusion through INaCa at resting potential in both cell groups. By this mechanism, the resting level of [Ca2+]i is maintained at 0.12 μmol/L in a nonfailing myocyte and at 0.15 μmol/L in a failing myocyte.
A linear Na+ background current, INa,b, is also incorporated into the model of a nonfailing myocyte for maintaining the resting level of [Na+]i (10 mmol/L in both cell groups). In a failing myocyte, Na+ ion extrusion by INaK balances the Na+ ion entry by INaCa so that incorporation of INa,b into this model is not necessary in that cell.
The model is written in Pascal and tested using a Turbo-Pascal compiler (Borland International) on an IBM-compatible computer with an Intel Pentium central processing unit. A fourth-order Runge-Kutta method with fixed time intervals is used for numerical integration of differential equations.
For the simulations in the present study, the fixed time interval for voltage-clamp simulations is 0.01 to 0.1 ms. The time interval for AP simulations is held at 0.0001 ms during the stimulus current and then set at 0.01 to 0.1 ms. APs are elicited in all simulations with 10 pA/pF of Ist for 0.7 ms. Standard software is used to convert the simulated data in ASCII format and to prepare the figures. Fitting of the voltage-clamp traces is performed with a commercial software using a nonlinear least-squares algorithm.
Simulations of Voltage-Clamp Experiments
An important test for the validity of an AP model is the accurate simulation of voltage-clamp experiments. Therefore, we have simulated the ionic currents ICa, Ito, IKr, IKs, and IK1 under voltage-clamp conditions using pulse protocols similar to those used in the experiments previously performed by our group13,27 and by Li et al.21 Figures 2⇓ and 3⇓ show ICa and [Ca2+]i transients in a nonfailing and a failing myocyte, respectively. Experimental data are shown as insets. Simulations of [Ca2+]i transients at various depolarizations were started with [Ca2+]NSR=[Ca2+]JSR=2.5 mmol/L in a nonfailing myocyte and [Ca2+]NSR=[Ca2+]JSR=1.0 mmol/L in a failing myocyte on the basis of experimental data.49 Under these conditions, simulated and experimentally found ICa and [Ca2+]i transients were similar in both cell groups. In particular, peak [Ca2+]i was maximal when ICa reached its maximum at +10 mV in both cell groups. Values of resting and peak [Ca2+]i (Table 2⇓) at +10 mV are in the range of experimental data.27 Although peak ICa was slightly greater in a failing than in a nonfailing myocyte, the triggered [Ca2+]i transient was smaller than in a nonfailing myocyte because of the lower Ca2+ content of the SR.
Figure 4⇓ shows Ito simulated in a failing myocyte. The corresponding experimental traces are depicted in Figure 4B⇓. The pulse protocol is shown as an inset. The current density of Ito at +40 mV (difference between peak current and maintained current at the end of the pulse) was 8.9 pA/pF. The time course of inactivation of Ito was largely independent of voltage. The time constant of its monoexponential decay in the voltage range of +10 to +80 mV was 13±0.9 ms. These values of Ito are in accordance with experimental data measured at 37°C.20 Figures 2⇑, 3⇑, and 4⇓ demonstrate that simulated ICa, [Ca2+]i transients, and Ito resemble the experimental recordings closely with regard to their magnitude and kinetics.
The simulations of voltage-clamp experiments of IKr and IKs are depicted in Figure 5⇓. Depolarizations from a holding potential of −60 mV to the various clamp pulse potentials generate a rapidly activating (Figure 5A⇓) and a slowly activating (Figure 5B⇓) K+ current through delayed rectifier channels with close approximation to experimental data.21 To validate simulations of IKr and IKs, current-voltage relations for IKtail of IKr and IKs are shown in Figure 5C⇓. The values are very close to experimental data.21 Activation voltage dependence was determined by normalizing IKtail at each test potential in Figure 5C⇓ to the current at the most positive potential. Results are shown in Figure 5D⇓. Curves are fitted to a Boltzmann distribution function. V0.5 and k values are consistent with experimental results (Table 3⇓).
Data of IK1 are shown in Figure 6⇓. Experimental traces of IK1 in failing myocytes are depicted in Figure 6A⇓. Currents were elicited from a holding potential of −30 mV to the indicated voltages. On the basis of these experimental data, IK1 was simulated using the same pulse protocol. The current-voltage relations of simulated IK1 in a nonfailing and a failing myocyte are shown in Figure 6B⇓. The whole-cell current slope conductance at the reversal potential of IK1 in a failing myocyte is smaller (0.23 nS/pF) than that in a nonfailing myocyte (0.4 nS/pF). In addition, the current density of IK1 at −70 mV (0.6 pA/pF [failing myocyte] versus 0.8 pA/pF [nonfailing myocyte]) and at −100 mV (−10 pA/pF [failing myocyte] versus −15 pA/pF [nonfailing myocyte]) is assumed to be smaller in a failing than in a nonfailing myocyte.
Simulated AP Is Prolonged in Heart Failure
After showing that simulated ionic currents and [Ca2+]i transients resemble experimental measurements, APs of a nonfailing and of a failing myocyte were simulated. To obtain a steady state in [Ca2+]i transients, 10 APs were elicited at a frequency of 1 Hz. Under these conditions, both APs are distinctly different (Figure 7⇓, top; nonfailing, dashed line; failing, solid line).
APD90 was significantly longer in a failing than in a nonfailing myocyte (548.8 versus 374.0 ms, respectively). In contrast to this, differences in APD25 and APD50 between both cell groups were smaller (APD25: nonfailing, 262.9 ms; failing, 305.7 ms; APD50: nonfailing, 310.2 ms; failing, 374.5 ms). Therefore, the prolongation of APD in heart failure was mainly due to the slower rate of repolarization in the late phase of AP in a failing compared with a nonfailing myocyte, which was also found in an experimental study on human myocardium.50 The ionic currents and the [Ca2+]i transients during the AP are shown in Figures 7⇑ and 8⇓ (nonfailing, dashed line; failing, solid line). At 4 mmol/L [K+]o and 140 mmol/L [K+]i, the resting membrane potential in this model was only slightly different in both cell groups (nonfailing, −89.7 mV; failing, −85.6 mV). A greater discrepancy in resting membrane potential between both cell groups was prevented by IK1. Although the current density of IK1 was reduced in a failing compared with a nonfailing myocyte, the increase of IK1 in these cells when the resting potential becomes more positive limits the depolarization of the cell membrane in a failing myocyte (Figure 7⇑, IK1).
After depolarization of the cell membrane by a suprathreshold stimulus, INa depolarized the membrane to an overshoot potential of +50 mV and inactivated immediately (not shown). Subsequently, ICa, Ito, IKs, and IKr were activated, and INaK increased. The fast activation of ICa (5 ms) was followed by a rapid incomplete inactivation to 12% of its peak value (Figure 7⇑). Repolarization was initiated by the activation of Ito. In both cell groups, Ito activated and inactivated rapidly (Figure 7⇑). Thereafter, IKr and, subsequently, IKs were completely activated and accelerated the repolarization, which was completed by the opening of IK1 channels (Figure 7⇑). When these repolarizing currents increased, ICa deactivated completely. During the plateau and repolarization phase of AP, INaK counteracted the depolarization of the cell membrane (Figure 8⇑).
The total current of the background currents, ICa,b and INa,b, during an AP was slightly higher in a nonfailing myocyte (Figure 8⇑), so that the alterations in APD in heart failure may even be underestimated.
In this model, INaCa was a repolarizing current during most of the AP plateau in a failing myocyte (Figure 7⇑). During the late plateau and repolarization phase, INaCa carried an inward current in its forward mode and extruded Ca2+ ions out of the myocyte. By this means, INaCa becomes a depolarizing current. Therefore, it slows down the rate of membrane repolarization during the late phase of membrane repolarization, especially in a failing myocyte, because of its greater size and the slowed decay of the [Ca2+]i transient. In this myocyte, the final repolarization phase was also prolonged because of the decrease of IK1 (0.68 versus 0.85 pA/pF in a nonfailing myocyte) and of INaK.
The [Ca2+]i transient during the AP was markedly different in both cell groups (Figure 8⇑). As a result of the higher Ca2+ content in the SR, the maximum of the Ca2+ transient was higher in a nonfailing than in a failing myocyte (nonfailing, 1100 nmol/L; failing, 569 nmol/L). The faster inactivation of ICa in the nonfailing myocyte was caused by enhanced Ca2+-dependent inactivation of this current and led, together with higher Ito and INaK, to a reduction of the plateau phase of the AP in this myocyte compared with a failing myocyte (nonfailing, +19.0 mV; failing, +29.2 mV).
In conclusion, these simulations of APs in both groups demonstrate that the prolongation of the AP in a failing myocyte is mainly due to a prolonged late repolarization phase caused by an enhanced activity of the Na+-Ca2+ exchanger and the slowed diastolic decay of the [Ca2+]i transient. The reduction of IK1 and INaK in heart failure additionally contribute to the difference in the late repolarization phase between nonfailing and failing myocytes.
Simulated APs Resemble Those Recorded in Human Ventricular Myocytes
APs in human ventricular myocytes of nonfailing and failing hearts measured in different laboratories show a great variability in duration and shape. Despite using comparable isolation procedures and recording APs under maintained conditions, a significant variability remains. APs measured in our laboratory (M. Lindner, unpublished data) vary more distinctly in failing than in nonfailing myocytes. Figure 9⇓ shows measured APs that represent the observed spectrum. It is obvious that simulated APs (Figure 7⇑) were generally similar to the measured APs in nonfailing (Figure 9A⇓) and failing (Figures 9B⇓ and 9C⇓) myocytes. However, a group of recorded APs in failing myocytes (Figure 9D⇓) characterized by a pronounced prolonged plateau phase showed significant differences to the simulation. Possible underlying factors will be discussed later.
Simulated and Experimental APD Restitution Curves Are Similar
In the human heart, similar to other mammalian hearts, increasing the pacing rate or shortening of the coupling interval of a premature beat leads to shortening of the APD.51,52 The reconstruction of this physiological phenomenon by this model serves a very important purpose, ie, validation. Therefore, APD restitution was simulated according to the experimental protocol of Morgan et al.51 For this purpose, paired stimuli were used to elicit a second AP (AP2) at variable times after the initiation of the first (AP1). Before each simulation with a different time interval, 10 APs were elicited at a frequency of 1 Hz to obtain a steady state in [Ca2+]i transients. For clarity, only APs at the following extrastimulus intervals are shown: 300, 400, 500, 600, 700, and 800 ms (Figure 10A⇓). APD restitution curves are depicted in Figure 10B⇓.
APD90 of the second AP obtained in simulations (Figure 10B⇑, circle) and in the experiment (Figure 10B⇑, square) are plotted as function of extrastimulus interval. The similarity of both curves substantiates the validity of this model.
The Effect of Ito on the APD in Human Myocardium Is Small
4-Aminopyridine is known to prolong the AP. From this result, it has been postulated that reduction of Ito may prolong the AP in cardiac myocytes.13,53,54 However, there are additional effects of 4-aminopyridine on other currents, especially on ICa23 and IK.55 With our model, we can investigate the possible contribution of Ito to the APD in human ventricular myocytes. To assess the effect of Ito on APD, we simulated the APs in both cell groups under conditions of various degrees of Ito inhibition (25%, 50%, 75%, and 100%). As described above, the simulations were preceded by 10 stimulations to obtain steady-state conditions. The simulated APs shown in Figure 11⇓ demonstrate that inhibition of Ito does not significantly prolong the AP in a nonfailing (Figure 11A⇓) or in a failing (Figure 11B⇓) myocyte.
IKr and IKs Control Repolarization in Human Myocardium
Despite their relative small current densities, IKr and IKs largely control repolarization of the AP in human ventricular myocytes. This effect of IKr and IKs on APD in guinea pig ventricular myocytes has already been demonstrated in a previous computer simulation study.56 Inhibition of IKr, the main mechanism of class III antiarrhythmic agents, has also been shown to prolong the AP in in vivo mapping studies.57–59
Carlsson et al60 have found a pronounced prolongation of the AP in nonfailing human ventricular muscles after inhibition of IKr by 10−6 mol/L H 234/09 (almokalant). At this concentration, almokalant significantly blocks only IKr.61
We attempted to imitate this effect of IKr on APD in human myocardium by varying gmax of IKr to 75%, 50%, 25%, and 0% of its original value. To obtain a steady state of the [Ca2+]i transient, 20 action potentials were elicited. Figure 12A⇓ shows the prolongation of the AP in a nonfailing cell by inhibition of IKr to various degrees (left). As in the experimental study,60 increased inhibition of IKr resulted in a progressive prolongation of AP. At 100% inhibition of IKr, APD90 was lengthened from 374.0 to 689.4 ms, which is in the range measured by Carlsson at al.60 This APD prolongation was, however, larger than that found in single human myocytes after 100% inhibition of IKr by Li et al.21 This discrepancy may be partially due to the different experimental conditions. In contrast to our simulations, the intracellular Ca2+ was buffered by using 5 mmol/L EGTA, significantly influencing ICa and, thereby, the AP. Inhibition of IKr also prolonged AP in a failing myocyte (Figure 12B⇓, left). In contrast to a nonfailing myocyte, 50% inhibition of IKr resulted in an incomplete repolarization of the cell membrane in a failing myocyte at 1.0 Hz (Figure 12B⇓, left; 50%). At 75% inhibition of IKr, even an EAD developed after 3 stimulations. Therefore, a failing myocyte is more sensitive than a nonfailing myocyte to IKr inhibition. Simulations with various degrees of inhibition of IKs have indicated that this current also has a significant impact on APD in both cell groups (Figure 12A⇓ and 12B⇓, right). However, inhibition of IKr had an even greater effect. In a nonfailing myocyte, 100% inhibition of IKs prolonged APD90 from 374.0 to 526.1 ms. In a failing myocyte, incomplete repolarization occurred at 100% inhibition of IKs.
Complete Inhibition of IKr Induces Development of Recurrent EADs in Failing Myocytes
The previous simulations reveal the critical role of IKr in repolarization and, thereby, in the electric stability of the cell membrane in a failing myocyte. Therefore, inhibition of IKr may facilitate the formation of EADs in failing myocytes. At 75% inhibition of IKr, we can observe an EAD after 3 stimulations. To investigate whether there are recurrent EADs in a failing myocyte without preconditioning stimulations, simulation of the action potential in a failing myocytes was performed while assuming complete inhibition of IKr. The computed AP, [Ca2+]i transient, and ICa and INaCa are shown in Figure 13⇓. It is obvious that 100% inhibition of IKr leads to recurrent EADs in a failing myocyte. IKr inhibition prolonged the time when the membrane potential was at ≈−30 mV. This allowed for sufficient time for reactivation of ICa and for generation of EADs. During the decay phase of EADs, INaCa transiently becomes a depolarizing current. However, the magnitude of INaCa (−0.05 pA/pF) was much smaller than that of ICa (−0.5 pA/pF). Therefore, it can be concluded that EADs are mainly carried by ICa in our model.
The Likelihood of Premature APs Is Enhanced in a Failing Myocyte
Premature APs can be triggered by DADs, which are caused by a spontaneous Ca2+ release from the SR.62,63 Since INaCa as a possible underlying current is increased in failing myocytes, it can be expected that premature APs occur more frequently in these myocytes than in nonfailing cells. However, there is evidence that the Ca2+ content of SR is decreased in failing myocytes, which would lead to a smaller Ca2+ increase after spontaneous Ca2+ release from the SR and, thereby, to a reduced driving force for the depolarizing INaCa in these cells.
Therefore, it remains unclear whether increased INaCa in failing myocytes is indeed combined with a higher occurrence of DADs. Additionally, it should pointed out that DADs are not, per se, arrhythmogenic. One could rather imagine that the generation of DADs would result in reduced excitability of the myocyte by affecting the availability of Na+ channels. The generation of DADs exerts a solely proarrhythmogenic effect if the depolarization reaches the threshold to open Na+ channels and trigger a premature AP. Consequently, we investigated whether the combined electrophysiological alterations in heart failure would enhance the likelihood of premature APs. For this purpose, a spontaneous Ca2+ release from the SR was simulated in both cell groups. As Luo and Rudy64 have shown, the recovery from the slow inactivation of Na+ channels, named factor j in their model, determines the recovery of the excitability after the AP. To investigate the influence of INaCa on the generation of premature APs separately, the Na+ channels should be completely recovered from their slow inactivation at the start of the spontaneous Ca2+ release from the SR. Therefore, a spontaneous Ca2+ release from SR was assumed to occur at least 250 ms after 90% repolarization of the last AP in both cell groups. After this diastolic time interval, the factor j=1 (eg, the availability of Na+ channels was ≈100%) in both cell groups. We assumed an equal Ca2+-independent mechanism of the spontaneous Ca2+ release from the SR in both cell groups. Simulations of spontaneous Ca2+ release from the SR were preceded by a train of stimulations at 2.0 Hz for elevating the Ca2+ content of the SR. After 11 stimulations, there were differences in Ca2+ homeostasis between both cell groups (Figures 14⇓ and 15⇓). Before spontaneous Ca2+ release occurred, diastolic [Ca2+]i was higher in a failing myocyte (275 nmol/L) than in a nonfailing myocyte (218 nmol/L). [Ca2+]NSR was increased from 2.9 to 3.4 mmol/L in a nonfailing myocyte and from 1.2 to 1.5 mmol/L in a failing myocyte. As expected, the postulated spontaneous Ca2+ release from the SR resulted in a greater increase of [Ca2+]i in a nonfailing myocyte (1394 nmol/L) than in a failing myocyte (713 nmol/L).
As a result, depolarizing peak INaCa was greater in a nonfailing than in a failing myocyte (−1.35 versus −1.0 pA/pF, respectively), although the activity of the Na+-Ca2+ exchanger was elevated in the failing myocyte. However, a premature AP was generated only in a failing myocyte (Figure 15⇑).
Results of simulations of ionic currents and of the APD restitution show that this model can reproduce important electrophysiological characteristics of human ventricular myocytes. Voltage-clamp simulations of the major ionic currents (Figures 2 to 6⇑⇑⇑⇑⇑) and the simulation of AP (Figure 7⇑) closely resemble experimental data obtained from human single myocytes. Simulations of the ionic currents during the time course of the action potential (Figures 7⇑ and 8⇑) and simulations of ionic current inhibition reveal the important role of IKr and IKs in repolarization in human ventricular myocytes. INaK may additionally contribute to repolarization in human ventricular myocytes and, together with IK1, stabilize the resting potential (Figure 8⇑). Furthermore, simulations of ionic currents during the AP demonstrate that INaCa is an important depolarizing current during the late phase of repolarization in a failing myocyte (Figure 7⇑) because of the enhanced activity of the Na+-Ca2+ exchanger and the slow decay of the [Ca2+]i transient in this cell (Figure 8⇑). In contrast to the results from an AP model of guinea pig ventricular myocytes,56 we have found that the influence of IKr on APD is greater than that of IKs. This discrepancy is due to a much smaller ratio of gmax of IKs to IKr in the present study (1.3) compared with the animal model (7.72). Our value is validated by simulations of current-voltage relations of tail currents of IKr and IKs, which are very close approximations to experimental data (Figure 5C⇑). Although the gmax of IKs is still greater than that of IKr in the present study, IKs contributes a smaller fraction of the total IK during a ventricular AP because of its slow activation kinetics (Figure 5B⇑).
The influence of IKr and IKs on APD may be one reason for the wide range of measured APDs in different publications (between 233 ms17 and 650 ms13 in single myocytes from nonfailing hearts). Yue et al65 have demonstrated that the expression of IK in cells isolated from the canine heart is very dependent on the isolation technique.
Among other factors, different expressions of IKr and IKs in single human myocytes, possibly due to different isolation procedures, could be a reason for this divergence of APD. However, the AP simulated in a nonfailing myocyte (Figure 7⇑) in our model very closely resembles the AP measured in single cells by our work group (Figure 9A⇑) and by Peeters et al.39 When the monophasic AP technique was used in in vivo studies,51,52 the observed shape and duration of AP in human hearts were very similar to our simulations.
In studies of various animal models of heart failure or hypertrophy, a reduction of Ito has been found, and this has been assumed to be an important factor causing AP prolongation in heart failure.54,66,67 In human ventricular myocytes, inhibition of Ito has been found to prolong AP.13 Our simulations of different degrees of inhibition of Ito (Figure 11⇑) suggest, however, that inhibition of Ito does not prolong AP in nonfailing or failing myocytes in human hearts. A reason for this discrepancy could be that 4-aminopyridine, which was used in the experimental study,13 also blocked IKs and IKr, which would lengthen the AP. Considering this problem in experimental conditions, we conclude from the simulations of our model that the influence of Ito on APD is negligible in human myocardium. Of course, this model cannot prove that Ito does not alter APD, but it strongly suggests that reduction of the current density of Ito found in failing myocytes of human hearts does not seem to contribute significantly to the APD prolongation in heart failure. This hypothesis should be tested experimentally using a highly specific blocker of Ito in the future. In a human atrial AP model, it has been demonstrated that the effect of Ito on the AP is largely dependent on the magnitude of IK.68
Therefore, the distinctly different influence of Ito on APD in human ventricular myocytes compared with other cell types and species may be caused mainly by the specific kinetics and magnitude of IKr and IKs found in human ventricular myocytes. In myocytes, where IK is found to be small, Ito may significantly determine the repolarization phase of the AP.69
Prolongation of the AP in human ventricular myocytes of failing hearts has been well established.13,22,24,26 Most authors have found the APD to range between 400 and 1260 ms. In most of these studies, intracellular Ca2+ was buffered using EGTA in the pipette solution. Therefore, the influence of INaCa on the prolongation of the AP can be only slight in these studies. Our simulations suggest that an APD of >600 ms can be explained only by assuming a significant reduction of IKr or IKs (Figure 12⇑). In those studies in which APD was very long, the authors could indeed detect only a small IKr and no IKs.13,24 As in nonfailing myocytes, the variability in shape and duration of APs in failing myocytes measured in our laboratory and in others could possibly be explained by a variable expression of IKr and IKs (Figure 9B⇑ to 9D). Further studies are necessary to show whether alterations of IKr and IKs in failing myocytes reflect real current changes in heart failure or whether they are caused by the isolation procedure.
Conclusions From the Simulated APs for Arrhythmogenesis in Heart Failure
From the results shown in Figures 13⇓, 14⇑, and 15⇑, we conclude that in heart failure one important mechanism for triggered arrhythmias could be DADs. In our model, DADs were initiated by a postulated spontaneous Ca2+ release from the SR. The resulting [Ca2+]i increase depolarized the cell membrane through INaCa in both cell groups. Therefore, as in animal myocytes, this indicates that INaCa may also significantly contribute to the generation of DADs in human myocytes.
Although the activity of the Na+-Ca2+ exchanger is enhanced in failing myocytes, INaCa is slightly larger in nonfailing than in failing myocytes, since the [Ca2+]i increase is much higher in this cell. Nevertheless, a premature AP can be triggered only in a failing myocyte as the repolarizing ionic currents, IK1 and INaK, are reduced in this cell. This indicates that a reduction of repolarizing currents (IK1 and INaK) rather than an increase of the depolarizing current (INaCa) seems to be responsible for the enhanced likelihood of triggered APs in failing myocytes. In our model, the increase of the Na+-Ca2+ exchanger activity in a failing myocyte can only partially compensate the smaller [Ca2+]i increase as a driving force after spontaneous Ca2+ release from the SR. It should be pointed out that the conclusion of these simulations is limited by the assumption that the spontaneous Ca2+ release from the SR is self-initiated and equal in both cell groups. This limitation is necessary because our understanding of the mechanisms involved in the spontaneous Ca2+ release from the SR is incomplete. At present, the mechanism of the spontaneous Ca2+ release from the SR is thought to be a Ca2+ overload of the cell.70,71 Since the Ca2+ content of the SR increases faster in a nonfailing than in a failing myocyte at 2.0 Hz (Figures 14⇑ and 15⇑), the spontaneous Ca2+ release from the SR after pacing burst and resulting DADs is expected to occur earlier and more frequently in a nonfailing than in a failing myocyte, assuming that spontaneous Ca2+ release occurs if the Ca2+ content of the SR reaches a threshold level. This is, however, in contrast to experimental animal studies in which DADs and triggered APs were observed more frequently in failing or hypertrophied animal myocytes.72,73 The development of this model is inadequate to resolve this discrepancy. It cannot predict the occurrence of spontaneous Ca2+ release from the SR in both cell groups. Nevertheless, it may help to elucidate whether there are differences in the induction of DAD-triggered APs between both cell groups. Indeed, the simulations clearly show that spontaneous Ca2+ release leads to a triggered AP only in a failing myocyte. From these triggered APs, triggered arrhythmias may arise.
Even if we assume a higher incidence of spontaneous Ca2+ release from the SR in nonfailing myocytes, the higher incidence of resulting DADs would not result in a higher incidence of triggered arrhythmias than in failing myocytes.
The conclusion about the role of INaCa in DADs and triggered APs does not mean that the Ca2+-dependent nonspecific cation channel has no role in the arrhythmogenesis in heart failure. In animal studies, there is evidence for its contribution in generating DADs.74,75 Luo and Rudy76 concluded from the results of their simulations that the contribution of Ins(Ca) and INaCa to the induction of DADs depends on the level of Ca2+ overload. However, at present, the existence of a nonspecific cation channel in human ventricular myocardium is unknown.
EADs are triggered during the plateau phase of an AP and are thought to be caused by reactivation of the ICa.9,10 The membrane potential has to remain at voltages positive to −35 mV until the L-type Ca2+ channels are able to recover from their inactivation and can open again. Detailed models have been developed addressing this issue. Nordin and Ming77 showed that current-induced EADs in guinea pig ventricular myocytes are mainly due to the L-type Ca2+ channel window current. Zeng and Rudy78 came to the same conclusion in their model simulating the effect of cesium, Bay K 8644, and isoproterenol on the AP. In various preparations, specific block of IKr has been found to result in EADs in nonfailing animal myocytes. However, EADs are not generated by inhibition of IKr block in a nonfailing myocyte in this model (Figure 12A⇑, left). Even if the gmax of IKr and IKs is changed simultaneously, corresponding to the approach of Zeng et al56 to generate EADs in their model, no EADs can be generated in a nonfailing myocyte (not shown). In contrast to this, 75% inhibition of IKr can lead to an EAD in a failing myocyte (Figure 12B⇑, left; 75%). When the inhibition of IKr is 100%, even recurrent EADs develop in this myocyte (Figure 13⇑).
As already demonstrated in previous models,77,78 reactivated ICa is also the underling inward current of these EADs. In conclusion, our results indicate that EADs are difficult to induce in human compared with animal myocytes. They can be generated only in failing myocytes after blocking IKr by at least 75% (Figure 12B⇑, left, and Figure 13⇑). This discrepancy in EAD formations between animal models and the present model is supported by experimental data. Vermeulen et al73 have observed that EADs occur only in papillary muscles of rabbit hearts but never in human papillary muscles. Further studies are required to assess the role of EADs in arrhythmias in human ventricular myocytes.
Besides triggered APs, reentry mechanisms may also contribute to the increased incidence of tachyarrhythmias in heart failure.79 Among many promoting factors, dispersion of refractoriness or wide variations in the duration of APs observed in hypertrophied myocardium80 can evoke reentry arrhythmias.81 Our model can also contribute to the investigation of this type of arrhythmia in heart failure by identifying the ionic currents that are important for the APD in human ventricular myocytes. The simulations demonstrate that IKs has an impact on AP in the human heart. Regional differences of IKs reported in canine left ventricle82 could partially explain the known heterogeneity of the AP in human left ventricle.83 This may also hold true to an increased extent in heart failure. In addition, the increase of INaCa and the alterations of [Ca2+]i handling may also not be uniform in heart failure, thereby increasing the inhomogeneity of the AP across the heart wall. Another factor determining dispersion of APs along the myocardial wall could be the passive electrical properties of the myocardium. Keung et al84 have demonstrated that these are different in normal and hypertrophied rat myocardium. Further studies are necessary to determine whether it also holds true in the human heart.
Limitations of the Model
A variety of models have been developed previously on the basis of the results of animal studies.31,56,85–88 Since major differences exist in the characteristics of ionic currents between human and animal myocytes, conclusions drawn from these models cannot easily be extrapolated to human heart cells. However, although the present model has the advantage of being based partially on human data, it also has several limitations.
Significant uncertainty remains concerning the magnitude of INaCa. At present, voltage-clamp data of this current in human ventricular myocytes are not available. Comparison of INaCa incorporated in the model with the measured data in the human atrium suggests that the magnitude of the simulated INaCa has been well estimated. Nevertheless, it should be pointed out that some results have to be interpreted with caution because of their dependence on INaCa. However, the simulations clarify that an increase of INaCa could play an important role in the arrhythmogenesis in heart failure and that its quantification in human ventricular myocytes is desirable.
To simulate [Ca2+]i transients, the approach of the LR model is used. Although major components of intracellular Ca2+ homeostasis are included in this model, this approach is only an approximation of the complex nature of the intracellular Ca2+ homeostasis, since some important features, such as the CICR from the SR or the mechanism of Ca2+ buffering, and their potential changes in heart failure are simplified. This is mainly due to our incomplete understanding of the exact mechanisms involved in these phenomena. Nevertheless, the simulated [Ca2+]i transients in both cell groups agree largely with experimental observations and provide an independent test of how well our model describes [Ca2+]i homeostasis.
Another assumption of the model that has not been investigated is that IKr and IKs are unaltered in heart failure. In addition, information about IKr dependence on [K+]o and IKs dependence on [Ca2+]i is not available at present.
Therefore, it should be stressed that further development of this model is needed for simulating alterations of the APs under various pathophysiological conditions, such as myocardial ischemia, as performed by Shaw and Rudy.89 Nevertheless, the conclusions from the simulations presented here, and also their limitations, highlight several important areas that deserve future experimental studies.
Selected Abbreviations and Acronyms
|[A̅]||=||maximum concentration of [A]|
|APD||=||action potential duration|
|APD25, APD50, APD90||=||APD at 25%, 50%, and 90% repolarization|
|CICR||=||Ca2+-induced Ca2+ release|
|I Ca||=||L-type Ca2+ current|
|I Ca,b||=||Ca2+ background current|
|I K||=||delayed rectifier K+ current|
|I K1||=||inward rectifier K+ current|
|IKr, IKs||=||rapid and slow component of IK|
|I Ktail||=||K+ tail current|
|I Na||=||fast Na+ current|
|I Na,b||=||Na+ background current|
|I NaCa||=||Na+-Ca2+ exchanger current|
|I NaK||=||Na+-K+ pump current|
|I ns(Ca)||=||Ca2+-dependent nonspecific cation current|
|I st||=||stimulus current|
|I to||=||transient outward K+ current|
|JSR||=||junctional sarcoplasmic reticulum|
|Kleak||=||rate of Ca2+ leakage out of NSR|
|Km,i||=||half-saturation concentration of channel i|
|k NaCa||=||scaling factor of INaCa|
|NSR||=||network sarcoplasmic reticulum|
|V0.5||=||50% activation voltage|
Formulations of Ionic Currents
Fast Na+ Current: INa
where m, h, and j are the activation gate, the fast inactivation gate, and the slow inactivation gate of INa, respectively, V is the membrane potential, ENa is the equilibrium potential for Na+, R is the universal gas constant, T is the absolute temperature, and F is the Faraday constant.
For V≥−40 mV For V<−40 mV For the total range of V
Slow Inward Current: ICa
where gCa,max is gmax for ICa, d is the activation gate of ICa, f is the inactivation gate of ICa, ECa is the equilibrium potential for Ca2+, fCa is a proportional factor for Ca2+-dependent inactivation of ICa, and KCa is half-maximum Ca2+ binding concentration for ICa.
Transient Outward Current: Ito
where gto,max is gmax for Ito, r and t are the activation gate and the inactivation gate of Ito, respectively, and Eto is the equilibrium potential for Ito.
Delayed Rectifier Current
Slowly Activating Current: IKs
where gKs,max is gmax for IKs, Xs is the activation gate of IKs, and EKs is the equilibrium potential for IKs.
Rapidly Activating Current: IKr
where gKr,max is gmax for IKr, Xr is the activation gate of IKr, rik is the inward-rectification factor of IKr, and EKr is the equilibrium potential for IKr.
Inward Rectifier Current: IK1
where gK1,max is gmax for IK1, K1∞ is the inactivation gate of IK1, and EK1 is the equilibrium potential for IK1.
Ca2+ Background Current: ICa,b
where G̅Ca,b is gmax for ICa,b, and ECa,b is the equilibrium potential for ICa,b.
Na+ Background Current: INa,b
where G̅Na,b is gmax for INa,b, and ENa,b is the equilibrium potential for INa,b.
Pump and Exchanger
Na+-K+ Pump: INaK
where fNaK is the voltage-dependence parameter of INaK, and ς is the [Na+]o-dependence factor of INaK.
Na+-Ca2+ Exchanger Current: INaCa
where ksat is the saturation factor of INaCa at very negative potentials, and η is the position of the energy barrier controlling voltage dependence of INaCa.
CICR of JSR
where Irel is the SR Ca2+ release current, and Grel is the rate constant of Ca2+ release from JSR.
Spontaneous Ca2+ Release of JSR
Ca2+ Uptake and Leakage of NSR: Iup and Ileak
where Iup is the SR Ca2+ uptake current, and Ileak is the leakage current.
Translocation of Ca2+ From NSR to JSR: Itr
where Itr is the SR Ca2+ translocation current.
Ca2+ Buffers in the Myoplasm: Troponin (TRPN) and Calmodulin (CMDN)
where Km,TRPN is the half-saturation concentration of TRPN, and Km,CMDN is the half-saturation concentration of CMDN.
Ca2+ Buffer in JSR: Calsequestrin (CSQN)
where Km,CSQN is the half-saturation concentration of CSQN.
This study was supported by the Deutsche Forschungsgemeinschaft (Be 1113/2–3) and the Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie (01 Kans 9502). Special thanks go to Dr M. Lindner for recording the APs.
- Received January 5, 1998.
- Accepted April 6, 1998.
- © 1998 American Heart Association, Inc.
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