Mouse hearts have a diversity of action potentials (APs) generated by the cardiac myocytes from different regions. Recent evidence shows that cells from the epicardial and endocardial regions of the mouse ventricle have a diversity in Ca2+ handling properties as well as K+ current expression. To examine the mechanisms of AP generation, propagation, and stability in transmurally heterogeneous tissue, we developed a comprehensive model of the mouse cardiac cells from the epicardial and endocardial regions of the heart. Our computer model simulates the following differences between epicardial and endocardial myocytes: 1) AP duration is longer in endocardial and shorter in epicardial myocytes, 2) diastolic and systolic intracellular Ca2+ concentration and intracellular Ca2+ concentration transients are higher in paced endocardial and lower in epicardial myocytes, 3) Ca2+ release rate is about two times larger in endocardial than in epicardial myocytes, and 4) Na+/Ca2+ exchanger rate is greater in epicardial than in endocardial myocytes. Isolated epicardial cells showed a higher threshold for stability of AP generation but more complex patterns of AP duration at fast pacing rates. AP propagation velocities in the model of two-dimensional tissue are close to those measured experimentally. Simulations show that heterogeneity of repolarization and Ca2+ handling are sustained across the mouse ventricular wall. Stability analysis of AP propagation in the two-dimensional model showed the generation of Ca2+ alternans and more complex transmurally heterogeneous irregular structures of repolarization and intracellular Ca2+ transients at fast pacing rates.
- cardiac myocytes
- action potential
- computer modeling
the mouse is an important animal for the study of the physiology of the heart and various pathophysiological and proarrhythmic conditions such as heart failure. The development and study of genetically modified mice have often led to unexpected or counterintuitive results in cardiac electrophysiology. The resolution of complex interactions often turned to the use of comprehensive models of mouse cardiac myocytes to explain experimental observations of action potential (AP) generation, propagation, Ca2+ dynamics, and arrhythmias (14, 35, 42, 44, 50, 56, 58, 67). Spatial diversity of molecular heterogeneity is now understood to be an important factor in arrhythmogenesis. One difference of key importance is thought to be the difference between endocardial and epicardial tissues (8, 18). In this article we develop models of epicardial and endocardial mouse myocytes, using our previous models of cells from the apex and septum regions of the heart (3, 4, 6) as a starting point. The new epicardial and endocardial cell models have different membrane current densities and distinct Ca2+ handling properties based on recent data from the different regions of the mouse heart (8, 18). These models reproduced the fundamental differences in the two cell types including some unexpected phenotypes. The model cells were coupled together into two-dimensional (2-D) tissues to explore the consequences of a transmural gradient on the stability of APs and Ca2+ transients.
Transmural heterogeneity is of critical importance in the generation of arrhythmias and ECG abnormalities in larger mammals. This has been less clear in the mouse, in part because of the absence of models that distinguish between epicardial and endocardial cells based on experimental data. Recently, voltage-clamp studies of major repolarization currents and Ca2+ handling in the mouse ventricular myocytes from two other regions of the mouse heart, epicardium and endocardium, have shown that there are differences in K+ repolarization currents and in Ca2+ handling mechanisms (8, 18). We have used these data to reproduce dynamic experimental studies of AP shape and duration from the isolated myocytes, epicardial and endocardial tissues (2, 8, 32). The model simulations of differences in AP shape and durations were predominantly due to the differential expression of three major K+ repolarization currents: rapidly inactivating transient outward K+ current (IKto,f), ultrarapidly activating K+ current (IKur), and noninactivating steady-state K+ current (IKss) (8). These current differences were capable of sustaining functional heterogeneity of repolarization in a 2-D sheet, roughly approximating the thickness of the left ventricular wall.
In contrast to our previous model of apex and septal cells, the new models incorporated experimental findings on the differences in transmural Ca2+ handling (18). These differences include diastolic and systolic intracellular Ca2+ concentration ([Ca2+]i), and [Ca2+]i transients, which are higher in paced endocardial and lower in epicardial myocytes, a Ca2+ release rate that is about two times larger in endocardial than in epicardial myocytes, and a Na+/Ca2+ exchanger turnover rate that is greater in epicardial than in endocardial myocytes. This allowed us to examine the interaction between membrane currents and intracellular Ca2+ handling as a mechanism for transmural differences in Ca2+ handling and the consequences of the interaction of heterogeneity of both Ca2+ and AP in a 2-D system.
Perturbation of the key components of the Ca2+ handling system revealed that intracellular components as opposed to sarcolemmal ion channel differences are the most sensitive factors regulating the differences in Ca2+ transient magnitude and behavior between the two myocyte types. Simulated pacing of isolated epicardial and endocardial myocytes revealed that there was a more complex response of the epicardial cells at rapid rates compared with endocardial cells. We also studied the consequences of transmurally inhomogeneous 2-D ventricular tissue that contained models of epicardial and endocardial myocytes. This multicellular modeling reproduced experimentally observed propagation velocities and predicted that rapid pacing rates would yield the development of Ca2+ alternans, irregular APs, and a complex transmurally inhomogeneous structure of [Ca2+]i transients. These studies suggest that molecular and electrotonic interactions can produce unexpected complex behavior related to potential arrhythmogenic mechanisms. These models should be a useful tool for others in understanding the complex interactions that occur during disease processes such as heart failure that are frequently studied on a molecular level in transgenic mice.
We investigated a transmural 2-D model tissue consisting of the mouse ventricular myocytes modified from Ref. 6 (Fig. 1A). Two different types of cells were developed to simulate AP propagation and Ca2+ handling in transmurally inhomogeneous model tissue: the epicardial and the endocardial cells. These cells differ in the set of K+ currents and Ca2+ handling systems. They are based on the model of Bondarenko et al. (6) with differences in parameters that are outlined in Table 1.
Epicardial and endocardial ventricular myocytes have different expression of K+ currents. These K+ channel expression levels are mainly responsible for differences in AP duration (APD) (8). Experimentally, differences were described in the magnitudes of three major repolarization currents: IKto,f, IKur, and IKss (8). We adjusted the values of the maximal current conductances for these currents to fit the experimental values of the current densities without changing the kinetic parameters (Table 1). Experimental data from Ref. 8 for the inactivation of total voltage-activated K+ currents are shown in Fig. 2, A and B. They compared well with the corresponding simulations for epicardial and endocardial model cells (Fig. 2, C and D).
The first step in modeling the differences in Ca2+ dynamics between epicardial and endocardial cells was to “tune” the previous model (6) and address some of the limitations to this portion of the system. The previously published models of apical and septal cells have been criticized as having disproportional levels of Ca2+ extrusion by the Na+/Ca2+ exchanger (INaCa) relative to the Ca2+ pump (6, 43). This limitation in the previous model has been addressed in the current model by a significant reduction of the maximum Ca2+ pump current from 1.0 pA/pF in the previous model to 0.17 and 0.085 pA/pF in the current model for epicardial and endocardial myocytes, respectively (Table 1). We also reduced the maximal conductance for the Ca2+ background current (Table 1) that played a compensatory role for Ca2+ extrusion via Ip(Ca) in the apical and septal cell models (6, 43).
We also improved the Markov model for the L-type Ca2+ current (ICaL) (Fig. 4 in Ref. 6) by simplifying the voltage dependence of the rate constants and tuning them to match the experimental data. While the model of the Ca2+ current in Ref. 6 matched the data, it had some arbitrary voltage dependence for the activation rate constant α(V) that had no biophysical interpretation. The Markov model structure was not changed. The rate constant for activation [α(V) and β(V)] and inactivation [γ(Ca2+) and Kpcf] were adjusted to match experimental data (6, 31, 33, 75). Simulated ICaL traces obtained by depolarization for 5 s to between −70 and +40 mV (in 10 mV increments) from a holding potential of −80 mV and under heavy buffer conditions (5–10 mM EGTA) are shown in Fig. 3A (simulations in Fig. 3, A–C, are shown for epicardial cells, which are not noticeably different from those for endocardial cells). They are similar to the corresponding current traces observed experimentally under similar conditions (41, 75). Normalized peak current-voltage relationships for ICaL are shown in Fig. 3B for normal (without buffer) and heavy buffer conditions by solid and dashed lines, respectively. For normal conditions, we used the same voltage-clamp protocol as for the heavy buffer conditions, except for the duration of the first pulse P1 (250 ms). Both simulations show similar voltage dependences for normalized peak current amplitude and compare well to the experimental data (31, 33, 75). The corresponding absolute maximum peak current amplitudes (7.9 and 8.1 pA/pF) are well centered within the range of experimental values 5.1–11.6 pA/pF as well (29, 53, 70). Markov models for ICaL were improved to fit not only steady-state inactivation relationships, as in our previous model (6), but also the voltage dependence of normalized ICaL conductance G/Gmax (Fig. 2C), which should reflect steady-state activation.
While the gating properties of the ICaL in isolation are similar for epicardial and endocardial myocytes, there are differences between them because of the amplitudes of the [Ca2+]i transients. These differences in transients are due to differences in Ca2+ handling mechanisms (18). The modeling of increased release is based on the experimental observation that the expression level of ryanodine receptors is about twofold higher in endocardial than in epicardial myocytes (18). To simulate this difference with our model, we used Ca2+ release rates ν1 = 1.5 and 2.9 ms−1 for epicardial and endocardial cells, respectively (Table 1). These values for ν1 were also chosen to match differences both in [Ca2+]i and to reproduce the average integrated experimental Ca2+ release efflux of about 39 ± 3 μM (40). The resulting simulations for the voltage dependences of peak [Ca2+]i are shown in Fig. 3D by solid and dashed lines for epicardial and endocardial myocytes, respectively. Corresponding experimental data (18) are shown in Fig. 3D by black and white diamonds. Relationships between voltage dependences of peak [Ca2+]i and ICaL reproduce graded Ca2+ release.
In addition to the differences in Ca2+ release as described above, epicardial and endocardial cells possess distinct diastolic Ca2+ levels and rates of Ca2+ extrusion by INaCa (18). Experimental data show that epicardial myocytes have a smaller level of diastolic Ca2+ during stimulation and about a twofold faster rate of Ca2+ extrusion via INaCa. However, both cell types have the same resting values of [Ca2+]i (18). To simulate these differences, we used two different scaling factors kNaCa for the Na+/Ca2+ exchanger in the epicardial and endocardial cell models, 292.8 and 146.4 pA/pF, respectively (Table 1). These values produced slightly different diastolic resting [Ca2+]i in the two myocyte models, 0.124 and 0.136 μM for epicardial and endocardial cells, respectively, which are close to the corresponding experimental resting values 0.150 ± 0.019 and 0.156 ± 0.021 μM (18).
The transmural 2-D model tissue consists of 9,100 mouse ventricular myocytes, a rectangular sheet of 130 myocytes × 70 myocytes (nx = L = 130, transmural direction; ny = N = 70, longitudinal direction, Fig. 1B). The number of cells and dimensions of 2-D tissue were chosen to match the experimental measurements of the mouse ventricle, with a length and thickness of about 7 and 1.6 mm, respectively (28, 59). Taking into account the cell volume 3.8×104 μm3 (cytosol volume is 2.584×104 μm3, or 68% of the total cell volume) (6, 39), we chose the cell length of 100 μm, the cell width of 30.4 μm, and the cell thickness of 12.5 μm (Fig. 1B). These myocyte dimensions are close to those obtained in different experimental studies (12, 45, 52, 55). As a result, our 2-D model represents the rectangular surface with a 7 mm length and 1.625 mm width. Each model myocyte is described by the equations from Ref. 6 with corresponding modifications (Table 1) and initial conditions plus intercellular currents: (1)where i is the cell number, Cm,i is the membrane capacitance of the ith cell, Vi is the membrane potential, Imemb,i is the total membrane ion current (Ref. 6 and Table 1), Istim,i is the stimulus current, Igap,i = ggap,xi (−Vi−1 + 2Vi − Vi+1) + ggap,yi (−Vi−nx + 2Vi − Vi+nx), where ggap,xi and ggap,yi are the gap conductances along axes x and y, respectively. Note that for the model myocytes on the tissue perimeter, the expression for Igap,i contains only two or three intercellular currents from the neighboring cells. In this article, we used values of ggap,xi = ggap,x and ggap,yi = ggap,y for all myocytes. The conductances in two directions were chosen to yield experimentally measured AP propagation velocities in the mouse hearts, ggap,x = 500 nS/pF (0.36 m/s) and ggap,y = 25 nS/pF (0.62 m/s) (27, 68). When we take into account that the mouse ventricular myocyte capacitance is about 100 pF (7, 8), these numbers correspond to resistances of ∼0.02 and 0.4 MΩ, respectively. Since there are no explicit measurements of transmural propagation velocity in the mouse heart, we assumed that this value is close to the measured minimal velocity, 0.36 m/s, measured for larger animals where transmural velocity varies from 0.33 to 0.41 m/s (47, 48, 73).
For the isolated myocytes, Istim was 60 pA/pF. In all cases, the stimulus pulse duration was equal to 0.5 ms. The amplitude of the stimulus current Istim,i for 2-D tissue was varied from 180 to 2,000 pA/pF to be equal approximately to 150% of the threshold value, depending on the intercellular conductance. Stimulation protocols will be described in detail in the text or in the figure legends. Inhomogeneous 2-D tissues consist of 65 columns of epicardial and 65 columns of endocardial cells. We also used the approximation that all conduction delay occurs at the gap junctions. This approximation was validated for model multicellular cardiac tissue in the detailed study of Shaw and Rudy (61).
Model equations were solved by a fourth-order Runge-Kutta method, with a time step of 0.0001 ms. Steady-state initial conditions were obtained by running the models until changes in all variables did not exceed 0.01%. The AP of the single ventricular myocyte, voltage-clamp experiments, and bifurcation diagrams were simulated on single processor (3.2 GHz Intel Xeon “Irwindale”). Thirty-six processors were used to simulate 2-D model activity: one of them was used for data input-output, and the remaining 35 processors performed numerical simulations, 260 model myocytes on each processor.
Models of AP and Ca2+ handling in epicardial and endocardial ventricular myocytes.
In this study, we developed models for epicardial and endocardial mouse ventricular myocytes. These models used our previously published model of the mouse apical ventricular myocyte (6) as a starting point. There are five major differences between epicardial and endocardial cells: 1) APD is longer in endocardial and shorter in epicardial myocytes; 2) diastolic and systolic [Ca2+]i and [Ca2+]i transients are higher in paced endocardial and lower in epicardial myocytes; 3) Ca2+ release rate is about two times larger in endocardial than in epicardial myocytes; 4) Na+/Ca2+ exchanger rate is greater in epicardial than in endocardial myocytes (18); and 5) epicardial and endocardial cell have different conductances for IKto,f, IKur, and IKss (8).
Simulated APs from epicardial and endocardial myocytes and the corresponding major repolarization currents are shown in Fig. 4. As expected from voltage-clamp data of K+ currents, the APD for the model epicardial cell is shorter than that for the endocardial cell [APD at 50% of repolarization (APD50) is equal to 3.7 and 5.4 ms, for epicardial and endocardial cells, respectively, Table 2]. Simulations of the time course of the major repolarization currents (Fig. 4, B and C) demonstrate that significantly larger outward K+ currents in epicardial myocytes produce a smaller APD. These data correspond reasonably well to the experimental APD50 that range from 2.6 to 6.2 for epicardial and from 4.5 to 9.9 ms for endocardial myocytes, respectively (Refs. 2, 8, 32, and Table 2). Similarly, simulated values for APD at 90% of repolarization (APD90) are close to the experimental values for isolated cardiac myocytes (8); however, they are somewhat smaller than the APD90 obtained from the mouse left ventricular tissues (2, 32). Such relative APD prolongation in cardiac tissue was also noted previously and may be due to reduced K+ repolarization currents (4). Other possibilities include difficulties associated with measuring APD90 under different experimental conditions (2, 8, 32).
To evaluate the relative contribution of AP waveform into [Ca2+]i, Dilly et al. (18) depolarized epicardial and endocardial ventricular myocytes with the stored endocardial and epicardial waveforms, respectively, with a stimulation frequency of 1 Hz. If the differences in [Ca2+]i were due exclusively to AP waveform, then [Ca2+]i transients of epicardial cell stimulated with endocardial AP waveform would resemble that of an endocardial cell. Similarly, [Ca2+]i transients of endocardial cell stimulated with an epicardial AP waveform would resemble that of an epicardial cell. Dilly et al. (18) found that this was not the case. While AP waveform contributed to the changes in [Ca2+]i transients, these changes were relatively small. We also stimulated epicardial and endocardial model cells with endocardial and epicardial AP waveforms, respectively, with a basic cycle length (BCL) of 1,000 ms. Simulation results shown in Fig. 5 demonstrated small changes in peak [Ca2+]i in clamped epicardial and endocardial myocytes. Simulated peak [Ca2+]i obtained from the epicardial cell model alone and from the epicardial cell model stimulated with endocardial AP waveform (Endo clamp) were 0.516 and 0.564 μM, respectively. These values were comparable with the experimental peak [Ca2+]i obtained for epicardial myocytes, 0.478 ± 0.078 and 0.707 ± 0.059 μM, respectively (18). Simulated peak [Ca2+]i obtained from the endocardial cell model alone and from the endocardial cell model stimulated with an epicardial AP waveform (Epi clamp) were 0.996 and 0.906 μM, respectively; they are close to the corresponding experimental values 0.886 ± 0.078 and ∼0.8 μM (18). Thus our simulations and the experimental data (18) show that the AP waveforms contribute to, but do not account for differences in, [Ca2+]i transients between epicardial and endocardial ventricular myocytes. The major contribution was due to differences in Ca2+ handling systems of these two myocyte types (18).
To determine the mechanisms underlying the differences in [Ca2+]i transients between epicardial and endocardial cells, we performed a sensitivity analysis, i.e., dependence of [Ca2+]i transients on changes in INaCa, Ip(Ca), the Na+/K+ pump (INaK) rates, background Ca2+ current (ICab), or ICaL, amplitudes, ryanodine release channel density (Jrel), or increased sarcoplasmic reticulum (SR) Ca2+ pump activity (Jup) (54, 64). We simulated changes in [Ca2+]i transients in response to a 15% increase and a 15% decrease in one of the components of the Ca2+ handling system (Fig. 6). In the myocyte model (epicardial or endocardial), we first increased (or decreased) one of the components of the Ca2+ handling system and then ran the perturbed model until steady state was achieved without stimulation. The myocyte model was then stimulated with a BCL of 1,000 ms, and the steady-state [Ca2+]i transient amplitude was calculated as the difference between its systolic and diastolic values. The difference between this [Ca2+]i amplitude and the [Ca2+]i amplitude for control epicardial or endocardial myocyte was plotted in Fig. 6, A and B. [Ca2+]i transient amplitudes are most sensitive to changes in Na+-K+ pump or Na+/Ca2+ exchanger activity, or ICab amplitude. However, a smaller but nonetheless significant contribution to changes in [Ca2+]i occurs in response to the variation of Ca2+ release or uptake rates or by changes in amplitude of ICaL.
To evaluate the absolute contribution of the particular component of the Ca2+ handling system to the difference in [Ca2+]i between the two myocyte types, we also calculated Δ[Ca2+]i for the epicardial cell when INaCa, Ip(Ca), ICab, INaK, or Jrel was changed from its epicardial to endocardial value (Fig. 6C). Our finding shows that there are three major players that are responsible for the difference: INaCa, ICab, and Ca2+ release rate from the SR (Jrel). The total increase in [Ca2+]i (“Total” bar in Fig. 6C) is equal to the sum of partial changes resulted from the changes of INaCa, Ip(Ca), ICab, INaK, and Jrel plus the contribution of Δ[Ca2+]i from the difference in APD between the two myocyte types (0.406 μM + 0.048 μM = 0.454 μM). This value is close to the Δ[Ca2+]i between epicardial and endocardial myocytes (0.470 μM). The result of this analysis suggests that each component of Ca2+ handling in the mouse ventricular myocyte contributes to the change in [Ca2+]i in a nearly additive fashion relative to the changes in the other components, at least near its normal operating point.
Epicardial and endocardial ventricular myocytes also showed different responses to changes in stimulation rate. Figure 7 shows AP amplitudes and APD50 for the epicardial and endocardial myocyte models as functions of the BCLs for the same stimulus amplitude Istim = 60.0 pA/pF (τstim = 0.5 ms). The AP amplitudes and APD50 were calculated after 48 s of stimulations over a 2-s time interval. The epicardial cell model showed a cascade of bifurcations in AP amplitude and APD50 with a decrease in BCL (Fig. 7, A and C). First, periodic activity became irregular when the BCL decreased from 54 to 53 ms. At a BCL of 52 ms, this chaotic activity was replaced by APs with four different APD50, i.e., period 4 locking. Further decreasing BCL to 51 and 48 ms yielded activities with three and two different APD50, respectively. Finally, at a BCL of 44 ms, the model epicardial cell failed to generate full-scale APs at all. Interestingly, this cascade of bifurcations for APD50 was accompanied by the same bifurcation scenario for [Ca2+]i, with identical BCLs for the transition from one type of activity to another. In contrast, the endocardial myocyte model showed only a single transition from periodic full-scale APs to low-amplitude perturbations of the membrane potential at a BCL of 61 ms (Fig. 7, B and D).
A distinctive and unusual property of mouse ventricular AP restitution is an increase in APD at the early stages of repolarization (APD at 30% of repolarization) when BCL decreases (34). Our model was able to reproduce this feature for the both myocyte types. As seen from Fig. 7, C and D, the APD50 increases gradually with the shortening BCL. Eventually, this trend is terminated in the epicardial cell when it starts to bifurcate and show complex behavior. In contrast, the increase in APD50 is terminated by a sharp decrease in endocardial cells. In addition, the amplitude of AP in both myocyte types gradually decreases when APD50 increases. The longer APD50 values at faster pacing rates are related to the recovery of major repolarization K+ currents in mouse ventricular myocytes (4). During a shorter BCL, the K+ currents do not recover completely and become decreased in amplitude. This causes the prolongation of APD50. The decrease of AP amplitude (Fig. 7, A and B) is due to the incomplete recovery of the Na+ current (4). The last effect is also observed experimentally (34).
Figure 8 shows the time dependence of the membrane potential for epicardial (Fig. 8, A–C) and endocardial (Fig. 8, D and E) myocytes. At a relatively long BCL of 80 ms, both cells demonstrate full-scale APs (Fig. 8, A and D). Epicardial cells have an interval of BCLs with irregular AP; the time-dependent behavior of one of them is shown in Fig. 8B. At relatively fast pacing rates with a BCL of 40 ms, both cells are unable to generate full-scale APs but only low-amplitude APs (Fig. 8, C and E). Such low-amplitude APs differ from the full-scale AP by considerably smaller excitatory currents that are unable to overcome the activation barrier for a full-scale AP but still maintain depolarization and by relatively small repolarization currents, such as IKto,f and IKur, which are active but need a longer time to cause repolarization. Such low-amplitude APs with increased depolarization can potentially be proarrhythmic for cardiac tissue.
Transmural gradients of repolarization and [Ca2+]i transients in heterogeneous mouse ventricular tissue.
We used our new models to investigate AP propagation and stability in a 2-D cardiac tissue model designed to mimic the transmural heterogeneity described for the ventricular wall of the mouse heart. Figure 9 shows a simulated distribution of APDs at 25%, 50%, and 75% of repolarization across the mouse ventricular wall. The APs in cardiac tissue are more prolonged than those for isolated epicardial and endocardial myocytes, consistent with experimental observations (2, 8, 32). Despite the small thickness of the ventricular wall in the mouse heart (∼1.6 mm), there is a predicted transmural gradient of repolarization at levels of 25% and 50% that is predominantly defined by the cellular properties. However, at 75% of repolarization, the gradient is mainly determined by the site of stimulation. This gradient is regulated by intercellular coupling (Fig. 9D): the reduction of transmural conductances increased the differences in APDs between epicardial and endocardial regions.
The small dimensions of the mouse heart and short APDs lead to some peculiarities in AP propagation and excitation-contraction coupling. One of them is that the [Ca2+]i transient is very delayed relative to the AP both in experiments and in our simulations (6, 49, 60). Figure 10 shows snapshots of AP and [Ca2+]i in 2-D transmurally heterogeneous tissue at three time points (10, 20, and 30 ms after Istim). The AP waveform at 10 ms showed a heterogeneity of repolarization with a relatively smooth transition from the epicardial to the endocardial region (Fig. 10A). The AP had already passed through the whole mouse ventricular tissue by 10 ms, whereas [Ca2+]i was still on the rising phase in the region of tissue experiencing the earliest excitation (Fig. 10D). By 30 ms, when repolarization is complete (Fig. 10C), we observed a maximum in simulated [Ca2+]i transients and significant differences in the magnitudes between epicardial and endocardial regions (Fig. 10F). Unlike the membrane potential, [Ca2+]i showed a relatively sharp transition between the two myocyte types and was determined predominantly by differences in Ca2+ handling mechanisms such as differences in SR Ca2+ load, Ca2+ extrusion rate, and the Ca2+ release rate, however, with some modulation by APD. Such differences are important for the stability of [Ca2+]i transients.
Stability of AP and [Ca2+]i transients in heterogeneous mouse ventricular tissue.
To investigate the stability of AP propagation and [Ca2+]i transients, we paced inhomogeneous 2-D model tissues with different BCLs. One set of simulations was with control intercellular conductances, ggap,x = 500 nS/pF and ggap,y = 25 nS/pF, and the second was where the intercellular conductances were reduced by factor 5 (ggap,x = 100 nS/pF, and ggap,y = 5 nS/pF). The two tissues, as well as isolated cardiac myocytes, demonstrated different susceptibilities to rapid pacing (Table 3). Epicardial myocytes showed a transition to irregular behavior of both APs and [Ca2+]i transients near threshold BCL, but endocardial cells failed to generate full-scale AP and [Ca2+]i below the threshold BCL. The threshold pacing rates in multicellular simulations were different (Table 3). When epicardial and endocardial myocytes were connected into inhomogeneous 2-D tissue, the thresholds for the instabilities of AP and [Ca2+]i transients in the tissue became different (Table 3). Alternans in [Ca2+]i transients appeared at relatively slower pacing rates (74 ms for normal and 76 ms for reduced conductances, respectively). APs during these stimulations do not show significant changes. However, when the BCLs were reduced to 64 and 67 ms for normal tissue and tissue with lowered conductances, respectively, the APs became unstable and demonstrated irregular behavior (Table 3).
The spatiotemporal development of [Ca2+]i alternans and corresponding behavior of APs are shown in Fig. 11 for row 35 in 2-D tissue with normal conductances (ggap,x = 500 nS/pF, and ggap,y = 25 nS/pF). At relatively long BCLs (≥75 ms), both APs and [Ca2+]i transients demonstrated periodic behavior (Fig. 11, A and E). A reduction of the BCL to 74 ms resulted in the appearance of [Ca2+]i alternans in the epicardial zone only, whereas the corresponding APs did not show significant changes (Fig. 11, B and F). The alternans are in phase in a whole epicardium. A further decrease in BCL to 72 ms also produced spatial differences in [Ca2+]i alternans, when the epicardium was divided into three clusters of cells within which [Ca2+]i transients oscillate in the same phase (Fig. 11G). However, the neighboring clusters of cells show the maxima and minima of [Ca2+]i in different stimulus cycles. At a smaller BCL of 70 ms, the number of such clusters increases and they create an irregular structure in the epicardium (Fig. 11H). Thus changes in the pacing rate resulted in the development of spatiotemporal [Ca2+]i alternans in multicellular cardiac tissue, which increased in complexity as the BCL decreased. It is interesting that the APs both in the epicardium and in the endocardium do not show a significant deviation from the periodic behavior at the same pacing rates (Fig. 11, A–D).
The spatial distributions of [Ca2+]i transients in 2-D transmurally inhomogeneous tissue at different pacing BCLs were particularly interesting. Figure 12, A–D, shows snapshots of simulated [Ca2+]i transients at 40 ms after the stimulus current was applied. At a BCL of 76 ms, [Ca2+]i showed different amplitudes in epicardial and endocardial regions, but without complex spatial structure, except for slightly larger values at the site of stimulation (Fig. 12A, row 1). During faster stimulation with a BCL of 74 ms (Fig. 12B), [Ca2+]i in the epicardial half demonstrated spatial (or discordant) alternans, in which two neighboring regions oscillate in opposite phases. Further decreasing BCL to 72 and 70 ms led to more complex curved spatial transmural structures of [Ca2+]i transients (Fig. 12, C and D). This spatial structure tends to center on the region near the site of the stimulation where the magnitude of [Ca2+]i is the smallest, and the structure did not appear in the endocardial region where [Ca2+]i transients were larger (Fig. 12, B and C).
To elucidate the mechanism of this transmural inhomogeneity of [Ca2+]i, we analyzed the spatial structure of AP amplitude and the duration along with [Ca2+]i transients (Figs. 9A and 12, E and F) based on the results of our previous study related to the stability of Ca2+ handling in mouse ventricular myocytes (4). The spatial distribution of the membrane potentials and APD at 25% of repolarization showed a gradient between epicardial and endocardial regions and within epicardial region itself (Figs. 9A and 12E). In the endocardial region, where [Ca2+]i values are relatively large, they produced stable ryanodine receptor gating and Ca2+-induced Ca2+ release from the SR (4). In the epicardial region, the Ca2+ release became unstable at small BCLs because of the smaller amount of released Ca2+ and shorter APD, compared with the endocardial region (Figs. 3 and 5). The Ca2+-induced Ca2+ release magnitude is determined by cellular mechanisms (there are no longitudinal Ca2+ fluxes between cardiac myocytes in the tissue) and by the Ca2+ influx through L-type Ca2+ channels, which are modulated by APD. Because the APD showed a spatial curvilinear structure in the epicardial region of the tissue (Figs. 9A and 12E), it modulated Ca2+ influx, Ca2+ release, and ultimately [Ca2+]i transients and their stability to the same degree along these curves (compare Fig. 12E and Fig. 12F). As result, a complex transmurally inhomogeneous spatial structure of [Ca2+]i transients emerged (Fig. 12F).
Differences in APs and Ca2+ handling in isolated epicardial and endocardial ventricular myocytes.
The diverse range of ventricular myocyte models reflects in large part the diversity of the different species (9, 13, 57). They are distinct in AP shape and duration, set of repolarization currents, and Ca2+ handling and reflect in part the different constraints imposed by size on cardiac function. Mouse ventricular myocytes are at the extreme end of the spectrum because of the very short APDs they generate. These short APs reflect the magnitude and types of repolarization currents (8, 26). There is further diversity in the different APs and ionic current densities in cells from different regions of the mouse heart as well (8, 74).
Ionic channel heterogeneity may be critical to the timing and spread of repolarization. Previously, we developed models of mouse ventricular myocytes from the apical and septal regions of the heart (6). Apical ventricular myocytes predominantly express Kv4 and Kv1.5 channels that encode the largest IKto,f, and IKur, IKto,f, and IKur define short apical APD50 of 4.0 ms (6, 74). In septal ventricular myocytes, the magnitude of IKto,f is considerably smaller than in apical myocytes that lead to the prolongation of APD (APD50 = 7.6 ms). These differences in repolarization led to functionally significant effects on the spatial distribution of refractoriness and Ca2+ transients from septum to apex. Similarly, IKto,f and IKur are larger in mouse epicardial than in endocardial ventricular myocytes, which manifest in shorter epicardial APD50 of 3.7 ms compared with endocardial APD50 of 5.4 ms (Table 2, Fig. 4, and Ref. 8). However, the magnitude and functional significance of such differences have not been clear.
It may seem surprising that abrupt discontinuities in AP properties occurred in the bifurcation analysis of the different cell types. However, such discontinuities are predicted because of the “all or none” threshold nature of full-scale AP generation. There have been relatively few attempts to reproduce such diagrams in living heart cells or tissues (see, for example, review in Ref. 5). The study of sheep and canine Purkinje fiber AP dynamics by Chialvo et al. (11) also shows small-amplitude APs and abrupt discontinuities in behavior. Thus the abrupt discontinuities are predictions that are consistent with observations in other species.
The small-amplitude electrical activity in Figs. 7 and 8 is not, properly speaking, an AP. The “upstroke” is predominantly due to the stimulus, but the repolarizing response is not a simple passive decay; inward and outward currents are both active, resulting in the slow repolarization, which fails to completely repolarize before the next stimulus. Such states can potentially be achieved during high-frequency stimulation of isolated cells or pathophysiologically during tachycardia in the heart, in which a repetitive stimulated region of tissue may become depolarized and “undulate” in response to electrotonic current flow (10, 62). This is also qualitatively similar to consistent stimulation during an EAD or with the development of reentrant arrhythmias because of programmed stimulation.
Two major functionally distinct regions are usually described for the transmural structure of the mouse ventricle, epicardium and endocardium. These differences arise from differential expression of not only K+ currents but also distinct Ca2+ handling (2, 8, 18, 32). In contrast, larger species have been described with a more complex structure of the ventricular wall that includes epicardial, endocardial, and midmyocardial layers (1, 19, 63). The mouse heart may also have these three distinct layers, but the resolution of these layers may lie beyond current experimental resolution.
Transmural heterogeneity of Ca2+ handling in the heart is less studied than the differences in AP and repolarization currents. However, regional differences in [Ca2+]i transients, ryanodine receptor, sarco(endo)plasmic reticulum Ca2+-ATPase (SERCA) pump, and Na+/Ca+ exchanger expression have been reported between epicardial and endocardial cells in different species (15, 18, 36). Several factors determine the magnitude of [Ca2+]i transients, such as APD, SR load, SR release rate, and SERCA pump activity. Our simulations for mouse ventricular myocytes show that the longer APD in endocardial cells slightly contributes to [Ca2+]i; however, the major contributing factors are about twofold larger SR release and slower Ca2+ extrusion by INaCa in endocardial cells compared with epicardial cells. The last contributes to an increase in [Ca2+]i transient through an increased SR Ca2+ load, resulting from the slower Ca2+ extrusion from the cell (20, 21, 66). The contributions of APD, Na+/Ca2+ exchanger rate, and Ca2+ release rate into [Ca2+]i transient magnitudes in mouse ventricular myocytes are similar to those determined experimentally (18). SR load is also larger in endocardial than in epicardial cells during pacing, both in experiments (18) and our simulations, in particular, at rapid pacing rates. Smaller SR load and larger Na+/Ca+ exchanger expression in epicardial than endocardial ventricular myocytes were also described for rats (23).
The mouse cellular models, therefore, are qualitatively similar in diversity to larger species, such as canine, where differences were observed in SERCA pump expression and activity, which were larger in the epicardial ventricular myocytes. However, no differences were found in ICaL and [Ca2+]i transient magnitudes and Na+/Ca+ exchanger expression (15, 36, 37) although a separate study (69) measured significantly larger ICaL in endocardial canine cells. Thus it is unclear to what extent heterogeneity in the Ca2+ handling systems in mice are analogous to those found in larger animals.
Transmural heterogeneity of repolarization in ventricular tissue.
Experimental data from the mouse hearts and from the hearts of other species show that the transmural heterogeneity of repolarization and Ca2+ handling is maintained in ventricular tissues (2, 16, 32, 36). The heterogeneity of repolarization and Ca2+ handling was also predicted by our simulation in the 2-D model of the mouse ventricular wall despite the very small wall thickness (about 1.6 mm) and is consistent with our previous simulations for apical and septal cell types (4). This is a critical point of contention since simulations of APs in the whole mouse heart using a model based on a modified rat AP did not predict significant transmural gradients of repolarization (59).
Electrotonic interactions are important in the mouse heart. Our simulations showed a prolongation of APD in ventricular myocytes in the propagating tissue model compared with isolated cells (Figs. 4 and 9). Experimentally, longer APDs were reported in mouse ventricles compared with isolated ventricular myocytes (2, 8, 32). Despite significant electrotonic interactions, both experimental data and our simulations demonstrate the existence and potential functional importance of heterogeneity of repolarization across the mouse ventricular wall. This supports the point of view that the APD is determined predominantly by the intracellular ionic currents. Changes in expression of major repolarization currents across the ventricle, i.e., IKto,f and IKur, determine the AP shape and duration across the wall. While APD contributes to the magnitude of [Ca2+]i transients, this contribution was not sufficient to account for the large differences in Ca2+ handling between epicardial and endocardial cells. Unlike APs, the sharp transition region for [Ca2+]i transients is maintained in simulated 2-D cardiac tissue (Figs. 9 and 10), and it is only weakly modulated by a relatively smooth transition for APs across the wall.
This weak modulation by electrotonic forces in an inhomogeneous cellular environment can influence cellular Ca2+ handling especially when integrated over several cycles. By influencing repolarization morphology and duration, Ca2+ entry and extrusion can be altered. This is particularly important at fast pacing rates and is predicted by our simulations to result in rate dependent Ca2+ instabilities arising out of heterogeneity.
The transmural distribution of [Ca2+]i transients in the mouse ventricle has not been reported. However, transmural [Ca2+]i transients have been investigated for the hearts of the other species (22, 36). Laurita et al. (36) concluded that the transmural heterogeneity of Ca2+ handling is most likely an intrinsic property of the cell, which is similar to the mechanism demonstrated in our simulations. In rats, [Ca2+]i transients across the ventricular wall had peak systolic and diastolic [Ca2+]i that were larger in paced endocardium than in epicardium (22). However, the magnitude of [Ca2+]i transients were similar, suggesting similar Ca2+ release rates in epicardial and endocardial rat myocytes. In mouse ventricular myocytes, the Ca2+ release rates differ by twofold and suggest larger [Ca2+]i transients in endocardial than in epicardial cells (18). However, the diastolic [Ca2+]i were larger in stimulated endocardial versus epicardial regions [both in rats and in mice (18, 22)], and these findings are matched by our simulations of paced 2-D mouse ventricular tissue.
Ca2+ alternans and stability of AP propagation in the 2-D model.
Our simulations of 2-D mouse ventricular tissue showed that the transmural heterogeneity of repolarization and Ca2+ handling can lead to different stability properties across the mouse ventricle. APs and [Ca2+]i transients in cardiac myocytes incorporated into a multicellular tissue demonstrated an increased susceptibility to instabilities and alternans compared with isolated myocytes (Table 3). Isolated epicardial and endocardial ventricular myocytes have the same threshold BCL for instability of both AP and [Ca2+]i, i.e., both Ca2+ and AP instabilities occurred concurrently in their respective cell types. However, when myocytes were connected in 2-D tissue, they developed [Ca2+]i alternans followed by AP instability with further reduction of BCL (Fig. 12, and Table 3). Ca2+ alternans first appeared in the epicardial region of the tissue where APD and [Ca2+]i transients are smaller because of instability of Ca2+ handling at the cellular level that was perturbed by electrotonic interactions (4). The fivefold reduction of intercellular conductances increased susceptibility to the alternans when paced with the same stimulus. This increase in threshold BCL for Ca2+ alternans was due to the increase in gradient of repolarization in 2-D tissue with the reduced intercellular coupling (Fig. 9D).
Our simulations of the relationship between cell-to-cell coupling and susceptibility to alternans are supported in part by experimental data on alternans of [Ca2+]i transients in rats and TNF-α transgenic mice (17, 38). Rat hearts that have had partial ryanodine-sensitive release inhibition show increased instability in Ca2+ transients (17). The phenotype of TNF-α-overexpressing transgenic mice is characterized by a more prolonged AP, smaller magnitude [Ca2+]i transients, and slower Ca2+ sequestration into the SR compared with wild-type littermates (38). In TNF-α mice, rapid heart pacing yielded significantly smaller conduction velocities and larger susceptibility to Ca2+ alternans compared with wild-type mice (38). Our simulations predict qualitatively similar results in the 2-D tissue with reduced conductances (smaller AP propagation velocity). Our simulations also show that Ca2+ alternans appears in a region with smaller [Ca2+]i transients (Fig. 12). While other contributions to the proarrhythmic behavior in the hearts from TNF-α mice play a role, our simulations are consistent with these experiments and provide a plausible mechanism for the development of alternans because of impaired Ca2+ handling (see also Ref. 4).
This mechanism of Ca2+ alternans is different from that thought to commonly occur in the larger species, where Ca2+ overload is an important proarrhythmic condition (30, 36, 46, 51). In the larger species, APDs are much longer than those for the mice and rats. Consequently, [Ca2+]i transients and APs largely overlap in time, and APD and [Ca2+]i are regulated by the magnitude and inactivation of ICaL, which, in turn, depend on [Ca2+]i. In adult mice, APD is not affected by ICaL to the similar extent, as in species with more prolonged APDs, and ryanodine receptors are the major players for alternans development (4, 17). In neonatal mouse hearts, however, the duration of the AP is significantly larger and Ca2+ channel inactivation may play a role (71). In contrast, in canine cardiac muscle, ICaL significantly inactivates during long APD and in addition because of a larger [Ca2+]i transient (24). During the next AP, this results in smaller ICaL amplitude, smaller APD, and smaller Ca2+ release. In humans, the importance of Ca2+ channel inactivation is underscored by the rare disease, Timothy Syndrome, in which a defect in the voltage-dependent component of the L-type Ca2+ channel inactivation leads to profound QT prolongation, arrhythmias, and sudden cardiac death at a young age (65).
The development of Ca2+ alternans occurs both in time and in space, yielding concordant and discordant alternans (72), and has several underlying mechanisms for development (see recent review in Ref. 25). The simulations of epicardial and endocardial muscle show that the discordant Ca2+ alternans can form complex 2-D structure, both in the transmural and in the longitudinal directions in the mouse ventricle. The transmural 2-D structure of the discordant alternans is largely determined by the heterogeneity of AP repolarization (Fig. 12), since APD is an important factor in alternans development in mouse tissue (4). Our simulations show that the complex structures of repolarization and [Ca2+]i transients can appear through instabilities of AP generation and Ca2+ handling that have cellular origin and are modulated by the heterogeneities in cardiac channel expression.
This work was supported in part by National Heart, Lung, and Blood Institute Grant R01-HL-59526 and National Science Foundation Knowledge and Distributed Intelligence Grant DBI-9873173 (to R. L. Rasmusson); Pittsburgh Supercomputer Center Grant MCB010020P; and a seed grant from Brain and Behavior Program at Georgia State University (to V. E. Bondarenko).
No conflicts of interest, financial or otherwise, are declared by the author(s).
This work was performed in part at the State University of New York's Buffalo's Center for Computational Research.
- Copyright © 2010 the American Physiological Society