Effects of I_Kr and I_Ks Heterogeneity on Action Potential Duration and Its Rate Dependence

Cell Model

The theoretical dynamic model of a mammalian ventricular AP (Figure 1A⇓), the LRd model, provides the basis for the simulations in this study.^{13 14} In this model, the AP is numerically constructed from ionic processes that are formulated on the basis of experimental data obtained mostly from the guinea pig. The model also accounts for processes that regulate ionic concentration changes. I_Ks was recently shown to display 2 (fast and slow) activation processes.¹⁵ The slow time constant is larger than the fast time constant by a factor of 3 to 5. We incorporated this property into the formulation of I_Ks by introducing an activation X_s2 gate, whose time constant is 4 times that of X_s1 (which corresponds to X_s in Reference 10¹⁰ ). The calcium dependence of I_Ks was modified on the basis of the recent study by Nitta et al.¹⁶ Formulation of I_Ks is provided in the Appendix. Heterogeneities of I_Kr and I_Ks densities were introduced by altering their maximum conductances, ḠKr and ḠKs. In view of recent findings regarding continuous Ca²⁺ release from the sarcoplasmic reticulum (SR), we have also modified the formulation of Ca²⁺ handling (Appendix). The modified formulation prevents nonphysiological discontinuities and oscillation(s) of [Ca²⁺]_i under extreme conditions. A detailed description of the cell model and its equations is provided in References 10, 13, and 14^{10 13 14} .

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Figure 1.

A, Schematic of LRd mammalian ventricular cell model. Detailed description is provided in References 10, 13, and 14. I_Kr and I_Ks, the currents that are modified to introduce cellular heterogeneity, are highlighted in the figure. B, Multicellular cardiac fiber composed of LRd cells interconnected by gap junctions. Spatial heterogeneities in density of I_Ks are introduced to simulate different cell types found in myocardium.

Simulation Protocols

Adaptation curves (the steady-state dependence of APD on rate) were obtained by pacing the cell at a constant cycle length (CL) for 5 minutes to achieve steady state, followed by a step increase to the next CL. APDs were obtained for CL of 300, 500, 800, 1000, 1500, 2000, and 5000 ms. APD was measured between the time of stimulus onset and 90% repolarization (APD₉₀). All computations were performed at a simulated temperature of 37°C.

Multicellular Fiber Model

The theoretical fiber¹⁷ (Figure 1B⇑) used in this study is composed of 190 ventricular cells, each of LRd formulation. The fiber contains an endocardial region (cells 1 to 80), M-cell region (cells 81 to 110), and an epicardial region (cells 111 to 190). Gap junction conductance (g_j) is homogeneous throughout the fiber and for different simulations varies from 2.5 μS (normal coupling¹⁸ ) to 0.025 μS (poor coupling). The fiber is stimulated or paced from cell 1.

Relative Density of I_Ks and I_Kr Has a Major Effect on APD and Its Rate Dependence

We have previously shown¹⁰ that modulation of I_Kr or I_Ks results in dramatic changes in the APD. Figure 2⇓ shows the effect of different I_Ks:I_Kr density ratios on APD and its rate dependence. In panel A, I_Ks density is varied, whereas I_Kr density is held constant. In panel B, I_Ks density is maintained, whereas I_Kr density is varied. Cell type 1 (CT1) serves as control. In panel A, I_Ks is decreased by 40% and 80% in CT2 and CT3 cells, respectively. In panel B, I_Kr is similarly decreased in CT4 and CT5 cells. At a basic cycle length (BCL) of 5000 ms, the increase of APD between CT3 and CT1 (I_Ks variations) is 70%, whereas between CT5 and CT1 (I_Kr variations) it is only 16%. This dominant effect of I_Ks is consistent with the experimental findings³ that transmural heterogeneity of I_Ks density (it is greatly reduced in the M cells) plays a major role in the transmural heterogeneity of APD (I_Kr density was found to be relatively uniform across the ventricular wall).

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Figure 2.

Effects of heterogeneities in I_Ks or I_Kr density on APD and its rate dependence. A, Effect of changes in density of I_Ks for constant I_Kr density. B, Effect of changes in density of I_Kr for constant I_Ks density. Shaded region indicates physiological heart rates.

Guided by these findings, the following simulations of APD heterogeneity will focus on the role of I_Ks. It should be mentioned, however, that the simulated variations of I_Kr in Figure 2B⇑ are likely to have greater effects on APD if they occurred on the background of a significantly smaller I_Ks. Such a situation can arise when I_Kr in M cells is suppressed by mutation (eg, LQT2) or by a specific blocking agent.

Figure 3⇓ shows the effect of pacing frequency on APD for 3 different I_Ks:I_Kr density ratios (I_Kr was kept constant). Figure 3A⇓ shows computed steady-state APs at fast and slow pacing frequencies (BCL of 300 and 5000 ms, respectively) in comparison to experimentally recorded APs (Figure 3B⇓) from the 3 regions of the guinea pig ventricular wall.⁵ The model accurately simulates the APD at the fast and slow rates for the different cell types on the basis of I_Ks heterogeneity. At a BCL of 300 ms, experimentally measured APDs⁵ are 102±21, 136±9, and 95±15 ms in epicardial cells, M cells, and endocardial cells, respectively; the corresponding simulated APDs computed from the model are 95, 129, and 96 ms. At a BCL of 5000 ms, measured APDs are 133±14, 185±24, and 135±13 ms in epicardial cells, M cells, and endocardial cells; corresponding simulated APDs are 138, 188, and 139 ms. These values indicate a much longer APD and a slightly more accentuated dependence of APD on stimulation frequency in the M cells compared with either epicardial or endocardial cells.

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Figure 3.

APs at fast and slow pacing rates for different densities of I_ks. Computed (A) and experimentally recorded (B) APs at BCL of 300 and 5000 ms, respectively.

The presence of transitional cells and heterogeneous APD even within the same cell type (see Figure 1B⇑ of Reference 3³ ) indicates that there is no unique value of I_Ks:I_Kr density ratio that is associated with each cell type. Instead, there is wide range of density ratios in the myocardium that result in a distribution of APDs and their rate dependence. Figure 4⇓ compiles the effects of different I_Ks:I_Kr density ratios on the adaptation of APD to rate changes (adaptation: ΔAPD=APD_(slow)−APD_(fast)). A decrease in ḠKs:ḠKr (introduced by decreasing ḠKs) increases ΔAPD. The slope of the curve is steep for ḠKs:ḠKr in the range from 5 to 35 (shaded region), indicating a strong effect of I_Ks:I_Kr heterogeneity on APD adaptation in this range. It is to be noted that these studies were conducted assuming uniform I_Kr density, consistent with the experimental findings.³ We choose to focus on this range in the mechanistic investigation of APD adaptation because in this range the effect of changing I_Ks density is most significant. Interestingly, Figure 5⇓ shows that for ḠKs:ḠKr ratio in this range, the adaptation curves are in very good agreement with experimentally recorded adaptation curves from the human left ventricle.⁷ Figure 5A⇓ also extends the simulated adaptation curves (dotted) to BCLs ranging from 300 to 1000 ms (a range of human basal heart rates). Experimental data from human myocytes for this range are currently unavailable.

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Figure 4.

Effect of different I_Ks:I_Kr density ratios on APD adaptation to rate changes. A decrease in ḠKs:ḠKr ratio (caused by decrease in I_Ks density) augments the adaptability (ΔAPD) of cells. Curve is steep for ḠKs:ḠKr in the range 5 to 35 (shaded region). ΔAPD is shown for BCL change from 5000 to 300 ms (upper curve) and from 1500 to 300 ms (lower curve).

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Figure 5.

Normalized APD adaptation curves. A, Simulated adaptation curves obtained with the use of 3 ḠKs:ḠKr ratios in the steep range of the curve of Figure 4⇑. B, Experimentally recorded APD adaptation curves from epicardial, midmyocardial, and endocardial regions of human ventricle. Dotted lines in A extend adaptation curves to physiological heart rates (BCLs of 300 to 1000 ms) not available experimentally.

Figure 6⇓ shows simulated APs of the 3 cell types at a BCL of 2000 ms (A), together with the associated I_Kr (B) and I_Ks (C). We label the simulated cell types with density ratios ḠKs:ḠKr=23:1, 15:1, and 7:1 as epicardial cells, endocardial cells, and M cells, respectively. It is important to emphasize that this classification is based only on I_Ks:I_Kr density differences and the resulting heterogeneity of APD and its rate dependence (the focus of this report). Other differences between these cell types (eg, the presence of a large I_to in epicardial cells of certain species and the associated “spike and dome” morphology of their AP) are not considered here. It is observed from the figure that the APD of the M cell is longer than that of the epicardial cell. I_Ks is significantly smaller in the M cell (Figure 6C⇓), starting from the early phase of the AP, and attains a much smaller peak magnitude during the course of the AP. I_Kr (Figure 6B⇓) appears to be smaller during the plateau phase of the AP in the M cell. However, it should be noted that the smaller I_Kr during this phase merely reflects the different time course of the AP.

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Figure 6.

Heterogeneity of APD in 3 cell types in relation to I_Kr and I_Ks. Simulated APs of 3 cell types at BCL of 2000 ms (A) with corresponding I_Kr (B) and I_Ks (C). Dotted lines denote magnitudes of I_Kr and I_Ks at membrane potential of 10 mV. Epi indicates epicardial; M, M cell; and Endo, endocardial.

Figure 7⇓ shows simulated APs of the 3 cell types for a wide range of BCLs, superimposed to facilitate comparison. At a BCL of 300 ms, all 3 cell types display relatively short APs of similar APD (105, 113, and 135 ms for epicardial cells, endocardial cells, and M cells, respectively). With the slowing of pacing rate, the APD of the M cell is prolonged much more than that of the epicardial or endocardial cells, demonstrating a steeper dependence of APD on rate (greater adaptation). At a BCL of 2000 ms the APD of epicardial cells, endocardial cells, and M cells are 168, 188, and 240 ms, respectively. An important observation is that the effect of I_Ks heterogeneity (reduced I_Ks in M cells) is to prolong phase 3 repolarization without significantly affecting the plateau.

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Figure 7.

Rate dependence of APD in 3 cell types. M cells prolong their APD dramatically compared with epicardial and endocardial cells with slowing of stimulation rate. APs at progressively decreasing rate (BCL=300, 500, 1000, and 2000 ms) are shown for each cell type.

I_Ks and I_NaCa Play an Important Role in APD Adaptation and in Greater Adaptability of M Cells

The ability of cardiac myocytes to adjust their APD to changes in pacing frequency is termed adaptation. The most prominent effect of decreasing BCL is shortening of APD (Figures 2⇑, 3⇑, and 7⇑). To investigate the mechanism of APD adaptation we examined selected transmembrane currents (I_Kr, I_Ks, I_Ca(L), and I_NaCa) that play a role during the plateau and repolarization phases of the AP. Figure 8⇓ shows simulated APs (panels A and E); corresponding ionic currents: I_Ks (D), I_Kr (F), I_Ca(L) (G), and I_NaCa (H); maximal conductance of I_Ks, ḠKs (B); and I_Ks activation, X_s1 · X_s2 (C). All parameters are shown for 2 different BCLs (300 ms and 2000 ms). The upstroke of the APs are aligned to facilitate comparison.

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Figure 8.

Mechanism of APD adaptation. A and E, Computed APs for slow pacing (bold line, BCL=2000 ms) and fast pacing (dotted line, BCL=300 ms) with corresponding ḠKs (B), I_Ks activation X_s1 · X_s2, (C), I_Ks (D), I_Kr (F), I_Ca(L) (G), and I_NaCa (H). Fast pacing increases I_Ks as the result of residual activation (arrows in D and C, respectively). Outward I_NaCa also increases (arrow in H) as the result of [Na⁺]_i accumulation.

I_Kr is similar at the 2 frequencies (panel F, differences secondary to APD differences should not be considered and currents should be compared at the same V_m). Due to the slow kinetics of I_Ks, there is residual activation at the onset of the upstroke (arrow in Figure 8C⇑). ḠKs, which depends on intracellular calcium, is not significantly altered (Figure 8B⇑) at the fast rate. For [Ca²⁺]_i=0.12 μmol/L (resting level), ḠKs is 0.6495 ms/μF. At a steady-state BCL of 300 ms, peak [Ca²⁺]_i is 1.5 μmol/L. For this concentration, the value of ḠKs is 0.6912 ms/μF, which is not significantly different from the resting value (an increase by only 6.5%). Therefore elevation of [Ca²⁺]_i at rapid rates has only a minor augmenting effect on I_Ks. The accumulation of I_Ks activation caused by incomplete deactivation is therefore the main cause of the larger I_Ks, which (together with increased outward I_NaCa; see below) acts to shorten the APD at a shorter BCL (arrow in Figure 8D⇑). The greater cumulative activation of I_Ks at the time of stimulation for the short BCL is reflected in the instantaneous jump (Figure 8D⇑) in the current at the onset of depolarization. The greater peak I_Ks at a longer BCL should not be considered in the comparison, as it is merely a consequence of the longer AP (because the membrane is depolarized for a longer duration, I_Ks is activated for a longer time and, in the absence of inactivation, reaches a greater peak current).

I_Ca(L) (Figure 8G⇑) has a larger plateau magnitude at the short BCL because of a larger driving force arising from a more negative plateau potential. Because I_Ca(L) is an inward current, this would act to prolong the AP rather than shorten it. Therefore this current cannot contribute to APD shortening. However, at the short BCL, I_NaCa (Figure 8H⇑), is predominantly outward (“reverse-mode”) during the plateau phase. I_NaCa is regulated by [Ca²⁺]_i and [Na⁺]_i. Elevation of [Ca²⁺]_i increases inward I_NaCa, whereas elevation of [Na⁺]_i increases outward I_NaCa. Fast pacing results in accumulation of [Ca²⁺]_i and [Na⁺]_i. Our simulations show that the net effect of ion accumulation is a larger outward I_NaCa than control conditions (no accumulation). The increased outward I_NaCa (arrow in Figure 8H⇑) acts to shorten the APD at the shorter BCL. Initially, the magnitude of this current appears to be greater than that of I_Ks (Figure 8D⇑) at the fast rate. However, this current diminishes relatively quickly, reverses direction, and becomes a relatively large depolarizing current. This direction reversal limits the role of I_NaCa in APD shortening. However, this current still remains a participant in the APD adaptation process.

The results of Figure 8⇑ provide insight into the processes that underlie adaptation. In Figure 9⇓, we attempt to explain the greater adaptability of cells with lower I_Ks density (eg, M cells). Transmembrane currents computed from epicardial and M cells are compared at 2 different frequencies, fast (BCL=300 ms) and slow (BCL=2000 ms). Figure 9⇓ shows plots of the APs (Figure 9A⇓), the activation kinetics of I_Ks (Figure 9B⇓), and I_Ks current (Figure 9C⇓). It is seen from Figure 9B⇓ that at a shorter BCL, X_s1 · X_s2 is greater in the M cell than in the epicardial cell, reflecting a greater degree of activation and of channel availability. Activation is greater because of the longer APD and shorter time (diastolic interval) between complete repolarization and the next stimulus, which results in less deactivation and greater residual activation (X_s1 · X_s2) at the time of the next stimulus. This results in a greater relative increase of I_Ks at smaller BCL in the M cell (128%) than in the epicardial cell (33%). The greater increase of channel availability on BCL shortening in the M cell is not evident from the plot of I_Ks itself (Figure 9C⇓), because I_Ks appears to be smaller in the M cell than in the epicardial cell. This simply reflects the smaller density of I_Ks in the M cell. The important observation is the greater percentage increase of I_Ks in the M cell (128% compared with 33% in the epicardial cell) on BCL shortening, which accounts for the greater adaptability. Due to a smaller total repolarizing current in the M cell (reflecting the smaller I_Ks density), any change in outward current magnitude, albeit small, will have a significant effect on the APD.

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Figure 9.

Mechanism of greater adaptability of M cells to changes in stimulation rate. A, Computed APs of epicardial (thin line; Epi) and M cell (bold line) at slow (BCL=2000 ms) and fast (BCL=300 ms) rates. B, X_s1 · X_s2 (I_Ks activation) displays greater residual value in M cell than in epicardial cell (compare black and white arrows). C, As a result of greater residual activation, there is a greater relative increase of I_Ks in M cell (128% compared with only 33% in epicardial cell).

I_NaCa (not shown in Figure 9⇑) also plays a role in the greater adaptability of M cells. The Na⁺-Ca²⁺ exchanger is transiently outward during the initial phase of the AP because of a transient increase in [Na⁺] influx through I_Na. However, at fast stimulation rates, accumulation of [Na⁺]_i increases the magnitude and duration of the transient outward component of I_NaCa (as seen in Figure 8H⇑), thereby increasing the total repolarizing current. As stated previously, because of a smaller total repolarizing current in the M cell, any change in this current (in this case due to an increased outward I_NaCa resulting from an elevated [Na⁺]_i) causes a significant change in APD.

Effect of Gap-Junction Coupling on APD

In the preceding sections we focused on understanding the mechanisms underlying the differences in APD and its rate dependence between isolated cells of the different types. However, in the intact heart, the cells are coupled through gap junctions and are subject to electrotonic interactions that affect their behavior. In this section we study the effects of intercellular coupling on APD and its rate dependence in the context of myocardial heterogeneity.

Figure 10A⇓ shows APDs along a multicellular fiber containing endocardial, M, and epicardial regions for different degrees of gap-junction coupling. The horizontal bold lines indicate APDs obtained from each cell type in isolation, in the absence of gap-junction coupling effects. For normal coupling of g_j=2.5 μS (velocity=56 cm/s, typical for propagation along fibers), APD differences (dispersion) between the 3 cell types are greatly reduced compared with the isolated cell case. For this value of gap-junction coupling, the APD difference between M cells and epicardial cells is only 18 ms (differences are computed between middle cells of the respective regions). This is reflected in the figure by the relatively flat APD curve (bold line) along the fiber. However, with reduced coupling, APD differences increase sharply. At g_j=0.25 μS (velocity=18 cm/s, typical for propagation transverse to fibers), the APD difference between M cells and epicardial cells increases to 44 ms. When cells are very poorly coupled, APDs approach isolated cell values, resulting in a marked dispersion of APD along the fiber. At g_j=0.025 μS (velocity=3.5 cm/s, observed under pathological conditions such as infarction), APD difference between M cells and epicardial cells in the fiber increases to 71 ms (90 ms in the isolated cells).

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Figure 10.

A, Effect of gap-junction coupling on APD heterogeneity. APDs along the multicellular fiber are shown (APD dispersion). Bold line corresponds to g_j=2.5 μS, thin line corresponds to g_j=0.25 μS, and dotted line corresponds to g_j=0.025 μS. Bold horizontal line in each region denotes single cell APD. B, Effect of different degrees of coupling on APD dispersion. ΔAPD shows steep dependence on gap junction conductance in normal physiological range (shaded region).

The importance of intercellular coupling is better illustrated in Figure 10B⇑, which compiles the effects of different gap-junction conductances on APD differences between M cells and epicardial cells in a fiber (ΔAPD between middle cells of each region). A 10-fold decrease in coupling (2.5 to 0.25 μS) increases dispersion of APD (ΔAPD) from 18 to 44 ms, demonstrating a steep dependence of ΔAPD on gap-junction coupling conductance in the physiological range. The simulations of Figure 10⇑ were limited to an action potential elicited by a single stimulus, a protocol that tends to maximize APD differences. This protocol approximates certain pathological conditions such as extreme bradycardia or a postpause AP. Under such circumstances, APD differences would indeed be maximized. In the normal beating heart, the myocardium is subject to periodic stimulation. The following section addresses the issue of APD dispersion during periodic beating by characterizing the rate dependence of APD heterogeneity in the multicellular fiber.

Figure 11⇓ shows the dispersion of APD along the fiber for 3 degrees of gap-junction coupling (top to bottom panels) during fast (BCL=300 ms, bold line) and slow (BCL=2000 ms, dotted line) pacing. It is observed that an increase in rate results in shortening of APD in all the 3 cell types. It is also observed that at the faster rate, APD differences along the fiber (dispersion) are reduced. The reduced dispersion is reflected in the flattening of the APD curve along the fiber at the fast rate compared with the slow rate. This behavior is similar to that of the isolated cells, in which at fast rates APDs of the 3 cell types converge (Figures 2⇑ and 7⇑). The mechanism of rate-dependent shortening of APD in the multicellular fiber is the same as in the isolated cells, that is, accumulation of I_Ks and a greater outward I_NaCa at fast rates.

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Figure 11.

Rate dependence of APD dispersion along multicellular fiber. APD dispersion along fiber for very poor gap-junction coupling (A) and for 2 values of coupling in normal physiological range (B and C). Thick curves correspond to fast pacing (BCL=300 ms); dotted curves correspond to slow pacing (BCL=2000 ms).

Small I_Ks Density Can Account for Distinct APD Characteristics of M Cells: I_Ks and I_NaCa Underlie APD Adaptation and Greater Adaptability of M Cells

Rate-dependent shortening of APD is an important property of cardiac myocytes. Abrupt shortening of APD caused by a premature stimulus (APD restitution) has been shown to result from incomplete deactivation of I_Kr and I_Ks.¹⁰ Steady-state shortening (APD adaptation) is also thought to be mainly due to incomplete deactivation and accumulation of ionic currents, in particular the slow delayed rectifier potassium current, I_Ks. Jurkiewicz et al¹⁹ have measured an increased I_Ks at fast heart rates, lending support to this hypothesis.

From simulations conducted in this study we have determined that I_Ks accumulation at fast rates can explain the shortening of APD during adaptation. Because of the slow deactivation kinetics of I_Ks, there is residual activation at the onset of the succeeding AP that accumulates at fast rates, increasing the probability of the channel being in an open state. This is reflected in an instantaneous jump in I_Ks, as observed in Figure 8D⇑ (arrows). The resulting greater I_Ks acts to shorten the APD at fast rates.

An increase in heart rate results in a progressive rise in [Na⁺]_i,²⁰ accompanied by an increase in the strength of contraction and shortening of the AP.²¹ The I_NaCa has been implicated in the sodium-induced changes in contraction as well as the rate-dependent shortening of AP.²¹ As seen in Figure 8H⇑, outward I_NaCa increases at fast rates during the initial plateau phase of the AP and contributes to the observed rate-dependent shortening of the AP. Other plateau currents (I_Ca(L) and I_Kr) do not play an important role in APD adaptation.

M cells have a steeper APD-rate relation than epicardial or endocardial cells. Investigation into this behavior has highlighted the importance of a reduced total repolarizing current in the M cells. As seen in the simulations, rapid pacing causes abbreviation of the APD as the result of increase in the repolarizing currents I_Ks and I_NaCa. This is true for all cell types, but the effect on APD is much greater in the M cells because it occurs on the background of a much smaller total repolarizing current.

Importance of Gap-Junction Coupling in Determining APD Heterogeneity in Multicellular Tissue

The unique repolarization properties of isolated M cells are widely accepted; however, controversy still exists over the extent to which heterogeneity in repolarization is expressed across the normal ventricular wall.^{22 23} This is an important issue with major implications to the mechanism of arrhythmias associated with delayed and presumably nonuniform ventricular repolarization (eg, torsade de pointes). The simulations conducted in this theoretical study, in which gap-junction coupling and ion channels could be simultaneously controlled and systematically varied, provide valuable insights into APD heterogeneity. Our study highlights the important influence of gap-junction coupling on the manifestation of APD heterogeneity in the multicellular tissue. Small changes in intercellular coupling have a profound effect on APD dispersion (Figure 10B⇑). This theoretical finding is consistent with a recent experimental observation made with optically recorded APs.²⁴ Small changes within the physiological range of gap-junction coupling (eg, directional differences caused by anisotropy, or presence of connective tissue septae) result in significant dispersions that are similar to values reported in the literature.^{23 25} In this regard, experiments designed to investigate the effects of gap-junction coupling (through application of uncouplers or coupling enhancers) could provide important information regarding APD heterogeneity in the intact tissue.

The results reported in this study were obtained mainly from a multicellular fiber consisting of equal segments of cell types. We have also conducted studies with a smaller/larger number of M cells. Results indicate that the smaller the M-cell region, the smaller the APD dispersion. However, increasing the M region beyond 30 cells did not enhance APD dispersion any further.

Significance of I_Ks:I_Kr Heterogeneity to Arrhythmogenesis

APD is an important factor in arrhythmogenesis. Heterogeneity of APD and the resulting dispersion of refractoriness across the ventricular wall could provide the substrate for unidirectional block and reentrant arrhythmias.²⁶ In addition, adaptation of APD can influence the degree of head-tail interaction during reentry and the stability of the reentrant circuit.²⁷ As seen in Figures 2⇑, 3⇑, and 7⇑, changes in the density of I_Ks could result in dramatic changes in APD and its rate dependence. For example, cells with decreased I_Ks increase APD disproportionately as stimulation rate is decreased. In more general terms, cells with smaller density of I_Ks are more responsive to pathologies or interventions (eg, the hereditary or acquired long QT syndrome) that cause APD prolongation. The results obtained from the multicellular fiber suggest that a baseline level of APD differences exists even in the normal myocardium. These differences can increase substantially in the presence of reduced intercellular coupling, even if the cellular repolarization properties remain normal. Conversely, the higher susceptibility of M cells to pathologies and interventions that prolong APD suggests that large dispersion can arise in the normally coupled myocardium. Of course, a combination of enhanced cellular heterogeneity and reduced coupling (eg, class III effects in an infarcted heart) can result in extreme levels of APD dispersion.

In the present study, we focus on intrinsic cellular heterogeneities and their interplay with gap-junction coupling in the context of dispersion of repolarization. In the presence of pathologies or interventions that slow repolarization, sufficient APD prolongation can generate EADs,^{8 28} preferentially in the M cells. Consequently, M cells could become a source of triggered activity and focal arrhythmias.⁸ The focal mechanism and the enhanced APD heterogeneity introduced by the M cells could act synergistically, with an EAD providing the trigger (a premature stimulus) to a substrate that is predisposed to the development of unidirectional block and reentry caused by large APD dispersion.²³ Although the study addresses the issue of I_Ks/I_Kr heterogeneity in ventricular myocardium and M cells, it is to be noted that Purkinje cells are also very prone to EAD generation.

Limitations of the Study

Of particular importance to myocardial electrophysiological heterogeneity is the variability in the density of I_to, I_Kr, and I_Ks, which are major players in determining the shape and duration of the AP. This study focuses on heterogeneities in I_Kr and I_Ks and their effects on APD and its rate dependence. Such heterogeneities have been observed in guinea pig ventricular myocardium, in which I_to is absent.⁵ The use of a guinea pig–type myocyte model in this study permitted us to investigate the effects of I_Kr and I_Ks variations without the complicating influences of I_to. Clearly, the role of I_to in influencing APD and its rate dependence also needs to be investigated, especially in the context of transmural heterogeneity of action potential morphology (“spike and dome” configuration¹ ).

It is well established that I_Ks decays more slowly than I_Kr in the guinea pig but decays more rapidly than I_Kr in canine ventricular myocytes. Therefore the conclusions of this study do not necessarily apply to canine myocytes. However, this study shows that rate-dependent shortening of APD results not only from I_Ks accumulation but also from a greater outward I_NaCa at rapid rates. It is likely that in the absence of I_Ks accumulation (eg, in the canine) the main mechanism of rate-dependent APD shortening is a larger outward I_NaCa at fast rates.

The role of gap-junction coupling in APD heterogeneity was studied in a 1-dimensional fiber. This is a simplified model that allowed us to investigate this phenomenon at the cellular and ionic channel levels and to establish the principle that APD dispersion is highly sensitive to the degree of intercellular coupling. Moreover, 1-dimensional propagation that can be simulated in a 1-dimensional model occurs frequently in the 3-dimensional heart when broad-plane waves are generated. Importantly, this is the situation during normal sinus rhythm in which planar waves created by the Purkinje network propagate from endocardium to epicardium. A similar situation occurs in the wedge preparation used extensively²⁵ to study myocardial heterogeneity. It is possible that the higher degree of connectivity in the 3-dimensional myocardium acts to reduce APD dispersion under conditions of tight coupling. However, the principle of strong dependence of APD dispersion on gap junction coupling should carry over to the 3-dimensional myocardium. It should also be emphasized that local dispersion on a cellular scale is involved in the generation of unidirectional block and possibly reentry. Even in the presence of complex 3-dimensional global wave fronts, local excitation can be 1-dimensional in nature. This is particularly true in the presence of structural complexities and pathologies, such as a narrow fiber connecting islands of surviving tissue in an infarct.

Formulation of I_Ks

CICR From Junctional SR (JSR)

Recent experimental evidence²⁹ has suggested that the apparent “threshold” for calcium release from the SR is due to the degree of recruitment of local SR release channels. It follows that there is always calcium release triggered by calcium entry, and the release becomes significant as calcium entry increases. In light of these findings we have modified the release criterion such that there is always a small release, which increases with the increase in calcium entry. This is achieved by adding a “tail” to the calcium entry–dependent term of the SR calcium release conductance, G_rel (see Reference 13). The modified formulation of the SR release current, I_rel, is given below. If Δ[Ca²⁺]_i,2>Δ[Ca²⁺]_{i, th} 2 ms after the time of V_max, If 0<Δ[Ca²⁺]_i,2<Δ[Ca²⁺]_i, 2 ms after the time of V_max, If Δ[Ca²⁺]_i,2≤0 2 ms after the time of V_max,

G_rel=0.0 Δ[Ca²⁺]_i,th=0.242 μmol/L; K_m,rel=0.8 μmol/L; τ_on=τ_off=2 ms; Ḡ_rel=18 ms⁻¹ for voltage clamp simulations; and Ḡ_rel=60 ms⁻¹ for action potential simulations.