INTRODUCTION

TOP ABSTRACT INTRODUCTION MODEL DEVELOPMENT RESULTS DISCUSSION CONCLUSION APPENDIX REFERENCES

Electrophysiological studies conducted over the past decade have revealed the so-called transmural heterogeneity, or differences in the action potential waveforms recorded in cells isolated from the epicardial and the endocardial tissues in the left ventricles of mammalian hearts (Antzelevitch et al., 1999). These include feline (Kimura et al., 1990), canine (Antzelevitch et al., 1991), rabbit (Fedida and Giles, 1991), rat (Clark et al., 1993), human (Nabauer et al., 1996), guinea pig (Bryant et al., 1997), and mouse (Guo et al., 1999; Nguyen-Tran et al., 2000). In the left ventricle, the electrophysiological properties of the epicardial and the endocardial cells differ primarily with respect to their repolarization characteristics, with the epicardial myocytes displaying shorter action potential durations (APD). The resulting transmural voltage gradient in the intact ventricular myocardium is thought to be partially responsible for the upright T-wave of the electrocardiogram (Franz et al., 1987). Epicardial and endocardial cells also respond differently to pharmacological agents and pathophysiological states. These heterogeneous responses can amplify the intrinsic electrical differences, and thereby contribute to the substrate, and/or a trigger for the development of re-entrant arrhythmias (Antzelevitch et al., 1999).

The adult rat has been widely used as an experimental model to investigate the electrical heterogeneity in the left ventricle under normal conditions (Clark et al., 1993; Shimoni et al., 1995) and pathophysiological states such as diabetes (Shimoni et al., 1995; Casis et al., 2000), thyroid dysfunction (Shimoni et al., 1995), cardiac hypertrophy (Bryant et al., 1999; Volk et al., 2001), and myocardial infarction (Qin et al., 1996; Yao et al., 1999). Endocardial myocytes isolated from healthy adult rats consistently have longer APD (Clark et al., 1993), more prominent rate-dependent effects in APD (Shimoni et al., 1995), and a larger peak overshoot (Shipsey et al., 1997; Volk et al., 2001), when compared with epicardial ones. The longer APD is important; in addition to the electrical implications it is also a significant inotropic variable in rat ventricular myocytes (Bouchard et al., 1995; Clark et al., 1996; Sah et al., 2001). It is therefore of great interest and potential significance to understand the ionic mechanisms and their interactions that underlie the intrinsic electrical heterogeneity in the healthy adult rat left ventricle.

Experimental evidence suggests that the Ca²⁺-independent transient outward K⁺ current (I_t) is an important determinant of the differences between epicardial and endocardial action potentials in most species (Campbell et al., 1995; Giles et al., 1996). The density of I_t is smaller and the recovery from inactivation kinetics slower in rat endocardial cells than epicardial ones (Clark et al., 1993; Shimoni et al., 1995). Recent experiments also suggest the existence of a transmural gradient of the Na⁺ current (I_Na) in the rat left ventricle, with higher densities reported in the endocardium (Ashamalla et al., 2001).

One way of obtaining additional, quantitative insight into the ionic basis of the observed epicardial-endocardial differences is to develop a mathematical model of the respective membrane action potentials. The main goal of this study was to mathematically reconstruct the action potentials from adult rat left ventricular epicardial and endocardial myocytes under normal conditions. Accordingly, the comprehensive, but incomplete, biophysical descriptors for the ionic currents involved in the genesis of these action potentials were utilized. The initial specific aim was to understand whether the experimentally observed differences in I_Na and I_t could account for the epicardial-endocardial action potential disparity. In addition, the consequences of the regional heterogeneity of the APD on ionic currents and antiporters other than I_t and I_Na and the effect on cardiac contractility were addressed.

	MODEL DEVELOPMENT

TOP ABSTRACT INTRODUCTION MODEL DEVELOPMENT RESULTS DISCUSSION CONCLUSION APPENDIX REFERENCES

The mathematical models for the epicardial and endocardial cells of the rat left ventricle are based on the classical formulation of Hodgkin and Huxley (1952) and are therefore somewhat similar to our previous computational work (Demir et al., 1994, 1997, 1999). The electrical equivalent circuit representing the sarcolemmal ion channels, pumps, and the Na⁺-Ca²⁺ exchanger in the adult rat left ventricular cell (epicardial and endocardial) is shown in Fig. 1 A. This circuit is coupled with a fluid compartment (Fig. 1 B), which describes the changes in Na⁺, K⁺, and Ca²⁺ ions in the myoplasm, and the Ca²⁺ ions in the sarcoplasmic reticulum (SR). Formulations for the equations used in the model for the epicardial cell are discussed briefly in the following subsections. The endocardial cell model is based on the epicardial formulation. The main differences between these two models are clearly outlined at the end of this section, and are justified by experimental data. The complete set of equations for these two models are provided in the Appendix.

View larger version (22K): [in this window] [in a new window]   FIGURE 1   (A) Electrical equivalent circuit of the adult rat epicardial/endocardial ventricular cell. (B) Fluid compartment model of the rat cardiac cell (epi and endocardial). The expressions for the intracellular Ca2+ dynamics are based on the work of Winslow et al., 1999.

Na⁺ current (I_Na)

The inward Na⁺ current (I_Na) is responsible for the initial upstroke of the action potential. Equations for I_Na are of the type first formulated by Beeler and Reuter (1977), and consist of a fast activation variable (m³), a fast inactivation variable (h), and a slow inactivation variable (j). The steady-state activation and inactivation curves used in the model (Fig. 2 A) are based on recent patch clamp experiments in rat ventricular myocytes (Lee et al., 1999). The basic kinetic characteristics of I_Na are similar in ventricular cells across different species (Hanck, 1995). Therefore, the time constants for activation (_m) (Fig. 2 B) and inactivation (_h, _j) (Fig. 2 C) were adapted from the guinea pig ventricular cell model (Luo and Rudy, 1994), and were scaled for room temperature (Colatsky, 1980). The maximum Na⁺ conductance (g_Na) was adjusted to generate an appropriate value for the action potential amplitude and the maximal upstroke velocity (dV/dt_max). The normalized peak current-voltage (I-V) relationship for I_Na is shown along with the experimental data in Fig. 2 D (Lee et al., 1999).

View larger version (15K): [in this window] [in a new window]   FIGURE 2   Parameters for the sodium current, INa (A) Steady-state activation () and inactivation (, ) ( and  represent data from Lee et al., 1999). (B) Activation time constant (m) (scaled from Luo and Rudy, 1994). (C) Inactivation time constants (h, j) (scaled from Luo and Rudy, 1994). (D) Normalized simulated current-voltage (I-V) characteristics of INa ( represents normalized data from Lee et al., 1999).

L-type Ca²⁺ current (I_CaL)

The inward L-type Ca²⁺ current (I_CaL) is responsible for the plateau phase of the action potential. The Ca²⁺ ions entering the cell through these channels provide the trigger for the Ca²⁺-induced Ca²⁺ release (CICR) from the SR (Bers and Perez-Reyes, 1999). Our formulation for I_CaL follows that of Nygren and co-workers (1998), and includes time- and voltage-dependent activation and inactivation, as well as Ca²⁺-dependent inactivation. Fig. 3 A shows the steady-state activation and inactivation curves used in the model, which are based on data from isolated rat cells (Katsube et al., 1998). The time constants for activation (_d) (Fig. 3 B) and inactivation (_f11, _f12) are shown (Fig. 3 C). The formulation of _d is based on a recent study in rat ventricular myocytes (Sun et al., 2000). At depolarized potentials inactivation is composed of both fast (_f11) and slow (_f12) components (Katsube et al., 1998). At more hyperpolarized potentials, the fast and the slow time constants have very similar values based on monoexponential fits to recovery from inactivation data at 50 mV and 80 mV (Meszaros et al., 1997; Nawrath and Wegener, 1997). Ca²⁺-dependent inactivation is modeled as a function of the Ca²⁺ concentration in the restricted subspace located between the junctional sarcoplasmic reticulum (JSR) and the T-tubules ([Ca²⁺]_ss). The reversal potential for I_CaL (E_CaL) was set to a constant value of +65.0 mV, as measured experimentally (Bouchard et al., 1995), instead of the Nernst potential for the Ca²⁺ ions, as in previous models (Lindblad et al., 1996; Nygren et al., 1998). It has been shown that there is a large variation in the density of I_CaL in rat ventricular myocytes, even among cells from the same region (Richard et al., 1993; Gomez et al., 1997). Therefore, a normalized I-V relationship for I_CaL (Fig. 3 D) was compared to experimental data (Richard et al., 1993). I_CaL influences the action potential morphology (plateau) in rats. Therefore, the maximum conductance value of I_CaL (g_CaL) was adjusted so that the simulated and experimentally recorded epicardial action potential waveforms were in close agreement. The value of g_CaL was further constrained by making sure that the influx of Ca²⁺ ions via I_CaL (Q_CaL) during a simulated action potential was comparable to experimentally measured values (Bouchard et al., 1995).

View larger version (16K): [in this window] [in a new window]   FIGURE 3   Parameters for the calcium current, ICaL (A) Steady-state activation () and inactivation (, ) ( and  represent data from Katsube et al., 1998). (B) Activation time constant (d) ( represents data from Sun et al., 2000). (C) Inactivation time constants (f11, f12) ( and  represent data for fast and slow inactivation from Katsube et al., 1998, data at 50 mV and 80 mV being recovery from inactivation kinetics, from Meszaros et al., 1997 and Nawrath and Wegner, 1997, respectively). (D) Normalized simulated current-voltage (I-V) curve for ICaL ( represents normalized data from Richard et al., 1993).

Ca²⁺-independent transient outward K⁺ current (I_t)

In most studies of the transient outward K⁺ current (I_t) in rat ventricular myocytes, Ca²⁺ channel blockers such as CdCl₂ or CoCl₂ are used to minimize the interference of I_CaL while recording I_t. However, divalent cations such as Co²⁺ and Cd²⁺ significantly alter the properties of I_t (Agus et al., 1991; Stengl et al., 1998a) by shifting the steady-state activation and inactivation characteristics and the voltage-dependence of the time constant for activation in the depolarized direction. Accordingly, to formulate equations for I_t under more physiological conditions, we have used experimental data for steady-state activation and inactivation (Fig. 4 A), which is free from this complication (Stengl et al., 1998a). Fig. 4, B and C show the time constants for activation and inactivation, respectively. The time constant for activation is based on experimental data obtained in the absence of divalent cations (Agus et al., 1991). Inactivation in I_t can be described as a sum of fast (s) and slow (s_slow) variables. The inactivation time constants display almost identical values at voltages positive to 0 mV (35 ms), and different values at voltages negative to 0 mV. This formulation is based on the fact that I_t is observed to inactivate with a monoexponential decaying time constant when elicited at depolarized potentials (Wettwer et al., 1993), whereas its recovery from inactivation kinetics is biexponential at more hyperpolarized potentials (Shimoni et al., 1995; Volk et al., 2001). Values for the fast and slow recovery time constants (at 90 mV), and their relative contributions to the total recovery of I_t were adapted from recent experiments in rat ventricular myocytes (Volk et al., 2001, Table 3). The biphasic recovery time course may suggest that I_t consists of a heterogeneous mixture of more than one type of current, as has been recently reported (Himmel et al., 1999). The differences in the properties of the components are not apparent during the onset of inactivation, but appear during reactivation, when the roles of recovery from inactivation are different. In Fig. 4 D the simulated I-V characteristics for I_t are shown along with experimental data (Clark et al., 1995). (Note that these characteristics were recorded in the presence of Cd²⁺, so that appropriate shifts in the activation and inactivation characteristics are used while generating the model results for the I-V characteristics.)

View larger version (14K): [in this window] [in a new window]   FIGURE 4   Parameters for the Ca2+-independent transient outward K+ current, It. (A) Steady-state activation () and inactivation (, slow) ( and  represent data recorded in the absence of divalent cations from Stengl et al., 1998a). (B) Activation time constant (r) ( represents data from Agus et al., 1991, recorded in the absence of divalent cations). (C) Inactivation time constants (s, sslow) ( represents data for fast inactivation from Wettwer et al., 1993. The data for the biphasic recovery from inactivation at 90 mV is obtained from Volk et al., 2001). (D) Simulated current-voltage (I-V) curve for It ( represents data from Clark et al., 1995).

Steady-state outward K⁺ current (I_ss)

The steady-state outward K⁺ current (I_ss) is characterized as a rapidly activating, very slowly inactivating current (Apkon and Nerbonne, 1991), and is also sometimes referred to as I_K in rat myocytes (Shimoni et al., 1994). Values for I_ss can be obtained at the end of a long (100-500 ms) depolarized voltage clamp pulse, when I_t is assumed to be completely inactivated. The steady-state activation and inactivation characteristics of I_ss (Fig. 5 A) are based on experimental data in rat ventricular myocytes (Weis et al., 1993). The time constant for activation (Fig. 5 B) is 10 times slower than the time constant for activation of I_t (Apkon and Nerbonne, 1991). The time constant for inactivation (Fig. 5 C) is constant (2.1 s), and is based on experimental measurements (Berger et al., 1998). In Fig. 5 D the model-generated I-V simulation of I_ss is compared with experimental recordings in rats (Clark et al., 1995).

View larger version (13K): [in this window] [in a new window]   FIGURE 5   Parameters for the steady-state K+ current, Iss (A) Steady-state activation (ss) and inactivation (ss) ( and  represent data from Weis et al., 1993). (B) Activation time constant (rss), which is 10 times slower than the corresponding one for It (cf. Apkon and Nerbonne, 1991). (C) Inactivation time constant (sss) (2.1 s), not dependent membrane potential (cf. Berger et al., 1998). (D) Simulated current-voltage (I-V) curve for Iss, recorded at the end of a 500-ms depolarizing voltage-clamp pulse ( represents data recorded at the end of a 500-ms depolarizing pulse, and is obtained from Clark et al., 1995).

Inwardly rectifying K⁺ current (I_K1)

The time-independent, inwardly rectifying K⁺ current (I_K1) strongly modulates the resting membrane potential, and determines the input resistance of the quiescent cell. I_K1 also contributes to the repolarization of the action potential by supplying an outward current during the late phase of repolarization. The formulation for I_K1 was adapted from earlier work (Oehmen, 1999), and is based on data recorded from Giles' laboratory (unpublished results). The I-V characteristics for I_K1 are displayed for different values of external K⁺ concentrations ([K⁺]_o) (Fig. 6 A).

View larger version (10K): [in this window] [in a new window]   FIGURE 6   (A) Current-voltage (I-V) curve for the inwardly rectifying background K+ current, IK1, based on data from Giles' laboratory. Data for external K+ ionic concentrations ([K+]out) of 10 mM (), 5.0 mM (), and 1.0 mM () are illustrated. (B) Simulated I-V relationship for INaCa. Experimental data () are digitized from studies in isolated rat ventricular myocytes from Stengl et al., 1998b.

Hyperpolarization-activated current (I_f)

A small hyperpolarization-activated inward current (I_f) was included in this model. The formulation for this current was adapted from our earlier work (Demir et al., 1994; Oehmen, 1999), and is based on data recorded in rat ventricular myocytes (Fares et al., 1998; Cerbai et al., 1996).

Background current (I_B)

The background current I_B is a sum of three linear background currents: an Na⁺ current (I_BNa), a Ca²⁺ current (I_BCa), and a K⁺ current (I_BK). These currents represent the small leak of ions across the sarcolemma, and their magnitudes are adjusted to achieve stability of the intracellular ionic concentrations (Demir et al., 1994).

Other ionic currents

A small persistent inward Na⁺ current (Saint et al., 1992), a tetrodotoxin-blockable calcium current I_Ca(TTX), which is generated by Na⁺ channels (Aggarwal et al., 1997), and a novel anionic background current (Spencer et al., 2000) have been reported in rat ventricular myocytes. We have not included these currents in this version of our model. The delayed rectifier K⁺ current (I_Kr) has been recently reported to be present in rat ventricular myocytes, albeit at a very small density (Pond et al., 2000). Therefore it was not included in the present model.

Na⁺-K⁺ pump (I_NaK) and the Ca²⁺ pump (I_CaP)

The Na⁺-K⁺ pump current (I_NaK) maintains the Na⁺ and K⁺ electrochemical gradient across the sarcolemma. The equation for I_NaK is based on earlier formulations (Luo and Rudy, 1994), and the maximum Na⁺-K⁺ pump current parameter (_NaK) was adjusted to achieve a stable internal Na⁺ ion concentration of 10.74 mM. Even with this adjustment, the magnitude of the model I_NaK in physiological concentrations of Na⁺ and K⁺ ions, and a clamp potential of 40 mV, was 0.2368 pA/pF (when normalized to 100 pF), and is within the experimentally measured range of 0.27 ± 0.05 pA/pF under identical conditions (Stimers and Dobretsov, 1998). The formulation for the Ca²⁺ pump (I_CaP) is based on our earlier description (Demir et al., 1994).

Na⁺-Ca²⁺ exchanger (I_NaCa)

The Na⁺-Ca²⁺ exchanger current (I_NaCa) plays a dual role in rat ventricular myocytes. It contributes to the late repolarization phase of the action potential, and also extrudes Ca²⁺ ions from the myoplasm. The equation for I_NaCa was based on our earlier work (Demir et al., 1994), and the scaling factor for I_NaCa (k_NaCa) was derived from a fit to the data obtained from the I-V characteristics in rat ventricular myocytes (Stengl et al., 1998b) (Fig. 6 B); k_NaCa was reduced by 20% in the whole-cell model simulations to achieve intracellular Ca²⁺ homeostasis on a beat-to-beat basis.

Other pumps and exchangers

Other active ionic mechanisms in rat include the Na⁺-H⁺ exchanger (Wallert and Frohlich, 1989) and the Na⁺-HCO cotransport mechanism (Aiello et al., 1998). We have not considered the contribution of these mechanisms to the cardiac action potential at the present time.

Intracellular and SR Ca²⁺ mechanism

The formulation for intracellular Ca²⁺ ion concentration ([Ca²⁺]_i) and its various regulatory processes was adapted from a recent model of the canine midmyocardial ventricular cell (Winslow et al., 1999). The parameters for the peak forward and reverse rates of SR uptake (v_maxf and v_maxr), and the "Ca²⁺ on rate for troponin high affinity sites" (k) were adjusted (see Appendix and Table 6 for the values in our rat ventricular cell model) to simulate a Ca²⁺ transient somewhat similar to the one observed in rats. The initial conditions for the Ca²⁺ subsystem variables in the rat model are also different from the canine ventricular cell model of Winslow et al., 1999. In particular, the values of the junctional SR and network SR are 0.066 mM, and comparable to the physiological values of 0.064 ± 0.006 mM, reported recently in rat ventricular myocytes (Trafford et al., 2001).

Myocyte ultrastructure

The volumes for the subcellular compartments in this model were determined from ultrastructural analysis carried out in rat ventricles (Page, 1978; Schaper et al., 1985), as described in the Appendix (Table 3). The volume and the capacitance of the cell model were assigned values of 16 pL and 100 pF, respectively, based on experimental measurements of 16 pL and 99 ± 8 pF (Bouchard et al., 1995). As a result, the model has a capacitance-to-volume ratio of 6.25 pF/pL, similar to experimentally reported values, 6.76 ± 0.62 pF/pL (Satoh et al., 1996).

Endocardial cell model

Voltage clamp measurements in the adult rat left ventricle have shown that the density of I_t is significantly smaller, and the reactivation kinetics are much slower in the endocardial cells than the epicardial myocytes (Clark et al., 1993; Shimoni et al., 1995). However, the voltage-dependence of steady-state activation, inactivation, and the activation kinetics of I_t are similar in both epicardial and endocardial cells (Benitah et al., 1993). Thus the formulation of I_t in the endocardial model includes different equations for the fast and slow inactivation time constants, and a reduced value for the maximum conductance parameter of I_t (g_t). The reduction in g_t was estimated from experimental I-V data (Fig. 7 A) in rat epicardial and endocardial cells (Shimoni et al., 1995). Rate dependence of I_t in rat ventricular myocytes has been assessed experimentally using two different methods: 1) as the peak amplitude relative to the zero current level (I_peak) (Shimoni et al., 1995); and 2) as the magnitude of the transient component, i.e., the difference between the peak and the steady-state current levels (I_peak  I_ss) (Shimoni et al., 1995; Volk et al., 2001).

View larger version (9K): [in this window] [in a new window]   FIGURE 7   (A) Simulated current-voltage (I-V) curve for It in epicardial (, solid line) and endocardial (, dashed line) myocytes, which were isolated from the adult rat left ventricle (cf. Shimoni et al., 1995). (B) Simulated recovery or reactivation characteristics for It as a function of the basic cycle length of stimulation in the endocardial cell. , Data obtained from the rate stimulation of It in rat endocardial myocytes. It was measured as the peak amplitude relative to zero (Shimoni et al., 1995).

When the formulations for I_t were based on data measured using the latter approach, the model was able to reproduce the epicardial action potential rate dependence (little or no prolongation of the APD), but failed to simulate the prominent rate dependence of the endocardial action potential. Therefore, the time constants for inactivation of I_t in the endocardial cell model were based on measurements of I_t (I_peak only) as a function of the basic cycle length of stimulation (Fig. 7 B) (Shimoni et al., 1995). The densities of I_CaL, I_K1, and I_ss have been found to be similar in the epicardial and the endocardial regions of the rat left ventricle (Clark et al., 1993; Shimoni et al., 1995). Accordingly, the formulations for these currents are identical in the epicardial and the endocardial cell models. Recent studies have reported that the density of I_Na is higher by 33% in rat left ventricular endocardial cells than epicardial ones (Ashamalla et al., 2001). Accordingly, g_Na was increased in the endocardial cell model. The membrane capacitance (C_m) was assigned the same value (100 pF) in the epicardial and the endocardial cell models, based on experimental observations (Clark et al., 1993). All other ionic mechanisms and parameters describing the SR Ca²⁺ handling were assumed to be identical in the epicardial and the endocardial models.

Basic assumptions for the models

The external ionic concentrations in both the models are assumed to be constant and all simulations were carried out at 22°C. All the results were noted in their "steady state," which was achieved by allowing the model to run for 20 s after a change in the initial conditions.

Computational aspects

The mathematical descriptors in the epicardial and/or the endocardial models consist of 28 nonlinear, first-order differential equations. A Runge-Kutta-Merson numerical integration algorithm, which includes an automatic step-size adjustment that is based on an error estimate, was used, as in our previous model simulations (Demir et al., 1994, 1997, 1999). The model equations were coded in C for the whole-cell simulations, and a SUN SPARC Ultra 60 workstation was used for all simulations. The software package Matlab was also utilized while developing equations for the individual ionic currents.

	RESULTS

TOP ABSTRACT INTRODUCTION MODEL DEVELOPMENT RESULTS DISCUSSION CONCLUSION APPENDIX REFERENCES

Simulated action potentials

A stimulus current of 0.6 nA was "applied" for 5 ms (at 1.0 Hz) to elicit the model epicardial and endocardial action potentials (Fig. 8 A). This is in accordance with the experimental protocol used to elicit action potentials in rat ventricular myocytes (Ward et al., 1997). The effects of the intracellular Ca²⁺ buffer EGTA have been modeled in the formulations for [Ca²⁺]_i, and were adapted from a recent model (Winslow et al., 1999), and mimic the experimental recording conditions, where 10 mM EGTA was present the pipette-filling solution (Shimoni et al., 1995). Fig. 8 B shows representative experimental epicardial and endocardial action potential recordings (Shimoni et al., 1995). There is close agreement between the simulated and the experimentally recorded epicardial action potential waveforms. Note also that the simulated and the recorded endocardial action potentials have a prolonged APD, and a more prominent plateau phase, when compared with epicardial action potentials. The less impressive similarity of the endocardial APs is due in part to the fact that the rat endocardial action potential configuration shows a great variation in duration and shape, despite similar recording conditions (Shimoni et al., 1995; see Fig. 4, C and D). These variations might be real; i.e., due to the regional variability of the action potential waveforms within the rat endocardial wall (the base, apex, or the septum regions) (Watanabe et al., 1983). A somewhat similar and significant variability in action potential morphology has been observed in myocytes isolated from the endocardial surface of the left ventricle in ferrets (Brahmajothi et al., 1999). Accordingly, instead of attempting to closely match the simulated and experimentally recorded rat endocardial action potential morphologies by "fine tuning" the model parameters, emphasis was placed on investigating the qualitative behavior of the simulated endocardial action potential.

View larger version (20K): [in this window] [in a new window]   FIGURE 8   (A) Model generated epicardial (solid line) and endocardial (dashed line) action potentials at 1 Hz and 22°C. (B) Experimentally recorded epicardial (solid line) and endocardial (dashed line) action potentials (Shimoni et al., 1995).

The characteristics of the simulated epicardial and endocardial action potentials are compared in Table 1. The resting membrane potentials (V_rest) for the epicardial and the endocardial cells are very similar, and in accordance with experimental recordings, which were 80.5 ± 0.5 mV and 81.3 ± 0.6 mV in the epicardial and the endocardial cells, respectively (Clark et al., 1993). The input resistances (R_in) of both the cells were almost identical (71 M), and are within the experimentally measured range of R_in values in rat ventricular cells, 62.8 ± 28.3 M (Shimoni et al., 1994). R_in was determined by the fraction V/I, where V was a short 5-mV hyperpolarizing pulse applied to the cell at V_rest, and I was the resultant change in membrane current. The peak overshoot of the simulated endocardial action potential is larger than the epicardial one by 11.33 mV, and lies between the mean differences in the peak overshoot between the epicardial and the endocardial cells noted experimentally, 10.6 mV (Volk et al., 2001), and 13.0 mV (Shipsey et al., 1997). Note from Fig. 8 B that the difference in the peak overshoot for the experimentally recorded epicardial and endocardial action potentials is only 3.8 mV. This is one of the reasons for the discrepancy between the simulated and this experimental endocardial action potential. The value of dV/dt_max is 24.76% higher in endocardial cells; this is qualitatively similar to the experimental measurements made in rat epicardial and endocardial cells (Qin et al., 1996) (The measurements in these studies were made at 37°C and under different recording conditions, and therefore cannot be compared quantitatively to model simulations.)

In Fig. 9 the frequency dependence of the simulated epicardial (panel A) and endocardial action potentials (panel B) is shown at two different frequencies, 0.5 Hz and 2.0 Hz. The model is able to reproduce 1) the more prominent rate dependence observed in the endocardial cells, and 2) exhibits little or no rate dependence in the epicardial cell simulations as observed experimentally (Shimoni et al., 1995). Note that the rate dependence of the action potentials was simulated under conditions mimicking the presence of 10 mM EGTA in the recording pipette (Shimoni et al., 1995).

View larger version (21K): [in this window] [in a new window]   FIGURE 9   (A) Steady-state effects of applying stimuli on the simulated epicardial action potential at 0.5 Hz (solid line) and 2.0 Hz (dashed line). (B) Steady-state effects of applying stimuli on the simulated endocardial action potentials at 0.5 Hz (solid line) and 2.0 Hz (dashed line).

Ionic mechanisms underlying the epicardial and endocardial action potentials

One approach for describing the ionic mechanisms underlying the epicardial and the endocardial action potentials is to separate the transmembrane currents that are responsible for the distinct properties of the epicardial and the endocardial action potentials (I_Na, I_t); and then describe the ion channels and antiporters whose behavior is significantly affected as a result of the transmural differences (I_CaL, I_ss, I_NaCa). Simulations for I_Na underlying the epicardial and the endocardial action potentials (stimulated at 1.0 Hz, in the absence of EGTA in the recording pipette, i.e., EGTA = 0 mM) are shown in Fig. 10 A. The larger density of I_Na in the endocardial cell results in a faster initial depolarization in the endocardial myocyte, and is also partially responsible for the increased peak overshoot.

View larger version (8K): [in this window] [in a new window]   FIGURE 10   Simulated action potentials and the underlying ionic currents. (A) INa in the epicardial action potentials (solid line) and endocardial action potentials (dashed line). (B) ICaL, It, and Iss during the epicardial action potentials (solid line) and endocardial action potentials (dashed line).

Fig. 10 B shows the ionic mechanisms primarily determining the repolarization differences between the epicardial and the endocardial cells (I_t, I_CaL, I_ss). The reduction of the main repolarizing current I_t in the endocardial cell is partially responsible for the increased peak overshoot, and is also the main reason for the prolongation of the APD and the more pronounced plateau phase of the endocardial action potential. The more prominent plateau phase results in an I_CaL waveform, which has a smaller peak magnitude and decays more slowly. This is accompanied by an increased activation of I_ss, which provides additional repolarizing current during the endocardial AP.

The slowed inactivation of I_CaL results in an increase in the Ca²⁺ ions entering the cell via I_CaL (Q_CaL) during an endocardial action potential, as compared to the epicardial one (Table 2). Q_CaL for the simulated epicardial action potential (14.32 pC) is in agreement with experimentally measured values, 16.2 ± 3.0 pC (Bouchard et al., 1995). Q_CaL is increased by 84.22% in the endocardial cell during an identical cardiac cycle, which agrees qualitatively with recent experimental results, which showed an 110.4% increase in the mean values of Q_CaL for rat myocytes of endocardial origin (Volk et al., 1999). Other transmembrane currents (I_B, I_K1, I_NaK, and I_NaCa) underlying the epicardial and endocardial action potentials are shown in Fig. 11. The magnitude of I_f underlying the action potential is <1 pA, and hence is not shown. The small differences among I_B, I_K1, and I_NaK underlying the epicardial and endocardial action potentials are primarily due to the different time-dependent behavior of the membrane voltages between these cells. In addition to contributing to the late phase of repolarization, I_K1 also opposes the initial depolarization of the cell membrane, and thus alters the threshold potential in both types of myocytes. I_NaCa underlying the epicardial and the endocardial action potentials differs in both magnitude and temporal behavior. This disparity is a result of the dependence of I_NaCa on membrane voltage and Ca²⁺ transient ([Ca²⁺]_i). This is discussed in detail in the subsequent section. A quantitative analysis of the net Ca²⁺ fluxes across the rat sarcolemma underlying the simulated epicardial and endocardial action potentials (Table 2) reveals that the Ca²⁺ homeostasis in these cells is maintained by an almost equal amount of Ca²⁺ influx (via I_CaL and I_BCa), and efflux via I_NaCa during each cycle. The small discrepancy between the two can be attributed to the flux carried by I_CaP. The model results thus conform to the experimental observation that the exchanger (I_NaCa) extrudes almost the same amount of Ca²⁺ ions that enter the cardiac cell via the L-type Ca²⁺ current during each beat (Bridge et al., 1990; Bouchard et al., 1995).

View larger version (15K): [in this window] [in a new window]   FIGURE 11   IK1, INaCa, IB, and INaK underlying epicardial action potentials (solid line) and endocardial action potentials (dashed line).

Ca²⁺ transients in the epicardial and endocardial cell models

Simulations describing the changes in [Ca²⁺]_i underlying the epicardial and endocardial action potentials are shown in Fig. 12 A. The predictions of our model for the diastolic and the peak systolic values of [Ca²⁺]_i in the epicardial cell (78.99 nM and 357.80 nM, respectively) are very similar to experimental values obtained from action potential voltage clamp measurements carried out in rat ventricular myocytes, which were 79.0 ± 4.7 nM and 283 ± 44.1 nM, respectively (Kaprielian et al., 1999). The difference between the peak systolic and diastolic values of the [Ca²⁺]_i ([Ca²⁺]_i), (which is sometimes considered as a more accurate estimate of the SR Ca²⁺ release, see Han et al., 1994), was 278.81 nM in the epicardial model. This is also comparable to experimental measurements obtained via field stimulation of rat cardiac myocytes, 254.0 ± 25.0 nM (McCall et al., 1998). The simulated epicardial and the endocardial Ca²⁺ transients exhibit waveforms qualitatively similar to those observed when rat myocytes were clamped by "short" and "long" action potentials, which were generated by stimulating a rat epicardial myocyte in the absence and presence of 3 mM 4-AP (a relatively selective blocker of I_t), respectively (Bouchard et al., 1995). The experimental action potentials, and the corresponding I_CaL and [Ca²⁺]_i waveforms, are shown in Fig. 12 B (Bouchard et al., 1995). The diastolic and peak systolic values of the simulated [Ca²⁺]_i for the endocardial cell are increased by 45.08% and 33.33%, respectively, compared to their epicardial counterparts (see Table 2). This is similar to experimental observations where the APD was prolonged as a result of blocking I_t (Bouchard et al., 1995), and even in rat epicardial-endocardial measurements, where an increase of 62% and 34.78% was reported for the mean diastolic and peak systolic values of [Ca²⁺]_i respectively (Figueredo et al., 1993). An increase in the amplitude of [Ca²⁺]_i also results in a corresponding increase in the magnitude of I_NaCa (see also Fig. 11), and is similar to the Na⁺-dependent "tail current" measurements (which were representative of I_NaCa) in rat ventricular myocytes (Bouchard et al., 1995; Clark et al., 1996). For the [Ca²⁺]_i during a simulated epicardial action potential of 278.81 nM, the corresponding peak value of I_NaCa was 18.32 pA (or 0.1832 pA/pF, when normalized to model C_m of 100 pF). This agrees closely with experimental results, where peak I_NaCa was 0.20 ± 0.03 pA/pF for a SR Ca²⁺ release of 257 ± 42 nM (Sham et al., 1995). Thus, even though the magnitude of I_NaCa has been adjusted (reducing the value estimated from the I-V plots by 20%) to maintain Ca²⁺ balance, the peak magnitude still remains within the range of physiological measurements.

View larger version (9K): [in this window] [in a new window]   FIGURE 12   (A) Simulated action potentials and the underlying changes in ICaL and [Ca2+]i for the epicardial (solid line) and endocardial (dashed line) myocytes. (B) Experimentally recorded action potentials and the underlying ICaL and [Ca2+]i in epicardial myocytes under control conditions (solid line), and in the presence of 3 mM 4-AP (dashed line) (from Bouchard et al., 1995). Changes in the ratio of fluorescence intensity of indo 1 at 410 nm to that at 500 nm (F410/500) were taken as a measure of changes in [Ca2+]i (Bouchard et al., 1995).

The simulated Ca²⁺ concentrations in the JSR ([Ca²⁺]_JSR) and [Ca²⁺]_ss for the epicardial and endocardial cells are shown in Fig. 13. [Ca²⁺]_ss has peak values of 27.6 µM and 22.6 µM for epicardial and endocardial cells, respectively. [Ca²⁺]_ss acts as the trigger for the Ca²⁺ release from the JSR. The value of [Ca²⁺]_JSR just before the application of the stimulus to evoke the action potentials is 40% higher in endocardial cells compared with epicardial cells (exact values were 66.07 µM and 92.34 µM for the epicardial and endocardial cells, respectively). The larger SR Ca²⁺ content, coupled with a larger depletion of SR during an action potential in the endocardial cell (58.5% depletion in the endocardial cell compared with 43.15% in the epicardial cell) results in a larger endocardial [Ca²⁺]_i amplitude. The increased Ca²⁺ content of SR in the presence of a longer APD in rat ventricular myocytes has been suggested earlier, based on experimental observations made via action potential voltage clamp measurements (Bouchard et al., 1995; Clark et al., 1996). Recent experiments have confirmed the occurrence of both an increased SR load and an increased fractional release of Ca²⁺ from the SR when the same rat ventricular myocyte was clamped with an action potential having a more prolonged APD (Sah et al., 2001).

View larger version (13K): [in this window] [in a new window]   FIGURE 13   Simulated action potentials [Ca2+]JSR and [Ca2+]ss in epicardial (solid line) and endocardial (dashed line) cells.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MODEL DEVELOPMENT RESULTS DISCUSSION CONCLUSION APPENDIX REFERENCES

This paper describes the first successful development of a comprehensive model of the membrane action potentials in adult rat left ventricular myocytes of epi and endocardial origin. When possible, data obtained via patch clamp experiments in isolated rat cells have been utilized. A number of somewhat similar models that simulate ventricular action potentials in different species such as canine (Winslow et al., 1999), guinea pig (Nordin, 1993; Luo and Rudy, 1994; Noble et al., 1998), human (Priebe and Beuckelmann, 1998), and frog (Riemer et al., 1998) have been published during the past decade. However, there are important functional differences between the action potential waveforms in rats and other mammalian ventricular cells. Recent findings suggest that the AP in mouse may be similar to the rat AP (Gussak et al., 2000), although the detailed ionic mechanisms underlying repolarization differ quantitatively between the rat and mouse (Fiset et al., 1997b). The rat action potential has a short APD, a somewhat "triangular" shape, and shows a prolongation of the APD with an increase in the stimulus frequency (Watanabe et al., 1983; Clark et al., 1993; Shimoni et al., 1995) as opposed to a longer APD, a "spike and a dome" configuration, and a decrease in the APD in response to an increased rate of stimulus in canine and guinea pig ventricular cells (Antzelevitch et al., 1999). As expected, therefore, a comparison of the ionic currents underlying the action potentials in different species shows that they display markedly different amplitudes and time-dependent behavior as well as transmural expression pattern. I_t is absent in guinea pig ventricular cells, and the regional electrical heterogeneity in the guinea pig ventricle is mediated via the delayed rectifier (I_K) and an Na⁺-dependent background current (Main et al., 1998). In addition to I_t, the transmural differences in the canine ventricle are found to be mediated via other ionic mechanisms, which include the slowly activating delayed rectifier (I_Ks) (Liu and Antzelevitch, 1995), a late sodium current (Eddlestone et al., 1996), and I_NaCa (Zygmunt et al., 2000). A new subpopulation of cells (midmyocardial "M" cells), which display a longer APD and steeper rate dependence of APD have been reported in canine (Sicouri and Antzelevitch, 1991) and guinea pig (Sicouri et al., 1996), although investigations in rat failed to provide evidence for the M cells (Shipsey et al., 1997).

The marked variation and diversity in the transmural electrophysiological characteristics across the ventricular myocardium among mammals indicates the need for the development of rigorous species-dependent and tissue-specific (epi and endocardial) models, based on detailed experimental data and analysis. This is a requirement for a correct understanding and interpretation of the integrated behavior of the biophysical processes underlying the cardiac electrical activity of the targeted animal model in normotensive and pathophysiological conditions. The main focus of this study was the development of computational models, which could be used as a basis for investigating the ionic mechanisms of transmural heterogeneity of dispersion in repolarization, and the excitation-contraction coupling between rat myocytes of epi and endocardial origin. This was accomplished by formulating equations for the main ionic currents responsible for the genesis of the action potential, and adopting the description for the intracellular and SR Ca²⁺ dynamics from the recent work of Winslow et al., 1999. The resulting models of the epicardial and endocardial myocytes of the adult rat are able to reconstruct many of their respective action potential properties, which include the similarities (V_rest, R_in), and prominent differences (APD, peak overshoot, dV/dt_max, rate dependence, Q_CaL, Q_NaCa, [Ca²⁺]_i, and [Ca²⁺]_JSR). These models also offer important additional insights regarding the ionic mechanisms that underlie the epicardial and endocardial action potential heterogeneity in the adult rat left ventricle, and illustrate the interesting and highly nonlinear interactions between the major time- and voltage-dependent channel mediated currents, currents due to antiporters and/or pumps, and the small background current.

Transmural gradient of I_Na

The density of I_Na has been recently reported to be distributed nonuniformly across the rat ventricular myocardium (Ashamalla et al., 2001). Thus the density of I_Na is almost identical in myocytes isolated from the left ventricular endocardium and the right ventricle, but is smaller (by 33%) in the left ventricular epicardial myocytes. The density of I_Na is seen to be the sole determinant of the initial upstroke (dV/dt_max), whose values are similar in the simulated left ventricular endocardial and right ventricular myocyte action potentials, but faster than that in the left ventricular epicardial action potential. The present study shows that when the transmural gradients in the densities of I_Na and I_t (increase in I_Na and decrease in I_t) were incorporated in the endocardial cell model, the simulations could account for the 10-13 mV difference in the peak overshoot observed between the epicardial and endocardial action potentials (Shipsey et al., 1997; Volk et al., 2001). In fact, if the density of I_Na was not increased in the simulation of the endocardial action potential (only I_t decreased), the peak overshoot of the endocardial action potential was greater than the epicardial one by 4.92 mV only. Apparently, changes in the densities of both I_t and I_Na can contribute almost equally (5 mV each) to the larger peak overshoot in the endocardial cell. Recent experiments have reported that the density of I_t in the isolated rat right ventricular myocytes is 25% greater than that in isolated rat left ventricular epicardial myocytes (Casis et al., 1998). When the left ventricular epicardial cell model was modified to simulate the action potential (not shown) in right ventricular myocytes (increased I_Na and increased I_t), the model's predicted values for V_rest, dV/dt_max, and peak overshoot were 80.42 mV, 181.91 V/s, and 41.44 mV, respectively. These values are in close agreement with experimental measurements carried out in rat myocytes isolated from the right ventricle, which were 80.74 ± 0.57 mV, 193.25 ± 13.94 V/s, and 37.47 ± 3.21 mV, respectively (MacDonnell et al., 1998).

The simulation results thus provide a mechanistic linkage between the transmural gradient of I_Na and the corresponding changes in the peak overshoot and dV/dt_max, and underline the integrative utility of our model when one considers the fact that all three characteristics (transmural gradient of I_Na, differences in peak overshoot, and dV/dt_max) were reported in a separate set of experiments, and from different laboratories.

Formulation and function of I_t

The available experimental data and our model simulations show that I_t is almost an order of magnitude larger than the other K⁺ currents (I_ss, I_K1) in both epi and endocardial myocytes; thus it is the dominant repolarizing current in both cell types. Our model simulations demonstrate that the differential density of expression and the differences in the reactivation kinetics of I_t clearly underlie the regional variations in APD and rate dependence, thus corroborating important experimental observations (Shimoni et al., 1995) with an in silico approach. This insight is important because during patch clamp experiments the properties of I_t are often studied in the presence of divalent cations, which alter the characteristics of I_t by shifting the steady-state activation and inactivation, and the voltage-dependence of the time constant for activation in the depolarized direction (Agus et al., 1991). The computational model allows one to investigate the role of I_t in a virtual cellular environment, and in interaction with other ionic currents, without altering the intrinsic properties of I_t. The half-maximal voltages of the steady-state activation (V_1/2,act) and inactivation (V_1/2,inact) for I_t in the rat models were 10.6 mV and 45.3 mV, respectively. Interestingly, these values are comparable to the corresponding values for the heterologously expressed voltage-gated K⁺ channel -subunits Kv4.2 and Kv1.4 which, along with some other - and -subunits, are deemed to constitute the putative molecular correlates of I_t (Oudit et al., 2001). V_1/2,act was 13.0 ± 2.0 mV (Diochot et al., 1999), and 7.7 ± 5.4 mV (Wickenden et al., 1999a) for Kv4.2 and Kv1.4, respectively; V_1/2,inact was 45.0 ± 3.0 mV (Fiset et al., 1997a) and 49.3 ± 1.4 mV (Wickenden et al., 1999a) for Kv4.2 and Kv1.4, respectively. Furthermore, the formulation of I_t consists of "fast" and "slow" inactivation variables, with the relative contribution of the "slow" variable to overall inactivation being significantly larger in the endocardial model (42%), as opposed to the epicardial cell model (11%). Thus the equations for I_t are analogous with the general emerging consensus that I_t in rat consists of a fast component (thought to be generated principally by Kv4.2 and/or Kv4.3), and a slower component (thought to be primarily encoded by Kv1.4), with a greater contribution of the slower component in the endocardium (Wickenden et al., 1999b; Oudit et al., 2001).

The model also makes a contribution to the methodology for data analysis. It clearly demonstrates that when the slower component of I_t contributes significantly to the overall inactivation, as is the case in the endocardial cell, I_peak rather than I_peak