Cardiac Electrophysiological Model for Simulation of Action Potential

The cardiac cell membrane contains various ionic channels, exchangers and pumps that allow specific ions to travel or be exchanged through the membrane (Fig. 1). Those ionic components in the cell membrane are utilized to maintain homeostasis of an intracellular and the extracellular environment; such gradient in ionic concentration causes an electrical voltage between the inside and outside of the cell, which is called a “membrane potential.”

Figure 1

A schematic diagram describing the components of a cardiac electrophysiological model. The model contains more than ten ionic currents, pumps and exchangers; this model is called the Kyoto model. Specific ions travel through ionic channels or are exchanged (more...)

The membrane potential and the gradient in ionic concentration both mediate a passive transport of ions through the cell membrane. In addition to the passive force driven by the membrane potential and the gradient in ionic concentration, each ionic channel in the cardiomyocyte opens or closes in response to the shift in the membrane potential, represented as a gating property of the ionic current. All of the current components on the membrane are formulated on the basis of a basic mathematical expression of an ionic current that include three parameters: conductance or a conversion factor, passive transport driven by membrane potential and gradient in ionic concentrations and gating properties.

The traveling of ions through the cell membrane causes either depolarization or hyperpolarization of the cell depending on the transport direction. In adult ventricular cells (Fig. 2), for example, opening of channels for I_Na (Na⁺ current) causes the membrane potential to rise to 40 mV; depolarization of the membrane then allows opening of channels for I_CaL (L-type Ca²⁺ current), which maintains the potential around 10 mV even after closing of the Na⁺ channels; the membrane repolarizes to the resting potential level by opening various channels for outward potassium currents. The entire tracing of the transient change in membrane potential is called “action potential.”

Figure 2

Simulated action potential of an adult ventricular cell with the Kyoto model and changes in I_Na, I_CaL and I_Kr accompanying the simulated action potential.

Developmental Changes in Electrophysiological Properties of Ventricular Cells

The action potential properties of the ventricular cell have been broadly studied in various species at various stages of development. The early embryonic ventricular cells generally have spontaneous action potential in mouse,¹ rat^2,3 and chick⁴; representative in vitro recording of spontaneous action potential in early embryonic rat ventricular cells is shown as an example in Figure 3A.

Figure 3

Action potential at different developmental stages in vitro. A) In vitro action potential recorded in ventricular myocytes of a 12-day fetal rat. [Reproduced from: Nagashima et al. J Mol Cell Cardiol 33(3):533-543; ©2001 with permission of Elsevier.] (more...)

The late embryonic and postnatal ventricular cells require external stimulation to fire the action potential.^5,6 Although several action potential parameters change among different stages, developmental changes in action potential duration (APD) are the most prominent; APD of guinea pig ventricular cells initially decreased in the neonatal stage and then increased until the adult period (Fig. B). Interestingly, the time courses of the changes are different among species; APD continues to decrease in postnatal development of mouse ventricular cells. The developmental changes and the species-specific differences in action potential are mediated by the ion channels of the cells.

Large amounts of data have been recorded via standard microelectrode techniques to describe electrophysiological properties of the ionic channels; several selected examples are reproduced from literature and shown in Figure 4. The basic property of an ionic current can be obtained by shifting the membrane potential from a given holding potential to an arbitrary potential; tracings of I_CaL and I_K1 (inward rectifier K⁺ current) are recorded from ventricular cells at different developmental stages, wherein membrane is depolarized from a holding potential of 50 to 0 mV for recording of I_CaL (Fig. 4A) and from a holding potential of –40 to –100, –90, –80 and –60 mV for recording of I_K1 (Fig. 4B). Current-voltage (I-V) curves of the currents can be drawn up by plotting selected points in the tracing that is obtained by shifting the membrane potential to different potentials (Fig. 4C, D).

Figure 4

Various ionic currents recorded in different developmental stages. A) Representative tracing of I_CaL recorded from ventricular myocytes of the mouse at 9.5-days postcoitum (dpc), 18-dpc and as an adult, wherein the membrane potential was depolarized from (more...)

Modeling Developmental Changes in Cardiomyocyte

As shown above, developmental changes in cardiomyocyte have been anecdotally reported at various levels. This chapter summarizes the basic concept presented in a recently published paper⁷ which aimed to integrate those anecdotally reported data from experiments in vivo or in vitro on the basis of comprehensive cardiac electrophysiological models. I-V curves of I_CaL and I_K1 (Fig. 4C, D) indicate that the activity level of the ionic current changes among different stages while voltage dependencies of the current do not change. On the basis of these data, we have assumed that developmental changes in the ionic currents can be represented quantitatively as the activities of the channels in the developing rodent relative to those in the adult. Application of the integrated model for simulation of action potential showed that action potential at different developmental stages can be reproduced with common sets of mathematical models, wherein quantitative changes in the ionic currents, pumps, exchangers and sarcoplasmic reticulum (SR) Ca²⁺ kinetics are expressed as relative activities.

Implementation of Cardiac Electrophysiological Models to E-cell Simulation Environment

Models for simulating the action potential at different developmental stages were constructed on the basis of the Kyoto and Luo-Rudy models, electrophysiological models of the guinea pig cardiomyocyte. ⁸ The structures of the Kyoto and Luo-Rudy models are very similar and both models have been developed for simulation of guinea pig ventricular cells. The latest version of the Luo-Rudy model⁹ was implemented in the E-Cell simulation environment version 3.1. All of the models are constructed on the basis of ElectrophysiologicalBaseProcess.hpp, which had been developed to facilitate further analysis via replacing mathematical equations of ionic currents from one electrophysiological model to the other and the models are available online at http://www.e-cell.org.*

General Approach to Modeling of Different Developmental Stages

All ionic currents, pumps, exchangers and SR Ca²⁺ kinetics are expressed in mathematical equations; as mentioned above, all of the equations include either a conversion factor (pA/mM) or conductance (pA/mV) as one of the parameters. For instance, I_CaL and I_K1 in the Kyoto model are expressed as follow:

I_CaL = P_CaL ·(CF_Ca + 0.000365·CF_K + 0.0000185·CF_Na)·p(open_CaL) (1)

I_K1 = G_K1 ·(V_m - E_K)·( f_o ⁴ + 8/3· f_o ³ f_B + 2· f_o ²· f_B ²)· y (2)

In Equation (1), CF_Ca, CFK and CF_Na represent constant field equations (mM) for Ca²⁺, K⁺ and Na⁺, respectively; the open probability of three gates in the L-type Ca²⁺ channel is expressed as p(open_CaL). Similarly in Equation (2), V_m represents the membrane potential (mV); E_K represents the equilibrium potential of K⁺ (mV); f_B and f_O represent the fractions of blocked state and those of open state, respectively; y represents the gating variable for a two-state gate. In order to simulate both ventricular cells and sinoatrial (SA) node cells by common mathematical equations, either the conversion factor (P_CaL in Eq. 1) or conductance (G_K1 in Eq. 2) was adjusted based on electrophysiological experiments of each cell.⁸

Various in vitro experimental data, including I-V curves and Western blot analyses, were utilized to estimate the relative activities of ionic currents, pumps, exchangers and SR Ca²⁺ kinetics. Those in vitro experimental studies that used guinea pigs were preferentially adopted, because both models were constructed using the adult guinea pig.^8,9 Although the guinea pig was the preferred experimental animal, data from the rat and mouse were also utilized, on the basis of the reported observation that the I-V relationships of the ionic channels are well conserved among different rodents.^10,11 In addition, the target stages for simulation of action potentials were set to early embryonic, late embryonic and neonatal, because plenty of literature was available for these stages. The early embryonic stage approximately represents the mouse at 9.5 days postcoitum (dpc) and the rat at 11.5 dpc; the late embryonic and neonatal stages correspond to 1-5 days before and after birth, respectively.

Ionic Currents

It was assumed that developmental changes in ionic currents are determined mainly by their quantitative changes (Fig. 5), which can be represented as the activities of the current in developing stages relative to that of those in the adult stage.

Figure 5

Schematic diagram for modeling rodent ventricular cells at different stages of development. The early embryonic stage approximately corresponds to 9.5-dpc mouse and 11.5-dpc rat. The late embryonic stage corresponds to 1-5 days before birth. The neonatal stage (more...)

The relative activities of ionic currents were either computed from I-V curves (Table 1) or estimated on the basis of qualitative observations (Table 2). It was confirmed that the I-V relationship did not change among different developmental stages for I_CaL,¹² I_CaT (T-type Ca²⁺ current),¹³ I_K1,^12,14 I_Kr (rapid component of delayed rectifier K⁺ current),¹⁵ I_Ks (slow component of delayed rectifier K⁺ current) ¹⁵ and I_Na.¹⁵

Table 1

Relative activities for ionic currents, as obtained from the literature.

Table 2

Estimated relative activities of ionic currents.

The relative activities were multiplied by the conversion factor or the conductance of the corresponding ionic current. In addition, all currents listed in Table 1 had to be normalized by the ratio of the cell capacitance (C_m) of individual myocytes at the corresponding developmental stages (Table 4) to that of adult ventricular cells (132 pF), because I-V relationships are usually reported as current density (pA/pF) and the Kyoto model presents current in pA. The ratios were 28/132 for the early embryonic ventricular cell model, 35/132 for the late embryonic ventricular cell model and 40/132 for the neonatal ventricular cell model.

Table 4

Cell capacitances and volumes of cell compartments.

The Luo-Rudy model includes all ionic currents listed in Figure 5 except I_to (transient outward current), I_KATP (ATP-sensitive K⁺ current) and I_ha (hyperpolarization-activated cation current), all relative activities except those of I_to, I_KATP and I_ha were thus implemented in the Luo-Rudy model by the same procedure used in the Kyoto model. Unlike the Kyoto model, all of the currents in the Luo-Rudy model are presented as current density (pA/pF), so it was not necessary to normalize the activity of the currents by the ratio of the C_m of individual myocytes at the corresponding developmental stages.

Background Ionic Currents

I_bNSC (background nonselective cation current) is known to have a higher current density in SA node cells than in ventricular cells.¹⁶ Because we found that I_bNSC plays an important role in the spontaneous action potential of both SA node cells and early embryonic ventricular myocytes, we scaled the current amplitudes at different stages according to the cell capacitances of the fetal and neonatal cells (Table 2).

I_KACh (ACh-activated K⁺ current) is known to have negligible effects on the action potential of ventricular cells during the course of development^15,17 and is not included in adult ventricular cell models.⁸ Hence, we excluded I_KACh from the models. Other background currents, including I_Kpl (nonspecific, voltage-dependent outward current), I_I(Ca) (Ca²⁺-activated background cation current) and I_Cab (background Ca²⁺ current) were assumed to have steady current densities; as such, these currents were normalized to the corresponding cell capacitance by the method described above.

Exchangers, Pumps and SR Ca²⁺ Kinetics

The relative activities of exchangers, pumps and SR Ca²⁺ kinetics were computed from a Western blot of SR-related proteins,^18,19 as listed in Table 3. Here, we assumed that the relative expression level of the proteins directly reflected the relative activities of Na⁺/Ca²⁺ exchange, SR Ca²⁺ pump, ryanodine receptor (RyR) channel and other SR Ca²⁺ kinetics. The average relative expression values of SR-related proteins in the early embryonic stage (0.04), late embryonic stage (0.30) and neonatal stage (0.30) were adopted for estimating those values for I_{SR, transfer} and I_{SR, leak}.

Table 3

Relative activities of exchangers, pumps and SR Ca²⁺ kinetics.

Cell Capacitance and Volume of Cell Compartments

C_m (cell capacitance) and volumes (V_i, V_rel, V_up) were computed on the basis of quantitative data obtained from the literature (Table 4). No significant differences were observed between the C_m of mouse ventricular cells (31 ± 3.3 pF) and that of guinea pig ventricular cells (34.5 ± 2.72 pF) in the late embryonic stage;^1,12 as such, the C_m values for mouse early embryonic ventricular cells (28 pF), guinea pig late embryonic ventricular cells (35 pF) and guinea pig neonatal ventricular cells (40 pF) were adopted.

The developmental change in V_i (cell volume accessible for ion diffusion) in rabbit ventricular cells is roughly proportional to that of cell capacitance.²⁰ In addition, a positive linear correlation has been found between membrane capacitance and cell volume in several species.²¹ Hence, cell volume was estimated by multiplying the adult V_i (8.0 × 10^–3 μL) by the corresponding C_m (28, 35 or 40 pF) over the adult C_m (132 pF).

In the Kyoto model, the volume fractions of V_rel (volume of SR release site) and V_up (volume of SR uptake site) were set to 2% and 5% of V_i, respectively.⁸ The SR-mediated Ca²⁺ transient is modeled by multiplying an estimated value called the “SA factor” by V_rel, V_up and SR-related currents in the Kyoto model.⁸ The same approach has been adopted for estimating V_rel and V_up in different developmental stages of ventricular cells; on the basis of quantitative changes in SR-related proteins, ¹⁸ the average relative expression values of those proteins in the early embryonic stage (0.04), late embryonic stage (0.30) and neonatal stage (0.30) were utilized for the estimation.

Simulation of Action Potential at Three Different Developmental Stages

On the basis of the assumption that developmental changes in ionic currents are determined mainly by their quantitative changes (Fig. 5), the developmental changes in ionic components were represented as the activities of the components in the developing rodent relative to those in the adult; the relative activities were multiplied by either the conversion factor or conductance in corresponding mathematical equations. All of the models were simulated for 200 s to confirm that the spontaneous action potentials were stable or that the membrane potential had reached a quasi-steady state. Hence, the simulation results presented in this chapter were recorded after simulating the corresponding models for 200 s. In addition, an external current (I_ext) was applied in the late embryonic and neonatal ventricular cell models in order to fire the action potential of the cells. Because the Luo-Rudy model requires “pacing” of the action potential, the model was simulated for 600 s as instructed in the report.⁹

Simulated Action Potential at Three Representative Developmental Stages

The implementation of relative activities of ionic components at early embryonic stage to both the Kyoto and Luo-Rudy models exhibited a spontaneous action potential (Fig. 6A,C). In the simulated action potential with the Kyoto model, the membrane slowly depolarized from the maximum diastolic potential (MDP) at –62.86 mV until it reached approximately –40 mV when spontaneous action potential was triggered. The membrane then started to repolarize after overshoot potential at 3.13 mV and completed the repolarization in less than 100 ms. The whole action potential was completed in a basic cycle length (BCL) of 492 ms. On the other hand, the membrane overshot to 13.74 mV from MDP at –71.16 mV in the simulation with the Luo-Rudy model; the whole action potential was completed in a BCL of 414 ms, which resulted from faster depolarization and repolarization of the membrane.

Figure 6

Simulated action potentials at different developmental stages with the Luo-Rudy model (B) in comparison with simulated action potential with the Kyoto model (A). A) Simulated action potential with the Kyoto model. B) Simulated action potential with the (more...)

The spontaneous action potential ceased in the later stages of development in simulation of the corresponding stages with both the Kyoto model (Fig. 6B) and the Luo-Rudy model (Fig. 6D). In simulation with the Kyoto model, both late embryonic and neonatal ventricular cells showed resting membrane potentials that were more negative (–83.60 mV) than the MDP of the early embryonic ventricular cell. Repolarization of the membrane occurred more slowly in the late embryonic ventricular cell than in the neonatal ventricular cell; the APD was 140 ms in the late embryonic ventricular cell and 117 ms in the neonatal ventricular cell. The qualitative characteristics of the cells at those stages were well reproduced in the simulation with the Luo-Rudy model as well; the late embryonic and neonatal ventricular cells showed more negative resting membrane potentials than the MDP of the early embryonic ventricular cell and shorter APD in neonatal ventricular cells than in late embryonic ventricular cells.

Evaluation of Individual Ionic Currents in Two Different Models

Comparison of the action potential tracings of early embryonic ventricular cell simulated with the Kyoto (Fig. 6A) and the Luo-Rudy model (Fig. 6C) clearly indicate that the Kyoto model reproduced the action potential recording in vitro (Fig. 3A) more consistently than did the Luo-Rudy model; the most prominent differences are the faster repolarization phase (RP) and diastolic slow depolarization (DSD) phase, both of which cause overall shortening of BCL. The differences in simulated action potential were determined by differences in mathematical equations of ionic currents, particularly I_Kr that plays a predominant role in repolarization of the membrane during action potential. The dynamic behaviors of IKr underlying the action potential were compared between the Kyoto and the Luo-Rudy models along with I_CaL and sums of I_CaL and I_Kr (Fig. 7). Apparently, activation of I_Kr in RP was faster in the LuoRudy model (Fig. 7B) than in the Kyoto model (Fig. 7A); the difference in I_CaL in the inactivation phase is canceled out by the fast activation of I_Kr, illustrated as sum of I_CaL and I_Kr. Hence, we made a working hypothesis that the difference in the mathematical equations of I_Kr in the two models undertakes more consistent simulated action potentials in the Kyoto model.

Figure 7

Simulated action potential, I_Kr and I_CaL, at early embryonic stage with the Kyoto model (A) the LuoRudy model (B). The simulated action potential can be divided into three phases: diastolic slow depolarization (DSD), depolarization phase (DP) and repolarization (more...)

In order to assess the working hypothesis, mathematical equations of I_Kr in the Luo-Rudy model was replaced with those of I_Kr in the Kyoto model. Figure 8 illustrates simulated results of adult ventricular cell with original Luo-Rudy model (A) and the Luo-Rudy model whose I_Kr is replaced with those of the Kyoto model (B); the conductance amplitude of I_Kr (G_Kr) was adjusted to achieve approximately the same APD. The replaced model was then utilized for simulation of spontaneous action potential in early embryonic ventricular cell (Fig. 9A). Apparently, the replacement of the mathematical equations made all quantitative characteristics of the action potential less consistent with those of the cells in vitro; such characteristics are more negative MDP (–75.94 mV), more positive overshoot (21.03 mV) and longer BCL (697 ms).

Figure 8

Simulated action potentials and tracings of I_Kr of adult ventricular cell with the original Luo-Rudy model (A) and the Luo-Rudy model whose I_Kr is replaced with those of the Kyoto model (B). Model equations of I_Kr in the Luo-Rudy model were replaced with (more...)

Figure 9

Simulated action potentials of early embryonic stage with various replacements. A) Simulated action potential with the Luo-Rudy model wherein the model equations of I_Kr were replaced with those of I_Kr in the Kyoto model. B) Simulated action potential (more...)

Another difference between the Kyoto and Luo-Rudy models is that the Kyoto model successfully incorporated five known types of background current components. One of the defined background current is called I_bNSC (background nonselective cation current), which shows ion selectivity of both K⁺ and Na⁺ and is determined by constant field equations of each ions. Because the balance of I_Kr and I_bNSC determines the MDP of the simulated spontaneous action potential of SA node cell with the Kyoto model²⁴ and the Luo-Rudy model lacks such nonselective current, model equations of I_bNSC were augmented in addition to replacing I_K equations (Fig. 9B). Varying the amplitude of I_bNSC (P_bNSC) arbitrarily showed that I_bNSC indeed plays an important role in both determining MDP and rate of DSD phase and all three quantitative characteristics, MDP (–71.15 mV), overshoot (18.20 mV) and BCL (510 ms), were thus improved compared to the I_Kr-replaced model without augmentation of I_bNSC.

Spontaneous Action Potential in Vitro Was Well Simulated with the Early Embryonic Models

The early embryonic ventricular cell model, which was constructed on the basis of the Kyoto model (Fig. 6A), reproduced well the spontaneous action potential that is generated by ventricular cells in 12-dpc rats.² Species-specific differences in spontaneous action potential waveforms have been observed between ventricular cells in 9.5-dpc mice¹ and those in 12-dpc rats.² The ventricular cells in 9.5-dpc mice generate a more hyperpolarized MDP (–71.2 ± 0.4 mV) than those in 12-dpc rats (–66.7 ± 3.6 mV). A spontaneous action potential is triggered when the membrane potential reaches approximately –60 mV in 9.5-dpc mice and approximately –40 mV in 12-dpc rats.^1,2 The simulated action potential in our early embryonic ventricular cell model constructed on the basis of the Kyoto model (Fig. 6A) was very similar to the action potential waveforms generated by the automatically beating cells in 12-dpc rats (Fig. 3A).² In addition, the MDP of the simulated action potential (–62.86 mV) was approximately consistent with that of the ventricular cells in 12-dpc rats. Hence, our early embryonic ventricular cell model could reproduce an action potential that was in reasonable agreement with those of previous studies.

The speed of the spontaneous action potential in the early embryonic stage has been a controversial issue. Unfortunately, the action potential of early embryonic guinea pig ventricular cells has not been reported. Early embryonic hearts have shown a large range of heart rates, from 61 to 219 min^–1 in 11.5-dpc rats.²² The BCL of the simulated action potential of our early embryonic ventricular cell model (492 ms) was roughly consistent with that of ventricular cells in 9.5-dpc mice, which is 510.8 ± 32.8 ms.¹ Although the BCL of the action potential of our model was approximately consistent with those of previous studies, it should be noted that the BCL of the simulated action potential might not be quantitatively accurate, because early embryonic hearts have a large range of rates in vivo.

Developmental Changes in APD Were Reproduced Q ualitatively with the Model

APD changes over the course of rodent development. Although the duration and shape of the action potential in the adult rat are totally different from those in the adult guinea pig because of differences in the I_to, shortening of the APD between the late embryonic stage and the neonatal stage has been observed in both guinea pigs² and rats.³ The rapid component (I_Kr) and slow component (I_Ks) of the delayed rectifier K⁺ current play important roles in repolarization and thus control the length of the APD; both I_Kr and I_Ks undergo very complex changes in their activities between the late embryonic and neonatal stages (Fig. 5). As described in the Methods sections, I estimated the relative activities of both I_Kr and I_Ks on the basis of the qualitative characteristics of these currents, including the changes in APD in response to a selective I_Kr blocker. Although qualitatively consistent, APD may not be quantitatively accurate because several relative activities were estimated on the basis of qualitative characteristics.

Evaluating the Role of Individual Ionic Currents for Better Simulation

All of the simulated results with the Luo-Rudy model indicate that the Kyoto model reproduced the action potential in the early embryonic stage more consistently than the LuoRudy model. The role of individual ionic currents in the better simulation with the Kyoto model was evaluated by replacing mathematical models of specific ionic currents between the Kyoto model and the Luo-Rudy model.

Dynamic behavior was improved by replacement of the mathematical models for I_Kr. This result indicated that the fast repolarization in the LuoRudy model is determined by fast activation of I_Kr. The formulations of I_Kr in the Luo-Rudy and Kyoto models are as follow:

I_Kr = G_Kr · (V_m - E_K) · X_r · R (3)

I_Kr = G_Kr · C_m · (V_m - E_K) · (0.6 · y1 + 0.4 · y2) · y3 (4)

Whereas I_Kr in the LuoRudy model (Eq. 3) is described by a time-dependent activation gate (X_r) and a time-independent inactivation gate (R),⁹ I_Kr in the Kyoto model (Eq. 4) is described by two activation gates (y1, y2) and by one inactivation gate (y3)⁸. In the Kyoto model, the equations were intentionally developed for two cell types, ventricular cells as well as SA node cells,²³ because no obvious difference has been observed in the electrophysiological properties of the currents in terms of their kinetics. In addition, the original paper²⁴ specifically mentioned that the balance of I_Kr and I_bNSC determines the MDP of the simulated spontaneous action potential of SA node cell with the Kyoto model; as such, the difference in background currents between the Kyoto and the Luo-Rudy models is one of the important differences contributing to the increased speed of the DSD phase. The Kyoto model may thus have been suitable for this study, because most currents in ventricular cells change quantitatively with similar kinetics throughout the stages of development.