Mathematical Model of an Adult Human Atrial Cell

The Role of K⁺ Currents in Repolarization

From the Department of Electrical and Computer Engineering (A.N., J.W.C., D.S.L.), Rice University, Houston, Tex, and the Department of Physiology and Biophysics (C.F., L.F., R.B.C., W.R.G.), University of Calgary (Canada) Medical School.

Correspondence to Dr J.W. Clark, Department of Electrical and Computer Engineering, Rice University, 6100 South Main, Houston, TX 77005-1892. E-mail jwc{at}rice.edu

	Abstract

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 Appendix 2 References

 
Abstract—We have developed a mathematical model of the human atrial myocyte based on averaged voltage-clamp data recorded from isolated single myocytes. Our model consists of a Hodgkin-Huxley–type equivalent circuit for the sarcolemma, coupled with a fluid compartment model, which accounts for changes in ionic concentrations in the cytoplasm as well as in the sarcoplasmic reticulum. This formulation can reconstruct action potential data that are representative of recordings from a majority of human atrial cells in our laboratory and therefore provides a biophysically based account of the underlying ionic currents. This work is based in part on a previous model of the rabbit atrial myocyte published by our group and was motivated by differences in some of the repolarizing currents between human and rabbit atrium. We have therefore given particular attention to the sustained outward K⁺ current (I_sus), which putatively has a prominent role in determining the duration of the human atrial action potential. Our results demonstrate that the action potential shape during the peak and plateau phases is determined primarily by transient outward K⁺ current, I_sus, and L-type Ca²⁺ current (I_Ca,L) and that the role of I_sus in the human atrial action potential can be modulated by the baseline sizes of I_Ca,L, I_sus, and the rapid delayed rectifier K⁺ current. As a result, our simulations suggest that the functional role of I_sus can depend on the physiological/disease state of the cell.

Key Words: human atrium • repolarization • sustained outward current • computer modeling • cardiac action potential

	Introduction

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 Appendix 2 References

 
Within the past 5 years, quite extensive voltage-clamp and action potential data from human atrial myocytes have been published from a number of laboratories. In most cases the human atrial tissue was obtained during open heart surgery, in which small pieces of the right atrial appendage are excised as part of the cannulation procedure for cardiopulmonary bypass. However, cardiac tissue can also be obtained from the free wall of the atrium during valve replacement procedures, from explanted failing hearts during heart transplantations, or from donor hearts that cannot be used for transplantation. Once a specimen of cardiac tissue is obtained, enzymatic dispersion techniques are used to isolate single cardiac cells, from which action potential and voltage-clamp data can be recorded. We have made recordings of the outward K⁺ currents, which are responsible for repolarization in human atrial myocytes.^{1 2} On the basis of these data, as well as other published results, we have developed the first comprehensive mathematical model of the electrophysiological responses of a representative human atrial cell.

There are a number of published mathematical models that simulate the electrophysiological responses in several different species and cardiac cell types. Examples include the Purkinje fiber model of DiFrancesco and Noble,³ the Hilgemann and Noble atrial model,⁴ the Earm and Noble model of the single atrial cell,⁵ the bullfrog atrial and sinus venosus models of Rasmusson and colleagues,^{6 7} the ventricular cell models of Luo and Rudy,^{8 9 10 11} the rabbit sinoatrial node cell model of Demir et al,¹² and the rabbit atrial cell model of Lindblad et al.¹³ Recently, the emphasis has shifted from general models, based on voltage-clamp data from several species,^{3 4} to more detailed models based on data from single isolated cells from a particular species. This is a direct reflection of the progress made in experimental work and the resulting availability of more comprehensive data. Our goal was to develop a model that is sufficiently accurate to have predictive capabilities for selected aspects of the electrophysiological responses in human atrium. Emphasis has been placed on the functional roles of the K⁺ currents during repolarization.

	Materials and Methods

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 Appendix 2 References

 
Basic Assumptions
Our model is of the general type first introduced by DiFrancesco and Noble³ and is based, in part, on the rabbit atrial model of Lindblad, Murphey, Clark, and Giles¹³ (hereafter referred to as the LMCG model). As shown in Fig 1, the model consists of a Hodgkin-Huxley–type electrical equivalent circuit for the sarcolemma coupled with a fluid compartment model. The dimensions of the human atrial myocyte are assumed identical to those of the rabbit atrial cell in the LMCG model. These dimensions (cylindrical geometry of 130-µm length and 11-µm diameter) are very close to the dimensions of human atrial myocytes (eg, 120-µm length and 10- to 15-µm diameter¹⁴ ). In addition, we have used a total cell capacitance of 50 pF,which agrees very well with our experimental observations for human atrial myocytes (51.9±3.5 pF, n=52).

View larger version (40K): [in this window] [in a new window]   Figure 1. Schematic representation of the mathematical model of the human atrial cell. A, Electrical equivalent circuit for the sarcolemma. B, Fluid compartment model, including intracellular, cleft, and extracellular spaces.

Membrane Currents
Fig 1A shows the electrical equivalent circuit for the sarcolemma of the human atrial cell. It includes each of the ionic currents that are known to contribute to the action potential in human atrial myocytes (I_Na, I_Ca,L, I_t, I_sus, I_K,r, I_K,s, and I_K1), the Ca²⁺ and Na⁺-K⁺ pump and Na⁺-Ca²⁺ exchanger currents responsible for maintaining intracellular ion concentrations (I_CaP, I_NaK, and I_NaCa), and the Na⁺ and Ca²⁺ background (leakage) currents (I_B,Na and I_B,Ca). Mathematical expressions describing the time and voltage dependence of the ionic currents have been developed on the basis of published voltage-clamp data recorded predominantly from human atrial myocytes. (See "Glossary" after Appendix 2 for terms used in text, figures, and tables.)

Na⁺ Current: I_Na
Voltage-clamp data for I_Na in human atrial and ventricular myocytes, recorded at 17°C have been published by Sakakibara and colleagues.^{15 16} These data suggest that the activation threshold is very close to the resting potential (-75 mV), which seems unrealistic, given that atrial and ventricular cells exhibit stable resting potentials and thresholds for activation near -55 mV. Moreover, the steady-state inactivation curves measured by Sakakibara and colleagues^{15 16} are such that I_Na would be completely inactivated (ie, no current available) at the resting potential. As pointed out by these authors, this is probably a result of time- and/or temperature-dependent shifts in the steady-state inactivation characteristics. Other results, such as the data from rabbit atrium published by Wendt et al¹⁷ (on which the I_Na description in the LMCG model is based) yield more positive (depolarized) steady-state activation and inactivation curves.

In developing a model of I_Na under physiological conditions, we have found it necessary to use indirect information about I_Na provided by action potential data in addition to voltage-clamp data. Thus, we have adjusted the steady-state activation curve (³) for I_Na so that the threshold at which an action potential is elicited agrees with experimental observations. Furthermore, the peak magnitude of I_Na was adjusted to match the maximum upstroke velocity of the action potential. (For a discussion of the relation between I_Na and action potential upstroke velocity, refer to Cohen and Strichartz.¹⁸ ) Fig 2A shows the steady-state activation (³) and inactivation () curves used to model I_Na. Compared with the data from human atrial myocytes obtained by Sakakibara et al¹⁵ (Fig 2A), there are significant positive shifts in both the ³ (+22.8 mV) and the (+32.2 mV) curves. Fig 2B shows a simulated peak current-voltage relationship (steps from a holding potential of -80 mV) for I_Na. The mathematical expressions for the kinetics of activation (_m) and inactivation (_h1 and _h2) are very similar to those of the LMCG model, as shown in Figs 2C and 2D. The processes of inactivation and recovery from inactivation are both described by the sum of a fast and a slow exponential. At plateau potentials, the fast component of inactivation has a time constant of 0.3 ms. The slow component of inactivation accounts for 10% of the total current and has a time constant of 3.0 ms at plateau potentials; ie, it is 10 times slower than the fast component, in agreement with the data of Sakakibara et al.¹⁵

View larger version (23K): [in this window] [in a new window]   Figure 2. Parameters for INa. A, Steady-state activation (3) and inactivation () ( and represent data from Sakakibara et al15 and dotted lines the expressions used in the LMCG model13 ). B, Simulated peak current-voltage relationship. C, Activation time constant (m). D, Inactivation time constants (h1 and h2).

L-Type Ca²⁺ Current: I_Ca,L
Several quite comprehensive studies of I_Ca,L in human atrial myocytes have been published.^{19 20 21 22 23 24 25} Overall, the results are quite consistent: I_Ca,L has an activation threshold near -40 mV, its peak at 0 mV, and an apparent reversal potential of +50 to +60 mV. The current density for I_Ca,L varies considerably among these studies, however, and is also known to be reduced in diseased or dilated cells.^{22 26} Table 1 shows published I_Ca,L densities compared with those used in the present model. The inactivation process for I_Ca,L is usually described as the sum of a fast and a slow component.^{19 20 21 22 23} In order to accurately measure the steady-state voltage dependence of inactivation, the duration of the "conditioning" prepulse in the voltage-clamp protocol (eg, see Li and Nattel²⁵ ) must be at least four to five times the time constant of the slower component of inactivation. If the prepulse duration is shorter, the slower component will not have reached its steady state when the availability of current is measured (second voltage-clamp pulse), and inactivation will appear incomplete. Ouadid et al²⁴ measured inactivation at room temperature (where the slower time constant exceeds 100 ms²³) and found that inactivation was incomplete (U-shaped inactivation curve) when the prepulse duration was 150 ms but that it became more complete if the prepulse duration was increased to 400 ms. The data obtained by Li and Nattel at 36°C yield almost complete inactivation characteristics for a prepulse duration of 150 ms, consistent with faster kinetics at this temperature. However, these authors also note that prolonging the prepulse results in more complete inactivation curves. On the basis of these results, we have modeled inactivation of I_Ca,L as a process involving two components with different time constants but identical, fully inactivating, steady-state voltage dependence. This formulation differs from the LMCG model¹³ as well as the Luo-Rudy model,⁹ which both use a single (voltage-dependent) component of inactivation with incomplete steady-state voltage dependence.

Ouadid et al²⁴ also report that the inactivation process is slowed considerably when Ba²⁺ is substituted for Ca²⁺ as charge carrier, indicating that inactivation is also Ca²⁺ dependent. This phenomenon is included in the Luo-Rudy model⁹ but not in the LMCG model.¹³ Our formulation differs from that of Luo and Rudy, however, in order to incorporate recent experimental results. First, there is now evidence that there exists a small restricted subsarcolemmal domain between the L-type Ca²⁺ channels and the peripheral junctional sarcoplasmic reticulum (SR) where Ca²⁺ concentration ([Ca²⁺]_d) may transiently reach much higher levels than in the cytosol as a whole.²⁷ This subsarcolemmal domain is not included in the Luo-Rudy model, which models Ca²⁺-dependent inactivation as a function of total cytosolic Ca²⁺ concentration. We have included such a subsarcolemmal domain and have modeled Ca²⁺-dependent inactivation as a function of [Ca²⁺]_d. Second, there is recent evidence to suggest that [Ca²⁺]_d modulates I_Ca,L inactivation by promoting a rapid mode of inactivation.^{28 29} Therefore, in our model, [Ca²⁺]_d determines the fraction of L-type channels that are in the rapidly inactivating mode, ie, the ratio of the fast to slow components of inactivation discussed above. Thus, in this model, an experiment with Ba²⁺ as charge carrier shifted this equilibrium so that all L-type channels were in the slow mode. With Ca²⁺ as charge carrier, however, the equilibrium is shifted toward the faster mode of inactivation by an amount determined by [Ca²⁺]_d. Hence, the inactivation of the total I_Ca,L follows a biexponential time course, where the relative contributions of the fast and slow exponentials are determined by [Ca²⁺]_d. We have chosen to model this [Ca²⁺]_d dependence as an instantaneous function of [Ca²⁺]_d, assuming that the shift between the two modes is rapid compared with the diffusion of Ca²⁺ out of the restricted domain.

Fig 3A shows the steady-state activation (d_L) and inactivation (f_L) curves used to model I_Ca,L. Assuming that activation can be measured more accurately at room temperature than at physiological temperature, we have used the activation curve (d_L) measured by Mewes and Ravens.²³ However, in order to fit the voltage-clamp data (peak currents) of Li and Nattel²⁵ (Fig 3B), we found it necessary to shift this activation curve by +3 mV. The expression for the inactivation curve (f_L) is identical to that reported by Li and Nattel for human atrial myocytes at physiological temperature. Furthermore, the reversal potential for I_Ca,L is set to a constant value of +60.0 mV, as measured by Li and Nattel, rather than to the Nernst potential for Ca²⁺ ions. As shown in Fig 3B, the simulated peak current-voltage relationship (steps from a holding potential of -80 mV) agrees well with voltage-clamp data. Figs 3C and 3D show the time constants of activation (_dL) and inactivation (_fL1 and _fL2) plotted against membrane voltage. We have used the inactivation and recovery time constant data obtained by Li and Nattel to formulate the expressions for _fL1 and _fL2 (see Fig 3D). These data were obtained in human atrial myocytes at physiological temperature. The expression for _dL (Fig 3C) is similar to that of the LMCG model.

View larger version (22K): [in this window] [in a new window]   Figure 3. Parameters for ICa,L. A, Steady-state activation (L) and inactivation (L) curves. B, Simulated peak current-voltage relationship, normalized to peak current ( represents data from Li and Nattel25 ). C, Activation time constant (dL). D, Inactivation time constants (fL1 and fL2), with data from Li et al25 (, recovery; , inactivation).

Transient and Sustained Outward K⁺ Currents: I_t and I_sus
Voltage-clamp experiments designed to study the outward currents responsible for repolarization in human atrial myocytes have identified a transient outward K⁺ current (denoted I_t), which activates rapidly on depolarization.^{1 30} In addition to this (Ca²⁺-independent) transient K⁺ current, a Ca²⁺-dependent transient outward current, which is activated by relatively large increases in [Ca²⁺]_i, is sometimes observed.³¹ We have chosen not to include this Ca²⁺-dependent current, since it has never been observed in our experimental work. After I_t has decayed (inactivated), a more slowly inactivating, "sustained," outward K⁺ current (denoted I_sus) is observed.^{2 22 32 33} The available data suggest that I_sus is a separate current from I_t and that it is also carried mainly by K⁺ ions.^{2 34 35}

The literature regarding the voltage dependence of I_t is somewhat inconsistent. In particular, the results regarding steady-state activation vary considerably. Shibata et al¹ report a half-activation voltage (V_1/2) of +1.0 mV for I_t in human atrium, whereas Näbauer et al,³⁶ Wettwer et al³³ (both in studies of human ventricle), and Le Grand et al²² (human atrium) report values of +16.7, +20.6, and +33.3 mV, respectively. These results seem unrealistic, since I_t is known to have a strong influence on early repolarization,^{1 2} and these values of V_1/2 would result in only a small amount of I_t current being activated during a normal action potential (peak at +20 to +30 mV). Among the possible explanations for this variability in V_1/2 are the sensitivity of this parameter to the concentration of divalent cations (Cd²⁺ and Co²⁺) used to block I_Ca,L^{33 37} and the possibility of a difference between human atrial and ventricular I_t. Therefore, we have based our model on the data of Shibata et al,¹ which were recorded from human atrium in the presence of a low (100 µmol/L) [Cd²⁺]. The data regarding the voltage dependence of steady-state inactivation are also variable between different studies. Fig 4A shows the steady-state activation () and inactivation () curves used to model I_t. The curve is based on a fit to data from Shibata et al ( in Fig 4A), and the curve is that reported by Firek and Giles.² Time constants of activation (_r) and inactivation (_s) are plotted against voltage in Fig 4C and 4D, respectively. Data regarding the time constant of inactivation² indicate that _s is 13 ms (at 33°C) at membrane voltages positive to 0 mV. At hyperpolarized potentials, the recovery of I_t from inactivation as measured in our laboratory appears to follow an exponential time course with a strongly voltage-dependent time constant, increasing from 15 ms at -100 mV to 387 ms at -60 mV. Our formulation for _s is based on a fit to experimentally obtained recovery time constant values at negative potentials and inactivation time constant values at positive potentials. The recovery of I_t from inactivation in human atrial cells has been shown to be considerably more rapid than in rabbit atrial cells.^{2 38} As a result, I_t magnitude and action potential waveshape are much less rate dependent in human atrial cells.

View larger version (23K): [in this window] [in a new window]   Figure 4. Parameters for It and Isus. A, Steady-state activation and inactivation curves for It ( and , respectively) ( represents data from Shibata et al1 ). B, Steady-state activation and inactivation curves for Isus (sus and sus, respectively) ( represents data from Wang et al35 ). C, Activation and inactivation time constants for It (r and s, respectively). D, Activation and inactivation time constants for Isus (rsus and ssus, respectively) ( represents data from Wang et al35 ).

Our model of I_sus is based on the data of Wang et al.³⁵ Fig 4B and 4D show the steady-state activation curve (r_sus) and time constant of activation (_rsus) for this current. On the basis of our recent data, we have also included a slow (300.0 ms) partial (40%) inactivation. The time constant for this inactivation process was obtained by fitting a biexponential function to the decaying outward current waveforms as described by Koidl et al.³⁹ In addition to providing an estimate of _ssus, this method also provides a more accurate separation of I_t and I_sus than when I_sus is estimated as the current at the end of the pulse.^{2 35 40} By combining our models of I_t and I_sus, we are able to produce current waveforms closely resembling those recorded from human atrial myocytes in response to voltage-clamp pulses. Fig 5 shows how I_t (panel A) and I_sus (panel B) combine (panel C) to produce waveforms similar to experimental results (panel D). In our experience, the size of these currents varies considerably between individual cells, and as reported by Amos et al,⁴⁰ there is also considerable variability in the ratio of I_t to I_sus. The sizes of I_t and I_sus in the model were chosen to provide good fits to action potential data and are well within this experimental variability. Table 2 compares I_t and I_sus densities reported in the literature with those used in this model.

View larger version (45K): [in this window] [in a new window]   Figure 5. It and Isus in response to the voltage-clamp protocol shown in the inset, normalized to cell capacitance. A, Simulated It waveforms. B, Simulated Isus waveforms. C, Sum of the It and Isus waveforms in panels A and B. D, Outward currents recorded from a human atrial myocyte.

Delayed Rectifier K⁺ Currents: I_K,r and I_K,s
Recent studies of the delayed rectifier K⁺ currents in human and rabbit atrial myocytes show that in both species the delayed rectifier current is generated by two distinct K⁺ conductances. These "rapid" (I_K,r) and "slow" (I_K,s) conductances have significantly different properties and can be separated experimentally on the basis of, for example, the sensitivity of I_K,r, but not I_K,s, to the antiarrhythmic drug E-4031.^{41 42 43} Since I_K,s is believed to contribute only a small fraction of the total delayed rectifier current during a normal atrial action potential,⁴¹ one could produce an acceptable fit to nominal action potential data using a model incorporating only I_K,r. However, because of the very slow activation characteristics of I_K,s, this current would be expected to be more significant at high heart rates, where it could build up progressively from cycle to cycle as a result of residual activation, ie, failure to decay completely between cycles. Therefore, we have chosen to include both I_K,r and I_K,s in our model, thus enabling it to simulate specific effects of antiarrhythmic drugs on I_K,r, the buildup of I_K,s at elevated heart rates, and the resulting changes, such as those in action potential duration (APD) and refractory period.

Our model of I_K,r is based on data from Wang et al,⁴⁴ Muraki et al,⁴¹ and Sanguinetti et al.⁴⁵ Figs 6A and 6C show the steady-state activation (_a) and inactivation (_i) curves and the time constant of activation (_pa) compared with the available data. Since the inactivation of I_K,r is very rapid compared with the activation, inactivation is modeled as being instantaneous. The expressions for I_K,s are based on data recorded in human atrial myocytes by Wang and colleagues.^{42 44} Figs 6B and 6D show the characteristics of this current.

View larger version (24K): [in this window] [in a new window]   Figure 6. Parameters for IK,r and IK,s. A, Steady-state activation and inactivation curves for IK,r (a and i, respectively). Data are from Wang et al44 (), Muraki et al41 (), and Sanguinetti et al45 (). B, Steady-state activation curve for IK,s (). Data are from Wang et al44 (). C, Activation time constant for IK,r (pa). Data are from Wang et al44 () and Muraki et al41 (). D, Activation time constant for IK,s (n). Data are from Wang et al44 ().

In order to verify our model of I_K,r we chose to simulate the "ramp clamp" experiment of Muraki et al⁴¹ in rabbit atrial cells (their Fig 7), in which the cell is subjected to an 0.8-V/s repolarizing ramp from an action potential peak potential down to -80 mV. This waveform is an approximation to the repolarization phase of an atrial action potential. The result (simulated response) is shown in Fig 7, along with data from Muraki et al. Note that the simulated waveform is very similar, although in order to fit action potential data, we have had to reduce the size of I_K,r to 15% of that shown in Fig 7. Given that it is often difficult to detect any delayed rectifier current at all in most human atrial myocytes,² it seems reasonable that this current should be assigned a very low density.

View larger version (14K): [in this window] [in a new window]   Figure 7. Current-voltage relationship for IK1 and IK,r normalized to peak IK1 (0.6 pA/pF) ( and represent data from Muraki et al41 ). The IK,r waveform is the response to an 0.8-V/s "ramp" repolarization from an action potential peak potential down to -80 mV.41

Time-Independent Currents: I_K1, I_B,Na, I_B,Ca, I_NaK, I_NaCa, and I_CaP
In the absence of reliable published data from human atrial cells for these currents, we have used expressions from the LMCG model¹³ with only minor scaling adjustments. One exception is the inward rectifier current, where we have found it necessary to make minor modifications to the rectifying characteristics in order to fit action potential data (slightly narrower outward "hump"). We have also adjusted the [K⁺]_c dependence of I_K1 to agree with data from our laboratory. Table 3 lists the changes made in the time-independent currents compared with those in the LMCG model.

Material Balance
As in the LMCG model,¹³ we have included a fluid compartment formulation to monitor and account for changes in ion concentrations. These concentration changes can be a result of current flow across the cell membrane or of redistribution of ions within the cell (eg, uptake of Ca²⁺ by the SR or binding of Ca²⁺ to an intracellular buffer). Our fluid compartment model is similar to the one in the LMCG model. It includes descriptions of extracellular and intracellular spaces, formulations for Ca²⁺ uptake and release and the buffering action of calsequestrin, and troponin and calmodulin buffers in the intracellular medium. Compartment volumes and other ultrastructural properties, as well as expressions describing the binding of Ca²⁺ to intracellular troponin and calmodulin buffers and to calsequestrin in the SR release compartment, are identical to those of the LMCG model, except as noted in the following sections.

Cleft Space
We have included a "cleft space" in our fluid compartment formulation, ie, a small restricted space surrounding the cell, in which accumulation or depletion of ions may occur (see Demir et al¹² ). The cleft space is modeled as an unstirred fluid layer; ie, ions can be exchanged between the cleft space and the extracellular medium (in which all concentrations are assumed constant) only through diffusion as a result of a concentration gradient. Ratios between diffusion time constants for the ions involved (_Na, _K, and _Ca) were calculated from values for ionic conductivity⁴⁶ and the composition of the extracellular solution ([Cl^-]_o=140 mmol/L, [Na⁺]_o=130 mmol/L, [K⁺]_o=5.4 mmol/L, and [Ca²⁺]_o=1.8 mmol/L). We have adjusted the size and diffusion properties of the cleft space so as to produce oscillations in cleft space [K⁺] ([K⁺]_c) similar to experimental data.^{47 48}

Electroneutral Na⁺ Influx
In order to achieve long-term stability in the ionic concentrations in the model, we have found it necessary to add a small (1.68-pA) electroneutral inward flux of Na⁺, denoted _Na,en. This flux could, for example, be accounted for in terms of electroneutral coupled transport mechanisms, such as Na⁺-H⁺ exchange and Na⁺-K⁺-2Cl cotransport. Modeling of these mechanisms, however, is beyond the scope of this work.

There are two major reasons for including this flux: First, the fact that long-term ionic homeostasis can be achieved with the addition of this small flux demonstrates that the sizes and other characteristics of the model elements are such that ionic homeostasis can reasonably be maintained. Second, if the ionic concentrations were allowed to change slowly from cycle to cycle (which would be the result if this flux were not included), the model would only be valid for short simulation times (seconds), for which this drift can safely be ignored. By ensuring long-term stability of the ionic concentrations, longer simulation times (minutes) become feasible and meaningful. Only with stable ionic concentrations can the model be used to simulate concentration changes as a result of rate changes or other interventions.

Sarcoplasmic Reticulum
Our formulation for the SR is very similar to that of the LMCG model. However, we have made one important modification in accordance with recent evidence demonstrating that Ca²⁺ can accumulate in a small domain between the sarcolemma and the peripheral junctional SR and trigger Ca²⁺ release.^{27 49} Specifically, we have removed the voltage-dependent term in the formulation for activation of SR Ca²⁺ release and replaced it with a term dependent on Ca²⁺ concentration in the restricted subsarcolemmal domain, [Ca²⁺]_d. The sole mechanism for SR Ca²⁺ release in our model is therefore Ca²⁺-induced Ca²⁺ release (CICR). Fig 1B includes a schematic representation of the model of the SR.

Stern⁵⁰ has shown that in order for a CICR model to be stable, ie, capable of producing a response that is graded by the amount of Ca²⁺ that enters the cell through I_Ca,L, the trigger Ca²⁺ has to be separated from that released from the SR. Anatomically, this can be understood in terms of the concept of "release units" discussed, for example, by Isenberg and Han.⁵¹ According to this concept, Ca²⁺ release from the SR is recruited stepwise by the all-or-none activation of individual release units, consisting of one or more L-type Ca²⁺ channel and associated SR release channels. The activation of a release unit results in CICR within that unit only. The released Ca²⁺ then diffuses into the myoplasm, without directly affecting other units. This phenomenon has been incorporated in our model as a lumped mechanism, where the SR release channel senses [Ca²⁺] in the restricted domain ([Ca²⁺]_d) but releases Ca²⁺ directly to the cytosol (Fig 1B), thus separating trigger Ca²⁺ from that released from the SR.

Parameter Values
A model of this type contains a large number of parameters that must be assigned values based on the available data. We have approached this part of the model development process in a two-step fashion, where the majority of the parameter values have been assigned in the first step, based on experimental studies of individual model components. This has the advantage that most of the parameters associated with an individual membrane current can be justified and assigned independently. Once the descriptions of individual membrane currents has been completed, one is left with a limited number of free parameters, most of which are scaling factors, such as the maximum conductance values for each ionic current. These remaining free parameters can then be determined using data for whole-cell responses (eg, action potential waveforms) or other constraints as indicated previously (eg, ionic homeostasis). It should be emphasized, however, that model development is very much an iterative process and that it has been necessary in some cases to modify individual current expressions to obtain acceptable fits to action potential data or (as in the case of I_Na) to use information from action potential recordings to resolve ambiguities in ion channel current data. The following sections describe the constraints and criteria used in order to assign values for the remaining free parameters.

The Quiescent Human Atrial Myocyte
In the absence of an external stimulus, a healthy human atrial cell is quiescent (does not contract or produce an action potential). In this quiescent state, the membrane potential comes to an equilibrium, "resting," potential at which the net ion flux across the sarcolemma is zero. The resting membrane potential varies somewhat among individual cells, ranging from -70 to -80 mV. The resting membrane potential is the result of a precise balance between the time-independent background, pump, and exchanger currents (I_K1, I_B,Na, I_B,Ca, I_NaK, I_CaP, and I_NaCa). Although the resting state of the cell may seem less interesting than the active state during an action potential, an accurate description of the resting conditions is, in our experience, essential for successful modeling of the action potential. Moreover, the model of the resting state of the cell determines very important threshold characteristics and subthreshold properties, such as the input resistance of the cell. Very accurate simulation of these passive characteristics is essential before the cell model is to be used in distributed simulations, eg, of the propagation of electrical activity from one cell to another. We have modeled the resting state of the cell by adjusting the magnitudes of these currents so as to produce zero net transmembrane current at a resting potential of 75 mV and the following intracellular ion concentrations: [Na⁺]_i8.5 mmol/L; [K⁺]_i130.0 mmol/L, and [Ca²⁺]_i60.0 nmol/L. The input resistance of the cell in this quiescent state is 600 M, which agrees with our experimentally observed values.

The Activated Human Atrial Myocyte
When the cell is stimulated, it produces an action potential, the shape of which depends on the relative sizes of the ionic currents involved. Action potential data recorded from isolated human atrial myocytes are (in our experience) quite variable from cell to cell. It is therefore questionable whether it is possible to define the "normal human atrial action potential" in a meaningful way. We have chosen to fit our model to an action potential waveform that is representative of what is most often recorded from isolated human atrial cells in our laboratory. By establishing this "nominal model," we have obtained a starting point from which the sensitivity of the action potential waveform to parameter perturbations may be studied (see "Results"). In addition to dictating the action potential waveform, the sizes (maximum conductance parameters) of the ionic currents also affect ionic homeostasis. The membrane currents involved in shaping the action potential therefore have to act in concert with those involved in the resting state to maintain constant ion concentrations at nominal stimulation rates. We have "tuned" our model so that ion concentrations remain constant from cycle to cycle at a stimulus frequency of 1 Hz.

	Results

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 Appendix 2 References

 
There are two major aims of the simulations presented in the following sections: (1) to establish the validity and usefulness of our model by demonstrating that the expressions that are based on fits to voltage-clamp data for individual ionic currents also are able to accurately reconstruct action potential data and (2) to use the model to investigate the functional roles of different ionic currents, to study the sensitivity of the action potential waveform to the sizes of those currents, and, finally, to predict the whole-cell response to, for example, selected channel blocking drugs.

Simulated Action Potential Waveform
Fig 8A shows a simulated action potential waveform (solid line) compared with an action potential (dotted line) recorded at a temperature of 33°C and a stimulus frequency of 0.5 Hz. There is close agreement between the waveforms (the discrepancy at the beginning of the upstroke is due to a stimulation artifact that is not simulated). As mentioned previously, the action potential waveform varies significantly among individual cells and in multicellular preparations from the human atrium, presumably because of variations in the magnitudes of the underlying ionic currents. Nevertheless, the ability of this model to accurately reproduce a recorded action potential, in combination with the previously demonstrated fits to voltage-clamp data, lends credibility to the model.

View larger version (28K): [in this window] [in a new window]   Figure 8. Model-generated action potential and membrane current waveforms. A, Model-generated action potential (solid line) compared with recorded data (dotted line). B and C, Model-generated waveforms for the ionic currents during the action potential (note the difference in scale between panels B and C).

In Fig 8, panels B and C show the behavior of the membrane currents during an action potential. The first current to respond to a depolarizing stimulus pulse (delivered at time=100 ms) is I_Na, which activates rapidly, resulting in a very large but transient inward current. Note that I_Na is too large to be shown on the scale of Fig 8B; its peak magnitude is -5.8 nA, which corresponds to a maximum upstroke velocity of 116 V/s. I_Na is primarily responsible for the upstroke (phase 0) of the action potential, but as seen from Fig 8B, a substantial amount of I_Na remains during the early peak phase of the action potential as a result of the second slower component of I_Na inactivation.

On depolarization of the cell, I_t, I_sus, and I_Ca,L are also activated. However, I_t and I_sus reach their peak magnitude faster than I_Ca,L, and their combined magnitude is thus larger than that of I_Ca,L early in the action potential. (This is because the peak of the action potential is close to the reversal potential for I_Ca,L.) The initial result on the action potential waveform is therefore a period of relatively rapid repolarization (phase 1), dominated by I_t. Since the time course of inactivation of I_Ca,L is slow compared with that of I_t, the net current gradually becomes dominated by I_Ca,L and I_sus. A situation where the repolarizing effect of I_sus (and the remaining I_t) is balanced by the depolarizing effect of I_Ca,L results. In the action potential waveform, the initial rapid repolarization (phase 1) is followed by a period during which the membrane potential levels off, forming a plateau (phase 2). Finally, as I_Ca,L slowly inactivates, the repolarizing effects of I_sus become dominant, and the action potential enters its final repolarization phase (phase 3). During this phase, I_sus is aided by the inward rectifier current, I_K1, and the delayed rectifier currents, I_K,r and I_K,s, in repolarizing the cell membrane back to the resting potential (phase 4).

Simulated Ionic Fluxes
The fluid compartment part of this model monitors ion concentrations in the intracellular and cleft spaces. Valid modeling of the action potential requires not only the reconstruction of the action potential waveform but also a demonstration that this can be accomplished under conditions of ionic homeostasis at nominal heart rates. Table 4 shows how our model has been tuned to achieve homeostasis at 1 Hz. The average charge transported across the sarcolemma for each ionic current has been computed by integrating each current over one cycle (1 s in the case of a quiescent, nonstimulated cell). Note that the sums of these average charges are zero for all ionic species at a stimulus rate of 1 Hz (Table 4). When the cell is quiescent, there is a small net loss of intracellular Na⁺ and gain of intracellular K⁺. The existence of such an ionic imbalance at quiescence is supported by the observation by Bénardeau et al⁵² that trains of depolarizing pulses that activate I_Na can be used to hyperpolarize the resting potential of human atrial cells after a period of quiescence. As suggested by these authors, the hyperpolarization and stabilization of the resting potential may be caused by activation of the Na⁺-K⁺ pump after Na⁺ entry during the train of pulses. In our model, at a 2-Hz stimulus rate there will be a net gain of intracellular Na⁺ and loss of K⁺.

Fig 9 illustrates the Ca²⁺ handling in the fluid compartment part of the model during the simulated action potential. A transient increase in [Ca²⁺]_i occurs early in the action potential, raising [Ca²⁺]_i from the very low diastolic levels (65 nmol/L) to a peak of 1.3 µmol/L (Fig 9B). This rise in [Ca²⁺]_i is primarily due to the rapid release of large amounts of Ca²⁺ from the SR (see Fig 9D). Several processes are responsible for the decline of the intracellular Ca²⁺ transient. The most potent of these is the rapid binding of Ca²⁺ to the intracellular Ca²⁺ buffers (troponin and calmodulin), which is particularly important in shaping the early portions of the Ca²⁺ transient. As seen in Fig 9C, the occupancies on these buffers increase rapidly as Ca²⁺ is released from the SR, thus "removing" large amounts of free Ca²⁺ from the cytosol. The uptake of Ca²⁺ by the SR also has a pronounced effect on the shape of the Ca²⁺ transient. This is also the primary pathway for actual removal of Ca²⁺ from the cytosol (as opposed to the "temporary storage" provided by the buffers), taking up intracellular Ca²⁺ as it dissociates from the buffers. In addition, some Ca²⁺ is removed from the cytosol via the Na⁺-Ca²⁺ exchanger, I_NaCa, and the Ca²⁺ pump, I_CaP. As seen in Table 4, I_NaCa and I_CaP, on average, remove the amounts of Ca²⁺ that were brought into the cell via I_Ca,L and I_B,Ca and thereby prevent a progressive buildup of cytosolic Ca²⁺ during repetitive stimulation.

View larger version (31K): [in this window] [in a new window]   Figure 9. Simulated waveforms for the fluid compartment model. A, Action potential waveform. B, Intracellular Ca2+ concentration, [Ca2+]i. C, Occupancies of the troponin (OTC and OTMgC), calmodulin (OC), and calsequestrin (OCalse) buffers. D, Sarcoplasmic reticulum release and uptake currents (Irel and Iup, respectively). E, Ca2+ concentrations in the sarcoplasmic reticulum release and uptake compartments ([Ca2+]rel and [Ca2+]up, respectively).

Increasing the stimulus rate from the baseline (1 Hz) results in a change in ion concentrations in the intracellular medium (Table 4) as well as in the cleft space surrounding the cell. For the ionic species that exist in relatively low concentrations in the extracellular medium (Ca²⁺ and K⁺), these changes can be significant. Fig 10 shows how the intracellular and cleft space concentrations change when the stimulus rate is increased abruptly from 1 to 2 Hz. Note the progressive shift in [K⁺]_c of 1 mmol/L.^{47 48} In contrast, the change in [Na⁺]_c is negligible because of its high baseline value of 130 mmol/L. If the simulation in Fig 10 is continued beyond the 20 s shown, [K⁺]_c will reach a peak value of 6.3 mmol/L, after which it will begin to decline as a result of increased I_NaK activity due to increased [Na⁺]_i and [K⁺]_c (simulation not shown). The asymptotic values for [K⁺]_c and [Na⁺]_i are 5.6 and 10.0 mmol/L, respectively (reached after 10 minutes). This behavior is consistent with experimental observations.⁴⁷

View larger version (43K): [in this window] [in a new window]   Figure 10. Ion concentrations in the intracellular and cleft spaces when the stimulus rate is increased from 1 to 2 Hz. A, Action potential waveform. B, Cleft space K+ concentration, [K+]c. C, Cleft space Ca2+ concentration, [Ca2+]c. D, Intracellular Na+ concentration, [Na+]i. E, Intracellular K+ and cleft space Na+ concentrations, [K+]i and [Na+]c.

Parameter Sensitivity of the Action Potential Shape
As mentioned in the previous section, there is considerable variation in action potential shape among individual cells from the human atrium. Our working hypothesis is that many of these differences in action potential waveshape can be explained in terms of differences in the magnitudes of the ionic currents (caused by previous drug treatment and/or natural variability). In order to investigate possible mechanisms of action potential shape variability, it is therefore necessary to have an understanding of how changes in the magnitudes of different ionic currents affect the action potential shape. Such an understanding is equally important, of course, in identifying suitable "targets" for drug action aimed at modifying the action potential shape.

A valid mathematical model provides a method for this sensitivity analysis⁵³ ; ie, it provides a method for studying how sensitive the state variables (eg, membrane voltage) are to perturbations in model parameters. We will restrict this analysis to a study of the sensitivity of the action potential waveform to changes in the sizes (maximum conductances) of the currents involved in shaping the action potential, although this type of analysis in principle can be used to study the sensitivity of any state variable to perturbations in any model parameter. Briefly, sensitivity analysis involves the computation of the partial derivative of the state variable of interest (in this case, membrane voltage) with respect to a model parameter. We have chosen to normalize these partial derivatives with respect to the nominal value of each parameter. Thus, we will present the results in terms of sensitivity functions, defined as:

which can be interpreted as a proportionality factor relating a relative change in a model parameter to a resulting change in membrane voltage. The computational aspects of this method are outlined in Appendix B (see also Paulsen et al⁵³ ).

Fig 11 shows the result of the sensitivity analysis, ie, the sensitivity functions for the membrane voltage (action potential) with respect to the parameters of interest. Sensitivity functions were computed for all maximum conductance parameters in the model as well as for the scaling parameters for I_NaK and I_NaCa. The sensitivity functions for I_Na and I_K,s have been omitted from Fig 11, since these currents were found to have negligible influence on the action potential shape (small sensitivity functions), except for the obvious importance of I_Na during the upstroke. Several of the sensitivity functions have maximum absolute values of 50 mV during the late repolarization phase of the action potential. In other words, a 10% change in either one of these parameters would alter the membrane voltage by 5 mV during this phase of the action potential. Although this estimate is based on a linearization around the nominal parameter value and thus is most accurate for small perturbations, it can provide an indication of the approximate change expected for larger perturbations. Overall, we can anticipate that changes in these parameters in the ±50% range will produce significant changes in the action potential waveform.

View larger version (29K): [in this window] [in a new window]   Figure 11. Results of the sensitivity analysis. A, Nominal action potential waveform. B, Sensitivity functions (for the action potential waveform) with respect to scaling factors for outward currents (t, sus, K1, NaK, and K,r). C, Sensitivity functions with respect to scaling factors for inward currents (Ca,L, B,Na, B,Ca, and kNaCa).

Perhaps more important than the absolute sensitivity values, however, is the information provided by the time course of the sensitivity functions. As the ionic conductances change during the action potential, the sensitivity functions indicate which currents have the greatest influence on the action potential shape at each point in time. For example, it is clear that the action potential waveform is sensitive to _t primarily early in the peak phase of the action potential but that _sus and _Ca,L rapidly become more important early in the plateau phase. Throughout the plateau phase, |_sus| (absolute value) and |_Ca,L| are larger than |_t|, and this portion of the action potential waveform is therefore particularly sensitive to perturbations in _sus and _Ca,L. Toward the end of repolarization, the action potential becomes very sensitive to the size of the time-independent currents involved in maintaining the resting potential. This analysis shows that (under the hypothesis that action potential changes can be explained in terms of changes in the magnitudes of ionic currents) _t, _sus, and _Ca,L are the most important model parameters in determining the action potential waveshape during the peak and plateau phases of the action potential.

Roles of I_t and I_sus in Repolarization
The effects on the human atrial action potential of blocking I_t with agents such as 4-aminopyridine, flecainide, and quinidine have been described in the literature.^{2 54 55 56} Partial block of I_t results in a slowing of the rate of repolarization of the action potential, particularly during the early repolarization phase (phase 1). This is, of course, consistent with the characteristics of I_t, and its role in the generation of the action potential as indicated by sensitivity analysis. In a recent study of the effects of some antiarrhythmic agents on I_t and I_sus, Wang et al⁵⁶ found that quinidine, in addition to blocking I_t, has a pronounced effect on I_sus at clinically relevant concentrations. Considering that our sensitivity analysis indicates a prominent role for I_sus in repolarization, the APD is expected to be quite sensitive to modulation of I_sus magnitude. A detailed account of the role of I_t and I_sus in the repolarization of the human atrial action potential is therefore essential for understanding the antiarrhythmic actions of quinidine.

Since pharmacological blocking agents used in experimental work usually affect more than one current and since their effects are often rate dependent, it is difficult to gain a quantitative understanding of the importance of a particular current (I_t or I_sus) in repolarization from experimental results alone. In a computer model, however, it is possible to alter the characteristics of one ionic current in a controlled fashion, while leaving all other currents unaffected. Such simulations can be a valuable complement to experimental work. Given that drugs that prolong the APD (class III drugs) have been shown to be effective in the treatment of atrial arrhythmias,⁵⁷ a thorough understanding of the influence of different ionic currents on the APD is needed.

Fig 12A shows the effects on the action potential of various degrees (30%, 60%, and 90%) of block of I_t. (In the present study, an x% block of I_t is simulated as an x% reduction in the maximum conductance, _t.) As observed experimentally, I_t block results in a broadening of the action potential peak during phase 1 of repolarization (refer to Fig 2 in Firek and Giles² ). In addition, because of the elevation of the action potential peak and plateau levels, the contribution of I_sus to repolarization is increased. The resulting prolongation of the APD is therefore only moderate, even for substantial reductions of I_t size. Fig 12B shows the effects of various degrees (15%, 30%, and 45%) of block of I_sus on the action potential. In contrast to I_t block, inhibition of I_sus primarily affects the plateau phase of the action potential, with little or no effect on the action potential peak. As a result, I_sus block produces a more pronounced prolongation of the APD than does I_t block. For example, 30% block of I_sus results in a 15% increase in APD at 90% repolarization (APD₉₀) compared with the 5% increase in APD₉₀ resulting from 30% I_t block. Many antiarrhythmic agents have effects on several ion channels. For example, according to Wang et al,⁵⁶ quinidine blocks both I_t and I_sus (in addition to its effects on Na⁺ channels). It is therefore of interest to study the effects on the action potential of combined I_t and I_sus block. Fig 12C shows the result of a simulation in which both I_t and I_sus have been reduced by 40% (approximately corresponding to the effect of 5 µmol/L quinidine at a stimulus rate of 1 Hz⁵⁶). As expected, the result is essentially a combination of the previously demonstrated effects of I_t and I_sus block, ie, a widened peak, an elevated action potential plateau, and a prolongation of APD₉₀ of 27%.

View larger version (33K): [in this window] [in a new window]   Figure 12. Effects of partial block of It and Isus. A, It block (30%, 60%, and 90%) has its main effect on the peak and produces only a moderate prolongation of the action potential. B, Isus block (15%, 30%, and 45%) has a strong effect on the action potential duration. C, Combined block (40%) of It and Isus widens the peak and prolongs the action potential, in a way similar to the effect of 4-aminopyridine (4-AP). D, Recorded action potential under control conditions and in the presence of 4-AP (data from Firek and Giles2 ).

Modulation of the Role of I_sus by Baseline I_Ca,L, I_sus, and I_K,r Sizes
As discussed in "Model Development," published data regarding the size of I_Ca,L and I_sus are quite variable. Both these currents (as well as I_t) are known to be depressed in diseased human atrial cells^{22 26} and modulated by adrenergic stimulation.^{26 58} Furthermore, a recent study⁵⁹ shows that the size of I_sus is significantly reduced in cells obtained from patients in chronic atrial fibrillation compared with patients in normal sinus rhythm. It is therefore likely that a range of sizes of these two currents contributes to the physiological (and pathophysiological) behavior of the human atrial cell. Similarly, I_K,r in our nominal model is very small, which is consistent with observations from our laboratory. Since results in other species⁶⁰ indicate that this may be a consequence of the cell isolation techniques used for human atrial myocytes,² this current may be considerably larger in vivo. Given these uncertainties in the actual sizes of several of the ionic currents, it is appropriate to investigate how the conclusions reached above are affected by our assumptions for these current sizes. We have chosen to focus on the role of I_sus in repolarization, since this current is an important determinant of APD. Starting from our nominal model, we have performed a large number of simulations for different combinations of increased/decreased I_Ca,L, I_sus, and I_K,r. All simulations were performed at a stimulation rate of 1 Hz, and 20 cycles were allowed after each change of parameter values in order for any initial transient behavior to die out before the APD prolongation was evaluated. Fig 13 shows how the APD prolongation resulting from a 50% reduction of I_sus depends on the baseline sizes of I_Ca,L and I_K,r. I_Ca,L and I_K,r sizes are expressed as percentages of those in the nominal model; ie, the nominal model corresponds to 100% of both currents. A 7-fold (700%) increase in I_K,r corresponds approximately to the size of I_K,r observed in rabbit atrial myocytes.⁴¹ It is clear from Fig 13 that the role of I_sus as a major determinant of APD depends strongly on the size of I_K,r current. The action potentials shown in the insets in Fig 13 provide an indication of the underlying mechanism. When I_sus is partially blocked, the action potential plateau is depolarized, which in turn increases the amount of I_K,r (and I_K,s) that is activated. This effect, which counteracts the APD-prolonging effect of I_sus block, becomes stronger as the size of I_K,r is increased. Similarly, Fig 14 shows how the APD prolongation as a result of 50% I_sus block depends on the baseline sizes of I_Ca,L and I_sus. Again, the APD-prolonging effect of I_sus block is strongly dependent on the baseline current densities. Generally, the APD prolongation as a result of partial I_sus block becomes larger as the baseline size of I_Ca,L increases, provided that the increase in I_Ca,L is balanced by a comparable increase in I_sus. Notice, however, in Fig 13 as well as in Fig 14, that when baseline I_Ca,L is increased without such a comparable increase in baseline outward current (I_sus or I_K,r), the APD-prolonging effect of I_sus block is instead reduced. The underlying mechanism in this case is that the action potential plateau, both before and after I_sus block, is prolonged and depolarized somewhat by the larger I_Ca,L. As a result, more I_K,r (and I_K,s) is activated, and the final repolarization phase becomes steeper and therefore less dependent on I_sus.

View larger version (62K): [in this window] [in a new window]   Figure 13. Prolongation of action potential duration at 90% repolarization (APD90) as a result of 50% reduction of Isus size, plotted as a function of the baseline ICa,L and IK,r sizes. (100% denotes the nominal sizes of ICa,L and IK,r.) Insets show examples of action potentials (before and after Isus block) corresponding to three points in the graph (middle inset is for the nominal model).

View larger version (58K): [in this window] [in a new window]   Figure 14. Prolongation of action potential duration at 90% repolarization (APD90) as a result of 50% reduction of Isus size, plotted as a function of the baseline ICa,L and Isus sizes. Insets show examples of action potentials (before and after Isus block) corresponding to three points in the graph (middle inset is for the nominal model).

	Discussion

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 Appendix 2 References

 
Mathematical models form an important complement to experimental work in attempts to elucidate the ionic mechanisms underlying the action potential and other electrophysiological phenomena in cardiac cells. At a minimum, these models can provide a means of integrating data obtained from many different experiments and laboratories so that biophysically based explanations of complex nonlinear phenomena such as action potential initiation (excitation) and repolarization can be given. Mathematical models also provide a way of reviewing data in the context of the normal behavior of a cell (during an action potential), even when some of the data may have been obtained under completely different experimental conditions, eg, a voltage-clamp experiment. Moreover, as illustrated in the last few sections of "Results," mathematical models can also have considerable predictive capabilities. After a model has been developed and carefully validated, it can be used to predict the response of the cell to selected drugs, experimental protocols, etc. Ideally, experimental work and model development should be carried out in close association, using the model to design and evaluate experiments and using experimental results to improve the model.

We have developed a mathematical model of the human atrial cell based primarily on data recorded from enzymatically isolated single human atrial cells. Our model is capable of accurately reconstructing a recorded human atrial action potential and illustrates the functional roles of the ionic currents. In addition, our model maintains ionic homeostasis at a nominal stimulus rate, demonstrating that the reconstruction of the action potential is accomplished using plausible current densities. We used the LMCG rabbit atrial model¹³ as a "starting point" for the model development. As a result, these two formulations are very similar in some aspects, particularly for the membrane currents where incomplete (or no) data from human atrial cells are available. However, there are some important differences between the electrophysiological responses of rabbit and human atrial cells; these provided the motivation for the development of this human atrial cell model. Perhaps most striking is the small rate dependence of I_t in human atrial cells compared with the very prominent rate dependence seen in rabbit atrial cells.^{13 38} This is mainly due to differences in the rates of recovery from inactivation of I_t in the two species. As discussed in "Membrane Currents," I_t in human atrial cells recovers from inactivation much more rapidly than does I_t in rabbit atrial cells. Another important difference between human and rabbit atrial cells is the sustained outward current (denoted I_sus in this article). Whereas this current in the rabbit atrium is believed to be carried mainly by Cl^- ions,¹³ there is convincing evidence that I_sus in human atrial cells is carried mainly by K⁺ ions.^{2 35} Both these differences are very important in studying the mechanisms of repolarization of the human atrial action potential and the effects of action potential–prolonging (class III) drugs that affect these currents.

Roles of I_t and I_sus in Repolarization
Both I_t and I_sus have important roles in the repolarization of the action potential of human atrium. In particular, I_sus, because of its noninactivating characteristics, is necessary for repolarization to the resting potential. This important role in repolarization makes I_t and I_sus potential targets for class III antiarrhythmic drugs, which are designed to prolong the APD. Indeed, a recent study of flecainide and quinidine,⁵⁶ both known to prolong the human atrial action potential,^{54 55} shows that both these drugs produce a partial block of I_t. Quinidine also blocks I_sus, which could explain its greater efficacy (compared with flecainide) in prolonging the APD.⁵⁵ Our simulations of partial I_t and I_sus block (see "Results") produce prolongations of the APD that are comparable to experimentally observed effects of flecainide and quinidine, ie, a 27% prolongation of APD₉₀ when I_t and I_sus are both reduced by 40%. For comparison, Wang et al⁵⁵ reported that 2.25 µmol/L quinidine increased APD₉₅ by 33% in human atrial cells at a stimulation rate of 1 Hz. It should be noted that only some of the known effects of quinidine have been modeled; therefore, our results cannot be directly compared with these experimental observations. For example, the effects of quinidine on I_Na, as well as the state dependence of I_t block by quinidine,⁶¹ would have to be included in a more comprehensive treatment of quinidine effects. Nevertheless, the experimental observations agree very well with our model predictions and provide an independent "test" of how well our model describes the roles of I_t and I_sus in repolarization.

Action potential generation involves a complex interaction among the ionic currents in a given cell type. The role of a particular ionic current in the action potential is therefore not determined solely by the characteristics of that current. We have investigated how the role of I_sus in repolarization is affected when the baseline densities of I_Ca,L, I_sus, and I_K,r are varied within ranges that are relevant to the physiological and pathophysiological behavior of the human atrial cell. Our results demonstrate that I_sus block will, in general, result in a prolongation of the action potential. The amount of prolongation, however, depends quite strongly on the baseline current densities. If the human atrial cell is assumed to have an I_K,r density comparable to that observed in the rabbit atrium⁴¹ or if the I_Ca,L and I_sus densities are reduced as observed in diseased cells,^{22 26} the APD prolongation resulting from I_sus block may be considerably smaller than indicated by our nominal model. The efficacy of a drug targeting I_sus would therefore be expected to depend critically on the disease state of the tissue. For example, based on the recent observation by Van Wagoner et al⁵⁹ that I_sus density is reduced in cells obtained from patients in chronic atrial fibrillation, the efficacy of an I_sus-blocking drug may be limited in these patients.

Limitations of the Model
When using our model to gain insight into the electrophysiological responses of the human atrial cell, it is important to be aware of certain limitations, which are summarized by the following items:

1. The Hodgkin-Huxley formalism along with its concept of independent activation and inactivation "gating" variables has some important limitations. Notably, the processes of inactivation and recovery from inactivation (and analogously activation and deactivation) are governed by a single time constant. Experimental observations, however, often indicate that inactivation and recovery from inactivation occur with different time constants, even at the same membrane potential. In order to overcome this problem, one would have to use a more complicated modeling formalism that treats inactivation and recovery as two separate processes. To reduce the computational requirements of the model, we have chosen a "compromise" solution, in which time constant values are determined by measured inactivation kinetics at depolarized membrane potentials and by measured recovery kinetics at hyperpolarized potentials. In cases in which the time constants of inactivation and recovery are very different, this compromise results in unconventional time constant expressions, such as those for I_Na in Fig 2D.

2. The available data regarding the intracellular Ca²⁺ transient and the Ca²⁺ handling in the SR have, with few exceptions,^{52 62} been recorded in cells from species other than humans. In addition, the understanding of the exact mechanisms involved in these phenomena is incomplete, and a quantitative model of SR Ca²⁺ release and uptake has not yet been developed. Our Ca²⁺-dependent formulation for SR Ca²⁺ release replaces the voltage-dependent formulation used in earlier models^{4 12 13} but is nevertheless only a qualitative description of this phenomenon. Other features of the SR function are the same as those in the LMCG model.¹³ Our SR formulation is therefore not based on human data.

3. The shape of the action potentials recorded from enzymatically isolated human atrial cells is variable, even at a fixed stimulus frequency. There are several reasons for this variability, including genuine heterogeneity between cells from different parts of the atrium, the disease states of the patients from whom the specimens are obtained, and the use of various drugs (eg, Ca²⁺ channel blockers and ß-blockers) by the donors. It is important to acknowledge that this variability exists and that the exact action potential waveform in the model is chosen because it is representative of the action potential shapes that are most often observed in our single cell records. As indicated by sensitivity analysis, the shape of the human atrial action potential as described by our model is quite sensitive to variations in the strengths of three ionic currents (I_t, I_sus, and I_Ca,L). Under the assumption that the currents involved and their kinetic properties are unchanged, our results therefore suggest that the experimentally observed variation in action potential shape is caused mainly by variations in these currents and can be explained within the framework of our model. However, it is also conceivable that regional diversity in the molecular basis of the currents in human atrium could give rise to regional differences in kinetics and pharmacological sensitivity. Most of the data on human atrial electrophysiology to date have been obtained from samples of the atrial appendage, and there are therefore little experimental data available regarding regional differences in human atrium. If and when data to suggest regional diversity become available, our model should provide a useful framework for predicting the consequences.

4. Our model of Ca²⁺-dependent inactivation of I_Ca,L uses a single lumped subsarcolemmal compartment in which Ca²⁺ accumulates and is therefore limited in its ability to simulate the effect of Ca²⁺ accumulation in restricted spaces close to each L-type Ca²⁺ channel. In contrast to voltage-dependent gating, where it is reasonable to assume that the controlling variable (membrane voltage) is spatially uniform (space-clamp conditions), this is in all likelihood not the case for subsarcolemmal Ca²⁺. Since inactivation is a nonlinear function of [Ca²⁺]_d, it is not strictly correct to use a formulation in which all channels are subject to one (average) Ca²⁺ concentration.

Notwithstanding these limitations, this model provides the most complete description available of the ionic mechanisms underlying the human atrial action potential, and it is based on the available data. As a result, it provides a very useful tool for investigating fundamental electrophysiological responses of the human atrial cell, such as excitability, refractoriness, and the action of channel blocking drugs.

	Appendix 1

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 Appendix 2 References

 
Tables 5 through 19 contain all the equations, parameter values, and initial conditions necessary to carry out the simulations presented in this article. Unless otherwise noted, the units are as follows: time in seconds (s), voltage in millivolts (mV), concentration in millimoles/liter (mmol/L), current in picoamperes (pA), conductance in nanosiemens (nS), capacitance in nanofarads (nF), volume in nanoliters (nL), and temperature in kelvin (K). The stimulus used to evoke an action potential consists of a rectangular current pulse (I_stim) with an amplitude of 280 pA and duration of 6 ms.

All simulations were performed by forward integration of the coupled system of differential equations using the CVODE solver package for ordinary differential equations. (CVODE was developed by S.D. Cohen and A.C. Hindmarsh at Lawrence Livermore National Laboratories, Livermore Calif) Sufficient accuracy was ensured by adjusting the temporal step size of the integration so that the local error in all state variables (as estimated by the CVODE algorithm) satisfied a relative error bound. Computer programs for the simulations were written in the C programming language under the UNIX operating system. Simulations were performed on Sun Microsystems Sparc workstations (Sparc 2, IPX) and on a Micron Millennia Pentium 166 PC running the Linux operating system. At a stimulus frequency of 1 Hz, one cycle (ie, 1 s of data) requires 0.9 s of CPU time on the Pentium 166 platform.

	Appendix 2

Top Abstract Introduction Materials and Methods Results Discussion Appendix 1 Appendix 2 References

 
This Appendix contains a brief derivation of the method used to study the parameter sensitivity of the action potential waveform. The reader is referred to Reference 53 for more detailed information.

Starting from a system of the general form,

we take the partial derivative of Equation 1 with respect to a parameter _k to obtain the following equation:

The time derivative of the state vector x(, t) can be written as follows:

where the time derivative of vanishes, since all parameters are assumed to be time invariant. Thus, from Equation 1, we have the following:

Interchanging the order of integration, we obtain the following equation:

where _k is the sensitivity of the state x to the parameter _k, ie,

Note, finally, that Equation 5 defines a costate system⁵³ of the same general form as Equation 1, ie,

where

The costate system for each parameter of interest can be integrated along with the original system, yielding the desired sensitivity functions.

Glossary
_Na, _K, _Ca I_Na

Na⁺ current

I_Ca,L

L-type Ca²⁺ current

I_t

Transient outward K⁺ current

I_sus

Sustained outward K⁺ current

I_K,s

Slow delayed rectifier K⁺ current

I_K,r

Rapid delayed rectifier K⁺ current

I_K1

Inwardly rectifying K⁺ current

I_B,Na

Background Na⁺ current

I_B,Ca

Background Ca²⁺ current

I_NaK

Na⁺-K⁺ pump current

I_CaP

Sarcolemmal Ca²⁺ pump current

I_NaCa

Na⁺-Ca²⁺ exchange current

_Na,en

Electroneutral Na⁺ influx

I_di

Ca²⁺ diffusion current from the diffusion-restricted subsarcolemmal space to the cytosol

I_up

Sarcoplasmic reticulum Ca²⁺ uptake current

I_tr

Sarcoplasmic reticulum Ca²⁺ translocation current (from uptake to release compartment)

I_rel

Sarcoplasmic reticulum Ca²⁺ release current

[Na⁺]_b

Na⁺ concentration in bulk (bathing) medium

[K⁺]_b

K⁺ concentration in bulk (bathing) medium

[Ca²⁺]_b

Ca²⁺ concentration in bulk (bathing) medium

[Na⁺]_c

Na⁺ concentration in the extracellular cleft space

[K⁺]_c

K⁺ concentration in the extracellular cleft space

[Ca²⁺]_c

Ca²⁺ concentration in the extracellular cleft space

[Na⁺]_i

Na⁺ concentration in the intracellular medium

[K⁺]_i

K⁺ concentration in the intracellular medium

[Ca²⁺]_i

Ca²⁺ concentration in the intracellular medium

[Mg²⁺]_i

Mg²⁺ concentration in the intracellular medium

[Ca²⁺]_d

Ca²⁺ concentration in the restricted subsarcolemmal space

[Ca²⁺]_up

Ca²⁺ concentration in the sarcoplasmic reticulum uptake compartment

[Ca²⁺]_rel

Ca²⁺ concentration in the sarcoplasmic reticulum release compartment

E_Na

Equilibrium (Nernst) potential for Na⁺

E_K

Equilibrium (Nernst) potential for K⁺

E_Ca

Equilibrium (Nernst) potential for Ca²⁺

E_Ca,app

Apparent reversal potential for I_Ca,L (differs from E_Ca)

P_Na

Permeability for I_Na

¯g_Ca,L

Maximum conductance for I_Ca,L

¯g_t

Maximum conductance for I_t

¯g_sus

Maximum conductance for I_sus

¯g_K,s

Maximum conductance for I_K,s

¯g_K,r

Maximum conductance for I_K,r

¯g_K1

Maximum conductance for I_K1

¯g_B,Na

Maximum conductance for I_B,Na

¯g_B,Ca

Maximum conductance for I_B,Ca

m

Activation gating variable for I_Na

h₁, h₂

Fast and slow inactivation gating variables for I_Na

d_L

Activation gating variable for I_Ca,L

f_L1, f_L2

Fast and slow inactivation gating variables for I_Ca,L

f_Ca

[Ca²⁺]_d-dependent ratio of fast (f_L1) to slow (f_L2) inactivation of I_Ca,L

k_Ca

Half-maximum Ca²⁺ binding concentration for f_Ca

r

Activation gating variable for I_t

s

Inactivation gating variable for I_t

s₁, s₂

Rapidly and slowly recovering inactivation gating variables for I_t

r_sus

Activation gating variable for I_sus

s_sus

Inactivation gating variable for I_sus

n

Activation gating variable for I_K,s

p_a

Activation gating variable for I_K,r

p_i

Inactivation gating variable (instantaneous) for I_K,r

¯m, ¯h₁, ...

Steady-state value of m, h₁, etc

F₁

Relative amount of "inactive precursor" in the I_rel formulation

F₂

Relative amount of "activator" in the I_rel formulation

_m

Activation time constant for I_Na

_h1, _h2

Fast and slow inactivation time constants for I_Na

_dL

Activation time constant for I_Ca,L

_fL1, _fL2

Fast and slow inactivation time constants for I_Ca,L

_r

Activation time constant for I_t

_s

Inactivation time constant for I_t

_s1, _s2

Rapidly and slowly recovering inactivation time constants for I_t

_rsus

Activation time constant for I_sus

_ssus

Inactivation time constant for I_sus

_n

Activation time constant for I_K,s

_pa

Activation time constant for I_K,r

O

Buffer occupancy

O_C

Fractional occupancy of the calmodulin buffer by Ca²⁺

O_TC

Fractional occupancy of the troponin-Ca²⁺ buffer by Ca²⁺

O_TMgC

Fractional occupancy of the troponin-Mg²⁺ buffer by Ca²⁺

O_TMgMg

Fractional occupancy of the troponin-Mg²⁺ buffer by Mg²⁺

O_Calse

Fractional occupancy of the calsequestrin buffer (in the sarcoplasmic reticulum release compartment) by Ca²⁺

R

Universal gas constant

T

Absolute temperature

F

Faraday's constant

C_m

Membrane capacitance

V

Membrane voltage

Vol_c

Volume of the extracellular cleft space

Vol_i

Total cytosolic volume

Vol_d

Volume of the diffusion-restricted subsarcolemmal space

Vol_up

Volume of the sarcoplasmic reticulum uptake compartment

Vol_rel

Volume of the sarcoplasmic reticulum release compartment

_Na, _K, _Ca

Time constant of diffusion of Na⁺, K⁺, and Ca²⁺ from the bulk medium to the extracellular cleft space

_di

Time constant of diffusion from the restricted subsarcolemmal space to the cytosol

¯I_NaK

Maximum Na⁺-K⁺ pump current

k_NaK,K

Half-maximum K⁺ binding concentration for I_NaK

k_NaK,Na

Half-maximum Na⁺ binding concentration for I_NaK

¯I_CaP

Maximum Ca²⁺ pump current

k_CaP

Half-maximum Ca²⁺ binding concentration for I_CaP

k_NaCa

Scaling factor for I_NaCa

Position of energy barrier controlling voltage dependence of I_NaCa

d_NaCa

Denominator constant for I_NaCa

¯I_up

Maximum sarcoplasmic reticulum uptake current

k_cyca

Half-maximum binding concentration for [Ca²⁺]_i to I_up

k_srca

Half-maximum binding concentration for [Ca²⁺]_up to I_up

k_xcs

Ratio of forward to back reactions for I_up

_tr

Time constant of diffusion ("translocation") of Ca²⁺ from sarcoplasmic reticulum uptake to release compartment

_rel

Scaling factor for I_rel

k_rel,i

Half-activation [Ca²⁺]_i for I_rel

k_rel,d

Half-activation [Ca²⁺]_d for I_rel

r_recov

Recovery rate constant for the sarcoplasmic reticulum release channel

	Acknowledgments

 
Mr Nygren was supported in part by a stipend from a Canadian Medical Research Council group grant. Drs Fiset and Firek are grateful for the support of Alberta Heritage Foundation for Medical Research fellowships. Dr Giles is supported as a medical scientist by the Alberta Heritage Foundation for Medical Research and receives ongoing support from the Canadian Medical Research Council and the Heart and Stroke Foundation of Canada.

Received June 9, 1997; accepted September 30, 1997.

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