Simulation of Action Potentials From Metabolically Impaired Cardiac Myocytes
Role of ATPSensitive K^{+} Current
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Abstract
The role of the ATPsensitive K^{+} current (I_{KATP}) and its contribution to electrophysiological changes that occur during metabolic impairment in cardiac ventricular myocytes is still being discussed. The aim of this work was to quantitatively study this issue by using computer modeling. A model of I_{KATP} is formulated and incorporated into the LuoRudy ionic model of the ventricular action potential. Action potentials under different degrees of activation of I_{KATP} are simulated. Our results show that in normal ionic concentrations, only ≈0.6% of the K_{ATP} channels, when open, should account for a 50% reduction in action potential duration. However, increased levels of intracellular Mg^{2+} counteract this shortening. Under conditions of high [K^{+}]_{o}, such as those found in early ischemia, the activation of only ≈0.4% of the K_{ATP} channels could account for a 50% reduction in action potential duration. Thus, our results suggest that opening of I_{KATP} channels should play a significant role in action potential shortening during hypoxic/ischemic episodes, with the fraction of open channels involved being very low (<1%). However, the results of the model suggest that activation of I_{KATP} alone does not quantitatively account for the observed K^{+} efflux in metabolically impaired cardiac myocytes. Mechanisms other than K_{ATP} channel activation should be responsible for a significant part of the K^{+} efflux measured in hypoxic/ischemic situations.
It is well known that myocardial hypoxia and ischemia cause profound changes in the electrophysiological properties of cardiac tissue. One of the major changes that occur in ventricular muscle cells during metabolic impairment is shortening of the AP.^{1} The reduction of APD initially causes a shortening of the refractory period,^{2} and this can facilitate the appearance of reentranttype arrhythmias.^{2} The reduction of APD during ischemia can partly be explained by an increased [K^{+}]_{o},^{3} ^{4} but additional factors seem to be involved in AP shortening. Indeed, during hypoxic perfusion, the AP is known to shorten in the absence of extracellular K^{+} accumulation.^{1} ^{5}
Since K_{ATP} channels were first described by Noma,^{6} their contribution to the shortening of the AP during hypoxia and ischemia and to other electrophysiological changes has been debated and is still not completely clarified. The major argument against the role of the K_{ATP} channels during early ischemia is that the value of [ATP]_{i} needed to open 50% of the channels is two orders of magnitude below the measured [ATP]_{i} bulk level in the first phase of ischemia.^{7} Thus, the fraction of channels activated during early ischemia is likely to be very low, and from this point of view, the current carried by K_{ATP} channels (I_{KATP}) might not seem to contribute significantly to the AP shortening and other ischemiarelated electrophysiological changes.^{4} ^{7} Moreover, because I_{KATP} channel blockers, such as glibenclamide, only partially prevent hypoxic/ischemic AP shortening, it has been suggested that currents other than I_{KATP} must significantly contribute to APD reduction.^{8} ^{9}
On the other hand, it has also been suggested that only a small number of K_{ATP} channels need to be activated to account for the changes observed in AP configuration. Following this “sparechannel hypothesis” suggested by Cook et al,^{10} several investigators have found indirect experimental evidence that supports this idea in the case of cardiac myocytes.^{11} ^{12} ^{13} ^{14} However, because of the difficulty of experimentally measuring the fraction of open K_{ATP} channels directly, there is still no direct proof of this hypothesis, and the quantitative importance of I_{KATP} channel opening in hypoxic/ischemic episodes is not yet completely established.
The contribution of I_{KATP} to cellular K^{+} loss during hypoxic/ischemic situations is not clear either. There exists experimental evidence that supports the idea that I_{KATP} channel activation largely^{14} ^{15} or partially^{16} ^{17} accounts for K^{+} loss from the cell in the first minutes of a hypoxic/ischemic episode. However, the ineffectiveness of K^{+} channel openers to enhance the rate of K^{+} loss^{18} ^{19} and the dissociation between K^{+} efflux, AP shortening, and intracellular ATP levels in hypoxia/ischemia,^{4} among others,^{20} are reasons against a major role of I_{KATP} in this phenomenon.
The main goal of the present study was to use a computer model to quantitatively study the influence of I_{KATP} on changes in AP configuration and cellular K^{+} loss in metabolically impaired conditions. For this purpose, we have formulated a detailed model of this current and have incorporated it into the LRII model^{21} of the guinea pig–type ventricular cardiac AP. We use this model to study the relationship between APD and intracellular nucleotide levels and ionic concentrations. The influence of I_{KATP} activation on reduction of APD during ischemia in the presence of high [K^{+}]_{o} has been theoretically elucidated. Finally, the contribution of I_{KATP} to the increase in the rate of K^{+} efflux in hypoxia/ischemia is also theoretically investigated.
Materials and Methods
Model of I_{KATP}
General Considerations
The mathematical model of I_{KATP} that we formulate here is based on different sets of published experimental data describing the dependence of the channel current density on ion concentrations ([K^{+}]_{o}, [Mg^{2+}]_{i}, and [Na^{+}]_{i}) and intracellular nucleotide levels ([ATP]_{i} and [ADP]_{i}). The parameters of the model are estimated, when necessary, using a linear leastsquares method to fit the experimental data. The complete set of equations of the model is given in Appendix 1.
Current Density
The general equation describing the current density (μA/cm^{2}) is as follows:where σ is the channel density, g_{0} is the unitary conductance of a fully activated individual channel, p_{o} is the open probability of a channel in the absence of ATP, f_{ATP} is the fraction of activated channels (relative current), and V_{m} is the membrane potential. The term E_{KATP} is the reversal potential of the channel, which is equal to the Nernst potential of K^{+} ions^{22} because of the specificity of the channel to K^{+} ions in cardiac muscle cells.
We have used in our simulations a value of 3.8 channels/μm^{2} for the channel density (derived from data in Reference 14), a value that is intermediate in the range of reported values for guinea pig ventricular myocytes. The open probability in the absence of ATP was assumed to be 0.91.^{17} The sensitivity of the results to these parameters is discussed later.
Unitary Conductance
The expression for the unitary conductance (g_{0}) of an individual K_{ATP} channel is as follows:where γ_{0} is the unitary conductance in the absence of intracellular Na^{+} and Mg^{2+}, f_{M} and f_{N} are nondimensional factors that account for the inward rectification of the channel, and f_{T} is a nondimensional temperature (T)dependent factor.
The value of γ_{0} is known to depend on [K^{+}]_{o}.^{22} ^{23} In the present model, we have used the formulation provided in Reference 22 (see Equation 10 of Appendix 1), which is widely accepted. On the other hand, K_{ATP} channels show inward rectification.^{22} ^{23} This property is the result of a voltagedependent block caused by Na^{+} and Mg^{2+} ions and obeys the laws of saturation kinetics.^{23} We used this approach to express both factors f_{M} (for Mg^{2+}) and f_{N} (for Na^{+}) in Equation 2 by means of Hilltype equations (see Equations 11 and 14 in Appendix 1). The halfmaximal saturation constants (K_{h,ion}) are given by Eyring rate theory:where the subscript “ion” stands for Mg^{2+} and Na^{+}, respectively, K^{0}_{h,ion} is the value of K_{h,ion} at zero membrane voltage, δ_{ion} is the electrical distance, z_{ion} is the valence of the considered ion, F is the Faraday constant, R is the gas constant, and T is the absolute temperature. The values of the parameters used in our simulations and the details of the equations are listed in Appendix 1.
It is known that increasing levels of extracellular K^{+} partially remove the voltagedependent block caused by Mg^{2+} ions.^{23} This fact was considered in our model by making K^{0}_{h,Mg} increase monotonically with [K^{+}]_{o} following a squareroot dependence (see Equation 13 in Appendix 1) to fit data from Horie et al.^{23}
Finally, a temperaturedependent term, f_{T}, was introduced by using a temperature coefficient Q_{10}=1.3 (see Equation 16 in Appendix 1).^{23}
Fig 1A⇓ illustrates the results obtained with the model of the unitary conductance in terms of the currentvoltage relationships of the channel. The symbols in both plots represent experimental values corresponding to different values of [K^{+}]_{o} (data duplicated from Reference 24). The solid lines represent the curves predicted by the model.
Nucleotide Dependence
It is well known that when intracellular ATP molecules bind to the channel protein, it becomes inactivated. Thus, f_{ATP} in a myocyte strongly depends on [ATP]_{i}. Experimental data^{6} ^{12} ^{14} ^{25} ^{26} are properly fitted by means of a Hilltype equation:where K_{m} is the halfmaximum inhibition constant and H is the Hill coefficient.
Several factors related to the metabolic state of the cell modulate the [ATP]_{i} dependence of f_{ATP} (for a review, see Reference 27). Among them, free cytosolic ADP is known to stimulate partially inactivated channels in the presence of Mg^{2+}.^{14} ^{25} ^{28} Both the halfmaximal inhibition constant (K_{m}) and the Hill coefficient of Equation 4 are dependent on free [ADP]_{i}. Using the data reported by Weiss et al,^{14} we have modeled the dependence of f_{ATP} on [ADP]_{i}. Specifically, we used the average values of K_{m} and Hill coefficients for individual membrane patches obtained by Weiss et al for different values of [ADP]_{i}. The mathematical expressions that result from the best fit are given in Appendix 1 (see Equations 18 and 19); they show a monotonic increase of K_{m} and a monotonic decrease of the Hill coefficient with [ADP]_{i}, respectively.
The fraction of open channels will depend, in this way, on intracellular concentrations of both ATP and ADP. Fig 1B⇑ graphically illustrates this dependence in an appropriate way to easily relate nucleotide levels to f_{ATP} values. Indeed, these curves can be used to “translate” a given value of f_{ATP} to the different combinations of [ATP]_{i}[ADP]_{i}, which, when present in the cell, give rise to such a value of f_{ATP}. Each curve in the figure can be regarded, then, as an “isoactivation” curve for I_{KATP}.
Model of the Ventricular AP
Once the model for I_{KATP} was formulated, we incorporated it into the LRII model described by Luo and Rudy.^{21} This mathematical model reproduces the AP of endocardial ventricular myocytes of guinea pig–type hearts with a high degree of electrophysiological detail. It includes mathematical descriptions of 12 different ionic currents, as well as intracellular Ca^{2+} buffering and the Ca^{2+}induced Ca^{2+} release process. The basic equation that relates V_{m} to ionic currents is the following:where C_{m} is the membrane capacitance, I_{stim} is the stimulus current, and ΣI_{ion} is the sum of all the ionic currents that cross the sarcolemma, namely, the fast Na^{+} current (I_{Na}), the current through the Ltype Ca^{2+} channels (I_{Ca,t}),* the delayed rectifier K^{+} current (I_{K}), the inward rectifier K^{+} current (I_{K1}), the plateau K^{+} current (I_{Kp}), the Na^{+}Ca^{2+} exchanger current (I_{NaCa}), the Na^{+}K^{+} pump current (I_{NaK}), a nonspecific Ca^{2+}activated K^{+} and Na^{+} current (I_{ns}), and the current carried by the sarcolemmal Ca^{2+} pump (I_{p(Ca)}). Mathematical details of this model can be found elsewhere.^{21}
In our simulations, extracellular ionic concentrations were held constant, unless otherwise noted (see “Extracellular K^{+} Accumulation” below). Intracellular concentrations changed dynamically as a result of ionic fluxes through the sarcolemma. The normal values of extracellular concentrations and initial intracellular concentrations are listed in the Table⇓. All simulations correspond to a temperature of 37°C.
Stimulation Protocol
In each simulation, unless otherwise noted (see “Extracellular K^{+} Accumulation” below), constant values were assigned to each relevant parameter of the model (ie, f_{ATP} and ionic concentrations), and in these conditions, the cell was stimulated with a constant BCL. In order to achieve steady state conditions and to avoid alternants in APD, 10 APs were elicited before recording the data. The stimulus consisted of rectangular current pulses 2 ms in duration and an amplitude 1.5 times the diastolic threshold.
APD
We defined APD in our simulations as the interval between the instant of maximum upstroke velocity of the AP, [dV/dt]_{max}, and the instant of 90% repolarization.
Calculation of J_{efflux}
When calculating the rate of K^{+} efflux from the cell, the basic LRII model was slightly modified so as to achieve zero net K^{+} efflux under basal normoxic conditions. Specifically, the maximum current density through the Na^{+}K^{+} pump was increased from 1.5 to 2.61 μA/μF, which is still in the range of measured values.^{29} This change affects AP morphology only slightly (small decrease of APD due to accelerated repolarization).
To compute J_{efflux}, we started by subtracting the K^{+} (inward) current carried by the Na^{+}K^{+} pump from the total K^{+} current (I_{K,T}) to obtain the total outward K^{+} current (I_{K,O}). Taking into account the 3:2 stoichiometry of the pump, this results in I_{K,O}=I_{K,T}+2I_{NaK} (where I_{NaK} is the Na^{+}K^{+} pump current). We then calculated the average outward current density (I_{out}) as the integral mean value of I_{K,O}. In order to compare the simulation results with experimental data, the current density value was translated to J_{efflux} (in μmol·g^{−1}·min^{−1}) using the following expression:
The assumed values of the parameters in Equation 6 were ρ=1 kg/L for the myocardial density, V_{ECW}=0.52 L/kg wet wt for the extracellular water content,^{30} and S_{v}=0.3 μm^{2}/μm^{3} for the surfacetovolume ratio of the myocyte. F stands for the Faraday constant. Details about derivation of Equation 6 can be found in Appendix 2.
Finally, ΔJ_{efflux} was calculated as the difference between the actual value of J_{efflux} and its control value (corresponding to f_{ATP}=0%).
Extracellular K^{+} Accumulation
We also carried out long simulations in which extracellular K^{+} accumulation was studied. In these simulations, the cell was paced with a BCL of 800 ms, and f_{ATP} was either abruptly or gradually increased. Extracellular concentrations were permitted to change dynamically as a result of ionic fluxes through the cellular membrane. A threecompartment model was assumed, and diffusion of ions from the extracellular cleft to the bulk extracellular medium was considered. Thus, ionic concentrations in the cleft can be described by the following equation:where [S]_{o} and [S]_{bulk} are the concentrations of the ionic species S in the extracellular cleft and in the bulk extracellular medium, respectively. The term z_{S} is the valence of the ionic species S, and I_{S,total} stands for the total current through the sarcolemma carried by the ionic species S. Finally, τ_{diff} (1 s) is the time constant associated with the diffusion of ions from the cleft to the bulk extracellular medium.
When simulating extracellular K^{+} accumulation during noflow ischemia, the diffusion term in Equation 7 was omitted (τ_{diff}) to account for the lack of flow.
The modified version of the Na^{+}K^{+} pump in the LRII model was also used in the simulations (see “Calculation of J_{efflux}” above).
Computation Methods
Programs were written in ACSL language using Gear stiff algorithm^{31} to solve the nonlinear system of differential equations that results from the AP model. Simulations were carried out in a SUN SparcStation 1 using doubleprecision variables. To ensure numerical accuracy, the maximum allowed time step was 10 μs. The maximum relative error allowed for every variable in each iteration was 10^{−6}.
Results
In all the simulations that are presented in this section, f_{ATP} is varied, and the effect of this variation on AP configuration, ionic currents, and K^{+} efflux is investigated. A given value of f_{ATP} can be related to [ATP]_{i} and [ADP]_{i} using the isoactivation curves shown in Fig 1B⇑.
Effects of K_{ATP} Channel Opening on AP Configuration and Ionic Currents
The effects of the progressive activation of I_{KATP} on the characteristics of the ventricular AP were first investigated using normal nonischemic values for the ionic concentrations. The values used are listed in the Table⇑.
Fig 2⇓ shows the results of these simulations. In Fig 2A⇓, a set of APs that correspond to different values of f_{ATP} is shown. It can be noted that AP configuration varies significantly when K_{ATP} channels become activated, even with very low values of f_{ATP}. When f_{ATP} increases, there is a marked reduction in APD, a moderate reduction of the plateau potential, and a slight diastolic hyperpolarization. Resting V_{m}, whose value is −86.5 mV in control (f_{ATP}=0%) conditions, decreases almost linearly to reach a value of −87.1 mV for f_{ATP}=2.5%. This would be in accordance with the slight diastolic hyperpolarization observed by Gasser and VaughanJones^{15} in myocytes exposed to hypoxic conditions, although other studies have reported opposite results.^{32} The cell becomes completely unexcitable for f_{ATP}≈3.1% for the standard stimulus used in the simulations (not shown).
The reduction in APD caused by increasing activation of I_{KATP} is represented in Fig 2B⇑, in which the results corresponding to two different BCLs are compared. For each BCL, the APD has been normalized to its maximum value, which corresponds to the complete inactivation of the K_{ATP} channels. For a BCL of 800 ms, the figure shows how the activation of ≈0.6% of the total population of channels is sufficient to account for a 50% shortening in APD. This figure increases to ≈0.7% when the pacing frequency is increased to a BCL of 475 ms.
The value of f_{ATP} needed to shorten APD to half its control value is in accordance with several experimental results.^{11} ^{13} ^{14} Moreover, the rate of change of APD with f_{ATP} agrees very nicely with indirect experimental findings by Nichols and Lederer.^{12}
Activation of I_{KATP} modifies the ionic sarcolemmal currents significantly. Fig 2C⇑ shows the evolution of the total K^{+} current (I_{K,T}) as activation of I_{KATP} progresses. The total time during which the K^{+} currents are flowing shortens in correspondence with the reduction in APD. Both the maximum peak of I_{K,T} and the amplitude of the K^{+} current “plateau” increase with f_{ATP}. The secondary peak of the K^{+} current in phase 3, mainly due to activation of the timeindependent I_{K1}, also increases, although only slightly, with f_{ATP}.
Fig 3⇓ depicts the relative contributions of I_{KATP} and the rest of the sarcolemmal K^{+} currents to I_{K,T}. The time courses of I_{KATP} and of the sum of all the other K^{+} currents (I_{KR}) are compared for six different values of f_{ATP}. The shape of I_{KATP} is a distorted version of the AP waveform, due to the inward rectification of the K_{ATP} channels, and presents a plateau whose level is proportional to the degree of channel activation. Regarding I_{KR}, both the initial peak during depolarization (due mainly to the plateau K^{+} current and to I_{K1}) and the secondary peak during repolarization (due basically to I_{K1}) are practically independent of f_{ATP}. Opening of K_{ATP} channels significantly depresses the plateau of I_{KR}. It is noticeable that for degrees of K_{ATP} channel activation over 0.4%, the overall contribution of I_{KATP} to I_{K,T} is higher than the contribution of all the rest of the K^{+} currents added together.
Effects of Changes on Ionic Concentrations
In the next set of simulations, [Mg^{2+}]_{i}, [Na^{+}]_{i}, and [K^{+}]_{o} are varied in turn while the other ionic concentrations remain at their control levels (see the Table⇑). The effects of changes in these concentrations, which modulate the activity of the K_{ATP} channels, on AP configuration and APD are further investigated.
Changes in [Mg^{2+}]_{i}
Myoplasmic free Mg^{2+}, which partially blocks K_{ATP} channels in a voltagedependent fashion, is known to increase from its control level (≈0.5 mmol/L) to ≈2.5 mmol/L in 6 to 9 minutes of global ischemia.^{33} To investigate the effects of increased intracellular Mg^{2+} level on AP configuration, we simulated APs under different K_{ATP} channel activation degrees for three different [Mg^{2+}]_{i} levels. The results are shown in Fig 4⇓. Each of the six sets of APs plotted in Fig 4A⇓ corresponds to a fixed value of f_{ATP}. It can be seen that increased levels of intracellular Mg^{2+} partially counteract the AP shortening caused by I_{KATP}. High [Mg^{2+}]_{i} also elevates the AP plateau level because of the enhanced inward rectification of the K_{ATP} channels.
The relationship between normalized APD and f_{ATP} for different [Mg^{2+}]_{i} levels is plotted in Fig 4B⇑. Note that the effect of intracellular Mg^{2+} on APD is more significant at higher values of f_{ATP}. The fraction of channels needed to be activated to reduce APD to 50% rises from ≈0.6% for [Mg^{2+}]_{i}=0.5 mmol/L to ≈0.8% for [Mg^{2+}]_{i}=1.5 mmol/L and ≈1.0% for [Mg^{2+}]_{i}=2.5 mmol/L. The current through K_{ATP} channels at V_{m}=0 mV is reduced from 80% to 46% of the maximum possible current when [Mg^{2+}]_{i} increases from 0.5 to 2.5 mmol/L. According to these results, the effect of an increased intracellular Mg^{2+} level during hypoxia/ischemia has a considerable effect on APD, reducing the K_{ATP}mediated shortening of the AP.
Changes in [Na^{+}]_{i}
Next, we investigated the effects of increased levels of intracellular Na^{+}. APs corresponding to different values of [Na^{+}]_{i} and different values of f_{ATP} are shown in Fig 5A⇓. Changes in [Na^{+}]_{i} have two different effects on AP configuration. The first one is independent of K_{ATP} channels and is due to the dependence on [Na^{+}]_{i} exhibited by several ionic channels, pumps, and exchangers in the sarcolemma.^{21} This direct effect tends to shorten the AP when intracellular levels of Na^{+} rise, even in the absence of K_{ATP} channel activation. Fig 5B⇓, which shows the dependence of APD on f_{ATP} for three different values of [Na^{+}]_{i}, illustrates this phenomenon. All APDs are referred to the value that corresponds to f_{ATP}=0% and [Na^{+}]_{i}=10 mmol/L. Note, indeed, that for any constant value of f_{ATP}, AP shortens as [Na^{+}]_{i} increases. On the other hand, as discussed previously, intracellular Na^{+} causes a partial voltagedependent block in K_{ATP} channels. This would tend to reduce the AP shortening caused by I_{KATP} activation, as happens with intracellular Mg^{2+}. To further investigate this effect, we constructed Fig 5C⇓ by normalizing APD values in a different way. Each value of APD corresponding to a given level of intracellular Na^{+} was normalized to the maximum APD value (corresponding to f_{ATP}=0%) found under that particular [Na^{+}]_{i}. In this way, the direct effect of intracellular Na^{+} previously mentioned is eliminated, while the K_{ATP}dependent effect is maintained and amplified. When APD values are normalized in this manner, it can be seen (Fig 5C⇓) that all the curves (APD versus f_{ATP}) fall reasonably well on a single curve, with maximum differences in APD values being in the range of 5% for all values of f_{ATP}. This means that the effect of [Na^{+}]_{i} on APD mediated by K_{ATP} channels is very small in the range of [Na^{+}]_{i} tested. Note that the current through K_{ATP} channels at V_{m}=0 mV is reduced from 87% to 63% of the maximum possible current when [Na^{+}]_{i} is increased from 10 to 20 mmol/L, which is much less significant than the reduction caused by Mg^{2+}.
Changes in [K^{+}]_{o}
Increases in [K^{+}]_{o} that take place in ischemic episodes are known to profoundly affect APD. The effects of high [K^{+}]_{o} on APD are mediated in part by an increase in the conductance of both the inward rectifier (g_{K1}) and the delayed rectifier (g_{K}) channels, something which tends to decrease APD. Similarly, [K^{+}]_{o} is also known to affect the conductance of K_{ATP} channels in a similar manner.^{22}
We used the model to investigate the effect of [K^{+}]_{o} on AP configuration for different degrees of activation of I_{KATP}. Fig 6⇓ depicts the results obtained. The upper left APs in Fig 6A⇓ correspond to complete inactivation of I_{KATP}, and it is seen, as expected, how APD reduces in response to increases in [K^{+}]_{o}. APs also exhibit diastolic depolarization, which is due to the increase in [K^{+}]_{o}. The other five sets of APs in Fig 6A⇓ show the effects of the progressive activation of I_{KATP} on AP configuration. As f_{ATP} increases, APD is further decreased, resting V_{m} is scarcely affected, and the absolute influence of [K^{+}]_{o} on APD is reduced.
Fig 6B⇑ shows the effect of [K^{+}]_{o} on the APDf_{ATP} dependence. APD values are normalized to the reference value corresponding to f_{ATP}=0% and [K^{+}]_{o}=5.4 mmol/L. The fraction of open K_{ATP} channels needed to produce a 50% reduction in APD is reduced from ≈0.6% to 0.55%, 0.48%, and 0.38% as [K^{+}]_{o} increases from 5.4 mmol/L to 7.5, 9.5, and 11.5 mmol/L, respectively.
However, if we normalize the values of APD for each value of [K^{+}]_{o} to their control value (f_{ATP}=0%) corresponding to that particular [K^{+}]_{o}, the results are different. As illustrated in Fig 6C⇑, the relative reduction of APD normalized in this way is independent of [K^{+}]_{o} (all the points fall reasonably well in one single curve). These results suggest that both high [K^{+}]_{o} and K_{ATP} channel activation tend to reduce APD, but the effects of these two factors seem to be independent of one another. Indeed, Fig 6C⇑ shows that for any value of [K^{+}]_{o} in the range of early ischemia, ≈0.6% of the total population of channels, when open, always cause a 50% reduction in APD from its control value independently of [K^{+}]_{o}. Similarly (although not shown in the figures), for a fixed value of f_{ATP} in the range of 0% to 2.5%, APD is reduced to 76% of its control value when [K^{+}]_{o} increases from 5.4 to 11.5 mmol/L, independently of the value of f_{ATP} considered.
Fig 6B⇑ can also be used to compare the separate effects of high [K^{+}]_{o} and K_{ATP} channel activation on AP shortening. It is seen that in the range of values chosen for [K^{+}]_{o} and f_{ATP}, the effect of K_{ATP} channel activation on AP shortening under conditions of normal [K^{+}]_{o} is more pronounced than that of extracellular K^{+} accumulation alone. In the absence of K_{ATP} channel activation, typical early ischemic levels of [K^{+}]_{o} of 11 to 12 mmol/L^{16} shorten the APD to ≈75% of its control value. On the other hand, activation of 0.6% of the total population of K_{ATP} channels, which might be a typical value in early ischemia (see “Discussion” and Reference 14), reduces APD to ≈50% in the presence of normal K^{+} levels.
Cellular K^{+} Loss
It is a wellknown phenomenon that cardiac myocytes lose K^{+} during metabolically impaired situations. In ischemic episodes, K^{+} loss begins at ≈15 s after the onset of ischemia, and net K^{+} efflux rate reaches a peak value in the range of 0.3 to 0.5 μmol/(g·min).^{4} ^{16} In substratefree hypoxia, net K^{+} loss averages 0.54 to 0.60 μmol/(g·min).^{14} ^{34} Finally, net K^{+} efflux rate seems to be higher, ≈0.9 μmol/(g·min) in hypoxia with glucose present.^{4}
The model presented here can be used to quantify the K^{+} loss caused by the activation of I_{KATP}. For this purpose, we simulate APs for different pacing frequencies and different [K^{+}]_{o} levels and quantify J_{efflux} and ΔJ_{efflux} using Equation 6 (see “Materials and Methods”).
Fig 7A⇓ shows the magnitude of ΔJ_{efflux} (shown as ΔJ_{T} in Fig 7⇓) as a function of f_{ATP} for two different values of BCL. As depicted in the figure, net increment in K^{+} loss shows a biphasic behavior with K_{ATP} channel activation. Indeed, ΔJ_{efflux} initially increases with f_{ATP}, reaching a maximum value of 0.08 μmol·g^{−1}·min^{−1} (BCL=800 ms) or 0.16 μmol·g^{−1}·min^{−1} (BCL=475 ms) for f_{ATP} of ≈1%. From this point, ΔJ_{efflux} decreases as I_{KATP} is further increased and even becomes negative for f_{ATP} >2.25% (BCL=800 ms) or 2.75% (BCL=475 ms).
The level of extracellular K^{+} modulates the rate of K^{+} loss from the cell, as demonstrated in Fig 7B⇑, in which ΔJ_{efflux} is plotted against f_{ATP} for two different values of [K^{+}]_{o}. It is noticeable how the maximum ΔJ_{efflux} decreases when [K^{+}]_{o} increases (0.081 μmol·g^{−1}·min^{−1} for [K^{+}]_{o}=5.4 mmol/L, 0.044 μmol·g^{−1}·min^{−1} for [K^{+}]_{o}=8.5 mmol/L). The degree of K_{ATP} channel opening for which the maximum takes place is also reduced (0.8% for [K^{+}]_{o}=5.4 mmol/L, 0.7% for [K^{+}]_{o}=8.5 mmol/L). Thus, in ischemic situations in which extracellular K^{+} accumulation takes place, cellular K^{+} loss through the K_{ATP} channels would be even smaller.
The rate of cellular K^{+} loss mediated by I_{KATP} obtained with the model is significantly lower than the values of total K^{+} efflux found experimentally. Panels C and D of Fig 7⇑ compare the simulation results with experimental measures of K^{+} loss in different situations.^{4} ^{14} ^{16} ^{34} In Fig 7C⇑, J_{efflux}, corresponding to a pacing frequency of 75 bpm (BCL=800 ms) obtained with the model, is compared with the experimental values obtained by Venkatesh et al^{34} in similar experimental conditions. In normoxia, both theoretical and experimental values agree nicely (1.11 versus 1.24 μmol·g^{−1}·min^{−1}, respectively). However, in substratefree hypoxia, the empirical J_{efflux} greatly exceeds the maximum theoretical K_{ATP}related J_{efflux} (1.79 versus 1.18 μmol·g^{−1}·min^{−1}, respectively).
Fig 7D⇑ compares the values of ΔJ_{efflux} caused by I_{KATP} activation, obtained with the model, with those obtained experimentally in different conditions. For the theoretical results, the maximum values of ΔJ_{efflux} in each situation have been chosen. The experimental values correspond to the peak of the K^{+} efflux rate during the ischemic episode. It can be seen that, with only one exception, experimental values of ΔJ_{efflux} increase with pacing frequency. The figure shows that theoretical ΔJ_{efflux} through K_{ATP} channels is in the order of 5 to 7 times less than experimental values obtained in similar conditions. Thus, I_{KATP} activation does not seem to quantitatively account for the entire observed hypoxic/ischemic cellular K^{+} loss. All these results will be discussed in the next section.
Extracellular K^{+} Accumulation
Many experimental studies have been published about the time course of [K^{+}]_{o} during ischemia. There is general agreement in that, during early ischemia, [K^{+}]_{o} initially increases and then plateaus at a level of ≈10 to 12 mmol/L.^{4} ^{14} ^{20} ^{30} ^{35} This behavior can be qualitatively reproduced by the model, as seen in Fig 8⇓. In Fig 8A⇓, noflow ischemia has been simulated by abruptly increasing f_{ATP} from 0% to 1.0%, while preventing K^{+} diffusion from the extracellular cleft to bulk extracellular medium (see “Materials and Methods”). It can be noted from the figure that [K^{+}]_{o} is approximately constant during normoxic perfusion (f_{ATP}=0%), because net K^{+} efflux is zero in normal conditions. However, when K_{ATP} current becomes activated, [K^{+}]_{o} rises until a steady state is reached (within minutes), when [K^{+}]_{o} increases in a linear manner. This reflects the constant value of J_{efflux} in this situation (constant slope of ≈0.13 mmol/L per second, which corresponds to a J_{efflux} of 0.075 μmol·g^{−1}·min^{−1}).
In Fig 8B⇑, noflow ischemia is simulated in a more realistic manner. K_{ATP} channels are progressively (and linearly) activated from 0% to 2.5% during 10 minutes. In this time frame, [K^{+}]_{o} increases from 5.4 to 8.0 mmol/L, reaching an approximately constant plateau.
Although the time course of [K^{+}]_{o} shown in Fig 8B⇑ is qualitatively similar to those obtained experimentally, the values of [K^{+}]_{o} reached are substantially smaller than the measured ones. Thus, again it is shown how, according to the model, ischemic K^{+} loss through K_{ATP} channels does not account for the total observed cellular K^{+} loss.
Discussion
The extent to which activation of I_{KATP} contributes to the reduction of APD and to other electrophysiological changes during metabolically impaired situations still remains unanswered from a quantitative point of view. We have used a computer modeling approach to the problem to elucidate this issue. Although computer models cannot provide real data, they can be used to make predictions and, in this case, can help us to understand the role of K_{ATP} channels in hypoxiaischemia from a theoretical point of view.
The cardiac action potential model described by Luo and Rudy,^{21} which has been used in the present study, is based on very recent patchclamp data and reproduces membrane dynamics with a great degree of electrophysiological detail. The inclusion of a new formulation of the K_{ATP} current in this model makes it possible to simulate metabolically impaired situations more comprehensively.
Model of I_{KATP}
In its original form, the LRII action potential model^{21} does not include a mathematical description of I_{KATP}. The first goal of the present study was to formulate a comprehensive model for this current. Our description of I_{KATP} is based on published experimental data regarding the main characteristics of the current.^{14} ^{17} ^{22} ^{23} We have integrated the available data regarding I_{KATP} dependencies on [K^{+}]_{o}, [Na^{+}]_{i}, [Mg^{2+}]_{i}, [ATP]_{i}, and [ADP]_{i} in a single set of equations. The model of I_{KATP} finally formulated satisfactorily reproduces the main electrical features of K_{ATP} channels (eg, see Fig 1A⇑ and compare with Fig 6A⇑ of Reference 24). Other factors not considered in the model have been ignored because of their presumed lack of a physiological role during the early phase of hypoxia/ischemia (eg, rundown of the channel^{24} ), lack of enough data to formulate a reliable model (eg, dependence on pH_{i}^{25} ^{36} ^{37} and on other nucleotides^{25} ^{38} and effects of [Mg^{2+}] on the fraction of open channels^{39} ), or lack of agreement between different authors (eg, dependence on lactate^{25} ^{40} ). It is to be noted that the effects of some of these factors, particularly the effects of acidosis, could be of considerable importance in hypoxic/ischemic situations.
To our knowledge, only a few authors have incorporated a model of I_{KATP} in an AP model and used it to study the effect of I_{KATP} activation in cardiac myocytes. Nichols and Lederer^{12} incorporated a formulation of I_{KATP} into the model of rat ventricular AP described by Noble.^{41} This formulation included only the dependence on [ATP]_{i}, although the dependence on [ADP]_{i} was implicitly considered. More recently, the incorporation of a model of I_{KATP} that considered dependencies on both [ATP]_{i} and [K^{+}]_{o} to an AP model has been reported.^{42} In a different context, Cook et al^{10} used a simple computer model to explain the sparechannel hypothesis for beta pancreatic cells.^{10} Our description of I_{KATP} is more comprehensive than these previous attempts, because it considers dependencies on intracellular ionic concentrations and intracellular ADP as well as on [K^{+}]_{o} and [ATP]_{i}.
The model used in the present study has several limitations. In its present form, it cannot be used to simulate true ischemia, for it lacks a description of other important ischemiarelated phenomena apart from I_{KATP} activation. Among them, the most important one might be the influence of acidosis on ionic currents. Also, intracellular ATP decline, free Mg^{2+} rise, and catecholamine release are known to affect other ionic currents, and this should also be considered in a more complete model. Specifically, a more detailed model of the Na^{+}K^{+} pump would be desirable to determine net K^{+} efflux during metabolic inhibition with more accuracy. In its present version, the pump current dependencies on both [Na^{+}]_{i} and [K^{+}]_{o} are considered,^{21} but the model lacks a description of its dependence on ATP and other metabolically related parameters. The influence of [Mg^{2+}]_{i} on APD through inward Ca^{2+} channels^{43} should also be considered. Finally, other pathophysiologically activated currents (such as the Na^{+}activated K^{+} current and the free fatty acid–activated K^{+} current) also deserve some attention.
Effect of I_{KATP} on APD
The theoretical results obtained with our model are in excellent agreement with the sparechannel hypothesis that was proposed by Cook et al^{10} for pancreatic cells and was later extended to cardiac cells, according to which only a very small fraction of the total population of K_{ATP} channels in a myocyte needs to be activated to account for the major electrophysiological changes observed in metabolic impairment. Indeed, according to our model, activation of <1% of the total number of K_{ATP} channels accounts for a 50% reduction in APD in all situations simulated. The value of ≈0.6% obtained for normal ionic concentrations correlates well with the values obtained experimentally using indirect methods, namely, 1%,^{11} 0.7%,^{13} and 0.41%.^{14}
The degree of K_{ATP} channel activation needed to account for a 50% reduction in APD might be easily achieved in early hypoxic and ischemic situations. For example, Weiss et al^{14} reported nucleotide levels of [ATP]_{i}=4.3 mmol/L and [ADP]_{i}=95 μmol/L after 10 minutes of substratefree hypoxia. This would correspond, according to Fig 1B⇑, to f_{ATP}=0.68%. In the same experimental study, 10 minutes of ischemia reduced intracellular ATP to 4.6 mmol/L and increased free cytosolic ADP to 63 to 99 μmol/L, which would yield a value between 0.57% and 0.63% for f_{ATP}. Thus, it is seen that even if intracellular ATP levels fall only modestly during early hypoxia/ischemia, activation of K_{ATP} channels may account for drastic reductions in APD. It is clear that the rise in free cytosolic ADP levels is a key factor to quantitatively explain the APD reduction. Indeed, if [ADP]_{i} was held constant, f_{ATP} would reach a hypoxic/ischemic value of only ≈0.2%, which is far less than the 0.6% needed to reduce APD to half its control value.
Our results also indicate that the fraction of open K_{ATP} channels that exist during normal perfusion causes some degree of “baseline” shortening in the AP, which would theoretically be reversed by applying a perfect K_{ATP} channel blocker. Indeed, using the normoxic values of intracellular ATP and ADP (6.8 mmol/L and 15 μmol/L, respectively) reported by Weiss et al,^{14} the normal value of f_{ATP} would be 0.11%. According to Fig 2B⇑, this would cause a reduction in APD to ≈88% to 91% of the value it would have in the complete absence of I_{KATP}. This result is in agreement with one experimental report^{44} but contradicts others regarding the inefficiency of sulfonylureas to prolong APD in normally perfused myocytes.^{17} ^{34} If we consider a new reference value for APD that corresponds to f_{ATP}=0.11%, then the fraction of open channels needed to be activated to reduce APD to 50% of its normal value would now be ≈0.7% instead of 0.6%.
Regarding the influence of intracellular cations on the AP shortening caused by I_{KATP}, the results of our model show that pathophysiological levels of Mg^{2+} exert a strong influence on APD, whereas the direct (K_{ATP}related) effect of Na^{+} is much less noticeable. Intracellular free Mg^{2+} is known to rise in early ischemia,^{33} and this would reduce the K_{ATP}mediated effects of ATP depletion on APD (Fig 4⇑). However, high levels of intracellular Mg^{2+} are known to significantly shorten APD by reducing Ca^{2+} inward currents.^{43} Thus, elevated free [Mg^{2+}]_{i} would have at least two opposite effects on APD, with K_{ATP}dependent effects partially counteracting Ca^{2+} current–dependent APD shortening.
As for the reduction of APD caused by increased Na^{+} levels (Fig 5B⇑), it is mainly due to an enhanced activity of the Na^{+}K^{+} pump as a response to high [Na^{+}]_{i}, being practically independent of K_{ATP} channel activity. This effect is not likely to be physiologically significant: the extent to which [Na^{+}]_{i} increases during early ischemia is not unanimously established,^{20} and whether the activity of the electrogenic Na^{+}K^{+} pump is enhanced or depressed during the first phase of ischemia and hypoxia is still not completely determined.^{20}
Effects of I_{KATP} Activation and High [K^{+}]_{o} in Ischemic AP Shortening
The question of the contribution of high extracellular K^{+} and of K_{ATP} channel activation to AP shortening is still being debated. Although it is generally accepted that I_{KATP} activation is the key factor in ischemic APD reduction,^{11} ^{12} ^{13} ^{14} ^{15} experimental evidence exists that questions this hypothesis.^{4} ^{7} ^{8} ^{9} According to a recent report by Yan et al,^{4} ischemic AP shortening would be due to high [K^{+}]_{o}, and the role of I_{KATP} in this matter would be irrelevant because K_{ATP} channels would not become activated at all. The reduction of APD following extracellular K^{+} accumulation is due to the [K^{+}]_{o}dependent change of the currentvoltage relation. Indeed, elevated levels of [K^{+}]_{o} produce an increase in both the delayed rectifier current (I_{K}) and the inward rectifier current (I_{K1}) because their conductance increases with [K^{+}]_{o} according to a squareroot law^{21} and rectification is partially relieved. This elevation in outward current accelerates repolarization, thus leading to a shortening in APD.
The results of Yan et al,^{4} however, show that hypoxia with high [K^{+}]_{o} produces an additional shortening of the AP that is not due to high [K^{+}]_{o} only (see Fig 7⇑ of Reference 4). This result has also been obtained in another experimental study.^{45} According to our theoretical results, a very low degree of K_{ATP} channel activation may easily account for this additional APD reduction. Indeed, both high [K^{+}]_{o} and I_{KATP} activation cooperate to shorten AP (see Fig 6⇑). In the complete absence of I_{KATP} activation, the LuoRudy model^{21} predicts a reduction of relative APD from 100% to 71% when [K^{+}]_{o} rises from 4.0 to 10.3 mmol/L, which is in very good agreement with the value reported by Yan. Given this [K^{+}]_{o}, <0.2% of the total population of K_{ATP} channels would need to open to account for the additional APD reduction (58%) observed by Yan. This degree of activation would in turn be achieved even with very modest variations in intracellular nucleotide concentrations.
In another study, Kodama et al^{45} observed that the reduction in APD caused by substratefree hypoxia with a high [K^{+}]_{o} was similar to that obtained under normal [K^{+}]_{o}. If the data of their Table 1⇑ regarding APD at 80% repolarization are normalized in the same manner as in Fig 6C⇑, it can be deduced that the normalized APD values for different [K^{+}]_{o} for 10 and 15 minutes of hypoxia are practically independent of the value of [K^{+}]_{o}. Our results illustrated in Fig 6C⇑ are in agreement with this observation. Moreover, their results regarding APD reduction in different degrees of hyperkalemia under normoxic and hypoxic conditions (see their Table 1⇑) nicely agree with the results of our model (partially depicted in Fig 6B⇑), suggesting that the hypoxiarelated AP shortening is mainly due to the activation of I_{KATP}.
Ischemic K^{+} Loss and Extracellular K^{+} Accumulation
Our results support the idea that K_{ATP} channel activation does not fully account for the observed cellular K^{+} loss during hypoxia/ischemia. The model predicts the existence of a net increment in J_{efflux} in hypoxia/ischemia, but its magnitude is significantly lower than that observed experimentally (Fig 7C and 7D⇑⇑). Qualitatively, though, our model predicts the wellknown plateau of [K^{+}]_{o} during the early phase of ischemia (see Fig 8B⇑). According to our results, this plateau is reached because of the biphasic behavior of ΔJ_{efflux} (Fig 7A and 7B⇑⇑). Initially, activation of K_{ATP} channel causes an increase in ΔJ_{efflux}, but after a certain value of f_{ATP} is reached, this trend changes and ΔJ_{efflux} declines until it reaches zero value. This would cause a stabilization of [K^{+}]_{o}, as shown in Fig 8B⇑.
However, our simulations show that the fraction of the total K^{+} loss attributable to I_{KATP} would be in the range of 1/5 (Fig 7D⇑), and so other mechanisms must account for the bulk of the observed K^{+} loss. Other possible mechanisms of K^{+} loss include changes in other currents during metabolic impairment, activation of other K^{+} channels during ischemia (such as the Na^{+}activated K^{+} channel), cotransport of lactate or Cl^{−} anions, or extracellular space shrinkage, among others (see Reference 20 for a review).
The results obtained with the model are in partial disagreement with one experimental result, which suggests that a degree of K_{ATP} channel activation of <0.5% would account for the observed hypoxic/ischemic K^{+} loss.^{14} However, the model predictions dealing with the participation of I_{KATP} in ischemic K^{+} loss agree nicely with other reported experimental values regarding the partial prevention of K^{+} loss in ischemia by glibenclamide. For example, the data from Hicks and Cobbe^{46} indicate that the glibenclamideprevented extracellular K^{+} accumulation during 30 minutes of global ischemia in rabbit septum reached 4.1 mmol/L, a value equivalent to an average ΔJ_{efflux} of 0.071 μmol·g^{−1}·min^{−1} (using the value V_{ECW}=0.52 L/kg wet wt reported by Weiss et al^{30} ), which is in the range of values predicted by our model (see Fig 7D⇑). In a study by Yan et al,^{4} glibenclamide reduced K^{+} efflux from 4.51 to 3.47 μmol/g wet wt in a 15minute period of hypoxia with high [K^{+}]_{o}. This yields a value of 0.069 μmol·g^{−1}·min^{−1} for the average ΔJ_{efflux} due to the glibenclamideblocked currents (mainly I_{KATP}), which is again in accordance with the predictions of the model.
Sensitivity of the Results to Model Parameters
One important issue regarding computer models that must always be taken into consideration is the sensitivity of the results to the values of the model parameters. In the model of I_{KATP} presented here, parameters are, in general, well matched to experimental measurements. The parameter that shows the greatest dispersion when measured experimentally is the [ATP] of halfmaximum inhibition of the channel (K_{m} in Equation 4).^{14} ^{26} However, its value does not influence our conclusions because the results are presented in terms of f_{ATP}.
Another parameter that could have influence in the quantitative results, because it multiplies f_{ATP} in Equation 1, is the K_{ATP} channel density (σ). The value chosen (3.8 channels/μm^{2}, derived from Reference 14) lies in the middle of the range of reported values for guinea pig ventricular myocytes. Figures as low as 0.55 channel/μm^{2} have been reported,^{47} and if this value were to be adopted, all the results regarding the value of f_{ATP} should be multiplied by a factor of 7, so the results of the present study would be compromised. However, all subsequent estimates of the parameter σ yielded considerably higher values. If the estimate of Nichols et al^{13} (≈5 channels/μm^{2}, which is the highest value reported for guinea pig cardiac cells) is taken into consideration, the values of f_{ATP} given in the present study would actually be 1.3 times smaller (eg, f_{ATP} needed for a 50% reduction in APD would now be 0.45%). Thus, all qualitative results would still withstand this examination.
As for the values obtained for K^{+} efflux, Equation 6 shows that the results are critically dependent on the chosen V_{ECW}, which has a rather uncertain value. The value chosen for V_{ECW} (0.52 L/kg wet wt) is typical for rabbit septa.^{30} Values as low as 0.2 have been reported for other animal species. If this value of 0.2 L/kg wet wt was adopted, the results obtained regarding K^{+} efflux rates would have been 66% higher. Even in this extreme case, the simulated values of ΔJ_{efflux} would still be on the order of 3 to 4 times lower than the reported experimental results.
Appendix 1: Formulation of I_{KATP}
General Equation
The general equation that describes the total current density through the K_{ATP} channels is the following:where σ is the channel density, g_{0} is the unitary conductance, p_{o} is the maximum channel open probability (in the absence of ATP), f_{ATP} is the fraction of open K_{ATP} channels, V_{m} is membrane potential, and E_{KATP} is the reversal potential.
The value chosen for the channel density was σ=3.8 channels/μm^{2}, and p_{o} was fixed at a value of 0.91.
Unitary Conductance
The expression for the conductance of a single fully open channel is as follows:
The term γ_{0} is the unitary conductance in the absence of intracellular Na^{+} and Mg^{2+} and depends on [K^{+}]_{o}:where γ_{0} is obtained in pS ([K^{+}]_{o} in mmol/L).
The term f_{M} in Equation 9 accounts for inward rectification caused by intracellular Mg^{2+} ions and is formulated by means of a Hill equation:where the halfmaximum saturation constant (K_{h,Mg}) depends on membrane potential and on [K^{+}]_{o}:with the value of the electrical distance (δ_{Mg}) being 0.32. F is the Faraday constant, R is the gas constant, and T is the absolute temperature. The factor K^{0}_{h,Mg} is given by the following:(both K^{0}_{h,Mg} and [K^{+}]_{o} in mmol/L).
The term f_{N} in Equation 9 accounts for inward rectification caused by intracellular Na^{+} ions and is again formulated by means of a Hill equation:where the value of the halfmaximum saturation constant (K_{h,Na}) depends on membrane voltage:The value adopted for electrical distance (δ_{Na}) is 0.35, whereas K^{0}_{h,Na} is 25.9 mmol/L.
Finally, the temperature (T) effect was introduced in Equation 9 according to the following expression:where Q_{10}, T, and T_{0} indicate temperature coefficient, absolute temperature, and reference temperature, respectively, with Q_{10}=1.3 and T_{0}=36°C.
Fraction of Activated K_{ATP} Channels
In the model, the term f_{ATP} in Equation 8 depends on concentrations of intracellular ATP and of free cytosolic ADP, according to the following expression:where both the maximuminhibition constant (K_{m}) and the Hill coefficient (H) depend on [ADP]_{i}. The equations that express these dependencies are as follows:(with K_{m} in μmol/L and [ADP]_{i} in μmol/L) and(with [ADP]_{i} in μmol/L).
Reversal Potential
The reversal potential of the K_{ATP} channel (E_{KATP}) is equal to the equilibrium potential for K^{+} and is thus given by the Nernst equation:
Appendix 2: Calculation of K^{+} Efflux
Calculation of the Average K^{+} Outward Current Density
The total instantaneous K^{+} current density (I_{K,T}) through the membrane is the sum of all the sarcolemmal currents carried by K^{+} ions. In the LRII model, this is expressed as follows:
Subtracting the inward current carried by the Na^{+}K^{+} pump from the total K^{+} current, we obtain the total outward K^{+} current (I_{K,O}):
The average outward current density (I_{out}) was then calculated as the integral mean value of I_{K,O}:
Derivation of Equation 6
The K^{+} efflux rate (J_{efflux}) can be defined as the number of moles of K^{+} leaving the cell (n_{out}) per unit time (Δt) and unit tissue weight (Δm):
The number of moles of n_{out} can be related to the electric charge carried by K^{+} ions leaving the cell (Q_{out}) by means of the Faraday constant (n_{out}=Q_{out}/F). Along with this, Q_{out} is related to the average K^{+} outward current density (I_{out}) aswhere A_{m} is the total membrane area of all the myocytes contained in the tissue of unit mass Δm.
Rearranging the equations, we obtain the following:
Now the total membrane area (A_{m}) can be related to the total cell volume (V_{cel}) by means of the surfacetovolume ratio of the myocyte (S_{v}): A_{m}=V_{cel}·S_{v}. Moreover, V_{cel} can be expressed as the difference between the total tissue volume (V_{t}) and the volume occupied by the extracellular water (V_{e}). Thus, we obtain the following:
The term Δm/V_{t} in the previous equation is the tissue density (ρ), and the term V_{e}/Δm is the extracellular water content per unit weight (V_{ECW}). This yields the following:
Finally, if we want to express J_{efflux} in μmol·g^{−1}·min^{−1} while having S_{v} in μm^{2}/μm^{3}, F in coulomb/mol, ρ in g/cm^{3}, V_{ECW} in mL/g, and I_{out} in μA/cm^{2}, then a unit conversion factor of 600 000 is needed in Equation 25. The resultant equation is identical to Equation 6 in the text.
Selected Abbreviations and Acronyms
AP  =  action potential 
APD  =  AP duration 
BCL  =  basic cycle length 
f_{ATP}  =  fraction of open K_{ATP} channels 
I_{K,O}  =  total outward K^{+} current 
I_{K,T}  =  total K^{+} current 
I_{KATP}  =  ATPsensitive K^{+} current 
I_{KR}  =  sum of K^{+} currents not including I_{KATP} 
I_{K1}  =  inward rectifier K^{+} current 
J_{efflux}  =  K^{+} unidirectional efflux rate 
ΔJ_{efflux}  =  net increment in K^{+} unidirectional efflux rate 
K_{ATP} channel  =  ATPsensitive K^{+} channel 
LRII model  =  phase II LuoRudy model 
V_{ECW}  =  extracellular water volume 
V_{m}  =  membrane potential 
Acknowledgments
This study was supported in part by the Conselleria de Educación y Ciencia de la Generalitat Valenciana (Programa de Formación, Perfeccionamiento y Movilidad de Profesores e Investigadores 94/4158). The authors would like to thank Dr Vicente Lopez Merino for helpful discussions.
Footnotes

Reprint requests to Dr José María Ferrero, Jr, Laboratorio Integrado de Bioingeniería, Universidad Politécnica de Valencia, Camino de Vera s/n, 46020 Valencia, Spain. Email [email protected]

*Note that the correct formulation of the term f_{∞}, which appears in the Ltype Ca^{2+} current formulation, is given in the text of the article by Luo and Rudy^{21} (page 1073) and not in the list of equations at the end of said article.
 Received November 27, 1995.
 Accepted May 10, 1996.
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 Simulation of Action Potentials From Metabolically Impaired Cardiac MyocytesJosé M. Ferrero, Javier Sáiz, José M. Ferrero and Nitish V. ThakorCirculation Research. 1996;79:208221, originally published August 1, 1996http://dx.doi.org/10.1161/01.RES.79.2.208
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 Simulation of Action Potentials From Metabolically Impaired Cardiac MyocytesJosé M. Ferrero, Javier Sáiz, José M. Ferrero and Nitish V. ThakorCirculation Research. 1996;79:208221, originally published August 1, 1996http://dx.doi.org/10.1161/01.RES.79.2.208
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