## Abstract

Repolarization of the action potential (AP) in cardiac muscle is a major determinant of refractoriness and excitability, and can also strongly modulate excitation–contraction coupling. In clinical cardiac electrophysiology, the Q-T interval, and hence action potential duration, is both an essential marker of normal function and an indicator of risk for arrhythmic events. It is now well known that the termination of the plateau phase of the AP and the repolarization waveform involve a complex interaction of transmembrane ionic currents. These include a slowly inactivating Na^{+} current, inactivating Ca^{2+} current, the decline of an electrogenic current due to Na^{+}/Ca^{2+} exchange and activation of three or four different K^{+} currents. At present, many of the quantitative aspects of this important physiological and pathophysiological process remain incompletely understood.

Recently, three mathematical models of the membrane AP in human ventricle myocyte have been developed and made available on the Internet. In this study, we have implemented these models for the purpose of comparing the K^{+} currents, which are responsible for terminating the plateau phase of the AP and generating its repolarization. In this paper, our emphasis is on the two highly nonlinear inwardly rectifying potassium currents, and . A more general goal is to obtain improved understanding of the ionic mechanisms, which underlie all-or-none repolarization and the parameter denoted ‘repolarization reserve’ in the human ventricle. Further, insights into these fundamental variables can be expected to provide a more rational basis for clinical assessment of the Q-T and Q-T_{C} intervals, and hence provide insights into some of the very substantial efforts in safety pharmacology, which are based on these parameters.

## 1. Introduction

Repolarization of the action potential (AP) in mammalian ventricles is a major determinant of action potential duration (APD) and cellular excitability. APD can also modulate excitation contraction coupling. The initiation of repolarization and the return of the voltage to the resting potential are due to a complex interaction between channel-mediated (Na^{+} and Ca^{2+} current inactivation, K^{+} current activation) and electrogenic currents due to Na^{+}/Ca^{2+} exchange and Na^{+}/K^{+} pump. Many of the general principles which are needed to rationalize and integrate the time- and voltage-dependent gating of cardiac K^{+} currents during repolarization have been described by Noble & Tsien (1972) under physiological conditions and by Carmeliet (1999) in the setting of ventricular ischaemia.

Standard electrophysiological studies and related voltage-clamp measurements have yielded substantial new information on the biophysical properties of the K^{+} currents in human ventricular cells. This experimental work has provided the basis for the development of three comprehensive mathematical models of the AP in isolated human ventricular myocytes (Bernus *et al*. 2002; Iyer *et al*. 2004; ten Tusscher *et al*. 2004). These models can be used to provide mechanistic insights into normal and abnormal repolarization waveforms. They may also be useful in electrophysiological studies of the characteristics of damaged or failing ventricular myocytes from human hearts. Mathematical models of heart tissue harbouring known genetic abnormalities, specifically Na^{+} and K^{+} channel mutations or channelopathies, have also been developed (Clancy & Rudy 2001; Lu *et al*. 2001).

The principal goals of the theoretical work which we have done is to explore further the fundamental concept in cardiac repolarization denoted ‘all-or-none repolarization’ and to provide an improved understanding of the term ‘repolarization reserve’. Both concepts have been applied as a measure of the safety factor for repolarization in normal hearts and in some studies of proarrhythmic potential of drugs.

An essential starting point was to make a detailed comparison of the contribution of the major repolarizing K^{+} currents in available models of human ventricular myocytes. Since the formulation of is the same in the ten Tusscher *et al*. model (2004) and in the model by Priebe & Beuckelmann (1998) and the minor differences in the dynamics of were not significant in the present model comparison, we have focused on a detailed comparison of the inwardly rectifying K^{+} currents in: (i) the ten Tusscher *et al*. model (2004) and (ii) the Iyer *et al*. model (2004). This initial emphasis on inwardly rectifying K^{+} currents was chosen, because mathematical modelling can identify and illustrate important functional consequences of the nonlinearities of these types of K^{+} conductances. Moreover, the inwardly rectifying K^{+} conductance (or Kir2.1) can be modulated during cardiac ischaemia and it is dramatically down regulated in patients diagnosed with Andersen-syndrome (Lopes *et al*. 2002). The second inwardly rectifying K^{+} current, which is expressed in mammalian myocytes, (or HERG) plays an important role in initiating repolarization. A number of well-characterized genetic defects in repolarization of human hearts arise as a result of abnormal gating or altered expression of this K^{+} conductance (Huang *et al*. 2001; Antzelevitch 2004). In combination, the current changes due to and HERG provide a significant fraction of the repolarizing current in human ventricle. This justifies their study in the context of attempting to obtain new insights into all-or-none repolarization and/or the parameter denoted ‘repolarization reserve’.

## 2. Methods

### (a) Mathematical models of human ventricle myocytes

The mathematical model of Iyer *et al*. was used, after adapting the Fortran code provided in the Internet. The recommended initial conditions: 1 Hz pacing ( mV, mM, mM and mM) were employed.

The model of ten Tusscher *et al*. is also provided via the Internet, encoded in C++. The initial conditions in this code were set to mV, , and mM. The was judged to be somewhat high when compared to values in the literature. Accordingly, the maximum Na^{+}/K^{−} pump current was adjusted to pA pF^{−1} and the initial conditions to mV, , and mM.

The external ion concentrations for both models were set to , and mM.

Iyer *et al*. have developed a model based on experimental data obtained from left ventriclular tissue. This may not fully apply to the ten Tusscher *et al*. model. Detailed information concerning how the experimental datasets were used to derive the model parameters are not always provided in these original papers.

### (b) Protocols

Several protocols were applied to analyse the behaviour of the inwardly rectifying K^{+} currents and assess their contributions to repolarization in human ventricular myocytes. Some of these, e.g. the current–voltage (*I–V*) relations generated using voltage clamp waveforms, are analogous to experimental manoeuvre and can be compared directly. Others, like the quasi-instantaneous *I–V* relations are important, but more difficult to compare directly to experimental data.

The steady-state *I–V* relations of selected K^{+} currents were simulated using a ramp function. The membrane voltage was clamped at −90 mV for 2 s and then slowly increased at a constant rate of 6 mV s^{−1} (see figure 3). A second type of *I–V* relation (see figure 2) was simulated by using a voltage clamp waveform very similar to a normal AP. This waveform included a small initial step to more negative values to start negative to the reversal potential of .

In figures 4 and 5, we present quasi-instantaneous *I–V* relations of the net current and selected K^{+} currents at selected times during a normal AP. At the denoted time points the gating states and the internal ion concentrations were held constant and the membrane voltage was changed (instantaneously) from −100 to +50 mV to derive the current densities. This approach has been used by Jack *et al*. (1975). It assumes that the *m* gate reacts instantaneously, i.e. maintains its steady-state value during the imposed voltage changes.

To simulate drug-induced block of selected K^{+} currents, we changed the conductance parameters of and , respectively. The initial conditions were chosen from the periodic solution at 1 Hz pacing. Inhibition of the currents was done instantaneously, and the first AP in the resulting computations is illustrated in this paper. (Except for prolongation, the waveforms did not change significantly even after 10 min of simulation.)

To enable a detailed comparison of the effect of in these two models, we substituted the formulation of the Iyer *et al*. model into the ten Tusscher *et al*. model. Its conductance was adjusted so that the current density showed the same amount of peak current in the *I–V* relation as in the normal ten Tusscher *et al*. model, i.e. it was set to nS pF^{−1}.

### (c) Numerical issues

All simulations were done in Matlab on a PC with Intel Pentium IV 3 GHz CPU. To solve the ordinary differential equations, the stiff solver ‘ode15s’ provided by Matlab was used. This is a variable order solver based on the numerical differentiation formulae, optionally using the backward differentiation formulae (Gear's method). Calculating one AP (i.e. 1 s for stimulation at 1 Hz) of the ten Tusscher *et al*. model took about 7 s, whereas the equivalent for the Iyer *et al*. model took 133 s, e.g. approximately 18 times larger, mainly due to the inclusion of Markov chain models for several channel-mediated currents.

## 3. Results

### (a) Net transmembrane ionic currents during repolarization

Figure 1 provides a comparison of the main repolarizing transmembrane ionic currents underlying the membrane APs generated using the ten Tusscher *et al*. model (left) and the Iyer *et al*. model (right). Figure 1*a* shows a membrane AP generated by each of these models. Figure 1*b–d* illustrates the corresponding main repolarizing currents. In each, the net current, , is denoted as a heavy black line (as a reference). The time axis is the same for all panels.

At a time corresponding to the start of the simulation, the net current has a large negative (depolarizing) value due to the rapid activation of (not shown) following the applied stimulus. After the initial peak of the AP, a Ca^{2+} independent transient outward K^{+} current, , is activated in both models. This current change dominates the net current, resulting in a net outward (repolarizing) value. This is soon offset by the activation of and becomes negative again. These dynamics create the notch after the initial upstroke of the AP in both models.

After these large changes in net current, the plateau develops. During this phase of the AP, the net current is small and almost constant measuring about 0.1–0.2 pA pF^{−1}. For comparison is about 5 times larger. Functionally, this means that even ‘small’ current changes can have a rather large influence on the AP waveform during this plateau phase.

There is no exact definition of the point which marks the end of the plateau phase or the beginning of the repolarization phase. This is sometimes denoted (final repolarization—as opposed to early repolarization). Initiation of repolarization can be detected by a change in . This approach is utilized here. Since the net current determines the slope of the AP waveform, we set the beginning of the repolarization phase (left border of the grey area in figure 1*a*) at a net outward current of 0.5 pA pF^{−1}. This provides a straightforward approach for separating the plateau from the final repolarization phase in both models.

During this final repolarization phase, increases substantially. This produces a large outward current, as is decreasing to zero and the Na^{+}/Ca^{2+} exchange current develops. The right border of the shaded area in figure 1*a* indicates the maximum outward current during this phase of repolarization.

### (b) Detailed comparison of K^{+} currents in these mathematical models of human ventricle

#### (i) The background inwardly rectifying K^{+} current:

Both models use the same basic formulation for This formulation assumes that the gating kinetics are sufficiently fast to be considered instantaneous. However, a very important difference between the two models is the function describing the open probability , and specifically its dependence on the membrane voltage. In the ten Tusscher *et al*. model essentially no channels are open at membrane potentials corresponding to the AP plateau and thus almost no outward current can be produced (figure 2*a*,*c*). Although the Iyer *et al*. model has only a small open probability in this important range of membrane potential, the resulting outward current is substantial (Iyer *et al*. 2004). In the type of formulation for , this is due to the relatively large electrochemical driving force (), which results in even a small amount of open channels producing a substantial current. We show below that the rather small difference in current in this region of membrane potential (between −10 and +20 mV) leads to very different qualitative behaviour of the simulated AP repolarization.

Most of the published data on in human ventricle has focused on the more negative range of membrane voltage. Thus, it is not possible to determine unequivocally which of these models would be more reliable for simulation of repolarization (see discussion in appendix A). To obtain a reliable estimate of the contribution of current changes of during the plateau phase (+20 to −10 mV) and define the contribution to repolarization of the AP additional data are needed. An experiment which measures the AP shape changes following block would be necessary (see Shimoni *et al*. (1992) and compare figure 6).

#### (ii) The ‘rapidly activating delayed rectifying’ K^{+} current:

The formulation of the so-called rapidly activating inward rectifier K^{+} current, HERG or is very different in these two mathematical models. Iyer *et al*. use a Markov chain model, whereas ten Tusscher *et al*. have employed a conventional Hodgkin–Huxley formalism. When an *in silico* voltage clamp AP waveform is applied as a forcing function, similar *I–V* relations for are obtained for both mathematical models (figure 2) despite the differences of the current densities assigned in these two models. This similarity holds also when a ramp function for the voltage clamp is used (figure 3). The main difference is that the peak in the *I–V* relation is about 5 mV more negative in the ten Tusscher *et al*. model.

In summary, and as illustrated in figures 2*b*,*c* and 3*b*,*c*, the current is very small in both models. However, it is important to note that is activated in a range of membrane potentials in which the net current is also very small. Moreover, the inwardly rectifying properties/mechanisms of can result in the deactivation current tails of , contributing a significant current change—during final repolarization. This HERG current is comparable in size to these due to the Na^{+}/K^{+} pump or the Na^{+}/Ca^{2+} exchanger (see Gintant 2000).

#### (iii) The slowly activating delayed rectifying K^{+} current:

There is a lack of agreement in the literature on the relative size or expression level of in healthy human ventricular cells (Veldkamp 1998). Apparently expression is very sensitive to the experimental myocyte isolation procedures, and is also influenced by the inhomogeneous expression of its molecular substituents minK and/or KvLQT1 in the human ventricle. This variability and intrinsic heterogeneity has made it very difficult to develop a mathematical model of for the human ventricular myocyte. In the ten Tusscher *et al*. model the maximal conductance for was chosen somewhat arbitrarily; determined by the goal to achieve an APD difference between epicardial and M-cell of about 100 ms (as has been demonstrated in ventricular wedge preparations from adult canine hearts). For this reason, density in epicardial myocytes in the ten Tusscher *et al*. model is relatively large. We have used the M-cell model in the ten Tusscher *et al*. paper as ‘normal’ ventricle myocyte. Comparison of the *I–V* relations (figures 2*b*,*c* and 3*b*,*c*) for makes it apparent that activates at much more negative membrane voltage in the ten Tusscher *et al*. model than in the Iyer *et al*. model.

### (c) Quasi-instantaneous *I–V* relations

To begin to investigate how a current or voltage change could alter repolarization (i.e. the net current) at selected time points during the AP, we have plotted *I–V* relation curves calculated as quasi-instantaneous values at pre-selected time points (figures 4 and 5). The continuum represented by the points denoted #1–5 provide the *I–V* relation of for a normal AP. Adopting an approach used in Jack *et al*. (1975) we have set the *m* gate of to change instantaneously to the respective voltage change for these calculations of the *I–V* curves, i.e. *m* has its steady-state value at each membrane potential. This is justified based on its much faster kinetics than the gating parameters for the other currents of interest. The resulting simulations demonstrate how small deviations of voltage from the resting potential can be ‘reset’ by the outward currents due to . However, larger changes exceed the AP threshold. The activation of then dominates this quasi-instantaneous *I–V* curve (#1).

At the beginning of the plateau phase (#2), the net current is very small (see above and figure 1). Note, however, that any rapid hyperpolarization at this time would reveal a counteracting inward (depolarizing) current. At progressively later times in the plateau or during final repolarization, this ability is reduced, because has inactivated substantially. Finally at (#4) an applied hyperpolarization would be expected to evoke even a faster repolarization (see also Gray *et al*. 2001).

Figure 5 shows the contribution of selected K^{+} currents to the phenomena described above. These *I–V* curves illustrate the computed changes in , , and the Na^{+}/K^{+} pump current as a consequence of an imposed voltage clamp corresponding to a ‘typical’ human ventricular AP. The time-dependent changes in current from (#1) diastole, through (#5) final repolarization, are shown as five superimposed quasi-instantaneous *I–V* curves.

### (d) Effects on AP waveform on changes in K^{+} current densities

To begin to investigate the effects of changes in and on the membrane potential waveform during repolarization, we have systematically changed the conductance of these two K^{+} currents and then computed corresponding AP waveforms. Figure 6 shows a simulation of 60% block of and , respectively. This amount of block of changed the APD, but had only a minor effect on the slope of the final repolarization.

In contrast, a 60% reduction of produced marked changes in the slope of late repolarization. As a consequence of having a substantial magnitude in the Iyer *et al*. model at membrane potentials during the plateau phase (see *I–V* relations in figure 2*a*) block of this K^{+} current lengthened the plateau phase. Thus , in this model, has a much larger effect on APD90 and the overall waveform than . The size of in the ten Tusscher *et al*. model at 0 mV is 0.025 pA pF^{−1}, in the Iyer *et al*. model, it is 0.442 pA pF^{−1} (net current and 0.42 pA pF^{−1}, respectively). These small expression levels of indicate that the ‘natural variability’ between cells may make it difficult to determine how much is present at plateau potentials. A standard deviation of about 10% of the mean is common in electrophysiological studies, although ramp voltage clamp potentials combined with signal analysis can provide very reliable estimates, of small, dynamic currents at a high signal-to-noise ratio. When we substituted the formulation of the Iyer *et al*. model into the ten Tusscher *et al*. model, we obtained a similar pattern of results from the ten Tusscher *et al*. model. Clearly, it is very important to determine whether can generate an outward current during the plateau phase (see §4) of the human ventricular AP.

Another important insight generated from computations using the ten Tusscher *et al*. model is that even though is small relative to (the peak current in the *I–V* relation is about 5% that of ), the effect of a selected percentage block of has a larger influence on APD than the same percentage of block of .

We also performed sensitivity analysis for both models. This analysis is based on the calculation of the derivative , i.e. the change of the membrane potential due to a change in a selected parameter *p*, for instance a reduction in an individual K^{+} channel conductance. This approach yielded similar insights to those obtained from altering a specific conductance in order to mimic ion channel block. An advantage of sensitivity analysis is that the quantitative output is strictly defined and can be used for comparisons of changes in parameters in each model. In contrast, due to the nonlinear *I–V* relationships such as those for and , the effects of two different percentage blocks lead to different, non-comparable results.

## 4. Discussion

### (a) Summary of main findings

We wish to acknowledge the availability and usefulness of both mathematical models of the human ventricle, which we have downloaded and used in this study. This modelling environment and previous mathematical models for human atrial myocytes (Courtemanche *et al*. 1998; Nygren *et al*. 1998) provide very useful platforms and potentially valuable approaches, for theoretical work on human cardiac electrophysiology and rhythm disturbances.

The starting point for our work consisted of a detailed comparison of membrane AP waveforms and underlying ionic currents in the ten Tusscher *et al*. and Iyer *et al*. models. This comparison is shown in figure 1. Our specific goal was to explore the functional role(s) of the two K^{+} currents which exhibit marked inward rectification, namely and . Although the approaches used to formulate differ substantially in these two mathematical models, the calculated current densities and kinetics are quite similar (see figures 2 and 3). In contrast, although the mathematical descriptions for in these two models are quite similar, both the current density and the waveform of inwardly rectifying *I–V* differ very substantially. As a consequence, in the Iyer *et al*. model, generates substantial outward current at membrane potentials corresponding to the plateau of the AP, while in the ten Tusscher *et al*. model the pronounced negative slope of the *I–V* curve for results in these K^{+} channels ‘shutting off’ completely at membrane potentials positive to approximately −5 mV. Note, however, (figure 2) that in both models the maximum outward current due to is much larger than and the electrogenic current due to Na^{+}/K^{+} pump current combined. is therefore a major determinant of final repolarization in human ventricle, as it is in rabbit ventricle (Shimoni *et al*. 1992) and ventricle myocytes from adult canine (Melnyk *et al*. 2002) and guinea pig (Miake *et al*. 2003). This is illustrated qualitatively in the computation shown in figure 6; a 60% inhibit of markedly alters the final repolarization AP waveform and thus prolongs APD in both of these models of the human ventricle membrane AP.

An important functional aspect of both models is that the human ventricular AP is characterized by relatively long (200 ms) plateau and that, at membrane potentials corresponding to the plateau, the myocyte, like that of most mammals, has a high resistance. This very important concept was first demonstrated by Weidmann (1956). Recent studies have shown that the resistance at membrane potentials corresponding to the end of the plateau phase is approximately 2.5–4 times larger than that measured in diastole in guinea pig ventricle (Zaniboni *et al*. 2000) and similar results have been obtained in myocytes isolated from rabbit ventricle (K. Spitzer & W. Giles, unpublished work). Our estimates of this important functional characteristic using the ten Tusscher *et al*. model yielded a ratio of approximately 160/40 MΩ and the corresponding ratio for Iyer *et al*. was 200/40 MΩ; both of which are in qualitative agreement with published data. This property, which arises mainly from the expression of inwardly rectifying K^{+} currents in human ventricle results in the plateau of the AP being very sensitive to any applied currents or intrinsic net current changes. Accordingly, it contributes to the phenomenon denoted all-or-none repolarization (Noble & Tsien 1972) and the related finding of time-varying threshold for repolarization during the AP waveform. The conceptual basis for this was developed by Noble & Hall (1963) and described in detail by Noble & Tsien in their 1972 review. Some of the fundamental mechanisms for human ventricle are illustrated in figures 4 and 5.

### (b) What is ‘repolarization reserve’?

The calculations in figures 1–3 demonstrate plausible schemes in which quite complex interactions between activating and inactivating ion channel-mediated currents, changes in the Na^{+}/Ca^{2+} exchange current, the intrinsic ion transfer characteristics of the background inwardly rectifying K^{+} current and the electrogenic Na^{+}/K^{+} pump current can combine to produce repolarization. Repolarization is initiated by a net outward current and subsequently is modulated or driven by activation of time- and voltage-dependent K^{+} currents. Although this working hypothesis may provide a qualitative explanation for AP repolarization in human ventricle, it does not fully account for the parameter: repolarization reserve. This concept has been utilized in studies of normal repolarization in ventricular myocardium and also as a descriptor of altered repolarization following hormonal changes (Wu & Andersen 2002) or during analyses of arrhythmias in the field of Safety Pharmacology.

In principle, the repolarization reserve could be defined operationally as the sum of the fully activated K^{+} currents , and/or under physiological conditions. In fact, since responds to imposed voltage changes almost instantaneously; at a normal plasma , this conductance mechanism generates its maximal current (delimited by its nonlinear ion transfer characteristics) during each AP. The time- and voltage-dependent properties of both and can result in residual activation, due to time-dependent decline of tail currents. This has been demonstrated experimentally (Gintant 2000; Berecki *et al*. 2005) and analysed using a mathematical model of guinea pig ventricle (Viswanathan *et al*. 1999). Note, however, the pronounced inward rectification of . This has the consequence that the outward current due to activation is significant only in a small range of membrane voltage (e.g. −20 to −40 mV). The relative slow deactivation (current tails) corresponding to can also contribute to significant repolarizing current (see Gintant 2000).

As noted is not the focus of this study. However, it is apparent that the intrinsic gating properties of this slowly activating, but relatively large K^{+} conductance and its relatively linear *I–V* relation will result in channels contributing substantially to repolarization reserve. would be expected particularly at relatively high heart rate or during pathophysiological situations including abnormal AP prolongation. A recent paper based on an improved formulation for *I*_{k,s} will be very useful in this context (Silva & Rudy 2005).

### (c) Implications for measurements of the Q-T interval of the electrocardiogram

The genesis of the T wave of the electrocardiogram and hence the unambiguous interpretation of the Q-T interval or its adjunct, the rate-corrected or Q-T_{C} interval continue to be topics of importance and debate in clinical cardiac electrophysiology. The insights gained from this and similar studies (Malfatto *et al*. 2003), make it apparent that attempts to relate changes in either Q-T or the Q-T_{C} intervals to alterations in the size(s) or the expression levels of any individual K^{+} current in human ventricle must always be done with considerable caution. The amount of change in *I*_{k,r} due to ‘pharmacological’ block does not scale linearly with Q-T change.

## 5. Limitations of our study

Both of these mathematical models of the membrane AP in human ventricle represent very valuable integrative tools for use as an adjunct to cellular electrophysiological experimentation; and research and education in clinical cardiac electrophysiology. The cardiac AP is sufficiently complex that intuition is of limited use, even in the setting of studies based on single, isolated myocytes. The example we have chosen, an analyses of the functional role(s) of the two inwardly rectifying K^{+} currents illustrates this point. However, further insights could be gained from this approach if:

the models were further refined based on more complete experimental measurements of , and in isolated myocytes from human ventricle at physiological heart rates;

more complete data on the intracellular Ca

^{2+}transient was available;the effects of tonic release of neurotransmitters or autonomic tone was included, and

insights gained from these studies of membrane APs could be evaluated further

*in silico*under conditions where one-, two- and three-dimensional activation and conduction were included.

There is reason to believe that much of the experimental data can be obtained. Preliminary multi-scale models of rabbit, canine and human ventricle have been developed. Accordingly, the challenge of relating cellular electrophysiology to clinical cardiac electrophysiology and Safety Pharmacology is being addressed (Noble 2002).

## Editors' note

Please see also related communications in this focussed issue by Biktasheva *et al*. (2006) and Marée *et al*. (2006).

## Acknowledgments

We are grateful for the financial assistance from the Office of the Deans of Medicine and Engineering at UCSD which provided the Postdoctoral Fellowship for M.F. In addition, this study was supported by the Canadian Institutes of Health Research and the Heart Stroke Foundation of Canada. The financial assistance of the Research Chair of the Heart and Stroke Foundation of Alberta and the N.W.T. is also gratefully acknowledged. Work in Oxford was supported by the Wellcome Trust and by the EU 6th Framework BioSim Consortium.

## Footnotes

One contribution of 13 to a Theme Issue ‘Biomathematical modelling I’.

- © 2006 The Royal Society