From the Hodgkin-Huxley axon to the virtual heart

Abstract

Experimentally based models of the heart have been developed since 1960, starting with the discovery and modelling of potassium channels. The early models were based on extensions of the Hodgkin–Huxley nerve impulse equations. The first models including calcium balance and signalling were made in the 1980s and have now reached a high degree of physiological detail. During the 1990s these cell models have been incorporated into anatomically detailed tissue and organ models to create the first virtual organ, the Virtual Heart. With over 40 years of interaction between simulation and experiment, the models are now sufficiently refined to begin to be of use in drug development.

Extending the Hodgkin–Huxley equations

Modelling excitable cells took a giant leap forward when Alan Hodgkin & Andrew Huxley (1952) published their equations for nerve conduction in the giant axon of the squid, an achievement for which they received the Nobel Prize for Physiology and Medicine in 1963. It was the first model to use mathematical reconstruction of experimentally determined kinetics of ion channel transport and gating, rather than abstract equations (e.g. Van der Pol & Van der Mark, 1928), and it correctly predicted the shape of the action potential, the impedance changes and the conduction velocity. As a brilliant demonstration of the need for biological modelling to respect the details of experimental results to achieve accurate predictions, I was fascinated by this work as a young medical student at University College London in the 1950s. As soon as Otto Hutter and I (Hutter & Noble, 1960; Hall et al. 1963) had demonstrated the existence of two kinds of potassium channels in Purkinje fibres of the heart, the inward rectifier i_K1 and the delayed rectifier i_K, I was keen to see whether the Hodgkin–Huxley model could be adapted to apply to the cardiac action potential and pacemaker rhythm.

Fitting equations to the data we had on the K⁺ channels was the first stage. For the delayed rectifier I used slowed-down versions of the Hodgkin–Huxley (HH) K⁺ current equations, representing the fact that a process that takes a few milliseconds in nerve requires hundreds of milliseconds in heart. For the inward rectifier, I fitted my own equations, including notably and successfully, the dependence of this current on external potassium (Noble, 1965). For the sodium current, I used the Hodgkin–Huxley equations since Weidmann (1956) had shown that the sodium inactivation process was very similar in the heart.

The next problem was how to solve the resulting differential equations. It took Andrew Huxley months with a Brunsviga mechanical calculator to solve for just a few milliseconds of nerve activity. At that rate, solving for hundreds of milliseconds of cardiac activity would have taken years. It was therefore necessary to use one of those huge valve machines, an early electronic computer. Fortunately, there was one in London University, though getting time on it was not easy, nor was the programming. Computer languages had hardly emerged from the endless gibberish of machine code (Noble, 2006b).

The first results, however, were disappointing. Despite the fact that the HH equations predict a steady plateau sodium current, later discovered experimentally (Attwell et al. 1979), it wasn't large enough during strong depolarizations. In retrospect, we can see that this calculation predicted either that the sodium current in heart differs from that in nerve or that other inward current channels were present. Both predictions are correct, but the demonstration of calcium channels had to wait until Reuter's pioneering work (Reuter, 1967). So, I opted in 1960 for modifying the sodium activation equations to generate current over a wider range of potentials. The inward current equations in the 1960 model are therefore seriously incomplete, not only by omitting calcium channels but also because we now know that there are late components of sodium current that are not predicted by the HH equations (Kiyosue & Arita, 1989; Maltsev et al. 1998; Sakmann et al. 2000; Zygmunt et al. 2001) and that these can play an important role in drug action (Bottino et al. 2006; Noble & Noble, 2006).

Nevertheless, the broad outline of the time course of potassium current was successfully reconstructed (Fig. 1). Moreover, the slow time course of decay of i_K readily allowed the equations to account for pacemaker rhythm at about the right frequency. As a first working model, it also illustrated an important property of the electrophysiology of repetitive activity: that pacemaker activity is an integrative characteristic of the system as a whole; there is no ‘molecular driver’. This kind of analysis is now one of the major features of the systems biology approach (Noble, 2006a, 2006b,). I think I can claim therefore to have been doing the ‘new’ subject of systems biology for around 45 years!

Figure 1. The first cardiac model.

A, the first analysis of potassium channel currents in the heart and their incorporation into a heart cell model. Redrawn from Hall et al. (1963). The continuous line shows the total membrane current recorded in a Purkinje fibre in a sodium-depleted solution. The inward-rectifying current was identified as i_K1, which is extrapolated here as nearly zero at positive potentials. The outward-rectifying current, i_K, is now known to be mostly formed by the component i_Kr. The horizontal arrow indicates the trajectory at the beginning of the action potential, while the vertical arrow indicates the time-dependent activation of i_K which initiates repolarization. B, sodium and potassium conductance changes computed from the first biophysically detailed model of cardiac cells (Noble, 1962). Two cycles of activity are shown. The conductances are plotted on a logarithmic scale to accommodate the large changes in sodium conductance. Note the persistent level of sodium conductance during the plateau of the action potential, which is about 2% of the peak conductance. Note also the rapid fall in potassium conductance at the beginning of the action potential. This is attributable to the properties of the inward rectifier i_K1, and it helps to maintain the long duration of the action potential, and in energy conservation by greatly reducing the ionic exchanges involved. The current i_K is then responsible for repolarization. The arrows correspond to those shown in the top panel. These insights into the main potassium current changes have been incorporated into all subsequent models of cardiac cells.

The 1962 model also revealed the nature of the balance evolution had struck in developing long action potentials in the heart. The inward rectifier, i_K1, is energy saving. By greatly reducing potassium ion flow during depolarization, it enables much smaller inward sodium and calcium currents to maintain the depolarization and so requires much less energy expenditure by ion pumps to restore the transmembrane gradients. This is important since the ATP consumption by ionic pumps is significant.

These insights have stood the test of time. They are also features of all subsequent cardiac cell models.

Developments of the cell models

We now know of course that there are many more ion channels and other transporters than were represented in that early model. I will review the key developments according to the decades in which they appeared.

The 1960s revealed three major changes. The first was Reuter's discovery of calcium current in the heart (Reuter, 1967). The second was the discovery with Dick Tsien that there are multiple components of slow changes in potassium currents (Noble & Tsien, 1968, 1969). These two developments formed the basis of the McAllister–Noble–Tsien Purkinje fibre model (McAllister et al. 1975), which was later developed by Beeler & Reuter (1977) into the first model of a ventricular cell, which in turn was a precursor to the Luo–Rudy models (Luo & Rudy, 1994, 1994). There was also a third important discovery in the 1960s, which was that of the sodium–calcium exchange in cardiac muscle (Reuter & Seitz, 1969). This was not, however, incorporated into electrophysiological models at that time since the early results suggested that the exchange was electrically neutral (two sodium ions for each calcium ion).

Almost as soon as these 1970s models had been formulated, problems developed. I have reviewed these problems in considerable detail in a previous review lecture (Noble, 1984) and in a recent essay (Noble, 2002b). Here, I will briefly refer to the key developments that became important for the development of later models. The first was DiFrancesco's (1981) discovery that the slow ionic current change in the pacemaker range of potentials in Purkinje fibres was not a pure potassium current. He showed that the channel responsible carries both sodium and potassium ions and is activated by hyperpolarization, not by depolarization. In fact, it is the same mechanism, i_f, as that already discovered in sinus node cells (see reviews by Irisawa et al. 1993; DiFrancesco, 1993,2006). The second was the progressive realization that not all of the ‘slow inward current’ was attributable to ionic current flow through calcium channels. Some of it was found to be carried by sodium–calcium exchange, for which the stoichiometry was revised upwards to 3: 1 instead of 2: 1, which made it electrogenic. These developments were used to develop the DiFrancesco–Noble model (DiFrancesco & Noble, 1985). This was the first model to take into account changes in ionic concentrations during electrical activity and to formulate a model of intracellular calcium signalling. These extensions were necessary not only to account fully for DiFrancesco's experimental findings on i_f but also to explain how the new model could account for previous experimental work on slow current changes at negative potentials.

Although DiFrancesco and I had detailed experimental evidence for most of the major developments in this model, there was one very significant gap. We needed to incorporate an electrogenic sodium–calcium exchanger but we did not know its voltage dependence. We chose to develop some equations that had been developed theoretically by Mullins (1981), and in doing this we had a remarkable stroke of luck. Figure 2B shows the voltage dependence given by these equations at various external sodium concentrations. Figure 2A shows the relations determined experimentally later by Kimura et al. (1986, 1987). It is easy to imagine the pleasure DiFrancesco and I experienced when these experimental results appeared.

Figure 2. The DiFrancesco-Noble model.

A, experimental results obtained by Kimura et al. (1987). B, current–voltage relations given by the equations for sodium–calcium exchange used in the DiFrancesco–Noble model. The curves show the relations at various external sodium concentrations. C, action and pacemaker potentials computed from the DiFrancesco–Noble modelling highlighting the roles played by activation of i_f during the pacemaker depolarization and of i_NaCa during the action potential.

It was the prediction from this model that there must be inward sodium–calcium exchange current flowing during the action potential that led to the development of the Hilgemann–Noble model (Hilgemann & Noble, 1987) and its single cell version (Earm & Noble, 1990). This model opened up the field to simulation of calcium handling by cardiac cells, an area that has now expanded into intense activity in a variety of laboratories (Eisner et al. 2000; Bers, 2001; Hinch, 2004; Hinch et al. 2006; Sobie et al. 2006). The DiFrancesco–Noble and Hilgemann–Noble models became the generic models from which all the modern models of excitation–contraction coupling derive (Luo & Rudy, 1994, 1994; Jafri et al. 1998; Noble et al. 1998; Winslow et al. 1999, 2000).

The Hilgemann–Noble model addressed a number of important questions concerning calcium balance, as follows.

1. How quickly is calcium balance achieved? Net calcium efflux is established as soon as 20 ms after the beginning of the action potential (Hilgemann, 1986), which was considered to be surprisingly soon. In the model this was achieved by calcium activation of efflux via the Na⁺–Ca²⁺ exchanger, thus revealing the time course of one of the important functions of this transporter in the heart. These results apply to the relatively short action potential of the atrium. The details vary according to the location of cells in the heart and which species we are dealing with.

2. Since the exchanger is electrogenic, where was the current that this would generate and did it correspond to the quantity of calcium that the exchanger needed to pump? Mitchell et al. (1984) provided the first experimental evidence that the action potential plateau is maintained by sodium–calcium exchange current. The Hilgemann–Noble model showed that this is precisely what one would expect, both qualitatively and quantitatively. Subsequent experimental and modelling work has fully confirmed this conclusion (Egan et al. 1989; LeGuennec & Noble, 1994; Eisner et al. 2000; Bers, 2001). In particular, Earm et al. (1990) performed the necessary experiments to show this phase of exchanger current during the late plateau of atrial cells (see Fig. 3).

3. Could a model of the SR that reproduced the major features of Fabiato's (1983) experiments be incorporated? The model followed as much of the Fabiato data as possible, but while broadly consistent with the Fabiato work it could not be based on that alone. Fabiato's experiments were heroic but they were done on skinned fibres, which removes many of the relevant membrane-based mechanisms. It is an important function of simulation to reveal when experimental data needs extending.

4. Were the quantities of calcium, free and bound, at each stage of the cycle consistent with the properties of the cytosol buffers? The great majority of the cytosol calcium is bound so that, although large calcium fluxes are involved, the free calcium transients are much smaller, as they are experimentally.

Figure 3. Predictions and experiments concerning the role of sodium–calcium exchange current during the late plateau of atrial cells.

A, the single atrial cell model (Earm & Noble, 1990) developed from the Hilgemann–Noble model showing the action potential (top), some of the ionic currents, including sodium-calcium exchange, and the calcium transient and contraction (bottom). B, experimental recordings of the sodium–calcium exchange current during voltage clamp at a potential in the range of the late plateau (Earm et al. 1990). The current trace is similar to that predicted by the model and, as a calcium chelator is diffused into the cell, the current change is eliminated.

The major deficiency of this model was that it could not account for graded release of calcium from the SR. Much more complex models, incorporating finer detail of the excitation–contraction process, including the communication between L-type calcium channels and the SR calcium release channels, are required to achieve this (Stern, 1992; Jafri et al. 1998; Rice et al. 1999; Winslow et al. 2000; Greenstein & Winslow, 2002).

Modelling of cardiac cells is now a highly active field. There are more than 30 curated cell models on the CellML website (http://www.cellml.org) that can be run with compatible software such as COR (http://cor.physiol.ox.ac.uk/). The days when such biological computation was the preserve of just a few with the relevant maths and computing expertise are over. We can look forward to simulation being as central to work in physiology as it is in physics and engineering, used by experimentalists as well as by theoreticians. It will be the iterative interaction between the two that will be important in refining our understanding and predictive ability.

Linking levels: building the virtual heart

I have been privileged to collaborate with several of the key people involved in extending cardiac modelling to levels higher than the cell. The earliest work was with Raimond Winslow who had access to the Connection Machine at Minnesota, a huge parallel computer with 64 000 processors. We were able to construct models in which up to this number of cell models were connected together to form 2D or 3D blocks of atrial or ventricular tissue. This enabled us to study the factors determining whether ectopic beats generated during sodium overload would propagate across the tissue (Winslow et al. 1993) and to study the possible interactions between sinus node and atrial cells (Noble et al. 1995).

At about the same time, I stayed for a period as a visiting professor at the University of Auckland where Peter Hunter, Bruce Smaill and their colleagues in bioengineering and physiology were constructing the first anatomically detailed models of a whole ventricle. These models include fibre orientations and sheet structure (Hooks et al. 2002; Crampin et al. 2004), and have been used to incorporate the cellular models in an attempt to reconstruct the electrical and mechanical behaviour of the whole organ.

This work includes simulation of the activation wave front (Smith et al. 2001; Noble, 2002a). This is heavily influenced by cardiac ultra-structure, with preferential conduction along the fibre–sheet axes, and the result corresponds well with that obtained from multielectrode recording from dog hearts in situ. Accurate reconstruction of the depolarization wavefront promises to provide reconstruction of the early phases of the ECG to complement work already done on the late phases (Antzelevitch et al. 2001) and as the sinus node, atrium and conducting system are incorporated into the whole heart model we can look forward to the first example of reconstruction of a complete physiological process from the level of protein function right up to routine clinical observation. The whole ventricular model has already been incorporated into a virtual torso (Bradley et al. 1997), including the electrical conducting properties of the different tissues, to extend the external field computations to reconstruction of multiple-lead chest and limb recording. Incorporation of biophysically detailed cell models into whole organ models (Noble, 2002c, 2002a; Trayanova 2006; Crampin et al. 2004) is still at an early stage of development, but it is essential to attempts to understand heart arrhythmias. So also is the extension of modelling to human cells (Nygren et al. 1998; Ten Tusscher et al. 2003).

Applications of the models

Understanding the electrophysiology of the heart is critical to resolving a major problem for the drug industry. This is that a large number of compounds target the proteins involved in cardiac repolarization, so causing arrhythmic side-effects that can be fatal. The cardiac models are now being used in drug development and in attempting to understand these problems (Noble & Colatsky, 2000; Bottino et al. 2006). Since I began with a problem in modelling the sodium current, it is appropriate therefore to finish this review lecture with an example of application to drug development that depends on understanding the role of sodium channels. There are compounds that have been developed recently that reduce or block the persistent component of sodium current and we have recently helped to understand the mechanism of action of one of these, Ranolazine (Belardinelli et al. 2006; Noble & Noble, 2006; Undrovinas et al. 2006). In addition to helping to prevent repolarization failure, block of persistent sodium current underlies the main area of application of this drug since it reduces sodium loading of cardiac cells during ischaemia and heart failure (Undrovinas et al. 2006). Figure 4 shows the basis of this action. Figure 4A is from the work of Boyett et al. (1987) showing the slow increase in intracellular sodium loading during repetitive activity. Figure 4B is a model simulation of the same effect using a ventricular cell model incorporating persistent sodium current (Sakmann et al. 2000). The predicted rise in sodium concentration is similar to that seen experimentally. Figure 4C shows that block of persistent sodium current would be expected to reduce sodium loading by nearly 50%. We have yet to repeat these computations in conditions corresponding to ischaemia, but such an effect may easily be large enough to prevent internal sodium rising to the point at which arrhythmias are triggered (Noble & Varghese, 1998).

Figure 4. Sodium loading during repetitive activity.

A, experimental recording of the rise in intracellular sodium during repetitive activity in a Purkinje fibre (from Boyett et al. 1987). B, computation of the same effect in a model ventricular cell in which the persistent sodium current has been incorporated (Sakmann et al. 2000). C, results of the same computation after block of the persistent sodium current. Sodium leading is reduced by 48%.

Concluding remarks

I started work on modelling cardiac cells just over 45 years ago. If we follow Moore's law on computing speed and power doubling every 18 months, that corresponds to 30 doubling periods, which is an increase in computing power of 2³⁰ or roughly 1 billion fold. The calculations that required 2 h of time on a huge valve computer in 1960 can now be done in microseconds, and much more complex cell models can be run faster than real time even on a small laptop. This brings closer the prospect of physiology becoming a quantitative mathematical science. Remember too that even the grandfather of experimental physiology, Claude Bernard, looked forward to such a day. In his seminal book, Introduction a l'étude de la médicine expérimentale, he wrote ‘Cette application des mathématiques aux phénomènes naturels est le but de toute science’ (This application of mathematics to natural phenomena is the aim of all science) (Bernard, 1865, 1984). But he also cautioned that, in 1865, it was too early to achieve that in physiology and medicine. Perhaps, 140 years later, we are progressively getting there.