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Physiology 19: 191-197, 2004; doi:10.1152/physiol.00004.2004
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Physiology, Vol. 19, No. 4, 191-197, August 2004
© 2004 Int. Union Physiol. Sci./Am. Physiol. Soc.

REVIEW

Modeling the Heart

Denis Noble

University Laboratory of Physiology Oxford OX1 3PT, UK
denis.noble{at}physiol.ox.ac.uk


   Abstract
 
Models of the heart have been developed since 1960, starting with the discovery and modeling of potassium channels. The first models of calcium balance were made in the 1980s and have now reached a high degree of physiological detail. During the 1990s, these cell models were incorporated into anatomically detailed tissue and organ models.


   Introduction
Top
 Introduction
Potassium Channels:...
 Calcium Balance: Calcium...
 Modeling Disease States
 Linking Levels: Putting Cell...
 References
 
Modeling excitable cells and systems took a giant leap forward when Alan Hodgkin and Andrew Huxley (21) published their equations for the squid giant axon, an achievement for which they received the Nobel Prize for Physiology or Medicine. It was the first model to use mathematical reconstruction of ion channel transport and gating, rather than abstract equations, and it correctly predicted the shape of the action potential, the impedance changes, and the conduction velocity. This was a brilliant demonstration of the need for biological modeling to respect the details of experimental results.

A "hole in one" is a rare phenomenon in theoretical biology. The squid axon is a relatively simple nervous mechanism. Most nerve cells are much more complex, and so are cardiac cells, with the added complexity of electromechanical coupling and mechanoelectric feedback (39). Modeling the heart has therefore been more like driving into the rough and then working out how to get back on the fairway and onto the green. The "hole," which I would identify as the ability to predict cardiac arrhythmias, is now on the horizon at least. We are at last reaping the rewards in terms of understanding disease states.

Even in the physical sciences, modeling has usually been an iterative process of interaction between mathematics and experimentation, involving successive approximations toward predictive capability. The iterative process generates insights on the way, even when models are wrong or incomplete. In fact, the mistakes can be as illuminating as the successes (34). Models of complex biological systems should be judged as much by the insights they have given as by their predictive power. The theme of this article will be to ask what kinds of questions were resolved with modeling that could not have been answered without it.


   Potassium Channels: Nature’s Pact with the Devil
Top
 Introduction
 Potassium Channels:...
Calcium Balance: Calcium...
 Modeling Disease States
 Linking Levels: Putting Cell...
 References
 
The first questions to be resolved were fundamental to cardiac electrophysiology. Compared with nerve, where the total duration is only a millisecond or two, the cardiac action potential is exceedingly long. Hundreds of milliseconds are required for repolarization to occur, and when the resting potential has been restored, it is not necessarily stable. In pacemaker regions like the sinoatrial node, the atrioventricular node, and the Purkinje fibers, it immediately starts to depolarize again to generate spontaneous rhythm.

What is responsible for these differences? The answer lies primarily in the nature of the potassium channels in cardiac cells. The first experimental results (5, 23) showed the existence of two kinds of channels: the inward rectifier, IK1, now known to be coded by Kir2.x, and the delayed rectifier IK. The delayed rectifier was later shown to be generated by multiple components (38, 48, 58), including the "rapid" IKr (coded by HERG and MinK) and "slow" IKs. We now know that there are many other K+ channels whose expression levels vary with spatial location (1). These include the transient outward Ito, which shapes the early phases of the action potential.

The discovery of the inward rectifier was a surprise. On depolarization it closes so that the conductance changes in precisely the opposite direction to that of the potassium channel in squid giant axon, and it was natural to ask whether this could account for the long-lasting nature of the cardiac action potential. The model incorporating this mechanism (35) showed unambiguously that it could. In fact, it is a major energy-saving device, because by reducing the membrane permeability, very small inward currents (generated by sodium and calcium ions) suffice to maintain a long plateau of depolarization. The energy cost of a long action potential generated in this way is an order of magnitude less than it would otherwise be. The 1962 model also correctly attributed repolarization to IK and identified its decay as an important contribution to pacemaker activity. FIGURE 1Go shows the voltage dependence of these two kinds of potassium channels and how they are predicted to change with time during electrical activity in cardiac cells.



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  		FIGURE 1. The first analysis of potassium channel currents in the heart and their incorporation into a heart cell model, and sodium and potassium conductance changes computed from the first biophysically detailed model of cardiac cells
A: solid line shows the total membrane current recorded in a Purkinje fiber in a sodium-depleted solution. IK1 is extrapolated here as nearly zero at positive potentials. IK is now known to be mostly formed by the component IKr. The horizontal arrow indicates the trajectory at the beginning of the action potential, whereas the vertical arrow indicates the time-dependent activation of IK that initiates repolarization. Figure redrawn from Ref. 18. B: reference membrane potential for C. C: two cycles of activity are shown (arrows correspond to those shown in A). The conductances are plotted on a logarithmic scale to accommodate the large changes in sodium conductance (GNa). Note the persistent level of GNa during the plateau of the action potential, which is ~2% of the peak conductance. Note also the rapid fall in potassium conductance (GK) at the beginning of the action potential. This is attributable to the properties of the IK1, and it helps to maintain the long duration of the action potential and to conserve energy by greatly reducing the ionic exchanges involved. IK is then responsible for repolarization. These insights into the main potassium current changes have been incorporated into all subsequent models of cardiac cells (35).

 
Evolution involves a playoff between competing factors, energy conservation and safety being two of them. Engineers are familiar with this concept in the context of constructing aircraft, bridges, spacecraft, elevators, and a host of other technologies. Evolution approaches these competing factors in similar ways (11). Energy conservation may have been one reason for the evolution of channels like IK1, but in the playoff the devil exacted a very severe price. What this model and all later models showed is that repolarization becomes a much more fragile process. Very small current changes can disturb and even prevent it. Moreover, the channel mainly responsible, IKr, is one of the most promiscuous receptors known to pharmacology. HERG or the G-coupled proteins controlling it interact with a large proportion of the output of the pharmaceutical industry, including drugs as diverse as antibiotics, antihistaminics, anticancer drugs, antihypertension drugs, etc. The list is almost endless, and many of these drugs interfere with repolarization in a way that generates potentially fatal arrhythmias. A major challenge for modeling now is whether it can help the industry to overcome these problems (32).

Although the model illustrated in FIGURE 1Go successfully represented the roles of IK1 and IK, it completely lacked calcium fluxes, and its representation of pacemaker activity was far too simple. There are many other mechanisms involved, including the hyperpolarizing-activated current If (12, 13) and various sodium and calcium channels (24). Modeling of pacemaker activity is currently a hot topic, with the roles of a number of ion transporters being reassessed (7, 47) and variations in expression levels being incorporated (59). A major new focus of this work is the mouse, which is of great importance in correlating genetic information with electrophysiology (10, 27) and for interpreting the effects of genetic modifications (37, 43).


   Calcium Balance: Calcium Channels, Sodium-Calcium Exchange, and the Sarcoplasmic Reticulum
Top
 Introduction
 Potassium Channels:...
 Calcium Balance: Calcium...
Modeling Disease States
 Linking Levels: Putting Cell...
 References
 
Cardiac physiology owes an immense debt to Harald Reuter, who first discovered calcium channels in the heart (44) and discovered the sodium-calcium exchanger (45), and to Alex Fabiato, who first demonstrated calcium-induced release of calcium from the sarcoplasmic reticulum (SR) (16).

The next major role for modeling was to integrate these three processes into the cellular web of interactions. Calcium currents were incorporated into models by McAllister et al. (30) and by Beeler and Reuter (2), but the first integrative models that addressed the key questions of calcium balance were those developed with DiFrancesco (13) and Hilgemann (20). These became the generic models from which all of the modern models of excitation-contraction coupling derive (25, 28, 29, 39, 56, 57). The Hilgemann-Noble model addressed a number of important questions concerning calcium balance:

  1. How quickly is calcium balance achieved? Net calcium efflux is estab-lished as soon as 20 ms after the beginning of the action potential (19), which was considered to be surpris-ingly soon. In the model this was achieved by calcium activation of efflux via the sodium-calcium exchanger, thus revealing the time course of one of the important functions of this transporter in the heart.
  2. If the exchanger is electrogenic, where is the current that this would generate and does it correspond to the quantity of calcium that the exchanger needs to pump? Mitchell et al. (31) 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 quan-titatively. Subsequent experimental and modeling work has fully confirmed this conclusion (3, 14, 15, 26).
  3. Could a model of the SR that reproduced the major features of Fabiato’s experiments be incorporated? The model followed as much of the Fabiato data as possible, but although it was broadly consistent with the Fabiato work it could not be based on that alone. Fabiato’s experiments were heroic, but they were done on fibers that had been skinned, which removes many of the relevant mechanisms. It is an important function of simulation to reveal when experimental data need 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.

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 (17, 25, 46, 57).


   Modeling Disease States
Top
 Introduction
 Potassium Channels:...
 Calcium Balance: Calcium...
 Modeling Disease States
Linking Levels: Putting Cell...
 References
 
Genetic mutations
Connecting genetics to physiological function at cellular and other levels is an exciting new development in which simulation is helping to unravel complex behavior resulting from simple changes at the genetic level. For example, we can now simulate long QT syndrome, which refers to the prolongation of the electrocardiogram interval between excitation of the ventricles (Q) and their repolarization (T) and is a phenomenon that can indicate life-threatening arrhythmias. (Q, T, and other cardiac waves are defined in FIGURE 2Go.) The simulations shown in FIGURE 3Go reveal the arrhythmogenic effects of two types of mutations in the sodium channel gene SCN5A (8). FIGURE 3AGo simulates a mutation that affects sodium channel inactivation and is associated with a congenital form of the long QT syndrome, LQT3. The simulations show that the mutant channel generates a persistent inward sodium current during the action potential plateau, and at long cycle lengths this generates an early afterdepolarization, thereby delaying repolarization and prolonging the QT interval. FIGURE 3BGo shows the effects in a guinea pig ventricular cell model of shifting the voltage dependence of sodium channel inactivation (40). A 12-mV shift (trace b) simply prolongs repolarization, but 18 mV produces an early afterdepolarization (trace c). This mimics part of the experimental effect of a missense mutation underlying the Brugada syndrome, an inherited condition in which sudden fatal ventricular fibrillation can occur in people who show no other manifestation of heart disease.



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  		FIGURE 2. A basic electrocardiogram
The form of the electrocardiogram is determined by the timing and form of action potentials as they spread through the heart. The interval between the Q and T waves is a measure of the time from excitation to repolarization of the ventricles.

 


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  		FIGURE 3. Connecting genetics to physiological function at cellular and other levels
Simulation is helping to unravel complex behavior resulting from simple changes at the genetic level. The simulations shown here reveal the arrhythmogenic effects of two types of mutations in the sodium channel gene SCN5A. A: simulation of the {Delta}KPQ mutation, a three-amino-acid deletion (lysine-proline-glutamine) that affects channel inactivation and is associated with a congenital form of long QT syndrome, LQT3. The simulations show that the mutant channel generates a persistent inward sodium current during the action potential plateau in the mutant cell, and at long cycle lengths this generates an early afterdepolarization, thus delaying repolarization and prolonging the QT interval. B: effects in a guinea pig ventricular cell model of shifting the voltage dependence of sodium channel inactivation. A 12-mV shift (trace b) simply prolongs repolarization, but 18 mV produces an early afterdepolarization (trace c). This mimics the experimental effect of a missense mutation underlying the Brugada syndrome, an inherited condition in which sudden fatal ventricular fibrillation can occur in people who show no other manifestation of heart disease. Redrawn from Refs. 8 and 40.

 
Congestive heart failure
The ability to track multiple changes in gene expression levels using cDNA arrays, real-time PCR, and other methods has revolutionized the range and quantity of data available on disease states. A major problem, however, is to distinguish those changes that are primary from secondary changes attributable to remodeling in response to the disease state. Computer modeling can help in this distinction because it can identify those changes that explain the disease effect, such as cardiac arrhythmia. An excellent example of this approach is the modeling of congestive heart failure. Experimental data show that Ito and IK1 are downregulated, leading to a prolongation of the action potential, whereas the SR calcium pump SERCA is also downregulated, leading to a reduction and slowing of the intracellular calcium transient. Sodium-calcium exchange is upregulated, which may be a secondary response to the other changes in calcium handling. Introducing these changes into cardiac cell models succeeds in reproducing failure of repolarization and multiple reexcitations of the kind that generate major arrhythmia (56) and in showing that the major factor involved is the changes in calcium balance.

Ischemia
Ischemia is a multifactorial process, starting with interruption of the blood supply (FIGURE 4Go), leading to rundown of metabolites like ATP involved in driving energy-using mechanisms, with consequent changes in ionic concentrations. These processes have been partially modeled (6). Both the metabolic and the ionic changes in turn modulate the activity of channels and transporters. Thus some potassium channels, such as the ATP-dependent potassium channel (IK ATP), are activated by the fall in ATP (41), thus shortening the action potential. Extracellular accumulation of potassium ions also contributes to this shortening through changes in IK1 and IKr. Meanwhile, the development of sodium and calcium overload may generate oscillatory changes in intracellular calcium that may initiate ectopic beating. These are the acute changes. Long-term changes create scar tissue that can contribute to determining the pathways for reentrant arrhythmia. Unraveling the ways in which all of these processes interact to generate life-threatening arrhythmia is a problem that requires modeling. The interactions are too complex to be understood without quantitative study. However, at present, all of the model studies are partial ones, and they have mostly been carried out at the cellular level, although Rudy and colleagues (49, 54) have done important studies on one-dimensional multicellular models and an ectopic focus has been simulated in two- and three-dimensional models (55). Compu-tations at the whole organ level have yet to appear. These will be important because reentrant arrhythmia and fibrillation are properties of the whole ventricle.



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  		FIGURE 4. Modeling the coronary circulation
This image is from a model of the coronary circulation developed by Smith, Pullan, and Hunter (see Refs. 9 and 51) in which a constriction (representing coronary occlusion) has been incorporated into a main branch. Flow velocity is color coded, with blue representing slow flow and red representing rapid flow. This simulation is part of a project to reconstruct the mechanisms of ischemic arrhythmias. Cellular and tissue models of ischemic conditions are also being developed to represent the processes that occur in the ischemic region and how they trigger life-threatening arrhythmias.

 

   Linking Levels: Putting Cell Models Into Tissue and Organ Models
Top
 Introduction
 Potassium Channels:...
 Calcium Balance: Calcium...
 Modeling Disease States
 Linking Levels: Putting Cell...
References
 
Life-threatening arrhythmias depend on molecular and cellular mechanisms, but they are fatal because of their actions at the level of the whole organ. Anatomically detailed models of the ventricles, including fiber orientations and sheet structure (9, 22), have been used to incorporate the cellular models in an attempt to reconstruct the electrical and mechanical behavior of the whole organ.

FIGURE 5Go shows frames from a simulation in which the spread of the activation wave front is reconstructed (33, 50). This is heavily influenced by cardiac ultrastructure, with preferential conduction along the fiber-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 (1), 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 (4), 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 (9, 33, 36, 53) is still at an early stage of development, but it is essential to attempt to understand heart arrhythmias. So also is the extension of modeling to human cells (42, 52).



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  		FIGURE 5. Spread of the electrical activation wavefront in an anatomically detailed cardiac model
As shown from left to right, the earliest activation occurs at the left ventricular endocardial surface near the apex. Activation then spreads in the endocardial-to-epicardial direction (outward) and from apex toward the base of the heart (upward). The activation sequence is strongly influenced by the fibrous-sheet architecture of the myocardium, as illustrated by the nonuniform transmission of excitation. Red, activation wavefront; blue, endocardial surface (33, 50).

 


   Acknowledgments
 
D. Noble is a British Heart Foundation Professor. His laboratory is also supported by the Medical Research Council, the Biotechnology and Biological Sciences Research Council, the Engineering and Physical Sciences Research Council, and the Wellcome Trust.


   References
Top
 Introduction
 Potassium Channels:...
 Calcium Balance: Calcium...
 Modeling Disease States
 Linking Levels: Putting Cell...
 References
 

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