Your heart beats about once a second throughout your life. These regular contractions pump blood to all parts of the body and are driven by electrical impulses from a natural pacemaker inside the heart. This pacemaker, known as the sino-atrial node, responds to signals from the brain that change the heart rate according to the body's needs, making it slower when resting and faster during exercise.
The heart essentially functions as a electromechanical pump. Each beat consists of two main actions: a synchronous contraction of the two upper chambers of the heart (the atria) drives blood into the lower chambers (the ventricles); and a synchronous contraction of the ventricles then ejects the blood into the circulatory system. As with all mechanical pumps, this two-stage contraction increases the pressure and ensures that blood can reach all of the capillaries in the body. Blood at a lower pressure is pumped from the right ventricle to the lungs to allow oxygen and carbon dioxide to enter and exit the blood stream.
The rhythmic contractions of the heart are triggered by waves of electrical activity that spread from the sino-atrial node throughout the heart muscle. This rhythm can be so regular that Galileo used his pulse to time the swings of a pendulum in the cathedral at Pisa. However, even the resting heart rate is not strictly periodic. There are small fluctuations in the time intervals between beats that are fractal in nature, and a loss in this variability is a sign of cardiac ill health. This variability is not essential for the heart to function as a pump: if the natural pacemaker of the heart fails, as is common in old age, its function can be replaced by an implanted electronic pacemaker.
However, a cardiac arrhythmia, in which the rhythm of electrical waves that drive the heart is broken, can be lethal. A loss in the synchronized rhythm of the heart causes different parts of the atrial or ventricular muscle to contract at different times, undermining the pumping action of the heart. An arrhythmia therefore leads to the mechanical failure of the heart.
Recent research has shown that cardiac arrhythmias can be explained in terms of nonlinear wave dynamics. This has made it possible to simulate what happens to the heart during an arrhythmia, and could help in the development of new strategies to treat the condition.
Loss of rhythm
Cardiac arrhythmias are detected from the electrical signals generated by the heart - either by recording an electrocardiogram from outside the body, or by measuring the electrical activity inside the surface of the heart using catheter electrodes.
One cause of arrhythmias is damaged tissue in the heart muscle, which can act as an abnormal pacemaker and cause the contractions of the heart to be driven by two different pacemakers operating at different rates. Another cause is a change in the pattern of electrical wave propagation. Electrical activity generated by the sino-atrial pacemaker usually spreads rapidly through the atria, generating an electrical "excitation" in the muscle that triggers the synchronous atrial contraction. This excitation is conducted from the atria to the ventricles through the atrio-ventricular node, but is transmitted so slowly that this node can be considered as a delay line between the excitations in the two parts of the heart. After passing through the atrio-ventricular node, the excitation is rapidly conducted to the ventricular muscle, where it triggers the other synchronous contraction.
An arrhythmia can occur if some of the atrial excitations do not propagate to the ventricles. This means that the atrial contractions outnumber those in the ventricles, resulting in an irregular and uneven pulse.
The electrical excitations that induce the contractions of the heart muscle are conducted, or propagated, through the cardiac tissue by the ohmic coupling between neighbouring cells in the cardiac muscle. Excitation is made possible because the fluids inside and outside the cells are weakly ionic: fluid inside the cells is rich in potassium ions, while fluid between the cells is rich in sodium ions.
The different compositions of the fluids inside and outside the cell are maintained by energy-dependent transport processes across the cell membrane - a biomolecular layer with different permeabilities for different ionic species. Ions diffusing across the membrane generate a membrane potential, which is usually close to the equilibrium potential for potassium ions: about -90 mV. Experiments on isolated pieces of cardiac tissue, or even on isolated cells, have made it possible to control the membrane potential and measure the corresponding ionic currents across the membrane.
The transport processes of ions across the cell membrane depend on the electrical energy supplied to the cell. This voltage-dependent flow can be quantitatively described by the "excitation equation" for a single cell. This equation consists of a complex system of higher-order ordinary differential equations, with about 20 dynamic variables and timescales ranging from a fraction of a millisecond to several hundred milliseconds. The excitation equation depends on the properties of the cell, in particular its permeability to different ions and its chemical and biological properties. This means that different parts of the heart are described by quantitatively different excitation equations.
Every cell in the cardiac muscle can become electrically excited when the electrical perturbation exceeds a certain threshold (figure 1). Such a perturbation alters the permeability of the membrane to sodium ions, allowing these ions to flow into the cell until the membrane potential approaches the equilibrium potential for sodium: about 70 mV. At this point, potassium ions start to flow out of the cell, reducing the membrane potential to about -90 mV. The cell recovers to its resting state.
This large-amplitude excursion in the membrane potential, known as the action potential, takes place every time a signal is sent out from the sino-atrial node. It can last for several hundreds of milliseconds, during which time the cell cannot be re-excited and is said to be in a refractory state. (In contrast, pacemaker cells have an unstable resting state and maintain their own rhythm. They respond to a perturbation by changing the phase of their oscillation.)
The ohmic coupling between cells means that an excitation can rapidly spread throughout the cardiac tissue. The ionic currents across the membrane depend on voltage, which yields a nonlinear relationship between the membrane voltage and current. Changes in the potential of one cell therefore generate a local current that can perturb the potential of neighbouring cells.
This knock-on effect can be represented as a lattice of coupled excitation equations, and in the continuum limit the "excitable medium" of the heart can be represented by a partial differential equation analogous to the nonlinear cable equation proposed for nerve impulse propagation by the physiologists Alan Hodgkin and Andrew Huxley in 1952. This is similar to the reaction-diffusion equations used to describe the coupling between chemical reactions and diffusion. For the heart, the diffusion term describes the spread of voltage with distance and the reaction term describes the mechanisms that generate the membrane ionic currents.
Re-entrant excitation and spiral waves
Such partial differential equations predict the formation of travelling waves in the excitable medium. Unlike solitons, these travelling waves are asymmetric, with the rate of rise greater than the rate of fall. The wave velocity depends on the rate of wave formation and on the curvature of the medium in either two or three dimensions. Two of these travelling waves will completely annihilate each other on collision.
The propagation of excitations in heart tissue can be described in terms of travelling waves with a velocity of about 0.5 m s-1 and an amplitude given by the action potential. Since the action potential lasts for several hundreds of milliseconds, a single wave extends over a distance of about 10 cm. This means that a normal human heart is hardly large enough to contain a single wave.
However, in medical conditions associated with a reduced blood supply to the heart muscle, or when the rate at which the waves form is particularly high, the duration of the action potential and the "wavelength" of the travelling wave are reduced. This can lead to re-entrant propagation, in which the same wave of activity repeatedly passes through the same tissue. Such repeated excitation of the atria or ventricles causes them to quiver and writhe, preventing the synchronous contractions that pump blood around the body.
Such re-entrant excitation was first observed in a ring of cardiac tissue before the First World War. However, it was not until 1946 that Norbert Wiener analysed the phenomenon.
A simple way to understand re-entrant excitation is to imagine a single wave propagating around a circular obstacle. The wave repeatedly travels along the same path at a frequency given by the wave velocity divided by the circumference of the obstacle. If the radius is gradually decreased, the frequency increases until the wavefront of a new wave catches up with the tail of the previous wave. At this point the rate at which the wave travels around the obstacle cannot increase any further because the new wave cannot re-excite regions that are still recovering from the previous wave.
If the radius is made even smaller, the wave is forced to adopt a spiral shape that continues to rotate around a central core (figure 2). The spiral cannot enter the central core because this region is in the refractory state and cannot be re-excited.
Spiral-wave solutions to reaction-diffusion equations are generated in two dimensions by a variety of special initial conditions. They do not require an obstacle to be present, and they can be produced and maintained in media that can be considered to be entirely homogeneous. In a homogeneous medium the spiral can either rotate around a circular core, or the tip of the spiral can "meander" around the central core in a motion that can be biperiodic, quasi-periodic and perhaps even chaotic (figure 3). In an inhomogeneous medium the tip of the spiral can also drift.
Spiral waves have been observed in chemical and physical experiments, and in numerical solutions of reaction-diffusion equations, and they are beginning to be understood mathematically. Re-entrant excitation in the heart can also be idealized as spiral waves that form in the heart wall in biophysical models that assume that the walls of the atria and ventricles are thinner than the wavelength of the re-entrant wave, and so are effectively two-dimensional. Such spiral waves have a spatial extent determined by the duration of the action potential, and a rotational frequency and motion that depend on the excitation model (the equivalent of the reaction-diffusion equation) and its parameters.
Although peculiar initial conditions are required to initiate a spiral wave, once formed, a stable spiral wave will invade and take over the entire medium. This is because the frequency at which spiral-wave sources emit waves is higher than for any other nonlinear travelling-wave source. The most frequent source of waves will come to dominate the medium because they will annihilate waves that are produced less often. This means that a re-entrant arrhythmia, once initiated in the heart muscle, can prove to be lethal.
Waves and arrhythmias
Since the early 1990s a number of numerical studies based on different biophysical models of the atrial and ventricular tissue have shown that spiral waves rotate about 10 times every second. This corresponds with the main frequency component of two forms of re-entrant arrhythmia: atrial flutter and ventricular tachycardia.
Atrial flutter, so named because it feels like the heart is fluttering, is incapacitating but not life threatening. It corresponds to a stable spiral wave rotating in the atrial muscle, and it allows the ventricles to maintain adequate blood circulation. Indeed, atrial flutter might clear itself. It is possible, for example, that the spiral wave drifts upwards until the tip meets the boundaries of the great veins. These regions cannot be excited, thus preventing any further rotation of the spiral wave. If this does not happen, the wave can be eliminated with drugs or, if this fails, by using electrical shocks to regain the regular motion of the heart.
Ventricular tachycardia is associated with a re-entrant wave in the ventricles and is lethal. This is because repeated re-excitation of the ventricles by the same wave propagating around the ventricular wall results in the loss of the pumping action of the heart. Blood pressure cannot be maintained, capillary beds collapse and death follows within minutes.
These two types of arrhythmia can lead to atrial or ventricular fibrillation, in which irregular contractions take over from the rhythmic motion of the heart. Indeed, fibrillation is so severe that it causes the heart surface to look like the surface of a bag of writhing worms. Fibrillation is triggered by electrical activity in the heart, but the mechanism for the breakdown into irregularity is not yet clear.
Some biophysical models suggest that fibrillation results from the fact that two-dimensional spiral waves are not stable. As these waves break down into smaller segments, the end of each broken wave can act as the tip of a new spiral source that can itself break down. In other models the two-dimensional spiral waves are stable but their extension into three dimensions, known as scroll waves, is unstable. Probably the most important cause of fibrillation is the anatomy of the heart. The orientation of the muscle fibre, and hence the velocity of the local scroll wave, changes through 120° from one side of the heart wall to the other. This change in orientation will readily cause a scroll wave travelling through the heart wall to break down (figure 4).
These ideas are largely based on computer simulations of electrical activity, both within idealized models that assume a homogeneous and isotropic medium, and in anatomically accurate models of the heart (see Winfree in further reading). This is because there has been no direct way to observe the spatio-temporal pattern of activity on the heart's surface, never mind within its walls. In the last few years, however, José Jalife and colleagues at the Health Science Center of State University of New York (SUNY) have used dyes sensitive to changes in voltage to monitor patterns of activity on both surfaces of isolated pieces of ventricular muscle, and they have achieved a spatial resolution of less than 1 mm and a temporal resolution of a few milliseconds. These experiments have observed re-entrant wave propagation and broadly confirm the beliefs fostered by computer simulation.
A collaboration between scientists at SUNY and Leeds University in the UK has used experiments and numerical simulations to show that the pattern of activity in the ventricle at least cannot be described by simple two-dimensional spiral waves. The results show that the activity on the epicardial surface (outside the heart) can be different from the pattern on the inner surface, indicating that the waves are essentially three-dimensional, and in most cases the patterns can be explained by small numbers of scroll waves propagating within the ventricular wall (figure 5). The presence of scroll waves in the human heart has also been inferred from clinical recordings of heart activity.
To sum up, simulations and observations now agree that ventricular tachycardia and fibrillation are caused by self-maintained re-entrant waves of electrical activity in the heart. If death is to be postponed, the only course of action is to defibrillate the ventricle immediately.
Single-shock defibrillation
During fibrillation, different parts of the heart are in different states: some are excited, some are returning to the rest state and others are ready to be excited. The re-entrant nature of the propagation maintains this irregularity: as the re-entrant excitation spreads into excitable areas, recovery processes produce further areas of excitable tissue for the excitation to spread into. To terminate this continuous process, all of the excitable tissue of the heart must be driven into the same state.
One of the most common methods of defibrillating the heart is to deliver a single large electrical shock to the heart. This dramatic piece of medical practice often works, but its physical basis is not clear. The voltage provided by the shock needs to have the same effect on different parts of the heart, but instead it decays exponentially with distance through the heart. Simple arguments on the scale of the heart suggest that the shock should excite the tissue near the negative electrode and favour recovery near the positive electrode, and so the excitation state of the cells will vary from one side of the heart to the other. Furthermore, the current flowing into the cell must be equal to the current flowing out, which suggests that the defibrillating shock has a different effect on different parts of the same cell.
There are two possible explanations for this, but we have no way of knowing which is correct. One explanation is that cardiac tissue is not a homogeneous medium but an inhomogeneous network of coupled cells. This would lead to a variation in activity that could be averaged out by the resistive coupling between the cells. Another explanation is based on the effects of the defibrillation shock on the position of the re-entrant wavefront. The shock essentially pushes the wavefronts as far forward along their direction of propagation as possible, until they are blocked by refractory tissue. The wavefronts then collapse backwards, while the wavebacks continue to move forwards. The two components of the travelling waves would then cancel each other out until all of the tissue is in a resting state.
The quantitative difference between these two pictures depends on role of the cell-to-cell coupling resistance compared with the resistance to current flow outside the cell. Neither of these parameters is known to sufficient precision even for normal tissue, and they are believed to change the conditions that lead to arrhythmias.
Despite these unknowns, defibrillation by a single large shock is widely used. It can also be incorporated into implanted defibrillators, which monitor the electrical activity of the heart and apply an electrical shock if they detect the high-frequency component of fibrillation.
Defibrillation at low amplitudes
The problem with a large shock is that it is damaging and painful, so there is considerable interest in reducing the intensity of the defibrillating shock. This could be achieved by altering the waveform or timing of the shock.
One possible approach, which exploits the mathematical physics of spiral waves, is to apply a series of small-amplitude perturbations at particular times. These times are determined by the time when a spiral wave arrives at a particular recording site. A stable spiral wave will respond to each perturbation by undergoing a small displacement by changing its phase (figure 6). Repeated perturbations at the same phase could direct this drift in the spiral wave source towards the boundaries of the cardiac tissue, at which point it can be extinguished. Numerical simulations by the Leeds group show that the spiral wave source can be moved by 0.5 cm s-1 using perturbations that provide about 10% of the electrical energy given by a single-shock defibrillation, and so could defibrillate a heart within a few tens of seconds. However, problems could arise from localized inhomogeneities that can interact with the spiral waves or prevent them from drifting altogether.
An alternative approach to defibrillation is to extend the linear components of the meandering motion of a spiral wave by using drugs to change the parameters of the excitation equation. Spiral waves would then be more likely to reach the boundary of the cardiac tissue, causing them to be extinguished. This approach is still at an exploratory stage, and numerical studies based on biophysical models of cardiac tissue are now investigating the factors that produce meander.
Waves in practice
The theory of wave propagation in cardiac tissue has developed from little more than a coupled system of differential equations describing the nonlinear responses of single cells. Yet these partial differential equations are pushing our understanding of cardiac arrhythmias beyond what can be revealed with experimental techniques alone. The hope is that this theory can now be exploited to develop better methods for treating cardiac arrhythmias.
