1.1 The Heart
The heart, being the central pumping organ for the transport of blood, is vital for survival. Here, some basic properties of the heart, especially those related to cardiac arrhythmia will be presented.
Starting with its large scale structure, the mammalian heart is composed of four chambers, the left and right atria, and the left and right ventricles as shown in Fig-ure 1.1. The overall blood flow through the body can be summarized as follows:
1. Oxygen enriched blood travels through the pulmonary veins into the left atrium.
2. During atrial contraction the blood is transported into the left ventricle.
3. As the ventricles contract blood is pressed out of the left ventricle into the aorta and through all of the body including the heart itself.
4. This blood gets de-oxygenated and reaches back to the heart into the right atrium through the superior and inferior vena cava.
5. Now, during the atria’s contraction, the right atrium pumps the de-oxygenated blood into right ventricle.
6. Finally, during the ventricles’ contraction, the right ventricle pushes the blood into the pulmonary artery to the lungs.
Superior Vena Cava
Aorta Pulmonary
Artery Pulmonary
Vein
Right Ventricle
Left Ventricle Right
Atrium
Left Atrium
Inferior Vena Cava
Mitral Valve
Aortic Valve Tricuspid
Valve Pulmonary
Valve
Fig. 1.1 Diagram of a human heart including the blood flow.1
1https://en.wikipedia.org/wiki/File:Diagram_of_the_human_heart_(cropped).svg
Since it has to provide blood flow to the whole body, the left ventricle performs the largest part of the pumping effort. Anatomically this means that the left ventricle has thicker walls and is generally larger than the right ventricle. Also the atria, which mostly assists the ventricles, is much smaller and has thinner walls by comparison.
The muscle tissue separating the left and right ventricles is called the septum and has a similar wall thickness as the rest of the left ventricle.
On smaller scales the heart is highly heterogeneous and does not have smooth walls on the inside (endocard). The so called trabeculae span the ventricles inside as ridges or bridges. Further heterogeneities are for example formed by the blood vessels which provide the heart with nutrients [4].
The cardiomyocytes are the main building blocks of the heart. They are elongated cells of about 100 µm in length and 10 µm in width and height, which contract along their major axis. The cells are aligned in a very regular and structured way. Sheets of cells with the same alignment wrap around the heart. At the same time their direction also changes from the inside (endocard) to the outside (epicard). This spatial organization is important for two properties of the heart. First, it explains why the wave propagation is anisotropic [5] – in the direction of cell alignment, waves travel twice as fast as in the orthogonal direction. Second, the complex cardiomyocyte alignment causes the twist like contraction observed during the heart beat. On the cellular level, more inhomogeneities are found. Clefts exist between cells, and other cell types such as fibroblasts are embedded into the tissue.
1.1.1 Electrical Conductance
The electrical conduction inside the heart is normally initiated by the sinoatrial node located in the right atrium. This sinoatrial node consists of pacemaker cells, which undergo periodic excitation. These pacemaker cells thus cause the normal (sinus) rhythm of the heart. After initial excitation, occurring at the sinoatrial node, waves first propagate throughout the atria causing their contraction. Between the atria and the ventricles lies the atrioventricular node. The atrioventricular node delays wave propagation of excitation waves from the atria. It then triggers the ventricular contraction through a fast conduction system and the Purkinje fibres. This causes a rapid depolarization of of the ventricles, with a wave that travels from the apex upward.
While the normal sinus rhythm as described above is associated with a fast activa-tion of the ventricles, this is different during arrhythmia, when a much less organized wave propagation with complex wave patterns can be found. Due to the fast conduc-tion system, in general the sinus rhythm also has a much faster propagaconduc-tion speed compared to activation, for example, by a local stimulus. However, cardiomyocytes do not require the fast conduction system. Instead they are directly coupled through gap junctions allowing for a wave to spread in all directions.
The wave spreading in the heart thus occurs due to an excited cell causing exci-tation in its neighbouring cells. In simulation studies this coupling of neighbouring cells is commonly described by a diffusion term in the membrane potential of the cells.
1.1 The Heart 5
0 200 400 600 800 1000
time [ms]
−80
−60
−40
−20 0 20 40
membranepotential[mV]
APD90 DI CL
0.3 0.4 0.5 0.6 0.7 0.8
cytosolCa2+ concentration[µM]
Fig. 1.2 Simulation of a rabbit ventricular myocyte paced at a frequency of 3 Hz and mod-elled using the Mahajan model [6]. The plot shows the membrane potential and cytosolic calcium concentration in the cytosol. The APD90, DI, and CL indicate the action potential duration at 90% repolarization, the diastolic interval, and the basic cycle length, respectively.
The coupling enables the organized excitation and contraction of the heart. However, it also allows for the complex – arrhythmic – wave patterns which will be described later.
1.1.2 The Cardiomyocyte and Action Potential
The main building blocks of the heart are its muscle cells, the cardiomyocytes. Like neurons, cardiomyocytes can be triggered by an external stimulus, and then will go through a fast depolarization and subsequent repolarization as shown in in the time course of the action potential depicted in Figure 1.2.
The blue curve in Figure 1.2 shows a cardiomyocytes action potential. After a fast upstroke or increase in membrane potential, a plateau phase is reached before the cells membrane potential falls again. A commonly used characteristic is the action potential duration (APD), typically defined as the duration of the action potential at a given percentage of the maximum depolarization. Thus, APD90 is be the time spent above 10% of the resting potential and within 90% of the maximum membrane potential (compare Fig. 1.2). The time after the APD until the next upstroke is called the diastolic interval (DI) since it is associated with the relaxation, the diastole, of the heart. The time interval between two consecutive upstrokes is referred to as the cycle length (CL).
The excitation of the cell is normally caused by neighbouring cells or an outside electrical current which increases the membrane potential above a threshold and by
that triggers the characteristic action potential described above. The heart itself is intrinsically paced periodically and can adept its frequency to external triggers like physical strain. Thus, it should be no surprise that the cardiac cells adapt to various pacing frequencies. A higher frequency of excitation will cause the action potential to shorten. Additionally, if a cell is paced too fast, it will not be able to follow the pacing anymore. This highlights two important characteristics of the cardiac system:
1. The APD restitution describes the shortening of the action potentials at higher frequencies/shorter CL. Furthermore, its APD restitution can explain some of the dynamical phenomena observed in the heart, including APD alternans:
When the heart is paced at a fixed frequency the APD of consecutive beats can differ [3].
2. The refractory period, is the time during which an external stimulus is not able to excite a cell again [7, 8]. A stimulus given during most of the action potential plateau will not be able to cause a new excitation.
The second property will be one of the main properties explaining how cardiac arrhythmia arises. The excitation of cells by an external stimulus, together with the refractory period, make the heart an excitable medium, which will be discussed in more detail in Section 1.2. Briefly, these properties are sufficient to explain wave patterns such as the spiral wave. In the cardiac sciences spiral waves are also often called rotors orreentry2.
A large number of ion channels are involved in the formation of cardiac action potential. One prominent ion current is the calcium release, which contributes both to the action potential and signals the contraction [9]. Another important ion is potassium. Increasing the extracellular potassium concentration ([K+]) increases the cells’ resting potential. This is clinically relevant since it is the active component of the cardioplegic solution used to preserve hearts in a low energy state for transportation.
Furthermore, many diseases or drugs are associated with specific ion currents and thus ionic currents are an important field in medical research.
However, the detailed ionic currents shall not be discussed here. From a dynamical system point of view, the general properties of the cell and the complexity arising through cell to cell coupling are more important. While the ionic currents are central, the properties leading to and sustaining cardiac arrhythmia are of a more fundamental nature and do not require a detailed understanding of the ionic currents in the cell.
1.1.3 Basic Mechanisms and Modelling of the Cardiac Cell
Cardiac cells can be modelled using electrical analogues. The single cell is charac-terized by its cell membrane capacitance 𝐶𝑚 as well as ionic currents through the cell membrane or within different parts of the cell, which cause the cell membrane potential 𝑉𝑚 to change [9]:
2Especially reentry may also be used to distinguish a spiral wave and wave propagation along a ring like structure.