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THE ELECTRICAL ACTIVITY OF THE HEART AND THE VAN DER
POL EQUATION AS ITS MATHEMATICAL MODEL
INTRODUCTION
The study of the heart has evolved over the years due to its importance to the
human body. As research expands new strategies on how to study the heart are
being discovered. A lot of this research has been based on the structure and or
physiology of the human heart. The van der pol equation invented by the Dutch
physicist Balthazar van der pol in 1920 made a great contribution to the research
on the actions of the heart in relation to its functions and disabilities. The van der
pol equation after its invention was later used by Johannes van der mark in
modeling the human heartbeat.
The heart is the organ that helps supply blood and oxygen to all parts of the body.
It is divided by a partition or septum into two halves, and the halves are in turn
divided into four chambers. The heart is situated within the chest cavity and
surrounded by a fluid filled sac called the pericardium. This amazing muscle
produces electrical impulses that cause the heart to contract, pumping blood
throughout the body. The heart and the circulatory system together form the
cardiovascular system.
The heart is a specialized muscle that contracts regularly and continuously,
pumping blood to the body and the lungs. The pumping action is caused by a flow
of electricity through the heart that repeats itself in a cycle. If this electrical activity
is disrupted - for example by a disturbance in the heart's rhythm known as an
it can affect the heart's ability to pump properly. This can lead to
dead. A death is described as sudden when it occurs unexpectedly, spontaneously
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and/or even dramatically. Some will be witnessed; some may occur during sleep or
during or just after exercise. Most sudden deaths are due to a heart condition and
are then called sudden cardiac death (SCD). Up to 95 in every 100 sudden cardiac
deaths are due to disease that causes abnormality of the structure of the heart. The
actual mechanism of death is most commonly a serious disturbance of the heart's
rhythm known as a 'ventricular arrhythmia' (a disturbance in the heart rhythm in
the ventricles) or 'ventricular tachycardia' (a rapid heart rate in the ventricles). This
can disrupt the ability of the ventricles to pump blood effectively to the body and
can cause a loss of all blood pressure. This is known as a cardiac arrest. If this
problem is not resolved in about two minutes, and if no-one is available to begin
resuscitation, the brain and heart become significantly damaged and death follows
quickly.
This serious and immediate effect on the whole human system whenever the heart
is infected has made the heart a critical sub-system of the human body. A lot of
research is therefore being continuously done on this critical system.
We are going to present a model that portrays the functioning of this critical
system using mathematical statements - THE VAN DER POL EQUATION
to be
precise. Mathematical statements are clearly applicable in this model since the
heart beat is a non-linear alternating and chaotic process that results from the
synchronization of many oscillatory cells. This motion is continuous with time,
hence dynamic.
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As explained previously, the human heart is a very critical system since it is not
only for us to work on but it is equally seen as the engine of a human being. That
is, the life of any man depends mostly on his heart.
This makes everybody eager to know more about this system. Its functions,
structure, effects and related side effects when affected.
The main objective of this work is to give a better understanding of the type of
behavior that the heart exhibits continuously as it beats and the various
contributions from specially developed structures.
There are also specific objectives such as having a detailed knowledge of the
structure of the heart, Understanding the functions of each cardiac structure in
relation to its structure and position, and seeing clearly that the heart can be
considered as an electrical system.
We expect that at the end of this work, one should be able to know more about the
heart and many other related concepts such as its anatomy, its physiology (or
functioning), its electrical activity and its complex dynamics, and the relationship
with other electrical systems.
The hearts relationship with other systems will be seen in chapter three. But we are
going to study the structure of the heart first since we most know the structure of a
system in order to have a better understanding of its functioning.
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CHAPTER ONE
THE HUMAN HEART
1.1 INTRODUCTION
The heart is a four-chambered organ with four main vessels, which either bring
blood to or carry blood away from the heart. The four chambers of the heart are the
right atrium, the right ventricle, the left atrium, and the left ventricle. The heart
weighs between 7 and 15 ounces (200 to 425 grams) and is a little larger than the
size of your fist. By the end of a long life, a person's heart may have beaten
(expanded and contracted) more than 3.5 billion times. In fact, each day, the
average heart beats 100,000 times, pumping about 2,000 gallons (7,571 liters) of
blood.
Your heart is located between your lungs in the middle of your chest, behind and
slightly to the left of your breastbone (sternum). A double-layered membrane
called the pericardium surrounds your heart like a sac. The outer layer of the
pericardium surrounds the roots of your heart's major blood vessels and is attached
by ligaments to your spinal column, diaphragm, and other parts of your body. The
inner layer of the pericardium is attached to the heart muscle. A coating of fluid
separates the two layers of membrane, letting the heart move as it beats, yet still be
attached to your body.
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Fig 1.0: The basic structure of the heart
The great vessels of the heart include the superior and inferior vena cava (fig 1.0),
which bring blood from the body to the right atrium; the pulmonary artery, which
transports blood from the right ventricle to the lungs; an
largest artery, which transports oxygen-rich blood from the left ventricle to the rest
of the body.
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If we remove some of the tough fibrous coating of the heart and great vessels, you
can get a better look at the heart beating. If you look carefully, you can see a series
of one-way valves that keep the blood flowing in one direction. If we inject dye
into the superior vena cava, you can watch it pass through the heart as it goes
through the cardiac cycle.
The blood first enters the heart into the right atrium. Blood passes from the right
atrium through the tricuspid valve and into the right ventricle. When the right
ventricle contracts, the muscular force pushes blood through the pulmonary
semilunar valve into the pulmonary artery,
The blood then travels to the lungs, where it receives oxygen. Next, it drains out of
the lungs via the pulmonary veins, and travels to the left atrium. From the left
atrium, the blood is forced through the mitral valve into the critically important left
ventricle. The left ventricle is the major muscular pump that sends the blood out to
the body systems. When the left ventricle contracts, it forces the blood through the
aortic semilunar valves and into the aorta, from here, the aorta and its branches
carry blood to all the tissues of the body.
1.2 ANATOMY OF THE HEART
The human heart fig 1.1 is considered as the most important part of the human
four components that work synchronously to maintain the flow of blood to the
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The four components of the heart are;
(1) The arteries, that is the tubes that carry oxygen to the heart muscle;
(2) The valves, which are the door-ways that connect the four chambers of the
heart;
(3) The muscle, which is the tissue that contracts and propels the blood to the
(4) The electrical system, which is the internal wiring that creates the rhythm and
pulse of the heart.
The heart is like a high-performance engine that requires regular maintenance and
servicing. When a part becomes rusty or a spark plug is out, for example, a car is
regular oil changes and general upkeep can lead to periodic breakdowns and
eventual engine failure. The heart has similar needs. When part of the heart has a
problem, we develop warning signs known as symptoms. Like the engine light in a
car, these signs are telling us that our heart needs urgent attention. Here, we are
going to concentrate on abnormal heartbeat.
A cardiologist, like a good mechanic, analyzes the symptoms, performs the
appropriate diagnostics, and troubleshoots to identify the ailing part.
The heart being the main part of the body that stimulates the functioning of all
other parts, have been given a lot of attention as far as research is concern.
Therefore many theses have been published about the heart ranging from structure,
functioning and especially diseases. The main function of the heart is to pump
blood unto other parts of the body and regulate blood circulation. The general
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structure of the heart as shown in fig 2.1 indicates that the heart is comprised of
many different structures. Each structure has its function contributing to the
heartbeat.
Fig 1.1: The general structure of the heart
The heart is situated between the two lungs and behind the sternum in the thorax. It
is surrounded by a tough sac, the pericardium, the outer part of which consists of
inelastic white fibrous tissue. The inner part consists of two membranes; the inner
and outer membranes. The membranes are attached to the heart and the fibrous
tissue respectively. Pericardial fluid is secreted between them and this reduces
friction between the heart wall and the surrounding tissues when the heart is
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beating. The inelastic nature of the pericardium as a whole prevents the heart from
being overstretched or overfilled with blood.
1.3 CARDIAC MUSCLE AND CORONARY ARTERIES
The walls of the heart are composed of cardiac muscle fibers, connective tissues
and tinny blood vessels. Each muscle fiber possesses one or two nuclei and many
large mitochondria. Each fiber is made up of many myofibrils. These contain actin
and myosin filaments which bring about contraction in the same way as skeletal
muscle. Cardiac muscle contract more slowly than skeletal muscle and does not
fatigue as easily.
The cardiac cycle refers to the sequence of events which take place during the
completion of one heartbeat. It involves repeated contraction and relaxation of the
heart muscle. Contraction is called systole and relaxation is called diastole.
Because the heart is composed primarily of cardiac muscle tissue that continuously
contracts and relaxes, it must have a constant supply of oxygen and nutrients. The
coronary arteries are the network of blood vessels that carry oxygen- and nutrientrich blood to the cardiac muscle tissue.
main artery.
Two coronary arteries, referred to as the "left" and "right" coronary arteries,
emerge from the beginning of the aorta, near the top of the heart.
The initial segment of the left coronary artery is called the left main coronary. This
blood vessel is approximately the width of a soda straw and is less than an inch
long. It branches into two slightly smaller arteries: the left anterior descending
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coronary artery and the left circumflex coronary artery. The left anterior
descending coronary artery is embedded in the surface of the front side of the
heart. The left circumflex coronary artery circles around the left side of the heart
and is embedded in the surface of the back of the heart.
Just like branches on a tree, the coronary arteries branch into progressively smaller
vessels. The larger vessels travel along the surface of the heart; however, the
smaller branches penetrate the heart muscle. The smallest branches, called
capillaries, are so narrow that the red blood cells must travel in single file. In the
capillaries, the red blood cells provide oxygen and nutrients to the cardiac muscle
tissue and bond with carbon dioxide and other metabolic waste products, taking
them away from the heart for disposal through the lungs, kidneys and liver.
When cholesterol plaque accumulates to the point of blocking the flow of blood
through a coronary artery, the cardiac muscle tissue fed by the coronary artery
beyond the point of the blockage is deprived of oxygen and nutrients. This area of
cardiac muscle tissue ceases to function properly. The condition when a coronary
artery becomes blocked causing damage to the cardiac muscle tissue itself is called
a myocardial infarction or heart attack.
Superior Vena Cava
The superior vena cava is one of the two main veins bringing de-oxygenated blood
from the body to the heart. Veins from the head and upper body feed into the
superior vena cava, which empties into the right atrium of the heart.
Inferior Vena Cava
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The inferior vena cava is one of the two main veins bringing de-oxygenated blood
from the body to the heart. Veins from the legs and lower torso feed into the
inferior vena cava, which empties into the right atrium of the heart.
Aorta
The aorta is the largest single blood vessel in the body. It is approximately the
diameter of your thumb. This vessel carries oxygen-rich blood from the left
ventricle to the various parts of the body.
Pulmonary Artery
The pulmonary artery is the vessel transporting de-oxygenated blood from the right
ventricle to the lungs. A common misconception is that all arteries carry oxygenrich blood. It is more appropriate to classify arteries as vessels carrying blood away
from the heart.
Pulmonary Vein
The pulmonary vein is the vessel transporting oxygen-rich blood from the lungs to
the left atrium. A common misconception is that all veins carry de-oxygenated
blood. It is more appropriate to classify veins as vessels carrying blood to the heart.
Right Atrium
The right atrium receives de-oxygenated blood from the body through the superior
vena cava (head and upper body) and inferior vena cava (legs and lower torso).
The sinoatrial node sends an impulse that causes the cardiac muscle tissue of the
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atrium to contract in a coordinated, wave-like manner. The tricuspid valve, which
separates the right atrium from the right ventricle, opens to allow the deoxygenated blood collected in the right atrium to flow into the right ventricle.
Right Ventricle
The right ventricle receives de-oxygenated blood as the right atrium contracts. The
pulmonary valve leading into the pulmonary artery is closed, allowing the ventricle
to fill with blood. Once the ventricles are full, they contract. As the right ventricle
contracts, the tricuspid valve closes and the pulmonary valve opens. The closure of
the tricuspid valve prevents blood from backing into the right atrium and the
opening of the pulmonary valve allows the blood to flow into the pulmonary artery
toward the lungs.
Left Atrium
The left atrium receives oxygenated blood from the lungs through the pulmonary
vein. As the contraction triggered by the sinoatrial node progresses through the
atria, the blood passes through the mitral valve into the left ventricle.
Left Ventricle
The left ventricle receives oxygenated blood as the left atrium contracts. The blood
passes through the mitral valve into the left ventricle. The aortic valve leading into
the aorta is closed, allowing the ventricle to fill with blood. Once the ventricles are
full, they contract. As the left ventricle contracts, the mitral valve closes and the
aortic valve opens. The closure of the mitral valve prevents blood from backing
into the left atrium and the opening of the aortic valve allows the blood to flow into
the aorta and flow throughout the body.
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Papillary Muscles
The papillary muscles attach to the lower portion of the interior wall of the
ventricles. They connect to the chordae tendineae, which attach to the tricuspid
valve in the right ventricle and the mitral valve in the left ventricle. The contraction
of the papillary muscles closes these valves. When the papillary muscles relax, the
valves open.
Chordae Tendineae
The chordae tendineae are tendons linking the papillary muscles to the tricuspid
valve in the right ventricle and the mitral valve in the left ventricle. As the
papillary muscles contract and relax, the chordae tendineae transmit the resulting
increase and decrease in tension to the respective valves, causing them to open and
close. The chordae tendineae are string-like in appearance and are sometimes
referred to as "heart strings."
Tricuspid Valve
The tricuspid valve separates the right atrium from the right ventricle. It opens to
allow the de-oxygenated blood collected in the right atrium to flow into the right
ventricle. It closes as the right ventricle contracts, preventing blood from returning
to the right atrium; thereby, forcing it to exit through the pulmonary valve into the
pulmonary artery.
Mitral Value
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The mitral valve separates the left atrium from the left ventricle. It opens to allow
the oxygenated blood collected in the left atrium to flow into the left ventricle. It
closes as the left ventricle contracts, preventing blood from returning to the left
atrium; thereby, forcing it to exit through the aortic valve into the aorta.
Pulmonary Valve
The pulmonary valve separates the right ventricle from the pulmonary artery. As
the ventricles contract, it opens to allow the de-oxygenated blood collected in the
right ventricle to flow to the lungs. It closes as the ventricles relax, preventing
blood from returning to the heart.
Aortic Valve
The aortic valve separates the left ventricle from the aorta. As the ventricles
contract, it opens to allow the oxygenated blood collected in the left ventricle to
flow throughout the body. It closes as the ventricles relax, preventing blood from
returning to the heart.
We can therefore see clearly that the heart and its components are wonderfully
developed for great functions. One of these great functions is the electrical
functioning of the heart which we will see in the next chapter.
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CHAPTER TWO
ELECTRICAL ACTIVITY OF THE HEART
2.1 INTRODUCTION
We have seen in chapter one that the heart is a muscles that works continuously.
Your heart's electrical system controls all the events that occur when your heart
pumps blood. The electrical system is also called the cardiac conduction system. If
you've ever seen the heart test called an EKG (electrocardiogram), you've seen a
graphical picture of the heart's electrical activity (fig 2.1). Your heart is a muscle
that works continuously, much like a pump. Each beat of your heart is set in
motion by an electrical signal from within your heart muscle. The electrical
activity is recorded by an electrocardiogram, known as an EKG or ECG. Each beat
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of your heart begins with an electrical signal from the sinoatrial node. The SA node
trium.
The heart is a highly complicated system comprising of an electrical system which
includes three important parts:
S-A node (sinoatrial node)
known as the heart's natural pacemaker, the
S-A node has special cells that create the electricity that makes your heart
beat.
A-V node (atrioventricular node)
the A-V node is the bridge between
the atria and ventricles. Electrical signals pass from the atria down to the
ventricles through the A-V node.
His-Purkinje system
signals
throughout
the His-Purkinje system carries the electrical
the
ventricles
to
make
them
contract.
The parts of the His-Purkinje system include: His Bundle (the start of the
system), Right bundle branch, Left bundle branch, and Purkinje fibers (the
end of the system)
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Fig 2.1: the electrical activity of the heart and the EKG result
2.2 THE HEARTBEAT
Most of the time, you may not be aware of your heartbeat. When running up and
down a flight of stairs, you may notice the pulse in your neck or chest becomes
strong and rapid. Your heartbeat is able to speed up and slow down because it is
wired with electrical tissue, similar to the wires that connect a stereo. Your heart
also has its own "pacemakers" that are like electrical outlets. They send signals that
tell the heart muscles to contract. This happens 24 hours a day, 365 days a year
without rest, even when you do not notice.
Without the electrical system, the heart would not contract and would not pump
blood. Blood would not circulate and the body would not receive the oxygen and
nutrients it needs. When blood flow stops to the brain, a person loses
consciousness in seconds and death follows within minutes.
2.3 THE CARDIAC CYCLE AND THE CONDUCTION SYSTEM
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2.3.1 THE CARDIAC CYCLE
The rhythmic contraction of the heart as seen above is called heartbeat. A cardiac
cycle is defined as a complete cardiac movement, including systole, intervening
pause, and diastole. The cardiac cycle begins with depolarization of the SA node
and atrial contraction. Each cycle requires a certain length of time for its
completion. Pressure, volume, electrical, and sound changes occur during each
cycle. Heart sounds are created primarily from turbulence in blood flow created by
valves closing, not from contraction.
Two separate networks of cardiac fibers regulate atrial and ventricular contraction.
The heart is innervated by the autonomic nervous system, which neither initiates
contraction nor affects the cardiac cycle.
The conduction system is composed of specialized muscle tissue and initiates and
conducts depolarization waves through the myocardium. Muscle fibers of the
ventricular walls are arranged in whirls that squeeze blood out of the ventricles
when they contract. The contraction phase is systole, while the filling phase is
diastole. Atria contract while ventricles relax, and the ventricles contract while
atria relax.
2.3.2 THE CARDIAC CONDUCTION SYSTEM
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Fig 2.2 Location of the nodal tissue within the heart
In order for the heart to pump efficiently, the individual myocardial fibers must
contract and relax in a coordinated, rhythmic fashion. This characteristic of the
healthy heartbeat is called synchrony. Synchrony is maintained by the heart's own
intrinsic electrical system, which originates and transmits electrical impulses
through a specialized conduction pathway. Just as the pulse is evidence of the
heart's mechanical activity, the electrocardiogram or ECG, is evidence of its
electrical activity. In the absence of synchrony, myocardial fibers contract in a
random, uncontrolled fashion called ventricular fibrillation and the heart can no
longer pump effectively. If no oxygenated blood reaches distant tissues, the patient
dies.
Anatomy
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The cardiac conduction system consists of highly specialized cells that
histologically resemble nerve tissue. When the anatomic components of this
system are affected by disease, abnormalities in the heart's electrical activity called
arrhythmias can result.
Five distinct anatomical structures that comprises the cardiac conduction
system:
The sinoatrial (SA) or sinus node is a small mass of tissue, about the
size of a match-head, located high in the wall of the right atrium near
the entrance of the superior vena cava. It is the heart's normal
pacemaker, automatically initiating impulses at a more rapid rate than
any other part of the conduction system.
The atrioventricular (AV) node is also located on the right side of the
heart, just beneath the surface of the interventricular septum. The AV
node and the conduction tissue surrounding it are known as the AV
junctional tissue.
The atrioventricular bundle, or bundle of His, is a band of nerve fibers
that originates at the AV node, and then passes along the
interventricular septum to the ventricles. Wilhelm His was the Swiss
physician who first described this tissue.
The right and left bundle branches are continuations of the bundle of
His. They proceed along the right and left sides of the interventricular
septum to the tips of the two ventricles.
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The Purkinje fibers (named after their discoverer, Johannes von
Purkinje) are the tree-like terminal branching of the right and left
bundles-thousands of fibrils extending between myocardial fibers for
about one-third to one-half the ventricular wall thickness.
The heart muscle grossly resembles skeletal muscle, yet it is structurally different.
Cardiac muscle cells are interconnected to form a syncytium, a multinucleate mass
of protoplasm produced by the merging of cells. This permits electrical excitation
waves to pass rapidly from one cardiac cell to the next. Cardiac muscle is
controlled by physiologic mechanisms under involuntary control, and mediated by
specific nerves.
Myocardial tissue has four main characteristics that integrate the heart's electrical
and mechanical activity:
Automaticity: the ability to initiate an impulse or stimulus. The
pacemaker cells of the cardiac conduction system spontaneously
depolarize in the absence of external stimulation. The AV-junctional
tissue and the His-Purkinje network also possess the property of
automaticity.
Excitability: the ability to respond to an impulse or stimulus. The cells
are electrically irritable because of an ionic imbalance across the cell
membranes. Cells thus respond to external stimuli from chemical,
mechanical or electrical sources. Atrial and ventricular myocardial
fibers respond to the impulse generated by the pacemaker cells of the
cardiac conduction system by depolarization and repolarization.
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Conductivity: the ability to transmit impulses to other areas. Both the
cells of the conduction system and the myocardial muscle fibers have
this property.
Contractility: the ability of cardiac cells to shorten, responding to
stimuli
with
mechanically
mechanical
to
electrical
action.
Myocardial
stimulation
by
fibers
respond
contracting.
The
simultaneous contraction of bands of myocardial fibers is the heart's
pumping action.
In order for the heart to pump efficiently, the myocardial muscle fibers must
contract and relax in a coordinated, rhythmic fashion, in synchrony. When
conductive tissue is damaged or deprived of oxygen, certain abnormal ventricular
contractions may occur. Asynchronous, random, uncontrolled contraction of the
ventricles is called ventricular fibrillation. In v. fibrillation, the ventricle flutters
and the blood cannot move out of the LV to oxygenate distant tissues. In
cardiopulmonary rescue, a defibrillator is applied in the hope of shocking the heart
back into a more normal rhythm. If a defibrillator cannot be used quickly, death
follows.
Automaticity
The property of cells of the conduction system to initiate pacing of electrical
impulses independent of the autonomic nervous system is called automaticity. The
SA node is the normal pacemaker. If the SA node is isolated from all neural or
hormonal control, this specialized tissue can generate impulses at rates higher than
100 per minute. Under autonomic control, the SA node paces the heart at a normal
rate of 60 to 100 impulses per minute.
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Other parts of the conduction system- the AV junctional tissue and the HisPurkinje network- also have the property of automaticity. The SA node is the
pacemaker, if it initiates impulses at a faster rate than other areas and if the impulse
is rapidly propagated throughout the conduction system. For instance, when AV
node function is impaired, or heart block occurs at this point in the system, other
cells in the ventricles may become secondary pacemakers-maintaining the vital
heartbeat, though usually at a different rate.
Thus, redundant automaticity is a protective mechanism that keeps the heart
pumping even in the absence of normal impulses originating from the SA node.
However, the ability of other cells along the conduction pathway to initiate
impulses can create problems. For instance, when conductive tissue is damaged or
deprived of oxygen (due to ischemia), it becomes irritable, and may cause certain
kinds of abnormal ventricular contractions. Ventricular tachycardia is a particularly
dangerous form of rapid heart rate that can easily convert to ventricular fibrillation
-the uncoordinated, random contraction of individual myocardial fibers that stops
any effective pumping action.
The AV node is the only normal conduction pathway through the atrioventricular
septum. When the excitation impulse reaches the AV node, it is delayed there for
0.08 to 0.16 second because of slow conduction along the delicate junction fibers
that connect the atrial myocardium with AV nodal tissue. During this delay, atrial
contraction is largely completed, so that when the impulse reaches the ventricles,
ventricular filling is complete.
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Fig 2.3 Sequence of cardiac excitation
After passing through the AV node, the impulse reaches the bundle of His and
again moves faster, passing through the right and left bundle branches and to the
terminal Purkinje fibers in 0.03 to 0.05 second. The Purkinje fibers penetrate the
ventricular wall from the endocardia surface, and only for a part of its thickness.
From the Purkinje fibers, the excitation impulse then continues through myocardial
cells outside the specialized conduction pathway, and a final 0.03 second is
required to reach the epicardial surface. This rapid, simultaneous spread of
excitation through the ventricles produces a coordinated contraction of both
ventricles, thus ensuring efficient pumping of blood to the pulmonary and systemic
circulations.
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2.4 THE HEART AS AN ELECTRICAL CIRCUIT
The S-A node normally produces 60-100 electrical signals per minute
this is
your heart rate, or pulse. With each pulse, signals from the S-A node follow a
natural electrical pathway through your heart walls. The movement of the electrical
signals causes your heart's chambers to contract and relax.
You may know or have heard of someone with an artificial pacemaker or other
implantable device to regulate the beat of the heart. Pacemakers and the wiring that
run through the heart coordinate contractions in the upper and lower chambers,
which makes the heartbeat more powerful so it can do its job effectively.
We normally have our own natural pacemakers that tell the heart when to beat. The
master pacemaker is located in the atrium (upper chamber). It acts like a spark plug
that fire in a regular, rhythmic pattern to regulate the heart's rhythm. This "spark
plug" is called the sinoatrial (SA), or sinus node. It sends signals to the rest of the
heart so the muscles will contract.
First, as soon as the signal is sent, the atrium contracts. Like a pebble dropped into
a pool of water, the electrical signal from the sinus node spreads through the atria.
Next, the signal travels to the area that connects the atria with the ventricles. This
electrical connection is critical. Without it, the signal would never reach the
ventricles, the major pumping chambers of the heart.
The electrical signal reaches another natural pacemaker called the atrioventricular
node (AV node). As the signal continues and crosses to the ventricles, it passes
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through a bundle of tissue called the AV bundle, also called the bundle of His. The
bundle divides into thin, wire-like structures called bundle branches that extend
into the right and left ventricles. The electrical signal travels down the bundle
branches to thin fibers. Lastly, these fibers send out the signal to the muscles of the
ventricles, causing them to contract and pump blood into the arteries.
In a normal heart, this coordinated series of electrical signals occurs about once
l system is responsible for
creating the signals that trigger the heart to beat and these signals prompt the
body. The process begins in the upper chambers of the heart (atria), which pump
blood into the lower chambers (ventricles). The ventricles then pump blood to the
body and lungs. This coordinated action occurs because the heart is "wired" to send
electrical signals that tell the chambers of the heart when to contract. In other
words the heart is a bit like the electrical wiring in your home. Hence we can
model the heart as an electrical circuit. This is shown in the next chapter.
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CHAPTER THREE
VAN DER POL EQUATION AS A MATHEMATICAL MODEL OF THE
ELECTRICAL ACTIVITY OF THE HEART
3.1 INTRODUTION
The description in chapters one and two brings us to the fact that the heart and its
electrical functions can be modeled as an electrical circuit. When this is done, we
can check the heart of any abnormality. Though we are talking about abnormality,
it is not really abnormal because the heart whether in good condition or not shows
changeable rates of rhythm depending on the activity (sleeping, moving, etc.) of
the person. This gives rise to a chaotic dynamical behavior. Most dynamical
systems occurring in nature shows nonlinear behaviors, but because of engineering
applications, essential approximation is made into linearity.
Examples of other nonlinear and complex systems include;
a) hurricane
b) cardiac arrhythmia(spiral motion)
c) spreading of epidemics
d) motion of fluid(turbulence) and
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e) onset of epilepsy
3.2 BACKGROUND HISTORY OF THE VAN DER POL EQUATION
Fig 3.1: BALTHASAR VAN DER POL
BALTHASAR VAN DERPOL was born on the 27th of Jan 1889 in Netherland. He
later graduated with a degree in Physics in 1916 from the University of Utrecht. He
started his radio work at Cambridge and was under two heads, experimental and
theoretical. We can say that Heaviside had 'invented' the ionosphere to explain
Marconi's remarkable transmission by radio over the Atlantic in 1901. In his
theoretical work van der Pol gave a quantitative proof that, if one neglected the
influence of a reflecting layer, experiment and theory were in flagrant
disagreement in the case of long-distance propagation. To do this van der Pol
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familiarized himself with the theoretical work on the diffraction of radio waves
round a conducting earth and used this to make a direct comparison between
(a) Signal strength predicted and
(b) Signal strength received,
in a practical case of radio transmission. The paper in question is in the Phil. Mag.
Sept. 1919 and is worth consulting. In this work van der Pol showed great
experimental skill. He made his own apparatus, as was customary in those days of
J J Thomson. His triode oscillator gave him waves of about 3 meters, which was
about as short as anyone else had produced in those days. All this work on the
dielectric constant of ionized air formed the basis of his Doctor's thesis. It was
round about this same period that van der Pol and Appleton collaborated in the
study of non-linear phenomena using triode circuits to cheek their theories.
Together they worked at
(a) Oscillation hysteresis and
(b) Forced vibrations in a non-linear system.
Van der Pol's work on relaxation oscillations was, of course, entirely his own. The
scientific work of Balthazar van der Pol covered pure mathematics, applied
mathematics, radio, and electrical engineering. In the field of mathematics, he
covered a number of theories, special functions, operational calculus and nonlinear
differential equations. In this last field he was a pioneer. To most mathematicians
the name of van der Pol is associated with the differential equation which now
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bears his name. This equation (the van der pol equation) first appeared in his article
published in the Philosophical Magazine in 1926.
The van der pol equation is an ordinary differential equation that can be derived
from the Rayleigh differential equation by differentiating and substituting
.
It is an equation describing self-sustaining oscillations in which energy is fed into
small oscillations and removed from large oscillations. This equation arises in the
study of circuits containing vacuum tubes and is given by
Which reduces to the equation of simple harmonic motion in a situation where
. That is, if
, we have
This equation serves as an example of a nonlinear oscillator with a partially
negative damping.
3.3 USES OF THE VAN DER POL EQUATION
The Van der Pol oscillator was originally proposed by the Dutch electrical
engineer and physicist Balthazar van der Pol while he was working at Philips. Van
der Pol found stable oscillations, which he called relaxation-oscillations and are
now known as limit cycles, in electrical circuits employing vacuum tubes. When
these circuits were driven near the limit cycle they become entrained, i.e. the
driving signal pulls the current along with it. Van der Pol and his colleague, van
der Mark, reported in the September 1927 issue of Nature that at certain drive
frequencies an irregular noise was heard. This irregular noise was always heard
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near the natural entrainment frequencies. This was one of the first discovered
instances of deterministic chaos.
From when it was intro
prototype for systems with self-excited limit cycle oscillations. The classical
experimental setup of the system is the oscillator with vacuum triode. The
investigations of the forced Van der Pol oscillator behavior have been carried out
by many researchers. The equation has been studied over wide parameter regimes,
from perturbations of harmonic motion to relaxation oscillations. It was much
attention dedicated to investigations of the peculiarities of the Van der Pol
oscillator behavior under external periodic (sinusoidal) force and, in particular, the
synchronization phenomena and the dynamical chaos appearing. The Van der Pol
equation is now concerned as a basic model for oscillatory processes in physics,
electronics, biology, neurology, sociology and economics. Van der Pol himself
built a number of electronic circuit models of the human heart to study the range of
stability of heart dynamics. His investigations with adding an external driving
signal were analogous to the situation in which a real heart is driven by a
pacemaker. He was interested in finding out, using his entrainment work, how to
stabilize a heart's irregular beating or "arrhythmias". Since then it has been used by
scientists to model a variety of physical and biological phenomena. For instance, in
biology, the van der Pol equation has been used as the basis of a model of coupled
neurons in the gastric mill circuit of the stomatogastric ganglion. The Fitzhugh
Nagumo equation is a planar vector field that extends the van der Pol equation as a
model for action potentials of neurons. In seismology, the van der Pol equation has
been used in the development of a model of the interaction of two plates in a
geological fault.
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3.4 VAN DER POL EQUATION AND THE HEART
To understand the van der pol equation well, there are certain concepts that are
very important to know. These concepts can be incorporated into the concepts of
dynamics and chaos.
3.4.1 DYNAMICAL SYSTEMS
The discipline of Dynamical Systems provides the mathematical language
describing the time dependence of deterministic systems. For the past four decades
there has been ongoing theoretical development. The work on dynamical systems
can be given in a phenomenological point of view, reviewing examples, some of
which have guided the development of the theory. The van der pol equation is just
one of such systems.
3.4.2 CHAOS
The central concept of the theory of dynamic systems is chaos, defined in terms of
unpredictability. The prediction principle we use is in terms of and based on past
observations. The unpredictability then is expressed in a dispersion exponent, a
notion related to entropy and Lyapunov exponents. Structural stability turns out to
be useful in proving that the chaoticity of the doubling and Thom maps is
persistent under small perturbations. The ideas on predictability are also used in the
development of reconstruction theory, where important dynamical invariants such
as the box-counting and correlation dimensions of an attractor, which often are
fractal, and related forms of entropy are reconstructed from time series based on a
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long segment of the past evolution. Also numerical estimation methods of the
invariants are discussed; these are important for applications, for example, in early
warning systems for chemical processes. Another central subject is formed by
multi- and quasi-periodic dynamics. Here the dispersion exponent vanishes and
quasiwith circle dynamics near rigid rotations and similar settings for area-preserving
and holomorphic maps. Persistence of quasi-periodicity is part of Kolmogorov
Arnold Moser (or KAM) theory. The major motivation to this subject has been the
fact that the motion of the planets, to a very good approximation, is multi-periodic.
Remember that the concept of chaos is a non-linear Physics concept that involves
synchronization. We can now study the details of the van der pol equation.
3.5 THE VAN DER POL EQUATION
The van der pol equation as shown in Equation (1) is an important special case of
the Lienard equation. Van der Pol's equation describes the auto-oscillations of one
of the simplest oscillating systems (the van der Pol oscillator). In particular,
equation (1) serves after making several simplifying assumptions as a
mathematical model of a generator on a triode for a tube with a cubic
characteristic. The character of the solutions of equation (1) was first studied in
detail by B. van der Pol.
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Fig 3.2: electrical circuit involving a triode
Since equation (1) is a specia
the system has a limit cycle by applying Lienard transform.
From Liénard transformation:
, where the dot indicates the time
derivative, the Van der Pol oscillator can be written in its two-dimensional forms
as
and
.
Another commonly used form based on the transformation
is leading
to
The forced van der pol oscillator takes the original function and adds a driving
function
to give a differencial equation of the form:
The circuit in fig 3.2 is resulting into a forced van der pol oscillator. This circuit
contains: a triode, a capacitor C, a resistor R, a coupled inductor set with self
inductance L and mutual inductance M. in the serial RLC circuit, there is a current
I, and towards the triode anode (plate) a current ia , while there is a voltage Ug , on
the triode control gride. The van der pol oscillator is forced by an AC voltage
source Es .
3.5.1 ITS SOLUTIONS AND MATHEMATICAL MODELING
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Equation (1) is not a linear differential equation and so the methods available for
the exact differential equations cannot apply to the van der pol equation. Therefore
we find ways of obtaining approximate solutions of the equation. This equation can
be solved numerically using the Euler, Taylor series, or Runge-Kutta integration
algorithm. Applying any of the above computational differential equation solver to
the equations, we obtained portraits similar to these bellow:
Fig: 3.3a Evolution of the limit cycle in the phase plane. Notice the limit cycle
begins as cycles and, with varying , become increasingly sharp, an example of a
relaxation oscil.
Fig: 3.3b phase portrait of the unforced van der pol oscillator showing a limit cycle
and the direction field.
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The above representations display many of those features supposed to occur in the
cardiac system such as limit cycle, synchronization, and chaos. We can write the
general form of the Vdp equation as
(7)
Where a, b and c are system parameters and (t) is an external forcing.
The qualitative features of an isolated VdP oscillator present a close similarity to
the features of the heart actuation potential. Both kinds of actuation potential
response (slow and fast) can be easily reproduced by the VdP oscillator.
Grudzinski and Zebrowski proposed a modified oscillator in order to simulate
important physiological features of the action potentials.
In general, the new equation has two fixed points and a dissipation term
asymmetric with respect to the voltage:
(8)
Here a, d, e, w1 and w2 are system parameters and (t) is an external forcing. As
already mentioned, the normal cardiac rhythm is primarily generated by the SA
node, which is considered as the normal pacemaker. Besides, the AV node is
another pacemaker. Each one of these presents an actuation potential that is
fundamental to the heart dynamics, but not necessarily the most expressive to
compose the ECG signal.
Activation (depolarization followed by repolarization) corresponds to a different
region of the heart and, as a consequence, generates currents of different
magnitudes. Therefore, the combination of activation waves coming from each
region of the heart is responsible for the ECG form and some of these signals may
be preponderant in this composition, like the waves originated in the atrium and
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ventricle. On the other hand, as these regions follow very close the activation of
the SA and AV nodes, their signature on the ECG signal is representative of the
ventricle, respectively. Under these assumptions, it is expected that coupled
oscillators, each one representing a different heart region signal, may represent the
general heartbeat dynamics. Usually, two oscillators are considered representing
the SA and AV nodes, however, it is observed that these two oscillators are not
enough to reproduce the ECG signal. This is because the signal of the first
oscillator corresponds to the activation of the SA node and atrium, and the signal
of the second oscillator corresponds just to the ventricle depolarization. Under this
assumption, it is possible to reproduce the P-curve but not the QRS complex,
because this interval mainly corresponds to the ventricle repolarization. This
observation motivate the inclusion of a third oscillator that represent the pulse
propagation through the ventricles, which physiologically represents the His
Purkinje complex, composed by the His bundle and the Purkinje fibers.
Fig: 3.6 Conceptual Model with three couple oscillators.
Fig 3.6 presents the conceptual model showing either the oscillators or the
coupling among them. In order to build a general model, bidirectional asymmetric
couplings are assumed among all oscillators. Moreover, external excitations are
incorporated to the system, considering a periodic driving term on each oscillator.
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This conceptual model may be represented by a set of differential equations as
follows:
(9)
These governing equations may be rewritten in a compact form as follows:
()
Where
(10)
Notice that the governing equations have a general form
(related to the
modified VdP oscillator), and K represents the coupling matrix. Time delays in
signal transmission are unavoidable and, since even small delays may alter the
system dynamics, it is necessary to understand how conduction delays change the
behavior of coupled oscillators. The inclusion of time delays in differential
equations can cause drastic changes and can make chaos emerges in a system that
would otherwise be described by a regular behavior. On this basis, the proposed
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mathematical model is changed in order to consider delay aspects in coupling
terms. Therefore, governing equations are changed as follows:
(11)
Where
and
represents the time delay. Notice that, actually, there
are different delays depending on the connection type. The general idea of these
coupled oscillators is that the ECG signal is built from the composition of these
signals as follows:
(12)
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Fig: 3.7 Normal EKG
Using experimental values from the normal EKG and our numerical simulations,
we then have a real and simulated ECG as shown in fig 3.8.
Fig: 3.8 Real and simulated normal ECG comparison.
By comparing numerical simulation with the real ECG, it is possible to observe
that they are in qualitative agreement. Fig 3.8 presents a comparison between the
real and the simulated ECG, showing that the model is able to capture the general
behavior of the real ECG, and the numerical simulation matches the real
measurements. These conclusions are clearer observing an enlargement of the
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simulated normal ECG presented in Fig 2.1, showing that it captures its general
characteristics, presenting the most important waves: P, QRS, T.
CONCLUSION
We have seen clearly that the heart is specially made for special functions and the
heart is made up of around half a billion cells. In general the heart is to a human
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body system as the car engine to a car. The heart having as main function to pump
oxygenated blood to all parts of the body has a complicated chain of activities even
involving electrical activities as if it is an electrified house. This just shows that the
heart is an electrical system. Without the electrical system, the heart would not
contract and would not pump blood. Blood would not circulate and the body would
not receive the oxygen and nutrients it needs. When blood flow stops to the brain, a
person loses consciousness in seconds and death follows within minutes. This
electrical system can be observed in various ways which are simplified as below;
The heartbeat originates in nodal tissue high in the right atrial wall called the
sinoatrial node, or pacemaker. Depolarization in the SA node is the event that
sparks off a heartbeat. From there, the wave of excitation spreads throughout the
muscles of both atria, which respond by contracting. The impulse passes through
another mass of nodal tissue, called the atrioventricular node (AV node), where the
impulse transmission is slowed. The AV node is located in the floor of the right
atrium near the interatrial septum. The AV node allows only 40-60 beats per
minute to pass on to the ventricles. When the impulse reaches the AV node it
enters a group of fibers called the AV bundle (Bundle of His) located in the upper
part of the interventricular septum. This divides into right and left bundle branches,
which innervate either side of the interventricular septum. The terminal branches
of the two bundle branches are the tree-like Purkinje fibers. When they reach the
terminals of the Purkinje fibers they leap across to the cardiac muscle fibers and set
in motion the molecular ratchets of actin and myosin, which slide past each other
to shorten the fiber. The impulses relayed to the individual myocardial cells result
in simultaneous contraction of both ventricles. All this happens in less than a
second.
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We can therefore conclude that a mathematical modeling of heart electrical
functioning is developed considering three modified VdP oscillators connected by
time delay coupling. Each oscillator represents one of the most important heart
natural pacemakers: sinoatrial node (SA), atrio-ventricular node (AV) and His
Purkinje complex (HP). The resulting differential difference equations are
integrated by considering an estimation of the time delayed system function with
carried out showing that
the proposed model is capable to capture the general heartbeat dynamics,
representing the normal ECG form with P, QRS and T waves. Moreover, external
pacemaker excitation is of concern representing other pathological behaviors as the
ventricular fibrillation. These results show that the three oscillator model captures
the general behavior of heart rhythms represented by ECG signals and may
encourage the identification of different pathological rhythms.
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