Linear Functionals in ECG and
VCG
Diplomarbeit
zur Erlangung des akademischen Grades
Diplom-Mathematiker/in
Westfälische Wilhelms-Universität Münster
Fachbereich Mathematik und Informatik
Institut für Numerische und Angewandte Mathematik
Betreuung:
Prof. Dr. Martin Burger
Eingereicht von:
Joanna Tendera
Münster, Februar 2013
i
Abstract
This thesis deals with the diagnostic method called vectorcardiogram, which is an
extension of the well known electrocardiogram. Based on the dipole theory, we express
the heart vector on the body surface as dot product of potential difference and lead
matrix. The linear functional strategy enables us to find a second representation of
the heart vector on the heart boundary. Thus, we analyze the heart vectors on the
body boundary and on the heart boundary for two different lead matrices and two
electrode configurations. Finally, we establish different vectorcardiogram parameters
and examine the relationship between the heart disease area and the corresponding
vectorcardiogram.
ii
Eidesstattliche Erklärung
Hiermit versichere ich, Joanna Tendera, dass ich die vorliegende Arbeit selbstständig
verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet
habe. Gedanklich, inhaltlich oder wörtlich übernommenes habe ich durch Angabe
von Herkunft und Text oder Anmerkung belegt bzw. kenntlich gemacht. Dies gilt in
gleicher Weise für Bilder, Tabellen, Zeichnungen und Skizzen, die nicht von mir selbst
erstellt wurden.
Alle auf der CD beigefügten Programme sind von mir selbst programmiert worden.
Münster, 21. Februar 2013
Joanna Tendera
iii
Acknowledgments
I want to thank everybody who made my studies and this thesis possible, especially:
• Prof. Dr. Martin Burger for giving me the opportunity to work on this interesting
topic, and for taking his time for assisting me with my problems and answering
all my questions.
• Meiner Familie, die immer an mich geglaubt hat.
• Matthias Gröne and Elin Sandberg for many helpful discussions and proofreading this thesis.
• all my friends who have supported me during the last years.
iv
Contents
1. Medical Background
1.1. Anatomy . . . . . . . . . . . . .
1.2. The Heart Cell . . . . . . . . .
1.3. Electric excitation of the Heart
1.4. Electrocardiogram . . . . . . .
1.5. 12-lead ECG . . . . . . . . . . .
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3. Vectorcardiography
3.1. Vectorcardiogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2. Linear Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4. Bidomain Modell
4.1. Bidomain Modell . . . . . . .
4.2. Forward and Inverse Problem
4.2.1. Forward Operator . . .
4.2.2. Inverse Operator . . .
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2. Heart Vector
2.1. Dipole Theory . . . . . . . . . . . . . . . . .
2.2. Dipole Approximation . . . . . . . . . . . .
2.3. Relation: Heart Vector Potential Difference .
2.4. Einthoven, Burger, van Milaan, Frank . . . .
2.5. Lead field . . . . . . . . . . . . . . . . . . .
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5. Linear Functional Strategy
5.1. Linear Functional Strategy and Heart Vector . . . . . . . .
5.1.1. Linear Algebra and the Linear Functional Strategy
5.2. Bidomain Model and Heart Vector . . . . . . . . . . . . .
5.3. Adjoint Bidomain Problem . . . . . . . . . . . . . . . . . .
5.3.1. Forward Adjoint Bidomain Problem . . . . . . . . .
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Contents
5.3.2. Inverse Adjoint Bidomain Problem . . .
5.4. Adjoint Operator . . . . . . . . . . . . . . . . .
5.4.1. Adjoint Operator: Heart-Torso Model .
5.4.2. Solution-Functional: Heart-Torso Model
5.4.3. Adjoint Bidomain Operator . . . . . . .
5.5. Solution-functional . . . . . . . . . . . . . . . .
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8. VCG and heart disease
8.1. Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2. VCG and diseased area . . . . . . . . . . . . . . . . . . . . . . . . . . .
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9. Conclusion and Outlook
79
A. Appendix
A.0.1. Transformation of Heart Vector into Solution Functional . . . .
A.1. Moore-Penrose-Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2. Functional Analysis Background . . . . . . . . . . . . . . . . . . . . . .
81
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List of Figures
90
Bibliography
93
6. Implementation
6.1. Finite Element Method . . . . . . . . . . . . . . .
6.2. Torso model . . . . . . . . . . . . . . . . . . . . .
6.3. Implementation Lead Matrix . . . . . . . . . . . .
6.4. Implementation Linear Functional Strategy . . . .
6.4.1. Solving the adjoint problem . . . . . . . .
6.4.2. Evaluation of the heart vector on the heart
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7. Results
7.1. Error Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2. Heart Vector with reciprocity Lead Matrix . . . . . . . . . . . . . . .
7.3. Heart Vector with Frank Lead Matrix . . . . . . . . . . . . . . . . . .
7.4. Heart Vector with Frank Lead Matrix and Frank potential differences
7.5. Reciprocity Lead Matrix vs. Frank Lead Matrix . . . . . . . . . . . .
1
Introduction
Despite every progress in modern medicine, cardiovascular diseases are among the
most widespread lifestyle diseases. Recalling the WHO, cardiovascular diseases cause
the most deaths globally. In the year 2008, 30% of all global deaths were caused by a
cardiovasular disease. The WHO prognosticates that in the year 2030 about 25 million
people will die from cardiovasular diseases. The reason why heart diseases are a lifestyle
disease are risk factors like unhealthy diet, physical inactivity, tabacco use and harmful
use of alcohol1 .
In first aid courses we learn that a quick identification of a cardiovascular disease
is very important. A first non-invasive diagnostic method is the electrocardiogram.
The electrocardiogram measures the potential difference on the body surface. The
cardiologist compares this data with standard values and can detect a cardiovascular
disease. The electrocardiogram pioneer Einthoven establised the concept of the heart
dipole. In the heart dipole theory it is assumed that the potential difference on the body
surface arises from a dipole or rather that the processes in the heart can be abstract
to one dipole. So the cardiologists want to display this heart dipole by using the
electrode configuration of the electrocardiogram. Till now the Einthoven triangle and
the electrical axis of the heart are very popular in medicine. The vectorcardiogram
results from this idea as a further diagnostic tool. The vectorcardiogram displays
the electrical axis of the heart as a function of time and space. Comparable to the
electrocardiogram, standard values are compared to the vectorcardiogram data. The
most common electrode configuration was investigated by Frank. In this thesis we
want to examine the vectorcardiogram in detail. We transform the dipole components,
which are recorded in the vectorcardiogram and measured on the body surface, on the
heart boundary. For this purpose we determine a matrix, which connects the heart
dipole and the measured potential differences. We compare the dipole components on
the heart boundary to the dipole components on the body boundary for the standard
Frank system as well as for our system. We also check the Frank system against our
system.
1
WHO[6]
Contents
2
This thesis is organized as follows. In order to comprehend how the heart works, we
begin in Chapter 1 with the electrical function of the heart. Furthermore, we describe
the standard 12-lead electrocardiogram in the first chapter. The vectorcardiogram is
acting on the assumption that the electrical activity of the heart can be approximate by
a dipole, Chapter 2 introduces the reader into the dipole assumption and describes the
relationship between the heart dipole and the electrocardigam’s potential difference.
In addition, the first vectorcardiogram lead systems are presented. Since the heart
dipole cannot only be interpreted as dot product, Chapter 3 describes the heart dipole
components as a linear functional. We introduce the Bidomain Model in Chapter 4
and take a look at the forward and inverse problem. In order to comprehend the heart
dipole, we want to find a representation for the heart dipole on the heart. We manage
the transformation of the heart dipole on the heart with the aid of the linear functional
strategy in Chapter 5. Chapter 6 deals with the finite element method, the basic model
geometry and the implementation of the linear functional strategy. The results of the
implementation are in Chapter 7, where we compare the heart dipoles determined on
the body surface and the heart surface as well as the two different lead matrices. In the
last chapter, Chapter 8, we have a look at diseased hearts and their vectorcardiograms.
Finally, we end with a conclusion and outlook in Chapter 9.
3
1. Medical Background
In this chapter, we want to describe the anatomy of the heart, its electrical excitation,
the resulting electrocardial signal and the position of the standard recordings of the
electrocardiogram. The anatomy part follows the ideas of [27], the electrical excitation
of the cells follow [28], the activation of heart can be found in [21] like the description
of the electrocardiographic signal. The standard recording positions follow the ideas
in [20].
1.1. Anatomy
Figure 1.1.: heart anatomy,[21]
The heart is the human’s vital pump. It is located between the two lungs behind the
sternum. The backside of the heart borders with the gullet and the aorta. At the
bottom, the heart is in direct contact with the diaphragm. The cardiac axis runs from
back top right to left down. Its size is comparable to a closed fist and the heart’s weight
1
Medical Background
4
is about 300 gram. When we have a close look at the heart, we can discover two equal
parts, which are divided by the cardiac septum. The right heart part is responsible for
absorbing the deoxygenated blood from the body and pumping it into the pulmonary
circulation. The left heart part distributes the oxygenated blood into the body. The
heart is a big muscle, which has four interior spaces: The two atriums and the two
ventricle. The atrium collects the blood and passes it into the ventricles through the
mitral valve on the left heart part and the tricuspid valve on the right heart part. After
that the left ventricle passes the blood through the pulmonary artery into the lungs
and the right ventricle passes the oxygenated blood through the aorta into the whole
body.
1.2. The Heart Cell
Before we have a look at the electric excitation of the heart, we examine the heart
cells and its excitation. The myocard consists of electrical excitable cells, which are
in a stable resting state. The inside of the cell is compared to the outside of the
cell negatively polarized, consequently a membrane potential of about -90mV to 70mV exists, the transmembrane potential. The transmembrane potential depends
on the cell type. The reason is the Nernst equilibrium of potassium K + . If a cell is
electrically activated, it communicates with the other cells through gap junctions. Ion
channels allow the transfer of ions from cell to cell. Each substance has an own ion
channel, which is open or closed subject to the transmembrane potential. When a heart
cell depolarize, the natrium ion channel opens and the potassium ion channel closes.
The natrium causes the cell to depolarize quickly, this is called upstroke. After that
the cell repolarize partially, caused by a potassium outflow. A long plateau follows,
because an equilibrium between the potassium and the slow inward calcium inflow
exists. Deactivation of the calcium inflows conduct to the last phase the repolarisation
of the whole cell. After that the cell achieves its resting potential.
Figure 1.2.: transmembrane potential of an excited cardiac muscle cell of a frog,[21]
1
Medical Background
5
1.3. Electric excitation of the Heart
We know how a single heart cell acts, when it is electrically excited. But the heart
consists of many cells, which are connected together. So if one cell is electrically excited,
the excitation is passed through the whole heart muscle. This excitation happens along
a certain order, which we will explain now.
Until now we always assumed that the cells were excited by an external stimulus, but
the heart posses cells, which can activate themselves. These cells form the sinus node
(SA node), which can be located in the right atrium at the super vena cava. The sinus
node with its self excitatory cells is the initial point of the hearts electric activation.
The atrium musculature passes the electric signal, caused in the SA node, through
the whole atrium, but cannot pass the signal directly to the ventricle. In this way
the propagation of the potential is delayed. The atrioventricular node (AV node) is
located at the border between the right atrium and the right ventricle and the AV node
makes it possible to activate the ventricles. But the ventricles are not activated like
the atriums by its musculature. In order to excite the ventricles we have the bundle
of His. The bundle of His consists of two branches, one for the right ventricle and the
other one for the left ventricle, they run along the cardiac septum and the branches
ends are called the Prukinje fibers. So the hearts conductivity system begins in the SA
node followed by the AV node, the bundle of His and ending in the Purkinje fibers.
Figure 1.3.: The conductivity system of the heart,[21]
1
Medical Background
6
1.4. Electrocardiogram
In the previous section, we have seen that the electric activation of the SA node propagates along a given order in the heart. In this process, there arise a low electrical
current flow, which is measurable at the heart surface but also at the body surface.
The arising potential difference can be detected with electrodes on the body surface.
We can interpret the resulting potential difference as the sum of the conductivitiy
systems potential differences, like Figure 1.4.a implies.
(a) The sum of the conductivity systems potential
differences,[21]
(b) explained electrocardiogram,[21]
Figure 1.4.: The Electrocardiogram
We want to have a closer look at the resulting potential difference, the electrocardiogram (ECG) Figure 1.4.b. The first deflection of the base line is the P wave, which
arises when the atriums are excited. If the whole atriums are excited the electrical
signal comes back to the base line. The activation of the ventricles results in the QRS
complex. The PQ interval can be interpreted as the excitation of the AV node and the
bundle of His. The Q spike is explained by the excitation of the septum. After the
activation of the ventricles the ECG signal returns to the base line. The last wave is
called T wave and represents the repolarization of the ventricles. In some cases a U
wave can be measured, which represents the recovery of the Purkinje fibres.
1
Medical Background
7
1.5. 12-lead ECG
In this section, we will explain the most used lead system, the 12-lead ECG. For
this purpose, three electrodes are fixed at the right arm, the left arm and the left
leg, respectively. Then we obtain the limb leads I, II and III of Einthoven. If we
interconnect two of this limb leads to an indifferent electrode, then we can introduce
three new leads, which are called augmented limb leads aVR, aVL, aVF. For the
augmented limb lead aVR, the left arm and the left leg electrode are connected, for
aVL the right arm and the left leg electrode and for aVF the right arm and the left arm
electrode. The precordial leads V1 , V2 , V3 , V4 , V5 and V6 complete the 12-lead ECG. In
order to describe the position of the precordial electrodes, we introduce the intercostal
space (ICR), which is the space between two ribs, in which the first ICR is the space
between the first rib and the second rib beginning at the clavicle. We introduce the
mid-clavicular line, which is the perpendicular line at the middle of the clavicle. The
mid axillary line is the perpendicular line beginning at the axilla. The precordial leads
are located like this: V1 and V2 are placed at the 4.ICR, V1 on the right side of the
sternum and V2 on the left side of the sternum. V4 is located at the interception point
of the mid-clavicular line and the 5.ICR. V3 is located between the electrode positions
V2 and V4 . The last two electrodes V5 and V6 are the interception points of the front
auxillary line and the 5.ICR and the mid axillary line and the 5.ICR, respectively. In
Figure 1.5 all 9 electrode positions are illustrated.
(a) Einthoven leads,[21]
(b) precordial leads,[21]
(c) Goldbergs augmented leads,[21]
Figure 1.5.: The electrode position of a 12-lead ECG
8
2. Heart Vector
The aim of this chapter is to obtain an idea of the heart vector concept. If we decode the
heart vector’s meaning, then we may interpret the vectorcardiogram in a mathematical
way. But first of all, we introduce the dipole concept, which is a key element of the heart
vector concept. After that we get to know the lead vector and lead fields, introduced
by Burger and van Milaan in [4]. The lead fields allow us to rewrite the lead voltage as
dot product of a heart vector and a lead vector. This representation helps to determine
a heart vector from an ECG.
2.1. Dipole Theory
One of the simplst source configuration is the point source or also called monopole.
This point source generates an electrical field with the potential ϕ (r0 ) at a measuring
point r0
ϕ (r0 ) =
1
I
,
4πσ r − r0
(2.1)
where r is the origin point of the source and I the current density.
But this source does not fit our problem. An extension of the point source is the
dipole. We consider two point sources with opposite charge. The positive monopole is
the source and the negative monopole the sink, the two monopoles are separated by a
small distance δ. The potential at a point Q with distance r from the middle of the
connecting line δ between the two poles, is given by the principle of superposition
I
ϕ (r) =
4πσ
1
1
−
r+ r−
,
(2.2)
with r+ = r +δ/2 and r− = r −δ/2. If we choose δ r it follows that r+ −r− ≈ δ cos ϕ
and r+ · r− ≈ r2 , like in Figure 2.1, so we obtain for the dipole potential (2.2)
2
Heart Vector
9
ϕ (r) =
I
δ cos θ,
4πσr2
(2.3)
with the angle θ between r and δ.
Figure 2.1.: Illustration of the geometry of a dipole,[3]
Let the displacement δ decrease and the current density I increase such that Iδ = H
remains finite, we rewrite the potential
ϕ (r) = H
cos θ
.
4πσr2
(2.4)
We receive on the one hand the mathematical dipole, by
δ→0
I→∞
and
Iδ = H
constant,
(2.5)
and on the other hand the dipole moment H.
A dipole is characterized by its dipole moment, which is described by a vector. In
Cartesian coordinates we can write the dipole moment
H = Hx ex + Hy ey + Hz ez ,
(2.6)
where Hx , Hy , Hz , are the dipole moments in x,y,z direction and ex , ey , ez , unit vectors
along the axis of the Cartesian coordinate system.
2
Heart Vector
10
2.2. Dipole Approximation
Our task is to define the heart vector in context of the dipole theory. We know that
the spreading action potential wave front in the heart is the result of a current source.
If we approximate this action potential wave front as a surface of dipoles, then we have
a surface of characterizing dipole moments. This entails us to our first simplification,
let us assume that we can represent this dipole surface with only one dipole which is
fixed in position. Then the dipole moment of this single dipole consists of the sum of
dipole moments of its dipole elements. We call this resulting dipole heart vector or
heart dipole H. It is given by
Z
H (t) =
Jdx,
(2.7)
Heart
where J is the dipole density.
The heart vector indicates the direction in which electricity is propagated by the heart,
it is a time varying quantity like the action potential.
2.3. Relation: Heart Vector Potential Difference
Now we are in a position to describe the relationship between the heart vector and
the potential. Let us recall how such a dipole generates a potential field. We know
that every little piece of the heart muscle contributes to the current field, so the total
action of the heart is the result of the actions of all small pieces. Consequently, the
current field is the sum of the electrical field strength in the different pieces of the heart
muscle. Actually we have a heart dipole exciting a current field. This current field in
the body can be measured between two points at the body surface. The principle of
superposition enables us to specify the potential difference Vij between two points i and
j to
V ij = Vxij ex + Vyij ey + Vzij ez ,
(2.8)
where Vxij , Vyij and Vzij are potential differences along x−, y− and z− axis, respectively.
The heart dipole produces a potential and on account of the dipole theory in Section
2.1 we can write the potential difference in x- direction as
2
Heart Vector
11
Vxij = ϕ (i)x − ϕ (j)x ,
r1 − r2
= H·
,
4πσ |r1 − r2 |3
= lx · Hx ,
(2.9)
similar equations occur for the potential difference in y- and z- direction. These constants lx , ly and lz , called lead vectors, do not depend on the direction and magnitude
of the heart vector but they depend on the shape, dimension and conductivity of the
body and the position of the electrodes used. The lead vector concept was introduced
r1 −r2
by Burger and van Milaan [4]. From dipole theory we obtain lx = 4πσ|r
3 with the
1 −r2 |
electrode leads at position r1 and r2 . We see that for a real person this lead vector is not
exact. Applying the lead vector representation (2.9) to the three potential differences
Vx , Vy , Vz and exploiting the linearity of the medium we obtain
V ij (t) = lx Hx (t) + ly Hy (t) + lz Hz (t) ,
= l · H (t) ,
(2.10)
(2.11)
where V ij is the dot product of the lead vector l = lx ex +ly ey +lz ez and the heart vector
H = Hx ex + Hy ey + Hz ez . We see that the heart vector is time-dependent, because the
potential difference is time-dependent too.
Until now we assumed that the dipole is fixed in location. If we allow that the source
location shifts then the lead vector concept fails. To encompass this problem we introduce the lead field, which is an extension of the lead vector concept and an idea of
McFee and Johnston [24]. Let us have a look at a family of lead vectors for various
dipole locations, then the lead field L (x, y, z) contains all this lead vectors. Exploiting the principle of superposition, the total lead voltage equals the sum of all the
contributions of each dipole,
V ij =
X
Hk · lk ,
k
with Hk dipole moments and k = 1, ..., n numbers of dipoles.
(2.12)
2
Heart Vector
12
So if we want to determine the heart vector we must measure simultaneously three
independent leads and we need three lead vectors. In hospitals, detecting three leads
at once is no problem and in the next section we choose three independent leads.
2.4. Einthoven, Burger, van Milaan, Frank
In the previous section, we explained the relationship between the heart vector and the
differential potential at the body surface. In this section, we want to introduce some
ideas how to determine the heart vector. We begin with Einthovens’ idea, following
[19]. He was the first one who recognized a relation between a heart vector and the
potential in the frontal plane. Einthoven assumed the body as a homogeneous, infinite
volume conductor with unit conductance. Einthoven used the readings VI , VII and VIII ,
and arranged the leads in a equilateral triangle, later called the Einthoven triangle.
The sides of the equilateral triangle are the connecting lines between right arm and left
arm, left arm and left foot and right arm and left foot. In that way, we obtain the lead
√ vector lI = (1, 0, 0) and lII = 12 , 21 3, 0 . Einthoven hypothesized that the potential
difference is the amplitude of the projection of the heart vector. So if the heart vector
is directed along the x− axis of the body, there would be a potential difference between
the right and left arm.
Figure 2.2.: Einthovens’ triangle
Einthoven took advantage of the geometry to gain the heart vector as follow: He drew
the amplitude of the reading VI in the middle of the connecting line between right and
left arm, in the same way he proceeded with the reading VII on the connecting line
between right arm and foot. Then he took the perpendicular of the origin and the
endpoint, the heart vectors’ origin results in the intersection point of the recordings’
origin perpendicular lines. The endpoint of the heart vector is the intersection point
of the recordings’s endpoint perpendicular lines, see Figure 2.2.
2
Heart Vector
13
In a mathematical way and with the lead vector concept we obtain by using Einthoven’s
triangle an overdetermined system of equations
VI = lI · H = Hx
√
3
1
Hy
VII = lII · H = Hx +
2
2√
1
3
VIII = lIII · H = − Hx +
Hy .
2
2
(2.13)
(2.14)
(2.15)
One solution for the heart vector coordinates is H = VI , 2·V√II 3−VI . It is clear that the
Einthoven lead vectors are not accurate. Burger and van Milaan extended Einthovens’
idea to a three dimensional geometry, see [4]. They examined the relationship between heart vector and lead in a glass phantom filled with electrolyte and an artificial heart. By these experiments they measured alternative lead vectors, given by
lI = (0.923, −0.298, 0.241) and lII = (0.202, 0.972, −0.121). Applying Kirchhoffs’ law,
VI + VIII = VII , we get the third lead vector lIII = (0.721, 1.270, 0.362). We see that this
problem is ill-conditioned, if we want to determine a three dimensional heart vector.
Many investigators used the leads’ geometry, like a tetrahedron, cube or rectangle, to
determine the heart vector.
The next one we will introduce is the lead system of Frank, see [15]. Einthoven, Burger
and van Milaan used the leads VI , VII and VIII to determine the heart vector, in Section 2.3 we noticed that three independent recordings are needed. Frank used seven
electrodes to minimize effects of dipole location variation. Five of seven electrodes are
placed at the transverse level, which is approximately the fifth interspace, electrode H
is on the back of the neck and electrode F is on the left leg. The electrodes on the
transverse level are placed as follows, E at the front and M on the back midline, I at
the right midaxillary line and A on the left midaxillary line and C at an angle of 45
degrees between the front midline and left midaxillary line.
Frank placed the heart dipole on the left side of the vertical plane, 14.8 per cent of the
thorax depth forward at the level of the fifth interspace, like dipole position 22 in [14],
see Figure 2.3.b.
So we receive for this dipole location the potential differences expressed in dipole components
2
Heart Vector
14
(a) Franks lead system, [15]
(b) Dipole position 22 equals second column and second row, [14]
Figure 2.3.: Franks lead system and dipole position
VA = 95Hx + 58Hz ,
VC = 131Hx − 113Hz ,
VE = −60Hx − 130Hz ,
VM = −32Hx + 80Hz ,
(2.16)
VI = −71Hx + 21Hz ,
VH = −24Hx − 76Hy + 35Hz ,
VF = −21Hx + 91Hy + 11Hz .
From node analysis of Figure 2.3.a we achieve
Vx = 0.610VA + 0.171VC − 0.781VI ,
Vy = 0.655VF + 0.345VM − 1.000VH ,
(2.17)
Vz = 0.133VA + 0.736VM − 0.264VI − 0.374VE − 0.231VC ,
with Vx , Vy and Vz the potential differences in x-, y− and z- direction. Inserting the
potential of equation (2.16) into the equations (2.17) received from node analysis we
obtain
2
Heart Vector
15
Vx = 136Hx − 0.2Hz ,
Vy = −0.8Hx + 136Hy − 0.2Hz ,
(2.18)
Vz = 136Hz .
Franks system was the first corrected orthogonal lead system, this means that the
length of all three lead vectors of orthogonal potential differences is nearly equal.
Next we want to introduce the SVCG lead system in which Franks’ lead system acts
as a reference system. In [10] Dower expressed the standard readings as a linear
combination of the orthogonal potentials Vx , Vy and Vz and obtained the Dower matrix.
The independent readings V1 to V6 , VI and VII can be rewritten
V = A · Vxyz ,
(2.19)
with V = (V1 , V2 , V3 , V4 , V5 , V6 , VI , VII )T , Vxyz = (Vx , Vy , Vz )T and A the 3x12 transfer
matrix. In order to acquire the vector Vxyz , we define the vector M = AT A and use
M −1 M = I, with I the identity matrix, then
Vxyz = IVxyz = M −1 M Vxyz = M −1 AT AVxyz = M −1 AT V,
(2.20)
where M −1 AT is called the inverse Dower matrix and is numerically given by


−0.172 −0.074
0.122
0.231 0.239 0.194
0.156 −0.010


M −1 AT =  0.057 −0.019 −0.106 −0.022 0.041 0.048 −0.227
0.887 ,
−0.229 −0.310 −0.246 −0.063 0.055 0.108
0.022
0.102
see [10].
Because of the fact that the Frank lead system acts as a reference system we use Frank’s
representation (2.18) and obtain the heart vector.
2
Heart Vector
16
2.5. Lead field
Knowing the three different lead vector systems of Einthoven, Frank, Burger and van
Milaan, we now want to explain how we can build a lead vector system. Weinstein formulated in [7] an idea for electroencephalography, we adopt his idea for the evaluation
of the heart vector. We have in mind that the lead vectors depend on
• the location of the source,
• the location of the electrodes and
• the shape and conductivity of the volume conductor.
So it is clear that theoretically we have to construct an individual lead vector system for
each patient, if we want to determine the heart vector. Therefor we use the Helmholtz
principle of reciprocity, which was applied by McFee and Johnston to electrocardiography, [24]. Let us consider a pair of electrodes A and B on the body surface and
a second pair of electrodes C and D in the heart region. If we inject a current I at
electrode C and remove it at electrode D we note a potential difference VAB between
the electrodes A and B. On the other hand if we inject a current I 0 at electrode A
and remove it at electrode B we can observe a potential difference VCD . Then the
reciprocity theorem states
VCD
VAB
= 0 .
I
I
(2.21)
Acting on the assumption that the potential difference on the body surface is caused
by a resulting dipole, we assume that the electrodes C and D are separated by a small
distance δ. Consequently, we can rewrite the potential difference VCD as product of
the electric field E and the small distance δ. Inserting this product in the reciprocity
theorem equation (2.21) and multiply both sides with the injected current I, we obtain
the potential difference VAB according to the heart vector,
VAB =
VCD · I
E · δI
E·H
=
=
.
0
0
I
I
I0
(2.22)
2
Heart Vector
17
Figure 2.4.: Illustration of the reciprocity theorem for electrocardiography. First one
shows the resulting potential difference VAB , second one the potential difference VCD , third one the lead field by injected current at electrodes A
and removed current at electrode B, [26]
.
So we can define the ratio of the electric field E and the injected current I 0 as lead
matrix L which contains the lead vectors. In order to receive the lead vectors we have
to inject a current I at an electrode A and remove it at electrode B. Then we can
measure the resulting potential field φ and by taking the gradient of the potential field
we obtain the electric field E. Dividing this electric field by the injected current we
have computed the lead field matrix.
18
3. Vectorcardiography
In the previous chapters, we introduced the mathematical dipole and pointed out the
relationship between the dipole and the electrical potential. In this chapter we get to
know the vectorcardiogram. We will see that the vectorcardiogram is a tool to visualize
the heart vector.
3.1. Vectorcardiogram
In section 2.2, we defined the heart vector as dipole moment of a point source. We
can describe a vector by its origin, magnitude and orientation. Assuming that the
heart vector is fixed in its origin and a time-dependent function in magnitude and
orientation, we can display the heart vector during a cardiac cycle. The tips of the
vector trace a loop in space, the vectorcardiogram. Projections of this path on three
orthogonal planes show changes in the frontal plane, (X, Y ), the sagittal plane, (Z, Y ),
and in the horizontal/transverse plane, (XZ). Plotting the heart vector during one
cardiac cycle we can see three loops. The first one, called P-loop, shows the excitation
in the heart atrium, Figure 3.1.a, the QRS-loop represents the excitation of the heart
ventricles, Figure 3.1.b.c, and the T-loop describes the regression of the heart ventricle,
Figure 3.1.d. In clinical application the largest dipole moment in the frontal plane is
called electrical axis of the heart. Till now it is determined by Einthovens triangle rule,
explained in Section 2.4.
3
Vectorcardiography
19
(a)
(b)
(c)
(d)
Figure 3.1.: Vectorcardiogram, [21]
3.2. Linear Functional
In order to analyse the heart vector in a more general case, we define the linear functional.
Definition 3.2.1 (linear functional). A continuous operator is a continuous linear
mapping between two normed vectorspaces. A linear functional F is a linear continuous
operator from a vectorspace X to its field of scalars <.
Our next aim is to rewrite the heart vector to a linear functional. Therefore we use the
relationship between the potential difference and the heart vector (2.10). We assume
that the lead vectors are well determined and we can invert them, then we obtain
3
Vectorcardiography
20
p (t) = l−1 · Vij .
(3.1)
It remains to rewrite the potential difference into Vij = u (ri )−u (rj ) with the potentials
u measured at the position ri and rj . Exploiting the dirac-delta function on the torso
boundary where we measure the potential to calculate the heart vector, we obtain
Z
l−1 uT δri − δrj ds.
p (uT (t)) =
(3.2)
∂B
So if uT is a continuous function then each entry of the heart vector p : C (∂B) → <
is a linear functional. Other linear functionals used in clinical applications are
• averaged activation time
Z
t1
Z
p1 [uT ] =
t0
l (x) uT (x, t) dσdt,
(3.3)
∂uT
(x, t) dσ,
∂t
(3.4)
∂B
• averaged propagation velocity/speed
Z
l (x)
p2 [uT ] =
∂B
with adequate weighting functions l (x).
21
4. Bidomain Modell
Now we want to introduce the Bidomain Model which describes the electrical properties
of the heart considering two potentials, the intracellular potential and the extracellular
potential. We follow the ideas of [18].
4.1. Bidomain Modell
First of all we divide the heart tissue into two separate domains, the first one is the
intracellular domain and the second one is the extracellular domain, the cell membrane
separates the two domains. Let us assume that all three of them, intracellular domain,
extracellular domain, and cell membrane are continuous, and that they fill the complete
volume of the heart muscle. The resistance of the cell membrane is very high so it selects
whether electrically charged molecules can pass or not. Hence the electrical current
will cross the membrane, this potential difference is called transmembrane potential.
It is the difference between the extracellular potential and the intracellular potential
and is different in every point of the heart. We define ui and ue as intracellular and
extracellular electrical potential, respectively. Then the transmembrane potential is
given by ue − ui = ν.
Figure 4.1.: schematic model of the heart and the torso and their normals, [23]
It remains to define the heart and torso domain, see Figure 4.1. Let H be the heart
region and T the surrounding torso. The entire body is then given as B = H ∪ T , the
4
Bidomain Modell
22
heart is bounded by ∂H and the body by ∂H and ∂T . The entire body boundary is
given by ∂T = ∂H ∪ ∂B. Let nH denote the outward heart surface normal, nT = nB
the outward body surface normal.
In order to explain the electrical behavior of the entire heart we assume that the human
body is a volume conductor. This assumption avoids the difficulties of modeling every
single tissue. Describing electric effects in a volume conductor leads to Maxwell’s
equation, which characterizes the relation between electric and magnetic fields, given
by
∇×E+
∂B
= 0,
∂t
(4.1)
where E and B are the strengths of the electric and magnetic fields, respectively.
The electrical activation in the heart is a fast process but the resulting variation in the
electric and magnetic fields are slow. Consequently, temporal variations can be ignored
and the Maxwell’s equation can be rewritten as
∇ × E = 0.
(4.2)
Neglecting temporal variations in field theory is equal to the assumption that the fields
are quasi static. From this information the electric field in the intracellular space, the
extracellular space and the extracardial space Ei , Ee , ET can be related to a scalar
potential
Ei = −∇ui ,
Ee = −∇ue ,
(4.3)
ET = −∇uT .
From physics we know that the electric current J, a flow of electric charge, in a conductor is described by Ohm’s Law of conductivity
J = M · E,
(4.4)
where M is the conductivity of the medium, a function of position. Inserting (4.3)
4
Bidomain Modell
23
gives
Ji = −Mi · ∇ui ,
Je = −Me · ∇ue ,
(4.5)
JT = −MT · ∇uT ,
with Mi , Me and MT conductivity tensors in the intracellular, extracellular and extracardial space. We noted that the cell membrane allows electrically charged molecules
to pass or not, so it acts like an insulator. Thus, we assume that there may be some
build up of charge in each domain, but every accumulation of charge on the one side
of the membrane excites an accumulation on the other side of the membrane of opposite charge. Because of the thickness of the membrane there is always a balance in
charge and the total charge accumulation is zero in any point. This balance of charges
describes the equation
∂
(qi + qe ) = 0,
∂t
(4.6)
where qi is intracellular charge and qe is extracellular charge.
The net current into a point is composed of the charge accumulation and the ionic
current Iion inflow or outflow over the membrane. The positive direction is defined
from intracellular to extracellular
∂qi
+ Iion ,
∂t
∂qe
=
+ Iion .
∂t
− ∇ · Ji =
(4.7)
−∇ · Je
(4.8)
Inserting (4.7) into (4.6), so the total current, Jtot = Ji + Je , is conserved and applying
(4.5) we obtain
0 = ∇ · (Ji + Je ) = ∇ · (Mi ∇ui ) + ∇ · (Me ∇ue ) .
Exploiting the transmembrane potential we can rewrite the total current (4.9)
(4.9)
4
Bidomain Modell
24
∇ · (Mi ν) + ∇ · ((Mi + Me ) ∇ue ) = 0.
(4.10)
To gain an equation for the torso T we use the fact that the total current is also
conserved in the body, so we obtain
0 = ∇ · Jtot = ∇ · (MT ∇uT ) .
(4.11)
Finally, we have to couple the heart and the body. For this purpose we need additional
assumptions. First, the extracellular domain is in direct contact with the extracardiac
domain, so the extracellular potential ue on the boundary of the heart is equal to the
extracardiac potential uT ,
ue = uT
on ∂H.
(4.12)
In contrast, the intracellular domain is completely insulated from its surroundings
(Mi ∇ui ) · nH = 0 in H.
(4.13)
Noticing that the heart is surrounded by the body, a volume conductor, the flow across
the boundary of the heart must be continuous, this results in the condition that the
normal of the total current must equal the normal of the extracardiac current
nH · (Me ∇ue ) = nH · (MT ∇uT ) = −nT · (MT ∇uT ) on ∂H.
(4.14)
The body is surrounded by air so the normal component of the extracardiac current
on the body equals zero
nT · (MT ∇uT ) = 0 on ∂B.
(4.15)
An additional boundary condition results when we integrate the conserved total current
4
Bidomain Modell
25
(4.9), if we apply Gauss theorem and exploit the fact that the intracellular domain is
insulated we obtain
nH · (Mi ∇ν) + nH · (Mi ∇ue ) = 0 on ∂H.
(4.16)
Inserting (4.16) into (4.14) we obtain the last boundary equation
nH · (Mi ∇ν + (Mi + Me ) ∇ue ) = nH · (MT ∇uT ) on ∂H.
(4.17)
4.2. Forward and Inverse Problem
Talking about electrocardiography and the bidomain model we cannot avoid the Forward and the Inverse Problem.
Definition 4.2.1 (Forward/Inverse Problem [29]):
A mathematical model Au = s is a mapping
A:X→Y
(4.18)
from a set of inputs (parameters) X to a set of outputs (data) Y. The forward or direct
problem calculates the outputs from the inputs, means we establish Au for u ∈ X. The
opposite case is called the inverse problem, means we find to an output s ∈ Y the input
u ∈ X, so that Au = s holds.
In this context we have to define the properly posed problem and the improperly posed
problem also called ill-posed problem.
Definition 4.2.2 (properly posed problem [1]):
The mathematical model Au = s corresponds to a properly posed problem, if
1. it has at least one solution (existence),
2. it has at most one solution (uniqueness) and
4
Bidomain Modell
26
3. the data (output) depends continuously on the solution (input)(continuous dependence),
where A : F1 → F2 denotes a mapping from some function space F1 to another function
space F2 , s is an indirect measurement and u some specific property of the phenomenon
of interest.
Definition 4.2.3 (improperly posed problem [1]):
The mathematical model Au = s corresponds to an improperly posed problem, if it fails
to satisfy at least one of the conditions for it to be a properly posed problem.
With these definitions let us have a look at the Bidomain Model. At first we will talk
about the Forward Problem, in this case the inputs are the conductivity of the entire
body and a source, which generates the electrical activity in the heart. Then we have
to determine a surface potential uT , the output. In mathematical terms we seek a
forward operator A which solves the problem
Aν = uT
A : H 1 (H) → L2 (∂T ) ν 7→ uT .
(4.19)
Here the transmembrane potential is linked to the surface potential via the Bidomain
equations (4.10) − (4.17). For further applications we split the forward operator, like
in [23], into
A = A2 ◦ A1 ,
(4.20)
where A2 is the operator used to map the extracellular potential of the heart boundary
to the extracardiac potential on the torso boundary,
1/2
A2 : H0 (∂H) → L2 (∂B) , ue |∂H 7→ uT |∂B .
(4.21)
The second operator A1 projects the transmembrane potential on the extracellular
potential on the heart boundary
1/2
A1 : H 1 (H) → H0 (∂H) , ν 7→ ue |∂H ,
(4.22)
4
Bidomain Modell
27
1/2
and we define H0 (∂H) as
1/2
H0
(∂H) :=
h∈H
1/2
Z
(∂H) |
hdσ = 0 .
(4.23)
∂H
The Inverse Problem deals with the evaluation of the inputs knowing the outputs. So we
have the measurements on the body surface but we want to know the transmembrane
potential. Therefore we would like to invert the forward operator, ν = A−1 uT but we
will show that inverting the forward operator is an improperly posed problem.
4.2.1. Forward Operator
Our next aim is to show that the Forward Problem is properly posed. First of all we
can remark, that the Bidomain Model can be split into two partial differential equations. The first one is the Poisson equation, which holds in the heart. The second one
is the Laplace equation valid in the torso. Both are linked together via the boundary
conditions. So the existence of a solution is assured, see [11]. In order to guarantee
R
a unique solution we add a normalization condition like ∂H uT dσ = 0, because the
solution of the Poisson equation with Neumann boundary condition is unique except
for an additive constant. At least we must show that the operator A is continuously
dependent. Therefore we will use operator splitting.
Theorem 4.2.1 (Continuity of the Forward Operator):
The forward Bidomain operator A : H 1 (H) → L2 (∂B) defined by (4.19) is continuous.
Proof. We split our proof into two parts, we show first that A1 is continuous, afterwards that A2 is continuous. Thereby A = A2 ◦ A1 and the linearity of Poisson and
Laplace equation is the Bidomain Forward problem a properly posed problem.
In order to prove that A1 is continuous we must define the trace of an operator.
Theorem 4.2.2 (trace operator [22]):
Assume ∂H has positive degree and is piecewise Lipschitz, then there exists a linear
continuous operator
T : H 1 (H) → H 1/2 (∂H)
with
T ν = ν|∂H = ue
∀ν ∈ C H
(4.24)
The operator T is surjective, means for all ue ∈ H 1/2 (∂H) exists a ν ∈ H 1 (H) with
4
Bidomain Modell
28
T ν = ue .
1/2
Application of the trace theorem implies that for ue ∈ H0 (∂H) ⊂ H 1/2 (∂H) exists
an ν ∈ H 1 (H), so we can identify the trace operator as our forward operator A1 and
conclude that A1 is continuous.
Now it remains to show that A2 is continuous, the idea of this proof is to use the
fundamental solution of the Laplace equation. Like in [11], we can rewrite uT as
Z
uT |∂B =
Z
MT ∇G (x, y) · nH · ue (y) dσ (y) =
∂H
k (x, y) · ue (y) dσ(y),
(4.25)
∂H
with G(x, y) the fundamental solution of the Laplace equation.
Remark that the fundamental solution is locally integrable, and the solution uT |∂B is
the conclusion of this fundamental solution and a function in C 2 (Ω),[11]. Then we can
rewrite the operator equation
Z
k (x, y) · uT (y) dσ (y) = uT .
A2 ue =
∂H
We are now in a position to estimate
kA2 ue k2L2 (∂B)
=
=
C.S
≤
≤
ue =uT on ∂H
=
Z
2
k
(x,
y)
·
u
(y)
dσ
(y)
T
2
∂H
L (∂B)
Z Z
k (x, y)2 · uT (y)2 dσ (y) dσ (x)
Z∂B Z∂H
Z
2
k (x, y) dσ (y)
uT (y)2 dσ (y) dσ (x)
∂BZ ∂H
∂H
2
C·
uT (y) dσ (y)
∂H
Z
C·
ue (y)2 dσ (y)
∂H
=
≤
C · kue k2L2 (∂H)
e · kue k2 1
C
H (∂H) .
(4.26)
4
Bidomain Modell
29
The previous transformation shows that the operator A2 is bounded, consequently
continuous. The forward operator A is a composition of continuous operators, therefore
also continuous.
4.2.2. Inverse Operator
In this section we examine the Inverse problem of the Bidomain Model. We would
like to define the inverse operator of Aν = uT with A : H 1 (H) → L2 (∂B). In order
to ensure that there exists a solution ν for every body potential uT , we introduce the
Moore-Penrose -Inverse A+ . With the aid of proposition A.1.2 we obtain:
if uT ∈ D(A+ ), with D(A+ ) = R(A) ⊕ R(A)⊥ , a unique minimum norm solution
ν + = A+ uT exists, which solves A∗ Aν = A∗ uT in N (A)⊥ .
Theorem 4.2.3:
The Moore-Penrose Bidomain Inverse A+ : D(A+ ) ∈ L2 (∂B) → H 1 (H), defines an
improperly posed problem.
Proof. To show that the inverse problem is improperly posed, we have to establish that
there exists no solution, or that the solution is not unique or that the data does not
depend continuously on the solution. So we split the Moore-Penrose Inverse like the
forward operator into two problems:
1/2
• find for every ue |∂H ∈ H0
a ν ∈ H 1 (H) that solves A1 ν = ue |∂H ,
1/2
• find for every uT ∈ L2 (∂B) a ue |∂H ∈ H0
that solves A2 ue |∂H = uT .
We will show that the second problem is not depending continuously on the data. For
this purpose, we use the next theorem which exploits that the fundamental solution is
continuous.
Theorem 4.2.4:
R
1/2
1/2
If ue ∈ H0 (∂H) and A2 : H0 → L2 (∂B) with A2 ue = ∂H k(x, y)ue (y)dσ(y) and
k(x, y) achieves
Z Z
k(x, y)2 dσ(x)dσ(y) = C ≤ ∞,
∂H
then A−1
2 is not continuous.
∂B
4
Bidomain Modell
30
Proof. Consider ej an orthonormal function system on H 1/2 (∂H) and have a look at
kA2 ej k2L2 (∂B)
Z
Z
2
Z
|A2 ej | dσ(x) =
∂B
∂B
Z Z
Z
2
=
k(x, y) dσ(y)
=
∂B
≤ C·
∂H
|k(x, y)ej (y)|dσ(y)2 dσ(x),
∂H
e2j (y)dσ(y)dσ(x),
∂H
kej (y)k2L2 (∂H)
e ≤ ∞.
≤C ·C
So kA2 ej k2L2 (∂B) is bounded. Exploiting the embedding of H 1/2 in L2 , we know that
orthonomal functions ej , which converge weakly in H 1/2 against zero, also converge
j→∞
A e
against zero in L2 . That results in kA2 ej kL2 (∂B) −→ 0. Let us define gj := kA2 ej k2 j2
,
L (∂B)
then kgj kL2 (∂B) = 1 holds. Examine
−1 A2 gj 1/2
H
(∂H)
=
=
!
A
e
−1
2 j
A
2
kA2 ej kL2 (∂B) 1/2
H
(∂H)
ej
kA2 ej kL2 (∂B) 1/2
H
(∂H)
j→∞
−→ ∞.
= ∞. Consequently A−1
We know kgj k = 1, it follows that A−1
2
2 is not continuous.
A−1
2 is not continuous, so the inverse Bidomain Operator is also not continuous and
unfortunately the inverse Bidomain model is improperly posed and needs regularization.
31
5. Linear Functional Strategy
In this chapter, we want to introduce the linear functional strategy to our problem.
We will follow the ideas of Anderssen [1],[2].
We know that most inverse problems arise in the context of indirect measurements and
we have seen that not only the potentials are a matter of interest but also quantities that
are given by integrals of the potentials. These quantities are often given as linear functionals, like moments. Then we can distinguish between two functionals, the first one
are the functionals defined on the problems solution, called solution-functionals, and
the second one are functionals defined on the measured data, the data-functionals. This
differentiation results in two mathematical problems. The forward problem in which
we have the data-functional and want to know the corresponding solution-functional
and the inverse problem, which determines the data-functional from the given solutionfunctional. We will show how we can link the solution-functional to the data-functional.
5.1. Linear Functional Strategy and Heart Vector
The linear functional strategy has its beginning in the evaluation of solution functionals. The first idea could be to solve the inverse problem to obtain the solution from
measurements and then to calculate the linear functional. Solving an inverse problem needs some kind of stabilization. The linear functional strategy offers another
approach. It transfers the solution functional to the data functional exploiting the
mathematical model of the problem. Therefore let us have a look at the following
proposition.
5
Linear Functional Strategy
32
Proposition 5.1.1 ([1]):
Consider an integral operator Ku = s (t), K : D(K) → R(K), with the domain D (K)
and range R (K) in a Hilbert space H with L2 -inner product and norm. Let K* denote
the adjoint of K with respect to the L2 -inner product. If the known θ which defines
1
Z
mθ (u) =
θ (x) u (x) dx
(5.1)
0
is contained in R (K ∗ ), then the required ϕ which defines
1
Z
mϕ (s) =
ϕ (x) s (x) dx
(5.2)
0
is determined by
K ∗ ϕ = θ.
(5.3)
Proof.
Ku=s
mϕ (s) = (ϕ, s) = (ϕ, Ku)
adjoint
=
(K ∗ ϕ, u) = (θ, u) = mθ (u)
So if we want to convey this proposition to the heart vector, we have the task to
evaluate the adjoint operator of the bidomain model. After that we can link the heart
vector, defined on the body surface, to a corresponding linear functionals.
5.1.1. Linear Algebra and the Linear Functional Strategy
In this section we want to explain how we can understand the Linear Functional Strategy in terms of linear algebra. Let us regard the linear operator K : H1 → H2 , mapping
from a Hilbert space H1 into another Hilbert space H2 . Is v ∈ H1 , then we can define
a linear functional
φv : H2 → < with φv (u) = (u, Kv).
(5.4)
5
Linear Functional Strategy
33
It is clear that φv (u) is an element of the dual space H2∗ . Then we can apply Riesz
representation theorem, which says that there exists an unique element in H2 such that
φv (u) = (K ∗ u, v) = (u, Kv).
(5.5)
In that way we can define the dual map K ∗ : H2∗ → H1∗ , which we rewrite as a
composition of the linear operator and the linear functional
K ∗ φv (u) = φv (u) ◦ K = ϕu (v).
(5.6)
This equation explains that the linear functional φv (u), as element in the dual space
of H2 , can be redefined with the dual map K ∗ as a linear functional on H1 . The
commutative diagram in Figure 5.1 exemplifies this result.
H1
/
K
K ∗ φv
H2
! φv
<
Figure 5.1.: Relation between dual map and the Linear functional strategy
5.2. Bidomain Model and Heart Vector
After explaining the relationship between the Linear functional strategy and the dual
map we want to apply this result. In Section 4.2 we have shown that the Bidomain
forward operator is a linear continuous operator and we know that the heart vector is
a linear functional so we can assign the Linear functional strategy. Therefore we can
redefine the heart vector on the body surface to an equivalent linear functional in the
heart. We obtain the following commutative diagram.
H 1 (H)
A
A∗ p(uT )
/
L2 (∂B)
% p(uT )
<
Figure 5.2.: Relation heart vector and adjoint heart vector
5
Linear Functional Strategy
34
In detail the diagram describes that the linear heart vector p(uT ) : L2 (∂B) → < with
R
p(uT ) = ∂B l−1 · uT dσ is an element of the dual space of L2 (∂B). The dual space
∗
(L2 (∂B)) can be identified with L2 (∂B) and we can write the heart vector as dual
pairing
Z
p(uT ) =
l−1 · uT dσ = (uT , p) .
(5.7)
∂B
Riesz representation theorem assures the existence of a unique element in L2 (∂B) such
that the dual pairing (5.7) obtains
p(uT ) = (ν, A∗ p) ,
(5.8)
∗
∗
with the adjoint Bidomain forward operator A∗ : (L2 (∂B)) → (H 1 (H)) . As we
∗
remarked above, the dual space (L2 (∂B)) can be identified with L2 (∂B) and the dual
∗
space (H 1 (H)) can be identified with the space H −1 (H), see [22].
When we reconsider the Diagram 5.2 we can define an other linear functional pe(v) but
this time on the domain of definition H 1 (H) and write it also as dual pairing
pe(ν) = (e
p, ν) .
(5.9)
∗
Like the heart vector this functional is linear and an element in the dual space (H 1 (H)) .
We are almost ready to describe the relationship between the heart vector on the body
surface p(uT ) and the linear functional pe(ν) in the heart. The adjoint Bidomain operator maps a linear functional into an other linear functional so we obtain by
A∗ p(uT ) = pe(ν),
(5.10)
a linear functional on H 1 (H). But in (5.9) we have defined such a linear functional on
H 1 (H), so we can rewrite the heart vector
p(uT ) = (uT , p) = (Aν, p) = (ν, A∗ p) = (ν, pe) = pe(ν).
(5.11)
5
Linear Functional Strategy
35
5.3. Adjoint Bidomain Problem
We are in a position to transform the heart vector in the heart by applying the adjoint
Bidomain operator on the heart vector defined on the body surface. Therefore we have
to solve the system A∗ p(uT ) = pe(ν). Like in the Bidomain Problem we also have a
forward and an inverse problem, these two problem are the topic of this section.
5.3.1. Forward Adjoint Bidomain Problem
We will pay attention to the forward adjoint Bidomain problem. Corresponding to the
definition (4.2.1) of a forward problem we define the forward adjoint Bidomain problem
as, calculating the heart vector on the heart by using the forward adjoint Bidomain operator and the heart vector. The forward adjoint Bidomain Problem means to evaluate
pe(ν) given p(uT ) by A∗ . Our next aim is to analyze if the forward adjoint Bidomain
problem is a properly posed problem.
Theorem 5.3.1:
The adjoint forward Bidomain problem is a properly posed problem.
Proof. The adjoint forward Bidomain problem is equivalent to solve a Poisson equation
on the heart and a Laplace equation on the body coupled by the boundary values.
Because of the Neumann boundary values the solution is unique except for an additive
constant. We obtain a unique solution when we add a normalization condition like
R
λ = 0. In order to establish that the problem is continuous, we recall the Diagram
∂H 2
5.2. So we see that we can rewrite A∗ p(uT ) as a composition of the forward Bidomain
operator and the heart vector
A∗ p(uT ) = p(uT ) ◦ A(ν).
(5.12)
We have shown in Section 4.2.1 that the forward Bidomain operator is continuous
and in Section 3.2 that the heart vector is continuous. Finally, the adjoint operator
A∗ is a continuous map, because it consists of two continuous maps. At the end we
have to show that the solution depends continuously on the data. For this sake define
p (uT ) = p(uT ) + (uT ), a disturbed functional on the body surface and apply the
adjoint Bidomain operator to the difference of disturbed and not disturbed functional,
exploiting the dual map and Hölders inequation lead us to
5
Linear Functional Strategy
36
|A∗ p (uT ) − A∗ p(uT )| = |p (Aν) − p(Aν)| = |p (uT ) − p(uT )|
= |p(uT ) + (uT ) − p(uT )| = |(uT )|
≤ kkL2 (∂B) kuT kL2 (∂B) .
So if kkL2 (∂B) tends to zero, |A∗ p (uT ) − A∗ p(uT )| tends also to zero, consequently the
forward adjoint Bidomain problem is a properly posed one.
5.3.2. Inverse Adjoint Bidomain Problem
After discussing the forward adjoint Bidomain problem we cannot circumvent the inverse adjoint Bidomain problem. We want to show that this problem is an improperly
posed problem. Therefore we must show similar to the inverse Bidomain problem, that
one of three the conditions defining a properly posed problem fails. So as to exploit the
properties of the inverse Bidomain problem, we also consider the failure of continuous
dependence on the data.
Theorem 5.3.2:
The inverse adjoint Bidomain Problem is an improperly posed problem.
Proof. Let us consider p (uT ) as a solution of A∗ p(uT ) = pe (ν) where pe (ν) = pe(ν)+(ν)
are the disturbed data with a perturbation . We obtain
|e
p (ν) − pe(ν)| = |(ν)| .
(5.13)
On the other hand we have to estimate the difference between the solution and the
solution evaluated of the disturbed data, therefore we use the relation between the dual
map and the linear functionals, shown in the diagram.
H 1 (H) o
pe(ν)
y
<
A+
L2 (∂B)
∗
(A+ )
pe(ν)
Figure 5.3.: Inverse adjoint Bidomain Problem
5
Linear Functional Strategy
37
So if the data error functional (ν) tends to zero, the solutions tend to infinity, because
the operator norm of the inverse Bidomain operator is not bounded. Consequently the
inverse adjoint Bidomain problem is improperly posed and needs regularization.
5.4. Adjoint Operator
This section deals with the evaluation of the adjoint operator and the transformation of
the heart vector into a solution functional. First of all, we will examine a simplification
of the Bidomain Model. We will assume that we know the potential at the heart
boundary. For this model we will determine the adjoint equations and show how to
link the heart vector to a solution functional. The second part of this section deals
with the Bidomain Model and its adjoint operator. Like in the first part we show how
we can rewrite the heart vector, after constructing the adjoint operator.
5.4.1. Adjoint Operator: Heart-Torso Model
We want to construct the adjoint operator for a simplified Bidomain Model. In order to
avoid the difficulties by modeling the intracellular and extracellular spaces we assume
that the heart is composed of homogeneous media and the potential u is equal to u∂H
on the boundary of the heart. It remains to model those reaction in the body and
the body surface, but this reactions are equal to the Bidomain ones, so we obtain the
following simplified model
∇ · (MT ∇u) = 0
nT · (MT ∇u) = 0
u = u∂H
in T,
(5.14)
on ∂B,
(5.15)
on ∂H.
(5.16)
Now we are in a position where we can form the Lagrangian for this model, therefore
we multiply the torso equation (5.14) by λ, a Lagrangian multiplier, and add the heart
vector, then the Lagrangian looks like
−1
Z
Z
∇ · (MT ∇u) · λdx +
L(u, l ) =
T
∂B
l−1 u(δri − δrj )ds.
(5.17)
5
Linear Functional Strategy
38
To obtain the adjoint operator we will apply Gauss’ theorem and exploit the boundary
condition ∂T = ∂B ∪ ∂H.
−1
Z
Z
Z
(MT ∇u · nT ) · λds + MT ∇u · ∇λdx +
∂T
Z
Z T
=
MT ∇u · nT · λds −
MT ∇u · nH · λds
∂B
Z ∂H
Z
l−1 u(δri − δrj )ds.
+ MT ∇u · ∇λdx +
l−1 u(δri − δrj )ds,
L(u, l ) =
T
∂B
∂B
Regarding the model we see that the boundary integral over the body surface vanishes
and a second application of Gauss theorem leads to
Z
−1
Z
L(u, l ) = −
Z ∂H
−
MT ∇uT · nH λds + ∇ · (MT ∇λ) · udx
Z
Z T
l−1 u(δri − δrj )ds.
MT ∇λ · nT udx + MT ∇u · ∇λdx +
∂B
T
∂T
Rewriting the torso boundary into the sum of body boundary and heart boundary and
inserting the heart potential (5.16) we can write the Lagrangian as
Z
−1
L(u, l ) = −
Z ∂H
−
Z
MT ∇uT · nH λds +
Z
MT ∇λ · nT uds +
∂B
∇ · (MT ∇λ) · udx
T
Z
MT ∇λ · nH u∂H ds +
∂H
l−1 u(δri − δrj )ds.
∂B
With the additional assumption λ = 0 on ∂H we can take the partial derivate of the
Lagrangian, then we obtain
∂L
=
∂u
Z
Z
∇ · (MT ∇λ)dx −
T
Z
l−1 (δri − δrj )ds.
MT ∇λ · nT ds +
∂B
If the partial derivative fulfills the optimal condition
system
(5.18)
∂B
∂L
∂u
= 0 we obtain the adjoint
5
Linear Functional Strategy
39
∇ · (MT ∇λ) = 0
in T,
(5.19)
MT ∇λ · nT = l−1 (δri − δrj ) on ∂T,
λ=0
(5.20)
on ∂H.
(5.21)
5.4.2. Solution-Functional: Heart-Torso Model
We are now in a position to rewrite our heart vector into a solution function, thereby
we obtain a representation of the heart vector, which is no longer dependent on the
body boundary potential therefore on the heart boundary potential. We start with
the heart vector representation (3.2) and insert the adjoint equation for the body
boundary (5.20), after that we exploit the relationship between the body boundary,
heart boundary and torso boundary and handle with care the normal directions.
Z
(5.20)
−1
l (δri − δrj ) · uds
Z
MT ∇λ · nT uds,
=
∂B
∂B∪∂H=∂T
Z∂B
Z
MT ∇λ · nH uds.
MT ∇λ · nT uds +
=
∂H
∂T
We transform the boundary torso integral with help of Gauss theorem and integration
by parts and see that the torso integral vanishes, because of the adjoint equation (5.19).
A second application of integration by parts and the model torso equation (5.14) leads
to a representation of the heart vector on the torso and heart boundary,
Z
−1
Z
Z
Z
l (δri − δrj ) · uds =
∂B
∇ · (MT ∇λ) · udx + MT ∇λ∇udx +
MT ∇λ · nH uds,
T
∂H
|
{z
}
=0
Z
Z
Z
= − ∇ · (MT ∇u) · λdx +
MT ∇u · nT λds +
MT ∇λ · nH uds.
T
∂T
∂H
|
{z
}
T
=0
Exploiting that the torso boundary is the union of the heart boundary and body
5
Linear Functional Strategy
40
boundary and inserting (5.21), the boundary condition for λ on the heart boundary,
and (5.15), the boundary condition of the heart torso model, leaves the last heart
boundary integral left.
Z
∂B
Z
Z
−1
l (δri − δrj ) · uds = −
|
Z
=
∂H
Z
MT ∇u · nT λds +
MT ∇u · nH λds +
∂B
{z
} |
{z
}
=0
MT ∇λ · nH uds,
∂H
=0
MT ∇λ · nH uds.
∂H
In the last step we obtain,
Z
−1
Z
l (δri − δrj ) · uds =
∂B
MT ∇λ · nH uH ds.
(5.22)
∂H
We obtained an alternative representation of the heart vector. So we can determine
the heart vector independent from whether we have the body surface potential or we
have the heart surface potential.
In Chapter 7, we evaluate the heart vector at both ways and will compare them.
5
Linear Functional Strategy
41
5.4.3. Adjoint Bidomain Operator
In order to obtain the adjoint operator of the bidomain model, we will consider the
bidomain forward problem. Here we assume that the transmembrane potential ν is
given in the heart, i.e.
ν=g
in H.
(5.23)
So as to gain the adjoint equations we examine the Lagrangian of the Bidomain Model,
which is composed of two state equations multiplied with a Lagrange multiplier λ and
a measurement. The first bidomain equation (4.10) and the torso equation, (4.11)
are the two state equations. They are multiplied with the Lagrange multiplier λ1 for
the heart area and λ2 for the torso. Our measurement is the heart vector. Then the
Lagrangian looks like
L ν, ue , uT , l
−1
Z
Z
∇ · ((Mi + Me ) ∇ue ) · λ1 dx
Z
l−1 uT δri − δrj ds
+ ∇ · (MT ∇uT ) · λ2 dx +
∂B
Z T
Z
=
Mi ∇ν · nH λ1 ds −
Mi ∇ν∇λ1 dx
∂H
H
Z
Z
(Mi + Me ) ∇ue ∇λ1 dx
(Mi + Me ) ∇ue · nH λ1 ds −
+
H
∂H
Z
Z
Z
+
MT ∇uT · nT λ2 ds − MT ∇uT ∇λ2 +
l−1 uT δri − δrj ds.
=
∇ · (Mi ∇ν) · λ1 dx +
HZ
∂T
H
T
∂B
In the first step we used integration by parts and the Gauss’ theorem. Next we want to
exploit ∂T = ∂H ∪∂B and the normal identity nB = nT = −nH to apply the Bidomain
boundary equation for the heart surface (4.17), and for the body surface (4.15).
5
Linear Functional Strategy
L ν, ue , uT , l
−1
42
Z
Z
Mi ∇ν · nH λ1 ds +
=
(Mi + Me ) ∇ue · nH λ1 ds
{z
}
∂H
∂H
|
=
R
MT ∇uT ·nH λ1
∂H
Z
Z
Mi ∇ν∇λ1 dx −
−
Z
(Mi + Me ) ∇ue ∇λ1 dx +
H
H
∂B
|
Z
=0
Z
−
MT ∇uT · nH λ2 ds −
Z∂H
+
Z
MT ∇uT · nB λ2
{z
}
MT ∇uT ∇λ2 dx
T
l−1 uT δri − δrj ds
∂B
Z
=
∂H
Z
−
((MT ∇uT ) · nH ) (λ1 − λ2 ) ds −
Mi ∇ν∇λ1 dx
H
Z
(Mi + Me ) ∇ue ∇λ1 dx − MT ∇uT ∇λ2 dx
ZH
T
l−1 uT δri − δrj ds.
+
∂B
After a second application of integration by parts and Gauss’ theorem we need the
symmetry of the conductivity tensors Mi , Me and MT to gain
L ν, ue , uT , l
−1
Z
=
Z
(MT ∇uT · nH ) (λ1 − λ2 ) ds +
∇ · (Mi ∇λ1 ) · νdx
H
Z
−
Mi ∇λ1 · nH · νds +
∇ · ((Mi + Me ) ∇λ1 ) · ue dx
H
∂H
Z
Z
−
(Mi + Me ) ∇λ1 · nH · ue ds + ∇ · (MT ∇λ2 ) · uT dx
∂H
T
Z
Z
−1
−
MT ∇λ2 · nT · uT ds +
l uT δri − δrj ds.
∂H
Z
∂T
It remains to repeat step number two to obtain
∂B
5
Linear Functional Strategy
L ν, ue , uT , l
−1
43
Z
=
Z
(MT ∇uT · nH ) (λ1 − λ2 ) ds +
∇ · (Mi ∇λ1 ) · νdx
H
Z
−
Mi ∇λ1 · nH · νds +
∇ · ((Mi + Me ) ∇λ1 ) · ue dx
∂H
H
Z
Z
(Mi + Me ) ∇λ1 · nH · ue ds + ∇ · (MT ∇λ2 ) uT dx
−
∂H
T
Z
Z
−
MT ∇λ2 · nT · uT ds −
MT ∇λ2 · nT · uT ds
∂B
∂H
Z
l−1 uT δri − δrj ds.
+
∂H
Z
∂B
In order to differentiate the Lagrangian function with respect to ν, ue and uT we assume
that λ1 equals λ2 on the heart boundary. In the beginning we assumed that we know
the potential in the heart, so we can rewrite the Lagrangian function to
L ν, ue , uT , l
−1
Z
=
Z
∇ · (Mi ∇λ1 ) · gdx −
Mi ∇λ1 · nH · νds
Z
(Mi + Me ) ∇λ1 · nH · ue ds
∇ · ((Mi + Me ) ∇λ1 ) · ue dx −
+
∂H
H
Z
Z
+ ∇ · (MT ∇λ2 ) uT dx −
MT ∇λ2 · nT · uT ds
T
∂B
Z
Z
l−1 uT δri − δrj ds.
MT ∇λ2 · nT · ue ds +
−
HZ
∂H
∂H
∂B
Now we are in the position to take the partial derivate of L with respect to ν, ue and
uT and obtain
Z
∂L
=
Mi ∇λ1 · nH ds,
∂ν
Z∂H
Z
Z
∂L
=
∇ · ((Mi + Me ) ∇λ1 ) dx −
(Mi + Me ) ∇λ1 · nH ds −
MT ∇λ2 · nT ds,
∂ue
H
∂H
∂H
Z
Z
∂L
=
∇ · (MT ∇λ2 ) dx −
MT ∇λ2 · nT ds
∂uT
TZ
∂B
+
l−1 δri − δrj ds.
∂B
(5.24)
5
Linear Functional Strategy
44
Since we have
∂L
= 0 and
∂ue
∂L
=0,
∂ν
∂L
= 0,
∂uT
(5.25)
we obtain the following adjoint problem
∇ · ((Mi + Me ) ∇λ1 ) = 0
in H
(5.26)
∇ · (MT ∇λ2 ) = 0
in T
(5.27)
on ∂H
(5.28)
on ∂H
(5.29)
on ∂B
(5.30)
on ∂H
(5.31)
Mi ∇λ1 · nH = 0
− (Mi + Me ) ∇λ1 · nH + MT ∇λ2 · nH = 0
MT ∇λ2 · nT = l−1 δri − δrj
λ1 = λ2
5.5. Solution-functional
The next part is about the identification of the heart vector to a corresponding solutionfunctional by taking advantage of the bidomain and adjoint bidomain system. In
Section 5.2 we identified
p(uT ) = (uT , p) = (Aν, p) = (ν, A∗ p) = (ν, pe) = pe(ν).
(5.32)
We want to have a precise look at the heart vector and the solution-functional, we
want to gain a representation like
Z
−1
Z
l uT ds =
∂B
wνdx,
(5.33)
H
in which we know w or we can calculate it. Therefore we use the adjoint equations
(5.26)-(5.31) and the bidomain equations (4.10)-(4.17), after certain applications of
integration by parts and Gauss’ theorem, we can rewrite the heart vector into
5
Linear Functional Strategy
Z
p(uT ) =
l
−1
45
Z
δri − δrj · uT ds =
ZH
∂B
=
H
∇ · (Mi ∇λ1 ) · ν δri − δrj dx (5.34)
w δri − δrj · νdx = pe(ν),
with w = ∇ · (Mi ∇λ1 ), see Appendix A.0.1 for the entire transformation.
This means, the first adjoint equation turns to the following Poisson equation
−∇ · (Me ∇λ1 ) = w
in H,
(5.35)
coupled to the adjoint torso equation ∇ · (MT ∇λ2 ) = 0 via the adjoint boundary
conditions, (5.28)-(5.31).
46
6. Implementation
This chapter is about the numerical realization of the Linear Functional Strategy. First
of all we will introduce the Finite Element Method and will model the torso containing
the heart. Then we evaluate the heart vector, we will use two different lead system.
We start with an electrode configuration of four electrodes, which are connected to
the orthogonal potential differences Vx and Vz . For this propose we have to evaluate
the lead matrix of this electrode configuration as explained in Section 2.5. The second
electrode configuration is the Frank lead system as explained in Section 2.4. Then we
have to solve the forward adjoint problem in order to obtain the solution functional.
6.1. Finite Element Method
The Finite Element Method (FEM) is a numerical technique for solving partial differential equations, we will follow the basic ideas of [5]. Because the FEM approximates
the weak formulation of a partial differential equation, we multiply the adjoint partial
differential equation (5.19) with a test function φ ∈ C ∞ (T ) and integrate over the torso
Z
∇ · (MT ∇λ) · φdx = 0.
(6.1)
T
Application of Gauss theorem and rewriting the torso boundary into the union of body
boundary and heart boundary leads to
Z
Z
∇ · (MT ∇λ) · φdx =
T
Z
MT ∇λnB · φds −
∂B
Z
MT ∇λnH · φds −
∂H
MT ∇λ∇φdx.
T
We want the heart boundary integral to disappear, so we restrict the test function to
φ = 0 on the heart boundary. Inserting the torso boundary condition, we obtain the
6
Implementation
47
weak formulation
Z
Z
l−1 (δri − δrj ) · φds.
MT ∇λ∇φdx =
T
(6.2)
∂B
In order to find an approximate solution, the next step is to divide the torso into small
areas, we will use triangles but also rectangles can be used like tetrahedrons or cuboids
in <3 . So we can rewrite the torso domain as union of a finite numbers of triangles Tj ,
for j = 1, ..., M ,
T =
M
[
Tj .
(6.3)
j=1
Comparing two triangles Tj and Ti for i 6= j, we have three opportunities in which way
they can be connected:
1. Ti ∩ Tj = {0}, the triangles have no connecting point.
2. Ti ∩ Tj = {Pi } the triangles share one point Pi and it is a node.
3. Ti ∩ Tj = {Ei } have more than one point in common and Ei is an edge.
So the edges Ei connect N different nodes Pi to triangles. Each node has a neighborhood N (Pi ) defined as
N (Pi ) =
[
Tj .
(6.4)
Pi ∈∂Tj
The next step is to construct a finite element for every gird node with the following
properties
ϕ ∈ C(T )
ϕi |Pk = δik
ϕi (x) = 0,
x∈
/ N (Pi )
(6.5)
ϕ|Tj ∈ PK (Tj ), Tj ∈ N (Pi ).
The expression PK (Tj ) is the set of polynoms with a smaller or equal degree to k on the
triangle Tj . If we choose the polynomial degree k to be equal one, we obtain piecewise
6
Implementation
48
linear elements. This piecewise linear elements are well-defined in <2 , because it is
enough to specify the function value in the grid nodes of a triangle.
Now we are in the position to approximate the solution λ, we write the discretized
solution λN as a linear combination of finite elements ϕ and coefficients λi ∈ <
N
λ =
N
X
λj ϕj ,
(6.6)
j=1
N is the number of grid points and λj = λ(xj ) is the evaluation of the solution at the
grid point xj . If we choose the finite elements ϕj , for j = 1, ..., N , as test functions in
the weak formulation (6.2) we obtain
N Z
X
j=1
Z
λj ∇ϕj · ∇ϕk dx =
T
l−1 (δri − δrj ) · ϕk ds.
(6.7)
∂B
In fact we have to solve a N xN system of equation, with the stiffness matrix K N ∈
<N xN
K
N
=
N Z
X
λj ∇ϕj · ∇ϕk dx for k = 1, ..., N ,
(6.8)
l−1 (δri − δrj ) · ϕk ds for k = 1, ..., N .
(6.9)
j=1
T
and the right hand vector FN
Z
FN =
∂B
In order to obtain the stiffness matrix we have to evaluate N 2 integrals and for the right
hand vector N integrals, but since we use the FEM, the finite elements vanish and we
only have to determine the integrals in the neighborhood of a grid node. Consequently,
the stiffness matrix K N is a sparse matrix.
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6.2. Torso model
This section will explain the model geometry on which we will solve the adjoint equation. We will use Frank’s lead system, so we describe the fifth interspace. In this
interspace we model the torso shape, the heart and the lungs. The torso shape resembles to the model in Franks’ paper [14], that models the transverse anatomic section
of a male torso. In our model the left- right distance is 66 units and the back front
distance 50 units. Into this torso we add the heart which lies in the middle front of
the torso. The heart slants from top right to down left and captures about 13% of the
torso surface. The heart is surrounded by the lungs, which require the most space in
the torso. Usually the lungs are surrounded by the rips, but since we want to describe
an interspace we can disregard the rips. Furthermore, we simplify our model by disregarding the sternum and the spine. Actually we obtain a torso model with the heart
and lungs, like in Figure 6.1.
Figure 6.1.: Torso geometry
Obviously the three areas, torso, heart, lungs do not have the same conductivities. So
we want to have a look at this topic. We assume in this model that the heart tissue is
diagonal isotrop, so the electric current prefers the x and z direction in equal measure.
The torso tissue containing fat and muscles has a larger conductivity coefficient than the
heart. We assume that the torso conductivity is twice as large as the heart conductivity.
The lung conductivity is smaller than the torso conductivity but larger than the one of
the heart. The torso and the lung conductivity are assumed to be anisotrop. Clearly
we use three different conductivity matrices, MT , ML , MH for the torso, lungs and
heart, respectively
MT =
1 0
0 0.95
!
, ML =
0.7 0
0 0.75
!
, MH =
0.5 0
0 0.5
!
.
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6.3. Implementation Lead Matrix
Our task is now to evaluate the lead matrix of our model geometry. In Section 2.5,
we explained how we receive the lead matrix with the help of Helmholtz principle of
reciprocity. So we have to choose a pair of electrodes A and B on the body and a
second pair in the heart. The second electrode configuration is our dipole origin and
the dipole top. If we inject a current at the electrode A and remove it at electrode B,
we can explain the potential difference between these electrodes VAB as product of the
potentials’ gradient and the heart vector. But we can rewrite this procedure as a linear
functional problem and apply the linear functional strategy, like in [30]. Therefore, we
define our linear functional as the potential difference VAB
Z
u(δA − δB )dr,
VAB =
(6.10)
∂B
with the Dirac distribution at the electrode points A and B and the potential u.
Applying the linear functional strategy to the state equations
∇ · (MT ∇u) = 0 in T
MT ∇u · nT = 0 on ∂B
and the measurement VAB , we obtain the adjoint system
∇ · (MT ∇w) = 0
MT ∇w · nT = 0
w = δA − δB
in T
on ∂B
(6.11)
on ∂B.
Note that we neglected the heart boundary, because we want to determine the potential
difference on the body not on the heart boundary. The next step is to define the
orthogonal potential differences at the body surface, for this purpose we have to choose
four electrode positions. We place the two electrodes at the intercept point of the xaxis and the torso model and the other two electrodes at the intercept point of the
z-axis and the torso model. Then the potential difference in the x-direction Vx is
the difference between the potential at the left electrode and right electrode and the
potential difference in the z-direction Vz equals the difference between the top electrode
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and the bottom electrode, see Figure 6.2.
Figure 6.2.: Torso geometry with coordinate system and dipole origin. x-axis from left
to right, z-axis from top to bottom. Electrode positions r1 to r3 . The
potential differences are then Vx = u(r3 ) − u(r1 ) and Vz = u(r4 ) − u(r2 )
Next we have to decide on the position of the dipole origin. Because we also want to
evaluate a vectorcardiogram on Franks lead system, we choose the dipole origin like
Frank does in his work [14]. Frank choose a dipole position at the fifth interspace
on the left side of the ventricle plane, 14.8 per cent of the thorax depth, compare
Figure 2.3.b in Section 2.4. We choose the dipole origin at the point rj = (2, −6),
like in Figure 6.2 implied. Now it remains to solve the adjoint problem (6.11) for the
two electrode configuration, for this propose we use the engineering analysis software
COMSOL Multiphysics. We proceed as follows:
1. Initialization of the adjoint problem for the lead matrix 6.11, with the model
geometry of Section 6.1
2. Inserting at electrode position r3 an unit current, remove it at electrode position
r1 , w(r3 ) = δr3 , w(r1 ) = −δr1
3. Solving the adjoint problem with the linear system solver UMFPACK
4. Evaluation of the gradient ∇w(rj ) at the dipole origin
In this way we obtain the first row of the lead field matrix. If we repeat the procedure
by changing the electrode position in step two to w(r4 ) = δr4 and w(r2 ) = −δr2 , we
obtain the second row of the lead field matrix. So the lead field matrix L looks like
L = ∇wx (rj )T , ∇wz (rj )
T
=
2.03408 0.13699
0.152482 2.71498
!
with rj the dipole origin, wx the solution of the adjoint problem 6.11 with the electrode
positions r1 and r3 and wz the solution of the adjoint problem with the electrode
position r2 and r4 .
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We want to have a closer look at step number three. So as to solve the adjoint lead
field matrix problem we build a mesh with 2143 mesh points and 4145 triangulars,
the finite elements were Lagrange quadratic. For further detail about the UMFPACK
solver take a look at [8]. In step four we evaluate the gradient of the solution w at the
dipole origin. In Comsol Multiphysics this part is not difficult, because we only have
to define this origin when we initialize the adjoint problem in step number one. Then
we can choose in the postprocessings of Comsol Multiphysics this point and evaluate
the gradient by a difference quotient.
6.4. Implementation Linear Functional Strategy
This section deals with the implementation of the Linear functional strategy explained
in Section 5.1. We showed that we are able to transform the heart vector, which is
defined on the torso boundary into a functional, which is defined on the heart boundary.
Therefore we have to solve the corresponding adjoint problem. We split our task into
two subtasks
1. solving the adjoint problem for each electrode position,
2. evaluation of the heart vector on the heart boundary.
In the following part of this section we will specify both steps. All calculations are
done with the software Comsol Multiphysics.
6.4.1. Solving the adjoint problem
In the previous section we have introduced the Bidomain Model and a simple hearttorso model for the electrical diffusion in the human heart. As the Bidomain Model
describes the heart as intracellular and extracellular areas, we will use the simple
heart-torso model to avoid difficulties. In the following section we have a look at this
heart-torso model for the potential uT
∇ · (M ∇uT ) = 0 in T
nT · (M ∇uT ) = 0 on ∂B
uT = g
on ∂H
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with the heart boundary potential g and the conductivity matrix M , which is different
for the heart, torso and lungs, see Section 6.1. The corresponding adjoint equations
are
∇ · (M λ) = 0
in T
nT · (M ∇λ) = l−1 (δri − δrj ) on ∂B
λ=0
on ∂H
with l−1 the inverse of the lead field matrix.
Let us have a closer look at the left side of the body boundary condition. We write
the x component of the heart vector as a linear functional
Hx (t) = (l−1 )11 · Vx + (l−1 )12 Vz
Z
Z
−1
=
(l )11 uT (δr3 − δr1 ) +
∂B
(6.12)
(l−1 )12 uT (δr4 − δr2 ).
(6.13)
∂B
So we obtain for the electrode positions r1 , r2 , r3 and r4 the boundary condition
nT · (M ∇λ) = (l−1 )11 (δr3 − δr1 ) + (l−1 )12 (δr4 − δr2 ) on ∂B.
(6.14)
Inserting this conditions in the model geometry, we have to solve two adjoint models,
one for the x- direction and a second one for the z- direction. We initialize the mesh
with 1600 mesh points, use 2952 triangles, and quadratic Lagrange finite elements.
Figure 6.3.: Mesh of model geometry, 1600 mesh points, 2952 triangles, 33760 number
of degrees of freedom
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(a) Solution of the adjoint problem (5.20) in the
x- direction with the torso boundary condition
nT ·(MT ∇λ) = 0.493491·(δr3 −δr1 )−0.0249229·
(δr4 − δr2 )
(b) Solution of the adjoint problem (5.20) in the
z- direction with the torso boundary condition
nT · (MT ∇λ) = −0.027741404 · (δr3 − δr1 ) +
0.370065 · (δr4 − δr2 )
Figure 6.4.: Solution of the adjoint problem
(a) heart boundary with navigation numbers 1-10
(b) Normale component M ∇λ · nT in x- direction,(c)
maximale value of 0.02052764 at the boundary
7, minimal value of -0.018597936 at the boundary 2, zero points at the boundaries 4 to 5 and
9
Normale component M ∇λ · nT in z- direction,
maximale value of 0.019060474 at the boundary
10, minimal value of -0.018989546 at the boundary 4, zero points at the boundaries 2 and 6
Figure 6.5.: Normal components of the adjoint problem
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6.4.2. Evaluation of the heart vector on the heart boundary
We have shown in Section 5.4.1 that the heart vector on the torso boundary can be
transformed to a linear functional on the heart boundary. This linear functional is a
boundary integral over the heart boundary with the integrand M ∇λ · nT u. We obtain
the first part of the integrand M ∇λ · nT by solving the adjoint equation (5.19)-(5.21).
This integrand M ∇λ · nT can be interpreted as weighthing function, like Figure 6.5
indicate. In order to obtain the potential u on the heart boundary we have to start
an extra simulation. So we will proceed as follows: First we will insert a dipole in the
heart area and will simulate one heart circle, in this way we obtain the potentials on
the heart boundary and on the torso boundary. After that we will determine the heart
vector on the heart boundary.
We are acting on the assumption that electrical activity in the heart can be described
by a single dipole, so we insert in the heart area a negative pole and a positive pole.
The negative pole is like in Section 7.2 described at the origin point (2,-6), we assume
that this origin does not change. As the potentials change during a heart cycle we
will change the position of the positive pole of the dipole. So we are changing also
the orientation and the magnitude of the heart vector. In detail we establish 28 dipole
positions, they are shown in Figure 6.6.
Figure 6.6.: Torso geometry with 28 dipole positions
For this 28 dipole positions we solve 28 forward problems (6.15)-(6.16)
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56
∇ · (M ∇u) = δri − δrj
nT · (M ∇u) = 0
in T
(6.15)
on ∂B
(6.16)
with rj = (2, −6) the negative pole position and ri for i = 1, ..., 28 the changing positive
pole position. In this way we obtain the potential at the heart boundary for 28 different
time steps.
Our next task is the evaluation of the heart boundary integral
Z
M ∇λ · nT uds.
H(t) =
(6.17)
∂H
We approximate the heart vector on the heart boundary by taking the weighted sum of
the integrand M ∇λ · nT u, evaluated at the integration points. This integration points
are those points which describe the heart boundary ∂H. We exploit the FEM mesh and
write the heart boundary as union of triangle boundary’s, then the integral over the
heart boundary equals the sum over the triangle boundary integrals. But the triangle
boundary integral is the length lij between two grid points i and j multiplied by the
weighted sum of the integrand. So we obtain
Z
M ∇λ · nT u(t)dt =
H(t) =
∂H
=
X
k
XZ
k
lij
X
M ∇λ · nT u(t)dt
(6.18)
Tk (H)
wj M (xkj )∇λ(xkj ) · nT (xkj )u(xkj ),
(6.19)
j
with Tk (H) triangles which describe the heart boundary, wj are the weights and xkj
the integration points. Evaluation of the sum (6.19) yield us the heart vector on the
heart boundary.
57
7. Results
In this chapter we want to compare the heart vector evaluated on the heart boundary
and the heart vector evaluated on the torso boundary. We will use two different lead
matrices, at first we will compare the two heart vectors for the lead matrix which we
have determined in Section 7.2 by applying the Helmholtz reciprocity principle. After
that we will check, if for the lead system of Frank, introduced in Section 2.4, the linear
functional strategy fits too. In the last step we will compare the two different lead
systems. In this context we examine the relative error re defined as
re =
H∂B − H∂H
,
H∂B
(7.1)
in which H∂B is the heart vector determined on the body surface, H∂H the heart vector
determined on the heart boundary. For every lead system we plot the corresponding
VCG, the heart vector components in x- and z- direction, the difference between the
heart vector components and the relative error in the x- and z- component. But first
of all we want to have a closer look at the relative error rd.
7.1. Error Estimation
In this section we have a closer look at the relative error re, introduced above. We
exploit the linearity of the heart vector and can write the disturbed heart vector on
δ
δ
the heart boundary H∂H
and the disturbed heart vector on the body boundary H∂B
as sum of the heart vector and a disturbed functional δH∂H ,δH∂B , respectively.
δ
H∂H
= H∂H + δH∂H ,
(7.2)
δ
H∂B
= H∂B + δH∂B .
(7.3)
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58
Let us examine the norm of the difference between the disturbed heart boundary heart
vector and the disturbed body boundary heart vector:
δ
δ H∂B − H∂H
= kH∂B + δH∂B − H∂H − δH∂H k
= kδH∂B − δH∂H k
≤ kδH∂B k + kδH∂H k .
(7.4)
In the fist step the H∂B and H∂H vanish, because applying the linear functional strategy
H∂B = H∂H . But we also can apply the linear functional strategy to the disturbed
functional δH∂B , then we obtain
δ
δ H∂B − H∂H
≤ 2 · kδH∂B k .
(7.5)
Dividing equation 7.5 by the heart vector H∂B and rewriting the heart vector and
the disturbed functional as dot product of lead matrix and potential difference u, we
receive
δ
δ δH∂B δ∂B · u H∂B − H∂H
≤ 2·
H∂B = 2 · L−1 · u H∂B
kδ∂B k
≤ 2 · −1 .
kL k
(7.6)
(7.7)
If we want to examine the relative error between the body boundary heart vector
evaluated with the reciprocity lead matrix Lrec and the Franks lead matrix LF , we
obtain
rec
F F H∂B − H∂B
= δH∂B ≤
H∂B H∂B
F δ ∂B
.
≤
kL−1 k
F
δ · u
∂B
kL−1 · uk
(7.8)
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59
F
We have rewritten the Frank heart vector H∂B
as sum of the reciprocity heart vector
rec
F
H∂B and a disturbance δH∂B . Application of the triangle inequality and the rewriting
the original dipole and the disturbance as dot product of the inverse lead matrix and
the potential, we received the inequation (7.8). So the relative error re depends on the
lead matrix of the disturbed functional H∂B and the original lead matrix L.
We want to introduce a second relative error, the relative difference rd, this relative
difference helps us to decide which lead system approximates the original dipole best
kH rec k H F −
.
rd =
kDk
kDk
(7.9)
So if rd is larger than 0, then the reciprocity lead matrix approximates the original
dipole better then Franks lead matrix. On the other hand if rd is smaller than 0,
Franks lead matrix fits better to the original dipole.
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60
7.2. Heart Vector with reciprocity Lead Matrix
In this part we use the lead matrix, which we have determined in Section 7.2
Lrec =
2.03408 0.13699
0.152482 2.71498
!
By the use of Lrec we receive that the heart vector components are apparently equal,
see Figure 7.2. Consequently the VCGS are similar, Figure 7.1. We see in Figure 7.2.b
that the difference between the x- components of heart vector on the heart boundary
and the heart vector on the torso boundary is larger than the difference in the zcomponent. The maximal difference in the x- component is 0.2853037 and in the zcomponent -0.0385622. The two bottom figures show the relative error depending on
the time. Figure 7.2.c shows the relative error of the x- component, which is maximal
2.24779 per cent. The relative error of the z- component is maximal 1.177413 per cent
and is shown in Figure 7.2.d.
Figure 7.1.: vector loop
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61
(a) the heart vectors x- and z- components
(b) difference between the heart vector components
(c) relative error x- component
(d) relative error z- component
Figure 7.2.: heart vector components
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62
7.3. Heart Vector with Frank Lead Matrix
Now we want to analyze Frank’s lead matrix LF rank in term of the linear functional
Strategy,
LF rank =
136 −0.2
0
136
!
.
We consider the different orthogonal potential differences 7.10-7.11
Vx = VA − VI
(7.10)
Vz = VM − VE .
(7.11)
As well as before we see that the heart vector components are apparently equal, Figure 7.4.a. The maximal difference in the x- component is 0.23576696 and in the zcomponent -0.0347652, Figure 7.4.b. In Figure 7.4.d we have a disparity, the maximal
relative error in the x- component is 2.368378 per cent in the zero point, in contrast
to the maximal relative error of the z- component, which is 1.00011 per cent.
Figure 7.3.: vector loop for potential differences Vx = VA − VI and Vz = VM − VE
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Results
63
(a) the heart vectors x- and z- components
(b) difference between the heart vector components
(c) relative error x- component
(d) relative error z- component
Figure 7.4.: heart vector components for potential differences Vx = VA − VI and Vz =
VM − VE
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64
7.4. Heart Vector with Frank Lead Matrix and Frank
potential differences
In Section 2.4 we not only introduced Frank’s lead matrix LF rank but also a lead
configuration, which uses five electrodes to evaluate the orthogonal potential differences
Vx and Vz . So we add into our torso model 6.1 a fifth electrode at an angle of 45 degrees
between the front midline and left midaxillary line and apply the linear functional
strategy on the heart vector

H∂B =
136 −0.2
0
136
!−1
·
!


0.610 0.171 −0.781
0
0

0.133 −0.231 −0.2646 0.736 −0.374 


VA
VC
VI
VM
VE








We receive a maximal difference in the x- component of -2.514794 and in the z- component of 1.861115. The relative error is in the x- direction 8.84696 per cent and in the
z- direction 99.885287 per cent. This errors arise, because we determine the orthogonal
potential differences Vx , Vz with one possible configuration for the five Frank electrodes
VA to VE . If we would like to correct our error, we have to solve an overdetermined
systems of equations.
Figure 7.5.: vector loop for potential differences Vx = 0.610VA + 0.171VC − 0.781VI and
Vz = 0.133VA + 0.736VM − 0.264VI − 0.374VE − 0.231VC
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65
(a) the heart vectors x- and z- components
(b) difference between the heart vector components
(c) relative error x- component
(d) relative error z- component
Figure 7.6.: heart vector components for potential differences Vx = 0.610VA +0.171VC −
0.781VI and Vz = 0.133VA + 0.736VM − 0.264VI − 0.374VE − 0.231VC
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66
7.5. Reciprocity Lead Matrix vs. Frank Lead Matrix
Now we want to compare the heart vectors, which we obtain by using the reciprocity
lead matrix Lrec and the Frank lead matrix LF . In order to decide which one approximates best the original dipole D we have introduced the relative difference rd in Section
7.1. With the aid of the relative difference we can decide whether the relative error
is larger for the reciprocity lead matrix or for the Frank lead matrix. Is the relative
difference positive then the reciprocity lead matrix fits best. If the relative difference
is negative, the Frank lead matrix approximates the original dipole best.
(a) relative difference x- component
(b) relative difference z- component
Figure 7.7.: relative difference for the x- and z- component for the reciprocity lead
matrix and Franks lead matrix with the potential difference Vx = VA − VI ,
Vz = VE − VM
We see in Figure 7.7 that the relative difference in the x- component is mostly positive
and in the z- component mostly negative. So if we want to approximate the original
dipole in the x- direction we obtain better results with the reciprocity lead matrix
than with Franks lead matrix. But on the other hand if we want to evaluate the zcomponent of the original dipole we have to use Franks lead matrix for better results.
Next we compare again the reciprocity lead matrix with the Frank lead matrix, but
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Results
67
this time we change for the Frank lead matrix the estimation of the potential difference.
We use now Vx = 0.610VA +0.171VC −0.781VI and Vz = 0.133VA +0.736VM −0.264VI −
0.374VE −0.231VC . We see in Figure 7.8 that the relative difference in the x- component
and in the z- component is positive, so the reciprocity lead matrix approximates the
original dipole D better than Franks lead matrix with Franks potential difference.
(a) relative difference x- component
(b) relative difference z- component
Figure 7.8.: relative difference for the x- and z- component for the reciprocity lead
matrix and Franks lead matrix with the potential difference Vx = 0.610VA +
0.171VC − 0.781VI and Vz = 0.133VA + 0.736VM − 0.264VI − 0.374VE −
0.231VC
At last we want to compare the Frank lead matrix with the two different potential
differences. After the previous comparisons we assume that the heart vector evaluated with the four electrodes is the better approximation and Figure 7.9 confirms this
assumption, because in both components the relative difference rd is positive.
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68
(a) relative difference x- component
(b) relative difference z- component
Figure 7.9.: relative difference for the x- and z- component for the Frank lead matrix
with the potential difference Vx = 0.610VA + 0.171VC − 0.781VI and Vz =
0.133VA + 0.736VM − 0.264VI − 0.374VE − 0.231VC and Vx = VA − VI ,
Vz = VE − VM
Finally, if we determine a lead matrix for our torso model and for the used electrode
configuration and if we know the dipole origin, we achieve the best results. Using the
Frank lead matrix, which resembles the reciprocity lead matrix, we have to differentiate between two lead configuration. If we use the same electrode configuration like
the reciprocity matrix, we obtain similar results. But if we use Franks electrode configuration, we obtain an inequality in the heart vectors, because we use one possible
representation of the orthogonal potential difference described by five electrodes. The
larges relative errors arise at the zero points.
69
8. VCG and heart disease
In this chapters we want to have a look at the relationship between the resulting
vectorcardiogram and heart diseases. For this purpose we simulate that a part of the
epicard is dead tissue and consequently not electric conductible any more. So we divide
the heart boundary ∂H into a vivid part ∂Hv and a dead part ∂Hd , then we obtain
for the potential u and the heart boundary potential g
∇ · (M ∇u) = 0 in T
(8.1)
M ∇u · nB = 0 on ∂B
(8.2)
u=g
(8.3)
on ∂Hv
u = 0 on ∂Hd .
(8.4)
We neglect the heart potential on a range from 2 per cent to 26 per cent of the circumference of the heart and calculate the heart vector with the reciprocity lead matrix.
Figures 8.1-8.4 show vectorcardiograms and the corresponding potential differences
with different heart disease sizes. We can see that the more parameters are in the
corresponding intervals, the more difficult it is to decide if a heart disease exists.
8.1. Parameters
In this section we will describe some parameters, which can help the doctors to decide
if a cardial disease exists or if the patient is healthy. We introduce 18 parameters with
an interval for health hearts. If less than 7 parameter lie in the corresponding intervals,
we can identify a heart disease in our model. Our parameters are all time independent,
we follow the ideas of [9] and [12].
8
VCG and heart disease
70
• maximal amplitude: Pamp , QRSamp , Tamp
Define the P-peak P p, QRS-peak QRSp and T-peak T p position in the vector
p
magnitude HV = Hx2 + Hz2 , with Hx , Hz the heart vectors in x- and z- direction, respectively. Then the maximal amplitude over the P-, QRS-, T-interval
is
Pamp = Hx (P p) + Hz (P p),
(8.5)
QRSamp = Hx (QRSp) + Hz (QRSp),
(8.6)
Tamp = Hx (T p) + Hz (T p).
(8.7)
• Areas: AP , AQRS , AT
define the area under the curve for the QRS-complex in x- direction with QRSx
and in z-direction with QRSz
AQRS =
p
QRSx2 + QRSz2 ,
(8.8)
analog for AP and AT .
• spatial ventricular gradient: SV G
It is the vectorial QRST-integral
SV G =
p
(QRSx + Tx )2 + (QRSz + Tz )2 .
(8.9)
• spatial mean QRS-T angle: SM QRS − T
The angle between the QRS area vector and the T area vector
SM QRS − T = cos
QRSx · Tx + QRSz · Tz
AQRS · AT
−1
.
(8.10)
• spatial QRS-T angle: SP QRS − T
The angle between the maximal QRS vector and the maximal T vector
SP QRS − T = cos
with |RP | =
Hx (QRSp) · Hx (T p) + Hz(QRSp) · Hz (T p)
|RP | · |T P |
−1
, (8.11)
p
p
Hx (QRSp)2 + Hz (QRSp)2 and |T P | = Hx (T p)2 + Hz (T p)2 .
8
VCG and heart disease
71
• modified SVG: mSV G
mSV G =
p
(QRSx + Tx + Px )2 + (QRSz + Tz + Pz )2 .
(8.12)
• spatial mean QRS-P angle: SM QRS − P
Angle between the QRSarea vector and the Parea vector
SM QRS − P = cos
QRSx · Px + QRSz · Pz
AQRS · AP
−1
.
(8.13)
• spatial QRS-P angle: SP QRS − P
Angle between the maximal QRS vector and the maximal P vector
SP QRS − P = cos
with |P P | =
Hx (QRSp) · Hx (P p) + Hz (QRSp) · Hz (P p)
|RP | · |P P |
−1
, (8.14)
p
Hx (P p)2 + Hz (P p)2 .
• vector-loop length: lP , lQRS , lT
Length of the three different vector loops,
lP =
p
(xi − xi+1 )2 + (zi − zi+1 )2 ,
(8.15)
analog for the QRS-loop and T-loop.
• relative Areas: relAP , relAQRS , relAT
relAQRS =
AQRS
.
lQRS
(8.16)
parameter
minimum
maximum
parameter
minimum
maximum
Pamp
2
2.05
QRSamp
14.25
14.5
Tamp
-1.1
-1
AP
6.3
6.36
AQRS
46.0
48.5
AT
15.05
15.2
SVG
56.0
58.0
SMQRST
54.5
56.5
SPQRST
62.7
64.7
mSVG
61.5
63.9
SMQRSP
14.3
15.0
SPQRSP
13.4
14.0
lP
0.493
0.495
lQRS
2.8
2.82
lT
0.994
0.988
relAP
12.6
13.0
relAQRS
16.6
16.85
relAT
15.0
15.6
8
VCG and heart disease
72
(a) The VCG of a healthy heart (blue) and the VCG of a 8% diseased heart (red)
(b) corresponding potential differences, above Vx , bottom Vx
Figure 8.1.: VCG and potential differences of a healthy heart (red) and a 8% diseased
heart (blue) with 7 of 18 parameters inside the corresponding interval
(a) The VCG of a healthy heart (blue) and the VCG of a 16% diseased heart (red)
(b) corresponding potential differences, above Vx , bottom Vx
Figure 8.3.: VCG and potential differences of a healthy heart (red) and a 16% diseased
heart (blue) with 0 of 18 parameters inside the corresponding interval
8
VCG and heart disease
73
(a) The VCG of a healthy heart (blue) and the VCG of a 11% diseased heart (red)
(b) corresponding potential differences, above Vx , bottom Vx
Figure 8.2.: VCG and potential differences of a healthy heart (red) and a 11% diseased
heart (blue) with 1 of 18 parameters inside the corresponding interval
(a) The VCG of a healthy heart (blue) and the VCG of a 26% diseased heart (red)
(b) corresponding potential differences, above Vx , bottom Vx
Figure 8.4.: VCG and potential differences of a healthy heart (red) and a 26% diseased
heart (blue) with 4 of 18 parameters inside the corresponding interval
8
VCG and heart disease
74
8.2. VCG and diseased area
In this last section we examine the vectorcardiograms of diseased hearts. We have
neglect on ten different boundary positions the heart potential of about 10 per cent.
We would like to find a relationship between the changes in the VCG and the location
which could not pass forward the electrical signal. For this purpose we divide the heart
like in Figure 8.5.
Figure 8.5.: partitioned heart into sectors of about 10%
We obtain the vectorcardiograms 8.6-8.15 corresponding to the parts one to ten. The
blue loops represent the healthy heart and the red loops the diseased one. At first
sight we recognize that the QRS-loop tip of the areas two and three shifts up and of
the areas four, five, six and seven down. The beginning of the QRS-loop differs from
the healthy vectorcardiogram for the areas one, two and three, but equals the healthy
vectorcardiogram at the ending. On the other hand the beginning remains equal and
the ending changes for the areas five, six and seven. For the areas eight, nine and ten
both beginning and ending change. The changes in the P-loop and in the T-loop are
smaller than in the QRS-loop. Beginning with the P-loop we notice that the right side
of the P-loop changes for the most part. We observe no change for the areas three,
four, five and six, small changes for the areas seven, eight, nine and ten. We notice the
biggest changes in the P-loop for the areas one and two. We divide the areas for the
T-loop into two groups, the fist one with small changes of the healthy heart contains
the areas four, five, six and seven. The second group contains the areas, which have
bigger changes compared to the healthy vectorcardiogram in the T-loop: one, two,
three, eight, nine and ten. Finally we have the following indicators: If the beginning of
the QRS-loop changes, the QRS-loop tip shifts up and we notice changes in the P-loop
8
VCG and heart disease
75
and the T-loop, a disease in the left atrium or at the top of the left ventricle can be
assumed. In our model the areas one, two and three. If the ending of the QRS-loops
changes, the QRS-loop tip shifts down, but we notice no significant changes in the Ploop and the T-loop, we can assume that a disease in the left ventricle exists, area five,
six and seven. If the beginning and the ending of the QRS-loop changes, the QRS-loop
tip does not change like the P-loop and the T-loop changes significant, we assume that
the right ventricle or the right atrium are diseased, area eight, nine and ten.
Figure 8.6.: part 1, 10.0676% diseased
Figure 8.7.: part 2, 10.04227% diseased
8
VCG and heart disease
Figure 8.8.: part 3, 10.22121% diseased
Figure 8.9.: part 4, 10.28912% diseased
Figure 8.10.: part 5, 10.10239% diseased
76
8
VCG and heart disease
Figure 8.11.: part 6, 9.988884% diseased
Figure 8.12.: part 7, 10.09401% diseased
Figure 8.13.: part 8, 10.1573% diseased
77
8
VCG and heart disease
Figure 8.14.: part 9, 10.0166% diseased
Figure 8.15.: part 10, 10.07034% diseased
78
79
9. Conclusion and Outlook
In the presented work, we examined the heart vector and the corresponding vectorcardiogram in detail. We took the Bidomain model as a basis for our theoretical analysis.
The basic assumptions for this work were: the processes in the heart can be described
by a single dipole called the heart vector and the heart vector can be expressed as a
dot product of the lead matrix and the potential difference on the body surface. By
representing the heart vector components on the body boundary as linear functionals,
we were able to transform this linear functionals on the heart region, applying the linear functional strategy. For these propose, we interpreted the heart vector on the body
boundary p(uT ) as an element of the space L2 (∂B)∗ and achieved a second equal representation for the heart vector on the dual space of H 1 (H) with the Riesz representation
theorem. Wanting to evaluate the heart vector on the heart region, we discussed the
forward problem and conclude that the forward adjoint bidomain problem is a properly posed problem. Thus applying the linear functional strategy avoids the problem
of regularization, which would arise when we want to calculate the heart vector on the
heart boundary on the direct way, because on the direct way we had to solve the inverse problem of electrocardiography and then determine the heart vector on the heart
region. With the aid of the Lagrangian, we obtain the adjoint bidomain model, so we
were able to find an integral representation of the heart vector on the heart region. In
view of the fact that the bidomain model describes the potential on a cellular level, we
implemented the linear functional strategy only up to the heart boundary and avoided
in this way the cellular level. We made a second simplification by modeling the torso
and the heart. We modeled a two dimensional torso and we omit the spine and the
breast bone as well as we assumed that the heart has one single conductivity like the
lungs and the torso tissue. In order to compare the two heart vectors, we calculated
the reciprocity lead matrix on the basic idea of Helmholtz reciprocity principle. This
resulted in the consistency of the heart vectors on the body boundary and on the heart
boundary apart from an error of the integration error range. But medicine systems
mostly use the electrode system of Frank, so applying the linear functional strategy to
the Frank lead matrix with the electrodes placed on the horizontal and the vertical line
9
Conclusion and Outlook
80
of the torso, we achieved similar results as with the reciprocity lead matrix. Next we
compared the heart vectors of the Frank lead matrix with the Frank electrode positions
and received an inequality of the heart vectors. This inequality is explained by the fact
that we had used five electrodes which were connected to describe two orthogonal potential differences, so we used one possible solution. Consequently, the reciprocity lead
matrix yield the best results, just as well is the Frank lead matrix with orthogonal
electrode positions. Furthermore, we tried to find out in what manner the heart vector
changes, if a heart disease exists. We established diagnostic intervals, if more than
eleven parameters lied outside the corresponding diagnostic interval, we could detect a
heart disease. Comparing vectorcaridiograms with a diseased heart boundary of 10%,
we identified that for the ventricle areas the vectorcardiograms changes apparently
more than for the atrium regions.
In order to continue to analyse the heart vector, we could extend our torso model, by
modeling the spine and the breast bone. It is possible to upgrade the model geometry
to the third dimension and evaluate the heart vector in three dimensions.
In Section 2.5, we saw that the lead vectors depend on the location of the dipole, the
electrode configuration, the shape and conductivity of the human torso. If we change
one of these four factors the lead vectors should change. Since the human torsos are
different and we do not know the dipole origin, it is interesting which effects can be
observed if we change the torso shape or the dipole origin while the lead vectors remain
equal. A related point are the electrode positions, because medicines fix the electrodes
with the help of reference points, so a single lead matrix for every patient is inexact.
A further point of interest is the examination of our results on real data. We would like
to test if we obtain with a reciprocity lead matrix for a real human also better results
than with the Frank lead matrix. In this context, we had to consider the electrode
positions and if we are able to find a better representation of the orthogonal potential
differences by the five Frank electrode positions.
An investigation point is the application of the entire bidomain model and the implementation of the heart vector representation in the heart region.
In order to confirm the dipole assumption, we should insert a dipole wavefront instead
of a single dipole in the forward calculation of the potential. Calculating a dipole which
represents the dipole wavefront and comparing this resulting dipole to the heart vector
on the body surface would explain whether we interpret the heart vector in a right
way.
81
A. Appendix
A.0.1. Transformation of Heart Vector into Solution Functional
Z
−1
l uT (δri − δrj )ds
(5.30)
Z
MT ∇λ2 · nT uT ds
=
∂B
∂T =∂B∪∂H
Z∂B
(5.29)(4.12)
Z∂T
PI
Z∂T
Z
MT ∇λ2 · nT uT ds +
=
MT ∇λ2 · nH uT ds
Z∂H
MT ∇λ2 · nT uT ds +
=
=
T
(Mi + Me )∇λ1 · nH ue ds
∂H
Z
∇ · (MT ∇λ2 ) · uT dx +
|
{z
}
MT ∇λ2 ∇uT dx
T
(5.27)
= 0
Z
+
(Mi + Me )∇λ1 · nH ue ds
Z
∇ · (MT ∇uT ) · λ2 dx +
MT ∇uT · nT λ2 ds
|
{z
}
∂T
Z∂H
=
−
T
(4.11)
= 0
Z
∂T =∂H∪∂B
=
(Mi + Me )∇λ1 · nH ue ds
Z
MT ∇uT · nT λ2 ds +
MT ∇uT · nT λ2 ds
{z
}
∂B |
∂H
+
Z
∂H
(4.15)
= 0
Z
(Mi + Me )∇λ1 · nH ue ds
Z
−
Me ∇ue · nH λ1 ds +
(Mi + Me )∇λ1 · nH ue ds
∂H
∂H
Z
Z
−
∇ · (Me ∇ue ) · λ1 dx −
Me ∇ue ∇λ1 dx
H
H
Z
+
(Mi + Me )∇λ1 · nH ue ds
Z ∂H
Z
∇ · (Mi ∇ν) · λ1 dx +
∇ · (Mi ∇ue ) · λ1 dx
HZ
Z H
−
Me ∇ue ∇λ1 dx +
(Mi + Me )∇λ1 · nH ue ds
+
(4.14)(5.31)
=
PI
=
(4.10)
=
Z∂H
H
∂H
A
Appendix
82
PI
Z
=
∂H
Mi ∇ν · nH λ1 + Mi ∇ue · nH λ1 ds
{z
}
|
(4.16)
= 0
Z
−
Z
Mi ∇ν∇λ1 dx −
ZH
−
ZH
=
PI
=
−
Z
Mi ∇ue ∇λ1 dx
ZH
Me ∇ue ∇λ1 dx +
(Mi + Me )∇λ1 · nH ue ds
∂H
Z
Mi ∇ν∇λ1 dx − (Mi + Me )∇ue ∇λ1 dx
H
H
(Mi + Me )∇λ1 · nH ue ds
Z
∇ · (Mi ∇λ1 ) · νdx −
Mi ∇λ1 · nH ν ds
{z
}
H
∂H |
Z∂H
(5.28)
= 0
Z
+
H
∇ · ((Mi + Me )∇λ1 ) · ue dx
{z
}
|
(5.26)
= 0
Z
Z
−
Z
∇ · (Mi ∇λ1 ) · νds
=
H
(Mi + Me )∇λ1 · nH ue ds
(Mi + Me )∇λ1 · nH ue ds +
∂H
∂H
A
Appendix
83
A.1. Moore-Penrose-Inverse
First of all we want to define define the range, the kernel and the orthogonal complement
Definition A.1.1 (kernel, range, orthogonal complement, [29]):
If X and Y are Hilbert spaces, M ⊂ X and A ∈ L(X, Y ) := {A : X → Y |A continuous and linear},
then we define
a) the kernel N (A) := {x ∈ X|Ax = 0},
b) the range R(A) := {y ∈ Y |∃x ∈ Xsuch that Ax = y},
c) the orthogonal complement of M, M ⊥ := {u ∈ X|(u, v) = 0∀v ∈ M }.
We want to solve the problem: find for every g ∈ Y a solution f ∈ X that solves
Af = g. Since this kind of problem are often ill-posed, we search an element f in X
that conforms
kAf − gkY ≤ kAφ − gkY
for every φ ∈ X.
(A.1)
Proposition A.1.1 ([29],p.21):
If g ∈ Y and A ∈ L(X, Y ), then the following statements are equivalent
a) f ∈ X achieves Af = PR(A) g, with PR(A) the orthogonal projection on the closure
of the range of the operator A,
b) f ∈ X minimizes the residuum: kAf − gkY ≤ kAφ − gkY for every φ ∈ X,
c) f ∈ X solves the equation A∗ Af = A∗ g.
The next Lemma ensures the existence.
Lemma A.1.1 ([29],p.21):
If g ∈ Y then count,
a) A solution of A∗ Aφ = A∗ g, for φ ∈ X, exists, if g ∈ R(A) ⊕ R(A)⊥ ,
b) The solution set L(g) = {φ ∈ X|A∗ Aφ = A∗ g} is closed and convex.
Now we select the element, which minimizes the residuum and has minimal norm.
A
Appendix
84
Lemma A.1.2 ([29],p.22):
For g ∈ R(A) ⊕ R(A)⊥ exists in the solution set L(g) an unique element f + with
minimal norm:
+
f < kφk
X
X
for every φ ∈ L(g)\ f + .
(A.2)
Now we can define the Moore-Penrose-Inverse and the minimum norm solution:
Definition A.1.2 ([29],p.22):
For ever g ∈ D(A+ ) the map A+ : D(A+ ) ⊂ Y → X with the domain of definition
D(A+ ) = R(A) ⊕ R(A)⊥ defines a unique element f + ∈ L(g) with minimal norm.
The map A+ is called Moore-Penrose-Inverse or pseudoinverse of the map A ∈
L(X, Y ). The element f + = A+ g is called the minimum norm solution of Af = g.
We characterize the minimum norm solution.
Proposition A.1.2 ([29],p.23):
If g ∈ D(A+ ), then f + = A+ g is the unique solution of the equation A∗ Af = A∗ g in
N (A)⊥ .
Proposition A.1.3 ([29],p.23):
The Moore-Penrose-Inverse A+ has the following properties
a) If R(A) is closed, then A+ is defined on the entire Y ,
b) R(A+ ) = N (A)⊥ ,
c) A+ is linear,
d) If the image of A is closed, R(A) = R(A), then A+ is continuous.
Proposition A.1.4 ([29],p.24):
The Moore-Penrose-Axiom define the Moore-Penrose-Inverse A+ of A ∈ L(X, Y )
clearly:
AA+ A = A,
A+ A = PR(A∗ ) ,
A+ AA+ = A+ ,
AA+ = PR(A) .
(A.3)
(A.4)
A
Appendix
85
A.2. Functional Analysis Background
In this section we want to introduce the Banach spaces, Hilbert spaces, Lebesgue
spaces Lp (Ω) and the Sobolev spaces W k,p (Ω). First of all we introduce the Banach
and Hilbert spaces, because we want to explain the Riesz representation theorem. In
order to define the Sobolev spaces we will explain the concept of weak derivatives of a
function. The Sobolev spaces are important for the theory of partial differential equations and for the finite element method explained in Section 6.1. So let us begin with
the definition of a norm.
Definition A.2.1 (norm, [11] p.635):
Let X be a real linear space. A map k.k : X → < is called norm on X, if for all
x, y ∈ X and λ ∈ <
• kxk ≥ 0 and kxk = 0 ⇒ x = 0,
• kλxk = |λ| kxk,
• kx + yk ≤ kxk + kyk.
Then a normed vector space, is a vector space equipped with a norm.
Definition A.2.2 (normed vector space, [31],I1):
Let X be a vector space and k.k : X → < the corresponding norm. The pair (X, k.k)
is called normed vector space.
Next we want to define Banach spaces, for this purpose we define the Cauchy sequence.
Definition A.2.3 (Cauchy sequence, [31],I1):
A sequence xn is called Cauchy sequence, if there exists for every > 0 an index
N () ∈ N such that kxn − xm k < .
Definition A.2.4 (Banach space,[11] p.635):
A normed vector space (X, k.k) is called Banach space, if every Cauchy sequence is
convergent.
Now we want to extend the Banach spaces to Hilbert spaces, so we have to define the
scalar product .
A
Appendix
86
Definition A.2.5 (scalar product/ inner product,[11] p.636):
Let X be a R vector space. A map h., .i : X × X → R is called scalar product or inner
product if for every xi , y ∈ X and λ ∈ R,
1. hx1 + x2 , yi = hx1 , yi + hx2 , yi,
2. hλx, yi = λ hx, yi,
3. hx, yi = hy, xi,
4. hx, xi ≥ 0,
5. hx, xi = 0 ⇔ x = 0.
Definition A.2.6 (Hilbert space, [11] p.636 ):
A Hilbert space is a Banach space whose norm is generated by a scalar product, such
p
that k.k = h., .i.
Talking about the linear functional strategy, we can not avoid the dual space.
Definition A.2.7 (dual space, [22]):
The dual space X ∗ of a normed vector space X, is the vector space of all continuous,
linear functionals on the vector space X. With
|l(x)|
= sup |l(x)| = inf {C > 0|l(x) < C kxk , ∀x ∈ X}
x∈X,kxk≤1
x∈X\{0} kxk
klk = sup
(A.5)
the dual space is also a normed vector space. If X is a Banach space or a Hilbert space
than X ∗ is also a Banach space or Hilbert space, respectively.
Now we are able to present Riesz representation theorem, which we applied in Section
5.1.1 for transforming the heart vector on the body boundary into a heart vector on
the heart boundary.
A
Appendix
87
Theorem A.2.1 (Riesz representation theorem, [22]):
Let X be a Hilbert space. For each linear functional l ∈ X ∗ , exists an unique element
xl ∈ X such that the following equation holds
l(y) = hxl , xi
∀x ∈ X.
(A.6)
On the other hand for every y ∈ X, is ly (x) = hy, xi a linear functional. In that way
X ∗ can be identify with X.
Finally we define the Lebesgue space in order to define the Sobolev spaces.
Definition A.2.8 (Lebesgue space and Lebesgue norm, [11] p.618):
Let 1 ≤ p < ∞. The Lebesgue space is defined as
p
L (Ω) =
Z
u : Ω → R|u measurable,
|u| dx < ∞ .
(A.7)
,
(A.8)
p
Ω
The Lebesgue norm for p 6= ∞ is defined as
Z
p
1/p
|u| dx
kukp =
Ω
in the case p = ∞ the norm is defined as
kuk∞ = esssup |u(x)| .
(A.9)
x
Proposition A.2.1 ([22]):
Lp (Ω) is a Banach space for all 1 ≤ p < ∞.
In the next step we introduce the weak derivation. For this purpose we introduce the
space C0∞ (Ω), the space of infinitely differentiable functions with compact support. If
we multiply a function φ ∈ C0∞ (Ω) with a function u ∈ C 1 (Ω) and apply integration
by parts we obtain
Z
Z
uφxi dx = −
Ω
uxi φdx,
(A.10)
Ω
for i = 1, ..., n. If we want to generalize this result, we have to introduce multi indexes.
A
Appendix
88
Definition A.2.9 (weak derivative, [11] p.242\ 243):
If k is a positive integer and α = (α1 , ..., αn ) is a multi index of order |α| = α1 +...+αn =
k and φ ∈ C0∞ , then ∂xα φ ∈ C0∞ is defined as
h∂xα φ, ϕi := (−1)|α| hφ, ∂xα ϕi ,
(A.11)
and is called the weak derivation of order |α| of the function φ.
Remark
For a function u ∈ C k (Ω) the weak derivation is defined as
Z
hφ, ϕi :=
uϕdx,
Ω
and u holds
Z
Z
h∂xα φ, ϕi = − hφ, ∂xα ϕi = −
(∂xi u)ϕdx.
u∂xi ϕdx =
Ω
Ω
Now we can define the Sobolev spaces, which exploit the weak derivation.
Definition A.2.10 (Sobolev space, Sobolev norm, [11] p.245):
For 1 < p < ∞ the Sobolev space is defined as
W 1,p (Ω) := u ∈ Lp (Ω)|∂xj u ∈ Lp (Ω, j = 1, ...d)
(A.12)
with the Sobolev norm
kuk1,p :=
d
X
∂x up p
kukpLp (Ω) +
j
L (Ω)
!p
(A.13)
j=1
Remark([11] p.245)
If p = 2, we usually write
H k (Ω) = W k,p (Ω),
(A.14)
for k = 0, 1, 2, ....
Theorem A.2.2 ([11],p.249):
For each k = 1, 2, ... and 1 ≤ p ≤ ∞, the Sobolev space W k,p (Ω) is a Banach space.
A
Appendix
89
Definition A.2.11 ([11], p.283):
We denote by H −1 (Ω) the dual space to H01 (Ω) := W01,2 (Ω) the closure of C0∞ (Ω) in
W 1,2 (Ω).
In the end we introduce the trace theorem.
Theorem A.2.3 (trace theorem, [11], p.258):
Assume Ω is bounded and ∂Ω is C 1 . Then there exists a bounded linear operator
T : W 1,p (Ω) → Lp (∂Ω)
such that
1. T u = u|∂Ω if u ∈ W 1,p (Ω) ∩ C(Ω),
2. kT ukLp (∂Ω) ≤ C kukW 1,p (Ω) ,
for each u ∈ W 1,p (Ω), with the constant C depending only on p and Ω.
(A.15)
90
List of Figures
1.1.
1.2.
1.3.
1.4.
1.5.
heart anatomy,[21] . . . . . . . . . . . . . . .
transmembrane potential of an excited cardiac
The conductivity system of the heart,[21] . . .
The Electrocardiogram . . . . . . . . . . . . .
The electrode position of a 12-lead ECG . . .
. . . .
muscle
. . . .
. . . .
. . . .
. . . . . . . . . .
cell of a frog,[21]
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .
3
4
5
6
7
2.1.
2.2.
2.3.
2.4.
Illustration of the geometry of a dipole,[3] . . . . . . . . . . . . . . . .
Einthovens’ triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Franks lead system and dipole position . . . . . . . . . . . . . . . . . .
Illustration of the reciprocity theorem for electrocardiography. First one
shows the resulting potential difference VAB , second one the potential
difference VCD , third one the lead field by injected current at electrodes
A and removed current at electrode B, [26] . . . . . . . . . . . . . . . .
9
12
14
3.1. Vectorcardiogram, [21] . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
4.1. schematic model of the heart and the torso and their normals, [23] . . .
21
5.1. Relation between dual map and the Linear functional strategy . . . . .
5.2. Relation heart vector and adjoint heart vector . . . . . . . . . . . . . .
5.3. Inverse adjoint Bidomain Problem . . . . . . . . . . . . . . . . . . . . .
33
33
36
6.1. Torso geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2. Torso geometry with coordinate system and dipole origin. x-axis from
left to right, z-axis from top to bottom. Electrode positions r1 to r3 . The
potential differences are then Vx = u(r3 ) − u(r1 ) and Vz = u(r4 ) − u(r2 )
6.3. Mesh of model geometry, 1600 mesh points, 2952 triangles, 33760 number of degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4. Solution of the adjoint problem . . . . . . . . . . . . . . . . . . . . . .
6.5. Normal components of the adjoint problem . . . . . . . . . . . . . . . .
6.6. Torso geometry with 28 dipole positions . . . . . . . . . . . . . . . . .
49
17
51
53
54
54
55
List of Figures
7.1.
7.2.
7.3.
7.4.
7.5.
7.6.
7.7.
7.8.
7.9.
vector loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
heart vector components . . . . . . . . . . . . . . . . . . . . . . . . . .
vector loop for potential differences Vx = VA − VI and Vz = VM − VE . .
heart vector components for potential differences Vx = VA − VI and
Vz = VM − VE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vector loop for potential differences Vx = 0.610VA + 0.171VC − 0.781VI
and Vz = 0.133VA + 0.736VM − 0.264VI − 0.374VE − 0.231VC . . . . . .
heart vector components for potential differences Vx = 0.610VA +0.171VC −
0.781VI and Vz = 0.133VA + 0.736VM − 0.264VI − 0.374VE − 0.231VC .
relative difference for the x- and z- component for the reciprocity lead
matrix and Franks lead matrix with the potential difference Vx = VA −VI ,
Vz = VE − VM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
relative difference for the x- and z- component for the reciprocity lead
matrix and Franks lead matrix with the potential difference Vx = 0.610VA +
0.171VC − 0.781VI and Vz = 0.133VA + 0.736VM − 0.264VI − 0.374VE −
0.231VC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
relative difference for the x- and z- component for the Frank lead matrix
with the potential difference Vx = 0.610VA + 0.171VC − 0.781VI and
Vz = 0.133VA +0.736VM −0.264VI −0.374VE −0.231VC and Vx = VA −VI ,
Vz = VE − VM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1. VCG and potential differences of a healthy heart (red) and a 8% diseased
heart (blue) with 7 of 18 parameters inside the corresponding interval .
8.3. VCG and potential differences of a healthy heart (red) and a 16% diseased heart (blue) with 0 of 18 parameters inside the corresponding
interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2. VCG and potential differences of a healthy heart (red) and a 11% diseased heart (blue) with 1 of 18 parameters inside the corresponding
interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4. VCG and potential differences of a healthy heart (red) and a 26% diseased heart (blue) with 4 of 18 parameters inside the corresponding
interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5. partitioned heart into sectors of about 10% . . . . . . . . . . . . . . . .
8.6. part 1, 10.0676% diseased . . . . . . . . . . . . . . . . . . . . . . . . .
8.7. part 2, 10.04227% diseased . . . . . . . . . . . . . . . . . . . . . . . . .
8.8. part 3, 10.22121% diseased . . . . . . . . . . . . . . . . . . . . . . . . .
8.9. part 4, 10.28912% diseased . . . . . . . . . . . . . . . . . . . . . . . . .
91
60
61
62
63
64
65
66
67
68
72
72
73
73
74
75
75
76
76
List of Figures
8.10. part
8.11. part
8.12. part
8.13. part
8.14. part
8.15. part
92
5, 10.10239% diseased .
6, 9.988884% diseased .
7, 10.09401% diseased .
8, 10.1573% diseased .
9, 10.0166% diseased .
10, 10.07034% diseased
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