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Linear Functionals in ECG and

VCG

Diplomarbeitzur Erlangung des akademischen Grades

Diplom-Mathematiker/in

Westfalische Wilhelms-Universitat Munster

Fachbereich Mathematik und Informatik

Institut fur Numerische und Angewandte Mathematik

Betreuung:

Prof. Dr. Martin Burger

Eingereicht von:

Joanna Tendera

Munster, 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 Erklarung

Hiermit versichere ich, Joanna Tendera, dass ich die vorliegende Arbeit selbststandig

verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet

habe. Gedanklich, inhaltlich oder wortlich ubernommenes habe ich durch Angabe

von Herkunft und Text oder Anmerkung belegt bzw. kenntlich gemacht. Dies gilt in

gleicher Weise fur Bilder, Tabellen, Zeichnungen und Skizzen, die nicht von mir selbst

erstellt wurden.

Alle auf der CD beigefugten Programme sind von mir selbst programmiert worden.

Munster, 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 Grone and Elin Sandberg for many helpful discussions and proof-

reading this thesis.

• all my friends who have supported me during the last years.

iv

Contents

1. Medical Background 3

1.1. Anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2. The Heart Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3. Electric excitation of the Heart . . . . . . . . . . . . . . . . . . . . . . 5

1.4. Electrocardiogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5. 12-lead ECG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2. Heart Vector 8

2.1. Dipole Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2. Dipole Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3. Relation: Heart Vector Potential Difference . . . . . . . . . . . . . . . . 10

2.4. Einthoven, Burger, van Milaan, Frank . . . . . . . . . . . . . . . . . . . 12

2.5. Lead field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3. Vectorcardiography 18

3.1. Vectorcardiogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2. Linear Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4. Bidomain Modell 21

4.1. Bidomain Modell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2. Forward and Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . 25

4.2.1. Forward Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2.2. Inverse Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5. Linear Functional Strategy 31

5.1. Linear Functional Strategy and Heart Vector . . . . . . . . . . . . . . . 31

5.1.1. Linear Algebra and the Linear Functional Strategy . . . . . . . 32

5.2. Bidomain Model and Heart Vector . . . . . . . . . . . . . . . . . . . . 33

5.3. Adjoint Bidomain Problem . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.3.1. Forward Adjoint Bidomain Problem . . . . . . . . . . . . . . . . 35

Contents v

5.3.2. Inverse Adjoint Bidomain Problem . . . . . . . . . . . . . . . . 36

5.4. Adjoint Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.4.1. Adjoint Operator: Heart-Torso Model . . . . . . . . . . . . . . 37

5.4.2. Solution-Functional: Heart-Torso Model . . . . . . . . . . . . . 39

5.4.3. Adjoint Bidomain Operator . . . . . . . . . . . . . . . . . . . . 41

5.5. Solution-functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

6. Implementation 46

6.1. Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6.2. Torso model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.3. Implementation Lead Matrix . . . . . . . . . . . . . . . . . . . . . . . . 50

6.4. Implementation Linear Functional Strategy . . . . . . . . . . . . . . . . 52

6.4.1. Solving the adjoint problem . . . . . . . . . . . . . . . . . . . . 52

6.4.2. Evaluation of the heart vector on the heart boundary . . . . . . 55

7. Results 57

7.1. Error Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

7.2. Heart Vector with reciprocity Lead Matrix . . . . . . . . . . . . . . . . 60

7.3. Heart Vector with Frank Lead Matrix . . . . . . . . . . . . . . . . . . . 62

7.4. Heart Vector with Frank Lead Matrix and Frank potential differences . 64

7.5. Reciprocity Lead Matrix vs. Frank Lead Matrix . . . . . . . . . . . . . 66

8. VCG and heart disease 69

8.1. Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

8.2. VCG and diseased area . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

9. Conclusion and Outlook 79

A. Appendix 81

A.0.1. Transformation of Heart Vector into Solution Functional . . . . 81

A.1. Moore-Penrose-Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

A.2. Functional Analysis Background . . . . . . . . . . . . . . . . . . . . . . 85

List of Figures 90

Bibliography 93

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.

1WHO[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]


1.3. 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]


1.4. Electrocardiogram

In the previous section, we have seen that the electric activation of the SA node prop-

agates 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 electrocardio-

gram (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.


1.5. 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 (r′) at a measuring

point r′

ϕ (r′) =1

4πσ

I

r − r′, (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

ϕ (r) =I

4πσ

(1

r+

− 1

r−

), (2.2)

with r+ = r+δ/2 and r− = r−δ/2. If we choose δ r it follows that r+−r− ≈ δ cosϕ

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

4πσr2δ cos θ, (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) = Hcos 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 = Hxex +Hyey +Hzez, (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

H (t) =

∫Heart

Jdx, (2.7)

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 = V ijx ex + V ij

y ey + V ijz ez, (2.8)

where V ijx , V ij

y and V ijz 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

V ijx = ϕ (i)x − ϕ (j)x ,

= H · r1 − r2

4πσ |r1 − r2|3, (2.9)

= lx ·Hx,

similar equations occur for the potential difference in y- and z- direction. These con-

stants 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

by Burger and van Milaan [4]. From dipole theory we obtain lx = r1−r24πσ|r1−r2|3

with the

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) = lxHx (t) + lyHy (t) + lzHz (t) , (2.10)

= l ·H (t) , (2.11)

where V ij is the dot product of the lead vector l = lxex+ lyey+ lzez and the heart vector

H = Hxex +Hyey +Hzez. 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 in-

troduce 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. Exploit-

ing the principle of superposition, the total lead voltage equals the sum of all the

contributions of each dipole,

V ij =∑k

Hk · lk, (2.12)

with Hk dipole moments and k = 1, ..., n numbers of dipoles.

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, 1

2

√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 (2.13)

VII = lII ·H =1

2Hx +

√3

2Hy (2.14)

VIII = lIII ·H = −1

2Hx +

√3

2Hy. (2.15)

One solution for the heart vector coordinates is H =(VI ,

2·VII−VI√3

). 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 be-

tween heart vector and lead in a glass phantom filled with electrolyte and an arti-

ficial 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 Sec-

tion 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 com-

ponents

2 Heart Vector 14

(a) Franks lead system, [15] (b) Dipole position 22 equals second col-umn 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 = ATA and use

M−1M = I, with I the identity matrix, then

Vxyz = IVxyz = M−1MVxyz = M−1ATAVxyz = M−1ATV, (2.20)

where M−1AT is called the inverse Dower matrix and is numerically given by

M−1AT =

−0.172 −0.074 0.122 0.231 0.239 0.194 0.156 −0.010

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 for-

mulated 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 electrocardio-

graphy, [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 ′ at electrode A

and remove it at electrode B we can observe a potential difference VCD. Then the

reciprocity theorem states

VABI

=VCDI ′

. (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 · II ′

=E · δII ′

=E ·HI ′

. (2.22)

2 Heart Vector 17

Figure 2.4.: Illustration of the reciprocity theorem for electrocardiography. First oneshows the resulting potential difference VAB, second one the potential dif-ference VCD, third one the lead field by injected current at electrodes Aand removed current at electrode B, [26]

.

So we can define the ratio of the electric field E and the injected current I ′ 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 func-

tional.

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

p (uT (t)) =

∫∂B

l−1uT(δri − δrj

)ds. (3.2)

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

p1 [uT ] =

∫ t1

t0

∫∂B

l (x)uT (x, t) dσdt, (3.3)

• averaged propagation velocity/speed

p2 [uT ] =

∫∂B

l (x)∂uT∂t

(x, t) dσ, (3.4)

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

∂t= 0, (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 con-

ductor 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)


gives

Ji = −Mi · ∇ui,

Je = −Me · ∇ue, (4.5)

JT = −MT · ∇uT ,

with Mi, Me and MT conductivity tensors in the intracellular, extracellular and extrac-

ardial 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 op-

posite 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

∂

∂t(qi + qe) = 0, (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

−∇ · Ji =∂qi∂t

+ Iion, (4.7)

−∇ · Je =∂qe∂t

+ Iion. (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) . (4.9)

Exploiting the transmembrane potential we can rewrite the total current (4.9)


∇ · (Miν) +∇ · ((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


(4.9), 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 For-

ward 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


3. the data (output) depends continuously on the solution (input)(continuous depen-

dence),

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 : H1 (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,

A2 : H1/20 (∂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

A1 : H1 (H)→ H1/20 (∂H) , ν 7→ ue|∂H , (4.22)


and we define H1/20 (∂H) as

H1/20 (∂H) :=

h ∈ H1/2 (∂H) |

∫∂H

hdσ = 0

. (4.23)

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−1uT 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 equa-

tions. 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

a unique solution we add a normalization condition like∫∂HuTdσ = 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 : H1 (H)→ L2 (∂B) defined by (4.19) is continuous.

Proof. We split our proof into two parts, we show first that A1 is continuous, after-

wards 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 : H1 (H)→ H1/2 (∂H) with Tν = ν|∂H = ue ∀ν ∈ C(H)

(4.24)

The operator T is surjective, means for all ue ∈ H1/2 (∂H) exists a ν ∈ H1 (H) with


Tν = ue.

Application of the trace theorem implies that for ue ∈ H1/20 (∂H) ⊂ H1/2 (∂H) exists

an ν ∈ H1 (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

uT |∂B =

∫∂H

MT∇G (x, y) · nH · ue (y) dσ (y) =

∫∂H

k (x, y) · ue (y) dσ(y), (4.25)

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 C2(Ω),[11]. Then we can

rewrite the operator equation

A2ue =

∫∂H

k (x, y) · uT (y) dσ (y) = uT . (4.26)

We are now in a position to estimate

‖A2ue‖2L2(∂B) =

∥∥∥∥∫∂H

k (x, y) · uT (y) dσ (y)

∥∥∥∥2

L2(∂B)

=

∫∂B

∫∂H

k (x, y)2 · uT (y)2 dσ (y) dσ (x)

C.S

≤∫∂B

∫∂H

k (x, y)2 dσ (y)

∫∂H

uT (y)2 dσ (y) dσ (x)

≤ C ·∫∂H

uT (y)2 dσ (y)

ue=uT on ∂H= C ·

∫∂H

ue (y)2 dσ (y)

= C · ‖ue‖2L2(∂H)

≤ C · ‖ue‖2H1(∂H) .


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 : H1(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) → H1 (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:

• find for every ue|∂H ∈ H1/20 a ν ∈ H1(H) that solves A1ν = ue|∂H ,

• find for every uT ∈ L2(∂B) a ue|∂H ∈ H1/20 that solves A2ue|∂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:

If ue ∈ H1/20 (∂H) and A2 : H

1/20 → L2(∂B) with A2ue =

∫∂Hk(x, y)ue(y)dσ(y) and

k(x, y) achieves ∫∂H

∫∂B

k(x, y)2dσ(x)dσ(y) = C ≤ ∞,

then A−12 is not continuous.


Proof. Consider ej an orthonormal function system on H1/2(∂H) and have a look at

‖A2ej‖2L2(∂B) =

∫∂B

|A2ej|2dσ(x) =

∫∂B

∫∂H

|k(x, y)ej(y)|dσ(y)2dσ(x),

=

∫∂B

∫∂H

k(x, y)2dσ(y)

∫∂H

e2j(y)dσ(y)dσ(x),

≤ C · ‖ej(y)‖2L2(∂H) ≤ C · C ≤ ∞.

So ‖A2ej‖2L2(∂B) is bounded. Exploiting the embedding of H1/2 in L2, we know that

orthonomal functions ej, which converge weakly in H1/2 against zero, also converge

against zero in L2. That results in ‖A2ej‖L2(∂B)

j→∞−→ 0. Let us define gj :=A2ej

‖A2ej‖L2(∂B),

then ‖gj‖L2(∂B) = 1 holds. Examine

∥∥A−12 gj

∥∥H1/2(∂H)

=

∥∥∥∥∥A−12

(A2ej

‖A2ej‖L2(∂B)

)∥∥∥∥∥H1/2(∂H)

=

∥∥∥∥∥ ej‖A2ej‖L2(∂B)

∥∥∥∥∥H1/2(∂H)

j→∞−→ ∞.

We know ‖gj‖ = 1, it follows that∥∥A−1

2

∥∥ = ∞. Consequently A−12 is not continuous.

A−12 is not continuous, so the inverse Bidomain Operator is also not continuous and

unfortunately the inverse Bidomain model is improperly posed and needs regulariza-

tion.

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 func-

tionals, 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 solution-

functional. 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 function-

als. 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 prob-

lem 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

mθ (u) =

∫ 1

0

θ (x)u (x) dx (5.1)

is contained in R (K∗), then the required ϕ which defines

mϕ (s) =

∫ 1

0

ϕ (x) s (x) dx (5.2)

is determined by

K∗ϕ = θ. (5.3)

Proof.

mϕ (s) = (ϕ, s)Ku=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 Strat-

egy 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)


It is clear that φv(u) is an element of the dual space H∗2 . 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∗ : H∗2 → H∗1 , 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.

H1K //

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.

H1(H) A //

A∗p(uT )%%

L2(∂B)

p(uT )<

Figure 5.2.: Relation heart vector and adjoint heart vector


In detail the diagram describes that the linear heart vector p(uT ) : L2(∂B)→ < with

p(uT ) =∫∂Bl−1 · uTdσ 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

p(uT ) =

∫∂B

l−1 · uTdσ = (uT , p) . (5.7)

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))∗ → (H1(H))

∗. As we

remarked above, the dual space (L2(∂B))∗

can be identified with L2(∂B) and the dual

space (H1(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 p(v) but

this time on the domain of definition H1(H) and write it also as dual pairing

p(ν) = (p, ν) . (5.9)

Like the heart vector this functional is linear and an element in the dual space (H1(H))∗.

We are almost ready to describe the relationship between the heart vector on the body

surface p(uT ) and the linear functional p(ν) in the heart. The adjoint Bidomain oper-

ator maps a linear functional into an other linear functional so we obtain by

A∗p(uT ) = p(ν), (5.10)

a linear functional on H1(H). But in (5.9) we have defined such a linear functional on

H1(H), so we can rewrite the heart vector

p(uT ) = (uT , p) = (Aν, p) = (ν,A∗p) = (ν, p) = p(ν). (5.11)


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 ) = p(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 op-

erator and the heart vector. The forward adjoint Bidomain Problem means to evaluate

p(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∫∂Hλ2 = 0. In order to establish that the problem is continuous, we recall the Diagram

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

pε(uT ) = p(uT ) + ε(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 Holders inequation lead us to


|A∗pε(uT )− A∗p(uT )| = |pε(Aν)− p(Aν)| = |pε(uT )− p(uT )|

= |p(uT ) + ε(uT )− p(uT )| = |ε(uT )|

≤ ‖ε‖L2(∂B) ‖uT‖L2(∂B) .

So if ‖ε‖L2(∂B) tends to zero, |A∗pε(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 in-

verse 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 pε(uT ) as a solution of A∗p(uT ) = pε(ν) where pε(ν) = p(ν)+ε(ν)

are the disturbed data with a perturbation ε. We obtain

|pε(ν)− p(ν)| = |ε(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.

H1(H)

p(ν)

L2(∂B)A+oo

(A+)∗p(ν)yy<

Figure 5.3.: Inverse adjoint Bidomain Problem


So if the data error functional ε(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 in T, (5.14)

nT · (MT∇u) = 0 on ∂B, (5.15)

u = u∂H 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

L(u, l−1) =

∫T

∇ · (MT∇u) · λdx+

∫∂B

l−1u(δri − δrj)ds. (5.17)


To obtain the adjoint operator we will apply Gauss’ theorem and exploit the boundary

condition ∂T = ∂B ∪ ∂H.

L(u, l−1) =

∫∂T

(MT∇u · nT ) · λds+

∫T

MT∇u · ∇λdx+

∫∂B

l−1u(δri − δrj)ds,

=

∫∂B

MT∇u · nT · λds−∫∂H

MT∇u · nH · λds

+

∫T

MT∇u · ∇λdx+

∫∂B

l−1u(δri − δrj)ds.

Regarding the model we see that the boundary integral over the body surface vanishes

and a second application of Gauss theorem leads to

L(u, l−1) = −∫∂H

MT∇uT · nHλds+

∫T

∇ · (MT∇λ) · udx

−∫∂T

MT∇λ · nTudx+

∫T

MT∇u · ∇λdx+

∫∂B


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

L(u, l−1) = −∫∂H

MT∇uT · nHλds+

∫T

∇ · (MT∇λ) · udx

−∫∂B

MT∇λ · nTuds+

∫∂H

MT∇λ · nHu∂Hds+

∫∂B


With the additional assumption λ = 0 on ∂H we can take the partial derivate of the

Lagrangian, then we obtain

∂L

∂u=

∫T

∇ · (MT∇λ)dx−∫∂B

MT∇λ · nTds+

∫∂B

l−1(δri − δrj)ds. (5.18)

If the partial derivative fulfills the optimal condition ∂L∂u

= 0 we obtain the adjoint

system


∇ · (MT∇λ) = 0 in T, (5.19)

MT∇λ · nT = l−1(δri − δrj) on ∂T, (5.20)

λ = 0 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.

∫∂B

l−1(δri − δrj) · uds(5.20)=

∫∂B

MT∇λ · nTuds,

∂B∪∂H=∂T=

∫∂T

MT∇λ · nTuds+

∫∂H

MT∇λ · nHuds.

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,

∫∂B

l−1(δri − δrj) · uds =

∫T

∇ · (MT∇λ) · udx︸︷︷︸=0

+

∫T

MT∇λ∇udx+

∫∂H

MT∇λ · nHuds,

= −∫T

∇ · (MT∇u) · λdx︸︷︷︸=0

+

∫∂T

MT∇u · nTλds+

∫∂H

MT∇λ · nHuds.

Exploiting that the torso boundary is the union of the heart boundary and body


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.

∫∂B

l−1(δri − δrj) · uds = −∫∂H

MT∇u · nHλds︸︷︷︸=0

+

∫∂B

MT∇u · nTλds︸︷︷︸=0

+

∫∂H

MT∇λ · nHuds,

=

∫∂H

MT∇λ · nHuds.

In the last step we obtain,

∫∂B

l−1(δri − δrj) · uds =

∫∂H

MT∇λ · nHuHds. (5.22)

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.


5.4.3. 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)

=

∫H

∇ · (Mi∇ν) · λ1dx+

∫H

∇ · ((Mi +Me)∇ue) · λ1dx

+

∫T

∇ · (MT∇uT ) · λ2dx+

∫∂B


)ds

=

∫∂H

Mi∇ν · nHλ1ds−∫H

Mi∇ν∇λ1dx

+

∫∂H

(Mi +Me)∇ue · nHλ1ds−∫H

(Mi +Me)∇ue∇λ1dx

+

∫∂T

MT∇uT · nTλ2ds−∫T

MT∇uT∇λ2 +

∫∂B


)ds.

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).


L(ν, ue, uT , l

−1)

=

∫∂H

Mi∇ν · nHλ1ds+

∫∂H

(Mi +Me)∇ue · nHλ1ds︸︷︷︸=∫∂H MT∇uT·nHλ1

−∫H

Mi∇ν∇λ1dx−∫H

(Mi +Me)∇ue∇λ1dx+

∫∂B

MT∇uT · nBλ2︸︷︷︸=0

−∫∂H

MT∇uT · nHλ2ds−∫T

MT∇uT∇λ2dx

+

∫∂B


)ds

=

∫∂H

((MT∇uT ) · nH) (λ1 − λ2) ds−∫H

Mi∇ν∇λ1dx

−∫H

(Mi +Me)∇ue∇λ1dx−∫T

MT∇uT∇λ2dx

+

∫∂B


)ds.

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)

=

∫∂H

(MT∇uT · nH) (λ1 − λ2) ds+

∫H

∇ · (Mi∇λ1) · νdx

−∫∂H

Mi∇λ1 · nH · νds+

∫H

∇ · ((Mi +Me)∇λ1) · uedx

−∫∂H

(Mi +Me)∇λ1 · nH · ueds+

∫T

∇ · (MT∇λ2) · uTdx

−∫∂T

MT∇λ2 · nT · uTds+

∫∂B


)ds.

It remains to repeat step number two to obtain


L(ν, ue, uT , l

−1)

=

∫∂H

(MT∇uT · nH) (λ1 − λ2) ds+

∫H

∇ · (Mi∇λ1) · νdx

−∫∂H

Mi∇λ1 · nH · νds+

∫H

∇ · ((Mi +Me)∇λ1) · uedx

−∫∂H

(Mi +Me)∇λ1 · nH · ueds+

∫T

∇ · (MT∇λ2)uTdx

−∫∂B

MT∇λ2 · nT · uTds−∫∂H

MT∇λ2 · nT · uTds

+

∫∂B


)ds.

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)

=

∫H

∇ · (Mi∇λ1) · gdx−∫∂H

Mi∇λ1 · nH · νds

+

∫H

∇ · ((Mi +Me)∇λ1) · uedx−∫∂H

(Mi +Me)∇λ1 · nH · ueds

+

∫T

∇ · (MT∇λ2)uTdx−∫∂B

MT∇λ2 · nT · uTds

−∫∂H

MT∇λ2 · nT · ueds+

∫∂B


)ds.

Now we are in the position to take the partial derivate of L with respect to ν, ue and

uT and obtain

∂L

∂ν=

∫∂H

Mi∇λ1 · nHds,

∂L

∂ue=

∫H

∇ · ((Mi +Me)∇λ1) dx−∫∂H

(Mi +Me)∇λ1 · nHds−∫∂H

MT∇λ2 · nTds,

∂L

∂uT=

∫T

∇ · (MT∇λ2) dx−∫∂B

MT∇λ2 · nTds

+

∫∂B

l−1(δri − δrj

)ds.

(5.24)


Since we have

∂L

∂ν=0,

∂L

∂ue= 0 and

∂L

∂uT= 0, (5.25)

we obtain the following adjoint problem

∇ · ((Mi +Me)∇λ1) = 0 in H (5.26)

∇ · (MT∇λ2) = 0 in T (5.27)

Mi∇λ1 · nH = 0 on ∂H (5.28)

− (Mi +Me)∇λ1 · nH +MT∇λ2 · nH = 0 on ∂H (5.29)

MT∇λ2 · nT = l−1(δri − δrj

)on ∂B (5.30)

λ1 = λ2 on ∂H (5.31)

5.5. Solution-functional

The next part is about the identification of the heart vector to a corresponding solution-

functional 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, p) = p(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

∫∂B

l−1uTds =

∫H

wνdx, (5.33)

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


p(uT ) =

∫∂B

l−1(δri − δrj

)· uTds =

∫H

∇ · (Mi∇λ1) · ν(δri − δrj

)dx (5.34)

=

∫H

w(δri − δrj

)· νdx = p(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 differ-

ential 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

∫T

∇ · (MT∇λ) · φdx = 0. (6.1)

Application of Gauss theorem and rewriting the torso boundary into the union of body

boundary and heart boundary leads to

∫T

∇ · (MT∇λ) · φdx =

∫∂B

MT∇λnB · φds−∫∂H

MT∇λnH · φds−∫T

MT∇λ∇φdx.

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

∫T

MT∇λ∇φdx =

∫∂B

l−1(δri − δrj) · φds. (6.2)

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⋃j=1

Tj. (6.3)

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 neighbor-

hood N(Pi) defined as

N(Pi) =⋃

Pi∈∂Tj

Tj. (6.4)

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 ∈ <

λN =N∑j=1

λjϕj, (6.6)

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∑j=1

∫T

λj∇ϕj · ∇ϕkdx =

∫∂B

l−1(δri − δrj) · ϕkds. (6.7)

In fact we have to solve a NxN system of equation, with the stiffness matrix KN ∈<NxN

KN =N∑j=1

∫T

λj∇ϕj · ∇ϕkdx for k = 1, ..., N, (6.8)

and the right hand vector FN

FN =

∫∂B

l−1(δri − δrj) · ϕkds for k = 1, ..., N. (6.9)

In order to obtain the stiffness matrix we have to evaluate N2 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 KN is a sparse matrix.

6 Implementation 49

6.2. Torso model

This section will explain the model geometry on which we will solve the adjoint equa-

tion. 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 resem-

bles 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 disre-

garding 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

).

6 Implementation 50

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

VAB =

∫∂B

u(δA − δB)dr, (6.10)

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 in T

MT∇w · nT = 0 on ∂B (6.11)

w = δA − δB 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 x-

axis 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

6 Implementation 51

and the bottom electrode, see Figure 6.2.

Figure 6.2.: Torso geometry with coordinate system and dipole origin. x-axis from leftto right, z-axis from top to bottom. Electrode positions r1 to r3. Thepotential 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.

6 Implementation 52

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 heart-

torso 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

6 Implementation 53

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

∇ · (Mλ) = 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)12Vz (6.12)

=

∫∂B

(l−1)11uT (δr3 − δr1) +

∫∂B

(l−1)12uT (δr4 − δr2). (6.13)

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 numberof degrees of freedom

6 Implementation 54

(a) Solution of the adjoint problem (5.20) in thex- direction with the torso boundary conditionnT ·(MT∇λ) = 0.493491·(δr3−δr1)−0.0249229·(δr4 − δr2)

(b) Solution of the adjoint problem (5.20) in thez- direction with the torso boundary conditionnT · (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,

maximale value of 0.02052764 at the boundary

7, minimal value of -0.018597936 at the bound-

ary 2, zero points at the boundaries 4 to 5 and

9

(c) Normale component M∇λ · nT in z- direction,

maximale value of 0.019060474 at the boundary

10, minimal value of -0.018989546 at the bound-

ary 4, zero points at the boundaries 2 and 6

Figure 6.5.: Normal components of the adjoint problem

6 Implementation 55

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 · nTu. 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)

6 Implementation 56

∇ · (M∇u) = δri − δrj in T (6.15)

nT · (M∇u) = 0 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

H(t) =

∫∂H

M∇λ · nTuds. (6.17)

We approximate the heart vector on the heart boundary by taking the weighted sum of

the integrand M∇λ · nTu, 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

H(t) =

∫∂H

M∇λ · nTu(t)dt =∑k

∫Tk(H)

M∇λ · nTu(t)dt (6.18)

=∑k

lij∑j

wjM(xkj)∇λ(xkj) · nT (xkj)u(xkj), (6.19)

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

the heart boundary Hδ∂H and the disturbed heart vector on the body boundary Hδ

∂B

as sum of the heart vector and a disturbed functional δH∂H ,δH∂B, respectively.

Hδ∂H = H∂H + δH∂H , (7.2)

Hδ∂B = H∂B + δH∂B. (7.3)

7 Results 58

Let us examine the norm of the difference between the disturbed heart boundary heart

vector and the disturbed body boundary heart vector:

∥∥Hδ∂B −Hδ

∂H

∥∥ = ‖H∂B + δH∂B −H∂H − δH∂H‖

= ‖δH∂B − δH∂H‖

≤ ‖δH∂B‖+ ‖δH∂H‖ . (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

∥∥Hδ∂B −Hδ

∂H

∥∥ ≤ 2 · ‖δH∂B‖ . (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

∥∥∥∥Hδ∂B −Hδ

∂H

H∂B

∥∥∥∥ ≤ 2 ·∥∥∥∥δH∂B

H∂B

∥∥∥∥ = 2 ·∥∥∥∥ δ∂B · uL−1 · u

∥∥∥∥ (7.6)

≤ 2 · ‖δ∂B‖‖L−1‖

. (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

∥∥∥∥Hrec∂B −HF

∂B

H∂B

∥∥∥∥ =

∥∥∥∥δHF∂B

H∂B

∥∥∥∥ ≤∥∥δF∂B · u∥∥‖L−1 · u‖

≤∥∥δF∂B∥∥‖L−1‖

. (7.8)

7 Results 59

We have rewritten the Frank heart vector HF∂B as sum of the reciprocity heart vector

Hrec∂B and a disturbance δHF

∂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

rd =‖Hrec‖‖D‖

−∥∥HF

∥∥‖D‖

. (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.

7 Results 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 z-

component. The maximal difference in the x- component is 0.2853037 and in the z-

component -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

7 Results 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

7 Results 62

7.3. Heart Vector with Frank Lead Matrix

Now we want to analyze Frank’s lead matrix LFrank in term of the linear functional

Strategy,

LFrank =

(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, Fig-

ure 7.4.a. The maximal difference in the x- component is 0.23576696 and in the z-

component -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

7 Results 63





Figure 7.4.: heart vector components for potential differences Vx = VA − VI and Vz =VM − VE

7 Results 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 LFrank 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- com-

ponent 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 andVz = 0.133VA + 0.736VM − 0.264VI − 0.374VE − 0.231VC

7 Results 65





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

7 Results 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 approxi-

mates 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 leadmatrix 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 z-

component 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

7 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.



Figure 7.8.: relative difference for the x- and z- component for the reciprocity leadmatrix 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 evalu-

ated 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.

7 Results 68



Figure 7.9.: relative difference for the x- and z- component for the Frank lead matrixwith 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 differen-

tiate 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 con-

figuration, 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 on ∂Hv (8.3)

u = 0 on ∂Hd. (8.4)

We neglect the heart potential on a range from 2 per cent to 26 per cent of the circum-

ference 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 Pp, QRS-peak QRSp and T-peak Tp position in the vector

magnitude HV =√H2x +H2

z , with Hx, Hz the heart vectors in x- and z- direc-

tion, respectively. Then the maximal amplitude over the P-, QRS-, T-interval

is

Pamp = Hx(Pp) +Hz(Pp), (8.5)

QRSamp = Hx(QRSp) +Hz(QRSp), (8.6)

Tamp = Hx(Tp) +Hz(Tp). (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 =√QRS2

x +QRS2z , (8.8)

analog for AP and AT .

• spatial ventricular gradient: SV G

It is the vectorial QRST-integral

SV G =√

(QRSx + Tx)2 + (QRSz + Tz)2. (8.9)

• spatial mean QRS-T angle: SMQRS − TThe angle between the QRS area vector and the T area vector

SMQRS − T = cos

(QRSx · Tx +QRSz · Tz

AQRS · AT

)−1

. (8.10)

• spatial QRS-T angle: SPQRS − TThe angle between the maximal QRS vector and the maximal T vector

SPQRS − T = cos

(Hx(QRSp) ·Hx(Tp) +Hz(QRSp) ·Hz(Tp)

|RP | · |TP |

)−1

, (8.11)

with |RP | =√Hx(QRSp)2 +Hz(QRSp)2 and |TP | =

√Hx(Tp)2 +Hz(Tp)2.


• modified SVG: mSV G

mSV G =√

(QRSx + Tx + Px)2 + (QRSz + Tz + Pz)2. (8.12)

• spatial mean QRS-P angle: SMQRS − PAngle between the QRSarea vector and the Parea vector

SMQRS − P = cos

(QRSx · Px +QRSz · Pz

AQRS · AP

)−1

. (8.13)

• spatial QRS-P angle: SPQRS − PAngle between the maximal QRS vector and the maximal P vector

SPQRS − P = cos

(Hx(QRSp) ·Hx(Pp) +Hz(QRSp) ·Hz(Pp)

|RP | · |PP |

)−1

, (8.14)

with |PP | =√Hx(Pp)2 +Hz(Pp)2.

• vector-loop length: lP , lQRS, lT

Length of the three different vector loops,

lP =√

(xi − xi+1)2 + (zi − zi+1)2, (8.15)

analog for the QRS-loop and T-loop.

• relative Areas: relAP , relAQRS, relAT

relAQRS =AQRSlQRS

. (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


(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% diseasedheart (blue) with 7 of 18 parameters inside the corresponding interval





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


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 P-

loop 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


79

9. Conclusion and Outlook

In the presented work, we examined the heart vector and the corresponding vectorcar-

diogram 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 lin-

ear 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 repre-

sentation for the heart vector on the dual space of H1(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 prop-

erly 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 in-

verse 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 po-

tential 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 imple-

mentation 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

∫∂B

l−1uT (δri − δrj)ds(5.30)=

∫∂B

MT∇λ2 · nTuTds

∂T=∂B∪∂H=

∫∂T

MT∇λ2 · nTuTds+

∫∂H

MT∇λ2 · nHuTds

(5.29)(4.12)=

∫∂T

MT∇λ2 · nTuTds+

∫∂H

(Mi +Me)∇λ1 · nHueds

PI=

∫T

∇ · (MT∇λ2) · uT︸︷︷︸(5.27)

= 0

dx+

∫T

MT∇λ2∇uTdx

+

∫∂H


= −∫T

∇ · (MT∇uT ) · λ2︸︷︷︸(4.11)

= 0

dx+

∫∂T

MT∇uT · nTλ2ds

+

∫∂H


∂T=∂H∪∂B=

∫∂B

MT∇uT · nTλ2︸︷︷︸(4.15)

= 0

ds+

∫∂H

MT∇uT · nTλ2ds

+

∫∂H


(4.14)(5.31)= −

∫∂H

Me∇ue · nHλ1ds+

∫∂H


PI= −

∫H

∇ · (Me∇ue) · λ1dx−∫H

Me∇ue∇λ1dx

+

∫∂H


(4.10)=

∫H

∇ · (Mi∇ν) · λ1dx+

∫H

∇ · (Mi∇ue) · λ1dx

−∫H

Me∇ue∇λ1dx+

∫∂H


A Appendix 82

PI=

∫∂H

Mi∇ν · nHλ1 +Mi∇ue · nHλ1︸︷︷︸(4.16)

= 0

ds

−∫H


Mi∇ue∇λ1dx

−∫H

Me∇ue∇λ1dx+

∫∂H


= −∫H


(Mi +Me)∇ue∇λ1dx∫∂H


PI=

∫H

∇ · (Mi∇λ1) · νdx−∫∂H

Mi∇λ1 · nHν︸︷︷︸(5.28)

= 0

ds

+

∫H

∇ · ((Mi +Me)∇λ1) · ue︸︷︷︸(5.26)

= 0

dx

−∫∂H

(Mi +Me)∇λ1 · nHueds+

∫∂H


=

∫H

∇ · (Mi∇λ1) · νds

A Appendix 83

A.1. Moore-Penrose-Inverse

First of all we want to define define the range, the kernel and the orthogonal comple-

ment

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

‖Af − g‖Y ≤ ‖Aφ− g‖Y 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: ‖Af − g‖Y ≤ ‖Aφ− g‖Y 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+∥∥X< ‖φ‖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.


If g ∈ D(A+), then f+ = A+g is the unique solution of the equation A∗Af = A∗g in

N(A)⊥.


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.


The Moore-Penrose-Axiom define the Moore-Penrose-Inverse A+ of A ∈ L(X, Y )

clearly:

AA+A = A, A+AA+ = A+, (A.3)

A+A = PR(A∗), AA+ = PR(A). (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 equa-

tions 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 ‖.‖ : X → < is called norm on X, if for all

x, y ∈ X and λ ∈ <

• ‖x‖ ≥ 0 and ‖x‖ = 0⇒ x = 0,

• ‖λx‖ = |λ| ‖x‖,

• ‖x+ y‖ ≤ ‖x‖+ ‖y‖.

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 ‖.‖ : X → < the corresponding norm. The pair (X, ‖.‖)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 ε > 0 an index

N(ε) ∈ N such that ‖xn − xm‖ < ε.

Definition A.2.4 (Banach space,[11] p.635):

A normed vector space (X, ‖.‖) 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 〈., .〉 : X ×X → R is called scalar product or inner

product if for every xi, y ∈ X and λ ∈ R,

1. 〈x1 + x2, y〉 = 〈x1, y〉+ 〈x2, y〉,

2. 〈λx, y〉 = λ 〈x, y〉,

3. 〈x, y〉 = 〈y, x〉,

4. 〈x, x〉 ≥ 0,

5. 〈x, x〉 = 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

that ‖.‖ =√〈., .〉.

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‖ = supx∈X\0

|l(x)|‖x‖

= supx∈X,‖x‖≤1

|l(x)| = inf C > 0|l(x) < C ‖x‖ ,∀x ∈ X (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) = 〈xl, x〉 ∀x ∈ X. (A.6)

On the other hand for every y ∈ X, is ly(x) = 〈y, x〉 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

Lp(Ω) =

u : Ω→ R|u measurable,

∫Ω

|u|p dx <∞. (A.7)

The Lebesgue norm for p 6=∞ is defined as

‖u‖p =

(∫Ω

|u|p dx)1/p

, (A.8)

in the case p =∞ the norm is defined as

‖u‖∞ = esssupx|u(x)| . (A.9)

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 C∞0 (Ω), the space of infinitely differentiable functions with compact support. If

we multiply a function φ ∈ C∞0 (Ω) with a function u ∈ C1(Ω) and apply integration

by parts we obtain ∫Ω

uφxidx = −∫

Ω

uxiφdx, (A.10)

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 ∈ C∞0 , then ∂xαφ ∈ C∞0 is defined as

〈∂xαφ, ϕ〉 := (−1)|α| 〈φ, ∂xαϕ〉 , (A.11)

and is called the weak derivation of order |α| of the function φ.

Remark

For a function u ∈ Ck(Ω) the weak derivation is defined as

〈φ, ϕ〉 :=

∫Ω

uϕdx,

and u holds

〈∂xαφ, ϕ〉 = −〈φ, ∂xαϕ〉 = −∫

Ω

u∂xiϕdx =

∫Ω

(∂xiu)ϕdx.

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)|∂xju ∈ Lp(Ω, j = 1, ...d)

(A.12)

with the Sobolev norm

‖u‖1,p :=

(‖u‖pLp(Ω) +

d∑j=1

∥∥∂xju∥∥pLp(Ω)

)p

(A.13)

Remark([11] p.245)

If p = 2, we usually write

Hk(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 H10 (Ω) := W 1,2

0 (Ω) the closure of C∞0 (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 C1. Then there exists a bounded linear operator

T : W 1,p(Ω)→ Lp(∂Ω) (A.15)

such that

1. Tu = u|∂Ω if u ∈ W 1,p(Ω) ∩ C(Ω),

2. ‖Tu‖Lp(∂Ω) ≤ C ‖u‖W 1,p(Ω),

for each u ∈ W 1,p(Ω), with the constant C depending only on p and Ω.

90

List of Figures

1.1. heart anatomy,[21] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2. transmembrane potential of an excited cardiac muscle cell of a frog,[21] 4

1.3. The conductivity system of the heart,[21] . . . . . . . . . . . . . . . . . 5

1.4. The Electrocardiogram . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5. The electrode position of a 12-lead ECG . . . . . . . . . . . . . . . . . 7

2.1. Illustration of the geometry of a dipole,[3] . . . . . . . . . . . . . . . . 9

2.2. Einthovens’ triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3. Franks lead system and dipole position . . . . . . . . . . . . . . . . . . 14

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] . . . . . . . . . . . . . . . . 17

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 . . . . . 33

5.2. Relation heart vector and adjoint heart vector . . . . . . . . . . . . . . 33

5.3. Inverse adjoint Bidomain Problem . . . . . . . . . . . . . . . . . . . . . 36

6.1. Torso geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

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) 51

6.3. Mesh of model geometry, 1600 mesh points, 2952 triangles, 33760 num-

ber of degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.4. Solution of the adjoint problem . . . . . . . . . . . . . . . . . . . . . . 54

6.5. Normal components of the adjoint problem . . . . . . . . . . . . . . . . 54

6.6. Torso geometry with 28 dipole positions . . . . . . . . . . . . . . . . . 55

List of Figures 91

7.1. vector loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.2. heart vector components . . . . . . . . . . . . . . . . . . . . . . . . . . 61

7.3. vector loop for potential differences Vx = VA − VI and Vz = VM − VE . . 62

7.4. heart vector components for potential differences Vx = VA − VI and

Vz = VM − VE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

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 . . . . . . 64

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 . 65

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

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 . 72

8.3. VCG and potential differences of a healthy heart (red) and a 16% dis-

eased heart (blue) with 0 of 18 parameters inside the corresponding

interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72



interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73



interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

8.5. partitioned heart into sectors of about 10% . . . . . . . . . . . . . . . . 74

8.6. part 1, 10.0676% diseased . . . . . . . . . . . . . . . . . . . . . . . . . 75

8.7. part 2, 10.04227% diseased . . . . . . . . . . . . . . . . . . . . . . . . . 75

8.8. part 3, 10.22121% diseased . . . . . . . . . . . . . . . . . . . . . . . . . 76

8.9. part 4, 10.28912% diseased . . . . . . . . . . . . . . . . . . . . . . . . . 76

List of Figures 92

8.10. part 5, 10.10239% diseased . . . . . . . . . . . . . . . . . . . . . . . . . 76

8.11. part 6, 9.988884% diseased . . . . . . . . . . . . . . . . . . . . . . . . . 77

8.12. part 7, 10.09401% diseased . . . . . . . . . . . . . . . . . . . . . . . . . 77

8.13. part 8, 10.1573% diseased . . . . . . . . . . . . . . . . . . . . . . . . . 77

8.14. part 9, 10.0166% diseased . . . . . . . . . . . . . . . . . . . . . . . . . 78

8.15. part 10, 10.07034% diseased . . . . . . . . . . . . . . . . . . . . . . . . 78

93

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