Principal component analysis of T-waves

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2007:230 CIV

M A S T E R ' S T H E S I S

Principal Component Analysis of T-waves

Ola Gustavsson

Luleå University of Technology MSc Programmes in Engineering

Electrical Engineering

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Principal Component Analysis of T waves

Ola Gustavsson

October 10, 2007

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Preface

This master’s thesis was carried out in the spring of 2007, at the Depart- ment of Biomedical Engineering and Informatics (Medicinsk teknik MT) at the University Hospital of Ume˚a. MT and the University of Ume˚a have a close cooperation in research and developement in the areas of biomedical mechanics, signal processing and sensor development.

My particular gratitudes goes to Urban Wiklund and Milos Kesek for introducing me to this area of study and for helping me understand it.

Your great support and positive attitudes have inspired me troughout this work. I am also very thankful for the help Marcus Karlsson and Urban Edstr¨om have given me in solving certain hardware and software problems that arose during the course of this study. One more helpful person has been Jennifer Frankel, my office mate, who gave me a lot of general feedback regarding this work.

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Abstract

The mechanical work of the heart is triggered by electrical activity in the muscle fibres. The potentials from individual muscle fibres interfere with each other to produce an electrical field which can be examined by recording potential differences on the body surface with an electrocardiogram (ECG). An ECG contains several wave-components, each describing one particular sequence of the heart cycle. The so-called T wave in the ECG contains information about the electrical recovery phase (repolarization).

Disturbances in the repolarization phase may increase the risk of fatal arrhythmias. Thus, markers which indicate repolarization disturbances would be of great clinical value.

In an earlier study, principal component analysis (PCA) of the T wave in a 12-channel ECG, showed differences between groups of patients with different degrees of repolarization disturbances.

However, the method is not sensitive enough to be used on single individuals. In order to make the method more sensitive, one hypothesis is to increase the number of ECG channels and the sampling frequency. Testing this hypothesis requires a software tool and unconventional measurement equipment.

This master’s thesis considers implementation and simulation tests of a software tool for PCA of the T wave in multi-channel ECGs. The practical aspect of this study consisted of building a connection block which gives extra surface ECG channels to an apparatus for invasive measurements.

Before PCA can be performed, the ECG signals must pass a number of preprocessing steps. Much of the effort in this study re- volved around defining the time instances when the T wave starts and ends. Three algorithms for defining the T wave onset and four algorithms for defining the T wave end where implemented and evaluated in a simulation study. In addition, a test was performed in which simulated data were reproduced according to an earlier simulation study. The reproduced data were analyzed by the implemented software tool and compared with the previous study.

The results from the simulation study suggested that a combination of a T-onset algorithm based on the heart rate interval and an area-based T-end algorithm is the most robust in sense of noise and wave-shape abnormalities. The test with the reproduced data showed similar results as the earlier simulation study.

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List of Abbreviations

ECG (Electrocardiogram)

LSI (Least square slope intersection) LQTS (Long QT syndrome)

MRI (Magnetic resonance imaging) NDPV (Non dipolar voltage)

PCA (Principal component analysis) PC (Principal component)

RMS (Root mean square) SI (Slope intersection) TA (T wave area)

TWRabs (T wave residue absolute) TWRrel (T wave residue absolute)

WTMM (Wavelet transform modulus maxima)

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1 Introduction

1.1 Background

An electrocardiogram (ECG) reflects the electrical activity of the heart, and the so-called T wave in the ECG contains information about the electrical recovery phase (repolarization). Disturbances in the repolarization phase may cause an increased risk of arrhythmia, and markers are needed to discriminate high-risk patients from low-risk patients.

Kesek [1] found differences in the principal components (PCs) of the T wave between healthy subjects and subjects with various types of ischemic heart disease. In that study, standard 12-lead ECG equipment with 240Hz sampling frequency was used. However, the method is not sensitive enough to be used on single individuals. A hypothesis to achieve better results is to increase the number of leads and sampling frequency. Testing this hypothesis required a software tool, and unconventional equipment for measuring multi-channel ECGs.

1.2 Purpose

The specific aim of this master’s thesis was to develop a software tool for PCA of T waves measured with up to 32-channel high-resolution ECGs.

This work consisted of:

• Implementation of detection rules for T waves.

• Adaption of hardware for invasive cardiological measurements to serve as expansion channels to standard surface ECG.

• Development of software for analysis of data in Matlab, including a graphical user interface.

• Evaluating the software in a simulation study.

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2 The ECG

The ECG was invented in the beginning of the 20th century and has developed into a standard routine in clinical work today. Certain heart diseases give rise to typical abnormalities in the ECG, which yields important diagnostic information. Some basic information useful for un- derstanding the ECG will be given in the following sections.

2.1 The electrical activation of cardiac muscle fibers

The information about the action potential, which is briefly presented here, is gathered from Chapter 20 in [11].

The myocardium consists of a great number of muscle fibers which are triggered to contract by electrical activation. The time pattern of the activation and repolarisation is known as an action potential and can be found in muscle fibers and heart cells. At resting state, the inside of a fiber has a potential difference of approximately –90 mV with respect to the outside. When a fiber is brought to a threshold by action potentials from surrounding fibers, its Na⁺ ion channels open. The concentration of Na⁺within the fibers rises rapidly and changes the potential difference to +20 mV. This is called depolarization and comprises the first phase of the action potential. When the inflow of the Na⁺ decreases, the depolarized state is maintained at a plateau due to an inflow of Ca²⁺, which causes the fiber to contract. After approximately 300 ms, the inflow of Ca²⁺stops, and K⁺channels open and an outflow of K⁺begins.

The potential difference is thus restored to its initial state –90 mV. This is called repolarization and is the final phase of the action potential.

0 50 100 150 200 250 300 350 400

−100

−80

−60

−40

−20 0 20 40

Membrane Potential [mV]

Time [ms]

Cardiac Action Potential

Figure 2.1: Shows a cardiac action potential

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2.2 ECG waves

The action potentials produced in myocardial fibers interfere with each other to produce a common electrical field. This field spreads through the body tissues and can be examined by recording the potentials on the body surface. The recorded signal, known as the ECG, has a characteristic waveform (Fig. 2.2). The information in this section is mainly a summary from [11].

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−1

−0.5 0 0.5 1 1.5

Time [s]

Amplitude [mV]

T

Q S

R

P

Figure 2.2: An ECG-complex.

The heart has four chambers, two upper chambers (atria) and two lower chambers (ventricles). The electrical activation of these chambers is reflected in the ECG as different waves. The P wave reflects the depolarization of the atria and the QRS complex represents the depolarization of the ventricles. Since the ventricles consist of more muscle fibers and produce a stronger potential field than the smaller atria, the repolarization phase of the atria is not visible in the ECG. This is due to the repolarization wave getting hidden behind the QRS complex [9]. The T wave represents the ventricular repolarization.

2.3 Repolarization inhomogeneity

One potential factor behind fatal arrhythmias is inhomogeneous repolarization, which may be manifested in duration differences of action potentials in the myocardium. The differences are known to be caused

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A prolongation of the action potentials in ventricular fibers may be observed in the ECG as a prolonged QT time (i.e. the time duration between the begining of the Q wave and the end of the T wave). Long-QT Syndrome (LQTS) is a diagnosis properly named after this fact. The diagnosis LQTS means a change of lifestyle for the affected person. Chronic medication and restricted hard physical exercise are often necessary.

2.4 Lead systems

Different electrode setups have been defined depending on practical con- siderations and what clinical information that has to be obtained. The 12-lead ECG is the most frequently used ECG lead system in clinical routine today and constists of a combination of both bipolar and unipolar leads. The following part is a summary of Chapter 6 in [9].

In the beginning of the 20th century, a physiologist named William Einthoven measured the potential differences between both arms (lead I), between the left leg and right arm (lead II ) and between the left leg and left arm (lead III ) (Eq. 2.1, 2.2, 2.3).

I = VLA− V^RA (2.1)

II = VLL− V^RA (2.2)

III = VLL− V^LA (2.3)

This set up of electrodes is called the Einthoven triangle and consti- tutes the first three components in a 12-lead ECG. The next three leads in the 12-lead ECG are called the augmented leads and are also derived from the Einthoven triangle. One augmented lead is defined as the potential difference between one electrode and the mean value of the other two as shown in (Eq. 2.4, 2.5, 2.6).

aV R = VRA− VLA− V^LL

2 (2.4)

aV L = VLA−VRA− V^LL

2 (2.5)

aV F = VLL−VLA− V^RA

2 (2.6)

The last six leads (precordial leads) are placed on the front left side of the chest as shown in Fig. 2.4. The potential difference is measured between these electrodes (VEn where n = 1, 2, . . . , 6) and the mean value of the potential of the left arm, right arm and left leg. This electrode setup is usually called the Wilson central terminal (see Eq. 2.7) and provides a stable reference during the whole heart cycle.

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Vn= VEn−VLA− V^RA− V^LL

3 (2.7)

Figure 2.3: Shows a 12-lead electrode setup.

2.5 The dipolar concept

At a given instant in time, groups of cells in the myocardium are depolarized or repolarized. The potential field generated by each group of cells can be summed into a resulting dipole vector describing the main direction of the electrical impulse [9]. The interpretation of the electrical source as a dipole vector is useful when describing ECG generation. A short description follows [3]: The ECG is proportional to the projection of the resulting vector onto a lead vector whose direction is determined by the position of the heart and electrodes (see Figs. 2.4 and 2.5). It is worth stressing that this model is only an approximation, although sur- prisingly accurate. The inhomogeneous conductivity of the torso and the fact that the source, i.e. the heart, has a significant size in comparison to the torso, are two limitations of this model [1].

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Figure 2.4: The dipolar source vector with its projection onto the Einthoven lead vectors. Inspired by Fig. 6.3 in [9] and Fig. 7 in [10].

The movement pattern of the electrical field has a characteristic look, contains three different loops, and each can be associated with one particular wave in the ECG (see Fig. 2.5). The smallest loop corresponds to the P wave, the middle-size loop to the T wave and the largest loop to the QRS complex.

Figure 2.5: Shows the movement pattern of the dipolar vector. Inspired by [3]

.

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3 Theory

3.1 Principal component analysis

This part is a breief summary of the theory behind principal component analysis from [4] and [5].

The relationship between a p-dimensional set of correlated variables {X¹, X2, . . . , Xp} can be examined by transforming the variables into a new set of uncorrelated variables {Y¹, Y2, . . . , Yp}, called principal components (PC). The new variables are linear combinations of the original variables and are calculated and ordered in a descending manner with respect to their variances σ_Y²₁ > σ_Y²₂ > . . . > σ_Y²_p. This means that the first PC corresponds to the maximal possible variance in the data, the second PC accounts for as much of the variance which the first PC does not explain, and so on. Consider the linear combination below.

Y1 = a11X1+ a12X2+ . . . + a1pXp = ~a^T1X Y2 = a21X1+ a22X2+ . . . + a2pXp = ~a^T2X

...

Yp = ap1X1+ ap2X2+ . . . + appXp = ~a^T_pX

(3.1)

The problem is to determine the constant vectors ~aj j = 1, 2, . . . , p such that:

• ~a¹ maximizes σ²_Y₁ = V ar(Y1)

• ~a²maximizes σ_Y²₂ = V ar(Y2) with the constraint that Cov(Y2, Y1) = 0...

• ~a^pmaximizes σ²_Y_p = V ar(Yp) with the constraint that Cov(Yp, Yn) = 0, where n = 1, 2, . . . , p − 1

Since the constant vectors ~aj in Eq. 3.1, contains an arbitrary scaling factor, an additional condition is that ~aj must be orthonormal, i.e.,

~ ajT

~

aj = 1. This condition is imposed to preserve the distances in the p space. It can be shown that the variances σY2

j and corresponding vector ~aj are equivalent to the eigenvalues λj and corresponding normalized eigenvectors ~ej of the covariance matrix (see Appendix A) of X [4], thus σY2

j = λj and ~aj = ~ej. Geometrically the procedure can be interpreted as a rotation of the p space (see Fig. 3.1)

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−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5 0 0.5 1 1.5 2

X1

X2

λ₁e λ₂e 1

2

Y2 Y

1

Figure 3.1: A bivariate plot with the two vectors λ1e1 and λ2e2 marked in the middle of the data swarm. One can see that the eigenvector associated with the greatest variance points in the direction that accounts for as much as possible of the variance. The second vector points in an orthogonal direction.

3.2 PCA of T waves

A consequence of repolarization inhomogeneity is that the T waves in a multi-channel ECG are less correlated with respect to each other (see Fig. 3.2). PCA is a powerful tool, which can reflect this by considering the set of correlated variables Xj, j = 1, 2, . . . , p in the previous section, as a set of multi-channel T-wave data {x¹(n), x2(n), . . . , xp(n)}. PCA can thereafter be performed as usual.

The rate in which the variance of the PCs falls off depends on the degree of independence between the T waves in multiple leads. As the T waves become less correlated, the fall-off rate of the ordered variances of the PCs becomes slower (Fig. 3.3). The calculated variances are in signal analysis terms interpreted as energy. Figure 3.4 shows an example of two correlated T waves and two less correlated T waves and their corresponding T loops (Fig. 3.5). The less correlated the T waves are the wider their corresponding T loop is.

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0 50 100 150 200

−0.5 0 0.5 1

Normal T−waves

time [ms]

Amplitude [mV]

0 50 100 150 200 250

−1

−0.5 0 0.5

Uncorrelated T−waves

time [ms]

Amplitude [mV]

Figure 3.2: The upper figure shows an example of normal T waves and the lower uncorrelated T-waves. These waves were simulated in a program called ECGSIM (see Section 6)

1 2 3 4 5 6 7 8

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

PC−number PC−value [uV2]

Normal T−waves Uncorrelated T−waves

Figure 3.3: Shows the variances of the principal components calculated from the T wave amplitude data in Fig. 3.2. The fall-off rate of the uncorrelated T wave data is slower than the correlated T wave data.

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0 50 100 150 200 250 0

0.5 1 1.5 2

Correlated T−waves

Time [ms]

Amplitude [mV]

0 50 100 150 200 250

−0.2 0 0.2 0.4 0.6 0.8

Less correlated T−waves

Time [ms]

Amplitude [mV]

Figure 3.4: The upper figure shows an example of two correlated T waves and the lower figure shows two less correlated T waves.

−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

−0.2 0 0.2 0.4 0.6 0.8 1 1.2

T−loops

Amplitude [mV]

Correlated T−waves Less correlated

T−waves

Figure 3.5: The T waves in Fig. 3.4 plotted against each other.

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An interpretation of the PCs related to the dipolar source model (see Section 2.5) is usually used. The dipolar content, i.e., the data describing the T wave loop, are reflected by the three largest PCs. If the T wave in a multi-channel ECG would be completely described by a single dipole, the power of the higher order PCs, i.e., λ4− λp, would be zero [1]. The higher order components reflect the nondipolar content of the T wave, i.e., the content which can not represent a pure dipolar source. Based on analysis of more than 100 ECGs [8] found that 98% of the content in a 12-lead ECG can be described by the three largest PCs.

The following parameters have been suggested for quantification of dipolar and nondipolar components [1].

T W Rabs = λ²4+ λ²5+ . . . + λp² [µV⁴] T W Rrel = _λ2 ^{T W Rabs}

1+λ²₂+...+λp² [−]

NDP V =√

λ4+ λ5+ . . . + λp [µV ] S2/S1 = λ2/λ1[-]

T wave residue absolute (TWRabs) and non dipolar voltage (NDPV) are direct measures of the nondipolar part of the signal. T wave residue relative (TWRrel) is a dimensionless measure which explains the ratio between the nondipolar content and the total content. S2/S1 is a measure of the width of the T wave loop. All these measures are assumed to increase by repolarization inhomogeneity.

3.3 Continuous wavelet transform

One of the algorithms described in Section 5.6 uses wavelet-based detection rules. The concept of wavelets is mathematically rather complex and is only explained briefly here. The theory is mainly a summary of chapter 10 in [6].

Wavelet expansion is a powerful mathematical tool for signal analysis which, in similarity to Fourier transformation, expands the observed signal x(t) to orthogonal base functions. In contrast to Fourier series expansion, which uses infinite complex sinusoids as basis functions, the Wavelet transform decomposes the signal in terms of small waves with few oscillations. Equation 3.2 shows the transformation formula.

W xs(τ ) =

Z _∞

−∞

x(t)ψ_s,τ^∗ (t)dt (3.2) where the bases ψ (t) functions are scaled (s) and translated (τ )

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versions of a prototype function, called a mother wavelet ψ(t) Eq. 3.3.

ψs,τ(t) = 1

√sψ

t − τ s

, s > 0 (3.3)

The constraint 1/√

s in Eq. (3.3) is to normalize the energy content of the wavelets such that

Z _∞

−∞|ψ^s,τ(t)|² =

Z _∞

−∞|ψ(t)|² = 1 (3.4)

Substituting (3.3) into (3.2) gives W xs(τ ) = 1

√s

Z _∞

−∞

x(t)ψ

t − τ s

dt, s > 0 (3.5) A major difference between wavelet bases and Fourier bases is that Wavelet bases are limited in time. This property provides possibilities to decompose a signal in both time and frequency simultaneously.

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4 Measurement equipment

Two sets of measurement equipment were available at the Department of Clinical Physiology at the University Hospital of Ume˚a: a newer EP tracer 102 and an older EP4, both manufactured by CardioTek Maas- trich. Both measurement systems consist of an ampifier which is connected to an ordinary personal computer. The amplifiers both have surface ECG channels and intracardiac channels. The latter are used for catheter measurements of the electical activity inside the heart. These intracardiac channels served as extra surface ECG channels in the present work. To connect conventional ECG cables to the intracardiac channels some modifications had to be made. This section presents information about the measurement systems and how the modifications were made.

4.1 EP tracer 102

The EP tracer 102 has a standard 12-lead surface ECG and 84 intracardiac inputs. The sample frequency is 1 kHz and the resolution is 12-bit, but the system stores the data in a 16-bit format. The Wilson central terminal stands as reference for both the precordial leads and the intracardiac channels.

4.2 EP4

The EP4 is an older product from CardioTek and has a 12-lead surface ECG, but in contrasst to the Ep-Tracer 102, it only has 10 intracardiac channels. The sample frequency is 1 kHz, and the resolution is 12-bit, but the system stores the data in an 8-bit format. The Wilson central terminal stands as reference for both the precordial leads and the intracardiac channels.

4.3 Cables

20 standard ECG cables with a minor modification of each connector were used. 2 mm banana connectors replaced the original connectors. A red connector was used for the signal and a black connector was used for the shielding (Fig. 4.3).

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Figure 4.1: ECG cables

4.4 Connection box

A connection box (Fig. 4.2) was built for this project, since the original boxes of the EP tracer 102 and EP4 systems had no abilities to connect the shielding of the ECG cables. Cables where drawn from the 25p Dsub connector in the front of the box to the connectors on the topside of the box. The Dsub connector can be connected directly to the amplifiers of the EP tracer 102 and EP4 systems.

Figure 4.2: Connection Box

4.5 Hardware-related problems

50 Hz interference from the power lines could be identified visually in the recorded signals (Fig. 4.3). The interference arose from the fact that the box was made of plastic. This problem was solved by painting the inside of the box with a layer of conducting paint and taping unshielded cables inside the box together with conducting tape. The tape and paint were thereafter connected to ground. The noise levels were significantly decreased by this measure (Fig. 4.4).

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4400 4600 4800 5000 5200 5400 5600 5800

−0.1 0 0.1 0.2 0.3 0.4

Time [ms]

Amplitude [mV]

Figure 4.3: 50Hz interference

5800 6000 6200 6400 6600 6800 7000

−0.1 0 0.1 0.2 0.3 0.4

Time [ms]

Amplitude [mV]

Figure 4.4: After measure.

4.6 Limitations in resolution

Since the storage space was a critical part of the acquisition of ECG data a few years ago, the resolution of the stored data in older systems was relatively low. The EP4 uses a 12-bit AD conversion, but stores the data in an 8-bit format. In the study of Milos Kesek [1], which stands as base for this thesis, a 16-bit AD converter was used. Thus, 8-bit is to be considered too low. A test presented in Section 6.2 shows that the low resolution of the signals affects the result of the PCA.

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5 Implementation

Before PCA can be performed, the ECG signals have to go through a number of processing steps (see Fig. 5.1), all of which are explained in detail below. All implementation was carried out in Matlab.

Figure 5.1: Program layout

5.1 Scaling

The signals recorded by the measurement equipment are stored in formats of 16-bit (EP tracer 102) and 8-bit format (EP4). The digital values must be transformed into proper units, [mV], in order to make the results comparable to other PCA evaluations.

A reference signal with known amplitude generated by a so-called ECG phantom was recorded with the EP tracer 102 and EP4 systems.

The digitized values of the Q tag Qdig and the R tag Rdig were matched with the known reference values ,Qref and Rref, respectively (se Fig. 5.2).

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0 100 200 300 400 500 600 700

−0.2

−0.1 0 0.1 0.2 0.3 0.4 0.5

Time [ms]

Amplitude [mV]

Rdig Rref

Qdig Qref

Figure 5.2: The recorded reference signal in one lead. The R and Q tag were measured in order to calculate a scaling factor.

A scaling factor was determined according to following calculations:

k = Rref − Q^ref

Rdig− Q^dig [mV /LSB] (5.1) The zero level was determined as:

δ = k2ⁿ

2 [mV ] (5.2)

where n was the number of bits (8-bit or 16-bit) of the recorded data.

A new corrected m×n data matrix X^mV was calculated from the digitized matrix Xdig according to:

X_mV = kXdig− δdiag(Im×m) (5.3)

5.2 Reference signal

In this application, a set of time instances in the ECG were defined. A global reference signal over the p-channel ECG, on which these instances were determined, was calculated as:

xvmag(n) =^qx²1(n) + x²2(n) + . . . + x²_p(n) (5.4) where x1(n), x2(n), . . . , xp(n) are the ECG channels. This reference

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5.3 R-peak detection

The occurrence time for each heartbeat, nk, has to be detected in order to perform an analysis of single heartbeats, or as in this application, to calculate a mean/median segment, as will be described later. A fix reference point common for all complexes is usually determined by the R peak, since it is the most dominant wave in the ECG complex (see Fig. 5.3). In this work, a detection algorithm used in an earlier study performed by Wiklund [2], was used.

This algorithm is based on the first difference x^′(n) = x(n + 1) − x(n) of the ECG signal x(n) and makes use of the typical morphology of the R peak, which has a strong positive flank followed by an equally strong negative flank or vice versa. The flanks are strongly represented in the difference signal, which make them easy to detect. The time instance of the R peak is defined as the maximum value of the ECG signal between the pair of flanks. The occurrence time of the positive flank is determined as when the difference signal exceeds 1.5 × sd(x^′(n)) and the negative flank as when the difference-signal falls below −1.5 × sd(x^′(n)), where sd(x^′(n)) stands for the standard deviation of the first difference of the ECG signal x(n). In this application the R peaks were determined on the vector magnitude.

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

0 2 4 6

Vector Magnitude

Time [s]

Amplitude [mV]

Figure 5.3: Shows the vector magnitude with the R peaks marked by the algorithm.

The algorithm returns a vector containing the sample time instances for the R peaks (5.5).

R^T = [n1, n2, . . . , nk] (5.5) The time between two heartbeats is called the RR interval since the time is taken between two R peaks. The RR interval for a single heartbeat can easily be determined from the elements of R^T according to:

RR (n ) = n − n (5.6)

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The time interval is determined as

RR(tk) = RRs(nk) fs

(5.7) where fs is the sample frequency.

5.4 Segmentation

Stochastic noise affects the result in most measurement applications.

This fact is also evident when performing PCA of T waves. A multi- channel T wave data vector X can be written as (5.8), where S is the actual ECG signals and N is a noise vector.

X= S + N (5.8)

The variance of the T waves is preserved in the PCA, and therefore the variance of the noise will affect the PCs. Consider equations 5.9 and 5.10

V ar(X) = V ar(S + N) (5.9)

if S ⊥ N

V ar(X) = V ar(S) + V ar(N) (5.10) Since the measurement parameters are variances, minimization of V ar(N) is necessary to make the measurement results more accurate.

Since the morphology of the complexes in an ECG is rather constant, noise reduction can be performed by calculating a mean, or alternatively, a median complex out of a number of windowed complexes in a recording. The window is defined as RR × 0.20 in the backward direction and RR × 0.80 in the forward direction relative to the R peak (see Fig. 5.4).

These window limits were chosen to involve as much as possible of the phases in a complex from the P wave to the T wave. The R peaks are globally determined over all leads on the vector magnitude. The shortest RR intervall between the chosen complexes is used as reference for the window.

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Figure 5.4: Shows the segments marked in one channel. The segment to the right is the mean/median segment

The segments, in Fig. 5.4 labeled gk(n), are the segments used in the mean/median (i.e., Eq. (5.11)/(5.12)) calculation of the new complex.

All segments are of the same length.

gmean(n) = 1 N

N −1X

k=0

gk(n), k = 1, 2, 3, ..., N (5.11)

gmedian(n) = median[g1(n), g2(n), g3(n), ..., gk(n)] (5.12) These calculations were performed on all channels with the vector magnitude included.

5.5 T-wave detection

Since statistical basic patient data may consist of a great number of subjects an automatic method for determination of the T wave’s onset and end is necessary for a convenient use of the analysis program. Conve- nience is not the only argument for automatic T-wave determination.

A consistent determination of the onset and end point is also desired since a manual decision would vary between different users and between measurements by the same operator. The following sections describe the algorithms which were used in this work.

Before the description of the algorithms begins, some fundamental time points, marked in Fig. 5.5, need to be explained.

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200 400 600 800 1000 1200

−0.5 0 0.5 1 1.5 2 2.5 3

time [ms]

Amplitude [mV]

RR

T_peak

T_end T_onset J

R−peak R−peak

Figure 5.5: Shows an ECG complex. The arrows point out a set of important time points in the complex that will be explained below.

5.5.1 J point

The J point defines the end of the QRS complex. Acar et al. [7] approximated this point as the first point after the R peak that falls below a threshold defined as 70% of the maximum value of the reference signal. The reference signal in [7] is calculated as the vector magnitude of the three largest principal components of the multiple ECG signals. A similar approach is used in this application with the difference that the reference signal is defined as the vector magnitude of all channels (see section 5.2).

5.5.2 T peak

The T peak is defined as the time instance at the first maximum point after the R peak. The maximum point is found on the vector magnitude segment. A time window, wherein the search for the maximum point is performed, is defined from J point+50 ms to the end of the vector magnitude complex.

5.5.3 T onset

Three different algorithms for approximating the T-onset point were implemented. Two of these algorithms are presented in this section and the third in section 5.6. The ideas for the following definitions were taken from [1] and [7].

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The definitions of the two algoritms are calculated as Tonset1 = J +RR

16 (5.13)

Tonset2 = J +1

3(Tpeak− J) (5.14)

5.5.4 T end

Several ways of determining the T-wave endpoint have been described in various articles. Xue and Reddy [8] explored five different ways to define the T end. The algorithms in this work are originally adapted to measure the T-wave endpoint for QT dispersion (QTd), which is another measure for repolarization disturbances. However, the procedure to find the T- wave endpoint works well in this application as well. In QTd the T-end is found on all channels, but in this application T-end is defined, globally over all leads, on the vector magnitude. Three algorithms, named sloe intersection (SI), least square slope intersection (LSI) and T-wave area (TA), are described in this section and the fourth (WTMM) is explained in section 5.6.

The SI algorithm uses the maximum slope as a reference to find T end.

An isoelectric line is defined as the median of the third quartile of the RR interval of the vector magnitude segment. The T end is defined as the intersection point between the maximum slope line and the isoelectric line. This algorithm is sensitive to noise, since it is based on finding the least value of the derivative within a predefined window of the observed signal. A fifth order Butterworth low-pass filter with break frequency at 50Hz is therefore applied on the ECG signal before the least value of the derivative is found.

−100 0 100 200 300 400 500 600 700 800

−0.5 0 0.5 1 1.5 2

Time [ms]

Amplitude [mV]

T−end Maximum−

Slope

Isoelectric line

Figure 5.6: Tangent-based T-end algorithm (SI).

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The LSI algorithm is identical to the SI algorithm, except that the maximum slope line is here interpolated around 10 data points in the forward and backward direction relative to the maximum slope point.

See Figure 5.5.4.

−100 0 100 200 300 400 500 600 700 800

−0.5 0 0.5 1 1.5 2

Time [ms]

Amplitude [mV]

T−end

Isoelectric line Least−Square fit Line

Figure 5.7: Least-square tangent-based T-end algorithm (LSI)

TA defines T end as the point were the magnitude value of the T wave falls under 1% of the integrated area from T onset according to 5.15

xvmag(n)

Pn

Tonset(xvmag(n) − isoelectricline) × δ < 1% (5.15) The isoelectric line is defined in the same way as for the SI algorithm and δ is the sample rate.

−100 0 100 200 300 400 500 600 700 800

−0.5 0 0.5 1 1.5 2

Time [ms]

Amplitude [mV]

< 1%

Isoelectric line

Figure 5.8: Area based T-end algorithm (TA)

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5.6 Wavelet based T-wave detection

The detection rules, named Wavelet Transform Modulus Maxima WTMM in this work, originate from [12] and [13]. Both authors propose methods for delineation of T waves in the ECG based on a multiscale wavelet approach. The local maxima lines of the wavelet coefficients across different scales are detected and used as base for the decision rules.

It can be shown [13] that if the wavelet is the first derivative of a smoothing function θ(t), than the wavelet transform (3.2) can be written as:

Ws(τ ) = −s d dτ

!Z _∞

−∞

x(t)θs(t − τ)dt (5.16) Where θs(t) = ^√¹_sθ(t). The smoothing function, θ(t), can be seen as the impulse response of a low-pass filter. Equation 5.16 means that the wavelet transform of the observed signal at scale s is proportional to the first derivative of the observed signal x(t) convolved with the smoothing function θs(t). The first order derivative of a Gaussian smoothing function (see Fig. 5.9) was chosen in this work. The implementation was carried out using the Continuous Wavelet Transform (CWT) package of Matlab.

−5 0 5

−50 0 50 100

Smoothing function θ(t)

−5 0 5

−1

−0.5 0 0.5 1

Wavelet ψ(t)

Figure 5.9: Gaussian Wavelet.

The wavelet-based algorithms suggested by [12] and [13] are both adapted to detect all time instances of the T onset and the T end in an ECG recording. The proposed algorithms support classification rules that involves several scales and sorting algoritms to find the modulus maxima lines. The complexity of the algorithms is thus quite high.

In this application, only one T wave is found on a vector magnitude segment. In addition, the noise level of the vector magnitude segment is lower than in an ECG, which makes the detection less complex. Hence, a simplification is motivated. Samar et al [14] propose a method in which only one wavelet scale is used and which still achieves good results. A

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The algoritm implemented in this work detects the local maximas of the absolute value of the Wavelet coefficients W10x(τ ). The T onset and the T end are defined as the points at which the coefficients fall below limits based on the Root Mean Square (RMS) of the wavelet coefficients W10x(τ ). The T onset point is defined as that point at which the value of W10x(τ ) falls under 0.25RMS(W10X(τ )) to the left of the first maximum and T end as the point at which Wsx(τ ) falls under 0.40RMS(W10X(τ )) to the right of the second maximum (see Fig. 5.10). These limits were suggested by [13]. The local maxima are found within a time window defined from the J point to the end of the complex.

0 100 200 300 400 500 600 700 800

0 1 2 3 4

Vector Magnitude

Amplitude [mV]

0 100 200 300 400 500 600 700 800

0 2 4 6 8 10

Time [ms]

|WSx(τ)|

0.40*RMS(W_Sx(τ)) 0.25*RMS(W

Sx(τ))

Time−Window

Tonset T

end

Figure 5.10: The upper graph shows the vector magnitude and the lower shows the absolute value of the wavelet coefficiens at scale S = 10. The singularities are marked as ”+” signs.

5.7 PCA step

When the T onset and the T end are found, all the T waves in the corresponding channels are ”cut-out” at the time instances defined by the T onset and the T end. The PCs are therafter found by performing Singular Value Decomposition directly on the T wave data matrix. This method was used in [1]. The different constellations of measurement parameters are, however, defined in the same way as in section 3.2.

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5.8 Graphical user interface

A graphical user interface was added for convenience. It provides possibilities for the user to mark the set of complexes which should be analyzed.

In addition, the mean/median complexes can be viewed together with the result of the different PCA parameters. The result parameters can be saved in a format which enables further analysis in Microsoft Excel.

The program supports file formats generated by ECGSIM (.ecg, .bsm), the EP4-system (.000) and the EP-Tracer 102 system (.000). Figure 5.11 shows the layout.

Figure 5.11: The main window of the GUI.

The numbers marked in Figure 5.11 are described below.

1. The selected complexes which can be marked by clicking the vector- magnitude graph.

2. The mean/median complex.

3. Result parameters.

4. Save buttons.

5. Drop down menu for selecting which channel that shall be presented

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6. Zoom buttons.

7. The patient data stored in the EP-Tracer files.

8. Time slider.

9. The three largest PCs in a 3D plot.

A settings panel was implemented to give the operator the possibility to choose which detection algorithm that would be used and the size of the time window for the analysis. The operator can also choose median or mean noise reduction (see Fig. 5.12).

Figure 5.12: An example of the settings panel

Below follows a description of the numbers marked in Figure 5.12.

1. Time-window duration.

2. Choice whether mean or median calculations shall be performed on the complexes.

3. Choice of T-onset algorithm.

4. Choice of T-end algorithm.

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6 Simulation study

Van Oosterom and Oostendorp [15] developed an interactive simulation tool for studying the genesis of the QRST waveforms based on action potential properties using a more reliable model than the dipolar source.

The program, ECGSIM (www.ecgsim.org), computes the forward trans- fer between the heart and the torso using two models, the bioelectrical electrical source, i.e., the heart., and a torso. The surface coordinates of the heart and the torso are derived from data provided by magnetic resonance imaging (MRI) of human subjects. The low conductivity of the lungs and the high conductivity in the ventricular cavities are taken into consideration, which makes the model rather realistic. Transmem- brane potentials can be changed interactively so that repolarization disturbances can be simulated. ECGSIM only generates single complexes.

Therefore, no preprocessing steps such as segmentation have to be performed when analyszing the data generated by ECGSIM. The sampling frequency is 1kHz.

Figure 6.1: ECGSIM.

6.1 Time duration differences of action potentials

ECGSIM provides the possibilities to manipulate the repolarization times in the nodes of the heart model. In [16] it is described how a pathological condition, similar to LQTS, is simulated by changing the standard deviation of the repolarization times of SDrep. Different levels of SDrep were obtained by changing the standard deviation of the action potential duration SDapd, so that SDrep were set in discrete steps of 20.8, 30.7, 40.6,

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be a normal state and 70.7 [ms] is considered to be a highly pathological state. Since ECGSIM only generates single complexes, the RR-interval was assumed to be 900 ms.

The different PCA parameters, described in Section 3.2, were calculated at all repolarization time steps. Two different electrode setups were compared in this test. The first set up consisted of the eight independent channels [I,II,V1,. . . ,V6] of a standard 12-lead electrode set up. Leads [III,avR,avL,avF] were omitted, since these can be derived from lead I and II (see section 2.4 equations 2.1-2.6). In the second electrode setup, the eight independent leads are used together with ten additional chest leads (see Fig. 2.4).

6.1.1 Results

Figure 6.2: TWRabs-parameter as a function of the standard deviaton of repolarization duration.

Figure 6.3: TWRrel-parameter as a function of the standard deviaton of repolarization duration.

Figure 6.4: NDPV-parameter as a function of the standard deviaton of repolarization duration.

Figure 6.5: S2/S1-parameter as a function of the standard deviaton of repolarization duration.

6.1.2 Analysis

One can clearly see in Fig. 7.2-7.5 that the 18-lead setup gives higher

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that the variance (energy) in the T waves in an 18-lead setup is totally greater than the 8-lead setup, since the variance of all the T waves in a multi-channel ECG is distributed over the principal components. It is not possible to draw any conclusions about the sensitivity of the measurement method. The question whether the 18-lead ECG will separate healthy subjects from sick subjects better than the 8-lead ECG is not possible to answer from this test. This question must be answered by experimental studies on healty and diseased populations.

According to this test, the S2/S1-parameter gets saturated, as the standard deviation of the repolarization times get higher (Fig. 6.5). This result was also found in [16], who used a similar parameter definition as S2/S1. TWRabs was shown to be unsensetive at lower lower SDrepvalues (Figs. 6.2 and 6.3). The NDPV and TWRrel parameters are rather linear from low to high SDrep values (Fig. 6.3 and 6.4).

6.2 Quantification test

The objective of this test was to assess whether the quantification level of the EP4 system was too high.

In the same way as in 6.1, the SDrep parameter in ECGSIM were changed. The high-resolution signals generated by ECGSIM were quan- tified into different levels to see how the PCA parameters were affected.

6.2.1 Results

A noteworthy pattern appeared with the TWRrel parameter as shown in Fig. 6.6. The influence on TWRabs, NDPV and S2/S1 can be viewed in Appendix B

6.2.2 Analysis

The intacardiac inputs of the EP4 has a quantification level of approximately 16uV/LSB. According to this test, this quantification value might be too large to guarantee an accurate TWRrel-parameter, because as the quantification levels gets higher the graph gets ”U” shaped and hence, one can not separate high-risk patients from low-risk patients.

6.3 Interference of noise

Noise not only affects the variance of the PCs. It also affects the accu- racy of the T-wave detection algorithms. The following test explores the robustness of the algorithms implemented in this work.

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10 20 30 40 50 60 70 80 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35

TWRrel(std.rep.time)

s.d. Repolarization [ms]

TWRrel [%]

0.625 [uV/LSB]

1.25 [uV/LSB]

2.5 [uV/LSB]

5.0 [uV/LSB]

10 [uV/LSB]

20 [uV/LSB]

Figure 6.6: Results from the the quantification test.

One vector magnitude segment was calculated from the 8 independent leads of a standard 12-lead ECG. It was generated by ECGSIM at default settings. The noise level was determined as:

SNR(dB) = 10log σ²_S σ_N²

!

(6.1) where σ²_S is the variance of the vector magnitude segment and σ_N² is the variance of the noise.

T-end and T-onset algorithms were tested with SNR= 80, 70, 60, 50, 40, 30, 20, 10 dB. The TA algorithm was given a value of T-onset=

156. Normal distributed noise was generated and repeatedly added to the vector magnitude, 1000 times for each noise level. The mean value and the standard deviation were calculated for every T-onset and T-end point for every noise level and for all algorithms.

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6.3.1 T end result

SI LSI TA WTMM

SNR(dB) mean std mean std mean std mean std

80 363 0 363 0 348 0 394 0

70 363 0 363 0 348 0 394 0

60 363 0 362.83 0.38 348 0 393.98 0.16

50 362.89 0.31 362.6 0.53 348.08 0.28 393.74 0.44

40 362.66 0.75 - - 348.73 0.48 393.56 0.50

30 408.06 154.48 - - 349.73 1.07 393.53 0.50 20 757.74 249.9 - - 351.29 2.83 401.46 71.65 10 947.01 165.32 - - 352.71 8.95 947.67 236.9

”-” indicates a failed test.

6.3.2 T onset result

J+RR/16 J+(T-peak-J)/3 WTMM

SNR(dB) mean std mean std mean std

80 156 0 162 0 152 0

70 156 0 162 0 152 0

60 156 0 162 0 152 0

50 156 0 161.97 0.17 152 0

40 156 0 161.72 0.45 152.01 0.09

30 156 0 161.61 0.52 152.23 0.42

20 156.03 0.16 161.57 1.32 152.32 0.69 10 156.28 0.46 161.53 2.62 150.71 2.54 6.3.3 Analysis

The T-onset algorithms all showed good performance at all the noise levels tested, which can be seen from the fact that the mean values and the standard deviations remained stable.

Both tangent-based T end algorithms were sensitive to noise, since both of them search for the minimum slope and make use of the derivative. The derivative becomes increasingly ill-defined with increasing noise. The SI algorithm did not give a reliable detection point below SNR=40 dB since the mean value rose against the end of the complex and the standard deviation increased too much. The LSI algorithm failed to detect the T-end point less than 50 dB, since the calculated tangent did not intersect the isoelectric line before the end of the complex.

The WTMM algorithm managed to give a reliable detection down to 30 dB, which is better than both SI and LSI. The TA algorithm showed the best result of all of the algorithms. It managed to keep a stable mean and standard deviation value down to 10 dB.

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0 200 400 600 800 1000

−2 0 2 4

Vector Magnitude +60dB

Amplitude [mV]

0 200 400 600 800 1000

−2 0 2 4

Amplitude [mV]

0 200 400 600 800 1000

−2 0 2 4

Time [ms]

Amplitude [mV]

Figure 6.7: The vector magnitude with three noise levels.

6.4 Infarction example

Pathological conditions may generate T waves that are hard to detect.

A typical example of this is when ECGs are taken in patients with acute myocardial infarction. In these cases, the T wave may interfere with the QRS complex, which results in an undefineable T onset point. This is also reflected in the vector magnitude. See Figure 6.8.

An infarction model, that was created with ECGSIM, was made by changing the initital values of node 16 in the heart model, from (resting potential = -90mV, LAT=42ms, APD=238ms) to (resting potential = -40mV, LAT=71ms, APD=301ms). The vector magnitude of [I,II,V1,. . . ,V6] was calculated whereupon the three implemented T-onset algorithms were tested on the resulting complex.

0 200 400 600 800 1000

0 1 2 3 4

Vector Magnitude

Time [ms]

Amplitude [mV]

Figure 6.8: Vector magnitude of an ECG with simulated infarction.

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6.4.1 Result

The time indicies of T-onset point of the complex in Fig. 6.8.

J+RR/16 J+(T-peak-J)/3 WTMM

169 112 323

6.4.2 Analysis

Tonset = J + (T p − J)/3, does not give a reliable T-onset point since it is approximated from the greatest value within a window defined from J+50 ms to the end of the complex. In this example, the greatest value occurs at J+50 ms due to the fact that the T wave has no distinct peak. This results in a T-onset point which is placed too far to the left (Tonset = 112). The WTMM algorithm places the T-onset point too far to the right (Tonset = 323). Tonset = J + RR/16 does not consider the T wave shape in its decision rules, since it uses the RR interval as base.

Hence this algorithm is unaffected by the wave shape abnormalties.

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7 Discussion

7.1 Limitations

Simulations are dependent on model properties and do not directly reflect the biological reality. Changes in the ECG could theoretically be caused by reasons other than repolarization inhomogeneity. The noise test is performed on one representative complex and the outcome may be different if other complexes are used.

7.2 Conclusions

The simulation study lead to three main conclusions:

• The test in section 6.1 suggests that either the TWRrel or the NDPV are the best choice of parameter since they have a linear response to repolarization inhomogeneity. This is a desirable property for discrimination between high and low levels of the nondipolar content in the ECG.

• The EP4-system is unsuitable as data acqusition equipment in an experimental study since the quantification levels of the recorded signals may lead to inaccurate TWRrel values.

• A combination of T-onset = J+RR/16 and T-end=TA was shown to be the best choice of detection algorithms in sense of robustness to noise and wave-shape abnormalities.

7.3 Improvements

Since noise affects the results, some method to quantify the noise levels in the different channels would be desirable. This might be done by calculating an SNR value of the vector magnitude segment. The tangent- based T-end algorithms may be improved by applying some other filter rather than the 5th order Butterworth filter that was used in this work.

The 50-Hz interference could probably be diminished by using a metal box instead of using a plastic box as was done in this study.

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References

[1] M. Kesek Traces of Repolarization Inhomogeneity in the ECG Up- psala: Acta Universitatis Upsalensis 2005

[2] U. Wiklund Modelling and Analysis of Heart Rate Variability Sig- nals: Wavelets and Autoregressive Methods in Clinical Enviroments Ph.D. Thesis Ume˚a: 2001

[3] J. Malmvuo, R. Plonsey Bioelectromagnetism Oxford: Oxford Uni- versity Press 1995

[4] C. Chatfield, A.J. Collins Introduction To Multivariate Analysis London: Chapman Hall 1980

[5] Richard A. Johnsson, Dean W. Wichern Applied Multivariate Sta- tistical Analysis 5^th ed.Upper Saddle River: Prentice-Hall 2002 [6] Alexander D. Poulaarikas The Transforms and Applications Hand-

book 2^nd ed.USA: CRC Press 2000

[7] B. Acar, G. Yi, K. Hnatkova, M. Malik Spatial, temporal and wave- front direction characteristics of 12-lead T-wave morphology Medical and Biological Engineering and Computation, 1999, 37, 574-584 [8] Q. Xue, S. Reddy Algorithms for Computerized QT Analysis Journal

of Electrocardiology, 1998, 30, 181-186

[9] L.S¨ornmo, P. Laguna Bioelectrical Signal Processing in Neurological Applications USA: Elsevier Academic Press 2005

[10] H.Shchaefer, H.G. Haas Electrocardiography USA: Waverly Press1962

[11] T.Grabowski Principles of Anatomy and Physiology 10^th ed. USA:

Von Hoffman Press, Inc 2003

[12] L. Cuiwei, Z. Chongxun, T. Changfeng Detection of ECG Char- acteristic Points Using Wavelet Transform IEEE Transactions on Biomedical Engineering 1995, 42, 21-29

[13] J. P. Martinez, R. Almeida, S. Olmos, A. P. Rocha, P. Laguna A Wavelet-Based ECG Delineator: Evaluation on Standard Databases IEEE Transactions on Biomedical Engineering 2004, 51, 570-581 [14] S. Krimi, K. Ouni, N. Ellouze T-wave Detection Based on an Ad-

justed Wavelet Transform Modulus Maxima International Journal of

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[15] O. V. Oosterom, T. F. Oostendorp ECGSIM: an interactive tool for studying the genesis of QRST waveforms Heart Online2004, 90, 165-168

[16] B. H. V. Huysduynen, C. A. Swenne, H. H. m. Draisma, M. L. An- toni, H. V. D. Vooren, E. E. V. D. Wall M. J. Schalij Validation of ECG Indices of Ventricular Repolarization Heterogeneity: A Com- puter Study J Cardiovasc Electrophysiol 2005, 16, 1097-1103

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A Appendix A

A.1 Covariance

Covariance is a measure of how two variables are associated with each other and is defined as:

Cov(X, Y ) = E[(X − E(X))(Y − E(Y ))]

A.2 Covariance Matrix

A set of variables X can be written as:

X=







X1

X2

... Xn







The covariance matrix of X is defined as:

Σ=







Cov(X1, X1) Cov(X1, X2) . . . Cov(X1, Xn) Cov(X2, X1) Cov(X2, X2) . . . Cov(X2, Xn)

... ... . .. ...

Cov(Xn, X1) Cov(Xn, X1) . . . Cov(Xn, Xn)







A.3 Eigenvalues

The eigenvalues λ of a matrix Σ can be found by solving the following equation.

det(Σ − λI) = 0

Since the covariance matrix is symmetric, the eigenvalues are real.

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B Appendix B

10 20 30 40 50 60 70 80

0 2 4 6 8 10 12 14 16

18x 10⁵ TWRabs(std.rep.time)

TWRabs [uV4]

0.625 [uV/LSB]

1.25 [uV/LSB]

2.5 [uV/LSB]

5.0 [uV/LSB]

10 [uV/LSB]

20 [uV/LSB]

Figure B.9: TWRabs results from the the quantification test.

10 20 30 40 50 60 70 80

5 10 15 20 25 30 35 40 45 50 55

NDPV(std.rep.time)

NDPV [uV]

0.625 [uV/LSB]

1.25 [uV/LSB]

2.5 [uV/LSB]

5.0 [uV/LSB]

10 [uV/LSB]

20 [uV/LSB]

Figure B.10: NDPV results from the the quantification test.

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10 20 30 40 50 60 70 80 5

10 15 20

S2/S1(std.rep.time)

S2/S1 [%]

0.625 [uV/LSB]

1.25 [uV/LSB]

2.5 [uV/LSB]

5.0 [uV/LSB]

10 [uV/LSB]

20 [uV/LSB]

Figure B.11: S2/S1 results from the the quantification test.

Principal component analysis of T-waves

M A S T E R ' S T H E S I S

Principal Component Analysis of T-waves

Ola Gustavsson

Principal Component Analysis of T waves

Ola Gustavsson

October 10, 2007

Preface

Contents

List of Abbreviations

1 Introduction

1.1 Background

1.2 Purpose

2 The ECG

2.1 The electrical activation of cardiac muscle fibers

2.2 ECG waves

2.3 Repolarization inhomogeneity

2.4 Lead systems

2.5 The dipolar concept

3 Theory

3.1 Principal component analysis

3.2 PCA of T waves

3.3 Continuous wavelet transform

4 Measurement equipment

4.1 EP tracer 102

4.2 EP4

4.3 Cables

4.4 Connection box

4.5 Hardware-related problems

4.6 Limitations in resolution

5 Implementation

5.1 Scaling

5.2 Reference signal

5.3 R-peak detection

5.4 Segmentation

5.5 T-wave detection

5.6 Wavelet based T-wave detection

5.7 PCA step

5.8 Graphical user interface

6 Simulation study

6.1 Time duration differences of action potentials

6.2 Quantification test

6.3 Interference of noise

6.4 Infarction example

7 Discussion

7.1 Limitations

7.2 Conclusions

7.3 Improvements

References

A Appendix A

A.1 Covariance

A.2 Covariance Matrix

A.3 Eigenvalues

B Appendix B