c 2005 MAIKO ARICHI ALL RIGHTS RESERVED

Transcription

1 c 25 MAIKO ARICHI ALL RIGHTS RESERVED

2 MONITORING ISCHEMIC CHANGES IN ELECTROCARDIOGRAMS USING DICKINSON-STEIGLITZ DISCRETE HERMITE FUNCTIONS A Thesis Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Master of Science Maiko Arichi August, 25

3 MONITORING ISCHEMIC CHANGES IN ELECTROCARDIOGRAMS USING DICKINSON-STEIGLITZ DISCRETE HERMITE FUNCTIONS Maiko Arichi Thesis Approved: Accepted: Advisor Dr. Dale Mugler Dean of the College Dr. Charles B. Monroe Faculty Reader Dr. Daniel B. Sheffer Dean of the Graduate School Dr. George R. Newkome Faculty Reader Dr. J. Patrick Wilber Date Department Chair Dr. Jianping Zhu ii

4 ABSTRACT Ischemic heart disease is one of the main common causes of death and an electrocardiogram (ECG) is used in the investigation of the heart disease. A method of real time detection of ischemic features from long term ECG signals based on Dickinson- Steiglitz discrete Hermite expansions is proposed. In this paper, the discrete Hermite functions were generated as eigenvectors of the extended tridiagonal matrix that commutes with the centered Fourier matrix. Each ECG complex was extracted from the long term ECG and expanded in terms of Hermite functions using a simple dot product. These coefficients were found to contain information about the shape of the corresponding ECG complex. The first 5 coefficients were used to reconstruct the ECG complex. These 5 coefficients were fed as input to train a committee Neural Network classifier to identify ST-segment and T-wave changes, which are one of the ischemic features in the ECG complex. The trained network was tested with long term ECG records from the MIT-BIH database. The performance was analyzed in terms of sensitivity and specificity. This results were compared with the ones that came from the same method used with a tridiagonal Hermite matrix (T matrix). iii

5 ACKNOWLEDGEMENTS I would like to express my gratitude to all those who gave me the possibility to complete this thesis. I am deeply indebted to my advisor Dr. Dale H. Mugler for his suggestions and encouragement helped me in all the time of research and for writing of this thesis. I own special thanks Dr. Narender P. Reddy for his invaluable suggestions during the project. I also thank the chair of the Applied Mathematics department, Dr. Jianping Zhu for his contribution and support. My special thanks to Raghavan Gopalakrishnan, who guided me to complete the project during every stage of the research. I would like to thank Koji Masuda for his love and support. I like to thank all graduate assistants of mathematics department, especially my friend, Ines Busuladzic, for their support and help. I would like to delicate this work to my parents who have been an indispensable source of love and affection. iv

6 TABLE OF CONTENTS Page LIST OF TABLES LIST OF FIGURES vii viii CHAPTER I. INTRODUCTION The Electrocardiogram Ischemic Heart Disease Artificial Neural Networks Objectives of the research II. TRIDIAGONAL MATRIX Continuous Hermite Function Discrete Hermite functions III. EXTENDED TRIDIAGONAL MATRIX Definition of an Extended Tridiagonal Matrix Dual sets of discrete Hermite functions Biomedical applications of Hermite functions Automated methods for ischemia monitoring v

7 IV. METHODOLOGY The MIT-BIH database ECG segmentation algorithm Generation of Hermite coefficients The ECG classification V. RESULTS Results from the ECG segmentation Results from the Hermite expansion of ECG signal Results from the feature extraction Results from ECG classification VI. DISCUSSION X matrix discrete Hermite functions A committee of Neural Networks BIBLIOGRAPHY APPENDICES APPENDIX A. MAIN PROGRAMS FOR THE GENERATION OF HERMITE COEFFICIENTS[34] APPENDIX B. MAIN PROGRAM FOR R-PEAK DETECTION AL- GORITHM[34] vi

8 LIST OF TABLES Table Page 3.1 The first 5 coefficients Cont. The first 5 coefficients Cont. The first 5 coefficients Performance analysis calculation of specificity, sensitivity and positive predictive value Results for beat classification Method I Results for beat classification Method I Results for beat classification Method I Results for beat classification Method II Results for beat classification Method II Results for beat classification Method II Results for beat classification from T matrix [22][37] Various Ischemic Monitoring Schemes (Source:Cardiovasc Rev Rep,23) [22][37] Result from the chi-square goodness of fit test (beat classification) vii

9 LIST OF FIGURES Figure Page 1.1 Components of an ECG signal(frank G. Yanowitz, M.D.,University of Utah School of Medicine) normal ECG signal [4] ischemia T wave and ST depression [1] Normal (Septal) Q waves and Significant Q wave [1] a biological neuron [5] an artificial neuron [5] an artificial neural network [5] The first few Hermite polynomials [6] The first six Hermite functions U k,b for k =, 1...5,b = The first six Hermite functions U k,b for k =, 1...5,b = The first nine Hermite functions X k,b for k =, 1...8,b = 1 Method I The first nine Hermite functions X k,b for k =, 1...8,b = 1 Method II X 55, X 6 X 65 at b=1 Method I X 55, X 6 X 65 at b=1 Method II viii

10 3.5 The first six even vectors in the dual sets of X matrix discrete Hermite functions The first six even eigenvalues of X matrix discrete Hermite functions the block diagonal of the Pan-Tompkins algorithm [35] Hermite expansion with the first 5 coefficients generated for a nonischemic ECG using Method I and II (PRD =.7624) Hermite expansion with all coefficients generated for a non-ischemic ECG using Method I and II(PRD = e-56) Hermite expansion with the first 5 coefficients generated for an ischemic ECG using Method I and II (PRD =.3472) Hermite expansion with all coefficients generated for an ischemic ECG using Method I and II(PRD = e-57) Feature extraction Hermite expansion of a non-ischemic ECG (C 2 =.481) Feature extraction Hermite expansion of a non-ischemic ECG (C 2 =.6) and (C 2 =.4) Simulated real time model of Hermite function based ischemic monitor for long term ECG signals [22] A non-ischemic ECG series Band pass filtered ECG series Derivative of the band pass filtered ECG Square of the derivative signal Moving averaged signal with its envelope Square pulse series with the band pass filtered ECG A non-ischemic ECG signal and its corresponding R-peak locations An ischemic ECG and its corresponding R-peaks Segmented ECG complexes ix

11 5.1 Hermite expansion of a non-ischemic ECG: (A) C 1 = (B) C 1 = 1. (C) C 1 = Training of a committee of Neural Network of T matrix case Training of a committee of Neural Network of X matrix case in method I and II x

12 CHAPTER I INTRODUCTION 1.1 The Electrocardiogram An electrocardiogram (ECG/EKG) is an electrical impulse recording of the heart and is used in the investigation of heart disease. These impulses are recorded as waves called P-QRS-T deflections [1]. Each cardiac cell is surrounded by and filled with a solution that contains, in part, sodium (Na+), potassium (K+), and calcium (Ca++). In its resting condition the interior of the cell membrane is considered negatively charged, with respect to the outside. When an electrical impulse is initiated in the heart, the inside of a cardiac cell rapidly becomes positive in relation to the outside of the cell. The electrical impulse causes this excited state and this change of polarity, is called depolarization. Immediately after depolarization, the stimulated cardiac cell returns to its resting state, which is called repolarization. The resting state is maintained until the arrival of the next wave of depolarization. This change in cell potential from negative to positive and back to negative is called an action potential. That action potential initiates a cardiac muscle contraction. The ECG is a measurement of the effect of this depolarization and repolarization for the entire heart on the skin surface, and is 1

13 also an indirect indicator of heart muscle contraction, because the depolarization of the heart leads to the contraction of the heart muscles. Although the phases of the ECG are due to action potentials traveling through the heart muscle, the ECG is not simply a recording of an action potential. During the heartbeat, cells fire action potentials at different times, and the ECG reflects patterns of that electrical activity [2] P-QRS-T wave Figure1.1 shows ECG waves and intervals as well as standard time and voltage measures on the ECG paper. Here are the definitions of P-QRS-T waves [2]. The intervals and segments in P-QRS-T wave provide important information about the function of the heart. Figure 1.1: Components of an ECG signal(frank G. Yanowitz, M.D.,University of Utah School of Medicine) 2

14 P wave represents the sequential activation (depolarization) of the right and left atria. Usually P waves are upright and slightly rounded. QRS complex represents the right and left ventricular depolarization (normally the ventricles are activated simultaneously). R wave is a positive deflection. Q wave is a negative deflection before an R wave. S wave is a negative deflection after an R wave. ST-T wave represents ventricular repolarization. T wave is normally upright and slightly rounded. U wave s origin is not clear. Probably, however, U wave represents afterdepolarizations in the ventricles, so it should be of the same direction as the T wave. PR interval is the time from the beginning of the P wave to the beginning of the QRS complex. PR interval represents the time interval from onset of atrial depolarization (P wave) to onset of ventricular depolarization (QRS complex). QRS duration represents duration of ventricular muscle depolarization. QT interval is the time from the beginning of the QRS complex to the end of the T wave. QT interval represents both ventricular depolarization and repolarization. RR interval represents the duration of ventricular cardiac cycle (an indicator of ventricular rate) 3

15 PP interval represents the duration of atrial cycle. (an indicator of atrial rate) ECG Monitoring Method Electrodes are placed on designated areas of the patient s body, and these various combinations of the electrodes are used for analysis of the heart condition. Each separate view of the heart is called an ECG lead. The two ECG monitoring methods are standard 12-lead ECG monitoring [1] and continuous ECG monitoring or holter monitoring [3]. 12-lead ECG consists of three standard leads, designated as lead I,II,III, and three augmented leads, designated as lead avr, avl and avf, that view the heart in the frontal plane, and six precordial or chest leads, designated V1 throughv2, that view the heart in the horizontal plane. Both the standard leads and the augmented leads are limb leads. The standard leads are called bipolar because they are composed of two electrodes-one that is negative and one that is positive-and the ECG records the difference in electrical potential between them. The standard 12-lead ECG records 12 different views of the same electrical activity on the ECG graph paper. Holter monitoring provides a continuous recording of heart rhythm during normal activity, and the monitor is usually worn for 24 hours. In holter monitoring, electrodes (small conducting patches) are placed on the chest and attached to a small recording monitor that can be carried or in a small pouch worn around the neck Normal ECG signal Here are the normal ECG waves [1]. 4

16 Normal sinus rhythm Each P wave is followed by a QRS complex. P wave rate is 6-1 beats per minute (bpm). If rate is less than 6 bpm, called sinus bradycardia. If rate is greater than 1, called sinus tachycardia. Normal P waves Height is less than 2.5 mm in lead II Width is less than.11 s in lead II Normal PR interval.12 to.2 s (3-5 small squares) Normal QRS complex less than.12 s duration ( 3 small squares) Normal QT interval Calculate the corrected QT interval by dividing the QT interval by the square root of the proceeding R-R interval. Normal is.42 s. Normal ST segment No elevation or depression 5

17 Figure 1.2: normal ECG signal [4] 1.2 Ischemic Heart Disease Ischemic heart disease or ischemia is a condition in which fatty deposits accumulate in the cells of the coronary arteries [1]. These fatty deposits build up gradually and irregularly and encircle the heart and the main source of its blood supply, whose process is called atherosclerosis. This process leads to narrowing or hardening of the blood vessels supplying blood to the heart muscle. This causes an inability to provide adequate oxygen to the heart muscle and result in damage to it. According to RxMed ( myocardial infarction causes 35 % of deaths in 6

18 men between the ages of 35 and 5. The death rate is higher for men than for women between the ages of 35 and 55. However, after the age 55, the death rate for men declines but the rate for women continues to climb Ischemic changes in ECG Ischemia is a lack of sufficient oxygenated blood to the left ventricle and is manifested on the ECG by symmetrically inverted T waves or ST depression (Figures 1.3 and 1.4). Abnormal waves of ECG, like T inversion or ST depression/elevation are caused by the abnormal electrical activity of the heart [2]. For example, inverted T wave is caused an abnormally fast resting heart rate (grater than 1 beats/ min). Infarction is necrosis or death of tissue in a portion of the ventricular myocardial wall, and follows the stages of ischemia if an adequate blood supply is not returned. In this case, the ECG signal is changed by significant Q waves. To be considered significant Q waves, they must either have a duration of.4 second or one third the height of the normal wave [1]. 7

19 Figure 1.3: ischemia T wave and ST depression [1] Figure 1.4: Normal (Septal) Q waves and Significant Q wave [1] 1.3 Artificial Neural Networks The brain has the ability to perform tasks such as pattern recognition, perception and motor control much faster than any computer - even though events occur in the 8

20 nanosecond range for silicon gates and milliseconds for neural systems. An artificial neural network(nn) is a model of biological neural systems that contains similar characteristics [5] Biological Neural System A biological Neural System is comprised of a mass of nerve cells, referred to as a neuron. A neuron consists of a cell body, dendrites and an axon. Neurons are massively interconnected by interconnections between the axon of one neuron and a dendrite of another neuron. This connection is found in the synapse. Signals propagate from the dendrites, through the cell body to the axon,once the signal reaches to a synapse, these signals are propagated to all connected dendrites. A signal is transmitted to the axon of a neuron only when the cell undergoes depolarization and repolarization. A neuron can either inhibit or excite the associated post-symptic neurons.[5]. Figure 1.5: a biological neuron [5] 9

21 1.3.2 Artificial Neuron (AN) An artificial neuron (AN) is a model of a biological neuron (BN). Each AN receives signals from the environment or other ANs, and gathers these signals. When the cell is activated, each AN transmits a signal to all connected ANs. Figure 1.6 represents an artificial neuron (AN). Input signals are inhibited or excited through negative and positive numerical weights associated with each connection in the AN. The firing of an AN and the strength of the exciting signal are controlled via a function, called an activation function. The AN collects all incoming signals, and computes a net input signal using the respective weights, then the net signal serves as input to the activation function which calculates the output signal of the AN. Figure 1.6: an artificial neuron [5] Basic Structure of an Artificial Network An artificial neural network (NN) is a layered network of ANs. An NN consists of an input layer, one or more hidden layers and an output layer. ANs in one layer are connected, fully or partially, to the ANs in the next layer. A typical NN structure is represented in Figure

22 Figure 1.7: an artificial neural network [5] The way to calculate the Net input signal is usually computed as the weighted (w i ) sum of all input signals (x i ) in Equation 1.1. These artificial neurons are referred to as summation units (SU) [5]. Net = i (x i w i ) (1.1) Once the net input signal is calculated, the function f Net,refered to as the activation function, receives it to determine the output of the neuron. Different types of activation functions can be used. The neural network is trained by a set of input data and the desired output, called targets using the back propagation algorithm. The back propagation algorithm, the best known training algorithm for the neural networks, is one in which the input data is continuously fed into the network and the predicted output of the network is compared with the desired output and the error generated. This process is done repeatedly until the error becomes insignificant. 11

23 The artificial neuron learning techniques are mainly classified into supervised and unsupervised learning, and the back propagation method is a type of supervised learning [5]. Supervised learning, where the neuron (or NN) is provided with a data set consisting of input vectors and a target(desired output) associated with each input vector. This data set is referred to as the training set. The aim of supervised training is then to adjust the weight values such that the error between the real output of the neuron and the target output is minimized. Unsupervised learning, where the aim is to discover patterns or features in the input data with no assistance from an external source. An artificial network has been used for a wide range of applications, including diagnosis of diseases, speech recognition, data mining, composing music, image processing, forecasting, robot control, credit approval, classification, pattern recognition, planning game strategies, compression and many others [5]. 1.4 Objectives of the research The objective of this research is to develop a scheme for monitoring ischemic changes in long term ECG signals based on discrete Hermite expansion using an extended tridiagonal matrix. Null hypothesis is that these is no difference between the methods using an extended tridiagonal matrix and using a tridiagonal matrix. 12

24 CHAPTER II TRIDIAGONAL MATRIX 2.1 Continuous Hermite Function The classical Hermite functions are basically Hermite polynomials multiplied by the Gaussian function. The Hermite polynomials, using the Rodrigues Formula [6], are given by H n (t) = ( 1) n dn t2 e dt n e t2, n =, 1, 2..., < t < (2.1) The first few Hermite polynomials are H (t) = 1, H 1 (t) = 2t, H 2 (t) = 4t 2 2, H 3 (t) = 8t 3 12t, H 4 (t) = 16t 4 48t (2.2) The Hermite polynomials also satisfy many functional relations, such as the three term recursion given by H n+1 (t) = 2tH n (t) 2nH n 1 (t), n = 1, 2... (2.3) 13

25 Figure 2.1: The first few Hermite polynomials [6] 2.2 Discrete Hermite functions The discrete Hermite functions are good approximations of the continuous Hermite functions [7]. There are two methods for the construction of discrete Hermite functions, the Dickinson-Steiglitz method [8] and the Grünbaum method [9]. According to the Dickinson-Steiglitz method, the discrete Hermite functions are eigenvectors of the extended tridiagonal matrix that commutes with the centered Fourier matrix as well. According to the Grunbaum method, the discrete Hermite functions are eigenvectors of a symmetric tridiagonal matrix that commutes with the centered Fourier matrix Fourier matrix The standard Fourier matrix, with indices i,k =,...n 1 is given by F [i, k] = e 2πj n (i)(k) (2.4) 14

26 The shifted Fourier matrix has the form given by F [i, k] = e 2πj n (i x)(k y) (2.5) where x and y are the horizontal and vertical shift parameters, with indices i, k =,...n 1. The centered Fourier matrix has the form given by 2πj n F a,b = e (i a)(k a) b 2 (2.6) where a is the shift parameter, b is called the dilation parameter. To obtain the primal centered case, it is used as a = (n 1)/2 with indices i, k =,..n 1. Working with the centered Fourier matrix has several advantages. One is that eigenvalues of the commutor are distinct. The eigenvectors of the dilated discrete Hermite functions are unique. Second is that the centered Fourier matrix makes the possible extension to the dilated case, when b is not equal 1 [11] Tridiagonal Matrix A tridiagonal matrix is a square matrix with nonzero elements only on the diagonal and slots horizontally or vertically adjacent to the main diagonal. The tridiagonal matrix that commutes with the centered Fourier matrix for two specific values of a was discovered by Grünbaum [9]. Grünbaum showed one for the standard Fourier matrix(2.4) when a = and another for the half way shifted Fourier matrix when a = n/2. The general form was discovered by Clary and Mugler [11]. 15

27 2.2.3 Dilated discrete Hermite functions A tridiagonal matrix T b is defined in the previous section. The main diagonal of the T b that commutes with the centered Fourier matrix is given by [11] 2 cos(πnτ) sin(πµτ) sin(π(n µ 1)τ), µ n 1 (2.7) and the off-diagonal of it are given by sin(πnτ) sin(π(n µ)τ) (2.8) where τ = 1/(nb 2 ). The set of eigenvectors of the T b are the dilated discrete Hermite functions U kb. The continuous Hermite functions [6] are defined for k by U k (x) = e x2 2 Hk (x) (2.9) The continuous Hermite function U k (x) are set of orthogonal polynomials. Therefore, the dilated discrete Hermite functions U k,b are similar to equation(2.9) with a dilation term. The index k stands for the number of zero crossings of the Hermite function. The effect of index b is that the function broadens with the value of b increasing and narrows down as its value decreases. Figure 2.2 shows the first six Hermite functions for a dilation parameter b = 1. Figure 2.3 show the first six Hermite functions for a dilation parameter b = 3. 16

28 .25 u u u2 value value value t t t value u3 u4 u value t t t value Figure 2.2: The first six Hermite functions U k,b for k =, 1...5,b = 1 u u1 u value value.5 value t t t u3 u4 u value value value t t t Figure 2.3: The first six Hermite functions U k,b for k =, 1...5,b = 3 17

29 CHAPTER III EXTENDED TRIDIAGONAL MATRIX 3.1 Definition of an Extended Tridiagonal Matrix An extended tridiagonal matrix is a square matrix that has all of its non-zero entries confined to the three middle diagonals and the two off-diagonal corners. The reason why this matrix is called Extended tridiagonal matrix is that the upper right and lower left corners can be regarded as cyclic extensions of the subdiagonal and the superdiagonal. The extended tridiagonal matrix that commutes with the Fourier matrix was discovered by Dickinson and Steiglitz [8] Discrete Hermite Functions An extended tridiagonal matrix X b is defined in the previous section. The main diagonal entries of the X b that commutes with the centered Fourier matrix are given by [11] 2 cos(π(2µ n + 1)τ), µ n 1 (3.1) and the subdiagonal and super diagonal entries are all 1, where τ = 1/nb 2. Only b = 1 works in this case. The eigenvectors of the X b are called discrete Hermite functions for the extended tridiagonal case. The biggest difference between the T matrix and the X matrix discrete Hermite functions is that X matrix discrete Hermite functions 18

30 have almost zero values in the middle, the bigger and bigger k becomes. The index k stands for the number of zero crossings of the Hermite function. The Matlab program produces a set of eigenvectors, but it is not unique, only up to multiples by plus or minus 1. Each set of eigenvectors is orthogonal and appropriate for a signal expansion. To obtain the closest analog to the continuous Hermite functions, two different methods were tried. In Method I, the maximum on the positive t-axis(time) was made to be more than the absolute value of the minimum, as is the case for continuous Hermite functions. In Method II, a small amount was added to the last element of the computed eigenvector, and the vector was multiplied by 1 if that sum was negative. Figure 3.1 shows the first nine X matrix discrete Hermite functions for a dilation parameter b = 1 using Method I and Figure 3.2 shows the first nine X matrix discrete Hermite functions for a dilation parameter b = 1 using Method II. Figure 3.3 and 3.4 shows the 55th, 6th and 65th X matrix discrete Hermite functions for a dilation parameter b = 1 using Method I and II. 19

31 X X 1 X 2 value value value t t t X X 3 4 X t t t value value value X 6 X 7 X 8 value value t value t t Figure 3.1: The first nine Hermite functions X k,b for k =, 1...8,b = 1 Method I 2

32 X X 1 X 2 value value value t X t X X t 5 value value value X t X t X 8 t value value t value t t Figure 3.2: The first nine Hermite functions X k,b for k =, 1...8,b = 1 Method II Coefficients The first 5 coefficients from the discrete Hermite expansion using T matrix and X matrix using Method I and Method II are always distinct. For example, one sample ECG signal is taken from the MIT-BIH database, with the number of samples at 25. An expansion using the T matrix vectors and one using the X matrix vectors is performed, then the norm using the difference of the expansions is calculated. For the example ECG signal, the norm =.3413 in using Method I and the norm = using Method II, which shows that the first 5 coefficients from an expansion using between T matrix and X matrix are different. Here is the first 5 coefficients from an expansion in the T matrix and the X matrix using Method I and Method II (Tables 3.1, 3.2 and 3.3). 21

33 3.2 Dual sets of discrete Hermite functions Several different sets of discrete Hermite functions, for example those from the T matrix and the X matrix, have been found with the basic property that they are eigenvectors of the centered Fourier matrix [11]. There exist a set of n orthogonal vectors for n-dimensions considered in the previous chapter from the Dickinson-Steiglitz method using eigenvectors of the X matrix when the initial function has eigenvalue 1. These discrete Hermite functions have two properties that are used in this construction of a related set, called the dual set [14]. 1. Each set of functions is connected to one of the four eigenvalues such as 1, i, 1, i of the Fourier matrix using the specific formula. F h k = i k h k, k =, 1,...N 1 (3.2) In the X matrix case, this property requires some reordering of the eigenvectors. h, the initial function, must be an eigenvector for the centered Fourier matrix with eigenvalue All even-indexed eigenvectors must be even functions and all odd-indexed eigenvectors must be odd functions. The Dickinson-Steiglitz method generates an orthogonal set of vectors. 22

34 3.2.1 Definition of the dual set The procedure to compute the dual set from a given set h k for k =,,, n 1 is the following 5 steps [14]. 1. Take the last half of each of the eigenvectors in the set. That is, define h2 k = h k (n/2 + 1 : n). Each of these vectors will be half the length of the original vectors, i.e. they will be length n/2. Since each eigenvector has either odd or even symmetry, the original set is completely defined by simply knowing the last half of each h k eigenvector, along with the appropriate symmetry for that vector. 2. Add a minus sign to every other element in each eigenvector, as in the formula h3 k [j] = ( 1) j h2 k [j] for j =, 1,..., n Reverse the order of each eigenvector, as in the flipud Matlab command for a column vector, or as described in the formula h4 k [j] = h3 k [n 1 j] for j =, 1,..., n Reverse the indexing of the vectors in the set, as specified by h5 k = h4 n 1 k for k =, 1,..., n Extend the even-indexed vectors to be even vectors, and odd-indexed vectors in the set to be odd vectors. The resulting set of n length n vectors is the dual set to the original set. The result is another set of orthogonal eigenvectors for the centered Fourier matrix. 23

35 The third step in this procedure connects with the production of a zero interval of values in the middle of the dual set, particularly for higher indices, because the discrete Hermite functions model the continuous Hermite functions, and have the property that low-indexed functions are concentrated near the middle of the interval, with zero values near the edge. The discrete Hermite functions generated by the X matrix show clearly that the flipping step moves that interval of zero values from the outside of the interval to the inside. The interesting evidence is that the eigenvalues and dual sets of the X matrix-generated discrete Hermite functions are numerically exactly the same, thus the X matrix-generated discrete Hermite functions are called self-dual as well(figure 3.5 and Figure 3.6)[14]. 3.3 Biomedical applications of Hermite functions In 1991 an adaptive system based on the Hermite functions was proposed to adaptively estimate and track the QRS complexes in the ECG signal for QRS detection by Laguna et al. [15]. In 1993, an orthogonal transformation based on Hermite functions was proposed for ECG data compression by Jane et al. [16]. In 1996, an Adaptive Hermite Model Estimation System (AHMES) was proposed for online beat-to-beat estimation of the QRS features by Laguna et al. [17]. Subsequently, in 1997, Braccini et al. showed that the ECG signals were decomposed in terms of Hermite coefficients and the information contained in the coefficients was used to classify the ECG signals using Self-Organizing Maps [18]. In 24

36 2, Lagerholm et al. introduced an integrated method for clustering ECG complexes using Hermite functions and Self-Organizing Maps [19]. In addition, the continuous Hermite functions were applied to other electrophysiological signals. In 1994, Hermite-Rodriguez and Associated Hermite basis functions were applied for the expansions of myoelectric signals(emg) which were constructed involving non-orthogonal vectors that are obtained by sampling the continuous Hermite functions by Loredana et al. [2]. Later, in 1995, Wahlberg et al. proposed that the Hermite functions were used for the feature extraction and clustering epileptic spikes in electroencephalogram(eeg) [21]. Using continuous Hermite functions to approximate the ECG complex has several difficulties. One is that a modern ECG recording is both digital and finite in length, whereas the Hermite functions are continuous and defined for all values of time. If the Hermite functions are simply sampled and the resulting vectors used for expansion, those vectors are not orthogonal. Thus, in 22, Mugler et al. proposed a new method to expand digital ECG signals using the discrete Hermite functions, including a dilation term. The dilated discrete Hermite U k,b is an orthonormal set, where the expansion of the signal may be found easily and efficiently, and is concentrated near the origin for small k indices, and expands outward with greater width as the index k increases. More recently, Gopalakrishnan et al. proposed to diagnose ischemic heart disease from long term ECG signals by expanding them in terms of dilated discrete Hermite functions [22] from the T matrix eigenvectors. This paper compares that technique to one used here that uses the X matrix eigenvectors. 25

37 3.4 Automated methods for ischemia monitoring According to recent studies, the continuous ECG monitoring was more provisional than the standard 12-lead ECG for monitoring ischemia [23] [24]. In the last decade, many techniques have been proposed for the automated detection of the ischemia from long term ECG signals which are based on the digital signal analysis, fuzzy logic methods and Artificial Neural Networks [23]. All these methods are used for classification of the ECG signals into either non-ischemic or ischemic. Senhadji et al. found a technique which used wavelets to examine the ECG signals and extract time and frequency information of the signal simultaneously and use them for classification of signals [25]. Jager et al. used the principal component analysis for ischemia episode detection [26]. Vila et al. monitored the heart rate in the time frequency domain and detected ischemia based on the ST-segment changes [27]. Pitas et al. detected ischemia using a mathematical model of ECG signals [28]. Fuzzy logic system is a frequently caused technique for classification of the ECG signals. For example, Vila et al. developed a fuzzy system for ischemic episode classification [29]. Recently, Artificial Neural Networks (ANN) have been used to perform diagnosis, because they are accurate and consistently [23]. Stamkopoulos et al. used a multilayer perceptron for the detection of myocardial ischemia for a single lead ECG signal by the Neural Network [3]. Silipo et al applied the multilayer perceptron for a two channel recording using data taken from the ST-segment [31]. Papadimitriou 26

38 et al. proposed an ANN system for the beat classification using information from the ST-T interval of the ECG signal [32]. 27

39 .2 X value t X value t X value t Figure 3.3: X 55, X 6 X 65 at b=1 Method I 28

40 .2 X value t X value t X value t Figure 3.4: X 55, X 6 X 65 at b=1 Method II 29

41 Table 3.1: The first 5 coefficients X matrix (Method I) T matrix X matrix (Method II)

42 Table 3.2: Cont. The first 5 coefficients X matrix (Method I) T matrix X matrix (Method II)

43 Table 3.3: Cont. The first 5 coefficients X matrix (Method I) T matrix X matrix (Method II)

44 dual dual2 dual value.15 value.5 value t t t 25 value dual value.5 dual8 value dual t t 25.2 t 25 Figure 3.5: The first six even vectors in the dual sets of X matrix discrete Hermite functions x x2 x value value value t t t 25.2 x6.2 x8.2 x value.5 value.5.5 value t 25 t 25 t 25 Figure 3.6: The first six even eigenvalues of X matrix discrete Hermite functions 33

45 CHAPTER IV METHODOLOGY The research includes four parts: 1. Obtaining the digital ECG signals and segmenting them automatically to get individual ECG complexes. 2. Expanding the ECG complexes in terms of the discrete Hermite functions and Hermite coefficients. 3. Classifying the heart beat either as ischemic or non-ischemic by using those Hermite coefficients. 4. Comparing the Hermite expansions using X matrix with one using T matrix in terms of sensitivity, specificity and positive predictive value. All algorithms are written using MATLAB software with signal processing toolbox and Neural Network toolbox. 4.1 The MIT-BIH database The MIT-BIH database provided by MIT and Boston s Beth Israel Hospital consists of ten different databases for various tests. Here the Long-Term ST database and the European ST-T database from the MIT-BIH database are used. 34

46 The Long-Term ST database recorded 86 lengthy ECG recordings from 8 human subjects. The database contains a variety of events of ST segment changes, including ST episodes, axis-related non-ischemic ST episodes, episodes of slow ST level drift, and episodes containing mixtures of these phenomena. The individual recordings of the Long-Term ST database are between 21 and 24 hours in duration, and contain two or three ECG signals which have been digitized at 25 samples per second. The European ST-T database is used for evaluation of algorithms for analysis of ST and T-wave changes. This database recorded 9 annotated excerpts of ambulatory ECG recordings from 79 subjects. The subjects were 7 men aged 3-84, and 8 women aged 55 to 71. Each record is two hours in duration and contains two signals, each sampled at 25 samples per second. The database includes 367 episodes of ST segment change, and 41 episodes of T-wave change, with durations ranging from 3 seconds to several minutes. 4.2 ECG segmentation algorithm The ECG signals were taken from the MIT-BIH database, from the long term ST database and the European ST-T database. The raw ECG signals were segmented to get individual ECG complexes with the R-peak as the center. The ECG segmentation algorithm had the following steps. 1. The text files of the raw ECG signals were taken from the MIT-BIH database. 35

47 2. The R- peak locations of the entire ECG signals were found. 3. Considering each R-peak separately, the R-R interval with the adjacent R- peaks was calculated, then the average value of the R-R intervals, say a, was calculated. 4. Taking half the time interval a/2 on either side of the R-peak with the R-peak as center, an ECG complex was extracted. 5. The sampling rate of the extracted ECG complex was 25. If the rate was not 25, it was made 25 by upsampling the signal by 25 times and then downsampling it by the original sampling rate of the signal. 6. The length of the ECG complex was not allowed to use over 25 samples. If the length of an ECG complex was more than 25 samples, the ends of either side of it were chopped off. 7. The DC component of the ECG complex was removed by first taking the Fourier transform of the signal and then making the first Fourier coefficient zero Pan-Tompkins algorithm The QRS complex has the largest slope in a cardiac cycle, by virtue of the rapid conduction and depolarization characteristics of the ventricles [35]. There are several techniques to apply for detecting the QRS complex. The Pan-Tompkins algorithm [35] is one of such technique. The algorithm is a real-time QRS detection based on 36

48 analysis of the slope, amplitude, and the width of QRS complexes. Figure 4.1 shows the steps of the algorithm. Figure 4.1: the block diagonal of the Pan-Tompkins algorithm [35] 1. The first step, the bandpass filter, contains two steps; the low-pass filter and high-pass filtering applied sequentially. At first the low-pass filter is applied over the ECG signal and followed by the high-pass filter. The basic idea of the low-pass filter is that the filter passes low frequencies fairly well, but attenuates, or blocks, high frequencies. The low-pass filter used in the algorithm is designed with stop band edge frequencies of 35 Hz and 45 Hz respectively. The high-pass filter is the opposite of the low-pass filter, which is that the filter passes high frequencies fairly well, but attenuates low frequencies. The high-pass filter used in the algorithm is designed with stop band edge frequencies of 5 Hz and 3 Hz respectively. Thus the ECG signal from the bandpass filter has all frequencies suppressed except for the QRS complexes. 2. The derivative procedure removes the low-frequency components of P and T waves, and provides the high-frequency components arising from the high slopes of the QRS complex. 37

49 3. Squaring process makes the result positive and emphasizes large differences resulting from QRS complexes.small differences arising from P and T wave are suppressed and the high-frequency components in the signal related of QRS complex are enhanced. 4. The algorithm performs smoothing of the output of the proceeding operations through a moving-window integration filter. In this process, a moving average is used over the squared signal to get the smooth pulse which corresponds to the QRS complex. An envelope, which is a range of frequencies of these smooth pulses is generated and a value of one-third the value of the envelope is set as a threshold. Whenever the pulse crosses the threshold value, a new square pulse is generated and overlapped with the QRS complexes. Within the index of the new square pulse, the bandpass filtered ECG signal is searched for the maximum value and that value is related to the location of the R-peak. 4.3 Generation of Hermite coefficients To generate of the Hermite coefficients, the signal i.e. ECG complexes generated as a result of segmentation process are used. Because they are orthogonal vectors, a simple dot product of the signal with the discrete Hermite function generates of the Hermite coefficients. 38

50 4.3.1 Generation of discrete Hermite functions The generated dilated discrete Hermite functions X k,b using Method I and Method II proposed in the previous chapter, where b = 1, are eigenvectors of an extended tridiagonal matrix that commutes with the centered Fourier matrix. This set of eigenvectors is orthonormal and the coefficients obtained by expanding a signal in terms of Hermite functions contain independent information. Give an ECG signal x of length n, the discrete Hermite expansion of x is an expansion of an n-dimensional digital signal in a particular orthonormal basis. The Hermite expansion y has the following form [22], y = C k,b X k,b (4.1) k n 1 with coefficients C k,b =< x, X k,b > (4.2) where b = 1. Coefficients are given by the standard inner products of the input ECG signal with the discrete Hermite functions Performance Measure The accuracy of the discrete Hermite function in representing an ECG signal was calculated using Percentage RMS difference (PRD) error. Let x i be the original ECG signal and y i be the discrete Hermite function representing x i. The PRD formula is given by P RD = ( i (x i y i ) 2 ) ( i (x 1 (4.3) i x) 2 ) 39

51 where x is the mean of the original ECG signal x i. The advantage of the PRD error is low complexity and inexpensive calculation. Figure 4.2 (Figure 4.4) shows a nonischemic (ischemic) ECG signal and the corresponding Hermite expansion(method I and Method II proposed in the previous chapter) using the first 5 coefficients. In both method, the exact same PRD error is gained, because a set of eigenvectors up to multiply by plus or minus 1 does not effect the Hermite expansion. Figure 4.3 (Figure 4.5) shows a non-ischemic (ischemic) ECG signal and the corresponding Hermite expansion using all the coefficients generated, and the small value of the PRD error shows perfect reconstruction. 4

52 1.2 original signal all coefficients hermite expansion 1.8 voltage time 1.2 original signal all coefficients hermite expansion 1.8 voltage time Figure 4.2: Hermite expansion with the first 5 coefficients generated for a nonischemic ECG using Method I and II (PRD =.7624) 41

53 original signal all coefficients hermite expansion 1.8 voltage time Figure 4.3: Hermite expansion with all coefficients generated for a non-ischemic ECG using Method I and II(PRD = e-56) 42

54 .4 original signal all coefficients hermite expansion.2.2 Voltage time.4.2 original signal all coefficients hermite expansion.2 Voltage time Figure 4.4: Hermite expansion with the first 5 coefficients generated for an ischemic ECG using Method I and II (PRD =.3472) 43

55 original signal all coefficients hermite expansion.2.2 Voltage time Figure 4.5: Hermite expansion with all coefficients generated for an ischemic ECG using Method I and II(PRD = e-57) Feature of ECG signals If the individual Hermite coefficients obtained for a particular ECG complex are varied, while keeping all other coefficients unaltered, one can see how changing the Hermite coefficients affects the ECG features. For example, the figures included here show a situation where the second Hermite coefficient is varied between ±.5 of the original value in steps of one. Figures 4.6 and 4.7 show the different second Hermite coefficient values C 2 and the corresponding Hermite representation on a non-ischemic ECG signal. C 2 =.481 is the normal coefficient value for this ECG signal in Figure 4.6. As the value is decreased to C 2 =.6, one sees the ischemic features like a depressed ST segment and as the value is increased to C 2 =.4, one sees an elevated ST-segment feature. 44

56 1.5 Coef #2 =.481 original signal hermite expansion 1.5 Voltage time Figure 4.6: Feature extraction Hermite expansion of a non-ischemic ECG (C 2 =.481) 45

57 1.5 Coef #2 =.6 original signal hermite expansion 1.5 Voltage Time Coef #2 = original signal hermite expansion 1.5 Voltage Time Figure 4.7: Feature extraction Hermite expansion of a non-ischemic ECG (C 2 =.6) and (C 2 =.4) 4.4 The ECG classification In order to find out the usability of Hermite coefficients to identify ischemic features and their ability to classify each ECG signal as non-ischemic or ischemic, a committee Neural Network classifier is developed. 46

58 4.4.1 Training of a committee of Neural Networks Five Neural Networks are developed and trained. Each network has three layers; input layer, hidden layer and output layer using feed forward network. The input layer has 5 neurons. The hidden layer has 35 to 55 neurons. The output layer has 3 neurons. The first 5 coefficients from the expansion of an ECG signal in discrete Hermite functions are used as input to the Neural Network. The outputs of the Neural Network are the presence/absence of ST-segment changes, the presence/absence of T- wave inversion and the presence/absence of ischemic ECG. Thus the targets should be the actual presence/absence of the ST-segment changes, T wave inversion and ischemic ECG. In both the hidden layer and the output layer, a tan-sigmoid function is used as the activation function. The conjugate gradient back propagation algorithm is used to train the network. ( trainscg in the MATLAB Neural Network) The Hermite coefficients used in the training process are extracted from 112 different ECG signals taken from the MIT-BIH database, of which 62 are non-ischemic ECG signals and 5 are ischemic ECG signals. The ischemic ECG signals are determined by two features. One is an elevated/depressed ST segment and another is an inverted T-wave. All possible combinations of these two features are shown to the network. The individual network results might vary in borderline ischemic cases, so the majority decision of the committee of trained neural network is used for the final classification. 47

59 4.4.2 Testing network A total of 24 long term ECG records from the European ST-T database, and the normal and ischemic episodes from the long term ST database were chosen to test the network, provided from the database. A total of 215 beats are used, of which 93 are non-ischemic beats and 122 are ischemic beats. The performance is analyzed as sensitivity, specificity and positive predictive value, based on the predicted network output for the beat and episode classification. Beat classification is based on whether the network is able to detect any ST segment changes and T wave changes in a particular beat. Episode classification is based on whether a particular episode is rightly classified as ischemic or non-ischemic, irrelevant to the prediction of ST-segment and T-wave change Simulated Real-time model A Hermite expansion that helps ECG segment classification is developed under the simulated real-time conditions for long term ECG monitoring applications. Figure 4.6 shows a simulated real time model of Hermite function based ischemic monitor for long term ECG signals. The long-term ECG signals are scanned continuously for R-peak location using a QRS detection algorithm, i.e. the Pan-Tompkins algorithm. R-peak locations are used to automatically segment the ECG signal centered at its R-peak with a window size equivalent to the corresponding R-R interval. The individual ECG 48