Spectral decomposition of electrocardiograms for the diagnosis of pulmonary hypertension and the estimation of invasively measured parameters

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Spectral decomposition of electrocardiograms for the diagnosis of pulmonary hypertension and the estimation of invasively measured parameters
Madsen, Henry M. ( author )
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Pulmonary hypertension ( lcsh )
Electrocardiography ( lcsh )
Pediatrics ( lcsh )
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )


Pediatric pulmonary hypertension (PH) is a serious disease that is best managed when diagnosed early and monitored closely. Diagnosis and prognosis of PH rely, in part, on cardiac catheterization- an invasive procedure that gives detailed pressure information. The objective of this thesis is to develop a relationship between 12-lead ECG and catheterization measurements. We obtained Fourier coefficients from spectral decomposition of each ECG lead, and evaluated a classification threshold and multivariate linear regression with the coefficients. We used ROC for PH versus control with a single Fourier coefficient, achieving an AUC of 0.83, and a classification accuracy of 82% over 300 trials, indicating that single ECG Fourier coefficients may be valid classifiers of disease states. We also created multivariate linear regression models to explore the correlation between ECG spectral information and measurements from right-heart catheterization. We found the following values for each correlation: VVCR (r2 =0.83, p = 2.6e-6), PVR (r2 = 0.85, p = 1.1e-6), RVP (r2 = 0.87, 4.4e-7), mPAP (r2 = 0.90, p = 8.2e-8), and RV contractility (r2 = 0.82, p = 4.3e-6). Our results suggest that ECG may offer more detailed prognostic and diagnostic capabilities than previously known, and should be thus developed in conjunction with echocardiogram.
Thesis (M.S.)--University of Colorado Denver
Includes bibliographical references.
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by Henry M. Madsen.

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HENRY M MADSEN B.A., Colorado College, 2014
A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment Of the requirements for the degree of Master of Science Bioengineering Program


This thesis for the Master of Science degree by Henry M Madsen has been approved for the Bioengineering Program by
Vitaly Kheyfets, Chair Stephen Humphries Richard Weir Johannes vonAlvensleben
Date: May 13, 2017

Madsen, Henry M (M.S., Bioengineering)
Spectral Decomposition of Electrocardiogram for The Diagnosis of Pulmonary Hypertension and The Estimation of Invasively Measured Parameters Thesis directed by Assistant Research Professor Vitaly Kheyfets
Pediatric pulmonary hypertension (PH) is a serious disease that is best managed when diagnosed early and monitored closely. Diagnosis and prognosis of PH rely, in part, on cardiac catheterization- an invasive procedure that gives detailed pressure information. The objective of this thesis is to develop a relationship between 12-lead ECG and catheterization measurements. We obtained Fourier coefficients from spectral decomposition of each ECG lead, and evaluated a classification threshold and multivariate linear regression with the coefficients. We used ROC for PH versus control with a single Fourier coefficient, achieving an AUC of 0.83, and a classification accuracy of 82% over 300 trials, indicating that single ECG Fourier coefficients may be valid classifiers of disease states. We also created multivariate linear regression models to explore the correlation between ECG spectral information and measurements from right-heart catheterization. We found the following values for each correlation: VVCR (r2 =0.83, p = 2.6e-6), PVR (r2 = 0.85, p = l.le-6), RVP (r2 = 0.87, 4.4e-7), mPAP (r2 = 0.90, p = 8.2e-8), and RV contractility (r2 = 0.82, p = 4.3e-6). Our results suggest that ECG may offer more detailed prognostic and diagnostic capabilities than previously known, and should be thus developed in conjunction with echocardiogram.
The form and content of this abstract are approved. I recommend its publication.
Approved: Vitaly Kheyfets

To Mom and Dad, thank you for everything!
To Helen and Tyler, the coolest people I know.
To Tracy, Mac, Mikal, and Tiffany, for all your advice.
To Terrance for being my brother.
To Alice for reminding me that theres always time for one more game of hide and seek. To Naomi and Miles- maybe one day you will be writing a thesis.
To Edwin and Cole, for being great friends.
To Robbie and Alex, who should move back to Colorado.
To all my other friends and family members... thank you!
To Patrick Rothfuss and Brandon Sanderson, for the books that kept me sane.

I would like to thank:
Dr. Vitaly Kheyfets for endless edits and great advice.
Dr. Weir for meeting and helping so often.
Dr. Humphries and Dr. vonAlvensleben for being a great committee. Melanie Dufva, for always being willing to help, especially last minute. Everyone else in the BioE department- you rock!

1. Introduction................................................................1
1.1 Purpose.............................................................1
1.2 Pediatric Pulmonary Hypertension...................................2
1.2.1 Important Differences between Pediatric and Adult PH .....3
1.2.2 Molecular Origins of Pulmonary Hypertension...............3
1.2.3 Biomechanical Considerations in Pulmonary Hypertension....4
1.3 Using ECG to Measure Cardiac Activity.............................5
1.3.1 Normal Electrical Activity of the Heart...................5
1.3.2 Abnormal Electrical Activity of the Heart.................7
1.3.3 Detecting Electrical Activity with 124ead ECG.............9
1.4 Previous Work in ECG Processing...................................13
1.4.1 Time-domain Analysis of ECG Signals.......................13 Advantages of Time-domain Analysis...............13 Previous Work in Time-domain ECG Analysis........14 Summary of Time-domain ECG Analysis..............17
1.4.2 Frequency-domain Analysis of ECG Signals..................18 Rationale for Frequency-domain Analysis..........18 Previous Work with Fourier-transform in ECG......19 Using Frequency and Phase to Localize Signals....20 Summary of Frequency-domain ECG Analysis.........21
1.4.3 Time-frequency-domain Analysis (TFA) of ECG Signals.......21
vii The Fourier Transform in TFA.........................22 Other TFA Techniques- Wavelets and Auto-Regression..23
1.4.4 Correlation of ECG Signals with Other Modalities..............24 Using ECG to Measure RVC.............................25 Using ECG to Measure mPAP............................27 Using ECG to Measure PVR.............................28 Using ECG to Measure RVP.............................29 Summary of Work with ECG and Invasive Parameters... 31
1.4.5 Statistical Classifiers for Disease State Diagnosis from ECG..31 Artificial Neural Networks and ECG...................31 Other Classifiers Used in ECG Literature.............32
1.4.6 Other Features Used in ECG Processing.........................34 Wavelets as ECG Features.............................35 Karhunen-Loeve Morphologies as ECG Features..........35 Hermite Polynomials as ECG Features..................36 Auto-Regressive Coefficients as ECG Features.........36 Pan-Tompkins QRS Detection Algorithm.................37 Hilbert Transform for QRS Complex Detection...........37
1.4.7 Methods for ECG Pre-Processing.................................38
1.5 Justification...........................................................39
1.6 Goals...................................................................40
2. Methods..........................................................................41
2.1 Measuring Cardiac Function with Right-Heart Catheterization.............41

2.1.1 Right-ventricular Pressure as Measured by RHC..................41
2.1.2 Mean Pulmonary Arterial Pressure as Measured by RHC...........42
2.1.3 Pulmonary Vascular Resistance as Measured by RHC and MRI... .42
2.1.4 Right-ventricular Contractility as Measured by RHC and MRI....42
2.1.5 Vascular Ventricular Coupling Ratio as Measured by RHC........44
2.2 ECG Format..............................................................45
2.3 Patient Selection.......................................................46
2.4 Uploading ECGs to MatLab................................................48
2.5 ECG Signal Pre-Processing...............................................49
2.6 Fourier Coefficients....................................................49
2.7 ROC Analysis............................................................54
2.8 Multivariate Linear Regression Analysis.................................56
3. Results.........................................................................58
3.1 ROC Analysis............................................................58
3.2 Multivariate Regression Analysis........................................61
4. Discussion......................................................................72
4.1 ROC Analysis............................................................72
4.1.1 Summary of Interpretation of ROC Classification................76
4.2 Multivariate Linear Regression Analysis.................................76
4.2.1 Multi-Variate Linear Regression for RVP.....................78
4.2.2 Multi-Variate Linear Regression for PVR.....................79
4.2.3 Multi-Variate Linear Regression for VVCR.....................81
4.2.4 Multi-Variate Linear Regression for mPAP.....................82

4.2.5 Multi-Variate Linear Regression for RVC.......................84
4.2.6 Summary of Interpretation of Linear Regression Analyses.......85
4.3 Strengths and Limitations..............................................86
4.4 Future Work............................................................87
5. Conclusion.....................................................................89
A. Figures and Tables.....................................................99
B. MatLab Code...........................................................101
B.l Main Script....................................................101
B.2 Function- pdf_cleanup..........................................103
B.3 Function- patient distribution.................................104
B.4 Function- ECG_Coeffs...........................................104
B.5 Function- input_output.........................................105
B.6 Function- ECG ROC..............................................106
B.7 Function- ECG MultiROC.........................................108
B.8 Function- linregress...........................................110
B.9 Function- singleleadvarselect..................................110
B. 10 Function- multistepregress...................................Ill
B. 11 Function- varselect..........................................112

2.1 General ECG findings for both CTL and PH populations..........................48
3.1 Results for uni-variate ROC analysis..........................................58
3.2 Multivariate linear regression results.......................................61
3.3 The multi-variate models and relevant Fourier coefficients...................61

1.1 The basic anatomy of the human heart.............................................5
1.2 The network of electrically conductive fibers, and their associated nodes........6
1.3 The correspondence between ECG waveform and electrical events in the heart.......7
1.4 The naming conventions for QRS morphology........................................8
1.5 The general placement of leads in a 12-lead electrocardiogram...................10
1.6 The three augmented leads and three limb leads of a 12-lead electrocardiogram...11
1.7 Landmarks of a normal electrocardiogram......................................14
2.1 (A) LV pressure curve, can be extrapolated to RV. (B) A pressure-volume loop.43
2.2 An example of the ECG tracings in our PDF files..............................45
2.3 Age matching for CTL and PH patients.........................................47
2.4 Half-power bandwidth of our populations......................................53
3.1 The distribution of the 50Hz coefficient on lead III for PH and CTL patients....59
3.2 The distribution of the 50Hz coefficient on lead III with outliers excluded,,,,.59
3.3 The ROC curve for the uni-variate analysis......................................60
3.4 Multivariate regression model for mPAP.......................................62
3.5 Bland-Altman plot for mPAP multivariate regression model.....................63
3.6 Multivariate regression for PVR..............................................64
3.7 Bland-Altman for PVR..........................................................65
3.8 Multivariate regression for RVP..............................................66
3.9 Bland-Altman for RVP..........................................................67
3.10 Multivariate regression for RVC.............................................68

3.11 Bland-Altman for RVC........................................................69
3.12 Multivariate regression for VVCR...............................................70
3.13 Bland-Altman for VVCR.......................................................71
4.1 Lead III on a normal confirmed EKG..............................................74
4.2 Lead III on an abnormal EKG.....................................................75
4.3 A depiction of the correspondence between ECG lead and anatomy..................75
4.4 The lateral (Figure 4.4, part A) and precordial (Figure 4.4, part B) leads for RVP....78
4.5 The lateral (Figure 4.5, part A) and precordial (Figure 4.5, part B) leads for PVR....79
4.6 Lateral (Figure 4.6, part A) andprecordial (Figure 4.6, part B) leads for VVCR..81
4.7 Lateral (Figure 4.7, part A) and precordial (Figure 4.7, part B) leads for mPAP.82
4.8 The lateral (Figure 4.8, part A) and precordial (Figure 4.8, part B) leads for RVC... 84

1.1 Purpose
Pulmonary hypertension (PH) is a condition that is clinically characterized by increased blood pressure and resistance in the arterial vasculature of the lung. These physical changes are the result of numerous pathological mechanisms, and they eventually culminate in right-ventricular (RV) failure. No definitive cure for PH has been found, but the disease can be managed with an early diagnosis. Early-detection of PH is difficult, however, because diagnosis requires invasive and expensive procedures. To this end, we have utilized the noninvasive and inexpensive electrocardiogram (ECG) modality to improve early detection and disease prognosis capabilities. We hope that this work will ultimately be translatable to the clinic, and, though our results are already promising, there is much room for improvement from experienced signal processing engineers.
The objective of this study was to discover diagnostic and prognostic ECG frequencies and leads in pediatric PH. The ECG frequency spectrum was obtained with Fourier transformation, and the information in the ECG was quantitatively expressed as the Fourier coefficients of this spectrum. The diagnostic capability of the ECG was analyzed by comparing ECG Fourier coefficient distributions between control and PH patient populations. The prognostic capability of the ECG was analyzed by comparing Fourier coefficients with parameters from right-heart catheterization (RHC) and MRI namely: vascular ventricular coupling ratio (VYCR), end systolic elastance as a

measurement of right-ventricular contractility (RVC), pulmonary vascular resistance (PVR), right-ventricular pressure (RVP), and mean pulmonary arterial pressure (mPAP).
Our hypothesis is two-fold: that ECG Fourier coefficients can be used to classify patients as pulmonary hypertensive or not, and that ECG Fourier coefficients can be used as predictors in a multi-variate linear model to approximate prognostic measurements currently obtained by invasive catheterization.
1.2 Pediatric Pulmonary Hypertension
Pulmonary hypertension (PH) is clinically defined by mPAP greater than 25 mmHg, a normal capillary wedge pressure less than 15 mmHg, and PVR above three Wood Units (1 Wood Unit = 1 Pa*s/m3) [1], PH can either be an idiopathic disease or a disease secondary to other conditions. The disease can form in children or adults. Regardless of its etiology, PH leads to increased ventricular afterload, decreased cardiac output, and eventually heart failure. The severity of PH is captured by the World Health Organization (WHO) standards.
There are five WHO categories for PH. Group I is the most prevalent category, accounting for approximately 90 percent of cases. This group, termed pulmonary arterial hypertension (PAH), includes idiopathic PAH, congenital PAH, familial PAH, and PAH associated with other diseases. Group II includes left-heart associated PH. Group III includes PH associated with lung disease and hypoxia. Group IV includes chronic thrombotic and embolic PH. Group V captures PH associated with other, miscellaneous conditions. The pediatric PH patients used in this study fall under Group I- PAH [2], Pediatric PH differs from adult PH in several aspects.

1.2.1 Important Differences between Pediatric and Adult PH
Most adult cases are secondary to other conditions, commonly diabetes, heart failure, and kidney failure. In most pediatric cases, including all cases in this paper, patients are either bom with congenital defects that lead to PH, or are born with idiopathic PH. This leads to important differences in cardiovascular biomechanics and histological findings between adult and child patients.
Because most pediatric PH patients are born with PH, whereas adults develop the disease over time, the tissues of pediatric patients have not undergone as extensive of adaptive changes. Over time, the pediatric patient will have structural and histological adaptations [90], Considering that our patient population is on the older end of pediatric populations (average PH patient age around 14 years), there are likely significant histological changes that have occurred. Pediatric patients often receive diagnosis and treatment much earlier in the etiology of PH than adult patients do.
1.2.2 Molecular Origins of Pulmonary Hypertension
The molecular origins of PH are not fully understood. The pathology of the disease involves thickening and/or stiffening of the arterial vessels. These vessels are made up of three layers: the innermost intima, the middle media, and the outermost adventitia. The intima consists of a layer of endothelial cells attached to a collagen IV basement membrane. The media is a collagen I matrix with concentric rings of elastic tissue, surrounded by smooth muscle cells. The adventitia has the most diverse cell population, mostly fibroblasts within a fibrous, proteinaceous matrix [3], For reasons that are not fully understood, the etiology of PH involves stiffening of the media and adventitia through a remodeling process. This remodeling relates to a change in the

function of local fibroblasts to become myofibroblasts, which behave more similarly to smooth muscle cells, as they secrete more alpha-actin [4], This altered fibroblast phenotype is associated with an inflammatory state. These molecular changes cause changes to the function and biomechanics of the cardiovascular system.
1.2.3 Biomechanical Considerations in Pulmonary Hypertension
The methodology of this thesis is more concerned with the biomechanics of PH than the molecular origins. The mechanical hallmarks of PH include increased mPAP, increased PVR, and increased right-ventricular hypertrophy, all leading to eventual heart failure. The heart goes through two distinct phases during the disease- the compensatory phase, and the failure phase. PH begins prior to the compensatory phase, when PVR, and thus mPAP, increase. This increase is associated with a stiffening of the pulmonary artery [5], During the compensatory phase, the right ventricle matches the increased resistance in its output by increasing RVC. This increase in RVC is achieved by hypertrophy of the myocardium, and by larger stroke volumes per the Frank-Starling mechanism [6], This interplay between a stiffened pulmonary artery and RVC is captured as VVCR, which is a powerful tool for the prognosis of PAH. As the right ventricle undergoes hypertrophy, the shape of the heart changes. One major change involves the flattening of the septum between the ventricles. Data from MRI studies have shown that this septal flattening ultimately decreases the ability of the right ventricle to contract fully [73], This decrease in RVC is compounded by the inability of the left ventricle to assist the right ventricle in contraction, which is an occurrence in a healthy heart, because of the altered septal morphology. As the stronger left ventricle loses the ability to transfer mechanical energy to the right ventricle, the right ventricle must work even harder [7, 8], The next stage of

PH occurs when the WCR ratio begins to fall, as the heart cannot overcome these systemic impositions and create strong enough contractions to match the impedance from stiffening and constricting vasculature. This stage ends in heart failure, and death.
Regardless of the molecular etiology or biomechanical consequences of PH, there exists no definitive cure. However, early detection improves patient prognosis and mortality. Thus, this paper is devoted to describing a novel method for detection of PH.
1.3 Using ECG to Measure Cardiac Activity
1.3.1 Normal Electrical Activity of the Heart
Each cardiac cycle consists of systole, or contraction, and diastole, or relaxation. The atria contract in systole to fill the ventricles, and the ventricles contract in diastole to supply blood to the lungs and systemic circulation. The atria and ventricles relax during diastole to refill with blood. These contractions are coordinated by a depolarization wave along the conductive fibers of the heart and the myocyte membranes. The basic anatomy the human heart is shown in Figure 1.1. The relaxation of heart is coordinated by the reversal of the electrical impulse along these membranes, termed cardiac repolarization. This pattern is very predictable in a normal heart.
Figure 1.1: The basic anatomy of the human heart. RA = right atrium, LA = left atrium, RV = right ventricle, LV = left ventricle, S = intra-ventricular septum. Note that the left ventricle has larger muscle mass than the right ventricle [75],

Cardiac depolarization begins with an electrical impulse generated in the SA node. This impulse spreads throughout the atria, making them contract. At the septal wall, between the atria and ventricles, there is a break in conductivity. The impulse must travel through the AV node to enter the ventricles. This causes a brief delay, allowing the ventricles to fill before contracting. Once the impulse has spread throughout the conductive fibers of the ventricle, it enters the ventricular myocardium near the apex of the septum, leaving the left bundle branch first. The depolarization then spreads outwards from the sub-endocardium to epicardium of the ventricles. Figure 1.2 shows a basic schematic of the network of conductive fibers in the human heart.
Figure 1.2: The network of electrically conductive fibers, and their associated nodes. SA = sinoatrial node, AV = atrioventricular node, RBB = right bundle branch, LBB = left bundle branch. Note that the right atrium houses the SA and the AV [75],
Cardiac repolarization occurs most quickly near the epicardium, slowing as it nears the sub-endocardium, in the opposite direction to the depolarization wave. The phases of electrical activity and their correspondence to the waveforms on a composite ECG can be found below, in Figure 1.3.

T im e [m s] 0 100 200 300 400 500 600 700
Figure 1.3: The correspondence between ECG waveform and electrical events in the heart. Note that the contribution of the pacemaker cells and conductive fibers is minimal compared to the much larger voltages measured from myocardial depolarization and repolarization [83],
1.3.2 Abnormal Electrical Activity of the Heart
ECG parameters for the diagnosis of right-ventricular hypertrophy are clinically useful in the prognosis of PH. These parameters were originally developed in 1949 for the left ventricle, by Sokolow and Lyon [14], These guidelines were developed for the right-ventricle, and a selection is discussed below [15],
Certain ECG parameters that are related to altered anatomy and physiology are indicative of a poor prognosis in PH patients. Two of the most well-established

parameters for prognosis, that has been corroborated by multiple research efforts, is an increased P-wave amplitude in lead II that corresponds with sudden death and right-heart failure, and the presence of a QR pattern in lead VI [9, 10], The P-wave in lead II is thought to correspond to right-ventricular hypertrophy and beat mechanics. The QR pattern, seen below in Figure 1.4, in lead VI is associated with right-ventricular strain [11].
Figure 1.4: The naming conventions for QRS morphology. Note that the QR complex is described by a downward Q-wave, followed by an upward R-wave, with no downward S-wave to follow [84],
A 1986 study aimed to correlate ECG parameters to right-ventricular hypertrophy, and did so with very good sensitivity and specificity [12], However, this study was performed on patients with heart diseases that could confuse the results. Thus, a study was undertaken to apply the methods of Butler et al. to a more clinically feasible population [13], This revealed sensitivity and specificity that were good, but below

clinical criteria. The parameter of interest in these studies was the vectorcardiogram of the QRS complex, and diagnosis of right-ventricular hypertrophy was made when QRS forces [sic] were maximally anterior and rightward, and minimally posterolateral. R-amplitude in lead I and P-amplitude in leads II, III, aVF, VI, or V2 were additional parameters. More than anything, these studies provided electrophysiological evidence to support the expected result that right-ventricular hypertrophy alters the distribution of forces in the beating heart.
ECG parameters have also been discovered that are prognostic of hemodynamics. These parameters include R-amplitude and R/S ratio in lead VI, which were found to correlated to pulmonary artery systolic pressure (PASP) [16],
1.3.3 Detecting Electrical Activity with 12-lead ECG
The ECG is designed to detect and record the depolarization and repolarization waves of the heart. The 12-lead ECG is performed by placing 10-leads at predetermined locations along the patients body. The chest leads VI through V6 account for 6 leads, and combinations of the RA, LA, RL, and LL leads account for the other 6 leads. These are combined according to the diagram to make leads I, II, III, aVR, aVL, and aVF. The general placement of these leads can be found in Figure 1.5. The placement of the augmented leads can be found in Figure 1.6.

Figure 1.5: The general placement of leads in a 12-lead electrocardiogram. There are only ten leads placed. Each of the six precordial leads (the chest leads VI through V6) have their own signal. The limb leads (RA, LA, RL, and LL) are combined in different ways to create the remaining six leads (three augmented leads and three limb leads) [76],
When no electrical activity is encountered by a lead, it records an isoelectric line. Depolarization waves moving towards the leads make a positive deflection, and depolarization waves moving away from the leads make a negative deflection. These

deflections are opposite for repolarizing current-i.e. repolarizing current moving towards the lead creates a negative deflection, and vice-versa.
Figure 1.6: The three augmented leads and three limb leads of a 12-lead electrocardiogram. Notice that the location of the positive electrode is important, as it dictates the expected direction of deflection on the ECG [77],
In traditional placement, lead VI sits over the right ventricle, and lead V6 sits over the left ventricle, with the other leads spanning the space on the front of the chest between VI and V6 at semi-regular intervals. The atria are at the back of the chest cavity and are not directly under any leads. This, combined with the smaller muscle mass of the atria, means that atrial depolarization causes a positive, but small, deflection, termed the P-wave. The P-wave is generally small because deflections are proportionate to muscle mass. In the ventricles, the left-to-right spread of current from the bundle branches creates a positive deflection in VI and a negative deflection in V6, termed the qQwave. The electrical current then enters the main muscle mass of the ventricles. In a normal

heart, the left-ventricular mass is much larger. Thus, lead VI records a large negative deflection and lead V6 records a large positive deflection. This large deflection forms the latter half of the QRS-complex. An abnormal morphology of the QRS complex is indicative of imbalances in the ventricles. Ventricular repolarization produces a positive deflection in V6 and a negative deflection in VI, termed the T-wave. The T-wave is sloping and less prominent, because repolarization is considerably longer in duration. The T-wave and QRS-complex are, furthermore, often deflected in the same direction in healthy hearts. Repolarization of the atria is not detectable on a standard 12-lead ECG because it is buried within the much larger QRS-complex.
It should be noted that one common difference in the placement of ECG electrodes concerns the chest leads, which may be placed in the above arrangement or in a right-sided arrangement. For our data set, lead V3 was often placed in a V3R arrangement. Thus, lead V3 was excluded from our analysis because it measured the electrical activity of the heart from an unreliable and changing perspective.
It should also be noted that, within the chest cavity, the heart is often in unique arrangements, i.e. no individuals anatomy is precisely that indicated in textbooks. This means that ECG data has very high variation between patients. This is inherent to the modality of ECGs, and means that large data sets are useful to reduce the statistical variations that this gives rise to. Thus, methods to normalize Fourier coefficients by the orientation of the heart could improve our methodology.

1.4 Previous Work in ECG Processing
1.4.1 Time-domain Analysis of ECG Signals Advantages of Time-domain Analysis
The time-domain analysis of ECG signals offers a significant advantage over other methods of feature extraction- being more highly understandable. Twelve-lead ECG signals are analyzed in the clinic by electrophysiologists, searching for identifiable morphologies that are inherently in the time-domain. The features identified by clinicians are manifold, and will not be discussed in entirety. See Figure 1.7 below for a depiction of the landmarks that electrophysiologists use to analyze ECGs in the clinic. Instead, this discussion will focus on features that have been analyzed with computer methods. The basis for clinical understanding of ECGs is in the time-domain, and most literature on interpreting ECG signals concerns time-domain features. Therefore, time-domain analysis of ECGs is largely limited to features that are already clinically understood. This offers strength in the interpretability of these analyses, but limits the possibility of novel findings. This thesis is focused on revealing novel diagnostic and prognostic information hidden within ECGs. Our analysis takes place in the frequency domain, thus, understanding of physiological relevance is sacrificed in the pursuit of new and interesting results. We believe this is an important pursuit in the well-established field of interpreting the progression of PH using time-domain ECG features.

Figure 1.7: Landmarks of a normal electrocardiogram, in terms of waves, segments, and intervals. The PR interval marks the time from the initiation of atrial depolarization to the initiation of ventricular depolarization. The PR segment marks the pause at the AV node. The QRS complex marks the beginning to end of ventricular depolarization, which includes atrial repolarization. The ST segment marks the pause between the end of ventricular depolarization and the beginning of ventricular repolarization. The QT interval marks the time from the initiation of ventricular depolarization to the completion of ventricular repolarization [85],
1.4,1,2 Previous Work in Time-domain ECG Analysis
The ECG is generally used after clinical suspicion of PH has been aroused, and allows clinicians to search for right heart disease as a cause [17], It should be noted that the progression of ECG markers throughout development of PH from the beginning of increased pulmonary resistance through cardiac failure has not been very well classified. This is because patients often do not present to the clinic for ECG testing until PH is well-developed. Early stage asymptomatic PH is not well-quantified in terms of ECG morphology, largely because ECG morphology remains unchanged until pulmonary vasculature pathologies have a notable impact on heart function. The understanding of the ECG in PH is improving as the risk groups for development of PH are becoming better defined [18], The changes in ECG morphology throughout progression of PH can

be approximated by comparing ECG at time of death to normal ECG and ECG at time of diagnosis.
ECG at time of PAH diagnosis is often inconclusive, but by time of death there are generally significant abnormalities. A 2013 study suggests that certain ECG parameters progress throughout the duration of the disease, namely: heart rate increasing, PR interval increasing, QRS duration increasing, R/S(V1) increasing, QTc duration increasing, and increasing right-deviation of QRS axis [19], This list of parameters is not exhaustive, but when combined with the parameters discussed in an earlier section, offers a thorough review of the most well-researched ECG time-domain features. Median heart rate is a self-explanatory parameter. PR interval represents the time it takes for an electrical signal to travel from the sinus node to the ventricles, and thus is a measurement of conduction speed and delay at the atrioventricular node. The duration of the QRS complex represents the time it takes for both ventricles to depolarize. An increased R/S ratio is indicative of many different pathologies, most relevant to this case being right-ventricular hypertrophy. QTc duration represents the time between the initiation of ventricular depolarization and the completion of ventricular repolarization. QRS axis rightward deviation can, again most relevantly, be indicative of right-ventricular hypertrophy [20], With such physiologically intuitive measurements of pathologies like right-ventricular hypertrophy, it should come as no surprise that PH patients with altered right heart anatomies and function can be recognized with ECG time-domain features.
Zhou et al. [22] have developed a method that uses quantile distributions of time-domain features such as R-peak and waveform slope, reduced in dimension by PCA, to classify diseased (cardiomyopathy, dysrhythmia, and bundle branch block) versus control

patients with accuracy nearing 90%. The classifier used in this case was an SDA classifier, though SVM and LASSO classifiers were also considered. Zhou also reports on features and classifiers used in previous works. Features that have been extracted from ECGs in previous works include heartbeat intervals, frequency features, cumulant features, Karhunen-Loeve morphologies, and hermite polynomials. Classifiers that have been used in previous works include linear discriminants, decision trees, neural networks, support vector machines, and Gaussian mixture models [22], These features and classifiers are discussed in brief later in the introduction. The weakness of many of these feature-classifier pairs are that they are very difficult to interpret. A discussion of these difficulties can be found in a subsequent section.
Zhou et al. [23] have reported a functional model that describes the morphology of ECGs in a highly understandable way. This methodology is effectively a form of statistical shape-modelling. The model shape-matches the T-wave of individual ECGs to a known shape, and develops a classifier based on deviations from this known shape. The deviations are organized in to categories, i.e. higher slope on leading edge, and these categories are linked to clinical diagnoses [23], Though this methodology has high physiological relevance, it is limited to non-novel features in the time-domain, and much more difficult to implement than our simple methodology. The model is a functional-data model, which measures statistical variations in functions that have been fit to the T-wave [24], Ease of implementation has been sacrificed for high understandability here.
Real-time ECG classification, while not the theme of this thesis, has been pursued by some research teams. The time-domain Pan-Tompkins QRS location algorithm has been applied to ECGs in real time, to obtain values for the delay of certain landmarks

such as T-wave onset, and these values have been shown to predict arrhythmias with over 97% accuracy. This paper is representative of much of the work done in the field of ECG classification, in that it pertains to classification of arrhythmias. Time-domain features are used in the case of real-time classification because they generally require less computing power and thus are more suitable to smartphone applications [25], This paradigm will likely shift in the future as mobile computing power increases. A shift away from time-domain analysis will likely be promising for the diagnostic capability of the ECG.
The ECG has long been thought to have less predictive power than echocardiography and Doppler ultrasound. The presence of right-ventricular hypertrophy as diagnosed by ECG time-domain signals was shown to not have a relationship with right-ventricular pressure as measured with ultrasound. When viewed in conjunction with studies presented later in the introduction, which indicate that right-ventricular pressure can be correlated to ECG when using non-time-domain features, this gives a basis for looking outside of the time-domain to improve the clinical usefulness of ECG [26],
1.4,1,3 Summary of Time-domain ECG Analysis
Easier to interpret because features are directly related to established clinical measurements taken by electrophysiologists
ECG features for early PH are not well understood because patients dont present to the clinic until heart function is significantly altered [18]
Time domain features change linearly from early to late disease stages

Features such as R-peak amplitude and waveform slope have been analyzed with quantiles to diagnose cardiac disorders such as bundle branch block with 90% accuracy
Functional models that describe changes from normal ECG morphologies are easy to interpret
1.4.2 Frequency-domain Analysis of ECG Signals Rationale for Frequency-domain Analysis
The power spectrum of ECG has been shown to quantitatively change in the minutes preceding sudden cardiac death. This gives an intuitive basis for the idea that the frequency-domain of ECG contains predictive power for the development of pathologies [27], The power spectral density of ECG, which can be estimated as the square of the magnitude of the frequency spectrum as obtained with a Fourier transform, was analyzed in 1991 to reveal that patients with myocardial infarctions and ventricular tachycardia had frequency components over 80Hz whereas control patients did not have such features in this area of the spectrum [28], Thus, patients who are near to cardiac failure, or who have suffered physiological and anatomical changes to the heart, have a frequency-domain signature that is identifiable. This thesis intends to strengthen the ability of ECG as a prognostic for patients who must undergo catheterization and MRI as part of their disease management plan, in the hopes that ECG may become a routine prognostic measurement that can be taken more often than MRI or catheterization measurements, likely in conjunction with echocardiogram measurements [29], In past literature, Fourier coefficients obtained from DFT are often used to suppress or alter the signal to noise ratio.
18 Previous Work with Fourier-transform in ECG
Pure Fourier-transform of ECGs has generally been limited to signal de-noising and signal compression. While the Fourier transform has indisputable value in these applications, the prognostic value of Fourier coefficients remains under-researched [30], To explain the predictive power of Fourier coefficients, it is useful to recognize that they represent a signature distinct to each persons biology. In fact, the coefficients are sensitive to short-term changes in an individuals cardiac output.
Linear discriminant analysis (LDA) and cross-correlation have been used to extract information from ECG Fourier coefficients. Pre-processed ECGs (discussed in a subsequent section) were converted to frequency-domain with FFT, and a pattern recognition algorithm involving LDA and correlation coefficients was used to train the data using the frequency-spectrum of well-defined ECG classes. This method produced classification accuracies on the order of 90% for ECG states such as resting, fear, exercising, smoking, etc [31], This study showed that the frequency-domain contains detailed information regarding heart activity. Subsequent research has showed that Fourier coefficients contain information that is related to an individuals anatomy and physiology, as well as activity state.
The concept that Fourier coefficients hold information unique to an individuals ECG is corroborated by human identification techniques that, in part, use Fourier coefficients. These coefficients were reduced in dimension to eliminate redundancy and then features that could identify an individual were selected with an artificial neural network (ANN) [32], Though this offers fairly definitive proof that ECG frequency

information is unique to the individual, it remains difficult to explain what the relevant Fourier coefficients represent in an ECG, much less in a living heart.
Frequency-features are inherently difficult to interpret, but their correlation to time-domain features indicates that this is possible. Fourier coefficients obtained from the discrete Fourier transform (DFT) have been shown to be useful in classifying QRS complex morphology [33], The coefficients have diagnostic power regardless of their time-domain analogues.
Most of the work on the diagnostic potential of the frequency-domain of ECGs concerns classification of arrhythmias. A grey-relational classifier was applied to coefficients from a fast Fourier transform (FFT) to classify the presence of heartbeat classes and thus cardiac arrhythmias, with promising diagnostic accuracy [34], ECG peaks identified by FFT, and used to train an ANN, showed promising results for the classification of cardiac arrhythmia diagnoses [35], These techniques are purely frequency-domain, and frequency related to time-domain, respectively. This duality indicates that the coefficients can serve as a stand-alone diagnosis, and can be physiologically interpreted if the extra computational effort is expended. Another way to increase efficacy and understandability while using more computational power involves combining frequency-domain features with features from other domains.
1.4,2,3 Elsing Frequency and Phase to Localize Signals
A 2012 study combined frequency-domain with phase-domain to successfully localize the location of scar tissue as a source of high-frequency arrhythmias. This study was motivated by the observation that localized high-frequency sources maintain ventricular fibrillation. The frequency information was obtained by Fourier analysis, and

the phase information was obtained by Hilbert analysis. The phase information was analyzed by tracking the propagation of a frequency signature across multiple leads, and using this information combined with an in-depth knowledge of the anatomical relation of ECG leads to localize the source of the frequency signature. This methodology could be useful in future studies to discover the physiological significance of our findings, i.e. unravelling the physical structure(s) that our predictive Fourier coefficients relate to [36], More work has been done in combining the frequency-domain with the time-domain.
1.4,2,4 Summary of Frequency-domain ECG Analysis
The frequency domain is generally harder to interpret than the time domain in terms of physiology
The frequency domain is quantitatively altered in many cardiac pathologies
Pure Fourier transform is generally reserved for signal de-noising, but is combined with other features for analysis
Fourier coefficients can predict physiological state such as resting, fear, smoking, etc.
Fourier coefficients have been used to develop recognition algorithms that are sensitive to individuals
The frequency and phase-domain have been combined to localize infarctions
1.4.3 Time-frequency-domain Analysis (TFA) of ECG Signals
The combination of information from multiple domains allows for more accurate diagnosis and prognosis from a smaller number of ECG signals. Time-frequency-domain analysis (TFA) consisting of taking the frequency spectrum in time-windows and comparing the ratios of frequency areas across windows revealed decent sensitivity and

specificity for arrhythmia-associated right-ventricular dysplasia. Right-ventricular dysplasia is a myocardial condition in which heart muscle tissue is replaced with fatty or fibrous deposits. These results concerning arrhythmias are in line with the work that has so far been discussed. The same paper goes on to indicate that combining the frequency-domain analysis with classical time-domain features improves diagnostic capability significantly [37], This combination of domains reduces the amount of data that is necessary to make an accurate classification.
1.4,3.1 The Fourier Transform in TFA
The simple method for obtaining time-frequency features by applying a Fourier transform along a moving time-window has been used outside of arrhythmia research. A similar TFA involving area ratios of Fourier-obtained frequency spectrums taken at windows post-QRS complex revealed that time-frequency domain features can have higher predictive power of the presence of myocardial infarctions when compared with traditional time-features such as QRS duration and late potential duration [38],
The TFA transform methodology of taking a Fourier transform along a moving time window is also known as the short-time Fourier transform (STFT). A 2017 study used the STFT, which combines the time and frequency information in a manner like the discrete wavelet transform (DWT), to obtain features from a single-lead ECG. These features were then reduced in dimension with the synchrosqueezing transform (SST), a dimensionality reduction algorithm tailored specifically to time-frequency domain problems [39], The time-frequency domain can also be used to improve the quality of time-domain ECG analysis.
22 Other TFA Techniques- Wavelets and Auto-Regression
TFA of ECGs is useful for improving signal quality, and thus time-domain feature prognostic ability. A TFA based on time-varying auto-regressive coefficients was used to automatically detect and remove artifacts from an ECG, to improve the performance of a QRS complex detection algorithm. This type of TFA analysis is well-suited to the identification and removal of high-frequency components [40], This paper offers autoregressive coefficients as a valid feature for ECG classification. Much of the literature concerning TFA involves DWT. One particularly efficacious example of DWT-based TFA is discussed.
Auto-regressive coefficients and wavelet coefficients from wavelet transform were obtained from a single-lead ECG. Thus, this study exemplifies the power of TFA to analyze smaller datasets. These features were combined and input in to a support-vector machine classifier to obtain over 99% classification accuracy for heartbeat conditions such as normal, left bundle branch block, right bundle branch block, etc. [41], So far, this review has focused on ECGs as classifiers. Another important use for ECGs involves correlating the signal to measurements obtained from MRI and catheterization.
1.4.4 Correlation of ECG Signals with Other Modalities
Previously literature shows that information from ECG has been correlated with parameters from right-heart catheterization (RHC) and MRI. The parameters discussed in the literature include RVC, right-ventricular pressure (RVP), pulmonary vascular resistance (PVR), and mean pulmonary arterial pressure (mPAP). No evidence was found for a previous link between ECG and vascular ventricular coupling ratio (VVCR).
23 Using ECG to Measure RYC
One important measure of cardiac function used in the prognosis of PH is RVC.
In this thesis, end-systolic elastance (Ees) was used as a measurement for RVC. This quantity was estimated from the single beat method of Brimioulle et al. [79], which computes Ees as the ratio of maximum minus end systolic pressure over stroke volume [(P max Pes )/SV], The pressures used in this calculation are measured with right heart catheterization, and stroke volume is measured via MRI [42, 43], ECG processing techniques have been developed to indirectly measure RVC, both expressed as Ees and in other forms [44], Some techniques rely solely on ECG data from the time and/or frequency domains, while others combine ECG data with information from other modalities.
ECG has been used by itself in correlations with contractility, both for the left and right ventricle. A 2007 Czech study found that QRS duration is negatively correlated with the left-ventricular contractility index (dP/dtmax), an invasive measure of contractility obtained via catheterization, with a high negative correlation coefficient (r = 0.81). This study was undertaken for the evaluation of a cardio-protective agent to be used in conjunction with cancer therapies [44], QRS duration represents the time it takes for both ventricles to fully depolarize. The correlation of QRS duration with contractility was not explained in physiological terms further than the observation that both parameters are concerned with the ventricles. One can speculate that, because left-ventricular contractility is measured as change in pressure per change in unit time, faster depolarization requires a larger pressure gradient to form to pump the same volume of

blood. Other time-domain features of the ECG have been correlated with more indirect
measurements of contractility.
Past experiments have indicated that the ECG time-parameter of QT duration is correlated with (left) ventricular contractility. This was decided in a pair of experiments on mongrel dogs where heart rate was increased, and then heart rate was kept constant while ventricular pressure was increased, both by exogenous electrical stimulation. In the former iteration, corrected QT interval remained constant. In the latter iteration, the QT interval was found to be altered. Because the first iteration held ventricular contractility constant relative to heart rate, and the second iteration altered contractility, these results indicate that changes in QT interval are associated with changes in ventricular contractility [45], It should come as no large surprise that QT duration is related to contractility. QT duration is a measure of the time it takes for the ventricles to fully depolarize and then repolarize, hence, both QT duration and contractility are measurements concerned with the ventricle. The following studies show that ECG parameters can be combined with measurements from other modalities to predict contractility.
ECG has been combined with echocardiography to reveal a correlation with (left-ventricular) contractility in healthy, non-PH rats. This problem is relevant to rats and small animals because less-invasive measurements such as catheterization are more difficult here, so fatal surgical procedures are often used to measure contractility. The QA interval, defined as the time from the Q wave to point A on the aortic pressure waveform, has shown that a combination of ECG and echocardiography data has a correlation with contractility as measured by surgical methods in rats. Though this study

was undertaken in rodents, the results offer promise that ECG contains some contractility-related information in humans as well. Again, QA interval is a measure of ventricular function, in this case the time from the initiation of ventricular depolarization to the pressure front of blood leaving the ventricles in to the aorta, and thus it should come as no great surprise that it is correlated with the contractility of the ventricles [46], This relationship has been shown with ECG and modalities other than echocardiography.
ECG and ballistocardiogram (BCG) have been combined to reveal a strong correlation with non-invasive measurements of contractility. BCG measurements represent the mechanical activity of the heart, and are recorded from the skin surface. Catheterization is one of the most effective ways to measure contractility, but is a highly invasive procedure. Preejection-period (PEP), which is a parameter taken from the ECG combined with the impedance-cardiogram (ICG), is a non-invasive surrogate measurement for contractility, where the ICG measures the electrical conductivity of the thorax to understand the activity of the heart. ECG and BCG were also shown to be combinable for correlation with PEP. This study was motivated by the observation that, while ICG is technically non-invasive, it is, in the words of the authors, far from unobtrusive. This correlation was found by taking the time from the ECG R peak to the J wave peak on the BCG, and using linear regression to find a coefficient of determination (r2 = 0.86) [47], This link between ECG-BCG measurements and contractility has been further studied. ECG and BCG can also be combined to estimate PEP using the ECG R-wave and the BCG I-wave peak, giving a correlation coefficient around 0.85 [48], Correlations between ECG parameters and measurements from other modalities are not limited to the ECG-contractility relationship.
26 Using ECG to Measure mPAP
One popular method for definitive diagnosis of PH, as mentioned earlier, involves mean pulmonary arterial pressure (mPAP) measurement by catheterization. A 2016 study by Kovacs et al. aimed to develop the ECG, in conjunction with other non-invasive measurements (the biomarker NT-proBNP), as a diagnostic surrogate for catheterization. The simple use of right-axis deviation (heart electrical axis >= 110) from the ECG was combined with several biomarkers, including NT-proBNP levels, to predict a probability that an individual has PH. The efficacy of the method was not particularly high, with the correct class being recognized in only half of patients at risk for PH. However, Kovacs study simultaneously indicates that there is likely simple time-domain feature based methods for accurately classifying PH versus control patients, and that the ECG likely has more diagnostic value within its data [21], This observation is corroborated by work on disease states outside of PH.
A 2003 Japanese study on patients with acute massive pulmonary thromboembolism (AMPTE) used simple statistics to evaluate the link between ECG abnormalities and mean pulmonary arterial pressure (mPAP), an important measurement in evaluating the severity of disease for AMPTE patients. In this thesis, mPAP was determined by right heart catheterization measurements. MPAP levels above 45mmHg were found to correlate (r = 0.82) with the number of ECG abnormalities in Japanese patients with Pulmonary thromboembolism (PTE). These ECG abnormalities were all clinically diagnosable time features, including: S(1)Q(3)T(3), negative T, clockwise rotation, and ST elevation [49], These parameters can be physiologically explained, respectively, as indicative of right heart strain and right ventricle hypertrophy,

physiological consequences of elevated pulmonary arterial pressure. Observe that, per the textbook guidelines mentioned earlier in the introduction, these ECG findings would indicate a greater muscle mass for the right ventricle than the left ventricle. We have seen that the measurement of catheter-based contractility, via PEP or Ees, can be correlated with ECG. Similarly, catheterization measurements of mPAP can be estimated from another parameter- PASP. Correlation between mPAP and PASP allows for interpretation of ECG-PASP results [50], This result is mentioned here because it may be useful for researchers who are more interested in diving specifically in to mPAP. An in-depth discussion of ECG-PASP correlation is not included herein.
1.4,4.3 Using ECG to Measure PVR
Another important measurement for PAH patients is pulmonary vascular resistance (PVR). PVR is the resistance component of impedance to the right ventricle, and is a relevant measurement in PAH that measures pre-capillary resistance. In this thesis, PVR was measured via right heart catheterization, and calculated as (PVR = (mPAP-PCWP)/CO), where PCWP is the pulmonary capillary wedge pressure and CO is the cardiac output. PCWP is a measurement of left atrial pressure, and CO is a measure of the volume of blood pumped by the heart per minute. CO is the product of heart rate and stroke volume (SV), where SV is measured by MRI.
Patients with PAH were shown, in a Chinese study from 2010, to have QRS axis correlated with mPAP (r = 0.75) and R(V1) + S(V5) correlated with PVR (r = 0.74). The deviation of the QRS axis is physiologically indicative of altered mPAP due to the reasons stated above. The R(V1) + S(V5) measurement shows dominance of the right

ventricle, and is thus indicative of right-ventricular hypertrophy, which would be observed at stages as the heart adapted to increased pulmonary vascular resistance [51], Using ECG to Measure RVP
Right-ventricular pressure is another parameter that has been thoroughly analyzed in conjunction with ECG features. The relationship between time-domain ECG features and right-ventricular hypertrophy (RVH) has been known for decades. Though RVH and RVP are obviously related, the measurement of RVH has proven easier. This thesis discusses RVP as right ventricular systolic pressure, measured by right heart catheterization. A 1994 publication in the Journal of Electrocardiology aimed to improve the sensitivity, or true positive rate, of the ECG as a diagnostic tool for right-ventricular hypertrophy (RVH) and for right-ventricular systolic pressure (RVP). The study was carried out in adult populations that had more well-defined cardiovascular anatomy. The results revealed an intuitive link between ECG and RVH, but left much to be desired in the link between ECG and RVP when RVP was below the high level of 45 mmHg [52], This link was found in subsequent studies.
The time-domain feature of QRS duration has been shown to have predictive power for right-ventricular pathologies in patients with a variety of congenital heart diseases. This parameter was observed to increase throughout the development of disease, especially in groups with congenital heart diseases that involved sub pulmonary pressure overloading of the right ventricle. Thus, patients with altered right-ventricular end diastolic volume (RVEDV) caused by chronically increased right-ventricular pressure also had altered QRS durations. Because RVEDV affects the pressure in the right ventricle, via the Frank-Starling mechanism, this finding sets the stage for a link

between ECG and RVP [53], A diagnosis of high or low RVP was subsequently found to be classifiable with simple time-domain features.
The time-domain features of P-wave amplitude(II), QRS axis orientation, and T axis orientation, were found to be valid classifiers to low or high right-ventricular pressure, as determined by ROC analysis. The AUC values were 0.80, 0.70, and 0.90, respectively [54], These parameters are all easily interpreted, respectively, as follows: atrial muscle mass increase necessary to compensate for higher than normal pressures in the ventricles, and the muscle mass discrepancy between the right and left half of the heart during both depolarization and repolarization of the ventricles. Some parameters that are more esoteric and archaic have also been shown to have correlation with RVP.
Ventricular gradients can be used to detect slight changes in RVP. Ventricular gradients express the difference in area contained by the QRS-complex and by the T-wave. More intuitively, they are a time-domain feature that measures the heterogeneity in ventricular action potential duration, and thus is a surrogate measurement of the variability of heart mechanics between beats. A 2008 study used ROC classifiers to link diagnosis of elevated right-ventricular pressure load with ventricular gradients (AUC = 0.99). This result indicates that altered RV pressure load is reflected in a change in the heterogeneity of the duration of ventricular action potentials [55], The studies presented so far demonstrate that the ECG contains information that we are currently using invasive and expensive methods to measure in the management of PH and other cardiovascular diseases. However, none of the past literature has drawn a direct link between ECG Fourier coefficients and the measurements of other modalities. Thus, our research is a beginning to filling this hole in the literature, and the methods that we describe can likely

be applied to many problems in cardiovascular medicine outside of PH. Regardless of the modality that is being compared to ECG, there are common statistical techniques that can be used to develop classifiers using ECG information. Summary of Work with ECG and Invasive Parameters
Previous literature has related ECG time-domain features to left-ventricular contractility
ECG has also been combined with echocardiography and ballistocardiogram in correlations with left-ventricular contractility
The ECG has been combined with neurohormone levels in correlations with mPAP
The number of clinically determined abnormalities in an ECG has shown to be a strong predictor of mPAP levels
ECG time-domain features which indicate right-ventricular strain, such as R(V1) + S(V5), have been correlated to PVR
Time-domain features such as P-wave amplitude, QRS axis, and T axis have been shown to be valid classifiers of RVP levels
Ventricular gradients, which measure the heterogeneity in ventricular waveforms, have also been correlated to RVP
1.4.5 Statistical Classifiers for Disease State Diagnosis from ECG
1.4,5.1 Artificial Neural Networks and ECG
The following discussion of classifiers can be found referenced in numerous machine-learning and statists texts [57], A 2010 IEEE study evaluated the effectiveness of logistic regression, decision trees, and artificial neural networks (ANNs) as classifiers

for detecting the presence of an identifiable heart condition. ANNs were found to have the highest classification accuracy and the lowest error rate. The neural networks, in brief, consist of many simple decision units connected by variable weightings. ANNs are suitable for regression problems wherein a model is iteratively achieved by reducing an error function. A starting number of inputs are combined to a smaller number of intermediaries, and this process is repeated until the number of classes matches the desired outputs. The disadvantage of this method is that trained networks are difficult to interpret, and thus it is hard to decide if the results are a mathematical fluke or have physiological relevance [56], Logistic regression was used in the methodology of this study and will thus be discussed in the methods section, though it is quite simple and does not warrant a lengthy explanation. Decision tree models use a flow-chart like structure to describe sequential decisions, often using simple rules like thresholding for an input value, and their possible outcomes. Random forests (RF) are a type of decision tree in which trees are separately trained via bootstrapping, and final decisions are computed using the majority vote of separate trees. These classifiers are like ANNs in that they are machine learning algorithms that are difficult to interpret, but differ in that ANNs iteratively compare regression models and decision trees form simpler if/else statements at nodes. Again, this is a method that can achieve very high efficacy results by sacrificing understandability with a mathematical black box that is not necessarily intuitive, but more so than an ANN.
1.4,5,2 Other Classifiers Used in ECG Literature
Other classifiers that have been used in ECG analysis include: stepwise discriminant analysis (SDA), support vector machines (SVM), least absolute shrinkage

and selection operator (LASSO), principal component analysis (PCA- a dimensionality reduction technique used in conjunction with many classifiers), linear discriminant analysis (LDA), and Gaussian mixture model classifiers. These methods of classification are described in brief. First, the general features of classifiers are discussed. For each classifier listed below, a geometric representation is a valid and useful tool for understanding and visualizing the function of the algorithm. Predictor variables reside in a hyperspace with dimensionality corresponding to the number of predictors-observations from the same class are clustered together. Supervised learning algorithms are those in which the correct clustering of classes is already known. These exemplary predictor-response pairs are used to estimate a function that matches predictors to responses with desired accuracy.
Stepwise discriminant analysis (SDA) is a form of supervised discriminant analysis in which multiple continuous predictors are used to select a categorical dependent variable by creating many functions and assigning each an eigenvalue based on its efficacy at the classification problem. The functions with the best eigenvalues are then selected as the most descriptive. A support vector machine (SVM) is a supervised classification algorithm that constructs a hyperspace from all inputs and outputs and finds a hyperplane that offers the best solution to categorize inputs in to output classes. LASSO uses regression to find the best fit for data using the fewest coefficients on predictors. PCA is a dimensionality reduction technique often used as a pre-processing step before regression or classification. PCA takes the data points of dimension corresponding to number of predictors and seeks to define a space of smaller dimension that maximizes the separation of the data points, by analyzing projections of the data points (i.e. projecting a

flashlight on to a 3D scene creates a 2D shadow-based representation). LDA is a dimensionality reduction technique that is generally performed prior to classification, that functions by projecting a dataset in to a lower dimension. In this manner, LDA is like PCA in that it removes correlated data by locating axes to maximize the variance in the data points, but differs in that LDA is supervised and PCA is unsupervised, while both maintain the ability to discriminate between established classes. LDA can do this because it is a supervised learning technique, whereas PCA is an unsupervised algorithm.
Gaussian mixture models perform supervised classification under the assumption that the distribution of data points comes from a finite number of Gaussian distributions, and arranges these Gaussian distributions to achieve maximum separation between clusters of data points, which are then defined as classes [58], A simpler Gaussian method, which tests for the deviation of signal parameters such as R-wave amplitude and QRS-duration against a golden standard with a Gaussian distribution for these values, can also be used to perform classification [59],
1.4.6 Other Features Used in ECG Processing
The description of the following mathematical techniques can be found in many graduate-level digital signal processing textbooks [60], Features that are not strictly time-domain that have been extracted from ECGs in previous works include wavelets, cumulant features, Karhunen-Loeve morphologies, hermite polynomials, and auto regression coefficients. The time-feature Pan-Tompkins algorithm and Hilbert transform for QRS detection are also discussed, because they are highly important to many analyses in non-time spaces.
34 Wavelets as ECG Features
Martis et al. described the use of higher order cumulant features and wavelets for the diagnosis of cardiac diseases [61], The researchers located and extracted the QRS complex using Pan-Tompkins algorithm, discussed below. A discrete wavelet transform (DWT) was then applied to capture both frequency and time information. The DWT was developed after the observation that the Fourier transform does not adequately capture sharp contrasts in time-domain data, because the transform is not localized in time. Wavelets can surpass this difficulty by localizing frequency data in time. Wavelets are finite-duration, zero-mean waveforms with pre-defined shapes that are scaled in duration and shifted along a time axis and then combined to represent a time-domain signal. The choice of wavelet shape and the scaling contain the frequency information, and the amount of shift contains the time information. A signal is decomposed in to wavelets by shifting wavelets along windows of defined length, scaling to capture frequency information in that window, and adding the wavelets together to achieve an accurate representation of the signal. The coefficients that define this transform can be mathematically analyzed in a manner like that of our methodology. The authors go on to use cumulant features to describe the wavelets. These cumulant features essentially capture the probability distribution of the wavelet coefficients. These distributions were then reduced in dimensionality with PC A and subjected to an SVM classifier.
1.4,6,2 Karhunen-Loeve Morphologies as ECG Features
Karhunen-Loeve morphologies (KLM) have been used to analyze the presence of acute myocardial infarctions [62], KLM uses eigenvector coefficients to describe the features of an ECG lead, and in this way, is KLM is mathematically like PC A. KLM was

applied, in this case, to the time-domain feature of J-point to T-wave delay. The covariance matrix, which represents the covariance, or linear association, between all combinations of data points, was calculated for these features, and the eigenvectors were identified and combined via KLM to achieve a lower dimensionality description of the data. Hermite Polynomials as ECG Features
Hermite polynomials are a set of orthogonal polynomials that are given weighting functions to represent a set of data points. These polynomials exist in a pre-defined set, that is the function describing each Hermite polynomial is a well-defined algorithm that can be derived from Taylor series and the Sheffer sequence. The Hermite polynomials are useful in ECG characterization because they have a similar geometrical shape to the QRS complex. Deviations from the expected Hermite polynomial values reveal alterations in QRS complex morphology. Auto-Regressive Coefficients as ECG Features
Auto-regressive (AR) coefficients are not easily explainable in intuitive terms. They arise from the transfer function of a linear time-invariant (LTI) filter, and can be thought of as representing the envelope of the power spectrum (a frequency-domain representation). The AR coefficients generate internal dynamics because they rely on previous values, hence the name auto-regressive. The number of previous values that have effect on the current value corresponds to the order of the AR model. The AR model is often calculated via the Burg maximum entropy method, which was developed in physics as a method to define the best possible quality frequency. This method is

analogous to dimensionality reduction techniques in that the final coefficients are those that are least-related to each other. Pan-Tompkins QRS Detection Algorithm
Pan-Tompkins algorithm was described in a seminal 1985 paper to detect the QRS complex [63], The algorithm relies on the recognition of slope, amplitude, and width features that are indicative of a normal QRS complex signature. Initially, the ECG is pre-processed to remove noise and artifacts- this process is further described in a subsequent section. The slope information is obtained using an LTI derivative filter. The results of this operation are squared to remove negative derivatives and emphasize the high frequency components that correspond to the useful ECG data. This information is passed through an integrating filter with a window that is approximately the width of the widest QRS complex and thus serves to separate the QRS complex information from the other information in the signal. The rising-edge of the result of the integration process is representative of the QRS complex, being localized in the time-domain by initiation and duration. Hilbert Transform for QRS Complex Detection
Another QRS detection algorithm utilizes the linear time-variant filter of the Hilbert transform to accurately localize the R-peak. This is done by using an odd-function filter, the output of which crosses the axis every time there is an inflection point in the ECG signal. The location of the inflection point is localized to the location of the R-peak because the transform is time-dependent, i.e. the current time-value is accounted for within the filter.

1.4.7 Methods for ECG Pre-Processing
Electrocardiograms represent a challenge for digital signal processing techniques, because of their spectral composition. The following information is from Luos review paper [64], As with other signals, the aim is to improve the signal to noise ratio (SNR) by reducing noise while preserving the useful signal information. This is difficult in the case of ECG signals because the power spectrums of signal and noise are overlapping. The most common sources of noise in ECG signals are low-frequency baseline wander due to motion artifact, mid-frequency noise due to powerline interference, and high-frequency noise due to muscle signal artifact. Each of these sources of noise is discussed in brief, because, in the authors view, commercial ECG machines have hardware and software that do an adequate job at limiting noise due to each of these sources. The basics of filtering are also briefly discussed.
ECG signal pre-processing generally takes a signal from the time-domain and represents it in the frequency-domain by its phase and magnitude responses to a given filter. Components of the frequency-domain representation can then be mathematically altered to affect the time-domain signal. In real-world applications, this generally involves a trade-off where signal quality is degraded by filtering because the frequency representation of the noise and useful signal cannot be perfectly separated. Distortions can also occur when the phase-information encoded in the frequency domain is altered.
The in-house signal processing of a 12-lead ECG device occurs mostly within a digital system. Powerline noise is attenuated with hardware, and then the signal is converted from analog to digital using a particular sampling rate- generally over-sampled. The sampling rate introduces an analog low-pass filter (LPF). A second, digital, LPF is

introduced later in a down-sampling process. A high-pass filter (HPF) is also introduced in the digital system to remove baseline wander and other low-frequency information that is not needed. Finite-impulse response (FIR) filters can be used to remove unwanted frequencies without altering phase information, but these filters take much longer to implement. Another approach is to use infinite-impulse response (HR) filters, which will distort phase, but perform their function much more quickly. By combining two HR filters in a row, the distortion to phase information can be removed. A notch filter is generally utilized to remove powerline interference, if necessary. Many different filters can be implemented for each purpose. This review does not further discuss the intricacies of digital filter design.
1.5 Justification
Pulmonary hypertension is a serious illness that has no known cure, and PH is severely understudied in pediatric populations. However, if diagnosed early enough and monitored carefully, the consequences of the disease can be mitigated. This diagnosis and mitigation requires unfortunately invasive procedures such as right-heart catheterization, as well as expensive MRIs. These procedures are done infrequently- at most once a year. Echocardiogram is taken more often, but is still relatively expensive. Thus, developing the ECG allows for more consistent and affordable monitoring of the disease. This means that pediatric PH patients face a disease with a large impact on quality of life- from time spent in hospitals, restrictions on physical activities, and high costs for drugs and tests, pediatric PH patients and their families need better options. The frequency spectrum of ECGs contains clinically useful information, that may serve as a surrogate measurement for pressures in the heart and surrounding vasculature.

1.6 Goals
Here, we propose a novel analysis of ECG data that may someday offer a cheap, routine prognostic modality for evaluating cardiac function and disease severity in patients with pulmonary hypertension.
Specific Aim 1: Develop a classifier using Fourier coefficients as predictors to discriminate between CTL and PH patients.
Specific Aim 2: Develop linear regression models to explore the correlation between ECG Fourier coefficients and VVCR, PVR, RVP, mPAP, and RVC.

2.1 Measuring Cardiac Function with Right-Heart Catheterization
The parameters of cardiovascular functions that we measured with RHC, and MRI in some cases, give a prognostic and diagnostic view of PH. The parameters, RVP, mPAP, PVR, VVCR, and RVC, are related through the etiology of PH. PVR is thought to be one of the earliest parameters to be altered in PH. PVR forms the resistive component of impedance. Increased PVR leads to an increase in mPAP. The combination of increased PVR and increased mPAP alters the mechanical properties of the pulmonary artery, known as arterial elastance, which is the reactive component of impedance. Alterations in arterial elastance, PVR, and mPAP are termed as alterations to right-ventricular afterload. An increase in RV afterload alters RVP and RVC, as the RV must work harder to pump blood through the lungs. Each component of this etiology was measured as follows.
2.1.1 Right-ventricular Pressure as Measured by RHC
Right-ventricular pressure (RVP), as reported by RHC, is a direct measurement of the pressure in the right ventricle during systole. The catheter is generally inserted through the iliac vein in the groin, in to the inferior vena cava, through the superior vena cava, entering the right atrium, and passing through the tricuspid valve in to the right ventricle. Once in the ventricle, a small balloon is inflated and used to measure the maximum pressure in the heart during contraction, or the maximum systolic pressure.

2.1.2 Mean Pulmonary Arterial Pressure as Measured by RHC
The mean Pulmonary Arterial Pressure (mPAP) is found by advancing the catheter from the right ventricle through the pulmonary valve and in to the common bundle of the pulmonary artery. The catheter is generally inflated in both the left and right branch of the pulmonary artery, and these pressures are averaged to give mPAP.
2.1.3 Pulmonary Vascular Resistance as Measured by RHC and MRI
Pulmonary Vascular Resistance (PVR) is a measure of the resistance in the blood vessels of the lungs. PVR is calculated as (mPAP PCW)/CO, where PCW is the pulmonary capillary wedge pressure and CO is the cardiac output. PCW is a surrogate measurement of left atrial pressure, obtained by threading a catheter far in to the pulmonary artery and inflating it to occlude blood flow. The pressure at the distal end of the catheter is then measured as an estimation of left arterial pressure. CO is calculated as heart rate multiplied by systolic volume (SV). Heart rate is a trivial measurement, and SV is obtained by subtracting the volume of a 3D-slice MRI reconstruction of the endocardium of the right ventricle during diastole from the volume of a reconstruction during systole.
2.1.4 Right-ventricular Contractility as Measured by RHC and MRI
Right-ventricular contractility (RVC) is a measurement of the mechanical functionality of the right ventricle. It is ideally measured by surgical techniques. In this thesis, RVC was non-invasively measured using the single-beat method of [79] as end systolic elastance (Ees), which is given as (Pmax Pes)/SV, where Pmax is the maximum of the sine wave fitted to the RVP curve, and Pes is estimated as the pressure 30ms before minimum dP/dt. These quantities can be found in Figure 2.1. In Figure 2.1,

the minimum of the RVP curve is the end diastolic pressure. A sine wave is fitted to the
isovolumetric region, which is seen in the pressure-volume loop of Figure 2.1. On the
RVP curve, the isovolumetric region is denoted by the dotted red line, which falls before
maximum dP/dt and after minimum dP/dt.
Figure 2.1: (A) Left ventricle pressure curve, which can be directly extrapolated to the right ventricle. Aortic valve opening point is maximum dP/dt, and aortic valve closing point is minimum dP/dt. The low point of the curve, marked as mitral valve closure, is

the diastolic pressure point. A sine wave is fitted between maximum and minimum dP/dt. (B) A pressure-volume loop. Isovolumetric regions are between points 1 and 2 and points 3 and 4. Events in the heart are noted in the figure. [89]
The surgical gold-standard for animals, which is not applied in human patients because of ethics standards, involves clamping of the vena cava. The superior vena cava is clamped to create a bolus of blood, which travels through the right atrium in to the right ventricle. This increases contractility (relative to normal volumes, per the Frank-Starling mechanism). The volume of the bolus is measured by MRI, and a catheter in the right ventricle measures pressures. The volume of the bolus is altered in different trials, and a pressure-volume loops is made from each trial. The slope between loops, connecting the point where the pulmonary valve closes, give Ees. The slope from a low-volume reading pulmonary valve closure point to a high-volume reading tricuspid valve opening point is the Ea.
2.1.5 Vascular Ventricular Coupling Ratio as Measured by RHC
Vascular ventricular coupling ratio (VVCR) is a measure of the interplay between right-ventricular mechanics and pulmonary artery stiffness. VVCR is non-invasively measured as Ea/Ees, where Ea is arterial elastance. This formula simplifies to Pes/(Pmax Pes), where Pes and Pmax are defined as above. VVCR can thus be calculated without the use of MRI data. The ideal surgical measurement for VVCR is carried out using the methods discussed above.

2.2 ECG Format
Twelve-lead ECGs were collected using a Philips TC70 machine. Some patients had lead V3R instead of lead V3, so lead V3 was not included in our analysis. The traces were converted to PDF using the Philips TracemasterVue program. These PDFs were uploaded to the EPIC database of Childrens hospital, Colorado. Our research team retrospectively retrieved these high-quality PDFs, which were created with vector-data. An example of the PDF files that we used is shown below, in Figure 2.1. There are generally two types of PDFs- vector-based and raster-based. The vector-based PDFs are higher quality than the raster-based. This can be seen by zooming in to a high magnification on a PDF. If the image turns fuzzy, it is a raster-based PDF. In this case, the image turns poor quality at high magnification because it has been saved as a bitmap at the original resolution. The vector-based PDFs are much higher quality, and zooming in to high magnification retains crisp contours and clean lines. This is because the image data is stored as vectors between precise points.
Figure 2.2: An example of the ECG tracings in our PDF files. Notice the rectangular wave form at the end of each line, which can be used for scaling of the data. The red

boxes can also be used for scaling, with each medium sized box (of which there are approximately fifty in the horizontal direction) representing 200 milliseconds horizontally and 0.5 millivolts vertically.
2.3 Patient Selection
Patients seen at Childrens Hospital Colorado, and retrospectively enrolled in our study, were classified as pulmonary hypertensive (PH) or control (CTL) per their ultimate clinical diagnosis. 32 patients were used in the final analysis. 11 of these patients were normal, healthy patients, and 21 were classified as PAH, WHO class I. They were age-matched as best as possible. This recruitment protocol was IRB approved. Each PH patient underwent ECG and catheterization on the same day. The distribution of ages can be found in Figure 2.2. The mean age for CTL patients was 16.5 years, and the mean age for PH patients was 14.3 years. A two-sample t-test determined that there was no statistical difference in the age distributions of these two groups.
All our patients were WHO class I (PAH), with idiopathic or congenital heart defects (CHD). Most of the patients are idiopathic. Fourteen patients were idiopathic, three had atrial-septal defects or ventricular septal defect, and one patient had patent ductus arteriosus, two patients had hereditary PH, one patient had connective tissue-disease related PH.

Years of age
Figure 2.3: Age matching for CTL and PH patients. Mean age for CTL patients was 16.5 years, with a standard deviation around 5 years. Mean age for PH patients was 14.3 years, with a standard deviation around 4.5 years. A two-sample t-test determined no statistical difference between the age distributions with a p-value threshold of 0.05.
The control population used in our study requires an in-depth description, and a justification. These individuals presented to the clinic with physical symptoms that aroused suspicion of PH. They were then given an ECG and an echocardiogram, and sent for right-heart catheterization if deemed necessary. Our distinction between PH and CTL patients comes from the final clinical diagnosis, that is from the findings of catheterization. Therefore, the ECG findings may or may not be normal for patients either in our control group or in our PH group. We believe that this is a strength for our classification analysis- that we are showing discriminatory ability at a level where traditional analysis of ECGs appears to be failing. Our analysis is not strictly related to time-domain representations of the ECGs, therefore, we are proving that ECGs can be

viewed in a manner that is more accurate than standard clinical electrophysiological interpretation. The ECGs of both our CTL and PH populations were independently verified by an electrophysiologist. The general ECG status of our patients is displayed in Table 2.1.
ECG Status Control ECG Findings PH ECG Findings
Total Number of Patients 11 21
% Abnormal ECG 64% 81%
% Normal ECG 36% 19%
% RVH 9% 52%
% RAD 27% 43%
% Inverted T Wave 18% 24%
Table 2.1: General ECG findings for both CTL and PH populations, independently verified by an electrophysiologist. RVH = right ventricular hypertrophy, RAD = right axis deviation, inverted T wave = catch-all for repolarization abnormalities. A more detailed description of each patients ECG can be found in the Appendix.
2.4 Uploading ECGs to MatLab
The ECG PDFs had to be converted to Excel files for MatLab. This was done through a multi-step process. First, the PDFs were converted to Scalable Vector Graphics (SVG) files to access the vector data behind the ECG tracings. This was done using the Adobe Illustrator program, and any program that can edit SVG files could be used. These SVG files were then converted to Comma Separated Value (CSV) files using a basic text editor. This was done to convert the ECG curve data in to a plain-text format that could ultimately be uploaded in to MatLab. The CSV files were opened in Excel, and the delimiter properties were finagled until the vector values stood out well from the other markings. In our case, the useful delimiters turned out to be commas, spaces, and quotation marks. After visual inspection of the Excel file, it was found that the relevant information was retained in a reliable pattern, beginning and ending around the same row for each patient. Excess information was removed, until only the relevant numbers and

some spare words that could not be easily removed remained. The files were then saved in XLSX format. The XLSX files were uploaded to MatLab. At this point, the XLSX files still contained some string fragments and the abscissa and ordinate coordinates alternated by the column. MatLab was used to separate the values accordingly, by saving even columns as x-coordinates and odd-columns as y-coordinates, and using the string fragments to locate the beginning and end of each lead. These string fragments were subsequently removed by locating and removing NaN values.
2.5 ECG Signal Pre-Processing
The ECG signals were saved in a cell array at this point, with rows arranged by patient and columns by lead. The time values for each lead were not uniformly spaced, because the vector data behind the pdf had a semi-random time distribution. This was remedied using cubic spline interpolation. The data was interpolated to 0.005 seconds, corresponding to a 200Hz sampling rate per EE times guidelines [66], This digital sampling rate was an effective low-pass filter. Through analysis of the power spectrum and visual inspection of the time-domain signal, we determined that there was no need to remove powerline noise. All signals were shortened to the duration of the shortest signal,
2.5 seconds, to facilitate analysis. The signals were de-trended to remove DC offset. This was done by using the detrend function in MatLab, which removes the best straight-line fit for data in a vector. This was verified to effectively remove the DC component of the frequency-spectrum.
2.6 Fourier Coefficients
The Fourier transform is a mathematical algorithm that relates a signal captured in time or space to a signal sampled in frequency. This is a useful tool in signal processing,

and was implemented in this thesis used Matlabs fast Fourier transform (fft) function.
The fft is more computationally efficient than computing the discrete Fourier transform, with the former requiring n*log(n) floating-point computations and the latter requiring n2 computations. The formula describing the fft is as follows:
Y(k)= i7=1^a)^nw"1)(fc_1) (i.i)
Wn = e(~2niVn (1.2)
and X(j) is the discrete-time series of n-length.
The basics of the Fourier transform are described below, with a focus on why we suspected that Fourier coefficients could contain useful information. The following discussion could be found by reading most introductory signal-processing, or mathematical modelling, texts [65],
The goal of the Fourier transform is to take any periodic function (i.e. all functions that can be represented as fit) = f(t + T), where T is the period), and express it as a sum of an infinite series of sine and cosine terms, or their complex analog. This series is written as
fit) =~+ £f=1iAkcoskwt + Bksinkwt). (1.3)
A and B are the Fourier coefficient values. Some basic definitions that are pertinent to the Fourier transform include: fundamental frequency (j = ), fundamental
angular frequency (w = 2 n /), and mean function value {A0/2). Each term in the summation represents a harmonic, or a sinusoidal wave that has frequency of kw.
To calculate the Fourier series, the angular frequency and the time-representation fit) must be known. Then, the Fourier coefficients can be calculated as such:

Ak = ^ f2T(f(t)coskwt)dt,k = 0,1,2 ... (1.4)
Bk = -f2T(f(t)sinkwt)dt,k = 1,2,3 ... (1.5)
The discrepancy in the index k accounts for the mean value of the function described as Ao. This algorithm is often implemented as the discrete Fourier transform (DFT) which replaces the integrals with summation operations. An accurate Fourier representation can be made of finite length iff(t) is known at discrete intervals, i.e. at t = sAt, where 5 is an integer and At is the time-step size. The number of terms in the Fourier series that are necessary to completely describe the time-function is directly related to the number of data points in the discrete signal representation. If the total number of points is N, the number of terms necessary is N/2 (add one if the N is odd). This arises because a sinusoidal wave can be described with a minimum of two points. The frequency of the highest-frequency term in this series corresponds to the Nyquist frequency from digital signal processing theorem. The frequency of the first, non-DC, harmonic is 1/T, where T is the period. The Nyquist frequency can be thought of in terms of the number of points required to accurately describe a sine wave, i.e. that there must be twice as many points as the highest frequency that you want to represent to fit a sine wave to the representation. In practice, signals should be oversampled, or the discrete representation should be taken with more points than considered necessary, to be able to compensate for random noise in the signal. This ultimately comes down to the frequency content of the signal of interest. For ECGs, the upper frequency limit recommended in the literature is around 150Hz [66],

To test whether the Fourier representation was performed with an appropriate amount of data, the variance between the obtained representation and the original signal should be calculated. By setting a maximum acceptable variance, or error, between the Fourier representation and the original signal, the minimum number of harmonics necessary can be selected.
A fast Fourier transform was performed to obtain Fourier coefficients for each signal. This was done using Matlabs fft function, as described in the methods section. This yielded 500 coefficients per lead, in accordance with the 2.5 second duration and 200 Hz sampling rate, in a two-sided spectrum format. The data was utilized in the two-sided spectrum format, because the single-sided spectrum is mathematically equivalent through a linear transformation. The two-sided spectrum includes both negative and positive frequencies, which are equivalent for a real-valued signal, such as an ECG. This format could also facilitate the physiological interpretation of results. The success of the transformation was verified by performing an inverse transform and visually and mathematically verifying that the original signal and the retrieved signal were identical.
The coefficients were further prepared for statistical analysis by removing the imaginary portion, by taking the absolute value of the signal. In MatLab, taking the absolute value of a complex Fourier coefficient serves to discard phase information and only retain magnitude information. The number of coefficients to be used for analysis was decided using the power spectrum. The half-power spectrum was analyzed to determine the appropriate number of coefficients necessary for analysis. This approach involves plotting Log10(H2) on the y-axis, where H is the magnitude, and plotting Log ^Frequency) on the x-axis. This gives a y-axis in decibels. A line is fitted to the

flat portion, and a line is fitted to the sloping roll-off portion. The intersection of a line three decibels below the flat portion with the roll-offline estimates the -3dB point (or half-power bandwidth). Our half-power band-width was at approximately 50Hz. We took a conservative approach and used 65Hz as our cutoff. The half-power bandwidth (-3dB) graph can be found in Figure 2.4.
Figure 2.4: The half-power bandwidth of our populations reveals that coefficients at frequencies greater than 50Hz have an insignificant influence on the signal. The y-axis is in decibels, or 20*logl0(H), where H is magnitude. The flat-portion of the graph is fitted with a dotted horizontal line at 20dB, and the roll-off portion is fitted with a sloping line. The intersection of a solid horizontal line -3dB from the flat-portion with the dotted roll-offline marks an estimate of the -3dB point, or half-power bandwidth.
The first 65 coefficients for each lead were organized in to a matrix. The coefficients corresponding to lead V3 were removed, because a significant portion of the patient population had V3R placement instead of V3 placement. The natural logarithm was applied to all Fourier coefficients, as it was found to increase statistical significance

in subsequent tests. It should also be noted that the frequency distribution for CTL and PH patients looks nearly identical to the naked eye, with some minor fluctuations but no broad trend. The -3dB cutoff for control could be slightly higher than that for PH, but we did not find the discrimination worthwhile considering our conservative 65Hz approach. It would be interesting to compare the distributions from our populations with distributions in future studies. Also, there is no discernible high-frequency signature differences as noted in the work of Voss et al., who observed a distinct lack of frequencies above 80Hz in healthy controls [28],
2.7 ROC Analysis
The receiver operating characteristic (ROC) curve is a tool used to describe the diagnostic performance of a test, or its accuracy in discriminating normal cases from diseased cases. The outcome of a given test applied to diseased and normal populations will likely have a distribution that is at least partially overlapping. Given this partial overlap, any value selected as some cutoff criteria will correctly classify the results as coming from a certain population a fraction of the time. This phenomenon leads to four relevant fractions from an ROC curve: false negative (FN), false positive (FP), true negative (TN), and true positive (TP). These fractions represent a number of patients, given in the following discussion as a, b, c, and cJ, respectively.
For a total population of n patients: a + c = n, and b + d = n. These variables can be used to describe several useful parameters, sensitivity and specificity, where: sensitivity = a/{a + b), and specificity = d./(c + d). The sensitivity represents the probability that a test result will be positive when the disease is present, and the specificity represents the probability that a test result will be negative when the disease is

not present. For a given threshold cutoff criteria, there is a tradeoff where FP decreases as specificity increases, while both TP and sensitivity will decrease. The ROC curve is a visualization of this tradeoff, in which sensitivity (TP) is plotted on the y-axis, and (100-specificity) (FP) is plotted on the x-axis [78], This test was performed using the MatLab perfcurve function.
Receiver-operating characteristic (ROC) curves were made for each Fourier coefficient from both the CTL and PH populations. These curves reveal the ability to predict whether a value comes from a control or PH population. This test works best for coefficients with means that are significantly different. Thus, a two-sample t-test was performed. The ROC analysis was performed on all coefficients (transformed with natural logarithm) that had a null hypothesis rejected by the t-test. This means that we only attempted to classify PH versus CTL using Fourier coefficients that had statistically different distributions for these two populations. The coefficient with the greatest area under the curve (AUC) value from the ROC test was selected as the best single coefficient for classification. The top five coefficients, selected by AUC, were saved for use in multivariate ROC testing. These coefficients were used to form a two-variable multivariate logistic regression model, which was tested with an ROC curve.
Boxplots with individual values marked were created for the top coefficient. This revealed the presence of one visual outlier for both the PH and CTL populations. The ROC test protocol described above was thus repeated after removing these visual outliers.

2.8 Multivariate Linear Regression Analysis
A stepwise regression model was used to find the correlation between coefficients and the invasive measurement values of RVC, VVCR, PVR, mPAP, and RVP. The process described here for RVC was replicated for the other outputs.
An output matrix was formed from the RVC measurement of each PH patient. An input matrix was formed by taking all logarithmic coefficients for PH patients, with rows corresponding to patients and columns corresponding to coefficient. These parameters were fed in to the step ir isefil M at Lab function. This function performs stepwise linear regression. In brief, the function begins with a constant model, defined as the average of the coefficients. It then finds the coefficient that will improve the coefficient of covariance the most when added to this constant model, and adds it to the model multiplied by an appropriate scaling factor. The coefficient with the next best coefficient of covariance is then added to the constant and first coefficient term, and the algorithm tests whether the model is a better fit with or without the first coefficient term included, i.e. the model does a check backwards and forwards at each step. This stepwise process thus tests whether adding or removing terms makes the model stronger. An artifact of our large number of inputs versus outputs was that the model would overfit and find an r-squared value of one. To correct for this, the p-parameter, or the p-value which coefficients must fall under to be included in the model, was adjusted until the model gave a few terms with finite confidence and an r-squared less than one. The unreliable nature of this model for large numbers of inputs was deemed to be acceptable, considering that our next step is an adequate check on these significant coefficients.

The stepwise model returned many relevant coefficients for each output. From these coefficients, we narrowed the model down to a three-variable multivariate model. This was done by testing all possible combinations of three inputs (obtained from our stepwise model), found using the nchoosek function, with the linear function. The best possible combination was chosen, and displayed alongside a least-squares line. A Bland-Altman plot was also created for each model, to describe any inherent biases in the data.

3.1 ROC Analysis
We performed ROC analysis with and without outliers present. For both cases, the ROC analysis gave an AUC of approximately 0.83. It should be noted that a total of nine Fourier coefficients yielded AUC greater than 0.70. The distribution in this case had a threshold value, at which the ROC curve delineates PH from CTL values, of 1.2755. To determine the efficacy of the classifier, we ran 100 iterations of a Monte-Carlo exclusion test. This test involved removing 1 CTL and 2 PH patients, in random configurations, 100 times, recreating the ROC analysis, and validating whether the classifier would correctly classify the 3 patients who were left out. This revealed 82% and 78% accuracy, respectively, for the outliers included and outliers excluded cases. These results can be found in Table 3.1, and Figures 3.1 through 3.3.
ROC Model AUC Threshold Coefficient Value Efficacy (%) Coefficients (Frequency, Lead #)
Uni-variate 0.83 1.28 82% 50 Hz, Lead III
Table 3.1: Results for uni-variate ROC analysis. The multi-variate model is given as y = 5.4 2.0*xl 2.2*x2, where xl is the 50Hz coefficient on lead III and x2 is the 39Hz coefficient on lead II. The efficacy matches the AUC well for the uni-variate model, but poorly for the multi-variate model.

Distribution of 50Hz, Lead III Coefficient with Outliers
Patient Distribution
Figure 3.1: The distribution of the 50Hz coefficient on lead III for PH and CTL patients. Notice the presence of one outlier in each distribution, circled in red, at a low coefficient value for PH and at a high coefficient value for CTL. Two-sample t-test reveals a two-sided p-value of 0.03, not significant at the Bonferroni-corrected threshold of 6e-05.
Distribution of 50Hz, Lead III Coefficient without Outliers
Patient Distribution
Figure 3.2: The distribution of the 50Hz coefficient on lead III for PH and CTL patients with outliers excluded. With outliers removed, the two-sample t-test gives a two-sided p-value that is significant at the Bonferroni-corrected threshold on 6e-05.

ROC Classifier for Single Fourier Coefficient
Figure 3.3: The ROC curve for the uni-variate analysis with the natural logarithm of the 50Hz coefficient from lead III is the same for the cases with outliers included or with outliers excluded. The AUC, or ability to discriminate PH from CTL, is 0.83. A test of classification accuracy, with 300 trials, confirmed accuracy of around 82%.
Briefly, the results for multivariate ROC analysis were as follows: AUC = 0.87, and cross-validated accuracy of 0.77. The discrepancy in these values indicates that a two-variable model introduces over-fitting. These results are not further discussed.

3.2 Multivariate Linear Regression Analysis
Parameter (Multi-lead Multi-variate) Correlation (r2) P
Contractility 0.82 4.33E-06
VVCR 0.83 2.57E-06
PVR 0.85 1.10E-06
mPAP 0.9 8.24E-08
RVP 0.87 4.40E-07
Table 3.2: Multivariate linear regression results. Correlation is given as the coefficient of covariance (r2), which indicates the degree of variability in the output that can be explained using the input model. P-values are given as two-sided p-values, and all are significant at the Bonferroni-corrected threshold p-value of 6e-05.
Parameter (Multi-lead Multi-variate) Model xl x2 x3
Contractility y = 0.59 + 0.07*xl -0.08*x2 -0.07*x3 62Hz on Lead 1 3Hz on Lead II 37Hz on Lead V5
VVCR y = 0.82 -0.03*xl -0.13*x2 + 0.06*x3 38Hz on Lead VI 14Hz on Lead V5 41Hz on Lead aVF
PVR y = 7.5 + 4.1*xl + 4.3*x2 7.1*x3 65Hz on Lead 1 19Hz on Lead V2 34Hz on Lead aVF
mPAP y = -19.4 + 13.7*xl + 17.1*x2 + 6.2*x3 57Hz on Lead 1 48Hz on Lead V6 65Hz on Lead V6
RVP y = -1.3 + 9.5*xl -11.3*x2 + 32.5*x3 45Hz on Lead V4 59Hz on Lead aVF 48Hz on Lead V6
Table 3.3: The multi-variate models and relevant Fourier coefficients are given for each parameter. Note that the natural logarithm has been applied to all Fourier coefficients.
Multivariate linear regression analysis was carried out to reveal the best tri-variate model for each output parameter. The results are presented as correlation plots alongside

Bland-Altman plots for each output. The r2 and two-sided p-values for the tri-variate models, as found in Table 3.2, are: RVC (r2 = 0.82, p = 4.3e-6), VVCR (0.83, 2.6e-6), PVR (0.85, l.le-6), mPAP (0.9, 8.2e-8), andRVP (0.87, 4.4e-7). The linear model coefficients can be found in Table 3.3. The Bland-Altman plots indicate the degree of skew of the distribution, and the percent error that our model has for diagnosis. The percent error is given for PVR and mPAP, the two diagnostic parameters, and it is calculated as the standard deviation found divided by the diagnostic threshold. A percent error around 20% is considered acceptable. The results of this analysis can be found in Figures 3.4 through 3.13.
Multi-Variate Linear Regression for mPAP
Figure 3.4: Multivariate regression model for mPAP. Linear model: y = -19.4 + 13.7*xl + 17.1*x2 + 6.2*x3, where xl is the 57Hz coefficient on lead I, x2 is the 48Hz coefficient

on lead V6, and x3 is the 65Hz coefficient on lead V6. Two-sided p-value (p = 8.24e-08) is significant at the Bonferroni-corrected threshold, and r2 value of 0.90 indicates that our model explains 90% of the variation in the mPAP of our measured population. Mean mPAP level of 44.89mmHg. Clinical diagnostic threshold for mPAP is 25mmHg.
30 -
Bland-Altman Plot for mPAP
20 -
Jj 10
o 0
9.8 (+1.96SD)
-0.00 [p=1.0]
-9.8 (-1.96SD)
10 20 30 40 50 60 70 80
Mean Linear Model Value & mPAP (mmHg)
Figure 3.5: Bland-Altman plot for mPAP multivariate regression model. The error between our linear model and mPAP as measured by catheterization has standard deviation of 5mmHg. Percent error (standard deviation / diagnostic threshold) of 20%. The distribution is uniform around zero, indicating that there is no bias in our linear model measurements.

Multi-Variate Linear Regression for PVR
Figure 3.6: Multivariate regression for PVR. Linear model: y = 7.5 + 4.1*xl + 4.3*x2 -7.1*x3, where xl is the 65Hz coefficient on lead I, x2 is the 19Hz coefficient on lead V2, and x3 is the 34Hz coefficient on lead aVF. Two-sided p-value (p = l.le-06) is significant at the Bonferroni-corrected threshold, and r2 value of 0.85 indicates that our model explains 85% of the variation in the PVR of our measured population. Mean PVR level of 10.2Pa*s/m3. Clinical diagnostic threshold for PVR is 3 Pa*s/m3 (also known as Wood Units).

Bland-Altman Plot for PVR
15 r

5 -
n -5
-10 -
4.8 (+1.96SD)
-0.00 [p=1.0]
-4.8 (-1.96SD)
0 5 10 15 20 25 30
Mean Linear Model Value & PVR (Pa*s/m3)
Figure 3.7: Bland-Altman for PVR. The error between our linear model and PVR as measured by catheterization has standard deviation of 2.45Pa*s/m3. Percent error (standard deviation / diagnostic threshold) of 82%. The distribution is uniform around zero, indicating that there is no bias in our measurements. The distribution is slightly clustered around lower values, indicating that some of our patients have abnormally high PVR levels.

Multi-Variate Linear Regression for RVP
Linear Model Value (mmHg)
Figure 3.8: Multivariate regression for RVP. Linear model: y = -1.3 + 9.5*xl 11.3*x2 + 32.5*x3, where xl is the 45Hz coefficient on lead V4, x2 is the 59Hz coefficient on lead aVF, and x3 is the 48Hz coefficient on lead V6. Two-sided p-value (p = 4.4e-07) is significant at the Bonferroni-corrected threshold, and r2 value of 0.87 indicates that our model explains 87% of the variation in the RVP of our measured population. Mean RVP level of 63mmHg.

Bland-Altman Plot for RVP
50 r
40 -
4 10
to 2 to >
D 0
|-10 c
1 -20
15 (+1.96SD)
----------------0.00 [p=1.0]
-15 (-1.96SD)
-40 -
-50 ------------1-----------1-----------1------------1-----------1
20 40 60 80 100 120
Mean Linear Model Value & RVP (mmHg)
Figure 3.9: Bland-Altman for RVP. Linear model has standard deviation of 7.65mmHg. Percent error for the low-side of the distribution (standard deviation / lowest measurement) of 30%. Percent error for the high-side of the distribution (standard deviation / highest measurement) of 7%. The distribution is uniform around zero, indicating that there is no bias in our measurements, except for one outlier that is below the 95% confidence interval.

RVC (mmHg/ml)
Multi-Variate Linear Regression for RVC
Linear Model Value (mmHg/ml)
Figure 3.10: Multivariate regression for RVC. Linear model: y = 0.59 + 0.07*xl -0.08*x2 0.07*x3, where xl is the 62Hz coefficient on lead I, x2 is the 3Hz coefficient on lead II, and x3 is the 37Hz coefficient on lead V5. Two-sided p-value (p = 4.3e-06) is significant at the Bonferroni-corrected threshold, and r2 value of 0.82 indicates that our model explains 82% of the variation in the RVC of our measured population. Mean RVC level of 0.39mmHg/ml.

Bland-Altman Plot for RVC
0.25 r
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E 0.1 E
dj T3 O
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-0.2 -
0.09 (+1.96SD)
------------------------------------0.00 [p=1.0]
-0.09 (-1.96SD)
-0.25 ---------1-----------1---------1----------1---------1
0.2 0.3 0.4 0.5 0.6 0.7
Mean Linear Model Value & RVC (mmHg/ml)
Figure 3.11: Bland-Altman for RVC. Linear model has standard deviation of 0.05mmHg/ml. Percent error for the low-side of the distribution (standard deviation / lowest measurement) of 25%. Percent error for the high-side of the distribution (standard deviation / highest measurement) of 8%. The distribution is uniform around zero, indicating that there is no bias in our measurements, except for one outlier that is above the 95% confidence interval.

Multi-Variate Linear Regression for VVCR
Figure 3.12: Multivariate regression for VVCR. Linear model: y = 0.82 0.03*xl -0.13*x2 + 0.06*x3, where xl is the 38Hz coefficient on lead VI, x2 is the 14Hz coefficient on lead V5, and x3 is the 41Hz coefficient on lead aVF. Two-sided p-value (p = 2.6e-06) is significant at the Bonferroni-corrected threshold, and r2 value of 0.83 indicates that our model explains 83% of the variation in the VVCR of our measured population. Mean VVCR level of 0.63Ea/Ees.

Bland-Altman Plot forVVCR
a 0.05
tj n
o u
E -0.05
> -0.1
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-0.00 [p-1.0]
-0.07 ( 1.96SD)
0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8
Mean Linear Model Value & WCR (Ea/Ees)
Figure 3.13: Bland-Altman for WCR. Linear model has standard deviation of 0.04Ea/Ees. Percent error for the low-side of the distribution (standard deviation / lowest measurement) of 9%. Percent error for the high-side of the distribution (standard deviation / highest measurement) of 5%. The distribution is uniform around zero, indicating that there is no bias in our measurements.

Using spectral decomposition techniques, we have discovered simple but strong correlations between the Fourier coefficients of ECG signals and parameters measured by MRI and catheterization. The Fourier coefficients may be a useful tool in the diagnosis and tracking of PH, and we suggest their development in conjunction with echocardiogram. Though our statistical techniques are easy to interpret, our frequency-domain analysis is not readily translatable into physiological information.
4.1 ROC Analysis
The results from our ROC analysis indicate that a simple threshold cutoff for one Fourier coefficient separates the PH and CTL cohorts with good accuracy. There is likely a valid multivariate model for this classifier, but we could not locate one that performed significantly better in all tests of efficacy than the best single coefficient. The single most promising coefficient was from lead III, at 50Hz. This coefficient yielded an AUC of around 83%. The AUC describes the accuracy of a test, and results in the 80%-90% range are generally deemed as good, but not excellent [80], The AUC measures the discrimination of a test, or the ability to accurately classify patients with and without the given disease. The AUC represents the percentage of correct diagnoses that would be expected for a population tested with the threshold value of the AUC. This interpretation arises because the population distributions of diseased and healthy individuals are slightly overlapping, thus, it is impossible to perfectly discriminate the populations based on these distributions. One confounding factor that likely decreased separation between PH and CTL populations is that all CTL patients presented to the hospital with PH-like

symptoms, and many were declared to have abnormal ECGs and arrhythmias. It is likely that, given CTL patients without these pathologies, our results would be stronger.
It should be noted that our methodology suffers from the curse of dimensionality, in that we have many times more predictors than observations. PCA is a simple technique that lessens the burden of this problem, and should be used in future work.
We cross-validated our data with a Monte-Carlo technique to validate the AUC findings. In this simulation, we created an ROC classifier with 30 of 33 patients, and then tested the discriminatory ability of the threshold value on the withheld patients. This process was repeated 100 times, to test a total of 300 combinations of patients. The result of this large number of trials was 82% accuracy, which supports the AUC value.
We also viewed the coefficient distribution behind the ROC curve. The distributions of the coefficient for PH and CTL patients were significantly different well below a p-value of 0.05. Because our test was performed with many more predictors than outputs, we can adjust this p-value to account for the large likelihood that a correlation may be found by chance. We lowered the p-value to reflect our lower acceptable likelihood that our results were achieved by chance. This is done using the Bonferroni correction, which adjusts the p-value by dividing it by the number of comparisons being made. In our case, we divided 0.05 by 715, to arrive at an adjusted p-value threshold of 6.9e-5. Thus, our threshold for statistical significance is much lower than the generally accepted p-value of 0.05. The p-value that we found for the distribution with obvious outliers included was nearly 0.03, and the p-value for the distributions with the outliers removed was 6.6e-6. Our simple ROC classifier shows that a single Fourier coefficient,

corresponding to a frequency of 50 Hz on lead III, could discriminate PH from CTL patients with good accuracy.
Figure 4.1 shows lead III from a CTL patient with a confirmed normal ECG. Figure 4.2 shows lead III from a PH patient with a confirmed abnormal ECG and likely right-ventricular hypertrophy. These figures correspond to the lowest and highest measurements for the 50 Hz coefficient on lead III (outliers excluded). Notable differences include an elevated heart rate in the PH patient, and the presence of a QR complex in the PH patient where a QRS complex is expected. Considering that lead III provides an inferior view of the right ventricle, as seen below in Figure 4.3, it appears that our classifier has picked up on right-ventricular pathologies. Previous literature indicates that the presence of a QR complex in lead VI is indicative of right-ventricular strain [11],
Figure 4.1: Lead III on a normal confirmed EKG, with a 50 Hz lead III coefficient value farthest from the PH distribution.

Figure 4.2: Lead III on an abnormal EKG, with a 50 Hz lead III coefficient value farthest from the CTL distribution.
Figure 4.3: A depiction of the correspondence between ECG lead and anatomical location in the human heart. Notice that the chest leads (VI through V6) view the heart from a horizontal cross-section, or transverse plane. Limb leads (I through III) and augmented leads (aVL, aVR, and aVF) view the heart from a vertical cross-section, or coronal plane [81].

4.1.1 Summary of Interpretation of ROC Classification
The 50Hz Fourier Coefficient on Lead III could discriminate between PH and CTL patients with 82% accuracy
Lead III corresponds to the inferior wall of the right ventricle
Comparison of CTL to PH Lead III EKG reveals that PH EKGs with high 50Hz coefficient values have QR complex morphology, indicative of right-ventricular strain
Our CTL population presented to the hospital with symptoms that aroused suspicion of PH. Using a CTL population that did not have PH-like symptoms would likely improve our classification accuracy.
Multivariate ROC models were tested, and found to suffer from overfitting. This can be taken as a symptom of the curse of dimensionality (ROC analysis utilized 32 observations and 715 predictors).
4.2 Multivariate Linear Regression Analysis We performed simple multivariate linear regression analysis to uncover correlations between ECG data and parameters measured by both catheterization and MRI. The right-ventricular fibrosis present in the compensatory phase of PH leads to electrophysiological changes that are evident when viewing ECG [87], This analysis yielded promising results, indicating that a linear combination of Fourier coefficients can explain upwards of 80% of the variation in these parameters. The physiological reasoning behind this correlation was probed, and the results indicate that this could be a valid and interesting experiment. In terms of the significance of our results, it is worth noting that the significance level is below the Bonferroni-corrected p-value.

We performed linear regression and created Bland-Altman plots to further explain the significance of our findings [67], Bland-Altman plots describe the mathematical relationship between variables by analyzing the difference between them, whereas linear regression describes a mathematical relationship by analyzing similarities. Thus, performing both Bland-Altman and linear regression analyses allows for more robust results. The plot shows the mean difference between measurements, and the limits within which 95% of the differences between measurements will fall. The measurement values that these limits define may or may not be acceptable physiologically or clinically, but that is difficult to illuminate in our case because the physiological significance of our results is largely unknown. Two common parameters that are used to describe Bland-Altman plots are the coefficient of variation and the coefficient of reproducibility. The Bland-Altman plot also shows the coefficient of variation (CV), which is an expression of the standard deviation in test-retest values as a percent of the average measurement value [68], The coefficient of reproducibility (RPC) is a measurement of the likely change in the value range in a repeated experiment, and it indicates by what percent of the mean value a measurement would have to change in a second trial to be considered different from the first trial at a confidence level of 95% [69], Neither CV or RPC are included in our analysis.

4.2.1 Multi-Variate Linear Regression for RVP
Figure 4.4: The lateral (Figure 4.4, part A) and precordial (Figure 4.4, part B) leads included in the linear model for RVP. Linear model: y = -1.3 + 9.5 *x 1 11.3*x2 + 32.5*x3, where xl = 45Hz on V4, x2 = 59Hz on aVF, and x3 = 48Hz on V6. Lead aVF corresponds to the right-ventricular wall, lead V4 corresponds to the septal wall near the valvular junction of the ventricles and atria, and lead V6 corresponds to the anterio-lateral wall of the left ventricle.
Our regression analysis indicated the following for RVP, that the tri-variate model, which includes leads V4, V6, and aVF, can predict 87% of the variability in RVP with a two-tailed p-value of 4.4e-7. The Bland-Altman analysis indicates that there is no bias in either measurement modality. The standard deviation in our RVP model was approximately 7.5mmHg. In our population, the lowest RVP measurement was 25mmHg and the highest RVP measurement was 105mmHg. Combining this with standard deviation, we find that the percent error is 30% for low measurements and 7% for high measurements. The anatomical correspondence of each lead can be found above, in

Figure 4.4. Lead aVF views the right-ventricular wall, lead V4 views the junction of the ventricles and atria, and lead V6 views the left-ventricular wall. Increased RVP is present when right-ventricular function is altered, as is expected in the compensatory phase of PH (which all our PH patients are in). Altered right-ventricular function, including right-ventricular hypertrophy, changes septal geometry, which thus affects left-ventricular function. The LV and RV function interdependently, and two often accompany each other regardless of the etiology of the disease [85], Pressure overload in the RV will alter valvular function- the RA must work harder to pump blood in to the RV, and alterations to RV function include fibrosis and stenosis of the valves [86], Overall, our ECG model indicates that the right ventricle, left ventricle, and the valvular junction are all important indicators of the extent of RVP change in pediatric PH.
4.2.2 Multi-Variate Linear Regression for PVR
Figure 4.5: The lateral (Figure 4.5, part A) and precordial (Figure 4.5, part B) leads included in the linear model for PVR. Linear model: y = 7.5 + 4.1*xl + 4.3*x2 7.1*x3, where xl = 65Hz on I, x2 = 19Hz on V2, and x3 = 34Hz on aVF. Lead aVF corresponds

to the right-ventricular wall, lead I corresponds to the lateral wall of the left ventricle, and lead V2 corresponds to the anterior wall of the right ventricle.
Our regression analysis indicated the following for PVR, that the tri-variate model, which includes leads I, V2, and aVF, can predict 85% of the variability in PVR with a two-sided p-value of 1. le-6. The Bland-Altman analysis indicated that there was no bias in either measurement modality. The Bland-Altman analysis also indicated that the error in measurement between our linear model and PVR as measured by RHC has a standard deviation of approximately 2.45 Pa*s/m3, which gives a percent error of 82% when compared to the diagnostic threshold of 3 Pa*s/m3. This means that a patient would have to have a PVR of around 5.45 Pa*s/m3 to feel confident in diagnosis. PVR is combined with mPAP and often other imaging measurements from imaging in the diagnosis of PH, so the error in our result does not make it clinically irrelevant. Figure
4.5 shows the anatomical locations relevant to the PVR model. Lead I views the left ventricle, lead aVF views the right ventricle, and lead V2 views the right ventricle. Increased PVR places stress on the RV as afterload is increased. Thus, alterations to RV function and electrophysiology are expected. The presence of a left-ventricular lead can be explained with the logic used to discuss our RVP findings. Also, it is plausible that increased pre-capillary impedance would lower venous return pressures to the left atrium. The LA would then fill less thoroughly, and the LV would have to work harder to pump a lower volume and lower pressure load of blood through the systemic circulation. In summary, our findings indicate that right-ventricular and left-ventricular electrophysiology are relevant predictors of elevated PVR.

4.2.3 Multi-Variate Linear Regression for VVCR
Figure 4.6: The lateral (Figure 4.6, part A) and precordial (Figure 4.6, part B) leads included in the linear model for VVCR. Linear model: y = 0.82 0.03*xl 0.13*x2 + 0.06*x3, where xl = 38Hz on VI, x2 = 14Hz on V5, and x3 = 41Hz on aVF. Lead aVF corresponds to the right-ventricular wall, lead VI corresponds to the posterio-lateral wall of the right ventricle, and lead V5 corresponds to the anterior wall of the left ventricle.
Our regression analysis indicated the following for VVCR, that the tri-variate model, which includes leads VI, V5, and aVF, can predict 83% of the variability in VVCR with a two-sided p-value of 2.6e-6. The Bland-Altman analysis indicates that there is no bias in either measurement modality. The standard deviation in our VVCR model was approximately 0.04Ea/Ees. In our population, the lowest VVCR measurement was 0.45Ea/Ees and the highest VVCR measurement was 0.8Ea/Ees. Combining this with standard deviation, we find that the percent error is 9% for low measurements and 5% for high measurements. VVCR shows the ratio of arterial function to ventricular function, i.e. it is a metric of coupling between the right ventricle and the pulmonary

artery. Figure 4.6 shows the anatomical locations relevant to the VVCR model. Lead VI views the right-ventricle, lead V5 views the anterior left ventricle, and lead aVF views the right ventricle. VVCR includes a measure of right-ventricular function, and the presence of right-ventricular leads in this model are intuitive. Considering the interplay between RV and LV function, noted in multiple previous works, it is not surprising that a left-ventricular lead is included in a measure of RV function [86, 73], In summary, right-ventricular and left-ventricular leads are likely included in the model for VVCR because of the mechanical interdependence of the RV and LV and the likely effect of precapillary pressures on left-ventricular function, as noted in the discussion on PVR.
4.2.4 Multi-Variate Linear Regression for mPAP
Figure 4.7: The lateral (Figure 4.7, part A) and precordial (Figure 4.7, part B) leads included in the linear model for mPAP. Linear model: y = -19.4 + 13.7*xl + 17.1*x2 + 6.2*x3, where xl = 57Hz on I, x2 = 48Hz on V6, and x3 = 65Hz on V6. Lead I corresponds to the lateral wall of the left ventricle, lead V6 corresponds to the anterio-lateral wall of the left ventricle.

Our regression analysis indicated the following for mPAP, that the tri-variate model, which includes leads I and V6, can predict 90% of the variability in measurement of mPAP with a two-sided p-value of 8.2e-8. The Bland-Altman analysis indicated that there was no bias in either measurement modality. The Bland-Altman analysis also indicated that the standard deviation of the error between our linear model and mPAP as measured by RHC is approximately 5mmHg, which gives a percent error of 20% when compared to the standard diagnostic threshold of 25mmHg. This indicates that a patient would have to measure at least 30mmHg in our linear model for a confident diagnosis to be made. Figure 4.7 shows the anatomical locations relevant to the mPAP model. Both lead I and lead V6 view the left ventricle. The exclusion of right-ventricular leads from our model is difficult to explain. The logic used in our discussion of PVR is useful here-that elevated mPAP, which accompanies an increased afterload to the heart, would ultimately affect left-ventricular function by creating a larger-than-normal pressure drop through the pulmonary circulation. In summary, our model for mPAP includes solely left-ventricular leads, indicating significant interplay between function of the left ventricle and pressure load to the right ventricle.

4.2.5 Multi-Variate Linear Regression for RVC
Figure 4.8: The lateral (Figure 4.8, part A) and precordial (Figure 4.8, part B) leads included in the linear model for RVC. Linear model: y = 0.59 + 0.07*xl 0.08*x2 -0.07*x3, where xl = 62Hz on I, x2 = 3Hz on II, and x3 = 37Hz on V5. Lead I corresponds to the lateral wall of the left ventricle, lead II corresponds to the intraventricular septum, and lead V5 corresponds to the anterior wall of the left ventricle.
Our regression analysis indicated the following for RVC, that the tri-variate model, which includes leads I, II, and V5, can predict 82% of the variability in RVC with a two-sided p-value of 4.3e-6. The Bland-Altman analysis indicates that there is no bias in either measurement modality. The standard deviation in our RVC model was approximately 0.05mmHg/ml. In our population, the lowest RVC measurement was 0.2mmHg/ml and the highest RVC measurement was 0.65mmHg/ml. Combining this with standard deviation, we find that the percent error is 25% for low measurements and 8% for high measurements. Figure 4.8 shows the anatomical locations relevant to the RVC model. Lead I views the left ventricle, lead II views the intra-ventricular septum,

variability in the frequency spectrum that may not exist for a longer signal with more peaks. Our use of the 12-lead ECG made our results more difficult to interpret, because our multivariate models contained a mixture of coefficients from different leads. The 12-lead ECG offers much more information than a signal-averaged or single-lead ECG. The 12-lead ECG was ideal for an early study to prove the utility of Fourier coefficients in prognosis and diagnosis, but ECG modalities with less information may prove easier to interpret in future studies [74], As to the speculative nature of the interpretation of our results, because the heart functions as a system, it is possible to relate the function of any portion of the heart to the function of any other portion. However, it is very difficult to know if the interpretation is the physiological reality. Essentially there is always a physiological explanation to be found, but it is difficult to know if the explanation is factual or just speculative. This difficulty, and the others listed above, could be overcome in future studies.
4.4 Future Work
Future work in this area should focus on strengthening the correlation between ECG and catheterization measurements, while increasing the physiological interpretability of results. These goals could be achieved in several ways. The correlations could likely be improved by obtaining larger patient populations and accounting for age, sex, and comorbidities. Also, considering the high degree of variability concerning anatomical orientation of the heart, data could be normalized by QRS angle. Classification performance could likely be improved by utilizing controls that do not have comorbidities, or by utilizing other classifiers listed in the introduction, such as ANNs. Dimensionality-reduction techniques such as PCA could likely improve our

results. Logistic regression would be a logical next step after verifying that thresholding serves as a simple classifier. These considerations could be taken while simultaneously improving physiological interpretability by using signal-averaged or single-lead ECGs. These ECGs measure less information, which would be easier to interpret if valid correlations were present.

The objective of this thesis was to develop non-invasive measurements for the diagnosis and prognosis of pediatric pulmonary hypertension. Pulmonary hypertension (PH) is a disease of the lung vasculature that ultimately affects the mechanics of the heart and leads to right-heart failure. Thus, measurement of pressures in the heart and surrounding vasculature are important for tracking the progression of the disease and allowing proper management. This is traditionally done using cardiac catheterization- an invasive procedure in which a needle is threaded from the groin in to the heart, often through heart valves. The electrocardiogram (ECG) is another, much less invasive measurement taken on pediatric PH patients. We aimed to develop the ECG as a diagnostic and prognostic capable of more accurately monitoring disease progression in routine appointments.
We performed spectral decomposition on pediatric ECG using the fast Fourier transform in MatLab. The Fourier transform can give a quantitative and detailed view of biological signals. Fourier coefficients contain detailed information about primary and secondary frequencies in ECG signals. Each coefficient represents the amplitude of a frequency in a representation of the signal by weighted sine waves of different frequencies. This analysis allowed us to turn each ECG lead from our 12-lead measurements in to a quantitative representation with 65 coefficients. We then created a simple ROC classifier to test the ability of Fourier coefficients to discriminate between pulmonary hypertensive (PH) and control (CTL) patients. We also performed multivariate linear regression analysis to determine the correlation between ECG Fourier

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