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Components of variability in digitized M-mode echocardiography : applied to monitoring of Adriamycin… Schumacher, Peter Michael 1986

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COMPONENTS OF VARIABILITY IN DIGITIZED M - M O D E ECHOCARDIOGRAPHY Applied to Monitoring of Adriamycin Cardiotoxicity by PETER MICHAEL SCHUMACHER B.A., The University of Toronto, 1983 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE (Bus. Admin.) in T H E FACULTY OF GRADUATE STUDIES (Faculty of Commerce) We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA May 1986 ©Peter Michael Schumacher, 1986 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Abs t r a c t . This work concerns the use of M-mode echocardiography, a medical diagnostic tech-nique based on sound wave imaging, in the assessment of the degree of impairment of heart function in children. One cause of left ventricular (LV) impairment is Adriamycin (doxorubicin), a drug used commonly in the treatment of cancer. Although many investigators have used echocardiography (ECHO) to monitor car-diac damage due to Adriamycin {ADM), the magnitude of long-term longitudinal or serial changes one can expect in a typical healthy child's readings has not been assessed. This knowledge might be useful in monitoring. We designed and carried out a study with 18 healthy children to: (i) quantify the longitudinal variation in ECHO parameters be-tween successive months, and compare this to short-term (minutes) serial variation and to between-subject variability; (ii) compare different ECHO measures with respect to these types of variability, as well as on the basis of measurement accuracy; (iii) compare short-term serial variation with that due to the measurement process, with the aim of increasing the precision of a reading; (iv) examine whether subject age and sex have any bearing on the readings. Our ECHO variables consist of 4 measures of change in cardiac dimension between systole and diastole, 6 peak velocities and 6 peak normalized velocities. We used a 4-factor hierarchal factorial design. To address (i) to (iii) above, the data were analyzed in a uni-variate fashion with a variance components model. It was found that: (a) between-subject variation accounts for the largest proportion of total variability; (b) within-subject monthly longitudinal variation is substantial, and approaches the magnitude of that between sub-jects; (c) variation between successive months is markedly larger than the short-term vari-ability; (d) tracing variation, a type of measurement error, is negligible in comparison with short-term variation, implying that the precision of a reading can be improved more by analyzing additional heartbeats than by replicating the analysis of each beat; (e) the LV shortening fraction is more stable over time and is subject to less measurement error than the peak systolic and peak diastolic velocities of the LV, the septal, or the posterior walls. To address (iv) above, we performed an Analysis of Covariance for each variable. Sub-ject sex was found to have no bearing on ECHO readings. There is some indication that some of the measures are linearly related to subject age. In addition, these data were combined with single ECHO measurements on 33 other ii healthy children to yield 95% prediction intervals for normal ECHO parameters; 95% bounds for normal monthly differences in the values were also obtained. Together with serial data on 25 children being treated with A D M , we attempted to identify which ECHO variables would be more useful in monitoring for LV dysfunction. As well as taking into account the relative sensitivities of the variables to ADM, we found the smallest set of measures which would collectively identify all children displaying abnormal (drops in) L V function. This was formulated and solved as an integer programming problem. Lastly, a Factor Analysis was used in an attempt to reveal the dimensionality of normal LV function. Apparently the major impact of ADM on LV function occurs during the first half of therapy. Many patients are marginally within normal bounds at mid-therapy, even though they may exhibit abnormally large changes since onset. Based on the above criteria, we tentatively recommend the following ECHO measures as potentially useful for monitoring of ADM cardiotoxicity: ejection fraction (shortening fraction may be used equivalently), systolic percent change of the septal and posterior wall thicknesses, peak (normalized) LV diastolic velocity, peak posterior wall systolic and normalized diastolic velocities, and peak septal wall diastolic velocity. iii To my parents. iv Tab le of Contents . Abstract ii Dedication iv List of Tables vii List of Figures viii Acknowledgement ix 1. Introduction 1 2. M-Mode Echocardiography 3 3. Adriamycin and Monitoring of its Detrimental Effects 10 Adriamycin 10 Monitoring 14 Endomyocardial Biopsy 14 Radionuclide Angiography 15 Echocardiography 19 Conclusions from Echocardiography Studies 29 Comparative Studies 31 Some Additional Remarks 38 4. An Experiment to Analyze Sources of Variability In Echocardiography Measures in Normal Children 39 Introduction 39 Identification of Objectives 41 Methods 47 Statistical Model I 51 Statistical Model II 53 Analysis and Results 55 Estimation and Testing of Model I Parameters 55 Results: Variance Components 68 Relative Magnitudes of Sources of Variability 70 Inter-Variable Comparisons 73 Homogeneity of Within-Subject Variability 75 Magnitude of Long-Term Serial Fluctuation 79 Normal Bounds 82 Model II - ANCOVAs for Sex and Age Effects 91 . . .cont'd v Conclusions for Chapter 4 95 5. Variable Selection 97 Introduction 97 Adriamycin Dataset 97 Conceptual Analysis 98 Relative Sensitivity of the Variables 99 Dimensionality of Cardiac Impairment . 106 Dimensionality of Normal LV Function 112 Combination of Chapter 5 Analyses 117 Conclusions for Chapter 5 119 6. Suggestions for Further Study 121 Bibliography 123 vi L i s t of T a b l e s . 4.1 - Variance Components: Estimates and their Standard Errors 68a 4.2 - Contributions to Total Variance, by Variable 70a 4.3 - P-Values for Tests of Variance due to Month and Beat Group 72a 4.4 - Coefficients of Variation, by Component and for Total Variability 73a 4.5 - Estimates of Monthly Serial Fluctuation, by Subject 75a 4.6 - Magnitude of Long-Term Serial Fluctuation 80a 4.7 - Means and Standard Deviations for the Two Normal Datasets 84a 4.8 - Range of Values Considered Normal for each Echocardiographic Variable . . 85a 4.9 - Bounds for Normal Monthly Differences in Echocardiographic Readings . . 88a 4.10 - P-Values for Sex and Age Effects on Echocardiographic Readings 92a 5.1 - Means and Standard Deviations for the Three Datasets 100a 5.2 - Number of Adriamycin Patients Exceeding Bounds 103a 5.3 - Loadings Matrix for Principal Factor Solution . 115a 5.4 - Loadings Matrix for Maximum Likelihood Solution 116a vii L i s t of F igures . 2.1 - Fragment of an M-Mode Echocardiogram 4a 2.2 - Anatomy of the Heart 4b 3.1 - Molecular Structure of Adriamycin (Doxorubicin) 10a 4.1 - Normal Scores Plot for SF 58a 4.2 - Normal Scores Plot for LVVa 58b 4.3 - Normal Scores Plot for PWNVd 58c 4.4 - Normal Scores Plot for LVVa 82a 4.5 - Normal Scores Plot for SF 82b 4.6 - Plot of SWNVd versus Age 93a 4.7 - Plot of PW% versus Age 93b 5.1 - Plot of Patient F's SF Values during Therapy 100b viii Acknow ledgement I would like to thank first of all my supervisor, Dr. Martin Puterman, for his continuing support and guidance. While keeping me on track, Marty encouraged me to explore other issues which I found personally interesting. Although some of these did not 'pan out', I learned much through this freedom. I am grateful for Marty's many helpful ideas. Special thanks also go to Dr. George Sandor and Dr. Michael Schulzer for their careful reading of the draft. Dr. Sandor met with us frequently, took the time to explain the medical aspects, and expressed a genuine interest in this work. All data are courtesy of his laboratory. Dr. Schulzer gave generously of his time; his valuable comments and suggestions are much appreciated. I would like to thank Ruby Popov for her ongoing contact with us and willingness to explain the details of data collection, and Mr. Eustace DeSouza for his explanation of echocardiography. Sherry Judah spent a considerable amount of time on the integer programming formu-lation. Thanks also go to Dr. Maurice Queyranne and Bernard Lamond in this respect. Dr. Harry Joe guided me on the analysis of more general ANOVA-type models. I bene-fitted from discussions with Dr. John Petkau on estimation and testing in the presence of imbalance. (John, it's finally done.) Brian Leroux assisted me with the interpretation of conditional expectation. Finally, I thank my parents for their unwavering encouragement. Partial support for this work was provided by The Vancouver Foundation. ix Chap te r 1. In t roduc t i on . This section will present a brief overview of the main body. Chapter 2 is a description of M-Mode echocardiography, including the physical principle behind it. A short illustrated anatomical description of the left ventricular region of the heart is presented. A sample fragment of an echocardiogram is displayed, and the sixteen variables are defined. How these are obtained from the echocardiogram with the aid of computer software is then explained. Chapter 3 is an in-depth description of Adriamycin: its chemical structure, side-effects, and particularly the proposed mechanisms of action on the heart. Morphological changes in heart tissue are briefly reviewed, as well as functional impairment. Factors potentiating the cardiotoxicity are listed. The need for an effective monitoring program is made clear. Three techniques currently in use (endomyocardial biopsy, radionuclide angiography, and echocardiography) are explained, and some clinicians' experiences with these methods are summarized. Nuclear angiography is dealt with in a little more detail than biopsy, since it is closer to echocardiography in nature. The primary focus is on experiences with echocardiography: case studies, aggregate statistical analyses, and variables which have been tried in monitoring. Several studies which compare the three techniques are examined. General conclusions are made. Chapters 4 and 5 describe our own work. Chapter 4 deals exclusively with components of variability in echocardiographic measures among healthy children. The components are between-subject, serial (short term and monthly) and tracing variation. Our designed longitudinal study is presented: motivation, objectives, design, model, analyses and con-1 elusions. Chapter 5 applies the results of Chapter 4 to serial data on children undergoing Adri-amycin therapy. An empirical argument is made on behalf of the need for monitoring of changes in readings, besides readings themselves. The more sensitive echocardiographic variables are identified. On the basis of several criteria for 'abnormality', we solve for the smallest subsets of the sixteen measures which would identify all abnormal patients. A data-driven approach is taken to identification of the number of dimensions required to describe the normal function of the LV. The combined results of Chapter 5 are used to make a final, albeit tentative, recommendation of echocardiographic variables likely to be useful in monitoring Adriamycin cardiotoxicity. Chapter 6 outlines some topics for further investigation. 2 Chap te r 2. M - M o d e Echocard iography. Mechan i cs . Echocardiography (Gr. r)xu = echo, napSla = heart) is a noninvasive medical technique whereby dynamic images of the chambers and walls of the heart are obtained by means of reflected ultra-high frequency sound waves, or ultrasound. The following description is abstracted from [1]. The patient reclines in the left oblique position. A transducer, about the size of a pencil, is placed on the chest wall in the third or fourth intercostal space (i.e., between the ribs) and at the left border with the sternum. The transducer is attached to the ultrasonoscope, which periodically generates a short electric pulse and sends it to a piezoelectric quartz crystal inside the transducer. This causes the crystal to vibrate and discharge a burst of ultrasound (frequency between 3 and 5 MHz), which is focussed and directed into the heart at the desired location. The sound waves, which travel at about 1540 m/sec in human soft tissue, are reflected by successive levels of cardiac tissue, and return to the transducer. Through a reverse phenomenon these waves are converted back into electric signals and passed to the ultra-sonoscope. Levels of tissue are distinguished by the radar principle: sound waves take longer to return from deeper levels. The transducer "acts as a transmitter less than 1 percent of the time and as a receiver more than 99 percent of the time, thereby minimizing the amount of ultrasound energy passing through the tissue."([l], p.3). The electric signals are further processed by the ultrasonoscope into visual images. There are a variety of 'modes' for visually displaying the echoes ([1], p.7-9). In cardiology, 3 two-dimensional and time motion mode are common. We are concerned here only with the latter, which is abbreviated 'M-mode'. Typically the echo images of the heart are displayed on a screen. Various tuning ad-justments (such as 'gain', or selective signal amplification) are controlled by the technician. When hef obtains the desired image, he can record it on photographic paper with a strip chart recorder. Paper speed is between 10 and 100 mm/sec. This permanent photographic record is referred to as the echocardiogram. See Figure 2.1. The main diagnostic advantage of echocardiography is that it provides a dynamic pic-ture of heart action. This is possible because there are about two thousand transmission-reception cycles per second, giving the impression of continuous-time imaging. In addition, with a sound frequency range of 3 to 5 MHz, physical structures about 0.1 mm apart can be resolved [l]. Another attractive property of this method is that ultrasound is harmless to the patient, especially in comparison with other techniques for studying heart function such as radionuclide angiography ([l], p.36). Echoca rd i og ram of the Left Ventr i c le . For an anatomical diagram of the heart see Figure 2.2, taken from [2]. Different cross-sections of the heart are imaged as the transducer is moved on the chest wall. We will be concerned here with the left ventricle (LV), the posterior LV wall (PW), and the interven-tricular septum (IVS) which divides the left and right ventricles. The diagnostic informa-tion provided by the echocardiogram is very sensitive to the cross-section displayed. Hence standard reference points of heart anatomy are identified in each subject. To obtain the LV echocardiogram, the transducer is first positioned to display the f Pronouns are used generic ally throughout. 4 Figure 2.1  Fragment of an Echocardiogram Right Endocardial _ ' Surface of IVS — Chest Wall .Right V e n t r i c l e I n t e r v e n t r i c u l a r Septum (IVS) L e f t Endocardial _ZL Surface of IVS L e f t V e n t r i c l e (LV) Anterior M i t r a l Valve L e a f l e t (AMVL) Z^'C' \.'*ttCjf2^ P o s t e r i o r Wall (PW) t I I i M Q-Wave of EKG EKG 4a Figure 2.2  Anatomy of the Heart P u l m o n a r y veins P u l m o n a r y trunk a n d a r te r ies Left a t r i u m A o r t i c va lve P u l m o n a r y valve M i t r a l va lve Left ventr ic le Interventricular septum 4b ventricles, with the right ventricle at the top of the screen. See Figure 2.1. The transducer is swept over a narrow arc, until the motion of the anterior mitral valve leaflet (AMVL) is identified. (The mitral valve controls blood flow between the left atrium and left ventricle.) "[The] ultrasonic beam is then directed slightly laterally and inferiorly away from the mitral valve (toward the cardiac apex and the left hip)" ([l], p.36). The ideal recording is made with the transducer directed "below the tip of the mitral valve, at the level of the chordae tendineae" ([3], p. 127). Due to such factors as patient respiration and body positioning, it is difficult to main-tain this ideal perspective for very long. For this reason the ultrasonographer must con-tinually sweep the beam over a small range, to relocate the tip of the AMVL. Thus the cross-section recorded will change slightly over time; when the echocardiogram is quanti-tatively analyzed later, only those heartbeats with the correct perspective are considered. As well as echo images of the heart, the echocardiogram contains a continuous elec-trocardiogram, to clearly mark the events of each heartbeat. In addition, the passage of time is recorded with line markers on the horizontal axis, and vertical markers are used to calibrate distance. As the reader will have gathered, obtaining a successful echocardiogram requires a high level of skill; entire books are devoted to procedure. We have given a very simple summary here, so that the reader may have a better understanding of our data. Quant i t a t i ve Ana lys i s of the Echoca rd iog ram. The measurements taken from the left ventricular echocardiogram essentially represent muscular contraction of this region of the heart. These variables fall into three categories: those pertaining to the LV internal dimension, to the thickness of the septal wall (the IVS), 5 and to the thickness of the posterior wall. (i) LV Internal Dimension. Each heartbeat is divided into two phases: the dilation phase or diastole, during which the LV fills with blood, and the contraction phase or systole, during which blood is expelled. Refer to Figure 2.1. The end of diastole is marked by the electrocardiographic Q wave, and the end of systole by the point of maximal posterior excursion (downward movement) of the left endocardial surface (lower boundary) of the IVS ([4], p.1-24, 1-25). Some authors use different definitions of these events. The one-dimensional 'vertical' span of the LV cavity at any point in time, from the left endocardial surface of the IVS to the endocardial surface of the PW, is termed the LV internal minor axis dimension (LVID). The length of this axis follows a cyclic progression over time. Denote its length at end-diastole by LVIDd and its length at end-systole by The first and most common M-mode echocardiographic variable is LV Shortening Frac-tion, defined as This measures the percent contraction of the LVID during systole. The reader can see that SF is monotonically related to the vigor of the LV pumping action. A three-dimensional analog to SF, the LV Ejection Fraction (EF), is defined as the difference in LV internal volume between end-diastole and end-systole, divided by the end-diastolic volume. Since the echocardiogram gives only a one-dimensional perspective of the heart, certain simplifying approximations are made to estimate EF. It is assumed that the LV is a prolate ellipse whose major axis is twice the length of the minor axis [1]. Based LVIDs. 6 on this geometry, a formula can be derived to estimate the volume of the LV at any point in time. Different authors use different 'corrections' to the formula; we used Table A-5 on p.294 of [5], which tabulates estimated EF for various values of LVIDs/LVIDd = 1 - SF. Consider again the length of the LVID as a continuous function of time. Since t h i s is theoretically a smooth curve, it has a first derivative j^LVID(t), which can be interpreted as the instantaneous rate of contraction (or dilation) of the LV as a function of time. Two common echocardiographic variables are the peak velocities of the LV during systole (abbre-viated LVVs) and during diastole (LVVd); each is defined as the maximum instantaneous rate of change of the LVID during the corresponding phase of the cardiac cycle. The events which delimit these phases are as defined above. Because the rate of change of the LVID is negative during systole, the maximum here is in the sense of absolute magnitude. The normalized velocity of the LV is defined as the ratio of instantaneous velocity to length of LVID at each point in time; i.e., at time to, it is The peak normalized velocities LVNVs and LVNVd are the maxima of this quantity over systole and again over diastole, analogous to the peak ordinary velocities. (ii) Septal Wall Thickness (SWT). See Figure 2.1. The SWT at any point in time is the distance between the left and right endocardial surfaces of the IVS, measured perpendicular to the horizontal time axis. Five septal wall measures are defined analogously to the LV measures. Denoting the SWT at end-systole and end-diastole by SWTs and SWTd respectively, the Septal Wall Percent Change is ±LVID{t) LVID{t0) 7 This measures the percent expansion of the S W during systole. The peak instantaneous rates of change of SWT{t) over systole and over diastole are the peak velocities, denoted SWVa and SWVd; the peak normalized velocities are SWNVa and SWNVd. (iii) Posterior Wall Thickness (PWT). From Figure 2.1, the PWT at any point in time is defined as the perpendicular distance between the endocardial and epicardial surfaces of the LV posterior wall. The Posterior Wall Percent Change is where the subscripts 's' and'd' refer to end-systole and end-diastole respectively. PW% represents the percent expansion of the PW during systole. The peak instantaneous rates of change of PWT(t) over systole and over diastole are the peak velocities PWVs and PWVd; the peak normalized velocities are PWNVs and PWNVd. In addition to these sixteen echocardiographic variables, others are sometimes used. But we will deal only with the ones defined above. Although we grouped the measures here on the basis of physical structure (LV, SW or PW), another classification can be made by variable type: 'percent-change measure', 'ordinary velocity', or 'normalized velocity'. This grouping will be retained throughout the thesis. H o w Var iab les are Measu red . A computer system (for example the Cardio 80 [4]) is used to analyze the echocardiogram to derive values for the variables described. The echocardiogram is placed on a magnetized tablet which is peripheral to the computer. It is aligned according to predefined reference points. Three successive heartbeats are selected for analysis on the basis of clarity of the 8 wall surfaces and correct perspective of the LV. An attached crosshairs stylus is used to enter the time and distance calibration and the points of the electrocardiographic Q-wave into the computer. The four curves delineating the septal and posterior walls and the LV (marked heavily in Figure 2.1) are traced with the stylus. Each curve is traced continuously over the three beats. This process is referred to as digitization of the echocardiogram. The computer calculates the thicknesses SWT and PWT and the dimension LVID for each time point as differences of the appropriate curves. It prints the values of these quantities at end-diastole and end-systole, averaged over the three curves. The techni-cian manually calculates the percent-change ratios to yield SF, SW% and PW%. EF is obtained using SF and the table mentioned earlier. A computer program estimates each velocity curve by means of a moving difference algorithm, takes the maximum over the appropriate phase of each heartbeat, and reports the average of these maxima over the three beats. 9 Chap te r 3. Ad r i amyc i n and Mon i t o r i n g of i ts De t r imen ta l Effects. In t roduc t ion . In this chapter we will first present a brief description of the antineoplastic drug Adri-amycin, including a summary of its cardiotoxic side-effects. Three major medical techniques (endomyocardial biopsy, radionuclide angiography and echocardiography) are currently em-ployed to assess the cardiac damage caused by Adriamycin. We will examine the success of each in turn. Although our own work involved only echocardiography, we felt that a more comprehensive review of the literature would give us a larger perspective on the at-tempts to monitor cardiac impairment due to Adriamycin. By familiarizing ourselves with the experiences of other workers we identified issues which have not been addressed; this motivated our own work. On another note, perhaps the reader may find the summary of some of the existing research reviewed in this chapter useful in its own right. 1. A d r i amy c i n . Adriamycin, also known as doxorubicin, is an Anthracycline drug effective in the treat-ment of a variety of cancers, both solid tumors and haematologic malignancies [6,7,8]. It is produced by a variant strain of Streptomyces peucetius, first found in 1957 in Italy, close to the Adriatic Sea [9]. Its molecular structure is shown in Figure 3.1, taken from [6]. Adriamycin intercalates between deoxyribonucleic acid (DNA) base pairs [9,10], thereby interfering with DNA synthesis. Thus it will select primarily for neoplastic (i.e., continu-ously cycling) cells [9,11]. It is known to have a long half-life in the body [11], as measured 10 Figure 3.1 Molecular Structure of Adriamycin (Doxorubicin) 10a by blood plasma levels, and is metabolized by the liver [12]. Unfortunately, Adriamycin has some severe side effects, including mucositis, alopecia, myelosuppression, hematologic suppression (e.g. of leukocytes and platelets), and congestive cardiomyopathy [11]. This thesis will focus on its cardiotoxic effects, which present the greatest threat to the patient [13]. We will now examine Adriamycin's microscopic effect on the heart. "Myocardial cells are extremely sensitive to certain of its pharmacological effects. The exact reason for this sensitivity is not known." [9]. On the other hand, "Two possible mechanisms of cardiac damage have been proposed: (1) uncoiling of DNA due to doxorubicin-DNA binding, and (2) damage to [lipid] membranes by doxorubicin-generated free radicals." [10]. (Square brackets inside a quotation delimit our own words.) These mechanisms are discussed further in [9]. While the changes induced by Adriamycin in the nuclei of human myocardial cells have been studied extensively (see [10]), there is also evidence of mitochondrial and tubular damage [9]. The observed myocardial cellular and morphologic changes in the heart can be found in [6], [14], and especially in [9]; the latter two give further references. In brief, these changes are: nuclear chromatin clumping, swelling and/or degeneration of mitochondria, loss of myofibrils, and swelling of the sarcoplasmic reticulum and T-tubules. Some tubular structures merge to form vacuoles. Our own work concerns Adriamycin-induced changes in heart function observed macro-scopically, rather than the cellular and morphologic changes observed microscopically. But the above short summary of cellular and morphologic activity should give the reader a feel for the cardiotoxic activity of Adriamycin at the most fundamental level. At the functional level, then, Adriamycin is observed to have a negative inotropic effect 11 [9]; that is, it lessens the force and vigor of the muscular contractions of the heart. This impairment is most severe in the left ventricle, but is present in all myocardial tissue [12]. In some cases it leads to congestive heart failure and death [15, 6,12]. While some of the immediate effects of the drug are reversible [9], this is not often true of the chronic cardiomyopathy, especially in older patients [16]. Although some cases have been treated with success [12], in general: If doxorubicin therapy is discontinued after left ventricular deterioration has occurred, the degree of dysfunction remains but is stable. This may persist for years without the development of congestive heart failure. [12] Even worse, "cases of heart failure appearing for the first time after completion of therapy have been reported." [17] Treatment is often not very helpful either: The patients may symptomatically improve with standard modes of ther-apy (sodium restriction, bed rest, diuretics, and digitalis), but the under-lying cardiomyopathy is generally felt to be irreversible and even fatal. [13] However, there have been experiences [18] of heart function improving some time after discontinuation of Adriamycin therapy. For those with further interest in the biology of Adriamycin cardiotoxicity, [9] is a comprehensive summary paper with discussions of pharmacokinetics, pathology, in vivo effects, proposed biochemical mechanisms of toxicity (see also [12]), possible prevention of cardiomyopathy with adjuvant drug therapy, and a survey of monitoring techniques. Close to two hundred further references can be found there. Adriamycin cardiotoxicity is cumulative [6,9,15,19], rather than schedule-dependent, although there is some evidence that altering the schedule of administration may influence 12 myocardial damage [8]. Although the literature seems to agree on a maximum cumulative dose of 550 mg/m 2 body surface area for adults, the toxic effects can be intensified in several ways, leading to severe cardiomyopathy at cumulative dosages well below this 'safe' level [20]. Mediastinal irradiation [13,15,19,20], concomitant administration of cyclophos-phamide (another antineoplastic agent) [15], hepatic dysfunction [11,15], or previous my-ocardial infarction [6] all serve to potentiate Adriamycin's cardiotoxic effects for various reasons. Moreover, the severity of cardiomyopathy (and incidence of congestive heart fail-ure) caused by a given cumulative dose of the drug varies widely across patients, even in the absence of any of the potentiating factors listed above [12]. (This is referred to as an idiosyncratic response.) If the cardiac damage is so severe that congestive heart failure or death is the result, one has obviously defeated the initial purpose of Adriamycin chemotherapy. Barring this fatal outcome, the drug remains popular among oncologists [7,8] because of its effectiveness against a broad spectrum of cancers: solid tumors, leukemias and lymphomas. Thus Adri-amycin is 'here to stay', at least for now. Coupling this with the idiosyncratic response to its cardiotoxic properties, one sees that "a total dose guideline appears inadequate and the need for monitoring . . . is obvious." [12]. A sensitive evaluation of myocardial function . . . can be used to follow an individual patient throughout the course of therapy. This may then allow the clinician to individualize the dose to that patient's requirements and tolerance rather than depending on a preset level that may be too much in some and too little therapeutically in others. [21] 13 2. Mon i t o r i n g of Ca rd i a c Impa i rment . Several techniques are in use to monitor the adverse cardiac effects of Adriamycin. See [6,9,12] for surveys. Before we examine these, it is profitable to keep in mind some reasonable criteria which an ideal monitoring scheme should satisfy. (i) Sensitivity, or high probability of detection of true (pathologic) changes in the patient's heart. This is perhaps the most important criterion. (ii) Specificity. When the monitoring program yields an abnormal finding, we should have reasonable assurance that it reflects Adriamycin-induced cardiac damage, and not something else. Or at least we should have a way to distinguish the two by other means. Examples of this will be presented later. (iii) Reliability, or consistent accuracy across patients. (iv) Safety and comfort of the patient. The reader will see that the existing techniques differ considerably with respect to these. (v) Elegance, simplicity and low cost. Clinicians should be able to use the procedure without laborious technical requirements of mathematics or statistics. It should be intu-itively reasonable and comprehendable. Moreover, the amount of work involved to obtain the essential diagnostic information should be at a minimum. We will now examine the major monitoring techniques in use. Endomyoca rd i a l B iopsy . This method involves removal of a sample of myocardial tissue from one of the two ventricles. One approach is to introduce a sheath into the right internal jugular vein, and pass a bioptome (a cutting instrument) through to the apex of the right ventricle [7,14]. Alternatively, the sheath and bioptome are introduced into the femoral artery, and passed 14 through to the apex of the left ventricle with the aid of fluoroscopic (X-ray) imaging [7]. In either case, specimens a millimeter or so in diameter are removed, and later analyzed by a pathologist, using both light and electron microscopy. This histopathological approach to assessment of cardiac damage uses essentially the same criteria of morphologic and cellular changes as outlined earlier in the description of Adriamycin's effects. Furthermore, a scale has been developed, whereby one can assign scores reflecting the degree of deviation from normalcy. See [9], and especially [7]. The advantage of the biopsy method is that it "reflects cardiac damage and not my-ocardial function which may be affected by the sympathetic nervous system, anemia, or metabolic or electrolyte problems." [9]. Studies have shown that "the histopathologic grade of the biopsy taken from the right ventricle, on a 0 to 3+ scale, correlates favorably with the total dose of doxorubicin." [12] The disadvantages are numerous. "[Few] centers are capable of doing a transvenous endomyocardial biopsy." [12]. The technique is invasive, uncomfortable for the patient, has some potential complications [7,9,14], and exposes the patient to X-ray radiation [14]. These drawbacks are of even greater concern in children. Since we would want to monitor repeatedly during the course of chemotherapy, these disadvantages are magnified. For further details on the biopsy method and how it fared in clinical studies, the reader is referred to [7] and [14]. Later we will summarize some studies which examined the agreement between biopsy, nuclear angiography and echocardiography. Radionuclide Angiography. Another technique that has been utilized in assessing cardiac damage is radionuclide angiography. Red blood cells are labelled i n vivo with a radioisotope of technetium. Counts 15 of gamma particle emissions from the left ventricle over short time intervals are obtained for several cardiac cycles. Electrocardiographic data are sometimes collected simultaneously [7,22,23]. The principal measure derived is the Left Ventricular Ejection Fraction [EF), which we encountered in Chapter 2. Here it is denned as the difference between end-diastolic and end-systolic counts, divided by end-diastolic counts [7]. This measures the percent of end-diastolic (full) volume of blood in the LV which is expelled by the heart during systole. Since Adriamycin depresses the vigor of LV action, one expects the ejection fraction to be abnormally low (< 50%) in Adriamycin patients. This has in fact been observed [9,12,16,22,23]. Some authors [7,12] suggest comparing a patient's ejection fraction to his baseline (pre-Adriamycin) value, instead of to a universal standard value of 50%. Also, there are indications [12,24] that even patients with abnormally low baseline values can still safely receive Adriamycin if left ventricular function is properly monitored. Others [7,9,12,22] have found that comparative angiograms between resting and exer-cising patients may be more successful at detecting abnormalities. The heart [may] compensate for mild or moderate injury through the sym-pathetic nervous system. It is possible that exercise stressing the heart may increase the sensitivity and specificity of radionuclide angiography by overcoming the effects of compensation for injury. [9] The main advantage in using radionuclide angiography instead of biopsy is that it is noninvasive. One must still label the red blood cells with a radioactive substance, however. The whole process, including data analysis, takes relatively little time: thirty minutes according to one author [24]. Let us examine how radionuclide angiography fares as a diagnostic technique. Saltiel 16 [12] summarizes another study in which angiographic ally determined ejection fractions had a perfect record in identifying those patients at risk of congestive heart failure: . . . they administered an additional doxorubicin dose to five patients who had angiographic ally defined doxorubicin-induced moderate cardiomyopa-thy; in all five congestive heart failure developed. Thereafter, six other patients with moderate cardiotoxic manifestation based on radionuclide ejection fraction had their doxorubicin therapy discontinued; in none did congestive heart failure develop, nor did they have any worsening of their ejection fraction. In addition, 13 patients were given more than 550 mg per sq m, as their ejection fractions were maintained above the requisites of discontinuation; congestive heart failure did not develop in any of these patients. [12] In another study by Dresdale et al. [16], nine patients in clinical congestive heart failure were then assessed by radionuclide angiography. All had highly abnormal (< 30%) ejection fractions. Also, thirty-two of sixty-one asymptomatic patients had abnormal angiographic results. All patients in this study had received at least 430 mg/m 2 Adriamycin. Dresdale's definition of abnormal includes not only ejection fractions less than 40%, but also takes into account comparisons between resting and post-exercise ejection fractions. A further study by Lenzhofer [23] examined fifty-three patients who received anti-neoplastic radiotherapy, cyclophosphamide, and Adriamycin, and forty-five others who received these drugs without radiotherapy. In each group a simple linear regression of angiographically determined resting ejection fraction against cumulative Adriamycin dose had a negative slope coefficient (p < .001 in each group). A study of thirty-seven patients by Lahtinen [25] found that radionuclide left ven-tricular ejection fraction "showed a significant decrease in left ventricular function with cumulative anthracycline doses." It seems from these four studies that nuclear angiographically determined ejection fraction is sensitive to Adriamycin-induced cardiac damage. But this conclusion is based 17 somewhat circumstantial evidence. Echocardiography. Echocardiography has some advantages over radionuclide angiography. There is no need to use in vivo radioactive labelling, and no intravenous injections are necessary. Also, patients need not be sedated during the session. These factors are of special importance in children. In addition, echocardiography yields many parameters of cardiac function, such as peak velocities. Possibly some of these are more informative measures for Adriamycin cardiomyopathy than angiographic ejection fraction. Since our own work involved echocardiography, we will examine the experiences of others with this method in some detail. Markiewicz [26] studied a total of 120 patients on Adriamycin. Only in thirty of these was it possible to gather serial echocardiograms at dosage levels of 0, 100, 200, 400 mg/m 2 Adriamycin, and one follow-up. In thirty-four of the remaining ninety (28% of all patients) the echocardiograms were "inadequate for evaluation of LV function." Mean age in the thirty patients with serial echos was 48.7 years. Mean total Adriamycin dose was 393.8 mg/m 2. Post-therapy echocardiographic follow-up was at a mean of 9.4 months. All pre-therapy shortening fractions were normal (> 25%); the values had a mean of 38.70%. "In the entire group of 30 patients, no significant difference in mean LV size or func-tion [shortening fraction] was noted between the pre-drug and the last treatment study". However, a case-by-case study revealed that in four patients of the thirty, shortening frac-tion dropped below normal during the course of Adriamycin therapy. The first patient, who had no previous heart disease but had received 3,300 rads mediastinal irradiation, exhibited a fall in shortening fraction to 22% after only 220 mg/m 2 Adriamycin. Therapy 19 was continued up to 450 mg/m 2; it was then discontinued because of "evidence of further deterioration of LV function." The patient did not develop heart failure. The second and third patients, both with no previous heart disease, had received 3,600 and 4,000 rads mediastinal irradiation respectively. These patients showed abnormal shortening fractions after 420 and 380 mg/m 2 Adriamycin respectively. Therapy was dis-continued in both cases. The second patient's shortening fraction returned to normal five months later, and the third patient "died 2 months later of terminal cancer without clinical CHF." The fourth patient was not irradiated, but had "a past history of diaphragmatic my-ocardial infarction and mild angina pectoris on effort." Shortening fraction became abnor-mal after only 220 mg/m 2 Adriamycin. Therapy was discontinued, but 3 months later he suffered clinical congestive heart failure and exhibited further deterioration of shortening fraction. However, 10 months later the patient "was free of CHF." Of the remaining twenty-six of thirty patients with serial echos, none experienced a fall in shortening fraction below 25%. None developed clinical congestive heart failure. There were seven patients in this group of thirty who had prior cardiovascular disease. Since this is a potentiating factor for Adriamycin cardiotoxicity, it merits a closer look here. Only one of these patients had his Adriamycin therapy stopped, and this was because of abnormal shortening fraction. Although he did develop congestive heart failure, he even-tually recovered. This may not have been the case had therapy been continued. Moreover, the other six patients continued to receive the beneficial antineoplastic effect of Adriamycin without cardiac incident; this was because their echo studies were normal. Echocardiogra-phy proved to be a valuable monitoring aid in this group of high-risk patients. 20 Of the ninety patients who did not have a complete set of serial echocardiograms done, none developed heart failure, and "all 170 adequate echocardiographic studies in this group showed normal LV size and function." Markiewicz' experience shows echocardiographic shortening fraction to be quite reliable in monitoring for cardiac damage caused by Adriamycin, even for those patients at high risk of toxicity because of previous cardiac disease. Considering all subjects now, no patient with a normal echocardiogram developed congestive heart failure. There were no abnormal echocardiograms except in four patients who were at increased risk because of potentiating factors. In these four, Adriamycin therapy was discontinued solely on the basis of abnormal echo results, and this action may have prevented heart failure in three cases. Ewy et al. [13] studied a total of 352 patients, mean age 55 years, almost all of whom had received cyclophosphamide in addition to Adriamycin. (Cyclophosphamide is thought to potentiate Adriamycin cardiotoxicity.) Initially the authors collected nonserial echocardiographic data on all patients who had received or were about to receive Adriamycin. These were then segregated according to whether the patient had received thoracic radiation as well. In each group, both the end-systolic LV minor axis length and the estimated ejection fraction were significantly different between pre-therapy patients and those who had received at least 450 mg/m 2 Adriamycin. In the irradiated group, such differences existed according to the end-diastolic LV minor axis length as well. Among patients who had received some but less than 450 mg/m 2 Adriamycin, the authors found "no consistent abnormality". Serial echocardiograms were obtained from fifty-eight patients. The data were split 21 into initial-dose (mean 16 mg/m2) and final-dose (mean 440 mg/m2) groups. These groups were not significantly different with respect to any of the variables mentioned above. (It appears that two-sample t-tests were performed here, when paired t-tests are clearly more appropriate. Perhaps the outcome of the tests has something to do with this.) So the aggregate analyses performed by these authors show mixed results for echocar-diography. Biancaniello [18] performed an in-depth echocardiographic study of forty-six children aged six months to twenty-four years, mean age ten. These patients were receiving Adri-amycin as well as other antineoplastic drugs. "Left ventricular performance was assessed before the first dose, prior to each subsequent dose, and every two to three months up to 40 months following the last dose of doxorubicin". Mean total Adriamycin dose was 367 mg/m 2. The echocardiographic variables used were shortening fraction {SF) and systolic time interval ratio, which is the ratio of LV pre-ejection period to ejection time. High values of this variable would indicate LV dysfunction. From past experience the author considered values of shortening fraction between 28% and 40% as normal, and changes of 5% or less in a single patient were not considered indicative of "a significant alteration in myocardial performance." This number was chosen "because the [change in] shortening fraction from repetitive studies performed in our lab within a short period of time under stable hemodynamic conditions did not exceed 5%." Hence Adriamycin therapy was temporarily discontinued ii SF "decreased 6% or more in patients whose function was low normal on the previous echocardiogram, and was rein-stated if the shortening fraction improved or remained stable." Also, because of experience with two patients who "developed symptomatic congestive heart failure when shortening 22 fractions dropped below 20%", Adriamycin therapy was subsequently discontinued in any patient whose SF value fell below 20%. Biancaniello is one of the few authors who considered an abnormally large serial change in shortening fraction as sufficient basis for discontinuation of Adriamycin therapy. Most others look only at readings on a given date. We also examined the possibility of using serial changes in echocardiographic readings as part of a monitoring program. The mean pre-therapy value of shortening fraction was 34%. Biancaniello classified patients into three groups "according to the lowest shortening fraction after cessation of doxorubicin therapy"; twenty-six patients had values above 28% (Group I), twelve had values between 20% and 28% (Group II), and eight below 20% (Group III). There was no significant difference in the doses of doxorubicin received by the patients in the three groups. Nor was there any significant difference, using the Student t test, in shortening fraction between patients receiving different chemotherapeutic regimens or mediastinal radiation. The mean LVSTI ratio [systolic time interval ratio] before therapy was normal for all patients. After therapy the ratio was significantly elevated for Group II and III patients, but there was no significant difference in the LVSTI ratios between Group II and Group III following therapy. One learns from this that the systolic time interval ratio appears correlated with short-ening fraction, but would not permit "separation of moderate from severe dysfunction." For this reason the author discounts its usefulness. (Goldberg [27] finds that "the ratio appears to increase mainly after a cumulative dose of 400 mg/m2.") It seems that the varying degree of Adriamycin-induced cardiac damage across the three groups can not be attributed to differences either in Adriamycin total dose or known enhancing factors. One would naturally suspect group differences in baseline shortening fraction, but this does not appear to be the case: mean initial baseline values are 34.1%, 35.1% and 31.1% respectively. As the author points out, there seems to be a great degree of 23 idiosyncratic cardiac response to Adriamycin, which precludes using standardized dosage formulae and observance of risk-enhancing factors to avoid cardiac damage. Biancaniello's findings underscore the need for a trustworthy monitoring technique, so that each patient can be given separate attention. Of course, these observations could suggest the alternate possibility that echocardiog-raphy (or shortening fraction in particular) is insensitive and nonspecific. After all, classi-fication of patients into categories on the basis of SF values has not suggested any reasons for the differing values. Further examination of the data makes this prospect unlikely. By basing decisions concerning termination of therapy on the shortening fraction or change thereof, these workers were able to limit congestive heart failure to two of forty-six children. Both of these had final SF values of less than 20%. (They developed CHF before the 20% criterion had been established.) Also, "Discontinuing doxorubicin therapy in patients with SF < 20% who.did not have clinical signs of congestive heart failure resulted in improved shortening fraction". Furthermore, Of the 12 patients who had shortening fraction between 20 and 28% (Group II), five returned to normal within two to 15 months . . . after cessation of doxorubicin therapy. Five died of their malignancy shortly after their last dose. The shortening fraction remained unchanged in two, 16 and 26 months after the last dose. Five of nine patients who received more than 500 mg/m 2 "had normal function after cessation of therapy", and showed no signs of heart failure. This suggests that shortening fraction is reliable even in the high dosage range. Biancaniello has found the echocardiographic technique, coupled with a simple moni-toring program and criteria for discontinuation of therapy, to be reasonably successful in preventing congestive cardiomyopathy in this group of children. His initially disappointing 24 results, that no 'correlation' was found between Adriamycin dose and SF category, may be taken as an indication that the potential usefulness of echocardiography should be studied on a case-by-case basis, rather than through aggregate statistical analyses. Bloom et al. [21] performed echocardiographic studies on twenty-six Adriamycin pa-tients (mean age seventeen years) prior to start of therapy, before each dose, then three months and one year after completion of therapy. Nine studies were performed on each patient. The variables measured were as follows. Mean rate of circumferential fiber shortening (mean Vcf) is the shortening fraction divided by the ejection time. It is interpreted as the mean rate of decrease of the left ventricular minor axis during systole, normalized (divided) by the axis length at end-diastole. Next is mean normalized interventricular septal velocity (mean ivsv), which is the posterior excursion of the LV end of the septum during systole, divided by the ejection time, and then normalized (divided) by the LV minor axis length at end-diastole. Correspondingly, mean normalized posterior wall velocity (mean pwv) is the anterior excursion of the endocardial surface of the posterior wall during systole, divided by the ejection time, and then normalized by the LV minor axis length at end-diastole. The three variables just described suffer from the fact that some of their components are measured at different times than others. This is because the transducer is just beyond the anterior mitral valve leaflet when echos used for measurement of LV internal dimensions are recorded, but is at the position of the aortic valve to record echos used to calculate LV ejection time. Hence internal dimensions and ejection times are recorded from different frames of the echocardiogram, yet these numbers are combined in the manner described 25 above to yield the mean velocities. This is not a great drawback if one assumes that the dynamics of the patient's heart are almost completely uniform throughout the session. The authors also computed the systolic time interval ratio, and estimated the ejection fraction. The data were analyzed statistically by performing univariate simple regressions of each variable against the cumulative dose to date. The authors found "a statistically significant drop for each [variable] compared to the control". Presumably they mean that the slope parameters of the regressions were all significantly negative. An important observation is that "mean heart rates tended to be higher at the onset of the study. This was probably due to tumour activity, recent operation, or general patient anxiety." Ejection times would be greater with a lower heart rate; since ejection time appears in the denominators of all three mean-velocity variables, perhaps this is the explanation for the significant decrease in these quantities over the course of therapy. The variables may not be sensitive to Adriamycin. (Adriamycin does not cause a chronic systematic drop in heart rate.) Others ([4], p.1-44) have suggested using empirically-derived equations to obtain a value for ejection time which is corrected for the effect of heart rate. The negative correlation between the systolic time interval ratio and cumulative Adri-amycin dose is interesting. Several other authors have found the reverse to be true. Bloom accounts for this by suggesting that abnormally high heart rates during the first part of therapy may be at fault. From graphs presented in the paper, it appears that all variables show a marked drift 26 toward normality following the cessation of Adriamycin therapy. The authors mention nothing about extensive cardiotonic therapy, so these improvements may have been spon-taneous. The echocardiographic results obtained were utilized in the prospective management of the indi-vidual patients . . . Sudden declines in these values [ejection fraction and mean Vcf] resulted in us withholding therapy until the parameters again improved. Fatal congestive heart failure was seen in only one patient. Elsewhere the authors imply that an abnormal reading in itself constituted sufficient reason for temporary discontinuation of Adriamycin. In summary, Bloom has found ejection fraction, the systolic time interval ratio, and three mean-wall-velocity variables all to decline significantly with increasing cumulative doses of Adriamycin. This decline may be spurious, i.e. a result of declining heart rates, in all but ejection fraction. In any case, ejection fraction and mean Vcf were used successfully in modifying therapy to prevent heart failure. Following cessation of therapy, functional parameters improved. Hutter et al. [28] performed between two and fifteen serial echocardiograms on each of thirty-six children receiving Adriamycin as well as other chemotherapy. They defined an abnormal echocardiogram to be one with a shortening fraction less than 28%, or one for which the shortening fraction dropped by more than 10% from a previous study (e.g., from 40% to 30%). For the group as a whole, no correlation could be made between the cu-mulative dose of doxorubicin administered and [SF]. Two asymptomatic patients, however, demonstrated depressed [SF] . . . at cumulative doxoru-bicin doses of less than 430 mg/sq m. So Hutter measured the peak systolic and diastolic velocities of the LV internal di-27 mension for these two patients and for eighteen others. (These measures were defined in Chapter 2 as LVVs and LVVd.) Hutter found both patients with abnormal (drops in) SF to have grossly abnormal values for these velocities. None of the other eighteen had abnomal velocities. There is some indication, then, that LVVs and LVVd are at least as sensitive to Adriamycin as SF. It should also be noted that these two patients were asyptomatic at the time; so these three variables were sensitive before the critical stage of congestive heart failure. Lewis et al. [29] performed echocardiograms in fifty-four children being treated with Adriamycin or with the closely related daunomycin. They found significant decreases from baseline shortening fraction only after at least 250 mg/m 2 had been administered. "Therapy should be discontinued when shortening fraction < 20%." It seems that either cardiac damage does not occur until after 250 mg/m 2, or it is not severe enough before this dosage level to be detected by shortening fraction. In a second paper [30], Lewis et al. found that shortening fraction in four severely af-fected children returned to normal within one half year following cessation of Anthracycline therapy. Goldberg et al. [27] compared shortening fraction and LV posterior wall thickness curves. "A prior published investigation of normals has shown that the left ventricular wall thins very rapidly in the first 20 - 40% of diastolic time, . . . and then at a much slower rate." Of eleven children in the current study who had received between 225 and 550 mg/m 2 Adriamycin, six exhibited PW thickness curves with uniform (i.e., abnormal) thinning rates throughout diastole; only one of these had an abnormal shortening fraction. None of seven patients who had received no Adriamycin exhibited abnormal thinning curves 28 or shortening fraction. Goldberg concludes that a time plot of the posterior wall thickness throughout dias-tole may be more informative than shortening fraction. In our work, we considered peak thickening and thinning rates of posterior wall thickness numerically. Conc lus ions f r om these Echocard iography Studies. We will now draw some general conclusions from these papers. Because of physical and ethical constraints, one does not often have the freedom to design studies in a way which would facilitate the most precise and pointed comparisons. Most studies above suggest certain conclusions based on only circumstantial evidence; when (asymptomatic) patients exhibiting abnormal values for some echocardiographic measures are taken off Adriamycin, their values (sometimes) improve. Many such patients did not manifest clinical heart failure, indicating that the echocardiographic measures are sensitive at a stage where dosage decisions can still affect cardiac function. One infers that the measures are good representatives of heart function, and will be useful in future monitoring. In some cases shortening fraction was found to be an adequate variable. Other workers found other measures to be superior. Furthermore, Biancaniello found that more than 500 mg/m 2 Adriamycin could be tolerated by some children, with no adverse symptomatology. Since these patients displayed normal shortening fraction and systolic time intervals after therapy, these measures may be useful in averting congestive heart failure even at high dose ranges. On the other hand, abnormal shortening fraction and LV peak velocities are not always associated with cardiomyopathy, as Hutter's study demonstrated. There is an apparent paradox that seems to occur frequently. Echocardiographic vari-29 ables may not correlate with cumulative dose of Adriamycin (Hutter and Biancaniello) when the data are analyzed in aggregate, and yet the same variables may be quite de-pendable for predicting those patients at imminent risk of heart failure (Biancaniello and Markiewicz). The established idiosyncratic cardiac response to Adriamycin explains some of this discrepancy, and has obvious implications for assessment criteria for a monitoring protocol: aggregate correlations, regressions, between-group t-tests and the like are not as important here as a focus on individual cases. Bloom's study brings to light the danger of confounding effects. Some of his measures were algebraically dependent on ejection time, which in turn depends on heart rate. In this way, haemodynamic fluctuation can cause spurious variability in these quantities. Con-versely, Hutter states that "[SF and] maximal [LV] velocities are independent of heart rate variations." 30 Compara t i ve Studies. Three studies now deserve special attention because they directly address the debate between the functional and structural or morphologic techniques. Functionally-based meth-ods are noninvasive, whereas structurally-based ones are sometimes thought to be more sensitive and specific for Adriamycin-induced cardiac damage. Nuclear angiography and echocardiography are examples of the first approach, and biopsy of the second. Although our work involves only echocardiography, we will spend some time with these three primarily nuclear angiographic studies for several reasons. Comparative studies be-tween biopsy and echocardiography are scanty in the literature. Both echocardiography and nuclear angiography yield ejection fraction, a prime functional measure. The studies we are about to examine compared angiographically determined ejection fractions with biopsy grades. Furthermore, one study below obtained echocardiograms as well as angiograms, and found little difference in ejection fractions estimated by either method. On this basis one could tentatively extrapolate the comparisons of biopsy with nuclear angiography to biopsy with echocardiographic ejection fraction, and by extension shortening fraction. In McKillop et al. [22] thirty-seven patients were assessed. Endomyocardial biopsies were graded on a three-point scale. Haemodynamic readings, such as cardiac blood pres-sures and output, were recorded using right heart catheterization. Ejection fractions were obtained angiographically. Also, shortening fraction was obtained with echocardiography. (Lahtinen [25] found a significant correlation (r = 0.44, p < 0.05) between radionuclide ejection fraction and echocardiographic shortening fraction.) McKillop found ejection fraction to be in agreement with biopsy. "Combined rest or 31 exercise [LV ejection fraction] data were abnormal in 27 of 33 patients with moderate or severe biopsy abnormalities, compared to three abnormal LVEF findings in eight patients with no or mild biopsy abnormalities." His results are somewhat discouraging for echocar-diography, however. Of nineteen patients predicted by biopsy and catheterization scores to be at moderate to high risk of congestive heart failure, only seven had abnormal (< 25%) echocardiographic shortening fractions. A study of 158 patients by Ewer et al. [14] investigated the degree of correlation between endomyocardial biopsy grades and ejection fractions. Each biopsy was graded on one of two scales. Higher scores on these 0(0.5)3 scales indicate increasingly abnormal morphology. The ejection fractions were determined by both nuclear angiographic and echocardiographic methods. Some patients had ejection fractions determined by both methods, though it is not clear whether both methods were used with a patient at the same point during therapy. A study of thirty-eight patients by Druck et al. [7] also compared biopsy grades and ejection fractions. The biopsies were graded on a scale which is the same as one of those used by Ewer. The ejection fractions were determined solely by radionuclide angiography. A Gated Nuclear Angiography (GNA) study "was considered abnormal if the rest ejection fraction was less than 55% or the response to exercise showed a fall greater than 5% of the rest value." Ewer "compared 55 sequential sets of cardiac biopsy grades and ejection fraction values. Only 16 sets (29%) exhibited the expected correlation between higher biopsy grade and a lower ejection fraction". These patients had received only moderate total cumulative dosages of Adriamycin. Although one can not separate angiography from echocardiography in the above statement, elsewhere the author states: "Correlation between ejection fraction 32 estimated by either echocardiographic methods or nuclear techniques and biopsy grade was poor." He is not more specific on this. When data are categorized by biopsy grade, it is apparent from Ewer's study that, at each grade, the mean of all ejection fractions determined angiographically agrees with the mean of all those determined echocardiographically. Furthermore, "Analysis of the data revealed a correlation between cumulative Adri-amycin dose and biopsy grade (p < 0.02). No similar relationship existed between cumula-tive Adriamycin dose and ejection fractions obtained at rest". These results weaken the case for noninvasive techniques, if one views biopsy as a 'gold standard' with respect to sensitivity. On the other hand, Druck found "poor correlation between the total cumulative dose of doxorubicin and the results of endomyocardial biopsy". However, when the cases are grouped according to biopsy grade, there is a trend: the mean dose rises with biopsy grade. Druck's statistical presentation from this point on is based on cross-classification instead of on correlation. "Excluding patients with known heart disease, all patients with abnormal GNAs had abnormal endomyocardial biopsies." No patient with a normal biopsy was found to be abnormal by GNA. Also, "Ten patients with normal GNAs had abnormal endomyocardial biopsies. Eight of these . . . had grade 0.5 pathologic changes", indicating they were close to normal. When these eight, and those with known heart disease, are excluded, then the sensitivity of the GNA method against biopsy is 63%. And when one includes exercise-rest GNA comparisons in the definition of abnormal, the sensitivity of GNA rises to 89%. Druck's study is encouraging for noninvasive techniques. 33 Not so for Ewer's study. He raises an important point: "It is the crucial intermediate range of Adriamycin doses which comprises the predominant levels represented in this series where the correlation between biopsy grade and ejection fractions is poor." We would certainly discard a monitoring technique which was useful only at the stage of congestive heart failure. To explain his findings, Ewer identifies two major points in the functional-structural debate. First: . . . biopsy quantitates structural changes specifically related to anthracy-clines whereas ejection fractions reflect global ventricular function that can be affected by other factors including nonanthracycline-related myocardial disease, infiltrative processes, ischemic or valvular disease, pericardial dis-ease, or metabolic state. This may be taken as a statement that functional techniques are less specific in their findings than biopsy. However, most Adriamycin patients, especially children, will not also suffer from the above disorders. In that case one can be reasonably certain that left ventricular dysfunction is the result of Adriamycin. The point about metabolic state is well taken, however; recall from the echocardiography papers that mean standardized velocities are related to heart rate, whereas measures such as SF and peak LV velocities are not [28]. Of course, the apparent greater specificity of the morphologic approach is balanced by the fact that functionally-oriented techniques are noninvasive, which is especially important in children. Ewer's second point: . . . anthracycline-related structural changes precede functional deteriora-tion, and in our group of patients, studied at moderate cumulative Adri-amycin dosages, the ejection fraction remained largely normal despite some degree of structural change. It is implied that functional abnormalities would have eventually occurred . . . with additional Adriamycin. 34 Again, the ominous prospect of not detecting cardiac damage until it is too late. On the other hand, Druck's patients encompassed the entire spectrum of Adriamycin therapy: doses ranged from 144 to 954 mg/m 2 [sic), with a mean of 426. Thus his encouraging results regarding sensitivity of angiographic ejection fraction apply to the higher intermediate range. We can now make some general observations. Ewer's study found both nuclear an-giography and echocardiography to yield similar ejection fractions at each biopsy grade. Lahtinen found correlation between angiographic ejection fraction and echocardiographic shortening fraction. And Biancaniello cites work by others who found . . . that the minor lateral dimensions of the left ventricle measured by ul-trasound correlates significantly with left ventricular dimensions and vol-umes measured by angiographic methods. In addition, [SF] correlates significantly with ejection fraction. The upshot is that since these angiographic and echocardiographic variables often agree, what we are examining here is not so much the potential usefulness of angiography or echocardiography as monitoring techniques, but the sensitivity of a functional technique versus that of a morphological one. It is advantageous at this point to recap earlier conclusions from the angiography papers, and to consider them together with results of these three comparative studies. The study cited by Saltiel and the work of Dresdale indicate that ejection fraction does well at identifying patients at risk of congestive heart failure. Dresdale's results also indicate that ejection fraction is abnormal in some asymptomatic Adriamycin patients, presumably with moderate but subclinical cardiac damage. Both Lenzhofer and Lahtinen found that ejection fraction correlates well with cumulative Adriamycin dose, although Ewer found otherwise. Taken together, these findings provide circumstantial evidence that ejection fraction reflects 35 to some degree the underlying morphologic changes wrought by Adriamycin. The three studies by Druck, Ewer, and McKillop address this question more directly, by examining the degree of agreement between biopsy and ejection fraction. The study by McKillop produced mixed results. Eighty-two percent of patients with moderate or severe biopsy grades had abnormal angiographic ejection fractions, which is a favorable statistic for adherents of the functional approach. But only thirty-seven percent of patients deemed by biopsy and haemodynamic measures to be at moderate to high risk of heart failure had abnormal shortening fractions. McKillop defined an abnormal shortening fraction as one below 25%, and hence did not allow for patients who had a higher-than-average baseline value, but dropped significantly after the start of Adriamycin therapy. This may account in part for the low sensitivity of shortening fraction here. Ewer states that only 29% of fifty-five series show a significant correlation between biopsy grade and ejection fraction. And these patients had received only moderate doses of Adriamycin, which might induce one to conclude that ejection fraction is an insensitive measure of moderate cardiac disorder. But perhaps the biopsy grades and ejection fractions within each series would have exhibited higher correlation had the pathological measure-ments been lagged behind the functional ones in the statistical analysis. After all, we recall that morphological changes precede functional ones, as Ewer, and also Saltiel, note. Maybe the reason why Druck's study found ejection fraction to be in closer agree-ment with biopsy grade, is because he summarized his results in the cross-classification framework of a simple two-way table, and did not formally calculate (lagged) correlation coefficients or other more sensitive statistics. Furthermore, the disagreement between the two methods in Druck's analysis may be a matter of timing: of the ten cases of normal 36 angiograms but abnormal biopsies, eight had a biopsy grade of 0.5, which is close to normal on this scale of range three. With continued Adriamycin therapy, and the attendant higher biopsy grades, maybe the ejection fractions would have become abnormal too. Some further thoughts on this last point are in order. It is a fact that Adriamycin will adversely affect the heart. Since it is the ability of the heart to function which is of primary concern, the fact that a functional technique used early in therapy may show no signs of cardiac impairment, while a morphological analysis may uncover some damage, is not a cause for great concern. In sum, as long as cardiac function has not been affected significantly, one would be inclined to continue with the chemotherapy as planned. 37 Some Add i t i o na l Remark s . A considerable number of intuitively plausible techniques have been explored in at-tempts to monitor Adriamycin cardiotoxicity. At the root of all efforts motivated by the functional approach is the fact that Adriamycin causes a general decrease in the vigor of the LV pumping action and consequently in its efficiency - volume of blood expelled with each stroke. These general functional changes are the direct result of deterioration of the endomyocardial cells. In light of these facts, the most popular functional variables to date, left ventricular ejection fraction and shortening fraction, are sensible choices. But in the search for better ways to assess Adriamycin damage with the aim of modifying therapy, one needs to go deeper, and consider other aspects of heart function. One can come up with many dy-namic measures: ratio of action time to relaxation time, electromechanical variables, wall velocities, percentage contraction of the walls, etc. The mechanics of the heart are complex; there is no general theory which describes the precise relationship between the morphologic changes brought about by Adriamycin and the functional aberrations manifested macroscopically. So one must search for optimal variables on a purely empirical basis. There are at least three directives: one must discover which variables change significantly with Adriamycin, do so early in therapy, and can be measured with sufficient accuracy to make it worthwhile. In addition to these goals, there are some basic questions pertaining to function of the normal heart, whose answers would shed light on monitoring approaches. This is the subject of the next chapter. 38 Chap te r 4. A n Expe r imen t to Ana l y ze Sources of Va r i ab i l i t y In Echocard iograph ic Measures i n N o r m a l Ch i l d r en . In t roduc t i on . We now turn to our own work. It became clear from our survey of the literature that attempts to monitor for LV dysfunction with M-mode echocardiography have been moder-ately successful. But in the words of Pollick et al. [31], "the variability and reproducibility of measurements has been assessed rarely." With the notable exception of [31], no com-prehensive effort has been made to isolate and quantify the sources of random 'noise', and suggest ways to minimize it. Besides reducing one's confidence in the readings, this noise will cause bounds for normal values to be excessively wide. The chances of identifying patients at risk of cardiac incident will consequently be lessened, even assuming there is an underlying echocardiographic 'signal' differentiating them from normals. This may account for some of the discouraging experiences clinicians have had. There may be other reasons why existing bounds are 'too wide'. Factors such as subject age or sex may have bearing on normal echocardiographic readings. It is possible, of course, that healthy subjects are so heterogeneous in their echocardio-graphic parameters that not much improvement can be made to the width of normal bounds. However, with repeat monitoring on any one individual, between-subject differences become irrelevant, and it is plausible that the magnitude of within-subject (longitudinal) changes over, say, one month could provide the basis for a more sensitive monitoring program. This presumes that 'random' serial variation in a normal subject's parameters is substantially 39 less than the more systematic decline in LV function exhibited by a patient on Adriamycin or similar cardiotoxic drugs. This assumption would have to be tested. Moreover, a comprehensive literature search failed to discover any studies which assess the magnitude of serial within-subject variability in healthy individuals' echocardiographic readings. The degree of homogeneity of subjects with respect to longitudinal variation, and whether some measures are more stable than others, do not appear to have been addressed. Since answers to these fundamental questions could prove useful for monitoring, we decided to carry out a study on normal children. Several other issues arose, some of which are not directly related to monitoring, but which would nevertheless be of interest to those who work with echocardiography. For example, what are the relative magnitudes of variation due to between-subject physiological differences, within-subject serial fluctuation, and technical factors such as tracing? We took the opportunity to make the most of our data, and address these questions as well. A detailed discussion of our goals follows now. 40 Identification of Objectives. Improvement of the Precision of a Reading. Echocardiographic readings are sometimes accused of being untrustworthy and imprecise, because they are subject to uncontrollable interference. In the discussion below, we first address this criticism, and look for ways to improve confidence in readings. Consider any single echocardiographic variable. Typically one would take a reading at several points throughout therapy. One should be confident that each recorded value could be duplicated with reasonable accuracy if the process were repeated immediately. If one could improve the precision of each recorded value, then the detection of true changes in LV function over longer periods of time would be more likely. Suppose one obtained multiple readings from the same scroll. The numbers recorded would in general not be the same. This variation may be attributed to three major sources: short-term physiological fluctuation of heart function over the course of the session; vari-ability due to the ultrasonoscope and its operator; and 'error' introduced by tracing of the echocardiogram. The variation due to the ultrasonoscope and operator is in part due to the fact that a human hand is holding the transducer against the chest wall. The technician constantly sweeps back and forth, as explained in a previous chapter. Although only those images which reflect the correct perspective of the LV will eventually be traced, this projection will never be exactly the same for two corresponding points in different sweep cycles. Slightly different cross-sections of the heart will be traced. In this way, another form of serial variation is introduced by the ultrasonographer. Since the starting point in obtaining a value for a variable is the echocardiogram (scroll) 41 itself, this technical variability is (at present) statistically inseparable from that due to short-term physiological changes. But one can perform repeat tracings of the same echocar-diogram, and hence the variability due to the tracer can be isolated from the technical / physiological variability. In recording a value for a given variable on a day, one can reduce the inherent variability by taking the average of several readings. The question then is: does analyzing curves of additional heartbeats, or tracing a single curve repeatedly, result in the greatest reduction in noise? In other words, which is greater: short-term inherent variability, or that due to tracing? This issue was addressed by Pollick et al. [31]. His conclusions were based on less-than-ideal data: the adult subjects were not all healthy, there was marked imbalance in design, and statistical parameter estimates from one group of subjects were then used with other data to derive estimates there. Given these facts, we felt it would be worthwhile to address the 'optimal-reading' issue again. It should be noted, however, that we can appreciate technicians' efforts required to produce even a dataset which does not conform to 'textbook' standards. Prediction Intervals. One would eventually want an interval of values considered normal for each echocardio-graphic variable, to use in monitoring. To maximize the chances of detection of LV dys-function, these prediction intervals should be as 'tight' as possible. If normal subjects can be segregated into more homogeneous groups, then intervals for normal values should be derived for each group separately. Since echocardiographic readings are physical measurements, subject age and sex are 42 two plausible bases for segregation. So it is of interest to examine whether these two factors have any bearing on the readings. For monitoring, one may wish to look not only at the readings themselves, but at changes in these over time. In the literature one finds reported normal ranges for readings, but not for serial changes in them. Our medical collaborators currently perform echocardiographic studies on Adriamycin patients at roughly six month intervals. In future, they may monitor more frequently; e.g., monthly. There is a need for a normal reference standard: a numeric interval for each echocardiographic variable, which gives the range of values for, say, monthly changes which one can expect in healthy children. Serial Variation. It would be interesting to compare the magnitude of serial variability in echocardiographic parameters over the course of the session, with that over a month's time. There do not appear to be any existing studies which inquire into the serial stability of these measures. If it happened to be the case, for example, that the degree of serial variation is about the same over the two time periods, then one could estimate normal bounds for monthly changes without waiting a month between studies. (Ultimately one would like to compare monthly and yearly fluctuation, to get an idea of when and if the longitudinal variance becomes constant.) Recalling earlier comments, it should be kept in mind that any estimates of serial physiologic variability will necessarily subsume technical variability due to the ultrasono-scope and its operator. However, it is conceivable that the contribution of this additional variation to the former is the same (absolutely) over minutes and over months (with the 43 same operator and machine). The implication is that although estimates of both pure serial physiologic variabilities will be biased upwards, we can still meaningfully compare the underlying true variances by looking at the estimates. In statistical terminology, the confounding factor has an uniform effect in both instances. Support for this assumption is provided by the fact that the technician takes great care to ensure that a consistent cross-section of the heart is travelled as the transducer sweeps. In fact, we observed that he spends a considerable amount of time sweeping back and forth, until he is satisfied that he has the correct perspective of the LV on the screen. Only then does he flip the switch to produce the scroll (hardcopy). It would follow that at any given session he will converge on the same set of images and continue to sweep over them. The remaining variability arises primarily from human limitations on visual resolution and hand-eye cooordination, and will be the same whether he does it now or in one month. Additional Topics of Interest. There are two fundamentally distinct groups of echocardiographic variables which we con-sider: percent-change measures, and peak velocities. We decided to compare these on the bases of long- and short-term serial physiological / technical variability, and on the basis of variability due to tracing. It is plausible a priori that the two groups should differ on these accounts. Our reasoning is as follows. The degree of wall movement during the cardiac cycle and the maximal force which the muscle musters to achieve this movement are two distinct physical attributes. Hence within-subject physiological variability may be different for percent-change measures and peak velocities. (Note however that we expect a higher correlation between degree of wall 44 movement and average force.) (The inseparability of technical variation due to the ultrasonographer from within-subject serial physiologic fluctuation is not a problem here. The former is expected to have the same effect on both percent-changes and velocities, since it affects only the pictorial echocardiogram; the variables are ex poet concepts, introduced at the tracing stage.) Readings for peak velocities incorporate data from the entire lengths of the relevant curves on the echocardiogram. The velocity curve is estimated instantaneously by the computer program, and then its maximum is reported. If the tracing error is such that the slope of the traced curve is exaggerated even for an instant, this 'velocity' may be greater than the true maximum. And so a deviation of the stylus from the line at any point can result in an erroneous value for the peak velocity. In this way spurious variability between successive readings could be introduced quite easily at the tracing stage. In comparison, the values produced for percent-changes are sensitive to tracing error only near the points of maximal and minimal excursion of the septal and posterior walls. It follows that erroneous values would occur less frequently than for the peak velocities. Hence with repeat traces the average tracing error of percent-change measures is expected to be small in comparison with that of the peak velocities. Examination of this hypothesis is worthwhile from a practical standpoint, since one would favor those echocardiographic variables prone to a minimum of tracing error. Lastly, it seems to be a common practice to normalize each velocity through instanta-neous division by the corresponding dimension (thickness or length); the peak normalized velocity is the maximum of (the absolute value of) this new quantity over the cardiac cycle, as descibed in an earlier chapter. Normalization attempts to correct for (suspected) corre-45 lation between instantaneous velocity and heart size, although non-normalized ('ordinary') peak velocities are also employed. If it is the case that ordinary peak velocities are not related to heart size, then it would seem that normalization introduces variation due to subject age, since a child's heart size is related to his age. This would manifest itself as greater between-subject variability for peak normalized vis-a-vis peak ordinary velocities. 46 Methods . Perhaps at this point a brief overview of our study will be advantageous to the reader. We tried various models prior to collection of any data; indeed, the one we finally settled on dictated the design. We performed the algebra for each model, and examined whether all our questions could be answered adequately. In other words: (i) whether the interpretation of the model conformed to our understanding of the physical processes; and (ii) whether everything of interest was estimable and/or testable. We felt that a Variance Components Analysis and an accompanying hierarchal model of random effects was suitable for examination of all but the age and sex effects. So for each echocardiographic variable, there is a four-factor successively nested random effects model. The factors are 'subject', 'month', 'beat', and 'trace'. Estimates of the variability due to each of these factors were used in several ways to address most of the questions we set out to answer. The possible effects of subject age and sex on readings was examined with an Analysis of Covariance for each variable. (i) Participants. Eighteen children (7 females and 11 males), ages 3 to 16 years (mean 9.6), took part in our study. No child was known to suffer from any cardiac defect or ailment. All partici-pants are children of staff at Children's Hospital, Vancouver, British Columbia. Data were collected on two visits approximately one month apart in the spring of 1985. With data on only eighteen subjects, our conclusions should be viewed conservatively. (ii) Echocardiographic Variables. These have been explained in a previous chapter. To refresh the reader's memory, they are: 47 Percent Changes: Posterior Wall Percent Change (PW%) Septal Wall Percent Change (SW%) Ejection Fraction (EF) Shortening Fraction (SF) Peak Ordinary Velocities of: LV Internal Minor Axis Dimension during Systole (LVVa) LV Internal Minor Axis Dimension during Diastole (LVVd) PW Thickness during Systole (PWVs) PW Thickness during Diastole (PWVd) SW Thickness during Systole (SWVs) SW Thickness during Diastole (SWVd) Peak Normalized Velocities: There are six of these as well, one corresponding to each peak velocity. Their abbreviations shall be distinguished from those of the above by the letter 'N' ; for example, LVNVs. (iii) Technical Details. A detailed account of the measurement process has been given in an earlier chapter. Subjects were placed in the left oblique position. M-mode echocardiograms were obtained by Mr. Eustace DeSouza, B.Sc, using a 3 or 5 MHz single crystal transducer with an A.T.L . 500 Ultrasonoscope; paper speed was 50 mm/second. Gains were adjusted to give the clearest septal and posterior wall images. Selection of heart cycles on the echocardiogram hardcopy for tracing, as well as iden-tification of curves taken to represent the physical limits of the septal and posterior walls 48 was made on the basis of clarity and definition of the wall surfaces. This was done by Mrs. Ruby Popov, R.N., who was responsible for all tracing and the recording of values. We explained the importance of random selection of curves, subject to clarity. Although it seems unlikely, it is possible that curves unsuitable for tracing differ systematically from the others with respect to information content. In this event bias would be introduced. LV dimensions and wall thicknesses were measured according to the standards of the American Society for Echocardiography. A Kontron Cardio 80 system (personal computer, digitizing stylus, magnetic tablet and software) was used to digitize the echocardiograms and produce values for all variables. Dr. George Sandor, M.B., Ch.B., supervised the collection of data. (iv) Structure and Collection of Data. Each child visited the laboratory on two occasions, approximately one month apart. On each visit, an echocardiogram comprising about thirty seconds of real time was ob-tained. On each echocardiogram, two distinct groups of three beats each were selected. These groups were chosen relatively far apart on the scroll, since they represent short-term (session) variability. Each group of three beats was digitized (traced) once. For each echocardiographic variable, an average of the three values was taken over each beat group. This average is the fundamental observation. (Using three beats instead of one for an observation is the standard practice in this laboratory; it is felt that this yields more 'representative' values.) All beat groups selected were traced once more. So, for a given variable, the data on 49 one child have the following layout: Subject i Month 1 Beat Group 1 Beat Group 2 Tr. 1 Tr. 2 Tr. 1 Tr. 2 3.23 3.11 3.83 3.70 Month 2 Beat Group 1 Beat Group 2 Tr. 1 Tr. 2 Tr. 1 Tr. 2 2.30 2.40 2.75 2.61 (v) Sources of Variability. There are four of these, mentioned earlier. In the notation below, there are four <r2's for each of the sixteen echocardiographic variables. (The formal definition of these symbols is deferred for the moment.) (a) Subject-to-subject variability ( t r | ) . This refers to the variation in readings between different subjects, due to physiological differences, etc. It is distinct from the variability exhibited over time by any single subject, and from variability due to the measurement process (except possibly for some subject-ultrasonographer interaction.) (b) Long-term (month-to-month) serial variability (o^)- This quantity represents within-subject serial fluctuation over one month's time due to normal physiologic changes. Also, it subsumes some variation attributable to the ultrasonographer. Month-to-month variation would not include growth-related fundamental changes in heart size and power, which would probably not be detected over such a short time period. They may become important over the duration of potentially cardiotoxic therapy - say two years. We will investigate this aging effect separately below. (c) Short-term serial variability (between groups of three consecutive beats each) (o^). This subsumes the same factors as (b), but pertains to the time period required to obtain 50 an echocardiographic image. The distinction between short- and long-term serial variabilities is somewhat artificial; they really refer to the same concept. But we distinguish them here because the magnitude of variation between beat groups has bearing on the trustworthiness of a reading, whereas variation between months would find application in setting normal standards for serial fluctuation. (d) Variability due to tracing of the echocardiogram (a-2-,). Due to human involvement, roundoff error, etc., repeat tracings of the same curve are expected to yield slightly different results. Sta t i s t i ca l M o d e l I. With sixteen variables among which one expects high correlation, it seems natural to think of a multivariate vector response. But we chose to stay with univariate analyses. Our medical co-workers are interested in each variable in its own right, as opposed to interpreting it as a component or facet of a multi-dimensional response. (This also rules out dimension-reduction techniques which create new variables as linear combinations of existing ones.) Moreover, there is some arbitrariness in our choice of variables. Since in future clinicians or researchers will work with only some of the variables that we have, a univariate focus is appropriate here so that our conclusions can be used by others. As the reader will see, the algebra of estimation and testing in our model is intricate enough. It is not clear that it would generalize easily to vector responses, even if this were desirable. We employed a four-factor successively nested random effects model for each echocar-51 diographic variable. (See SchefK [32].) The cr2's above are the variances of the respective model terms. Formally, let F,yjt< be the reading which is produced by the Ph trace of the kth beat group chosen on the itk subject's scroll for the 3 th month. The model is: Yiju = n+ Si + Mj(i) + Bk(ij) + Ti(ijk). Here 5",-, t = 1,...,18 are NID(0,(Tg), representing the random subject effect. Condi-tional on subject t, the Mj^, j = 1,2 are NID(0,erj^), representing the random month effect. Because (i) each subject's Adfy(,-j's have this same distribution and (ii) Af,-(,-) 's for different subjects are assumed independent, it follows that the My(,)'s are unconditionally NID(oyM). Note that each subject i has his own Afy(,-) 's, although all subjects were studied during the same two months. We posit here that 'month' should be nested within 'subject', not crossed with it, because there is nothing special about, say, March 1985 which is common to all subjects. Because the subjects are of different ages, the two successive months may be thought of as being randomly chosen from each subject's lifetime. Note also that the monthly variance parameter is <rj^ , not <7j^[t]; we assume that monthly longitudinal variation is the same for all subjects. Later we shall examine whether this assumption is reasonable, given the data. Analogous to the Afyj.-j's, the i?jt(,-y)'s, k = 1,2 are NID(0, o-g), conditionally on and unconditionally. These terms represent the random effect due to the group k of three beats selected on the echocardiogram. The same sort of reasoning can be applied here. Each subject's heart beats many times on a day. Of all these, we randomly select two groups of three consecutive beats each. Since the groups are selected from an echocardiogram spanning only a few minutes, we can interpret o~\ as a measure of serial variation over 52 the course of an echocardiographic session. (The only other restriction on sampling is that fuzzy images were not candidates for selection.) Since two types of variability of interest are temporal in nature, the reader may have thought that a temporal stochastic-process model would be a more natural choice than a nested discrete-effects model such as ours. We felt that, with the appropriate interpretation, our model preserves the essentials of reality, allows us to examine the questions of interest formally, and most of all requires far less data than a 'time series' analysis would. Finally, T .^-yjt), t — 1, 2 are NID(0, a2*.), representing the 'trace' effect. Then a\ is the variance of observations produced by repeat tracings of the same group of beats. All four effects in the model are assumed independent. Hence the variance of a single observation is o\OTAL = E[Yijki — fi]2 = o~% + aj^ + a\ + a\. An important point should be kept in mind. The variance parameter is the vari-ability explained by monthly fluctuation, over and above that explained by beat-to-beat variation. It is the net monthly component of variance. So when testing whether monthly variability is the same as that over the session, we test Ho'.o~\f = 0, not flo:(TM = ° B -The reader can convince himself of this by noting that if aj^ is zero, then the Af.(yj in the model are all identically zero. Dropping this term from the model is tantamount to stating that 'month' has no effect on readings for any given subject. Similar comments apply to a%; it is the variability in readings resulting from different beat groups on the same echocardiogram, net of the variation in these numbers attributable to the fact that they arise from different traces. Sta t i s t i ca l Model II. To examine age and sex effects for any given variable, we first took the means over 53 the eight readings on each child, yielding eighteen observations per variable. We then performed an ANCOVA for each variable, with sex as the categorical variable and age as the fixed continuous covariate. Both of these were treated as fixed effects. Homogeneity of slopes was first examined. We decided to analyze the averages of the eight readings for each subject because the alternative would have been to deal with a mixed model with four random and two fixed effects. Where an age effect existed, we further tested whether it could be better approximated by a quadratic relationship. 54 Ana lys i s and Resu l ts . E s t ima t i on and Test ing of M o d e l I Parameters . Because the model terms are random and nested, the observations on a given subject for any particular variable are not i.i.d.. One could still write down sums of squares as in a fixed effects model, demonstrate algebraically that an ANOVA-type decomposition exists, and put one's trust in a 'quickie' Expected Mean Squares Table, as in Montgomery [33], p. 147. This would lead to unbiased estimators of the variance parameters as linear combinations of the mean squares. Although many of our questions can be addressed with (point) estimates alone, we are interested in testing as well. We want to know for example whether <rj^  is zero for each variable. Since the observations are not independent, Cochran's Theorem [32] does not apply, and therefore we can't know without additional work whether the sums of squares are multiples of independent chi-square random variables. (Besides allowing us to perform tests, this last distributional statement, if true, will imply some optimality properties of our estimators. We will return to this later.) Graybill and Hultquist [34] deal with a general class of models of which ours is a member. They give conditions under which, in their terminology, an ANOVA 'exists'. Unfortunately these conditions involve calculation of eigenvalues of a matrix whose elements are linear combinations of our er2's. An application of a theorem by Albert [35] shows that ANOVA-type .F-tests for model variance parameters exist, with much more tractable algebra. Consider one echocardiographic variable. Let Y be a random 144-vector corresponding to observations on it; i.e, there are 18 groups (one per subject) of 8 observations each, and within each group the components are ordered as in the data layout earlier, read left to 55 right. Define random vectors as follows: Xs = (Si, S2,..., Sis)' X M = ( A f i ( i ) , M 2 ( i ) , . . . , A / i ( i 8 ) , M 2 ( i 8 ) ) ' Xfl = (01(11), 02(11), ..01(12) . 02(12) i • • • , 01(18,1)' 02(18,1). 01(18,2), 02(18,2) )' X T = (Ti (ui ) , . . . , r 2 (i8 ( 2 ,2)) ' Then the random X r ' s correspond to the subject, month, beat-group and trace effects, respectively. Note that the vectors are of different dimensions. We now define some 'design' matrices. J„ is the identity of order n, Jn is the n x n matrix of ones, 1„ is an n-vector of ones, and '®' denotes the Kronecker or direct product of two matrices. Let Gs = Is, Us = hs®Gs GM = h® I4, HM = lis ® GM GB = I4 ® I21 Ho = I\& ® Go-Then the model presented earlier for an observation Yijkt is equivalent to Y = All 1 4 4+ Y, H'Xr + X T - (*) r e ( S M , B } (This belongs to the Eisenhart Model II class.) Under the normality and independence assumptions made earlier, the components of each X R for r G {S,M,B,T} are i.i.d. uni-variate N(0,a 2), and components in different vectors are independent. Thus the X r ' s are independent multivariate normal (with different dimensions), and hence so are the HfX/s, which all have dimension 144 x 1. It follows that Y is multivariate normal with mean film; denote its covariance matrix by W. It is trivial from (*) to show that W = <r 2,Ilii + E <r 2HrHr'. This reduces to / i s ® ( 4 7 8 + £ a 2GTGT'), r€{SM,B} re{S,M,J?} 5C because sets of observations on different subjects are independent. (For definitions and results on Kronecker products used implicitly here, see Anderson [36].) The advantage of this multivariate recasting (not a reparameterization) of the original nested effects univariate model will become apparent shortly. Note that we did not have to make any further distributional assumptions in order that Y be multivariate normal. This is not a trivial consequence, since the Y.jjt/'s are not i.i.d.. Assessing Normality. Before moving on to sums of squares and estimation, we pause here to have a look at the underlying assumptions of normality. It will be seen that the estimators we derive are still optimal within a fairly large class, even without assuming normality. But the appropriateness of the tests and the bounds for normal echocardiographic values do rely on this assumption. Because the observations Yij^i are not independent (W ^  I), we cannot check normality with, for example, normal scores plots of them. But by making use of the known covariance structure, an alternate route is available. Partition Y as (Yi ' , Y2',..., Yxg')', where each Y,- has eight components, and corre-sponds to the observations on one subject. Then from Y ~ Nmdilm, W), it follows that the Y,'s are i.i.d. JV 8(/il 8 , V), where W = I 1 8 ® V. Hence Z,- = V - 4 ( Y . - - / i l 8 ) - NID(Os,h), i = l, . . . ,18. (**) In other words, the eight components z.j. of each Z,- are independent and N(0,1). Further-more, since the Z,-'s themselves are independent, we have 18x8 = 144 independent N(0,1) random variables 2,*. Since all this follows directly from the initial assumptions of normality and indepen-57 dence concerning the 5,'s, Af^.-j's, etc. of the hierarchal model, we could check those assumptions by obtaining normal scores plots of the z.-jt's, if we knew V and p. It will be argued now that using estimates in place of these parameters and proceeding with normal scores plots, is not unreasonable. Consider V. Although it is an 8 x 8 matrix, there are really only four unknown param-eters attached to it. For recall that re{S,M,B} The Gr are design matrices of known constants. Contrast this with the general unstructured case where the covariance matrix would have (8 x 9)/2 = 36 different parameters, and the estimate of each would have only 17 degrees of freedom associated with it. Also, each component of the mean vector is the same: p. In the general case, this is not true. Intuitively then, since 144 data points are being used to estimate only five parameters, we expect that the estimates will be very close to the true values. (The actual 'degress of freedom' associated with each variance component estimate will be presented later.) And so it is not unreasonable to replace V by its estimate {a^Is + Yl ^GTGT') and p re{S,M,B} by the sample mean Y in (**). The data for some of the echocardiographic variables (one from each of the three groups) were transformed in this way, and for each variable normal scores plots were obtained for the 144 (Denoting the ranks by r,-, the scores used are those due to Blom [37]: 3>-1[(r,- — 3/8)/(n + 1/4)].) Figures 4.1, 4.2 and 4.3 present the normal scores plots for SF, LVVa and PWNVd respectively. Each is quite close to a straight line, indicating that the assumption of normality is upheld. 58 00 0» (0 u o o w o 3 . 0 + 2 . 0 + 1 . 0 + 0 . 5 • - 0 . 5 + - 2 . 0 - 3 . 0 + -+ - -- 4 . 0 Figure 4.1 - Normal Scores P l o t f o r SF L E G E N D : A - 1 0 8 S . B = 2 O B S . E T C . A A A B A B A A C D A B B E A -BD D C F B B B B A D G D B B D B B B C B D D A A A A A A - 3 . 5 - 3 . 0 - 2 . 5 - 2 . 0 - 1 . 5 - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 o 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 SF (transformed) Figure 4.2 - Normal Scores P l o t f o r LWs cn oo tr 0) u o o w o z 3.0 + 2.0 + 0.0 + -1.0 + A A A B A A B A A A A A A A A B B C A B A F 4CB C C A DC A C D D C C C A B A B A B A A A A A B B A A DA C B CA B A -2.0 + - 3 . 0 + -3.00 1.50 3.00 LWs (transformed) Figure 4.3 - Normal Scores P l o t f o r pWNVd I 3 . 0 + 2 . 0 + A A A A A A A A B B A D 1 . 0 + AC B A A A AD D B 0 . 5 + A C B A C C A E A A D A A 0 . 0 + B B B B B C B A B C A A C B 0 . 5 + A B D A B A B B B A DA 1 . 0 + A A A A ABA A B B A 1 . 5 + A A A I AA A • A 2 . 0 + A A 2 . 5 i e 3 . 0 + A A A A - 2 . 0 0 - 1 - 2 5 - 0 . 5 0 0 . 2 5 1 . 0 O 1 . 7 5 2 . 5 0 PWNVd (transformed) Also, skewness and kurtosis estimates were computed for the 2 , * . They are as follows: Variable N Mean Std. Dev. Skewness Kurtosis SF 144 -0.0004 1.0000 0.0036 1.1586 LVVs 144 0.0000 1.0000 0.0093 -0.0895 PWNVd 144 0.0000 1.0000 -0.0466 0.2190 The skewness and kurtosis are small for all three variables. This supports the assump-tion of normality. (Note that standardizing the original observations does not affect the skewness and kurtosis estimates.) Given that many of the echocardiographic variables are ratios and maxima, one might think that the normal is an unlikely distribution for these data. But given these favorable diagnostics, it seems reasonable to prceed under this assumption. We will later examine the properties of our estimators under more general circumstances. 59 Sums of Squares. We now write the intuitive sums of squares and express them as quadratic forms. Abbre-18 2 2 2 viating E E E E by E . w e h a v e t = l j=l fc=l «=1 i,j,k,l SSTOTAL= [Yjjki - Y"....)2 - 2 ^?fc« ~ Y44 y- 2-Y ' Y - Y ' P r o r Y , with PTOT The sum of squares for subjects is SSS= J2(Y'--Y-)2 i,j,k,l 144 = Y ' P S Y , With PS = (hi ® \Jt) ~ J ^ - W For months, one gets SSM= (Yij..-Yi..f i,j,k,l = Y'PMY, where M — ^18 1 -1 -1 1 For beat groups, where SSB = ^2(Yiik.-Yii..)2 i,],k,t = Y'PBY, PB = hs ® |/2 ® Q [ _Xr "* ® J^j J Finally, for the effect due to trace, SST = £ (Y-yM - Y,yfc.)2 = Y ' P r Y , 60 where F r ^ h ^ ( h 9 \ [ \ - 1]) It is easy but tedious to verify that the sums of squares (in summation notation) for subject, month, beat and trace add up to the corrected SSTOTAL- A check on our quadratic forms is provided by the fact that Ps + PM + PB + PT + PTOT = ha, or in other words t h a t Y ' Y = £ Y ' P . Y . ie{S,M,B,T,TOT} These five P-'s (known as kernels of the quadratic forms) are symmetric, idempotent (P? = Pi), and pairwise orthogonal (P.-P* = Oi44> if * 7^ k). Thus in Albert's terminology they form a complete set of orthogonal projections. Thus far we have merely written the original model and sums of squares in terms of matrix algebra; a clue to the independence of the sums of squares Y ' P , Y is given by the mutual orthogonality of the kernels. But there are further conditions which must obtain in order that the SS be multiples of chi-squares. Theorem 1 of Albert gives necessary and sufficient conditions for this in the case of general W = Cov(Y). Essentially, each column of each kernel must be an eigenvector of W, with a single eigenvalue per kernel. In a Corollary to Theorem 1, Albert gives the conditions when W has the structure of our model: W = o\lm + £ a2HrHT'. re{SM,B} Then the Y'P,Y's are multiples of independent chi-squares iff there are scalars Afcr such that HTHr'Pk=XkTPk, Vke{S,M,B,T,TOT}; VrG{5 ,M,P} . Again, a set of eigenvector relationships. It turns out that this condition is indeed 61 satisfied by our Hr and Pjt matrices. The Ajtr's are as follows: r = 5 M B / k = S 8 4 2 M 0 4 2 B 0 0 2 T 0 0 0 TOT V 8 4 2 , Then with 7* = a\ + Ylre{s,M,B} ^ t r 0 ? . e a c h Y'PkY/lk ia chi-square with degrees of freedom rank(Pk), and noncentrality parameter / i 2 l i 4 4 ' P f c l i 4 4 / 7 f c . Evaluating these, we get: SSS *°h + 2*1 + o2 X?r'(0) j ) T u r SSD 1aB + or 2 SSr X ? . ' ( 0 ) xie'(O) X?2'(0) Y ' P r o r Y tog + 4»j, + 2a2 + ex2 ~ ^ '("^ /^ S + 4*2, + 2 * | + 4 ) , and these are mutually independent. (Note that the numerator of the last quantity is y 2 „/144 , not SSTOTAL-) Estimation. With these results in hand, we return to our objectives. We want to know the relative magnitudes of variation due to the four effects subject, month, beat and trace. In other words, for a given variable, how much does each effect contribute to the total variability of an observation {PTOTAL =  as + AM + A& + .°T)? A. meaningful interpretation of this 62 contribution is the ratio of an estimate of the component variance parameter to an estimate of a'TOTAL • Estimators of the four components are derived as follows. The mean squares MSK - Y'PkY/rank{Pk) have expectations 7* for k £ {S,M, B, T} which are known linear combinations of the a2's. Thus we find linear combinations of the mean squares whose expectations are the individual <r 2 's. This leads to the unbiased estimators: &l = ± (MSS - M S M ) &h=\ { M S M - M S B ) &B = \ (MSB - M S T ) b\ = MST-By the way, the expectations of the four mean squares agree with the results of a 'quickie' EMS table. But we could not appeal to Cochran's Theorem to establish that they are distributed as multiples of chi-squares, since not all observations are i.i.d.. This is why additional algebra (following Albert) was necessary. Finally, we estimate &TOTAL nnbiasedly by <r?> 4- <7j^ f ~t~ ~t~ Oy. Properties of the Estimators. We have employed what is formally referred to as the 'Analysis of Variance Method' of estimating the variance parameters c 2 . Besides being unbiased, our estimators possess some other desirable properties. By Theorem I of Graybill and Wortham [38], under the normality assumptions we made, if an ANOVA exists, then all our df's (including ^OTAL) have minimum variance in the class of all unbiased estimators. (By the existence of an ANOVA is meant that the sums of squares are multiples of independent chi-squares.) 63 Many of our conclusions rest on 'eyeball' comparisons of estimates alone, and do not involve formal tests. For this we need not assume normality. But we still want the estimates to be 'close' to the true parameter values. So if the assumption of normality does not hold (our normal scores plots do not 'prove' it), what can be said about the variance of the estimators? In Theorem II of [ 3 9 ] , Graybill states without proof that in the general balanced nested (hierarchal) classification, the analysis of variance method gives the best (minimum vari-ance) quadratic unbiased estimators (BQUE's) of the variance parameters. And an appli-cation of his Theorem IV assures us that our o\OTAL is also BQUE. These statements presume no distributional assumptions beyond that the terms in our univariate model have means zero and finite third and fourth moments. A more rigorous presentation is made in Graybill and Hultquist [34] for the model in its Eisenhart form. Their Theorem 7 guarantees that once we have demonstrated the existence of an analysis of variance under the assumption of normality, then under some additional conditions (satisfied by our HT matrices), our <72's are BQUE even when the assumption of normality is replaced by one about finite moments as above. In short, if the assumption of normality is violated, then only the validity of the tests is called into question; the estimates are at least still BQUE. This has some appeal; variances are inherently 'quadratic'. Standard Errors. We derive the variances of the of's, and provide estimates of these. From ~ X2ank{pk)^ k G {S, M, B, T}, it is easy to show that H Var[MSk] rank(Pk)' 64 where 7* = E[MSk\ = <rT'+ £ ^ f c r ^ 2 - Also, different MSjt's are independent. Following Searle [40], p. 415-417 with our own notation, if we define the vector M as {MSS,MSM,MSB,MST)', then D = Cov{M) is diag{2il/rank{Pk)). Next let £ = {^S^M^B'^T)'' •8 4 2 1" 0 4 2 1 0 0 2 1 ' .0 0 0 1. Then E[M] = L S . The estimators of the variance components are obtained through £ = L - 1 M . And Cov[£] = L~1D(L~1)'. (Since the same mean square appears in more than one variance component estimate, the latter matrix is not diagonal.) Now C7ou[]S] depends on unknown parameters a2 through the 7fc's in D. The latter are the expected mean squares, so to estimate Cou[S] we could just replace them by the observed mean squares mst- Unfortunately "These estimators have no known desirable properties. They are not even unbiased." ([40], p. 417.) We can make them at least unbiased by using 2 + rank(Pk) in place of rank(Pk) in D (p. 417). We will not be concerned with other optimality properties of the estimators. As Searle points out, "The quadratic nature of the elements of . . . [Cot;[S]] . . . makes estimation of them in any optimal manner not easy." Hence standard errors of the CT^'S are given by square roots of the diagonal elements of L - 1 D ( L - 1 ) ' , where for D we take diag{2[msk]2/[2 + rank[Pk)]). A standard error for crTOTAL follows easily from V"ar[or 0 r > 1 L] = VarflVS] = JVCoufS]!^. Since it is often desirable to look at the square roots of variances, as they are on the same scale as the observations, one might want standard errors for these 'standard deviations'. Unfortunately these are more difficult to obtain. 65 Confidence Intervals. With the exception of o\, the estimates of variance parameters have unknown distributions, because they are differences of multiples of chi-squares. Moreover, the multiples themselves are unknown, since they involve the of's. So exact confidence intervals for the of's cannot be obtained in a straightforward manner. Searle on p. 413 provides approximate intervals due to Graybill. We did not pursue these. We can get some idea of plausible values for the <72's from the point estimates together with their standard errors. Testing. We turn now to testing. To determine for each variable whether monthly serial fluctuation is significantly larger than that over the course of an echocardiographic session, we test Ho'. O~M = 0 versus Hi'.arj^ > 0. That this is the appropriate null hypothesis, as opposed to H0: = aB, is evident if one looks at the estimator aj^ earlier. From there one sees that should be interpreted as representing variation in the data attributable to the month effect over and above that explained by the beat-group-to-beat-group effect. From the distributional results above, a test of^ = 0 is provided by comparing 5 5 B / 3 6 to an Fi8,36 distribution. Similarly, one examines whether serial variation over the course of an echocardiographic session is significantly larger than that due to the tracer, by testing Ho.crB — 0 versus H\.aB > 0. This is carried out by comparing S S B / 3 6 SST/72 to an 7*36,72 distribution. 66 Since there is obviously some variability introduced by the tracer, it is not very mean-ingful to ask whether aT is identically zero. (The trace effect Tijkt is like a replication effect.) We chose to inquire instead how large the tracing variability is, relative to the other components. This is quantified by the ratio ^xl^TOTAL-Instead of testing formally whether the magnitude of between-subject variation f j | is nonzero, we looked at the ratio < 7 § / ^ r o r x L • 67 Resu l ts : Var iance Components . PROC NESTED of The Statistical Analysis System (5.45) Version 82 was used for all computations. The estimates of <r|, crj ,^ a\, o~\ and o~TOTAL for each variable are presented in Table 4.1. In parentheses beside each estimate <r2 is an estimate of its standard error (y/vTr[a?]). A Negative Estimate for a Variance Component. Unfortunately, <r§ is less than zero for SW%. This happened because MSM is greater than MSs here. Such things can be shown to have nonzero probabilities of happening (Searle [40], p. 415). Searle suggests several courses of action, "few of them satisfactory". We review some of these. First off, it makes some sense to replace the negative value for CT| with zero in the estimate of o~TOTAL. We have done this in Table 4.1, although "such a truncation procedure disturbs the properties of the estimates as otherwise obtained. For example, they are no longer unbiased." (p. 407) It is also reasonable to interpret the negative estimate as strong evidence that «T| = 0 for SW%; i.e., that serial and tracing variation in readings on different subjects over time swamp the variation attributable to between-subject differences. But one is then at a loss to explain why it is the case for only SW%. (Although discussions with the technical personnel revealed that it is difficult to identify the boundaries of the septal wall.) Another option is to regard the model as wrong. We don't think this is called for, though, for two reasons. None of the O t h e r sixty-three estimates are negative. And the model of nested effects agrees well with one's intuitive understanding of the mechanism behind the data. The normality assumptions are not at fault here, because the estimation does not rely 68 TABLE 4 .1 V a r i a n c e Components E s t i m a t e s and T h e i r S t a n d a r d E r r o r s V a r i a b l e E s t i m a t e s ( S t a n d a r d E r r o r s ) T o t a l Be tween-S u b j e c t M o n t h l y S e r i a l W i t h i n - S u b j e c t Between Bea t Groups T r a c i n g PERCENT CHANGES PW% SW% EF SF 568 (112) 355 ( 48) 13 .5 ( 3 . 2 ) 10 .5 ( 2 . 5 ) 221 (111) - 77 ( 44) 6 .1 ( 3 . 1 ) 5 .1 ( 2 . 5 ) 132 ( 68) 178 ( 83) 5 .3 ( 2 . 0 ) 3 .8 ( 1 . 4 ) 83 ( 36) 138 ( 36) 1.4 ( 0 . 4 ) 1.2 ( 0 . 3 ) 132 ( 22) 40 ( 6 . 5 ) 0 . 7 ( 0 . 1 ) 0 . 4 ( 0 . 1 ) VELOCITIES LVVs 1 . 18 ( • 27) 0 . 58 ( . 2 7 ) 0 . 28 ( . 14) 0 , .30 ( . 07 ) 0 .02 ( . 003) LVVd 3 .26 ( . 8 0 ) 2 . .03 ( . 80 ) 0 . 36 ( . 2 7 ) 0 . 84 ( . 2 0 ) 0 . .03 ( . 004) PWVs 0 .32 ( . 0 8 ) 0 . 19 ( . 0 8 ) 0 . 05 ( .03) 0 . 07 ( . 02 ) 0 . .01 ( . 002) pwvd 2 . .03 ( . 4 5 ) 0 . 86 ( . 4 5 ) 0 . 65 ( . 29 ) 0 . .50 ( . 12) 0 . 02 ( . 003) SWVs 0. .25 ( . 05 ) 0 . .03 ( . 0 4 ) 0 . 1 1 ( . 0 5 ) 0 . . 10 ( . 02 ) 0, .01 ( . 002) SWVd 0. .60 ( 11) 0 . 05 ( • 11) 0 . 31 ( . 1 4 ) 0 . 23 ( . 0 5 ) 0 . .01 ( . 002) NORMALIZED VELOCITIES LVNVs 0 . . 14 ( . 0 4 ) 0 . 09 ( . 0 4 ) 0 . ,03 ( . 01 ) 0 . 02 ( . 006) 0 . 0 0 2 ( .0003) LVNVd 0 . 48 ( . 13) 0 . ,34 ( . 13) 0 . ,04 ( . 03 ) 0 . 09 ( . 02 ) 0 . 0 0 7 ( .001 ) PWNVs 0. .71 ( . 15) 0 . ,35 ( . 15) 0 . .03 ( . 07 ) 0 , ,31 ( . 07 ) 0 . 0 2 ( . 004) PWNVd 2. .29 . 4 9 ) 0 , 89 ( . 4 9 ) 0, .70 ( . 34 ) 0 , .68 ( . 16 ) 0 . 0 2 ( . 004) SWNVs 0. .45 ( . 0 8 ) 0, .08 ( .08 ) 0, , 17 ( . 09 ) 0 . . 18 ( . 04 ) 0 . 0 2 ( . 003) SWNVd 1 . .08 ( . 2 3 ) 0 . ,41 ( . 2 3 ) 0, .32 ( . 16 ) 0 . 33 ( . 0 8 ) 0 . 0 2 ( . 003) 68a on them. The estimates are best quadratic unbiased under much more general assumptions. One could also blame the method of estimation, and do something like maximum likelihood with non-negativity constraints on the variance estimators. We did not pursue this. But it is worth noting that the nonzero probability of a negative variance estimate shows that this 'ANOVA-type' method of estimation is not optimal in all respects. The sample size of eighteen on which &g is based may be the source of the trouble. One question remains: do the estimates of the three other variance components still make sense for this variable? We do not know the answer, and hence view all 'findings' for SW% with suspicion. The reader will see that the negative estimate leads to consistently conspicuous results for SW% in the upcoming analyses. Precision of the Estimates. If we define a measure of the precision of the estimates of the variance components as the ratio of the estimate to its standard error, then the average precision over all sixteen variables ranges from 2 for variability due to subjects, on up to 6 for variability due to tracing. These numbers merely reflect the fact that a"g is estimated with the least degrees of freedom, and b\ with the most. For almost all variables, the precision of the estimate of &TOTAL 1 3 intermediate in range (average precision 4.5). Recall that we estimated o T O T A L as o% + off + oB + oT. So the variance of bTOTAL will involve covariances between component estimates, some of which are negative. For example: Oov[»l,ait] = Cov\l{MSs-MSM), ]{MSM - MSD)} O 4 = -~Var[MSM] < 0. 69 Hence the standard error of the estimate of O\OTAL 13 reduced. Although one would have a better idea of how close a point estimate of a variance component is to the true value if one knew its distribution, the standard errors give some information. Average precision between two and six means that the point estimates can be used to make conclusions about the relative magnitudes of sources of variability with reasonable confidence. A curious aside is that for each variable, the estimated standard errors of <r| and °TOTAL a r e a l m o s t identical, even though the variance estimates themselves are not. We checked this algebraically. It turns out that 64(17 + 2) 64(18 + 2) 16(36 + 2) 4(72 + 2)' where the 7's are the expected mean squares in terms'of the <r's. Also, Var[(r|] is the sum of only the first two terms above. That the other two are small in comparison follows from the fact that 7s > 7M > 7c > 7r always. This last relation can be seen from earlier algebra. Re la t i ve Magn i tudes of Sources of Var iab i l i t y . Table 4.2 gives the estimates of the component cr2's and of (^XOTAL f ° r e a c n variable, together with the ratios (xlOO) of the component estimates to o\OTAL, in parentheses. These percentages are the proportions of total variability attributable to each component. The variance estimates in Table 4.2 should not be used to make t'nier-variable compar-isons on the basis of any one component or on total variability, because different variables are measured on different scales. The proportions of a2rOTAL accounted for by each com-ponent, although free of scale effects, are not as meaningful for inter-variable comparisons as 'coefficients of variation', which we shall examine later. 70 TABLE 4.2 C o n t r i b u t i o n s t o T o t a l V a r i a n c e , by V a r i a b l e V a r i a b l e E s t i m a t e s o f V a r i a n c e Components T o t a l B e t w e e n - W i t h i n - S u b j e c t S u b j e c t M o n t h l y Between T r a c i n g S e r i a l Bea t Groups PERCENT CHANGES PW% 568 221 (39) 132 (23) 83 (15) 132 (23) sw% 355 - 77 ( 0) 178 (50) 138 (39) 40 (11) EF 12 I.5 6.1 (46) 5.3 = (39) 1 .4 ( 10) 0.7 ( 5) SF 10.5 5.1 (48) 3.8 (36) 1 .2 (11) 0.4 ( 4) VELOCITIES L W s 1 . 18 0 .58 (49) 0 .28 (24) 0 .30 (25) 0.02 ( 2) L W d 3. 2G 2 .03 (62) 0 . 36 (11) 0 .84 (26) 0.03 C O PWVs 0. 32 0 . 19 (58) 0 .05 (16) 0 .07 (22) 0.01 ( 4) PWVd 2. 03 0 .86 (42) 0 .65 (32) 0 .50 (25) 0.02 ( D SWVs 0. 25 • 0 .03 (14) 0 . 11 (42) 0 . 10 (38) 0.01 ( 6) SWVd 0. 60 0 .05 ( 8) 0 .31 (52) 0 .23 (38) 0.01 ( 2) NORMALIZED VELOCITIES LVNVs 0. 14 0 .09 (64) O .03 (18) 0 .02 (17) 0.002 ( 1) LVNVd 0. 48 0 .34 (72) 0 .04 ( 8) 0 .09 (18) 0.007 ( 2) PWNVs 0. 71 0 .35 (49) 0 .03 ( 5) 0 .31 (43) 0.02 ( 3) PWNVd 2. 29 0 .89 (39) 0 . 70 (31) 0 .68 (30) 0.02 ( D SWNVs 0. 45 0 .08 (18) 0 . 17 (38) 0 . 18 (40) 0.02 ( 4) SWNVd 1 . 08 0 .41 (38) 0 . 32 (30) 0 . 33. (30) 0.02 ( 2) 70a It is evident from Table 4.2 that, for all variables except SW%, SWVa, SWVd and SWNVa, the between-subject component constitutes the greatest proportion of total vari-ability. But with one or two exceptions, this proportion is not much more than 50%. In other words, there is considerable serial fluctuation in values for any given subject. Next to between-subject variability, the within-subject monthly serial variation is the largest component for ten of the sixteen variables, and usually represents a sizeable frac-tion of cr TOTAL- This fraction is somewhat higher for septal wall velocities and normal-ized velocities than for the other measures. Correspondingly, the variation attributable to between-subject differences is lower for the SW variables. The reader is cautioned against the following subtle trap: the fact that PM/<7TOTAL and &B/&TOTAL a r e 30% for SWNVd does not mean that variation between months is no greater than that between beat groups during one session. What is true is that variation between beat groups net of the tracing variability equals variation between months net of the beat and tracing variation. There is no special significance attached to this. By the same token, the relative magnitudes of these ratios for LVNVd and PWNVs reported in Table 4.2 do not pose a conceptual problem. Variability between beat groups on the same echocardiogram is also sizeable for all variables. In particular, it is usually much larger than variability due to tracing. Finally, in comparison with the first three components, variability due to tracing is negligible for all but PW% and SW%. Optimal Measurement. Since tracing variability is (with two exceptions) very small compared to the short-term serial fluctuation, spurious 'noise' can be reduced more by averaging readings from several 71 beat frames each traced once than by taking the average of several tracings of one frame. Our results in this respect are consistent with Pollick [31]. And since short-term serial fluctuation is itself non-negligible, one expects the noise reduction gained by such averaging to be substantial. There appear to be problems in obtaining readings for SW% and PW%. The tracing variability is large. This is consistent with the nurse's difficulty in identifying the right endocardial surface of the septal wall and the epicardial surface of the posterior wall on the echoc ardiogr am. Tests. A glance at the estimates of a]^ and erg and their standard errors for each variable in Table 4.1 suggests that tests of HQ-.O^ = 0 and HQ:<TB = 0 may not be necessary. However, we perform the F-tests for the sake of completeness. The p-values are presented in Table 4.3. Short-term serial variation is without question significantly greater than tracing variability for all measures. Serial fluctuation over one month's time is significantly greater than the short-term variation for all variables except LVVd and PWNVs. For LVVd, the p-value is 0.0584; given the many assumptions of the model, we do not take this as firm indication of increased variation over a month. For PWNVs, the p-value is 0.3062; this would usually justify the conclusion that monthly and session variability are the same for this variable. This is not easy to interpret, given the results for the other correlated measures. If it had turned out that the p-values for all velocities were small, but not those for the percent-change measures, then we could interpret these in a credible way. Besides the general interest one has in characterizing the serial fluctuation of cardiac 72 TABLE 4.3 P-Values f o r F-Tests of Va r i a n c e due to Month and Beat Group Var i a b l e Month Beat PERCENT CHANGES PW% SW% EF SF 0.0045 0.0012 0.0000 0.0000 0.0017 0.0000 0.0000 0.0000 VELOCITIES LWs LWd PWVs PWVd SWVs SWVd 0.0041 0.0584 0.0149 0.0006 0.0020 0.0004 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 NORMALIZED VELOCITIES LVNVs LVNVd PWNVs PWNVd SWNVs SWNVd 0.0016 0.0493 0.3062 0.0023 0.0043 0.0031 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 72a measures, one should also keep in mind to what practical use these results would be put. If monthly variation were equal to that over a session for all measures, then bounds for normal monthly changes could be estimated from data spanning a shorter period. But this is not the case for virtually all variables, so one may as well wait a month for PWNVa too. Ldgistically, it is impractical to collect data one some echocardiographic variables now and on others later. Perhaps multivariate tests of monthly variation, by variable group, would be more meaningful. Setting this aside, we face the ubiquitous problem of simultaneous inference with these sixteen (or thirty-two) tests. 'Remedies' such as the Bonferroni correction are not relevant here, since given the above comments we are not interested in 'rejecting the null' for some variables but not for others. Indeed, it could be argued that a dichotomous acceptance / rejection setting is not appropriate here. We prefer to regard the p-values as statistics which summarize the data. Collectively they indicate that monthly variation is (significantly) greater than that over a session. In te r -Va r i ab le Compar i sons . Here we compare different variables on the basis of individual components. The scale effects were removed by calculating coefficients of variation, by component and for total variability. For each variable, these are 100 times the square roots of the component and total variance estimates, each then divided by the mean reading on that variable over all subjects. They can be interpreted as 'noise to signal ratios', by component. See Table 4.4. It is apparent that EF and SF exhibit significantly less variability than any other measure, on the basis of each of the components as well as on total variability. PW% is 73 TABLE 4.4 V a r i a b l e C o e f f i c i e n t s of V a r i a t i o n (X 100) By Component and f o r T o t a l V a r i a b i l i t y (St. Dev. / Mean over a l l Subjects) X 100 T o t a l PERCENT CHANGES PW% SW% EF SF 21 . 1 9 32.54 5.66 8.73 Between-Subject 1 3.22 0.00 3.80 6.08 Within-Subject Monthly Between T r a c i n g S e r i a l Beat Groups 10.21 23.04 3.55 5.25 8.10 20.29 .1.82 2.95 10.21 1 0.92 1 .29 1 .70 VELOCITIES L W s 13.28 LWd 15.61 PWVs 19.05 PWVd 21.11 SWVs 21.46 SWVd 25.91 9, 12, 14, 13, 7, 7, 31 31 68 74 43 48 .6, 5, 7, 1 1 , 1 4, 18, 47 19 53 94 23 62 6 7 8 1 0 13 16 70 92 91 48 57 04 1 .73 50 37 10 29 34 NORMALI ZED VELOCITIES LVNVs 16.41 LVNVd 19.63 PWNVs 21.50 PWNVd 20.59 SWNVs 24.94 SWNVd 31.11 13, 16, 15, 12, 10, 16 52 09 84 51 19.17 7.60 5.67 4.42 1 1.38 15.33 16.94 6.20 8.50 14.20 1 1 .22 15.77 1 7.20 1 .96 2.37 3.61 1 .92 5.26 4.23 73a much 'noisier' than EF or SF, on any component. Most likely this arises from the difficulty in identifying the epicardial surface of the posterior wall. Pollick's experiences were similar. SW% displays considerably more serial and tracing variability than the other measures. The truncated negative estimate of <r| again gives rise to difficulties of interpretation. We predicted that the percent-change measures would have lower tracing variability than the velocities. We also argued that, since percent-change measures and velocities are conceptually different, they may differ with respect to serial variability. These predictions are borne out by SF and EF, but not by PW% and SW%. For ordinary and normalized velocities, the septal wall measures seem noisier than the posterior wall measures, which are in turn noisier than the left ventricle variables. This is true for all components except the between-subject one. Almost every normalized velocity shows slightly more variation between subjects and between beat groups (though not between months) than the corresponding ordinary ve-locity does. This also holds for total variability. These findings for the between-subject component are possibly due to the fact that normalization involves division of the velocity by a dimension (i.e, length measurement.) With subjects varying in age from three to sixteen years, there would be quite a range of heart sizes as well. Except in cases where the coefficients of variation on the basis of some component differ greatly between variables, all the above findings should be interpreted conservatively. For these are ratios of point estimates only; whether apparent differences are really signif-icant is another matter. However, the patterns consistently repeated across variables and components give some indication that the differences are 'real'. 74 Homogene i ty of W i t h i n -Sub j e c t M o n t h l y Ser ia l Var iab i l i t y . In the preceding analyses it turned out that for any echocardiographic parameter the between-subject variability constituted a large proportion of total variability. If we think of each child varying randomly with time around his own mean level for each variable, then the above implies that the mean levels differ considerably across subjects. But what about the serial variability exhibited by each subject around his mean? Is it the same across subjects? We implicitly made thi3 assumption in the hierarchal model, since there was only one aM parameter for each variable. We can make only a crude assessment of the degree of homogeneity of within-subject monthly variability, for we have data at only two months. The replicates at each month do not help us. Consider a single variable. Taking the average of the four observations at each month j for the ith subject, we denote this by V\/. . . A measure Vi of the ith subject's 2 _ serial variability is then Tjzriy Yl {Yij.. ~ Y,\..)2. Table 4.5 gives the V,'s for one variable from each of the three groups. It is apparent that for each variable the numbers are vastly different across subjects, even if we regard extremes (Subject 14) as outliers. The apparent inhomogeneity is due in part to the fact that each estimate of variability is based on two datapoints. But we can do no better with these data. It was presumed in our hierarchal model earlier that all subjects sampled M ;(,) ran-domly from N(0ytj1t). In light of the above, perhaps a more realistic assumption would have been that each subject first samples a variance parameter aM\i\ once from some pop-ulation with distribution function 7M, then samples My(,j monthly from N(0,aM[i]). Let £ M be EjKI(aM\i)), the mean within-subject monthly variability. We could generalize the beat-group variability in a parallel way: £*(,-y) ~ NID(0,aB) 75 TABLE 4.5 Estimates of Monthly S e r i a l F l u c t u a t i o n , by Subject Subject Estimate SF L W s LVNVd 1 3.1250 0.43478 0.063012 2 2.9767 0.02820 0.000903 3 0.6760 0.48020 0.147153 4 1 .0719 0.12878 0.000450 5 10.6195 2.22078 0.088200 6 3.2017 0.00945 0.024753 7 1.1864 0.01015 0.253828 8 10.2826 0.11520 0.079003 9 0.6198 0.11520 0.056953 10 10.4206 0.04425 0.037128 1 1 1.0544 0.41633 0.012800 12 9.7939 0.37628 " 0.143113 1 3 0.4306 0.10580 0.042778 14 23.6523 2.10638 0.427812 1 5 1.5537 0.00001 0.037813 16 0.2360 0.69325 0.006328 1 7 0.0427 0.13005 0.009453 18 0.1226 0.50753 0.100128 75a becomes #fc(ty) ~ NID(0,<rB[i]), with irB[t\ ~ 7B- Let £ B be the mean within-subject beat-group variability. Let us now establish some distributional results, which we shall make use of later. Conditional on a single subject's ^ [ t ] , the My(,j are i.i.d. iVfO.ffj^i]). But what about unconditionally? Fortunately, we do not have to know anything about ?M (beyond that £ivf = / td7M(t) can be defined) to get the unconditional mean and variance of Afy(,-j. From E{Mj(i} \ o\f[i)} — 0, Vtrj^Ji], we get that the unconditional mean is zero. The unconditional variance is Var{MM) = EalMVar{MM \ < T & [ * ] } = ^[.1(^1*1) = (it-Here we have made implicit use of the relation Var(S) = E[Var{S | T)] + Var[E{S | T)]. Thus E\Mj(i}\ = 0, and Var[M}-^] = where these expectations are taken over all subjects t and all months j. A similar argument would show that E[Bk^] = 0 and Var[Bk(ii)] = £|. Hence E\Yiikt) = p, and Var[Yijkl} = a § + & + t% + aT-If subjects are indeed so inhomogeneous with respect to serial variability, what are the implications for the preceding analyses? In particular, what did we estimate with 'o^'? The estimator was ^ ~(MSM ~ MS&), which can be rewritten as ,=i [y=i y=ifc=i Denote the terms in braces by A{. Conditioning on the t t f c subject, E[A{] — o-]^[i]. (We get this by expanding A{ and using the fact that E[T2] = [E[T])2 + Var[T].) Thus Ai is an estimator of the ith subject's variance parameter o^ t ] - So | ( M 5 M — MSB) is really ^ ]C;=i ^MM) i-e-> 3 1 1 unbiased estimator of . So the estimate of 'the' 76 serial monthly component of variance should be interpreted as an estimate of the mean of all subjects' monthly variance parameters. Hence all statements even implicitly involving month-to-month fluctuation refer to a hypothetical average child, and not to any individual subject. Similar qualifications apply to aB. Here | ( M S B —MST) estimates £ B . Thus the point estimates of variance components, the statements about relative mag-nitudes of components of variability, and the comparisons among variables with respect to individual components are all still valid, provided we acknowledge that these pertain to a hypothetical average child. The degree to which actual subjects differ from this population mean is simply greater than we had assumed. The assumption of i.i.d. normality of the random effects Afy(,j and must be drawn into question. However, our normal scores plots (Figures 4.1 - 4.3) and table of sample skewness and kurtosis indicated that normality was not a bad assumption for the (transformed) Yiyju's. Possibly there is an application of a version of the Central Limit Theorem here. Suppose normality does not hold. Specifically, assume again that the A4y(,-) are i.i.d. N(0,aM[i]) conditional on <7"jvfK]> and that the latter come randomly from a population with distribution 7M- We can combine these two statements, so that My(,j has uncon-ditional distribution function M . We already know that M has mean zero and variance According to Graybill and Hultquist [34], Theorem 7, our estimator of £ M is best quadratic unbiased if At has finite third and fourth moments and the My(,) are i.i.d.. This last assumption is not strictly true, though, since Af^,) and M2(i) are now not independent, although there are eighteen independent pairs (Afi(,-),M2(,-)). In any case, our estimator of 77 £ M is unbiased, and is intuitively sensible. Similar comments apply to (B . Are the F-tests still meaningful? First, we should rewrite our null hypotheses as Ho' (M = 0 and Ho'- £B= U - TO what extent we really tested these is a difficult question. Some light is shed on it by the following. We established above that E [ \ { M S M - M S B ) ) = e M 4 E[^(MSB-MST)} = eD-Together with E[MST] = &T> rearrangement yields ElMSM] = 4tlt + 2tl+ol E\MSB) = 2ZB+°T-The implication is that the F-ratio MSM/MSB will be large if £M > 0, and MSB/MST will be large if £B > 0. The F-tests still have some validity. Putting this together with the fact that (i) normality seemed reasonable when assessed, and (ii) the p-values for the F-tests were in general very small, we retain our conclusions that variation between months is much greater than that between beat groups, which is itself much greater than tracing variability. 78 Magn i t ude of L o n g - T e r m Ser ia l F l u c tua t i on . For a given subject, one thinks of a hypothetical mean level for each variable around which his readings vary with time. Over very short intervals, the values will be correlated. But readings set one month apart are probably independent. Working under this assump-tion, we wish to quantify the serial within-subject variability of each echocardiographic measure over the very long term, for a typical healthy child. Suppose that each recorded value results from one trace of a group of three successive beats. Recalling our model for observations on a given variable Yijkl = f l + Si+ My(.) + + Tl(i]k), we see that at each month j we sample new M, B and T terms. Since we are following a particular subject t, we sample 5,- only once, 'observing' Successive monthly observations are conditionally independent and normally distributed, with mean fi + and variance ^MW + ^ B M + OT-Let us interpret this. The variance above is taken to mean the variance of the population of all monthly observations on this subject, from now to infinity. We are implicitly assuming here that the variance of the underlying continuous-time process has stabilized by one month; in other words, if we took observations at yearly intervals, they would have the same variance as the monthly ones. Of course, this last assumption cannot be tested with our data. At the time of writing, almost a year after the initial data were gathered, new echocardiograms are being obtained from the same eighteen subjects. Analysis of these will answer the question of whether monthly and yearly serial fluctuation are the same. Even if our assumption is correct, we are still stretching things a bit by using data 79 from only two months to predict the variability over an infinite number of months. We realize this limitation. The magnitude of long-term serial fluctuation we are predicting is for informative purposes only; it will not be used in any clinical situation. (Moreover, we are really using 18 x 2 = 36 months to estimate the serial variability of a hypothetical average subject's readings.) So with this qualification recognized, we predict that for a given echocardiographic variable, 100(1 — a)% of all future monthly observations on an average healthy child will fall within fl + Si ± *f ^<7J[f[t] + ff|[i]+<4-Now suppose each observation represents the average of two groups (on the same echocardiogram) of three successive beats each, traced once. Then the above interval becomes li +ti ± z% yJolf[i] + X-o\ [i] + Now Si, oM[i\ and aB\i\ are not directly observable for any subject. But since we stipulated that 5,- ~ N(0, <ri|), it follows that is zero for a hypothetical typical child who represents the population average. And for this child, o-M[i] = £M and o~B\i\ = £B, the population means. We estimated the intervals for each variable by replacing ft with the sample average, and each variance parameter with its estimate. For a = 0.05, the ninety-five percent prediction intervals for each echocardiographic variable are presented in Table 4.6. Beside each interval is recorded 100 times the ratio of half-width to midpoint, facilitating comparisons among variables. It is clear that, whether one or two beat groups are analyzed each month, there is quite 80 TABLE 4.6 Magnitude of Long—Term S e r i a l F l u c t u a t i o n V a r i a b l e I n t e r v a l s C o n t a i n i n g 95% of Future Monthly Observations For a H y p o t h e t i c a l T y p i c a l C h i l d , And Percent D e v i a t i o n on E i t h e r Side of Midpoint One Beat Group Two Beat Groups PERCENT CHANGES PW% (75 .5, 148.5) 32 .6 (81 .7, 142.3) 27 . 1 sw% (21 .4, 95 .4) 63 .3 (26 .4, 90 .4) 54 .9 EF (59 .6, 70 .2) 8. 22 (60 .0, 69 .8) 7. 59 SF (32 .6, 41 .7) 12 .3 (32 • 9, 41 .3) 1 1 .3 VELOCITIES L W s (6. 66, 9. 70) 18 .6 (6. 88, 9. 48) 1 5 .9 LVVd (9. 43, 1 3 .8) 18 .7 (9. 85, 13 .4) 1 5 . 1 PWVs (2. 26, 3. 68) 23 .8 (2. 38, 3. 56) 19 .8 PWVd (4. 63, 8. 87) 31 .4 (4. 88, 8. 62) 27 .7 SWVs (1 . 41 , 3. 25) 39 .7 (1 . 53, 3. 13) 34 .2 SWVd (1 . 54, 4. 44) 48 .6 (1. 70, 4. 28) 43 .0 NORMALIZED VELOCITIES LVNVs (1 . 85, 2. 75) 19 .4 (1 . 90, 2. 70) 17 .7 LVNVd (2. 80, 4. 26) 20 .6 (2. 95, 4. 1 1 ) 16 .5 PWNVs (2. 74, 5. 10) 30 .0 (3. 05, 4. 79) 22 . 1 PWNVd (5. 03, 9. 67) 31 .6 (5. 34, 9. 36) 27 .3 SWNVS (1 . 50, 3. 88) 44 .3 (1. 67, 3. 71) 37 .9 SWNVd (1 . 74, 4. 94) 48 .0 (1. 96, 4. 72) 41 .3 80a a large range in long-term serial fluctuation among the variables. EF is the most stable over time (8.2% above and below the serial mean with one beat group per month, 7.6% with two), followed closely by SF (12.3% and 11.3%). Except for the anomalous SW% (where CT| = 0), SWVd and SWNVd are the most variable over time, with values approaching 50%. Even among either the ordinary or normalized velocities, there are substantial differ-ences in degree of serial fluctuation. But any given ordinary velocity is predicted to exhibit about the same long-term serial variation as its normalized counterpart. Lastly, analyzing an additional beat group each month reduces the serial variation over months, but only marginally. 81 N o r m a l Bounds . The practicing cardiologist's prime interest would most likely be in bounds for each variable, which could be used to identify patients at risk of serious cardiac impairment. Besides bounds for the readings themselves, bounds for monthly changes could also be useful. Bounds for Echocardiographic Readings. In addition to the dataset currently being discussed (which we shall refer to as the 'serial data'), we had access to existing data on thirty-three normal children ranging in age from 1.0 years to 21.0 years. Mean age was 9.0 years, median 8.5 years. The variables recorded are the same as in the serial dataset. But there is only one value per child for each variable, and so we refer to this dataset as the 'one-time data'. Each observation in this latter dataset represents an average of three tracings of a group of three consecutive beats. The technician operating the ultrasonoscope and the nurse performing the tracing were the same as for the serial data. We assessed normality with normal scores plots. For LVVs and SF, the plots are presented in Figures 4.4 and 4.5. They do not deviate systematically from a straight line. Since most of our objectives involved components of variance, particularly serial vari-ability, we made little use of the one-time data. However, here we have an opportunity to do so. Bounds for normal values will then be based on fifty-one instead of eighteen normal subjects. The only problem is that observations in the two datasets represent different numbers of tracings. We decided to construct normal bounds for readings resulting from three instead of one tracing, even though our findings reported above indicate that repeat tracings would not reduce noise substantially, and therefore are not recommended. The 82 Figure 4.4 - Normal Scores P l o t f o r LWs oo to pi co a) u o o o - 0 . 5 + A AA A A A - 2 . 0 + - 2 . 5 + 6 . 5 6 . 8 7 . 1 7 . 4 7 . 7 8 . 0 8 . 3 8 . G 8 . 9 9 . 2 9 . 5 9 . 8 1 0 . 1 1 0 . 4 LWS L E G E N D : A - 1 O B S . B - 2 O B S . E T C . 1 . 8 + Figure 4.5 - Normal Scores P l o t f o r SF 0 . 9 + CO to tr CO a> u o o O - 0 3 3 - 1 . 5 + 2 9 3 0 3 1 3 2 3 3 3 4 35 3 6 3 7 3 9 4 0 4 1 4 2 4 3 4 4 SF alternative would have been to ignore an entire dataset. Since tracing variability was a small fraction (< 5%) of total variability for all but PW% and SW%, one would not be grossly misled if one used our bounds even for readings based on one tracing. So the bounds we are about to construct for each variable will apply to an observation obtained by averaging three tracings of one group of three consecutive beats. These bounds will pertain to any healthy subject at any time. If we think of all such observations for a given variable as coming from some parent population P with mean p, and variance r 2 , then r 2 incorporates both between-subject and within-subject variation in healthy children. Consider the one-time data. The observations V,- on any one variable represent 'snap-shots' of the subjects at a particular instant. In terms of our conceptual framework for the underlying process, the variation in the numbers is explained by the fact that subjects have different serial mean levels, and also that each subject varies around that level over time. _ 33 We estimate fi by the sample average Y., and r 2 by the sample variance ^ £ (Yi — Y.)2. •=i Denote these estimators by £(33) and f 2 3 3 j , to distinguish them from the estimators we will obtain from the serial data. Now consider the serial data for a given variable, and the model for each observation: Yijki = fi + Si + Mj(i) + 0fc(t'y) + V^(tj'Jb) • Here we explicitly allow that each subject t has his own serial variance parameters crM[i] and crB[i\. (This is true of the one-time data as well; the sample variance there subsumes all types of variation.) Using earlier results, the population variance corresponding to r 2 above is 61=1 6 where each £ 2 is a mean of subjects' individual serial variance parameters, as denned earlier. 83 We estimate fi by the sample mean V... . . Denote this by ji(is)- And fa = ^(MSM — MSB) and £ B = |(AfSjj — MST); O% and a\ are estimated as before. Thus we set f 2 l g j equal to b% + fa + || + For each variable, the sample means /i(ig) and /i(33) and the 'standard deviations' ^ / f 2 1 8 j and \JTfss) a r e presented in Table 4.7. With one or two exceptions, the estimates from the two datasets are quite close. In particular, the standard deviations never differ by more than a factor of 1.5. To pool the estimates, we use weights which reflect the number of subjects in each dataset. For the variances, one usually uses (n^ — l ) / ( « i + r»2 — 2) as weights, to reflect the fact that one degree of freedom is lost because E V ^ M = nfc^fc.- It is not clear that this is appropriate for the serial data, where the variance r 2 is estimated in a different way. But we thought it reasonable, and so we pooled the estimates in the following way: 18. 33/ ^Pooled = 5^(18) + 5~y^(33) «2 17 2^ 32 ^2 rP0oUd = ^ r ( 1 8 ) + ^gr(33)-If future patients have 'healthy' hearts (i.e., similar to our normal subjects), then their readings on a given variable will come from the same parent population P with mean /* and variance r 2. The normal scores plots for the one-time data give us some assurance that P is characterized by a normal distribution. We decided earlier that the assumption of normality was not unreasonable for the serial data, notwithstanding the possibility that the serial variance parameters are different across subjects. So we assume P is normal with some confidence, and hence predict that ninety-five percent of all observations on healthy children lie in fi ± lSGy/r 2. For each variable, the interval is estimated by substituting fiPooUd for fi and Tpoolcd for r 2 . 84 Table 4.7 Means and 'Standard D e v i a t i o n s ' f o r the Two Normal Datasets PERCENT CHANGES VELOCITIES NORMALI ZED VELOCITIES Based on Three T r a c i n g s Normal Dataset I ( S e r i a l ) (18 s u b j e c t s ) Normal Dataset II (One—time) (33 s u b j e c t s ) V a r i a b l e Mean Std. Mean Std. PW% SW% EF SF 112.48 58.39 64.94 37. 1 3 21.91 18.15 3.61 3.20 80.79 65.42 63.42 36.55 22.31 20.76 5.36 4.-54 L W s LWd PWVs PWVd SWVs SWVd 8.18 1 1 2 6 2 3 57 ,97 75 ,35 ,05 1 .08 1 .80 0.56 1 .42 0.49 0.77 8.21 11.15 3.12 5.17 2.90 3.79 0 1 0 1 0 98 41 56 07 55 0.83 LVNVs LVNVd PWNVs PWNVd SWNVs SWNVd 28 53 92 35 71 38 0.37 0.69 0.84 1.51 0.66 1 .03 2.54 3.67 4.45 6.36 3.99 4.83 0.38 0.60 0.82 1 .39 0.99 1.12 * Between-subject component estimate negative, so zero used i n s t e a d . 84a When using these bounds for monitoring, the clinician would compare a patient's read-ing on a given variable to the appropriate interval. If it lies outside, he can conclude that there is evidence the patient's heart is different from those in a normal group. Since both high and low readings are considered abnormal here, these are two-sided bounds. They are presented in Table 4.8. But it is known that for all sixteen measures, a low reading is associated with cardiac deterioration. So the clinician may be interested in only a 'one-sided test'. In this case he would want to know the number below which 5% (not 2.5%) of normal subjects' readings for a variable are likely to fall. Conversely, if he suspects that the patient's heart may be supranormal, he would want a 5% 'test' for this, looking only at high values. The estimated intervals are then /2± 1.645v/r2. See Table 4.8. The reader's attention is drawn to the distinction between the one-sided and two-sided intervals. One uses the latter when looking for any abnormality, whereas the former is appropriate for abnormalities in a prespecified direction. For positively correlated echocardiographic variables, if a patient is (ab)normal accord-ing to one of these then he is likely to be so on others too. In practice, clinicians would work with only a few measures. Our suggestions regarding selection of variables are forthcoming in the next chapter. Furthermore, the serial correlation in a given subject's readings means that successive 'tests' on him are not independent. The intervals are still valid for each trial, though. Bounds for Monthly Differences in Echocardiographic Readings. Examination of Table 4.8 reveals that the intervals for readings are quite wide. This has nothing to do with our sample sizes, since we treated the means and variances as known, 85 TABLE 4.8 Range of Values Considered Normal f o r Each V a r i a b l e Based on Average of Three T r a c i n g s of Three Consecutive Beats VELOCITIES NORMALI ZED VELOCITIES One-Sided Two-Sided PERCENT CHANGES PW% SW% EF SF (55.5, (30.2, (56.0, (30.0, 129) 95.7) 71.9) 43.5) (48.5, 135) (24.0, 102) (54.5, 73.4) (28.7, 44.8) LWs LWd PWVs PWVd SWVs SWVd (6.53, (8.74, (2.15, 9.87) 13.9) 3.99) (3.75, 7.71) (1.83, 3.58) (2.20, 4.86) (6.21, 10.2) (8.25, 14.4) (1.97, 4.16) (3.37, 8.09) (1.67, 3.74) ( 1 .94, 5.12) LVNVs LVNVd PWNVs PWNVd SWNVs SWNVd (1.83, 3.07) (2.58, 4.66) (2.90, 5.62) (4.35, 9.07) (2.08, 5.00) (2.53, 6.11) (1.71, 3.19) (2.38, 4.86) (2.64, 5.88) (3.90, 9.52) (1.79, 5.28) (2.18, 6.45) 85a and used percentage points from the normal distribution. The width of the bounds may be attributed to the inhomogeneity of healthy subjects with respect to echocardiographic parameters. The bounds in Table 4.8 are appropriate for the following setting. One chooses a healthy subject at random, and takes one reading on him. Ninety-five percent of the time, the reading will fall within the corresponding bound. But suppose one is taking repeat readings on only one healthy subject. Here it is not true that 95% of all readings on this subject would fall within the bounds. One reason is that implicit in the bounds is the between-subject component of variability t r | . This inflates the width. One can get around this deficiency by looking at the serial change in a subject's read-ings. We define this to be the difference in readings taken at two successive months. We estimated bounds for these using only the serial data, obviously. Since we found that noth-ing would be gained by repeat tracing, these bounds will be for observations resulting from one trace only. But because there is significant variation between beat groups during any one session, clinicians may want to average over two such groups to obtain a more mean-ingful reading for that day. So for each variable we present one- and two-sided prediction intervals for differences in readings between two consecutive months, where each reading results from one trace of one (respectively two) groups of three successive beats. Consider any single echocardiographic variable. Using the model for a given variable in the serial dataset, we have for the monthly difference in one-beat-group readings on a typical subject t : Yi2u - Yini = [M2(i) - M 1 ( l ) ] + \ B m ) - B1{{1)] + [T1(i2l) - T 1 ( , n ) ] . (*) We imagine first having sampled aM[i] from 7M and aB[i\ from 7B- Then at each month we 86 sample the M, B and T terms independently from iV^cTj^ft]), iV(0,cr ,^[t]) and N(0,oT) respectively. Conditional on this subject, the mean of the difference above is zero, and its variance is 2a^[i] + 2<7B[t] + 2oT. In general these variances are unknown. Later we consider estimating them, but for now we take another approach. Using Var(U) = E[Var(U \ V)] -f Var[E(U \ V)}, we get 2^M + 2^B + 2oT for the unconditional variance of (*), and zero for the unconditional mean. Here we have integrated over all subjects. Under the assumption of normality for each of the three differences in (*), we expect ninety-five percent of healthy children's monthly changes in readings on this variable to lie in 0 ± 1.96\/2f£f + 2 £ B + 2a\. The endpoints of this interval are two-sided bounds. If one is looking specifically for cardiac deterioration, one determines whether the change in readings (month two minus month one) lies below — 1.645\/2fj^ -f 2 £ B + la2*, for 95% of all normal children's monthly changes will lie above this number. Alternately, one can use the positive counterpart of the above quantity to examine whether positive changes in echocardiographic values indicate significant improvement, or merely random fluctuation. Thus the above number is the one-sided bound. If at each month one averages values from two groups of three beats each, where each 2 2 is traced once, then the monthly difference is | E Yf'2fci ~ § X) Yuki, o r k=i k=i 2 2 2 2 [M2(i) - M 1 ( t ) ] + - \ E **(.•!)] + i\ E r K « * ) -\fl TM.ifc)]- (**) k-1 k=l k=l k-1 Conditioning on this subject, the mean of (**) is zero, and the variance is 2aj^[i] + ^I^BK]) + 2(|o,2f.)( or just 2t7j^ [t] -f trB[t] + a\. (Note that this variance calculation is straightforward because all terms above are independent.) Unconditionally over all subjects, (**) has mean zero and variance 2£M -f £ B + aT. 87 The one- and two-sided bounds are 1.645^2^ + + a\ and 1.96^2^ + (B + <rT re-spectively. All bounds were estimated using £ M , £ B and a\, as denned earlier. See Table 4.9. Let us state explicitly how these are to be interpreted by the clinician. We consider the case where two beat groups are averaged each month. Suppose SF is the variable of interest. The clinician records a reading at each of two successive months, and takes the difference (month two minus month one). Call this number 6. If he wishes to know simply whether this change is typical of normal children, he looks at the bound 5.95. If the observed change S lies between —5.95 and +5.95, there is little cause for concern, based on this information alone. (Clearly a value close to either limit should not be ignored.) Suppose the patient is on Adriamycin therapy, and one wishes to examine whether his heart has deteriorated significantly since last month; i.e., whether SF has dropped sub-stantially. If 6 is positive, then this information gives no indication of cardiac deterioration, and bounds are not needed. Now suppose S is negative. If it is less than —5.00 (i.e., negative and greater in magni-tude than 5.00), the clinician may interpret this as an abnormally large change. Only 5% of normal children would lie in this range. We do not intend these bounds to be taken as absolute cutoff points, such that if an observed difference in readings is within the bound, one concludes there is no cause for concern. They should be used as guidelines only; values close to the bound but not beyond it, are still an indication that something may be amiss. Indeed, why choose 95%? A more informative number may be an estimate of the proportion of healthy children whose monthly differences lie beyond the observed 6. (This 88 Table 4.9 Bounds f o r Normal Monthly D i f f e r e n c e s In E c h o c a r d i o g r a p h i c Readings V a r i a b l e One Beat Group Two Beat Groups One-Sided Two-Sided One-Sided Two-Sided PERCENT CHANGES PW% 43.43 51.74 36.08 42.99 SW% 43.83 52.22 37.96 45.23 EF 6.32 ' 7.53 5.86 6.98 SF 5.43 6.46 5.00 5.95 VELOCITIES L W s 1.81 2.16 1.55 1.85 LWd 2.58 3.07 2.07 2.47 PWVs 0.86 1.02 0.71 0.85 PWVd 2.51 2.99 2.21 2.64 SWVs 1.09 1.30 0.94 1.12 SWVd 1.72 2.05 1.52 1.81 NORMALI ZED VELOCITIES LVNVs 0.52 0.62 0.45 0.54 LVNVd 0.85 1.01 0.69 0.82 PWNVs 1.41 1.67 1.04 1.24 PWNVd 2.76 3.28 2.39 2.84 SWNVs 1.41 1.67 1.20 1.43 SWNVd 1.91 2.27 1.64 1.96 88a is like a p-value in the context of hypothesis testing.) We could theoretically construct tables and charts which aid the clinician in calculating this proportion, but have chosen not to do so. First of all, the numbers would rely heavily on the functional form of the underlying distribution of changes, which we have assumed to be normal for construction of the bounds. With our sample size of eighteen, we do not put excessive faith in our normal scores plots. Second, bur small sample provides only rough estimates of the variance parameters. But we will outline here the procedure for estimating the proportion. The reader is again cautioned that the estimate is very rough. Consider again 6, the observed difference in values for SF, where the values at each month represent the average over two beat groups each traced once. Consider A as the random quantity before the observation is made. If this patient is 'like' our normal subjects, then _ ^ #(0,1). Our estimate of \ / ^ A ' s V^fjvf +$} + <*r- For SF, we can get this number from Table 4.9 as (5.00/1.645) = (5.95/1.96) = 3.04. Since the patient did not contribute to this estimate, we treat the variance as known, and hence (A/3.04) ~ N(Q, 1). By dividing the observed change 6 by 3.04 and referring the result to standard JV'(0,1) tables, the clinician gets an idea of what proportion of normal subjects would lie beyond 6. This is information on the status of the patient. Extensions. By using the monthly difference in readings on a given subject as a diagnostic variable, we have removed the between-subject variability cr|, reducing the variance. Thus we expect this method to be more sensitive than one based on readings alone. However, further 89 improvements are theoretically possible. It is not true, for example, that 95% of monthly differences in a given healthy subject's readings would lie within our bounds, for we used estimates of population averages $M and f jj instead of the individual subject's parameters <rM[i] and crB[i]. For subjects with crM[i] > %M and trB[i] > $B, we would want wider bounds. The converse is true if the inequalit/M are reversed. The probability statements made earlier are still meaningful since o priori we have no knowledge of a subject's variance parameters, and so we treat them as random; this is algebraically equivalent to using the P's as naive estimates. (Another way to look at it is that the fraction of monthly differences lying outside the bounds will be lower than 5% for some healthy children and higher than 5% for others; on average it is 5%.) But what if we could get an idea of a patient's normal serial variance parameters? This would have to be done before Adriamycin therapy started. In general, therapy begins soon after cancer is diagnosed, so we do not have the luxury of several months' observations on this patient, from which we would get £jjf[t]. But it is plausible that the magnitude of a patient's normal random variation over a few days is related to his monthly variability. If one knew the functional form of this relationship, which could be estimated with serial data on normal subjects, then one could project an estimate of o~M[i] from o,|,aj([t], and use this to construct 'personal' bounds for monthly differences to use throughout therapy. Even though this projected estimate of the subject's normal monthly variation is rough, it is probably better than using the population mean $M. Even more generally, the above procedure could be modified to incorporate the em-pirical distribution 7M of normal subjects' ^[I'I'S in estimating a given patient's monthly variability. This is related to empirical Bayes methods. 90 M o d e l I I - A N C O V A s for Sex and Age Effects. We examined possible systematic variation of the echocardiographic measures with subject age and/or sex using an Analysis of Covariance for each variable. Ideally one might like to retain the hierarchal four-factor model, and augment it with 'age' and 'sex' factors. But this introduces unwarranted complications, since the factors S, M, B and T are random; observations are not i.i.d., which is an assumption of standard ANCOVA. So instead we averaged out the effects of the four random factors by working with the means over the eight observations on each subject. This was done for each variable, yielding eighteen datapoints for each ANCOVA. The ANCOVA model (with 'age' as continuous linear covariate and 'sex' as classificatory fixed effect) is standard, so we do not write it out formally here. SAPs PROC GLM was used to obtain the sums of squares and tests. In this context the ANCOVA is an instance of the General Linear Model (GLM) with i.i.d. normal errors. Each effect is tested individually by looking at the reduction in error sum of squares due to adding the effect to a reduced model. Parameter estimates are those which minimize the sum of squares for error. Central to ANCOVA is the assumption of homogeneity of slopes: that the effect (if any) of subject age on the response, is the same for both males and females. This assumption can be examined conveniently here by including an 'interaction' term in our model. A test for the importance of this term in explaining variability should look at the reduction in error sum of squares due to adding it to a model already containing the main effects 'sex' and 'age'. For each of the sixteen echocardiographic measures, the F-test for the interaction term 91 was not significant at test size 0.05. The lowest p-value was 0.149. We conclude that the age effect is the same for both sexes, and hence one can pool all eighteen observations to estimate the common slope for 'age'. The ANCOVA model was refitted for each variable, with no interaction term this time. Here a Secision was necessary regarding how the effects were to be tested. For the factors 'sex' and 'age' are not orthogonal; i.e., the reduction in error sum of squares due to adding one effect to a reduced model depends on whether the reduced model includes the other effect, or just the grand mean. SAS gives four types of sums of squares (SAS for Linear Models [41], pp. 103-105, 189-190). We will not go into any detail on this. We chose Type II sums of squares, which correspond to testing for the significance of each effect after the other has been corrected for. This is the whole point of using an ANCOVA instead of two one-way ANOVAs: the precision of the test for one factor is increased by first accounting for variability due to the other. Table 4.10 displays the p-values corresponding to the F-tests for 'sex' and for 'age'. It is seen that subject sex has no bearing on any echocardiographic variable. However there is evidence that some of the measures may vary systematically with subject age. In Table 4.10, all flagged variables show indication of an age effect (p < 0.05). Double-starred measures (p < 0.01) exhibit a stronger age influence. According to the signs of the estimated slopes, the age effect is positive for LVVs and negative for PW% and SWNVd. Our medical collaborators stated that it is generally felt that peak velocities decrease with age; this is consistent with the sign for SWNVd, but not for LVVs. We are inclined to acknowledge the existence of an age effect sooner for SWNVd, since its p-value is 0.007 and the slope coefficient has the expected sign. The 92 TABLE 4.10 P-Values for Sex and Age Effe c t s on Echocardiographic Readings Variable Sex Effe c t Age Effe c t PERCENT CHANGES PW% 0.48 0.002 ** SW% 0.30 0.44 EF 0.28 0. 11 SF 0.32 0.13 VELOCITIES LWs 0.12 0.04 LWd 0.92 0.37 PWVs 0.67 0.11 PWVd 0.87 0.07 SWVs 0.59 0.07 SWVd 0.09 0.43 NORMALIZED VELOCITIES LVNVs 0.58 0.09 LVNVd 0.94 0.08 PWNVs 0.97 0.05 PWNVd 0.62 0.10 SWNVs 0.83 0.06 SWNVd 0.35 0.007 ** 92a unexpected sign and the p-value of 0.04 for LVVa gives rise to some doubt here. Recall that normalization of the velocities supposedly corrects for heart size, and by extension for subject age in this paediatric population. This normalization does not seem to be necessary, given the p-values for age in Table 4.10. They are not in general lower for ordinary velocities than for normalized ones. With these many tests, the probability is much greater than 5% that at least one p-value is less than 0.05 by chance when in fact no age effect exists. Calculation of an exact individual test size to guarantee a family error rate of 0.05 is not possible since the variables are correlated. The Bonferroni correction would set the individual test size at 0.05/16 = 0.0031, in which case only PW% can be said to vary with subject age. Figures 4.6 and 4.7 are plots of subject mean SWNVd and PW% values against age. These variables have the lowest p-values for age effect. The plot symbols 'F' and ' M ' represent subject sex. A mild age effect is evident from the plots. For each of PW%, LVVa and SWNVd, a regression model with only linear and quadratic age terms (and no sex effect) was fitted. Whether the quadratic term explained significantly more variation in the data than that accounted for by the linear term, was examined with F-tests. The p-values are 0.61, 0.53 and 0.40 respectively. We conclude that any age effect is satisfactorily approximated by a linear relationship. Let us consider the application of these findings. When monitoring an Adriamycin patient's status, one should compare his readings to bounds based on normal subjects who are in all other respects 'like' the patient. If one uses a measure which is correlated with heart size, then tighter bounds are possible if one first segregates normal children according to their ages, and computes separate bounds for each such category. 93 Figure 4.6 r- Plot of SWNVd versus Age P L O T OF S W N V d A G A I N S T A G E . S Y M B O L I S S U B J E C T S E X . E A C H P O I N T I S A V E R A G E OF 8 O B S E R V A T I O N S ON THAT S U B J E C T . 5 . 0 + 4 . 0 + LO 3 . 0 + 2 . 0 + 1 . 0 + 0 . 0 + 9 1 0 11 A G E ( Y E A R S ) 12 13 14 15 16 17 Figure 4.7 - p l o t of PW% versus Age P L O T OF PWX A G A I N S T A G E . S Y M B O L I S S U B J E C T S E X . E A C H P O I N T I S A V E R A G E OF 8 O B S E R V A T I O N S ON T H A T S U B J E C T . PW% 1 7 0 + 1 6 0 1 5 0 1 4 0 1 3 0 1 2 0 F F F 10 LO 1 1 0 + I 1 0 0 + I 9 0 + 8 0 + I 7 0 + I 6 0 + I 5 0 + I 4 0 + 3 0 + I 2 0 1 0 I 2 3 4 5 6 7 8 9 1 0 11 12 1 3 14 15 16 17 A G E ( Y E A R S ) We interpret Table 4.10 to mean that there may be something to be gained through such segregation; although only three measures had age p-values less than 0.05, seven of the others are 0.10 or less. Our bounds reported earlier are based on all subjects because we had only eighteen to begin with. 94 Conc lus ions for Chap te r 4. Our data support the following conclusions regarding M-mode LV echocardiographic percent-change and velocity measures in healthy children. (1) Between-subject differences are larger than both within-subject serial fluctuation and tracing variability, for almost all variables we considered. But the two serial components (month and beat group), representing within-subject physiological variation and also vari-ability introduced by the ultrasonographs, comprise substantial fractions of total variabil-ity. There is some indication that subjects may be inhomogeneous with respect to degree of serial monthly fluctuation. But for the average child, variability between successive months is significantly greater than that between beat groups during one session. And the latter is significantly greater than variability due to tracing, which is itself a negligible fraction of total variability. (2) In taking a reading, a greater reduction in extraneous 'noise' is possible by averaging values from several heartbeats each traced once, rather than vice versa. (3) SF and EF exhibit less variation than any other measure, on the basis of any component as well as on total variability. The results for SW% were difficult to interpret. PW% shows as much variation as the (normalized) velocities on all components save tracing variability, where it is conspicuously noisy. There is no clear separation of percent-change measures and (normalized) velocities on the basis of either tracing or serial variability. Between-subject, between-beat-group and total variability is in general slightly less for an ordinary velocity than for the corresponding normalized velocity. On the basis of any component, including total, there is a wide range in variability among the (normalized) 95 velocities. With respect to serial, tracing and total variability, septal wall velocities show more variation than posterior wall velocities, which are in turn more variable than velocities of the left ventricle. (4) We predict considerable long-term random fluctuation in monthly echocardiographic readings for a typical healthy child. This is not much reduced by analyzing two instead of one group of three beats each month. EF and SF are the most stable over time. About ninety-five percent of an average child's monthly EF (SF) readings would fall within ± 8% (12%) of his long-term mean. For (normalized) velocities, this bound ranges from 15% to 49%, depending on the variable and on the number of beat groups analyzed; and it is about the same for any normalized velocity and its ordinary counterpart. (5) For each echocardiographic variable, estimated 95% bounds for readings on healthy children are presented in Table 4.8. Bounds for monthly changes in readings on an average healthy child are given in Table 4.9. (6) Subject sex has no bearing on echocardiographic readings in healthy children. There is evidence that some variables (especially PW%, LVVa and SWNVd) may vary sys-tematically with subject age. This effect is linear, and is the same for both females and males. 96 Chap te r 5. Var iab le Se lect ion. In t roduc t i on . In this chapter we analyze echocardiographic data on children who had undergone an-tineoplastic therapy with Anthracyclines (e.g., Adriamycin), with the intent of determining which variables are likely to be useful for monitoring of such patients. Adriamycin Dataset. These data consist of repeated echocardiographic measurements on twenty-five children (12 females, 13 males) with various malignancies, who were concurrently undergoing ther-apy with Anthracyclines (23 Adriamycin, 1 daunorubicin, 1 both). These children ranged in age from 1.9 to 15.6 years at start of therapy. Mean and median age are 10.5 years. Twelve patients had received previous chemotherapy. Three of these also received radio-therapy, but none in the mediastinal region. Primary neoplasms are as follows: 8 osteogenic sarcoma, 5 Ewing's sarcoma, 3 Wilms' tumor, 3 non-Hodgkins' lymphoma, 2 acute myel-ogenous leukemia, 1 acute lymphoblastic leukemia, 2 rhabdomyosarcoma, and 1 malignant histiocytosis. It had initially been planned by the clinicians that data would be collected on each child at five points: pre-therapy (immediatiely prior to start of therapy), mid-therapy (mean 6.7 months after pre-therapy), post-therapy (mean 2.3 months after cessation of therapy, and mean 6.1 months after mid-therapy), early followup (mean 7.9 months after cessation), and late followup (mean 16.5 months after cessation). For various reasons (e.g., patient unavailable for session), much data are missing. Only five children had five complete sets 97 of observations. The breakdown is as follows: 25 children with pre-therapy readings, 24 with mid-therapy, 24 with post-therapy, 9 with early followup, and 15 with late followup. Mean cumulative dose of Adriamycin at the mid-therapy echocardiogram was 250.8 mg/m 2. Mean cumulative dose at cessation was 409.6 mg/m 2. As well as the sixteen echocardiographic measures of the previous chapter, fourteen additional variables were recorded. But we restricted our attention to the six ordinary velocities, six normalized velocities, and four percent-change measures. The values recorded for the echocardiographic variables were arrived at in a manner similar to that already described, with the exception that one group of three beats was traced three times, and the overall average recorded. This work was done prior to our involvement. Data were collected between December 1976 and June 1983. Concep tua l Ana lys i s . In the absence of a 'gold standard' (such as endomyocardial biopsy grades) with which to correlate the echocardiographic measures, it is reasonable to favor those variables for monitoring which in some sense respond most to increasing cumulative doses of Adriamycin. This is the criterion of sensitivity. Medically speaking, there is a higher cost associated with failing to identify a patient at risk of cardiac incident, than with falsely concluding that a patient is at such risk. In statistical terms, it is more important to avoid Type II than Type I errors, where the 'null hypothesis' is 'patient not at risk'. Although we have data for early and late followup, this information is not relevant here because at this stage Adriamycin would have been discontinued. So we will restrict attention to pre-, mid- and post-therapy data. If one thinks of the left ventricle as a mechanical pump, then it seems likely that only 98 a few (<< 16) 'dimensions' are needed to describe its action. In this case the sixteen echocardiographic variables will be correlated, and this should be taken into account in the selection of a subset for monitoring. There are at least two possible approaches here. Loosely speaking, one could find the smallest subset of variables which preserves the information on Adriamycin patients conveyed by the entire set. On the other hand, one could search empirically for a few dimensions which capture the covariation in healthy subjects' parameters, and ensure that each dimension is represented in a subset of variables used in monitoring. We pursued both courses of action; our analyses are gathered under the heading of dimensionality. To the reader familiar with multivariate statistical methods, dimension-reduction tech-niques such as Factor Analysis or Principal Components may come to mind. While these methods are conceptually useful for summarizing the variation in the data, our clinical co-workers would like to be able to interpret the measures used for monitoring. One can understand what is physically meant by a drop in shortening fraction; this may not be possible with linear combinations of measures. So we chose to select some of the existing variables for monitoring, instead of inventing new ones. Re la t i ve Sens i t iv i ty of the Var iab les . Two factors could explain differing degrees of sensitivity of echocardiographic variables to Adriamycin. First, some aspects of LV function may be impaired more severely than others. This has its basis purely in physiology, and has nothing to do with measurement. Second, the imprecision inherent in a reading will affect the chances of detecting LV dys-function. This noise differs among variables, as we have found in the previous chapter (Table 4.4). 99 But a direct empirical examination of the relative sensitivities of the variables to Adri-amycin combines the two factors above, thereby facilitating an immediate comparison. We define a 'sensitive' variable to be one according to which more Adriamycin patients are classified as abnormal at either mid- or post- therapy. Because of the repeated nature of the monitoring process, there are two definitions of abnormality, both of which are reasonable: an abnormal reading for a variable, or an abnormally large change in its value over time. Both of these implicitly assume that an interval of normal values or normal changes has been previously defined. Also, we are interested primarily in deviations in the direction which is physiologically associated with functional deterioration of the heart. Consider first the 'abnormal reading' definition. Table 4.8 of the previous chapter gives bounds for echocardiographic readings (based on three traces), above which 95% of observations on all normal subjects are expected to fall. These were based on the combined observations of the serial and one-time normal datasets. Before comparing Adriamycin patients' mid- and post-therapy values to the bounds, we should be satisfied that the three groups are relatively homogeneous at pre-therapy. See Table 5.1, which gives the sample mean and standard deviation of each variable, for each dataset. (For the serial dataset, the 'standard deviation' for a variable is really yjffg + £M + £% + \<?T, in the notation of the previous chapter.) The values are in general close, except that both means and standard deviations for the Adriamycin patients are slightly higher. Figure 5.1 is a plot of Patient F's shortening fraction values during therapy. The dashed lines mark the one- and two-sided bounds for SF from Table 4.8. This is a simple and at first glance reasonable way to monitor for cardiac dysfunction. However, there are 100 Table 5.1 Means and Standard D e v i a t i o n s f o r the Three Datasets Adriamycin Data Normals I Normals II Pre - Therapy ( S e r i a l ) (One- Time) (25 s u b j e c t s ) (18 s u b j e c t s ) (33 s u b j e c t s ) V a r i a b l e Mean Std. Mean Std. * Mean Std. PERCENT CHANGES PW% 87.88 26.99 112.48 21.91 80.79 22.31 SW% 56.52 25.04 58.39 18.15 ** 65.42 20.76 EF 63.80 5.07 64.94 3.61 63.42 5.36 SF 36.92 4.25 37. 1 3 3.20 36.55 4.54 VELOCITIES L W s 9.46 1 .65 8.18 1 .08 8.21 0.98 LWd 1 2.43 2.70 1 1 .57 1 .80 11.15 1.41 PWVs 3.71 0.69 2.97 0.56 3.12 0.56 PWVd 6.85 1 .80 6.75 1 .42 5.17 1 .07 SWVs 3.14 0.94 2.35 0.49 2.90 0.55 SWVd 3.89 1 .27 3.05 0.77 3.79 0.83 NORMALI ZED VELOCITIES LVNVs 2.83 0.57 2.28 0.37 2.54 0.38 LVNVd 3.97 1 .01 3.53 0.69 3.67 0.60 PWNVs 4.71 1 .01 3.92 0.84 4.45 0.82 PWNVd 7.71 1 .97 7.35 1 .51 6.36 1 .39 SWNVs 3.99 1.15 2.71 0.66 3.99 0.99 SWNVd 4.63 1.19 3.38 1 .03 4.83 1.12 * Based on three t r a c i n g s . ** Between -subject component estimate n e g a t i v e , so zero used i n s t e a d • -1 00a Figure 5.1 - P l o t of Patient F's SF Values during Therapy I 5 5 + 5 0 + 2-sided 4 5 + . — : |_ 1-sided 4 0 + 3 5 + F 1-sided 3 0 + : I 2-sided 2 5 + 2 0 + 1 5 + + + P R E M I D P O S T Date i n Therapy drawbacks. Before the onset of therapy, Patient F was considerably above the pre-therapy Adriamycin group mean for SF of 36.9%, with a value of 42%. By mid-therapy he had fallen to 32%. This is quite an appreciable drop; possibly this patient is particularly sensitive to Adriamycin, even though at mid-therapy he still displays adequate cardiac performance. We would want to be able to identify him at this stage and watch his progress carefully, or even modify his Adriamycin dosage. As can be seen in Figure 5.1, Patient F's heart did in fact deteriorate by the post-therapy date to a subnormal level; shortening fraction was 26%. This reading was taken two months after cessation of therapy. But because Patient F's mid-therapy reading was within the normal range, he would not have been identified at that stage. Hence comparison of pre- and mid-therapy readings becomes important. This brings us to the second definition of abnormality: a large change in a variable over time. Monitoring according to serial changes in readings has some conceptually attractive properties. We are interested in repeated monitoring; serial changes implicitly use pre-vious information on the patient in question, and so make more efficient use of available data. Instead of assuming naively that a patient is initially an average member of a nor-mal population, this method incorporates baseline data on him. This becomes especially important if a patient displays supranormal cardiac performance at the onset of therapy, then deteriorates drastically but is still above the lower normal bound for single readings. However, it should be kept in mind that we do not have to choose one definition of abnormality and discard the other. Biancaniello [18] defined abnormality in this dual way. Our goal here is to outline an intuitively reasonable classification method(s) having a good chance of flagging patients in particular danger. Soon we will present some results 101 concerning the relative sensitivities of methods based on the two definitions. But both are meaningful, and our medical collaborators have stated that they see no conceptual problem in using them simultaneously. We look more closely at the method based on serial changes. Is the change in Patient F's shortening fraction from 42% to 32% 'significant'? More generally, which of the variables consistently exhibit (significantly) large changes over time in the Adriamycin dataset? Any sort of ANOVA-type analysis of these data is inappropriate, because the estimated error would not represent what really should be the reference standard: magnitude of random serial fluctuation in healthy subjects, over the same time period. Recall that in the previous chapter we derived bounds for normal monthly differences in readings. These were based on one tracing; using the same algebra as before, we con-structed them for readings based on three tracings, to correspond to the Adriamycin data. Unfortunately, the readings in the latter dataset are spaced at roughly six month inter-vals. For the normal bounds to be applicable, we must tentatively make the assumption that normal serial fluctuation in echocardiographic measures over one month and over six months are equal. We are currently collecting data to check this assumption. Since Adriamycin patients' values for all sixteen measures are theoretically expected to decrease as therapy progresses, one-sided 95% bounds are appropriate here. The es-timated bound for SF is —5.28; that is, ninety-five percent of monthly changes in SF for a hypothetical average healthy child will lie above —5.28. Furthermore, monthly serial variation will be less than this for some healthy children and greater for others; using the .conditional expectation results of Chapter 4, ninety-five percent of all healthy children's monthly differences in SF will lie above —5.28. 102 Thus in some sense Patient F's change of -10 is 'significant'. We can not state conclu-sively that this reflects the impact of Adriamycin, since it is theoretically possible that: (i) this patient exhibits exceptionally large random serial variation, and (ii) he is resistant to the cardiotoxic effects of Adriamycin. To rule out this alternative, we would have to have serial data on Patient F prior to therapy. The one-sided 95% bounds for negative monthly changes in each variable are reported in Table 5.2, together with the number of Adriamycin patients whose differences (mid-minus pre-therapy, post- minus mid-therapy, and post- minus pre-therapy) are negative and exceed each bound in magnitude. Also presented in Table 5.2 are the one-sided 95% lower bounds for readings. These are taken from Table 4.8. The counts are the number of patients whose values fall below the bound at pre-, mid- or post-therapy respectively. Patient E had no post-therapy echocardiogram; patient O had no mid-therapy echocar-diogram. These subjects are excluded from Table 5.2. Consider the results for the serial changes. If we interpret these counts as the number of patients whose changes are abnormal, then we would have to assume that normal serial variation is the same over one month and over six (or twelve) months. To the extent that this does not hold, the counts are inflated. And we also grant that there may be some patients who exceed the bounds merely because they display exceptionally large random serial variation, although it is unlikely that this is true of all twenty-three patients. Apart from these comments, there is another qualification. For any single variable, five percent (ideally) of healthy children would lie beyond the bound. But with sixteen correlated measures, the expected proportion of patients wrongly classified as abnormal by at least one measure is greater than five percent. 103 TABLE 5.2 Number of Adriamycin Patients Exceeding Bounds Bound for Readings Number of Patients Below Bound Bound for Changes Number of Patients Below Bound Pre Mid Post Pre Mid Pre to to to Mid Post Post PERCENT CHANGES PW% 55.5 1 3 5 -37.5 5 2 2 SW% • 30.2 2 2 3 -42.2 1 3 3 EF 56.0 1 4 7 -6.12 9 4 10 SF 30.0 1 4 7 -5.28 9 3 10 VELOCITIES LWs 6.53 0 1 2 -1 .79 5 2 6 LWd 8.74 0 3 5 -2.56 7 3 8 PWVS 2.15 0 0 0 -0.83 4 3 5 PWVd 3.75 0 2 2 -2.50 1 5 4 SWVS 1 .83 1 0 . 0 -1 .07 1 2 4 SWVd 2.20 1 1 1 -1.71 2 3 3 NORMALIZED VELOCITIES LVNVs 1.83 LVNVd 2.58 PWNVs 2.90 PWNVd 4.35 SWNVs 2.08 SWNVd 2.53 0 2 1 0 1 0 0 4 0 2 0 0 5 6 1 3 0 1 0.52 •0.83 •1 .37 •2.74 •1 .38 -1 .89 5 7 3 2 5 3 4 4 4 4 3 3 10 8 4 4 5 6 103a In light of these reservations, we have some hesitation in claiming that the counts truly reflect the effects of Adriamycin in an absolute sense. On the other hand, since the purpose of this analysis is to rank the variables according to sensitivity, one could argue that although the 'true' counts might be lower, the same ranking of variables would be observed. Having stated these cautions, we try to interpret the counts. Looking at the serial changes, it appears that the cardiotoxic effects of Adriamycin are stronger during the first half of therapy, especially according to SF, EF, LVVd and LVNVd. These four measures seem to be the most sensitive for pre- to mid-therapy changes. For mid- to post-therapy changes, there is not much difference among variables in the number of patients identified. For pre- to post-therapy changes, SF, EF, LVVd, LVNVs and LVNVd appear most sensitive. It is noteworthy that all these measures are derived from LV internal dimensions, rather than from septal and posterior wall thicknesses. This may be due to the difficulty in identifying the wall surfaces distal to the LV chamber; the 'true' values of the wall measures may indeed be sensitive. But it is the observed ones that are relevant. Some measures such as PWVd and PWNVd identify more patients between mid- and post-therapy than between pre- and mid-therapy. This is in opposition to most of the other measures, and hints that some aspects of heart function may deteriorate at different points in therapy than others. Turning now to the counts based on bounds for the readings themselves, we note first that only one or two patients were outside the bounds at pre-therapy. This gives us additional assurance that the Adriamycin patients are similar to the normal subjects at 104 onset of therapy. The sensitive variables suggested by the counts based on mid-therapy readings are: EF, SF, LVVd and LVNVd. But PW% may also be useful. We remarked earlier that fewer patients lay beyond the bounds according to mid-to post-therapy changes than according to pre- to mid-therapy changes. Also, more patients are abnormal with respect to pre-to-mid changes, than by mid-therapy readings. Taken together, these statements suggest that although Adriamycin's major impact on LV function occurs during the first half of therapy, it is often not sufficient to depress the echocardiographic readings below the bounds for normal values. Thus examination of changes would be more useful here. The comparatively mild further impairment during the latter half of therapy causes the readings to drift below these bounds by the post-therapy date. It appears that PW%, SF, EF, LVVd, LVNVs and LVNVd are most sensitive according to counts for post-therapy readings. Except for PW%, this subset is similar to that suggested by the counts for pre- to post-therapy changes. This analysis has produced some useful results concerning choice of variables and meth-ods for monitoring. But in a very important sense Table 5.2 is too much of a summary. We do not know which children were classified as abnormal by different variables. Although nine patients were identified as having experienced abnormally large drops in SF during the first half of therapy, whereas only four were detected by PWVa, we can not conclude from this that SF is superior to PWV a for monitoring. It may be that some of the four children detected by PWV a were not identified by SF. After all, the former is a measure of posterior wall velocity, whereas the latter represents the change in LV internal size during 105 systole. This brings us to the next issue. Dimens iona l i t y of Ca rd i a c Impa i rment . Patients in general are known to demonstrate idiosyncratic responses to Adriamycin: they differ with respect to the degree of cardiac impairment. It would not be surprising then if they experienced different types of depression of LV function; i.e., impairment may not be unidimensional. In this event it is not sufficient to select only the most sensitive echocardiographic measures for monitoring; we could miss some patient who is impaired in a different way. However, we could ask: Of our sixteen variables, what is the smallest subset such that all children who are classified as abnormal by some measure in the entire set, are done so by at least one in the subset? This is an explicit optimization problem, known as a 'set-covering problem'. Not only will the solution favor the more sensitive variables, but it will take into account the covariance structure among the measures. For if two variables are highly correlated we would expect that a child who is classified as abnormal according to one, will also be abnormal according to the other. Choosing a minimal set with the above constraint will eliminate this duplication. Conversely, if two different children experience LV deterioration on two independent (uncorrelated) dimensions of LV function, then variables representing each dimension would be in an optimal subset. Of course, there are five solutions to this problem, depending on whether we look at pre- to mid-, mid- to post-, or pre- to post-therapy changes, or readings at mid- or post-therapy. It would be instructive to obtain all five optimal subsets and compare them. Earlier we pointed out several serious qualifications that must be kept in mind when interpreting the bounds as dividing points between a 'normal L V and an 'abnormal L V . 106 However we will adhere to this interpretation implicitly for the time being, and state that our results here should be viewed conservatively. We can formulate the optimization problem in graph-theoretic terms. Let each of the sixteen echocardiographic variables be represented by a node u,- belonging to a set Vi , and each of the twenty-three patients by a node py belonging to V2. For a given criterion, for example pre- to mid-therapy changes in readings, let there be an arc (i,j) if patient j was classified as abnormal according to variable t. Call the set of arcs E. Thus the bipartite which is connected to Vi, is also connected to Vi". This constraint can be written and possibly t = k. Note that the problem is feasible, since we can always let Vi* equal Vi. To obtain a solution we reformulate this as an integer programming problem. To simplify things, V2 can be restricted to those patients classified as abnormal by at least one of the sixteen variables. Let graph is G = (Vj \,V2,E). The problem is to choose the smallest subset Vi* of Vx such that each node py 6 V2 Vpy, ]«,• e s . t . {i,j)eE => 3ukeV!* s.t. {k,j)zE, otherwise. And define the adjacency matrix A by Then the problem is to ie mtmmtze 1=1 subject to 16 i=l 107 and Xi = 0,1. We are grateful to Sherry Judah, M.A., Maurice Queyranne, Ph.D. and Bernard Lamond, Ph.D. of the Faculty of Commerce for this integer programming formu-lation. The program LINDO was used to obtain solutions. It employs a modified linear pro-gramming algorithm, and then checks that the solution is also optimal for the integer program. This was always the case. The optimal subsets of echocardiographic variables are as follows. Mid-therapy: PW%, SW%, EF, LVVd, LVNVd, PWNVd. Post-therapy: SW%, EF, LVVd, SWVd, SWNVd. Pre- to mid-therapy changes: EF, LVVd, PWVs, SWVd, PWNVd. Mid- to post-therapy changes: SW%, EF, LWs, SWVs, LVNVs, LVNVd, PWNVs, PWNVd, SWNVs. Pre- to post-therapy changes: EF, LWs, LVVd, PWNVd, SWNVs, SWNVd. The subset corresponding to any one of the five criteria has the property that if one of the variables were deleted, then at least one patient abnormal on that criterion would go undetected. In this sense the solutions are minimal subsets. There are fewer variables in the solution for pre- to mid-therapy changes than in that for mid- to post-therapy changes; this is no doubt related to the fact that more patients were classified as abnormal by their changes during the first half of therapy than by changes during the second. An interesting result is that only diastolic, not systolic, velocities are included in the solutions for mid- and post-therapy readings. Where the criterion is changes in readings, both types are included. 108 Each of the five solutions would substantially reduce the number of variables needed for effective monitoring; the largest optimal subset contains only nine of sixteen variables. But which subset do we use? If we were looking only at readings and not at changes in them, we could use PW%, SW%, EF, LVVd, LVNVd and PWNVd at mid-therapy and SW%, EF, LVVd, SWVd and SWNVd at post-therapy. But clearly we cannot use one subset for pre-to-mid changes and another subset for mid-to-post changes, unless we measure all twelve variables in their union at mid-therapy. Only EF is common to all five solutions. The only variables which occur in none of the solutions are PWVd and SF. Although the latter is one of the most sensitive according to Table 5.2, EF has been chosen in its place. These two are highly correlated, which we know because EF is estimated directly from SF values using a table described in an earlier chapter. If we were to choose a single subset of variables to recommend for monitoring, the union of the optimal subsets for 'mid' and 'pre-to-mid' makes some sense. For it is most important to assess a patient's status before the course of Adriamycin therapy is completed. There are eight variables in this set: PW%, SW%, EF, LVVd, PWVs, SWVd, LVNVd and PWNVd. The only measures which appeared most sensitive in Table 5.2 but which are not in this set are SF and LVNVa. One can make a rudimentary assessment of how many patients would go 'undetected' if only these eight variables were used. Specifically, according to each of the five criteria for abnormality, how many patients in the Adriamycin dataset were not identified by at least one of these eight measures, who were identified by some other measure? On the basis of mid-therapy readings, thirteen patients were outside some bound, and all thirteen were 109 identified by the optimal subset of eight variables, obviously. For post-therapy readings, fourteen children were beyond at least one bound, and all fourteen were identified by the optimal subset. For pre- to mid-therapy changes, nineteen of nineteen were identified. For mid- to post-therapy changes, twelve of seventeen were identified. The remaining five were abnormal according to only one variable, different for each child (LVVs, SWVs, LVNVs, PWNVs, SWNVs). And for pre- to post-therapy changes, nineteen of twenty-two patients were identified by the optimal subset. Two of the remaining three were abnormal on only one measure each (LVVs and SWNVd), and the third was so on two measures (LVNVs and SWNVs). Our clinical colleagues state that an abnormal reading on only one or two measures is usually insufficient for discontinuation of therapy. It appears that not much information would be sacrificed by using this optimal subset of eight variables in place of all sixteen, whatever criterion is used. There is some doubt as to whether the eight patients 'missed' are truly abnormal, since seven of them are outside only one bound. We recommend the set of eight measures as a suggestion only; given the data, it is the best that we can do. The ultimate choice rests with clinicians. Hopefully Table 5.2 together with the five optimal subsets will be useful to them. Additional Reservations. We have been assuming throughout this discussion that if a patient is outside some bound, this can be interpreted as a cardiotoxic response to Adriamycin. Even apart from the qualifications made earlier, there is ideally a five percent 'error rate' attached to each bound: about five percent of healthy children will fall beyond a given bound at some time by chance. With twenty-three patients, sixteen correlated variables and five criteria, this could happen quite frequently. It was argued earlier that this would not greatly affect our 110 conclusions about relative sensitivities of variables to Adriamycin. However, the choice of optimal subsets of variables is a different matter. A detailed examination of the data reveals that it is sometimes the case that a patient lies beyond only a single bound. For example, both Patient Q's and Patient V's mid-therapy readings for SW% fall below the lower bound, but they are in the normal range for each of the remaining fifteen measures. Furthermore, they are the only two subjects whose mid-therapy readings for SW% are less than the lower bound. Although in light of these facts we might want to eliminate SW% from consideration as a useful monitoring variable, the constraints imposed on the optimization problem force SW% into the solution for mid-therapy readings. The problem lies not in the formulation of the objective and constraints. Indeed, it is precisely this type of situation that motivates the whole analysis. We want to take account of the multidimensionality in LV (dys)function, and not just choose those variables classifying the most patients as abnormal. We want to identify all patients at risk of heart trouble, even when this may not be indicated by, say, EF. The problem arises because one cannot know from these data alone whether a low reading or a large drop in a reading for an individual patient at a given date reflects cardiac impairment, or whether the patient just had a 'bad day'. In the absence of other information, we must treat all incidences of 'patient beyond bound' as real for the purposes of our analysis. It could be that Patients Q and V suffered impairment of an aspect of heart function that is quantified best (or only) by SW%, and that this is rare among patients. At least we are erring in the right direction; it is better to assume that the patient may be in trouble, than that the reading is not representative. In practice, clinicians can repeat the echocardiogram on the following day. Ill Dimens iona l i t y of N o r m a l L V Func t i on . The above attempts to incorporate the dimensionality of LV dysfunction into variable selection led to some concrete suggestions. There were many reservations. Here we consider another type of dimensionality: that of normal LV function. It will be seen that the analysis here is on somewhat firmer ground. We look at healthy subjects' readings on the sixteen echocardiographic measures, and try to empirically extract a few orthogonal dimensions which summarize LV function. This yields an additional constraint on variable selection: each dimension should be represented by at least one variable when monitoring for LV dysfunction. We used the two datasets of serial and one-time readings on healthy subjects. The natural statistical method to use is Factor Analysis. The presumption of this model is that responses are independent, and so for the serial data we first averaged the eight vector responses for each subject. These were then pooled with observations in the one-time dataset, for a total of fifty-one responses. So that the analysis will not be distorted by the scale effects among variables, we factor the correlation matrix instead of the covariance matrix. Formally, we are implicitly standardizing the variables by subtracting off the sample means and dividing by the sample standard deviations. These new variables have covariance matrix equal to the correlation matrix of the unstandardized variables. Before we proceed, we should have a look at the correlations among variables, to verify that a Factor Analysis is appropriate. Below are the sample squared multiple correlations of each variable with the others. 1 1 2 PW% SW% EF SF LVVa 0.650 0.550 0.994 0.994 0.915 0.905 PWVa PWVd SWVa SWVd LVNVa LVNVd 0.776 0.900 0.951 0.941 0.937 0.948 PWNVa PWNVd SWNVa SWNVd 0.789 0.910 0.967 0.966 AH correlations are high; we anticipate that a few common factors will explain much of the variation. Because Factor Analysis is a standard procedure, we will not go into too much detail. The reader is referred to Johnson and Wichern [42]. Here we just define our notation. The common orthogonal factor model with m < 16 factors is Y = LF + E , where Y is the 16 x 1 (standardized) response vector, F is the m x 1 random vector of common factors, L is the 16 x m matrix of factor loadings, and E is a vector of errors, or 'specific factors'. Under the usual assumptions, the population covariance (and correlation) m matrix $ of Y has the form LV + where * = diag^fo) = CW(E). Let /»,- = £ £ y 2 denote the ith communality, or variance of the ith variable accounted for by the m factors. Shortly we will make use of <j>a = 1 = -f i[>i-There are several approaches to estimating factor loadings. We chose the Method of Principal Factors and the Maximum Likelihood Method. Method of Principal Factors. The diagonal elements of the sample correlation matrix R are replaced by initial commu-nality estimates hi0; we took the squared multiple correlations for these. Denoting this 'reduced' correlation matrix by R\, it is factored as LV + where the yh column of L 113 is the product of the square root of the j largest eigenvalue of i?, and its corresponding eigenvector. This factorization gets closer to Ry. as more factors are retained. Since this method is close to the Principal Components solution, it makes sense to look at the size of successive ranked eigenvalues of R? in choosing the number of factors to retain. Each positive eigenvalue is equal to the sum of squares of the corresponding column of the loadings matrix, and thus is interpreted as the collective variance explained by that factor. SASPs PROC FACTOR [37] was used for computation. We encountered a minor prob-lem. Although R is always positive semidefinite, it turned out that the smallest five eigen-values of our sample reduced Rr were negative. But since none of these exceeded —0.093, we will ignore them. The total of all sixteen eigenvalues is 14.09; the largest five are 5.44, 2.97, 1.87, 1.52 and 1.05, with a total of 12.86, or about 91% of 14.09. Alternately, 12.86 is 80% of 16, the total sample variance (trace of R). It looks as if five factors might be sufficient. The analysis was repeated, constraining the number of factors at five. Let us look at some 'goodness of fit' measures. The final communality estimate for the ith variable is hi = £ 3 y = i A j 2 ; if th e true specific variance ipi is small, the corresponding communality should be close to unity. The smallest estimated communality is 0.472 (SW%), and the largest is 0.985 (EF); ten of the sixteen are above 0.80. Since the common factor model should account for correlations among variables, we can look at the residual correlations, or difference between the observed ones r,y and the 'predicted' ones (LL)ij. Many of these are zero or close to it; the largest is 0.13. These diagnostics indicate that a five-factor model is adequate. We turn now to inter-114 pretation. For orthogonal factors, the loadings matrix is determined only up to an orthogonal rotation. So we tried a few rotations to simplify the structure of L. The residual correla-tions, estimated communalities and variance explained by the live factors collectively are all invariant under orthogonal rotations, although the variance attributable to each individual factor changes. We tried an orthomax rotation with the parameter T set to zero; this tries to simplify the rows of L. We also tried a varimax rotation, which attempts to make each loading within any column of L either very large or very small. Since the factors are orthogonal, this should ideally force each variable to load highly on only a single factor. The varimax rotation was easier to interpret. See Table 5.3. Loadings have been multiplied by one hundred and rounded to the nearest integer. The rows have been arranged to group together variables whose highest loadings are on the same factor. This loadings matrix has a simple and suggestive structure. Each variable has one high loading on a single factor (flagged with an asterisk), and in general much lower ones on the others. Moreover, the grouping suggests names for the factors: 'septal wall', 'normalized LV velocity', 'ordinary LV velocity', 'percent change of LV internal dimension', and 'posterior wall' factors. Note that PWNVs and PWVs are associated with the 'LV velocity' factors. The variance explained collectively by the five factors has been distributed more evenly among them: 3.26, 2.55, 2.53, 2.37, 2.14. In a sense, each factor is now 'equally important'. Maximum Likelihood Solution. Since we have no prior notion of which method of estimation of L is preferable, it is a good idea to try another solution. Here we maximize the normal likelihood of the data, with the 115 TABLE 5.3 Loadings Matrix f o r P r i n c i p a l F a c t o r S o l u t i o n A f t e r Varimax R o t a t i o n V a r i a b l e F a c t o r s I II III IV V SWVs 78 * 1 6 49 -1 -14 SWVd 78 * 30 21 24 -6 SWNVs 85 * 41 -6 -2 -5 SWNVd 75 * 55 -22 8 -3 SW% 66 * -9 3 17 5 LVNVs 1 2 86 * . 21 31 -8 LVNVd 18 71 * 14 36 41 PWNVs 36 70 * 1 1 -7 14 L W s 9 4 89 * 13 -4 LWd 16 -2 67 * 26 48 PWVs 2 25 74 * 4 0 EF 14 1 5 15 95 * 12 SF 17 19 17 94 * 5 PWVd -26 -10 49 0 73 PWNVd 3 29 -4 -1 91 PW% -5 -3 -1 1 41 55 1 15a constraint that R = LV + Another constraint imposed to ensure a unique solution is that L'^/~1L be diagonal. Although this method implicitly assumes a multivariate normal distribution for the data, it is equivalent to some other 'nonparametric' solutions for L ([37], p.113), so it is valid for descriptive purposes. The number of factors extracted was constrained at five. We again used the squared multiple correlations as prior communality estimates. The likelihood is maximized by means of a recursive algorithm; during iteration, it happened that some communality estimates exceeded unity. This 'ultra-Heywood' situation was rectified by setting the offending com-munality estimates to one, and proceeding. The final estimates were all less than unity, with the exception of EF whose communality was identically one. The residiial correlations were about the same as those of the principal factor solution; some were slightly larger, some slightly smaller. The largest residual was 0.23. Estimated communalities were in general somewhat lower than before, although a few were a bit higher. We performed an equamax orthogonal rotation. See Table 5.4 for 100 times the rotated loadings, with Factors III and IV interchanged. The ordering of the rows is the same as in Table 5.3. The loadings are strikingly close to those of Table 5.3; this is significant, as both the initial method of extraction and the type of rotation are different. It means that the factor pattern is robust to the method of estimation, and strengthens our faith in interpreting the solution. ie „ The variance explained by each rotated factor ( £ Uj2) is 2.88, 2.73, 2.37 (IV), 2.42 ' 1=1 (III) and 2.16 respectively. Total variance explained is 12.56, or 79% of 16, the total sample variance. This agrees with the previous solution. 116 TABLE 5.4 Loadings M a t r i x f o r Maximum L i k e l i h o o d S o l u t i o n A f t e r Equamax R o t a t i o n V a r i a b l e F a c t o r s I II IV II I V SWVs 76 * 22 53 0 -13 SWVd 73 * 35 20 25 -4 SWNVs 81 * 49 -1 0 -7 SWNVd 68 * 62 -21 9 -5 SW% 64 * -3 1 17 0 LVNVs 2 90 * 27 32 -8 LVNVd 1 4 69 * 1 1 37 42 PWNVs 30 66 * 8 -2 1 4 LWs 4 7 90 * 1 1 0 LWd 17 -3 60 * 24 51 PWVs 2 1 4 69 * 8 6 EF 1 3 13 1 5 96 * 1 1 SF 1 6 17 17 95 * 5 PWVd -22 -15 42 1 79 PWNVd 3 29 -6 1 89 PW% -9 0 -12 41 48 1 16a Which solution we choose is immaterial, since they coincide. We have been fortunate in that we can assign each variable to a single factor. Comb ina t i on of Chap te r 5 Ana lyses . We have identified three criteria for selection of echocardiographic variables for moni-toring. The dominant criterion is that most of the information on patients conveyed by the sixteen measures should be conveyed by the subset as well. This was the motivation behind the integer programming optimization problems. Second, each chosen variable should be relatively sensitive to increasing cumulative doses of Adriamycin, but should not duplicate information conveyed by any other chosen measure. Third, each of the five dimensions underlying normal LV function (as defined by the sixteen measures) should be represented in the optimal set of variables. Not all three criteria can be satisfied. Recalling the integer programs, SW% was included in the solution for mid-therapy status, even though it is not one of the more sensitive measures according to number of children identified. Let us begin with the union of the 'optimal subsets' for pre- to mid-therapy changes and mid-therapy readings. The eight variables in this set (call it V) are: PW%, SW%, EF, LVVd, PWVs, SWVd, LVNVd and PWNVd. The only measures which appeared most sensitive in Table 5.2 but which are not in V are SF and LVNVs. We can discard SF because it is merely a transformation of EF; all patients identified at any time (mid-therapy, pre to mid changes, etc.) by SF were also identified by EF. Also, not much would be gained by inclusion of LVNVs; there are only two instances where a patient was identified by it but not by one of the measures in V: Patient J's mid- to post-therapy change in LVNVs was below the bound, but he was normal according to all other sixteen 117 variables. And Patient B's pre- to post-therapy change in LVNVs was abnormal, but he was normal by all other variables here except SWNVs. We examined earlier how many patients would not have been detected if only vari-ables in "V were considered. It was found that most children would be identified, and the remainder were outside bounds for only one or two variables. Returning to the loadings matrices in Tables 5.3 or 5.4, it is seen that each factor has a high loading coefficient for at least one variable in V. Since under the factor model each loading is the correlation between variable X; and factor Fj, each factor is represented by at least one variable in "V. 118 Conc lus ions for Chap te r 5. (1) The changes over time in readings for echocardiographic variables are likely to be useful as diagnostic aids in addition to the readings alone. We will not say 'more useful', although more patients were identified by changes than by readings, since this could be explained by increased random serial variation over six months vis-a-vis one month. However, patients who exhibit supranormal LV function at onset of therapy may not be identified as abnormal by mid-therapy readings, even if they exhibit drastic reductions in LV function. In such cases, examination of the serial change in readings is indicated. (2) It appears that the major impact of Adriamycin on LV function occurs during the first half of therapy, although many patients are still marginally within normal bounds for readings at mid-therapy. (3) Based on the number of patients classified as abnormal at mid- or post-therapy, the variables EF, SF, LVNVd, LVVd, and possibly PW% and LVNVs are the most sensitive to Adriamycin of the sixteen variables we considered. This is true whether one examines readings or serial changes in them. These variables are also in approximate order of sensi-tivity. (4) If the cardiotoxic effect of Adriamycin involves more than one aspect of the LV, then it may be that some patients at risk would not be identified by the measures listed above. Although selection of an optimal sensitive subset of variables then becomes difficult, we tentatively recommend the following: EF, LVVd, LVNVd, PW%, PWVs, PWNVd, SW% and SWVd. (5) It appears that there are five orthogonal dimensions to normal LV function as rep-resented by the sixteen echocardiographic variables. These may be named 'septal wall', 119 'posterior wall', 'ordinary LV velocity', 'normalized LV velocity', and 'percent change of LV internal dimension'. As a final note, we repeat that our recommendations here are suggestive only. The analyses of this chapter should be viewed as expository rather than definitive. Many sta-tistical assumptions have been made. Furthermore, there is much subjectivity involved in the interpretation of results, and clinicians may draw other conclusions from our findings. 120 Chap te r 6. Suggestions for Fu r the r S tudy . We have presented some preliminary Endings which we hope will be useful in improv-ing the echocardiographic technique. With respect to the application of this method to monitoring of cardiac dysfunction, there are many more issues worth examining. A few are suggested below. Our analysis of normal variability was based on only eighteen subjects. Although we feel comfortable with the conclusions about relative magnitudes of components of variability, the bounds for normal monthly changes should really be based on more subjects. Also of interest is whether the serial within-subject variation would be the same over, say, six months or one year as over one month. Presumably the variance stabilizes at some point. We plan to look into this in the future. If longitudinal variation increases beyond one month's time, then the normal ranges for serial changes in readings should be estimated from data spanning a time period comparable to the spacings at which Adriamycin patients are evaluated. There was some indication, based on only a crude assessment, that healthy children are inhomogeneous with respect to long-term longitudinal variability. More data are required to resolve this properly. If it turns out to be the case, then one could try to explain the differences by subject age, sex, etc. If such a systematic explanation is possible, then healthy subjects should be segregated into more homogeneous groups before the ranges of normal longitudinal variation are established. This would increase the chances of detection of LV dysfunction if such bounds are used clinically. More generally, it was suggested at the end of Chapter 4 that 'personal' bounds for 121 normal serial variation could be constructed for each patient if one had an idea of his long-term serial variance parameter under normal conditions. Since Adriamycin patients usually begin therapy shortly after diagnosis, one can not follow them for a month or more under 'normal' circumstances to estimate this. But one may be able to gather data spanning a week prior to commencement of therapy. 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