Statistical and deterministic approaches to designing transformations of electrocardiographic leads

Journal of Electrocardiology Vol. 35 Supplement 2002

Statistical and Deterministic Approaches to Designing Transformations of Electrocardiographic Leads B. Milan Hora´cˇek, PhD, James W. Warren, BSc, Dirk Q. Feild, MS,* and Charles L. Feldman, ScD†

Abstract: Two different approaches can be used to investigate the relationships among electrocardiographic leads: a statistical one, based on the analysis of recorded electrocardiograms (ECGs), and a deterministic one, based on physical principles that govern the current flow in irregularly shaped volume conductors such as the human body. The purpose of this study was to compare these two approaches. For the statistical investigation, the data set consisted of 120-lead ECGs recorded in a population including normal subjects (n ⫽ 290), post-myocardial-infarction patients (n ⫽ 497), patients with a history of ventricular tachycardia but no evidence of a previous myocardial infarction (n ⫽ 105), and patients with a single-vessel coronary artery disease who underwent coronary angioplasty (n ⫽ 91). Lead transformations of interest were obtained by fitting the multiple-regression model to this data set by the least-squares method. For the deterministic investigation, we used a boundary-element model of the human torso to simulate body-surface potentials in response to three orthogonal unit dipoles placed consecutively at 1,239 ventricular source locations, and the resulting body-surface potential distributions (instead of the recorded ECGs) were then fitted by the multipleregression model. The results suggest that the lead transformations should be preferably designed by statistical analysis of recorded ECGs. Regression models with a small number of predictors (eg, those based on three ECG leads) are the most reliable; those using more predictors are fraught with the danger of collinearity when predictors are highly correlated (as occurs in the standard 12-lead ECG). Model-derived deterministic transformations are compatible with statistically derived ones, provided that the distributed character of the cardiac sources is taken into account. We conclude that statistical associations among electrocardiographic leads can be reliably quantified in sufficiently large and diverse databases of recorded data; the causality of these associations can be supported by appropriate deterministic models based on the laws of physics. Key words: Electrocardiographic lead systems, body-surface potential mapping, myocardial infarction, acute ischemia.

From the Faculty of Medicine of Dalhousie University, Halifax, Nova Scotia, Canada, *Philips Medical Systems, Oxnard, CA, and the †Brigham & Women’s Hospital, Boston, MA. Supported by grants from the Canadian Institutes of Health Research, the Heart and Stroke Foundation of Nova Scotia, and Philips Medical Systems. Reprint requests: B. Milan Hora´cˇek, PhD, Department of Physiology & Biophysics, Sir Charles Tupper Medical Bldg., 5859 University Ave, Dalhousie University, Halifax, Nova Scotia B3H 4H7, Canada; e-mail: [email protected]. Copyright 2002, Elsevier Science (USA). All rights reserved. 0022-0736/02/350S-0006$35.00/0 doi:10.1054/jelc.2002.37154

41

42 Journal of Electrocardiology Vol. 35 Supplement 2002 In 1962, Burger et al. (1)—assuming that the electrical activity of the heart can be represented by a single stationary dipole— envisaged numerical transformations among electrocardiographic lead systems. They showed that such transformations can be described by a system of linear algebraic equations in which dimensionless coefficients completely define them. To obtain these coefficients, one need only measure for a given subject the voltages in the constituent leads at several (at least three) instants of time and then solve the equations for the unknown coefficients to obtain an individual transformation. To obtain generally applicable transformation coefficients, one needs to study a sufficiently large number of cases and calculate a global transformation, which gives, on average, for each individual case the best possible fit. Burger et al. (1) accomplished this by pooling the equations pertaining to each lead from all subjects in their study population (n ⫽ 169) and solving the overdetermined system by the method of least-squares. They used 5 time instants of the QRS complex to sample the X, Y, and Z leads of each orthogonal system studied; therefore, for the entire population, they had to solve 2,535 equations for all three of the vector components, and “for these calculations a simple electric calculator was utilized.” Some 15 years earlier, Burger and van Milaan (2), using an electrolytic-tank model of the human torso to study the relationships among electrocardiographic leads, had introduced the notion of the lead vector to define the relationship between a dipolar source and the response voltage in a lead. They showed that the entire body-surface potential distribution from any fixed-location dipolar source relates to that source through a set of lead vectors, one for each site on the body surface; the tips of these lead vectors form Burger’s image surface (2– 4). The image surface reflects the influence of body shape, body composition, and the dipole-source location on the body-surface potential distribution. Since the heart’s electrical activity can be adequately approximated by the distributed current dipoles, a set of image surfaces pertaining to all possible locations of the heart’s dipolar sources provides the information that is both necessary and sufficient if one wishes to construct a surface ECG for any known source distribution. Thus, such a set of image surfaces defines in the deterministic sense the relationships among all possible electrocardiographic leads. Attempts to make practical use of lead transformations in the clinical setting have been ongoing since the 1960s. Cady et al. (5) proposed a system for deriving the 12 standard leads from the scalar

components of the Frank leads (6), whereas Dower (7) derived those leads from the Frank orthogonal leads, by using coefficients determined by Frank in the electrolytic-tank model of the human torso (3). Among many other early studies that followed were those of Mizuno and Yasui (8) and Wolf et al. (9), who investigated lead transformations for specific disease groups. In our recent study (10), investigating the transformation from EASI* leads (11) into the 12 standard leads, we were encouraged to find that a very small amount of diagnostic information was dissipated in the transformation process for the population that consisted of normal subjects and patients with remote myocardial infarction. The purpose of the present study was to focus on the methods for deriving the coefficients for lead transformations, with a specific aim at comparing the statistical analysis of recorded electrocardiographic signals and the deterministic modeling of the genesis of these signals, with due regard to the human torso’s geometry, its electrical inhomogeneities, and the distributed character of the cardiac sources (4). It is widely recognized that statistical models, even though they can quantify associations among variables well, need to be corroborated by deterministic models that obey the laws of physics, because causality cannot be established by statistical analyses (12). Accordingly, the objective of this study was to corroborate statistical models for transformations of electrocardiographic leads by deterministic modeling.

Materials and Methods Study Population For the statistical derivation of general transformation coefficients, we used a study population consisting of 892 individuals: 290 normal subjects, 497 patients who previously suffered a myocardial infarction (MI), and 105 patients with a history of spontaneous ventricular tachycardia (VT) but no evidence of a previous MI. One averaged QRST complex (sampled at 2-ms intervals) was used for each individual. The total number of time-instants for which the relationships among leads were analyzed was N ⫽ 197,182. The pertinent clinical characteristics of this data set are in Table 1. The diagnosis of MI was based on non-ECG evidence in the acute phase—including prolonged,

*EASI is patented technology that is available under license from Philips Medical Systems, Andover, MA; royalty-free licenses are available for research use by not-for-profit entities.

Design of ECG-Lead Transformations •

Hora´cˇek et al. 43

Table 1. Characteristics of Patient Population for Deriving Lead Transformations

Subjects, no. (%) Age, y* Sex, M/F —, % QRS duration, ms QT interval, ms QTc interval, ms Heart rate, bpm LVEF, %

Normals

MI/noVT/noQ

MI/noVT/Q

MI/VT

noMI/VT

290 (32.5) 36 ⫾ 12 163/127 56.2/43.8 99 ⫾ 10 419 ⫾ 32 424 ⫾ 23 62 ⫾ 10 —

36 (4.0) 61 ⫾ 11 26/10 72.2/27.8 106 ⫾ 17 438 ⫾ 28 436 ⫾ 34 60 ⫾ 10 68 ⫾ 11†

282 (31.6) 58 ⫾ 11 235/47 83.3/16.7 105 ⫾ 15 443 ⫾ 43 457 ⫾ 41 65 ⫾ 13 51 ⫾ 18†

179 (20.1) 62 ⫾ 10 155/24 86.6/13.4 122 ⫾ 25 460 ⫾ 55 494 ⫾ 54 71 ⫾ 14 33 ⫾ 12†

105 (11.8) 55 ⫾ 14 79/26 75.2/24.8 118 ⫾ 26 442 ⫾ 51 476 ⫾ 48 71 ⫾ 14 45 ⫾ 17†

MI indicates myocardial infarction; VT, ventricular tachyarrhythmia; MI/noVT/noQ, patients with a non-Q MI and no history of VT; MI/noVT/Q, patients with a Q-wave MI and no history of VT; MI/VT, patients with a Q-wave MI and a history of VT; noMI/VT, patients with a history of VT and no evidence of previous MI. *Values with plus/minus are mean ⫾ SD. † LVEF was not available for all patients.

ischemic-type chest pain, and a peak creatine kinase enzyme level more than twice the upper limit of normal—and the presence of diagnostic 12-lead ECG changes. The members of 2 of the patient subgroups presented with electrocardiographically documented sustained VT in the absence of a reversible cause, such as electrolyte imbalance, a proarrhythmic drug effect, or an MI within the previous two weeks. To show a specific collinearity problem in transforming the 12-standard leads into vessel-specific leads for detecting acute myocardial ischemia, we used a study population consisting of 91 patients who had single-vessel coronary artery disease for which they had undergone percutaneous transluminal coronary angioplasty (PTCA). In 32 of these patients, the lesion was in the left anterior descending (LAD) coronary artery; in 36, it was in the right coronary artery (RCA); and in 23, it was in the left circumflex (LCx) coronary artery. Two averaged QRST complexes, sampled at 2-ms intervals, were used for each patient; one represented a baseline state and the other an ischemic state. The total number of time-instants used in the regression analysis was N ⫽ 55,108. The pertinent clinical characteristics of this data set were described in more detail elsewhere (13). All subjects who participated in the clinical studies were informed of the study’s procedures, in accordance with the ethical guidelines approved by the institutional ethics committee. Data Acquisition and Processing We recorded 120-lead simultaneous ECGs for each subject. The lead array had three limb leads and 117 unipolar chest leads (13). Recordings were made for 15 consecutive seconds, while the subjects were supine and had a normal sinus rhythm. Analog ECGs were amplified, filtered (bandpass from

0.025 to 125 Hz), multiplexed, and digitized at a rate of 500 12-bit samples/second/channel (with 2.5-␮V resolution for the least-significant bit). Subsequent data processing was performed off line on an IBM RS/6000 computer (IBM Corp, Armonk, NY). From the 15-s recordings, individual complexes were identified and sorted into families based on QRS morphology. The complexes were averaged and the baseline was corrected to yield a single averaged PQRST complex for each lead. The fiducial marks of the ECG waves were determined automatically by computer algorithms; in addition, we checked and edited on screen the computerdetermined marks and then stored the edited values. We plotted the averaged complexes in a format which resembled the layout of the electrodes on the chest and then edited these plots, eliminating leads with artifacts. To replace rejected or missing leads, we performed a three-dimensional interpolation (14), based on a numerical torso model described below, which produced potentials at 352 node locations on the torso (13). Torso Model A boundary-element model of a realistic threedimensional human torso (Fig. 1) containing lungs and intracavitary blood masses of different conductivities was used to calculate the lead vectors and deterministic coefficients for the transformation of electrocardiographic leads. It consisted of a system of 1,368 linear algebraic equations arising from a discrete approximation to the integral equation formulation of the appropriate boundary-value problem. This torso model is a modification of one described previously (15), with the outer surface modified to simplify the identification of commonly used ECG electrode sites on the model’s surface

44 Journal of Electrocardiology Vol. 35 Supplement 2002

Fig. 1. A three-dimensional wire diagram of the Dalhousie torso model (15). After Frank (3), the numbered transverse levels from neck to waist are 1⬙ apart, and the positions around transverse sections correspond to Frank’s equiangular division (although our division uses, up to the midclavicular line, sagittal planes). There are 352 nodes— constituting vertices of 700 triangles —at which electrocardiographic potentials are interpolated from the set of potentials recorded at 117 measurement sites. A corresponding two-dimensional diagram of the Dalhousie torso model is shown elsewhere (13). Xmarks in the heart region show locations where orthogonal unit dipoles were placed to calculate bodysurface potential distributions that were used as input data in the regression analysis.

(13). The torso’s boundary surfaces were tessellated into triangular area elements. By using 352 nodes (13) as vertices, we triangularized the outer boundary into 700 elements; the internal surfaces comprised 668 elements. The algebraic system was solved by the Gauss-Seidel iterative method. A method of multiple deflations was applied to ensure the fast convergence of the iterative method (16). Three systems of equations—for unit dipoles oriented along the x, y, and z axes at a given source location—were solved simultaneously. Each triplesolution of the 1368-equation system required about 10 s of CPU time on our RS/6000 computer. Since the solutions yielded the mean potential for each triangle, the potentials had to be averaged from those of adjacent triangles to obtain potential values at 352 nodes. Three models were studied: a homogeneous torso, a torso with lungs of one-fourth body con-

ductivity, and the latter plus intracavitary blood masses of three-times body conductivity. Three orthogonal unit dipoles were placed in the total of 1239 source locations within the torso’s ventricular region, and a body-surface potential distribution was calculated for each dipole; for any specified source location, this provided data for an image surface specific for that particular location of the dipole source (3,4,15). In this manner, 3,717 simulated body-surface potential distributions were obtained in each model; these represented all possible potential distributions arising from dipolar sources of arbitrary orientation (which, under the assumption of linearity, are the linear combination of distributions produced by orthogonal dipoles) anywhere in the ventricular myocardium. These distributions were used as input data for the regression analysis seeking “deterministic” lead transformations.

Design of ECG-Lead Transformations •

Regression Analysis The objective of the regression analysis was to fit a regression model both to the recorded and simulated sets of data, with the aim of obtaining either a statistical or deterministic estimate Vˆ of the instantaneous voltage V at a given response lead by fitting the regression equation

e៮ rms. For a given individual, recorded and estimated sampled voltages of a given lead can be expressed as ˆ (consisting of eleK-dimensional vectors V and V ments Vj, Vˆ j representing recorded or estimated voltages, in ␮V, at time instants j ⫽ 1, 2, . . . , K). For a population consisting of n individuals, e៮ rms (in ␮V) is defined as

Vˆ ⫽ ␤ˆ 0 ⫹

冘

(4)

(1)

␤ˆ i V i

to the recorded or simulated voltages Vi in predictor leads i ⫽ 1, . . . , k; thus the problem was formulated as one of finding the best-fitting transformation coefficients ␤ˆ i, for i ⫽ 0, 1, . . . , k (linear regression with intercept) or for i ⫽ 1, . . . , k (linear regression without intercept) when ␤ˆ 0 was set to zero. We used the least-squares-solution approach to this problem, which yields such “least-square estimates” of ␤ˆ 0, ␤ˆ 1, . . . , ␤ˆ k that the error sum of squares for all N available data points of the observed data

冘 共V ⫺ Vˆ 兲 ⫽ 冘 共V ⫺ ␤ˆ ⫺ ␤ˆ V N

N

2

i

i

i⫽1

i

0

1

1i

. . . ⫺ ␤ˆ 1Vki兲2

i⫽1

is a minimum. Observed data points were either extracted from the recorded ECGs of the subjects described in the Study Population above, or from simulated voltages produced by the deterministic model, as described in the Torso Model above; throughout this paper, we refer to the former method as a statistical approach and to the latter as a deterministic approach. To perform least-squares solutions to the linear-regression problems, we used a general-purpose procedure for regression (PROC REG) from the SAS System (17). Goodness of Fit Once the transformation coefficients that best fitted the available data were known, they were applied as constant coefficients to the time-varying ECG signals. Thus, for example, the time-varying voltage V(t) of the desired lead was estimated as (3)

1 e៮ rms ⫽ n i⫽1

K

2

j

j

j⫽1

and R៮ is defined as

i⫽1

(2)

冑

冘 冘 共V ⫺ Vˆ 兲 , n

k

Hora´ cˇ ek et al. 45

Vˆ 共t兲 ⫽ ␤ˆ ES V ES 共t兲 ⫹ ␤ˆ AS V AS 共t兲 ⫹ ␤ˆ AI V AI 共t兲

from the time-varying voltages VES(t), VAS(t), and VAI(t) recorded in the predictor leads ES, AS, and AI of the EASI lead system. The goodness of fit of the transformations was assessed by the mean correlation coefficient R៮ and by the mean root-mean-square error

R៮ ⫽

(5)

冘

n ˆ V䡠V 1 ˆ . n i⫽1 VV

Collinearity Analysis When fitting a regression model with a larger number than three predictor leads to a particular set of electrocardiographic data, it is advisable to quantify the degree of collinearity among the predictor leads. One way to examine collinearity among leads is to consider what happens if each predictor lead is the response variable in a multiple regression in which the independent variables are all of the remaining leads. For k predictor leads, there are k such regression models. To assess collinearity, one needs the associated R2-values based on fitting these models. If any of these multiple R2-values equals 1.0 (when one of the predictor leads is an exact linear combination of the others), a perfect collinearity is said to exist among the set of predictors (12). A near collinearity arises when the multiple R2 of one predictor lead with the remaining predictors is nearly 1. Denoting R2j the squared multiple correlation based on regressing predictor-lead voltage Vj on the remaining k ⫺ 1 predictors, the variance inflation factor (VIF), computed as (6)

VIF j ⫽

1 j ⫽ 1, 2, . . . ,k , 共1 ⫺ R 2j 兲

can be used to assess collinearity. The larger the value of VIFj, the more questionable is the contribution of the associated predictor lead. Usually, a value to be concerned with is VIFj ⬎ 10.0; this corresponds to R2j ⬎ 0.90 or, equivalently, Rj ⬎ 0.95. To show the relevance of the collinearity concept in a multiple-regression analysis involving more than three ECG leads in predicting the desired lead, we assessed the degree of collinearity among the 12 standard ECG leads (i e, in 8 “independent” leads of

46 Journal of Electrocardiology Vol. 35 Supplement 2002 the 12-lead ECG system). First, using the PTCA set of recorded ECGs, we computed a Pearson correlation matrix among predictor leads I, II, and V1–V6 and bipolar vessel-specific response leads LAD, LCx, and RCA (13). Next, we applied a multiple-regression analysis using all eight predictor leads and computed VIFj for each; the predictor lead with the largest VIFj ⬎ 10.0 was dropped from the regression equation and the analysis was repeated; by this process, predictor leads which contributed mostly redundancy were gradually eliminated until no VIFj values exceeded 10. Thus, the most important predictor leads of the standard 12-lead ECG were identified by this method.

Results Lead Field Determined by the Torso Model Three components of the lead field were determined at 1,239 ventricular sites for 352 unipolar body-surface leads in the homogeneous torso, in the torso with lungs, and in the torso with lungs and intracavitary blood masses. Mean lead-field values with standard deviations were calculated for selected leads from these 1,239 vectors and listed for all three configuration of the torso in Table 2. The components of the lead field were calculated in absolute units for unit-dipole sources in the torso of unit conductivity and then normalized by the length of lead vector for lead I and tabulated in relative units. By multiplying the values listed in Table 2 by a factor of 2.52511/0.2, the lead field in absolute units of ⍀/m for unit current dipoles in absolute units of A ⫻ m can be restored (assuming a torso conductivity of 0.2 S/m). The mean leadfield vectors for leads I, II, and III form a characteristic Burger’s triangle, with the length of the vectors for leads II and III exceeding that for lead I, as would be expected from geometric lead theory (2,4). The Wilson central terminal exhibits the shortest length of the mean lead-field vector, with only the z-component having a value that is not negligible. The unipolar precordial leads V2–V4 have the largest magnitude of the mean lead field, matched or exceeded only by the bipolar vesselspecific leads LAD, LCx, and RCA (13). The lead field for Frank’s orthogonal leads X, Y, and Z (6) shows excellent orthogonality and reasonably uniform sensitivity for all three leads. These findings are similar in all 3 configurations of the torso model; however, closer examination reveals that

both lungs and intracavitary blood masses exert a noticeable effect on the lead field in specific leads. Transformations Derived from the Recorded Data Using recorded ECGs from the design set comprising 892 subjects, statistical transformation coefficients ␤ˆ ES, ␤ˆ AS, and ␤ˆ AI for deriving 352 unipolar electrocardiographic leads from leads ES, AS, and AI of the EASI lead system were computed by means of the least-squares method. Table 3 shows the transformation coefficients (to 5 decimal places for computer implementation) for 18 commonly used electrocardiographic leads; note that different coefficients are listed for leads using a standard Wilson central terminal (using electrodes on the wrists and the left ankle) and for those using Mason-Likar substitutes for the standard limb leads (18). The recorded voltage V in a given lead at a given time instant of the cardiac cycle is estimated as a response variable Vˆ ⫽ ␤ˆ ESVES ⫹ ␤ˆ ASVAS ⫹ ␤ˆ AIVAI, where VES, VAS, and VAI are voltages recorded in predictor leads ES, AS, and AI, respectively. Mean correlation coefficient R៮ and mean rms error e៮ rms were calculated between actually recorded and EASI-derived ECGs; these columns in Table 3 indicate how good an approximation of the given lead can be expected (on average) from the ៮ transformation. In general, based on the values of R and e៮ rms, estimates can be expected to be good. Table 3A gives transformation coefficients for deriving orthogonal and vessel-specific leads from EASI leads; these coefficients were determined by the same method as those in Table 3, and the results are displayed in a similar format, but without the distinction between coefficients for standard limb leads and Mason-Likar substitutes (which is meaningless in the listed leads). Transformations Derived from the Model The deterministic transformation coefficients ␤ˆ ES, ␤ˆ AS, and ␤ˆ AI for deriving 352 unipolar electrocardiographic leads from leads ES, AS, and AI of the EASI lead system were computed by fitting a multiple-regression model to 3,717 potential distributions, generated by three orthogonal dipoles at 1,239 source locations, by means of the leastsquares method; this was conducted for potential distributions generated in the homogeneous torso model and in the models that included inhomogeneities. Table 4 lists these deterministic transformation coefficients, together with their corresponding

c៮ y ⫺0.27 ⫾ 0.16 1.16 ⫾ 0.11 1.44 ⫾ 0.21 ⫺0.45 ⫾ 0.09 ⫺0.85 ⫾ 0.17 1.30 ⫾ 0.15 ⫺0.03 ⫾ 0.03 ⫺0.22 ⫾ 0.52 ⫺0.55 ⫾ 0.91 ⫺0.26 ⫾ 0.79 0.07 ⫾ 0.47 0.07 ⫾ 0.28 0.03 ⫾ 0.16 0.03 ⫾ 0.12 0.05 ⫾ 0.11 0.06 ⫾ 0.11 0.05 ⫾ 0.20 0.09 ⫾ 0.12 0.07 ⫾ 0.08 ⫺0.17 ⫾ 0.16 1.11 ⫾ 0.06 0.17 ⫾ 0.35 0.29 ⫾ 0.86 0.46 ⫾ 0.25 ⫺0.18 ⫾ 0.14 ⫺0.73 ⫾ 0.60 1.24 ⫾ 0.69 2.00 ⫾ 0.50

0.95 ⫾ 0.16 0.58 ⫾ 0.10 ⫺0.37 ⫾ 0.16

⫺0.76 ⫾ 0.10 0.66 ⫾ 0.15 0.11 ⫾ 0.11

0.07 ⫾ 0.03 ⫺1.07 ⫾ 0.59 ⫺0.13 ⫾ 0.72 0.88 ⫾ 0.88 1.37 ⫾ 0.70 1.30 ⫾ 0.50 0.85 ⫾ 0.23

0.56 ⫾ 0.12 0.33 ⫾ 0.10 0.08 ⫾ 0.08

⫺0.98 ⫾ 0.39 ⫺0.84 ⫾ 0.21 ⫺0.68 ⫾ 0.10

1.18 ⫾ 0.11 0.12 ⫾ 0.04

0.12 ⫾ 0.11

⫺0.63 ⫾ 0.54 0.99 ⫾ 0.25 1.42 ⫾ 0.17

0.67 ⫾ 0.76 0.43 ⫾ 0.52 ⫺0.44 ⫾ 0.41

I (38,51) II (38,344) III (51,344)

aVR (38,51,344) aVL (38,51,344) aVF (38,51,344)

CT (38,51,344) V1 (169,CT) V2 (171,CT) V3 (173,CT) V4 (194,CT) V5 (196,CT) V6 (198,CT)

V7 (199,200,CT) V8 (200,201,CT) V9 (201,202,CT)

V3R (167,188,CT) V4R (187,CT) V5R (186,CT)

X (165, 175, 178) Y (17, 182, 344) Z (165, 170, 175, 178, 182)

ES (44,170) AS (44,178) AI (165, 178)

LAD (174,221) LCx (150,221) RCA (129,342)

⫺2.24 ⫾ 0.82 2.14 ⫾ 0.81 1.06 ⫾ 0.57

⫺1.75 ⫾ 1.21 0.17 ⫾ 0.08 0.11 ⫾ 0.09

1.30 ⫾ 0.34

⫺0.15 ⫾ 0.05 0.07 ⫾ 0.06

⫺0.66 ⫾ 0.26 ⫺0.36 ⫾ 0.09 ⫺0.12 ⫾ 0.02

0.29 ⫾ 0.14 0.40 ⫾ 0.18 0.45 ⫾ 0.22

0.21 ⫾ 0.06 ⫺1.21 ⫾ 0.71 ⫺2.26 ⫾ 1.26 ⫺2.21 ⫾ 1.15 ⫺1.44 ⫾ 0.75 ⫺0.46 ⫾ 0.24 0.13 ⫾ 0.11

⫺0.01 ⫾ 0.04 0.09 ⫾ 0.04 ⫺0.08 ⫾ 0.05

0.07 ⫾ 0.04 ⫺0.04 ⫾ 0.05 ⫺0.11 ⫾ 0.05

c៮ z

2.58 ⫾ 0.96 2.62 ⫾ 0.93 2.37 ⫾ 0.66

2.07 ⫾ 1.34 1.14 ⫾ 0.25 1.44 ⫾ 0.18

1.35 ⫾ 0.39

1.21 ⫾ 0.12 1.12 ⫾ 0.05

1.20 ⫾ 0.45 0.93 ⫾ 0.22 0.70 ⫾ 0.10

0.64 ⫾ 0.17 0.54 ⫾ 0.19 0.48 ⫾ 0.22

0.23 ⫾ 0.05 1.69 ⫾ 0.95 2.50 ⫾ 1.44 2.58 ⫾ 1.36 2.08 ⫾ 0.95 1.41 ⫾ 0.53 0.88 ⫾ 0.25

0.89 ⫾ 0.10 1.09 ⫾ 0.17 1.31 ⫾ 0.14

1.00 ⫾ 0.16 1.31 ⫾ 0.11 1.50 ⫾ 0.20

c៮ 

0.70 ⫾ 0.77 0.45 ⫾ 0.58 ⫺0.19 ⫾ 0.42

⫺0.46 ⫾ 0.58 1.06 ⫾ 0.29 1.33 ⫾ 0.17

0.03 ⫾ 0.17

1.12 ⫾ 0.13 0.27 ⫾ 0.07

⫺0.93 ⫾ 0.44 ⫺0.79 ⫾ 0.24 ⫺0.64 ⫾ 0.10

0.51 ⫾ 0.12 0.26 ⫾ 0.05 ⫺0.01 ⫾ 0.08

0.07 ⫾ 0.03 ⫺1.03 ⫾ 0.64 ⫺0.09 ⫾ 0.73 0.94 ⫾ 0.92 1.45 ⫾ 0.72 1.32 ⫾ 0.53 0.83 ⫾ 0.24

⫺0.79 ⫾ 0.07 0.52 ⫾ 0.09 0.26 ⫾ 0.08

0.87 ⫾ 0.10 0.70 ⫾ 0.08 ⫺0.17 ⫾ 0.10

c៮ x

⫺0.87 ⫾ 0.66 1.57 ⫾ 0.72 2.20 ⫾ 0.48

0.29 ⫾ 0.94 0.77 ⫾ 0.29 ⫺0.03 ⫾ 0.12

0.22 ⫾ 0.38

⫺0.07 ⫾ 0.15 1.34 ⫾ 0.12

⫺0.04 ⫾ 0.21 0.07 ⫾ 0.13 0.06 ⫾ 0.08

0.14 ⫾ 0.11 0.13 ⫾ 0.10 0.10 ⫾ 0.09

⫺0.01 ⫾ 0.04 ⫺0.40 ⫾ 0.52 ⫺0.75 ⫾ 0.97 ⫺0.32 ⫾ 0.88 0.15 ⫾ 0.54 0.21 ⫾ 0.30 0.16 ⫾ 0.16

⫺0.63 ⫾ 0.06 ⫺0.84 ⫾ 0.12 1.47 ⫾ 0.11

⫺0.14 ⫾ 0.11 1.40 ⫾ 0.09 1.54 ⫾ 0.14

c៮ y

⫺2.35 ⫾ 0.91 2.22 ⫾ 0.91 1.24 ⫾ 0.60

⫺1.84 ⫾ 1.27 0.09 ⫾ 0.07 ⫺0.01 ⫾ 0.09

1.42 ⫾ 0.40

⫺0.26 ⫾ 0.29 0.12 ⫾ 0.09

⫺0.73 ⫾ 0.34 ⫺0.39 ⫾ 0.14 ⫺0.11 ⫾ 0.04

0.23 ⫾ 0.10 0.36 ⫾ 0.13 0.47 ⫾ 0.20

0.21 ⫾ 0.07 ⫺1.32 ⫾ 0.81 ⫺2.37 ⫾ 1.31 ⫺2.35 ⫾ 1.25 ⫺1.54 ⫾ 0.79 ⫺0.56 ⫾ 0.26 0.05 ⫾ 0.10

0.02 ⫾ 0.07 ⫺0.02 ⫾ 0.05 0.00 ⫾ 0.10

⫺0.03 ⫾ 0.04 ⫺0.02 ⫾ 0.11 0.01 ⫾ 0.10

c៮ z

Torso With Lungs

2.74 ⫾ 1.06 2.87 ⫾ 1.03 2.60 ⫾ 0.66

2.13 ⫾ 1.41 1.35 ⫾ 0.29 1.33 ⫾ 0.18

1.48 ⫾ 0.44

1.16 ⫾ 0.15 1.38 ⫾ 0.12

1.22 ⫾ 0.52 0.90 ⫾ 0.26 0.66 ⫾ 0.10

0.59 ⫾ 0.14 0.47 ⫾ 0.13 0.49 ⫾ 0.20

0.23 ⫾ 0.06 1.80 ⫾ 1.04 2.67 ⫾ 1.50 2.75 ⫾ 1.47 2.23 ⫾ 1.00 1.49 ⫾ 0.56 0.86 ⫾ 0.24

1.01 ⫾ 0.08 1.00 ⫾ 0.11 1.50 ⫾ 0.11

0.89 ⫾ 0.10 1.57 ⫾ 0.10 1.56 ⫾ 0.14

c៮ 

0.55 ⫾ 0.78 0.50 ⫾ 0.57 ⫺0.12 ⫾ 0.44

⫺0.47 ⫾ 0.51 0.99 ⫾ 0.36 1.22 ⫾ 0.29

0.06 ⫾ 0.19

1.02 ⫾ 0.23 0.27 ⫾ 0.12

⫺0.87 ⫾ 0.40 ⫺0.73 ⫾ 0.21 ⫺0.58 ⫾ 0.09

0.48 ⫾ 0.15 0.25 ⫾ 0.07 0.00 ⫾ 0.08

0.07 ⫾ 0.04 ⫺0.98 ⫾ 0.60 ⫺0.17 ⫾ 0.64 0.77 ⫾ 0.87 1.29 ⫾ 0.81 1.21 ⫾ 0.63 0.77 ⫾ 0.30

⫺0.73 ⫾ 0.12 0.47 ⫾ 0.16 0.26 ⫾ 0.15

⫺0.85 ⫾ 0.58 1.49 ⫾ 0.72 2.04 ⫾ 0.57

0.19 ⫾ 0.82 0.72 ⫾ 0.26 ⫺0.01 ⫾ 0.12

0.24 ⫾ 0.34

⫺0.05 ⫾ 0.12 1.23 ⫾ 0.24

⫺0.09 ⫾ 0.26 0.02 ⫾ 0.18 0.03 ⫾ 0.13

0.14 ⫾ 0.10 0.13 ⫾ 0.09 0.10 ⫾ 0.10

0.00 ⫾ 0.05 ⫺0.44 ⫾ 0.49 ⫺0.76 ⫾ 0.86 ⫺0.35 ⫾ 0.73 0.12 ⫾ 0.43 0.20 ⫾ 0.25 0.16 ⫾ 0.13

⫺0.59 ⫾ 0.15 ⫺0.77 ⫾ 0.18 1.36 ⫾ 0.28

⫺0.12 ⫾ 0.11 1.30 ⫾ 0.28 1.42 ⫾ 0.29

c៮ y

⫺2.25 ⫾ 1.00 2.20 ⫾ 1.15 1.24 ⫾ 0.77

⫺1.80 ⫾ 1.48 0.14 ⫾ 0.17 0.03 ⫾ 0.16

1.38 ⫾ 0.55

⫺0.21 ⫾ 0.17 0.15 ⫾ 0.20

⫺0.75 ⫾ 0.44 ⫺0.40 ⫾ 0.21 ⫺0.13 ⫾ 0.09

0.24 ⫾ 0.13 0.36 ⫾ 0.15 0.46 ⫾ 0.22

0.21 ⫾ 0.08 ⫺1.33 ⫾ 0.98 ⫺2.30 ⫾ 1.53 ⫺2.22 ⫾ 1.34 ⫺1.43 ⫾ 0.83 ⫺0.50 ⫾ 0.29 0.08 ⫾ 0.13

⫺0.02 ⫾ 0.13 ⫺0.02 ⫾ 0.14 0.04 ⫾ 0.20

0.00 ⫾ 0.12 0.04 ⫾ 0.21 0.04 ⫾ 0.21

c៮ z

c៮ 

2.60 ⫾ 1.12 2.83 ⫾ 1.21 2.48 ⫾ 0.82

2.04 ⫾ 1.55 1.27 ⫾ 0.32 1.24 ⫾ 0.30

1.45 ⫾ 0.58

1.06 ⫾ 0.25 1.29 ⫾ 0.25

1.21 ⫾ 0.54 0.87 ⫾ 0.25 0.62 ⫾ 0.10

0.57 ⫾ 0.19 0.47 ⫾ 0.16 0.49 ⫾ 0.22

0.23 ⫾ 0.07 1.80 ⫾ 1.10 2.61 ⫾ 1.62 2.54 ⫾ 1.50 2.02 ⫾ 1.09 1.36 ⫾ 0.66 0.80 ⫾ 0.30

0.96 ⫾ 0.13 0.92 ⫾ 0.19 1.41 ⫾ 0.29

0.82 ⫾ 0.18 1.48 ⫾ 0.28 1.46 ⫾ 0.30

Torso With Lungs and Blood Masses

0.80 ⫾ 0.16 0.66 ⫾ 0.15 ⫺0.14 ⫾ 0.20

c៮ x

*Leads are specified by node numbers pertaining to torso sites (13). Mean values of the lead-field components c៮ x, c៮ y, and c៮ z are in relative units, scaled by c៮ I; absolute units can be restored as indicated in the text.

Lead (Nodes*)

c៮ x

Homogeneous Torso

Table 2. Mean Lead Field with Standard Deviations as Determined at 1,239 Ventricular Sites for Selected Electrocardiographic Leads

Design of ECG-Lead Transformations • Hora´ cˇ ek et al. 47

48 Journal of Electrocardiology Vol. 35 Supplement 2002 Table 3. Coefficients for Deriving Common Leads from EASI Leads as Determined from Recorded ECGs Lead

␤ˆ ES

I II III aVR aVL aVF V1 V2 V3 V4 V5 V6 V7 V8 V9 V3R V4R V 5R

0.04942 0.03050 ⫺0.01892 ⫺0.03987 0.03407 0.00573 0.60849 1.21563 0.96319 0.58888 0.23912 ⫺0.00509 ⫺0.09543 ⫺0.13050 ⫺0.15349 0.36983 0.20773 0.07644

Standard Limb Leads ␤ˆ AS ␤ˆ AI ⫺0.22834 1.06113 1.28947 ⫺0.41611 ⫺0.75827 1.17445 ⫺0.41241 ⫺1.16755 ⫺0.71894 ⫺0.16262 0.14396 0.37505 0.33708 0.26664 0.24496 ⫺0.05348 0.09856 0.23074

0.75383 ⫺0.70265 ⫺1.45648 ⫺0.02552 1.10420 ⫺1.07881 0.04633 1.09816 1.12614 0.96043 0.77168 0.31029 0.10343 0.01626 ⫺0.11560 ⫺0.30203 ⫺0.42939 ⫺0.52022

៮ R

e៮ rms

␤ˆ ES

0.952 0.901 0.803 0.947 0.845 0.851 0.972 0.978 0.897 0.820 0.893 0.986 0.962 0.902 0.830 0.945 0.928 0.938

52 71 90 44 64 77 69 93 161 177 111 33 45 57 56 65 52 34

0.04257 ⫺0.02461 ⫺0.06718 ⫺0.00898 0.05487 ⫺0.04589 0.60519 1.21233 0.95988 0.58557 0.23581 ⫺0.00840 ⫺0.09874 ⫺0.13381 ⫺0.15679 0.36652 0.20442 0.07313

Mason and Likar (M-L) Leads ៮ ␤ˆ AS ␤ˆ AI R ⫺0.23677 1.17795 1.41472 ⫺0.47059 ⫺0.82575 1.29634 ⫺0.42538 ⫺1.18051 ⫺0.73191 ⫺0.17559 0.13099 0.36208 0.32412 0.25367 0.23199 ⫺0.06645 0.08559 0.21777

0.95894 ⫺0.62063 ⫺1.57957 ⫺0.16915 1.26925 ⫺1.10010 ⫺0.04320 1.00863 1.03661 0.87091 0.68215 0.22076 0.01390 ⫺0.07327 ⫺0.20512 ⫺0.39156 ⫺0.51891 ⫺0.60975

0.957 0.912 0.725 0.959 0.818 0.816 0.974 0.979 0.901 0.806 0.876 0.979 0.929 0.823 0.748 0.953 0.947 0.959

e៮ rms 71 93 125 56 91 104 70 95 161 177 110 37 48 60 60 67 54 37

Node numbers for all leads are in Table 2; M-L leads, substitutes recommended by Mason and Likar (18) for standard limb leads. Lead voltage V is estimated as Vˆ ⫽ ␤ˆ ESVES ⫹ ␤ˆ ASVAS ⫹ ␤ˆ AIVAI, where VES, VAS, and VAI are voltages recorded in predictor leads ES, AS, and AI, respectively, of the EASI lead system; R៮ , mean correlation coefficient between recorded and EASI-derived ECGs; e៮ rms, mean root-mean-square error (rounded to the nearest microvolt) in approximating recorded ECGs by EASI-derived ECGs.

measures of the goodness of fit, R៮ and e៮ rms, for 18 common leads and for 2 configurations of the torso model. A comparison with the Mason-Likar section of Table 3 reveals that the transformation coefficients for all precordial (V1–V6), posterior (V7–V9), and right-sided (V3R–V5R) leads agree in sign, with the exception of small-valued ␤ˆ AI in leads V1 and V8; moreover, the values of all coefficients are very similar. The coefficients for the six limb leads (Mason-Likar) also agree well, showing sign disagreement only in the small values of ␤ˆ ES and in ␤ˆ AI for lead aVR. However, a comparison of the measures of fit (R៮ , e៮ rms) reveals the superiority of the coefficients derived from recorded ECGs; interestingly though, this supremacy is evident only in a minority of leads (III, aVL, and V4). The coefficients derived from potential distributions generated in the inhomogeneous torso fare better, in terms of

Table 3A. Coefficients for Deriving Orthogonal and Vessel-specific Leads from EASI Leads as Determined from Recorded ECGs Lead

␤ˆ ES

␤ˆ AS

␤ˆ AI

៮ R

e៮ rms

X Y Z LAD LCx RCA

0.10543 0.02271 ⫺0.63799 1.00813 ⫺1.15192 ⫺0.74408

⫺0.11637 1.02447 0.47157 ⫺1.18405 1.72290 2.33548

0.90516 ⫺0.83675 ⫺0.42287 1.38845 ⫺1.31026 ⫺2.19712

0.991 0.910 0.960 0.863 0.970 0.923

30 61 65 201 117 146

Leads are specified by node numbers given in Table 2; lead voltage is estimated as shown in Table 3.

goodness of fit, than those derived in the homogeneous model in some leads (III, aVL, V8, V9) but worse in others (V1–V5). Collinearity Analysis To assess the degree of collinearity among eight “independent” leads of the 12-lead ECG, we used recorded ECGs of patients whose controlled acute ischemia was studied during balloon-inflation PTCA. A Pearson correlation matrix among predictor leads I, II, V1–V6 and bipolar vessel-specific response leads LAD, LCx, and RCA for these patients shows that lead I correlates highly with leads V5 and V6, and lead II with lead V6; moreover, all precordial leads V1–V6 exhibit a strong correlation with their immediate neighbors (Table 5). The consequencies of these correlations are apparent when multiple-regression analysis, with intercept, is applied to the same data (Table 5A). Initially, all 8 “independent” leads of the 12-lead ECG were used as predictors; for that many predictors, the variance inflation factor (VIF) was greater than 10.0 in 6 of 8 predictor leads. After lead V4 was dropped from the regression equation and the analysis was repeated, the value of VIF was still excessive in five leads out of seven. Near collinearity was eventually eliminated when four out of eight “independent” leads (in the order V4, V3, V6, and V1) were dropped. The resulting regression equations, with intercept, specify transformations from predictor

Design of ECG-Lead Transformations •

Hora´ cˇ ek et al. 49

Table 4. Coefficients for Deriving Common Leads from EASI Leads as Determined from Simulated Body-Surface Potential Distributions Lead

␤ˆ ES

I II III aVR aVL aVF V1 V2 V3 V4 V5 V6 V7 V8 V9 V3R V4R V 5R

⫺0.00928 0.03630 0.04558 ⫺0.01351 ⫺0.02743 0.04094 0.62359 1.08419 0.77361 0.39896 0.13510 ⫺0.00004 ⫺0.04271 ⫺0.07236 ⫺0.09397 0.27489 0.12733 0.03439

Homogeneous Torso (M-L Leads) ៮ ␤ˆ AS ␤ˆ AI R ⫺0.25764 1.62152 1.87916 ⫺0.68194 ⫺1.06840 1.75034 ⫺0.60967 ⫺1.47199 ⫺0.87827 ⫺0.01736 0.30924 0.29326 0.24714 0.24983 0.24835 ⫺0.07684 0.05070 0.06430

0.84689 ⫺0.78803 ⫺1.63492 ⫺0.02943 1.24091 ⫺1.21148 ⫺0.00216 1.36368 1.42442 1.01145 0.71993 0.40876 0.21483 0.04336 ⫺0.14242 ⫺0.47689 ⫺0.54427 ⫺0.49099

0.950 0.911 0.565 0.957 0.561 0.778 0.968 0.972 0.849 0.717 0.844 0.976 0.906 0.758 0.611 0.939 0.940 0.951

e៮ rms

␤ˆ ES

81 143 193 66 129 165 88 115 187 205 134 50 61 70 67 78 62 44

0.00088 0.01437 0.01349 ⫺0.00763 ⫺0.00630 0.01393 0.67540 1.06178 0.74734 0.38007 0.12694 ⫺0.00007 ⫺0.04153 ⫺0.07463 ⫺0.10667 0.32642 0.15915 0.04767

Inhomogeneous Torso (M-L Leads) ៮ ␤ˆ AS ␤ˆ AI R ⫺0.14138 1.55998 1.70136 ⫺0.70930 ⫺0.92137 1.63067 ⫺0.66706 ⫺1.41354 ⫺0.88163 ⫺0.11916 0.19523 0.23682 0.23506 0.24638 0.23691 ⫺0.13365 0.03686 0.07497

0.76199 ⫺0.77274 ⫺1.53474 0.00537 1.14837 ⫺1.15374 0.04372 1.37472 1.54229 1.25609 0.89469 0.45142 0.19118 ⫺0.02613 ⫺0.24089 ⫺0.44269 ⫺0.54014 ⫺0.50256

0.952 0.912 0.597 0.956 0.674 0.784 0.971 0.971 0.807 0.686 0.834 0.973 0.907 0.777 0.733 0.946 0.947 0.954

e៮ rms 77 131 169 66 114 146 96 122 205 227 143 48 57 64 63 79 60 43

Transformation coefficients ␤ˆ ES, ␤ˆ AS, and ␤ˆ AI were calculated by fitting a multiple-regression model to 3,717 potential distributions generated by three orthogonal dipoles at 1,239 source locations in the homogeneous torso and in the inhomogeneous torso with lungs and intracavitary blood masses. R៮ and e៮ rms defined in Table 3.

leads I, II, V2, and V5 into 3 vessel-specific leads LAD, LCx, and RCA. Note that, even in the fourpredictor regression model, the coefficient ␤ˆ V5 is small enough to warrant neglecting the contribution of lead V5 to both the LCx and RCA leads; in addition, ␤ˆ I is small enough that, in deriving the LCx lead, the contribution of lead I can also be neglected.

Discussion The physical laws governing the current flow arising from the electrical activity of distributed cardiac sources in a volume conductor, such as the human body, have a mathematical form that can be translated into a discretized computer model. Except for the anisotropy of extracardiac tissues, the human-torso model used in this study considers everything that Burger et al. (19) deemed important for an understanding of the properties of electrocardiographic leads: 1) The dipolar nature of the cardiac electrical sources, 2) The distribution of these sources over the heart, 3) The shape of the boundaries of the thorax and the position of the heart relative to it, and 4) The electrical properties of the conducting tissues outside the heart (including inhomogeneities and anisotropy). Burger et al. (19) stated that:

The correct way to investigate the influence of the dipole distribution over the heart, ie, in a part of the thorax that is not at all small, is to use a model and move the artificial dipole in it. By experiments of this kind it is possible to study the effect of dipole position. Then an attempt can be made to design a system, the leads of which will not depend too much on the dipole position, so that they can be used to find the total dipole, irrespective of the distribution of its local constituents.

We did exactly that. In Table 2, we present the properties of electrocardiographic leads in terms of the mean lead field (4,20,21); thus, we take into account Burger’s dictum regarding the distribution of dipoles over the heart region. Closer examination of the lead-field values produced by an inhomogeneous model reveals that both lungs and intracavitary blood masses exert a noticeable effect, which can in some specific leads be understood in terms of relationships derived by theoretical analysis (22). With recorded ECGs from the large design set, statistical transformation coefficients for deriving the desired electrocardiographic leads from three given leads were computed by means of the leastsquares method suggested by Burger et al. (1). One has to bear in mind that such transformations (e g, those presented in Tables 3 and 3A) rest on the assumption of the dipole hypothesis. So we should ask again: Can we assume that the electrical activity of the heart takes place in a region whose dimensions are much smaller than those of the whole thorax? In the 1950s, Schmitt et al. (23), and later many others, used the mirror-pattern cancellation

50 Journal of Electrocardiology Vol. 35 Supplement 2002 Table 5. Pearson Correlation Matrix for Standard And Vessel-Specific Leads Lead

I

II

V1

V2

V3

V4

V5

V6

LAD

LCx

RCA

I II V1 V2 V3 V4 V5 V6 LAD LCx RCA

1. 0.57 ⫺0.39 ⫺0.01 0.37 0.66 0.82 0.85 0.20 0.27 0.07

1. ⫺0.34 ⫺1.11 0.24 0.55 0.71 0.82 ⫺0.04 0.50 0.53

1. 0.88 0.60 0.20 ⫺0.13 ⫺0.39 0.71 ⴚ0.88 ⫺0.76

1. 0.88 0.55 0.23 ⫺0.07 0.92 ⴚ0.87 ⴚ0.82

1. 0.88 0.64 0.36 0.95 ⫺0.61 ⫺0.63

1. 0.92 0.73 0.74 ⫺0.22 ⫺0.27

1. 0.93 0.44 0.13 0.04

1. 0.12 0.42 0.32

1. ⫺0.79 ⫺0.79

1. 0.94

1.

I, II, V1–V6 are standard leads; LAD, LCx, and RCA are vessel-specific leads for detecting acute myocardial ischemia, identified by using the same PTCA data set as in a previous study (13). Values in bold face are those for which R ⬎ 0.8.

test that involved searching for ECG patterns on the thoracic surface that were identical in shape, but opposite in sign and not necessarily of the same amplitude. The results of these cancellation experiments were generally considered to support the dipole hypothesis in qualitative terms. A more general method of testing the dipole hypothesis— which is of interest in the context of the present paper—was suggested by Becking and reported by Burger (24). According to Becking’s cancellation method, one can test the dipole hypothesis by determining how well an ECG in an arbitrary lead could be synthesized by a linear combination of ECGs in three reference leads. Becking-cancellation experiments most often utilized corrected orthogonal leads as reference leads (25); however, it is

possible to use any set of three uncorrelated reference leads and to perform a Becking-cancellation experiment. When the leads ES, AS, and AI of the EASI lead system are used as reference leads, the Becking-cancellation method—whether executed by using the analog devices of the 1950s (25) or their current computational equivalents—yields transformation coefficients from the EASI lead system to any desired lead. Burger et al. (19) cautioned that investigations of a linear relation of four leads, by analog means, in his laboratory showed significant departure from dipolarity in body-surface potential distributions. Therefore, this old issue deserves to be reexamined. With today’s digital technology, it is incomparably easier to perform the calculations proposed in alge-

Table 5A. Transformations From Standard to Vessel-Specific Leads: Sources of Collinearity Lead

␤ˆ I

␤ˆ II

␤ˆ V1

LAD LCx RCA VIF LAD LCx RCA VIF LAD LCx RCA VIF LAD LCx RCA VIF LAD LCx RCA VIF

0.09 (0.09) ⫺0.21 (⫺0.21) ⫺0.60 (⫺0.59) 6.68 0.12 ⫺0.25 ⫺0.66 6.54 0.11 ⫺0.25 ⫺0.65 6.54 ⫺0.33 ⫺0.03 ⫺0.67 5.14 ⫺0.16 0.10 ⫺0.67 3.58

⫺0.37 (⫺0.38) 0.62 (0.62) 1.19 (1.18) 4.42 ⫺0.41 0.66 1.25 4.18 ⫺0.28 0.69 1.19 3.95 ⫺0.71 0.90 1.18 2.52 ⫺0.69 0.91 1.18 2.50

0.11 (0.12) ⫺0.31 (⫺0.30) ⫺0.16 (⫺0.13) 14.32 0.09 ⫺0.28 ⫺0.11 14.10 0.15 ⫺0.34 0.00 12.55 ⫺0.33 ⫺0.25 ⫺0.00 12.09 0. 0. 0. 0.

␤ˆ V2

␤ˆ V3

␤ˆ V4

⫺0.29 (⫺0.31) 1.56 (1.57) ⫺0.67 (⫺0.69) ⫺0.57 (⫺0.59) ⫺0.39 (⫺0.37) 0.80 (0.78) 0.03 (⫺0.01) ⫺1.49 (⫺1.46) 1.27 (1.23) 92.85 283.43 353.29 ⫺0.09 1.03 0. ⫺0.82 0.24 0. ⫺0.36 ⫺0.49 0. 54.04 61.51 0. 0.69 0. 0. ⫺0.64 0. 0. ⫺0.73 0. 0. 13.33 0. 0. 0.96 0. 0. ⫺0.77 0. 0. ⫺0.72 0. 0. 10.64 0. 0. 0.78 0. 0. ⫺0.91 0. 0. ⫺0.73 0. 0. 1.47 0. 0.

␤ˆ V5 0.96 (0.98) ⫺1.11 (⫺1.09) ⫺0.42 (⫺0.38) 225.50 0.47 ⫺0.53 0.49 57.71 1.33 ⫺0.33 0.08 19.24 0.65 0.01 0.06 6.05 0.65 0.01 0.06 6.04

␤ˆ V6

␤ˆ 0

⫺0.94 (⫺0.96) ⫺4.42 (0.0) 1.06 (1.04) ⫺4.74 (0.0) 0.04 (0.0) ⫺10.80 (0.0) 56.77 0. ⫺0.74 ⫺5.24 0.82 ⫺3.77 ⫺0.34 ⫺9.26 40.26 0. ⫺1.36 ⫺5.31 0.67 ⫺3.78 ⫺0.04 ⫺9.23 28.97 0. 0. ⫺12.16 0. ⫺0.41 0. ⫺9.43 0. 0. 0. ⫺8.93 0. 2.07 0. ⫺9.42 0. 0.

LAD, LCx, and RCA are vessel-specific leads for detecting acute myocardial ischemia (13); voltage in a response lead is estimated as Vˆ ⫽ ¥␤ˆ iVi ⫹ ␤ˆ 0, where Vi are voltages (␮V) recorded in the predictor leads, ␤ˆ i are dimensionless coefficients, and ␤ˆ 0 is an intercept (␮V); VIF is a variance inflation factor indicating the presence of a collinearity problem when its value exceeds 10 (bold face). The present study used the same data set and the same method of multiple-regression analysis as a recent one (13), with one notable difference that the regression without intercept used previously [values in parenthesis are those previously published (13)] was replaced by regression with intercept for the purposes of calculating VIF, which accounts for the small differences in the coefficient values.

Design of ECG-Lead Transformations •

braic form by Becking and Burger. The results presented in Tables 3 and 3A not only provide numerical coefficients for transforming voltages recorded in predictor leads into an estimate of lead voltage V in the desired lead, but, by showing e៮ rms (in ␮V), also provide an estimate of the mean nondipolar content that cannot be represented as a linear combination of the predictor leads. Is this residual error large or small? On average, it is quite acceptable in remote leads; it is large (in absolute terms) in precordial leads V3–V5. In Table 3, different transformation coefficients are listed for leads using standard limb leads (with electrodes on the wrists and the left ankle) and for those using Mason-Likar substitutes of these leads ៮ that are smaller than 0.8 (18); the only 2 values of R belong to derived Mason-Likar leads. It should be empasized at this point that the part of Table 3 that is more relevant for clinical applications of the EASI leads is the left side (for standard limb leads), because it lists coefficients that would be most appropriate to use in clinical practice; the right side of Table 3 is provided only for investigational purposes (e g., for comparing recorded and derived leads in monitoring, stress-testing and cath-lab applications where it is not practical to record standard limb leads). Another indirect indication of the extent of nondipolar content in the given set of leads is provided by measures of collinearity. The notion of collinearity is applied to electrocardiographic leads to indicate that one of the predictor leads is a linear combination of the others, which is exactly the idea behind the Becking-cancellation method (24). We have shown an example of the application of this concept in Table 5A. Eight “independent” leads of the 12-lead ECG that we examined show strong correlations; eg, all precordial leads exhibit a strong correlation with their immediate neighbors (Table 5). When multiple-regression analysis was applied (Table 5A), the variance inflation factor red-flagged six out of eight predictor leads as being linear combinations of the others. Near collinearity was only eliminated when we dropped four out of eight “independent” leads (V4, V3, V6, and V1 in that order) from the regression analysis. The resulting regression equations (on the bottom of Table 5A) specify transformations from predictor leads I, II, V2, and V5 into three bipolar leads LAD, LCx, and RCA. In two of these leads—LCx and RCA—the contribution of lead V5 is small; this suggests that three predictor leads can adequately estimate them. On the other hand, adequate estimation of the bipolar lead LAD (with one electrode near V3) requires 4 predictor leads.

Hora´ cˇ ek et al. 51

The principal result of our previous investigation (10) of the transformation between 3 bipolar EASI leads and the 12 standard leads was that, although there are differences—in terms of goodness of fit— between the recorded and derived 12-lead ECG signals, these differences in the signals affect diagnostic classification to a surprisingly insignificant degree, as is evident from the comparison of ROC curves. Both our previous and current results with regard to goodness of fit (Tables 3 and 3A) are fairly consistent with what one might expect. These results reveal that proximity to the electrodes of the EASI lead system assures better reproducibility of a given derived lead. The derived precordial leads approximate their recorded counterparts better than do the derived limb leads (or posterior and right-sided unipolar leads). The derived lead V6 is the best because of its proximity to electrode A, and leads V1 and V2 closely follow, because of their proximity to electrode E. The derived leads V3–V5 are not reproduced as well. However, we have shown (10) that the diagnostic performance of the classifier, characterized by the ROC curves, is excellent. Thus, our previous results indicate that EASIderived 12-lead signals reproduce faithfully the diagnostic information conveyed by the actually recorded 12-lead ECGs. On this ground, the 12 standard leads derived from the EASI leads can be considered acceptable for clinical use. This can perhaps be attributed to the fact that the diagnostic information in body-surface potential distributions that are associated with acute ischemia and chronic myocardial infarction is concentrated in the dipolar component; this would not be surprising, considering the regional character of ischemia and infarction.

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