Quadrupole components of the human surface electrocardiogram

Quadrupole components surface electrocardiogram R. D. S. R.

of the human

M. Arthur, Ph.D.* B. Geselowitz, Ph.D.** A. Briller, M.D. F. Trod, Ph.D. Philadelphia, Pa.

T

he most important theoretical construct in electrocardiography is the heart vector or dipole. According to dipole theory the electrical sources in the heart, which give rise to the surface electrocardiogram (ECG), may be approximated by a dipole whose magnitude and orientation vary during the cardiac cycle. In the past several decades investigators have developed lead systems for determining the dipole components,1-4 and have attempted to evaluate the approximation provided by the dipole.5-7 Other studies have been directed to extensions of the dipole model to provide in turn a better approximation of the surface ECG and a more detailed description of the cardiac sources. In 1955 Frank reported the results of an experiment designed to give a direct evaluation of the fixed dipole model.6 Frank chose a subject and constructed a plaster cast of his torso. He then filled the torso phantom with water to provide a homo-

geneous conducting medium and placed an artificial dipole source in the fluid. In essence his experiment involved finding a location for the dipole where its magnitude and orientation could be adjusted so that surface potentials at selected points on the phantom agreed well with ECG’s measured at corresponding points on his subject at several instants during QRS. Frank found good quantitative agreement between the QRS complexes on his subject and the potentials from a fixed location dipole in the homogeneous model. This type of experiment is extremely tedious for subject and investigator. Therefore workers in a number of laboratories turned to simpler tests of the dipole hypothesis. These included the mirror pattern or cancellation study,5 a synthesis technique,7 and the method of factor analysis or principal components.8*g These experiments share the inability to provide a definitive quantitative measure of the

From

the Biomedical Engineering Division. Moore School of Electrical Engineering, and the Department of Medicine. School of Medicine, University of Pennsylvania, Philadelphia, Pa. This work was supported in part by United States Public Health Service Grants HE-08805, HE-5239. S-TOl-GM00606. and FR-15. Submitted for publication July 29, 1971. Reprint requests to: Dr. S. A. Briller, Dept. of Medicine, Cardiovascular-Pulmonary Division, Hospital of the Uni. versity of Pennsylvania, 3400 Spruce St., Philadelphia. Pa. 19104. *Present address: Box 1127. Washington University, St. Louis. MO. 63130. A portion of this material formed part of the Ph.D. dissertation in Biomedical Engineering submitted by Dr. Arthur to the University of Pennsylvania. #Present address: Pennsylvania State University. University Park. Pa. 16802.

Vol. 83, No. 5, pp. 663-677

May, 1972

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dipolarity of the heart generator. They test instead for a generator with three degrees of freedom, of which the dipole is only one special case. Therefore even if perfect mirror patterns are found, one cannot conclude unambiguously that the equivalent source is a dipole. On the other hand, failure to cancel is clearly inconsistent with a dipole, but the inconsistency cannot be analyzed. Briefly stated, all these experiments indicated that significant components beyond 3 were measurable. More recently interest has been rekindled in obtaining complete body surface isopotential maps. l”,rl If the equivalent source is well represented by a single dipole in a homogeneous conductor, then the isopotential map usually exhibits one maximum and one minimum.20 The maps, however, consistently show multiple maxima and minima at certain instants during QRS. Again no quantitation of dipolarity is obtainable from these studies. The dipole may be looked upon as the first term of an infinite series representation of a source distribution. The second term in the series is the quadrupole, the third term is the octupole, etc. This series is the multipole expansion and may be considered to be a canonical form for the equivalent generator incorporating all available surface potential information.12-l4 The present experiment was designed as part of an investigation to determine more quantitatively the nature of the human surface ECG. Specifically, the equivalent heart dipole and quadrupole were determined during the entire cardiac cycle. The experiment was designed to answer a number of questions including the following: (1) How much of the surface ECG is accounted for by the heart dipole alone? (2) What improvement occurs when the quadrupole term is added? Multipole

expansion

Consider for the moment electric sources in an unbounded homogeneous conductor. Let the conductivity of the medium be g and let the current source dipole moment per unit volume be Ji, so that Jidv is the current dipole source in a small volume dv, for example in a region of

the myocardium. If the origin of a spherical coordinate system is placed in the region containing the sources, then the potential I” at any point whose spherical coordinates are r, 8, 4 must be of the form

VL’S n 47rg n=O z=o b,,,, sin m +) P:

r f-+1

(cos 0)

(a,,

cos m $I + (1)

provided r is greater than the radius of a sphere which contains all the sources (Pz are associated Legendre polynomials). As far as measurements of potential are concerned, then, the source is completely characterized by the multipole coefficients. Note that the contribution of b,, is always 0, and therefore for each n there are 2n+l coefficients. For n= 1 the form of the potential in equation (1) is just that of a dipole source whose x, y, and z components are air, bri, and alo respectively. For n=2, the contribution to the potential falls off as the third power of the distance from the origin. The five coefficients a20, ~21, b21, a22, bzz comprise the quadrupole. In the case of the bounded volume conductor, the multipole coefficients may be determined from a knowledge of the potential V on the surface of the conductor.13 The result is expressed most compactly by introducing a complex function

where 6m0 is the Kronecker delta (1 for m=o, o for mfo) and i = 4 - 1. It may then be shown14that a,,+ib,, = jgVV$,,,*dS = J-J”*V$,,,,,dv (3) where dS is a vector element of the surface of the volume. The second equality indicates how the multipole coefficients are related to the sources, Ji, and provides the basis for characterizing the cardiac sources in a more meaningful way, given the multipole coefficients and knowledge of the electrophysiology and anatomy. In the present study, dipole and quadrupole terms were considered. From equation (3), the individual components may be written as follows: Dipole:

alo = jgVd.S, all = .fgVdS, b,~ = JgVdS,

(4)

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Qaadrupole: alo = JgV(2zdS,-xdS,-y&S,) a21 = SgVWS,+xdSz) b21= jgV(zdS,+ydS,) a22 = Pi.fgVWS,-yfZ.S,) bzz= J4~gV(ydSz+xdS,)

(5)

The multipole coefficients depend on the origin selected. The mathematical development of the theory, however, predicts a relationship between the coefficients evaluated at different origins.r5 The dipole term may be shown to be independent of the origin, while the quadrupole will change. If the origin is displaced to a point (x0, yo, z,) the new quadrupole components, indicated by primes, are a’20 = azo-2z,alo+x,all+y,bll a’21 = a21-zoall--xoa~0 b’zl = btl-zabtl--yoa10 a’22 = a~~-M(xoa~l-yobd b’22 = btz-- M(yoall+xobd

(6)

These shift equations provide a powerful tool for evaluating the consistency of the results obtained. In our experiment, two origins were used, and all coefficients were determined independently for each origin. The results were then compared with those predicted by the shift equations. MOihOd

Torso geometry. The experiment to be described was performed on a normal male subject. First it was necessary to obtain an accurate mathematical description of the configuration of the torso surface at a constant phase of respiration. All measurements were made with the subject in a supine position. A specially designed anthropometric table was used to measure the surface geometry. l6 The table permitted determination of the Cartesian coordinates of any torso point. The coordinate system is shown in Fig. 1,a. A respiration indicator was attached to the table. The tip of the arm of the indicator rested on the subject’s torso. The arm had a 1 mm. travel between two contacts. A measurement was made only when the arm was touching neither contact. The subject’s position was fixed by the location of eight permanent body features (tiny moles, nipple centers, etc.). The location of each of the eight features always fell within a 1.5 mm.3 volume as the sub-

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ject’s weight ranged from 175 to 183 pounds. On several occasions when the weight exceeded 183 pounds the permanent sites could not be positioned within this tolerance and no measurements were made. 2,800 torso points were measured from the neck to the bottom of the rib cage at a density of about one point per square centimeter. Contours were plotted at 44 levels from the base of the neck to the abdomen from the coordinates of the 2,800 measured torso sites. About 900 points were selected from these contours to describe the surface. These points were used to construct zones of triangular elements representing the regions between the contours. Fig. 1,~ shows the frontal projection of the contours utilized in the torso model with the triangular elements filled in for two zones. The dotted lines indicate caps added to close the surface. The model was described by 1,426 elements in 24 zones. The azimuthal angle of the centroid of each element in the measured region was found for a longitudinal axis through the center of the model. These angles were employed to obtain the projection of the torso model surface shown in Fig. 1,b. Appropriate tests were made to check the geometry before any solution was attempted.” Teleroentgenograms of the chest were taken with the subject in both the supine and the lateral positions with opaque markers at known locations on the torso surface. The actual coordinates of the heart center were determined after correction was made for divergence of the x-ray beam. ECG recordings. ECG’s were recorded four at a time on an FM tape recorder (Ampex FR-1100). Lead II was always included to permit subsequent time alignment. Preamplification was provided by high gain, high input impedance differential units (Tektronix type E in type 127 power supply). In order to synchronize and calibrate events on the four data channels, a 1 mv. pulse was added to a right arm (reference) electrode, common to all leads. The subject was placed on the anthropometric table in the same position used for the geometric measurements. Coordinates

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b

a

LJUIL ti,BK RPRL RNl,tlL . . . . . . . . . . . .. .. .. . .. .. .,. . -..-. .-;,.;::J . . . . . . : : :. ‘::. : : . . . . ............. . .. ..... ........... .- ....... .............. .... .............. ............... ....... _ ..... .. : .... ........... ................... ..... _ . . . : ......................................... -: ..T........ ........ ..... ............... ......... ............... . . . . :::: : .......... .... .......... . ...............:.. :. :.. : .. ........... ........ ... ..* .. .. : . ..... .I”-. ......... ....~........~ :: :: :: :: :: ;::;m$;$;’ ::...:::. :: :...:. 1: :: :: :: :: ::::::w&f:-;:: :: :: :: :: : :::.:.s::::::: :.j ..::. : ; .: . :: :: 5

1

::

**.’ :

......

. . ...... ..#.-a. . . ........................ ................ ...................... ...................... ........................ . ...................... .

: .

I,

e 8 l ....... . . .

. .

.

.

:

.

. .

.

.

. .

. .

.. : : .

MAP Fig. 1, a and b. Torso model and map. a, The origin of the coordinate system is placed at the x-ray heart center on a frontal projection of the torso model. The horizontal lines are contours defined by torso measurements. Two zones of the 1,426 triangular elements which describe the model are shown. All triangle vertices fall on one of the contours. The dashed lines indicate caps added to complete the torso surface. b, Projection of the torso surface shows the location of the centroids of the triangular model elements. The vertical position is given by the longitudinal coordinate, Y. The horizontal coordinate is the azimuthal angle about a centrally located longitudinal axis. Anatomical references are riaht midaxillary line (IZMAL), anterior midline (ANT&U.), left midgxillary line (LMAZ), and mid-back (MBK). of the site to be measured were determined. Stainless steel hex-wrench screws with 5 mm. diameter heads were used as electrodes on the body surface. An electrode was secured to each site by passing the screw shaft through the center of a rubber square which was then glued to the subject with appliance cement. At least ten successive ECG’s taken at a constant phase of respiration were recorded from each of 284 torso sites. Each set of four simultaneously recorded signals was played back at a four-to-one reduction in speed with flutter compensation. Signals were transmitted one at a time over a data phone (Bell analog data set 602A) to a remote computer facility (IBM 1710) for processing. Realignment of the four signals originally recorded simultaneously was accomplished by using the trailing edge of the calibration pulse as a fiducial mark. Thereafter the peak of the R wave in Lead II was used to align all sets. An interface amplifier was required between the phone output and the input of the analog to digital converter of the

computer system. The analog system responded (3 db.) to frequencies from 0.05 to 800 Hz. The random noise level for the whole system referred to the input was about 90 pv. peak-to-peak. The effective sampling rate was 1,600 samples per second during P and QRS and 800 samples per second during T. The high frequency response of the analog system was reduced to 250 Hz during T to avoid aliasing errors. Each sample was quantized to within x /.N. P, QRS, and T were each characterized by 190 samples, so that about 120 msec. during P and QRS, and 240 msec. during T were sampled. The levels of the 1 mv. signal and the baseline along with 190 samples of P, QRS, and T from five successive complexes were stored for each torso ECG. The conversion process was repeated five times to determine the effect of the conversion rate and noise on the procedure used to achieve time coherence. The peak of R could be determined to within 1 msec. The processing technique was found to be satisfactory because the RMS value of the probable

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Table I. Dipole and quadrupole components calculated from appropriate surface integrals of the transfer impedancesfor unit component sourcesat the x-ray heart center Source Original source alI alI= 1 bn = 1 al0= 1 a20 = 1 a21 = 1 bzj = 1 a22= 1 by! = 1

0.991 0.000 -0.001 0.000 0.000 0.000 0.000 0.000

1

bit

0.001 1.000 .-0.001 0.000 0.000 0.000 0.000 0.000

/

calculated

ato 0.000 0.001 0.994 0.000 0.000 0.000 0.000 0.000

errors associated with the processing was 0.012 mv. compared to 0.027 mv. for the probable errors associated with variations in the successive complexes. Multipole determination. Two techniques were used to compute the dipole and quadrupole. The first uses the surface integrals of equations (4) and (5) in a discrete form. For this purpose each triangular element of the torso model was represented by an outward normal at the centroid of the triangle with a magnitude equal to the area of the triangle. Since ECG’s were recorded at only 284 sites, a linear interpolation scheme was used to specify the potentials at the centroids of the 1,426 torso elements. The second technique utilizes the transfer impedances which relate each unit multipole current source to the potential it produces on the surface. Transfer impedances were calculated on a DEC PDP-6 digital computer using the “one solid angle” approximation and the deflation technique described by Barnard and co-workers.18 Surface potentials were calculated for each of three unit (1 ma. - cm.) dipole and five unit (1 ma. - cm.2) quadrupole current sources located at the x-ray heart center and at a second origin in the region of the atria (at 1.5, -3.5, and 0.6 cm. with respect to the heart center). Iteration was continued until the variation between successive solutions was less than 1 in the fifth significant digit. Each solution was verified by solving the inverse problem using equations (4) and (5). Table I shows the results for the sources at the x-ray

1

from

ato

0.021 -0.038 0.042 0.991 -0.002 -0.003 -0.002 -0.004

equations

1

(4) and

a21 1

-0.022 -0.002 -0.006 0.000 0.982 0.002 0.002 0.000

bzl

0.051 0.007 0.020 0.000 0.000 0.999 -0.001 0.002

(5)

/

a22 -0.004 -0.005 0.023 0.001 -0.001 0.001 0.987 -0.002

/

br’p

0.016 -0.009 -0.016 0.000 0.000 0.000 -0.001 0.990

heart center. Similar results were obtained at the second origin. The desired condition -i.e., the inverse solution yielding unity on the diagonal and zero elsewhere-is very nearly met for all components. Once the eight transfer impedance solutions were available, the 3 dipole and 5 quadrupole components were obtained from the electrocardiographic data at 284 sites utilizing the criterion of least squares. Torso sites at which the electrocardiograms were recorded did not necessarily correspond to sites (centroids) where computer solutions of the transfer impedance had been obtained. In these cases linear interpolation was used to obtain the transfer impedance. Results

Transfer impedance. Figs. 2 and 3 show the transfer impedances for sources at the x-ray heart center in the form of isoimpedance contours, i.e., isopotential contours generated by unit multipoles. Note that the dipole maps have one maximum and one minimum; the quadrupole maps have two maxima and two minima. A complete listing of the transfer impedances, as well as a detailed description of the torso model, is available” but is not included here. Multipole components. Results obtained with the first technique (surface integration) were unsatisfactory and will not be presented here in detail. The major difficulty with this technique is that it requires integration of the potential over the entire body surface. In our torso model the

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all

bll JO

DEW

ft)tRL

I’

RNl,NL

VRL

VK

Fig. 2. Dipole transfer impedances. Isoimpedance contours relate the three orthogonal components of a dipole current source at the x-ray heart center (H) to the resulting surface potentials. The + and signs indicate maxima and minima, respectively. Contours calculated for a torso resistivity of 500 ohm-cm. are: X(-0.25, -0.12, 0.00, 0.12, 0.25, 0.37, 0.50, and 0.75 ohm/cm.); Y (-0.25, -0.12, 0.00, 0.12, 0.37, 0.50, 0.62, and 0.75 ohm/cm.); Z (-1.0, -0.50, -0.25, -0.12, 0.00, and 0.12 ohm/cm.).

cephalic and caudal parts are capped (truncated) to form a closed surface. Potentials on the caps, which do not correspond to a real surface of the body, were estimated by interpolation. To test the validity of this approach, a different truncation (cap) was used. The solution changed significantly especially for the quadrupole components. We therefore concluded that the attempt to utilize interpolated potentials on the top and bottom caps in evaluating the surface integrals led

to significant errors, and that surface integration did not provide a practical basis for obtaining the multipolar components. Figs. 4 and 5 show dipole and quadrupole components of the multipole source at the x-ray heart center obtained from the second technique, which utilizes transfer impedance data. All 284 ECG’s were utilized for solution during QRS. The components during P and T waves were computed to fit 212 and 256 of the ECG’s respectively, since several waveforms could not be used because of a poor signal to noise ratio. From equations (3), (4), and (S), it is evident that the numerical value of each multipole component is proportional to g, the conductivity of the volume conductor. In the present case g-l was assigned the value of 500 ohm-cm. Other choices for g will lead to different scale factors; waveforms and relative amplitudes, however, will be unaffected. Three tests were made to determine the accuracy of these solutions. In each test the multipole components were computed in two different ways, which ideally should give the same solution. As a measure of error, the peak difference between the two solutions is reported as a percentage of the peak amplitude of the component. First the question of whether a sufficient number of ECG’s was used was investigated by selecting various subsets of the 284 recordings of the QRS complex to obtain solutions for comparison. As an example, when 33 EC(Ys were chosen (about g the density of the full set), peak differences were less than 4 per cent for the dipole components and less than 8 per cent for the quadrupole components. It was therefore assumed that additional measurements beyond 284 would not have affected the results. The second test concerned the use in the present experiment of only the dipole and quadrupole terms of the infinite series multipole expansion. To estimate the effect of this truncation, the dipole term was computed, first ignoring and then including the transfer coefficients for the quadrupole term in the least squares solution. Figure 6,n shows the two solutions at the heart center together with the difference between them during QRS. The largest

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Fig. 3. Quadrupole transfer impedances. Isoimpedance contours relate the five orthogonal components of a quadrupole current source at the x-ray heart center (H) to the resulting surface potentials. The + and - signs indicate maxima and minima, respectively. Contours calculated for a torso resistivity of 500 ohm-cm. are: A (-0.12, -0.03, 0.00, 0.06, and 0.37 ohm/cm.*); B (-0.75, -0.37, 0.00, 0.12, 0.25, 0.50 and 0.75 ohm/cm.e); C (-0.07, -0.03, -0.01, 0.00, 0.02, 0.05, and 0.09 ohm/cm.2); D (-0.25, -0.12, -0.06, 0.00, 0.12, and 0.30 ohm/cm.e); E (-0.04, -0.03, -0.02, -0.01, 0.00, 0.01, and 0.02 ohm/cm.*).

peak difference observed was 10 per cent. From mathematical theory, the dipole component should not change when the origin is shifted. This fact underlies the third test. Figure 6,b shows the dipole components at the heart center and at the second origin in the atria1 region. The typical peak difference for a dipole at these two origins is 3 per cent of the peak amplitude of the average dipole moment. The quadrupoie components, on the other hand, depend upon location and therefore must be compared using the shift relations given in equation (6). The solutions were shifted from the two origins utilized in the experiment to the midpoint between them. Ideally the results should be identical. Fig. 7 shows the shifted components and the difference in the shifted components

from the average during QRS. The typical peak difference for the quadrupole components is 13 per cent. Reconstructed ECG’s. We turn now to the question of how much of the surface ECG is accounted for by the dipole alone and by the dipole together with the quadrupole. The contribution of the dipole and quadrupole to any surface lead can be calculated from knowledge of the transfer coefficients. For example, the contribution of the dipole to Lead i is given by = TxiX +

TyiY

f TziZ

(7)

where TX{, TY~, and Tz~ are the appropriate transfer coefficients for the X, Y and 2 components of the dipole. Similarly, for the quadrupole contribution,

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a10 b-4 15 msec

P

QRS

T

Fig. 4. Dipole components at the x-ray heart center. Transfer impedances were utilized to match surface ECG’s on a least squares error basis at 212, 284, and 256 sites during P, QRS, and T, respectively. The gaps between the time segments processed during P, QRS, and T are consistent with the time scale shown. Each segment contains 190 intervals. The ordinate is scaled to indicate sources in a torso with a resistivity of 500 ohm-cm. VQi

=

Ta&zo

= TA,A

i- TsiCh + Tcibsl + Tu;~nn f TEib22 + TBiB + TciC + TDiD + TEG (8)

where the symbols A, B, C, D, E have been associated with azo, a21, b21, a22, and b22, respectively. ECG’s were reconstructed at 36 torso sites during QRS. Fig. 8 shows the dipole and dipole plus quadrupole reconstruction, i.e., Voi and VD; + VQ; as well as the measured ECG at six of those sites for sources at the x-ray heart center. The differences between the reconstructed and measured quantities are also depicted. At most sites the addition of the quadrupole significantly improved the reconstruction and reduced the error to almost zero throughout QRS. Fig. 8,f depicts the worst case among the 36. The dipole and quadrupole components were also utilized to compute the potential at all the centroids of the torso model at ten instants during P, QRS, and T. These reconstructed potentials were then used to generate equipotential plots.

The projection scheme used is similar to a Mercator projection. Each point on the torso surface is identified by its azimuthal angle and its distance along a vertical (head to foot) axis. The torso surface is then projected using the azimuthal angle for the abscissa and the vertical position for the ordinate. Greatest distortion is introduced at the upper part of the map which includes the shoulder region. None of the isopotential maps showed multiple maxima or minima. Plots from three instants during QRS are shown in Fig. 9 along with maps constructed from the measured electrocardiograms. For all instants of time examined during QRS equipotential maps synthesized from dipole plus quadrupole were closer to measured maps than when dipole information alone was employed. A quantitative evaluation of the dipole and quadrupole contributions to the surface electrocardiogram can be found by evaluating the root mean square (RMS) reconstruction error. The mean square errors at each instant are found as follows

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I

b22

2

macm

2 I-+

P

QRS

15msec

T

Fig. 5. Quadrupole components at the x-ray heart center. Transfer impedances were utilized to match surface ECG’s on a least squares error basis at 212, 284, and 256 sites during P, QRS, and T, respectively. The gaps between the time segments processed during P, QRS, and T are consistent with the time scale shown. Each segment contains 190 intervals. The ordinate is scaled to indicate sources in a torso with a resistivity of 500 ohm-cm.

(9) 1V D+Q= $ $= 1 z

(Vi- VDi-

VQi)’

where Vi is the measured electrocardiogram and the summation is over all sites where

recordings were obtained. Note that it is these errors which are minimized under the least squares error criterion, The RMS value is simply the square root of the mean square. To provide a reference for evaluating the significance of these errors, the RMS value of the measured surface electrocardiograms was also computed.

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b

a

Z H

15 msec

LOCATION

TRUNCATION

Fig. 6, a and b. Dipole consistency during QRS. a, Least squares dipole components calculated for a multipole expansion containing only a dipole term compared to components calculated for an expansion containing both dipole and quadrupole terms. b, Comparison of dipole components calculated for sources at the x-ray heart center with those calculated for sources at a second origin in the atria. The difference between the components is also included in each case. The mean square of the measured electrocardiograms at each instant is given by N j72 = 1_ z N i=l

V?

(10)

Potentials were reconstructed at all the torso sites where ECG’s had been measured. Although not all were examined individually, they were employed to calculate the RMS value of the reconstruction error. The RMS dipole and dipole plus quadrupole errors, along with the RMS probable error of the data is shown in Fig. 10 for sources at the x-ray heart center

(origin A) and at a location in the atria (origin 23). In order to characterize the reconstruction errors by a single quantity, the RMS value of each of the curves in Fig. 10 was computed. Table II contains these spatiql-temporal RMS values, as well as the corresponding value for the measured ECG’s. The numbers in parentheses show the reconstruction and probable errors as a percentage of the RMS of the measured ECG’s. Discussion

SchubertI determined dipole and quadrupole components of a normal human

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b

a

I-I

15 msec

I

2 macm 2

Fig. 7. Quadrupole consistency during QRS. Quadrupole components calculated for sources at the x-ray heart center compared to components calculated for sources at a second origin in the atria. The components are shifted to the midooint between the two origins for comparison. The difference between the shifted components and the average value is also included in each case. -

ECG and reported an improved fit with dipole plus quadrupole as compared to dipole alone. He obtained the transfer impedance with a probe in a torso tank model and limited his study to the QRS complex in 20 leads. In the present experiment a computer solution of the boundary value problem was employed, P, QRS, and T waves were included, and 284 leads were recorded and analyzed. Schubert’s results show substantial discrepancies for the dipoles evaluated at two origins as well as for the dipole calculated with and without transfer impedances for the quadrupole. In the present experiment no significant discrepancy of this sort appeared (Fig. 6). The use of two origins to obtain independent solutions for the multipole components provides a check on the consistency

of the results. This feature of multipole theory in the determination of cardiac sources from the surface ECG is important and useful and appears to be unique. The quadrupole contribution decreased reconstruction errors in ECG’s over most of the torso. Even when the error was not negligible, the quadrupole addition brought out details of the ECG not apparent with the dipole alone (Figure 8,~). An examination of equipotential maps throughout QRS revealed none with multiple maxima or minima. Presumably the quadrupole would be more significant in subjects with such surface distributions. Both ECG’s and equipotential maps showed that the quadrupole provided the greatest improvement at sites along a line through the precordial region from the left

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a

b

DIPOLE D8Q ACTUAL ERRORS

I Imv I

H

15 msec

Fig. 8, a through f. Reconstructed ECG’s during QRS. Surface ECG’s calculated from dipole and dipole plus quadrupole components along with the measured ECG and the differences between reconstructed and measured values at six torso sites. a, Left anterior axillary line in the seventh intercostal space, b, Precordial site V4, c, Third intercostal space above the left nipple, d, Right anterior axillary line on the fourth rib, e, Right mid-clavicular line on the eighth rib, f, Right dorsal site at the level of the xiphoid process.

Table II. Spatial-temporal RMS value in millivolts of measured ECG’s, dipole and dipole plus puadrupole reconstruction errors, and the probable error of the data for multipole sources at two origins. Figures in parenthesesshow percentage of value for measured ECG’s. Origin

A (x-ray

P

/

heart

center)

Origin

B (in atria1

region)

P

1

1

Variables

Measured Dipole

ECG’s reconstruction

1

T

0.052 ‘fygf

0.388 y;($

0.267 ‘y’

(46%) 0.015 (29%) 0.014 (27%)

b’.:?? (14%) 0.032

0.048

error

Dipole plus quadrupole reconstruction error Probable error of measurement

QRS

(22%)

(8%)

(18%) 0.019 (7%)

0.052 (100%) 0.021 tY2 (29%) 0.014 (27%)

QRS

T0.267 (100%) 0.064 (24%) 0.049

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b

DIPOLE

D+Q

MEASURED

-P40 MSEC

4

73 MSEC

Fig. 9. Reconstructed equipotential maps during QRS. Surface potentials computed from dipole components, dipole plus quadrupole components, and map constructed from measured electrocardiograms are shown at the three instants indicated on the Lead II waveforms.

shoulder to the right side. In an investigation utilizing factor analysis to study ECG’s of normal subjects, Horan and coworkers reported that principal factors four through eight made the greatest contribution in this same region.9 These observations would lend support to a conjecture that the first two principal factors tend to be closely correlated with the two dipole components in the plane of the loop. A definite answer, however, would require a concurrent study of principal factors and multipole components on a series of subjects. Horan, Flowers, and Brodyg used RMS values of the errors to estimate the principal factor content of the ECG. For example they report that for three factors the RMS error is 14.6 per cent of the RMS value of the electrocardiograms. The per cent resynthesis is therefore 85.4 per cent. The corresponding per cent resynthesis reported for 8 factors is 99.7 per cent.

Per cent resynthesis for dipole and dipole plus quadrupole in the present experiment can be obtained in a similar way from the data presented in Table II. For example, for the QRS complex (origin A), the per cent resynthesis for dipole alone is 77 per cent, while for dipole plus quadrupole it is 86 per cent. Note that in the latter case the RMS error (14 per cent) approaches the probable error of the data (8 per cent). In the case of the P wave the dipole plus quadrupole error is very close to the probable error, which is much higher on a percentage basis than for QRS or T because the amplitude of P is smaller. Dipole reconstruction errors and dipole plus quadrupole reconstruction errors are larger than the corresponding errors reported by Horan, Flowers, and Brodyg for 3 factors and 8 factors, respectively. This result is to be anticipated since the three factor result is the best that can be obtained with a generator possessing three

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Fig. IO, u throughf. Root mean square reconstruction errors. RMS values of the dipole and dipole plus quadrupole reconstruction errors are shown during P, QRS, and T, together with the probable error of the data. In each case the upper trace is the error associated with dipole alone, the middle trace is the error associated with dipole plus quadrupole, and the lowest trace is the probable error of the data. a, c, and e, Sources at the x-ray heart center. b, d, and f, Sources at an origin in the atrial region.

degrees of freedom. Since the dipole is a special case of such a generator, the reconstruction error cannot be less than that obtained with the first three principal factors. A similar statement may be made for a comparison of dipole plus quadrupole with 8 principal factors. From Fig. 10 and Table II it is apparent that during QRS the dipole alone provides a better fit at origin A, the heart center. The same observation is true to a lesser extent for the T wave. During P, however, the dipole located at origin B in the region of the atria provides a better fit. This result is clearly consistent with location of

electric sources in the region of the ventricles during the inscription of QRS, and in the atria during P. Table II shows that the reconstruction errors for dipole plus quadrupole are essentially independent of the origin. The results presented here are based on the arbitrary assumption of a homogeneous torso. One could also work with equivalent sources in a particular inhomogeneous volume conductor. Since our generator uses multipoles in a homogeneous torso, any attempt to relate these equivalent sources to the actual distribution of myocardial activity would require an accounting of inhomogeneities.

Volume 83 Number5

Quadrupole components of human surface KG

Summary

The equivalent heart dipole and quadrupole were determined on a normal male subject for two origins in the heart region. Determinations were based on a detailed measurement of the torso surface geometry, a digital computer solution of transfer impedances relating unit dipole and quadrupole components to surface potentials they generate, and the measurement of 284 ECG’s. Addition of the quadrupole contribution gave a better fit to the surface ECG. The RR% error during QRS was 0.091 mv. for dipole alone and 0.054 mv. for dipole plus quadrupole representing respectively 23 per cent and 14 per cent of the total RMS value of the recorded ECG’s. Similar values for P were 0.024 mv. (46 per cent) for dipole and 0.015 mv. (29 per cent) for dipole plus quadrupole. For T the results were 0.058 mv. (22 per cent) and 0.048 mv. (18 per cent). We are deeply indebted to Drs. H. L. Gelernter and J. C. Swihart and to Mrs. M. A. K. Angel1 of the International Business Machines Corporation, Yorktown Heights, N. Y., for their conceptual and computational support of this work during its earliest moments. REFERENCES Becking, A. G., Burger, H. C., and van Milaan, J. B.: A universal vectorcardiograph, Br. Heart J. 12:339, 1950. 2. Frank, E.: An accurate clinically practical system for spatial vectorcardiography, Circulation 13:737, 19.56. 3. Schmitt, 0. H., and Simonson, E.: The present status of vectorcardiography, Arch. Intern. Med. 96:574, 1955. 4. McFee, R., and Parungao, A.: An orthogonal lead system for clinical electrocardiography, ,4h1. HEART T.62:93. 1961. O.-H., Ledine, R. B., and Simonson, 5. Schmitt, E.: Electrocardiographic mirror pattern studies AM. HEART J. 45:416, 1953. 6. Frank, E.: Absolute quantitative comparison of instantaneous QRS equipotentials on a normal subject with dipole potentials on a homogeneous torso model, Circ. Res. 3:243, 1955.

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Okada, R. H., Langner, P. H., Jr., and Briller, potentials from S. A.: Synthesis of precordial the SVEC III vectorcardiographic system, Circ. Res. 7:185, 1959. Scher. A. M.. Younc. A. C.. and Meredith, W. n/i.: Fact& analy& of el&trocardiograms. Test of electrocardiography theory: Normal leads, Circ. Res. 8:519, 1960. Horan, L. G., Flowers, N. C., and Brody, D. A.: Principal factor waveforms of the thoracic QRS complex, Circ. Res. 15:131, 1964. Taccardi, B.: Distribution of heart potentials on the thoracic surface of normal human subjects, Circ. Res. 12:341, 1963. Spach, M. S., Silberberg, W. P., Boineau, J. P., Barr, R. C., Long, E. C., Gallie, T. M., Gabor, M. A., and Wallace, A. G.: Body surface isopotential maps in normal children, ages 8 to 14 years, AM. HEART J. 72:640, 1966. Yeh, G. C. K., Martinek, J., and de Beaumont, H.: Multipole representation of current generators in volume conductors, Bull. Math. Biophys. 20:203, 1958. Geselowitz, D. B.: Multipole representation for an equivalent cardiac generator, Proc. IRE 48:75, 1960. Brody, D. A., Bradshaw, J. C., and Evans, J. W.: A theoretical basis for determining heartlead relationships of the equivalent cardiac multipole, IRE Trans. Biomedical Electronics BME-8:139, 1961. Geselowitz, D. B.: Two theorems concerning the quadrupole applicable to electrocardiography, IEEE Trans. on Biomedical Engineering BME-12:164, 196.5. Briller, S. A., Gelernter, H., Geselowitz, D. B., and Swihart, J. C.: The human quadrupole image surEace, Circulation 28:694, 1963. Arthur, R. M.: Evaluation and use of a human dipole plus quadrupole equivalent cardiac venerator. Ph.D. dissertation. Philadelohia. ?968, University of Pennsylvania. . Barnard, A. C. I,., Duck, I. M., Lynn, M. S., and Timlake, W. P.: The application of electromagnetic theory to electrocardiography. II Numerical solutions of the integral equations, Biophys. J. 7:463, 1967. Schubert, R. W.: An experimental study of the multipole series that represents the human electrocardiogram, IEEE Trans. on Biomedical Engineering BME-15:303, 1968. Rush, S.: Inhomogeneities as a cause of multiple peaks of heart potential on the body surface: Theoretical studies, IEEE Trans. on Biomedical Engineering BME-18:115, 1971.