Download Principal Factor Waveforms of the Thoracic QRS Complex

Principal Factor Waveforms of the
Thoracic QRS Complex
By Leo G. Horan, M.D., Nancy C. Flowers, M.D., and
Daniel A. Brody, M.D.
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• Concern has been expressed repeatedly
in the electrocardiographic literature as to
whether the three leads of a spatial vectorcardiogram contain more or less information
than that found in the standard twelve-lead
electrocardiogram.1"4 Frequently the adequacy
of the report of the twelve-lead electrocardiogram has also been questioned and addition
of one or more exploratory leads has been
recommended.5"7 These questions are two
aspects of the more fundamental problem:
What is the minimum number of leads which
can contain all of the electrocardiographic
information available on the body surface?
A theoretical approach to the answer would
be to determine the minimum number of
unique building blocks capable of synthesizing any waveform detectable on the body surface. For example, by means of principal factor analysis Scher et al. derived a minimum
number of mathematically uncorrelated QRS
waveforms found in an assortment of electrocardiographic leads varying from 8 to 32 in
number.8 A practical answer would be to find
those body sites especially sensitive to these
elemental waveforms so that placement of
leads would insure adequate acquisition of
information. In the present study we have extended the use of factor analysis to between
150 and 180 exploratory leads systematically
spaced over the thoracic surface. This was
done in the hope both of deriving a clearer
appreciation for the total scope of informaFrom the Section of Cardiology, Department of
Medicine, University of Tennessee, Memphis, Tennessee.
Supported by Grants HE-D1362-11 and 5-K6-HE14, 032-02 from the National Institutes of Health,
U. S. Public Health Service, and by a grant from
the Tennessee Heart Association.
Received for publication January 20, 1964.
Circulation Research, Vol. XV, August 1964
tion available and of obtaining indications as
to whether the topography of lead sensitivity
is ascertainable in individual subjects.
Factor analysis was chosen as a mathematical means of comparing each QRS complex with all the others. This permitted first
the removal, from the entire population of
QRS waveforms, of the most common basic
QRS pattern. Then similarly, the second most
frequently appearing pattern was removed
and the process was repeated until all of the
distinctive waveform information was accounted for. The net result was a consolidation of
the information in the large number of QRS
complexes into a considerably smaller number of individually dissimilar waveforms.
Methods
Two subjects were chosen for study. Systematic
spacing of 153 unipolar leads over the surface
of the chest of a dog for equipotential mapping
has been previously described.9 A 17-kg male dog
was placed in normal standing position in a
Pavlov stand, was supported by straps under the
trunk, and was anesthetized with sodium pentobarbital (30 mgAg I V )- A Crass kymograph C4J
camera was employed to photograph the oscilloscopic display of four simultaneous leads on
film moving at the rate of 250 mm/sec. A similar
set of QRS waveforms was measured from the
recordings of 180 electrode sites (with 5.0 rather
than 3.0 cm spacing and with eight simultaneous
channels *) for a normal healthy 37-year-old man
in the sitting position (fig. 1A). For both subjects
attention was given to the phase of respiration;
records were obtained between the end of expiratory movement and the beginning of inspiration. All records were examined carefully to
ascertain the time of earliest departure from the
base line of any QRS complex with regard to the
control lead. This instant was marked on a
* Tektronix 565 oscilloscope with two 3A74 fourtrace amplifier plug-in units and eight 122 low-level
preamplifiers.
131
132
HORAN, FLOWERS, BRODY
*
12
tit
*
13
25
FIGURE 1A
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A representative set of simultaneously recorded QRS waveforms from a normal human subject
and a computer plot of their digitized representations. Original waveforms were recorded on
film moving at 250 mm/sec. The plot was made by machine interpolation at 2-msec intervals
between the 10-msec data points obtained by direct measurement of an enlarged image from
the original film. A loss of precision to one significant figure has occurred in the plot because
in this figure the vertical scale has been kept small to make the plot visually comparable to
the original record. In subsequent illustrations the vertical scale has been increased so that the
curves are smoother and approach clinically familiar configurations.
permanent tracing of the control lead Vf on crosssection paper as time zero or the onset of ventricular depolarization. Optically enlarged images
of the constant control lead (Vf) were successively superimposed onto the permanent tracing
of the control lead. In this way the entire set of
QRS waveforms for each subject was matched
temporally. Measurements of potential were then
made of all QRS complexes at five-msec intervals
for the dog and ten-msec intervals for the man.
The total duration of QRS complex was about 36
msec in the canine subject and about 87
msec in the human. (About indicates difficulty in
measuring the exact duration of the QRS complex
because very gentle terminal slopes of the QRS
and slight displacement of the ST-segment combined to obscure the exact endpoint in the critical
anterior midchest leads.)
From the measurements at five- or ten-msec
intervals, maps of instantaneous isopotential
distribution were constructed by bilinear interpolation and plotting with the aid of a digital
computer (fig. 2). 9 Because of the multipolar
patterns in evidence on these maps (as will be
discussed later) we turned to principal factor
analysis to determine if more components than
those necessary to describe a dipole were easily
detectable. The general plan for processing the
data is indicated in the flow chart (fig. IB).
Digitized forms of each QRS complex at each
electrode site were constructed by interpolation
between the measured time values (fig. 1A). For
each four successive points a third degree curve
was fitted and the central interval retained for
calculation of interpolated points (to each msec
in the dog and to every second msec in the man).
This interpolative procedure permitted the development of digitized QRS waveforms from seven
to eight measured points. Representative interpolated waveforms were superimposable upon
those digitized at 1-msec intervals from the
original with less than 1% error. These surface
potential data were organized as a large matrix
consisting of 153 or 180 leads (rows) with 40 or
45 successive potential values in time (columns).
Such a matrix of data of potential values was designated matrix V. In terms of matrix algebra the
equation
(180X180) (180X45)
(180x45) (1)
A
X
=
V
states that the potential values throughout the
QRS and over the surface of the body may be resynthesized as the product of a set of factors, X,
and their distributive coefficients, A. If each of
these factors is unique in that none contains information found in the others, the equation represents an orthogonal transformation and the
factors may be designated as principal.10 The
CircmUtion Ruurcb,
Vol. XV, August 1964
133
PRINCIPAL FACTOR WAVEFORMS: QRS COMPLEX
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B
FIGURE IB
A flow-chart presenting diagrammatically the Iwndling of data in the study of principal factor
waveforms of the QRS complex in a human subject. The boxes containing labels in capital
letters indicate major processes done by the digital computer. From 180 systematically spaced
points on the thoracic surface (a) high-fidelity recordings of the QRS complex (b) were obtained
and, after time-justification, were measured at 10-msec intervals. These digital measurements
(c) were fed into the digital computer which fitted third degree curves between the data points
so that the digitized representation of the QRS complex (d) closely simulated one obtained by
sampling at 2-msec intervals. All 180 digitized QRS waveforms were then factored by the
method of principal factor analysis. Eight component waveforms or principal factors (e) were
found. Each digitized QRS waveform was then re-examined for its content of each of the principal factors. From this assay it was possible to determine the degree to which successive addition of factors satisfactorily reproduced the original waveform at any site (f) and to determine
the distribution over the subject's chest of each of the principal factors (g).
principal factors to be determined are at the
outset represented as an unknown number of
waveforms (rows) with successive temporal
variation in amplitude (columns). The coefficients
for 180 electrode sites (rows) will specify the
proportions contributed by the respective principal factors to each waveform at each site (columns). Details are provided in the Appendix.
Upon diagonalization, the effective size of the
transforming or eigenvector matrix, A"1, and
consequently the number of significant principal
factors may be greatly reduced. In this instance
because the eigenvalues beyond the eighth approach zero, only the first eight rows of A-1 need
be considered.
Therefore,
(8X180) (180X45) ( 3 X 4 5 )
(2)
V
= X
A-1
(In actual practice the size of the memory of
Circulation Rtiurcb,
Vol. XV, A*g*u 1964
the digital computer was limiting. Therefore,
factoring was done in successive stages, i.e., 20
waveforms at a time. Subsequently these factors
were factored by combining sets of 8 factors into
a group of 16 and so on pyramidally until the
final factors obtained represented the principal
factors of the whole original matrix.)
Thus, as indicated in the Appendix, the eigenvector matrix is the inverse of the matrix of
coefficients for each principal factor necessary to
resynthesize the recorded waveform found at
each electrode site. The resynthesis was then
performed by substitution into equation 1 in a
stepwise fashion in order to examine the contribution of each factor as it was added.
The root mean square of the deviation of the
original waveform from the base line was accepted as an estimate of total amount of information present in the waveform, and the root mean
HORAN, FLOWERS, BRODY
134
.-175 •._
HUUAN
gu
-100.".,.
-1BO
I-
075
OS
FIGURE 2
An isopotential map of the chest of a normal human subject at 40 msec after the onset of ventricular depolarization. Original electrode sites are indicated by intersection of grid-lines. The
isopotential contour lines are spaced at intervals of 025 mv. Note region of positioity over the
left anterior and lateral chest where the maximum divides at times into two submaxima.
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square (RMS) of the deviation of a resynthesized
waveform from the original was accepted as an
estimate of the error or remainder, after any
given number of factors made their contribution
to the resynthesis. Thus the appraisal of the
relative inadequacy of resynthesis was developed
by using a ratio of these HMS values (deviation of
resynthesized from original/original's deviation
from base line) and converting to percentage.
The difference between per cent of error and
100X was used as a convenient, though mathematically imprecise, index of the adequacy of resynthesis (fig. 3 ) .
Results
COMPARISON BETWEEN MAN AND DOG OF THE
SURFACE DISTRIBUTION OF POTENTIAL DURING
VENTRICULAR DEPOLARIZATION
Information =
error
= \JId
% r.jynihesls = '"formation-error
Information
x 1 O
Q
FIGURE 3
Diagrammatic representation of estimate of error in
the process of resynthesis of waveforms. The solid-line
curve represents the original or known waveform to be
resynthesized. The broken line represents an attempted
resynthesis. The deviation d between the real and calculated curves for a given interval of observation
(2 msec) is the instantaneous error, and the deviation
D from the base line represents the instantaneous information in the original recording. The relative error
Figure 2 presents a map of potential distribution of the thoracic surface of a man
at 40 msec after the onset of ventricular depolarization. The extremes of intensity of surface potential were more limited for the man
than for the dog, varying between ±2.5 mv
in the man as compared to ± 4 mv in the
dog.9 Duration of the QRS complex in the
for the whole waveform may be estimated by the ratio
of root sum squares, i.e., per cent error = y ~%&*/
X 100. Similarly, the adequacy of resynthesis
V
in per cent may be estimated by dividing the difference between "information" and "error" by the desired "information" and multiplying the result by 100.
CircmUtion Ristxrch, Vol. XV, Amgmst 1964
PRINCIPAL FACTOR WAVEFORMS: QRS COMPLEX
135
human subject was slightly greater than twice
that in the dog. The principal evidence of
multipolar activity occurred between 40 and
50 msec after onset in the man as seen in
figure 2 with again strong suggestion of multiple islands of activity late in the QRS complex
at about the time of the right precordial R
prime.
THE PRINCIPAL FACTORS AND THE
PROBLEM OF RESYNTHESIS
PRINCIPAL FACTORS
OF
QRS
WAVEFORMS
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FIGURE 4
Digital computer plot of the configuration of the eight
principal factors derived by factor analysis of 180 QRS
waveforms systematically spaced over the thoracic
surface of a normal man. The digits of the plot have
been joined by a bold black line. Note the relatively
greater magnitude of factors one and two as compared with the remainder and the resemblance of factor one to the common configuration of lead Ve. See
text for details.
Circulstion Rtjurcb,
Vol. XV, August 1964
Figure 4 shows the eight final principal factors for the human subject and figure 5 illustrates the role of each in the progressive
resynthesis of a selected set of QRS complexes
recorded from the anterior chest. The effect
of reconstructing the original pattern of surface distribution for given instants after the
onset of ventricular depolarization by successive addition of the contributions of each
of the principal factors was also examined.
Gross distinctions between stages of resynthesis were less obvious than with individual waveforms. Usually upon addition of
the third principal factor, very close resemblance to the original waveform became
apparent. However, a substantial percentage
of information found in the final waveform
remained in factors four through seven in the
dog and four through eight in man (fig. 5).
The eigenvalues corresponding to the principal factors tapered very sharply with an almost abrupt cutoff or diminution in value on
the order of 10"3 between seven and eight in
the dog. In the man, an abrupt diminution on
the order of 10"4 occurred after the eighth
eigenvalue. The abrupt cutoff very probably
reflected a limitation by the number of raw
data points obtained for each set of QRS waveforms. For example, five sets of eight simultaneous leads (I, Vf, and Vi through Vo) for
different human subjects were digitized at
2.5-msec intervals. Principal factors for each
of these sets were determined and showed
heavier concentration of information in factors
one and two. Then the points intervening between the 10-msec intervals were discarded
and new points at 2-msec intervals were determined by interpolation, i.e., there were
now eight original data points. The resulting
interpolated waveforms were factored. The
contribution of the respective eigenvalues did
HORAN, FLOWERS, BRODY
136
D5
E 12
2 5 msec
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FIGURE 5
F 17
not vary more than 2%. When interpolation
was made from only six points (using a 12.5msec sampling interval) there was little difference in eigenvalues until after the sixth
when again an abrupt cutoff appeared. Because eigenvalues are estimates of variance
these findings indicate that in these normal
subjects evidence was not exaggerated or suppressed by interpolation. Similarly, the number of sampling points was limiting only in
that it did not permit the detection of effect
of principal factors of higher order than the
number of sampling points. As noted in table
Digital computer plots of the resynthesis of three QRS
waveforms showing the effect of successive addition
of the contributions of the eight •principal factors. For
orientation it may be noted that familiar clinical leads
which are near the thoracic sites from which the
waveforms were detected are respectively (a) Vr, (b)
V3, and (c) Ve. The number of each 2-msec interval
represents the cumulative effect of all factors up to
and including the numbered factor. Thus, the ones
show only the effect of principal factor one, but the
threes report the resynthesized waveform after addition of principal factors one, two, and three. Complete resynthesis of the original wave is achieved on
each line after addition of factor eight (solid line).
Whenever the effect of two or more factors resulted
in a coincident point the computer was instructed to
report only the final number. Thus in many instances,
for example, an eight may have replaced a preceding
six and seven.
1, the rough estimate of the adequacy of resynthesis obtained by subtracting the more
precise estimate of error from 10035 (fig. 3)
was well predicted by the square root of
the appropriate eigenvalue.
SURFACE DISTRIBUTION OF THE PRINCIPAL FACTORS
Certain regions of the thorax showed a relatively high percentage of contribution to the
waveform from one or more principal factors
as compared with the others. Figure 6 shows
the surface distribution of three of the eight
principal factors for the man. Note the inCircmUtiou Rtsttrcb,
Vol. XV, August 1964
PRINCIPAL FACTOR WAVEFORMS: QRS COMPLEX
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tensity of factors one and two over the anterior
chest and their resemblance to the patterns of
distribution of two orthogonally oriented vectors.9 The distribution of intensity of the
third factor and the remaining factors cannot
be so simply described but the pattern demonstrated by the third factor is typical in that
each of the later factors showed many roughly
symmetrical small whorls and islands. Similar
maps showing the degree to which successive
addition of the factors reproduced all of the
information available were of interest. Figure
7 presents a synopsis of a series of maps representing the degree of reproduction of information on the chest of the man. It can
be seen that the first principal factor accounts
in high degree for much of the information
present in the region of maximum excursion in
potential values. By the time factor three was
incorporated, a majority of waveforms in the
region of large amplitude had been satisfied
but there were distinctive holdouts of information which were accounted for gradually by
the successive addition of factors through
eight (or, in a similar analysis of data from
the dog, seven).
Discussion
LIMITATIONS OF THE METHOD
This work must be considered merely as
preliminary to the task of systematic, compre-
137
hensive acquisition of surface electrocardiographic information, its subsequent reduction
to nonredundant form, and, finally, its evaluation for physiological and clinical significance.
Only limited conclusions can be drawn from
observations, however extensive, from just two
subjects, each of a different species. The following discussion relates to difficulties in the
successive stages of instrumentation, observation, measurement, and computation that have
been encountered and to the obstacles that
need to be overcome as this method is extended both to other normal and to abnormal subjects.
As previously indicated,9 reconstruction of
maps of distribution of surface potential from
multiple records is plagued by the possible introduction of artifacts from nonsimultaneous
sampling. This is particularly important during
the early and late phases of the QRS complex
when regions of low potential gradient exhibit
islands or small local variations which may
arise either from multipolar behavior of the
cardiac generator, from noise, or from an error
in matching introduced by movement of the
animal between samples. We have taken great
pains to reduce somatic tremor, to record during the resting postexpiratory phase of breathing, and to match the waveforms to a common
timebase by superimposing the constant con-
TABLE l
Adequacy of Resynthesis
Principal
factor
Eigenvalue
Square
root of
eigenvalue
Dog 1
2
3
4
5
6
7
6891.56
543.07
90.66
8.43
5.57
3.07
1.05
83.0
23.3
9.5
2.9
2.4
1.8
1.0
Man 1
2
3
4
5
6
7
8
832.34
170.07
9.53
3.35
2.05
1.23
.76
.50
28.8
13.0
3.1
1.8
1.4
1.1
.9
.7
CirctUlion Resttrcb, Vol. XV, Ant*"
1964
Average
error in %
of total
Information
% Rciynthwis
(100*/«-average
error)
67.0
85.8
93.5
95.9
97.8
99.2
100.0
47.0
26.9
7.2
5.3
4.0
2.3
.7
53.0
73.1
92.8
94.7
96.0
97.7
99.3
56.6
82.3
88.3
91.9
94.7
96.9
98.6
100.0
46.8
20.6
14.6
11.9
9.1
6.9
3.6
.3
53.2
80.4
85.4
88.1
90.0
93.1
96.4
99.7
Cumulative %
of square root
of
eigenvalue
138
HORAN, FLOWERS, BRODY
trol lead when each set of measurements was
made. The decision to utilize the time of
earliest departure from the base line of the
midsternal leads as time zero was arbitrary but
was consistendy adhered to throughout. Two
precautions are suggested concerning die fu-
1HTENSITT Of PRINCIPAL MCTOA MO. 1
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FIGURE 6
Contour maps of the distribution of the first three of the eight principal factors in terms of
their strength of contribution to the local configuration of the QRS waoeform. Note that maps
for factor one and factor two resemble the patterns of distribution of effect of a simple vector3
but that the third factor cannot be so simply described. See text for discussion.
CircuUsion Resetrcb, Vol. XV, August 1964
PRINCIPAL FACTOR WAVEFORMS: QRS COMPLEX
ture development of techniques to settle this
question about possible multipolar effects in
the regions of low gradient. First, there must
be widespread multichannel recording over
large areas of the chest so that simultaneity is
guaranteed. Second, serious consideration
should be given to averaging techniques using
long magnetic tape recordings. Repetitive summing of waveforms by such methods would
allow for the cancellation of both random
PER CENT ERROR NRESVNTHES1S_
.•WI-EN ONLY TACTOR ONE APPLIED
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FIGURE 7
a. An equipercentile contour map of the chest demonstrating the error remaining after attempting
resynthesis of the QRS waveforms with principal factor one alone. For example, within the
boundary marked "10%" the error was 10% or less, i.e., the resynthesized waves very closely
approached the original configuration; however, within the region enclosed by the "90%" line
there was an error of 90% or greater and the resynthesized waveform very poorly resembled
the recorded QRS complex.
b. Composite map of such 10% equipercentile lines from the successive addition of the effect of
each principal factor. The black area (1) indicates the region within which factor one brought
the resynthesized waveform close to the original, i.e., with less than 10% error. The slant-lined
regions (2) indicate the extension of the boundary of close approximation by the addition of the
contribution of factor two to the resynthesized waveform and similarly the dotted regions
(3) for factor three. A bold black line demarcates the area of over 90% restoration produced by
the first three factors alone from those remaining areas which required factors four through eight.
CircuUaion Ktsctrch, Vol. XV, Amgmii 1964
140
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noise and unrelated cyclic phenomenaX1 thus
bringing out only distinctive and real differences in potential gradient. The significance
of respiratory changes and the effect of averaging on such variation would have to be experimentally assessed.
A limitation imposed by economy and time
is the sampling interval of 5 msec in the dog
and 10 msec in man. These intervals produced
effective sampling rates of 200/sec and 100/sec
respectively. Such sampling rates are perfectly
satisfactory as long as there are no portions of
the QRS waveform of higher frequency than
100 cycles/sec in the dog and 50 cycles/sec in
man. Comparison of original and reconstructed waveforms, after the computer filled in the
interval between samples, indicated to us that
there was no significant information above this
level in these normal subjects. When study is
extended, however, to abnormal subjects, it
will be necessary to prepare for the known
occurrence of such phenomena as the high frequency components in the electrocardiogram
associated with myocardial infarction.12 The
use of magnetic tape recordings with automatic analog-to-digital conversion at sampling
rates of 1,000/sec or higher should effectively
remove this Limitation and have the additional
advantages of removing observer error in
measurement and speeding up acquisition of
data.
There remains one serious impediment to
extending this study to a wider spectrum of
subjects: the time required for electronic data
processing on the digital computer is long.
After the computer program was fully developed and ran smoothly, it required about 90
minutes to factor 20 leads. Thus the total computer time needed to obtain eight principal
factors from 180 leads (including intermediate
steps in the pyramid) for the data from the
human subject was about 24 hours. According
to the availability of the digital computer,
this processing was done at the rate of one and
a half to three hours per day. An equal amount
of time was required for the resynthesis of
these waveforms.
The fact that only seven or eight data points
were measured per waveform accounts for
little or no saving of computation time. In-
HORAN, FLOWERS, BRODY
deed, the interpolative procedure was employed to simulate the introduction to the computer of data directly digitized at 2-msec
intervals. This limitation of sampling points
however, does restrict the interpretation to be
placed on the cutoffs of factors after the
seventh and eighth.
Interpolation by fitting with curves of third
degree equations instead of simple linear interpolation caused the waveforms to approach
more nearly the original in configuration but
the theoretical expectation of more final principal factors than seven (in the dog) and eight
(in the man) was not seen. Sampling with a
greater number of points per curve may allow
more factors to be detected but is not Likely
to extend the number of significant factors all
the way to the new number of samples per
curve. Thus, the number of factors found in
these two subjects is probably somewhat less
than if sampling were carried out to 1-msec
intervals. The greater the number of sampling
points the greater the degree to which individual variation in waveform is conserved. The
crucial point is whether this individual variation in waveform results from noise (including
observer error) or significant signal. If noise,
then the limiting of the number of samples has
reduced the degree to which we have been
deceived by spurious variation in contour; if
signal, then limiting of the number of samples
has made our estimate of the quantity of
waveform information, that is present, conservative. We expect that improvement in instrumentation, which will tend to eliminate
both noise and observer error, may reduce the
number of significant principal factors. On the
other hand, in the abnormal population high
frequency components, as noted in subjects
with myocardial infarction, and local irregularities of contour, as noted in subjects with
conducton defects, may be expected to increase the number of factors. Our tentative
conclusion is that within the limits of our experimental design there were at least seven
significant principal factor waveforms in the
dog and eight in the man.
INTERPRETATION OF MULTIPOLAR PATTERNS OF
SURFACE POTENTIAL DISTRIBUTION
The presence of multipolar patterns of
Circmltsion Research, Vol. XV, Angmlt 1964
141
PRINCIPAL FACTOR WAVEFORMS: QRS COMPLEX
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surface potential distribution at various instants during ventricular depolarization is now
well documented both for man and dog.9'18'14
It is possible that an occasional finding of
more than one positive maximum or more than
one negative maximum of potential may be
arrifactual, especially when found in a region
of low gradient where the signal has fallen
close to the noise level. However, with this
exception, multipolar patterns have been demonstrated on repeated examination of individual subjects and during comparable times in the
QRS complex in many subjects.9'18 It is important to consider the various possible causes
of instantaneous surface "multipolarity." The
biophysical possibilities are (1) the formation
of reflections or images at each interface between intrathoracic regions of differing resistivity, e.g., between blood and myocardium
or between myocardium and lung; (2) conductive focussing of the effect of the wave of
activation in selected portions of the myocardium; (3) proximity effects, i.e., where a
small fragment of the wave of activation in a
portion of the myocardium very close to the
chest wall is oriented differently from the large
remainder elsewhere; (4) qualitative differences in portions of the wavefronts of myocardial activation. Failure to elicit distinctive
multipolar patterns upon implantation of artificial dipolar (bipolar) current sources in the
canine myocardium suggested to us that the
first of the aforementioned possibilities is unlikely.0 As to the second, the theoretical possibilities of conductive focussing as a contributor
are good both because of the theoretical effect
of the blood mass on the electrocardiogram16-17
and because of experimental evidence supporting the concept of focussing of the septal contribution.18 Pure proximity effects also seem
likely, first because the demonstration of instantaneous surface multipolarity is more striking in the dog than in man, and second because of the appearance of multipolar patterns
during the time of the right precordial R
prime when it is suspected that the right
aspect of the interventricular septum or the
right free wall near the pulmonary valve is
activated. Finally, we do not know that the
wave of activation sweeping through the
Circulation Rurtrcb,
Vol. XV, A*gnit 1964
myocardium remains uniform in density of
charge either over its surface or in time; we
think, however, it is reasonable to assume uniformity until experimental evidence suggests
otherwise. We agree with Taccardi that the
evolution of the surface pattern should be
related if possible to the underlying events in
the heart.14 We therefore suspect that the appearance of the markedly multipolar pattern
which reached its peak between 40 and 50
msec in the human subject may well relate to
the separation of components of the wave of
activation in the right free wall and interventricular septum from the left free wall by
epicardial breakthrough at the apex. Similarly,
we suspect that the appearance of a multipolar
pattern in the terminal portion of the QRS
complex or of the reappearance of a small
island of positivity on the anterior precordium
late in ventricular depolarization relates to the
firing of basilar portions of the right ventricle
or interventricular septum.
INTERPRETATION OF PRINCIPAL FACTORS
By derivation the principal factors are expressions of the limits of information contained
in the entire population of waveforms about
the chest. As Scher et al. have pointed out,8
finding no more than three principal factors of
significance—from which all the information
from the body surface could be reproduced—
would be compatible with the reduction of the
cardiac generator to an equivalent dipole.
They, of course, made the proviso that the
discovery of three factors was not proof of
dipolar equivalence but that any number of
three-factor generators were possible, among
them a dipole. Because the principal factors
are mathematically uncorrelated or orthogonal
in terms of signal space, if only three were significant one could consider them also orthogonal in three-dimensional space and merely
determine the proper rotational transformation of axes to find the XYZ components of the
equivalent dipole or vector. However, more
than three significant principal factors were
observed. In fact, there appeared to be at least
seven and eight (the upper limit set by the
number of sampling points). The attempt to
translate the information contained in the
HORAN, FLOWERS, BRODY
142
principal factors into biologically useful terms
should begin with careful reconsideration of
the conceptual tools by which we relate surface potential to the interplay between the
real or an equivalent cardiac generator and
the conductive characteristics of the body.
The relationship between current and voltage along a line conductor (or in a network of
lines) is described by Ohm's law, E = IR or
RI. Potential drop E along the line is the
algebraic product of the current I and the resistance of the conductor R. In a volume conductor the relationship is more complex; in the
case where the current source is a dipole, surface potentials may be described by a series of
expressions similar to those employed by
Burger et al.:20
V! = auXi + a12x2 + a13xH
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t> 3 =
(3)
+ as2x2 +
The relationship between the potential v1
across a given lead and the spatially orthogonal components of the heart vector or dipole
{xu x-2, xs) at a specified instant is defined in
terms of the orthogonal components of the lead
vector (aii,a]2,fiis) of the lead. This algebraic
expression can be stated more briefly in vector
form:
V = A-X
(4)
In a very rough way equation 4 is analogous
to Ohm's law in that it states that lead
potential (mv) is the scalar product of the
lead vector (mv/ma • cm) and the heart vector (ma • cm). The lead vector has been
sometimes treated as a statement of the impedance characteristics of the conducting
medium for the given lead and called transfer
impedance.19 Viewed in the latter way the
units of the lead vector may be simplified to
ohm/cm. Now the relationship for any number
of leads to the heart vector and their respective lead vectors may be stated in matrix
form
( n x l ) (nX3) (3X1)
VA
X
(5)
in which V represents a column of individual
lead voltages, (vuv2, v3 . . . vn), X represents
a column describing the three spatial or Carte-
sian components of the instantaneous heart
vector, and A is a matrix of an appropriate
number of sets of lead vectors (each with
three coefficients). This instantaneous relationship can be expanded for the time-varying
dipole by replacing the units in the matrix dimensions by a given number of consecutive
instants in time thus converting V and X
respectively from single columns into rectangular matrices.
RELATIONSHIP BETWEEN PRINCIPAL FACTORS AND
THE EQUIVALENT CARDIAC GENERATOR
Because the simplification of the cardiac
generator into a dipole of fixed location but
varying orientation and intensity is not sufficient to explain the surface potentials found
in the living subject a more complex equivalent generator must be considered. Two of the
alternative physical models proposed are (1)
the dipole with a shifting locus21 or (2) an
instantaneous multipolar equivalent generator.22' 2!t In the first instance quadripolar
or octapolar components may arise as harmonic additions to the suspected potential at
a given surface point because of proximity
effect or the effect of changing relative distance from the moving dipole. Now that we
know that there are not only proximity effects
but also instantaneous multipolar surface potential patterns, the shifting dipole has ceased
to be a satisfactory equivalent cardiac generator.
However, there are practical as well as conceptual difficulties in substituting a three-dimensional model of an equivalent cardiac multipole as X and a set of lead tensors as A in
equation 5. 22 ' 23 We can employ an equivalent
generator from whose components the surface
leads can be satisfactorily constructed, a mathematical model composed of principal factors.
Unfortunately, this turns out to be an unvisualisable eight-dimensional "model" in signal
space instead of a three-dimensional model in
the everyday Cartesian space of sensory perception.24 Increase of data relating to lead
parameters 25 and exploration of the constancy
of factors between individual subjects and
among normal and abnormal populations may
eventually lead to a suitable transformation of
Circulation Research, Vol. XV, Angut 1964
PRINCIPAL FACTOR WAVEFORMS: QRS COMPLEX
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the eight-dimensional provisional generator
into a three-dimensional model directly relevant to the actual electrophysiologic process in
the heart.
In extending the analogy from the lead vector concept to the present findings it is pertinent to note that just as the principal factors resemble the components of the heart
vector, their coefficients of distribution resemble the components of the lead vector. We
may then look upon these coefficients as descriptions of the resistive or impedance characteristics of the volume conductor with respect to the various principal factors. Indeed,
if the major significant component of the
equivalent generator is a tensor of first
rank, i.e., a dipole or vector, and if the successive orientations in time of the surface
distribution of heart vector all lie within a
plane, then it is possible that the surface distribution of factors one and two (fig. 7) describes the two axes of the plane of the
vector loop. In such event, the major dipolar
component is described by factors one and
two,20 and the remaining factors may describe
(1) any departures from the plane of the loop,
and (2) any contributions to the equivalent
generator from higher order components.
Thus, knowledge of the relative strength of
the distributive coefficients over the body surface may be expected to furnish a very meaningful guide to the choice of leads so that
all the factors or components of information
will be well represented in the body surface
electrocardiogram.
In terms of anatomic distribution, the lesser
factors played a relatively greater role in accounting for the waveforms in the regions of
the low right chest and over the left shoulder.
If one views the distribution of factors one
and two as describing an ecliptic for the major
or dipolar component then those regions are
the regions off the ecliptic (i.e., farthest from
the plane of the vector loop) and therefore,
where noise or somatic tremor would be expected to reach a relative maximum and signal
to reach a relative minimum. Here again is the
problem discussed under limitations: the region where multipolar effects should be detectable by virtue of not being swamped by
CircnUiioH Rtsunch, Vol. XV, August 1964
143
the first order (dipolar) signal is also the
region of low gradient where one has to be
extremely careful not to read noise and artifact as information. However, careful examination of data exemplified in figure 5 showed
symmetric, progressive development of the
pattern of distribution. For example, the effect
exerted by factor 5 (figs. 5a, b, and 7b) strongly suggested to us the detection of significant
information from a higher order component.
Summary
High-speed, high-fidelity recordings from
150 to 180 systematically spaced points over
the thoracic surface of both a dog and a man
have yielded detailed maps of QRS waveforms. With the aid of digital computer
analysis, these waveforms have been resolved
into the minimum number of mathematically
uncorrelated waveforms necessary for adequate resynthesis of all the original QRS complexes. Under the conditions imposed by the
experimental design, seven (in the dog) and
eight (in the man) such waveforms, or principal factors, were found. Maps were constructed for both the contribution of each
principal factor to the various regions of the
chest and also for the degree of reproducibility of the original waveforms by successive
addition of the principal factors.
The first three factors accounted for an
average of 92.8% of the available information
on the chest in the dog and 85.4% in the man.
A resynthesis of over 99% of the information
was achieved for both subjects when all remaining principal factors were also employed.
This necessity for more than three factors or
generating functions for reproduction of all
the known surface potentials was interpreted
as strong support to the equivalent multipole
concept of the electrocardiographic generator.
In addition, the reduction of total surface information into a compact set of waveforms of
specified distribution opens an experimental
avenue to the examination of such concepts
in living subjects.
Acknowledgment
We are indebted to Mr. Wallace Marquardt and
Mr. Irvin Jaynes of the Department of Medical Illus-
144
HORAN, FLOWERS, BRODY
tration for excellent photographic reproduction of the
maps and charts, to Miss Martha Hutchison for technical assistance, and to Dr. Charles Sheppard for the
use of the facilities of the University Computer Service
(supported by U. S. Public Health Service Grant FR
0001).
References
1. ABILDSKOV, J. A.: The relation of precordial and
orthogonal leads. Circulation 27: 58, 1963.
2.
4a. HUCENHOLTZ,
P.
G.,
WHJPPLE,
G.
H., AND
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LEVINE, H. D.: A clinical appraisal of the vectorcardiogram in myocardial infarction. I. The
cube system. Circulation 24: 808, 1961.
HUCENHOLTZ, P. G., FORKNER, C. E., JR., AND
LEVINE, H. D.: A clinical appraisal of the vectorcardiogram in myocardial infarction. II. The
Frank system. Circulation 24: 825, 1961.
5.
COELHO, E . , AArRAN, S. S., E S A , A. B . , MENDES,
J. C. F., AND TAVARES, V.: Electrocardiograph-
ic and vectorcardiographic alterations in chronic
cor pulmonale. Am. J. Cardiol. 10: 20, 1962.
6. HOLZMANN, M.: Clinical Electrocardiography,
chapt. 5. London, Staples Press, 1952.
7.
MYERS, G. B., KLEIN, H. A., AND STOFER, B. E.:
Correlation of electrocardiographic and pathologic findings in lateral infarction. Am. Heart
J. 37: 374, 1949.
8. SCHEH, A. M., YOUNG, A. C , AND MEREDITH,
W. M.: Factor analysis of the electrocardiogram. Test of electrocardiographic theory:
normal hearts. Circulation Res. 8: 519, 1960.
9. HORAN,
L. G., FLOWERS,
N. C ,
12.
17.
HORAN, L. C ,
ANDREAE, R. L., AND YOFFEE,
H. L.: The effect of intracavitary carbon dioxide on surface potentials in the intact canine
chest. Am. Heart J. 61: 504, 1961.
18.
HORAN, L. G., HANSEN, F. L., AND BOUSQUET,
R. M.: Relationship between the anatomical
orientation of the interventricular septum and
the manifest orientation of ventricular depolarization in dogs. Circulation Res. 10: 859, 1962.
19. SCHMITT, O. H.: Lead vectors and transfer impedance. In Electrophysiology of the Heart.
N. Y. Acad. Sci. 65: 1092, 1957.
20.
BURGER, H. C , AND VAN MILAAN, J. B.: Heart
vector and leads. Part I. Brit. Heart J. 8: 157,
1946.
21. PIPBERCER, H. V.: Application of corrected electrocardiographic lead systems in man. Am. J.
Med. 25: 539, 1958.
22.
BRODY, D. A., BRADSHAW, J. C ,
AND EVANS,
J. W.: A theoretical basis for determining
heart-lead relationships of the equivalent cardiac multipole. IRE Trans. Bio-Med. Electron.
8: 139, 1961.
23. CESELOWITZ, D. B.: Multipole representation for
an equivalent cardiac generator. Proc. IRE 48:
75, 1960.
24.
YOUNG, T. Y., AND HUGCINS, VV. H.: The intrin-
sic component theory of electrocardiography.
IRE Trans. Bio-Med. Electron. 9: 214, 1962.
AND BRODY,
D. A.: Body surface potential distribution:
Comparison of naturally and artificially produced signals as analyzed by the digital computer. Circulation Res. 12: 373, 1963.
10. HARMAN, H.: Modern Factor Analysis. Chicago,
University of Chicago Press, 1960.
11. DAWSON, C. D.: A summation technique for
detecting small signals in a large irregular
background. J. Physiol. 115: 2, 1951.
NELSON, C. V., CHATTERJEE, M., AND ANGELAKOS,
E. T.: Further studies on the effect of the intracavitary blood on the electrocardiogram.
Proc. New Engl. Cardiovascular Soc. 16: 20,
1957-58.
PIPBERGER, H. V., FREIS, E. D., TABACK, L., AND
MASON, H. L.: Correlation of the clinical information in the standard 12-lead ECC and in
a corrected orthogonal 3-lead ECG. Am. Heart
J. 61: 34, 1961.
4b.
16.
BURCH, G. E., HORAN, L. C , ZISKIND, J., AND
CRONVICH, J. A.: A correlative study of postmortem electrocardiographic and spatial vectorcaxdiographic data in myocardial infarction.
Circulation 18: 325, 1958.
3.
13. TACCARDI, B.: Distribution of heart potentials on
dog's thoracic surface. Circulation Res. 11:
862, 1962.
14. TACCARDI, B.: Distribution of heart potential on
the thoracic surface of normal human subjects.
Circulation Res. 12: 341, 1963.
15. BRODY, D. A.: A theoretical analysis of intracavitary blood mass influence on the heart-lead
relationship. Circulation Res. 4: 731, 1956.
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ARZBAECHER, R. C , AND BRODY, D. A.: Electro-
cardiographic lead tensor measuring system.
Proc. Sixteenth Ann. Conf. Eng. Med. Biol.
5: 34, 1963.
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HORAN,
L. G., FLOWERS, N. C ,
AND BRODY,
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Appendix
FACTOR ANALYSIS OF ELECTROCARDIOGRAPHIC
WAVEFORMS
Principal factor analysis as applied to waveform
CircmUtion Rtsetrcb, Vol. XV, August 1964
PRINCIPAL FACTOR WAVEFORMS: QRS COMPLEX
analysis is a method of examining the degree of correlation between all the members of a given population of waveforms. Correlation between any two members of the set may be estimated by summing the
products of their amplitudes for a series of instantaneous measurements. Stated more intuitively, two
waveforms correlate highly if the result of multiplying amplitudes instant-by-instant and totalling the
result yields a large number and poorly if the result
approaches zero. (Obviously, if the amplitudes of
waveforms are referred to a common size the number
is more meaningful.) When each member of the population is thus compared with every member (including itself) a family or matrix of numbers results which
describes the correlations throughout the population
in detail. Principal factor analysis involves what may
be appreciated in a geometric and intuitive fashion
as rotating the matrix of numbers until most of the
information lies on the fewest number of reference
axes, i.e., until the correlations between all the waveforms may now be described by how much each contains of a small number of reference waveforms.
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Oscillographic tracings of QRS complexes are analog
displays of electrocardiographic potentials as continuous functions of time, v^t), where the subscript, i,
indicates reference to a given member of a set of
leads. The initial goal of factor analysis as employed
in this study is to derive from the experimentally
observed set of leads, vt(t), another set of time functions, Xj(t), whose individual waveforms are uncorrelated with each other. Not only are the uncorrelated waveforms to be determined, but also the
distributive coefficients, Oy, which satisfy the transformation relationship
vl(t)=a(}
x,(t)
(1)
The correlation coefficient between any two leads,
v( and Vj, of the entire set of waveforms V, is defined
as
h*t — -
vt(t) vAt) dt
(2)
where htj is the correlation coefficient and T is the
interval of time occupied by the deflections. If a
given htj is zero the two waveforms are said to be
orthogonal in time, and therefore uncorrelated. Factor
analysis of the experimentally derived leads is accomplished by determining a set of distributive coefficients such that each of the Xj(t) in equation 1 is
orthogonal with respect to all other members.
In digitized form the <th lead of the original leads
becomes the set, c,(f): t ; , ^ ) , v{(t2), ©,((3),
v t
i( m)- Tn^ can be expressed more como((tt),
pactly as the row matrix, vli)k, where the parentheses about first subscript indicate that our attention
is focussed for the moment on a set of values which
represent a single given lead. Upon removing the
parentheses the entire set of digitized lead dpta assumes the matric form, V =: v(k.
CircuUtion Rtsurcb, Vol. XV, Auguil 1964
145
If the ith row of V is postmultiplied by the /th column of its transpose we obtain
k— m
fc=l
which is approximately equal to the correlation coefficient given in equation 2. Upon removing the particularizing parentheses we obtain
H — VVT
(3)
where V is the transpose of V, and H is the set of
all correlation coefficients in matric form. With the
lead data cast into digitized, matric form equation 1
becomes V = A X, from which equation 3 becomes
H = WT = A(XXJ)AT
(4)
Since both H and XXT are necessarily square symmetric matrices it follows that A is a rotatory matrix
with its transpose identical to its inverse, that is,
AT = A-1. Therefore premultiplying the members of
equation 4 by AT and postmultiplying by A gives
XXT — A-iHA.
The major objective of factor analysis is to determine a transformation such that the individual waveforms of the derived leads are uncorrelated with each
other. In other words a rotatory transformation matrix,
A, is sought such that
A-iHA = A
(5)
where A is a diagonal matrix. After the required matrix diagonalization has been accomplished, the elements of the derived, uncorrelated waveforms are determined from the relationship X = A-W.
Diagonalization as employed here is essentially a
rotatory transformation of axes in multidimensional
signal space. The amount by which the axes are
rotated is specified by A or A-1, which is commonly
referred to as an eigenvector matrix. The individual
diagonal elements of A are known as eigenvalues. The
magnitude of each eigenvalue is a measure of the
mean square deflection of the corresponding principal
factor waveform. When the number of leads analyzed
exceeds the number of principal factor waveforms
some of the eigenvalue magnitudes are essentially
zero. It is in this way that redundancy of pattern is
recognized in the analysis of numerous leads. It is the
number of non-zero eigenvalues which specifies the
number of principal factor waveforms.
In computer applications diagonalization is accomplished by a series of plane rotations in multidimensional signal space. Each rotation is designed to reduce
the maximum absolute valued off-diagonal element
of the correlation matrix to zero (Jacobi's method).
Although this element does not remain zero with
succeeding rotations the process is a convergent one,
and eventually all off-diagonals can be reduced to or
below a specified level of acceptable near-zero values.
The eigenvector matrix is simultaneously built up during the computational process by cumulative multiplication of a unit or identity matrix by the plane rotatory matrices.
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Principal Factor Waveforms of the Thoracic QRS Complex
LEO G. HORAN, NANCY C. FLOWERS and DANIEL A. BRODY
Circ Res. 1964;15:131-145
doi: 10.1161/01.RES.15.2.131
Circulation Research is published by the American Heart Association, 7272 Greenville Avenue,
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Copyright © 1964 American Heart Association, Inc. All rights reserved.
Print ISSN: 0009-7330. Online ISSN: 1524-4571
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