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On electrophysiological activity of the normal heart
’ On Electrophysiological Activity of the Normal Heart by
P. s. THIRY
and R. M. ROSENBERG
Department of Mechanical Engineering University of California, Berkeley, California
ABSTRACT:
The heart is divided into eleven componenta, each being modeled by a dipole of
known and fixed location and direction; these dipoles are shown to result from the propagation of the tranemembrane potentials over the surface of the myocardial cella. The components of the left ventricular wall are composed of “Durrer layers”, and these in combination with the
“contiguity effect” result in a positive ventricular gradient and
cardiograms
generated by the eleven dipoles yield high-fidelity
12-M
T-wave.
Electro-
ECG’s.
1. Introduction
In a recent paper (l),a mathematical model of electrical heart activity was described which consists of a finite number of dipoles of known and fixed location and direction, but of variable strength. One of the stated assumptions in that paper is “. . . that the moment of each dipole is proportional to the monophasic action potential of the heart component modeled by it”, i.e. modeled by that dipole. In this paper, we abandon that assumption; in fact, we remove the question of dipole strength from the domain of assumptions and make it, instead, the subject of a detailed analytical study on the level of the cell. In constructing the present model we come to grips with the “puzzle of the T-wave”. Heart muscle, like muscle in general, contracts when the extremities of the muscle fiber are subjected to an electrical potential difference, and it regains its original length when the potential is removed. The heart is known to generate electrical potentials which cause the muscle to contract and, thus, the heart to beat. These potentials are produced by a periodic exchange of sodium and potassium ions across the boundaries or membranes of the myocardial cells, and the resulting transmembrane action potentials can be, and have been, measured by placing microelectrodes on either side of the cell membrane. These potentials look somewhat as shown in Fig. 1. A nonzero transmembrane potential is a potential difference between neighboring points and, thus, an electrical dipole with direction normal to the membrane, and with moment proportional to the transmembrane action potential at the dipole location. With the body regarded as a volume conductor, the dipoles produced by this electrical heart activity generate a potential at every point in the body as well as on the skin surface. These skin surface potentials, measured at certain selected points, are recorded and are the electrocardiograms or ECG’s. A typical recording is shown in Fig. 2.
377
P. X. Thiry and R. M. Rosenberg In it, the activation of the atria is totally represented by the P-wave; so-called QRS complex and the T-wave are due to ventricular activity.
the
-6O-
Depolarization Period
FIG. 1. Typical transmembrane
action potential.
The “puzzle of the T-wave” is this: The QRS complex occurs during so-called “ventricular depolarization”, and the T-wave during “repolarization”. Now, it is known (a property which will be examined in detail in the next sections) that if the ECG response is positive (R-wave up) during depolarization, it should be negative during repolarization (T-wave down). However, as can be seen from Fig. 2, the T-wave of clinical skin surface ECG’s is in general up, not down.
FIG. 2. Normal Lead II electrocardiogram.
For more than forty years, efforts have been made to understand the mechanism of repolarization and of the T-wave. It has been suggested that the sequence of repolarization follows a path through the ventricle different however, there does not appear to be any from that of depolarization; anatomical evidence to support this view. Other theories are that the T-wave polarity is governed by the repolarization sequence across the ventricular wall (2), that it is produced by differences in repolarization time between distant areas of the ventricle (3), and between adjacent areas (4). A brilliant investigation, associated with the names of Burgess, Harumi and Abildskov (5, 6) and summarized recently by Burgess (7), explains the unexpected polarity of the T-wave by means of experimental observations by van Dam and Durrer (4) who discovered that the action potential duration
378
Journal of The Franklin Institute
On Electrophysiologicak Activity of the Normal Heart differs for some adjacent layers of the ventricle. Here, Burgess and her collaborators echoed the remarkable statement made already by Wilson in 1934 (8) that “if two elements of a muscle fiber differ in their state of activity, one being nearer to or farther away from the resting state than the other, there must be an electromotive force across and normal to the plane that separates them”. In the Burgess model, electrical cardiac activity is treated as a traveling wave phenomenon, resulting in a very elegant formulation of the problem. The Burgess group tested their model in three ways: once, by comparing T-wave forms of normal ECG’s, then, when ventricular recovery periods were altered by local heating of prescribed areas of the ventricle and, finally, when infarction was produced by stopping the coronary blood supply to The comparisons between observed and certain areas in the ventricle. predicted T-wave forms were impressive in all cases. Our model is also based on Wilson’s observation and on the experimental results of van Dam and Durrer (4). Nevertheless, it is so different from that of the Burgess group as to hardly resemble it. In our model, the heart is divided into a number of components, each being modeled by a dipole of fixed location and direction while, in the Burgess model, the heart is considered as a whole, and dipoles are computed from the known wavefront advance through the heart. We have retained the separation into components of our earlier model in order to preserve our conduction mechanism model described elsewhere (10). In this way, we can introduce partial signal conduction block of any degree, or complete block, at any point or points of the conduction system. We also feel that the existence of highly localized conditions in the myocardium, such as lesions, is easily modeled when the heart is subdivided into components, but becomes difficult in the wavefront model because of the treatment of the line integrals which must be computed when the wavefront does not coincide with the boundaries of the lesions.
II.
The Single
Cell
As stated above, the transmembrane action potential is the result of a periodic ion exchange across the cell boundary. This exchange does not take place simultaneously at all points of the cell membrane; rather, it starts at one point and proceeds in the direction of signal propagation to all other points of the membrane. In fact, from the known propagation velocity through the myocardium and the size and shape of the cell, one can readily compute the propagation velocity along the cell surface. Ideally, one would wish to build a mathematical model of the cell membrane and of the process of ion penetration, and then obtain the transmembrane potential as the output of that model. Unfortunately, this mechanism is not sufficiently well understood to permit construction of a model, and cell membrane action remains for the present one of the intriguing
Vol.297,No. 5,May 1974
379
P. 8. Thiry ad
R. M. Rosenberg
mysteries of biology. We are therefore obliged to accept the transmembrane potential as observed, and to give it some functional representation. Not all myocardial tissue is alike, and the transmembrane potentials of cells belonging to different types look different. However, in this paper we are primarily concerned with tissue contributing to the skin surface potentials, and for that tissue the transmembrane potential is well represented by*
f(t) z 0
in O
f(t) = k > 0
in t,Qt
f(t) = k l-(t-~~o~P)2] [
in t,+p
f(t) = k
p,+p+
loo-t)2 5000
\
I
(1)
in t,+~+50Gt
The graph off(t) is shown in Fig. 3. The numerical values which appear in (1) have been chosen so as to give a realistic action potential with f(t) in millivolts (mV) and t in milliseconds (msec). The symbol k is an “amplitude constant”, and p gives the duration of the “depolarized state”. Since the depolarization and repolarization durations are fixed, p is a measure of the action potential duration.
T
I
k
FIG. 3. Graph of mathematical
representation of transmembrane
action potential.
The particular representation of f(t) as given in (1) is not essential to our model. Any other choice resulting in a graph similar to Fig. 3 would do just as well; we simply find (1) a convenient and adequate representation. We shall now calculate the dipole generated when the action potential propagates over the surface of a spherical cell of diameter 2r in the direction *For convenience, we have taken the resting potential as the datum. This is equivalent to adding a constant to realistic action potentials; this addition has no effect on the resultant dipole of the cell.
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Journal of The Franklin Inst,itute
On Electrophysiological
Activity of the Normal Heart
of a diameter A, B, as shown in Fig. 4. If the (known) speed of signal propagation in the myocardium is v, the speed of propagation (assumed uniform) over the cell surface is 7, and 27 = (v/r)?W = 7rv.
(2)
Let the cell membrane be activated at the time t,; then, a surface element located at C (see Fig. 4) will be activated at the time (3)
t = t,+P+,
where /I is the angle from OA to OC. We obtain the resultant dipole for the spherical cell at any time t by integrating the dipole per unit area f(t) over the cell surface. The moment of f(t) is at each point proportional to the transmembrane action potential, and f(t) is normal to the cell surface.
FIG.
4.
Signal propagation along surface of spherical cell.
The resultant dipole denoted by D(t) is for all time parallel to AB, and if e, is the unit vector from A to 0 (i.e. positive in the direction of propagation), a simple calculation shows that the resultant dipole is given by
D(t) = nrz[sin2~f(t-t,-~)d~e,,
(4)
and its moment is D(t) = ~r2~sin2~f(t-t0-~)
db
(5)
withf(t) as given in (1). For the numerical values v = O-6 m/see, r = 0.003 m, p = 200 msec and k = 100 mV, the graph of D(t) looks as shown in Fig. 5 in which the amplitude of the first peak is nPk.
Vol. 297, No. 5, May 1974
381
P. S. Thiry and R. M. Rosenberg In analogyto ECG terminology we shall refer to the first peak as the R-wave and the (negative) second peak as the T-wave. It is easy to show that the R-wave in Fig. 5 is a translated, inverted cosine wave. Consider the surface
in the /I, T,f-space; it is shown in Fig. 6. One sees that f = 0 in 0 < r < BY/T, and f=k in @-/~
s vlr
D(t) = 77r2k
sin2/3d/3 = +rzk(l-cos:T).
0
For T in nr/q
FIG. 5. Resultant
ZZZ. The
Gradient
of a Single
dipole moment
of spherical
cell.
Cell
The fact that the R- and T-waves have opposite the much stronger result that, in fact,
polarity
follows from
(7)
where r2 is the time at which the repolarization phase of the transmembrane potential is completed at point A (see Figs. 4 and 6). In analogy to terminology introduced originally by Wilson et al. (8) and frequently used in cardiology, we shall refer to G as the “cell gradient”; hence, (7) states that the cell gradient vanishes. Wilson’s “ventricular gradient” is the time integral of the &RX complex and the T-wave. It is a sort of global measure of the vector quantity D(t) over ventricular activity which makes his nomenclature seem apt. In general, the ventricular gradient of the normal heart is positive, not zero which is another description of the “puzzle of the T-wave”. For the proof of (7), consider Fig. 7, and denote f(r, 0) in O
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Journal
of The Franklin
Institute
Activity of the Normal Heart
On Electrophysiological
phase of a transmembrane potential. We shall show that jfD(~) dr is independent of f&T) and depends on k only. From this demonstration, (7) follows directly when k is replaced by -k. f
T
\ \ \ \ \ \ \
Td?)
.\
P FIG.
FIG.
Except
7.
6.
Transmembrane
potential in space-time
domain.
Arbitrary depolarization phase in space-time
for a constant,
domain.
we have D(T) =.(l”f
(T-F)
sin2jId/3,
(8)
and D(0) = 0 because f( -pr/q) = 0 for 0 < /3 < rr. Moreover, D(T) = 0 for 7 2 E = D + (‘rrr/q) because in T 2 E, f- k and (8) is zero in that case. Then
Now, the first double integral in (9) vanishes becausef= 0 on OAB. Moreover, the third double integral is a function of k only, say K(k), because f 2 k on CDE. Finally, the second double integral is
(10)
Vol. 297, No. 5, lCIaJ1974
383
P. 8. Thiry and R. H. Rosenberg With r-pr/~
= 9, the integral
which is independent
of j3. Hence, (10) is I OBCD= @(f, D)
llsin Z/3d/3 = 0. s0
(11)
It follows that %(,)dr s0
= K(k).
(12)
This completes the proof. It is surprising that the only contribution to the integral (12) is made in the time interval which begins when the depolarization is completed at A, and which ends when the depolarization is completed at B. It should be pointed out that, while the time integral of the R-wave is independent of the shape or duration of the depolarization phase of the transmembrane potential, the R-wave itself depends strongly on both. In particular, the R-wave has the shape of an inverted cosine wave (see Eq. (6)) only when the depolarization of the transmembrane potential is a step function.
IV.
Some
Electrophysiological
Observations
When constructing a mathematical model of electrical heart activity, one needs to know both the time at which cellular electrical activity begins at every point of the myocardium, and the shape and duration of the transmembrane action potentials for every point in the heart. Probably, the most authoritative information on both of these questions is contained in work done at Amsterdam, and associated largely with the name of Durrer. In some recent experimental work on the excitation sequence in the isolated human heart, Durrer et aJ. (9) have obtained detailed information on the distribution, in time and space, of the depolarization phase through the myocardium. From control experiments on dogs, they found that the essential difference between observations in situ and in the isolated heart was in the propagation velocity, but the isochrones* remained similar in both cases. Thus, we have utilized their excitation sequence combined with propagation velocities observed in the heart in situ (9, 11). The question of action potential durations was examined by van Dam and Durrer (4) in an earlier paper where they studied the duration of the “functional refractory period” of the dog heart in situ. Their observations permit the conclusion that the action potential duration in the left ventricular wall is longest in the endocardial layers, shortest in the middle layers and of * Isochrones are the surfaces containing all points of contemporaneous excit&ion.
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Journal
of The Franklin
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On Electrophysiological
Activity of the Normal Heart
intermediate duration in the epicardial layers ; for simplicity, we shall call these “Durrer layers”. It would lead us too far afield to discuss here the relation between the duration of the refractory period and of the action potential; anyone interested in this question is referred to Hoffman and Cranefield (11)and van Dam and Durrer (4). We merely point out that these time intervals are closely related and, when the functional refractory period is longer in one location of the myocardium than in another, so is the action potential duration. We shall, in fact, make the assumption that these periods are directly proportional ; i.e. when the ratio of functional refractory periods at two different locations is n, so is the ratio of action potential durations. We shall show that it is the difference in action potential durations between contiguous Durrer layers of myocardial tissue which is responsible for the positive polarity of the T-wave but, to our knowledge, Burgess et al. (S-7) were the first to utilize that property to explain the T-wave mechanism.
V.
The Dipole
of Two
Cells
Consider two spherical cells, Cell I and II with their centers lying in the direction of signal propagation as shown in Fig. 8(a) ; let the points B of minimum distance between them be connected by a wire in which the velocity of signal propagation is infinite. This wire is a model of the so-called “intercalated disc”, a small high-conductance element forming the electrical connection between neighboring cells along the conduction path. The conductance of the intercalated discs is on the order of 400 times that of the cell membrane (12,13).
FIG.
8. Two noncontiguous
cells and their dipole moment.
If the transmembrane action potentials of the two cells were identical in all respects their combined dipole moment D(t) would simply be the sum of two curves like that in Fig. 5, but with the curve of Cell II shifted to the right in the amount n-r/q relative to that of Cell I. However, we shall suppose that the action potential duration of Cell II is shorter than that of Cell I. Thus, these cells belong to different Durrer layers, and we imagine them to lie immediately next to the boundary between these layers. The dipole moments of these cells are shown in Fig. 8(b). In that diagram it was assumed that the difference in action potential durations satisfies p, -pI1 > n’r/v. This
Vol. 297,No. 6, May 1974
385
P. S. Thiry and R. M. Rosenberg has the effect that the T-wave of Cell II begins earlier It is not at all essential to our argument that this be the is pr-prr > 0. Evidently, the cell gradient of each cell, sum, vanishes in accordance with (7), and the T-wave of these two cells is necessarily negative.
than that of Ceil I. case ; all we require and hence of their of the combination
-. (b)
****-- (L contiguity effect ’ FIG. 9. Two contiguous cells and their dipole moment.
Now let the length of the wire connecting Cells I and II tend to zero, so that the two cells become contiguous as shown in Fig. 9(a). In that case, a new effect appears which we call the “contiguity effect”. To see this, consider the two transmembrane action potentials at the points B, shown in Fig. 10. Evidently, between B in Cell I and B in Cell II there exists a positive potential difference during the time interval
where t, is the time of activation of point B. The interval of existence of this nonzero potential difference extends over the repolarization times of both
f(t)
---I Cell I Cell II
FIG. 10. Action potential at B of Cells I and II.
cells. This potential difference between two neighboring points is a dipole having positive moment, and it would be absent if the cells were not
386
Journal of The Franklin Institute
On Electrophysiological
Activity of the Normal Heart
contiguous. The resultant dipole moment of shown in Fig. 9(b). Under the assumption pr the contiguity effect to the T-wave is positive result in a positive T-wave; for pI -p,, < 0, the effect is negative. In either case, the contiguity in nonzero two-cell gradients.
the two contiguous cells is -p,, > 0, the contribution of and, if sufficiently large, can contribution of the contiguity effect will, in general, result
VI.
Volume
Elements
of the Myocardium
To model electrical cardiac activity, we decompose the myocardium into a number of volume elements which are then idealized as being spherical in shape. Every element is considered to consist of a small number M of contiguous electrophysiologically different Durrer layers and each electrically homogeneous Durrer layer is imagined to consist of a much larger number N of “sublayers”. The boundaries between the layers and the sublayers are normal to the direction of signal propagation. Every volume element is represented by an electrical dipole of known and fixed location and direction, but of variable moment. The dipole moment of the entire volume element is the sum of the dipole moments of all sublayers and of the contiguity effects (if any; i.e. if 1M> 1). Each of the dipole moments has the general shape shown in Fig. 5, but the amplitude of each is proportional to the volume of the sublayer; i.e. to the number of cells in it. Of course, the dipoles of the sublayers must be added with the proper time relation retained between them. During depolarization this sum is the envelope of the R-waves of the sublayers, but during repolarization it is the sum of overlapping portions of the T-waves of the sublayers and of the contiguity effects. Consider the left ventricle. In view of the results of van Dam and Durrer, we divide the left ventricle wall and the septum into three layers as shown in Fig. 11. A typical volume element and its spherical idealization is shown in Fig. 12. Evidently, the volumes of the sublayers increase in the direction of signal propagation until the intermediate layers are reached, and then they decrease as one passes toward the right-hand boundary of the volume element. As stated above, the dipole moment of each sublayer is proportional to its volume because it is proportional to the number of cells in that sublayer. It follows that the R-wave, i.e. the dipole moment of the volume element during the depolarization activity has a shape such as that shown in Fig. 13. This curve is the envelope of the dipole moments of the individual sublayers, as indicated in Fig. 12, and it is assumed to have the equation D(t) = A sing
t
= *A [ 1- cos 37i(tGh’3)]
Vol.
297,
No.
5, May
1974
in O
in -&A 6 t < A.
(13)
387
P. S. Thiry and R. M. Rosenberg This representation is such as to produce a much sharper rise to the maximum value than the subsequent descent back to zero. The reason for the representation (13) is that the endocardial layers start being activated through a
FIG 11. Durrer layers in left ventricle.
dense network of so-called “Purkinje fibers” in which the speed of the signal propagation is several times that in the myocardium (11).It is this effect which is modeled by (13) in the first third of the depolarization time of the volume element. The behavior in the latter two-thirds of the depolarization Direction of signal propagation
Al-l
Spherical idealization Volume element FIG. 12. Volume element of left ventricle and its spherical idealization.
time is similar to that found for a single spherical cell, and it reflects our view that the spherical volume element behaves qualitatively like a spherical cell. It should be pointed out, however, that our results are not sensitive to the particular choice of mathematical representation (13) ; other forms could
388
Journal of The Franklin Institute
On Electrophysiological
Activity of the Normal Heart
have been chosen as well including those possessing symmetry with respect to the maximum. For computation purposes, it is only necessary that some representation be made, and we chose (13) because it seemed convenient and realistic.
FIG. 13. R-wave
of volume
element
of left ventricle.
The action potential duration in the endocardium is longer than in the intermediate layer. Hence, the contiguity effect from the interface between them produces a dipole moment in, rather than opposing, the direction of signal propagation. Moreover, the action potential duration in the intermediate layer is shorter than in the epicardium; therefore, the contiguity effect from that interface produces a dipole moment in the direction opposite to that of signal propagation. The effects are added to the T-waves of the sublayers to obtain the dipole moment of a volume element of the left ventricle wall. The characteristic radial direction of the signal propagation across the ventricular wall which occurs in the thicker left ventricle is observed neither in the atria, nor in the right ventricle; moreover, the only experimental evidence of different lengths of action potential which exists is that across the left ventricle wall. Therefore, we have divided the remainder of the myocardium into volume elements each of which is considered electrically homogeneous; in other terms, in them M = 1, and no contiguity effects arise. The dipoles are taken to have the direction of signal propagation in the myocardium of each element, and the dipole moments are functions of the volume of the elements. The heart has been divided into eleven volume elements. This choice is based on the maps of isochrones published by Durrer et al. (9) as well as on the known propagation pattern of the impulse from the sino-atria1 node through the specialized conduction system of the heart to the different portions of the myocardium (14, 15). The electrical activity of the cells comprising the specialized conduction system do not contribute significantly
Vol.
297, 19
No. 6, May
1974
389
P. S. Thiry and R. M. Rosenberg to the electrocardiogram, since they represent a negligible volume in comparison with the myocardium elements which are considered (1). The eleven volume elements have been labeled as follows (see Fig. 14) : right atrium (R.A.), Ieft atrium (L.A.), Ieft posterior septum (L.P.S.), left anterior septum (L.A.S.), right septum (R.S.), left posterior ventricle (L.P.V.), left anterior ventricle (L.A.V.), middle left posterior ventricle (M.L.P.V.), middle left anterior ventricle (M.L.A.V.), right posterior ventricle (R.P.V.), and right anterior ventricle (R.A.V.).
Right
ventricle
Septum
Left ventricle
Section X-X
FIG. 14. Volume components of heart model.
As before (l), we define in the body a Cartesian coordinate system with origin at the centroid of the so-called “Einthoven triangle”. This is an equilateral triangle in the frontal plane with apex down and base at the Following normal practice in cardiology, we take the shoulder joints. positive x-axis horizontally to the left, the positive y-axis vertically down and the positive z-axis in the dorsal direction. Each volume element and the dipole Di(t) associated with it is characterized by (i) the dipole direction obtained from experimental results (9) and introduced through the direction cosines li, md, n, in the above coordinate system,
390
Journal of The Franklin Institute
On Electrophysiological
Activity of the Normal Heart
(ii) the amplitude Ai of its R-wave, a measure of the volume of the element, (iii) the depolarization time X, of each element obtained from experimental data (4), (iv) the number Mi of Durrer layers and the action potential duration of each and (v) the starting time of depolarization tsi which is the time of start of activation of the element relative to that of the sino-atria1 node. The numerical values of all these parameters are given in Table I. TABLE I value of parameters Volume element
i 1 2 3 4 5 6 7 8 9 10 11 -
VII.
Component R.A. L.A. L.P.S. L.A.S. R.S. L.P.V. L.A.V. M.L.P.V. M.L.A.V. R.P.V. R.A.V.
li
-
-
-
mi
0.5942 0.6944 0.7544 0.2824 0.7424 0.3215 0.6298 0.8867 0.0842 0.6189 0.6626
The Skin Surface
-
0.6957 0.7091 0.6330 0.9237 0,3462 0.9390 0.7323 0.3582 0.9622 0.6409 0.7359
ni
Ai
Mi
- 0.4035 - 0.1219 -0.1736 - 0.2588 0.5736 - 0.1219 0.2588 - 0.2924 0.2588 - O-4540 0.1392
0.225 0.313 0.630 0.476 0.321 2.368 1.186 1.564 1.699 0.626 0.387
1 1 3 3 1 3 3 3 3 1 1
P 70 70 200-185-190 20&185-190 200 200-185-190 200-185-190 20&185-190 20&18&190 200 200
Xi
t*i
69 66 60 51 60 54 60 60 60 69 69
25 45 130 130 133 140 145 140 145 150 150
Electrocardiograms
The ECG’s are recordings called “leads” of potentials relative to ground, or differences between such potentials, at certain selected points on the skin surface. A complete, standard ECG consists of twelve leads. Six of them are obtained from the potentials at the three corners of the Einthoven triangle, and the six remaining ones are potentials at six points of the rib cage. The disposition of these nine points on the body surface is shown in Fig. 15, and
FIG. 15. Skin surfwe electrode locations.
their Cartesian coordinates in the coordinate system described above are listed in Table II where 6 is the length of the side of the Einthoven triangle, and it is used as the unit of length.
Vol. 297, No. 6, May 1974
391
P. S. Thiry and R. M. Rosenberg TABLE
II
Skin surface x-component
points
y-component
- 0.5b
z-component
- 0.288673 - 0.288676
0.5b
0.57735b 0.0 0.0 O.lb 0.25b 0.25b 0.25b
0.0 - 0.12b - 0.06b 0.0 0.2b 0.33b 0.42b
-
0.0 0.0 0.0 0.3b 0.2753 0.3b 0.275b 0.2b 0.0
In computing the skin surface potentials, we regard the body as an electrically homogeneous, infinite volume conductor in which the skin surface points are simply points in that volume conductor. Admittedly, this is a simplification of a very complex situation because the body with the lungs, varieties of tissue, bone and voids is surely not a homogeneous volume conductor. A number of investigations have been made [see, for instance (16, 17)] which take certain of these inhomogeneities into account in more or less realistic ways. This leads, in general, to a three-dimensional boundary value problem with very complicated boundaries, and to overwhelmingly complex computations. To avoid these complexities, we consider an “equivalent body” which is imagined electrically homogeneous and immersed in an infinite, electrically homogeneous medium having the same electrical properties as the equivalent body. The equivalence is such that the potentials induced by a dipole of given location, direction and moment induces identical potentials at identical points in the actual body, and in its equivalent. Under these assumptions, consider a point Pj having position vector rj relative to the origin of the Cartesian coordinate system. The potential induced at Pj by m dipoles of different directions and moments, located at 0, is
= Ko~AiSi(t)ei-rj/r~,
(14)
i=l
where K, is an electrical constant taking into account the conductivity properties of the body, e, is the unit vector in the direction of the dipole Di(t), Di(t) = &6,(t) is the moment of that dipole, and it has maximum amplitude Ai. We denote as electrocardiographic constants of our model the quantities Cij = K, Ai ei *r*/r,“. (15)
39.2
JournalofThe FranklinInstitute
1 2 3 4 5 6 7 8 9 10 11
Component \
Lead
0.463 0.753 - 1.646 0.466 0.825 2.637 2.588 4.804 0.495 - 1.342 - 0.888
I
-
-
-
0.701 1.042 0.373 1.086 0.079 7.989 1.312 0.721 5.152 1.875 1.299
II
-
-
-
0.238 0.289 2.020 1.552 0.746 5.352 3.899 4.083 4.657 0.533 0.410
III
- 0.582 - 0.898 0.637 0.310 - 0.452 - 5.313 - 0.638 - 2.762 - 2.824 1.608 1.093
aVR
Modified
-
-
-
III
0.113 0.232 1.333 1.009 0.786 1.357 3.244 4.443 2.081 0.405 0.239
aVL
-
-
-
0.470 0.666 1.196 1.319 0.333 6.671 2.606 1.681 4.905 1.204 0.854
aVF
electrocardiographic
TABLE
0.332 - 0.434 2.664 0.617 - 2.485 - 0.142 - 5.387 - 0.867 - 4.420 - 1.149 1.391
Vl
constan,ts
0.760 -0.114 2.628 l-158 - 2.912 1.510 - 5.795 1.908 - 5.808 - 2.462 1.354
v2
-
-
-
1.356 1.064 2.299 0.222 2.098 9.770 5.659 2.566 0.998 3.965 0.390
v3
1.208 1.456 0.462 - 0.654 - 0.409 10.476 - 2.024 3.499 4.209 - 3,405 - 1.432
V4
-
-
-
1.043 1.387 0.363 0,421 0.144 8.898 0.329 4.208 3.786 2.932 1.492
V5
-1.532
0.816 1.257 -0.856 -0.458 0.619 7.499 0.828 3.788 4.015 -2.253
V6
03: u
i?
q e
g
b c si ;: g 3 g $ c;* E k
9
P. 8. Thiry and R. M. Rosenberg In Table III we list the values of the modified electrocardiographic
bij = (b’/Ko) Cij, bik = bi,-6..
constants
(16)
23
where the cii, b and K,, have already been defined. The ECG leads generated by our model are shown next to a normal 12-lead clinical electrocardiogram in Fig. 16. A comparison between them shows that the model has reproduced the features of the clinical ECG in all essential details. In particular, we point to the following gross features common to the clinical and the computed ECG’s. In both, Leads III, aVL and aVF are relatively inactive, and aVR looks a great deal like Lead II turned upside down, and both V5 and V6 resemble Leads I and II except for much larger R-waves in the former. However, even in the finer details do we find good agreement. Thus, while Vl and V2 are skin surface potentials of points lying near each other (see Fig. 16), they do not resemble each other very much. In particular, Vl has a negative T-wave and a diphasic P-wave; this latter feature is not easy to see on first glance, but it is present in the computed and the clinical trace. In contradistinction, V2 has positive Pand T-waves and an abnormally large S-wave. Also, V5 has a much larger T-wave than VS even though these are skin surface potentials of neighboring points. The features just described are not necessarily typical of all normal ECG’s ; they are characteristic for a so-called “45 degree heart”. The resultant of all the component dipoles of the heart has a predominant direction when viewed from the frontal plane, and at no time does it deviate very much from that direction. When it does, the strength of the resultant dipole is relatively small. This predominant direction determines the electrical orientation of the heart. If we wish to generate the normal ECG’s or model for an orientation other than that shown, we achieve this very simply by an appropriate coordinate axis rotation. The model described here is suitable for generating ECG’s of hearts with lesions and/or conduction defects. Some of that work will be described elsewhere. Acknowledgement We are indebted to C. Ablow, Stanford Research Institute, Menlo Park, California, for the demonstration of Eq. (7), given on p. 382 because we consider his proof much “prettier” than our own earlier one. References
(1) R. M. Rosenberg, C. H. Chao and J. Abbott,
“A new mathematical model of electrical cardiac activity”, Math. Biosci., Vol. 14, p. 367, 1972. (2) L. H. Nahum and H. E. Hoff, “The nature of the precordial electrocardiogram”, Am. J. Physiol., Vol. 155, p. 215, 1948. (3) E. W. Reynolds and C. R. van der Ark, “An experimental study on the origin of T-waves based on determinations of effective refractory period from epicardial and endocardial aspects of the ventricle”, Circ. Rerr., Vol. 7, p. 943, 1969.
394
Joumal of The Franklin Institute
On Electrophysiological
~
Vol.
297,
No.
5, May
1974
-
Activity of the Normal Heart
.
395
P. X. Thiry and R. M. Rosenberg (4) R.
Th. van Dam and D. Durrer, “Experimental study on the intramural distribution of the excitability cycle and on the form of the epicardial T-wave in the dog heart in situ”, Am. Heart J., Vol. 61, p. 537, 1961. (5) K. Harumi, M. J. Burgess and J. A. Abildskov, “A theoretic model of the T-wave”, Circulation, Vol. 34, p. 659, 1966. (6) M. J. Burgess, K. Harumi and J. A. Abildskov, “Application of a theoretic model to experimentally induced T-wave abnormalities”, Circulation, Vol. 34, p. 669, 1966. in “Advances in Electro(7) M. J. Burgess, “Physiologic basis of the T-wave”, cardiography” (ed. by R. C. Schlant and J. W. Hurst), p. 367, Grune & Stratton, New York, 1972. (8) F. N. Wilson, A. G. MacLeod, P. S. Barker and F. D. Johnston, “The determination and significance of the areas of ventricular deflections of the electrocardiogram”, Am. Heart J., Vol. 10, p. 46, 1934. (9) D. Durrer, R. Th. van Dam, G. E. Freud, M. J. Janse, F. L. Meijler and R. C. Arzbaecher, “Total excitation of the isolated human heart”, Circulation, Vol. 41, p. 899, 1970. (10) M. Zloof, R. M. Rosenberg and J. Abbott, “A computer model for atrioventricular blocks”, Math. Biosci., Vol. 18, p. 87, 1973. (11) B. Hoffman and P. Cranefield, “Electrophysiology of the Heart”, McGraw-Hill, New York, 1960. (12) A. M. Scher, “Electrical correlates of the cardiac cycle”, in “Physiology and Biophysics” (ed. by T. C. Ruth and H. D. Patton), p. 565, Saunders, Philadelphia, 1965. (13) R. Plonsey, “Bioelectric Phenomena”, McGraw-Hill, New York, 1969. (14) T. N. James, “The specialized conducting tissue of the atria”, in “Mechanisms and Therapy of Cardiac Arrhythmias” (ed. by L. S. Dreifus and W. Likoff), Grune & Stratton, New York, 1966. “Atrioventricular and intraventricular (15) H. H. Hecht and C. E. Kossmann, conduction”, Am. J. Cardiol., Vol. 31, p. 232, 1973. (16) R. H. Okada, “Potential produced by an eccentric current dipole in a finite length conducting cylinder”, IRE Trans. Med. Electron., Vol. 7, p. 14, 1956. model of the genesis (17) H. L. Gelernter and J. C. Swihart, “A mathematical-physical of the electrocardiogram”, Biophys. J., Vol. 4, p. 285, 1964.
396
Journal
of The Franklin
Institute
P. s. THIRY
and R. M. ROSENBERG
Department of Mechanical Engineering University of California, Berkeley, California
ABSTRACT:
The heart is divided into eleven componenta, each being modeled by a dipole of
known and fixed location and direction; these dipoles are shown to result from the propagation of the tranemembrane potentials over the surface of the myocardial cella. The components of the left ventricular wall are composed of “Durrer layers”, and these in combination with the
“contiguity effect” result in a positive ventricular gradient and
cardiograms
generated by the eleven dipoles yield high-fidelity
12-M
T-wave.
Electro-
ECG’s.
1. Introduction
In a recent paper (l),a mathematical model of electrical heart activity was described which consists of a finite number of dipoles of known and fixed location and direction, but of variable strength. One of the stated assumptions in that paper is “. . . that the moment of each dipole is proportional to the monophasic action potential of the heart component modeled by it”, i.e. modeled by that dipole. In this paper, we abandon that assumption; in fact, we remove the question of dipole strength from the domain of assumptions and make it, instead, the subject of a detailed analytical study on the level of the cell. In constructing the present model we come to grips with the “puzzle of the T-wave”. Heart muscle, like muscle in general, contracts when the extremities of the muscle fiber are subjected to an electrical potential difference, and it regains its original length when the potential is removed. The heart is known to generate electrical potentials which cause the muscle to contract and, thus, the heart to beat. These potentials are produced by a periodic exchange of sodium and potassium ions across the boundaries or membranes of the myocardial cells, and the resulting transmembrane action potentials can be, and have been, measured by placing microelectrodes on either side of the cell membrane. These potentials look somewhat as shown in Fig. 1. A nonzero transmembrane potential is a potential difference between neighboring points and, thus, an electrical dipole with direction normal to the membrane, and with moment proportional to the transmembrane action potential at the dipole location. With the body regarded as a volume conductor, the dipoles produced by this electrical heart activity generate a potential at every point in the body as well as on the skin surface. These skin surface potentials, measured at certain selected points, are recorded and are the electrocardiograms or ECG’s. A typical recording is shown in Fig. 2.
377
P. X. Thiry and R. M. Rosenberg In it, the activation of the atria is totally represented by the P-wave; so-called QRS complex and the T-wave are due to ventricular activity.
the
-6O-
Depolarization Period
FIG. 1. Typical transmembrane
action potential.
The “puzzle of the T-wave” is this: The QRS complex occurs during so-called “ventricular depolarization”, and the T-wave during “repolarization”. Now, it is known (a property which will be examined in detail in the next sections) that if the ECG response is positive (R-wave up) during depolarization, it should be negative during repolarization (T-wave down). However, as can be seen from Fig. 2, the T-wave of clinical skin surface ECG’s is in general up, not down.
FIG. 2. Normal Lead II electrocardiogram.
For more than forty years, efforts have been made to understand the mechanism of repolarization and of the T-wave. It has been suggested that the sequence of repolarization follows a path through the ventricle different however, there does not appear to be any from that of depolarization; anatomical evidence to support this view. Other theories are that the T-wave polarity is governed by the repolarization sequence across the ventricular wall (2), that it is produced by differences in repolarization time between distant areas of the ventricle (3), and between adjacent areas (4). A brilliant investigation, associated with the names of Burgess, Harumi and Abildskov (5, 6) and summarized recently by Burgess (7), explains the unexpected polarity of the T-wave by means of experimental observations by van Dam and Durrer (4) who discovered that the action potential duration
378
Journal of The Franklin Institute
On Electrophysiologicak Activity of the Normal Heart differs for some adjacent layers of the ventricle. Here, Burgess and her collaborators echoed the remarkable statement made already by Wilson in 1934 (8) that “if two elements of a muscle fiber differ in their state of activity, one being nearer to or farther away from the resting state than the other, there must be an electromotive force across and normal to the plane that separates them”. In the Burgess model, electrical cardiac activity is treated as a traveling wave phenomenon, resulting in a very elegant formulation of the problem. The Burgess group tested their model in three ways: once, by comparing T-wave forms of normal ECG’s, then, when ventricular recovery periods were altered by local heating of prescribed areas of the ventricle and, finally, when infarction was produced by stopping the coronary blood supply to The comparisons between observed and certain areas in the ventricle. predicted T-wave forms were impressive in all cases. Our model is also based on Wilson’s observation and on the experimental results of van Dam and Durrer (4). Nevertheless, it is so different from that of the Burgess group as to hardly resemble it. In our model, the heart is divided into a number of components, each being modeled by a dipole of fixed location and direction while, in the Burgess model, the heart is considered as a whole, and dipoles are computed from the known wavefront advance through the heart. We have retained the separation into components of our earlier model in order to preserve our conduction mechanism model described elsewhere (10). In this way, we can introduce partial signal conduction block of any degree, or complete block, at any point or points of the conduction system. We also feel that the existence of highly localized conditions in the myocardium, such as lesions, is easily modeled when the heart is subdivided into components, but becomes difficult in the wavefront model because of the treatment of the line integrals which must be computed when the wavefront does not coincide with the boundaries of the lesions.
II.
The Single
Cell
As stated above, the transmembrane action potential is the result of a periodic ion exchange across the cell boundary. This exchange does not take place simultaneously at all points of the cell membrane; rather, it starts at one point and proceeds in the direction of signal propagation to all other points of the membrane. In fact, from the known propagation velocity through the myocardium and the size and shape of the cell, one can readily compute the propagation velocity along the cell surface. Ideally, one would wish to build a mathematical model of the cell membrane and of the process of ion penetration, and then obtain the transmembrane potential as the output of that model. Unfortunately, this mechanism is not sufficiently well understood to permit construction of a model, and cell membrane action remains for the present one of the intriguing
Vol.297,No. 5,May 1974
379
P. 8. Thiry ad
R. M. Rosenberg
mysteries of biology. We are therefore obliged to accept the transmembrane potential as observed, and to give it some functional representation. Not all myocardial tissue is alike, and the transmembrane potentials of cells belonging to different types look different. However, in this paper we are primarily concerned with tissue contributing to the skin surface potentials, and for that tissue the transmembrane potential is well represented by*
f(t) z 0
in O
f(t) = k > 0
in t,Qt
f(t) = k l-(t-~~o~P)2] [
in t,+p
f(t) = k
p,+p+
loo-t)2 5000
\
I
(1)
in t,+~+50Gt
The graph off(t) is shown in Fig. 3. The numerical values which appear in (1) have been chosen so as to give a realistic action potential with f(t) in millivolts (mV) and t in milliseconds (msec). The symbol k is an “amplitude constant”, and p gives the duration of the “depolarized state”. Since the depolarization and repolarization durations are fixed, p is a measure of the action potential duration.
T
I
k
FIG. 3. Graph of mathematical
representation of transmembrane
action potential.
The particular representation of f(t) as given in (1) is not essential to our model. Any other choice resulting in a graph similar to Fig. 3 would do just as well; we simply find (1) a convenient and adequate representation. We shall now calculate the dipole generated when the action potential propagates over the surface of a spherical cell of diameter 2r in the direction *For convenience, we have taken the resting potential as the datum. This is equivalent to adding a constant to realistic action potentials; this addition has no effect on the resultant dipole of the cell.
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Journal of The Franklin Inst,itute
On Electrophysiological
Activity of the Normal Heart
of a diameter A, B, as shown in Fig. 4. If the (known) speed of signal propagation in the myocardium is v, the speed of propagation (assumed uniform) over the cell surface is 7, and 27 = (v/r)?W = 7rv.
(2)
Let the cell membrane be activated at the time t,; then, a surface element located at C (see Fig. 4) will be activated at the time (3)
t = t,+P+,
where /I is the angle from OA to OC. We obtain the resultant dipole for the spherical cell at any time t by integrating the dipole per unit area f(t) over the cell surface. The moment of f(t) is at each point proportional to the transmembrane action potential, and f(t) is normal to the cell surface.
FIG.
4.
Signal propagation along surface of spherical cell.
The resultant dipole denoted by D(t) is for all time parallel to AB, and if e, is the unit vector from A to 0 (i.e. positive in the direction of propagation), a simple calculation shows that the resultant dipole is given by
D(t) = nrz[sin2~f(t-t,-~)d~e,,
(4)
and its moment is D(t) = ~r2~sin2~f(t-t0-~)
db
(5)
withf(t) as given in (1). For the numerical values v = O-6 m/see, r = 0.003 m, p = 200 msec and k = 100 mV, the graph of D(t) looks as shown in Fig. 5 in which the amplitude of the first peak is nPk.
Vol. 297, No. 5, May 1974
381
P. S. Thiry and R. M. Rosenberg In analogyto ECG terminology we shall refer to the first peak as the R-wave and the (negative) second peak as the T-wave. It is easy to show that the R-wave in Fig. 5 is a translated, inverted cosine wave. Consider the surface
in the /I, T,f-space; it is shown in Fig. 6. One sees that f = 0 in 0 < r < BY/T, and f=k in @-/~
s vlr
D(t) = 77r2k
sin2/3d/3 = +rzk(l-cos:T).
0
For T in nr/q
FIG. 5. Resultant
ZZZ. The
Gradient
of a Single
dipole moment
of spherical
cell.
Cell
The fact that the R- and T-waves have opposite the much stronger result that, in fact,
polarity
follows from
(7)
where r2 is the time at which the repolarization phase of the transmembrane potential is completed at point A (see Figs. 4 and 6). In analogy to terminology introduced originally by Wilson et al. (8) and frequently used in cardiology, we shall refer to G as the “cell gradient”; hence, (7) states that the cell gradient vanishes. Wilson’s “ventricular gradient” is the time integral of the &RX complex and the T-wave. It is a sort of global measure of the vector quantity D(t) over ventricular activity which makes his nomenclature seem apt. In general, the ventricular gradient of the normal heart is positive, not zero which is another description of the “puzzle of the T-wave”. For the proof of (7), consider Fig. 7, and denote f(r, 0) in O
382
Journal
of The Franklin
Institute
Activity of the Normal Heart
On Electrophysiological
phase of a transmembrane potential. We shall show that jfD(~) dr is independent of f&T) and depends on k only. From this demonstration, (7) follows directly when k is replaced by -k. f
T
\ \ \ \ \ \ \
Td?)
.\
P FIG.
FIG.
Except
7.
6.
Transmembrane
potential in space-time
domain.
Arbitrary depolarization phase in space-time
for a constant,
domain.
we have D(T) =.(l”f
(T-F)
sin2jId/3,
(8)
and D(0) = 0 because f( -pr/q) = 0 for 0 < /3 < rr. Moreover, D(T) = 0 for 7 2 E = D + (‘rrr/q) because in T 2 E, f- k and (8) is zero in that case. Then
Now, the first double integral in (9) vanishes becausef= 0 on OAB. Moreover, the third double integral is a function of k only, say K(k), because f 2 k on CDE. Finally, the second double integral is
(10)
Vol. 297, No. 5, lCIaJ1974
383
P. 8. Thiry and R. H. Rosenberg With r-pr/~
= 9, the integral
which is independent
of j3. Hence, (10) is I OBCD= @(f, D)
llsin Z/3d/3 = 0. s0
(11)
It follows that %(,)dr s0
= K(k).
(12)
This completes the proof. It is surprising that the only contribution to the integral (12) is made in the time interval which begins when the depolarization is completed at A, and which ends when the depolarization is completed at B. It should be pointed out that, while the time integral of the R-wave is independent of the shape or duration of the depolarization phase of the transmembrane potential, the R-wave itself depends strongly on both. In particular, the R-wave has the shape of an inverted cosine wave (see Eq. (6)) only when the depolarization of the transmembrane potential is a step function.
IV.
Some
Electrophysiological
Observations
When constructing a mathematical model of electrical heart activity, one needs to know both the time at which cellular electrical activity begins at every point of the myocardium, and the shape and duration of the transmembrane action potentials for every point in the heart. Probably, the most authoritative information on both of these questions is contained in work done at Amsterdam, and associated largely with the name of Durrer. In some recent experimental work on the excitation sequence in the isolated human heart, Durrer et aJ. (9) have obtained detailed information on the distribution, in time and space, of the depolarization phase through the myocardium. From control experiments on dogs, they found that the essential difference between observations in situ and in the isolated heart was in the propagation velocity, but the isochrones* remained similar in both cases. Thus, we have utilized their excitation sequence combined with propagation velocities observed in the heart in situ (9, 11). The question of action potential durations was examined by van Dam and Durrer (4) in an earlier paper where they studied the duration of the “functional refractory period” of the dog heart in situ. Their observations permit the conclusion that the action potential duration in the left ventricular wall is longest in the endocardial layers, shortest in the middle layers and of * Isochrones are the surfaces containing all points of contemporaneous excit&ion.
384
Journal
of The Franklin
Institute
On Electrophysiological
Activity of the Normal Heart
intermediate duration in the epicardial layers ; for simplicity, we shall call these “Durrer layers”. It would lead us too far afield to discuss here the relation between the duration of the refractory period and of the action potential; anyone interested in this question is referred to Hoffman and Cranefield (11)and van Dam and Durrer (4). We merely point out that these time intervals are closely related and, when the functional refractory period is longer in one location of the myocardium than in another, so is the action potential duration. We shall, in fact, make the assumption that these periods are directly proportional ; i.e. when the ratio of functional refractory periods at two different locations is n, so is the ratio of action potential durations. We shall show that it is the difference in action potential durations between contiguous Durrer layers of myocardial tissue which is responsible for the positive polarity of the T-wave but, to our knowledge, Burgess et al. (S-7) were the first to utilize that property to explain the T-wave mechanism.
V.
The Dipole
of Two
Cells
Consider two spherical cells, Cell I and II with their centers lying in the direction of signal propagation as shown in Fig. 8(a) ; let the points B of minimum distance between them be connected by a wire in which the velocity of signal propagation is infinite. This wire is a model of the so-called “intercalated disc”, a small high-conductance element forming the electrical connection between neighboring cells along the conduction path. The conductance of the intercalated discs is on the order of 400 times that of the cell membrane (12,13).
FIG.
8. Two noncontiguous
cells and their dipole moment.
If the transmembrane action potentials of the two cells were identical in all respects their combined dipole moment D(t) would simply be the sum of two curves like that in Fig. 5, but with the curve of Cell II shifted to the right in the amount n-r/q relative to that of Cell I. However, we shall suppose that the action potential duration of Cell II is shorter than that of Cell I. Thus, these cells belong to different Durrer layers, and we imagine them to lie immediately next to the boundary between these layers. The dipole moments of these cells are shown in Fig. 8(b). In that diagram it was assumed that the difference in action potential durations satisfies p, -pI1 > n’r/v. This
Vol. 297,No. 6, May 1974
385
P. S. Thiry and R. M. Rosenberg has the effect that the T-wave of Cell II begins earlier It is not at all essential to our argument that this be the is pr-prr > 0. Evidently, the cell gradient of each cell, sum, vanishes in accordance with (7), and the T-wave of these two cells is necessarily negative.
than that of Ceil I. case ; all we require and hence of their of the combination
-. (b)
****-- (L contiguity effect ’ FIG. 9. Two contiguous cells and their dipole moment.
Now let the length of the wire connecting Cells I and II tend to zero, so that the two cells become contiguous as shown in Fig. 9(a). In that case, a new effect appears which we call the “contiguity effect”. To see this, consider the two transmembrane action potentials at the points B, shown in Fig. 10. Evidently, between B in Cell I and B in Cell II there exists a positive potential difference during the time interval
where t, is the time of activation of point B. The interval of existence of this nonzero potential difference extends over the repolarization times of both
f(t)
---I Cell I Cell II
FIG. 10. Action potential at B of Cells I and II.
cells. This potential difference between two neighboring points is a dipole having positive moment, and it would be absent if the cells were not
386
Journal of The Franklin Institute
On Electrophysiological
Activity of the Normal Heart
contiguous. The resultant dipole moment of shown in Fig. 9(b). Under the assumption pr the contiguity effect to the T-wave is positive result in a positive T-wave; for pI -p,, < 0, the effect is negative. In either case, the contiguity in nonzero two-cell gradients.
the two contiguous cells is -p,, > 0, the contribution of and, if sufficiently large, can contribution of the contiguity effect will, in general, result
VI.
Volume
Elements
of the Myocardium
To model electrical cardiac activity, we decompose the myocardium into a number of volume elements which are then idealized as being spherical in shape. Every element is considered to consist of a small number M of contiguous electrophysiologically different Durrer layers and each electrically homogeneous Durrer layer is imagined to consist of a much larger number N of “sublayers”. The boundaries between the layers and the sublayers are normal to the direction of signal propagation. Every volume element is represented by an electrical dipole of known and fixed location and direction, but of variable moment. The dipole moment of the entire volume element is the sum of the dipole moments of all sublayers and of the contiguity effects (if any; i.e. if 1M> 1). Each of the dipole moments has the general shape shown in Fig. 5, but the amplitude of each is proportional to the volume of the sublayer; i.e. to the number of cells in it. Of course, the dipoles of the sublayers must be added with the proper time relation retained between them. During depolarization this sum is the envelope of the R-waves of the sublayers, but during repolarization it is the sum of overlapping portions of the T-waves of the sublayers and of the contiguity effects. Consider the left ventricle. In view of the results of van Dam and Durrer, we divide the left ventricle wall and the septum into three layers as shown in Fig. 11. A typical volume element and its spherical idealization is shown in Fig. 12. Evidently, the volumes of the sublayers increase in the direction of signal propagation until the intermediate layers are reached, and then they decrease as one passes toward the right-hand boundary of the volume element. As stated above, the dipole moment of each sublayer is proportional to its volume because it is proportional to the number of cells in that sublayer. It follows that the R-wave, i.e. the dipole moment of the volume element during the depolarization activity has a shape such as that shown in Fig. 13. This curve is the envelope of the dipole moments of the individual sublayers, as indicated in Fig. 12, and it is assumed to have the equation D(t) = A sing
t
= *A [ 1- cos 37i(tGh’3)]
Vol.
297,
No.
5, May
1974
in O
in -&A 6 t < A.
(13)
387
P. S. Thiry and R. M. Rosenberg This representation is such as to produce a much sharper rise to the maximum value than the subsequent descent back to zero. The reason for the representation (13) is that the endocardial layers start being activated through a
FIG 11. Durrer layers in left ventricle.
dense network of so-called “Purkinje fibers” in which the speed of the signal propagation is several times that in the myocardium (11).It is this effect which is modeled by (13) in the first third of the depolarization time of the volume element. The behavior in the latter two-thirds of the depolarization Direction of signal propagation
Al-l
Spherical idealization Volume element FIG. 12. Volume element of left ventricle and its spherical idealization.
time is similar to that found for a single spherical cell, and it reflects our view that the spherical volume element behaves qualitatively like a spherical cell. It should be pointed out, however, that our results are not sensitive to the particular choice of mathematical representation (13) ; other forms could
388
Journal of The Franklin Institute
On Electrophysiological
Activity of the Normal Heart
have been chosen as well including those possessing symmetry with respect to the maximum. For computation purposes, it is only necessary that some representation be made, and we chose (13) because it seemed convenient and realistic.
FIG. 13. R-wave
of volume
element
of left ventricle.
The action potential duration in the endocardium is longer than in the intermediate layer. Hence, the contiguity effect from the interface between them produces a dipole moment in, rather than opposing, the direction of signal propagation. Moreover, the action potential duration in the intermediate layer is shorter than in the epicardium; therefore, the contiguity effect from that interface produces a dipole moment in the direction opposite to that of signal propagation. The effects are added to the T-waves of the sublayers to obtain the dipole moment of a volume element of the left ventricle wall. The characteristic radial direction of the signal propagation across the ventricular wall which occurs in the thicker left ventricle is observed neither in the atria, nor in the right ventricle; moreover, the only experimental evidence of different lengths of action potential which exists is that across the left ventricle wall. Therefore, we have divided the remainder of the myocardium into volume elements each of which is considered electrically homogeneous; in other terms, in them M = 1, and no contiguity effects arise. The dipoles are taken to have the direction of signal propagation in the myocardium of each element, and the dipole moments are functions of the volume of the elements. The heart has been divided into eleven volume elements. This choice is based on the maps of isochrones published by Durrer et al. (9) as well as on the known propagation pattern of the impulse from the sino-atria1 node through the specialized conduction system of the heart to the different portions of the myocardium (14, 15). The electrical activity of the cells comprising the specialized conduction system do not contribute significantly
Vol.
297, 19
No. 6, May
1974
389
P. S. Thiry and R. M. Rosenberg to the electrocardiogram, since they represent a negligible volume in comparison with the myocardium elements which are considered (1). The eleven volume elements have been labeled as follows (see Fig. 14) : right atrium (R.A.), Ieft atrium (L.A.), Ieft posterior septum (L.P.S.), left anterior septum (L.A.S.), right septum (R.S.), left posterior ventricle (L.P.V.), left anterior ventricle (L.A.V.), middle left posterior ventricle (M.L.P.V.), middle left anterior ventricle (M.L.A.V.), right posterior ventricle (R.P.V.), and right anterior ventricle (R.A.V.).
Right
ventricle
Septum
Left ventricle
Section X-X
FIG. 14. Volume components of heart model.
As before (l), we define in the body a Cartesian coordinate system with origin at the centroid of the so-called “Einthoven triangle”. This is an equilateral triangle in the frontal plane with apex down and base at the Following normal practice in cardiology, we take the shoulder joints. positive x-axis horizontally to the left, the positive y-axis vertically down and the positive z-axis in the dorsal direction. Each volume element and the dipole Di(t) associated with it is characterized by (i) the dipole direction obtained from experimental results (9) and introduced through the direction cosines li, md, n, in the above coordinate system,
390
Journal of The Franklin Institute
On Electrophysiological
Activity of the Normal Heart
(ii) the amplitude Ai of its R-wave, a measure of the volume of the element, (iii) the depolarization time X, of each element obtained from experimental data (4), (iv) the number Mi of Durrer layers and the action potential duration of each and (v) the starting time of depolarization tsi which is the time of start of activation of the element relative to that of the sino-atria1 node. The numerical values of all these parameters are given in Table I. TABLE I value of parameters Volume element
i 1 2 3 4 5 6 7 8 9 10 11 -
VII.
Component R.A. L.A. L.P.S. L.A.S. R.S. L.P.V. L.A.V. M.L.P.V. M.L.A.V. R.P.V. R.A.V.
li
-
-
-
mi
0.5942 0.6944 0.7544 0.2824 0.7424 0.3215 0.6298 0.8867 0.0842 0.6189 0.6626
The Skin Surface
-
0.6957 0.7091 0.6330 0.9237 0,3462 0.9390 0.7323 0.3582 0.9622 0.6409 0.7359
ni
Ai
Mi
- 0.4035 - 0.1219 -0.1736 - 0.2588 0.5736 - 0.1219 0.2588 - 0.2924 0.2588 - O-4540 0.1392
0.225 0.313 0.630 0.476 0.321 2.368 1.186 1.564 1.699 0.626 0.387
1 1 3 3 1 3 3 3 3 1 1
P 70 70 200-185-190 20&185-190 200 200-185-190 200-185-190 20&185-190 20&18&190 200 200
Xi
t*i
69 66 60 51 60 54 60 60 60 69 69
25 45 130 130 133 140 145 140 145 150 150
Electrocardiograms
The ECG’s are recordings called “leads” of potentials relative to ground, or differences between such potentials, at certain selected points on the skin surface. A complete, standard ECG consists of twelve leads. Six of them are obtained from the potentials at the three corners of the Einthoven triangle, and the six remaining ones are potentials at six points of the rib cage. The disposition of these nine points on the body surface is shown in Fig. 15, and
FIG. 15. Skin surfwe electrode locations.
their Cartesian coordinates in the coordinate system described above are listed in Table II where 6 is the length of the side of the Einthoven triangle, and it is used as the unit of length.
Vol. 297, No. 6, May 1974
391
P. S. Thiry and R. M. Rosenberg TABLE
II
Skin surface x-component
points
y-component
- 0.5b
z-component
- 0.288673 - 0.288676
0.5b
0.57735b 0.0 0.0 O.lb 0.25b 0.25b 0.25b
0.0 - 0.12b - 0.06b 0.0 0.2b 0.33b 0.42b
-
0.0 0.0 0.0 0.3b 0.2753 0.3b 0.275b 0.2b 0.0
In computing the skin surface potentials, we regard the body as an electrically homogeneous, infinite volume conductor in which the skin surface points are simply points in that volume conductor. Admittedly, this is a simplification of a very complex situation because the body with the lungs, varieties of tissue, bone and voids is surely not a homogeneous volume conductor. A number of investigations have been made [see, for instance (16, 17)] which take certain of these inhomogeneities into account in more or less realistic ways. This leads, in general, to a three-dimensional boundary value problem with very complicated boundaries, and to overwhelmingly complex computations. To avoid these complexities, we consider an “equivalent body” which is imagined electrically homogeneous and immersed in an infinite, electrically homogeneous medium having the same electrical properties as the equivalent body. The equivalence is such that the potentials induced by a dipole of given location, direction and moment induces identical potentials at identical points in the actual body, and in its equivalent. Under these assumptions, consider a point Pj having position vector rj relative to the origin of the Cartesian coordinate system. The potential induced at Pj by m dipoles of different directions and moments, located at 0, is
= Ko~AiSi(t)ei-rj/r~,
(14)
i=l
where K, is an electrical constant taking into account the conductivity properties of the body, e, is the unit vector in the direction of the dipole Di(t), Di(t) = &6,(t) is the moment of that dipole, and it has maximum amplitude Ai. We denote as electrocardiographic constants of our model the quantities Cij = K, Ai ei *r*/r,“. (15)
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JournalofThe FranklinInstitute
1 2 3 4 5 6 7 8 9 10 11
Component \
Lead
0.463 0.753 - 1.646 0.466 0.825 2.637 2.588 4.804 0.495 - 1.342 - 0.888
I
-
-
-
0.701 1.042 0.373 1.086 0.079 7.989 1.312 0.721 5.152 1.875 1.299
II
-
-
-
0.238 0.289 2.020 1.552 0.746 5.352 3.899 4.083 4.657 0.533 0.410
III
- 0.582 - 0.898 0.637 0.310 - 0.452 - 5.313 - 0.638 - 2.762 - 2.824 1.608 1.093
aVR
Modified
-
-
-
III
0.113 0.232 1.333 1.009 0.786 1.357 3.244 4.443 2.081 0.405 0.239
aVL
-
-
-
0.470 0.666 1.196 1.319 0.333 6.671 2.606 1.681 4.905 1.204 0.854
aVF
electrocardiographic
TABLE
0.332 - 0.434 2.664 0.617 - 2.485 - 0.142 - 5.387 - 0.867 - 4.420 - 1.149 1.391
Vl
constan,ts
0.760 -0.114 2.628 l-158 - 2.912 1.510 - 5.795 1.908 - 5.808 - 2.462 1.354
v2
-
-
-
1.356 1.064 2.299 0.222 2.098 9.770 5.659 2.566 0.998 3.965 0.390
v3
1.208 1.456 0.462 - 0.654 - 0.409 10.476 - 2.024 3.499 4.209 - 3,405 - 1.432
V4
-
-
-
1.043 1.387 0.363 0,421 0.144 8.898 0.329 4.208 3.786 2.932 1.492
V5
-1.532
0.816 1.257 -0.856 -0.458 0.619 7.499 0.828 3.788 4.015 -2.253
V6
03: u
i?
q e
g
b c si ;: g 3 g $ c;* E k
9
P. 8. Thiry and R. M. Rosenberg In Table III we list the values of the modified electrocardiographic
bij = (b’/Ko) Cij, bik = bi,-6..
constants
(16)
23
where the cii, b and K,, have already been defined. The ECG leads generated by our model are shown next to a normal 12-lead clinical electrocardiogram in Fig. 16. A comparison between them shows that the model has reproduced the features of the clinical ECG in all essential details. In particular, we point to the following gross features common to the clinical and the computed ECG’s. In both, Leads III, aVL and aVF are relatively inactive, and aVR looks a great deal like Lead II turned upside down, and both V5 and V6 resemble Leads I and II except for much larger R-waves in the former. However, even in the finer details do we find good agreement. Thus, while Vl and V2 are skin surface potentials of points lying near each other (see Fig. 16), they do not resemble each other very much. In particular, Vl has a negative T-wave and a diphasic P-wave; this latter feature is not easy to see on first glance, but it is present in the computed and the clinical trace. In contradistinction, V2 has positive Pand T-waves and an abnormally large S-wave. Also, V5 has a much larger T-wave than VS even though these are skin surface potentials of neighboring points. The features just described are not necessarily typical of all normal ECG’s ; they are characteristic for a so-called “45 degree heart”. The resultant of all the component dipoles of the heart has a predominant direction when viewed from the frontal plane, and at no time does it deviate very much from that direction. When it does, the strength of the resultant dipole is relatively small. This predominant direction determines the electrical orientation of the heart. If we wish to generate the normal ECG’s or model for an orientation other than that shown, we achieve this very simply by an appropriate coordinate axis rotation. The model described here is suitable for generating ECG’s of hearts with lesions and/or conduction defects. Some of that work will be described elsewhere. Acknowledgement We are indebted to C. Ablow, Stanford Research Institute, Menlo Park, California, for the demonstration of Eq. (7), given on p. 382 because we consider his proof much “prettier” than our own earlier one. References
(1) R. M. Rosenberg, C. H. Chao and J. Abbott,
“A new mathematical model of electrical cardiac activity”, Math. Biosci., Vol. 14, p. 367, 1972. (2) L. H. Nahum and H. E. Hoff, “The nature of the precordial electrocardiogram”, Am. J. Physiol., Vol. 155, p. 215, 1948. (3) E. W. Reynolds and C. R. van der Ark, “An experimental study on the origin of T-waves based on determinations of effective refractory period from epicardial and endocardial aspects of the ventricle”, Circ. Rerr., Vol. 7, p. 943, 1969.
394
Joumal of The Franklin Institute
On Electrophysiological
~
Vol.
297,
No.
5, May
1974
-
Activity of the Normal Heart
.
395
P. X. Thiry and R. M. Rosenberg (4) R.
Th. van Dam and D. Durrer, “Experimental study on the intramural distribution of the excitability cycle and on the form of the epicardial T-wave in the dog heart in situ”, Am. Heart J., Vol. 61, p. 537, 1961. (5) K. Harumi, M. J. Burgess and J. A. Abildskov, “A theoretic model of the T-wave”, Circulation, Vol. 34, p. 659, 1966. (6) M. J. Burgess, K. Harumi and J. A. Abildskov, “Application of a theoretic model to experimentally induced T-wave abnormalities”, Circulation, Vol. 34, p. 669, 1966. in “Advances in Electro(7) M. J. Burgess, “Physiologic basis of the T-wave”, cardiography” (ed. by R. C. Schlant and J. W. Hurst), p. 367, Grune & Stratton, New York, 1972. (8) F. N. Wilson, A. G. MacLeod, P. S. Barker and F. D. Johnston, “The determination and significance of the areas of ventricular deflections of the electrocardiogram”, Am. Heart J., Vol. 10, p. 46, 1934. (9) D. Durrer, R. Th. van Dam, G. E. Freud, M. J. Janse, F. L. Meijler and R. C. Arzbaecher, “Total excitation of the isolated human heart”, Circulation, Vol. 41, p. 899, 1970. (10) M. Zloof, R. M. Rosenberg and J. Abbott, “A computer model for atrioventricular blocks”, Math. Biosci., Vol. 18, p. 87, 1973. (11) B. Hoffman and P. Cranefield, “Electrophysiology of the Heart”, McGraw-Hill, New York, 1960. (12) A. M. Scher, “Electrical correlates of the cardiac cycle”, in “Physiology and Biophysics” (ed. by T. C. Ruth and H. D. Patton), p. 565, Saunders, Philadelphia, 1965. (13) R. Plonsey, “Bioelectric Phenomena”, McGraw-Hill, New York, 1969. (14) T. N. James, “The specialized conducting tissue of the atria”, in “Mechanisms and Therapy of Cardiac Arrhythmias” (ed. by L. S. Dreifus and W. Likoff), Grune & Stratton, New York, 1966. “Atrioventricular and intraventricular (15) H. H. Hecht and C. E. Kossmann, conduction”, Am. J. Cardiol., Vol. 31, p. 232, 1973. (16) R. H. Okada, “Potential produced by an eccentric current dipole in a finite length conducting cylinder”, IRE Trans. Med. Electron., Vol. 7, p. 14, 1956. model of the genesis (17) H. L. Gelernter and J. C. Swihart, “A mathematical-physical of the electrocardiogram”, Biophys. J., Vol. 4, p. 285, 1964.
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Journal
of The Franklin
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