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Single dipole, multiple dipole, and dipole-quadrupole models of the double-layer in a circular lamina
J. E L E C T R O C A R D I O L O G Y , 3 (2) 95-110, 1970
Experimental Studies
Single Dipole, Multiple Dipole, and Dipole-quadrupole Models of the Double-layer in a Circular Lamina* BY KENNETH M. KEMPNER, M.S.~, AND JOSEPH GRAYZEL, M.D.~
SUMMARY
INTRODUCTION
Within a conductive circular lamina, the boundary potentials produced by centric and eccentric double-layers, long and short in length, were compared with those produced by single dipoles, combinations of dipoles, and a dipolequadrupole, respectively. Excellent agreement existed between respective boundary potentials of a single dipole and double-layer, centric or eccentric, provided the single dipole was located at the center of the double-layer. Two dipoles more accurately modeled the double-layer, but only if the two dipoles were optimally located, in which case their spacing or separation reflected the spatial extent of the double-layer. Arrays of three or more dipoles represented a slight further improvement only if the individual dipoles were optimally weighted. Lastly, the centric dipolequadrupole was not as good a model of the eccentric layer as the single dipole located at the center of the layer. The accuracy of various models of the double-layer is examined in terms of the even and odd symmetries of the respective generator configurations.
As early as 1954, G a b o r and Nelson t showed that an equivalent cardiac dipole could be determined from body-surface potentials. The solution of two sets of simultaneous equations specified the components and location, respectively, of the dipole. Also, the same technique permitted determination of a multiple dipole model, which improved the accuracy with which the real surface potentials were duplicated by the equivalent model. Recent efforts to refine the model of the heart generator have generally focussed on multipolar representations, in particular the quadrupole, which is the next ordered pole above the dipole. Considering the electrocardiographic problem as essentially the solution of Poisson's equation for the human torso, the classical technique of separation of variables in spherical coordinates yields a series in Legendre polynomials for the general solution 2. If the boundary is simple and wellbehaved, the series solution converges. Since current streamlines run tangent to the body-surface boundary, the boundary condition OV/On = 0 allows the coefficients of the Legendre polynomial series to be determined. These coefficients are the components of a multipole at the origin which produces a distribution of boundary potentials which matches the real body-surface voltages. To cite some examples, this general approach was employed by Yeh for current generators in a sphere 3, by Hlavin and Plonsey for a turtle heart 4, and for the cardiac generator by Geselowitz 5 who also noted that the dipole components calculated by G a b o r and Nelson I were identical to the dipole terms in the general multipolar expansion. The general solution tot the human torso consists of an infinite number of multipoles 2. Also, the higher the order of a multipole, the more rapidly its contribution to field potential diminishes with distance from the source. Hence, at
* From the Division of Bioengineering, Columbia University. t Formerly NIH predoctoral fellow in Bioengineering, Columbia University; presently electronic engineer, Division of Computer Research and Technology, NIH, Bethesda, Md. Formerly Asst. Prof. Biomedical Engineering, Columbia University; presently Chief of Cardiology and Director, Cardio-Pulmonary Laboratory, Bergen Pines County Hospital, Paramus, N. J., and Senior Research Associate in Bioengineering, Columbia University. This investigation was supported by research fellowship GM-32, 785 and Research Grant HE-10579 from the NIH, USPHS, and a grant from the New York Heart Association. Requests for reprints should be addressed to: Joseph Grayzel, M.D., Cardiology Section, Bergen Pines County Hospital, Paramus, N.J. 07652. 95
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KEMPNER A N D G R A Y Z E L
reasonable distances from the origin the dipole alone is an acceptable, or perhaps adequate representation of the generator. A practical need is to define what is meant by a "reasonable distance" or an "adequate" model of the cardiac generator. The literature abounds with studies of cardiac electrical models in various conducting media. The dipole has been examined in a planar circular lamina 6-~~ a planar elliptical lamina u, a cylinder v-',l:~,a sphere ~4-~8 and in a prolate spheroid xg. The double-layer has been studied in a circular lamina '-'~ Rarely have several generator types been placed within the same conducting medium in an attempt to compare their resultant boundary potentials. Frank compared the dipole with a disc-shaped double-layer2L The multipolar representation within a sphere of an eccentric dipole and of a double-layer were studied by Yeh and Martinek '-'~. We chose the plane circular lamina for its simplicity in providing a practical physical method for experimental verification of analytically derived results, yet affording precise comparisons among electric generators which are meaningful for three-dimensional man. Recent suggestions for refinement of the fixedlocation dipole model have included a single moving dipole, two or more dipoles, and a dipolequadrupole. During the physiologic spread of myocardial excitation, the wave-front of depolarization is relatively thin and is generally considered to be a double-layer current source. Therefore, the present study examined the degree of accuracy with which double-layers of different size and location are modelled by a single dipole, multiple dipoles, and a dipole-quadrupole in the same conducting medium. While the present study employs the forward method, i.e., generators are placed within the medium and the resultant boundary potentials computed and compared, such comparison suggests the relative virtues or shortcomings of generators considered for the inverse problem, i.e., the measurement of boundary potentials from which parameters of a particular generator form are determined. MATERIAL AND METHOD The physical model consisted of a homogeneous circular conductive paper lamina, 18 inches in diameter, 0.008 inch thick, and with a bulk conductivity of 0.0106 mho/inch. An analytic counterpart of this physical model was developed by deriving the field equations in the above lamina
for each generator studied, and evaluating the boundary potentials with the aid of a digital computer. The generator configurations placed within this lamina were single dipoles, combinations of these dipoles, double-layers, and a dipole-quadrupole. F o r all of these, the distance between source and sink, or pole separation, was 0.5 inch. The physical paper lamina provided experimental data which principally served to verify the analytic results for basic generator configurations --single dipoles and double-layers. Once verified, the computed boundary potentials were employed for further calculations, comparisons, and plotting since the computed results are more accurate and precise. Single Dipoles. Nine dipole locations were employed, spaced at 3/4 inch intervals along the lamina's horizontal diameter and symmetrical about the center. Thus, the dipoles lie along the central 6 inches of the lamina's 18-inch diameter, as shown in Fig. 1. These nine dipole locations are consecutively numbered from left to right, permitting the simple notation DI to designate a dipole at the first, most leftward location, D5 the centric dipole, D9 the most rightward, and other locations correspondingly. A plexiglas template was used in cutting the circular paper lamina from a roll and locating the two poles of each dipole. Each pole point was permanentized with a tiny drop of silver paint. Through each dipole, in turn, a constant current of several milliamperes was introduced, monitored with an ammeter, while voltages around the lamina's boundary were measured at 5~ intervals with respect to the boundary point at zero degrees. The boundary potentials were divided by the dipole's current to yield values corresponding to one milliampere of source current. Various symmetries confirmed the accuracy of the experimental method. For example, all dipoles shown (Fig. 1) possess odd symmetry about the horizontal diameter, potential around the lower semicircular boundary being the negative mirrorimage of the upper semicircle. Boundary points at 0 ~ and 180 ~ are at zero potential, hence no voltage difference should exist between them. In addition, the centric dipole D5 produces right-left symmetry around the boundary, and such symmetry also exists between symmetrical dipole p a i r s - - l , 9 ; 2,8; 3,7; 4,6. The accuracy of such symmetries among the experimental data support the accu-
DIPOLES AND QUADRUPOLES
LAYER POSITION L(I- 5) SHOWN 9
IN A CIRCULAR
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racy of experimental techniques, such as locating and orienting the dipoles, homogeneity and isotropicity of the paper, designation of the boundary points, and reproducability of results since the entire group of experiments was performed over several months. The analytic analogue for the foregoing consisted of an equation for the potential around the boundary of a circular lamina due to a current dipole located on and normal to the horizontal diameter (X-axis). The derivation is given in Appendix A, and consists of first deriving an equation for the dipole in an infinite lamina and then employing the method of images to yield the equation for a finite lamina, which is I la s i n ~ 1 I In 1 + - (r~) 2 I VD(q~) = _7c~b
(1)
where VD is the potential on the boundary in millivolts. $ is the degrees of angle between the line segment from origin to boundary point and the positive X-axis. I is the dipole current in milliamperes. is the bulk conductivity of the medium, mho/inch. b is the thickness of the lamina in inches. a is the radius of the lamina in inches. 1 is the pole separation in inches. Xo is the x-coordinate or displacement of the dipole in inches.
r,
is the distance, in inches, from the dipole to the boundary point.
and (r~) 2 = a2 _ 2axo cos ~b + x02 VD (q~) is the boundary potential with respect to a point at zero potential. Since our dipoles were located on and perpendicular to the lamina's horizontal diameter, this diameter is the zero isopotential line and its end-points at zero and 180 degrees are at zero potential. Recalling that the experimental voltages were measured with respect to the boundary point at zero degrees, these values should match those calculated. Equation (1) was programmed in A L G O L for a Burroughs 220. Multiple Dipoles. The distribution of potential around the lamina resulting from two or more dipoles acting simultaneously was obtained by superposition of the individual dipoles' respective boundary values, appropriately weighted according to each dipole's current strength. Arrays of two, three and five dipoles were studied, with details and results presented below. Double Layers. Three double-layers were studled--(1) a centric double layer 6 inches long, positioned along the lamina's horizontal diameter extending from dipole location #1 to #9; this layer is noted L 1-9 to indicate the relation of its end points to the dipole locations studied; (2) a short, centric double-layer, L 3-7, three inches in length; (3) a short, eccentric double-layer, L 1-5, also
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KEMPNER A N D G R A Y Z E L
three inches in length. The separation of a doublelayer was 0.5 inch, as was the pole separation for dipoles. The geometry of the short, eccentric layer L 1-5 is shown in Fig. 1. The three physical models were prepared by cutting a circular lamina and painting two fine silver stripes, 88 inch above and below and parallel to the horizontal diameter, the stripes running the extent of the particular double-layer under study. The double-layer was energized by contacting each stripe along its entire length with a heavy, weighted brass plate on end, and passing the measured current between the plates. As before, boundary potentials were measured with respect to the boundary point at zero degrees. The analytic analogue was an equation for boundary potential around a circular lamina due to a current double-layer along the horizontal diameter. This equation is obtained by integrating equation (1) along the diameter between limits which are the end-points of the particular doublelayer. This equation, derived in Appendix B, is
_
4q 2 + h l t a n - ' .
\qtan-I
+ ,q)ll
x~s,j
....
(2)
where VL is the boundary potential in millivolts, produced by the double-layer I K--
7rablo lo is the length of the double-layer in
inches is the layer separation in inches p = (a2 - 2axcosq~q-x 2) q = asin~b l
S =
ll = 1/2. In the centric dipole-quadrupole generator, the dipolar and quadrupolar currents may be varied independently. The net distribution of boundary potential is obtained by superposition. R ES U LTS
Double-layer vs Single Dipole.
VL(dp) = Kxln(1 + lq~ '~ P /1~, ~-Kln[(pT~q)e
considered as a model for the electric doublelayer of the heart. Both the dipolar and quadrupolar contributions were calculated from equation (1). The dipole is simply D5, which had been calculated. The centric quadrupole's geometry, shown in Fig. 6, indicates how its boundary potential distribution may be computed from equation (1) by representing the quadrupole with two dipoles of opposite orientation, each displaced a distance one-half the pole separation either side of the origin and each carrying unit current. This produces equal dipole and quadrupo[e magnitudes in our system. Quadrupole moment, M~ = 21ld, and since d ---l = 0.5, M~ = 1/2. Also, dipole moment, Ma ----
acosdp
xl, x~ are coordinates of the left and fighthand end-points, respectively, of the double-layer a, b, or, and q~are the same as for eq. (1). Equation (2) was programmed in F O R T R A N IV for the IBM 7094. Centrie dipole-quadrupole. The distribution of boundary potential around the lamina was also studied for the centric dipole-quadrupole combination since this configuration can produce asymmetric potential distributions and has often been
Very good or excellent agreement was observed between the boundary potentials produced by a double-layer and those produced by a single dipole located at the mid-point of the layer, the better match occurring for the shorter layer. Comparisons are shown in Fig. 3 for the long centric layer and the short centric layer, and in figure 4 for the short eccentric layer. The long centric layer possessed a linear dimension one-third the diameter of the conductive lamina, a ratio somewhat larger than might exist in real man at any instant of ventricular excitation, yet the rms error of the single centric dipole was less than 3 ~o of the maximum potential on the boundary (Fig. 3A). The shorter layer, whose length is one-sixth the diameter of the conductive lamina provides a more realistic proportion. F o r both centric (Fig. 3B) and eccentric (Fig. 4) positions of this layer, the single dipole at the layer's midpoint provided an excellent match of boundary potentials, with an rms error less than 1 ~o- Clearly, this error is within the accuracy of clinical measurement in almost all situations.
Double-layer vs symmetrical dipole pair. Referring to Fig. 1, the nine dipole locations permit four symmetrical p a i r s - - l , 9 ; 2,8; 3,7; 4,6. With one-half the current flowing through each
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DIPOLES AND QUADRUPOLES IN A CIRCULAR LAMINA
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DEGREES
Fig. 2. Boundary potentials for single dipoles and centric quadrupole.
dipole in a pair, the boundary potentials of each pair, respectively, were compared with those of the long, centric double-layer (Fig. 5). The dipole pair 1,9 had an rms error of 5 %. This error diminished as the two dipoles moved toward each other and was a minimum for dipole pair 3,7. As the spacing between the pair further decreased, the error rose again, and for the centric dipole was nearly 3 % as mentioned above. The rms errors of interest appear in Table 1, from which it is readily seen that the optimum pair 3,7 provides a truly excellent match for the double-layer and an rms error only one-fourth that of the centric dipole, though the latter was already a very good model for the long double-layer.
Double-layer vs multiple dipoles. As the number of dipoles simulating a doublelayer is increased, the multiple-dipole array should produce boundary potentials which approach those of the double-layer and, in the limit, be indistinguishable. The manner in which this occurs was studied under two experimental conditions-equal weighting of dipoles in the array, and optimum weighting for minimizing the rms error of boundary potentials compared to those produced by the double-layer. Equal weighting of dipoles. The long, centric double-layer, L 1-9, was simulated by (a) three dipoles at positions 1,5,9, (b) five dipoles at positions
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KEMPNER AND GRAYZEL I
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A. Long double-layer, L(1-9)
Fig 3.
Comparmon of boundary potentials for centric double-layer vs centric dipole, i
1,3,5,7,9, and by (c) nine dipoles at positions 1 to 9. Also, (d) the short, eccentric double-layer, L 1-5, was simulated by three dipoles at positions 1,3,5, and (e) the short centric double-layer, L 3-7, was simulated by three dipoles at positions 3,5,7. The three-dipole array, D 159, with equal weighting produced boundary potentials in very good agreement with the long, centric doublelayer, but was no better than the single centric dipole alone, D5, both having virtually identical rms errors of 2.6 Yo- The five-dipole array reduced the rms error in half to 1.3 ~o, and the nine-dipole array further reduced rms error to 0.65 ~o, thus providing excellent simulation of the long doublelayer. The three-dipole array, D 135, with equal weighting provided an excellent match for the short eccentric double-layer, L 1-5, but no better than the single dipole, D3, located at the midpoint of the layer, both yielding an rms error of 0.70%. Similarly, for the short, centric doublelayer, L 3-7, the single centric dipole, D5, and the
three-dipole array D 357, were equally excellent with an identical rms error of 0.64 ~o. Optimum Weighting of Dipoles. Alternatively, an algorithm was programmed to compute the optimum weighting of individual dipoles in an array such that rms error was minimized. This algorithm is described in Appendix C. The optimum coefficients for such a "best fit" always summed to yield unit total current and always reduced rms error to 0.1 ~o or less. Thus, a graph of boundary potentials for a double-layer and any of the dipole arrays described above were essentially identical when optimum dipole weighting was employed. Table 1 summarizes the foregoing simulations of double-layers by single dipoles, equally weighted dipole arrays, and optimally weighted arrays.
Eccentric Double-Layer vs Centric Dipole-Quadrupole. As presented above, an excellent match occurred between boundary potentials of the short
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DIPOLES AND QUADRUPOLES IN A CIRCULAR LAMINA
eccentric layer, L 1-5, and the single dipole at the layer's midpoint, D3; the rms error was 0.70~o. A similar comparison between this short, eccentric layer and the optimum centric dipole-quadrupole, D5Q5, revealed a poorer match, with an rms error of 2.56 ~o which is more than three times the error of the single eccentric dipole. These comparisons are graphed in figure 4 and summarized in Table 1, where the relative current strengths through each source-sink pair of the particular generator are presented. I n modeling the layer L 1-5 of unit current strength, both the eccentric dipole D3, and the dipole component of D5Q5 possessed unit current for the best respective matches of boundary potentials. That the dipole component of an equivalent multipolar generator does not depend upon the source's location has been indicated elsewhere 23. The present comparison illustrates this, as well as the fact that the dipolar component is related to the true current strength of the doublelayer, while the quadrupole is related to displacement of the multipolar model from the actual current source. DISCUSSION With regard to developing a practical model of the heart generator, two aspects of our results appear significant. First, very good or excellent agreement existed between respective boundary potentials of a single dipole and a double-layer, centric or eccentric, provided the single dipole was located at the center of the double-layer. Second, definite limitations exist for the fixed-location dipole-quadrupole as a model for the eccentric double-layer.
good model. Also, best agreement is always attained when the current through dipole and double-layer are identical. Thus, a single dipole whose location varies in accordance with the spread of ventricular excitation appears to be an excellent, meaningful model of the heart's doublelayer; its moment reflects the ianic current flowing across the thickness of the real double-layer and its location is in the region of the heart undergoing depolarization. Only the spatial extent of the instantaneous double-layer is not revealed by the dipole model, but the quadrupole also fails in this regard, as we shall indicate later.
Multiple Dipoles. Two dipoles provided a more accurate model of the double-layer than than did a single dipole, but only if the two dipoles were optimally located. Comparisons with dipole pairs were made for the long double-layer, in which case the optimum locations for the two dipoles were at the center of the right and left halves of the layer, i.e. at points one-fourth and three-fourths along the layer's length. Since our long double-layer was twice the length of the short double-layer, the optimum locations for two dipoles treats the long layer as TABLE 1 Double Layer 1-9
Single Dipole Considering the dimensions of the human ventricles and chest cage, the largest double-layer at any instant of ventricular excitation could be approximately one-sixth the transverse diameter or one-third the antero-posterior diameter of the chest. These ratios correspond to our short double-layer and long double-layer, respectively, within the circular lamina. For the short double-layer, centric or eccentric, a single dipole at the center of the layer was an excellent model, rms error being less than 1 ~o, and no further refinement of this dipole model seems indicated or realizable. For the long doublelayer, the rms error was less than 3 ~o, still a very
3-7
1-5
Dipole Model D D D D D D D D
5 4,6 3,7 2,8 1,9 1,5,9 1,3,5,7,9 1 to 9 1 D 5 9 1 3 D 5 7 9 D 5 D 3,5,7 3 D 5 7 D3 D 1,3,5 1 D3 5 DIPOLE D5 Q-POLE Q5
Relative Current 1.0013 0.5 each " " " 0.33 each 0.20 each 0.11 each 0.166 / 0.668 ~ 0.166 0.083 0.336 I 0.162 / 0.336 / 0.083 1.0O0 0.33 each 0.168 1 0.665 0.168 / 1.001 0.33 each 0.167 / 0.666 ~ 0.168 J 1.0O 3.01 ;
RMS Error ~o of Max 2.6 2.1 0.7 1.8 5.3 2.64 1.31 0.65 0.11
<0.03 0.64 0.64 <0.04 0.70 0.70 <0.04 2.56
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KEMPNER AND GRAYZEL
two adjacent short layers, and places a dipole at the center of each. Interestingly, the rms error for the two-dipole model of the long double-layer was virtually identical to the rms error for the singledipole model of the short double-layer. The two-dipole model of the long double-layer possessed an rms error less than 1%, compared with a 2.6 % rms error for the single-dipole model. In addition to this improvement of an already small error, the real advantage of the two-dipole model is its ability to reflect the spatial dispersion or extent of the double-layer. At their optimum locations, the separation of the two dipoles was one-half the length of the long, uniform doublelayer being modelled. Though arrays of three or more dipoles were also compared to the double-layers, our interest in these models is small, having observed the accuracy of single-dipole and two-dipole models. Of note, however, was that these more extensive arrays possessed increased accuracy only if the individual dipoles were optimally weighted. Though optimum weighting was never equal weighting,
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the optimum array always proved to have a total current of unity, equal to current through the double-layer. The foregoing observations of multiple-dipole arrays indicate that the Baule-McFee24 method for experimentally producing a double-layer is not rigorously correct. In that method a double-layer is obtained by an array of "physical" dipoles, with each pole pair carrying identical current owing to current-limiting resistors. Our results with analytically derived dipole arrays and double-layers indicate that the number of such pole pairs, or physical dipoles, per unit length of the double-layer must be large for precise simulation. Otherwise, optimum weighting of the dipoles should be employed to avoid significant errors at certain boundary points related to the location of the generator.
Dipole-Quadrupole. It was of considerable interest to find that the centric dipole-plus-quadrupole was not as good a mode[ of the short eccentric layer as the single eccentric dipole located at the center of the layer. I
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B. Short double-layer, L(3-7) Fig. 3. Comparison of boundary potentials for centric double-layer vs centric dipole.
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DIPOLES AND QUADRUPOLES IN A CIRCULAR LAMINA !
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A N G L E ~ A B O U T ORIGIN
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Fig. 4. Comparison of boundary potentials between eccentric double-layer L(1-5), eccentric dipole D3 at the middle of this layer, and the optimum centric dipole-quadrupole D5Q5.
And this short layer, L 1-5, was only slightly eccentric, its center being 1.5 inches from the center of the lamina with radius 9 inches, yielding an eccentricity of 1.5/9 or 0.16. Furthermore, examination of our results indicates that the quadrupole is limited in the amount of eccentricity it can simulate, beyond which the centric dipole-quadrupole model deteriorates. This property of the quadrupole is now examined. Graphed in Fig. 2 are the potentials around the upper semi-circular boundary of the lamina for dipoles 1,2,3,4, and 5, respectively, and also for a centric quadrupole. For D5, the centric dipole, the maximum potential occurs at the midpoint, 90~ around the boundary. As the dipole location becomes increasingly eccentric the maximum shifts around the boundary. Note, however that the centric quadrupole produces a maximum at 135 ~ and thus, can not simulate a simple distribution whose maximum is further clockwise. For D1, with eccentricity 0.33, the maximum is at 125 ~, and for a dipole with eccentricity greater than 0.5 the maximum will lie beyond 135 ~ Relying on the
excellent relation between a double-layer and a dipole located at its center (rms error < 1 ~o), we recognize that the boundary potentials induced by a double-layer with eccentricity greater than 0.5 will not be closely modelled by a centric dipoleplus-quadrupole, and as the layer's eccentricity increases beyond 0.5, the simulation deteriorates unless higher-order multipoles are added. Schubert 25performed a related analysis within a sphere, and demonstrated that simulation of a dipole whose eccentricity was 0.4 with a centric dipolequadrupole incurred an rms error of approximately 20 ~o. Considering the location of the heart within the chest, double-layers with eccentricities* greater than 0.5 are to be expected. The right ventricular free wall and the antero-septal region of the left ventricle lie immediately beneath the sternum and anterior rib cage, and are highly eccentric. The * Eccentricity of a source in a circular or spherical medium has been defined quantitativelyas the distance of the source from the center of the medium divided by the radius of the medium.
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A N G L E q~ A B O U T ORIGIN D E G R E E S Fig. 5. Comparison of boundary potentials between tile long centric double-layer L(I 9) and symmetrical pairs of dipoles. The best match for this double-layer is provided by the dipole pair D 3,7 (see text).
eccentricity of the left ventricular free wall is normally about 0.5, which increases with enlargement of the heart in disease states. In view of the foregoing geometry, the limited multipolar expansion truncated at the quadrupole would probably be a poor model of the cardiac generator. On the other hand, extending the multipolar expansion to the octapole, let alone higher-order poles, represents an imposing computational exercise applied to man. Schubert 2~ suggested that the error associated with the centric dipole-quadrupole would be reduced if the multipolar series were expanded about the unique "best" origin, which generally corresponds to the electrical center of a distributed source. A theorem developed by Geselowitz2~ locates this optimum origin, which proves to be nothing more than the point where a dipole alone provides the best simulation of boundary potentials. Thus, we have returned to our earlier observation that an excellent model of the double-layer is a single dipole at its center. But even having determined the optimum loca-
tion for the dipole, examination of the symmetries of double-layers, dipoles, and quadrupoles reveals why the quadrupole at this "best" location fails to make a significant contribution to the model. Pertinent generator configurations are shown in Fig. 6. Both the double-layer and dipole have even symmetry about the Y-axis and odd symmetry about the X-axis. It is evident that any discrepancies between the distribution of potentials produced by a uniform double-layer and its optimally located dipole model, respectively, are properly corrected only by addition to the dipole of another generator with the same symmetries as the doublelayer. The rectangular quadrupole has odd symmetries about both axes, and if rotated 45 ~ it has even symmetry about both axes. The linear quadrupole has even symmetry about both axes. Thus, the quadrupole fails. The particular octapole shown has the desired symmetries--even about the Y-axis and odd about the X-axis--and is a candidate for improving the dipole model of the double-layer. Review the foregoing in terms of the boundary
DIPOLES AND QUADRUPOLES IN A CIRCULAR LAMINA
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J
Fig. 6. Geometry of relevant electric generators.
potentials induced by the long double-layer, L I-9, and its model, D5, respectively, shown in Fig. 3A. Compared to L 1-9, the maximum induced by D5 is slightly greater, as are potentials between 60 ~ and 120~. Outside this central region on either side the dipole's potentials fall below those of the layer until the X-axis is reached at 0~ and 180~ where potentials are zero and equal. From 180~ to 360 ~ the relationships are the negative mirror image of those from 180 ~ to 0 ~ Improvement of this match by addition of a higher-order multipole to the dipole requires a distribution with the correct symmetries. The rectangular quadrupole oriented along the axes induces boundary potentials, shown in Fig. 2, which are zero at 90 ~ and 270 ~ as well as at 0 ~ and 180~ and therefore, if added to D5 would not affect the maximum or minimum. Owing to the quadrupolar distribution's odd symmetry about the boundary point at 90 ~, any improvement of the dipole's match in one quadrant is associated with a corresponding deterioration in the other. Rotating the rectangular quadrupole 45 ~, as shown in Fig. 6, could improve the match
around the upper half of the l~/mina, but now the even symmetry about the X-axis produces a corresponding deterioration around the lower half of the boundary. The linear quadrupole, with even symmetry about both axes, suffers a similar limitation. Thus, the quadrupole plays no role in describing the uniform double-layer if the optimum origin for the multipolar expansion is employed. To date, the dipole-quadrupole has been applied to man by measuring potentials from living, heterogeneous humans, and then computing multipolar coefficients for a homogeneous torso. If the best origin could be chosen, the existence of quadrupolar coefficients would result from distortions of the real-life field produced by the torso's inhomogeneities, such as lung and blood, which are not accounted in the computation. If the best origin for the multipolar expansion is not employed, the quadrupolar coefficients also reflect the displacement or eccentricity of the source with respect to the mathematical origin. The octapole, however, added to the dipole
106
KEMPNER AND GRAYZEL
could improve the match. Its symmetry is o d d about the X-axis and even about the Y-axis. Inspection of the particular octapole in Fig. 6 reveals that, suitably weighted and added to the dipole, it makes the m a x i m u m less positive, the m i n i m u m less negative, and improves the match around the sides. Also, examining our results for the single dipole DS, dipole pairs of varying separation (Fig. 5), and the short layer L 3-7 vs the long layer L 1-9, we observe the more disperse the source, the greater the departure of boundary potentials from those of the single dipole, and the greater will be the moment of the octapole in a dipole-octapole model. Thus, the octapole reflects spatial dispersion of the uniform double-layer, and represents the next multipole capable of providing additional information concerning the double-layer itself, as distinguished from the elt'ccts of eccentricities and torso inhomogeneities. It appears that the first improvement of the fixed-dipole model should be that of a moving dipole. A subsequent refinement could be addition of the octapole to the dipole at its optimum instantaneous location, or substituting for the moving dipole a moving dipole pair with o p t i m u m separation. Acknowledgment: The authors wish to thank Miss Diane Blackman for assisting us with programming and for providing plots of all data. REFERENCES 1. Gabor, D., and Nelson, C. V.: Determination of the resultant dipole of the heart from measurements on the body surface. J. of Appl. Physics 25 : 413, 1954. 2. Byerly, W. E.: An Elementary Treatise on Fourier's Series and Spherical, Cylindrical, and Ellipsoidal Harmonics with Applications to Problems in Mathematical Physics. Dover, N. Y., pp. 195-218, 1959. 3. Yeh, G. C. K. : Eccentric multipole representations of current generators in a spherical volume conductor. Bull. Mathematical Biophysics 23: 263, 196l. 4. Hlavin, J. M., and Plonsey, R.: An experimental determination of a multipole representation of a turtle heart. IEEE Trans. Biomed. Electronics 10: 98, 1963. 5. Geselowitz, D. B. : Multipole representation for an equivalent cardiac generator. Proc. IRE 48: 75, 1960. 6. Bayley, R. H. : The electric field produced by an eccentric dipole in a homogeneous circular conducting lamina. Circulation Res. 7: 272, 1959. 7. Bayley, R. H. : Unipolar measurements in the electric field produced by an arbitrary dipole in a circular homogeneous lamina. Circulation Res. 7: 537, 1959.
8. Bayley, R. H., and Berry, P. M. : The electric field produced by the eccentric current dipole in the nonhomogeneous conductor. Am. Heart J. 63: 808, 1962. 9. Nelson, C. V., and Gastonguay, P.: A definition for zero potential. Circulation Res. 7: 1039, 1959. 10. Toyoshima, H., Yamada, T., and Otani, K.: The Electrical Analysis of So-called Electrical Heart Axis by Vectorcardiography. Jap. Circ. J. 16: 323, 1953. 1l. Bayley, R. H.: Measurements of unipolar potentials in the electrical field produced by an arbitrary dipole in the elliptical homogeneous lamina. Am. Heart J. 59: 737, 1963. 12. Okada, R. H. : Potentials produced by an eccentric current dipole in a finite-length circular conducting cylinder. [RE Trans. Med. Electronics ME-7: 14, 1956. 13. Okada, R. H. : An experimental study of multiple dipole potentials and the effects of inhomogeneities in volume conductors. Am. Heart J. 54: 567, 1957. 14. Wilson, F. N., and Bayley, R. H.: The electric field of an eccentric dipole m a homogeneous spherical conducting medimn. Circulation I : 84, 1950. 15. Frank, E.: Electrical potential produced by two point current sources in a homogcneous conducting sphere. J. Appl. Physics 23: 1225, 1952. 16. Nelson, C. V.: A convenient expression for the potential at the surface of a sphere due to an eccentric dipole. Digest 1961 International Conf Med. Electronics, p. 238. 17. Bayley, R. H., and Berry, P. M.: Body surface potentials produced by the eccentric dipole in the heart wall of the nonhomogeneous volume conductor. Am. Heart J. 65: 209, 1963. 18. Yeh, G. C. K., and Martinek, J.: Comparison of surface potentials due to several singularity representations of the human heart. Bull. Mathematical Biophysics 19: 293, 1957. 19. Yeh, G. C. K., and Martinek, J.: The potential of a general dipole in a homogeneous conducting prolate spheroid. N. Y. Acad. Sci. 65: 1093, 1957. 20. Bayley, R. H., and Berry, P. M.: The arbitrary electromotive double layer in the eccentric 'Heart' of the nonhomogeneous circular lamina. IEEE Trans. Biomed. Eng. BME-II, 137, 1964. 21. Frank, E. : A comparative analysis of the eccentric double-layer representation of the human heart. Am. Heart J. 46: 364, 1953. 22. Yeh, G. C. K., and Martinek, J.: Multipole representation of an eccentric dipole and an eccentric double-layer. Ball. of Mathematical Biophysics 21: 33, 1959. 23. Geselowitz, D. B.: Two theorems concerning the quadrupole applicable to electrocardiography. 1EEE Trans. Biomed. Eng. BME-12: 164, 1965. 24. Baule, G., and McFee, R.: Theory of magnetic detection of the heart's electrical activity. J. of Appl. Physics 36: 2066, 1965. 25. Schubert, Roy W.: Surface potential errors introduced by the truncation of the infinite series that represents an eccentric dipole, IEEE Trans. Biomed. Engr. BME-13, 101, 1966. 26. Nelson, C. V.: Effect of the finite boundary on potential distribution in volume conductors. Circulation Res. 3: 236, 1955.
107
DIPOLES AND QUADRUPOLES IN A CIRCULAR LAMINA
27. Plonsey, R.: Current dipole images and reference potentials. IEEE Trans. Biomed. Electronics. BME-10: 3, 1963. 28. Plonsey, R., and Collin, R.: Principles and applications of electromagnetic fields. McGraw-Hill, New York, pp. 69-70, 1961. APPENDIX A BOUNDARY POTENTIALS OF AN ECCENTRIC DIPOLE IN A CIRCULAR LAMINA
Consider a current source of strength I in an infinite conductive lamina of thickness b and conductivity or. Current streamlines are radial from the source and surfaces of constant current density Jr, are concentric circular cylinders through the thickness of the lamina, where
or
lizing the circular symmetry of the system 26-28. The boundary may be removed if an appropriate image dipole with a pole separation a21/xo ~, is located at a distance a2/xo from the center of the circular lamina, where a is the radius of the lamina and x0 is the displacement of the interior dipole from the center of the lamina (Fig. 7b). The following relations were determined from the geometry of the system: sin q~i -
ri
sin qS~ 2
/2
2
2
1.~ =
I Jr -- 27rl"b -- cr Er
(A 1)
I Er = 2~rab
(A2)
-
(2ra2/x0)
V(+~ =
~
1
I
r sin q) r8
cos ~ +
Jb
Y
P
T
ro
-2~Trcrbdr = -2-~crb In -- (A3)
I
]
r,
Similarly, for a current sink V( I -
(.2/x0)2
r~ = r -- 2rxo cos 4) + xo 2
The potential difference between a p~int at a distance rl from the source and a reference point r0 is obtained by integrating eq (A2), yielding r~
r sin 4)
PX
(A4)
f L-- 1, Inr~ I"2
Referring to Fig. 7a, the potential resulting from the source-sink pair is V(+_~ =
V(+) +
I V(_) = ~ I n r 2
(A5) 1.1
The field of this source-sink pair approaches that of a dipole as r / l -+ o~, where / is the pole separation. The following relations then become valid: rl-- r~--
89
Fig. 7. A. Geometry for single dipole in an infinite lamina.
s
r -- r l ~ / s i n 4 )
ri
rs
Utilizing these relations in eq (A5) yields
Vj =
, ( -2-Tr~bbIn 1 +
--x2~ ,
(a6)
which is the potential at a distance r from a dipole in the infinite conducting lamina. If a circular boundary is imposed on the conducting lamina, the problem may be solved conveniently by employing the method of images uti-
Xo
B. Geometry for single dipole and its image dipole in a finite circular lamina.
108
KEMPNER AND GRAYZEL
At the boundary of the lamina, r
=
a
and
Integrating equation (B1) by parts yields:
a ri
=
--- rs Xo
v~ (49)
.
Calculating the potential led due to the dipole and its image yields:
VD(a, 49) = Ve(a, 49) + V~(a, 49)
Z
(
= KxIn
K
f[
1 q- ( a - - [3x q - x 2)
"
~
+
.r(2x - ~)x
1 (~ + ~ -- 5 x + X2 ) (~ -- ~X + 22 )
la sin 49)(A7)
dx
(B2) Where
where r,
2
K = I/zco-blo
2
= a -- 2axo cos 49 -}-x0 ~
2
o l : a
Equation (A7) is equation (1) appearing earlier in the manuscript. Though several equations pertaining to dipoles and doublets in a circular lamina have been published s,~, we derived eq. (A7) in order to provide a closed-form solution which is readily integrated with respect to dipole position, thereby providing a comparable equation for the double-layer. A further simplification can be accomplished by utilizing the fact that In (1 -+- u) ~ u if u -~ 0, which rcduccs cq. (A7) to V ~ ( a , qg) ~
___lla sin 49 rccrbrs 2
/3 = 2acos49 y = la sin 49
Expanding the integrand by partial fractions and integrating results in the following expression:
...... ~, ........ )~' v,.O) = K,: In 1 + (~ _ ~." + x") x~ + K t , ( ( x + A)A(x + B)"(x -- C) ~
(A8)
This cxpression is identical to the final result of Plonsey ~-7 in an analysis of dipole images. At x0 = 0 and 49 = 90 ~ this approximation produces a boundary potential value which is about 3 ~o higher than the solution derived here, per eq. (A7). The agreement is generally better for other wflues of x0 and 49.
x(~-D)"),il
(B3)
Where A = --~/2
q- ~ 1 ~ / 4 _
(a -t- 7)
~- - - C ~ Jr- jC2
B = --~/2 -- 4~'~/4 _ (o, + ~,) =
--
C1
=
+~/2
--
JC2
APPENDIX B c
B O U N D A R Y P O T E N T I A L S OF A D O U B L E LAYER IN A CIRCULAR LAMINA
",J~V4
-
= q- C1 -- jCa
The double-layer shown in Fig. 8 has a pole separation l, and a length l0 = Ix2 -- xll. The differential contribution to the boundary potential of the double-layer is that of a single dipole whose pole strength is the current density along the layer,
z/z0
o = +~/2
+ +74
-
= -]-C1 -}- jCa
Introducing polar coordinates as follows: (x q- A ) = Rle+~~
I
dVL
-
(49 )
-
wcrblo
/n(1 -}-la sin~_)dx r. 2
where 2
r~ =
a2
-- 2ax cos 49 q- x 2
(B1)
(x q- B) = R t e - i ~ (x-
C) = R2e+Y~
(x-
D) = R2e-i~
allows the reduction of the boundary potential
109
DIPOLES AND QUADRUPOLES IN A CIRCULAR LAMINA
Y <~
E
205 202.9
03
,~ 200
+Z/.~ ~
..3
Z hi I
o~- 195(~
I
I
I
I
I
9
2 4 6 lAYER LENGTH (INCHES) Fig. 9. The maximum boundary potential induced by a double-layer as a function of the layer's length. IX.
~o = Ixz-x, I Fig. 8. Geometry for deriving the double-layer from the single dipole. APPENDIX C ALGORITHM FOR CALCULATING THE OPTIMUM D I P O L E C O E F F I C I E N T S F O R ~BEST F I T " WITH A DOUBLE-LAYER
expression to the final form: Let, DL(4)) represent the double-layer boundary potential distribution and Di(r represent the boundary potential distribution of the ith dipole in a multiple dipole approximation of the doublelayer. For N dipoles, the total squared error around the boundary is:
VL(4)) = K x ln 1 + i
+ Kin
-
-
e 2 q t a n -1
~lq~"+ lq tan-1 \
7--s-]
q
(B4)
f]
Where
2
2
q = a sin q5 S ~
DL(4,)-
aiOi(~) i=1
where a~ are the coefficients in the linear combination of multiple dipoles, e2 will be a minimum when
I 7rablo p = a -- 2 a x c o s q 5 + x
e2 = ~ %
0e 2 ----= Oaj
0For j=
1,2,.-.
N,
or
a cosq~ =
This expression is evaluated between the end points of the double-layer to provide the exact boundary potential value at any given boundary point. In order to check this solution, the potential at q~ --- 90~ was computed for centric layers varying in length from 6 inches to 0.2 inches. The results graphed in Fig. 9, indicated that for smaller layer lengths the potential at q~ = 90 ~ increases, as would be expected. In the limit, as layer length approaches zero, the potential becomes equal to that of the centric dipole.
,=1
which simplifies to: N
~_~ D j D L ( ~ )
= ~
ai ~ , Dj(dp)Di(c~)
i=i
For N dipoles, a system of N simultaneous equations of the above type specify the coefficients ai, for the minimum squared error. This reduces to a matrix equation of the following form:
110
KEMPNER A N D GRAYZEL
[B] = [A][X] Where [Bj
Z)~ (~)Dr~(~)
=
I I I r
[A]
DN(dp)DL(dp)
=
D1 (*)O1(*)
~ D1 (,)DN(~)
~_, DN(dp)DI(dp) _
~_~ Ds(ck) DN(O)
r
ep
Ix1 = I ~ I aN
This best-fit analysis was carried out in F O R T R A N IV utilizing double precision arithmetic on an IBM 70~)4 computer. The system of equations was solved with a Gaussian elimination scheme with partial pivoting.
Experimental Studies
Single Dipole, Multiple Dipole, and Dipole-quadrupole Models of the Double-layer in a Circular Lamina* BY KENNETH M. KEMPNER, M.S.~, AND JOSEPH GRAYZEL, M.D.~
SUMMARY
INTRODUCTION
Within a conductive circular lamina, the boundary potentials produced by centric and eccentric double-layers, long and short in length, were compared with those produced by single dipoles, combinations of dipoles, and a dipolequadrupole, respectively. Excellent agreement existed between respective boundary potentials of a single dipole and double-layer, centric or eccentric, provided the single dipole was located at the center of the double-layer. Two dipoles more accurately modeled the double-layer, but only if the two dipoles were optimally located, in which case their spacing or separation reflected the spatial extent of the double-layer. Arrays of three or more dipoles represented a slight further improvement only if the individual dipoles were optimally weighted. Lastly, the centric dipolequadrupole was not as good a model of the eccentric layer as the single dipole located at the center of the layer. The accuracy of various models of the double-layer is examined in terms of the even and odd symmetries of the respective generator configurations.
As early as 1954, G a b o r and Nelson t showed that an equivalent cardiac dipole could be determined from body-surface potentials. The solution of two sets of simultaneous equations specified the components and location, respectively, of the dipole. Also, the same technique permitted determination of a multiple dipole model, which improved the accuracy with which the real surface potentials were duplicated by the equivalent model. Recent efforts to refine the model of the heart generator have generally focussed on multipolar representations, in particular the quadrupole, which is the next ordered pole above the dipole. Considering the electrocardiographic problem as essentially the solution of Poisson's equation for the human torso, the classical technique of separation of variables in spherical coordinates yields a series in Legendre polynomials for the general solution 2. If the boundary is simple and wellbehaved, the series solution converges. Since current streamlines run tangent to the body-surface boundary, the boundary condition OV/On = 0 allows the coefficients of the Legendre polynomial series to be determined. These coefficients are the components of a multipole at the origin which produces a distribution of boundary potentials which matches the real body-surface voltages. To cite some examples, this general approach was employed by Yeh for current generators in a sphere 3, by Hlavin and Plonsey for a turtle heart 4, and for the cardiac generator by Geselowitz 5 who also noted that the dipole components calculated by G a b o r and Nelson I were identical to the dipole terms in the general multipolar expansion. The general solution tot the human torso consists of an infinite number of multipoles 2. Also, the higher the order of a multipole, the more rapidly its contribution to field potential diminishes with distance from the source. Hence, at
* From the Division of Bioengineering, Columbia University. t Formerly NIH predoctoral fellow in Bioengineering, Columbia University; presently electronic engineer, Division of Computer Research and Technology, NIH, Bethesda, Md. Formerly Asst. Prof. Biomedical Engineering, Columbia University; presently Chief of Cardiology and Director, Cardio-Pulmonary Laboratory, Bergen Pines County Hospital, Paramus, N. J., and Senior Research Associate in Bioengineering, Columbia University. This investigation was supported by research fellowship GM-32, 785 and Research Grant HE-10579 from the NIH, USPHS, and a grant from the New York Heart Association. Requests for reprints should be addressed to: Joseph Grayzel, M.D., Cardiology Section, Bergen Pines County Hospital, Paramus, N.J. 07652. 95
96
KEMPNER A N D G R A Y Z E L
reasonable distances from the origin the dipole alone is an acceptable, or perhaps adequate representation of the generator. A practical need is to define what is meant by a "reasonable distance" or an "adequate" model of the cardiac generator. The literature abounds with studies of cardiac electrical models in various conducting media. The dipole has been examined in a planar circular lamina 6-~~ a planar elliptical lamina u, a cylinder v-',l:~,a sphere ~4-~8 and in a prolate spheroid xg. The double-layer has been studied in a circular lamina '-'~ Rarely have several generator types been placed within the same conducting medium in an attempt to compare their resultant boundary potentials. Frank compared the dipole with a disc-shaped double-layer2L The multipolar representation within a sphere of an eccentric dipole and of a double-layer were studied by Yeh and Martinek '-'~. We chose the plane circular lamina for its simplicity in providing a practical physical method for experimental verification of analytically derived results, yet affording precise comparisons among electric generators which are meaningful for three-dimensional man. Recent suggestions for refinement of the fixedlocation dipole model have included a single moving dipole, two or more dipoles, and a dipolequadrupole. During the physiologic spread of myocardial excitation, the wave-front of depolarization is relatively thin and is generally considered to be a double-layer current source. Therefore, the present study examined the degree of accuracy with which double-layers of different size and location are modelled by a single dipole, multiple dipoles, and a dipole-quadrupole in the same conducting medium. While the present study employs the forward method, i.e., generators are placed within the medium and the resultant boundary potentials computed and compared, such comparison suggests the relative virtues or shortcomings of generators considered for the inverse problem, i.e., the measurement of boundary potentials from which parameters of a particular generator form are determined. MATERIAL AND METHOD The physical model consisted of a homogeneous circular conductive paper lamina, 18 inches in diameter, 0.008 inch thick, and with a bulk conductivity of 0.0106 mho/inch. An analytic counterpart of this physical model was developed by deriving the field equations in the above lamina
for each generator studied, and evaluating the boundary potentials with the aid of a digital computer. The generator configurations placed within this lamina were single dipoles, combinations of these dipoles, double-layers, and a dipole-quadrupole. F o r all of these, the distance between source and sink, or pole separation, was 0.5 inch. The physical paper lamina provided experimental data which principally served to verify the analytic results for basic generator configurations --single dipoles and double-layers. Once verified, the computed boundary potentials were employed for further calculations, comparisons, and plotting since the computed results are more accurate and precise. Single Dipoles. Nine dipole locations were employed, spaced at 3/4 inch intervals along the lamina's horizontal diameter and symmetrical about the center. Thus, the dipoles lie along the central 6 inches of the lamina's 18-inch diameter, as shown in Fig. 1. These nine dipole locations are consecutively numbered from left to right, permitting the simple notation DI to designate a dipole at the first, most leftward location, D5 the centric dipole, D9 the most rightward, and other locations correspondingly. A plexiglas template was used in cutting the circular paper lamina from a roll and locating the two poles of each dipole. Each pole point was permanentized with a tiny drop of silver paint. Through each dipole, in turn, a constant current of several milliamperes was introduced, monitored with an ammeter, while voltages around the lamina's boundary were measured at 5~ intervals with respect to the boundary point at zero degrees. The boundary potentials were divided by the dipole's current to yield values corresponding to one milliampere of source current. Various symmetries confirmed the accuracy of the experimental method. For example, all dipoles shown (Fig. 1) possess odd symmetry about the horizontal diameter, potential around the lower semicircular boundary being the negative mirrorimage of the upper semicircle. Boundary points at 0 ~ and 180 ~ are at zero potential, hence no voltage difference should exist between them. In addition, the centric dipole D5 produces right-left symmetry around the boundary, and such symmetry also exists between symmetrical dipole p a i r s - - l , 9 ; 2,8; 3,7; 4,6. The accuracy of such symmetries among the experimental data support the accu-
DIPOLES AND QUADRUPOLES
LAYER POSITION L(I- 5) SHOWN 9
IN A CIRCULAR
97
LAMINA
90 ~ o
POLE LOCATION
\
!
•0 o
\ §
"x 0
g.
/OOo
1
is0 --~--~///////'l///////.O//////~/~///////~
I
2
3
-- 9
5
4
+0
+0
-0
6
+l
[
I -0
7
I
0 0
"*X
-0
8
9
1 Fig. 1. Geometry of single dipoles and double-layers.
racy of experimental techniques, such as locating and orienting the dipoles, homogeneity and isotropicity of the paper, designation of the boundary points, and reproducability of results since the entire group of experiments was performed over several months. The analytic analogue for the foregoing consisted of an equation for the potential around the boundary of a circular lamina due to a current dipole located on and normal to the horizontal diameter (X-axis). The derivation is given in Appendix A, and consists of first deriving an equation for the dipole in an infinite lamina and then employing the method of images to yield the equation for a finite lamina, which is I la s i n ~ 1 I In 1 + - (r~) 2 I VD(q~) = _7c~b
(1)
where VD is the potential on the boundary in millivolts. $ is the degrees of angle between the line segment from origin to boundary point and the positive X-axis. I is the dipole current in milliamperes. is the bulk conductivity of the medium, mho/inch. b is the thickness of the lamina in inches. a is the radius of the lamina in inches. 1 is the pole separation in inches. Xo is the x-coordinate or displacement of the dipole in inches.
r,
is the distance, in inches, from the dipole to the boundary point.
and (r~) 2 = a2 _ 2axo cos ~b + x02 VD (q~) is the boundary potential with respect to a point at zero potential. Since our dipoles were located on and perpendicular to the lamina's horizontal diameter, this diameter is the zero isopotential line and its end-points at zero and 180 degrees are at zero potential. Recalling that the experimental voltages were measured with respect to the boundary point at zero degrees, these values should match those calculated. Equation (1) was programmed in A L G O L for a Burroughs 220. Multiple Dipoles. The distribution of potential around the lamina resulting from two or more dipoles acting simultaneously was obtained by superposition of the individual dipoles' respective boundary values, appropriately weighted according to each dipole's current strength. Arrays of two, three and five dipoles were studied, with details and results presented below. Double Layers. Three double-layers were studled--(1) a centric double layer 6 inches long, positioned along the lamina's horizontal diameter extending from dipole location #1 to #9; this layer is noted L 1-9 to indicate the relation of its end points to the dipole locations studied; (2) a short, centric double-layer, L 3-7, three inches in length; (3) a short, eccentric double-layer, L 1-5, also
98
KEMPNER A N D G R A Y Z E L
three inches in length. The separation of a doublelayer was 0.5 inch, as was the pole separation for dipoles. The geometry of the short, eccentric layer L 1-5 is shown in Fig. 1. The three physical models were prepared by cutting a circular lamina and painting two fine silver stripes, 88 inch above and below and parallel to the horizontal diameter, the stripes running the extent of the particular double-layer under study. The double-layer was energized by contacting each stripe along its entire length with a heavy, weighted brass plate on end, and passing the measured current between the plates. As before, boundary potentials were measured with respect to the boundary point at zero degrees. The analytic analogue was an equation for boundary potential around a circular lamina due to a current double-layer along the horizontal diameter. This equation is obtained by integrating equation (1) along the diameter between limits which are the end-points of the particular doublelayer. This equation, derived in Appendix B, is
_
4q 2 + h l t a n - ' .
\qtan-I
+ ,q)ll
x~s,j
....
(2)
where VL is the boundary potential in millivolts, produced by the double-layer I K--
7rablo lo is the length of the double-layer in
inches is the layer separation in inches p = (a2 - 2axcosq~q-x 2) q = asin~b l
S =
ll = 1/2. In the centric dipole-quadrupole generator, the dipolar and quadrupolar currents may be varied independently. The net distribution of boundary potential is obtained by superposition. R ES U LTS
Double-layer vs Single Dipole.
VL(dp) = Kxln(1 + lq~ '~ P /1~, ~-Kln[(pT~q)e
considered as a model for the electric doublelayer of the heart. Both the dipolar and quadrupolar contributions were calculated from equation (1). The dipole is simply D5, which had been calculated. The centric quadrupole's geometry, shown in Fig. 6, indicates how its boundary potential distribution may be computed from equation (1) by representing the quadrupole with two dipoles of opposite orientation, each displaced a distance one-half the pole separation either side of the origin and each carrying unit current. This produces equal dipole and quadrupo[e magnitudes in our system. Quadrupole moment, M~ = 21ld, and since d ---l = 0.5, M~ = 1/2. Also, dipole moment, Ma ----
acosdp
xl, x~ are coordinates of the left and fighthand end-points, respectively, of the double-layer a, b, or, and q~are the same as for eq. (1). Equation (2) was programmed in F O R T R A N IV for the IBM 7094. Centrie dipole-quadrupole. The distribution of boundary potential around the lamina was also studied for the centric dipole-quadrupole combination since this configuration can produce asymmetric potential distributions and has often been
Very good or excellent agreement was observed between the boundary potentials produced by a double-layer and those produced by a single dipole located at the mid-point of the layer, the better match occurring for the shorter layer. Comparisons are shown in Fig. 3 for the long centric layer and the short centric layer, and in figure 4 for the short eccentric layer. The long centric layer possessed a linear dimension one-third the diameter of the conductive lamina, a ratio somewhat larger than might exist in real man at any instant of ventricular excitation, yet the rms error of the single centric dipole was less than 3 ~o of the maximum potential on the boundary (Fig. 3A). The shorter layer, whose length is one-sixth the diameter of the conductive lamina provides a more realistic proportion. F o r both centric (Fig. 3B) and eccentric (Fig. 4) positions of this layer, the single dipole at the layer's midpoint provided an excellent match of boundary potentials, with an rms error less than 1 ~o- Clearly, this error is within the accuracy of clinical measurement in almost all situations.
Double-layer vs symmetrical dipole pair. Referring to Fig. 1, the nine dipole locations permit four symmetrical p a i r s - - l , 9 ; 2,8; 3,7; 4,6. With one-half the current flowing through each
99
DIPOLES AND QUADRUPOLES IN A CIRCULAR LAMINA
250
I
I
I
I
I
I
I
I
I
I
200
150 < E E d
I00
Point of MAXIMUM P O T E N T I A L DIPOLE MAGNITUDE LOCATION NO. (mV/rnA) (degrees)
tzLd t-0 rl ,,~ t23 z ::3 o 138
I 2 3 4 5
50
227.3 215.9 2 08.5 204.3 20 2.9
125 ~ 120 ~
IIO ~ I00 ~ 90 ~
CENTRIC QUADRUPOLE
0
I I i
-50
t
0o
I 500
45 ~ I
ANGLE
1, 60 ~
I
I 90 ~
I
qb A B O U T ORIGIN
I 120 ~
155" I
I 150 ~
180 ~
DEGREES
Fig. 2. Boundary potentials for single dipoles and centric quadrupole.
dipole in a pair, the boundary potentials of each pair, respectively, were compared with those of the long, centric double-layer (Fig. 5). The dipole pair 1,9 had an rms error of 5 %. This error diminished as the two dipoles moved toward each other and was a minimum for dipole pair 3,7. As the spacing between the pair further decreased, the error rose again, and for the centric dipole was nearly 3 % as mentioned above. The rms errors of interest appear in Table 1, from which it is readily seen that the optimum pair 3,7 provides a truly excellent match for the double-layer and an rms error only one-fourth that of the centric dipole, though the latter was already a very good model for the long double-layer.
Double-layer vs multiple dipoles. As the number of dipoles simulating a doublelayer is increased, the multiple-dipole array should produce boundary potentials which approach those of the double-layer and, in the limit, be indistinguishable. The manner in which this occurs was studied under two experimental conditions-equal weighting of dipoles in the array, and optimum weighting for minimizing the rms error of boundary potentials compared to those produced by the double-layer. Equal weighting of dipoles. The long, centric double-layer, L 1-9, was simulated by (a) three dipoles at positions 1,5,9, (b) five dipoles at positions
100
KEMPNER AND GRAYZEL I
I
I
I
I
I
I
I
!
I
I
200
<
150
E
...I < ~I00 Z hl
8 < 121 5 0 Z
0m
0 O*
I 30*
I
ANGLE
I 60* ~
I
ABOUT
I 90*
I
I I 120"
ORIGIN
I 150"
I 180*
DEGREES
A. Long double-layer, L(1-9)
Fig 3.
Comparmon of boundary potentials for centric double-layer vs centric dipole, i
1,3,5,7,9, and by (c) nine dipoles at positions 1 to 9. Also, (d) the short, eccentric double-layer, L 1-5, was simulated by three dipoles at positions 1,3,5, and (e) the short centric double-layer, L 3-7, was simulated by three dipoles at positions 3,5,7. The three-dipole array, D 159, with equal weighting produced boundary potentials in very good agreement with the long, centric doublelayer, but was no better than the single centric dipole alone, D5, both having virtually identical rms errors of 2.6 Yo- The five-dipole array reduced the rms error in half to 1.3 ~o, and the nine-dipole array further reduced rms error to 0.65 ~o, thus providing excellent simulation of the long doublelayer. The three-dipole array, D 135, with equal weighting provided an excellent match for the short eccentric double-layer, L 1-5, but no better than the single dipole, D3, located at the midpoint of the layer, both yielding an rms error of 0.70%. Similarly, for the short, centric doublelayer, L 3-7, the single centric dipole, D5, and the
three-dipole array D 357, were equally excellent with an identical rms error of 0.64 ~o. Optimum Weighting of Dipoles. Alternatively, an algorithm was programmed to compute the optimum weighting of individual dipoles in an array such that rms error was minimized. This algorithm is described in Appendix C. The optimum coefficients for such a "best fit" always summed to yield unit total current and always reduced rms error to 0.1 ~o or less. Thus, a graph of boundary potentials for a double-layer and any of the dipole arrays described above were essentially identical when optimum dipole weighting was employed. Table 1 summarizes the foregoing simulations of double-layers by single dipoles, equally weighted dipole arrays, and optimally weighted arrays.
Eccentric Double-Layer vs Centric Dipole-Quadrupole. As presented above, an excellent match occurred between boundary potentials of the short
101
DIPOLES AND QUADRUPOLES IN A CIRCULAR LAMINA
eccentric layer, L 1-5, and the single dipole at the layer's midpoint, D3; the rms error was 0.70~o. A similar comparison between this short, eccentric layer and the optimum centric dipole-quadrupole, D5Q5, revealed a poorer match, with an rms error of 2.56 ~o which is more than three times the error of the single eccentric dipole. These comparisons are graphed in figure 4 and summarized in Table 1, where the relative current strengths through each source-sink pair of the particular generator are presented. I n modeling the layer L 1-5 of unit current strength, both the eccentric dipole D3, and the dipole component of D5Q5 possessed unit current for the best respective matches of boundary potentials. That the dipole component of an equivalent multipolar generator does not depend upon the source's location has been indicated elsewhere 23. The present comparison illustrates this, as well as the fact that the dipolar component is related to the true current strength of the doublelayer, while the quadrupole is related to displacement of the multipolar model from the actual current source. DISCUSSION With regard to developing a practical model of the heart generator, two aspects of our results appear significant. First, very good or excellent agreement existed between respective boundary potentials of a single dipole and a double-layer, centric or eccentric, provided the single dipole was located at the center of the double-layer. Second, definite limitations exist for the fixed-location dipole-quadrupole as a model for the eccentric double-layer.
good model. Also, best agreement is always attained when the current through dipole and double-layer are identical. Thus, a single dipole whose location varies in accordance with the spread of ventricular excitation appears to be an excellent, meaningful model of the heart's doublelayer; its moment reflects the ianic current flowing across the thickness of the real double-layer and its location is in the region of the heart undergoing depolarization. Only the spatial extent of the instantaneous double-layer is not revealed by the dipole model, but the quadrupole also fails in this regard, as we shall indicate later.
Multiple Dipoles. Two dipoles provided a more accurate model of the double-layer than than did a single dipole, but only if the two dipoles were optimally located. Comparisons with dipole pairs were made for the long double-layer, in which case the optimum locations for the two dipoles were at the center of the right and left halves of the layer, i.e. at points one-fourth and three-fourths along the layer's length. Since our long double-layer was twice the length of the short double-layer, the optimum locations for two dipoles treats the long layer as TABLE 1 Double Layer 1-9
Single Dipole Considering the dimensions of the human ventricles and chest cage, the largest double-layer at any instant of ventricular excitation could be approximately one-sixth the transverse diameter or one-third the antero-posterior diameter of the chest. These ratios correspond to our short double-layer and long double-layer, respectively, within the circular lamina. For the short double-layer, centric or eccentric, a single dipole at the center of the layer was an excellent model, rms error being less than 1 ~o, and no further refinement of this dipole model seems indicated or realizable. For the long doublelayer, the rms error was less than 3 ~o, still a very
3-7
1-5
Dipole Model D D D D D D D D
5 4,6 3,7 2,8 1,9 1,5,9 1,3,5,7,9 1 to 9 1 D 5 9 1 3 D 5 7 9 D 5 D 3,5,7 3 D 5 7 D3 D 1,3,5 1 D3 5 DIPOLE D5 Q-POLE Q5
Relative Current 1.0013 0.5 each " " " 0.33 each 0.20 each 0.11 each 0.166 / 0.668 ~ 0.166 0.083 0.336 I 0.162 / 0.336 / 0.083 1.0O0 0.33 each 0.168 1 0.665 0.168 / 1.001 0.33 each 0.167 / 0.666 ~ 0.168 J 1.0O 3.01 ;
RMS Error ~o of Max 2.6 2.1 0.7 1.8 5.3 2.64 1.31 0.65 0.11
<0.03 0.64 0.64 <0.04 0.70 0.70 <0.04 2.56
102
KEMPNER AND GRAYZEL
two adjacent short layers, and places a dipole at the center of each. Interestingly, the rms error for the two-dipole model of the long double-layer was virtually identical to the rms error for the singledipole model of the short double-layer. The two-dipole model of the long double-layer possessed an rms error less than 1%, compared with a 2.6 % rms error for the single-dipole model. In addition to this improvement of an already small error, the real advantage of the two-dipole model is its ability to reflect the spatial dispersion or extent of the double-layer. At their optimum locations, the separation of the two dipoles was one-half the length of the long, uniform doublelayer being modelled. Though arrays of three or more dipoles were also compared to the double-layers, our interest in these models is small, having observed the accuracy of single-dipole and two-dipole models. Of note, however, was that these more extensive arrays possessed increased accuracy only if the individual dipoles were optimally weighted. Though optimum weighting was never equal weighting,
I
I
I
I
I
I
I
I
I
the optimum array always proved to have a total current of unity, equal to current through the double-layer. The foregoing observations of multiple-dipole arrays indicate that the Baule-McFee24 method for experimentally producing a double-layer is not rigorously correct. In that method a double-layer is obtained by an array of "physical" dipoles, with each pole pair carrying identical current owing to current-limiting resistors. Our results with analytically derived dipole arrays and double-layers indicate that the number of such pole pairs, or physical dipoles, per unit length of the double-layer must be large for precise simulation. Otherwise, optimum weighting of the dipoles should be employed to avoid significant errors at certain boundary points related to the location of the generator.
Dipole-Quadrupole. It was of considerable interest to find that the centric dipole-plus-quadrupole was not as good a mode[ of the short eccentric layer as the single eccentric dipole located at the center of the layer. I
I
I
I
I
I
I
I
l
I
I
I
200
150 K
..3
~-Iz
I00
iii t-o r >-
l::b z
50
o r 0
I
0~
30 ~
60 ~
ANGLE
~
ABOUT
90 ~ ORIGIN
120 ~
150 ~
180 ~
DEGREES
B. Short double-layer, L(3-7) Fig. 3. Comparison of boundary potentials for centric double-layer vs centric dipole.
103
DIPOLES AND QUADRUPOLES IN A CIRCULAR LAMINA !
I
I
I
I
I
l
I
I
I
I
I
I
I
I
I
I
I
I
I
I
200
150
E > E ._1 I-
I00
5
I-0 (3.
>n.., (E3 z
50
D 0
rn I
0 0~
50 o
60 ~
90 ~
A N G L E ~ A B O U T ORIGIN
120 ~
150 ~
180 ~
DEGREES
Fig. 4. Comparison of boundary potentials between eccentric double-layer L(1-5), eccentric dipole D3 at the middle of this layer, and the optimum centric dipole-quadrupole D5Q5.
And this short layer, L 1-5, was only slightly eccentric, its center being 1.5 inches from the center of the lamina with radius 9 inches, yielding an eccentricity of 1.5/9 or 0.16. Furthermore, examination of our results indicates that the quadrupole is limited in the amount of eccentricity it can simulate, beyond which the centric dipole-quadrupole model deteriorates. This property of the quadrupole is now examined. Graphed in Fig. 2 are the potentials around the upper semi-circular boundary of the lamina for dipoles 1,2,3,4, and 5, respectively, and also for a centric quadrupole. For D5, the centric dipole, the maximum potential occurs at the midpoint, 90~ around the boundary. As the dipole location becomes increasingly eccentric the maximum shifts around the boundary. Note, however that the centric quadrupole produces a maximum at 135 ~ and thus, can not simulate a simple distribution whose maximum is further clockwise. For D1, with eccentricity 0.33, the maximum is at 125 ~, and for a dipole with eccentricity greater than 0.5 the maximum will lie beyond 135 ~ Relying on the
excellent relation between a double-layer and a dipole located at its center (rms error < 1 ~o), we recognize that the boundary potentials induced by a double-layer with eccentricity greater than 0.5 will not be closely modelled by a centric dipoleplus-quadrupole, and as the layer's eccentricity increases beyond 0.5, the simulation deteriorates unless higher-order multipoles are added. Schubert 25performed a related analysis within a sphere, and demonstrated that simulation of a dipole whose eccentricity was 0.4 with a centric dipolequadrupole incurred an rms error of approximately 20 ~o. Considering the location of the heart within the chest, double-layers with eccentricities* greater than 0.5 are to be expected. The right ventricular free wall and the antero-septal region of the left ventricle lie immediately beneath the sternum and anterior rib cage, and are highly eccentric. The * Eccentricity of a source in a circular or spherical medium has been defined quantitativelyas the distance of the source from the center of the medium divided by the radius of the medium.
104
KEMPNER A N D G R A Y Z E L
I
2oo!
I
I
I
I
i
I
J
I
150
.......... .....
E ._l
l-z
I00
L (I-9) D (4,6) D (3,7) D (2,8) D (1,9)
ILl
I-o CL
r~ z
50
o rn
0 ~
0~
I
250*
I
613"
I
90*
I
I
120"
_J____l
150"
I
I
180"
A N G L E q~ A B O U T ORIGIN D E G R E E S Fig. 5. Comparison of boundary potentials between tile long centric double-layer L(I 9) and symmetrical pairs of dipoles. The best match for this double-layer is provided by the dipole pair D 3,7 (see text).
eccentricity of the left ventricular free wall is normally about 0.5, which increases with enlargement of the heart in disease states. In view of the foregoing geometry, the limited multipolar expansion truncated at the quadrupole would probably be a poor model of the cardiac generator. On the other hand, extending the multipolar expansion to the octapole, let alone higher-order poles, represents an imposing computational exercise applied to man. Schubert 2~ suggested that the error associated with the centric dipole-quadrupole would be reduced if the multipolar series were expanded about the unique "best" origin, which generally corresponds to the electrical center of a distributed source. A theorem developed by Geselowitz2~ locates this optimum origin, which proves to be nothing more than the point where a dipole alone provides the best simulation of boundary potentials. Thus, we have returned to our earlier observation that an excellent model of the double-layer is a single dipole at its center. But even having determined the optimum loca-
tion for the dipole, examination of the symmetries of double-layers, dipoles, and quadrupoles reveals why the quadrupole at this "best" location fails to make a significant contribution to the model. Pertinent generator configurations are shown in Fig. 6. Both the double-layer and dipole have even symmetry about the Y-axis and odd symmetry about the X-axis. It is evident that any discrepancies between the distribution of potentials produced by a uniform double-layer and its optimally located dipole model, respectively, are properly corrected only by addition to the dipole of another generator with the same symmetries as the doublelayer. The rectangular quadrupole has odd symmetries about both axes, and if rotated 45 ~ it has even symmetry about both axes. The linear quadrupole has even symmetry about both axes. Thus, the quadrupole fails. The particular octapole shown has the desired symmetries--even about the Y-axis and odd about the X-axis--and is a candidate for improving the dipole model of the double-layer. Review the foregoing in terms of the boundary
DIPOLES AND QUADRUPOLES IN A CIRCULAR LAMINA
DIPOLE
DOUBLE LAYER
Y
t
OCTAPOLE YAXlS
I +
I I
105
@ @(D @ XAXB 0 @(D (3
I
AXIAL
ORIENTATION
RECTANGULAR QUADRUPOLES - - ]
LINEAR
[--
A XIAL ORIENTATION
--I I
QIQ O(Z)
QUADRUPOLES
45>45> C) ROTATED
45~
l
AXIAL ORIENTATION
J
Fig. 6. Geometry of relevant electric generators.
potentials induced by the long double-layer, L I-9, and its model, D5, respectively, shown in Fig. 3A. Compared to L 1-9, the maximum induced by D5 is slightly greater, as are potentials between 60 ~ and 120~. Outside this central region on either side the dipole's potentials fall below those of the layer until the X-axis is reached at 0~ and 180~ where potentials are zero and equal. From 180~ to 360 ~ the relationships are the negative mirror image of those from 180 ~ to 0 ~ Improvement of this match by addition of a higher-order multipole to the dipole requires a distribution with the correct symmetries. The rectangular quadrupole oriented along the axes induces boundary potentials, shown in Fig. 2, which are zero at 90 ~ and 270 ~ as well as at 0 ~ and 180~ and therefore, if added to D5 would not affect the maximum or minimum. Owing to the quadrupolar distribution's odd symmetry about the boundary point at 90 ~, any improvement of the dipole's match in one quadrant is associated with a corresponding deterioration in the other. Rotating the rectangular quadrupole 45 ~, as shown in Fig. 6, could improve the match
around the upper half of the l~/mina, but now the even symmetry about the X-axis produces a corresponding deterioration around the lower half of the boundary. The linear quadrupole, with even symmetry about both axes, suffers a similar limitation. Thus, the quadrupole plays no role in describing the uniform double-layer if the optimum origin for the multipolar expansion is employed. To date, the dipole-quadrupole has been applied to man by measuring potentials from living, heterogeneous humans, and then computing multipolar coefficients for a homogeneous torso. If the best origin could be chosen, the existence of quadrupolar coefficients would result from distortions of the real-life field produced by the torso's inhomogeneities, such as lung and blood, which are not accounted in the computation. If the best origin for the multipolar expansion is not employed, the quadrupolar coefficients also reflect the displacement or eccentricity of the source with respect to the mathematical origin. The octapole, however, added to the dipole
106
KEMPNER AND GRAYZEL
could improve the match. Its symmetry is o d d about the X-axis and even about the Y-axis. Inspection of the particular octapole in Fig. 6 reveals that, suitably weighted and added to the dipole, it makes the m a x i m u m less positive, the m i n i m u m less negative, and improves the match around the sides. Also, examining our results for the single dipole DS, dipole pairs of varying separation (Fig. 5), and the short layer L 3-7 vs the long layer L 1-9, we observe the more disperse the source, the greater the departure of boundary potentials from those of the single dipole, and the greater will be the moment of the octapole in a dipole-octapole model. Thus, the octapole reflects spatial dispersion of the uniform double-layer, and represents the next multipole capable of providing additional information concerning the double-layer itself, as distinguished from the elt'ccts of eccentricities and torso inhomogeneities. It appears that the first improvement of the fixed-dipole model should be that of a moving dipole. A subsequent refinement could be addition of the octapole to the dipole at its optimum instantaneous location, or substituting for the moving dipole a moving dipole pair with o p t i m u m separation. Acknowledgment: The authors wish to thank Miss Diane Blackman for assisting us with programming and for providing plots of all data. REFERENCES 1. Gabor, D., and Nelson, C. V.: Determination of the resultant dipole of the heart from measurements on the body surface. J. of Appl. Physics 25 : 413, 1954. 2. Byerly, W. E.: An Elementary Treatise on Fourier's Series and Spherical, Cylindrical, and Ellipsoidal Harmonics with Applications to Problems in Mathematical Physics. Dover, N. Y., pp. 195-218, 1959. 3. Yeh, G. C. K. : Eccentric multipole representations of current generators in a spherical volume conductor. Bull. Mathematical Biophysics 23: 263, 196l. 4. Hlavin, J. M., and Plonsey, R.: An experimental determination of a multipole representation of a turtle heart. IEEE Trans. Biomed. Electronics 10: 98, 1963. 5. Geselowitz, D. B. : Multipole representation for an equivalent cardiac generator. Proc. IRE 48: 75, 1960. 6. Bayley, R. H. : The electric field produced by an eccentric dipole in a homogeneous circular conducting lamina. Circulation Res. 7: 272, 1959. 7. Bayley, R. H. : Unipolar measurements in the electric field produced by an arbitrary dipole in a circular homogeneous lamina. Circulation Res. 7: 537, 1959.
8. Bayley, R. H., and Berry, P. M. : The electric field produced by the eccentric current dipole in the nonhomogeneous conductor. Am. Heart J. 63: 808, 1962. 9. Nelson, C. V., and Gastonguay, P.: A definition for zero potential. Circulation Res. 7: 1039, 1959. 10. Toyoshima, H., Yamada, T., and Otani, K.: The Electrical Analysis of So-called Electrical Heart Axis by Vectorcardiography. Jap. Circ. J. 16: 323, 1953. 1l. Bayley, R. H.: Measurements of unipolar potentials in the electrical field produced by an arbitrary dipole in the elliptical homogeneous lamina. Am. Heart J. 59: 737, 1963. 12. Okada, R. H. : Potentials produced by an eccentric current dipole in a finite-length circular conducting cylinder. [RE Trans. Med. Electronics ME-7: 14, 1956. 13. Okada, R. H. : An experimental study of multiple dipole potentials and the effects of inhomogeneities in volume conductors. Am. Heart J. 54: 567, 1957. 14. Wilson, F. N., and Bayley, R. H.: The electric field of an eccentric dipole m a homogeneous spherical conducting medimn. Circulation I : 84, 1950. 15. Frank, E.: Electrical potential produced by two point current sources in a homogcneous conducting sphere. J. Appl. Physics 23: 1225, 1952. 16. Nelson, C. V.: A convenient expression for the potential at the surface of a sphere due to an eccentric dipole. Digest 1961 International Conf Med. Electronics, p. 238. 17. Bayley, R. H., and Berry, P. M.: Body surface potentials produced by the eccentric dipole in the heart wall of the nonhomogeneous volume conductor. Am. Heart J. 65: 209, 1963. 18. Yeh, G. C. K., and Martinek, J.: Comparison of surface potentials due to several singularity representations of the human heart. Bull. Mathematical Biophysics 19: 293, 1957. 19. Yeh, G. C. K., and Martinek, J.: The potential of a general dipole in a homogeneous conducting prolate spheroid. N. Y. Acad. Sci. 65: 1093, 1957. 20. Bayley, R. H., and Berry, P. M.: The arbitrary electromotive double layer in the eccentric 'Heart' of the nonhomogeneous circular lamina. IEEE Trans. Biomed. Eng. BME-II, 137, 1964. 21. Frank, E. : A comparative analysis of the eccentric double-layer representation of the human heart. Am. Heart J. 46: 364, 1953. 22. Yeh, G. C. K., and Martinek, J.: Multipole representation of an eccentric dipole and an eccentric double-layer. Ball. of Mathematical Biophysics 21: 33, 1959. 23. Geselowitz, D. B.: Two theorems concerning the quadrupole applicable to electrocardiography. 1EEE Trans. Biomed. Eng. BME-12: 164, 1965. 24. Baule, G., and McFee, R.: Theory of magnetic detection of the heart's electrical activity. J. of Appl. Physics 36: 2066, 1965. 25. Schubert, Roy W.: Surface potential errors introduced by the truncation of the infinite series that represents an eccentric dipole, IEEE Trans. Biomed. Engr. BME-13, 101, 1966. 26. Nelson, C. V.: Effect of the finite boundary on potential distribution in volume conductors. Circulation Res. 3: 236, 1955.
107
DIPOLES AND QUADRUPOLES IN A CIRCULAR LAMINA
27. Plonsey, R.: Current dipole images and reference potentials. IEEE Trans. Biomed. Electronics. BME-10: 3, 1963. 28. Plonsey, R., and Collin, R.: Principles and applications of electromagnetic fields. McGraw-Hill, New York, pp. 69-70, 1961. APPENDIX A BOUNDARY POTENTIALS OF AN ECCENTRIC DIPOLE IN A CIRCULAR LAMINA
Consider a current source of strength I in an infinite conductive lamina of thickness b and conductivity or. Current streamlines are radial from the source and surfaces of constant current density Jr, are concentric circular cylinders through the thickness of the lamina, where
or
lizing the circular symmetry of the system 26-28. The boundary may be removed if an appropriate image dipole with a pole separation a21/xo ~, is located at a distance a2/xo from the center of the circular lamina, where a is the radius of the lamina and x0 is the displacement of the interior dipole from the center of the lamina (Fig. 7b). The following relations were determined from the geometry of the system: sin q~i -
ri
sin qS~ 2
/2
2
2
1.~ =
I Jr -- 27rl"b -- cr Er
(A 1)
I Er = 2~rab
(A2)
-
(2ra2/x0)
V(+~ =
~
1
I
r sin q) r8
cos ~ +
Jb
Y
P
T
ro
-2~Trcrbdr = -2-~crb In -- (A3)
I
]
r,
Similarly, for a current sink V( I -
(.2/x0)2
r~ = r -- 2rxo cos 4) + xo 2
The potential difference between a p~int at a distance rl from the source and a reference point r0 is obtained by integrating eq (A2), yielding r~
r sin 4)
PX
(A4)
f L-- 1, Inr~ I"2
Referring to Fig. 7a, the potential resulting from the source-sink pair is V(+_~ =
V(+) +
I V(_) = ~ I n r 2
(A5) 1.1
The field of this source-sink pair approaches that of a dipole as r / l -+ o~, where / is the pole separation. The following relations then become valid: rl-- r~--
89
Fig. 7. A. Geometry for single dipole in an infinite lamina.
s
r -- r l ~ / s i n 4 )
ri
rs
Utilizing these relations in eq (A5) yields
Vj =
, ( -2-Tr~bbIn 1 +
--x2~ ,
(a6)
which is the potential at a distance r from a dipole in the infinite conducting lamina. If a circular boundary is imposed on the conducting lamina, the problem may be solved conveniently by employing the method of images uti-
Xo
B. Geometry for single dipole and its image dipole in a finite circular lamina.
108
KEMPNER AND GRAYZEL
At the boundary of the lamina, r
=
a
and
Integrating equation (B1) by parts yields:
a ri
=
--- rs Xo
v~ (49)
.
Calculating the potential led due to the dipole and its image yields:
VD(a, 49) = Ve(a, 49) + V~(a, 49)
Z
(
= KxIn
K
f[
1 q- ( a - - [3x q - x 2)
"
~
+
.r(2x - ~)x
1 (~ + ~ -- 5 x + X2 ) (~ -- ~X + 22 )
la sin 49)(A7)
dx
(B2) Where
where r,
2
K = I/zco-blo
2
= a -- 2axo cos 49 -}-x0 ~
2
o l : a
Equation (A7) is equation (1) appearing earlier in the manuscript. Though several equations pertaining to dipoles and doublets in a circular lamina have been published s,~, we derived eq. (A7) in order to provide a closed-form solution which is readily integrated with respect to dipole position, thereby providing a comparable equation for the double-layer. A further simplification can be accomplished by utilizing the fact that In (1 -+- u) ~ u if u -~ 0, which rcduccs cq. (A7) to V ~ ( a , qg) ~
___lla sin 49 rccrbrs 2
/3 = 2acos49 y = la sin 49
Expanding the integrand by partial fractions and integrating results in the following expression:
...... ~, ........ )~' v,.O) = K,: In 1 + (~ _ ~." + x") x~ + K t , ( ( x + A)A(x + B)"(x -- C) ~
(A8)
This cxpression is identical to the final result of Plonsey ~-7 in an analysis of dipole images. At x0 = 0 and 49 = 90 ~ this approximation produces a boundary potential value which is about 3 ~o higher than the solution derived here, per eq. (A7). The agreement is generally better for other wflues of x0 and 49.
x(~-D)"),il
(B3)
Where A = --~/2
q- ~ 1 ~ / 4 _
(a -t- 7)
~- - - C ~ Jr- jC2
B = --~/2 -- 4~'~/4 _ (o, + ~,) =
--
C1
=
+~/2
--
JC2
APPENDIX B c
B O U N D A R Y P O T E N T I A L S OF A D O U B L E LAYER IN A CIRCULAR LAMINA
",J~V4
-
= q- C1 -- jCa
The double-layer shown in Fig. 8 has a pole separation l, and a length l0 = Ix2 -- xll. The differential contribution to the boundary potential of the double-layer is that of a single dipole whose pole strength is the current density along the layer,
z/z0
o = +~/2
+ +74
-
= -]-C1 -}- jCa
Introducing polar coordinates as follows: (x q- A ) = Rle+~~
I
dVL
-
(49 )
-
wcrblo
/n(1 -}-la sin~_)dx r. 2
where 2
r~ =
a2
-- 2ax cos 49 q- x 2
(B1)
(x q- B) = R t e - i ~ (x-
C) = R2e+Y~
(x-
D) = R2e-i~
allows the reduction of the boundary potential
109
DIPOLES AND QUADRUPOLES IN A CIRCULAR LAMINA
Y <~
E
205 202.9
03
,~ 200
+Z/.~ ~
..3
Z hi I
o~- 195(~
I
I
I
I
I
9
2 4 6 lAYER LENGTH (INCHES) Fig. 9. The maximum boundary potential induced by a double-layer as a function of the layer's length. IX.
~o = Ixz-x, I Fig. 8. Geometry for deriving the double-layer from the single dipole. APPENDIX C ALGORITHM FOR CALCULATING THE OPTIMUM D I P O L E C O E F F I C I E N T S F O R ~BEST F I T " WITH A DOUBLE-LAYER
expression to the final form: Let, DL(4)) represent the double-layer boundary potential distribution and Di(r represent the boundary potential distribution of the ith dipole in a multiple dipole approximation of the doublelayer. For N dipoles, the total squared error around the boundary is:
VL(4)) = K x ln 1 + i
+ Kin
-
-
e 2 q t a n -1
~lq~"+ lq tan-1 \
7--s-]
q
(B4)
f]
Where
2
2
q = a sin q5 S ~
DL(4,)-
aiOi(~) i=1
where a~ are the coefficients in the linear combination of multiple dipoles, e2 will be a minimum when
I 7rablo p = a -- 2 a x c o s q 5 + x
e2 = ~ %
0e 2 ----= Oaj
0For j=
1,2,.-.
N,
or
a cosq~ =
This expression is evaluated between the end points of the double-layer to provide the exact boundary potential value at any given boundary point. In order to check this solution, the potential at q~ --- 90~ was computed for centric layers varying in length from 6 inches to 0.2 inches. The results graphed in Fig. 9, indicated that for smaller layer lengths the potential at q~ = 90 ~ increases, as would be expected. In the limit, as layer length approaches zero, the potential becomes equal to that of the centric dipole.
,=1
which simplifies to: N
~_~ D j D L ( ~ )
= ~
ai ~ , Dj(dp)Di(c~)
i=i
For N dipoles, a system of N simultaneous equations of the above type specify the coefficients ai, for the minimum squared error. This reduces to a matrix equation of the following form:
110
KEMPNER A N D GRAYZEL
[B] = [A][X] Where [Bj
Z)~ (~)Dr~(~)
=
I I I r
[A]
DN(dp)DL(dp)
=
D1 (*)O1(*)
~ D1 (,)DN(~)
~_, DN(dp)DI(dp) _
~_~ Ds(ck) DN(O)
r
ep
Ix1 = I ~ I aN
This best-fit analysis was carried out in F O R T R A N IV utilizing double precision arithmetic on an IBM 70~)4 computer. The system of equations was solved with a Gaussian elimination scheme with partial pivoting.