IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL 39. NO 9. SEPTEMBER 1992

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Calculating Endocardial Potentials from Epicardial Potentials Measured During External Stimulation Patrick D. Wolf, Member, IEEE, Anthony S. L. Tang, Raymond E. Ideker, and The0 C. Pilkington, Fellow, IEEE

Abstract-This paper presents a boundary integral method for calculating the potential field generated by external stimulation at locations within the heart using realistic heart geometry and samples of the potential taken from the epicardial surface. This method assumes the heart is homogeneous and isotropic. To test the method we made epicardial and endocardial measurements in dogs during transthoracic pacing stimuli. From the epicardial potential measurements we predicted the endocardial potential values and compared them with the measured data. Despite the seemingly gross assumptions, the mean correlation coefficient between the measured and predicted potentials for three dogs and eleven stimulation electrode configurations was 0.985, and the mean rms error was 17%.

INTRODUCTION LECTRICAL stimulation of the heart is used routinely in the clinical environment for pacing, cardioversion, and defibrillation. Despite its routine use, the actual mechanism responsible for the effectiveness of these therapies is not known. In order to better understand how electric fields interact with cardiac tissue in these instances, it is necessary to determine the potential field in the heart during the applied stimulus. Recently, our group and others have developed the capability to measure these potentials [ 131, [ 141. We have measured both pacing and defibrillation potentials in small areas of the heart close to stimulating electrodes. The measurements were made with closely spaced electrodes fixed in well defined two and three dimensional arrays. Using the measured values as node voltages in a finite element mesh that included resistive anisotropy, potential gradients and current densities were calculated for the tissue within the grid. Studies in dogs using this technique have determined the potential gradient and current densities required for pacing and the voltage gradient which produces electrical block near high-current electrodes [4], [ 161.

E

Manuscript received February 14, 1991; revised February 6 , 1992. This work was supported in part by research grants HL-40092, HL-42760, HL44066, HL-28429, HL-33637 from the National Heart, Lung, and Blood Institute, National Institutes of Health, Bethesda, Maryland, and the National Science Foundation Engineering Research Center Grant CDR8622201. P. D . Wolf is with the Departments of Biomedical Engineering and Medicine, Duke University, Durham, NC 27706. A.S.L. Tang is with the Department of Medicine, University of Ottawa, Ontario, Canada. R. E. Ideker is with the Departments of Medicine and Pathology, Duke University, Durham, NC 27706. T. C . Pilkington is with the Departments of Biomedical Engineering and Electrical Engineering, Duke University, Durham, NC 27706. IEEE Log Number 9201963.

Even though significant information has been gained from potential measurements made in vivo with closely spaced arrays of electrodes, to study the more complex problems of defibrillation and external cardiac pacing it is necessary to characterize the field throughout the myocardium. Doing this with only 128 measuring points means the spatial sampling frequency within the heart is low and so too is the resolution of our model mesh. If all of our measuring points could be utilized to sample the potential on the epicardial surface and a model could be used to determine interior potentials then optimum use would be made of our measuring bandwidth. The whole heart has a very complex conductivity structure. Four separate irregularly shaped, blood-filled cavities are surrounded by anisotropic muscle layers where the direction of the anisotropy is continuously changing in at least two dimensions. Rather than attempt to model this structure we wanted to measure how much error was associated with assuming the heart was a homogeneous and isotropic volume conductor. By measuring the error directly we could judge if the error generated by these assumptions would be acceptable for the needs of a particular modeling study. To characterize the error associated with our model, we measured in dogs the potentials from two sites on plunge electrodes distributed evenly in the heart while pacing externally with thoracic patch electrodes. Potentials recorded from the epicardial site on the plunge needles were used as boundary conditions for the interior potential problem which was solved at each of the endocardial recording sites. We then compared the predicted endocardial potentials to the endocardial potentials measured in the animals. The purpose of the study was to show how well the endocardial potentials could be predicted from the measured epicardial potentials for the specific case of an externally applied field. Two applications of this type of modeling are 1) determining the field in the heart during transthoracic defibrillation and 2) determining the field generated in the heart by emergency external pacing. In this paper we use the actual measured potentials on the surface of the heart as boundary conditions for calculating potentials inside. Claydon [2] used torso normal gradient boundary conditions to predict the epicardial potentials resulting from an applied current. Oostendorp showed the importance of using constant potential bound-

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ary conditions on the electrodes when calculating heart potentials from electrodes placed on the body surface [7]. These studies used boundary integrals to calculate the field in the heart resulting from a given set of ideal torso boundary conditions and so are useful for screening electrode configurations before they have been tested in an animal. The method discussed here is used to determine the characteristics of the intracardiac field once the epicardial potentials have been measured experimentally.

METHODS The details of how the measured potentials and heart geometry were obtained appear elsewhere [12] and will only be summarized here. Three mongrel dogs, 20-25 kg, were anesthetized with pentobarbital. Through a median sternotomy the heart was exposed and suspended in a pericardial cradle. Thirty left-ventricular plunge needles with potential measuring electrodes located 1 and 9 mm from the epicardial surface, 22 right-ventricular plunge electrodes with recording sites 1 and 5 mm from the epicardium, and eight atrial wire loops with a single epicardial recording site were secured to the heart. The chest was closed and the pleural space evacuated. Eight model 412 R2 ECG-Defib electrodes (Darox Corporation, Niles, IL) were attached to the torso of the animal in the positions diagrammed in Fig. 1. The surface area of each electrode was 41 square cm. A Physio Control (Redmond, WA) emergency external pacing device was used to deliver stimuli of 80-190 mA through four different electrode combinations: left-lower right-upper (LLRU), anterior posterior (ATPT), left-lower left-upper LLLU), and left right (LTRT). The resulting potentials on the myocardial electrodes were measured and stored using our cardiac mapping system [ 141, [ 151. Following the experiment, the electrode positions were marked and the heart was fixed in formalin and encased in a gelatine cube [ 5 ] . The cube was sliced into approximately 30 slices, each 2 mm thick, and the electrode positions demarcated on each slice using a digitizing pad. Using the x-y coordinates obtained from the digitizer and the z coordinate obtained from the slice number, the electrode locations were found. The epicardial surface was triangulated by hand. The resulting heart surface made from the faces of the triangles is shown for one dog in Fig. 2(a). Each triangle was subdivided into six subtriangles using the centroid and the midpoint of each side as vertices. Two of the subtriangles were associated with each vertex. Each triangle which had a given electrode as a vertex contributed two subtriangles to the region assigned to that electrode [8], [lo]. Fig. 2(b) shows the subregion of the surface associated with each electrode for the heart surface triangulated in panel (a). All of the subtriangles associated with a given vertex were assigned the potential measured on the electrode at the vertex. This is equivalent to saying that a constant shape function was used for the potential on the boundary elements. The heart was considered as a uniform, isotropic me-

Fig. 1 . Diagram of the animal torso shown form the anterior with the positions o f the patch electrodes indicated: RU right upper, LU left upper, RT right, LT left, AT anterior, PT posterior (hidden), RL right lower, and LL left lower.

dium with homogeneous conductivity. Because the pacing shocks were given during electrical diastole, the heart region was modeled as source free. Using these assumptions and using the derivation shown in the appendix, the vector of potentials qCat a set of interior points was calculated from -

@c = z., (1) where &H was the vector of measured epicardial potentials and Z was the forward transfer coefficient matrix based entirely on the positions of the electrodes. Using the discretized heart surface described above, the forward transfer coefficients (the elements of the Z matrix) were calculated. Solid angles were calculated for each subtriangle using the equation found in Barr [ 11. The self term, or the solid angle subtended by an element of surface from a point on that surface, was calculated by subtracting the sum of the remaining solid angles from 4 a. Elements of the (G") matrix were calculated using a seven point Gauss quadrature integration for a triangle [IO]. Multiplying the Z matrix by the measured epicardial voltage vectors obtained during the experiments produced a set of predicted endocardial values for each source electrode combination. Dog 6415 had 60 epicardial points, dog 8799 had 61, and dog 6598 had 62. Dog 8799 had no data for the left-lower left-upper electrode configuration. The measured and calculated endocardial potentials were compared using linear regression and the root mean square (rms) error. Because a substantial mean potential was present for each shock measurement, the rms error was calculated after the mean potential had been subtracted from each value. The slopes from all groups were compared against a mean of 1 .O using the t-statistic.

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WOLF e r a / : PREDICTING E N D O C A R D I A L POTENTIALS

(a)

(b)

Fig. 2. (a) Anterior surface of the heart from dog 6415 showing a gray scale proportional to the measured potentials (volts). The potentials shown were generated by the LLRU electrode configuration. Also shown are the triangles used to discretize the surface. A measuring electrode was positioned at the vertex of each triangle. (b) Discretized surface of the heart shown in (a) showing the surface region associated with each measuring electrode. This heart is also shaded with the gray level proportional to the measured voltage.

TABLE I CORRFLATION C O E F ~ I C I RMS E ~ T ,ERROR A N D RECREWOL SLOPEFOR THE C O M P A R I SBFI O ~W F F N MEASURFD VOLT4GEb A Y 1 1 VOLTAGES PRFDICTED BY T H F MODELFOR T H E FOLRELECTRODt CONFIGURATIONS I N EACHDoc Electrode LLLU

LLRU

LTRT

PTAT

Mean

0.986 0.978

0.988 15.8 1.014

0.992 14.8 1.058

0.985 18.9 I .044

0.988 16.5 1.023

0.971 23.3 1.004

0.971 23.4 I .O06

0.985 22.2 1.125

0.988 15.9 1 .OS9

0.979 21.2 1.049

0.990 15.4 1.051

0.992 13.3 1.045

0.987 16.1

-

1.032

0.990 14.9 1.043

0.978 19.9 0.991

0.983 18.2 1.024

0.990 16.8 1.076

0.987 16.5 1.045

0.985 17.0 1.038

Dog

Measure

6415

Corr. Coef. rms Error ( % ) Slope (V/V)

16.6

6598

Corr. Coef. rms Error ( % ) Slope ( V / V )

8799

Corr. Coef. rmsError(%) Slope (V /V)

Mean

-

Corr. Coef. r m s Error ( % ) Slope ( V / V )

~

~

represents missing data

RESULTS

> E

1

The linear correlation between the measured and predicted endocardial potentials is summarized in Table I. Table I shows the correlation coefficients, the rms error, and the calculated regression slope for each dog and pacing electrode combination. Dog and electrode means are also shown. The mean correlation coefficient for all dogs was 0.985; the range was from 0.971 to 0.992. For all dogs, all cases, the mean rms error was 17.0%; the range extended from 13.3 to 23.4%. The mean slope for all cases was 1.038 which was significantly different than 1 .O ( p < 0.01) when compared using the r statistic. There was a range of slopes from 0.978 to 1.125. Fig. 3 is a regression plot for dog 6415 during a shock through the

%

400 .i.

...*

9 200

.*

e-

n

4

-200

0

200

400

MEASURED VOLTAGE (rnV) Fig. 3. Regression plot showing the calculated potentials as a function of the measured potentials for an LTRT shock configuration in dog 6415. The correlation coefficient was 0.992 and the r m s error was 14.8% forthis comparison.

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39. NO. 9, SEPTEMBER 1992

916

RV

RV LV

LV

-2500

RV

RV -2300

LV

LV

w -2200

(c) (d) Fig. 4. Isopotential maps of the endocardial surface of dog 8799. The view of the heart is from the anterior with the right ventricle on the left and the left ventricle on the right. Panels (a) and (c) are the measured potentials, panels (b) and (d) are the corresponding calculated potentials. Panels (a) and (b) are the potentials due to a shock given through the RTLT configuration, the isopotential lines are 50 mV apart on these maps. Panels (c) and (d) are from a shock given from the LLRU configuration. the lines on these maps are 100 mV apart.

LTRT electrode combination. Our model predicted the endocardial potentials with 14.8% error for this shock. To give the reader a better feel for the amount of error represented by value of 17% rms error we present the isopotential maps shown in Fig. 4. The panels show maps for one dog and two different shock electrode configurations. For the LTRT configuration the error was 13.3%; for the LLRU configuration the error was 15.4%. It can be seen that there is very little qualitative difference between the measured and predicted maps. Fig. 5 shows the distribution of weighting coefficients for a left-ventricular endocardial point. The absolute value of the coefficient is shown on the y axis; the distance from the field point to the measuring electrodes is shown on the z axis. Only 17 electrodes have coefficients greater than 1 % of the total weighting. The greatest magnitude coefficient always weighted the voltage on the closest electrode. In all cases this was the epicardial electrode on the same plunge needle as the endocardial voltage being calculated. Because the weighting coefficient of the epicardial voltage most proximal to the endocardial potential being calculated was so significant, we tested the degree of correlation between these two measured potentials. We repeated the statistical analysis done for the measured versus predicted endocardial potentials on the measured endocardial versus measured epicardial potentials. The result of this comparison appears in Table 11. It can be seen that the correlation coefficients are again high with a mean of 0.974 and a range of 0.938 to 0.992. The calculated slopes were less than 1 as expected with a mean of 0.786 and a range from 0.71 to 0.875. This group of slopes was also significantly different than 1.0. The nns error re-

8

IO

20 30 40 50 60 DISTANCE TO FIELD POINT [mm)

Fig. 5. Plot of the magnitude of the 2 matrix coefficients for electrode number 9 for dog number 6415. The coefficients are plotted as a function of the distance between electrode 9 and the location of the measured voltage weighted by the coefficient. Electrode 9 was a left ventricular plunge electrode.

flected this error in slope with an average value of 28.3% and a range from 22 % to 35 % . These errors are considerably greater than those obtained when all the surface potentials were utilized in the boundary integral model. The fact that the correlation coefficients between measured epicardial and measured endocardial voltages were high suggests that the shape of the distributions are very similar. The value of slope and the resulting rms error show that potentials on the endocardium are lower reflecting an increased distance from the source. This relationship is similar to what would be expected from measurements made in a homogeneous and isotropic volume conductor.

~

WOLF et

U[.:

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PREDICTING ENDOCARDIAL POTENTIALS

TABLE I1 CORRELATION COEFFICIENT, RAMS ERROR A N D REGRESSION SLOPEFOR THE COMPARISON BETWEEN MEASURED EPICARDIAL VOLTAGES AND MEASURED ENDOCARDIAL VOLTAGES FOR THE FOUR I N EACHDoc ELECTRODE CONFIGURATIONS ~~

~

~

Electrode

Dog

Measure

LLLU

LLRU

LTRT

PTAT

Mean

6415

Corr. Coef. rms Error ( % ) Slope (V /V)

0.979 22.0 0.875

0.982 22.5 0.846

0.987 24.1 0.796

0.978 28.0 0.775

0.981 24.1 0.823

6598

Corr. Coef. r m s Error (%) Slope ( V / V )

0.938 35.6 0.801

0.949 33.5 0.795

0.983 30.4 0.729

0.978 29.4 0.754

0.962 32.2 0.770

8799

Corr. Coef. rms Error (%) Slope (V /V)

-

0.972 28.5 0.789

0.992 24.6 0.775

0.976 33. I 0.710

0.980 28.7 0.758

Corr. Coef rms Error ( % ) Slope (V/V)

0.959 28.8 0.838

0.967 28.1 0.810

0.987 26.3 0.767

0.977 30. I 0.747

0.974 28.3 0.786

Mean

-

-represents missing data

DISCUSSION We measure the voltage in the heart during externally applied shocks to characterize the field and determine its effect on the tissue. To make optimal use of our measurements we apply them to volume conductor models where the tissue is represented as a purely resistive medium and then solve Laplace's equation to interpolate between samples and to determine other characteristics of the field. If the conductivity model of the heart under certain conditions could be assumed to be homogeneous and isotropic it would greatly simplify our models. For this reason we wanted to determine with what error we could predict the voltages inside the heart during externally applied stimuli. We used the boundary integral technique as described in the methods to predict the endocardial potentials from the measured epicardial data and a purely geometric transfer coefficient matrix. We then compared the predicted values with potentials measured on the endocardium. As the results indicate, the endocardial potentials can be predicted with relatively small error despite the assumptions of a homogeneous isotropic heart. These results are particularly surprising in light of the experimental error which can be expected. We estimate an error of 3-5 mm in the measured position of the electrodes and 1-2 % error in the recorded potentials. The position error is due in part to the technique for obtaining the geometry in the fixed specimen and in part to the unknown changes in geometry produced by the formalin fixing process. Below we discuss some of the sources of error and possible reasons for the good correlations that we obtained with such idealistic assumptions.

Discretization and Numerical Technique Considering the discretization of the heart surface shown in Fig. 2 it appears the level of discretization, es-

pecially in the atria, could contribute to some of the error in our results. The boundary element technique we use to solve the problem assumes constant or step shape functions for the potential and normal gradient on the elements. Assuming that these values are constant over a region also results in some modeling error. We quantitated the amount of error due to the numerical technique and the level of discretization as follows. The relative positions of the electrodes from dog 8799 were placed in a 12 cm radius homogeneous and isotropic sphere. The heart shape was placed in the first octant of the sphere which was centered at the origin. Spherical cap electrodes with constant current density and an area of 40 square cm were positioned on the surface of the sphere; the remainder of the boundary was considered isolated. Using Legendre polynomials (a = 1000) we solved this Neumann problem for the potentials at all of the electrode sites. These data were then treated as sets of measured data and processed in the same manner as the actual measured data. Three configurations of electrodes were considered: 1) polar, with electrodes placed at the top and bottom poles of the sphere, 2) right-left, with the electrodes placed on the right and left sides of the sphere, and 3) left-front with one electrode on the left pole and the second 90" away on the front of the sphere. 100 mA of current was used in the simulation and the conductivity of the sphere was 2 mS/cm. The range of voltages calculated at the electrode positions was approximately 1 V . This was similar to the range we measured in the animal experiments. The mean correlation coefficient for the Legendre calculated versus the model predicted endocardial voltages was 0.9997, the mean rms error was 2 . 4 % .The largest error (2.95% rms) was associated with the left-front configuration. In this configuration the spatial gradients were the highest. This simulation was intended to remove all other sources of error and to test the numerical accuracy of the

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algorithm and the level of discretization. Although it is possible that the potential field in the canine thorax is more spatially complex than the fields tested here, the gradients for the left-front configuration appeared as high as any seen in the animal data. We conclude from this analysis that the discretization level and the numerical technique contribute about 3% rms to the total error.

Anisotropy It is known that the resistivity of heart muscle is different along and across fibers. The results of studies in the tissue bath 191 and in animals have shown a ratio of about 3 to 1 for transverse versus longitudinal resistivities. In all of these studies, however, the conductivities were measured over relatively short distances, usually in the same planes as the tissue anisotropy. Frazier et al. [14] showed that for epicardial stimulation the isopotential lines in a subepicardial plane of recording electrodes produced the elliptical pattern characteristic of an anisotropic conductor; but at the level of the endocardium the isopotential lines were almost circular, as would be expected for an isotropic medium. The explanation for this phenomenon is most likely that the rotation of the muscle layers through the wall [ 1 11 cancels the anisotropic effect. Because the direction of the tissue anisotropy is continuously changing, its effect over distances greater than a few millimeters appears to be smoothed. This is not to say that the microscopic effects of anisotropy can be neglected. The anisotropy is a major determinant of stimulation threshold [4]. But when calculating gross voltage changes over distances large compared to the cell structure, the isotropic assumption appears to generate a limited amount of error. Blood Cuvity The most surprising result generated by this model is that the predicted endocardial potentials are good even when the inhomogeneity of the blood cavity is neglected. To estimate the magnitude of the effect of neglecting the high conductivity of blood we created a simpler two concentric spheres model to compare the endocardial potentials calculated for the homogeneous and nonhomogeneous cases. The spheres represented a 3.0 cm radius heart with a 2.0 cm radius cavity in an infinite torso. The conductivities were 6.67, 3 . 3 3 , and 2.87 mS/cm for the blood, heart muscle and torso, respectively. Simulated “measured” epicardial and endocardial potentials were calculated at radii of 2.9, 2.5, and 2.1 cm for the spheres in a uniform field. These distances represented the recording levels of the epicardial and endocardial points on the right and left type of plunge electrodes. The measured epicardial potentials were then applied to the same locations on a homogeneous sphere of conductivity 3.33 mS/cm. Using these voltage boundary conditions, the potentials at the locations of the endocardial electrodes were calculated. For this simple model the ratio of the homogeneous endocardial potentials to the inhomoge-

neous endocardial potentials was 1.17 for the left plunges and 1.05 for the right plunges. When a thin walled shell (0.6 cm) was used to calculate the inhomogeneous values, the ratio for right ventricular plunges was 1.10. If the heart did fit the concentric sphere model, we would expect a slope for the predicted versus actual data in Table I to be close to 1.13; our value for mean slope was 1.04. We would also expect different slopes for the left plunge and right plunge groups. We used an analysis of covariance to determine if there were different slopes within each of the dog stimulation subgroups. The analysis showed there to be no significant reduction in the variance of the predicted minus measured potentials residual when the data were split into the two plunge categories. The slopes were not significantly different for the left versus right but the correlation coefficients were higher and the rms errors lower for the right plunge data. We must conclude that for the purpose of calculating endocardial potentials from epicardial potentials, the heart looks more like a homogeneous volume than an inhomogeneous volume with a single large blood-filled cavity. One possible reason for the small sensitivity to the cavity may be the effect of the septa which actually separate the blood cavities into four different blood pools. The presence of the septa would have the effect, in the spherical model, of increasing the average resistivity of the blood cavity. thus narrowing the difference between the homogeneous and inhomogeneous cases.

Sensitivity and Limitations cf the Model The results presented here are generated from a very simple model of the conductivity structure of the heart. The error associated with these assumptions is reasonable considering that we are comparing a model to measurements obtained in a living animal. While more complex models may be developed for other in vivo comparisons, we feel these data serve as a benchmark for future studies of this type. This experiment may simply show the robustncss of the Dirichlet boundary condition problem when the volume is assumed homogeneous and the measurements are made at a relatively large distance from any sources. Another possible conclusion is that predicting the potential so close to recording sites is not a sensitive measure of these assumptions and that 17% error in the voltage data would lead to much greater errors if the actual electric field were calculated. Unfortunately, we do not have the data to compare measured and calculated electric fields directly. Still a third possibility is that the scale of the anisotropy and inhomogeneity effects is at a level lower than that of the whole organ. If it is possible to neglect these effects when modeling the heart at the organ level it would certainly reduce the computational complexity of many problems. We believe these data indicate that if one is willing to accept this level of error the homogeneous, isotropic assumption can be used. We would emphasize, results presented here apply to the specific conditions of this experiment. Most impor-

WOLF et a l . : PREDICTING ENDOCARDIAL POTENTIALS

tantly , computations of the endocardial potentials are limited to fields generated by external stimulation and cannot be applied to compute endocardial potential fields generated by intrinsic cardiac sources during cardiac activity. The principal reason for this limitation is the assumption in the derivation that Laplace's equation holds in the volume surrounded by the epicardial electrodes. If sources are included in this region then the assumption is invalid and a new derivation would be required. Of equal importance, we believe, is the proximity of the sources to the anisotropic region in this model. The sources in this transthoracic pacing study had relatively large surface area and were applied to the body surface. These relatively uniform sources probably reduced the significance of the myocardial conductivity on the final distribution of current in the myocardium thereby reducing the structure's overall contribution to the potentials measured on the endocardium. The effect of the heart's conductivity would become more significant as the sources are moved closer to the heart and would, we believe, be critical in a model used to predict potentials generated by sources located on or in the myocardium. The importance of the conductivity structure of the heart for intrinsic sources was studied by Cuppen and van Oosterom who found that a homogeneous heart and torso model could not be used to satisfactorily reconstruct simulated heart isochronal lines from body surface potentials when an inhomogeneous model was used to generate the simulated forward data. Their model included unique lung and blood cavity conductivities but did not consider anisotropy [3]. One final comment on conductivity-the nature of this calculation does not require a specific value of conductivity to predict the endocardial potentials. It is not clear to us what value of conductivity would be most accurate to use if the heart were modeled as a homogeneous isotropic structure in a torso model. This presents a bit of a quandary for someone wishing to model the heart because on the one hand these results suggest a homogeneous model may be sufficient for many problems, but on the other hand neither the model nor the results imply a specific value of heart conductivity. CONCLUSION Using measured epicardial potentials as boundary conditions for a simple model of the heart appears to be a useful tool for determining the potential field generated by external stimulation at the level of the endocardium. We conclude from our results that the effects of tissue anisotropy and the blood cavity inhomogeneity cause less than 20% rms error in potential calculations within the heart and could, in some cases, be neglected when making field calculations at the organ level.

APPENDIX The heart is considered as a uniform, isotropic medium with homogeneous conductivity. Green's second identity

919

is

1"

(AV'B - BV'A) d V

=

i,

(AVB

-

BVA) . ti d S .

In this equation A and B are scalar functions of position, S is the surface surrounding the volume V , and ti is the outward pointing surface normal. For this development A is the function 1 / r and B is the electric potential 9.Making these substitutions and defining H as the epicardial surface of the heart:

(2) In this equation, 9o is the potential at some observation point inside the heart, aHis the potential on the surface of the heart and r is the distance from the observation point to the element of integration on the epicardium. The normal component of voltage gradient can be obtained in the following manner [6]: the obsevation point approaches a point on the surface from the inside. In the limit as it approaches the surface the following equation holds:

(3) where pr.v. denotes the Cauchy principal value and K contains the contribution to the solid angle from the singular point. Discretizing the surface and assuming the potential does not vary over the discretized element (or equivalently, using a constant shape function) the above integrals can be converted to summations. The equation for point j with the summations substituted becomes

0 = K9j

+

l

N

C

-

47r

r o p i * ti; + ; 7 A S j

i=l i#j

TO-i

+ -47rI=i -1c - v9jr o -- Lti, AS,.

(4)

When N of these equations are written, one for each measured surface potential, a system of equations results which can be solved for the normal gradient. Making the following substitutions: 1 r,-, ti, QHH,l = 3 As, 47r r , - , ~

N

Q",,

=

K,

=

1

-

C a",, r=l 1

*J

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL. 39, NO. 9, SEPTEMBER 1992

920

and solving for

rH:

(111 D. D. Streeter, Jr., “Gross morphology and fiber geometry of the heart, ’’ in Handbook of Physiology; Section 2; The Cardiovascular

A potential at any point within the surface can now be calculated using (2). When there are M unknown interior cavity potentials the following substitutions allow (2) to be written as summations in matrix notation:

[I21

(131

[I41 1151

Substituting for

(7)

r yields

[I61

System, R. M. Berne, Ed. Bethesda, MD: American Physiological Society, 1979, p. 61. A. S . L. Tang, P. D. Wolf, F. J. Claydon 111, W. M. Smith, T . C . Pilkington, and R. E. Ideker, “Measurement of defibrillation shock potential distributions and activation sequences of the heart in three dimensions,” Proc. IEEE, vol. 76, p. 1176, 1988. F. X . Witkowski and P . A . Penkoske, “A new system design for transmural directly coupled (DC) cardiac mapping in vivo,” in Proc. 10th Annu. Conf. IEEE EMBS, 1988, p. 99. P. D. Wolf, J . M . Wharton, C. D. Wilkinson, W . M. Smith, and R. E. Ideker, “A method of measuring cardiac defibrillation potentials, ACEMB86, 1986, p. 4. P. D. Wolf, D. L. Rollins, W. M. Smith, and R. E. Ideker, “A cardiac mapping system for the quantitative study of internal defibrillation,” in Proc. loth Annu. Conf. IEEE EMBS, 1988, p. 217. S . Yabe, W. M. Smith, J. P. Daubert, P. D. Wolf, D. L. Rollins, and R. E. Ideker, “Conduction disturbances caused by high current density electric fields,” Circ. Res., vol. 66, p. 1190, 1990.

This can be shortened to

(9)

REFERENCES [ I ] R. C. Barr, M. Ramsey 111, and M. S . Spach, “Relating epicardial to body surface potential distributions by means of transfer coefficients based on geometry measurements, ” IEEE Trans. Biomed. Eng., vol. BME-24, 1977. [2] F. J . Claydon 111, T . C. Pilkington, A. S . L. Tang, M. N. Morrow, and R. E. Ideker, “A volume conductor model of the thorax for the study of defibrillation fields,” IEEE Trans. Biomed. Eng., vol. BME35, p. 981, 1988. 131 J. J. M. Cuppen and A. Oosterom, “Model studies with the inversely calculated isochrones of ventricular depolarization,” IEEE Trans. Biomed. Eng., vol. BME-31, p. 652, 1984. [4] D. W. Frazier, W. Krassowska, P-S. Chen, P. D. Wolf, E. G. Dixon, W. M. Smith, and R. E. Ideker, “Extracellular field required for excitation in three-dimensional anisotropic canine myocardium, Circ. Res., vol. 63, p. 147, 1988. [5] C. Laxer, R. E. Ideker, and T . C . Pilkington, “The use of unipolar epicardial QRS potentials to estimate myocardial infraction,” IEEE Trans. Biomed. Eng., vol. BME-32, p. 64, 1985. 161 W. C . Metz and T . C. Pilkington, “The utilization of integral equations for solving three-dimensional, time-invariant, conservative fields,” Intern. J . Eng. Sei.., vol. 7, p. 183, 1969. 171 T . Oostendorp and A. van Oosterom, “The potential distribution generated by surface electrodes in inhomogeneous volume conductors of arbitrary shape,” IEEE Trans. Biomed. Eng., vol. 38, p. 409, May 1991. [8] T . C . Pilkington, M . N. Morrow, and P. C . Stanley, “A comparison of finite element and integral equation formulations for the calculation of electrocardiographic potentials-11, ” IEEE Trans. Biomed. Eng.. vol. BME-34, p. 258, 1987. [9] M. S . Spach, R. C . Barr, G. A. Serwer, J. M. Kootsey, and E. A. Johnson, “Extracellular potentials related to intracellular action potentials in dog Purkinje system,” Circ. Res., vol. 30, no. 5 , p. 505, 1972. [ l o ] P. C . Stanley, T . C . Pilkington, and M. N . Morrow, “The effects of thoracic inhomogeneities on the relationship between epicardial and torso potentials,” IEEE Trans. Biomed. Eng., vol. BME-33, p. 273, 1986. ”

Patrick D. Wolf (M’89) was born in Altoona, PA, in 1956. He received the B.S. degree in electrical engineering and an M.S. degree in bioengineering from the Pennsylvania State Univeristy, University Park. From 1978 until 1983 he worked as a Biomedical Engineer in the Department of Surgery at the Milton S . Hershey Medical Center supporting research in coronary blood flow and cardiac dynamics. In 1983 he joined the Basic Arrhythmia Lab at Duke University Medical Center where he is currently designing instrumentation for research in electrical therapy of the heart. He is also a student in the Ph.D. program at Duke University and is a fellow in the Engineering Research Center for Emerging Cardiovascular Technology. His professional interests include electronic circuit design, cardiac mapping and modeling of electric fields in the heart.

Anthonj S. L. Tang received the M D degree at the University ot Toronto, Ont , Canada, and received Medical Specialty Fellowship in Internal Medicine and Cardiology He received cardiac electrophysiology training at Duke University. NC He is now Assistant Professor of Medicine at the University of Ottawa, Ont., Canada His research interests include mechanism and treatment of cardiac arrhythmias. He is a member of the Canddian Cardiovascular Society and a Fellow of the American College ot Cardiology

Raymond E. Ideker, for a photo and biography, see p. 279 of the March 1992 issue of this TRANSACTIONS.

Theo C. Pilkington (S’54-M’59-SM’S4-F’87) for a photograph and biography, see p. 279 of the March 1992 issue of this TRANSACTIONS.