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A note on the “Brody-effect”
J. ELECTROCARDIOLOGY 11 (1), 1978, 87-90
Annotations A Note on the "Brody-Effect" BY YORAM RUDY, M.Sc.* AND ROBERT PLONSEY, PH.D.t
By application of the image principle to the spherical model, Brody concluded t h a t the images ~Augment the manifest strength of radially oriented doublets by a factor of 2 R3/ d3. '' (Where R is the radius of the sphere, and d i s the distance of the dipole from the center of the sphere.) That is, the potential measured will be (1+2 R3/d 3) times the potential due to a single dipole in a homogeneous medium. In the limit as d-~R (the dipole is at the interface), the effect is to triple the strength of the dipole, i.e., the images are two dipoles of the same strength and at the same location as the original source. This factor of 3.0 will also describe the situation when d = R and R --* :% in coutradiction with the factor of 2.0 obtained directly from the solution of the semiinfinite plane wall case. The purpose of this communication is to clarify this discrepancy; in fact to elucidate the nature of the image e n h a n c e m e n t of a dipole s o u r c e in t h e aforementioned spherical model. The discrepancy which occurs in the Brody paper appears in a different form in the paper of Rush and Nelson 3 which gives, instead, an enhancement factor of (1 +R2/d2). Here, in the limit as d o R , the effect is to double the strength of the source. (Which is true for the plane wall case). The authors conclude that "the effects are in the same direction as those of the plane wall case, though not as great," i.e., according to them, the m a x i m u m factor is 2.0 (when the dipole is at the interface), and in t h a t case the image is a single dipole of the same strength and at the same location as the original one. Rush and Nelson correctly mention a "third bipole" t h a t appears as an image, but leave the reader with the impression t h a t its contribution to the potential is negligible because of its small pole strength. The aforementioned ambiguities can be resolved by noting t h a t the correct system of images for the case of a dipole of strength M within the medium of finite conductivity but at the interface with the perfectly conducting
SUMMARY The effect of a perfectly conducting sphere simulating the intracavitary blood mass on a dipole source located at the interface with the outer tissue (myocardium) is studied, utilizing image theory. The resulting enhancement factor is found to be a function of the field point location and is not a constant, as previously reported by Brody and by Rush and Nelson. Of the several thoracic inhomogeneities the high conductivity intracavitary blood is considered to be most influential, and its effect best understood. In order to analyze this effect, Brody 1 used an idealized model in which the intracavitary blood mass was represented by a perfectly conducting sphere immersed in an infinite homogeneous medium (the surrounding tissues of the thorax). The effect of this perfectly conducting sphere on a dipole source located in the outer tissue is determined by utilizing image theory. 2
From the Department of Biomedical Engineering, Case Western Reserve University, Cleveland, Ohio. *Graduate student tProfessor of Biomedical Engineering Supported by N.I.H. Grant HL 10417 from the National Institutes of Health. The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. w1734 solely to indicate this fact. Reprint requests to: Yoram Rudy, Dept. of Biomedical Engineering, Case Western Reserve University, Cleveland, OH 44106. 87
88
RUDY AND PLONSEY
Z
Fig. 1. The system of images for a dipole of strength M located within the region of finite conductivity at the interface with the spherical perfectly conducting region. The primary dipole is not shown. The potential is calculated at the spherical surface of radius W which represents the torso boundary. "Realistic" values of R (4.0 cm) and F (5.0 cm) were chosen in the numerical calculations for Table 1.
sphere, (see Fig. 1), consists of: 1. A dipole of s t r e n g t h M at the same location as the real source; 2. A point source of strength M/R also at t h e same location; 3. A point source of s t r e n g t h - M / R at the center of the sphere. As R --*~, the two point sources vanish and the potential is doubled, which is the right beh a v io r in the plane wall case. On the other hand, if the potential is measured at a distance E > > R from the source, the two point sources appear as a third dipole of strength M/R x R = M, i.e., equal to the s t r engt h of the original source. Except in these limiting cases the field of the separated positive and negative point sources (The Rush-Nelson bipole) cannot be r ead ily characterized but depends on the specific geometry. Its contribution is n e i t h e r negligible nor, strictly, dipolar. In order to illustrate the effect of this correct system of images on the potential distribution, the potentials were calculated on a hypothetical spherical surface (representing t h e torso) w i t h i n which t he s pher i c a l int r a c a v i t a r y blood mass is located eccentrically, as shown in Fig. 1. The geometrical p a r a m e t e r s were chosen to reflect "realistic" torso parameters. The results are shown in Table 1 as a function of W, for various angular field positions 0. (Note t h a t as W increases, E
which is gi ven by W - ( F + R ) increases as well.) For d e t a i l s of t h e c a l c u l a t i o n s see Appendix 1. It is clear t h a t the e n h a n c e m e n t factor is not a constant and varies with 0 and W. (For comparison, if t he two point sources were app r o x i m a t e d by a second i m a g e dipole of strength M at (F+R, O), the e n h a n c e m e n t factor would have been a constant, having the value of 3.0 for every 0 and W.) Fig. 2 shows the e n h a n c e m e n t factor for 0 = 0 ~ (location of the peak potential), as a function of W (or E). The e n h a n c e m e n t factor for the case of two image dipoles at (F+R, 0) is also included and has the constant value of 3.0 (broken line). For small values of W, i.e., when E is small and the field point is very close to the image dipole, the factor is close to 2.0, i.e., the potential is doubled relative to the case of a dipole in a homogeneous medium, and the two point sources do not play an i m p o r t a n t role. For higher values of W, as the field point g e t s f a r t h e r a w a y from t he source, t he f a c t o r gets larger, and approaches asymptotically t h e v a l u e of 3.0. T h i s w as e x p e c t e d , of course, since at a great distance the two point sources appear as a second image dipole of strength M. The case o f W = 12.0 cm represents a realisJ. ELECTROCARDIOLOGY, VOL. 11, NO. 1, 1978
BRODY-EFFECT
89
TABLE|
0~
W=12,,E=3.
W = 9.5, E = .5
W - 9,1, E - . 1 *enh. Potential factor
W = 20., E =11.
enh. factor
Potential
enh. factor
W = 100., E =91.
Potential
enh, factor
Potential
enh. factor
102.439
2.024
4,444
2.111
.158
2.428
1.432x102
2.733
2.384x104
2.957
8.986x102
-8.278
.168 3.276 2~575x10-2 11.122
,101
2.554
1.319x10-2
2.754
2.315x10-4
2.959
1.972x10 "8
2.999
.041
2.905
1.043x10-2
2.818
2,171x10-4
2.963
1.882x10 "8
2.999
3.610 2.351
7.242x103 7.337x104
2.916 4.831
1.944x10-4 9,415x10-5
2.971 3.020
.159
60~
-3.946x10-3 -1,005x10-2
1.362 2.674
-8.727x10 "3
2.630
1.555x10-2 3.203x10-3
9.o~
-8.837x10-3
3,058
-8.199xlQ-3
3,041
-5.230x1~ ~3
2,96t
-1.580x10-3
2,853
-1,583xlff 5
2.781
120~
-7.760x163 -7.159xlff 3 -6.970x10-3
3.205 3.266 3,283
-7.336x10 -3 -6.817x10 "3 -6,649x103
3.195 3.258 3,275
-5.289x10-3 -5.135x10"3 -5.088x10"3
3.142 3.215 3.235
-2.305x10"3 -2.521x10-3 -2.568x10-3
3.061 3.138 3,160
-1.024x104 -1.544x10~ -1.715x10-4
2.997 3.029 3.038
150~ 1~80~
enh. factor 2.999
Potential 2.003x10"8
Potential
1.212x10-2
20~ 30~
W = 10000, E =9991.
1.481xi0"3
.788
1.734x10 -8 2~999 9.996x10"9 3.000 -1.597xlff 11 2,777 -l.00Oxl0 -8 -I,730x10"8
2.999 3.000
-1.996x10 -8
3.000
The potentials generated by the system of images shown in Fig. 1, at the hypothetical spherical surface of radius W. The *enhancement factors are also included in the table. The calculations were made for R = 4 cm and F = 5 cm.
tic t o r s o d i m e n s i o n . I n t h i s c a s e , t h e enh a n c e m e n t factor is far f r o m b e i n g c o n s t a n t , a n d for O = 0 ~ it h a s t h e v a l u e of 2.428 ( r a t h e r t h a n t h e idealized 3.0). C o n s e q u e n t l y , t h e two p o i n t sources c a n n o t be r e p r e s e n t e d b y a dipole a t ( F + R , 0). On the o t h e r h a n d , since t h e a u g m e n t a t i o n factor is not close to 2.0, t h e s e two p o i n t sources c a n n o t be neglected either. W e c a n c o n c l u d e t h e r e f o r e , t h a t in t h e r e a l i s t i c case (E = 12.0 cm), t h e c o n t r i b u t i o n of t h e two p o i n t sources is i m p o r t a n t . However, no s i m p l i f i c a t i o n is possible a n d t h e y m u s t be included in t h e c o m p u t a t i o n of the p o t e n t i a l d i s t r i b u t i o n as two separate sources a n d not as a second i m a g e dipole. As a result,
the p o t e n t i a l s a t t h e torso will be e n h a n c e d , c o m p a r e d to t h e h o m o g e n e o u s case, b u t not s i m p l y by a c o n s t a n t factor.
APPENDIXI The potential values in Table 1 were calculated by a superposition of the contributions of the dipole and the point sources shown in Fig. 1. The resulting expression for field points on the spherical surface of radius W is: r = M (W cos O - F - R ) + M/R M/R R13 R1 R2 where: R I = ( W L 2 W ( F + R ) cos O+(F+R)2)I/~ R2=(W2-2WF cos O+F2)1/2 and we chose M = 1. For small values of W in the range 0~ ~
ENHANCEMENT FACTOR 0=O ~ 3,0
T W O IMAGE D I P O L E S ........................................
2,8
2_,6
2,4 / 2.,2
2,0
of
IMAGES
/ I
t
IO
I
I
12_ 2
I
I
14 4
I
I
16 6
I
I
18 8
IO
J. ELECTROCARDIOLOGY, VOL. 11, NO. !, 1978
I
20 W (crn) E(cm)
Fig. 2. The enhancement factor due to the correct system of images (for O = 0~ The factor due to two image dipoles at (F+R, 0) is also shown (...... ).
90
RUDY AND PLONSEY
the field points are very close to the sources and the potential, as well as the enhancement factor, varies markedly with small changes of O. REFERENCES 1. BRODY, D A: A theoretical analysis of intracavitary blood mass influence on the heartlead relationship. Circ Res 4:731, 1956
2. JACKSON, J D: Classical Electrodynamics.
J Wiley, New York, 1962, p 26 3. RUSH, S ANDNELSON,C V: The effects of electrical inhomogeneity and anisotropy of thoracic tissues on the field of the heart. In The Theoretical Basis of Electrocardiology, C V NELSONAND D B GESELOWITZ,eds. Oxford University Press, New York, 1976, pp 323-354
J. ELECTROCARDIOLOGY, VOL. 11, NO. 1, 1978
Annotations A Note on the "Brody-Effect" BY YORAM RUDY, M.Sc.* AND ROBERT PLONSEY, PH.D.t
By application of the image principle to the spherical model, Brody concluded t h a t the images ~Augment the manifest strength of radially oriented doublets by a factor of 2 R3/ d3. '' (Where R is the radius of the sphere, and d i s the distance of the dipole from the center of the sphere.) That is, the potential measured will be (1+2 R3/d 3) times the potential due to a single dipole in a homogeneous medium. In the limit as d-~R (the dipole is at the interface), the effect is to triple the strength of the dipole, i.e., the images are two dipoles of the same strength and at the same location as the original source. This factor of 3.0 will also describe the situation when d = R and R --* :% in coutradiction with the factor of 2.0 obtained directly from the solution of the semiinfinite plane wall case. The purpose of this communication is to clarify this discrepancy; in fact to elucidate the nature of the image e n h a n c e m e n t of a dipole s o u r c e in t h e aforementioned spherical model. The discrepancy which occurs in the Brody paper appears in a different form in the paper of Rush and Nelson 3 which gives, instead, an enhancement factor of (1 +R2/d2). Here, in the limit as d o R , the effect is to double the strength of the source. (Which is true for the plane wall case). The authors conclude that "the effects are in the same direction as those of the plane wall case, though not as great," i.e., according to them, the m a x i m u m factor is 2.0 (when the dipole is at the interface), and in t h a t case the image is a single dipole of the same strength and at the same location as the original one. Rush and Nelson correctly mention a "third bipole" t h a t appears as an image, but leave the reader with the impression t h a t its contribution to the potential is negligible because of its small pole strength. The aforementioned ambiguities can be resolved by noting t h a t the correct system of images for the case of a dipole of strength M within the medium of finite conductivity but at the interface with the perfectly conducting
SUMMARY The effect of a perfectly conducting sphere simulating the intracavitary blood mass on a dipole source located at the interface with the outer tissue (myocardium) is studied, utilizing image theory. The resulting enhancement factor is found to be a function of the field point location and is not a constant, as previously reported by Brody and by Rush and Nelson. Of the several thoracic inhomogeneities the high conductivity intracavitary blood is considered to be most influential, and its effect best understood. In order to analyze this effect, Brody 1 used an idealized model in which the intracavitary blood mass was represented by a perfectly conducting sphere immersed in an infinite homogeneous medium (the surrounding tissues of the thorax). The effect of this perfectly conducting sphere on a dipole source located in the outer tissue is determined by utilizing image theory. 2
From the Department of Biomedical Engineering, Case Western Reserve University, Cleveland, Ohio. *Graduate student tProfessor of Biomedical Engineering Supported by N.I.H. Grant HL 10417 from the National Institutes of Health. The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. w1734 solely to indicate this fact. Reprint requests to: Yoram Rudy, Dept. of Biomedical Engineering, Case Western Reserve University, Cleveland, OH 44106. 87
88
RUDY AND PLONSEY
Z
Fig. 1. The system of images for a dipole of strength M located within the region of finite conductivity at the interface with the spherical perfectly conducting region. The primary dipole is not shown. The potential is calculated at the spherical surface of radius W which represents the torso boundary. "Realistic" values of R (4.0 cm) and F (5.0 cm) were chosen in the numerical calculations for Table 1.
sphere, (see Fig. 1), consists of: 1. A dipole of s t r e n g t h M at the same location as the real source; 2. A point source of strength M/R also at t h e same location; 3. A point source of s t r e n g t h - M / R at the center of the sphere. As R --*~, the two point sources vanish and the potential is doubled, which is the right beh a v io r in the plane wall case. On the other hand, if the potential is measured at a distance E > > R from the source, the two point sources appear as a third dipole of strength M/R x R = M, i.e., equal to the s t r engt h of the original source. Except in these limiting cases the field of the separated positive and negative point sources (The Rush-Nelson bipole) cannot be r ead ily characterized but depends on the specific geometry. Its contribution is n e i t h e r negligible nor, strictly, dipolar. In order to illustrate the effect of this correct system of images on the potential distribution, the potentials were calculated on a hypothetical spherical surface (representing t h e torso) w i t h i n which t he s pher i c a l int r a c a v i t a r y blood mass is located eccentrically, as shown in Fig. 1. The geometrical p a r a m e t e r s were chosen to reflect "realistic" torso parameters. The results are shown in Table 1 as a function of W, for various angular field positions 0. (Note t h a t as W increases, E
which is gi ven by W - ( F + R ) increases as well.) For d e t a i l s of t h e c a l c u l a t i o n s see Appendix 1. It is clear t h a t the e n h a n c e m e n t factor is not a constant and varies with 0 and W. (For comparison, if t he two point sources were app r o x i m a t e d by a second i m a g e dipole of strength M at (F+R, O), the e n h a n c e m e n t factor would have been a constant, having the value of 3.0 for every 0 and W.) Fig. 2 shows the e n h a n c e m e n t factor for 0 = 0 ~ (location of the peak potential), as a function of W (or E). The e n h a n c e m e n t factor for the case of two image dipoles at (F+R, 0) is also included and has the constant value of 3.0 (broken line). For small values of W, i.e., when E is small and the field point is very close to the image dipole, the factor is close to 2.0, i.e., the potential is doubled relative to the case of a dipole in a homogeneous medium, and the two point sources do not play an i m p o r t a n t role. For higher values of W, as the field point g e t s f a r t h e r a w a y from t he source, t he f a c t o r gets larger, and approaches asymptotically t h e v a l u e of 3.0. T h i s w as e x p e c t e d , of course, since at a great distance the two point sources appear as a second image dipole of strength M. The case o f W = 12.0 cm represents a realisJ. ELECTROCARDIOLOGY, VOL. 11, NO. 1, 1978
BRODY-EFFECT
89
TABLE|
0~
W=12,,E=3.
W = 9.5, E = .5
W - 9,1, E - . 1 *enh. Potential factor
W = 20., E =11.
enh. factor
Potential
enh. factor
W = 100., E =91.
Potential
enh, factor
Potential
enh. factor
102.439
2.024
4,444
2.111
.158
2.428
1.432x102
2.733
2.384x104
2.957
8.986x102
-8.278
.168 3.276 2~575x10-2 11.122
,101
2.554
1.319x10-2
2.754
2.315x10-4
2.959
1.972x10 "8
2.999
.041
2.905
1.043x10-2
2.818
2,171x10-4
2.963
1.882x10 "8
2.999
3.610 2.351
7.242x103 7.337x104
2.916 4.831
1.944x10-4 9,415x10-5
2.971 3.020
.159
60~
-3.946x10-3 -1,005x10-2
1.362 2.674
-8.727x10 "3
2.630
1.555x10-2 3.203x10-3
9.o~
-8.837x10-3
3,058
-8.199xlQ-3
3,041
-5.230x1~ ~3
2,96t
-1.580x10-3
2,853
-1,583xlff 5
2.781
120~
-7.760x163 -7.159xlff 3 -6.970x10-3
3.205 3.266 3,283
-7.336x10 -3 -6.817x10 "3 -6,649x103
3.195 3.258 3,275
-5.289x10-3 -5.135x10"3 -5.088x10"3
3.142 3.215 3.235
-2.305x10"3 -2.521x10-3 -2.568x10-3
3.061 3.138 3,160
-1.024x104 -1.544x10~ -1.715x10-4
2.997 3.029 3.038
150~ 1~80~
enh. factor 2.999
Potential 2.003x10"8
Potential
1.212x10-2
20~ 30~
W = 10000, E =9991.
1.481xi0"3
.788
1.734x10 -8 2~999 9.996x10"9 3.000 -1.597xlff 11 2,777 -l.00Oxl0 -8 -I,730x10"8
2.999 3.000
-1.996x10 -8
3.000
The potentials generated by the system of images shown in Fig. 1, at the hypothetical spherical surface of radius W. The *enhancement factors are also included in the table. The calculations were made for R = 4 cm and F = 5 cm.
tic t o r s o d i m e n s i o n . I n t h i s c a s e , t h e enh a n c e m e n t factor is far f r o m b e i n g c o n s t a n t , a n d for O = 0 ~ it h a s t h e v a l u e of 2.428 ( r a t h e r t h a n t h e idealized 3.0). C o n s e q u e n t l y , t h e two p o i n t sources c a n n o t be r e p r e s e n t e d b y a dipole a t ( F + R , 0). On the o t h e r h a n d , since t h e a u g m e n t a t i o n factor is not close to 2.0, t h e s e two p o i n t sources c a n n o t be neglected either. W e c a n c o n c l u d e t h e r e f o r e , t h a t in t h e r e a l i s t i c case (E = 12.0 cm), t h e c o n t r i b u t i o n of t h e two p o i n t sources is i m p o r t a n t . However, no s i m p l i f i c a t i o n is possible a n d t h e y m u s t be included in t h e c o m p u t a t i o n of the p o t e n t i a l d i s t r i b u t i o n as two separate sources a n d not as a second i m a g e dipole. As a result,
the p o t e n t i a l s a t t h e torso will be e n h a n c e d , c o m p a r e d to t h e h o m o g e n e o u s case, b u t not s i m p l y by a c o n s t a n t factor.
APPENDIXI The potential values in Table 1 were calculated by a superposition of the contributions of the dipole and the point sources shown in Fig. 1. The resulting expression for field points on the spherical surface of radius W is: r = M (W cos O - F - R ) + M/R M/R R13 R1 R2 where: R I = ( W L 2 W ( F + R ) cos O+(F+R)2)I/~ R2=(W2-2WF cos O+F2)1/2 and we chose M = 1. For small values of W in the range 0~ ~
ENHANCEMENT FACTOR 0=O ~ 3,0
T W O IMAGE D I P O L E S ........................................
2,8
2_,6
2,4 / 2.,2
2,0
of
IMAGES
/ I
t
IO
I
I
12_ 2
I
I
14 4
I
I
16 6
I
I
18 8
IO
J. ELECTROCARDIOLOGY, VOL. 11, NO. !, 1978
I
20 W (crn) E(cm)
Fig. 2. The enhancement factor due to the correct system of images (for O = 0~ The factor due to two image dipoles at (F+R, 0) is also shown (...... ).
90
RUDY AND PLONSEY
the field points are very close to the sources and the potential, as well as the enhancement factor, varies markedly with small changes of O. REFERENCES 1. BRODY, D A: A theoretical analysis of intracavitary blood mass influence on the heartlead relationship. Circ Res 4:731, 1956
2. JACKSON, J D: Classical Electrodynamics.
J Wiley, New York, 1962, p 26 3. RUSH, S ANDNELSON,C V: The effects of electrical inhomogeneity and anisotropy of thoracic tissues on the field of the heart. In The Theoretical Basis of Electrocardiology, C V NELSONAND D B GESELOWITZ,eds. Oxford University Press, New York, 1976, pp 323-354
J. ELECTROCARDIOLOGY, VOL. 11, NO. 1, 1978