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Qualitative effects of thoracic resistivity variations on the interpretation of electrocardiograms: The low resistance surface layer
Qualitative effects of thoracic resistivity on the interpretation of electrocardiograms: resistance surface layer Richard McFee, Ph.D.* Stanley Rush, Ph.D.*” Syracuse, N. Y., and Burlington,
A
lthough the torso is often assumed in electrocardiographic studies to be homogeneous, it is not. In a previous paper’ we have considered the resistivity changes which produce the most drastic modification of surface electrocardiograms (ECG’s). These are those variations which occur in and around the heart itself, i.e., the differences in conductivity between heart muscle, blood, and lung, as well as the lower resistivity of heart muscle in the direction of its fibers compared to that in a transverse direction. In this article, we discuss what we consider to be the second most important resistivity change, namely, that occurring in the vicinity of the surface electrodes. This is due to the relatively low resistance surface layer formed by the muscles girdling the thorax. As in our first paper, we use elementary models, and neglect the perturbing effects of other resistivity changes, e.g., the ribs, spine, sternum, liver, blood, blood vessels, pleural membranes, etc. Experimental
Observed electrical characteristics oj the layer. Fig. 1 shows a schematic
surface
This
study was supported by Research National Institutes of Health. Received for publication July 25, 1967. *Department of Electrical Engineering, **Department of Electrical Engineering,
48
American Heart Journal
Grants
Syracuse University
variations The low
Vt.
representation of this low resistance layer. The skin and surface fat overlie the muscle which, in turn, overlies the lung. This muscle layer shows variations in thickness from approximately 1 to 3 cm. in anatomic cross section drawings.2 In the region of the chest surface of greatest interest to us, over the heart, the “pectoralis major” muscles tend to run from the left shoulderarm region toward the caudal rib cage and sternum. Of the muscles between the ribs, the “intercostalis externus” are oriented more or less vertically, and the “intercostalis internus” horizontally. Other nearby muscles, “pectoralis minor” and “serratus anterior,” have still different orientations. Thus, the successive layers of muscle are by no means well aligned. Electrical effects of this low-resistance layer are seen in measurements3~4of the over-all resistivity of the trunk made with currents flowing from head and shoulders to feet. Measurements of normal male adults have yielded values of 463 and 489 ohm-cm. which differ approximately by a factor of four from the value of 2,000 ohm-cm. found for dog lung. Comparisons of measurements of this type made on a HE-08607
and
HE-09831
from
the
National
Heart
Institute
of the
University, Syracuse, N. Y. of Vermont. Burlington, Vt.
July, 1958
Vbl. 76, No. 1, pp. 48-61
Qunlitdive
dog,4 first with the chest intact and second with the chest opened and the lungs insulated by wrapping them with a plastic film, have yielded a mean value for the resistivity of the surface layer of 281 ohm-cm. The relative amplitudes of ECG’s registered from human subjects also show effects similar to those which would be produced by a low-resistance layer. The average potential differences between a grid of electrodes covering the chest and another the back, were compared with the voltage in the chest-back lead in the axial system for six subjects.5 It was found in all cases that the ratio of the peak amplitude of the axial lead to that of the grid lead was smaller than the sensitivity ratio determined with a homogeneous tank “torso,” the same leads, and a dipole located at a position corresponding to the anatomical center of the heart. This sort of result would be expected were the “effective” thickness of the surface layer many times its actual thickness. Theoretical churacteristics of the surface layer. The effects on the ECG of a layer of muscle can to some degree be predicted
Fig. 1. Schematic layer.
cross
section
of
torso
surface
ejects of thoracic rcsistivity
vtrricrtions
49
theoretically. So far as chest electrodes are concerned, it is convenient here to make the simplifying assumption that the chest surface is flat, rather than curved and infinite in extent6 At first glance, this model may appear to be overly simplified. However, one can easily extend results obtained with it to a box-shaped conductor, through the use of multiple reflections of images in the box sides. Furthermore, experiments with rectangular tanks with and without rounded edges have shown that in most leads the effect of rounding is to change relative sensitivities by only a few per cent. In addition, a quantitative estimate of the effect of curvature is possible in the case of a spherical surface layer. In the section entitled “Effect of curvature of the surface layer” it is found that the predictions obtained with the flat surface layer theory agree reasonably well with those obtained from the exact analysis of the sphere. A recently published survey by Geddes and Baker’ of the resistivity of various tissues includes a number of measurements of muscle conductivity which show that the low-frequency resistivity of muscle along the fibers is much less than across the fibers. Measurements by Burger and van biilaan8 Burger and van Dongen,9 and by Rush and associates4 have indicated that the former is about 150 ohm-cm. and the latter 2,300 ohm-cm. So far as currents flowing parallel to the surface of the chest are concerned, this disparity between the high and low resistivity is greatly reduced
m=-!L. h+L
Fig. 2. Resistor matrices placed in parallel.
showing
how
two
perpendicular
anisotropic
sheets
yield
an
isotropic
sheet
when
by the fact that successive laminas of muscle are oriented in different directions. In fact, if we use the simplifying assumption that the laminas are thin and of uniform thickness and run alternately at right angles to one another, we find that the resistivity parallel to the surface (in the limit as the thickness of the laminas approaches zero) no longer depends on the direction.“’ This is easily visualized from Fig. 2, where two two-dimensional resistor matrices, representing two adjacent anisotropic conducting sheets, are connected in parallel, the result being an isotropic resistance matrix. This shows that the conductivity al,, of the melded layers is given by the means of the high (Q) and low (UI) conductivity, i.e., by (T,,, = $1 (uh+uJ. This may be expressed in terms of corresponding resistivities, p, as pm = 2~1~h/(pl+~h). Taking the high and low resistivities of muscle to be 1.50 and 2,300 ohm-cm., respectively, we obtain an average isotropic resistivity pm tangential to the surface of 280 ohm-cm. (the very close correspondence between this and the experimental value 281 ohm-cm., although fortunate, does not mean that our model is accurate within one half of 1 per cent!). The conclusion that the muscle may be considered homogeneous in directions parallel to the surface is supported by the regular nature of constant potential lines on the body surface arising from passing current from head to foot. No indications
of the underlying anisotropy are seen.8 The component of current normal to the surface encounters a common high resistivity from all the muscles. With the mean value pm representing the tangential resistivity, and the high value Ph representing the resistivity normal to the surface of the body, a section of the muscle perpendicular to the surface can be represented by the resistance matrix shown in Fig. 3,A. Now, so far as the distribution in the network of currents introduced into it from outside is concerned, it would make no difference were this matrix stretched by a factor of dh/m as is shown in Fig. 3,B. This stretching has the effect of making the voltage drop per centimeter in the h resistors, for a given surface current density, smaller. It thus decreases the effective resistivity of the medium represented by the network in the direction of the “h” resistors. The stretching increases the resistivity in the direction of the m resistors by the same factor. Since P,,, dz/~,~ = Ph dpmiph = \I=, the stretching has the effect of making the two resistivities the same. Furthermore, if the matrix in Fig. 3,B is sufficiently fine grained, it will make no difference to external measurements if the number of resistors in the direction perpendicular to the surface is increased provided that the resistivity of each resistor is reduced in proportion. Similarly, the number of resistors in a direction parallel
Surface m
___ Surface m
m
___
h m
---
Surface
m
m
__-
---
f
---
h
_--
h
f
h ---
m
___
m h
___
-__
f
h
~ A
m
f
B
f
---
f f
f ___ ~
---
f f
f m
h
T
f
f m
m h
f
f
---
f f
f
__-
c
Fig. 3. Resistor matrices showing how the anisotropic surface layer represented is equivalent to a thicker isotropic layer. Each matrix represents a thin slice of the conductor perpendicular to the plane of the surface. .1, Actual matrix; B, stretched matrix (stretch factor is 4 h/m); C, isotropic square-celled matrix equivalent to stretched matrix.
Qualitative
to the surface can be increased providing that the resistance of each is increased in proportion. The result of these manipulations is the isotropic square-celled matrix shown in Fig. 3,C. This matrix represents a homogeneous isotropic surface layer of resistivity l/prnphand of a thickness dph/pm times its original thickness. The result above was derived on a field theoretic basis for an infinite anisotropic slab by Kunz and Moran.” According to this analysis, a muscle layer 1 cm. thick, having a resistivity perpendicular to its surface of 2,300 ohmcm., and 280 ohm-cm. parallel to its surface, will act as if it were a layer d2,300/280 = 2.85 N 3 cm. thick having a resistivity of 1/2300.280 = 800 ohm-cm. Of course, if the muscle fibers do not run precisely parallel to the surface, Ph will be lower and there will be less increase iu the effective thickness. Efect oj low-resistance surface layer on the lead jield of an exploring electrode. I’er-
haps the simplest situation to which these results can be applied is that shown in Fig. 4. Here the lead consists of an exploring electrode on the chest surface coupled with an ideal indifferent electrode presumed to be located at infinity. If the surface layer is anisotropic, we have already seen how its apparent thickness will be increased, and its effective resistivity changed. Our analysis gave a value of 800 ohm-cm. and a thickness of 3 cm. for this layer of augmented thickness. As 800 ohm-cm. is substantially smaller than lung resistivity, we determine here the effect of this residual difference. The analysis in Appendix I shows that the field produced in the lung by the surface electrode may be considered to be produced by a number of sources in an infinite homogeneous conductor. These are of progressively diminishing size and increased remoteness located on a line passing
Exploring
e.fects qf thorn&
resistivity
4. Simple
model
for determining
A more accurate approximation for the jield of un exploring electrode on a low-resistance surj’uce layer. In regions of the
lung close to the exploring electrode, the approximations used for points remote front the exploring electrode can be improved upon substantially. For example, in Appendix II we compute the field in the “lung” (2,000 ohm-cm.) at a point P located 5 cm. below the undersurface of a 1 cm. thick anisotropic muscle layer whose effective isotropic thickness is 3 Clll. and resistivity 800 ohm-cm. The calculated lead field density at this point, as well as others at 3, 4, 6, and 7 cm. from the low resistance layer, are plotted as heavy dots in Fig. 5. By waq- of contrast, curve A shows the field which would exist in this region were the surface layer
Electrode..&
the lead field
51
through the exploring electrode perpendicular to the surface (Fig. 11,B). At points in the lung relatively remote from the electrode, these sources may be grouped together into a single equivalent source whose source strength is the same as that of the actual exploring electrode and which is located at a distance from the bottom surface of the surface layer (Fig. 4) increased over the actual distance by a factor given by the ratio (~l,,~~/~l~,.\.~~)of the resistivity of the lung to that of the effective surface layer. These statements regarding the lead field of the exploring electrode have a simple interpretation in electrocardiography. First, a low-resistance surface layer tends to make the voltages measured on the surface smaller than they would be in a homogeneous conductor, because of an increase in effective distance to the EMF’s in the heart. The increased distance also acts to reduce “proximity” effects and to reduce the surface manifestations of multiplicity in the sources which in reality comprise the heart’s electrical generator.
Electrode
Fig.
variations
of an exploring
electrode.
at Infinity
52
Am. Hrart .I. .Ild,v, 1968
.McFee and Rush
I ampere
.006 I-
t \
“E y .005“a
\field
meowed this line
along
5
Thorax
E e 5 0
ossumed
_ Points show values _ anisotropic surface
Accurate for large distances
111
I 11
I!
3
11
I II
4 Distance
I I II!
5 from
Underside
I III
6 of
I IIII
7
Surface
8
cm
Loyer
Fig. 5. Figure representing the main conclusions of this article. It shows how an anisotropic low-resistance surface layer (muscle) can be represented by a thicker surface layer having the same resistivity as the underlying medium (lung). The dots show the field calculated for an anisotropic 1 cm. surface layer. Curve A is the field which would exist with a 1 cm. surface layer of 2,000 ohm-cm. resistivity. This curve shows that lead sensitivity estimated with homogeneous models may be three times greater than actual sensitivities. Curves B accurate rule and C are similar to A except for layer thickness. Curve C is the basis of the simple, reasonably of thumb that a 1 cm. thick muscle layer over the heart is equivalent to 5 cm. of lung. Curve D represents the best approximation to the actual values (dots) and is described in text.
1 cm. thick and of the same resistivity as the underlying medium. We note that the actual field 5 cm. below the underside of the surface layer, a position roughly corresponding to the center of the heart, is only 38 per cent of the field that would exist there were the entire conductor homogeneous. Obviously, the assumption that the thorax acts as if it were homogeneous leads to gross errors. Curve B shows the field which would be prodoced if the thickness of the equivalent isotropic 3 cm. surface layer were increased by a factor of 2% to 7% cm., and its resis-
tivity
made
the same
as the
rest
of the
conductor. This curve gives a better estimate of the actual field than curve A, but is by no means precise. In curve C the surface layer thickness is taken to be 5 cm. rather than 7% cm. The approximation here is quite respectable. Curve D is that obtained with a 3 cm. surface layer whose resistivity has been changed to match the rest of the medium, and whose lead field source strength has been reduced to two thirds of an ampere. This curve gives an excellent approxima-
Qualitutive
tion of the actual field directly under the electrode. (It can be shown theoretically that, if the field is a good match to the actual field on this axis, it will be a good match off the axis also.) The reduced lead field strength implies that the voltages that will be measured in the region of the heart with an exploring electrode directly over the heart will be almost identical to two thirds of the voltages which would be measured were that surface layer replaced with a 3 cm. surface layer having the same resistivity as the underlying medium. We have assumed that a high degree of anisotropy exists in the surface layer. As a limiting contrary case, we might suppose that the anisotropy is negligible and that the resistivity in all directions is 280 ohm-cm. This is the value estimated from experimental data for current flow in a tangential direction. The field is computed as outlined in Appendix II. It is found in a manner similar to that of curve D of Fig. 5, that the actual field can be very well approximated by assuming that the surface layer thickness has been increased from 1 to 2.5 cm., giving it the same resistivity as the rest of the medium, and by decreasing the lead field source strength of the electrode from 1 to 0.7 amperes. These values do not differ much from the 3 cm. and 0.67 amperes shown in curve D, which were determined for the anisotropic layer. Eflect of curvature of the surface layer. 110 these conclusions, derived from flat surface layers, also apply to curved layers? To gain insight into this question, we derive, in Appendix III, an equation for the lead field at the center of a homogeneous sphere enclosed in a spherical shell of lower resistivity into which current flows via electrodes located on opposite sides of the sphere. (This is the inverse of a problem solved by Bayley and Berry.12) The sphere and its surface layer form a crude model of the body whose dimensions have been chosen to match the flat surface layer case already considered. We assume that the inner sphere has a radius of 5 cm. and resistivity of 2,000 ohm-cm. The shell is taken to be anisotropic, 1 cm. thick, and also is assumed to be representable with good approximation by a layer 3 cm. thick of 800 ohm-cm. isotropic resistivity.
effects of thoracic resistivity
variations
53
We find from Equation (26) in Appendix III that the introduction of a unit current will produce a field of 0.00435 amperes per square centimeter at the sphere’s middle. On the other hand, were the sphere and shell of the same resistivity, and of total radius 6 cm., the central field strength would be 0.0135. The actual value 0.00435 is 32 per cent of this. Thus, we find in the sphere roughly the same degree of reduction (38 per cent) found with a flat surface layer. The reduction is somewhat greater for the sphere in part because some of the lead field current which otherwise would pass through the center has been drawn away to the low resistance sides of the sphere. In any case, the values are enough alike to indicate that what is true for a flat layer is also true for a curved layer. Exact compensation for the low-resistance surface layer in ideal leads. Through the use of electrodes covering the entire surface of a homogeneous volume conductor connected together via appropriate resistor networks, it has been shown13J4 that it is possible to construct leads whose lead fields are “ideal.” For example, they may be (1) uniform, corresponding to heart vector leads, or (2) appear to radiate out from a point, corresponding to unipolar leads, or (3) have a point of zero field corresponding to “null,” “cancellation,” or “quadripole” leads. Furthermore, it has also been shown14 that these and other types of lead fields may be produced in a region corresponding to the heart even if the latter has a resistivity different from that of the rest of the body and is spherical or ellipsoidal in shape. Finally, leads may, in principle, be constructed under these assumed conditions in which the exploring electrode appears to be within the volume conductor, close to the “heart,” despite the fact that all electrodes are on the body surface. These objectives may still be achieved when there is an effectively homogeneous low-resistance surface layer whose inner surface is ellipsoidal in shape. We will present here examples which should, along with the references cited, serve to convey the idea behind the mathematical approach, which is based on the lead field concept.
54
McFee
and Rush
Consider a sphere bounded by a lowresistance surface layer in the form of a spherical shell of uniform thickness. Suppose there exists in the sphere proper a uniform lead field. Then, as was discussed in the appendix of the previous article,’ the lead field in the outer spherical shell will be that of a dipole plus a uniform lead field. (The sources are outside the shell.) If one constructs a lead to introduce or remove the part of this lead field currelit which arrives or leaves at the outer surface of the spherical shell, then the uniqueness theorem requires that this external lead produce the desired uniform field lvithin the sphere. The sanle argument ma>- be applied to lead fields of various types in the inner the unipolar lead sphere, as for example type, the null lead type, etc. These ma). be broken down into spherical harmonic components’” (uniform, dipolelike, etc.) each of which propagates into the spherical shell a fixed combination of conlponents, which when summed, will yield a surface field on the outer surface of the spherical shell which can be matched by a properly designed lead. That lead will therefore produce the desired field in the sphere. It is a small generalization of this procedure, conceptually speaking, to drop the assumption that the outer shell is uniform in thickness with a spherical outer surface. Even if the outer surface is irregular, the identical procedure may be followed to construct a lead which introduces the required lead field surface current. Furthermore, this type of approach can be extended to ellipsoidal shapes by using ellipsoidal harmonics’4 rather than spherical harmonics. Fig. 6 illustrates this possibility. The procedure is the same when all surfaces are ellipsoids or spheres (escept the outer surface which may have any shape whatever), and the conductivit> in each region is homogeneous and isotropic. A$proximatc compensation for the lowresistawe sllrfc’ftlce Iuyer. As a practical matter, what needs to be done to adapt ideal leads to the presence of a low-resistance surface layer is to provide extra lead field current for the portions of this layer that run parallel to the direction of the lead field. This extra current is needed
Fig. 6. Cross section of a heterogeneous conductor showirlg how a uniform lead field can nevertheless he produced inside all ellipsoidal homogeneous “heart.”
,p
= 400
p = 200
Fig. 7. Model illustrating the shunting effect of low resistance la~cr 011 the neck-foot lead.
Fig. 8. Cross section of trapezoidal model indicating how, with vectorcardiographic leads, extra lead tield current is needed (with “low-resistance” surface layer of uniform thickness) only at points where the direction of the layer changes.
because the resistance of the surface is less. Thus, if the tangential potential gradient in the layer is to match that in the body proper, additional current must be supplied. As an example, consider the box-shaped conductor shown in Fig. 7 which is covered with a low-resistance surface layer of uniform thickness. A lead for inducing a uniform vertical field in this box must provide five times as much current density for the sides of the box as for the center. Hence, a
?‘olume Number
76 1
Qzditutive
properly designed lead would employ resistors for electrodes feeding the surface layer parallel to the lead field having five times the conductance of those feeding other parts of the conductor. In a conductor having a trapezoidal cross section, as shown in Fig. 8, in a case where one wishes to produce a uniform lead field in the body proper, the lead field current in the layer may be divided into two components: (1) a component of current flowing perpendicularly across the layer, which supplies the normal component of the lead field flowing into the body proper, and (2) a component of current flowing along the layer, parallel to its surface. This component maintains the same tangential potential gradient in the surface layer as is required by the uniform lead field in the body proper. When the field desired in the body proper is nonuniform, the same approach may be employed provided that the surface layer is thin enough so that the normal component of the field on its outer surface may be considered the same as that on its inner surface. It is clear that, when the internal field is uniform, component 2 of lead field current, i.e., the total current flowing tangential to the low-resistance surface layer, needs to be augmented or diminished by extra current from the surface electrodes only in regions where the surface layer changes direction, i.e., curves. In general, a curved layer may be represented as a series of straight sections. The extra surface current required may be introduced at the points where the various segments meet. Discussion
The effect of the low-resistance surface layer may be summarized qualitatively in the following manner. In the neighborhood of the electrodes, the layer tends to spread the lead field current out over a larger area, thus making the electrode seem to be further away and the field within the body weaker. Along the sides of the body, the low-resistance layer tends to shunt the lead field from the central portion of the body, once again making the internal lead field seem weaker. Both of these effects can be predicted if one views the low-resistance layer as an effectively
efects of thorcrcic resistivity
variations
.55
thicker layer having the same resistivity as the underlying tissue. The weakening of the lead field in the neighborhood of the heart is quite substantial. We have already considered several cases in which the field is reduced to about one third the value it would have were the surface layer resistivity the same as that of the underlying medium. Consider further a head-foot lead (Fig. 7). If one assumes that the resistivity of the lungs is 2,000 ohm-cm., that the tangential resistivity of the surface layer is 280 ohmcm., and that the cross sectional areas of the two with regard to head-foot lead field currents are 400 and 180 cm.?, respectively, then the current which flows in the lung will be (400/2,000)/[(400/2,000) + (180/280)] of the total, i.e., 23.7 per cent rather than the 66 per cent it would be were the trunk homogeneous. Thus, the low-resistance surface layer here also diminishes the sensitivity of this lead by a factor of about three, as compared to what would exist were the body homogeneous. It is not our intention to arrive at quantitative conclusions using the simple models considered in this paper, but they do indicate that the effect of the low-resistance layer is to reduce the internal lead fields to values substantially less than one half of those found with homogeneous models. Furthermore, the reduction factors are not necessarily the same in the different leads. Clearly this is a consideration that should not be ignored in electrocardiographic interpretation or in lead system design. Conclusion
The surface of the body is covered by a layer of muscle whose thickness varies greatly from point to point, as well as from subject to subject. Because the resistivity of the muscle in the direction of the fibers is low, about the same as blood, and across the fibers high, the same as lung, and because it differs in any case from that of lung, this layer has a marked effect on the interpretation of ECG’s. Studies with simple theoretical models in this paper indicate that this layer may be considered as a rough approximation to have the same resistivity as the under-
56
Am. Heart
.McFee and Rush
J.
July. 1365
lying tissue (lung) provided that it is also considered to have a thickness of the order of three to seven times its actual thickness (see Fig. 5). A muscle layer 1 cm. thick, as might be typical for the region over the heart, appears to be about 5 cm. thick. As a result of this increased effective distance between the heart and the surface electrodes, the sensitivity of the lead is reduced to a value generally. less than one half the value which would exist were there no difference in resistivity between the surface layer and lung.* Furthermore, proximity effects are substantially reduced also. As the sensitivity reduction factor depends upon the lead considered, this means that heterogeneous models must be employed for accurate determination of sensitivity factors of vectorcardiographic leads.
11.
*An opposite effect is produced by the low resistance of blood and heart muscle relative to lung which tends to increase the sensitivity (on the average), Fig. 1.1
General equations for the lead jield of an exploring electrode(unit point current source) on a j&t low-resistance surface layer. If there is a unit source in a semi-infinite homogeneous conducting medium bounded by a conducting or insulating plane (Fig. 9, A), then the effect of the plane can be taken into account by pretending that the medium is infinite in all directions and that there is an image source located as shown in Fig. 9, B. Note that, if the source is on the surface, it and its image merge and the effective strength of the source is doubled. If the infinite medium is separated into two halves, one with resistivity (II and the other ,B (Fig. 10, A) with a source in the a: region, then the field in the cx region will be that of the original source plus an image of strength c = @-CX)/(CX+~) times the original source, as shown in Fig. 10, H. In the /3 region the strength of the original source will appear to be reduced by a factor k = (l-c). Here c is the “reflection factor” and k is the “transmission factor.” An electrostatic analogue of this problem is treated by Slater and Frank.15 Straightforward use of these results shows that in the case of Fig. 11, A the field in the ,6 region will appear to originate from an infinite number of sources representing the original source of doubled strength attenuated by a factor k, its
REFERENCES 1.
2.
.z
4.
5.
6.
7.
8.
9.
10.
McFee, R., and Rush, S.: Qualitative effects of thoracic resistivity variations on the interpretation The “Brody” of electrocardiograms: effect, AK HEART J. 74:642, 1967. Eycleshymer, H. C., and Schoemaker, D. M.: A cross-section anatomy, New York, 1911, Appleton-Century-Crofts. Schmitt, 0. H.: Lead vectors and transfer impedance, Ann. New York Acad. SC. 65:1109, 1957. Rush, S., Abildskov, J. A., and McFee, Ii.: Resistivity of body tissues at low frequencies, Circulation Res. 12:40, 1963. McFee. R.. and Paruneao. A.: An orthoeonal lead system for clinic;1 electrocardiography, AK HEART J. 62:93, 1961. Bayley, R. H.: Electrocardiographic analysis, Vol. 1, Biophysical principles of electrocardiography, New York, 19.58, Paul B. Hoeber, Inc., p. 211. Geddes, I,. A., and Baker, L. E.: The specific resistance of biological material. A compendium of data for the biomedical engineer and physiologist, Med. & Biol. Eng. 5:271, 1967. Burger, H. C., and van Milaan, J. B.: Measurement of the specific resistance of the human body to direct current, Acta med. scandinav. 114584, 1943. Burger, H. C., and van Dongen, Ii.: Specific electrical resistance of body tissues, Phys. Med. & Biol. 5:431, 1961. Rush, S.: A principle for solving a class of anisotropic current flow problems and applications to electrocardiography, IEEE Tr. Biomed. Eng. 14:18, 1967.
Kunz, K. S., and Moran, J. H.: Some effect. of formation anisotropy on resistivity measures ments in boreholes, Geophysics 23:770, 195812. Bayley, R. H., and Berry, P. M.: Changes produced in the resultant “heart” vector by the non-homogeneous volume conductor with normal specific resistivities, Proc. of the Long Island Jewish Hosp. Symposium on vectorcardiography, Amsterdam, 1965, North Holland Publishing Co., p. 109. 13. McFee, Ii., and Johnston, J. D.: Electrocardiographic leads III, Circulation 9:868, 1954. 14. McFee, R.: Analysis and synthesis of electrocardiocrranhic leads, Ph.D. Dissertation (Electrical Engineering) Ilniversity of Michigan, Ann Arbor, 1954. J. C., and Frank, N. H.: Electromag1.5. Slater, netism. ed. 1. New York. 1947. McGraw-Hill Book Co., Inc., p. 47. ’ ’ 16. Smythe, W. R.: Static and dynamic electricity, ed. 2, New York, 19.50, McGraw-Hill Book Co., Inc., p. 239. Y
Appendix
I
I
Volzlme h’umbcr
76 1
Qualitative
Mnite Homogeneous Conductor
Fig.
9. An
insulating
surface
effects oJ’ thoracic resistivity
produces
an “image.”
Ham
Current Source
k=l-C=&
c
B
10. A change in resistivity produces an “image.” a region ; C, equivalent situation as viewed from
A
p=B
c= (P-a) (P+a)
A Fig. from
57
Insulaling Surface
p=a
Semi -%inite Conductors
vclrintions
A, Actual 0 region.
situation;
B, equivalent
situation
as viewed
: I
Conducting Slab
\
Semi-infinite Homogeneous Conductor
Fig. 11. Image sources for a unit source layer. d, Actual situation; B, equivalent
of current situation
connected as viewed
p=p
Everywhere
to the outside from B region.
q= (I-p)
Fig. 12. Image sources for a unit source of current located inside a conductor A, Actual situation; B, equivalent situation as viewed from fl region.
surface
/
of a low-resistance
surfaqe
a-P p=------a+P near a low-resistance
surface
layer.
58
Am. Heart
McFee and Rush
July,
g = 2k + 2kc + 2kc2 + 2kc3 + . . . = 2k/l-c Z in2
= 2k(x) = 2k(l
+ 2kc(x
- c) x -
= 2
(1)
- 2d) + 2kc2 (x - 4d) + . . .
+ c + c2 + . . .) x - 4dkc(l
= (2k/l
= 2klk
I. 1968
[4dkc/(l
attenuated doubly reflected image 2kc, the re-reflected image 2kc2, etc., the entire conglomeration being located in an apparent infinite homogeneous medium of resistivity p. This apparent infinite series of sources will itself appear as a single apparent source, if the point of measurement is sufficiently far away. The strength, g, of this approximate apparent source will be equal to the sum of the strength of all its components as in Equation (1). We will define its “effective position,” x, to the left of the first source k, as that position about which the dipole moment Zi,?,, of the series of sources is zero. Here i, is the strength of the nth source in amperes, and < its position vector relative to an arbitrary coordinate system. (If a unit source were placed at this position, its remote lead field would then match the actual field with minimum error.) Thus we find, as in Equation (2).
+ 2c + 3c” + . . .)
- c)‘]
(2)
replaced the c and k factors, defined in Fig. 10 by p and q factors, respectively. Analysis of the reflections and re-reflections involved in this situation leads to the infinite series of sources shown in Fig. 12, B. The first image, generated at the ar:p boundary, is of strength p. The second, due to the transmitted effect q of the internal unit source, is totally reflected at the outer boundary and transmitted back to the internal medium with additional attenuation k, yielding effective source strength qk. This second image is also responsible for the third and higher order images (qk)c, (qk)?, etc. The electrical center of this group of sources can be determined once again by choosing x, its distance measured to the left of the qk source, so that the dipole moment 2(i,?,) of the cluster is zero. This requires
&at
0 = M = p(x + 2d) + qkx + [qkc(x
- ad)
If we set this to zero, we obtain + qkc’ (x - 4d) + qkc3 (x - 6d) + . . . l(6) 0 = x -
(2&/k)
(3)
This gives for x x = d(Q-PI/P
(4)
Thus the effective center of the infinite series of charges is at distance x + d = d(QIP)
(5) to the right of the boundary between the surface slab and the underlying medium. This means the effective slab thickness has been increased by (a/p.). In the same way, one can treat the case analogous to esophageal leads where the current source is in the underlying medium, as shown in Fig. 12, A. Here the roles of the resistivities of the two media are reversed and to avoid confusion we have
This may be rewritten
as
0 = fix + 2pd + qk(l
+ c + c2 + . . .) .Y
- 2qkdc(l which
+ 2c + 3c2 + 4c3 + . . . )
in turn may be simplified
0 = px + 2dp + qk/(l 2gkdc/(l
-
(7)
to yield
- c) x -
c)’ = -2d
The distance between this effective electrical center and the internal source, which we will designate as 2s’, will be twice the distance s’ from the source to the effective reflecting surface which would
Qualitative
exist were the medium homogeneous with resistivity 6. This distance is 2s’ = x + 2d + 2s = 2d
(9)
effects of tl torn&c resistivity
0P
J=
[2(4/7)(3/7)2/41r(20)21
o(
It is clear from this equation that the lower resistivity of the surface layer makes it appear as if the surface is thicker by a
59
(amperes), and r the distance (centimeters). In our case, we must sum the effects of many sources and thus obtain for a unit source
Thus the field in the underlying medium acts as if there were a reflecting (insulating) surface at a distance s’=s+d
variations
+ . . .
(12)
The value of this is found to be 0.00167 amps per square centimeter, and is plotted in Fig. 5 along with the values for the other points I’, through PT.
factor of (t), the entire conducting medium being considered homogeneous with the same resistivity as the underlying medium.
Appendix
Ill
Field of point current sources on the of concentric spheres. In this appendix we determine the lead field at the center of a conducting sphere of radius “a” (simulating the rib cage and interior) which is surrounded by a concentric conducting shell of outer radius “b” (simulating the skeletal muscle, fat, and skin) which in turn is imbedded in an insulating medium (air). The field arises from a current source and sink lying at the surface of the shell at opposite ends of a diameter. The resistivity of the inner sphere is ur and of the outer shell ~2. Both are assumed isotropic. The derivation is an extension of that given in Smythe l6 for a single sphere. The potential @rinside the solid sphere, medium (1)s satisfies Laplace’s Equation and in spherical coordinates is surface
Appendix
II
Sample calculation of the lead field under an exploring electrode on a flat, isotropic, low-resistance surface layer. We assume first that the layer is 3 cm. thick, has a resistivity of 800 ohm-cm., and covers a semi-infinite medium of resistivity 2,000 ohm-cm. We are interested in the field at points Pa, Pq, Pb, Ps, and P7 which are located 3, 4, 5, 6, and 7 cm., respectively, from the bottom surface of the low-resistance layer, directly under the exploring electrode. As a sample calculation, let us consider the field at Pj. As shown in Appendix I, we may consider this point to be part of an infinite homogeneous conductor of 2,000 ohm-cm. provided that we consider the sources of the field to be those shown in Fig. 11, L3. Determining k and c with the equations shown in Fig. 10, we find for them values of (4/7) and (3/7), respectively. The nearest source is located (5+3) cm. from P5 and according to Fig. 11, B has strength 2k or 2(4/7) amperes. The next most distant source is located (5+3+6) or 14 cm. from Pb and has strength 2kc or 2(4/7)(3/7) amperes. The field density at Pr, can be determined using the equation for the field of a source in an infinite homogeneous conductor J = I/4mz
(11) where J is the current density (amps per square centimeter), I the source strength
a1
= $J
n=o
C,
rn
P, (cos 0)
(13)
r is the spherical coordinate radius and P,(cos 0) the Legendre Polynomial. The electrodes are located on the line, 8 = 0. In the shell, a slightly more general solution of Laplace’s Equation is required and can be expressed
$I2= 2 -4,rn+ -& P, (COSe) n=ll ( 1
(14)
At the boundary r = a, the potentials and normal current densities are continuous. Thus, cp1(a) = fD2(a)
(15)
except at the electrodes. We note that because of the symmetry only odd Legendre Polynomials are required. Thus at the surface the normal current density from Equation (14) is
in which ~71 and (TP are conductivities in media (1) and (2). The insertion of Equations (13) and (14) into the boundary conditions of Equations (15) and (16) leads to the coefficient relationships
--(17)
I3232
ar
on the left-hand
(cosfq
S)]" d(cos e) +1(cos 1+b'[Pzn
side, in which
be recognized as [1/2sb2], while the integral Thus, Equation (20) leads to
on the right is available
(21), (18), and (17) can be used to find the coefficients B a+1
- C2n+1u
as [4nf3]-r.
(21) as
.~n+3(!2y5)(~%2)
2az(n + 1) + a1(2n +1, u2(4n + 3)
:lan+l = Czn+l-----c
-- I (4% + 3Y b2” + 1 b”” + 3 [2az(n + 1) + u1(2n + l)] - (u, - uz) 2 (n + lVf3 zn+l - 27r (2% + 1) [ Use of Equation
(24) in Equation
(13) gives
(20)
-a$ is zero except at the point electrode, can
.-1zn+ 1 (2x + 1) b”” - (Lp&J!!$=L& Equations
(19)
Pz n + 1 (cos e) d(cos e)
2(x + 1) = uz .-12n+l (an + 1) P - b2n+yBzn+I
The integral
P&+1
To obtain a relation among the A’s, B’s, and total current I, both sides of Equation (19) are multiplied by Pzn+r (cos 0) and integrated with respect to cos 0 from 0 to 1. This procedure gives us Equation (20).
The remaining boundary condition is found from the normal component of current at r = b. This is zero everywhere
i
1
(22)
(23)
1 -’
(24)
Qditutive
effects of thorucic resistivity
vuricltions
61
in which
D = p&qy-q
[1 _ ($+“I
When the gradient of Equation (25) is taken, converted multiplied by the conductivity, r1, to obtain the-current quantity is
to rectangular coordinates density, Jz, at the origin,
and that
(26) From Equation (26), the current densities and a sphere alone, u1 = ~2, can be found.
at the origin for a sphere and shell, G-~f
~2,
A
lthough the torso is often assumed in electrocardiographic studies to be homogeneous, it is not. In a previous paper’ we have considered the resistivity changes which produce the most drastic modification of surface electrocardiograms (ECG’s). These are those variations which occur in and around the heart itself, i.e., the differences in conductivity between heart muscle, blood, and lung, as well as the lower resistivity of heart muscle in the direction of its fibers compared to that in a transverse direction. In this article, we discuss what we consider to be the second most important resistivity change, namely, that occurring in the vicinity of the surface electrodes. This is due to the relatively low resistance surface layer formed by the muscles girdling the thorax. As in our first paper, we use elementary models, and neglect the perturbing effects of other resistivity changes, e.g., the ribs, spine, sternum, liver, blood, blood vessels, pleural membranes, etc. Experimental
Observed electrical characteristics oj the layer. Fig. 1 shows a schematic
surface
This
study was supported by Research National Institutes of Health. Received for publication July 25, 1967. *Department of Electrical Engineering, **Department of Electrical Engineering,
48
American Heart Journal
Grants
Syracuse University
variations The low
Vt.
representation of this low resistance layer. The skin and surface fat overlie the muscle which, in turn, overlies the lung. This muscle layer shows variations in thickness from approximately 1 to 3 cm. in anatomic cross section drawings.2 In the region of the chest surface of greatest interest to us, over the heart, the “pectoralis major” muscles tend to run from the left shoulderarm region toward the caudal rib cage and sternum. Of the muscles between the ribs, the “intercostalis externus” are oriented more or less vertically, and the “intercostalis internus” horizontally. Other nearby muscles, “pectoralis minor” and “serratus anterior,” have still different orientations. Thus, the successive layers of muscle are by no means well aligned. Electrical effects of this low-resistance layer are seen in measurements3~4of the over-all resistivity of the trunk made with currents flowing from head and shoulders to feet. Measurements of normal male adults have yielded values of 463 and 489 ohm-cm. which differ approximately by a factor of four from the value of 2,000 ohm-cm. found for dog lung. Comparisons of measurements of this type made on a HE-08607
and
HE-09831
from
the
National
Heart
Institute
of the
University, Syracuse, N. Y. of Vermont. Burlington, Vt.
July, 1958
Vbl. 76, No. 1, pp. 48-61
Qunlitdive
dog,4 first with the chest intact and second with the chest opened and the lungs insulated by wrapping them with a plastic film, have yielded a mean value for the resistivity of the surface layer of 281 ohm-cm. The relative amplitudes of ECG’s registered from human subjects also show effects similar to those which would be produced by a low-resistance layer. The average potential differences between a grid of electrodes covering the chest and another the back, were compared with the voltage in the chest-back lead in the axial system for six subjects.5 It was found in all cases that the ratio of the peak amplitude of the axial lead to that of the grid lead was smaller than the sensitivity ratio determined with a homogeneous tank “torso,” the same leads, and a dipole located at a position corresponding to the anatomical center of the heart. This sort of result would be expected were the “effective” thickness of the surface layer many times its actual thickness. Theoretical churacteristics of the surface layer. The effects on the ECG of a layer of muscle can to some degree be predicted
Fig. 1. Schematic layer.
cross
section
of
torso
surface
ejects of thoracic rcsistivity
vtrricrtions
49
theoretically. So far as chest electrodes are concerned, it is convenient here to make the simplifying assumption that the chest surface is flat, rather than curved and infinite in extent6 At first glance, this model may appear to be overly simplified. However, one can easily extend results obtained with it to a box-shaped conductor, through the use of multiple reflections of images in the box sides. Furthermore, experiments with rectangular tanks with and without rounded edges have shown that in most leads the effect of rounding is to change relative sensitivities by only a few per cent. In addition, a quantitative estimate of the effect of curvature is possible in the case of a spherical surface layer. In the section entitled “Effect of curvature of the surface layer” it is found that the predictions obtained with the flat surface layer theory agree reasonably well with those obtained from the exact analysis of the sphere. A recently published survey by Geddes and Baker’ of the resistivity of various tissues includes a number of measurements of muscle conductivity which show that the low-frequency resistivity of muscle along the fibers is much less than across the fibers. Measurements by Burger and van biilaan8 Burger and van Dongen,9 and by Rush and associates4 have indicated that the former is about 150 ohm-cm. and the latter 2,300 ohm-cm. So far as currents flowing parallel to the surface of the chest are concerned, this disparity between the high and low resistivity is greatly reduced
m=-!L. h+L
Fig. 2. Resistor matrices placed in parallel.
showing
how
two
perpendicular
anisotropic
sheets
yield
an
isotropic
sheet
when
by the fact that successive laminas of muscle are oriented in different directions. In fact, if we use the simplifying assumption that the laminas are thin and of uniform thickness and run alternately at right angles to one another, we find that the resistivity parallel to the surface (in the limit as the thickness of the laminas approaches zero) no longer depends on the direction.“’ This is easily visualized from Fig. 2, where two two-dimensional resistor matrices, representing two adjacent anisotropic conducting sheets, are connected in parallel, the result being an isotropic resistance matrix. This shows that the conductivity al,, of the melded layers is given by the means of the high (Q) and low (UI) conductivity, i.e., by (T,,, = $1 (uh+uJ. This may be expressed in terms of corresponding resistivities, p, as pm = 2~1~h/(pl+~h). Taking the high and low resistivities of muscle to be 1.50 and 2,300 ohm-cm., respectively, we obtain an average isotropic resistivity pm tangential to the surface of 280 ohm-cm. (the very close correspondence between this and the experimental value 281 ohm-cm., although fortunate, does not mean that our model is accurate within one half of 1 per cent!). The conclusion that the muscle may be considered homogeneous in directions parallel to the surface is supported by the regular nature of constant potential lines on the body surface arising from passing current from head to foot. No indications
of the underlying anisotropy are seen.8 The component of current normal to the surface encounters a common high resistivity from all the muscles. With the mean value pm representing the tangential resistivity, and the high value Ph representing the resistivity normal to the surface of the body, a section of the muscle perpendicular to the surface can be represented by the resistance matrix shown in Fig. 3,A. Now, so far as the distribution in the network of currents introduced into it from outside is concerned, it would make no difference were this matrix stretched by a factor of dh/m as is shown in Fig. 3,B. This stretching has the effect of making the voltage drop per centimeter in the h resistors, for a given surface current density, smaller. It thus decreases the effective resistivity of the medium represented by the network in the direction of the “h” resistors. The stretching increases the resistivity in the direction of the m resistors by the same factor. Since P,,, dz/~,~ = Ph dpmiph = \I=, the stretching has the effect of making the two resistivities the same. Furthermore, if the matrix in Fig. 3,B is sufficiently fine grained, it will make no difference to external measurements if the number of resistors in the direction perpendicular to the surface is increased provided that the resistivity of each resistor is reduced in proportion. Similarly, the number of resistors in a direction parallel
Surface m
___ Surface m
m
___
h m
---
Surface
m
m
__-
---
f
---
h
_--
h
f
h ---
m
___
m h
___
-__
f
h
~ A
m
f
B
f
---
f f
f ___ ~
---
f f
f m
h
T
f
f m
m h
f
f
---
f f
f
__-
c
Fig. 3. Resistor matrices showing how the anisotropic surface layer represented is equivalent to a thicker isotropic layer. Each matrix represents a thin slice of the conductor perpendicular to the plane of the surface. .1, Actual matrix; B, stretched matrix (stretch factor is 4 h/m); C, isotropic square-celled matrix equivalent to stretched matrix.
Qualitative
to the surface can be increased providing that the resistance of each is increased in proportion. The result of these manipulations is the isotropic square-celled matrix shown in Fig. 3,C. This matrix represents a homogeneous isotropic surface layer of resistivity l/prnphand of a thickness dph/pm times its original thickness. The result above was derived on a field theoretic basis for an infinite anisotropic slab by Kunz and Moran.” According to this analysis, a muscle layer 1 cm. thick, having a resistivity perpendicular to its surface of 2,300 ohmcm., and 280 ohm-cm. parallel to its surface, will act as if it were a layer d2,300/280 = 2.85 N 3 cm. thick having a resistivity of 1/2300.280 = 800 ohm-cm. Of course, if the muscle fibers do not run precisely parallel to the surface, Ph will be lower and there will be less increase iu the effective thickness. Efect oj low-resistance surface layer on the lead jield of an exploring electrode. I’er-
haps the simplest situation to which these results can be applied is that shown in Fig. 4. Here the lead consists of an exploring electrode on the chest surface coupled with an ideal indifferent electrode presumed to be located at infinity. If the surface layer is anisotropic, we have already seen how its apparent thickness will be increased, and its effective resistivity changed. Our analysis gave a value of 800 ohm-cm. and a thickness of 3 cm. for this layer of augmented thickness. As 800 ohm-cm. is substantially smaller than lung resistivity, we determine here the effect of this residual difference. The analysis in Appendix I shows that the field produced in the lung by the surface electrode may be considered to be produced by a number of sources in an infinite homogeneous conductor. These are of progressively diminishing size and increased remoteness located on a line passing
Exploring
e.fects qf thorn&
resistivity
4. Simple
model
for determining
A more accurate approximation for the jield of un exploring electrode on a low-resistance surj’uce layer. In regions of the
lung close to the exploring electrode, the approximations used for points remote front the exploring electrode can be improved upon substantially. For example, in Appendix II we compute the field in the “lung” (2,000 ohm-cm.) at a point P located 5 cm. below the undersurface of a 1 cm. thick anisotropic muscle layer whose effective isotropic thickness is 3 Clll. and resistivity 800 ohm-cm. The calculated lead field density at this point, as well as others at 3, 4, 6, and 7 cm. from the low resistance layer, are plotted as heavy dots in Fig. 5. By waq- of contrast, curve A shows the field which would exist in this region were the surface layer
Electrode..&
the lead field
51
through the exploring electrode perpendicular to the surface (Fig. 11,B). At points in the lung relatively remote from the electrode, these sources may be grouped together into a single equivalent source whose source strength is the same as that of the actual exploring electrode and which is located at a distance from the bottom surface of the surface layer (Fig. 4) increased over the actual distance by a factor given by the ratio (~l,,~~/~l~,.\.~~)of the resistivity of the lung to that of the effective surface layer. These statements regarding the lead field of the exploring electrode have a simple interpretation in electrocardiography. First, a low-resistance surface layer tends to make the voltages measured on the surface smaller than they would be in a homogeneous conductor, because of an increase in effective distance to the EMF’s in the heart. The increased distance also acts to reduce “proximity” effects and to reduce the surface manifestations of multiplicity in the sources which in reality comprise the heart’s electrical generator.
Electrode
Fig.
variations
of an exploring
electrode.
at Infinity
52
Am. Hrart .I. .Ild,v, 1968
.McFee and Rush
I ampere
.006 I-
t \
“E y .005“a
\field
meowed this line
along
5
Thorax
E e 5 0
ossumed
_ Points show values _ anisotropic surface
Accurate for large distances
111
I 11
I!
3
11
I II
4 Distance
I I II!
5 from
Underside
I III
6 of
I IIII
7
Surface
8
cm
Loyer
Fig. 5. Figure representing the main conclusions of this article. It shows how an anisotropic low-resistance surface layer (muscle) can be represented by a thicker surface layer having the same resistivity as the underlying medium (lung). The dots show the field calculated for an anisotropic 1 cm. surface layer. Curve A is the field which would exist with a 1 cm. surface layer of 2,000 ohm-cm. resistivity. This curve shows that lead sensitivity estimated with homogeneous models may be three times greater than actual sensitivities. Curves B accurate rule and C are similar to A except for layer thickness. Curve C is the basis of the simple, reasonably of thumb that a 1 cm. thick muscle layer over the heart is equivalent to 5 cm. of lung. Curve D represents the best approximation to the actual values (dots) and is described in text.
1 cm. thick and of the same resistivity as the underlying medium. We note that the actual field 5 cm. below the underside of the surface layer, a position roughly corresponding to the center of the heart, is only 38 per cent of the field that would exist there were the entire conductor homogeneous. Obviously, the assumption that the thorax acts as if it were homogeneous leads to gross errors. Curve B shows the field which would be prodoced if the thickness of the equivalent isotropic 3 cm. surface layer were increased by a factor of 2% to 7% cm., and its resis-
tivity
made
the same
as the
rest
of the
conductor. This curve gives a better estimate of the actual field than curve A, but is by no means precise. In curve C the surface layer thickness is taken to be 5 cm. rather than 7% cm. The approximation here is quite respectable. Curve D is that obtained with a 3 cm. surface layer whose resistivity has been changed to match the rest of the medium, and whose lead field source strength has been reduced to two thirds of an ampere. This curve gives an excellent approxima-
Qualitutive
tion of the actual field directly under the electrode. (It can be shown theoretically that, if the field is a good match to the actual field on this axis, it will be a good match off the axis also.) The reduced lead field strength implies that the voltages that will be measured in the region of the heart with an exploring electrode directly over the heart will be almost identical to two thirds of the voltages which would be measured were that surface layer replaced with a 3 cm. surface layer having the same resistivity as the underlying medium. We have assumed that a high degree of anisotropy exists in the surface layer. As a limiting contrary case, we might suppose that the anisotropy is negligible and that the resistivity in all directions is 280 ohm-cm. This is the value estimated from experimental data for current flow in a tangential direction. The field is computed as outlined in Appendix II. It is found in a manner similar to that of curve D of Fig. 5, that the actual field can be very well approximated by assuming that the surface layer thickness has been increased from 1 to 2.5 cm., giving it the same resistivity as the rest of the medium, and by decreasing the lead field source strength of the electrode from 1 to 0.7 amperes. These values do not differ much from the 3 cm. and 0.67 amperes shown in curve D, which were determined for the anisotropic layer. Eflect of curvature of the surface layer. 110 these conclusions, derived from flat surface layers, also apply to curved layers? To gain insight into this question, we derive, in Appendix III, an equation for the lead field at the center of a homogeneous sphere enclosed in a spherical shell of lower resistivity into which current flows via electrodes located on opposite sides of the sphere. (This is the inverse of a problem solved by Bayley and Berry.12) The sphere and its surface layer form a crude model of the body whose dimensions have been chosen to match the flat surface layer case already considered. We assume that the inner sphere has a radius of 5 cm. and resistivity of 2,000 ohm-cm. The shell is taken to be anisotropic, 1 cm. thick, and also is assumed to be representable with good approximation by a layer 3 cm. thick of 800 ohm-cm. isotropic resistivity.
effects of thoracic resistivity
variations
53
We find from Equation (26) in Appendix III that the introduction of a unit current will produce a field of 0.00435 amperes per square centimeter at the sphere’s middle. On the other hand, were the sphere and shell of the same resistivity, and of total radius 6 cm., the central field strength would be 0.0135. The actual value 0.00435 is 32 per cent of this. Thus, we find in the sphere roughly the same degree of reduction (38 per cent) found with a flat surface layer. The reduction is somewhat greater for the sphere in part because some of the lead field current which otherwise would pass through the center has been drawn away to the low resistance sides of the sphere. In any case, the values are enough alike to indicate that what is true for a flat layer is also true for a curved layer. Exact compensation for the low-resistance surface layer in ideal leads. Through the use of electrodes covering the entire surface of a homogeneous volume conductor connected together via appropriate resistor networks, it has been shown13J4 that it is possible to construct leads whose lead fields are “ideal.” For example, they may be (1) uniform, corresponding to heart vector leads, or (2) appear to radiate out from a point, corresponding to unipolar leads, or (3) have a point of zero field corresponding to “null,” “cancellation,” or “quadripole” leads. Furthermore, it has also been shown14 that these and other types of lead fields may be produced in a region corresponding to the heart even if the latter has a resistivity different from that of the rest of the body and is spherical or ellipsoidal in shape. Finally, leads may, in principle, be constructed under these assumed conditions in which the exploring electrode appears to be within the volume conductor, close to the “heart,” despite the fact that all electrodes are on the body surface. These objectives may still be achieved when there is an effectively homogeneous low-resistance surface layer whose inner surface is ellipsoidal in shape. We will present here examples which should, along with the references cited, serve to convey the idea behind the mathematical approach, which is based on the lead field concept.
54
McFee
and Rush
Consider a sphere bounded by a lowresistance surface layer in the form of a spherical shell of uniform thickness. Suppose there exists in the sphere proper a uniform lead field. Then, as was discussed in the appendix of the previous article,’ the lead field in the outer spherical shell will be that of a dipole plus a uniform lead field. (The sources are outside the shell.) If one constructs a lead to introduce or remove the part of this lead field currelit which arrives or leaves at the outer surface of the spherical shell, then the uniqueness theorem requires that this external lead produce the desired uniform field lvithin the sphere. The sanle argument ma>- be applied to lead fields of various types in the inner the unipolar lead sphere, as for example type, the null lead type, etc. These ma). be broken down into spherical harmonic components’” (uniform, dipolelike, etc.) each of which propagates into the spherical shell a fixed combination of conlponents, which when summed, will yield a surface field on the outer surface of the spherical shell which can be matched by a properly designed lead. That lead will therefore produce the desired field in the sphere. It is a small generalization of this procedure, conceptually speaking, to drop the assumption that the outer shell is uniform in thickness with a spherical outer surface. Even if the outer surface is irregular, the identical procedure may be followed to construct a lead which introduces the required lead field surface current. Furthermore, this type of approach can be extended to ellipsoidal shapes by using ellipsoidal harmonics’4 rather than spherical harmonics. Fig. 6 illustrates this possibility. The procedure is the same when all surfaces are ellipsoids or spheres (escept the outer surface which may have any shape whatever), and the conductivit> in each region is homogeneous and isotropic. A$proximatc compensation for the lowresistawe sllrfc’ftlce Iuyer. As a practical matter, what needs to be done to adapt ideal leads to the presence of a low-resistance surface layer is to provide extra lead field current for the portions of this layer that run parallel to the direction of the lead field. This extra current is needed
Fig. 6. Cross section of a heterogeneous conductor showirlg how a uniform lead field can nevertheless he produced inside all ellipsoidal homogeneous “heart.”
,p
= 400
p = 200
Fig. 7. Model illustrating the shunting effect of low resistance la~cr 011 the neck-foot lead.
Fig. 8. Cross section of trapezoidal model indicating how, with vectorcardiographic leads, extra lead tield current is needed (with “low-resistance” surface layer of uniform thickness) only at points where the direction of the layer changes.
because the resistance of the surface is less. Thus, if the tangential potential gradient in the layer is to match that in the body proper, additional current must be supplied. As an example, consider the box-shaped conductor shown in Fig. 7 which is covered with a low-resistance surface layer of uniform thickness. A lead for inducing a uniform vertical field in this box must provide five times as much current density for the sides of the box as for the center. Hence, a
?‘olume Number
76 1
Qzditutive
properly designed lead would employ resistors for electrodes feeding the surface layer parallel to the lead field having five times the conductance of those feeding other parts of the conductor. In a conductor having a trapezoidal cross section, as shown in Fig. 8, in a case where one wishes to produce a uniform lead field in the body proper, the lead field current in the layer may be divided into two components: (1) a component of current flowing perpendicularly across the layer, which supplies the normal component of the lead field flowing into the body proper, and (2) a component of current flowing along the layer, parallel to its surface. This component maintains the same tangential potential gradient in the surface layer as is required by the uniform lead field in the body proper. When the field desired in the body proper is nonuniform, the same approach may be employed provided that the surface layer is thin enough so that the normal component of the field on its outer surface may be considered the same as that on its inner surface. It is clear that, when the internal field is uniform, component 2 of lead field current, i.e., the total current flowing tangential to the low-resistance surface layer, needs to be augmented or diminished by extra current from the surface electrodes only in regions where the surface layer changes direction, i.e., curves. In general, a curved layer may be represented as a series of straight sections. The extra surface current required may be introduced at the points where the various segments meet. Discussion
The effect of the low-resistance surface layer may be summarized qualitatively in the following manner. In the neighborhood of the electrodes, the layer tends to spread the lead field current out over a larger area, thus making the electrode seem to be further away and the field within the body weaker. Along the sides of the body, the low-resistance layer tends to shunt the lead field from the central portion of the body, once again making the internal lead field seem weaker. Both of these effects can be predicted if one views the low-resistance layer as an effectively
efects of thorcrcic resistivity
variations
.55
thicker layer having the same resistivity as the underlying tissue. The weakening of the lead field in the neighborhood of the heart is quite substantial. We have already considered several cases in which the field is reduced to about one third the value it would have were the surface layer resistivity the same as that of the underlying medium. Consider further a head-foot lead (Fig. 7). If one assumes that the resistivity of the lungs is 2,000 ohm-cm., that the tangential resistivity of the surface layer is 280 ohmcm., and that the cross sectional areas of the two with regard to head-foot lead field currents are 400 and 180 cm.?, respectively, then the current which flows in the lung will be (400/2,000)/[(400/2,000) + (180/280)] of the total, i.e., 23.7 per cent rather than the 66 per cent it would be were the trunk homogeneous. Thus, the low-resistance surface layer here also diminishes the sensitivity of this lead by a factor of about three, as compared to what would exist were the body homogeneous. It is not our intention to arrive at quantitative conclusions using the simple models considered in this paper, but they do indicate that the effect of the low-resistance layer is to reduce the internal lead fields to values substantially less than one half of those found with homogeneous models. Furthermore, the reduction factors are not necessarily the same in the different leads. Clearly this is a consideration that should not be ignored in electrocardiographic interpretation or in lead system design. Conclusion
The surface of the body is covered by a layer of muscle whose thickness varies greatly from point to point, as well as from subject to subject. Because the resistivity of the muscle in the direction of the fibers is low, about the same as blood, and across the fibers high, the same as lung, and because it differs in any case from that of lung, this layer has a marked effect on the interpretation of ECG’s. Studies with simple theoretical models in this paper indicate that this layer may be considered as a rough approximation to have the same resistivity as the under-
56
Am. Heart
.McFee and Rush
J.
July. 1365
lying tissue (lung) provided that it is also considered to have a thickness of the order of three to seven times its actual thickness (see Fig. 5). A muscle layer 1 cm. thick, as might be typical for the region over the heart, appears to be about 5 cm. thick. As a result of this increased effective distance between the heart and the surface electrodes, the sensitivity of the lead is reduced to a value generally. less than one half the value which would exist were there no difference in resistivity between the surface layer and lung.* Furthermore, proximity effects are substantially reduced also. As the sensitivity reduction factor depends upon the lead considered, this means that heterogeneous models must be employed for accurate determination of sensitivity factors of vectorcardiographic leads.
11.
*An opposite effect is produced by the low resistance of blood and heart muscle relative to lung which tends to increase the sensitivity (on the average), Fig. 1.1
General equations for the lead jield of an exploring electrode(unit point current source) on a j&t low-resistance surface layer. If there is a unit source in a semi-infinite homogeneous conducting medium bounded by a conducting or insulating plane (Fig. 9, A), then the effect of the plane can be taken into account by pretending that the medium is infinite in all directions and that there is an image source located as shown in Fig. 9, B. Note that, if the source is on the surface, it and its image merge and the effective strength of the source is doubled. If the infinite medium is separated into two halves, one with resistivity (II and the other ,B (Fig. 10, A) with a source in the a: region, then the field in the cx region will be that of the original source plus an image of strength c = @-CX)/(CX+~) times the original source, as shown in Fig. 10, H. In the /3 region the strength of the original source will appear to be reduced by a factor k = (l-c). Here c is the “reflection factor” and k is the “transmission factor.” An electrostatic analogue of this problem is treated by Slater and Frank.15 Straightforward use of these results shows that in the case of Fig. 11, A the field in the ,6 region will appear to originate from an infinite number of sources representing the original source of doubled strength attenuated by a factor k, its
REFERENCES 1.
2.
.z
4.
5.
6.
7.
8.
9.
10.
McFee, R., and Rush, S.: Qualitative effects of thoracic resistivity variations on the interpretation The “Brody” of electrocardiograms: effect, AK HEART J. 74:642, 1967. Eycleshymer, H. C., and Schoemaker, D. M.: A cross-section anatomy, New York, 1911, Appleton-Century-Crofts. Schmitt, 0. H.: Lead vectors and transfer impedance, Ann. New York Acad. SC. 65:1109, 1957. Rush, S., Abildskov, J. A., and McFee, Ii.: Resistivity of body tissues at low frequencies, Circulation Res. 12:40, 1963. McFee. R.. and Paruneao. A.: An orthoeonal lead system for clinic;1 electrocardiography, AK HEART J. 62:93, 1961. Bayley, R. H.: Electrocardiographic analysis, Vol. 1, Biophysical principles of electrocardiography, New York, 19.58, Paul B. Hoeber, Inc., p. 211. Geddes, I,. A., and Baker, L. E.: The specific resistance of biological material. A compendium of data for the biomedical engineer and physiologist, Med. & Biol. Eng. 5:271, 1967. Burger, H. C., and van Milaan, J. B.: Measurement of the specific resistance of the human body to direct current, Acta med. scandinav. 114584, 1943. Burger, H. C., and van Dongen, Ii.: Specific electrical resistance of body tissues, Phys. Med. & Biol. 5:431, 1961. Rush, S.: A principle for solving a class of anisotropic current flow problems and applications to electrocardiography, IEEE Tr. Biomed. Eng. 14:18, 1967.
Kunz, K. S., and Moran, J. H.: Some effect. of formation anisotropy on resistivity measures ments in boreholes, Geophysics 23:770, 195812. Bayley, R. H., and Berry, P. M.: Changes produced in the resultant “heart” vector by the non-homogeneous volume conductor with normal specific resistivities, Proc. of the Long Island Jewish Hosp. Symposium on vectorcardiography, Amsterdam, 1965, North Holland Publishing Co., p. 109. 13. McFee, Ii., and Johnston, J. D.: Electrocardiographic leads III, Circulation 9:868, 1954. 14. McFee, R.: Analysis and synthesis of electrocardiocrranhic leads, Ph.D. Dissertation (Electrical Engineering) Ilniversity of Michigan, Ann Arbor, 1954. J. C., and Frank, N. H.: Electromag1.5. Slater, netism. ed. 1. New York. 1947. McGraw-Hill Book Co., Inc., p. 47. ’ ’ 16. Smythe, W. R.: Static and dynamic electricity, ed. 2, New York, 19.50, McGraw-Hill Book Co., Inc., p. 239. Y
Appendix
I
I
Volzlme h’umbcr
76 1
Qualitative
Mnite Homogeneous Conductor
Fig.
9. An
insulating
surface
effects oJ’ thoracic resistivity
produces
an “image.”
Ham
Current Source
k=l-C=&
c
B
10. A change in resistivity produces an “image.” a region ; C, equivalent situation as viewed from
A
p=B
c= (P-a) (P+a)
A Fig. from
57
Insulaling Surface
p=a
Semi -%inite Conductors
vclrintions
A, Actual 0 region.
situation;
B, equivalent
situation
as viewed
: I
Conducting Slab
\
Semi-infinite Homogeneous Conductor
Fig. 11. Image sources for a unit source layer. d, Actual situation; B, equivalent
of current situation
connected as viewed
p=p
Everywhere
to the outside from B region.
q= (I-p)
Fig. 12. Image sources for a unit source of current located inside a conductor A, Actual situation; B, equivalent situation as viewed from fl region.
surface
/
of a low-resistance
surfaqe
a-P p=------a+P near a low-resistance
surface
layer.
58
Am. Heart
McFee and Rush
July,
g = 2k + 2kc + 2kc2 + 2kc3 + . . . = 2k/l-c Z in2
= 2k(x) = 2k(l
+ 2kc(x
- c) x -
= 2
(1)
- 2d) + 2kc2 (x - 4d) + . . .
+ c + c2 + . . .) x - 4dkc(l
= (2k/l
= 2klk
I. 1968
[4dkc/(l
attenuated doubly reflected image 2kc, the re-reflected image 2kc2, etc., the entire conglomeration being located in an apparent infinite homogeneous medium of resistivity p. This apparent infinite series of sources will itself appear as a single apparent source, if the point of measurement is sufficiently far away. The strength, g, of this approximate apparent source will be equal to the sum of the strength of all its components as in Equation (1). We will define its “effective position,” x, to the left of the first source k, as that position about which the dipole moment Zi,?,, of the series of sources is zero. Here i, is the strength of the nth source in amperes, and < its position vector relative to an arbitrary coordinate system. (If a unit source were placed at this position, its remote lead field would then match the actual field with minimum error.) Thus we find, as in Equation (2).
+ 2c + 3c” + . . .)
- c)‘]
(2)
replaced the c and k factors, defined in Fig. 10 by p and q factors, respectively. Analysis of the reflections and re-reflections involved in this situation leads to the infinite series of sources shown in Fig. 12, B. The first image, generated at the ar:p boundary, is of strength p. The second, due to the transmitted effect q of the internal unit source, is totally reflected at the outer boundary and transmitted back to the internal medium with additional attenuation k, yielding effective source strength qk. This second image is also responsible for the third and higher order images (qk)c, (qk)?, etc. The electrical center of this group of sources can be determined once again by choosing x, its distance measured to the left of the qk source, so that the dipole moment 2(i,?,) of the cluster is zero. This requires
&at
0 = M = p(x + 2d) + qkx + [qkc(x
- ad)
If we set this to zero, we obtain + qkc’ (x - 4d) + qkc3 (x - 6d) + . . . l(6) 0 = x -
(2&/k)
(3)
This gives for x x = d(Q-PI/P
(4)
Thus the effective center of the infinite series of charges is at distance x + d = d(QIP)
(5) to the right of the boundary between the surface slab and the underlying medium. This means the effective slab thickness has been increased by (a/p.). In the same way, one can treat the case analogous to esophageal leads where the current source is in the underlying medium, as shown in Fig. 12, A. Here the roles of the resistivities of the two media are reversed and to avoid confusion we have
This may be rewritten
as
0 = fix + 2pd + qk(l
+ c + c2 + . . .) .Y
- 2qkdc(l which
+ 2c + 3c2 + 4c3 + . . . )
in turn may be simplified
0 = px + 2dp + qk/(l 2gkdc/(l
-
(7)
to yield
- c) x -
c)’ = -2d
The distance between this effective electrical center and the internal source, which we will designate as 2s’, will be twice the distance s’ from the source to the effective reflecting surface which would
Qualitative
exist were the medium homogeneous with resistivity 6. This distance is 2s’ = x + 2d + 2s = 2d
(9)
effects of tl torn&c resistivity
0P
J=
[2(4/7)(3/7)2/41r(20)21
o(
It is clear from this equation that the lower resistivity of the surface layer makes it appear as if the surface is thicker by a
59
(amperes), and r the distance (centimeters). In our case, we must sum the effects of many sources and thus obtain for a unit source
Thus the field in the underlying medium acts as if there were a reflecting (insulating) surface at a distance s’=s+d
variations
+ . . .
(12)
The value of this is found to be 0.00167 amps per square centimeter, and is plotted in Fig. 5 along with the values for the other points I’, through PT.
factor of (t), the entire conducting medium being considered homogeneous with the same resistivity as the underlying medium.
Appendix
Ill
Field of point current sources on the of concentric spheres. In this appendix we determine the lead field at the center of a conducting sphere of radius “a” (simulating the rib cage and interior) which is surrounded by a concentric conducting shell of outer radius “b” (simulating the skeletal muscle, fat, and skin) which in turn is imbedded in an insulating medium (air). The field arises from a current source and sink lying at the surface of the shell at opposite ends of a diameter. The resistivity of the inner sphere is ur and of the outer shell ~2. Both are assumed isotropic. The derivation is an extension of that given in Smythe l6 for a single sphere. The potential @rinside the solid sphere, medium (1)s satisfies Laplace’s Equation and in spherical coordinates is surface
Appendix
II
Sample calculation of the lead field under an exploring electrode on a flat, isotropic, low-resistance surface layer. We assume first that the layer is 3 cm. thick, has a resistivity of 800 ohm-cm., and covers a semi-infinite medium of resistivity 2,000 ohm-cm. We are interested in the field at points Pa, Pq, Pb, Ps, and P7 which are located 3, 4, 5, 6, and 7 cm., respectively, from the bottom surface of the low-resistance layer, directly under the exploring electrode. As a sample calculation, let us consider the field at Pj. As shown in Appendix I, we may consider this point to be part of an infinite homogeneous conductor of 2,000 ohm-cm. provided that we consider the sources of the field to be those shown in Fig. 11, L3. Determining k and c with the equations shown in Fig. 10, we find for them values of (4/7) and (3/7), respectively. The nearest source is located (5+3) cm. from P5 and according to Fig. 11, B has strength 2k or 2(4/7) amperes. The next most distant source is located (5+3+6) or 14 cm. from Pb and has strength 2kc or 2(4/7)(3/7) amperes. The field density at Pr, can be determined using the equation for the field of a source in an infinite homogeneous conductor J = I/4mz
(11) where J is the current density (amps per square centimeter), I the source strength
a1
= $J
n=o
C,
rn
P, (cos 0)
(13)
r is the spherical coordinate radius and P,(cos 0) the Legendre Polynomial. The electrodes are located on the line, 8 = 0. In the shell, a slightly more general solution of Laplace’s Equation is required and can be expressed
$I2= 2 -4,rn+ -& P, (COSe) n=ll ( 1
(14)
At the boundary r = a, the potentials and normal current densities are continuous. Thus, cp1(a) = fD2(a)
(15)
except at the electrodes. We note that because of the symmetry only odd Legendre Polynomials are required. Thus at the surface the normal current density from Equation (14) is
in which ~71 and (TP are conductivities in media (1) and (2). The insertion of Equations (13) and (14) into the boundary conditions of Equations (15) and (16) leads to the coefficient relationships
--(17)
I3232
ar
on the left-hand
(cosfq
S)]" d(cos e) +1(cos 1+b'[Pzn
side, in which
be recognized as [1/2sb2], while the integral Thus, Equation (20) leads to
on the right is available
(21), (18), and (17) can be used to find the coefficients B a+1
- C2n+1u
as [4nf3]-r.
(21) as
.~n+3(!2y5)(~%2)
2az(n + 1) + a1(2n +1, u2(4n + 3)
:lan+l = Czn+l-----c
-- I (4% + 3Y b2” + 1 b”” + 3 [2az(n + 1) + u1(2n + l)] - (u, - uz) 2 (n + lVf3 zn+l - 27r (2% + 1) [ Use of Equation
(24) in Equation
(13) gives
(20)
-a$ is zero except at the point electrode, can
.-1zn+ 1 (2x + 1) b”” - (Lp&J!!$=L& Equations
(19)
Pz n + 1 (cos e) d(cos e)
2(x + 1) = uz .-12n+l (an + 1) P - b2n+yBzn+I
The integral
P&+1
To obtain a relation among the A’s, B’s, and total current I, both sides of Equation (19) are multiplied by Pzn+r (cos 0) and integrated with respect to cos 0 from 0 to 1. This procedure gives us Equation (20).
The remaining boundary condition is found from the normal component of current at r = b. This is zero everywhere
i
1
(22)
(23)
1 -’
(24)
Qditutive
effects of thorucic resistivity
vuricltions
61
in which
D = p&qy-q
[1 _ ($+“I
When the gradient of Equation (25) is taken, converted multiplied by the conductivity, r1, to obtain the-current quantity is
to rectangular coordinates density, Jz, at the origin,
and that
(26) From Equation (26), the current densities and a sphere alone, u1 = ~2, can be found.
at the origin for a sphere and shell, G-~f
~2,