![]() ![]() ![]() | (11.8) |
where | ![]() | = dipole source density [µA/cm2] |
s | = conductivity [mS/cm] | |
![]() | = spatial derivative of transmembrane voltage [mV/cm] |
Miller and Geselowitz used published data to evaluate action potential waveforms at various sites throughout the heart as well as times of activation. They could thus estimate Vm(x,y,z,t) and as a result, could evaluate the "actual" dipole moment per unit volume at all points. For simplicity the heart was divided into a finite number of regions, and the net dipole source strength in each region found by summing
idV in that region.
In determining the surface potential fields the authors considered the number of dipole elements to be a small set (of 21) and evaluated the contribution from each. This part of their work constituted a relatively straightforward solution of the forward problem (dipole source in a bounded volume conductor). The reconstructed electrocardiograms showed very reasonable qualities.
![]() ![]() ![]() | (11.11) |
The lead vector is a three-dimensional transfer coefficient which describes how a dipole source at a fixed point Q inside a volume conductor influences the potential at a point within or on the surface of the volume conductor relative to the potential at a reference location. The value of the lead vector depends on:
We tacitly assume that the potential at the reference is zero and hence does not have to be considered. Note that the value of the lead vector is a property of the lead and volume conductor and does not depend on the magnitude or direction of the dipole .
It can be shown that in an infinite, homogeneous volume conductor the lead vector is given by the sum of components along lines connecting the source point with each of the two electrode points (each scaled inversely to its physical length). The same also holds for a spherical, homogeneous volume conductor, provided that the source is at the center.
11.4.2 Extending the Concept of Lead Vector
In the previous section we considered the lead voltage to be measured relative to a remote reference - as it is in practice in a so-called unipolar lead. In this section, we consider a bipolar lead formed by a lead pair (where neither electrode is remote), and examine the corresponding lead vector, as illustrated in Figure 11.6.
For each location P0 . . . Pn of P, that lies within or at the surface of the volume conductor, we can determine a lead vector
0 . . .
n for the dipole
at a fixed location, so that, according to Equation 11.11, we have
![]() ![]() ![]() | (11.12) |
Then the potential difference between any two points Pi and Pj is
![]() | (11.13) |
This describes the voltage that would be measured by the lead whose electrodes are at Pi and Pj. To what lead vector does this lead voltage correspond? Consider first the vector ij formed by
![]() ![]() ![]() ![]() | (11.14) |
Now the voltage between the points Pi and Pj given by Equation 11.13 can also be written, by substitution from Equation 11.12, as follows:
![]() ![]() ![]() ![]() ![]() ![]() ![]() | (11.15) |
hence identifying i j as the lead vector for leads Pi - Pj. From this result we can express any bipolar lead voltage V as
![]() ![]() | (11.16) |
where is a lead vector. We note that Equation 11.16 for bipolar leads is in the same form as Equation 11.11 for monopolar leads. But Equations 11.14-11.16 can be interpreted as that we may first determine the lead vectors i and j corresponding to unipolar leads at Pi and Pj, respectively, and then form their vector difference, namely ij. Then the voltage between the points Pi and Pj, as evaluated by a bipolar lead, is the scalar product of the vector ij and the dipole , as shown in Figure 11.6 and described by Equation 11.16.
Fig. 11.6. Determination of the voltage between two points at or within the surface of a volume conductor.
(A) The potentials Fi and Fj at Pi and Pj due to the dipole
may be established with scalar products with the lead vectors
i and
j , respectively.
(B) For determining the voltage Vi j between Pi and Pj, the lead vector
i j =
i -
j is first determined.
(C) The voltage Vi j is the scalar product of the lead vector
i j and the dipole
.
11.4.3 Example of Lead Vector Application: Einthoven, Frank, and Burger Triangles
As an example of lead vector application, we introduce the concept of Einthoven triangle. It represents the lead vectors of the three limb leads introduced by Einthoven (1908). Einthoven did not consider the effect of the volume conductor on the lead vectors. The effect of the body surface on the limb leads was published by Ernest Frank (1954), and the effect of the internal inhomogeneities was published by Burger and van Milaan (1946). The corresponding lead vector triangles are called Frank triangle and Burger triangle. In this section we discuss these lead vector triangles in detail.
Einthoven Triangle
PRECONDITIONS:
SOURCE: Two-dimensional dipole (in the frontal plane) in a fixed location
CONDUCTOR: Infinite, homogeneous volume conductor or homogeneous sphere with the dipole in its center (the trivial solution)
In Einthoven's electrocardiographic model the cardiac source is a two-dimensional dipole in a fixed location within a volume conductor that is either infinite and homogeneous or a homogeneous sphere with the dipole source at its center.
Einthoven defined the potential differences between the three pairs of these three points to constitute the fundamental lead voltages in electrocardiography. These are designated VI, VII, and VIII and are given by
Since
Fig. 11.7. Einthoven triangle. Note the coordinate system that has been applied (the frontal plane coordinates are shown). It is described in detail in Appendix A.
For the lead vectors we obtain:
Frank Triangle
PRECONDITIONS:
Ernest Frank measured the lead vectors of the scalar leads by constructing an electrolytic tank model of the human torso (Frank, 1954). The following values were obtained for the three lead vectors of the standard leads. Note that only the relative values of these lead vectors have any meaning because the measurement procedure was not calibrated.
We noted earlier that since VI + VIII = VII, a condition dictated by Kirchhof's law, the corresponding lead vectors must form a closed triangle. One can confirm from Equation 11.21 that, indeed,
Burger Triangle
PRECONDITIONS:
Lead vector concept was first introduced by H. C. Burger and J. B. van Milaan (1946, 1947, 1948) (Burger, 1967), who also used an inhomogeneous electrolyte tank model of the human torso to measure the lead vectors of standard leads.
Einthoven first recognized that because the limbs are generally long and thin, no significant electrocardiographic currents from the torso would be expected to enter them. Accordingly, Einthoven realized that the potential at the wrist was the same as at the upper arm, while that at the ankle was the same as at the upper thigh. Einthoven consequently assumed that the functional position of the measurement sites of the right and left arm and the left leg corresponded to points on the torso which, in turn, bore a geometric relationship approximating the apices of an equilateral triangle. He further assumed that the heart generator could be approximated as a single dipole whose position is fixed, but whose magnitude and orientation could vary. The location of the heart dipole relative to the leads was chosen, for simplicity, to be at the center of the equilateral triangle. (In matter of fact, the Einthoven assumptions and model were not truly original, but were based on the earlier suggestions of Augustus Waller (1889).)
Because of the central location of the heart dipole in the Einthoven model, the relationship between potentials at the apices of the triangle are the same whether the medium is considered uniform and infinite in extent, or the volume conductor is assumed to be spherical and bounded. For the unbounded case, we can apply Equation 8.12, which may be written FP =
·
r /(4psr 2 ) from which we learn that the lead vector for a surface point P is
r /(4psr 2 ) - that is, along the radius vector to P. Point P is, according to Einthoven, at the apices of the equilateral triangle. Consequently, if the right and left arms and left foot are designated R, L, and F, respectively, then the three corresponding lead vectors
R,
L, and
F are the radius vectors between the origin and the corresponding points on the equilateral triangle, as illustrated in Figure 11.7. From the aforementioned, the potentials at these points are:
FR =
R ·
FL =
L ·
(11.17) FF =
F ·
VI = FL - FR =
L ·
-
R ·
= (
L -
R ) ·
=
I ·
VII = FF - FR =
F ·
-
R ·
= (
F -
R ) ·
=
II ·
(11.18) VIII = FF - FL =
F ·
-
L ·
= (
F -
L ) ·
=
III ·
R,
L, and
F are equal in magnitude and each is in the direction from the origin to an apex of the equilateral triangle, then
I,
II, and
III must lie along a leg of the triangle (since
I =
L -
R, etc.) For example
I is seen to lie oriented horizontally from the right arm to the left arm.
In summary, VI, VII, and VIII are the three standard limb leads (or scalar leads) in electrocardiography. From Equation 11.18 one can confirm that the three lead vectors
I,
II, and
III also form an equilateral triangle, the so-called Einthoven triangle, and these are shown in Figure 11.7.
The limb lead voltages are not independent, since VI + VIII - VII = 0 , as can be verified by substituting for the left side of this equation the component potentials from Equation 11.18, namely (FL - FR) + (FF - FL) - (FF - FR), and noting that they do, in fact, sum to zero. The above relationship among the standard leads is also expressed by
I·
+
III·
-
II·
= 0, according to Equation 11.18. Since
is arbitrary, this can be satisfied only if
I +
III -
II = 0, which means that the lead vectors form a closed triangle. We were already aware of this for the Einthoven lead vectors, but the demonstration here is completely general.
From the geometry of the equilateral (Einthoven) triangle, we obtain the following values for the three lead voltages. Please note that the coordinate system differs from that introduced by Einthoven. In this textbook, the coordinate system of Appendix is applied. In this coordinate system, the positive directions of the x-, y-, and z-axes point anteriorly, leftward, and superiorly, respectively.
(11.19)
I =
II = 0.5
- 0.87
(11.20) III = - 0.5
- 0.87
SOURCE: (Three-dimensional) dipole in a fixed location
CONDUCTOR: Finite, homogeneous
I = - 14
+ 76
+ 27
II = 16
+ 30
- 146
(11.21) III = 30
- 46
- 173
I +
II -
III = 0 and hence form a closed triangle. This triangle is called the Frank triangle, and it is illustrated in Figure 11.8.
SOURCE: Dipole in a fixed location
CONDUCTOR: Finite, inhomogeneous
The lead vectors, which they measured, are given below. Since these vectors must necessarily form a closed triangle (just as Einthoven and Frank triangles), this triangle has been called Burger triangle; it is shown in Figure 11.8. The absolute values of the lead vectors have no special meaning since no calibration procedure was carried out. The lead vectors obtained were
I = - 17
+ 65
+ 21
II = 15
+ 25
- 120
(11.22) III = 32
- 40
- 141
We may compare the three triangles described so far (i.e., the Einthoven, Frank, and Burger) by normalizing the y-component of each I vector to 100. This means that the values of the Einthoven triangle components must be multiplied by 100, those of the Frank triangle by 100/76 = 1.32, and those of the Burger triangle by 100/65 = 1.54. (The reader can confirm that in each case
I = 100 results.) The resulting lead vector components are summarized in Table 11.1.
One may notice from the table that in the measurements of Frank and Burger, the introduction of the boundary of the volume conductor has a great influence on the lead vectors. As pointed out earlier, the lead vector also depends on the dipole location; thus these comparisons may also reflect differences in the particular choice that was made. Figure 11.8 illustrates the Einthoven, Frank, and Burger triangles standardized according to Table 11.1.
Lead | Triangle | cx | cy | cz |
cI | Einthoven Frank Burger | -18 -26 | 100 100 100 | 36 32 |
cII | Einthoven Frank Burger | 21 23 | 50 40 38 | -87 -192 -185 |
cIII | Einthoven Frank Burger | 39 49 | -50 -61 -62 | -87 -228 -217 |
The shape of the Frank and Burger triangles was recently investigated by Hyttinen et al. (1988). Instead of evaluating the lead vectors for a single dipole location, they examined the effect of different dipole positions within the heart. According to these studies the shape of the Frank and Burger triangles varies strongly as a function of the location of the assumed heart dipole
. They showed that the difference between the original Frank and Burger triangles is not necessarily so small if the dipole is placed at other locations. Figures 11.9 and 11.10 illustrate the variation of the Frank and Burger triangles as functions of the source location. Tables 11.2A and 11.2B compare the lead vectors for the Einthoven, Frank, and Burger triangles from two source locations.
Fig. 11.8. Einthoven (E), Frank (F), and Burger (B) triangles. Note that the Einthoven triangle lies in the frontal plane, whereas the Frank and Burger triangles are tilted out of the frontal plane. Only their frontal projections are illustrated here.
Fig. 11.9. Variation of the Frank triangle as a function of dipole location. The black circle in the miniature lead vector triangles arising from the Frank torso are superimposed on the site of the dipole origin. (From Hyttinen et al., 1988.).
Fig. 11.10. Variation of the Burger triangle as a function of source location. (From Hyttinen et al., 1988.).
Lead | Triangle | cx | cy | cz |
cI | Einthoven Frank Burger | -2.8 -31.2 | 100 100 100 | -1.8 -6.4 |
cII | Einthoven Frank Burger | 16 97 | 50 53 46 | -87 -88 -162 |
cIII | Einthoven Frank Burger | 19 135 | -50 -47 -57 | -87 -86 -163 |
Lead | Triangle | cx | cy | cz |
cI | Einthoven Frank Burger | -6.6 -3.0 | 100 100 100 | 12 -8.2 |
cII | Einthoven Frank Burger | 23 33 | 50 44 62 | -87 -117 -217 |
cIII | Einthoven Frank Burger | 30 30 | -50 -60 -39 | -87 -130 -209 |
![]() ![]() | (11.23) |
From Figure 11.12 it is easy to see the following relationship between the lead vectors 1, 2, and s:
![]() ![]() | (11.24) |
Therefore, the voltage, measured in the real space from the point P must fulfill the requirement:
![]() ![]() | (11.25) |
The point that fulfills this requirement in the real space can be found in the following way: We connect between the points P1 and P2 two resistors in series having the resistance ratio of a/b. The point P is at the interconnection of these resistors. (We must choose Ra and Rb large enough so that the current through this pathway has a negligible effect on 1 and 2.).
Fig. 11.12. Determination of the point P in the real space corresponding to an image space point P' located inside the image surface.
11.5.4 Application of the Image Surface to the Synthesis of Leads
In this section, we examine how the image surface concept can be applied to the identification of an unknown dipole inside the volume conductor from measurements on the surface. Our initial task is to synthesize an orthonormal lead system for the measurement of the dipole. The concept "orthonormal" denotes that a lead system is both (1) orthogonal and (2) normalized; that is, the three measured components of the dipole are orthogonal and their magnitudes are measured with equal sensitivity. That means, the lead voltage corresponding to equal-value components of the dipole source is the same.
To begin, we construct the image surface of the volume conductor in relation to the known location of the dipole. Now we want to find two points on the surface of the volume conductor such that the voltage between them is proportional only to the y-component of the dipole. Mathematically, this can be formulated as:
![]() ![]() ![]() | (11.26) |
and we seek a lead vector 21, which both lies in the image space and is oriented solely in the direction of the y-axis. This corresponds to identifying any pair of points on the image surface that are located at the intersections of a line directed parallel to the y-axis. The voltage measured between those points in real space is consequently proportional only to the y-component of the dipole. To obtain the largest possible signal (in order to minimize the noise), we select from all image space points that fulfill the requirement discussed above, the one with the longest segment (maximum lead vector), as illustrated in Figure 11.13.
If we want to measure all three orthogonal components of the dipole source, we repeat this procedure for the z and x directions. Because the resulting (maximum) lead vectors are usually not of equal length, we equalize the measured signals with a resistor network to obtain both an orthogonal and a normalized lead system. Such a normalizing procedure is described in Figure 11.13 for a two-dimensional system. In this case two resistors, RA and RB, form a simple voltage-divider, and the output voltage is reduced from the input by RB/(RA + RB). We choose this ratio to compensate for a lead vector amplitude that is too large. Note that the assumed voltage-divider behavior requires that the voltage-measuring circuit (amplifier) has a sufficiently high input impedance for negligible loading..
11.5.5 Image Surface of Homogeneous Human Torso
We consider here the image surface of the homogeneous human torso, as determined by Ernest Frank (1954). Frank constructed a tank model having the form of the thorax. It was oriented upside down because it was easier to insert and manipulate the source dipole from the larger opening of the model at the level of the abdomen. The model was filled with a salt solution and therefore formed a finite, homogeneous model.
Frank adopted the following coordinate system for the model: The model was divided into 12 levels with horizontal planes at increments of 5 cm (2 inches). The center of the heart was located on level 6 about 4 cm to the front of the plane located at the midline of the right and left arms, and about 2.5 cm to the left of the sagittal plane located at the midline of the model. On each horizontal plane, 16 points were established by drawing 8 lines through the midline of the model (within increments of 22.5°). The intersections of these lines on the surface of the model were labeled with letters A through P in a clockwise direction starting from the left side, as shown in Figure 11.14. Note that the coordinate axis nomenclature used here is not the same as that adopted by Frank, since the consistent coordinate system of the Appendix A has been used.
Figures 11.15, 11.16, and 11.17 illustrate the image surface measured by Frank in the three projections - the frontal, sagittal, and transverse planes. The figures also show the points corresponding to the Einthoven limb leads, which in this case form the Frank triangle.
11.5.6 Recent Image Surface Studies
In recent years the image surface for the human torso has been investigated using computer models. In these models one can introduce not only the effect of body shape but also inhomogeneities such as the lungs, intracavitary blood masses, surface muscle layers, and so on. One such study is that of Horá ek (1971), who included the effect of body shape, lungs, and intracavitary blood. Horá ek observed that the lungs and the intracavitary blood masses can substantially distort the image surface and consequently cause variations in the body surface potential distribution. However, because of the complexity of the effect, no simple universal statement can be made to describe the influence of the inhomogeneities.
Fig. 11.14. The Frank torso model and coordinate system. (The latter has been related to correspond with the system adopted in this text and discussed in the Appendix A.).
Fig. 11.15. The image surface of the Frank torso model in frontal view.
Fig. 11.16. The image surface of the Frank torso model in sagittal view.
Fig. 11.17. The image surface of the Frank torso model in transverse view.
A modified Horá ek model that includes the skeletal muscle was developed and studied by Gulrajani and Mailloux (1983). The latter authors chose to examine the effects of modifications introduced by inhomogeneities in terms of effects on body surface potentials rather than on the image surface per se.
![]() ![]() | (11.27) |
where the left-hand side of the equation denotes case 1 and the right-hand side case 2.
Demonstration of the Consistency of the Reciprocity Theorem
It is easy to demonstrate that Equation 11.27 does not depend on the area of the double layer source. This is illustrated by the following examples.
If we make the area of the double layer K times larger, the current IL through the galvanometer in case 1 is now (by the application of superposition) K times larger - that is, KIL. In case 2, the electromotive force Vd in the galvanometer wire remains the same, because it represents the (unchanged) voltage over the double layer source in case 1. Therefore, it still produces the same current density in the source area. But because the source area is now K times larger, the total current through it is also K times larger - that is, Kir. Consequently, Equation 11.27 becomes
![]() ![]() | (11.27A) |
and dividing both sides by K returns it to the expression arising from the original area. (In the above one should keep in mind that the original area A and KA are assumed to be very small so that ir and Vd can be considered uniform.)
11.6.4 Lead Field Theory: the Historical Approach
In this section we derive the basic equation of the lead field from the original formulation of Helmholtz (expressed by Equation 11.27) based on a description of current double layer source and lead voltage. As stated before, Helmholtz described the source as a voltage double layer element Vd, whose effect is evaluated by a measured lead current IL. Alternatively, as is done presently, the source may be described as a current dipole layer element i, whereas the signal is the lead voltage VL produced by it. We can directly obtain these expressions from those of Helmholtz by application of the principle of duality. The result, illustrated in Figure 11.21, is discussed below..
![]() | |
![]() | ![]() |
![]() |
Fig. 11.21. Derivation of the equation for lead field theory.
Since Helmholtz's theorem applies to a discrete source, we make the following assumptions:
The lateral extent of the voltage double layer element Vd is differential - that is, Ds.
The separation of the poles of the corresponding dipole element The conductivity at the source point is s.
The resistance of the galvanometer circuit between the measurement points equals R.
iDs is Dd, where
i is an applied current density so that
iDd has the dimensions of a double layer source.
In case 1 and case 2 we may further evaluate the following expressions. For case 1:
Instead of examining the reciprocal current density ir at the (removed) source point we can evaluate the related lead field current density The required voltage source Vd in the circuit connected to the conductor can be achieved if we use a reciprocal current source Ir = Vd/R, since then we have Vd = IrR.
L = ir/Ds. These are connected by ir = Ds ·
L. The dot product is required here because the current ir is the component of the reciprocal current flowing through the source area in the direction of the source
i. This can also be written
ir = DsL ·
i/|
i|.
![]() ![]() | (11.28) |
where the left-hand side of the Equation 11.28 denotes case 1 and the right-hand side case 2 in the Helmholtz procedure, respectively. Solving for the lead voltage VL in Equation 11.28, we obtain
![]() ![]() | (11.29) |
where Ds Dd = Dv, which is the volume element of the source. (In the limit, Dv
where
11.6.5 Field-Theoretic Proof of the Reciprocity Theorem
A brief explanation of Helmholtz's reciprocity theorem was given in Section 11.6.3, without offering a mathematical proof. That explanation was based on the ideas of the original publication of Helmholtz (1853). The field-theoretic proof of the reciprocity theorem as described by Plonsey (1963) is presented below.
Proof of the Reciprocity Theorem
Consider an arbitrary volume v bounded by the surface S and having a conductivity s (which may be a function of position). If F1 and F2 are any two scalar fields in v, the following vector identities must be satisfied:
If we subtract the second equation from the first one, integrate term by term over the volume v, and use the divergence theorem, we obtain
Since
(Thus IF is a flow source, as defined earlier in Equation 8.35.) We assume further that F2 is the scalar potential produced solely by current caused to cross the surface S with a current density J [A/m2]. Usually we assume that J flows from conducting electrodes of high conductivity compared with s, so that the direction of J is normal to the bounding surface. In this case J can be specified as a scalar corresponding to the flow into v. (The scalar potential F2 is later identified as the reciprocal electric scalar potential due to the reciprocal current Ir fed to the lead.) Since the current J is solenoidal, it satisfies
The scalar fields F1 and F2 satisfy the following equations:
since IF is a source of F1 and
since the field F2 is established by the applied current J.
For the source J at the surface, the current must be solenoidal everywhere in v; hence:
We may rewrite Equation 11.33 by substituting Equations 11.38 and 11.37 into its left-hand side, and Equations 11.38 and 11.36 into its right-hand side, obtaining
which is the desired form of the reciprocity theorem.
The Reciprocity Theorem of Helmholtz
The reciprocity theorem of Helmholtz can be derived from Equation 11.40 in the following way. Consider that F2 (the reciprocal electric scalar potential) arises from a particular distribution J, where an inflow of a unit (reciprocal) current is concentrated at point
where
where
If we choose I0 to be unity, then this equation shows that the voltage between two arbitrary surface points a and b due to a unit current supplied internally between points 1 and 2 equals the voltage between these same points 2 and 1 due to a unit current applied externally (reciprocally) between points a and b. This is essentially the reciprocity theorem of Helmholtz.
dv.) By extending Equation 11.29 throughout all source elements, and choosing the reciprocal current to be a unit current Ir = 1 A, we may write:
(11.30) LE denotes an electric lead field due to unit reciprocal current. Note that although
i was originally defined as a current density, it may also be interpreted as a volume dipole density, as is clear in Equation 11.30 and by their similar dimensions. Equation 11.30 is the most important equation in the lead field theory, as it describes the lead voltage (the electric signal in a lead) produced by an arbitrary volume source described by
i(x,y,z). It may be stated in words as follows:
To determine the lead voltage produced by a volume source, we first generate the lead field in the volume conductor by feeding a unit (reciprocal) current to the lead. Every element of the volume source contributes to the lead voltage a component equal to the scalar product of the lead field current density and the volume source element divided by the conductivity.
If the volume conductor is homogeneous throughout the source region, we may move the coefficient 1/s outside the integral and write:
(11.31) According to Equation 11.31, the lead field has an important property: it equals the lead sensitivity distribution. This means that at each point of the volume conductor, the absolute value of the lead field current density equals to the magnitude of the lead sensitivity, and the direction of the lead field current equals the direction of the lead sensitivity. It should be noted that the lead field fully takes into account the effect of the volume conductor boundary and internal inhomogeneities; hence these have an effect on the form of the lead field. (The concept of secondary sources is contained within lead field theory through the effect of the inhomogeneities on the form of the lead field.)
Lead field theory is a very powerful tool in bioelectromagnetism. It ties together the sensitivity distribution of the measurement of bioelectric sources, distribution of stimulation energy, and sensitivity distribution of impedance measurements, as is explained later. In general, if the lead and the volume conductor are known, the distribution of the lead sensitivity may be determined, based upon lead field theory. On the other hand, if the source and the volume conductor are known, the distribution of the actual field may be determined directly without using the lead field concept. All this holds for corresponding biomagnetic phenomena as well.
· F1 (s
F2 ) = F1
· (s
F2 ) + s
F1 ·
F2
(11.32) · F2 (s
F1 ) = F2
· (s
F1 ) + s
F2 ·
F1
(11.33) F1 ·
F2 =
F2 ·
F1 , these terms cancel out in deriving Equation 11.33 from 11.32. The derivation of Equation 11.33 is well known in the physical sciences; it is one of a number of forms of Green's theorem.
Now we assume that F1 is the scalar potential in volume v due to sources within it specified by the equation
IF = -
·
i
(11.34)
(11.35)
· (s
F1 ) = - IF
(11.36)
s
F2 · d
= J dS
(11.37) Since -s
F2 carries the direction of the current (=
L ) and d
is the outward surface normal, Equation 11.37 shows that for our chosen signs J is positive for an inflow of current. No current due to the source IF crosses the boundary surface (since in this case it is totally insulating), and hence
F1 · d
= 0
(11.38)
· (s
F2) = 0
(11.39)
(11.40) b on the surface and an outflow of a unit (reciprocal) current at point
a on the surface, where
a and
b are position vectors shown in Figure 11.22. Note that this is opposite to the current flow direction given in Plonsey (1963). (With this sign notation, current dipoles in the direction of the lead field current produce a positive signal in the lead, as will be seen later in Equation 11.50 (Malmivuo, 1976).)
The above can be expressed mathematically:
J =
s(
b -
) -
s(
a -
)
(11.41) s is a two-dimensional unit Dirac delta function on the bounding surface, and hence the magnitude of both inflow and outflow is unity.
Consider IF to consist of a point source of current I0 at
1 and a point sink of equal magnitude at
2, where
1 and
2 are the position vectors shown in Figure 11.22. Now IF can be written:
IF = I0[
v(
1 -
) -
v(
2 -
)]
(11.42) v is a three-dimensional Dirac delta function.
Substituting Equations 11.41 and 11.42 into Equation 11.40, we obtain
(11.43)
Deriving the Equations for Lead Field
The value of F2 (r1) can be specified in terms of the field at
Note that since the field F2 is established by currents introduced at the surface into a source-free region, it is well behaved internally and a Taylor series can always be generated. If we let (
Denoting the voltage between the points a and b as
and substituting Equations 11.45 and 11.46 into Equation 11.43, we obtain
Because the impressed current sources are totally contained within S, the integrand is zero everywhere on S, and the first term on the right-hand side of Equation 11.49 is zero; thus we obtain
The quantity -
where
11.6.6 Summary of the Lead Field Theory Equations
In this section we summarize the equations of the lead field theory for electric leads. (Equations for magnetic leads are given in the next chapter.) We consider the situation in Figure 11.23, where two disklike electrodes in a volume conductor form the bipolar electric lead.
This reciprocal current field s is defined as the lead field.
If the volume conductor is homogeneous, the conductivity s may be taken in front of the integral operation, and we obtain
2 by means of a Taylor series expansion:
F2 (
1) = F2 (
2) +
F2 · (
1 -
2) + . . .
(11.44) 1 -
2 ) approach zero and the current I0 approach infinity, such that their product remains constant, then a dipole moment of I0(
1 -
2 ) =
0 is created. Under these conditions the higher-order terms in Equation 11.44 can be neglected, and we obtain
I0[F2 (
2) - F2 (
1)] = - I0
F2 · (
1 -
2 ) = -
F2 ·
0
(11.45)
VLE = F1 (
a ) - F1 (
b )
(11.46)
VLE = -
F2 ·
0
(11.47) Note that -
F2 corresponds precisely to a description of the sensitivity distribution associated with this particular lead, and is in fact the lead vector (field). Since no assumption has been made concerning the volume conductor, we have found a powerful method for quantitatively evaluating lead vector fields of arbitrary leads on arbitrary shaped inhomogeneous volume conductors.
The actual bioelectric sources may be characterized as a volume distribution
i with dimensions of current dipole moment per unit volume. Equation 11.47 may be generalized to the case of such a volume distribution of current dipoles with a dipole moment density of
i to obtain
(11.48) The quantity F2 was earlier defined as the reciprocal electric potential field in the volume conductor due to unit reciprocal current flow in the pickup leads a and b and is designated in the following as FLE. Plonsey (1963) has termed this potential field as the lead field in his field-theoretic proof of the reciprocity theorem. In this text, however, the term "lead field" denotes the current density field due to reciprocal application of current in the lead. They are related, of course, by
LE = -s
FLE.
Using the vector identity
· (FLE
i ) =
FLE ·
i + FLE
·
i and the divergence theorem, we obtain from Equation 11.48
(11.49)
(11.50) ·
i is the strength of the impressed current source and is called the flow (or flux) source IF as defined in Equation 8.35. Thus Equation 11.50 can be expressed as
(11.51) McFee and Johnston (1953) designated the vector field
LE = -s
F = s
LE the lead field. Here the symbol ELE denotes the reciprocal electric field due to unit reciprocal current, and the conductivity of the volume conductor. Using this formulation, we may rewrite Equation 11.48 as:
(11.52) LE is the lead field arising from unit reciprocal current (the reader should review the definition of J in Equation 11.41). But Equation 11.52 corresponds precisely to Equation 11.30 (assuming Ir = 1 [A]). Consequently, Equation 11.52 confirms Equation 11.30, which is the equation characterizing lead field theory, introduced earlier.
To determine the lead field, a unit reciprocal current Ir is fed to the lead. It generates a reciprocal electric potential field FLE in the volume conductor (this potential field was defined as F2 in Section 11.6.5 in the proof of the reciprocity theorem). If the electrodes are parallel and their lateral dimensions are large compared to their separation, FLE is uniform in the central region. The negative gradient of this electric potential field FLE is the reciprocal electric field,
LE :
LE =
FLE
(11.53) The reciprocal electric field is related to the reciprocal current field by the conductivity of the medium:
LE = s
LE
(11.54) Now, when we know the lead field
LE, we can remove the reciprocal current generator (of unit current) from the lead. The electric signal VLE in the lead due to current sources
i in the volume conductor is obtained from the equation
(11.30)
(11.31) Section 11.6.1 introduced the concept of isosensitivity surface and its special case half-sensitivity surface which bounds a half-sensitivity volume. The isosensitivity surfaces, including the half-sensitivity surface, are surfaces where the lead field current density
LE is constant. In a homogeneous region of a volume conductor, where s is constant, the isosensitivity surfaces are, of course, surfaces where the reciprocal electric field
LE is constant. In certain cases the isosensitivity surfaces coincide with the isopotential surfaces. These cases include those, where all isopotential surfaces are parallel planes, concentric cylinders, or concentric spheres. Then the surfaces where the electric field is constant (i.e. where two adjoining isopotential surfaces are separated by a constant distance) have the same form as well. But in a general case, where the isopotential surfaces are irregular so that two adjacent surfaces are not a constant distance apart the surfaces of constant electric field do not have the same form.
As summarized in Figure 11.23, as a consequence of the reciprocal energization of an electric lead, the following three fields are created in the volume conductor: electric potential field FLE (illustrated with isopotential surfaces), electric field
LE (illustrated with field lines) and current field
LE (illustrated with current flow lines and called the lead field). In addition to these three fields we defined a fourth field of surfaces (or lines): the field of isosensitivity surfaces. When the conductivity is isotropic, the electric field lines coincide with the current flow lines. In a symmetric case where all isopotential surfaces are parallel planes, concentric cylinders, or concentric spheres, the isopotential surfaces and the isosensitivity surfaces coincide.
In an ideal lead field for detecting the equivalent dipole moment of a volume source (see the following section) the isopotential surfaces are parallel planes. To achieve this situation, the volume conductor must also be homogeneous. Thus, in such a case from the aforementioned four fields, the electric field lines coincide with the lead field flow lines and the isopotential surfaces coincide with the isosensitivity surfaces..
Fig. 11.23. Basic form of a bipolar electric lead, where
Ir | = unit reciprocal current; | |
FLE | = reciprocal electric scalar potential field; | |
![]() | = reciprocal electric field; | |
![]() | = lead field: | |
VLE | = voltage in the lead due to the volume source ![]() | |
s | = conductivity of the medium. |
11.6.7 Ideal Lead Field of a Lead Detecting the Equivalent Electric Dipole of a Volume Source
PRECONDITIONS:
SOURCE: Volume source
CONDUCTOR: Infinite, homogeneous
In this section we determine the desired form of the lead field of a detector that measures the equivalent (resultant) electric dipole moment of a distributed volume source located in an infinite homogeneous volume conductor.
As discussed in Section 7.3.2, a dipole in a fixed location has three independent variables, the magnitudes of the x-, y-, and z-components. These can be measured with either unipolar or bipolar electrodes locating at the coordinate axes. The vectorial sum of these measurements is the dipole moment of the dipole.
Because a volume source is formed from a distribution of dipole elements, it follows from the principle of superposition that the dipole moment of a volume source equals to the sum of the dipole moments of its dipole elements. This can be determined by measuring the x-, y-, and z-components of all the elementary dipoles and their sums are the x-, y-, and z-components of the equivalent dipole moment of the volume source, respectively. To introduce the important equations we show this fact also in mathematical form.
The equivalent electric dipole moment of a volume source may be evaluated from its flow source description. It was shown in Equation 8.35 (Section 8.5) that the flow source density IF is defined by the impressed current density (Plonsey, 1971) as
![]() ![]() ![]() | (8.35) |
The resultant (electric) dipole moment of such a source can be shown to be
![]() ![]() | (11.55) |
This dipole moment has three components. Because = x
+ y
+ z
, these three components may be written as:
![]() ![]() | (11.56) |
We consider the x-component of the dipole moment. Noting Equation 8.35 and using the vector identity
· (x
i) = x
·
i +
i ·
x, we obtain
![]() ![]() | (11.57) |
Using the divergence theorem, we may rewrite the first term on the right-hand side of Equation 11.57 as
![]() ![]() | (11.58) |
Since there can be no impressed current density on the surface, this term vanishes. Therefore, and because x =
(
x/
x) +
(
y/
y) +
(
z/
z) =
i , we obtain for the x-component of the dipole moment
![]() ![]() | (11.59) |
In fact, recalling the dual identity of The lead field current density is given by The lead field current density is uniform throughout the source area.
Three such identical, mutually perpendicular lead fields form the three orthogonal components of a complete lead system.
i as a dipole moment per unit volume, we can write Equation 11.59 directly.
Equation 11.59 can be described as follows: one component of the equivalent electric dipole (moment) of a volume source may be evaluated from the sum of corresponding components of the distributed dipole elements of the volume source independent of their location. A comparison of Equation 11.59 with Equation 11.49 identifies FL with x. Consequently, we see that this summation is, in fact, accomplished with a lead system with the following properties (see Figure 11.24):
L =
x =
(so that it is everywhere in the x direction only).
Fig. 11.24. Ideal lead field (sensitivity distribution) for detecting the electric dipole moment of a volume source. Each component is uniform in one direction throughout the source region, and the components are mutually orthogonal.
(A) Lead field current density vector presentation.
(B) Lead field current flow line presentation.
This is the physiological meaning of the measurement of the electric dipole. (See the text for details.)
Physiological Meaning of Electric Dipole
The sensitivity distribution (i.e., the lead field), illustrated in Figure 11.24, is the physiological meaning of the measurement of the (equivalent) electric dipole of a volume source.
The concept "physiological meaning" can be explained as follows: When considering the forward problem, the lead field illustrates what is the contribution (effect) of each active cell on the signals of the lead system. When one is considering the inverse problem, the lead field illustrates similarly the most probable distribution and orientation of active cells when a signal is detected in a lead..11.6.8 Application of Lead Field Theory to the Einthoven Limb Leads
PRECONDITIONS:
SOURCE: Volume source
CONDUCTOR: Infinite, homogeneous
To build a bridge between lead field and lead vector and to clarify the result of Equation 11.59 illustrated in Figure 11.24, we apply lead field theory to the Einthoven limb leads.
Previously, in Section 11.4.3, the Einthoven triangle was discussed as an application of the lead vector concept. The volume source of the heart was modeled with a (two-dimensional) dipole in the frontal plane. It was shown that the signal in each limb lead VI, VII, and VIII is proportional to the projections of the equivalent dipole on the corresponding lead vectors.
Instead of modeling the volume source of the heart with the resultant of its dipole elements, we could have determined the contribution of each dipole element to the limb leads and summed up these contributions. In this procedure one can use the lead field theory to illustrate the lead fields - that is, the sensitivities of the limb leads. The idealized lead fields of the limb leads are uniform in the directions of the edges of the Einthoven triangle. Figure 11.25 illustrates the sensitivity distribution of the (ideal) Einthoven limb leads within the area of the heart.
This is the physiological meaning of the measurement of the Einthoven limb leads (see the previous section).
Fig. 11.25. The ideal lead field (sensitivity distribution) of Einthoven limb leads VI, VII, and VIII. This is the physiological meaning of the measurement of the limb leads.
11.6.9 Synthesization of the Ideal Lead Field for the Detection of the Electric Dipole Moment of a Volume Source
Synthesization of the Ideal Lead Fields in Infinite, Homogeneous Volume Conductors
PRECONDITIONS:
SOURCE: Volume source
CONDUCTOR: Infinite, homogeneous
We begin the discussion on synthesis of ideal lead fields for detecting the equivalent dipole moment of a volume source by discussing the properties of unipolar and bipolar leads in infinite, homogeneous volume conductors.
If the dimensions of a distributed volume source are small in relation to the distance to the point of observation, we can consider it to be a lumped (discrete) dipole. The detection of such an electric dipole is possible to accomplish through unipolar measurements on each coordinate axis, as illustrated on the left hand side of Figure 11.26A. If the dimensions of the distributed volume source are large in relation to the measurement distance, the lead field of a unipolar measurement is not directed in the desired direction in different areas of the volume source and the magnitude of the sensitivity is larger in the areas closer to the electrode than farther away. This is illustrated on the right hand side of Figure 11.26A.
The quality of the lead field both in its direction and its magnitude is considerably improved when using a bipolar lead, where the electrodes are located symmetrically on both sides of the volume source, as illustrated in Figure 11.26B. (Note also that in the bipolar measurement the difference in potential between the electrodes is twice the unipolar potential relative to the center.)
The quality of the lead field of a bipolar lead in measuring volume sources with large dimensions is further increased by using large electrodes, whose dimensions are comparable to the source dimensions. This is illustrated in Figure 11.26C.
ELECTRODE CONFIGURATION | LEAD FIELD OF ONE COMPONENT |
![]() |
Fig. 11.26. Properties of unipolar and bipolar leads in detecting the equivalent electric dipole moment of a volume source.
(A) If the dimensions of the volume source are small compared to the measurement distance the simplest method is to use point electrodes and unipolar leads on the coordinate axes.
(B) For volume sources with large dimensions the quality of the lead field is considerably improved with the application of bipolar leads.
(C) Increasing the size of the electrodes further improves the quality of the leads.
Synthesization of the Ideal Lead Fields in Finite, Homogeneous Volume Conductors
PRECONDITIONS:
SOURCE: Volume source
CONDUCTOR: Finite, homogeneous
Using large electrodes is in practice impossible. In the following we describe a method to design a lead to detect the equivalent electric dipole moment of a volume source in a finite, homogeneous volume conductor of arbitrary shape (Brody, 1957).
According to Section 11.6.7, such a lead, when energized reciprocally, produces three orthogonal, uniform, and homogeneous lead fields. We consider the construction of one of them. This may be done according to the following steps:
Suppose that the volume conductor has the arbitrary shape shown in Figure 11.27A and that our purpose is to synthesize an ideal lead field in the y direction within this region.
We extend the volume conductor in the direction of the y-axis in both directions so that it forms a cylinder limited by two planes in the zx direction and having the cross section of the original volume conductor (Figure 11.27B).
Then we plate the end planes of the cylinder with a well-conducting material. If electrodes are connected to these plates and a reciprocal current is fed to them, an ideal lead field is created in the volume conductor (Figure 11.27B).
Thereafter the extension of the volume conductor is slit as described graphically in Figure 11.27C, generating isolated "fibers." These cuts do not modify the form of the lead field because they are made along the flow lines which are nowhere intersected, as is clear in Figure 11.27C.
Each of the volume conductor "fibers" may now be replaced with discrete resistances of equal resistive value, as illustrated in Figure 11.27D.
11.6.10 Special Properties of Electric Lead Fields
Two special properties of the lead fields are summarized as follows:
If the volume conductor is cut or an inhomogeneity boundary is inserted along a lead field current flow line, the form of the lead field does not change. Only the intensity of the field changes in relation to the conductivity.
The reciprocity theorem may be applied to the reciprocal situation. This means that it is possible in electrolytic tank models to feed a "reciprocally reciprocal" current to the dipole in the conductor and to measure the signal from the lead and interpret the result as having been obtained by feeding the reciprocal current to the lead and measuring the signal from the dipole.
The latter is easily proved by imagining that the lead field is a result of the mapping of the behavior of the lead vector as a function of the source location, as discussed in Section 11.6.2. This mapping is done by feeding unit currents in each coordinate direction at each point of the source area and by measuring the corresponding voltages at the lead, as explained in Section 11.4.1.
The benefit of this "reciprocally reciprocal" arrangement is that for technical reasons, the signal-to-noise ratio of the measurement may be improved while still having the advantage of the interpretations associated with the lead field current distribution.
The special properties of electric lead fields are discussed in more detail in connection with magnetic lead fields.
11.6.11 Relationship Between the Image Surface and the Lead Field
In this section, the relationship between the image surface and the lead field is described with the aid of Table 11.3 and Figure 11.28.
The source in the concept of the image surface is a dipole. This can be a discrete dipole (at a point), or it can be a dipole element of a distributed volume source. In the lead field concept, the source may be a distributed volume source or a discrete dipole. The conductor in both cases was previously considered to be finite (and inhomogeneous). However, the theory holds for infinite volume conductors as well.
Source location in the image surface was fixed and the measurement points were variable and forming a continuum. In characterizing the lead field, we note that the situation is the opposite: The measurement points are fixed while the source point varies (continuously). This means that in the image surface the lead vectors are mapped as a function of the measurement point, but in the lead field the mapping is a function of the source point. The image surface takes into account field points lying on a surface, whereas in the lead field the source point may lie within a three-dimensional volume.
Geometrically, in the image surface concept, the ends of the lead vectors form the image surface. In the lead field concept, the field of lead vectors establish the lead field.
The equations for the application of the lead vector and image surface (Equation 11.16) and the lead field (Equations 11.30 and 11.31) are, in principle, of the same form. The main difference is that the equation for the lead field is in integral form. This comes from the fact that it is applied to a volume source.
An important consequence of the reciprocity theorem of Helmholtz is that the lead field is identical to the current field resulting from feeding the (unit) reciprocal current to the lead.
| ||||||||||||||||||||||||||||||||||
Note: | (1) There is similarity between the variables: ![]() ![]() ![]() ![]() ![]() ![]() | |||||||||||||||||||||||||||||||||
(2) It follows from the reciprocity theorem that ![]() in the volume conductor due to feeding the reciprocal current Ir of 1 A to the lead. |
![]() ![]() ![]() | (8.35) |
and the resulting (electric) dipole moment of such a system is evaluated from the definition (Equation 11.55)
![]() ![]() | (11.55) |
where | ![]() | = the radius vector |
dv | = the volume element |
The dipole moment has three components, as was illustrated by Equation 11.56. We now examine the x-component of this dipole moment. We develop it in the following way: The explanation for each step is given on the right-hand side of the column.
px | ![]() | from Equation 11.57 |
![]() | because IF = ![]() ![]() | |
![]() | from Equation 7.3 we have ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
![]() | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
![]() | ![]() | |
![]() | because the volume integral may be transformed to a surface integral by integrating with respect to x | |
![]() | because the surface integral may be written in a more convenient form by using a vectorial surface element dx whose absolute value is dS = dy dz, and which is directed outward and normal to the surface defined by dy dz. |
which expresses the resultant dipole moment of a volume source in an arbitrary volume conductor.
Summing Equation 11.61 and similar expressions for py and pz and replacing the potential F with voltage V, we finally obtain the vector equation
(11.62) We now explain in detail the meaning of Equation 11.62, as illustrated in Figure 11.29.
Figure 11.29A illustrates the homogeneous volume conductor including the volume source. In the illustration the Gabor-Nelson theorem is discussed in two dimensions. The equivalent dipole moment of the volume source is
. The vectorial surface element d
is a vector attached to the surface element. It is directed outward and normal to the surface element, and its absolute value equals the area of the surface element. For clarity the volume conductor is divided into 12 surface elements, S1 through S12. (When applying Equation 11.62, of course, one assumes that the number of surface elements is infinite.) A vector d
1 through d
12 is attached to each surface element. The volume source produces a potential, F1 through F12, at each surface element.
It is obvious that because the surface is closed, the sum of the vectorial surface elements is equal to zero; that is, Sd
i = 0 (Figure 11.29B).
If we multiply each vectorial surface element d
i by the corresponding potential Fi (or actually with the voltage Vi measured at each surface element in relation to an indifferent reference), the sum of these products, SVid
i, is no longer zero. It is clear that when one is considering the surface potential due to the dipole
along the surface elements of increasing index, one finds that it is at its maximum at the surface elements S1 and S2. Then it decreases and reaches the value zero somewhere between the surface elements S4 and S5. Thereafter the surface potential turns to negative polarity and reaches its maximum at the surface element S7. Thereafter the (negative) surface potential decreases to zero and increases again to the positive maximum in the area of S1. Therefore, the sum SVid
i is not zero; and according to the Equation 11.62, if the number of the surface elements is infinite, one obtains
. For clarity, the length of
is shown longer in Figure 11.29C than in Figure 11.29A..
Fig. 11.29. Illustration of the Gabor-Nelson equation for evaluating the resultant (equivalent) dipole of a volume distribution lying in a bounded homogeneous volume conductor. The integral is shown approximated by discretizing the surface into 12 elements. The calculation of the integral is explained in detail in the text.
11.7.2 The Location of the Equivalent Dipole
Next we describe the procedure for finding the position of the resultant dipole. If we actually had an equal point source and sink, +I and -I, located at points
![]() ![]() |
the second moment of the source distribution is in the x direction and given by
![]() ![]() | (11.63) |
where upper-case X denotes the x-coordinate of the dipole location and lower-case x is the variable in this coordinate.
Integrating by parts with respect to x and again replacing the potential F with the measured voltage V gives
Two similar equations arise by cyclic permutation of the coordinates. We can also derive three new equations of the type
in the same manner as Equations 11.61 and 11.66 were derived. We can now eliminate the unknown volume integral of V from the equations of the type 11.66, and, together with the three equations of the type 11.68, we are left with the five equations for the three quantities X, Y, and Z. Any three of these five equations can be used for finding the location of the dipole, and the other two for checking how well the assumption of one dipole accounts for the observation. One can also use the method of least squares to obtain the best fit.
11.8 SUMMARY OF THE THEORETICAL METHODS FOR ANALYZING VOLUME SOURCES AND VOLUME CONDUCTORS
We have discussed six different theoretical methods for analyzing volume sources and volume conductors. Two of them are used for solving the forward problem, and the other four for solving the inverse problem. These methods are:
In various cases we had the following sources:
These sources have been located in volume conductors that were:
In the limit Dx
0 and IDx
px we obtain
(11.64) We now transform this second moment integral, following the same steps as with the first moment, namely
Xpx
(11.65)
(11.66) In a similar way, we obtain equations for Ypy and Zpz. It is obvious that we cannot determine any of these by surface measurements alone because the second term in each expression requires a volume integral of the potential V. However, in the same way as one obtains Equation 11.66, one can show that
(11.67)
(11.68)
Solid angle theorem
Miller-Geselowitz model
Lead vector
Image surface
Lead field
Gabor-Nelson theorem
Double layer
Distributed dipole
Dipole (in a fixed location)
Moving dipole
Dipole moment of a volume source
Multiple dipole
Multipole
Infinite, homogeneous
(Infinite, inhomogeneous, not discussed)
Finite, homogeneous
Finite, inhomogeneous
The application of each method is limited to certain source-conductor combinations, as expressed in the sets of preconditions in connection with the discussion of each method. We summarize these preconditions in Figure 11.30. The former one of these shows the application areas for the two methods used in solving the forward problem, and the latter one for those used in solving the inverse problem.
Figure 11.30A is quite obvious. The preconditions of the two methods are shown by locating the methods in the corresponding location in the source-conductor plane. The application area of the solid angle theorem is shown to be both the infinite homogeneous volume conductor, as derived first by Helmholtz, and the finite homogeneous and inhomogeneous conductors, where it can be extended with the concept of secondary sources.
Figure 11.30B needs some clarification, and certainly some details of this figure could perhaps be presented also in some other way.
The source for the lead vector and image surface methods is dipole. In Sections 11.4 and 11.5, these methods were discussed, for simplicity, in connection with finite conductors. There is, however, no theoretical reason, that would restrict their application only to finite conductors, but they are applicable to infinite conductors as well. Therefore, their application area is shown for both finite and infinite conductors, but more light shaded in infinite conductors.
The same holds also for the lead field theory. Section 11.6.4 did not discuss the application of the lead field theory for a multiple dipole or multipole source. The lead field theory may, however, be applied also in connection with these sources. Therefore, they are included into the application area but with lighter shading.
The application area of the Gabor-Nelson theorem is clear. It can be applied for solving the dipole moment of a single dipole or a volume source in a finite homogeneous volume conductor. It also gives the location of this dipole moment..
Fig. 11.30. (A) The source-conductor combinations where the solid angle theorem and Miller-Geselowitz model may be applied in solving the forward problem. (B) The source-conductor combinations where the lead vector, image surface, and lead field methods as well as Gabor-Nelson theorem may be applied in solving the inverse problem.