12
Theory of Biomagnetic Measurements
The current density throughout a volume conductor gives rise to a magnetic field given by the following relationship (Stratton, 1941; Jackson, 1975):
(12.01) 
where r is the distance from an external field point at which is evaluated to an element of volume dv inside the body, dv is a source element, and is an operator with respect to the source coordinates. Substituting Equation 7.2, which is repeated here,
(7.02) 
into Equation 12.1 and dividing the inhomogeneous volume conductor into homogeneous regions v_{j} with conductivity σ_{j}, we obtain
(12.02) 
If the vector identity Φ = Φ + Φ is used, then the integrand of the last term in Equation 12.2 can be written σ_{j} [Φ (1/r)]  Φ (1/r). Since Φ = 0 for any Φ, we may replace the last term including its sign by
(12.03) 
We now make use of the following vector identity (Stratton, 1941, p. 604):
(12.04) 
where the surface integral is taken over the surface S bounding the volume v of the volume integral. By applying 12.4 to Equation 12.3, the last term in Equation 12.2, including its sign, can now be replaced by
(12.05) 
Finally, applying this result to Equation 12.2 and denoting again the primed and doubleprimed regions of conductivity to be inside and outside a boundary, respectively, and orienting d_{j} from the primed to doubleprimed region, we obtain (note that each interface arises twice, once as the surface of v_{j} and secondly from surfaces of each neighboring region of v_{j} )
(12.06) 
This equation describes the magnetic field outside a finite volume conductor containing internal (electric) volume sources ^{i} and inhomogeneities (σ"_{j}  σ'_{j} ). It was first derived by David Geselowitz (Geselowitz, 1970).
It is important to notice that the first term on the righthand side of Equation 12.6, involving ^{i}, represents the contribution of the volume source, and the second term the effect of the boundaries and inhomogeneities. The impressed source ^{i} arises from cellular activity and hence has diagnostic value whereas the second term can be considered a distortion due to the inhomogeneities of the volume conductor. These very same sources were identified earlier when the electric field generated by them was being evaluated (see Equation 7.10). (Just, as in the electric case, these terms are also referred to as primary source and secondary source.)
Similarly, as discussed in connection with Equation 7.10, it is easy to recognize that if the volume conductor is homogeneous, the differences (σ"_{j}  σ'_{j} ) in the second expression are zero, and it drops out. Then the equation reduces to the equation of the magnetic field due to the distribution of a volume source in a homogeneous volume conductor. This is introduced later as Equation 12.20. In the design of highquality biomagnetic instrumentation, the goal is to cancel the effect of the secondary sources to the extent possible.
From an examination of Equation 12.6 one can conclude that the discontinuity in conductivity is equivalent to a secondary surface source _{j} given by _{j} = (σ"_{j}  σ'_{j} )Φ where Φ is the surface potential on S_{j}. Note that _{j} is the same secondary current source for electric fields (note Equation 7.10) as for magnetic fields.
where  I_{r}  = reciprocal current 
Φ_{LM}  = reciprocal magnetic scalar potential field  
_{LM}  = reciprocal magnetic field  
_{LM}  = reciprocal magnetic induction field  
_{LM}  = reciprocal electric field  
_{LM}  = lead field  
V_{LM}  = voltage in the lead due to the volume source ^{i} in the volume conductor  
μ  = magnetic permeability of the medium  
σ  = conductivity of the medium  
= radius vector. 
An unbounded homogeneous medium is required for the conductivity to be dual to the magnetic permeability, where the latter is uniform in the body and in space. As in electric measurements, it is possible to create compound magnetic leads by connecting any number of detectors together.
We investigate now the nature of the magnetic lead field _{LM} produced by reciprocal energization of the coil of the magnetic detector with a current I_{r} at an angular frequency . Using the same sign convention between the energizing current and the measured voltage as in the electric case, Figure 11.23, we obtain the corresponding situation for magnetic measurements, as illustrated in Figure 12.2.
The reciprocal magnetic field _{LM} arising from the magnetic scalar potential Φ_{LM} has the form:
_{LM} = Φ_{LM}  (12.07) 
The reciprocal magnetic induction _{LM} is
_{LM} = μ_{LM}  (12.08) 
where µ is the magnetic permeability of the medium. We assume µ to be uniform (a constant), reflecting the assumed absence of discrete magnetic materials.
The reciprocal electric field _{LM} arising from the reciprocal magnetic induction _{LM} (resulting from the energized coil) depends on the field and volume conductor configuration. For a magnetic field that is axially symmetric and uniform within some bounded region (cylindrically symmetric situation), 2πrE_{φ} = π^{2}B_{z} within that region ( Φ and z being in cylindrical coordinates), or in vector notation:
(12.09) 
In this equation is a radius vector in cylindrical coordinates measured from the symmetry axis (z) as the origin. As before, harmonic conditions are assumed so that all field quantities are complex phasors. In addition, as noted before, I_{r}(ω) is adjusted so that the magnitude of B_{LM} is independent of ω. The 90degree phase lag of the electric field relative to the magnetic field,is assumed to be contained in the electric field phasor. The field configuration assumed above should be a reasonable approximation for practical reciprocal fields established by magnetic field detector.
The result in Equation 12.9 corresponds to the reciprocal electric field _{LE} = Φ produced by the reciprocal energization of an electric lead (described in Equation 11.53 in the previous chapter).
The magnetic lead field current density may be calculated from Equation 12.9. Since
_{LM} = σ_{LM}  (12.10) 
we obtain for the magnetic lead field _{LM}
(12.11) 
As before, the quantity _{LM} is the magnetic induction due to the reciprocal energization at a frequency ω of the pickup lead.
This magnetic lead field _{LM} has the following properties:
Based upon Equation 11.30 and noting that also in the magnetic case the reciprocal current I_{r} is normalized so that it is unity for all values of ω, we evaluate the voltage V_{LM} in the magnetic lead produced by a current dipole moment density ^{i} as (Plonsey, 1972)
(12.12) 
This equation is similar to Equation 11.30, which describes the sensitivity distribution of electric leads. The sensitivity distribution of a magnetic measurement is, however, different from that of the electric measurement because the magnetic lead field _{LM} has a different form from that of the electric lead field _{LE}.
In the material above, we assumed that the conducting medium is uniform and infinite in extent. This discussion holds also for a uniform cylindrical conducting medium of finite radius if the reciprocally energized magnetic field is uniform and in the direction of the symmetry axis. This comes about because the concentric circular direction of _{LM} in the unbounded case is not interfered with when the finite cylinder boundary is introduced. As in the infinite medium case, the lead field current magnitude is proportional to the distance r from the symmetry axis. On the axis of symmetry, the lead field current density is zero, and therefore, it is called the zero sensitivity line (Eskola, 1983; Eskola and Malmivuo, 1983).
The form of the magnetic lead field is illustrated in detail in Figure 12.3. For comparison, the magnetic lead field is illustrated in this figure with four different methods. Figure 12.3A shows the magnetic lead field current density in a perspective threedimensional form with the lead field flow lines oriented tangentially around the symmetry axis. As noted before, because the lead field current density is proportional to the radial distance r from the symmetry axis, the symmetry axis is at the same time a zero sensitivity line. Figure 12.3B shows the projection of the lead field on a plane transverse to the axis. The flow lines are usually drawn so that a fixed amount of current is assumed to flow between two flow lines. Thus the flow line density is proportional to the current density. (In this case, the lead field current has a component normal to the plane of illustration, the flow lines are discontinuous, and some inaccuracy is introduced into the illustration, as may be seen later in Section 13.4.) Figure 12.3C illustrates the lead field with current density vectors, which are located at corners of a regular grid. Finally, Figure 12.3D shows the magnitude of the lead field current density J_{LM} as a function of the radial distance r from the symmetry axis with the distance from the magnetometer h as a parameter. This illustration does not show the direction of the lead field current density, but it is known that it is tangentially oriented. In Figure 12.3E the dashed lines join the points where the lead field current density has the same value, thus they are called the isosensitivity lines.
The relative directions of the magnetic field and the induced currents and detected signal are sketched in Figure 12.2. If the reciprocal magnetic field _{LM} of Equation 12.11 is uniform and in the negative coordinate direction, as in Figure 12.2, the form of the resulting lead field current density _{LM} is tangential and oriented in the positive direction of rotation. It should be remembered that harmonic conditions have been assumed so that since we are plotting the peak magnitude of _{LM} versus _{LM}, the sign chosen for each vector class is arbitrary. The instantaneous relationship can be found from Equation 12.11, if the explicit phasor notation is restored, including the 90degree phase lag of _{LM}.


(12.13) 
Using the vector identity (Φ_{LM}^{i }) = Φ_{LM} (^{i }) + Φ_{LM}(^{i }), we obtain from Equation 12.13
(12.14) 
Applying the divergence theorem to the first term on the righthand side and using a vector expansion (i.e., (^{i }) = ^{i }  ^{i }) on the second term of Equation 12.14, and noting that = 0, we obtain
(12.15) 
Since ^{i} = 0 at the boundary of the medium, the surface integral equals zero, and we may write
(12.16) 
This equation corresponds to Equation 11.50 in electric measurements. The quantity Φ_{LM} is the magnetic scalar potential in the volume conductor due to the reciprocal energization of the pickup lead. The expression ^{i} is defined as the vortex source, _{v} :
_{v} = ^{i}  (12.17) 
In Equation 12.16 this is the strength of the magnetic field source.
The designation of vortex to this source arises out of the definition of curl. The latter is the circulation per unit area, that is:
(12.18) 
and the line integral is taken around ΔS at any point in the region of interest such that it is oriented in the field to maximize the integral (which designates the direction of the curl).
If one considers the velocity field associated with a volume of water in a container, then its flow source must be zero if water is neither added nor withdrawn. But the field is not necessarily zero in the absence of flow source because the water can be stirred up, thereby creating a nonzero field. But the vortex thus created leads to a nonzero curl since there obviously exists a circulation. This explains the use of the term "vortex" as well as its important role as the source of a field independent of the flow source.
Table 12.1. The equations for electric and magnetic leads  
Quantity  Electric lead  Magnetic lead  
Field as a negative gradient of the scalar potential of the reciprocal energization 
_{LE} =  Φ_{LE}  (11.53)  _{LM} =  Φ_{LM}  (12.7)  
Magnetic induction due to reciprocal energization 
_{LM} = μ_{LM}  (12.8)  
Reciprocal electric field *)  _{LE}( =  Φ_{LE})  (11.53) 

(12.9)  
Lead field (current field)  _{LE} = σ_{LE}  (11.54)  _{LM} = σ_{LM}  (12.10)  
Detected signal when: I_{RE} = 1 A, dI_{RM}/dt = 1 A/s 
(11.30)  (12.12)  
*) Note: The essential difference between the electric and magnetic lead fields is explained as follows: The reciprocal electric field of the electric lead has the form of the negative gradient of the electric scalar potential (as explained on the first line of this table). The reciprocal electric field of the magnetic lead has the form of the curl of the negative gradient of the magnetic scalar potential. (In both cases, the lead field, which is defined as the current field, is obtained from the reciprocal electric field by multiplying by the conductivity.) Numbers in parentheses refer to equation numbers in text. 
The magnetic dipole moment of a volume current distribution with respect to an arbitrary origin is defined as (Stratton, 1941):
(12.19) 
where is a radius vector from the origin. The magnetic dipole moment of the total current density , which includes a distributed volume current source ^{i} and its conduction current,
= ^{i}  σΦ  (7.2) 
is consequently
(12.20) 
Assuming σ to be piecewise constant, we may use the vector identity Φ = Φ + Φ = Φ (because = 0), and convert the second term on the righthand side of Equation 12.20 to the form:
(12.21) 
We now apply Equation 12.4 to 12.21 and note that the volume and hence surface integrals must be calculated in a piecewise manner for each region where σ takes on a different value. Summing these integrals and designating the value of conductivity σ with primed and doubleprimed symbols for the inside and outside of each boundary, we finally obtain from Equation 12.20:
(12.22) 
This equation gives the magnetic dipole moment of a volume source ^{i} located in a finite inhomogeneous volume conductor. As in Equation 12.6, the first term on the righthand side of Equation 12.22 represents the contribution of the volume source, and the second term the contribution of the boundaries between regions of different conductivity. This equation was first derived by David Geselowitz (Geselowitz, 1970).
This section develops the form of the lead field for a detector that detects the equivalent magnetic dipole moment of a distributed volume source located in an infinite (or spherical) homogeneous volume conductor. We first have to choose the origin; we select this at the center of the source. (The selection of the origin is necessary, because of the factor r in the equation of the magnetic dipole moment, Equation 12.22.)
The total magnetic dipole moment of a volume source is evaluated in Equation 12.20 as a volume integral. We notice from this equation that a magnetic dipole moment density function is given by the integrand, namely
(12.23) 
Equation 12.14 provides a relationship between the (magnetic) lead voltage and the current source distribution ^{i}, namely
(12.24) 
Substituting Equation 12.23 into Equation 12.24 gives the desired relationship between the lead voltage and magnetic dipole moment density, namely
(12.25) 
This equation may be expressed in words as follows:
One component of the magnetic dipole moment of a volume source is obtained with a detector which, when energized, produces a homogeneous reciprocal magnetic field _{LM} in the negative direction of the coordinate axis in the region of the volume source.
This reciprocal magnetic field produces a reciprocal electric field _{LM} = ½_{LM} and a magnetic lead field _{LM} = σ_{LM} in the direction tangential to the symmetry axis.
Three such identical mutually perpendicular lead fields form the three orthogonal components of a complete lead system detecting the magnetic dipole moment of a volume source.
Figure 12.4 presents the principle of a lead system detecting the magnetic dipole moment of a volume source. It consists of a bipolar coil system (Figure 12.4A) which produces in its center the three components of the reciprocal magnetic field _{LM} (Figure 12.4B). Note, that the region where the coils of Figure 12.4A produce linear reciprocal magnetic fields is rather small, as will be explained later, and therefore the Figures 12.4A and 12.4B are not in scale. The three reciprocal magnetic fields _{LM} produce the three components of the reciprocal electric field _{LM} and the lead field _{LM} that are illustrated in Figure 12.5. It is important to note that the reciprocal magnetic field _{LM} has the same geometrical form as the reciprocal electric field _{LE} of a detector which detects the electric dipole moment of a volume source, Figure 11.24.
Similarly as in the equation of the electric field of a volume source, Equation 7.9, the second term on the righthand side of Equation 12.22 represents the contribution of the boundaries and inhomogeneities to the magnetic dipole moment. This is equivalent to the effect of the boundaries and inhomogeneities on the form of the lead field. In general, a detector that produces an ideal lead field in the source region despite the boundaries and inhomogeneities of the volume conductor detects the dipole moment of the source undistorted.
Fig. 12.4. The principle of a lead system detecting the magnetic dipole moment of a volume source.
(A) The three orthogonal bipolar coils.
(B) The three components of the reciprocal magnetic field _{LM} in the center of the bipolar coil system.
The region where the coils produce linear reciprocal magnetic fields is rather small and therefore Figures 12.4A and 12.4B are not to scale.
As in the case of the detection of the electric dipole moment of a volume source, Section 11.6.9, both unipolar and bipolar leads may be used in synthesizing the ideal lead field for detecting the magnetic dipole moment of a volume source. In the case of an infinite conducting medium and a uniform reciprocal magnetic field, the lead field current flows concentrically about the symmetry axis, as shown in Figure 12.3. Then no alteration results if the conducting medium is terminated by a spherical boundary (since the lead current flow lines lie tangential to the surface). The spherical surface ensures lead current flow lines, as occurs in an infinite homogeneous medium, when a uniform reciprocal magnetic field is established along any x, y, and zcoordinate direction.
If the dimensions of the volume source are small in relation to the distance to the point of observation, we can consider the magnetic dipole moment to be a contribution from a point source. Thus we consider the magnetic dipole moment to be a discrete vector. The evaluation of such a magnetic dipole is possible to accomplish through unipolar measurements on each coordinate axis as illustrated on the left hand side of Figure 12.6A. If the dimensions of the volume source are large, the quality of the aforementioned lead system is not high. Because the reciprocal magnetic field decreases as a function of distance, the sensitivity of a single magnetometer is higher for source elements locating closer to it than for source elements locating far from it. This is illustrated on the right hand side of Figure 12.6A. In Figure 12.6 the dashed lines represent the reciprocal magnetic field flux tubes. The thin solid circular lines represent the lead field flow lines. The behavior of the reciprocal magnetic field of a single magnetometer coil is illustrated more accurately in Figures 20.14, 20.15, and 22.3.
The result is very much improved if we use symmetric pairs of magnetometers forming bipolar leads, as in Figure 12.6B. This arrangement will produce a reciprocal magnetic field that is more uniform over the source region than is attained with the single coils of the unipolar lead system. (Malmivuo, 1976).
Just as with the electric case, the quality of the bipolar magnetic lead fields in measuring volume sources with large dimensions is further improved by using large coils, whose dimensions are comparable to the source dimensions. This is illustrated in Figure 12.6C..
MAGNETOMETER CONFIGURATION  LEAD FIELD OF ONE COMPONENT  
A  UNIPOLAR LEADS, SMALL COILS  
B  BIPOLAR LEADS, SMALL COILS  
C  BIPOLAR LEADS, LARGE COILS  
To describe the behavior of the reciprocal magnetic field and the sensitivity distribution of a bipolar lead as a function of coil separation we illustrate in Figure 12.7 these for two coil pairs with different separation. (Please note, that the isosensitivity lines are not the same as the reciprocal magnetic field lines.) Figure 12.7A illustrates the reciprocal magnetic field as rotational flux tubes for the Helmholtz coils which are a coaxial pair of identical circular coils separated by the coil radius. With this coil separation the radial component of the compound magnetic field at the center plane is at its minimum and the magnetic field is very homogeneous. Helmholtz coils cannot easily be used in detecting biomagnetic fields, but they can be used in magnetization or in impedance measurement. They are used very much in balancing the gradiometers and for compensating the Earth's static magnetic field in the measurement environment. Figure 12.7B illustrates the reciprocal magnetic field flux tubes for a coil pair with a separation of 5r. Figures 12.7C and 12.7D illustrate the isosensitivity lines for the same coils.
Later in Chapter 20, Figure 20.16 illustrates the isosensitivity lines for a coil pair with a separation of 32r. Comparing these two bipolar leads to the Helmholtz coils one may note that in them the region of homogeneous sensitivity is much smaller than in the Helmholtz coils. Due to symmetry, the homogeneity of bipolar leads is, however, much better than that of corresponding unipolar leads.
The arrangement of bipolar lead must not be confused with the differential magnetometer or gradiometer system, which consists of two coaxial coils on the same side of the source wound in opposite directions. The purpose of such an arrangement is to null out the background noise, not to improve the quality of the lead field. The realization of the bipolar lead system with gradiometers is illustrated in Figure 12.8. Later Figure 12.20 illustrates the effect of the second coil on the gradiometer sensitivity distribution for several baselines..
Fig. 12.7. Flux tubes of the reciprocal magnetic field of
(A) the Helmholtz coils having a coil separation of r
(B) bipolar lead with a coil separation of 5r.
The isosensitivity lines for
(C) the Helmholtz coils having a coil separation of r
(D) bipolar lead with a coil separation of 5r.
(Note, that the isosensitivity lines are not the same as the flux tubes of the reciprocal magnetic field.).
In summary, we note the following details from the lead fields of ideal bipolar lead systems for detecting the electric and magnetic dipole moments of a volume source:
The lead system consists of three components.
For a spherically symmetric volume conductor, each is formed by a pair of electrodes (or electrode matrices), whose axis is in the direction of the coordinate axes. Each electrode is on opposite sides of the source, as shown in Figure 11.24.
For each of these components, when energized reciprocally, a homogeneous and linear electric field is established in the region of the volume source (see Figure 11.25). Each of these reciprocal electric fields forms a similar current field, which is called the electric lead field _{LE}. (Note the similarity of Figure 11.25, illustrating the reciprocal electric field _{LE} of an electric lead, and Figure 12.7, illustrating the reciprocal magnetic field _{LM} of a magnetic lead.)
The lead system consists of three components.
In the spherically symmetric case, each of them is formed by a pair of magnetometers (or gradiometers) located in the direction of the coordinate axes on opposite sides of the source, as illustrated in Figure 12.6C (or 12.8).
For each of these components, when energized reciprocally, a homogeneous and linear magnetic field is established in the region of the volume source, as shown in Figure 12.4.
Each of these reciprocal magnetic fields forms an electric field, necessarily tangential to the boundaries. These reciprocal electric fields give rise to a similar electric current field, which is called the magnetic lead field _{LM}, as described in Figure 12.5.
Superimposing Figures 12.8, 12.4, and 12.5 allows one more easily to visualize the generation and shape of the lead fields of magnetic leads.
(12.26) 
whereas for the tangential component Jit it is:
(12.27) 
In these expressions the component lead fields are assumed uniform and of unit magnitude.
We note from Equations 12.26 and 12.27 that the total sensitivity of these two components of the electric lead to radial and tangential current source elements ^{i} is equal and independent of their location. The same conclusion also holds in all three dimensions..
The special properties of electric lead fields, listed in Section 11.6.10, also hold for magnetic lead fields. Magnetic lead fields also have some additional special properties which can be summarized as follows:
If the volume conductor is cut or the boundary of an inhomogeneity is inserted along a lead field current flow line, the form of the lead field does not change (Malmivuo, 1976). This explains why with either a cylindrically or spherically symmetric volume conductor, the form of the symmetric magnetic lead field is unaffected. There are two important practical consequences:
Because the heart may be approximated as a sphere, the highly conducting intracardiac blood mass, which may be considered spherical and concentric, does not change the form of the lead field. This means that the Brody effect does not exist in magnetocardiography (see Chapter 18).
The poorly conducting skull does not affect the magnetic detection of brain activity as it does with electric detection (see Figure 12.10).
Magnetic lead fields in volume conductors exhibiting spherical symmetry are always directed tangentially. This means that the sensitivity of such magnetic leads in a spherical conductor to radial electric dipoles is always zero. This fact has special importance in the MEG (see Figure 12.11).
If the electrodes of a symmetric bipolar electric lead are located on the symmetry axis of the bipolar magnetic field detector arranged for a spherical volume conductor, the lead fields of these electric and magnetic leads are normal to each other throughout the volume conductor, as illustrated in Figure 12.12 (Malmivuo, 1980). (The same holds for corresponding unipolar leads as well, though not shown in the figure.)
The lead fields of all magnetic leads include at least one zero sensitivity line, where the sensitivity to electric dipoles is zero. This line exists in all volume conductors, unless there is a hole in the conductor in this region (Eskola, 1979, 1983; Eskola and Malmivuo, 1983). The zero sensitivity line itself is one tool in understanding the form of magnetic leads (as demonstrated in Figure 12.13).
The reciprocity theorem also applies to the reciprocal situation. This means that in a tank model it is possible to feed a "reciprocally reciprocal" current to the dipole in the conductor and to measure the signal from the lead. However, the result may be interpreted as having been obtained by feeding the reciprocal current to the lead with the signal measured from the dipole. The benefit of this "reciprocally reciprocal" arrangement is that for technical reasons the signaltonoise ratio may be improved while we still have the benefit of interpreting the result as the distribution of the lead field current in the volume conductor (Malmivuo, 1976)..
Fig. 12.10. The poorly conducting skull does not affect the magnetic detection of the electric activity of the brain.
Fig. 12.11. Magnetic lead fields in volume conductors exhibiting spherical symmetry are always directed tangentially. The figure illustrates also the approximate form of the zero sensitivity line in the volume conductor. (The zero sensitivity line may be imagined to continue hypothetically through the magnetometer coil.).
Fig. 12.12. If the electrodes of a symmetric bipolar electric lead are located on the symmetry axis of the bipolar magnetic field detector arranged for a spherical volume conductor, these lead fields of the electric and magnetic leads are normal to each other throughout the volume conductor.
Fig. 12.13. Zero sensitivity lines in volume conductors of various forms. The dimensions are given in millimeters (Eskola, 1979, 1983; Eskola and Malmivuo, 1983). As in Figure 12.11, the zero sensitivity lines are illustrated to continue hypothetically through the magnetometer coils.
(12.28) 
where F and V denote flow and vortex, respectively. By definition, these vector fields satisfy ^{i}_{F} = 0 and ^{i}_{V} = 0.
We first examine the independence of the electric and magnetic signals in the infinite homogeneous case, when the second term on the righthand side of Equations 7.10 and 12.6, caused by inhomogeneities, is zero. These equations may be rewritten for the electric potential:
(12.29) 
and for the magnetic field:
(12.30) 
Substituting Equation 12.28 into Equations 12.29 and 12.30 shows that under homogeneous and unbounded conditions, the bioelectric field arises from ^{i}_{F} , which is the flow source (Equation 7.5), and the biomagnetic field arises from ^{i}_{V} , which is the vortex source (Equation 12.17). Since the detection of the first biomagnetic field, the magnetocardiogram, by Baule and McFee in 1963 (Baule and McFee, 1963), the demonstration discussed above raised a lot of optimism among scientists. If this independence were confirmed, the magnetic detection of bioelectric activity could bring much new information not available by electric measurement.
Rush was the first to claim that the independence of the electric and magnetic signals is only a mathematical possibility and that physical constraints operate which require the flow and vortex sources, and consequently the electric and magnetic fields, to be fundamentally interdependent in homogeneous volume conductors (Rush, 1975). This may be easily illustrated with an example by noting that, for instance, when the atria of the heart contract, their bioelectric activity produces an electric field recorded as the Pwave in the ECG. At the same time their electric activity produces a magnetic field detected as the Pwave of the MCG. Similarly the electric and magnetic QRScomplexes and Twaves are interrelated, respectively. Thus, full independence between the ECG and the MCG is impossible.
In a more recent communication, Plonsey (1982) showed that the primary cellular source may be small compared to the secondary cellular source and that the latter may be characterized as a double layer source for both the electric scalar and magnetic vector potentials.
(12.31) 
where F = da is the magnetic flux evaluated by the integral of the normal component of the magnetic induction across the surface of the loop. For a circular loop the integral on the lefthand side of Equation 12.31 equals 2πrE, where r is the radius of the loop, and we obtain for the current density
(12.32) 
where σ is the conductivity of the medium. The current density is tangentially oriented. Now the problem reduces to the determination of the magnetic flux linking a circular loop in the medium due to a reciprocally energizing current in the coaxially situated magnetometer coil..
Fig. 12.14. Geometry for calculating the spatial sensitivity of a magnetometer in a cylindrically symmetric situation.
The basic equation for calculating the vector potential at point P due to a current I flowing in a thin conductor is
(12.33) 
where  μ  = magnetic permeability of the medium 
r_{p}  = distance from the conductor element to the point P 
This equation can be used to calculate the vector potential at the point P in Figure 12.14. From symmetry we know that in spherical coordinates the magnitude of is independent of angle Φ. Therefore, for simplicity, we choose the point P so that Φ = 0. We notice that when equidistant elements of length d_{1} at +Φ and Φ are paired, the resultant is normal to hr. Thus has only the single component A_{φ}. If we let dl_{φ} be the component of d_{1} in this direction, then Equation 12.33 may be rewritten as
(12.34) 
The magnetic flux F_{21} may be calculated from the vector potential:
(12.35) 
With the substitution φ = π  2α , this becomes
(12.36) 
where
(12.37) 
and K(k) and E(k) are complete elliptic integrals of the first and second kind, respectively. These are calculated from equations 12.38A,B. (Abramowitz and Stegun, 1964, p. 590)
(12.38a) 
(12.38b) 
The values K(k) and E(k) can also be calculated using the series:
(12.39a) 
(12.39b) 
The calculation of K(k) and E(k) is faster from the series, but they give inaccurate results at small distances from the coil and therefore the use of the Equations 12.38A,B is recommended.
Substituting Equation 12.36 into Equation 12.32 gives the lead field current density J_{LM} as a function of the rate of change of the coil current in the reciprocally energized magnetometer:
(12.40) 
Because we are interested in the spatial sensitivity distribution and not in the absolute sensitivity with certain frequency or conductivity values, the result of Equation 12.40 can be normalized by defining (similarly as was done in Section 12.3.1 in deriving the equation for magnetic lead field)
(12.41) 
and we obtain the equation for calculating the lead field current density for a singlecoil magnetometer in an infinite homogeneous volume conductor:
(12.42) 
where all distances are measured in meters, and current density in [A/m^{2}].
If the distance h is large compared to the coil radius r_{1} and the lead field current flow line radius r_{2}, the magnetic induction inside the flow line may be considered constant, and Equation 12.42 is greatly simplified. The value of the magnetic flux becomes πr^{2}. Substituting it into Equation 12.32, we obtain
(12.43) 
The magnetic induction may be calculated in this situation as for a dipole source. Equation 12.43 shows clearly that in the region of constant magnetic induction and constant conductivity, the lead field current density is proportional to the radial distance from the symmetry axis. Note that this equation is consistent with Equation 12.11.
Fig. 12.15. The lead field current density distribution of a unipolar singlecoil magnetometer with a 10 mm coil radius in a cylindrically symmetric volume conductor calculated from Equation 12.42. The dashed lines are the isosensitivity lines, joining the points where the lead field current density is 100, 200, 300, 400, and 500 pA/m^{2}, respectively, as indicated by the numbers in italics.
Figure 12.16 illustrates the isosensitivity lines for a unipolar singlecoil magnetometer of Figure 12.15; the coil radius is 10 mm, and the volume conductor is cylindrically symmetric. The vertical axis indicates the distance h from the magnetometer and the horizontal axis the radial distance r from the symmetry axis. The symmetry axis, drawn with thick dashed line, is the zero sensitivity line. The lead field current flow lines are concentric circles around the symmetry axis. To illustrate this, the figure shows three flow lines representing the current densities 100, 200, and 300 pA/m^{2} at the levels h = 125 mm, 175 mm and 225 mm. As in the previous figure, the lead field current density values are calculated for a reciprocal current of I_{R} = 1 A/s.
The effect of the coil radius in a unipolar lead on the lead field current density is shown in Figure 12.17. In this figure, the lead field current density is illustrated for coils with 1 mm, 10 mm, 50 mm, and 100 mm radii. The energizing current in the coils is normalized in relation to the coil area to obtain a constant dipole moment. The 10 mm radius coil is energized with a current of dI/dt = 1 [A/s].
Fig. 12.16. The isosensitivity lines for a unipolar singlecoil magnetometer of Figure 12.15; the coil radius is 10 mm, and the volume conductor is cylindrically symmetric. The vertical axis indicates the distance h from the magnetometer and the horizontal axis the radial distance r from the symmetry axis. The symmetry axis, drawn with a thick dashed line, is the zero sensitivity line. Thin solid lines represent lead field current flow lines.
Fig. 12.17. Lead field current density for unipolar leads of coils with 1 mm, 10 mm, 50 mm, and 100 mm radii. The energizing current in the coils is normalized in relation to the coil area to obtain a constant dipole moment.
Fig. 12.18. Lead field current density for unipolar leads realized with differential magnetometers of 10 mm coil radius and with 300 mm, 150 mm, 100 mm, and 50 mm baseline.
Fig. 12.19. The lead field current density distribution of a bipolar lead in a cylindrically symmetric volume conductor realized with two coaxial singlecoil magnetometers with 10 mm coil radius. The distance between the coils is 340 mm. The dashed lines are the isosensitivity lines, joining the points where the lead field current density is 500 and 1000 pA/m^{2}, respectively, as indicated with the numbers in italics.
Fig. 12.20. The isosensitivity lines for the bipolar lead of Figure 12.19. The coil radii are 10 mm and the distance between the coils is 340 mm. The vertical axis indicates the distance h from the first magnetometer and the horizontal axis the radial distance r from the symmetry axis. The symmetry axis, drawn with thick dashed line, is the zero sensitivity line. Lead field current flow lines encircle the symmetry axis and are illustrated with thin solid lines.
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