2121ch20

20Volume Conductor

Theory

Robert PlonseyDuke University

20.1 Basic Relations in the Idealized HomogeneousVolume Conductor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-1

20.2 Monopole and Dipole Fields in the Uniform Volumeof Infinite Extent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-3

20.3 Volume Conductor Properties of Passive Tissue . . . . . . . 20-420.4 Effects of Volume Conductor Inhomogeneities:

Secondary Sources and Images . . . . . . . . . . . . . . . . . . . . . . . . . 20-5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20-7

This chapter considers the properties of the volume conductor as it pertains to the evaluation of electricand magnetic fields arising therein. The sources of the aforementioned fields are described by �J i , a functionof position and time, which has the dimensions of current per unit area or dipole moment per unit volume.Such sources may arise from active endogenous electrophysiologic processes such as propagating actionpotentials, generator potentials, synaptic potentials, etc. Sources also may be established exogenously,as exemplified by electric or magnetic field stimulation. Details on how one may quantitatively evaluate asource function from an electrophysiologic process are found in other chapters. For our purposes here, weassume that such a source function �J i is known and, furthermore, that it has well-behaved mathematicalproperties. Given such a source, we focus attention here on a description of the volume conductor as itaffects the electric and magnetic fields that are established in it. As a loose definition, we consider thevolume conductor to be the contiguous passive conducting medium that surrounds the region occupiedby the source �J i . (This may include a portion of the excitable tissue itself that is sufficiently far from �J i tobe described passively.)

20.1 Basic Relations in the Idealized HomogeneousVolume Conductor

Excitable tissue, when activated, will be found to generate currents both within itself and also in allsurrounding conducting media. The latter passive region is characterized as a volume conductor. Theadjective volume emphasizes that current flow is three-dimensional, in contrast to the confined one-dimensional flow within insulated wires. The volume conductor is usually assumed to be a monodomain(whose meaning will be amplified later), isotropic, resistive, and (frequently) homogeneous. These aresimply assumptions, as will be discussed subsequently. The permeability of biologic tissues is important

20-1

© 2006 by Taylor & Francis Group, LLC

20-2 Biomedical Engineering Fundamentals

when examining magnetic fields and is usually assumed to be that of free space. The permittivity is amore complicated property, but outside cell membranes (which have a high lipid content) it is also usuallyconsidered to be that of free space.

A general, mathematical description of a current source is specified by a function �J i (x , y , z , t ), namely,a vector field of current density in say milliamperes per square centimeter that varies both in space andtime. A study of sources of physiologic origin shows that their temporal behavior lies in a low-frequencyrange. For example, currents generated by the heart have a power density spectrum that lies mainlyunder 1 kHz (in fact, clinical ECG instruments have upper frequency limits of 100 Hz), while most otherelectrophysiologic sources of interest (i.e., those underlying the EEG, EMG, EOG, etc.) are of even lowerfrequency. Examination of electromagnetic fields in regions with typical physiologic conductivities, withdimensions of under 1 m and frequencies less than 1 kHz, shows that quasi-static conditions apply. That is,at a given instant in time, source–field relationships correspond to those found under static conditions.1

Thus, in effect, we are examining direct current (dc) flow in physiologic volume conductors, and thesecan be maintained only by the presence of a supply of energy (a “battery”). In fact, we may expect thatwherever a physiologic current source �J i arises, we also can identify a (normally nonelectrical) energysource that generates this current. In electrophysiologic processes, the immediate repository of energy isthe potential energy associated with the varying chemical compositions encountered (extracellular ionicconcentrations that differ greatly from intracellular concentrations), but the long-term energy source isthe adenosine triphosphate (ATP) that drives various pumps that create and maintain the aforementionedconcentration gradients.

Based on the aforementioned assumptions, we consider a uniformly conducting medium of conduct-ivity σ and of infinite extent within which a current source �J i lies. This, in turn, establishes an electricfield �E and, based on Ohm’s law, a conduction current density σ �E . The total current density �J is the sumof the aforementioned currents, namely,

�J = σ �E + �J i (20.1)

Now, by virtue of the quasi-static conditions, the electric field may be derived from a scalar potential �[Plonsey and Heppner, 1967] so that

�E = −∇� (20.2)

Since quasi-steady-state conditions apply, �J must be solenoidal, and consequently, substitutingEquation 20.2 into Equation 20.1 and then setting the divergence of Equation 20.1 to zero show that�must satisfy Poisson’s equation, namely,

∇2� =(

1

σ

)∇ · �J i (20.3)

An integral solution to Equation 20.3 is [Plonsey and Collin, 1961]

�p(x′, y ′, z ′) = − 1

4πσ

∫

v

∇, �J i

rdv (20.4)

where r in Equation 20.4 is the distance from a field point P(x ′, y ′, z ′) to an element of source at dv(x , y , z),that is,

r =√(x − x ′)2 + (y − y ′)2 + (z − z ′)2 (20.5)

1Note that while, in effect, we consider relationships arising when ∂/∂t = 0, all fields are actually assumed to vary intime synchronously with �J i . Furthermore for the special case of magnetic field stimulation, the source of the primaryelectric field, ∂ �A/∂t , where �A is the magnetic vector potential, must be retained.


Volume Conductor Theory 20-3

Equation 20.4 may be transformed to an alternate form by employing the vector identity

∇ ·[(

1

r

)�J i]≡(

1

r

)∇ · �J i + ∇

(1

r

)· �J i (20.6)

Based on Equation 20.6, we may substitute for the integrand in Equation 20.4 the sum ∇ · [(1/r)�J i] −∇(1/r) · �J i , giving the following:

�p(x′, y ′, z ′) = − 1

4πσ

{∫

v∇ ·

[(1

r

)�J i]

dv −∫

v∇(

1

r

)�J idv

}(20.7)

The first term on the right-hand side may be transformed using the divergence theorem as follows:

∫

v∇ ·

[(1

r

)�J i]

dv =∫

s

(1

r

)�J i · d�S = 0 (20.8)

The volume integral in Equation 20.4 and Equation 20.8 is defined simply to include all sources. Con-sequently, in Equation 20.8, the surface S, which bounds V , necessarily lies away from �J i . Since �J i thus isequal to zero on S, the expression in Equation 20.8 must likewise equal zero. The result is that Equation 20.4also may be written as

�p(x′, y ′, z ′) = − 1

4πσ

∫

v

∇ · �J i

rdv = 1

4πσ

∫

v

�J i · ∇(

1

r

)dv (20.9)

We will derive the mathematical expressions for monopole and dipole fields in the next section, but basedon those results, we can give a physical interpretation of the source terms in each of the integrals on theright-hand side of Equation 20.9. In the first, we note that −∇ · �J i is a volume source density, akin tocharge density in electrostatics. In the second integral of Equation 20.9, �J i behaves with the dimensionsof dipole moment per unit volume. This confirms an assertion, above, that �J i has a dual interpretationas a current density, as originally defined in Equation 20.1, or a volume dipole density, as can be inferredfrom Equation 20.9; in either case, its dimension are mA/cm2 = mA · cm/cm3.

20.2 Monopole and Dipole Fields in the Uniform Volume ofInfinite Extent

The monopole and dipole constitute the basic source elements in electrophysiology. We examine the fieldsproduced by each in this section.

If one imagines an infinitely thin wire insulated over its extent except at its tip to be introducing acurrent into a uniform volume conductor of infinite extent, then we illustrate an idealized point source.Assuming the total applied current to be I0 and located at the coordinate origin, then by symmetry thecurrent density at a radius r must be given by the total current I0 divided by the area of the sphericalsurface, or

�J = I0

4πr2�ar (20.10)

and �ar is a unit vector in the radial direction. This current source can be described by the nomenclatureof the previous section as

∇ · �J i = −I0δ(r) (20.11)

where δ denotes a volume delta function.



One can apply Ohm’s law to Equation 20.10 and obtain an expression for the electric field, and ifEquation 20.2 is also applied, we get

�E = −∇� = I0

4πσ r2�ar (20.12)

where σ is the conductivity of the volume conductor. Since the right-hand side of Equation 20.12 is afunction of r only, we can integrate to find�, which comes out

� = I0

4πσ r(20.13)

In obtaining Equation 20.13, the constant of integration was set equal to zero so that the point at infinityhas the usually chosen zero potential.

The dipole source consists of two monopoles of equal magnitude and opposite sign whose spacingapproaches zero and whose magnitude during the limiting process increases such that the product ofspacing and magnitude is constant. If we start out with both component monopoles at the origin, then thetotal source and field are zero. However, if we now displace the positive source in an arbitrary direction�d , then cancellation is no longer complete, and at a field point P we see simply the change in monopolefield resulting from the displacement. For a very small displacement, this amounts to (i.e., we retain onlythe linear term in a Taylor series expansion)

�p = ∂

∂d

(I0

4πσ r

)d (20.14)

The partial derivative in Equation 20.14 is called the directional derivative, and this can be evaluated bytaking the dot product of the gradient of the expression enclosed in parentheses with the direction of d(i.e.,∇() · �ad , where �ad is a unit vector in the �d direction). The result is

�p = I0d

4πσ�ad · ∇

(1

r

)(20.15)

By definition, the dipole moment �m = I0�d in the limits as d → 0; as noted, m remains finite. Thus, finally,

the dipole field is given by [Plonsey, 1969]

�p = 1

4πσ�m · ∇

(1

r

). (20.16)

20.3 Volume Conductor Properties of Passive Tissue

If one were considering an active single isolated fiber lying in an extensive volume conductor (e.g., anin vitro preparation in a Ringer’s bath), then there is a clear separation between the excitable tissue andthe surrounding volume conductor. However, consider in contrast activation proceeding in the in vivoheart. In this case, the source currents lie in only a portion of the heart (nominally where ∇Vm �= 0).The volume conductor now includes the remaining (passive) cardiac fibers along with an inhomogeneoustorso containing a number of contiguous organs (internal to the heart are blood-filled cavities, whileexternal are pericardium, lungs, skeletal muscle, bone, fat, skin, air, etc.).

The treatment of the surrounding multicellular cardiac tissue poses certain difficulties. A recently usedand reasonable approximation is that the intracellular space, in view of the many intercellular junctions,can be represented as a continuum. A similar treatment can be extended to the interstitial space. Thisresults in two domains that can be regarded as occupying the same physical space; each domain is separatedfrom the other by the membrane. This view underlies the bidomain model [Plonsey, 1989]. To reflect the



TABLE 20.1 Conductivity Valuesfor Cardiac Bidomain

S/mm Clerc [1976] Roberts [1982]

gix 1.74× 104 3.44× 10−4

giy 1.93× 10−5 5.96× 10−5

gax 6.25× 10−4 1.17× 10−4

gay 2.36× 10−4 8.02× 10−5

underlying fiber geometry, each domain is necessarily anisotropic, with the high conductivity axes definedby the fiber direction and with an approximate cross-fiber isotropy. A further simplification may bepossible in a uniform tissue region that is sufficiently far from the sources, since beyond a few spaceconstants transmembrane currents may become quite small and the tissue would therefore behave as asingle domain (a monodomain). Such a tissue also would be substantially resistive. On the other hand,if the membranes behave passively and there is some degree of transmembrane current flow, then thetissue may still be approximated as a uniform monodomain, but it may be necessary to include some ofthe reactive properties introduced via the highly capacitative cell membranes. A classic study by Schwanand Kay [1957] of the macroscopic (averaged) properties of many tissues showed that the displacementcurrent was normally negligible compared with the conduction current.

It is not always clear whether a bidomain model is appropriate to a particular tissue, and experimentalmeasurements found in the literature are not always able to resolve this question. The problem is that if theexperimenter believes the tissue under consideration to be, say, an isotropic monodomain, measurementsare set up and interpreted that are consistent with this idea; the inherent inconsistencies may never cometo light [Plonsey and Barr, 1986]. Thus one may find impedance data tabulated in the literature for anumber of organs, but if the tissue is truly, say, an anisotropic bidomain, then the impedance tensorrequires six numbers, and anything less is necessarily inadequate to some degree. For cardiac tissue, it isusually assumed that the impedance in the direction transverse to the fiber axis is isotropic. Consequently,only four numbers are needed. These values are given in Table 20.1 as obtained from, essentially, the onlytwo experiments for which bidomain values were sought.

20.4 Effects of Volume Conductor Inhomogeneities: SecondarySources and Images

In the preceding we have assumed that the volume conductor is homogeneous, and the evaluation offields from the current sources given in Equation 20.9 is based on this assumption. Consider what wouldhappen if the volume conductor in which �J i lies is bounded by air, and the source is suddenly introduced.Equation 20.9 predicts an initial current flow into the boundary, but no current can escape into thenonconducting surrounding region. We must, consequently, have a transient during which charge pilesup at the boundary, a process that continues until the field from the accumulating charges brings the netnormal component of electric field to zero at the boundary. To characterize a steady-state condition withno further increase in charge requires satisfaction of the boundary condition that ∂�/∂n = 0 at thesurface (within the tissue). The source that develops at the bounding surface is secondary to the initiationof the primary field; it is referred to as a secondary source. While the secondary source is essential forsatisfaction of boundary conditions, it contributes to the total field everywhere else.

The preceding illustration is for a region bounded by air, but the same phenomena would arise if theregion were simply bounded by one of different conductivity. In this case, when the source is first “turnedon,” since the primary electric field �Ea is continuous across the interface between regions of differentconductivity, the current flowing into such a boundary (e.g., σ1 �Ea) is unequal to the current flowingaway from that boundary (e.g., σ2 �Ea). Again, this necessarily results in an accumulation of charge, anda secondary source will grow until the applied plus secondary field satisfies the required continuity of



current density, namely,

−σ1∂�1

∂n= −σ2∂

�2

∂n= Jn (20.17)

where the surface normal n is directed from region 1 to region 2. The accumulated single source densitycan be shown to be equal to the discontinuity of ∂�/∂n in Equation 20.17 [Plonsey, 1974], in particular,

Ks = Jn

(1

σ1− 1

σ2

)σ

The magnitude of the steady-state secondary source also can be described as an equivalent double layer,the magnitude of which is [Plonsey, 1974]

�K ik = �k(σ

′′k − σ ′k)�n (20.18)

where the condition at the kth interface is described. In Equation 20.18, the two abutting regions aredesignated with prime and double-prime superscripts, and �n is directed from the primed to the double-primed region. Actually, Equation 20.18 evaluates the double-layer source for the scalar function = �σ ,its strength being given by the discontinuity in at the interface [Plonsey, 1974] (the potential is necessarilycontinuous at the interface with the value �k called for in Equation 20.18). The (secondary) potentialfield generated by �K i

k , since it constitutes a source for with respect to which the medium is uniform andinfinite in extent, can be found from Equation 20.9 as

SP = σP�

SP =

1

4π

∑

k

∫

k�k(σ

′′k − σ ′k)�n · ∇

(1

r

)ds (20.19)

where the superscript S denotes the secondary source/field component (alone). Solving Equation 20.19for�, we get

�SP =

1

4πσP

∑

k

∫

k�k(σ

′′k − σ ′k)�n · ∇

(1

r

)ds (20.20)

where σP in Equation 20.19 and Equation 20.20 takes on the conductivity at the field point. The total fieldis obtained from Equation 20.20 by adding the primary field. Assuming that all applied currents lie in aregion with conductivity σa , then we have

�SP =

1

4πσa

∫�J i · ∇

(1

r

)dv + 1

4πσP

∑

k

∫

k�k(σ

′′k − σ ′k)�n · ∇

(1

r

)ds (20.21)

(If the primary currents lie in several conductivity compartments, then each will yield a term similar tothe first integral in Equation 20.21.) Note that in Equation 20.20 and Equation 20.21 the secondary sourcefield is similar in form to the field in a homogeneous medium of infinite extent, except that σP is piecewiseconstant and consequently introduces interfacial discontinuities. With regard to the potential, these justcancel the discontinuity introduced by the double layer itself so that�k is appropriately continuous acrosseach passive interface.

The primary and secondary source currents that generate the electrical potential in Equation 20.21 alsoset up a magnetic field. The primary source, for example, is the forcing function in the Poisson equationfor the vector potential �A [Plonsey, 1981], namely,

∇2 �A = −�J i (20.22)



From this it is not difficult to show that, due to Equation 20.21 for �, we have the following expressionfor the magnetic field �H [Plonsey, 1981]:

�H = 1

4π

∫�J i ×∇

(1

r

)dv + 1

4π

∑

k

∫�k(σ

′′k − σ ′k)�n ×∇

(1

r

)dS (20.23)

A simple illustration of these ideas is found in the case of two semi-infinite regions of different con-ductivity, 1 and 2, with a unit point current source located in region 1 a distance h from the interface.Region 1, which we may think of as on the “left,” has the conductivity σ1, while region 2, on the “right,”is at conductivity σ2. The field in region 1 is that which arises from the actual point current source plusan image point source of magnitude (ρ2 − ρ1)/(ρ2 + ρ1) located in region 2 at the mirror-image point[Schwan and Kay, 1957]. The field in region 2 arises from an equivalent point source located at the actualsource point but of strength [1 + (ρ2 − ρ1)/(ρ2 + ρ1)]. One can confirm this by noting that all fieldssatisfy Poisson’s equation and that at the interface� is continuous while the normal component of currentdensity is also continuous (i.e., σ1∂�1/∂n = σ2∂�2/∂n).

The potential on the interface is constant along a circular path whose origin is the foot of the perpen-dicular from the point source. Calling this radius r and applying Equation 20.13, we have for the surfacepotential�S

�S = ρ1

4π√

h2 + r2

(1+ ρ2 − ρ1

ρ2 + ρ1

)(20.24)

and consequently a secondary double-layer source �KS equals, according to Equation 20.18,

�KS = ρ1

4π√

h2 + r2

(1+ ρ2 − ρ1

ρ2 + ρ1

)(σ2 − σ1)�n (20.25)

where �n is directed from region 1 to region 2. The field from �KS in region 1 is exactly equal to that from apoint source of strength (ρ2−ρ1)/(ρ2+ρ1) at the mirror-image point, which can be verified by evaluatingand showing the equality of the following:

�P = 1

4πσ1

∫�KS · ∇

(1

r

)ds = (ρ2 − ρ1)/(ρ2 − ρ1)

4πσ1R(20.26)

where R in Equation 20.26 is the distance from the mirror-image point to the field point P , and r inEquation 20.26 is the distance from the surface integration point to the field point.

References

Clerc, L. 1976. Directional differences of impulse spread in trabecular muscle from mammalian heart.J. Physiol. (Lond.) 255: 335.

Plonsey, R. 1969. Bioelectric Phenomena. New York, McGraw-Hill.Plonsey, R. 1974. The formulation of bioelectric source–field relationship in terms of surface

discontinuities. J. Frank Inst. 297: 317.Plonsey, R. 1981. Generation of magnetic fields by the human body (theory). In S.-N. Erné,

H.-D. Hahlbohm, and H. Lübbig (Eds.), Biomagnetism, pp. 177–205. Berlin, W de Gruyter.Plonsey, R. 1989. The use of the bidomain model for the study of excitable media. Lect. Math. Life Sci.

21: 123.Plonsey, R. and Barr, R.C. 1986. A critique of impedance measurements in cardiac tissue. Ann. Biomed.

Eng. 14: 307.Plonsey, R. and Collin, R.E. 1961. Principles and Applications of Electromagnetic Fields. New York,

McGraw-Hill.



Plonsey, R. and Heppner, D. 1967. Consideration of quasi-stationarity in electrophysiological systems.Bull. Math. Biophys. 29: 657.

Roberts, D. and Scher, A.M. 1982. Effect of tissue anisotropy on extracellular potential fields in caninemyocardium in situ. Circ. Res. 50: 342.

Schwan, H.P. and Kay, C.F. 1957. The conductivity of living tissues. NY Acad. Sci. 65: 1007.


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