Download Chapter 9 - Volume Conductor Theory

Plonsey, R. “Volume Conductor Theory.”
The Biomedical Engineering Handbook: Second Edition.
Ed. Joseph D. Bronzino
Boca Raton: CRC Press LLC, 2000
9
Volume Conductor
Theory
9.1
9.2
9.3
9.4
Robert Plonsey
Duke University
Basic Relations in the Idealized Homogeneous
Volume Conductor
Monopole and Dipole Fields in the Uniform
Volume of Infinite Extent
Volume Conductor Properties of Passive Tissue
Effects of Volume Conductor Inhomogeneities:
Secondary Sources and Images
This chapter considers the properties of the volume conductor as it pertains to the evaluation of electric
and magnetic fields arising therein. The sources of the aforementioned fields are described by J i, a
function of 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 action potentials, 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 a source function from an electrophysiologic process are found in other chapters. For our
purposes here, we assume that such a source function J i is known and, furthermore, that it has wellbehaved mathematical properties. Given such a source, we focus attention here on a description of the
volume conductor as it affects the electric and magnetic fields that are established in it. As a loose
definition, we consider the volume conductor to be the contiguous passive conducting medium that
surrounds the region occupied by the source J i. (This may include a portion of the excitable tissue itself
that is sufficiently far from J i to be described passively.)
→
→
→
→
9.1 Basic Relations in the Idealized Homogeneous
Volume Conductor
Excitable tissue, when activated, will be found to generate currents both within itself and also in all
surrounding conducting media. The latter passive region is characterized as a volume conductor. The
adjective volume emphasizes that current flow is three-dimensional, in contrast to the confined onedimensional 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 are
simply assumptions, as will be discussed subsequently. The permeability of biologic tissues is important
when examining magnetic fields and is usually assumed to be that of free space. The permittivity is a
more complicated property, but outside cell membranes (which have a high lipid content) it is also usually
considered to be that of free space.
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→
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 and
time. A study of sources of physiologic origin shows that their temporal behavior lies in a low-frequency
range. For example, currents generated by the heart have a power density spectrum that lies mainly under
1 kHz (in fact, clinical ECG instruments have upper frequency limits of 100 Hz), while most other
electrophysiologic sources of interest (i.e., those underlying the EEG, EMG, EOG, etc.) are of even lower
frequency. Examination of electromagnetic fields in regions with typical physiologic conductivities, with
dimensions 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 these
can be maintained only by the presence of a supply of energy (a “battery”). In fact, we may expect that
wherever a physiologic current source J i arises, we also can identify a (normally nonelectrical) energy
source that generates this current. In electrophysiologic processes, the immediate repository of energy is
the potential energy associated with the varying chemical compositions encountered (extracellular ionic
concentrations that differ greatly from intracellular concentrations), but the long-term energy source is
the adenosine triphosphate (ATP) that drives various pumps that create and maintain the aforementioned
concentration gradients.
Based on the aforementioned assumptions, we consider a uniformly conducting medium of conductivity σ and of infinite extent within which a current source J i lies. This, in turn, establishes an electric
field E and, based on Ohm’s law, a conduction current density σ E . The total current density J is the sum
of the aforementioned currents, namely,
→
→
→
→
→
r
r r
J = σE + J i
(9.1)
Now, by virtue of the quasi-static conditions, the electric field may be derived from a scalar potential Φ
[Plonsey and Heppner, 1967] so that
r
E = −∇Φ
(9.2)
→
Since quasi–steady-state conditions apply, J must be solenoidal, and consequently, substituting Eq. (9.2)
into Eq. (9.1) and then setting the divergence of Eq. (9.1) to zero show that Φ must satisfy Poisson’s
equation, namely,
( )
r
∇2 Φ = 1 σ ∇ ⋅ J i
(9.3)
An integral solution to Eq. (9.3) is [Plonsey and Collin, 1961]
(
)
1
Φ p x ′, y ′, z ′ = −
4 πσ
r
∇⋅ J i
dv
v
r
∫
(9.4)
where r in Eq. (9.4) 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 ′) + ( y − y ′) + ( z − z ′)
2
2
2
(9.5)
Note that while, in effect,→we consider relationships arising when ∂/∂t = 0, all fields are actually assumed to vary
in time synchronously with
J i. Furthermore
for the special case of magnetic field stimulation, the source of the
→
→
primary electric field, ∂ A /∂t, where A is the magnetic vector potential, must be retained.
1
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Equation (9.4) may be transformed to an alternate form by employing the vector identity
[( ) ] ( )
( )
r
r
r
∇⋅ 1 r J i ≡ 1 r ∇⋅ J i +∇ 1 r ⋅ J i
(9.6)
→
→
Based on Eq. (9.6), we may substitute for the integrand in Eq. (9.4) the sum ·[(1/r) J i] – (1/r)· J i,
giving the following:
(
)
Φ p x ′, y ′, z ′ = −
∫ [( ) ]
∫ ( )
ri
ri 
1 
 ∇ ⋅ 1 r J dv − ∇ 1 r J dv 
v
4 πσ  v

(9.7)
The first term on the right-hand side may be transformed using the divergence theorem as follows:
∫ ∇ ⋅ [(1 r ) J ] dv = ∫ (1 r ) J
ri
ri
v
r
⋅ dS = 0
(9.8)
s
The volume integral in Eq. (9.4) and (9.8) is defined simply to include all sources. Consequently, in
Eq. (9.8),the surface S, which bounds V, necessarily lies away from J i. Since J i thus is equal to zero on S,
the expression in Eq. (9.8) must likewise equal zero. The result is that Eq. (9.4) also may be written as
→
(
)
1
Φ p x ′, y ′, z ′ = −
4 πσ
r
∇⋅ J i
1
dv =
v
r
4 πσ
∫
→
∫ J ⋅ ∇ (1 r )dv
ri
(9.9)
v
We will derive the mathematical expressions for monopole and dipole fields in the next section, but based
on those results, we can give a physical interpretation of the source terms in each of the integrals on the
right-hand side of Eq. (9.9). In the first, we note that – · J i is a volume source density, akin to charge
density in electrostatics. In the second integral of Eq. (9.9), J i behaves with the dimensions of dipole
moment per unit volume. This confirms an assertion, above, that J i has a dual interpretation as a current
density, as originally defined in Eq. (9.1), or a volume dipole density, as can be inferred from Eq. (9.9);
in either case, its dimension are mA/cm2/ = /mA·cm/cm3.
→
→
→
9.2 Monopole and Dipole Fields in the Uniform
Volume of Infinite Extent
The monopole and dipole constitute the basic source elements in electrophysiology. We examine the
fields produced by each in this section.
If one imagines an infinitely thin wire insulated over its extent except at its tip to be introducing a
current 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 the
current density at a radius r must be given by the total current I0 divided by the area of the spherical
surface, or
r
I r
J = 0 2 ar
4 πr
(9.10)
→
and a r is a unit vector in the radial direction. This current source can be described by the nomenclature
of the previous section as
()
r
∇ ⋅ J i = − I 0δ r
where δ denotes a volume delta function.
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(9.11)
One can apply Ohm’s law to Eq. (9.10) and obtain an expression for the electric field, and if Eq. (9.2)
is also applied, we get
r
E = −∇Φ =
I0
4 πσr 2
r
ar
(9.12)
where σ is the conductivity of the volume conductor. Since the right-hand side of Eq. (9.12) is a function
of r only, we can integrate to find Φ, which comes out
φ=
I0
.
4 πσr
(9.13)
In obtaining Eq. (9.13), the constant of integration was set equal to zero so that the point at infinity has
the usually chosen zero potential.
The dipole source consists of two monopoles of equal magnitude and opposite sign whose spacing
approaches zero and whose magnitude during the limiting process increases such that the product of
spacing and magnitude is constant. If we start out with both component monopoles at the origin, then
the total 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
monopole field resulting from the displacement. For a very small displacement, this amounts to (i.e., we
retain only the linear term in a Taylor series expansion)
→
Φp =
∂  I0 
d
∂d  4 πσr 
(9.14)
The partial derivative in Eq. (9.14) is called the directional derivative, and this can be evaluated by taking
the dot product of the gradient of the expression enclosed in parentheses with the direction of d [that
is, ( )· a d , where a d is a unit vector in the d direction]. The result is
→
→
→
Φp =
→
( )
I 0d r
ad ⋅∇ 1 r
4 πσ
(9.15)
→
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]
Φp =
( )
1 r
m ⋅∇ 1 r .
4 πσ
(9.16)
9.3 Volume Conductor Properties of Passive Tissue
If one were considering an active single isolated fiber lying in an extensive volume conductor (e.g., an
in vitro preparation in a Ringer’s bath), then there is a clear separation between the excitable tissue and
the surrounding volume conductor. However, consider in contrast activation proceeding in the in vivo
heart. 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 inhomogeneous
torso containing a number of contiguous organs (internal to the heart are blood-filled cavities, while
external are pericardium, lungs, skeletal muscle, bone, fat, skin, air, etc.).
The treatment of the surrounding multicellular cardiac tissue poses certain difficulties. A recently used
and reasonable approximation is that the intracellular space, in view of the many intercellular junctions,
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TABLE 9.1 Conductivity Values for
Cardiac Bidomain
S/mm
Clerc [1976]
Roberts [1982]
gix
giy
gax
gay
1.74 × 10–4
1.93 × 10–5
6.25 × 10–4
2.36 × 10–4
3.44 × 10–4
5.96 × 10–5
1.17 × 10–4
8.02 × 10–5
can be represented as a continuum. A similar treatment can be extended to the interstitial space. This
results in two domains that can be regarded as occupying the same physical space; each domain is
separated from the other by the membrane. This view underlies the bidomain model [Plonsey, 1989]. To
reflect the underlying fiber geometry, each domain is necessarily anisotropic, with the high conductivity
axes defined by the fiber direction and with an approximate cross-fiber isotropy. A further simplification
may be possible in a uniform tissue region that is sufficiently far from the sources, since beyond a few
space constants transmembrane currents may become quite small and the tissue would therefore behave
as a single 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
the tissue may still be approximated as a uniform monodomain, but it may be necessary to include some
of the reactive properties introduced via the highly capacitative cell membranes. A classic study by Schwan
and Kay [1957] of the macroscopic (averaged) properties of many tissues showed that the displacement
current was normally negligible compared with the conduction current.
It is not always clear whether a bidomain model is appropriate to a particular tissue, and experimental
measurements found in the literature are not always able to resolve this question. The problem is that if
the experimenter believes the tissue under consideration to be, say, an isotropic monodomain, measurements are set up and interpreted that are consistent with this idea; the inherent inconsistencies may never
come to light [Plonsey and Barr, 1986]. Thus one may find impedance data tabulated in the literature
for a number of organs, but if the tissue is truly, say, an anisotropic bidomain, then the impedance tensor
requires six numbers, and anything less is necessarily inadequate to some degree. For cardiac tissue, it is
usually 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 9.1 as obtained from, essentially, the only
two experiments for which bidomain values were sought.
9.4 Effects of Volume Conductor Inhomogeneities:
Secondary Sources and Images
In the preceding we have assumed that the volume conductor is homogeneous, and the evaluation of
fields from the current sources given in Eq. (9.9) is based on this assumption. Consider what would
happen if the volume conductor in which J i lies is bounded by air, and the source is suddenly introduced.
Equation (9.9) predicts an initial current flow into the boundary, but no current can escape into the
nonconducting surrounding region. We must, consequently, have a transient during which charge piles
up at the boundary, a process that continues until the field from the accumulating charges brings the
net normal component of electric field to zero at the boundary. To characterize a steady-state condition
with no further increase in charge requires satisfaction of the boundary condition that ∂Φ/∂n = 0 at the
surface (within the tissue). The source that develops at the bounding surface is secondary to the initiation
of the primary field; it is referred to as a secondary source. While the secondary source is essential for
satisfaction 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 the
region were simply bounded by one of different conductivity. In this case, when the source is first “turned
on,” since the primary electric field Ea is continuous across the interface between regions of different
conductivity, the current flowing into such a boundary (e.g., σ1 E a) is unequal to the current flowing
→
→
→
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→
away from that boundary (e.g., σ2 E a). Again, this necessarily results in an accumulation of charge, and
a secondary source will grow until the applied plus secondary field satisfies the required continuity of
current density, namely,
− σ1∂ Φ1 ∂n = − σ 2∂Φ2 ∂n = J n
(9.17)
where the surface normal n is directed from region 1 to region 2. The accumulated single source density
can be shown to be equal to the discontinuity of ∂Φ/∂n in Eq. (9.17) [Plonsey, 1974], in particular,
Ks = J n(1/σ 1 – 1/σ 2)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]
(
)
r
r
K ki = Φk σ k′′− σ ′k n
(9.18)
where the condition at the kth interface is described. In Eq. (9.18), the two abutting regions are designated
with prime and double-prime superscripts, and n is directed from the primed to the double-primed
region. Actually, Eq. (9.18) evaluates the double-layer source for the scalar function ψ = Φσ, its strength
being given by the discontinuity in ψ at the interface [Plonsey, 1974] [the potential is necessarily
continuous at the interface with the value Φk called for in Eq. (9.18)]. The (secondary) potential field
generated by K ik, since it constitutes a source for ψ with respect to which the medium is uniform and
infinite in extent, can be found from Eq. (9.9) as
→
→
ψ SP = σ P Φ SP =
1
4π
∑ ∫ Φ (σ′′− σ′ )n ⋅∇(1 r ) dS
r
k
k
k
k
k
(9.19)
where the superscript S denotes the secondary source/field component (alone). Solving Eq. (9.19) for Φ,
we get
Φ SP =
1
4 πσ P
∑ ∫ Φ (σ′′− σ′ )n ⋅∇(1 r ) dS
r
k
k
k
k
k
(9.20)
where σP in Eqs. (9.19) and (9.20) takes on the conductivity at the field point. The total field is obtained
from Eq. (9.20) by adding the primary field. Assuming that all applied currents lie in a region with
conductivity σa , then we have
Φ SP =
1
4 πσ a
∫
( )
r
J i ⋅∇ 1 r dv +
1
4 πσ P
∑ ∫ Φ (σ′′− σ′ )n ⋅∇(1 r ) dS
r
k
k
k
k
k
(9.21)
[if the primary currents lie in several conductivity compartments, then each will yield a term similar to
the first integral in Eq. (9.21)]. Note that in Eqs. (9.20) and (9.21) the secondary source field is similar
in form to the field in a homogeneous medium of infinite extent, except that σP is piecewise constant
and consequently introduces interfacial discontinuities. With regard to the potential, these just cancel
the discontinuity introduced by the double layer itself so that Φk is appropriately continuous across each
passive interface.
The primary and secondary source currents that generate the electrical potential in Eq. (9.21) also set
up a magnetic field. The primary source, for example, is the forcing function in the Poisson equation
for the vector potential A [Plonsey, 1981], namely,
→
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r
r
∇2 A = − J i
(9.22)
From this it is not difficult to show that, due to Eq. (9.21) for Φ, we have the following expression for
the magnetic field H [Plonsey, 1981]:
→
r
1
H=
4π
∫
( )
r
1
J i × ∇ 1 r dv +
4π
∑ ∫ Φ (σ′′− σ′ )n × ∇(1 r ) dS
r
k
k
k
(9.23)
k
A simple illustration of these ideas is found in the case of two semi-infinite regions of different
conductivity, 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 plus
an 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 actual
source point but of strength [1 + (ρ2 – ρ1)/(ρ2 + ρ1). One can confirm this by noting that all fields satisfy
Poisson’s equation and that at the interface Φ is continuous while the normal component of current
density is also continuous (i.e., σ1∂Φ1/∂n = σ2∂Φ2 /∂n).
The potential on the interface is constant along a circular path whose origin is the foot of the
perpendicular from the point source. Calling this radius r and applying Eq. (9.13), we have for the surface
potential Φs
ΦS =
 ρ2 − ρ1 
1 +

4 π h2 + r 2  ρ2 + ρ1 
ρ1
(9.24)
→
and consequently a secondary double-layer source K s equals, according to Eq. (9.18),
r
KS =
 ρ2 − ρ1 
r
1 + ρ + ρ  σ 2 − σ1 n
2
2 
4π h + r
2
1
(
ρ1
)
(9.25)
→
→
where n is directed from region 1 to region 2. The field from Ks in region 1 is exactly equal to that from
a point source of strength (ρ2 – ρ1)/(ρ2 + ρ1) at the mirror-image point, which can be verified by evaluating
and showing the equality of the following:
ΦP =
(
)(
ρ2 − ρ1 ρ2 _ ρ1
r
1
K S ⋅∇ 1 r dS =
4 πσ1
4 πσ1R
∫
( )
)
(9.26)
where R in Eq. (9.26) is the distance from the mirror-image point to the field point P, and r in Eq. (9.26)
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.
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Plonsey R. 1981. Generation of magnetic fields by the human body (theory). In S-N Erné, H-D Hahlbohm, 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, Barr RC. 1986. A critique of impedance measurements in cardiac tissue. Ann Biomed Eng
14:307.
Plonsey R, Collin RE. 1961. Principles and Applications of Electromagnetic Fields. New York, McGraw-Hill.
Plonsey R, Heppner D. 1967. Consideration of quasi-stationarity in electrophysiological systems. Bull
Math Biophys 29:657.
Roberts D, Scher AM. 1982. Effect of tissue anisotropy on extracellular potential fields in canine myocardium in situ. Circ Res 50:342.
Schwan HP, Kay CF. 1957. The conductivity of living tissues. NY Acad Sci 65:1007.
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