FIGURE 10.1 (A) A schematic diagram of a suspension of spherical cells; the ... parameters is an old and interesting problem in electromagnetic theory [Cole, ...
Roth, B. J. “The Electrical Conductivity of Tissues.” The Biomedical Engineering Handbook: Second Edition. Ed. Joseph D. Bronzino Boca Raton: CRC Press LLC, 2000
10 The Electrical Conductivity of Tissues
Bradley J. Roth Oakland University
10.1
10.1 10.2 10.3 10.4
Introduction Cell Suspensions Fiber Suspensions Syncytia
Introduction
One of the most common problems in bioelectric theory is the calculation of the potential distribution, Φ (V), throughout a volume conductor. The calculation of Φ is important in impedance imaging, cardiac pacing and defibrillation, electrocardiogram and electroencephalogram analysis, and functional electrical stimulation. In bioelectric problems, Φ often changes slowly enough so that we can assume it to be quasistatic [Plonsey, 1969]; that is, we ignore capacitive and inductive effects and the finite speed of electromagnetic radiation. (For bioelectric phenomena, this approximation is usually valid for frequencies roughly below 100 kHz.) Under the quasistatic approximation, the continuity equation states that the divergence, ∇•, of the current density, J (A/m2), is equal to the applied or endogenous source of electrical current, S (A/m3):
∇ • J = S.
(10.1)
In regions where there are no such sources, S is zero. In these cases, the divergenceless of J is equivalent to the law of conservation of current that is often invoked when analyzing electrical circuits. Another fundamental property of a volume conductor is that the current density and the electric field, E (V/m), are related linearly by Ohm’s Law,
J = g E,
(10.2)
where g is the electrical conductivity (S/m). Finally, the relationship between the electric field and the gradient, ∇, of the potential is
E = –∇ Φ .
(10.3)
The purpose of this chapter is to characterize the electrical conductivity. This task is not easy, because g is generally a macroscopic parameter (an “effective conductivity”) that represents the electrical properties of the tissue averaged in space over many cells. The effective conductivity can be anisotropic, complex (containing real and imaginary parts), and can depend on both the temporal and spatial frequencies.
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Before beginning our discussion of tissue conductivity, let us consider one of the simplest volume conductors: saline. The electrical conductivity of saline arises from the motion of free ions in response to a steady electric field, and is on the order of 1 S/m. Besides conductivity, another property of saline is its electrical permittivity, ε (S s/m). Permittivity is related to the dielectric constant, κ (dimensionless), by ε = κ εo, where εo is the permittivity of free space, 8.854 × 10-12 S s/m. (The permittivity is also related to the electric susceptibility, χ, by ε = εo + χ.) Dielectric properties arise from a bound charge that is displaced by the electric field, creating a dipole. Dielectric behavior also arises if the applied electric field aligns molecular dipoles (such as the dipole moments of water molecules) that normally are oriented randomly. The DC dielectric constant of saline is similar to that of water (about κ = 80). The movement of free charge produces conductivity, whereas stationary dipoles produce permittivity. In steady state the distinction between the two is clear, but at higher frequencies the concepts merge. In such a case, we can combine the electrical properties into a complex conductivity, g′:
g′ = g + iωε ,
(10.4)
where ω (rad/s) is the angular frequency (ω = 2πf, where f is the frequency in Hz) and i is – 1 . The real part of g′ accounts for the movement of charge that is in phase with the electric field; the imaginary part accounts for out-of-phase motion. Both the real and the imaginary parts of the complex conductivity may depend on the frequency. For many bioelectric phenomena, the first term in Eq. (10.4) is much larger than the second, so the tissue can be represented as purely conductive [Plonsey, 1969]. (The imaginary part of the complex conductivity represents a capacitive effect, and therefore technically violates our assumption of quasistationarity. This violation is the only exception we make to our rule of a quasistatic potential.)
10.2
Cell Suspensions
The earliest and simplest model describing the effective electrical conductivity of a biological tissue is a suspension of cells in a saline solution [Cole, 1968]. Let us consider a suspension of spherical cells, each of radius a (Fig. 10.1a). The saline surrounding the cells constitutes the interstitial space (conductivity σe), while the conducting fluid inside the cells constitutes the intracellular space (conductivity σi). (We
FIGURE 10.1 (A) A schematic diagram of a suspension of spherical cells; the effective conductivity of the suspension is IL/VA. (B) An electric circuit equivalent of the effective conductivity of the suspension.
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shall follow Henriquez [1993] in denoting macroscopic effective conductivities by g and microscopic conductivities by σ.) The cell membrane separates the two spaces; a thin layer having conductivity per unit area Gm (S/m2) and capacitance per unit area Cm (F/m2). One additional parameter—the intracellular volume fraction, f (dimensionless)—indicates how tightly the cells pack together. The volume fraction can range from nearly zero (a dilute solution) to almost 1 (spherical cells cannot approach a volume fraction of 1, but tightly packed nonspherical cells can). For irregularly shaped cells, the cell “radius” can be difficult to define. In these cases, it is easier to specify the surface-to-volume ratio of the tissue (the ratio of the membrane surface area to tissue volume). For spherical cells, the surface-to-volume ratio is 3f/a. We can define operationally the effective conductivity, g, of the cell suspension by the following process (Fig. 10.1a): Place the cell suspension in a cylindrical tube of length L and cross-sectional area A (be sure L and A are large enough so the volume contains many cells). Apply a DC potential difference V across the two ends of the cylinder (so that the electric field has strength V/L) and measure the total current, I, passing through the suspension. The effective conductivity is IL/VA. Deriving an expression for the effective conductivity of a suspension of spheres in terms of microscopic parameters is an old and interesting problem in electromagnetic theory [Cole, 1968]. For DC fields, the effective conductivity, g, of a suspension of insulating spheres placed in a saline solution of conductivity σe is
g=
( )σ .
2 1− f 2+f
(10.5)
e
For most cells, Gm is small enough so that the membrane behaves essentially as an insulator at DC, in which case the assumption of insulating spheres is applicable. The net effect of the cells within the saline is to decrease the conductivity of the medium (the decrease can be substantial for tightly packed cells). The cell membrane has a capacitance of about 0.01 F/m2 (or, in traditional units, 1 µF/cm2), which introduces a frequency dependence into the electrical conductivity. The suspension of cells can be represented by the electrical circuit in Fig. 10.1b: Re is the effective resistance to current passing entirely through the interstitial space; Ri is the effective resistance to current passing into the intracellular space; and C is the effective membrane capacitance. (The membrane conductance is usually small enough so that it has little effect on the suspension behavior, regardless of the frequency.) At low frequencies, all of the current is restricted to the interstitial space, and the electrical conductivity is given approximately by Eq. (10.5). At large frequencies, C shunts current across the membrane, so that the effective conductivity of the tissue is again resistive:
g=
( ) ( ) (2 + f ) σ + (1 − f ) σ
2 1 − f σ e + 1 + 2f σ i e
σe .
(10.6)
i
At intermediate frequencies, the effective conductivity has both real and imaginary parts, because the membrane capacitance contributes significantly to the effective conductivity. In these cases, Eq. (10.6) still holds if σi is replaced by σ i∗ , where
σ∗i =
σ i Ym a , σ i + Ym a
with
Ym = Gm + i ω C m .
(10.7)
Figure 10.2 shows the effective conductivity (magnitude and phase) as a function of frequency for a typical tissue. The increase in the phase at about 300 kHz is sometimes called the “beta dispersion”.
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FIGURE 10.2 The magnitude and phase of the effective conductivity as a function of frequency, for a suspension of spherical cells: f = 0.5; a = 20 µm; σe = 1 S/m; σi = 0.5 S/m; Gm = 0; and Cm = 0.01 F/m2.
10.3
Fiber Suspensions
Some of the most interesting electrically active tissues, such as nerve and skeletal muscle, are better approximated as a suspension of fibers rather than as a suspension of spheres. This difference has profound implications because it introduces anisotropy: The effective electrical conductivity depends on direction. Henceforth, we must speak of the longitudinal effective conductivity parallel to the fibers, gL, and the transverse effective conductivity perpendicular to the fibers, gT. (In theory, the conductivity could be different in three directions; however, we consider only the case in which the electrical properties in the two directions perpendicular to the fibers are the same.) The conductivity is no longer a scalar quantity, but is a tensor instead, and therefore must be represented by a 3 × 3 symmetric matrix. If we choose our coordinate axes along the principle directions of this matrix (invariably, the directions parallel to and perpendicular to the fibers), then the off-diagonal terms of the matrix are zero. If, however, we choose our coordinate axes differently, or if the fibers curve so that the direction parallel to the fibers varies over space, we have to deal with tensor properties, including off-diagonal components. When the electric field is perpendicular to the fiber direction, a suspension of fibers is similar to the suspension of cells described above (in Fig. 10.1a, we must now imagine that the circles represent crosssections of cylindrical fibers, rather than cross-sections of spherical cells). The expression for the effective transverse conductivity of a suspension of cylindrical cells, of radius a and intracellular conductivity σi , placed in a saline solution of conductivity σe , with intracellular volume fraction f, is [Cole, 1968]
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FIGURE 10.3 An electrical circuit representing a one-dimensional nerve or muscle fiber: ri and re are the intracellular and extracellular resistances per unit length (Ω/m); rm is the membrane resistance times unit length (Ω m); and cm is the membrane capacitance per unit length (F/m).
gT =
(1 − f ) σ + (1 + f ) σ (1 + f ) σ + (1 − f ) σ e
∗ i
e
∗ i
σe ,
(10.8)
where Eq. (10.7) defines σi∗. At DC (and assuming that Gm = 0) Eq. (10.8) reduces to
gT =
1− f σe . 1+ f
(10.9)
When an electric field is parallel to the fiber direction, a new behavior arises that is fundamentally different from that observed for a suspension of spherical cells. Let us return for a moment to our operational definition of the effective conductivity. Surprisingly, the effective longitudinal conductivity of a suspension of fibers depends on the length L of the tissue sample used for the measurement. To understand this phenomenon, we must consider one-dimensional cable theory [Plonsey, 1969]. A single nerve or muscle fiber can be approximated by the circuit shown in Fig. 10.3. Adopting the traditional electrophysiology definitions, we denote the intracellular and extracellular resistances per unit length along the fiber by ri and re (Ω/m), the membrane resistance times unit length by rm (Ω m), and the capacitance per unit length by cm (F/m). The cable equation governs the transmembrane potential, Vm :
λ2
∂2 Vm ∂x
2
=τ
∂Vm + Vm , ∂t
(10.10)
where τ is the time constant, rmcm, and λ is the length constant, r m ⁄ ( r i + r e ) . For a truncated fiber of length L (m) with sealed ends, and with a steady-state current I (A) injected into the extracellular space at one end and removed at the other, the solution to the cable equation is
Vm = I re λ
( ), cosh(L 2λ ) sinh x λ
(10.11)
where the origin of the x-axis is at the midpoint between electrodes. The extracellular potential, Ve , consists of two terms: One is proportional to x, and the other is re /(ri+re) times Vm . We can evaluate Ve at the two ends of the fiber to obtain the voltage drop between the electrodes, ∆Ve
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∆Ve =
ri re r I L + e 2λ tanh L 2λ . ri + re ri
(
)
(10.12)
Two limiting cases are of interest. If L is very large compared to λ , the extracellular voltage drop reduces to
∆Ve =
ri re LI ri + re
L >> λ .
(10.13)
The leading factor is the parallel combination of the intracellular and extracellular resistances. If, on the other hand, L is very small compared to λ , the extracellular voltage drop becomes
∆Ve = re L I
L
