21The Electrical

Conductivity ofTissues

Bradley J. RothOakland University

21.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-121.2 Cell Suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-221.3 Fiber Suspensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-321.4 Syncytia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-6Defining Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-11Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-11Further Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21-12

21.1 Introduction

One of the most important problems in bioelectric theory is the calculation of the electrical potential, (V), throughout a volume conductor. The calculation of is important in impedance imaging,cardiac pacing and defibrillation, electrocardiogram and electroencephalogram analysis, and functionalelectrical stimulation. In bioelectric problems, often changes slowly enough that we can assume it isquasistatic [Plonsey, 1969]; we ignore capacitive and inductive effects and the finite speed of propagationof electromagnetic radiation. (Usually for bioelectric phenomena, this assumption is valid for frequenciesbelow about 100 kHz.) Under the quasistatic approximation, the continuity equation states that thedivergence,∇·, of the current density, J (A/m2), is equal to the applied or endogenous source of electricalcurrent, S (A/m3):

∇ · J = S. (21.1)

In regions of tissue where there are no sources, S is zero. In these cases, the divergenceless of J is equivalentto the law of conservation of current that is often invoked when analyzing electrical circuits. Anotherproperty of a volume conductor is that the current density and the electric field, E (V/m), are relatedlinearly by Ohm’s Law,

J = g E, (21.2)

21-1

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21-2 Biomedical Engineering Fundamentals

where g is the electrical conductivity (S/m). Finally, the relationship between the electric field and thegradient,∇, of the potential is

E = −∇. (21.3)

The purpose of this chapter is to characterize the electrical conductivity. This task is not easy, because gis generally a macroscopic parameter (an “effective conductivity”) that represents the electrical propertiesof the tissue averaged over many cells. The effective conductivity can vary with direction, can be complex(contain real and imaginary parts), and can depend on the temporal and spatial frequencies.

Before discussing the conductivity of tissue, consider one of the simplest and most easily understoodvolume conductors: saline. The electrical conductivity of saline arises from the motion of free ions inresponse to a steady electric field, and is on the order of 1 S/m. Besides conductivity, another propertyof saline is its electrical permittivity, ε (S sec/m). This property is related to the dielectric constant, κ(dimensionless), by ε = κε0, where ε0 is the permittivity of free space, 8.854 × 10−12 S sec/m. Dielectricproperties arise from bound charge that is displaced by the electric field, creating a dipole. They can alsoarise if the applied electric field aligns molecular dipoles (such as the dipole moments of water molecules)that are normally oriented randomly. The DC dielectric constant of saline is similar to that of water (aboutκ = 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. Insuch a case, we can combine the electrical properties into a complex conductivity, g ′:

g ′ = g + iωε, (21.4)

where ω (rad/sec) is the angular frequency (ω = 2π f , where f is the frequency in Hz) and i is√−1. The

real part of g ′ accounts for the movement of charge in phase with the electric field; the imaginary partaccounts for out-of-phase motion. Both the real and the imaginary parts of the complex conductivity maydepend on the frequency. For many bioelectric phenomena, the first term in Equation 21.4 is much largerthan the second, so the tissue can be represented as purely conductive [Plonsey, 1969]. (The imaginarypart of the complex conductivity represents a capacitive effect, and therefore technically violates ourassumption of quasistationarity. This violation is the only exception we will make to our general rule of aquasistatic potential.)

21.2 Cell Suspensions

The earliest and simplest model describing the electrical conductivity of a biological tissue is a suspensionof cells in a saline solution [Cole, 1968; Peters et al., 2001]. Let us consider a suspension of spherical

(conductivity σe), while the conducting fluid inside the cells constitutes the intracellular space (conduct-ivity σi). (We shall follow Henriquez [1993] in denoting macroscopic effective conductivities by g andmicroscopic conductivities by σ .) The cell membrane separates the two spaces; a thin layer having con-ductivity 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 are packed together.The volume fraction can range from nearly zero (a dilute solution) to almost 1. (Spherical cells cannotapproach a volume fraction of 1, but tightly packed, nonspherical cells can). For irregularly shaped cells,the “radius” is difficult to define. In these cases, it is easier to specify the surface-to-volume ratio of thetissue (the ratio of the membrane surface area to tissue volume). For spherical cells, the surface-to-volumeratio is 3f /a.

We can define operationally the effective conductivity, g , of a cell suspension by the following process(Figure 21.1a): Place the suspension in a cylindrical tube of length L and cross-sectional area A (be sureL and A are large enough so the volume contains many cells). Apply a DC potential difference V across

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cells, each of radius a (Figure 21.1a). The saline surrounding the cells constitutes the interstitial space

The Electrical Conductivity of Tissues 21-3

(a) (b)A

a

V

I

V

I

Re Ri

C

L

FIGURE 21.1 (a) A schematic diagram of a suspension of spherical cells; the effective conductivity of this suspensionis IL/VA. (b) An electric circuit equivalent of the effective conductivity of this suspension.

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 microscopicparameters is an old and interesting problem in electromagnetic theory [Cole, 1968]. For DC fields, theeffective conductivity, g , of a suspension of insulating spheres placed in a solution of conductivity σe is

g = 2(1− f )

2+ fσe. (21.5)

For most cells, Gm is small enough so that the membrane behaves as an insulator, and hence the assumptionof insulating spheres is applicable. The net effect of the cells is to decrease the conductivity of the solution(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), whichcauses the electrical conductivity to depend on frequency. The electrical circuit in Figure 21.1b representsthe suspension of cells: Re is the effective resistance to current passing entirely through the interstitialspace; Ri is the effective resistance to current passing into the intracellular space; and C is the effectivemembrane capacitance. (The membrane conductance is usually small enough so that it has little effect,regardless of the frequency.) At low frequencies all of the current is restricted to the interstitial space, andthe electrical conductivity is given approximately by Equation 21.5 above. At high frequencies, C shuntscurrent across the membrane, so that the effective conductivity of the tissue is:

g = 2(1− f )σe + (1+ 2f )σi

(2+ f )σe + (1− f )σiσe. (21.6)

At intermediate frequencies, the effective conductivity has both real and imaginary parts, because themembrane capacitance contributes significantly to the effective conductivity. In these cases, Equation 21.6still holds if σi is replaced by σ ∗i , where

σ ∗i =σiYma

σi + Ymawith Ym = Gm + iωCm . (21.7)

tissue. The increase in the phase at about 300 kHz is sometimes called the “beta dispersion.”

21.3 Fiber Suspensions

Many of the most interesting electrically active tissues, such as nerve and skeletal muscle, are better approx-imated as a suspension of cylinders rather than a suspension of spheres. This difference has profound

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Figure 21.2 shows the effective conductivity (magnitude and phase) as a function of frequency for a typical

21-4 Biomedical Engineering Fundamentals

FIGURE 21.2 The magnitude and phase of the effective conductivity as a function of frequency, for a suspension ofspherical cells: f = 0.5; a = 20 µm; σe =1 S/m; σi = 0.5 S/m; Gm = 0; and Cm = 0.01 F/m2.

implications because it introduces anisotropy. The effective electrical conductivity depends on direction.Henceforth, we must speak of the longitudinal effective conductivity parallel to the cylindrical fibers,gL, and the transverse effective conductivity perpendicular to the fibers, gT. (In theory, the conductivitycould be different in three directions; however, often the electrical properties in the two directions per-pendicular to the fibers are the same.) In general, the conductivity is no longer a scalar quantity, but is atensor instead, and must be represented by a 3× 3 symmetric matrix. If our coordinate axes lie along theprinciple 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 withtensor properties, including off-diagonal components.

When the electric field is applied perpendicular to the fiber direction, a suspension of fibers is similar

cross sections of cylindrical fibers, rather than spherical cells). The expression for the effective transverseconductivity of a suspension of cylindrical cells, of radius a and intracellular conductivity σi, placed in asolution of conductivity σe, with intracellular volume fraction f , is [Cole, 1968]

gT = (1− f )σe + (1+ f )σ ∗i(1+ f )σe + (1− f )σ ∗i

σe, (21.8)

where Equation 21.7 defines σ ∗i . At DC, and for Gm = 0, Equation 21.8 reduces to

gT = 1− f

1+ fσe. (21.9)

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to the suspension of cells described above (in Figure 21.1a, we must now imagine that the circles represent

0.7

0.6

0.5

0.4

0.1 100

Frequency (kHz)

100,000

g (m

ag)

S/m

20

10

00.1 100

Frequency (kHz)

100,000

g (p

hase

) (°

)

The Electrical Conductivity of Tissues 21-5

re

rm

ri

Cm

FIGURE 21.3 An electrical circuit representing a one-dimensional nerve or muscle fiber: ri and re are the intracellularand extracellular resistances per unit length (/m); rm is the membrane resistance times unit length (m); and cm isthe membrane capacitance per unit length (F/m).

When an electric field is applied parallel to the fiber direction, a new behavior arises that is fundamentallydifferent from that observed for a suspension of spherical cells. Return for a moment to our operationaldefinition of the effective conductivity. Surprisingly, the effective longitudinal conductivity of a suspensionof fibers depends on the length L of the tissue sample used for the measurement. To understand thisdependence, we must consider one-dimensional cable theory [Plonsey, 1969]. The circuit in Figure 21.3aapproximates a single nerve or muscle fiber. Adopting the traditional electrophysiology nomenclature, wedenote the intracellular and extracellular resistances per unit length along the fiber by ri and re (/m),the membrane resistance times unit length by rm(m), and the capacitance per unit length by cm (F/m).The cable equation governs the transmembrane potential, Vm :

λ2 ∂2Vm

∂x2= τ ∂Vm

∂t+ Vm , (21.10)

where τ is the time constant, rmcm , and λ is the length constant,√

rm/(ri + re). For a truncated fiber oflength L (m) with sealed ends, and with a steady-state current I (A) injected into the extracellular spaceat one end and removed at the other, the solution to the cable equation is

Vm = Ireλsinh(x/λ)

cosh(L/2λ), (21.11)

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

Ve = rire

ri + reI

[L + re

ri2λ tanh

(L

2λ

)]. (21.12)

If L is very large compared to λ, the extracellular voltage drop reduces to

Ve = rire

ri + reLI L >> λ. (21.13)

The leading factor is the parallel combination of the intracellular and extracellular resistances. If, on theother hand, L is very small compared to λ, the extracellular voltage drop becomes

Ve = reLI L << λ. (21.14)

In this case, the leading factor is simply the extracellular resistance alone. Physically, there is a redistributionof current into the intracellular space that occurs over a distance on the order of a length constant. Ifthe tissue length is much longer than a length constant, the current is redistributed completely between

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21-6 Biomedical Engineering Fundamentals

the intracellular and extracellular spaces. If the tissue length is much smaller than a length constant, thecurrent does not enter the fiber, but instead is restricted to the extracellular space. If either of these twoconditions is met, then the effective conductivity (IL/AVe, where A is the cross-sectional area of thetissue strand) is independent of L. However, if L is comparable to λ, the effective conductivity dependson the size of the tissue being studied.

Roth et al. [1988] recast the expression for the effective longitudinal conductivity in terms of spatialfrequency, k (rad/m). This approach has two advantages. First, the temporal and spatial behaviors areboth described using frequency analysis. Second, a parameter describing the size of a specific piece of tissueis not necessary: the spatial frequency dependence becomes a property of the tissue, not the measurement.The expression for the DC effective longitudinal conductivity is

gL = (1− f )σe + f σi

1+ f σi

(1− f )σe

1

1+(

1

λk

)2

(21.15)

To relate the effective longitudinal conductivity to Equation 21.13 and Equation 21.14 above, note that1/k plays the same role as L. If kλ 1, gL reduces to (1 − f )σe + f σi, which is equivalent to theparallel combination of resistances in Equation 21.13. If kλ 1, gL becomes (1− f )σe, implying that thecurrent is restricted to the interstitial space, as in Equation 21.14. Equation 21.15 can be generalized to alltemporal frequencies by defining λ in terms of Ym instead of Gm

the magnitude and phase of the longitudinal and transverse effective conductivities as functions of thetemporal and spatial frequencies.

The measurement of effective conductivities is complicated by the traditionally used electrode geometry.Typically, one uses a four-electrode technique [Steendijk et al., 1993], in which two electrodes inject current

electrical properties of skeletal muscle and found that the effective conductivity depended on the interelec-trode distance. Roth [1989] reanalyzed Gielen et al.’s data using the spatial frequency dependent model

contains typical values of skeletal muscle effective conductivities and microscopic tissue parameters.

21.4 Syncytia

Cardiac tissue is different from the other tissues we have discussed in that it is an electrical syncytium: thecells are coupled through intercellular junctions. The bidomain model describes the electrical propertiesof cardiac muscle [Henriquez, 1993]. It is essentially a two- or three-dimensional cable model that takes

concept of current redistribution, discussed earlier in the context of the longitudinal effective conductivityof a suspension of fibers, now applies in all directions. Furthermore, cardiac muscle is markedly aniso-tropic. These properties make impedance measurements of cardiac muscle difficult to interpret [Plonseyand Barr, 1986; Le Guyader et al., 2001]. The situation is complicated further because the intracellularspace is more anisotropic than is the interstitial space (in the jargon of bidomain modeling, this con-dition is known as “unequal anisotropy ratios”) [Roth, 1997]. Consequently, an expression for a singleeffective conductivity for cardiac muscle is difficult, if not impossible, to derive. In general, one mustsolve a pair of coupled partial differential equations simultaneously for the intracellular and interstitialpotentials.

The bidomain model characterizes the electrical properties of the tissue by four effective conductivities:giL, giT, geL, and geT, where i and e denote the intracellular and interstitial spaces, and L and T denote the

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Table 21.2 lists nerve effective conductivities.

[Roth et al., 1988]. Figure 21.4 shows

and two others measure the potential (Figure 21.5). Gielen et al. [1984] used this method to measure the

and found agreement with some of the more unexpected features or their data (Figure 21.6). Table 21.1

into account the resistance of both the intracellular and the interstitial spaces (Figure 21.7). Thus, the

The Electrical Conductivity of Tissues 21-7

0.7

(a) (b)

(c) (d)

0.6

0.5

0.4

0.3

0.21

1

101

101102

103104

105106

102 103

k, 1/m

k, 1/m

104 105 106

szb

szb

1/Ωm

v 1/se

c

11

101

101102

103104

105106

102 103

k, 1/m104 105 106

v 1/se

c

1.0

90°

0°

5°

10°

15°

20°

60°

30°

0°

0.50.4

0.3

0.1

0.2

1

1

101

101102

103104

105106

102 103 104 105 106

k, 1/m1 101102103104105106

sr

b

sr

b

1/Ωm

v 1/se

c

1101

102103

104105

106

v 1/se

c

Phaseof

Phaseof

FIGURE 21.4 The magnitude (a), (c) and phase (b), (d) of the effective longitudinal (a), (b) and transverse (c), (d)effective conductivities, calculated using the spatial, k, and temporal,ω, frequency model. The parameters used in thiscalculation are Gm = 1 S/m2; Cm = 0.01 F/m2; f = 0.9; a = 20 µm; σi =0.55 S/m; and σe = 2 S/m. (From Roth, B.J.,Gielen, F.L.H., and Wikswo, J.P., Jr. 1988. Math. Biosci. 88: 159–189. With permission.)

V

I

s

FIGURE 21.5 A schematic diagram of the four-electrode technique for measuring tissue conductivities. Current, I ,is passed through the outer two electrodes, and the potential, V , is measured between the inner two. The interelectrodedistance is s.

directions parallel to and perpendicular to the myocardial fibers. We can relate these parameters to themicroscopic tissue properties by using an operational definition of an effective bidomain conductivity,similar to the operational definition given earlier. To determine the interstitial conductivity, first dissect acylindrical tube of tissue of length L and cross-sectional area A (one must be sure that L and A are large

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21-8 Biomedical Engineering Fundamentals

FIGURE 21.6 The calculated (a) amplitude and (b) phase of gL (solid) and gT (dashed) as a function of frequency, foran interelectrode distance of 0.5 mm; (c) and (d) show the quantities for an interelectrode distance of 3.0 mm. Circlesrepresent experimental data; gL (filled), gT (open). (From Roth, B.J. 1989. Med. Biol. Eng. Comput., 27: 491–495. Withpermission.)

TABLE 21.1 Skeletal Muscle

Macroscopic effective conductivities (S/m)

gL gT Note Reference

0.35 0.086 10 Hz, IED= 3 mm Gielen et al. [1984]0.20 0.092 10 Hz, IED= 0.5 mm0.52 0.076 20 Hz, IED= 17 mm Epstein and Foster [1983]0.70 0.32 100 kHz, IED= 17 mm0.67 0.040 0.1 sec pulse Rush et al. [1963]

Microscopic tissue parameters

σi σe f Cm Gm References(S/m) (S/m) (F/m2) (S/m2)

0.55 2.4 0.9 0.01 1.0 Gielen et al. [1986]

Note: IED = interelectrode distance.

enough so the volume contains many cells, and that the dissection does not damage the tissue). Next,apply a drug to the tissue that makes the membrane essentially insulating (i.e., the length constant is muchlonger than L). Finally, apply a DC potential difference, V , across the two ends of the cylinder and measurethe total current, I . The effective interstitial conductivity is IL/VA. This procedure must be performedtwice, once with the fibers parallel to the axis of the cylinder, and once with the fibers perpendicular to it.To determine the effective intracellular conductivities, follow the above procedure but apply the voltagedifference to the intracellular space instead of the interstitial space. Although the procedure would be

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1(a)

0.5

0.3

0.2

0.1

1 10 100 1,000 10,000

S/m

40(b)

20

0

–20

Pha

se (

°)1(c)

0.5

0.3

0.2

0.1

S/m

40(d)

20

0

–20

Pha

se (

°)

f (Hz)1 10 100 1,000 10,000

f (Hz)

1 10 100 1,000 10,000

f (Hz)1 10 100 1,000 10,000

f (Hz)

The Electrical Conductivity of Tissues 21-9

TABLE 21.2 Nerve

Macroscopic effective conductivities (S/m)

gL gT Note Reference

0.41 0.01 Toad sciatic nerve Tasaki [1964]0.57 0.083 Cat dorsal column, 10 Hz Ranck and BeMent [1965]

Microscopic tissue parameters

σi (S/m) σe (S/m) f Cm (F/m2) Gm (S/m2) Reference

0.64 1.54 0.35 1 0.44 Roth and Altman [1992]

Note: Volume fraction of myelin = 0.27, Gm is proportional to axon diameter; theabove value is for an axon with outer diameter of 6.5 µm.

FIGURE 21.7 A circuit representing a two-dimensional syncytium (i.e., a bidomain). The lower array of resistorsrepresents the intracellular space, the upper array represents the extracellular space, and the parallel resistors andcapacitors represent the membrane.

extraordinarily difficult in practice, we can imagine two arrays of microelectrodes that impale the cells atboth ends of the cylinder and maintain a constant potential at each end.

Expressions have been derived for the effective bidomain conductivities in terms of the microscopictissue parameters [Roth, 1988; Henriquez, 1993; Neu and Krassowska, 1993]. The effective conductivitiesin the direction parallel to the fibers are simplest. Imagine that the tissue is composed of long, straight

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21-10 Biomedical Engineering Fundamentals

TABLE 21.3 Cardiac Muscle

Macroscopic effective conductivities (ventricular muscle) (S/m)

giL giT geL geT Reference

0.17 0.019 0.62 0.24 Clerc0.28 0.026 0.22 0.13 Roberts et al.0.34 0.060 0.12 0.080 Roberts and Scher

Microscopic tissue parameters

σi σe f a b G Reference(S/m) (S/m) (µm) (µm) (µS)

1 1 0.7 100 10 3 Roth [1988]0.4 2 0.85 100 7.5 0.05 Neu and Krassowska [1993]

fibers (like skeletal muscle) and that the intracellular space of these fibers occupies a fraction f of thetissue cross-sectional area. If the conductivity of the interstitial fluid is σe, then the effective interstitialconductivity parallel to the fibers, geL, is simply

geL =(1− f

)σe. (21.16)

If we neglect the resistance of the gap junctions, we obtain a similar expression for the effectiveintracellular conductivity parallel to the fibers in terms of the myoplasmic conductivity, σi: giL = f σi.When the gap junctional resistance is not negligible compared to the myoplasmic resistance, the expressionfor giL is more complicated:

giL = 1

1+ (πa2σi)/(bG)f σi, (21.17)

where G is the junctional conductance between two cells (S), b is the cell length (m), and a is the cellradius (m).

The effective interstitial conductivity perpendicular to the fibers is identical with the DC transverseeffective conductivity for skeletal muscle given in Equation 21.9

geT = 1− f

1+ fσe. (21.18)

The effective intracellular conductivity perpendicular to the fibers is the most difficult to model, but areasonable expression for giT is

giT = 1

1+ (bσi)/Gσi. (21.19)

of the microscopic tissue parameters are also given in Table 21.3, although some are quite uncertain(particularly G).

If the intercellular junctions contribute significantly to the intracellular resistance, the bidomain modelonly approximates the tissue behavior [Neu and Krassowska, 1993]. For sufficiently large junctionalresistance, the discrete cellular properties become important, and a continuum model no longer representsthe tissue well. Interestingly, as the junctional resistance increases, cardiac tissue behaves less like asyncytium and more like a suspension of cells. Thus we have come a full circle. We started by consideringa suspension of cells, then examined suspensions of fibers, and finally generalized to syncytia. Yet, whenthe intercellular junctions in a syncytium are disrupted, we find ourselves again thinking of the tissue as asuspension of cells.

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Table 21.3 contains measured values of the bidomain conductivities (see also Roth [1997]). Typical values

The Electrical Conductivity of Tissues 21-11

Defining Terms

Anisotropy: Having different properties in different directions.Bidomain: A two- or three-dimensional cable model that takes into account the resistance of both the

intracellular and the extracellular spaces.Cable theory: Representation of a cylindrical fiber as two parallel rows of resistors (one each for the

intracellular and extracellular spaces) connected in a ladder network by a parallel combination ofresistors and capacitors (the cell membrane).

Conservation of current: A fundamental law of electrostatics, stating that there is no net currententering or leaving at any point in a volume conductor.

Conductivity: A parameter (g ) that measures how well a substance conducts electricity. The coefficientof proportionality between the electric field and the current density. The units of conductivity aresiemens per meter (S/m). A siemens is an inverse ohm, sometimes called a “mho” in the olderliterature.

Interstitial space: The extracellular space between cells in a tissue.Ohm’s law: A linear relation between the electric field and current density vectors.Permittivity: A parameter (ε) that measures the size of the dipole moment induced in a substance by

an electric field. The units of permittivity are siemens second per meter (S sec/m), or farads permeter (F/m).

Quasistatic: A potential distribution that changes slowly enough that we can accurately describe it bythe equations of electrostatics (capacitive, inductive, and propagation effects are ignored).

Spatial frequency: A parameter governing how rapidly a function changes in space; k = 1/(2π /s), wheres is the wavelength of a sinusoidally varying function.

Syncytium (pl., syncytia): A tissue in which the intracellular spaces of adjacent cells are coupled throughintercellular channels, so that current can pass between any two intracellular points without crossingthe cell membrane.

Volume conductor: A three-dimensional region of space containing a material that passively conductselectrical current.

AcknowledgmentsI thank Dr. Craig Henriquez for several suggestions and corrections, and Barry Bowman for carefullyediting the manuscript.

References

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21-12 Biomedical Engineering Fundamentals

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Further Information

Mathematics and Physics

Jackson, J.D. Classical Electrodynamics, 3rd ed., John Wiley & Sons, New York, 1999. (The classic graduatelevel physics text.)

Purcell, E.M. Electricity and Magnetism, Berkeley Physics Course, Vol. 2, McGraw-Hill Book Co., NewYork, 1963. (A wonderfully written undergraduate physics text full of physical insight.)

Schey, H.M. Div, Grad, Curl and All That, 3rd ed., Norton, New York, 1997. (An accessible and usefulintroduction to vector calculus.)

Bioelectric Phenomena and Tissue Models

The texts by Cole and Plonsey, cited above, are classics in the field.Gabriel, C., Gabriel, S., and Corthout, E. 1996. The dielectric properties of biological tissues: I. Literature

survey. Phys. Med. Biol. 41: 2231–2249.Geddes, L.A. and Baker, L.E. 1967. The specific resistance of biologic material — a compendium of

data for the biomedical engineer and physiologist. Med. Biol. Eng. Comput. 5: 271–293. (Measuredconductivity values for a wide variety of tissues.)

© 2006 by Taylor & Francis Group, LLC

The Electrical Conductivity of Tissues 21-13

Plonsey, R. and Barr, R.C. Bioelectricity, A Quantitative Approach, 2nd ed., Plenum Press, New York, 2000.(An updated version of Plonsey’s “Bioelectric Phenomena,” and a standard textbook for bioelectricitycourses.)

Polk, C. and Postow, E. (Eds.) CRC Handbook of Biological Effects of Electromagnetic Fields, CRC Press,Boca Raton, FL, 1986.

Journals: IEEE Transactions on Biomedical Engineering, Medical and Biological Engineering and Computing,Annals of Biomedical Engineering.

© 2006 by Taylor & Francis Group, LLC