Analysis and design of electrodes and circuits for transfer impedance measurements in biological media

ANALYSIS AND DESIGN OF ELECTRODES
AND CIRCUITS FOR TRANSFER
IMPEDANCE MEASUREMENTS
IN BIOLOGICAL MEDIA








By
ALLEN HARRIS FLASTERSTEIN


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY










UNIVERSITY OF FLORIDA
1968



































Copyright by

Allen Harris Flasterstein

1968





































To

Eric, Ian and Grandma Ess















ACKNOWLEDGEMENTS


The author wishes to express his appreciation to Doctor Jack R.

Smith for sponsoring and directing his research activities; and to

Doctor Arnold H. Nevis who has contributed much toward the author's

development in the biological impedance field.

The author also wishes to acknowledge Doctors Melvin J.

Fregly, Earnest B. Wright, Charles V. Shaffer and Theodore S. George

who, as members of his supervisory committee, have offered helpful

suggestions and have been especially patient in awaiting the conclusion

of this work.

Finally, special appreciation is due William K. Converse, whose

laboratory skills and other assistance have been of special value;

and Barbara Rucker for doing such a fine job in typing the final

manuscript.

This investigation was supported by Public Health Service

Fellowship No. 4-Fl-GM-25, 522-04, from the National Institute of

General Medical Sciences.

















TABLE OF CONfTEt;,S


PAGE

ACKNOWLEDGEMENTS . . .. iv

LIST OF TABLES . . .. . viii

LIST OF FIGURES. . . ... x

ABSTRACT . . .. . xiii

CHAPTER

1 INTRODUCTION
Problems and Contributions. . 1
Network Classification. . 2
Complex Impedance . . 3
Tissue Impedance. . . 6
Literature Review . . 14
Two-Electrode Measurements. . ... 15
Three-Electrode Measurements. . ... 20
Four-Electrode Measurements . ... 22
Summary . . .. ... 36

2 MATRIX AND NETWORK IE.:. ITION OF FOUR-ELEC_' :D, SYSTEMS
Voltage and Current Conventions . ... 39
The Impedance Parameter Matrix. . ... 40
Analysis of the m Parameters. . .. 42
The Effective Interface Impedances. ... 45
The h Coefficients. . .. 48
The Equivalent Network. . ... 55
Examples of Impedance Parameter Matrices. .. 58
Concluding Remarks. . ... 63

3 ERRORS IN THE IDENTIFICATION OF TRANSFER IMPEDANCE BY THE
CLASSICAL METHOD
Introduction. . . ... ... 64
Loading Error . . ... .. 65
Errors Due to Admittance Classes 1, 2, and 3. 68
Errors Due to Ground Admittance ... 84
Common Mode Rejection Error . ... 87
Current Control Error . ... 90
Noise in the Classical System . ... 97
Instrumentation Noise . ... 98









TABLE OF CONTENTS (Continued)


PAGE

Environmental Noise . .. 99
Electrode-Tissue Noise. . .. 102

4 THE VIRTUAL-GROUiND CONFIGURATION
Introduction. . . .. 105
Common Mode Rejection Error . .. 108
Single-Ended Operation. . .108
Differential Operation. . .. 111
Loading Error . . 113
Cable and Amplifier Input Admittance. ... 113
Ground Admittance . .. 118
Current Control Error . .. 119
Stability Considerations. . .. 121
Stabilization Techniques. . ... 125
Noise in the Virtual-Ground System . ... 133
Instrumentation Noise . .. 133
Environmental Noise . .. 135
Electrode-Tissue Noise. . .. 137
The Negative-Feedback Filter Loop .. 137

5 AN EXPERIMENTAL SYSTEM BASED ON THE VIRTUAL-GROUND
CONFIGURATION
Introduction . . 147
The Physical Layout . . .. 148
Connection to the Electrode-Tissue System .148
Ground Connection . .. 152
Output Connection . .. 152
Input Connection . .. 153
Circuit Details . . 153
The Response Amplifier . .. 153
The Virtual-Ground Amplifier. . ... 155
The Feedback Filter . .. 156
The Current Control Circuit . .. 157
Experimental Studies . .. 161
Performance Tests . .. 162
Four-Electrode Impedance Measurements .. 173

RECOMMENDATIONS FOR FUTURE STUDY . ... 185
Electrode Systems . .. 185
The Virtual-Ground System . .. 185
Information Retrieval . .. 186
Biological Studies . 186

APPENDICES

1 Reciprocity Relations of the Impedance Parameter
Matrix. . ... 188









TABLE OF CONTENTS (Continued)


PAGE

2 The Voltage Follower Amplifier. . ... 190

3 The Voltage Inverter Amplifier. . ... 192

4 Principles of the Lock-in Amplifier ... .193

REFERENCES ................... ......... 196
















LIST OF TABLES


Table Page

2.1 Definitions of the m Parameters . 41

2.2 Approximate Values of X and Q for Stainless Steel
Electrodes in Isotonic Saline. . ... 59

2.3 Matrix Elements for Example 1 . .. 61

2.4 Matrix Elements for Example 2. . ... 62

3.1 Sign of Magnitude and Phase Errors in m11. . 74

3.2 Limits of Class 1, 2, and 3 Conductance for the
Examples of Chapter 2 . . .. 78

3.3 Limits of Class 1, 2, and 3 Susceptance and Capacitance
for the Examples of Chapter 2. . ... 82

3.4 Limits of Ground Admittance for the Examples of Chapter 2. 86

3.5 Minimum Values of Common Mode Rejection Ratio for the
Examples of Chapter 2 . . ... 90

3.6 Minimum Values of Current Source Resistance for the
Examples of Chapter 2. . . .. 95

4.1 Values of BV for the Examples of Chapter 2 .. 116

4.2 Limits of Cable and Amplifier Input Admittance for the
Examples of Chapter 2. . . .. 117

4.3 Limits of Ground Admittance for the Examples of Chapter 2. 119

4.4 Minimum Values of Current Source Resistance for the
Examples of Chapter 2. . . .. 121

4.5 Limits of 5 for Given Values of . 131

5.1 Frequency Response Errors. . ... 166

5.2 Common Mode Rejection Errors . .. 169

5.3 Current Control Test Results . .. 172


viii









Table Page

5.4 Test Results of an Electrode Assembly in Isotonic Saline 177

5.5 Specific Impedance of Rat Cerebral Cortex. ... 182
















LIST OF FIGURES


Figure Page

1.1 Frequency Dispersion in Tissues. . .. 8

1.2 Four-Electrode Bridge . . 23

1.3 Voltage Clamp Scheme. . . ... 27

1.4 Four-Electrode Comparison Scheme. . ... 30

2.1 Voltage and Current Conventions for the Electrode-Tissue
System. . . ... . 39

2.2 Electrode-Tissue Interface in an Arbitrary Current Field.. 43

2.3 Parallel Wire Four-Electrode System . .... 46

2.4 Electrode-Tissue System Showing the Effective Interface
Impedances. . . .. .. 48

2.5 Electrode Current Density in the System of Figure 2.3 51

2.6 Frequency Dispersion of the Geometrical Factor for the
System of Figure 2.3. . . ... 52

2.7 Equivalent Network for the Hypothetical System. 55

2.8 Complete Equivalent Network of the Electrode-Tissue System. 58

2.9 Four-Electrode System with Hemispherical Electrodes .. 60

3.1 Basic Block Diagram for Impedance Measurements. ... 65

3.2 Schematic Diagram of the Electrode-Tissue and Loading
Admittance Systems. . .... 67

3.3 Equivalent Network for Loading Admittance Classes 1, 2,
and 3 . .. . 69

3.4 Phasor Relations of aL in the Complex Plane . 74

3.5 Input Circuit of a Differential Amplifier .. 83









LIST OF FIGURES (Continued)


Figure Page

3.6 Approximation of a Current Source . .. 91

3.7 Phasor Relations of aQ in the Complex Plane ... 94

3.8 Noise Sources in the Classical Method. . 98

4.1 Basic Connections of the Virtual-Ground Configuration 106

4.2 Nominal Gain and Phase Shift Characteristics of Operational
Amplifiers. . . ... 107

4.3 Virtual-Ground System Showing Loading Admittances and
Noise Sources . . .. 109

4.4 Simplified Equivalent Circuit for the Virtual-Ground
Amplifier Feedback Loop . .... 124

4.5 Stabilization Techniques for the Virtual-Ground Amplifier
Feedback Network. .. . . 126

4.6 Typical Magnitude and Phase Characteristics of a Lag
Network . . ... .. 129

4.7 Electrode-Tissue System with Stabilizing Resistors. ... 133

4.8 Virtual-Ground System with Negative-Feedback Filter Loop. 138

4.9 Simple Low-Pass Filter. . .... 142

4.10 Simple High-Pass Filter . .... 143

4.11 Overall Transfer Function Characteristics Using Several
Methods of Noise Filtering. . ... 145

5.1 Close-up Photograph of the Experimental Virtual-Ground
System ................. .. 149

5.2 Complete View of the Experimental Arrangement .. 150

5.3 The Response Amplifier Circuit. . ... 154

5.4 Principles of Current Control . .... 158

5.5 The Current Control Circuit . ... 159

5.6 Block Diagram for Experimental Studies .. .. 163









LIST OF FIGURES (Continued)


Figure Page

5.7 Attenuator Circuit for Frequency Response Test. ... 165

5.8 Circuit for Common Mode Rejection Test. . ... 167

5.9 Circuit for Current Control Test. . .. 171

5.10 Electrode Assembly for In Situ Impedance Measurements 175

5.11 .Impedance of Rat Cerebral Cortex In Situ. ... 179

5.12 Polar Plot of the Impedance of Rat Cortex .. 183

Al Voltage and Current Conventions for Reciprocity Relations 188

A2 Voltage Follower Amplifier. . ... 190

A3 Voltage Inverter Amplifier. . . 192

A4 Block Diagram of the Lock-in Amplifier. . ... 194


xii







Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy


ANALYSIS AND DESIGN OF ELECTRODES AND CIRCUITS FOR
TRANSFER IMPEDANCE NEASUPEIlEUTS IN BIOLOGICAL MEDIA

By

Allen Harris Flasterstein

August, 1968

Chairman: Dr. J. R. Smith
Co-chairman: Dr. T. S. George
Major Department: Electrical Engineering

Attempts by many investigators to measure the electrical

impedance properties of animal tissues in situ at frequencies of

physiological interest (0 to 200 kHz) have met with only moderate

success. The effects of electrode polarization have been reduced or

eliminated by using separate voltage pickup and current electrode

pairs. However, the practical limitations of this method have not

been clearly defined, nor have appropriate analytical techniques been

applied to the analysis and design of the electrode and instrumenta-

tion systems. These deficiencies and the inadequacy of existing

instrumentation techniques have hindered progress in the impedance

art.

In the dissertation abstracted here, the ensemble consisting

of four electrodes in an electrolytic medium (or biological tissue)

is depicted as a linear four terminal device and described by means

of a 3 x 3 matrix of complex impedance parameters. The matrix elements

are dissected into electrolyte and interface components which are

defined in terms of the field geometry, and the impedance characteris-

tics of the electrolyte and the electrode-electrolyte interfaces.


xiii









Although the matrix description is more direct and often simpler to

use, an equivalent network has also been evolved. The matrix elements

of two different electrode configurations are calculated and used to

illustrate major analytical results.

The immediate objective of most four-electrode impedance measure-

ments is to identify what has been defined as the first element in the

impedance parameter matrix. This element is estimated experimentally

by determining the ratio of an arbitrary measure of the nominal voltage

response to an arbitrary measure of the nominal exciting current under

the assumption that the current in the voltage-response electrodes

is zero. The parameters defined in the impedance matrix permit a

comprehensive analysis to be made of the errors involved in this

determination. On the basis of such an analysis, the classical method

of grounding one current electrode and recording differentially from

the potential electrodes has been found to be impractical for small

electrode systems at low and high frequencies.

To overcome many of the limitations of the classical method, a

new technique, called the virtual-ground system, has been developed.

In this system, one potential electrode is maintained close to ground

potential by means of negative feedback. Three classes of error--

loading, common mode rejection, and current control--are analyzed for

the classical and virtual-ground systems. In addition, several types

of noise and methods of noise filtering are considered in each system.

The results of the error and noise analyses demonstrate the advantages

of the virtual-ground system and provide a set of design formulae. To

complement the virtual-ground system, a current control circuit has


xiV










been adopted from the literature and analyzed for application to broad

band impedance measurements.

A successfully operating system using the virtual-ground system

and the adopted current control circuit has been built at modest cost

and is described in the dissertation. It has been tested with small

electrode assemblies in saline and with similar assemblies implanted

in the cerebral cortex of live rats. Several experiments are reported

to verify system performance and indicate the quality of results

already obtained.
















CHAPTER 1


INTRODUCTION


Problems and Contributions


Attempts by many investigators to measure the electrical

impedance properties of animal tissues in situ at frequencies of

physiological interest (0 to 200 kHz) have met with only moderate

success. The effects of electrode polarization have been reduced or

eliminated by using separate voltage pickup and current electrode

pairs. However, the practical limitations of this method have not

been clearly defined, nor have appropriate analytical techniques been

applied to the analysis and design of the electrode and instrumenta-

tion systems. These deficiencies and the inadequacy of existing

instrumentation techniques have hindered progress in the impedance

art.

In the present chapter, definitions and assumptions are

explained which are basic to the study of biological impedance

phenomena; the nature of tissue impedance is reviewed according to

current knowledge; and a review of selected literature is presented,

which reveals various measurement techniques and common sources of

error. The material in Chapters 2, 3, 4 and 5 represents an original

application of engineering principles toward the general fulfillment

of the needs expressed in the opening paragraph. In Chapter 2,


~









special parameters are defined which lead to an unprecedented under-

standing of the electrode-tissue system and provide an appropriate

basis for the analysis and design discussions that follow. In Chapter

3, the classical method for transfer impedance measurements is

thoroughly investigated, resulting in useful design criteria and

sophistication in understanding the major types of measurement error.

In Chapter 4, a new instrumentation technique is proposed and analyzed

in depth. The new technique is shown to offer major advantages over

existing methods. An experimental system which incorporates this

technique and several other special advances is described in Chapter 5

along with pertinent experimental results.


Network Classification


To make impedance measurements more intelligible at the present

state of the art, one usually invokes the restrictions of time

invariance and linearity on the medium of interest.

Time invariance implies that the impedance properties of the

medium do not vary during the critical period of measurement. The

critical period may just include one to several repetitions of the

applied signal waveform or, for example, it may include a whole

sequence of separate measurements at different sinusoidal frequencies.

Time invariance does not exclude the measurement of impedance changes

provided that the changes are slow compared to the time required for

a single measurement. Certain impedance characteristics of tissues

in situ are known to vary spontaneously and under various stimuli in

periods ranging from many hours to several seconds.










A system is linear if the response to a forcing function

comprised of several components is equal to the sum of the responses

which would be obtained from each component acting alone (Dern and

Walsh, 1963). Further, if the amplitude of a given forcing function

is doubled, then the amplitude of the response function should also

double while the waveform remains the same. This restriction appears

to be satisfied in practice if the charge and current density in the

tissue are kept sufficiently low. Hence, currents and voltages are

typically measured in microamperes and microvolts. At these levels,

several different sources of noise may be significant. This topic is

considered later in the work.

When the linearity condition is satisfied, it is almost certain

that the medium will also be bilateral and passive. A system is

bilateral if the negative of the response to a given forcing function

is the response to the negative of that forcing function. Most tissues

are not strictly passive in the network sense (Raisbeck, 1954) because

they contain electrochemical sources of energy. In the linear range,

however, the electrical activity of these sources appears to be indepen-

dent of the applied signal. Since measurement techniques attempt to

weed out the effects of background noise, the final results ideally

correspond to those of a passive medium.


Complex Impedance


If a sinusoidal forcing function is applied to a time invariant

linear bilateral network, every waveform in the network will also be

sinusoidal with the same frequency once the initial transients die









out, i.e., when the steady state is reached. Only the phase and

amplitude may differ from place to place. This convenient property

leads to the definition of complex impedance,



Z E -II e (1.1)
'II

where IVj and III are the peak amplitudes of an arbitrary steady state

voltage-current pair and 8 is the algebraic phase difference between

them, measured with respect to the voltage waveform.

The factor eje is known as the polar form of the complex number

given by


ej8 = cos 8 + j sin 8 (1.2)


where e is the natural logarithm base and j is the imaginary operator

/Ji. The cosine and sine terms are known as the real and imaginary

parts of the complex number, respectively.

If V and I occur at the same terminal pair, Z is called a driving

point impedance. When they occur at different terminal pairs, Z is

known as a transfer impedance. In either case, it is an inherent

physical parameter of the network. Sometimes, it is more convenient

to use the reciprocal of the complex impedance called the complex

admittance. For the same quantities as before, the complex admittance

is defined as



IVI






5



The concepts of complex impedance and complex admittance will be used

frequently in the present work. In addition, all voltage and current

signals in the electrode-tissue system will be written in complex form.

The complex form of the signal



v(t) = IVI Cos( t + 6) (1.4)



is defined as



V v= IVej (1.5)



Explanations of the use of complex notation may be found in basic texts

in electrical engineering (e.g., Van Valkenburg, 1955).

The theory of linear time invariant systems shows that if the

sinusoidal steady state response of a system is known at a sufficient

number of frequencies, the response to other waveforms, periodic or

transient, can be determined analytically and vice versa. This

equivalency is especially welcomed when spectral information is desired

and the time for measurements is limited. Although useful results have

been obtained from nonsinusoidal methods (Smith, 1967), much of the

available information is lost due to inadequate resolution and band-

width in the measuring system. Improved data processing techniques

using digital computation are certain to overcome some of the difficulty.

The remaining problem lies in the extraction and amplification of the

raw analog data which is a major theme in the present work.


~









Tissue Impedance


Animal tissues fall into four major classifications: epithelial,

connective, muscular and nervous (Ham, 1965); and each of these may

be further divided into a number of specific subclasses. The constitu-

tion and structure of tissues, even within the same classification, may

vary markedly. Furthermore, the electrical properties of relatively

few tissue types, such as muscular and nervous, have been studied in

detail. The following discussion presents a brief synopsis of the

highlights of tissue impedance studies. It pertains most directly to

muscular and nervous tissues but is also applicable, with some modifi-

cations, to other tissues.

In tissue, the applied current may travel between cells in the

extracellular fluid and across cell membranes into the intracellular

fluid and subcellular structures. At frequencies of the order of 200

kHz and below, the electrical properties of the fluids are, for all

practical purposes, independent of frequency and are characterized by

resistivity values of 100 to 200 ohm-cm (Schwan, 1963). However,

membranes (cellular and subcellular) which are less than 200 Angstroms

thick are associated with relatively high transverse resistance (1 to

iP00 ohm-cm2) and high transverse capacitance (, 1 uf/cm2).

The charging and discharging of adjacent membrane-fluid interfaces

(similar to plates of a fixed capacitor) is one of several mechanisms

which impart frequency dependent characteristics to the overall tissue

impedance. The characteristics due to this specific mechanism in

tissue have been termed 8 dispersion by Schwan (Schwan, 1957, 1959,

1963; Schwan and Cole, 1960). The membraneous nature of tissue is also










responsible, in less evident ways, for other dispersion phenomena at

lower frequencies than the B dispersion. These phenomena have been

termed a dispersion (in tissues) by Schwan and have been associated

with several different mechanisms which are still under debate. Further

discussions of a and B dispersion are given in later paragraphs.

In addition to microscopic heterogeneities measured in Angstroms

and microns, tissue properties may vary grossly in distances ranging

from one to many millimeters. Many tissues are also anisotropic (Rush

et al, 1963; Nicholson, 1965; Ranck and Be Ment, 1965). These

characteristics and the close boundaries of some tissues in situ may

be expected to influence the impedance determined with different

electrode configurations in complex ways. Hence, the concept of

specific impedance, defined for a one-centimeter cube of tissue in a

uniform field, is an idealization which requires special qualification

in each case. To be sure, data obtained with different electrode

configurations in the same tissue may not simply differ by a constant

geometrical factor and a comparison of results is at best approximate.

In practice, one tries to use an electrode system which measures average

properties of a large population of cells, but avoids the effects of

gross inhomogeneities. There are, however, applications in which

careful localization is not an important factor.

For an arbitrary electrode system, let the measuredd complex

impedance be written


Z = R(f) + jX(f)


(1.6)





8



where R and X designate resistance and reactance, respectively, and

both are functions of frequency. It has been reasonably well established

that two fairly distinct frequency dispersions occur in many, if not

all, tissues in the 0 to 200 kHz frequency band (Schwan, 1957; Fatt,

1963; Ranck, 1963a). Using Schwan's terminology, the a and 8 disper-

sions are characterized approximately by circular arcs in the complex

impedance plane as indicated in Figure 1.1. The arcs usually overlap








^2 increasing
Frequency


v f-






f



0
1 2 3 4 5

Resistance (relative)


Figure 1.1. Frequency Dispersion in Tissues


and have centers below the R axis. In results so far reported, the

overlap region generally occurs at frequencies below 2 kHz. The low

frequency for maximum reactance, f may'vary from several hundred

Hz to below 20 Hz depending on tissue and experimental conditions.









Likewise, the high frequency for maximum reactance, f,, may vary from

roughly 1 kHz to well above 200 kHz.

Scales have been provided in Figure 1.1 to illustrate relative

features of the dispersion phenomena. It is evident that a high degree

of measurement accuracy is required to resolve points on the a disper-

sion arc. On this arc, the reactance is usually less than one fifth

as large as the resistance with corresponding phase angles of less than

12 degrees. In addition, the resistance usually changes by less than

20 percent over the arc. The accuracy required to measure a dispersion

is difficult to achieve in vitro, but it is especially difficult to

achieve in situ.

In vitro, the a and 8 dispersions are both sensitive to age

and environmental factors. The a dispersion appears to be more closely

related to in vivo properties of the tissue and breaks down rapidly

after excision. The p dispersion may not change significantly for many

hours after excision. It disappears only after cellular structure

completely deteriorates. A detailed discussion of a and a dispersion

is beyond the scope of the present work. However, a brief introduction

to the mechanisms which underlie these phenomena is of interest and is

given in the following sections.


B dispersion


The explanation of a dispersion follows from principles of

classical electric field theory first presented by Maxwell in 1873.

The theory has been applied to suspensions of conducting spherical,

elipsoidal and cylindrical particles surrounded by poorly conducting









shells (membranes) to simulate various biological materials. In brief,

8 dispersion is a structural phenomenon depending on the distribution

of the membranes and conducting fluids, the resistivity of the fluids,

and the dielectric properties of the membrane material.

At frequencies just above the 8 arc, the membrane impedance is

negligible in comparison to the effective resistance of the cell

interior, and the tissue behaves like a suspension of conducting bodies

in a similar conducting medium. As the 8 arc is entered, the membrane

impedance grows comparable to the effective resistance of the cell

interior. Toward the low frequency end of the arc, the membrane

impedance has grown much larger than the effective resistance of the

cell interior and the tissue begins to approach (but does not necessarily

attain) a condition in which the cells appear like empty holes in the

extracellular fluid.

While B dispersion is primarily associated with the cell

membrane proper, the membraneous structures within the cell may also

contribute to the dispersion phenomena by a similar mechanism. However,

these effects are smaller and tend to occur at higher frequencies (1

MHz to 10 MHz). Dispersions due to proteins and other macromolecular

matter in the fluids also occur in the MHz range, but these mechanisms

are distinctly different from that of the B dispersion.


a dispersion


The additional resistance rise associated with a dispersion

originates from one or more mechanisms which cannot follow the rapid

electrical changes that occur at frequencies above the a dispersion









range. Hence, this dispersion, as 8 dispersion, is characterized by

capacitive properties (negative reactance). But, contrary to $ disper-

sion, the underlying phenomenon does not necessarily involve the charge

and discharge of adjacent faces of a dielectric material. At least

five different mechanisms have been proposed to explain a dispersion

(Schwan, 1957; Schwan and Cole, 1960; Fatt, 1963; Ranck, 1963b). These

are summarized in the following paragraphs.


Membrane permeability.--The term permeability refers to the

physical transport of ions through the cell membrane and is envisioned

to occur at specific sites in the membrane. If the net ion current is

regulated by the cell such that it is related to the membrane voltage

through a differential equation in time, then the permeability is

electrically equivalent to an impedance characterized by resistive and

capacitive elements, even though no physical capacitance exists.

Inductive elements are also possible. Evidence for the permeability

mechanism is based mainly on studies of large nerve axons (Cole, 1955;

Schmitt, 1955).


Membrane structure.--The membrane is envisioned as a double

layered structure of electrically different materials (lipid and protein).

This structure results in a frequency dispersion of the total membrane

impedance, which is characterized by a bounded increase in the effective

parallel resistance and capacitance of the membrane as the frequency

is decreased. Schwan (1957) has predicted that this dispersion should

occur in the frequency range of 10 Hz to 1 kHz, which is in agreement

with the typical range for a dispersion.






12


Intracellular channels.--Fatt (1963) has developed a theory of

dispersion based on the structural organization of the muscle fibre.

He considers two parallel current paths. One is formed by the bulk

impedance of the cell membrane in series with the bulk intracellular

fluid and is primarily responsible for 8 dispersion. The other path

originates at specific sites (pores) in the cell membrane and continues

through a system of intracellular channels (the sarcotubular system

of the endoplasmic reticulum). It is primarily responsible for a

dispersion. For convenience, let the parallel paths be designated by

a and B corresponding to their respective dispersion phenomena.

Each path is represented by a resistor in series with a capacitor.

Since the channels making up the a path collectively occupy a minute

cross section of the membrane surface and cell interior, the effective

resistance of the a path is much higher (order of 100 times) than that

of the 8 path. On the other hand, a special mechanism is postulated

for the channel entrance which permits the effective capacitance of the

a path to be much larger (order of 25 times) than that of the B path.

Hence, at high frequencies, the a path appears as a high resistance

across the 8 path and has little effect on the 8 dispersion. At low

frequencies, the 8 path appears as a large reactance across the a path

and has little effect on the a dispersion. The a dispersion is much

smaller than the 6 dispersion because the effective impedance of the

extracellular fluid that shunts the cell remains constant, while the a

path phenomenon occurs at a much higher impedance level than that of

the a path.

With a constant resistor and capacitor representing the two










paths, the a and 8 arcs are centered on the real axis, corresponding

to one time constant each. In practice, the arcs are centered below

the axis, corresponding to a distribution of time constants. To

account for this fact, Fatt postulates and investigates the existence

of distributed electrical coupling between the channel system and the

bulk intracellular fluid.

Cable properties.--Ranck (1963b) has developed a theory of a

dispersion based on the cable-like properties of cell processes in

cortical tissue. He postulates that at low frequencies (below 5 kHz),

the signal current is constrained by the cell membranes to flow

primarily along the cell processes (axons and dendrites). The tissue

then appears as a system of cables consisting of intracellular and

extracellular fluid conductors separated by a leaky dielectric material

(the cell membrane). Using the cable equation and certain assumptions

to account for geometrical and material aspects of the tissue, Ranck

shows, in quantitative terms, that the cable mechanism may well give

rise to the experimentally observed a dispersion (Ranck, 1963a).

At higher frequencies (above 5 kHz), the membrane impedance has

dropped sufficiently so that the signal current is no longer constrained

to paths along the cell processes. The current then flows primarily in

directions transverse to the cell membranes. The mechanism for 8 disper-

sion occurs according to the principles of Maxwell as previously

discussed.


Relaxation of ionic atmospheres.--This mechanism, suggested by

Schwan, is based on impedance studies of suspensions of small non-

conducting particles (polystyrene, glass, kaolin, etc.) in electrolytic










media. These suspensions give rise to low and high frequency dispersion

phenomena which are similar to the a and a dispersion of tissue. At

low frequencies, the particles are perfectly insulating. Hence, the

low frequency dispersion can only arise through a variation in the

effective impedance of the current paths between the particles. The

mechanism lies in the displacement from equilibrium position of the

ionic layers which are known to collect around particles in electrolyte

suspension. The displacement is accompanied by a change in the stored

electrostatic energy of the layers and a purturbation of the field

around the particle. These phenomena are frequency dependent and are

reflected as reactive and resistive components in the suspension

impedance.

Many biologically significant phenomena have been reported to

influence the impedance properties of certain tissues in vivo. In

view of these findings and the possible mechanisms that underlie

impedance phenomena, the measurement and analysis of tissue impedance

have become an important research tool (e.g., Adey et al, 1962;

Aladjalova, 1964; Cole, 1962; Geddes and Baker, 1967; Nevis and Collins,

1967; Shalit and Mahler, 1966; Van Harreveld, 1966). Impedance measure-

ments are also of potential interest in clinical applications. Unfor-

tunately, inadequacies and uncertainties in experimental technique have

hindered their development for widespread use. Some of the problems are

resolved in the present work.


Literature Review


Three principal classes of electrode systems have evolved to










meet the varied demands of biological impedance studies. In each case,

two or more electrodes (the current electrodes) are required to supply

signal current to the tissue under study. The classes are distinguished

according to the method of observing the voltage signal. Thus, in two-

electrode systems, the voltage is observed directly at the current

electrodes; in three-electrode systems, the voltage is observed between

a potential electrode and one of the current electrodes; in four-

electrode systems, the voltage is observed between two potential

electrodes.

The major reason for using three-electrode and four-electrode

systems is that the potential electrodes nominally do not draw current

so that the observed voltage response is not influenced by interface

impedance. Higher order systems also permit greater freedom in

selecting the current field configuration, which sometimes leads to

more versatile experiments and better anatomical localization of

observed impedance phenomena. Furthermore, they facilitate control

of the signal current level and are especially convenient for monitoring

ongoing biological activity during the impedance measurement.

To provide further insight into the nature and limitations of

biological impedance measurements, some of the major works in this

field are reviewed in the following sections. An effort is made to

indicate historical background and reveal common relationships between

the three classes of electrode systems.


Two-Electrode Measurements


Early studies of the conductivity of biological materials may










be traced as far back as the nineteenth century. However, the earliest

major work to influence modern tissue impedance research was that of

Fricke and Morse (1925), who first applied field theory to the inter-

pretation of experimental impedance data. Thus, from measurements of

suspensions of red blood cells, the authors were able to estimate the

resistive and capacitive properties of the cell membranes. The suspen-

sions were contained in an electrolytic cell and measured by means of

a wheatstone bridge. The electrolytic cell was shaped like an hour-

glass with large platinized platinum electrodes sealed into either

end. Although this design was selected to minimize electrode polariza-

tion effects, the authors were well aware that the low frequency

utility of the cell was still quite limited. The wheatstone bridge

was operable from 800 Hz to 4.5 MHz, but no results were reported for

frequencies below 3600 Hz.

A large body of impedance work has been done on excised tissues

as well as cell suspensions,using methods similar to that of Fricke and

Morse. Various improvements have appeared with respect to the bridge

system (Cole and Curtis, 1937; Schwan and Sittel, 1953). In addition,

a number of techniques have evolved for extending the low frequency

capability of the electrolytic cell (Schwan, 1963). The three principal

methods are summarized below.


Distance variation method.--In the distance variation method

(Fricke and Curtis, 1937), measurements are made at two different

electrode separations. By subtracting the smaller from the larger

result, polarization effects are eliminated and the impedance of the

difference volume is obtained.










Substitution method.--In the substitution method (Cole and

Curtis, 1937), the sample with unknown impedance properties is replaced

by a sample with known impedance properties. The electrode polariza-

tion impedance is then calculated from measurements on the known sample

and subtracted from measurements on the unknown sample to give the

desired impedance. Unfortunately, the method is not completely straight-

forward because polarization impedance is influenced by the presence

of cells in the sample (Schwan, 1963). One way to avoid this discrepancy

is to connect the electrodes indirectly to the sample through a fixed

intermediate electrolyte (Schwan, 1954; and Fatt, 1963).


Frequency variation method.--Theoretical (e.g., Fricke, 1932) and

experimental (e.g., Smith, et al., 1967) studies have shown that electrode

polarization impedance may be approximated as a negative power function

of frequency with a constant phase angle, i.e., Z = Mf-e-Jo, where M,

a, and 6 are real positive constants and a < 1, 8 < 90 degrees.

Several schemes based on this fact have been given to distinguish

between electrode polarization and issue impedance (Schwan, 1963; Smith,

et al., 1967). In essence, the polarization impedance is first measured

at low frequencies where the sample impedance is negligible. These

data are used to estimate the polarization impedance at higher

frequencies so that it may be discounted from the sample measurements.

In all of the techniques for eliminating the effects of electrode

polarization, the sample impedance is estimated as the difference

between two quantities which are assumed to share a common polarization

impedance. As the frequency is decreased, the common polarization

impedance increases with respect to the sample impedance. Hence, a










small relative error in the quantities to be difference leads to an

increasingly larger error in the result. In practice, a satisfactory

degree of accuracy can only be achieved at frequencies where the common

impedance is less than the impedance to be estimated.

In 1950, the status of in situ impedance measurements was

reviewed (Benjamin, et al., 1950) and many of the technical difficul-

ties involved in such measurements were discussed. The authors

concluded that reliable in situ data were not yet available at physio-

logical frequencies and called for greater effort in that direction.

For obvious reasons, methods of correcting for electrode polarization

are more difficult to implement in situ than in vitro. Few, if any,

two-electrode studies in situ are beyond serious criticism.

The earliest of the more precise in situ studies was reported

by Schwan and Kay (1956, 1957). They measured resistive and capacitive

properties of various thoracic tissues in anesthesized dogs. The

electrode system consisted of a cylindrical probe with one electrode

forming the tip and one electrode forming part of the shank. The

electrode material was platinized platinum and the surface area of each

electrode was 0.3 cm2. Measurements were performed with a wheatstone

bridge at frequencies from 10 Hz to 10 kHz.

To correct for electrode polarization, the polarization impedance

was measured as a function of frequency in isotonic saline. These

results were modified by a masking factor to estimate the polarization

impedance in the tissue at each frequency. The masking factor was

assumed to be independent of frequency and was determined by comparing

the in-saline and in-tissue polarization impedances at 10 Hz. Special










precautions were necessary to assure that the fragile electrode surfaces

were not altered between saline and tissue measurements. The accuracy

of the series polarization resistance and capacitance calculated for

in-tissue conditions was stated as about 10 percent. The error in

the result for tissue capacitance was given as 10 percent at 100 Hz

and 100 percent at 10 Hz. In other words, the reactive part of the

polarization impedance was equal to that of the tissue at 100 Hz and

was ten times that of the tissue at 10 Hz. For the tissue resistance,

the error was given as 3 percent at 10 Hz. The higher accuracy

reflects the fact that at 10 Hz the tissue resistance is much larger

(more than 30 times) than the tissue reactance.

Schwan and Kay reported that the resistance of certain tissues

rises about 20 percent beginning at a frequency below 1 kHz and

continuing more sharply toward lower frequencies. This finding and

certain of the values found for tissue resistance have since been

disputed by others (Burger and van Dongen, 1960-61; Rush, et al, 1963),

who, on the basis of four-electrode measurements, claimed the resistivity

to be independent of frequency in this range. Several reasons were

proposed by Rush et al. for the discrepancy. Important among these

were erroneous assumptions in the correction for polarization impedance.

Further evidence (Schwan, 1963, Figure 10; Smith et al., 1967) indicates

that contrary to what was assumed, the masking factor may well have been

dependent on frequency, especially below 500 Hz. If this is true, then

the anomalous change in slope of the capacitance curve (Schwan and Kay,

1957, Figure 1) at frequencies below 500 Hz is also in question. It has

been reasonably well established that a small variation in resistivity










does occur in many tissues at low frequencies (a dispersion). Therefore,

in view of the many uncertain factors, it is suspected that none of

the conflicting reports are completely reliable.

The work of Schwan and Kay illustrates some of the problems and

uncertainties encountered in two-electrode measurements in situ. It

probably also demonstrates the best accuracy that can be achieved

under such conditions--at least in the light of current knowledge.

Many experiments (especially chronic ones) require smaller electrodes

and the error due to polarization is correspondingly larger. In any

case, manual and computational correction procedures are time consuming

and difficult to carry out with the degree of caution necessary. Further-

more, the validity of the key assumptions on which the accuracy of two-

electrode results depend is difficult, if not impossible, to verify

in specific experiments.


Three-Electrode Measurements


Three-electrode systems are useful where there is extreme

assymmetry in the current electrode configuration. They have been used

primarily to study the membrane properties of individual cells in situ

and in vitro. The current is applied between a microelectrode (0.5 to

5 micron tip) in the cell body and a large reference electrode in the

external medium. It passes across the membrane of the cell body and

in the case of neurons, also through the dendritic branches (Rall, 1959).

Almost all of the potential difference between the interior of the

cell body and the reference electrode occurs across the cell membrane.

The reference electrode can usually be designed to produce a









negligible polarization component in the observed voltage. However,

the impedance of the microelectrode is quite large (megohms) and

cannot be neglected. To overcome this problem, a second microelectrode

may be added in the cell body to record potential (Fessard and Tauc,

1956; Combs et. al., 1955, 1959). The high impedance of the intra-

cellular electrodes makes such systems especially sensitive to

parasitic coupling between electrodes. Fessard and Tauc used completely

separate micropipettes to minimize this hazard. For smaller cells,

the double-barrelled micropipette of Combs et. al. is more convenient,

but the problem of parasitic coupling is intensified.

The foregoing methods have been used only with pulse waveforms

to determine membrane time constants and for stimulating and observing

active responses of neurons. A systematic application of sinusoidal

techniques would be desirable both from the standpoint of defining

measurement errors and obtaining more precise information about the

electrical properties of the cell membrane.

Three-electrode systems have also been used with a distance

variation technique to study tissue impedance in anesthetized animals.

(Van Harreveld, et. al., 1963; Nicholson, 1965). The current field is

set up by means of a fixed pair of current electrodes and the voltage

is measured with respect to one current electrode as a function of

potential electrode position. Regardless of the particular procedure

and field configuration used, the calculation of specific impedance

requires the evaluation of the voltage difference between probe posi-

tions. Hence this method is subject to similar limitations with

respect to the ratio of common to difference voltage as were discussed










earlier for two-electrode systems. Another limitation of this technique

is that the measurements at each electrode position are made at

different time intervals.


Four-Electrode Measurements


It is convenient to classify four-electrode measurements

according to the manner in which the potential electrodes are used.

In this section, a method is labeled continuous if the potential

electrodes take part in the impedance measurement simultaneously and

discontinuous if separate measurements are required at each electrode.


Discontinuous methods


The only method of current interest which deals separately with

each potential electrode is a bridge balancing scheme. The basic

technique was originally described by Shedlovsky (1930a, b) for

measuring conductance in nonbiological electrolytes. Recently it was

applied to biological media (Songster, 1967; Hill, unpublished) and

modified to include capacitive components and a broader frequency

range (5 Hz to 200 kHz). The main features are illustrated in Figure

1.2. The electrode-tissue system is represented by the simplified

equivalent network, Za Z, and Zc (see Figure 2.8 for the complete

equivalent network). The current electrode interface impedances are

included in Za and Zc. Zb represents the tissue impedance to be

determined. The bridge balances are achieved by means of Z1 and Z2,

and the detector may be switched from position 1 to position 2 as needed.

The measurement procedure begins with Z2 set at zero and proceeds

in two successive steps.


























balanced
source


Figure 1.2. Four-Electrode Bridge


Step 1: Adjust Z1 to null point 1.

Step 2: Adjust Z2 to null point 2.


If ti and t2 denote the times at which step 1 and step 2 are performed,

then the balance conditions may be written


at tl: Z1 1 + Za(t) = Zb(tl) + Zc(tl)


at t2: Z2 2 + Ze(t2) = Zb(t2) + Z(t2) + Z1
222 c 2 b 2 a 2 1










where the e are uncertainties in the adjustment of Z and Z Note
1 2
that each condition actually represents two equations, one for the

resistive parts and one for the reactive parts of the impedances.

The balance equations may be added and the result arranged to

give


Zb(tl) + Zb(t2) = Z2 + (AZc AZa) (El E2)


where AZc and AZa are the changes in Zc and Za, if any, that occur

from time tl to time t2. Ideally, AZ AZa, E1 and C2 should be zero

so that Z2/2 is equal to Zb (or its average value if it varies). In

practice, these deviations are finite and must fall within limits much

smaller than Zb if the method is to be successful.

At least two factors are of major concern in evaluating the

suitability of the bridge method for a particular biological study:

bridge resolution and constancy of Za, Zb and Zc with time. At

frequencies below 100 Hz (sometimes higher), Za and Zc may be of the

same order as or very much larger than Zb, depending primarily on the

electrode design. For example, with large platinized platinum electrodes

in vitro, these impedances will usually be on the same order as or

smaller than Zb. They will also tend to remain constant in the tlt2

interval. However, with small stainless steel electrodes, in situ,

Za and Z may well be 1000 or more times as large as Zb and are likely

to vary during the measurement interval. The first example is easily

within bridge capabilities, but the electrode system requires careful

preparation. In the second case, very high bridge resolutions (greater

than 1 part in 105) would be required to make the e much smaller than Zb.










Even then, the slightest variations in Z and Z with time could
a c
produce errors greater than 100 percent. The problem is compounded

by the fact that as the frequency is decreased, it usually takes longer

to achieve a bridge balance.

Clearly, there is a whole range of experiments with relative

impedance values lying between the extremes discussed above. For many

of these, the bridge method is not suitable on the basis of resolution

and constancy arguments alone. On-line bridge methods are also limiting

in that only one frequency at a time may be measured and each measure-

ment takes several or more minutes to perform. Hence, impedance cannot

be monitored continuously, nor can the instantaneous impedance spectrum

be determined.

An advantage of the bridge method is that at balance, the potential

electrode in use is approximately at ground potential. Hence, the input

impedance of the null amplifier need not be especially large. This

feature is shared with the method presented in Chapters 4 and 5 of the

present work.


Continuous methods


The use of a four-electrode method in biological impedance

measurements was first reported by Burger and van Milaan (1943). They

measured various segments of the human body and used the results to

estimate the specific resistance of certain body tissues. The current

was supplied in manually switched d.c. pulses from a battery in series

with a large fixed resistor. To find the transfer resistance of the

electrode-tissue system, the voltage between potential electrodes was

compared to the voltage across a known resistor connected in series










with the current electrodes. The voltages were amplified by means of

a single-ended amplifier and read on a galvanometer. Since the current

source and the electrode-tissue system were not independently grounded,

the amplifier could be switched at will to the terminal pairs of

interest.

Rush et. al. (1963) used a method similar to that of Burger

and van Milaan to generate the current waveform. However, the voltage

response was sensed by means of a differential amplifier. The

amplifier common (ground) was connected to a remote point on the animal.

The common mode signal of this configuration was of the same order of

magnitude as the differential signal. Hence, the requirements on the

input impedance and common mode rejection ratio of the amplifier were

not difficult to satisfy.

For reasons of convenience, reliability and accuracy, it is

usually desirable to use ground as a common reference for the current

electrode and potential electrode circuits. This is especially true

when sophisticated measurement techniques are employed and battery

operation becomes impractical. The methods discussed in the remainder

of this section use electronic current generators and common ground

connections.

Among the earliest applications of four-electrode methods were

the now famous voltage clamp experiments of Hodgkin, Huxley, and Cole.

In the original experiments (Cole, 1949; Marmont, 1949), the membrane

voltage was sensed with the same pair of electrodes that were used to

provide the membrane current (a two-electrode method). Later, Hodgkin,

Huxley, and Katz (1952) eliminated the error introduced by the current









electrode polarization voltages and some of the intervening medium

by adding a second pair of electrodes to measure the membrane voltage

(a four-electrode method).

The basic voltage clamp scheme has been carried out in various

ways. One of the later arrangements used by Moore and Cole (1963) is

depicted in simplified form in Figure 1.3. The potential amplifier


I I


EZIf III


Figure 1.3.


Voltage Clamp Scheme


senses the membrane voltage differentially by means of two micropipette

electrodes. The output, Vo, is compared to the control signal, Vc,

in the control amplifier which adjusts the current of the internal

current electrode to maintain V = V The external current electrode
o c










consists of three separate cylindrical pieces. Current is measured

only in the center piece where the field is uniform and calculable.

The input of the current sensing circuit is essentially at ground

potential. The external current electrodes in this system have a

relatively large area and are usually platinized platinum (sometimes

Ag-AgCl).

The common mode signal to the potential amplifier is the

voltage occurring between the external potential electrode and ground.

The differential signal is essentially the membrane voltage. The

highest ratio of common mode to differential signal occurs at the peak

of the membrane current. A typical ratio is 20 my/60 my or 1/3 (Moore

and Cole, 1963, Figure 10). Hence, for a 1 percent common mode error,

the rejection ratio of the potential amplifier need only be about 33,

which is not difficult to achieve. The low common mode signal found

here is in contrast to the relatively high ratios that occur in a

large number of four-electrode applications, especially those concerned

with in situ tissues. Higher common mode to differential ratios not

only require higher common mode rejection ratios, but also higher input

impedances in the amplifiers. These topics are discussed in detail

along with other problems in chapters to follow.

In 1955, Freygang and Landau reported a clever four-electrode

scheme to measure the specific resistance of the cerebral cortex in

anesthetized animals. One current electrode was placed in a well of

physiological fluid overlying an exposed section of cerebral cortex.

The other electrode was placed in the pharynx. In this way, a fairly

uniform current density was established in the cortex under the well.









The current electrodes were made of platinized platinum and had fairly

large contact areas to minimize polarization impedance. The voltage

across the cortex was sensed by means of two micropipette electrodes

which were connected to a differential amplifier. The applied current

was a square wave derived from a phase splitter with an output impedance

greater than 20 kilohms. Pulse durations of 0.3 ms to 0.7 ms were used.

In a typical run, the cortical voltage was 20 my or lesswhile

the voltage between current electrodes was about 1.8 volts. The

common mode rejection ratio of the differential amplifier was given as

2,000. Although the current was derived from a balanced source, the

common mode signal to the amplifier was not necessarily inconsequential

because of probable differences in the impedance of the current paths

above and below the region of the potential electrodes. To illustrate

this point, suppose the common mode signal was one fourth the voltage

between current electrodes or approximately 0.45 volts. Then the

common mode rejection error would be somewhat greater than one percent

of the cortical voltage. The fundamental frequencies of the applied

square waves were greater than 700 Hz. On the basis of amplifier

rejection ratio alone, it is unlikely that reliable measurements could

have been made at much lower or much higher frequencies with the system

described.

Most of the four-electrode studies published to date have

occurred since 1960. In the first of these works, Burger and van Dongen

(1960-1961) modified the original four-electrode method of Burger and

van Milaan (1943) to include sinusoidal frequencies from 20 Hz to 5 kHz.

The basic scheme is shown in Figure 1.4, where the rectangle with four




















V P


-- ------



R
VR





Figure 1.4. Four-Electrode Comparison Scheme


terminals represents the electrode-tissue system. The procedure was

to adjust R until the amplitude of VR was equal to that of Vp. Then

the magnitude of the transfer impedance of the four-electrode system

would be approximately R divided by the gain of the differential

amplifier. The common mode signal to the amplifier includes one

current interface voltage as well as VR and some contribution from the

tissue. This signal is likely to have been much greater than the

differential signal in some of the experiments reported. The authors

make almost no mention of system performance or measurement difficulties

encountered. Also no technical data on the apparatus were given. This

omission is unfortunate in view of the significant differences noted









between their results and those of Schwan and Kay (1956) which this

investigation was meant to clarify.

Ranck (1963a) used an arrangement similar to that shown in Figure

1.4 to measure transfer impedance in rabbit cerebral cortex in acute

experiments. The voltage and current signals were connected to the

vertical and horizontal sweeps of an oscilloscope, respectively. The

magnitude and phase of the impedance were determined from the parameters

of the resulting eliptical pattern observed on the oscilloscope screen.

The electrode system included three small platinum electrodes (20 to

40 micron diameter) arranged in a linear array on the exposed cortical

surface. These correspond, in order, to the upper three electrodes

of Figure 1.4. The fourth electrode was located remotely on the

animal. The cortical electrodes were spaced at least 300 microns apart,

but the array never exceeded 1.3 mm in length.

Ranck's highly assymmetrical electrode system is reminiscent of

the arrangements used for three-electrcde measurements on individual

cells and possesses similar advantages. The polarization impedance of

the remote electrode can easily be made negligible, and the voltage

gradient decreases rapidly away from the cortical array. Hence, the

common mode signal to the differential amplifier is of the same order

as the differential signal. The common mode rejection ratio of the

amplifier was given as at-least 1,000, which in this case appears quite

satisfactory. Requirements on amplifier input impedance were also not

too difficult to meet.

The most informative discussion of errors in a four-electrode

impedance investigation was given by Ranck in a later paper (Ranck,









1966). This paper also was first to report the application of four-

electrode technique to chronic impedance studies in unrestrained animals.

The basic signal-sensing scheme was as shown in Figure 1.4, except

that the current monitoring resistor was replaced by an operational

amplifier circuit with negligible input impedance (similar to that

shown in Figure 1.3). As before, the voltage and current signals were

displayed as a Lissajous pattern on an oscilloscope. In addition, they

were connected to a special instrument, the JB-5 lock-in amplifier,

which, with auxiliary circuits, was calibrated to display the in-

phase and quadrature components of the tissue voltage response with

respect to the applied current.

The lock-in-amplifier uses tuned circuits and special averaging

techniques which greatly increase the signal-to-noise ratio of the

displayed signal. Hence, Ranck was able to measure voltage responses

of the order of 20 microvolts in the presence of EEG activity of the

order of 300 microvolts. This is not feasible with the Lissajous

pattern method unless the signal frequency is far removed from the

noise frequencies and a noise filter is used. Low current densities

in the tissue (hence small voltage responses) are desirable to minimize

the possible influence of the test current on the tissue impedance.

This effect has yet to be ascertained.

The main electrode -assembly was chronically implanted in

selected sites of the rat brain. The submerged part consisted of four

parallel wires (75 micron) spaced uniformly in a linear array 1 to 1.5

mm across. Each wire was roughly 1 mm long and was insulated to within

about 100 microns of the tip. The wires were made of platinum and the









tips were platinized to reduce polarization impedance. Two field

configurations were used. In the linear configuration, the brain

electrodes were connected in the order shown in Figure 1.4. In the

radial configuration, a remote electrode was used as the current return

in place of an end head electrode. The latter configuration did not

possess the advantage of a modest common mode signal as found in the

radial configuration. Ranck made reference to this fact and noted

greater difficulties and poorer results with the linear configuration.

Ranck's discussion of errors has been a source of inspiration

for some of the more general treatments given in the present work.

In particular, he called attention to the errors caused by parasitic

resistance (leakage) and capacitance which provide undesirable coupling

between the current and potential electrode conductors. In the linear

configuration, Ranck found, for example, that at frequencies below 100

Hz, the parasitic resistance had to be greater than 108 ohms to realize

a voltage sensing error of less than 1 percent. For frequencies below

about 10 Hz, the requirement became greater than 1010 ohms. He noted

the difficulty of maintaining resistances of this magnitude on the head

of animals in chronic experiments. These requirements also came

uncomfortably close to the nominal input resistance of the response

amplifier, which was given as greater than 109 ohms.

Ranck neglected the phase angle of the electrode polarization

impedance in his error analysis. But, he did make use of this

phenomenon in testing the electrode assemblies before implantation.

Hence, each assembly was checked for leakage in 0.2 percent saline at

1.5 Hz and the phase angle of the voltage response was taken as the










most sensitive indicator of error. The phase angle, if any, was due

to the effect of electrode polarization capacitance. The reliability

of the results obtained with this method of electrode testing is

questionable for reasons explained in the following paragraphs.

As Ranck indicated, the error due to a given amount of leakage

between a current electrode and a potential electrode is proportional

to the product of the electrode impedances divided by the transfer

impedance to be measured. Mathematically



IE Z-- IZpl
IZTl

where ZQ, Zp and-ZT designate the current electrode, potential

electrode, and transfer impedances, respectively. At the test frequency

of 1.5 Hz, ZQ and Zp were largely determined by interface impedance.

The interface impedance is not a sensitive function of saline concentra-

tion, since it depends mainly on a hydrogen transfer mechanism with

the water. Therefore, IZQI and IZpl would be expected to have much

higher values in the tissue than in the saline test solution due to

the masking effect of cellular components (Schwan, 1963).

The 0.2 percent saline used for the electrode tests produced

approximately the same IZTI as was found in the tissue. However,

since IZQI and |Zpj were probably much lower in the saline than in the

tissue, the simulated error would likewise be expected to be much

smaller, but by the same factor squared. Suppose that isotonic saline

(0.9 percent) were used for the electrode tests. Then, the ratio of IZQI

to IZTI in the saline would more closely simulate that found in the









tissue (assuming the cell masking factor was roughly proportional to

the ratio of tissue to tissue fluid impedance). In this case, the

simulated error would still be too low, but only by the factor in |Zp[

to the first power. The foregoing considerations are rather significant

because they imply that electrode test results obtained in pure saline

do not provide a reliable indication of leakage errors.

The choice of test solution is also important with respect to

errors caused by imperfect common mode rejection in the differential

amplifier. The ratio of common mode signal to differential signal

is approximately given by the ratio of IZQI to IZTI. As previously

indicated, the ratio found in tissues is more closely simulated in

isotonic saline than in 0.2 percent saline. In the latter case, the

ratio would be too low to give a reliable indication of the common

mode rejection error.

The phase angle of the voltage response was also the most

sensitive indicator of error at higher frequencies where polarization

impedance was negligible and electrode impedance was essentially

resistive. Here, the phase angle, if any, was due entirely to capacitive

coupling between the current and potential electrode conductors. The

input capacitance of the response amplifier was given as 30 picofarads

which is equivalent to about 5 megohms at 1 kHz. This reactance

significantly loaded the potential electrodes and, in the radial configu-

ration, caused a phase error in the apparent response of approximately

9 degrees)while the magnitude error was less than 2 percent.

The effects of parasitic resistance and capacitance were greatly

magnified in the linear configuration by the high ratio of common mode










signal to differential signal which varied from approximately 40 at 1

kHz to 600 at 1.5 Hz. This high ratio was also troublesome with

respect to the common mode rejection capability of the amplifier. To

obtain common mode error signals of less than 1 percent, the common

mode rejection ratio would have had to exceed limits ranging from 4,000

at 1 kHz to 60,000 at 1.5 Hz. The common mode rejection ratio of the

system used was given as only better than 1,000. On the basis of the

foregoing discussion and in concurrence with some of Ranck's remarks,

it may be concluded that the results obtained with the linear configura-

tion were unreliable at nearly all frequencies. Much of this problem

lay in the extreme demands that this configuration made on the instru-

mentation components. Better performance could have been achieved with

the instrumentation technique described in Chapters 4 and 5 of the

present work.


Summary


In a broad sense, electrode polarization leads to a similar

problem in every type of electrode system, which becomes worse as the

electrode dimensions are reduced and the frequency is lowered. Briefly

stated, the problem is to find an unknown quantity, the tissue impedance,

within arbitrary accuracy limits, by taking the difference between two

quantities which may become nearly equal at low frequencies. The

differencing process may be done by manual and computational means

(as in two-electrode and three-electrode systems) or continuously and

automatically by electronic means (as in four-electrode systems). The

experimental evidence indicates that the former methods are not











reliable unless the common quantity is significantly smaller than the

difference quantity. In contrast, the four-electrode electronic

methods are reliable and capable of high accuracy even when the common

quantity is much larger than the difference quantity. This subject is

considered in detail in Chapters 3 and 4 under the title of common

mode rejection.

The advantages of the four-electrode method are obtained at the

cost of increased complexity and additional sources of error. However,

a properly designed and well-understood four-electrode experiment can

be no more difficult to perform than a two-electrode experiment, all

factors considered. The basis for this statement may be found in the

remaining chapters of this report, which are summarized in the

following paragraphs.


Chapter 2.--Chapter 2 is concerned with basic analytical

techniques for describing, analyzing and designing the electrode-

tissue system. Special matrix parameters are defined for the system,

and the concept of interface impedance is clarified. Certain boundary

effects caused by interface impedance are explained and techniques for

correcting them are suggested. Further insight is provided by means

of a convenient equivalent network, which is derived in terms of the

matrix parameters of the system. To conclude the chapter, the matrix

parameters are derived from field theory for two different electrode

arrangements and illustrated numerically.


Chapter 3.--Chapter 3 is concerned with the most common method

for amplifying the signal of the potential electrodes. Four major










types of measurement error are defined and analyzed. Limit expressions

are derived for the parameters of the instrumentation system and

associated paraphernalia. The results are illustrated numericallyusing

the examples of Chapter 2. Besides being of value in itself, Chapter

3 provides important definitions and background for Chapter 4.


Chapter 4.--In Chapter 4, a new instrumentation technique--the

virtual-ground system--for extracting the signal of the potential

electrodes is introduced. The four major types of measurement error

defined in Chapter 3 are analyzed for the new system. Limit expressions

are derived for the system parameters and numerical results are

illustrated using the examples of Chapter 2. These results are

compared with those of the common method of Chapter 3. Two additional

topics are treated: the stabilization of the virtual-ground system for

broad band measurements and the operation of the negative feedback

filter loop for noise reduction.


Chapter 5.--In this chapter, an experimental system using the

virtual-ground configuration is described. The system includes a

current control scheme which was taken from the literature and adapted

for broadband impedance measurements. Several experiments are reported

to verify system performance and indicate the quality of results already

obtained in investigations on live animals.















CHAPTER 2


MATRIX AND NETWORK DESCRIPTION OF FOUR-ELECTRODE SYSTEMS


Voltage and Current Conventions


It is convenient to assign the symbols Ql, P1, P2, and Q2 to

the terminals of a four-electrode system, where Q designates the current

or excitation electrodes and P designates the potential or response

electrodes. Let the complex terminal voltages and currents describing

the state of the system be as defined in Figure 2.1.


, t


I
01,


SQ


-I P1

V P IP2 P2
p P2
P2


S2


Q2


Figure 2.1.


Voltage and Current Conventions for the
Electrode-Tissue System










According to the laws of network theory


V2 + Vp + V1 VQ = 0 (2.1)


and


IQI + IQ2 + Ip + IP2 = 0 (2.2)


which shows that only three voltages and three currents may be defined

independently. Note that the V and I quantities usually possess in-

phase and quadrature components which vary with frequency. Equations

(2.1) and (2.2) each imply two equations, one for the in-phase and one

for the quadrature components, with all quantities being evaluated at

the same frequency.


The Impedance Parameter Matrix


In a linear system with sinusoidal excitation, the complex

voltages and currents are related by a set of linear simultaneous equa-

tions with complex coefficients. It follows from Equations (2.1) and

(2.2) that three independent equations are sufficient to describe the

four-electrode system. Using vector and matrix notation with Vp(f)

VQ(f), IQ1(f), Ipl(f), and Ip2(f) as the working variables, the system

equations may be written



V mll m12 -m13 IQ1


VQ m21 m22 m23 Il (2.3)


V1 m"31 -m32 -m33 IP2









where the m coefficients are, by definition, parameters of the electrode-

tissue system and have the dimensions of complex impedance. This

particular matrix form was chosen to facilitate the inclusion of inter-

face effects and to simplify the analysis of errors given in later

chapters. The minus signs are included so that the real part of each

parameter will be positive in a passive system. For example, in a

purely resistive system, all the parameters would be real and positive.

The m parameters of a given system may be determined experi-

mentally or estimated, a priori, from known properties of the tissue

and electrodes. In either case, one refers to the basic definitions

listed in Table 2.1, which may be obtained by inspection from Equation

(2.3).



TABLE 2.1

DEFINITIONS OF THE m PARAtF-ETFRS


Constraint Ipl = Ip2 = 0 IQI = Ip2 = 0 IQl = Ip = 0


Excitation Ql-Q2 Pl-Q2 P2-Q2

Vp Vp -Vp
mll = -- m12 = 13
IQl Ipl IP2

VQ VQ VQ
m21 -- m22 -- m23 --
IQl Ipl IP2


V1 -V1 -V1
m31 32 1 -m33 -
IQI Imp Ip2

The immediate objective of most impedance measurements is to









identify the transfer impedance, mll. This parameter is estimated by

determining the ratio of an arbitrary measure of Vp (the nominal voltage

response) to an arbitrary measure of IQl (the nominal exciting current)

under the assumption that Ipl and Ip2 are zero. As shown in later

chapters, the other m parameters may play a significant role in

determining the accuracy of the result.

In a system composed of bilateral elements, the Reciprocity

Theorem from network theory shows that only six of the nine m parameters

are independent. Three of them may be eliminated by means of the

following relations (Appendix 1).


m22 = "23 + mll (2.4)


m32 = m12 + m33 mll (2.5)


m21 = m31 + m23 + mll (2.6)


The m parameters are determined by the impedance characteristics of the

tissue and the electrode-tissue interfaces and by the system geometry.

These factors are considered in the remainder of the chapter.


Analysis of the m Parameters


The significance of the m parameters may be determined from a

study of the general field relationships that exist within the

electrode-tissue system. Consider a metal electrode in an electrolytic

medium in the presence of an arbitrary current density field as shown

in Figure 2.2. Let s denote an imaginary surface which surrounds, and

is arbitrarily close to, the electrode such that it includes essen-

tially only that part of the tissue which is involved in interfacial





43





/ / /

/ / /
S/ /







electrode surface s
/ / / T /
/ / /- n/
















Figure 2.2. Electrode-Tissue Interface in an
Arbitrary Current Field


phenomena. Then, the potential of the electrode with respect to an

arbitrary point of reference in the tissue mass may be written


V =U -E f(2.7)


where E is the average potential on s with respect to the reference

point and U is the average potential difference across the interface.

It is apparent that is a function of the current density field and

the tissue impedance, while U is a function of the normal current

density on s and the interface impedance.

At any position on s, the potential difference across the inter-

face may, in principle, be written as the product of the complex

impedance per unit area, X(s), and the normal component of the current

density, Jn(s), at that point. Then the average potential difference








is given by



U = XJn ds (2.8)
As
where the integral is taken over all of s, A is the total surface area

of s and Jn is taken as positive out of s.

A more convenient form of Equation (2.8) is obtained if the
following substitutions are made


X = X + n(s) (2.9)

n = Jn + v(s) (2.10)

where X and Jn are the average values of interface impedance and normal

current density on s, and n(s) and v(s) are deviations from the averages,

respectively. Note that by definition


Sn (s) ds = 0 (2.11)

and

Sv (s) ds = 0 (2.12)


If I designates the total electrode current in the external circuit,

then from Equations (2.10) and (2.12)


J = (2.13)
n A
With Equations (2.9) through (2.13), Equation (2.8) may be rewritten


U = X I + 1 nv ds (2.14)
A A Js










The first term on the right in Equation (2.14) expresses the

average potential difference across the interface due to current flowing

through the electrode and the external circuit. The second term

expresses the average potential difference across the interface due to

non-uniformity in the interface impedance. For example, if the inter-

face impedance were truly uniform, the integral would be zero because

n(s) would be zero over all of s. In most cases, the major determining

factor in n(s) is the inherent inhomogeneity of the electrode surface

(Flasterstein, 1966b).

The relative importance of the terms in Equation (2.14) depends

on how the electrode is used. For the current electrodes of a four-

electrode system, the second term is undoubtedly negligible compared

to the first. However, for the potential electrodes, where I is

nominally zero, the second term may be significant (Schwan, 1963). In

cases where there is a choice of position, the potential electrodes

should be placed where the current density intercepted by the electrodes

will be low (in a region of low potential gradient) and the voltage

between the electrodes is high. In this way, v is kept to a minimum

and the inhomogeneity term in Equation (2.14) is more likely to be

negligible in comparison to the response voltage. For example, in the

parallel wire configuration shown in Figure 2.3, the potential electrodes

are placed outside rather than between the current electrodes, where

the same potential difference may be recorded and the current density

is smaller.


The Effective Interface Impedances

In a system of four electrodes, Equation (2.7) holds for each
















P1 Ql Q2 P2

IQI




surface of medium








Figure 2.3. Parallel Wire Four-Electrode System


electrode. Using the appropriate subscripts and summing the electrode

potentials according to Kirchhoff's Voltage Law and Figure 2.1, the

vector equivalent of Equation (2.7) is found to be


Vp (UPl Up2) Ep


V = (UQl UQ2) + EQ (2.15)


V1 (UQ up) E


where Ep, EQ, and E1 are the average potential differences between the

s surfaces of electrode pairs Pl-P2, Ql-Q2, and Q1-P1, respectively.

The present investigation is primarily concerned with the effects

of the first term on the right in Equation (2.14). To simplify the

presentation, the term due to inhomogeneity will be neglected from this










point on. With this omission and adding the appropriate subscripts

in Equation (2.14), the average interfacial potential differences of

the system in Figure 2.1 become


-- _PQl2
Q1



Upl Apl l Zplp
Pl



U ~ I Z I
P2 ~A P2 P2 P2
P2
- XQ2
U2 Q2 -I Z I
Q2 A Q2 Q2IQ2
Q2


(2.16)



(2.17)



(2.18)


(2.19)


x
where, for brevity, the ratio is replaced by Z for each interface.
A
Henceforth, ZQ1, Zp1, Zp2, and ZQ2 are referred to as the effective

interface impedances.

Using Equations (2.16) through (2.19), Equation (2.15) may be

written


Vp


V -


v1


where IQ2

effective

sentation

matically


0 ZPI -ZP2


ZQl+ZQ2 Q2 ZQ2


IQl


L ZQ1 -ZPI 0 I LP2J LEli

has been eliminated by means of Equation (2.2). Through the

interface impedances, Equation (2.20) gives explicit repre-

to the interface potential differences and is shown diagra-

in Figure 2.4.


(2.20)





















Q1 Q1


P- ZQ


Vp IP 2



V I E
IP2 P



V2 IQ2 Q2 2E

Q2-


Q1


---



P2

Q----

Q2
!T2


Figure 2.4. Electrode-Tissue System Showing the
Effective Interface Impedances


The h Coefficients


In Figure 2.4, QI, PI, P2 and Q2 may be viewed as the terminals

of a hypothetical electrode-tissue system whose terminal voltages are

equal to the average surface voltages of the real system. These

voltages may be related to the electrode currents by means of a matrix

equation as follows:









h12 -h13


h21 h22 h23


(2.21)


E1 h31 -h32 -h33 P2

where the definitions of the h coefficients follow from those given for

the m parameters in Table 2.1 with the appropriate substitutions.

From Equations (2.20) and (2.21), it is evident that the

impedance parameter matrix in Equation (2.3) is given by


m12 -m13


m22 m23


-m32 -m33


0 ZpI -Zp2


ZQ1+ZQ2 ZQ2 ZQ2


ZQ1 -ZP1


hll h12 -h13


h21 h22 h23


h31 -h32 -h33


(2.22)


Note that for all frequencies


mll = h1l





m33 = h33


(2.23)


(2.2,L)


At low frequencies (e.g., below 100 Hz), the elements in the interface

impedance matrix are much larger than those in the h matrix in all

except the 11 and 33 positions. At high frequencies (e.g., above


m21


m31


hll










100 kHz), the elements in the interface impedance matrix tend to become

small compared to those-in the h matrix. However, in many cases, the

interface impedances may not become negligible before the errors of

measurement exceed permissible limits. This possibility is embodied

in the results of succeeding chapters.

It may appear that interface effects are expressed entirely by

the interface impedance matrix. This statement is only approximately

true. The geometry of the current field is determined mainly by the

electrode configuration. However, it is also influenced in varying

degree by the frequency-dependent boundary conditions imposed by the

impedance of the electrode-tissue interfaces. This factor is not likely

to be important except with respect to its direct effect on hll. For

the sake of completeness and to demonstrate the complexity of the

four-electrode field problem, a brief discussion of possible boundary

effects is in order. The examples that follow are of a hypothetical

nature and have not been studied in experimental or analytical detail.

The existence of boundary condition effects is evidenced by certain

anomalous behavior of electrodes in saline which is difficult to explain

in terms of other sources of error.


Examples of boundary condition effects


To indicate how interfacial boundary conditions may affect hll,

consider again the electrode configuration of Figure 2.3. The exposed

tips of the potential probes are located near the surface of the tissue

mass where the current density field is fairly planar. Assume that a

sinusoidal current of constant amplitude, IQl, excites the system. At

high frequencies, where the interface impedance is small, the current









density distribution along Ql and Q2 is determined primarily by the

configuration of the metalic boundaries and is indicated qualitatively

in Figure 2.5 (high). The rise toward the electrode tips is due to








4I
high

= low






t depth
tips of tips of
Pl and P2 Ql and Q2


Figure 2.5. Electrode Current Density in the System of
Figure 2.3


fringing into the tissue mass. At low frequencies, the interface

impedances of Ql and Q2 are large compared to the tissue impedance,

and they tend to smooth the current density distribution, as shown in

the same figure (low). Since the total current is constant, the areas

between the low and high frequency graphs are equal, and there is more

current at low frequencies in the field of Pl and P2 than at high

frequencies. The importance of this effect should depend on the rela-

tive dimensions of the current electrodes and become smaller as the

electrode length is increased.










A useful expression for hll is


hll = rE


(2.25)


where r is a geometrical factor and j is the specific impedance of the

tissue. Since r reflects the current density distribution, it varies

with frequency and for the configuration of Figure 2.3 takes the

general form shown in Figure 2.6. At low frequencies, where the effect


log frequency


Figure 2.6.


Frequency Dispersion of the Geometrical
Factor for the System of Figure 2.3


of interface impedance is much greater than that of the tissue impedance,

the graph is flat. When the impedances are comparable, T decreases

with frequency and assymptotically approaches a value corresponding to

zero interface impedance.

Since the electrode-tissue system is assumed to be linear and

passive, it should obey the phase-magnitude relationships of network









theory (Bode, 1945). Hence, the magnitude of hll cannot vary with

frequency without a concomitant change in its phase angle as well.

This implies that there can be no frequency dependence of IFI unless F

is a complex number. Indeed, careful consideration of the phenomenon

illustrated by Figure 2.5 indicates that the phase angle of the current

density may vary along the electrode length even though the phase of

the total current is fixed. The phase of the current toward the tips

of Ql and Q2 would tend to be positive, while that toward the tips of

P1 and P2 would tend to be negative, with respect to the phase of the

total current. Hence, in agreement with network theory, a decrease

in hll| (or Irl) with frequency would be accompanied by a negative

phase angle in hll (or r). In view of these considerations, the

ordinates in Figures 2.5 and 2.6 should be interpreted as the magnitude

of the quantities indicated. Further, the areas between the high and

low curves in Figure 2.5 are not equal since phase angles must be taken

into account.

Experimentally, F is often estimated by testing the electrode

system in saline of known specific resistance. The frequency dependence

of r may be expected to vary with the ratio of interface impedance to

saline impedance. For example, as saline concentration is increased,

this ratio increases, and the dispersion in F may be expected to occur

at higher frequencies. If the frequency dependence of F is significant

in a given system, it becomes necessary to use a saline concentration

which produces a similar ratio of interface impedance to specific

impedance as occurs in the tissue to be measured. This problem is

complicated by the masking effect of cells in the tissue on the inter-

face impedance. Hence, rather than use saline of approximately the










same specific impedance as the tissue, the saline impedance should be

like that of the extracellular fluid which is much lower.

Another possible boundary condition effect on hll is due to

non-uniformity of the interface impedance along the electrode surface.

Non-uniformity at the potential electrode interfaces has already been

discussed in connection with Equation (2.14). When it occurs at the

current electrode interfaces, it may alter the current distribution in

varying and indeterminate ways. For example, the low frequency plot

in Figure 2.5 might actually have an irregular shape which varies

spontaneously with time. Further, if the phase angle of the interface

impedance varies over the electrode surface, F may become a complex

number introducing an erroneous phase angle in hll.

The guard electrode principle.--In the system of Figure 2.3,

the constancy and definition of the current field in the potential

electrode region may be improved by using two electrically separate

sections for one of the current electrodes. The two sections should

appear continuous except for a minute break occurring at an arbitrary

point below the potential electrode tips. The controlled exciting

current is caused to flow through the upper section, while the potential

of the lower section is made to follow the potential developed by the

upper section by means of external circuitry. From the field viewpoint,

the two sections appear as a single electrode, but with the current in

the region of interest under direct control. The guard electrode

principle, as it is commonly called, may be applied in many different

configurations (see, for example, Van Harreveld, et al., 1963; Graham,

1965b).










The Equivalent Network


The significance of interface impedance and geometrical factors

is enhanced if the four electrode system is viewed in terms of an

equivalent network. Consider the hypothetical system described by

Equation (2.21). As indicated earlier for the m parameters, in a

system obeying reciprocity, only six of the nine h coefficients are

independent. Such a system may be represented by a network with six

independent impedance elements. A convenient form for this network is

shown in Figure 2.7 where W denotes complex impedance. This


Ip2


Figure 2.7.


Equivalent Network for the Hypothetical
System


configuration was chosen because it places in evidence certain implicit

properties of the electrode-tissue system. For example, Wpl and Wp2

may be interpreted as electrode impedances which characterize the effect










of drawing current through the potential probes. The remaining four

elements split the exciting current into two parts: one passing

through W1, Wp, and W2, which generates the response voltage between

points 1 and 2, and one passing through WQ, which, in a sense, is

wasted. In many geometries, the effect of WQ is negligible, and the

network simplifies to a double tee form.

The matrix equation governing the network of Figure 2.7 is

W W2
Ep ( YWp (YWp + + Wpl) -(WP2 + -W) 1Q


Q Y(W1 + 2 + ( + 2 + ) 2 1l

w wi w
E! YW1 1(W + ----P + Wpl -! I
11 D 2 D P P D 2 P2

(2.26)
where
WQ
Y (2.27)
S+ W + Wp + WQ

and

WI + W2 + Wp
D = W1 + W2 + Wp + W (2.28)
E1+ 2+ WP+tWQ 1-y

Setting the matrices of Equations (2.21) and (2.26) equal, the following

expressions are found for the network elements

h
WP = h12 hll(l+ h33 (2.29)
23
h33
W = h h 33 (2.30)
P2 13 11 3l

S31
Wp = hl1 (2.31)










W1 = 1 h31 (2.32)


1 3 1
W = h3 (2.33)
2 y 23


1 h23h31
W 1 23 (2.34)
Q Y h33



and


1 h33h21
1-= 33 (2.35)
Y h23h31



In general, the network elements are frequency dependent through the

tissue impedance and the interfacial boundary conditions.

As noted earlier, the immediate objective of four-electrode

measurements is to identify hll. In terms of the equivalent network



hl = YWp (2.36)



where Y, as defined in Equation (2.27), is the fraction of IQ1 which

flows through Wp when Ipl and Ip2 are zero. As shown in the examples

at the end of the chapter, the value of Y depends on the electrode

configuration, but it may not be greater than unity.

To complete the equivalent network, the effective interface

impedances are added as indicated in Figure 2.8. At low frequencies

(e.g., below 100 Hz), Zpl and Zp2 are much larger than Wpl and Wp2,

respectively. At high frequencies, the interface impedances should

become negligible in the limit. However, as indicated for the matrix
















Pzpl wpl 1 wQ Q











2--- Zp2 Wp2 2 Z2 Q2








Figure 2.8. Complete Equivalent Network of the Electrode-
Tissue System


parameters of Equation (2.22), this condition may not occur before the

errors of measurement exceed permissible limits.

In the topics treated later in this work, the matrix parameters

are used almost exclusively because they are more basic, easier to

manipulate, and lead to more concise results. However, some phenomena

are more readily understood when viewed in terms of the equivalent

network.


Examples of Impedance Parameter Matrices


The major results of this chapter are illustrated in the

examples that follow. To simplify the computation, the configurations









are idealized, and the effect of interfacial boundary conditions on the

h coefficients is neglected. It is assumed that all electrodes have

the same average interface impedance per unit area (X), and the

impedance medium is homogeneous with respect to the dimensions of the

main field region. The specific impedance of the medium is denoted by



The h coefficients are found from the basic definitions in Table

2.1 with appropriate modifications (compare Equations [2.3] and [2.21]).

The effective interface impedances follow from the definitions in

Equations (2.16) through (2.19). Finally, the network elements may

be calculated from Equations (2.29) through (2.35), if desired. For

convenience, the results are tabulated as matrices following the form

of the right hand terms in Equation (2.22).

In each example, numerical values are illustrated for the matrix

elements at low and high frequencies. The computations make use of

the approximate values of X and 4 shown in Table 2.2, which are based

on observations by the author using stainless steel electrodes in 0.9

per cent (isotonic) saline. The factor e-je is defined in Chapter 1.


TABLE 2.2

APPROXIMATE VALUES OF X AND 4 FOR STAINLESS STEEL ELECTRODES
IN ISOTONIC SALINE

Frequency X ohm-cm2 4 ohm-cm

3 Hz 1,000 e-j720 60 ejO0

100 kHz 0.5 e-j720 60 ej00



Here, it indicates a phase angle of minus 72 degrees. The complex

number equivalent is









e-j720 = cos 720 jsin 720

= 0.31 j0.95


The examples are calculated for the system shown in Figure 2.9, which

consists of four hemispherical electrodes set into the surface of a

semi-infinite impedance medium. Electrode radii and separation are

represented by r and d with appropriate subscripts, respectively.


I
p1l


V



I


semi-infinite medium


Figure 2.9. Four-Electrode System with Hemispherical
Electrodes





61



Example 1


The approximate element expressions in Table 2.3 are determined

for the following geometrical conditions


dl = dp = d d
1 P 2


and


rQl = rPI = rP2 = rQ2 = r<

The numerical results correspond to d=0.2 cm and r=0.01 cm.


TABLE 2.3

MATRIX ELEMENTS FOR EXAMPLE 1

Interface Impedance


Frequency


h Coefficients


All 0 1/2 -1/2 1/2d 1/2r -1/2r

Y/Tr2 1 1/2 1/2 ipi 1/r 1/2r 1/2r

1/2 -1/2 0 1/2r -l/2r -1/6d



3 Hz 0 1.6 -1.6 47.7 955 -955

106e-j72 3.2 1.6 1.6 1910 955 955

1.6 -1.6 0 955 -955 -15.9



100 kHz 0 800 -800 47.7 955 -955

e-j72 1600 800 800 1910 955 955

800 -800 0 955 -955 -15.9



In this example, Y is approximately unity at all frequencies.










Example 2


The geometrical conditions for this example are


d1 = dp d d2>>d
1 P 2


rQ = rP = rP2 r<

rQ2 =2d


The approximate element expressions are given in Table 2.4, where the

numerical results correspond, as before, to d=0.2 cm and r=0.01 cm.


TABLE 2.4

MATRIX ELEMENTS FOR EXAMPLE 2

Interface Impedance


Frequency


h Coefficients


All 0 1/r2 -1/r2 1/2d 1/r -1/r

/27r 1/r2 1/4d2 /4d2 /2n 1/r 3/2d l/d

1/r2 -1/r2 0 1/r -1/r -1/2d



3 Hz 0 1.6 -1.6 23.8 955 -955

106e-j72 1.6 0.001 0.001 955 71.4 47.7

1.6 -1.6 0 955 -955 -23.8



100 kHz 0 800 -800 23.8 955 -955

e-j72 800 0.5 0.5 955 71.4 47.7

800 -800 0 955 -955 -23.8


In this example, Y is 2/3 at all frequencies.









Concluding Remarks


The analysis in this and succeeding chapters uses methods of

linear mathematics which assume that the electrode-tissue system is

linear throughout. In practice, while the exciting and observed

variables may be linearly related, the voltages and currents at the

current electrode interfaces may not necessarily be so, depending on

the current densities and frequencies encountered. However, this

discrepancy detracts little from the value of the linear approach in

understanding and estimating system performance. A linear analysis

also serves as a basis for the inclusion of nonlinear effects.















CHAPTER 3


ERRORS IN THE IDENTIFICATION OF TRANSFER IMPEDANCE BY
THE CLASSICAL METHOD


Introduction


The classical mode of operation in four-electrode measurements

is to ground one current electrode and record differentially from the

potential electrodes. The purpose of this chapter is to define and

analyze the errors of this method and provide a basis of comparison

for the more advanced technique in the next chapter.

Errors in the identification of mll include four major types.

Briefly, they are

1. loading error

2. common mode rejection error

3. current control error

4. noise

Each of these errors is discussed and the first three are analyzed in

detail in this chapter.

A basic block diagram for impedance measurements is shown in

Figure 3.1. The signal-processing and display systems are irrelevant

to the present study and have been omitted. The connecting system

comprises all the conductors, connectors, and supporting materials that

link the electrode-tissue system to the instrumentation. The electrode-

tissue system includes the tissue and only those parts of the metal















exciting
system


amplifying
system


connecting
system


electrode-
tissue
system


Figure 3.1. Basic Block Diagram for Impedance Measurements


electrodes which are in contact with the tissue. The exciting system

usually includes a signal generator and means for controlling or sensing

the applied current. The amplifying system amplifies the voltage response.


Loading Error


The term loading admittance will be used to refer to the

extraneous admittances in an impedance experiment that permit

undesirable currents to flow in the electrode-tissue system. For

clarity, it may be viewed as consisting of four arbitrarily defined

classes of components:










1. transverse admittance

2. cable admittance

3. amplifier input admittance

4. ground admittance


Transverse admittance.--Transverse admittance occurs in the

connecting system and denotes parasitic current paths between any or

all of the signal conductors. It arises from fixed and indeterminate

properties of the insulators which support the electrodes and various

connectors. Indeterminate properties are primarily due to bulk and

surface contamination of the insulators by such media as blood,

saline, water, and solder flux. Since it has a somewhat amorphous

origin, transverse admittance is appropriately described by means of

a 3 by 3 matrix, which may be converted to an equivalent network.

If the transverse admittance obeys reciprocity, the matrix and the

network each contain six independent elements.


Cable and amplifier input admittance.--These classes of

admittance are commonly specified in terms of finite elements between

ground and the signal conductors. Elements may also occur between

the conductors directly.


Ground admittance.--Ground admittance encompasses all stray

current paths occurring between the mass of the tissue and ground.

In a practical sense, it may arise unintentionally by contact with

grounded objects such as equipment, people, and wet surfaces. For

present purposes, the manner in which the ground admittance is

distributed need not be known. It is hereafter represented as a










finite element, connected between ground and an arbitrary point in

the tissue.

The first three classes of loading admittance may be viewed as

a four-terminal network in parallel with the electrode-tissue system.

This network is indicated by Ql', PI', P2', and Q2' in Figure 3.2. In

the same figure, ground admittance is represented by Y .
0


IQ


Figure 3.2.


Schematic Diagram of the Electrode-Tissue and
Loading Admittance Systems


The desired parameter of the electrode-tissue system, mll, is

estimated as the ratio of the voltage response, Vp, to the source

current, I According to the definition of mll in Table 2.1, the

result is based on the assumption that Ipl and IP2 are zero, and that

IQ, IQ1 and -IQ2 are equal. This assumption is valid only if the






68


loading currents IQ, I, I2 and I are zero. Each non-zero loading

current causes an error in the estimated value of m11, which, by

definition, is the loading error of that current.

The errors of admittance classes 1, 2, and 3 are analyzed

together in the next section. For clarity, the error due to ground

admittance is treated separately.


Errors Due to Admittance Classes 1, 2, and 3


It is convenient to represent the first three classes of

loading admittance by the equivalent network defined in Figure 3.3

which contains six independent elements. The current vector of this

network is


Il Y3 YQ (Y1+Y3)" V


= Y Y -(Y+Y4) VQ (3.1)

Ip2 -(Yp+Y+Y3) Y2 -(Y2+Y3) V


With reference to Figure 3.2, the current vector of the

electrode-tissue system is given by



II I IQ
Ql Q Ql

I = 0 I' (3.2)
1 1 i (3.2)
PI P1

1 0 It
P2 P2











I'
Q' Q1


Figure 3.3.


Equivalent Network for
Classes 1, 2, and 3


Loading Admittance


and from Equation (2.3), the voltage vector is


m21 m22


m -m
31 32


It follows from Equations (3.2) and (3.3) that


-m13


m23


-m
33


IQl


Ip1

I
P2


(3.3)


m12









VP = mlQ ll ml l + ml3Ip2 (3.4)
P QllQ 11Q 1


where the terms involving I 1, IP and IP2 are, by definition, loading

errors in Vp which ideally should be zero. Dividing these terms by

Vp and substituting for the currents from Equation (3.1), the relative
loading errors in Vp may be written


aQl V M Q + (1 + Vp)Y3 + (3.5)
P 1 P P P


S-ml2IPl V V V
pl V -m12 )Y4 V Y1 + Y (3.6)



and


m[Ip' V V V1
a V 2 m3 V( 1)Y (1 + V)Y3 Y (3.7)
P2 Vp 13 VP VP 2 P 3 P



The operating condition of primary interest in four-electrode

measurements occurs when aQ1, cPI and a P2 are quite small. Under this

condition, it is possible to approximate the voltage ratios in the

bracketed terms of Equations (3.5), (3.6), and (3.7) by expressions

involving only the m parameters. The procedure is justified by the

fact that when the a are small, the errors in the approximating

expressions will also be small as explained in the next paragraph.

According to Equation (3.3), Vp, VQ, and V1 each include a

contribution from Ipl and Ip2. If one considers the geometrical and









interfacial properties of four-electrode systems, it is not difficult

to see that V1 and especially V are, in most cases, less sensitive than

Vp with respect to Ipl and Ip2. In the analysis that follows, the

error in Vp due to these currents is required to be less than 0.05Vp

in magnitude. It is reasonable to assume, therefore, that, under this

condition, the corresponding errors in V and V1 will similarly be

less than 0.05 VQ and 0.05 Vl, respectively.

To determine the voltage ratios, the errors in Vp, VQ and V1

due to Ipl and Ip2 will be neglected in accord with the foregoing

discussion. With this understanding, Equation (3.3) yields



Vp = mllQl (3.8)





VQ = m2 = (m31 + "23 + mll)IQl (3.9)



and



V1 = m31Ql (3.10)



where the dependent parameter, m21, has been replaced by means of

Equation (2.6). It follows that the voltage ratios are



V1 m31
V m (3.11)
P 11










VQ 31 + 23 + mll
Vp mll


(3.12)


Using Equations (3.11) and (3.12), the expressions for the

relative errors become


aQl -(mll + 31 + m23)YQ + (l + m31 3 + m31




-m12
pI ~ 1 [(mll + 23)Y m1Y + mllY]
Pl m 11 11 23 4 31 1 1 P




"13
P2 = ~ m23 Y (ml + m31)Y mllY]
P2 m11232 11 31 3 11 P


(3.13)





(3.14)






(3.15)


Magnitude and phase errors


Before continuing with the analysis of loading error, it is

appropriate to determine how the error is manifested in the experimental

results. The experimental estimation of the desired impedance

parameter, m11, is defined by


^ _-V
mll IQ


(3.16)


If the total relative loading error in Vp due to admittance classes

1, 2, and 3 (i.e., the sum of aQ1, a1p and aP2) is designated by aL,





73



then, from Equation (3.4)



m = --_ (3.17)
m11- 1 a



It follows from Equation (3.17) that the relative magnitude
A
error in mll is



M m 1 (3.18)
Imlll 1i aL



and the phase error is




S i 11 = 1 a ) (3.19)



To study the dependence of the magnitude and phase errors on aL,

consider the phasor diagram shown in Figure 3.4. If aL is allowed to

rotate in the complex plane, the algebriac signs of M and D vary as

indicated in Table 3.1. For a given La l, the relative magnitude

error becomes maximum when the angle of aL is 0 or 180 degrees, and

the phase error becomes maximum when the angle of aL is (90 sin-1ja L)

degrees. For small |aLl, the maximum errors may be written



M = +aL (3.20)


with zero phase error, and


























1-a


Figure 3.4.


Phasor Relations of aL in the Complex Plane


TABLE 3.1

SIGN OF MAGNITUDE AND PHASE ERRORS IN m l


Quadrant of aL--


M


*More precisely, the sign
through


I II III IV


+- +


+ +



of M changes when the angle of cL passes


cos-l( IaL)] degrees,
7 a


which for IaL <0.05 is within 2.1 degrees of the quadrature axis.


[90 1
2









S= 57.3 IaL degrees (3.21)


2LI
with approximately relative magnitude error.
2
In the computations that follow, |aLI will be limited to less

than 0.05, corresponding to magnitude and phase errors within 5 percent

and 2.9 degrees of zero, respectively. In practice (e.g., when

measuring very low frequency dispersion phenomena) a smaller limit for

IaLl may be necessary. In that case, it is a simple matter to modify
all of the results by the appropriate constant factor.


Low frequency errors


If reasonable care is taken in the construction and maintenance

of the connecting system, low frequency errors do not become significant

until the low frequency approximations of the m parameters indicated

in connection with Equation (2.22) become valid. At low frequencies,

the Y coefficients will normally be pure conductances. Making the

appropriate substitutions in Equations (3.13), (3.14), and (3.15), and

denoting Y by G for conductance, the low frequency relative loading

errors become



aQ = (ZQ + ZQ2)GQ + ZQlG3 + ZQlGl] (3.22)


-Zpl
11h


2 h [Z2G2 Z G -hllG (3.24)
~P2 h11 Q2 2 Qi 3 11 (










Since the effective interface impedances are generally much larger

than hll, aQl will ordinarily be negligible compared to apl and aP2'

For the same reason, the terms including Gp are likely to be negligible

also.

When the angle of hll is small, the phase angle of aL is given

approximately by the sum of the angles of two interface impedances

plus 0 or 180 degrees, depending on the final sign of the result. For

example, for the phase angles given in Table 2.2, the angle of aL may

fall in the first or third quadrant. In this case, the magnitude and

phase errors are both positive or both negative as shown in Table 3.1,

respectively.

If electrolytic residue contaminates the connecting system,

metal-electrolyte interfaces may form at the signal conductor surfaces

and act in series with the residue mass. In this case, the Y

coefficients may be primarily conductive, or they may include large

susceptive components. The outcome depends on the relative importance

of the effective impedances of the residue mass and the interfaces

formed. Clearly, unless these impedances are quite high, the analysis

of small errors developed here is not valid.

If the expressions for aQ1l aPI and aP2 are combined, it is

evident that the total relative error, aL, comprises six terms, each

multiplied by a different G coefficient. Since the G coefficients are

independent finite parameters of the physical system (see Figure 3.3),

it is desirable to determine what constraints they should satisfy to

assure a certain level of experimental accuracy. The simplest approach

is to place a suitable upper limit on the magnitude of each of the six

terms in aL. Since these terms possess different algebraic signs and










phase angles, the resulting upper limit on Ja L can only be estimated

in the general case. The terms including GQ, Gp, G3, and G4 carry

negative signs, while those of G1 and G2 carry positive signs.

Further, the G3 and G4 terms tend to cancel the G1 and G2 terms; and

the terms including GQ and Gp are usually much smaller than the others.

Accordingly, the upper limit for jaLl may be determined by setting

the terms of G3 and G4 (or G1 and G2) at their upper limits and

neglecting the remaining terms. If, for example, the magnitude of

each of the six terms in aL is limited to less than 0.025, then |aLl

may be assumed to be less than 0.05.

Combining Equations (3.22), (3.23) and (3.24) and applying the

0.025 limit as described yields the following requirements for the G

coefficients:



G 1 > 40IZQ + ZQ21 (3.25)




-1 ZPlZQ2
G > 40 (3.26)
4 hll




G > 40 h (3.27)
2 hll




G 1 > 40 p2 (3.28)
3 h


~









-1 IZPl ZQl
G > 40
1 1 h1





-1 Zp Zp
G- > 40Z + Z I
P Pl P2


(3.29)






(3.30)


where the reciprocals are used in order to express the results in

familiar ohmic dimensions. Table 3.2 illustrates the requirements on

the G coefficients at a frequency of 3 Hz for the numerical examples

at the end of Chapter 2.


TABLE 3.2

LIMITS OF CLASS 1, 2, AND 3 CONDUCTANCE
CHAPTER 2


FOR THE EXAMPLES OF


Example 1

> 130 M


> 2.2 1012


> 2.2 1012


> 2.2 1012


> 2.2 1012


> 130 M


Example 2

64 M


2.6 109


2.6 109


4.4 1012


4.4 1012


130 M


ohms at 3 Hz


High frequency errors


Although, in the limit, the interface impedances become negli-

gible with frequency, this condition may not occur before the error due


-1
GQ
Q

G-
4

G-


G-1
3

G-1


G-1
P









to parasitic capacitance in the Y coefficients exceeds permissible

limits. Thus, in the analysis that follows, both components of the

m parameters are included and the Y coefficients are represented by

ej90S, where S is the susceptance (magnitude) and eJ90 is the 90-

degree phase factor indicating pure capacitance. The high frequency

relative loading errors are



aQl = -ej90 [(ZQ1 + h31 + h11 + ZQ2 + h23)SQ


+ (ZQ1 + h31 + hll)S3 + (ZQ1 + h31)S1l (3.31)



*p = -ejgO ZPI + h12
I = -ej9 p+ h12 [(hll + ZQ2 + h23)S


(ZQ1 + h31)S1 + hllSp] (3.32)


P2 = ej90 2l [(ZQ2 + h23)S2 (ZQ + h31 + hll)S3



hllSp] (3.33)


In addition to the ej90 factor, the phase angle of the total error is

clearly a function of the .phase angles in the interface impedance

modified by the h parameters. One is led to the conclusion that acL

may fall in any quadrant of the complex plane, depending on the

strengths of the various factors.

The combination of Equations (3.31), (3.32) and (3.33) to form










aL produces six terms, each containing a different S coefficient.

These terms are completely analogous to those of the G coefficients

studied earlier and may be treated in the same manner. At high

frequencies, the terms of SQ and Sp are more significant than were

their low frequency counterparts, but they may still be assumed to

be smaller than the other high frequency terms. Hence the 0.025 limit

is applied again to each term and aLI is reasoned to be less than

0.05. Accordingly, the requirements for the S coefficients are



SQ1 > 40 ZQ + h31 h + + ZQ2 + h231 (3.34)




S +h ZQ2 + h23 (3.3)
S-1 > 40 Z + h 1 + (3.35)
4 Pl 12 h




S-1 > 40 Z + h ZQ2 + h23 (3.36)
2 P2 13 hh




1 ZQ1 + h311
S > 40 ZP2 + h3 + l 1 + 11 (3.37)





-1 ZQ1 + h31
S> 40 Z + hl2 h h (3.38)




S1 4
Sp > 40ZP + h2 + ZP2 + hl3 (3.39)
1 IFl 12-h21 1










The results of these relations for the examples in Chapter 2 are

summarized in Table 3.3. To show the effect of interface impedance

and the importance of the h parameters at high frequencies, the limits

were also computed with the interface impedances neglected. The

limits are expressed in ohms at 100 kHz in the upper part of the

table and in picofarads of capacitance in the lower part, where



C = (2 105)-1S (3.40)



and 2n 105 is the radian frequency at 100 kHz.


Transverse admittance errors.--Connectors and implanted electrodes

are usually supported by means of a potting material such as acrylic

or epoxy resin. This support introduces interelectrode capacitances

of between 0.5 and 10.0 picofarads. The choice and handling of support

materials requires special consideration, especially when limits on

the reciprocal interelectrode conductances are above 109 ohms.

With respect to the results in Tables 3.2 and 3.3, the limits in

Example 1 are precarious for G 1, G2 G3 G1 C4, C2, C3 and Cl.

In Example 2, the limits are precarious for G3 GI C3 and Cl. These

results demonstrate that for more reliable performance at both low and

high frequencies, it is advantageous to minimize the current density

at the current electrodes. If a larger electrode were used for Ql in

Example 2, all of the G and C limits would have been within convenient

range.


Cable and amplifier input admittance errors.--The input circuit










TABLE 3.3

LIMITS OF CLASS 1, 2, AND 3 SUSCEPTANCE AND CAPACITANCE
FOR THE EXAMPLES OF CHAPTER 2*


Example 1 Example 2
-1
S > 120k 60k
(78k) (40k)
-1
S1 > 1.8M 170k
(800k) (110k)

S-1 1.7M 110k
(760k) (76k) ohms at 100 kHz

S31 1.8M 3.4M
(840k) (1.6M)
-1
S 1.7M 3.4M
(720k) (1.6M)

-1
S1 > 110k 110k
(76k) (76k)


13
(20)

0.90
(1.9)

< 0.90
(2.0)

< 0.85
(1.8)

0.95
(2.2)

14
(20)


26
(39)

9.0
(14)

14
(20)

0.46
(1.0)

0.46
(1.0)

14
(20)


picofarads


*Values in parentheses are obtained by neglecting interface impedance.










of a typical differential amplifier may be represented by three admittance

elements, as indicated in Figure 3.5 (Graham, 1965a). Two of the elements,


Figure 3.5. Input Circuit of a Differential Amplifier


Ya and Yb' are generally equal. In most cases, a foot or more of

shielded cable is required to connect the amplifier to the P terminals

of the electrode-tissue system. If the two cable conductors are

enclosed in a single shield, a component of the cable admittance adds

to Ya, Yb' and Yd to form the major part of Yq, Y2, and Yp (see Figure

3.3), respectively. If separate shields are used, the cables only

contribute to Y4 and Y2.

The results of Tables 3.2 and 3.3 for the elements of Yq, Y2'

and Yp illustrate requirements on cable and amplifier input admittance

parameters. With respect to the amplifier, the limits in Example 2









are not difficult to meet, especially with field effect transistor

input stages. In Example 1, the limits on G41 and G21 are borderline,

while those on C4 and C2 may be very difficult to satisfy in practice.

With respect to the cables, the requirements in Example 1 on Cq and C2

are too stringent to meet unless special provision is made to compensate

the cable capacitance. One such technique is described in Chapter 5.

Another source of cable admittance is the cable connecting

terminal Q1 to the exciting system. It usually constitutes the major

part of YQ. With reference to Table 3.2, the requirements on G1 are

easy to meet. In Table 3.3, however, special care may be necessary to

satisfy the requirement on Cq in Example 1. The Ql cable is frequently

not shielded.


Errors Due to Ground Admittance


Since the error due to ground admittance depends on how the

stray ground current is distributed with respect to the main current

field, a detailed analysis in the general case is not feasible.

However, it is possible to characterize, in a general way, the

susceptibility of a system to stray ground current. For that purpose,

assume that all of the ground admittance, denoted as Y in Figure
g
3.2, is concentrated at a point of maximum voltage magnitude in the

tissue mass. For this extreme condition, the voltage and current are

indicated by Vgmax and Igmax respectively, and may be expressed in

the following manner


gI lla F (3.41)
I l I Q gg









where



max (3.42)
g IQI



If the loading errors are small, F is primarily a function of

electrode-tissue system parameters (as shown in the next paragraph)

and may be viewed as a figure of merit for the system. The larger F
g
is, the greater is the ratio of stray ground current to exciting

current for a given Y and the greater is the chance for error.

The maximum voltage in the tissue mass corresponds to points

adjacent to the Ql electrode and is given by



Vgmax = VQ ZQ1IQl (3.43)



If VQ is eliminated by means of Equation (3.9), the expression for

Vgmax becomes
gmax



Vgmax = (m31 + 23 + mll ZQ)IQl (3.44)


which is valid when the loading error in V is small. Substituting

Equation (3.44) into Equation (3.42) and replacing the m parameters

by their constituents from Equation (2.22) yields



Fg = |h31 + hll + ZQ2 + h231 (3.45)



Let the relative loading error in Vp due to ground admittance