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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. 4FlGM25, 52204, 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 TwoElectrode Measurements. . ... 15 ThreeElectrode Measurements. . ... 20 FourElectrode Measurements . ... 22 Summary . . .. ... 36 2 MATRIX AND NETWORK IE.:. ITION OF FOURELEC_' :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 ElectrodeTissue Noise. . .. 102 4 THE VIRTUALGROUiND CONFIGURATION Introduction. . . .. 105 Common Mode Rejection Error . .. 108 SingleEnded 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 VirtualGround System . ... 133 Instrumentation Noise . .. 133 Environmental Noise . .. 135 ElectrodeTissue Noise. . .. 137 The NegativeFeedback Filter Loop .. 137 5 AN EXPERIMENTAL SYSTEM BASED ON THE VIRTUALGROUND CONFIGURATION Introduction . . 147 The Physical Layout . . .. 148 Connection to the ElectrodeTissue System .148 Ground Connection . .. 152 Output Connection . .. 152 Input Connection . .. 153 Circuit Details . . 153 The Response Amplifier . .. 153 The VirtualGround Amplifier. . ... 155 The Feedback Filter . .. 156 The Current Control Circuit . .. 157 Experimental Studies . .. 161 Performance Tests . .. 162 FourElectrode Impedance Measurements .. 173 RECOMMENDATIONS FOR FUTURE STUDY . ... 185 Electrode Systems . .. 185 The VirtualGround 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 Lockin 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 FourElectrode Bridge . . 23 1.3 Voltage Clamp Scheme. . . ... 27 1.4 FourElectrode Comparison Scheme. . ... 30 2.1 Voltage and Current Conventions for the ElectrodeTissue System. . . ... . 39 2.2 ElectrodeTissue Interface in an Arbitrary Current Field.. 43 2.3 Parallel Wire FourElectrode System . .... 46 2.4 ElectrodeTissue 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 ElectrodeTissue System. 58 2.9 FourElectrode System with Hemispherical Electrodes .. 60 3.1 Basic Block Diagram for Impedance Measurements. ... 65 3.2 Schematic Diagram of the ElectrodeTissue 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 VirtualGround Configuration 106 4.2 Nominal Gain and Phase Shift Characteristics of Operational Amplifiers. . . ... 107 4.3 VirtualGround System Showing Loading Admittances and Noise Sources . . .. 109 4.4 Simplified Equivalent Circuit for the VirtualGround Amplifier Feedback Loop . .... 124 4.5 Stabilization Techniques for the VirtualGround Amplifier Feedback Network. .. . . 126 4.6 Typical Magnitude and Phase Characteristics of a Lag Network . . ... .. 129 4.7 ElectrodeTissue System with Stabilizing Resistors. ... 133 4.8 VirtualGround System with NegativeFeedback Filter Loop. 138 4.9 Simple LowPass Filter. . .... 142 4.10 Simple HighPass Filter . .... 143 4.11 Overall Transfer Function Characteristics Using Several Methods of Noise Filtering. . ... 145 5.1 Closeup Photograph of the Experimental VirtualGround 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 Lockin 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 Cochairman: 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 electrodeelectrolyte 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 fourelectrode 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 voltageresponse 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 virtualground 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 controlare analyzed for the classical and virtualground 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 virtualground system and provide a set of design formulae. To complement the virtualground 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 virtualground 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 electrodetissue 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 voltagecurrent 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 electrodetissue 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 ohmcm (Schwan, 1963). However, membranes (cellular and subcellular) which are less than 200 Angstroms thick are associated with relatively high transverse resistance (1 to iP00 ohmcm2) and high transverse capacitance (, 1 uf/cm2). The charging and discharging of adjacent membranefluid 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 onecentimeter 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 cablelike 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 threeelectrode 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 threeelectrode and fourelectrode 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. TwoElectrode 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 = MfeJo, 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, twoelectrode 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 insaline and intissue 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 intissue 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, 196061; Rush, et al, 1963), who, on the basis of fourelectrode 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 twoelectrode measurements in situ. It probably also demonstrates the best accuracy that can be achieved under such conditionsat 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. ThreeElectrode Measurements Threeelectrode 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 doublebarrelled 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. Threeelectrode 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 twoelectrode systems. Another limitation of this technique is that the measurements at each electrode position are made at different time intervals. FourElectrode Measurements It is convenient to classify fourelectrode 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 electrodetissue 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. FourElectrode 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. Online 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 fourelectrode 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 electrodetissue 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 singleended amplifier and read on a galvanometer. Since the current source and the electrodetissue 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 fourelectrode 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 twoelectrode 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 fourelectrode 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 AgAgCl). 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 fourelectrode 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 fourelectrode 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 fourelectrode studies published to date have occurred since 1960. In the first of these works, Burger and van Dongen (19601961) modified the original fourelectrode 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. FourElectrode Comparison Scheme terminals represents the electrodetissue 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 fourelectrode 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 threeelectrcde 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 atleast 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 fourelectrode 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 signalsensing 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 JB5 lockin 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 lockinamplifier uses tuned circuits and special averaging techniques which greatly increase the signaltonoise 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 andZT 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 twoelectrode and threeelectrode systems) or continuously and automatically by electronic means (as in fourelectrode 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 fourelectrode 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 fourelectrode method are obtained at the cost of increased complexity and additional sources of error. However, a properly designed and wellunderstood fourelectrode experiment can be no more difficult to perform than a twoelectrode 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 techniquethe virtualground systemfor 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 virtualground 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 virtualground 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 FOURELECTRODE SYSTEMS Voltage and Current Conventions It is convenient to assign the symbols Ql, P1, P2, and Q2 to the terminals of a fourelectrode 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 ElectrodeTissue 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 inphase 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 fourelectrode 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 PARAtFETFRS Constraint Ipl = Ip2 = 0 IQI = Ip2 = 0 IQl = Ip = 0 Excitation QlQ2 PlQ2 P2Q2 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 electrodetissue 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 electrodetissue 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. ElectrodeTissue 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 nonuniformity 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 FourElectrode 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 PlP2, QlQ2, and Q1P1, 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. ElectrodeTissue 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 electrodetissue 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 thosein 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 frequencydependent boundary conditions imposed by the impedance of the electrodetissue 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 fourelectrode 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 electrodetissue system is assumed to be linear and passive, it should obey the phasemagnitude 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 nonuniformity of the interface impedance along the electrode surface. Nonuniformity 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 electrodetissue 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 1y 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 fourelectrode 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 eje is defined in Chapter 1. TABLE 2.2 APPROXIMATE VALUES OF X AND 4 FOR STAINLESS STEEL ELECTRODES IN ISOTONIC SALINE Frequency X ohmcm2 4 ohmcm 3 Hz 1,000 ej720 60 ejO0 100 kHz 0.5 ej720 60 ej00 Here, it indicates a phase angle of minus 72 degrees. The complex number equivalent is ej720 = 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 semiinfinite impedance medium. Electrode radii and separation are represented by r and d with appropriate subscripts, respectively. I p1l V I semiinfinite medium Figure 2.9. FourElectrode 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 106ej72 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 ej72 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 106ej72 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 ej72 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 electrodetissue 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 fourelectrode 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 signalprocessing 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 electrodetissue 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 electrodetissue 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 fourterminal network in parallel with the electrodetissue 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 ElectrodeTissue and Loading Admittance Systems The desired parameter of the electrodetissue 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 nonzero 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 electrodetissue 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 Sml2IPl 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 fourelectrode 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 fourelectrode 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 sin1ja L) degrees. For small aLl, the maximum errors may be written M = +aL (3.20) with zero phase error, and 1a 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 cosl( 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, metalelectrolyte 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 G1 3 G1 G1 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) S1 > 40 Z + h 1 + (3.35) 4 Pl 12 h S1 > 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 12h21 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) S1 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 electrodetissue 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 electrodetissue 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 