*Written as part of my Master’s thesis. Not revised since 2012.*

Bioimpedance is the historically encumbered term used to describe the study of tissues subject to external electrical excitation. It is distinct within the field of electrophysiology in that is primarily concerned with ascertaining the electrical properties of biological tissues (whether living, dying, or dead) by inducing a current, measuring the response, and inferring the underlying physiological mechanisms. The response typically measured is the difference in electrical potential, voltage, between two points. Applying the complex version of Ohm’s law, the indirect measure of impedance can be found (Z = V/I). Because biological systems are bound by their chemistry and chemistry is indivisibly related to electrical properties, by properly measuring electrical characteristics such as impedance we may come to understand living systems in a unique and useful way.

More broadly, when studying impedance we are interested in measuring the total opposition an object (whether it be a circuit, a tissue, etc.) has against an alternating current (AC) at a particular frequency.[2] The vector representation of impedance, **Z**, is composed of a real part – resistance, *R* – and an imaginary part – reactance, *X* – on the complex vector plane (see Figure 1 – 1). In addition to being expressed in rectangular coordinates (*R* + *jX*) it may also be shown in the polar form consisting of magnitude and phase angle, |**Z**| ∠ *θ*. The mathematical relationship between these quantities is

Admittance, **Y**, is the reciprocal of impedance and thus characterizes how easily current can flow through an object. Just as with impedance it is composed of both a real and imaginary part, referred to as conductance, *G*, and susceptance, *B*, respectively. In the same vein it is also possible to express admittance in rectangular (*G* + *jB*) and polar coordinates (|**Y**| ∠ *θ*). An analogous mathematical relationship to that of impedance exists between the components of admittance and thus no further derivation is here needed. The units of impedance are ohms (Ω) and those of admittance are siemens (S). As the two quantities are merely reciprocals of one another, they are really describing the same phenomenon – the ability of current to flow through something – in two different ways. The concept of ‘immittance’ was coined by H.W. Bode[1] and used to generalize the separate functions developed for impedance and admittance as a way of mitigating the problem of switching from one frame of reference to the next.[3] Given the need for practical relationships when observing this phenomenon (provoked especially by the need to use units for instruments during measurements), it has not enjoyed popular usage.

Though this study of passive properties of tissue is usually described in terms of impedance (as a result of the voltage being dependent on the current injected into the sample), the term ‘bioimpedance’ itself is a bit of a misnomer in that both the impedance and admittance of tissues can be and frequently are measured, explored, and investigated within the field. A broader and perhaps more correct term might be ‘bioimmittance’ – as previously mentioned, ‘immittance’ is the generic term used to refer to the fundamental phenomenon that both impedance and admittance describe.[4] Given the entrenched character of the term and the experiments described within the body of this thesis, however, ‘bioimpedance’ will be used generally throughout this discussion.[2]

Measuring biological impedance alone does little good. Bioimpedance is an indirect measure of meaningful physiological happenings. Take for instance the well-known account of Galler’s experiment measuring the resistance across the body of living and dead frogs. [5], [6] Though the deterioration of tissues at death is potentially a biologically significant quality it is not necessarily measureable, while the decrease in impedance across living and dead specimens can be measured readily, it is not necessarily relevant. Describing the passive electrical properties of biological materials – bioimpedance – therefore acts as a means by which to measure physiological events for which there is no known direct mechanism of transduction. [7], [8] Through context, theory, and modeling, however, certain realms of biology and electrical engineering can be synthesized. With this synthesis, engineers and scientists have found much to discover and draw upon. Whitman was right to sing the body electric.[9]

From the analysis of single cells [10] and cultures [11] to tissues [12] and organs [13] to whole bodies [14] and across populations [15], bioelectrical impedance has proven its value in quantifying physiology time and again. Generally the application of bioimpedance is seen in three contexts: *diagnostic* (plethysmography [16], cardiography, [17], [18], electrocardiography [19], electrogastrography [20], electromyography [21], [22], brain imaging [23], [24], body composition analysis [25-27], bodily fluid analysis for both composition [26] and flow [28], skin hydration [29], breathing pattern measurement [30], diagnosing metabolic syndromes [31]); *therapeutic* (electroshock and electrotheraphy [19], electrosurgery [32], [33], radiofrequency thermo-ablation [34]); and *for research* (needle guidance [35], bacterial growth measurement [36], DNA hybridization monitoring [37], cell locomotion tracking [38], hemodynamic analysis [39], real-time detection of neuronal activity [40]). This is in no way suggests that this is the full extent of bioimpedance as there are many other fields which utilize impedance measurements immensely (including meat quality [41] and food processing [42]) and others that use it uniquely (fingerprint detection [43]), rather only to hone our attention in on diagnostic and therapeutic applications for this research. The study of bioimpedance, as I have hopefully shown in above, is simply too broad to grapple with all at once.

However, it is necessary that we are knowledgeable of the field before we begin our discussion. It is, after all, one thing to say that the research associated with biological impedance is capable of telling us all these great things, and quite another thing entirely to show *how* these things can come about. Without being sufficiently versed in the language, thoughts, and history of the field, little can be understood going forward. We must first know where we are to know where we are going. Let us first begin with how we will speak on this matter, before covering the historical and scientific background.

## Terminology

Because bioimpedance as a field has had to draw on the concepts of many seemingly disparate disciplines, a certain inconsistency in terminology has developed over time. Combined with the accumulation of jargon of only historical relevance, these factors often unintentionally obfuscate discussions on the subject. In fact, this discussion began with the claim that ‘bioimpedance’ itself is an historically encumbered as a term and showed how it has snuck its way into being used far outside of its literal bounds. All of this is meant to suggest that as with any field of study there exists a lot of idiosyncratic tools, techniques, and modes of thinking that are used either out of tradition within the field or through deference to influential researchers. For instance, the most recognizable phrase in the field, *Cole-Cole*, which was originally used to described the complex permittivity plot (ε′ vs. ε″) [44], is often misappropriated to refer to any complex immittance plot with a depressed semi-circular arc. It has even been used to refer any complex impedance plot[3] regardless of the nature of its locus.[45] To avoid confusion, general terms will be sought out and used, even when these terms are not necessarily the first ones that researchers in the field might think to use. Following the theme this thesis in general, this was done to lay the groundwork of an ultimately ‘more correct’ presentation of the material.

## Historical Derivation

If a line is to be drawn at where bioimpedance as a field began, it might rightfully be placed at the feet of Emil du Bois-Reymond in the mid-nineteenth century. In addition to his many other discoveries (including the action potential of nerves which has earned him the reputation as the ‘father of experimental electrophysiology’), du Bois-Reymond was the first to show that as the frequency of a current passed through a specimen of animal tissue increased, the tissue’s ‘resistance’ decreased.[45] Though the concept of complex impedance was unavailable to him at the time, du Bois-Reymond discovered the principle notion of bioimpedance, namely that changes in the frequency of the current passed through a specimen result in measureable changes, providing information about the tissue under investigation.

For nearly fifty years, du Bois-Reymond’s work had no expansion, until in 1907 Lapicque proposed a three-component circuit, consisting of a parallel resistor and capacitor pair in series with an additional resistor.[46] This basic circuit (Figure 1 – 2a), made popular in the literature by Lapicque (and later by Philippson in 1921[47], [48]) lies at the root of much of modern bioimpedance measurements. Being of such importance, it will be useful here to demonstrate the basic rudimentary principles involved in impedance loci. And as it is one of the simplest circuits available for impedance studies, the derivation is relatively easy and straightforward. Our primary focus here will be to show how the characteristic semicircular impedance locus arises on the resistance-reactance plane. While this circuit has somewhat fallen into disuse for biological tissues (as early as 1925, in fact [49]), it nevertheless serves as an appropriate stepping off point for both its historical significance and simple mathematical expression.

It will serve the reader to recall the behavior of three simple components – the resistor, the capacitor, and the inductor – whose impedance can be represented by

respectively, where *R* is the resistor’s resistance value (units of ohms), *j* is the imaginary unit, is the angular frequency, *C* is the capacitor’s capacitance value (units of farads), and *L* is the inductor’s inductance value (units of henrys). These impedance terms follow the same rules of series and parallel electric component combinations that are likely familiar to the reader and will not be further expounded upon here.[4] The presence of the ‘imaginary’ operator in the capacitor and the inductor terms causes both to be reactive in nature, while the resistor, whose behavior is entirely ‘real’, acts in an entirely resistive manner.

Returning to the circuit under analysis, we have three components: a resistance, *R*_{1}, in series with parallel pair of resistive and capacitive elements, *G* and *C*, respectively. Considering the frequency-dependent nature of the capacitive component of the circuit, it will serve us to consider two limiting cases as we begin.[6] At very low frequencies, the reactance of *C* becomes very large and hence the overall impedance becomes

while at very high frequencies with the reactance of *C* becoming negligible, the impedance becomes

These two limiting cases are important for physical and physiological interpretations. Assuming that the tissue under investigation can be thought of as a volume containing cells within an interstitial fluid (Figure 1 – 3), the current has two routes it can take.[50] At low frequencies, when the impedance due to the capacitance of the cell membrane is high, the current passes around the cells. At high frequencies, however, the membrane capacitance lowers the impedance and allows the current to pass through the cells. Thus *R*_{0} represents the impedance of the extracellular solution and *R*_{∞} represents the impedance of both the extracellular solution and the internal resistance of the cell.[5]

**Figure 1-3.** The possible paths of current through a volume of cells within an extracellular fluid. When injected at (a) low frequency, the current tends to works its way around cells and through the extracellular fluid because the cell membrane’s capacitive element dissuades its path. At (b) high enough frequencies the current is able to penetrate through the cell membrane with little effort.

Focusing again on the circuit itself, it can be seen without much effort that the impedance is of the form

To find the resistance and reactance components (again, the real and imaginary components of impedance, respectively) we multiply the numerator and denominator of the right most term of (1.8) by the complex conjugate of the denominator.

The impedance equation of (1.10) can be further simplified by introducing a characteristic time constant, τ, as

and substituting our limiting cases, (1.6) and (1.7) to arrive at

We can then break out resistance, *R*, and reactance, *X*,

As can be seen from the above equations, the resistance and reactance terms are functions of frequency, *ω*, causing ** Z** to continually change along the curve in the

*R-X*plane as the frequency changes from 0 to ∞. The curve that impedance traces on this plane is known as an impedance locus and its shape can be obtained analytically by first recollecting from equation (1.1) that

If we then consider *R* and *X* to be functions of the parameter , we can solve (1.14) in terms of in two ways to get

which can be substituted into the numerator and denominator of (1.14), respectively, to yield

Squaring both sides of (1.17) and rearranging leads to the expression

which to some individuals readily springs to mind a circle. For those of us who wish to see it more explicitly, a completion of the quadratic expression of *R* in (1.18) is required. A moment’s inspection[6] reveals that this can be accomplished by adding (*R*_{0}+*R*_{∞})^{2}/4 to both sides of the equation. We then arrive at the final analytical expression for the impedance locus:

Equation (1.19) represents a circle with a radius (*R*_{0}–*R*_{∞})/2 centered on the R axis at (*R*_{0}+*R*_{∞})/2. Since the right side cannot change its sign in physical and biological systems, the impedance locus is actually only a semicircle in the +*R*,–*X* region of the resistance-reactance plane.

Similar analyses to that shown above can be performed to understand the relationships between resistance and reactance, angular frequency and resistance, and angular resistance and reactance. A graphical representation of these behaviors can be seen in Figure 1 – 4. The simple *Mathematica* script written to produce these figures is presented in Appendix A, along with further exploration of the system as the parameters of the resistors and capacitor are varied.

In the early 1910s, independently of Lapicque, Rudolf Höber conducted a series of experiments measuring the impedance of red blood cells, culminating in his postulation on the existence of cell membranes to explain the high resistance seen in intact cells at lower frequencies.[45], [51] Philippson expanded on this work in 1921, finding the same frequency-dependent nature of impedance across various tissues.[47] As previously mentioned, he used the equivalent circuit put forth by Lapicque (as seen in Figure 1 – 2) to interpret his results, arguing that *R*_{1} represented the resistance of the cell interior and *G* and *C* were the resistance and capacitance of the membrane, respectively. [45], [47] He thus ignored the possibility of current flowing around and between cells. Upon calculating the value of the cell membrane capacitance at each frequency he found that it decreased with increasing frequency, *f*, via the following the relationship:

In equation (1.20), *C*_{m} is the membrane capacitance, *C*_{1} is the value of *C*_{m} at *f* = 1, and *m* is a constant. Incorporating this capacitive relationship – which he referred to as a ‘polarization capacitance’ and is similar to that for metal-electrolyte interactions [47],[8], [51] – he arrived at the following expression for polarization impedance

where *B* = 1/2π*C*_{1} and *α* = 1 – *m*. Philippson claimed that this polarization impedance experienced the same phase shift as a purely capacitive element (as evidenced by the presence of *j*). Unfortunately, Philippson appears to have only measured |*Z*| values and was not able to realize his mistake: for linear systems, the Kramers-Kronig relationship requires that *j* and *ω* (where *ω* = 2π*f*) be raised to the same power. [52], [53] This mistaken assumption and correspondingly incorrect formulation would later be corrected through better measuring techniques and modeling.

Fricke and Morse in 1925 proposed a slightly more intuitive equivalent circuit on their studies of blood.[54] Instead of a resistive element in series with a parallel combination of a resistor and a capacitor (which is the simplest circuit relevant to impedance measurement [2]), they fit their data to an equivalent circuit made of a resistor in parallel with a capacitor and resistor in series (see Figure 1 – 5a). The difference is subtle, but important. In this case, the branch composed of the lone resistor, *R*_{1}, was thought to represent the resistance of the suspension medium, while the branch consisting of the resistor, *R*_{i}, and capacitor, *C*_{m}, was thought to describe the resistive properties of the suspending cells and the capacitance of the membranes, respectively (Figure 1 – 5b). By utilizing this convention they inferred two of the most important aspects of bioimpedance and outlined what was to become a standard approach across bioimpedance. The first observation is that at low frequencies, the membranes have high reactance, causing current to flow through the extracellular fluid (the branch containing *R*_{1}) whereas at higher frequencies when the reactance negligible, the current flows through both the surrounding fluid and the cells’ interiors (both branches). The second and most significant portion for our analysis here is that for suspensions two branches of a parallel circuit could be employed to describe the suspending fluid and the suspended element[7] regardless of how complicated they become.[55]

**Figure 1-5.** (a) Fricke and Morse’s equivalent circuit for (b) a suspension of cells. The parallel branch on the left is meant to represent the extracellular component made of an electrically resistant fluid, *R _{1}*. On the right is represented the cellular contribution of the intracellular resistance,

*R*, and the cellular membrane,

_{i}*C*.

_{m}Applying the same analysis as we did for the first circuit we derived (Figure 1 – 2a), it follows that

Unlike the first case, impedance is not uniquely defined by a single time constant. This makes it analysis somewhat difficult. Fortunately, however, if we consider the admittance, this situation can be resolved. This is somewhat hastened by considering the left most branch of the parallel circuit as a conductive element (*G* = 1/*R*_{1}). Working with this definition the admittance becomes

which is uniquely defined by

to become

It is left as an exercise for the reader to show how this behavior will produce a similar semicircular arc in the conductance-susceptance plane.

Fricke and Morse found their model to accurately fit the data obtained on suspensions of red blood cells, but unsatisfactory in representing the more complex behavior of other tissues. It would be Cole (in 1928) who synthesized the electrical model (and thus the equivalent circuit) of Fricke and Morse with the known analogous behavior observed by Philippson, proposing that the membrane impedance itself varies with frequency.[56] His first spark of insight was to assume that cellular membranes were not merely capacitive elements but had both resistance, *R*_{m}, and reactance, *X*_{m}, terms represented by

such that

where A is a constant. Though I have used them regularly throughout this paper already, Cole’s original theoretical analysis paper would be one of the first times that anyone thought to plot the electrical properties of biological tissues on a complex impedance plane.[8] By doing so, Cole was able to see that this slight modification in his formulation predicted that the semicircular locus of the total impedance would be ‘depressed’ in the resistance-reactance plot (see Figure 1 – 6). The high-frequency intercept on the real axis would then be given by

In this theoretical paper he examined the two types of three-component equivalent circuits we have discussed thus far, though his configuration consisted of a capacitor and both an ideal, lumped resistor and a frequency dependent resistor). Cole alternated between considering his model purely descriptive versus interpreting it as Philippson had done with direct correlations between components and tissues. With the release of his experimental paper testing the impedance of a suspension of sea urchin eggs (specifically those of the genus *Arbacia*) later that year, he empirically validated his results.[57]

In 1929, Debye published a now classic book, *Polar Molecules*, in which he derived a model of complex permittivity based on the assumption that molecules could be regarded as spheres in a continuous medium. His theory on the dielectric behavior of dilute suspensions made of dipoles culminates in a complex dielectric constant:

Where *ϵ ^{*}* is the complex permittivity,

*ϵ*

_{0}and

*ϵ*

_{∞}are the dielectric constants at zero and infinite frequencies respectively, and

*τ*

_{0}is a characteristic time constant generally referred to as the relaxation time. Though Debye’s work was not largely concerned with biological materials as it was generally unknown at the time that many organic molecules are polar [51], it has since become somewhat of a staple of researchers working in electrical impedance spectroscopy. An equivalent circuit is suggested in this modeling and is comprised of a resistor,

*τ*

_{0}/(

*ϵ*

_{0}–

*ϵ*

_{∞}), in series with a capacitor, (

*ϵ*

_{0}–

*ϵ*

_{∞}), both in parallel with a high-frequency capacitor

*ϵ*

_{∞}where each of these representations should be normalized to the permittivity of free space.

Over a decade later Cole and Cole would synthesize and slightly modify this model and plot the complex permittivity (the real Cole-Cole plot as pointed out in the *Terminology *section) to produce the characteristic depressed loci semi-circle curve with which the reader is no doubt by now thoroughly familiar. The researcher began by first noting that several experimental results justify an altered equation the purely dissipative resistor of Debye’s model with a constant-phase angle impedance, *Z _{CPA}*, to arrive at

where

with

From here, it is a simple matter of moving from the complex permittivity plot to the complex impedance plot to arrive at

Which has long been considered the gold standard of bioimpedance measurements as it accounts for most of the facets we have introduced here, including an equivalent circuit comprised of parallel circuit whose branches represent the components of the sample under examination, the incorporation of polarizing mechanisms from both the measurement system and the sample, a range of possible relaxation times, and the characteristic depressed loci, semi-circle curve seen in the resistance-reactance plane.

[1] Bode also briefly considered the clunkier chimeric ‘adpedance’ as a possible term.

[2] ‘Biological impedance’, ‘tissue impedance’, and ‘electrochemical impedance’ will all be used to varying degrees, with deference to the terminological baggage of each, to relieve the text’s tedium. For further clarification of the reasoning behind the use and disuse of terms used throughout this thesis, please consult the *Terminology* portion of the Introduction.

[3] To make matters worse, complex impedance plots have also been referred to as ‘Argand diagrams’ – a plot of any complex number on the real and imaginary axis – and ‘Nyquist plots’ – complex plane plots usually reserved for transfer function response.

[4] Any introductory electrical engineering textbook could be used to help readers unsure of these formulations.

[5] The astute reader will no doubt notice that with this principle in mind, the circuit we are working on becomes untenable in terms of physiological relevance. The situation will be rectified shortly.

[6] And by consulting the excellent derivation by Schanne and Ruiz P.-Ceretti.

[7] Readers are advised to retain this fact as it will become relevant when modeling tofu.

[8] The names of these plots have a contentious history. Often these plots have been referred to as ‘Cole-plots’, though many argue that this unnecessary. Please consult the *Terminology* section for why I choose to avoid ‘proper name’-specific terms throughout the course of this thesis.

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