Electrical Impedance

The flow of direct current through an electrical circuit is impeded only by the resistance presented by the circuit. In systems utilizing alternating current, which is a sinusoidal signal, the opposition to a current is the impedance, which takes into account both the magnitude of the opposition to the applied current and the phase shifts between the current and voltage caused by the components of the circuit [74, 75] . Electrical impedance (Z), measured in Ohms (Ω), is the total opposition to current flow in an alternating circuit and is defined in terms of three individual components: resistance (R), inductance (L) and capacitance (C). The angular frequency of the current is represented by ω, and j is the square root of (-1) [22, 74, 76]. Impedance may be expressed as a complex number (Equation 1.1).

Z = R + j(ωL − 1/ωC)

Equation 1.1: Equation for the determination of electrical impedance (Z). R – resistance, j – square root of -1, ω – angular frequency, L – inductance and C – capacitance.

Resistance is a property that opposes current flow and conductance is the inverse of resistance. Capacitance is the opposition of a change in voltage or electrical potential across an object and acts to store energy. A capacitor consists of two conductors; each oppositely charged and separated by a dielectric material. Permittivity is a property of the dielectric material and reflects the ability of charges in the material to move in response to an electric field [77]. Capacitance is a function of the permittivity and the physical geometry of the object. The capacitance formula for a two-plate capacitor is:

C = εA/d

Equation 1.2: Formula for the calculation of value of a two plate capacitor. C – capacitance, ε – permittivity, A – area of each plate, d – distance between plates. Reference [77].

The modulus of the impedance, |Z|, is the ratio of the magnitudes of the voltage and current (V/I). For the purpose of this thesis only the modulus of the impedance will be investigated as the study is only interested in

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assessing the difference in the resistive element of impedance to provide a relative measurement capable of differentiating between biological tissue types.

The measurement of electrical impedance has been investigated for breast analysis since the early 20th century when Fricke and Morse first described altered capacitance values of breast cancer tissue compared to normal breast tissue in 1926 [78]. In the last two decades the utility of impedance in discrimination between biological samples has been extensively investigated. The term bio-impedance is used to describe the response of a living organism to an externally applied electric current, and is a measure of the opposition to the flow of that electric current through tissues [31, 74]. Each cell type in the human body is composed of different chemical and physical elements which result in the production of distinctive impedance signals generated by each variant [79, 80]. This is also true of diseased, cancerous tissues which have been shown to have a lower impedance value when compared to healthy cells of the same type [15, 18, 79]. Most benign lesions have been shown to exhibit impedance similar to healthy tissues. This finding could provide a potential method of differentiating between benign and cancerous lesions during screening of breast cancer patients [81-83].

The differential electrical properties associated with cancerous tissues result from increased cellular water and salt content, amount of extracellular fluid, altered membrane permeability, changed packing density and orientation of cells [84-87]. Tissues within the human body provide two forms of resistance to an applied electrical current. Capacitive R (reactance) results from the lipid bilayer cell membrane while resistive R is due to the extra- and intracellular fluid present in the tissue of interest [88]. A number of equivalent circuit models of biological tissues have been proposed to explain this behaviour. The most commonly used simple equivalent circuit arranges intracellular fluid R (Ri) and cell membrane capacitance (Cm) in series with extracellular fluid R

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Figure 1.5: A simple equivalent circuit model of biological tissues. Ri – intracellular resistance, Cm – cell membrane capacitance, Re – extracellular resistance. Figure is

Author’s own.

Another equivalent circuit model of cell and tissue behaviour can be described as an ([(CR)(CR)](CR)) circuit (Figure 1.6). In this system both the resistance and capacitance of the extracellular fluid, intracellular fluid and the cell membrane are taken into account [79].

Figure 1.6: An equivalent circuit model of biological cell behaviour. Ri – intracellular

resistance, Re – extracellular resistance, Rm – cell membrane resistance, Ci – intracellular

capacitance, Ce – extracellular capacitance, Cm – cell membrane capacitance. Figure is

Author’s own.

At zero (to low) frequency, the applied current does not penetrate the cell membrane, which acts as an insulator, and as a result the current is transferred exclusively across the extracellular fluid. At very high frequencies the cell membrane is charged as a near perfect capacitor which results in the dielectric properties of the tissue resulting from the combined impedance of the extra- and intracellular fluid [80, 88]. The electrical properties of tissues vary according to applied frequency as seen from α-, β- and γ-dispersion [84, 89, 90]. α- (10 Hertz [Hz] – 10 kHz) and β-dispersions (10 kHz – 10 MHz) are of relevance to medical applications as the changes between pathological and normal tissue typically occur in these ranges [84, 91]. The α-dispersions

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are associated with electrical double-layers and surface ionic conduction effects at the membrane boundaries [89]. The β-dispersion results from a combination of the capacitive shorting-out of membrane resistance and rotational relaxations of bio-macromolecules [89]. The γ-dispersion arises from the relaxation of bulk water in the tissue [89].

There are two branches of electrical impedance measurement which have evolved gradually in the last number of years: electrical impedance tomography (EIT) and invasive impedance detection.

1.10 Electrical Impedance Tomography