Obtaining electrical equivalent circuits of biological tissues using the current interruption method, circuit theory and fractional calculus

Abstract

We have adapted the current interruption method to obtain electrical equivalent circuits of rat tissues (liver viscera and triceps surae muscle). This paper provides a comprehensive and in-depth explanation of the procedure, and its suitability has been assessed. Experimental data are interpreted using circuit theory and fractional calculus. Histological analysis was also carried out. The technique proposed has a clear physical meaning and we have tried to make this communication accessible to potential readers with multidisciplinary backgrounds. To the best of the authors' knowledge, there is no previous report of adapting the current interruption method to obtain electrical equivalent circuits of tissues using a four-electrode arrangement, circuit theory and fractional calculus.

---

1. Introduction

Impedance Spectroscopy (IS) is a well-established technique used in determining the physiological state of biological tissues from their electrical properties,^1,2 which involves the electrochemical processes for the intracellular-space/cell-membrane/extracellular-space system. In this introduction we review succinctly some basic concepts that will serve to expose the reader to the current interruption method for the electrical characterization of tissues.

Fig. 1(A) shows a schematic of single eukaryotic cell. Intracellular (pink color) and extracellular (blue color) spaces are electrolyte solutions containing primarily Na⁺, K⁺ and Cl⁻. Cell membrane (green color), about 7 nm thick, separates intra- and extracellular compartments regulating the ionic concentration differences (cellular homeostasis).²

Fig. 1 Cell, tissue and electrical equivalent circuit (EEC). (A) Single eukaryotic cell. (B) EEC of the cell shown in (A). (C) Biological tissue. (D) EEC of the tissue shown in (C). (E) EEC of (D) in the time domain and (F) frequency domain.

Let us consider an electric current flowing from left to right through the single cell of Fig. 1(A). In general, a portion of this current will cross the cell membrane and will enter the intracellular fluid. The remaining current will flow through extracellular fluid. Both current paths are implemented in the electrical equivalent circuit (EEC) of Fig. 1(B) using two parallel branches. The upper branch (capacitance C_M in series with a resistance R_I) models the current path through the cell membrane and the intracellular space. The lower branch (resistance R_E) considers the current path through the extracellular compartment. Note that cell membrane acts as an electrical capacitor by involving a charge separation. Intra- and extracellular environments (electrolyte solutions) are represented by the intra- and extracellular resistances, R_I and R_E, respectively. EEC of Fig. 1(B) was initially proposed by Fricke and Morse.³

Fig. 1(C) shows a portion of tissue which involves a group of similar cells. Thus, the EEC of Fig. 1(D) comprises an array of interconnected EECs (each EEC corresponds to a single cell) which provides a modified Fricke and Morse EEC. The capacitor C_M is replaced by a fractional capacitor,⁴ or more generally known as constant phase element (CPE_M) where Q_M and α are the CPE_M parameters (see below).^2,5 CPE_M takes into account the space distribution of the electrical tissue properties. While the EEC of Fig. 1(B) exhibits one single time-constant, the EEC shown in Fig. 1(D) involves a distribution of time constants.

Fig. 1(E) redraws the EEC of Fig. 1(D). i(t) is the current supplied to the circuit (excitation). Currents and voltages have been labeled in each element. i_M(t) and i_E(t) are the currents through the CPE_M and R_E, respectively. v_M(t), v_I(t) and v_E(t) are the voltages across the CPE_M, R_I, and R_E, respectively. Eqn (1) expresses i_M(t) as a function of v_M(t),

where Q_M has the units of (farads) × (seconds)^α−1 with v(t) in volts, i_M(t) in amperes and t in seconds. α is a dimensionless quantity 0 < α < 1. d^α/dt^α is the fractional-order derivative of order α. If we let α = 1, eqn (1) involves a ordinary derivative and CPE_M is an ideal capacitor (Q_M = C_M).⁴

It is important to point out that the CPE_M and the resistors R_I and R_E are LTI (linear time-invariant) circuit elements. Indeed, the parameters Q_M, α, R_I, and R_E do not vary with time (time-invariant property) and do not depend on voltages across and/or currents through the corresponding circuit element under consideration (linear property). It follows that the EEC of Fig. 1(E) is a LTI circuit.⁶

Fig. 1(F) shows the EEC of Fig. 1(E) transformed into the frequency-domain which is usually used in IS. Thus, i(t), i_M(t), i_E(t), v_M(t), v_I(t), and v_E(t) are replaced by their corresponding phasors Ī, Ī_M, Ī_E, V̄_M, V̄_I, and V̄_E, respectively. The impedances of the circuit elements, Z_CPEM(jω), R_I, and R_E, are also indicated.

Specifically, the impedance of the CPE_M (ratio V̄_M/Ī_M) is written as⁵

where ω is the angular frequency (ω = 2πf where f is frequency) and j is the imaginary unit. From eqn (1) and (2), we observe how the α fractional-order time derivative of a sinusoidal waveform is equivalent to multiplying the corresponding phasor by (jω)^α.

The total impedance between the endpoints of the EEC of Fig. 1(F) is found be

In eqn (3), we find that the coefficient of (jω) – physical frequencies – is the distributed time-constant (τ) of the EEC of Fig. 1(E) when a current is supplied to it; that is,

Observe that if α = 1 (CPE_M is an ideal capacitor), eqn (3) yields the impedance of the EEC of Fig. 1(B) and its time constant is written as (R_I + R_E)C_M where (R_I + R_E) is the resistance seen by C_M (a current source is replaced by an open circuit). The inverse of the time constant provides the characteristic frequency ω_o (known as frequency pole in first-order (α = 1) LTI circuits)⁷

It is worth remembering that impedance Z(jω) is a complex number whose real and imaginary parts are the resistance R(ω) and the reactance X(ω), respectively. That is, Z(jω) = R(ω) + jX(ω). Note that at sufficiently low and high frequencies, eqn (3) yields approximately R_E and R_IR_E/(R_I + R_E), respectively. The same conclusion can be stated from the EEC of Fig. 1(F). Suppose that a sinusoidal current i(t) is connected to the EEC of Fig. 1(E) and a sinusoidal steady-state is reached (refer to Fig. 1(F)). At sufficiently low frequencies, the impedance magnitude of the upper branch |Z_CPEM(jω) + R_I|, will become large enough to cause |Z_CPEM(jω) + R_I| ≫ R_E, so almost all the current i(t) will flow through the extracellular space (resistor R_E), resulting in i_E(t) ∼ i(t). At sufficiently high frequencies, R_I ≫ |Z_CPEM(jω)|, the current will flow through both intra- and extracellular media according to the current-divider rule (circuit consisting of R_I and R_E).

Interestingly, the expression in eqn (3) is of the same form as that proposed by Cole⁸

where the high- and low-frequency limit resistances are R_∞ = R_IR_E/(R_I + R_E) and R₀ = R_E, respectively. The four parameters R_∞, R₀, τ (or ω_o) and α are known as Cole parameters.

Of specific interest in IS is the Nyquist plot (−X(ω) vs. R(ω) in the complex plane). Nyquist plot of eqn (3) sketches a depressed semicircle (the center of its corresponding circle lies below the real axis).^1,2 This depressed semicircle intersects the real axis at two points: R_IR_E/(R_I + R_E) and R_E at the frequencies ω → ∞ and ω = 0, respectively (see above).

It is easy and intuitive to see that changes in a biological tissue will be reflected in the corresponding EEC. For instance, consider an ischemic tissue. Cell swelling causes a reduction of the extracellular volume which yields an increase of the extracellular resistance R_E in the EEC of Fig. 1(D).⁹ Next, to determine the parameters of the EEC shown in Fig. 1(D), electrical measurements are taken on the tissue under study.

One simple way of obtaining the electrical properties of biological tissues is using the four-electrode or Kelvin method,^1,2 as shown in Fig. 2(A). Four electrodes are placed into the tissue in a straight line. An electric current i(t) is injected from electrode 1 (EL1) and collected at electrode (EL4). It determines a potential distribution in the tissue, resulting in a voltage v(t) between the two inner electrodes (EL2 and EL3). This setup (voltage-measuring electrodes placed in line and between the current-injecting electrodes) may usually provide a good signal to noise ratio. Fig. 2(B) shows the EEC of the four-electrode method. EEC_ELi and EEC_ELi/T are the electrical equivalent circuits of the electrode i and (electrode i)/(surrounding tissue) interface, repectively. EEC_Tij is the equivalent circuit of the tissue portion carrying the current i(t) and placed between the two equipotential surfaces (perpendicular to current flow lines) which contain the electrodes i and j. Note that in circuit theory, we find voltages and currents which can also be measured using voltmeters and ammeters. Nevertheless, field theory should be regarded if we want to analyze the potential or current distributions in tissue or in the electrode/tissue interface.¹⁰

Fig. 2 Four-electrode method. (A) Four electrodes are placed into the tissue. (B) EEC of the four-electrode arrangement shown in (A). (C) EEC of the tissue portion which is measured using the four-electrode method.

Simple EECs of ELi and ELi/T interface comprise a resistor and a capacitor (or CPE) in parallel with a resistor, respectively. Electrodes may also involve more complex processes such as anomalous diffusion.¹¹ Typical EEC_Tij is shown in Fig. 1(D). Note that in sinusoidal steady-state analysis, EEC_ELi, EEC_ELi/T, and EEC_Tij are replaced by their corresponding impedances Z_ELi(jω), Z_ELi/T(jω), and Z_Tij(jω), respectively.

Since an ideal voltage-measuring device draws no current (open circuit), it follows from Fig. 2(B) that the measured voltage v(t) between terminal 2 and terminal 3 is equal as that of EEC_T23. This makes the four-electrode method suitable for characterizing the average electrical properties of the tissue portion which carries the current i(t) and contributes to the voltage v(t) between terminals 2 and 3. Fig. 2(C) shows the EEC_T23 of the tissue portion which is measured. Note that the measured voltage v(t) is indeed the voltage v_E(t) across the extracellular space (resistance R_E). We shall henceforth use v_E(t) and v(t) interchangeably. Ideally, v(t) does not take into account the effects of electrode polarization (voltages across EEC_ELi/T). Errors in four-electrode configuration have been analyzed elsewhere.¹² Note that Fig. 1(E) is repeated here in Fig. 2(C) for the reader's convenience.

As the integrity of the cell membrane is a critical point to maintain the cell's life, the question arises as to what frequency range should be used to characterize its electrical properties. Cellular structure of tissues exhibits a characteristic frequency, ω_o/(2π), ranging from several kHz up to MHz. That range is known as β dispersion.^13,14 Thus, if a specific tissue has a characteristic frequency ω_o/(2π), an ideal situation is to cover an useful frequency range on both sides of ω_o to obtain accurate values for the parameters of the EEC. It should be pointed out that the tissues also exhibit other characteristic frequencies (α or γ dispersions) corresponding to different relaxation processes which may be evidenced in an impedance spectrum.^13,14

We use entirely the LTI circuit theory.⁶ Hence, the i(t)–v(t) relationship for the tissue (see Fig. 2(A)) should be described by a linear differential equation with constant coefficients. Although values of current, electric field or voltage have been proposed to ensure linear condition, they result in a large variability depending on the type of tissue and its state.^13–15 Additionally, the properties of a biological tissue can vary with time (for instance, during an ischemic event) and, thus, the measurement acquisition time should be sufficiently short to assure that the tissue is “frozen” in time (the electrical properties of the tissue do not change during the measurement acquisition time). This requirement is very critical for some processes such as electroporation.¹⁵

An important consideration is the current waveform i(t) to be injected. Characteristic current values should be sufficiently small to ensure the linearity (see above) but sufficiently large to allow an acceptable signal to noise relation without inflicting damage to the tissue (heating, pH changes or accumulation of toxic electrochemical products). The current density (the current per unit area) distribution in the tissue should be adequate to not inflict damage.¹⁰

Typically, experimental impedance measurements are obtained in a sequential way: impedance at each frequency is measured using a single sinusoidal-signal which may be time-consuming. The use of excitation signals different from that sinusoidal-signal for analyzing electrical properties of tissue has been barely reported in the scientific literature.^15–20 Some advantages such as a drastic reduction of the measurement acquisition time and/or an optimum frequency spectrum which contains sinusoidal components of interest were emphasized.¹⁷

EECs of biological tissues have been obtained from the experimental transient-response to a step function.^19,20 Basically, a step excitation corresponds to a switch which closes at a specific instant and connects a dc battery to a given circuit.⁶ For instance, the body composition was obtained from a step-voltage response using the two-electrode method.¹⁹ Cole parameters were determined in apple samples using a step-current response, fractional calculus, and also the two-electrode arrangement.²⁰

In this paper, we have obtained EECs of rat tissues using the current interruption method (see below) which has a clear physical meaning. Cell membranes are constant current charged. When the current is cut-off, cell membranes begin to discharge through the surrounding intra/extracellular spaces. The analysis of the discharge voltage curve involves circuit theory and fractional calculus. Importantly, this discharge voltage curve depends only on the tissue's electrical properties.

The current interruption method has been used in electrochemistry to estimate ohmic losses.²¹ We have adapted this method for obtaining the parameter values of the EEC shown in Fig. 1(D) using a four-electrode arrangement.

2. Current interruption method using a four-electrode arrangement

At this point, we explain the current interruption method and how the values for R_E, R_I, α, and Q_M are found. Fig. 3(A) and (B) show the waveforms of the current i(t) injected into the tissue and the resulting extracellular voltage v(t), respectively (refer to Fig. 2(A)).

Fig. 3 Current interruption method. (A) Waveform of the current injected into the tissue. (B) Waveform of the resulting extracellular voltage. (C) EEC of Fig. 1(E) at t = 0⁻. (D) EEC of Fig. 1(E) at t = 0⁺. (E) EEC of Fig. 1(E) for t > 0.

Firstly, a dc current I_DC (excitation) is injected into the tissue between the two outer electrodes for a sufficiently long time T (T ≫ τ, see below), so that the voltage (response) between the two inner electrodes reaches a steady state (constant value). At time t = 0, the current is abruptly cut off. Fig. 3(C) shows the EEC of Fig. 1(E) at t = 0⁻ (just before the current is interrupted). CPE_M is considered to be fully charged (dc steady-state) and acts as an open circuit. Thus all the current I_DC flows through the extracellular space (resistor R_E), yielding a voltage drop of R_EI_DC. Since no current flows through the cell membrane (upper branch), no voltage appears across the intracellular space (resistor R_I). It follows that the voltages across CPE_M and R_E are equal. Voltages and currents at t = 0⁻ are indicated on the circuit diagram in Fig. 3(C).

Just after interrupting the current (that is, at t = 0⁺), the energy previously stored in the cell membranes (CPE_M) begins to be released and dissipated in the surrounding intra/extracellular spaces. Fig. 3(D) shows the EEC of Fig. 1(E) at t = 0⁺. It is assumed that the voltage across the CPE_M cannot change instantaneously, so that the voltage across it just before (t = 0⁻) and just after (t = 0⁺) interrupting the current are equal. At t = 0⁺, the CPE_M voltage drives a current R_EI_DC/(R_I + R_E) through R_E and R_I resistors, giving voltages of R_E²I_DC/(R_I + R_E) and R_IR_EI_DC/(R_I + R_E), respectively (refer to Fig. 3(D)).

Next, the voltage relaxation (free or natural response) is measured for t > 0. Importantly, the natural response obtained for t > 0 depends only on the tissue's electrical properties. Fig. 3(E) shows the EEC of Fig. 1(E) for t > 0. Let us consider that CPE_M is an ideal capacitor (α = 1 and Q_M = C_M). From circuit theory, it can be shown that the expression for v(t) is written as v(t) = [R_E²I_DC/(R_I + R_E)]e^−t/τ, where the time constant τ is (R_I + R_E)C_M which is as that of eqn (4) with α = 1 and Q_M = C_M. Nevertheless, CPE_M involves a fractional derivative – see eqn (1) – and the Mittag–Leffler function is used to include a fractional relaxation.

where E_α(z) is the one-parameter Mittag–Leffler function,²² defined as

Note that if α = 1, eqn (8) is the Maclaurin series for the exponential function, resulting in E₁(−t/τ) = e^−t/τ.

The theoretically predicted waveform of v(t) has been sketched and clearly labeled in Fig. 3(B). Note that R_IR_EI_DC/(R_I + R_E) is the magnitude of the jump discontinuity at t = 0. Fig. 3(B) also shows that the fractional relaxation (eqn (7)) exhibits a faster (slower) decay than the exponential one, for short (large) times.

The values of the parameters of the EEC shown in Fig. 1(D) are determined by performing the following procedure:

(i) R_E is found from the previous steady-state (value v(0⁻)) as

(ii) R_I is obtained from the value of v(0⁺) as

(iii) The transient-voltage response is fitted to eqn (7); α and τ are then determined. Finally, Q_M is calculated using eqn (4). The fitting procedure was performed using a routine in MATLAB.²³

We have assumed that the current is interrupted instantaneously (from t = 0⁻ to t = 0⁺). Nevertheless a finite (nonzero) fall time (t_f) exists in a real-life setup. The tissue will “feel” that the current is abruptly set to zero if t_f ≪ τ. It should also be noted that a transient recorder with a sufficiently sampling rate is needed in order to more accurately determine the critical points (jump discontinuity and the initial fast decay of the fractional relaxation) of the experimental waveform of v(t).

3. Experimental

Adult male Wistar rats were provided by a commercial supplier (Harlan Ibérica, Spain) and used at the age of ca. 25 weeks (n = 3, 351 ± 43 g). All experimental protocols adhered to the regulations of the European Commission (directives 2010/63/EU and 86/609/EEC) and the Spanish Government (RD53/2013) for the protection of animals used for scientific purposes. All the experiments were made at a room temperature.

Surgical procedures of organ and muscle extraction were performed under intraperitoneal (IP) analgesia with xylazine (10 mg kg⁻¹) and anesthesia with sodium pentobarbital (55 mg kg⁻¹) mixed with atropine (0.05 mg kg⁻¹). Eyes were covered with Lubrithal™ gel to prevent corneal abrasion and dehydration. In all cases, extreme care was taken to minimize muscle tissue damage. When deeply anesthetized, the skin and muscles over the breastbone were cut and the sternal manubrium was grasped to gain access to the thoracoabdominal union. The abdominal cavity was opened by cutting through the abdominal wall in the midline from the tip of the processus xiphoideus to the pecten ossis pubis. The liver was removed and imbibed in saline solution. After that, a right Achilles tenotomy was performed; the triceps surae was dissected, removed and immersed in saline solution just prior to carry out the electrical measurements. Rats were sacrificed immediately after the liver and triceps surae extraction with an IP lethal dose of sodium pentobarbital (110 mg kg⁻¹). Finally, organs were carefully examined looking for any findings of gross damage and properly prepared for subsequent electrical characterization.

We analyzed three samples of livers and other three of triceps surae muscles from three rats. Electrical measurements were taken on each sample using the current interruption method. Just after that, conventional bioimpedance measurements were obtained. This protocol is then repeated twice more in each one of the livers and muscles, to guarantee the reproducibility and repeatability of experimental data. We selected these two particular tissues because of their opposite characteristics related to cellular homogeneity and electrical excitability. So, we chose the liver as an example of high degree of cellular homogeneity but bad electrical excitability and the striated muscle (triceps surae), as an example of low degree of cellular homogeneity but good electrical excitability.

Electrical measurements were performed using four platinum needle electrodes. Each electrode (refer to Fig. 4(A)) was 10 mm long and 0.3 mm in diameter (Grass Technologies, subdermal needle electrode, model F-E2-48). The electrodes were placed into the tissue in a straight line and equally spaced (5 mm). Fig. 4(B) and (C) show the four-electrode arrangement for liver and triceps surae, respectively.

Fig. 4 Four-electrode setup. (A) Platinum needle electrode. (B) Four platinum needle electrodes are placed into the liver and (C) into the triceps surae. (D) Representative 10× optical microscopy images of the liver after hematoxylin-van Gieson staining, showing a normal tissue structure. Black scale bar = 100 μm. (E) Representative 40× transverse section of triceps surae muscle after ATPase staining, showing the normal aspect of the three types of muscle fibers: 1 (black ones); 2A (white ones); and 2B (gray ones). White scale bar = 100 μm.

An AutoLab PGSTAT302N Potentiostat/Galvanostat (EcoChemie), combined with the ADC10M module, was used to implement the current interruption method. The instrument was controlled by a computer and driven by NOVA software. A dc current of 100 μA was injected between the two outer electrodes for 1 ms and just after that, the current is abruptly switched off. The voltage between the two inner electrodes was recorded between 1 ms before and 100 or 50 μs after switching off the current. A sampling rate of 10 MHz was used.

A Solartron 1260 frequency response analyzer connected to a Solartron 1294 impedance interface system were used to obtain the impedance measurements. Both instruments were controlled by a computer and driven by SMaRT software. Sinusoidal signals of 100 μA amplitude were injected between the two outer electrodes. The frequency-values were logarithmically spaced (5 steps per decade) between 1 MHz and 1 kHz.

After making electrical measurements, all samples were placed in paraformaldehyde 4% at 4 °C for immersion-fixation and later, three days more in sucrose (30% in phosphate buffered saline, PBS) at 4 °C for cryo-protection. Tissue pieces were mounted on plastic containers, quick-frozen in optimal cutting temperature compound (Tissue Tek, Hatfield, PA) and cut in transverse sections of 10 μm by using a Microm HM550 cryostat. Livers and muscles were initially examined using hematoxylin-van Gieson (Fig. 4(D)) and ATPase (Fig. 4(E)) stainings, respectively. Digital images were collected with a fluorescence Olympus BX51 microscope with 2×, 10×, 20×, and 40× objectives.

4. Results and discussion

No significant acute damage was observed in the organs microscopically evaluated. Neither alteration of normal tissue structure, nor signs of inflammation or necrosis were found in the analyzed sections of the livers (Fig. 4(D)) and muscles (Fig. 4(E)). Thus, the electrical measurements corresponded to normal rat tissues and importantly, the technique presented has not acutely damaged them.

Fig. 5(A) and (B) show the resulting voltage for the liver and triceps surae, respectively. The shape of the waveforms is similar to that predicted theoretically (refer to Fig. 3(B)), showing: (i) the previous steady state; (ii) the jump discontinuity. Values of v(0⁻) and v(0⁺) have been indicated by arrowheads; and (iii) the voltage relaxation for t > 0. Simulated data are also shown in Fig. 5(A) and (B). There is an excellent agreement between the experimental and simulated data. Average errors of 0.46% and 0.62% were obtained for liver and triceps surae, respectively. The following function was used: , where v_exp(t_i) and v_sim(t_i) are the experimental and simulated voltages at time t_i, respectively, v^max_exp is the maximum value of v_exp(t_i), and N is the total number of points.

Fig. 5 Experimental and simulated results. Waveforms of the resulting extracellular voltage for (A) the liver and (B) the triceps surae, using the current interruption method. Nyquist plots for (C) the liver and (D) the triceps surae.

Table 1 specifies the parameter values of the EEC shown in Fig. 1(D), using the current interruption method. They have been obtained by applying the procedure outlined in Section 2. Additionally, the values of the time constants associated with each of the two tissues have also been listed in Table 1.

Table 1 Parameter values of the EEC shown in Fig. 1(D), obtained using the current interruption method and the conventional IS technique. Distributed time constants are also indicated

Biological tissue	Measurement method	RE (Ω)	RI (Ω)	QM (nF s^α−1)	α	τ (μs)
Liver viscera	Current interruption method	1160.6	106.6	361.1	0.74	30.7
	Impedance spectroscopy measurements	1146.1	107.8	412.6	0.73	31.5
Triceps surae muscle	Current interruption method	231.3	25.9	1432.3	0.73	19.8
	Impedance spectroscopy measurements	238.7	25.3	1260.3	0.72	14.8

---

Note that we should check whether end up with a consistent solution. The distributed time constants for the liver and triceps surae are about 30 μs and 20 μs, respectively. Thus we see that a previous steady state is reached because 1 ms ≫ 30 μs and 1 ms ≫ 20 μs. A sampling rate of 10 MHz allowed more accurate values for v(0⁻) and v(0⁺) and to measure the time taken for the experimental setup to change the current from the previous steady-state value to zero. This fall time (t_f) was of about 1 μs. Thus, the tissue “feels” that the current is abruptly switched off because 1 μs ≪ 30 μs and 1 μs ≪ 20 μs. For clarity's sake, only the samples taken every 1 μs are shown in Fig. 5(A) and (B). The previous steady state is shown during the 20 μs preceding the current interruption.

Typical Nyquist plots for the liver and triceps surae are shown in Fig. 5(C) and (D), respectively. Well defined depressed semicircles can be observed. Experimental impedance data were fitted to the EEC shown in Fig. 1(D) using ZView software. The parameter values obtained are listed in Table 1. Impedance plots also show simulated data which have been obtained using the EEC of Fig. 1(D) and the parameter values given in Table 1. As can be seen there is an excellent agreement between the experimental and simulated results. Average errors of 0.27% and 0.51% were obtained for liver and triceps surae, respectively. We have used the following expression: , where Z_exp(jω_i) and Z_sim(jω_i) are the experimental and simulated impedance data at the frequency ω_i, respectively, Z^max_exp is the maximum impedance magnitude of experimental data, and N is the total number of points. Note that the calculated average errors are normalized to the maximum value in the data set.²⁴

Table 1 shows that the parameter values obtained using the current interruption method are remarkably close to those found from IS measurements. We note minor discrepancies of less than 3% for R_E, R_I, and α parameters and of the order of 10% for Q_M parameter. Determination of Q_M is more prone to errors because it depends on R_E, R_I, α, and τ. In turn, α and τ are obtained from the fitting procedure (see Section 2). Even so, a reasonable estimate for Q_M could be obtained.

The measurement acquisition time (1.1 ms and 1.05 ms for the liver and triceps surae, respectively) using the current interruption method is much shorter than that of the conventional impedance spectroscopy technique (∼35 s). Thus, it may allow monitoring of fast changes in a biological tissue. Nevertheless, as we have seen earlier, specific requirements on current interruption time and sampling rate should be fulfilled.

5. Conclusions

We have obtained EECs of rat tissues using the current interruption method and also impedance spectroscopy measurements (for comparative purposes). Accurate estimates of EEC parameters have been obtained using circuit theory and fractional calculus. Extra/intracellular resistances are obtained at time instants just before and just after interrupting the current. In order to determine the cell membrane parameters (CPE parameters), the transient-voltage response is fitted to the fractional relaxation equation. Importantly, this discharge voltage curve depends only on the tissue's electrical properties. Because of its ability to perform very fast measurements, this method (used here for the first time) may allow a real-time monitoring of the physiological state of tissues. Specific requirements on current interruption time and sampling rate have to be fulfilled. The histological images analysis showed that the technique presented did not produce acute damage to the tissues studied.

Acknowledgements

This work was funded by Junta de Comunidades de Castilla-La Mancha (Project PEII-2014-021-A). E. Hernández-Balaguera expresses his gratitude to the Universidad de Castilla-La Mancha for the Pre-PhD contract granted to him.

References

1. B. Rigaud, J. P. Morucci and N. Chauveau, Crit. Rev. Bioeng., 1996, 24, 257–351.
2. S. Grimnes and O. G. Martinsen, Bioimpedance and Bioelectricity Basics, Academic Press, London, 2015.
3. H. Fricke and S. Morse, J. Gen. Physiol., 1925, 9, 153–167.
4. R. L. Magin, Crit. Rev. Bioeng., 2004, 32, 105–193.
5. E. T. McAdams, J. Jossinet and A. Lackermeier, Innov. Tech. Biol. Med., 1995, 16, 662–670.
6. F. F. Kuo, Network Analysis and Synthesis, John Wiley & Sons, New York, 1966.
7. A. S. Sedra and K. C. Smith, Microelectronic Circuits, Oxford University Press, New York, 2004.
8. K. S. Cole, Cold Spring Harbor Symp. Quant. Biol., 1940, 8, 110–122.
9. E. Gersing, Bioelectrochem. Bioenerg., 1998, 45, 145–149.
10. G. R. Hernández-Labrado, J. L. Polo, E. López-Dolado and J. E. Collazos-Castro, Med. Biol. Eng. Comput., 2011, 49, 417–429.
11. G. R. Hernández-Labrado, R. E. Contreras-Donayre, J. E. Collazos-Castro and J. L. Polo, J. Electroanal. Chem., 2011, 659, 201–204.
12. S. Grimnes and O. G. Martinsen, J. Phys. D: Appl. Phys., 2007, 40, 9–14.
13. H. P. Schwan, Adv. Biol. Med. Phys., 1957, 5, 147–209.
14. H. P. Schwan, Proceedings of 16th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, Baltimore, United States, 1994.
15. U. Pliquett, Proceedings of the 11th International Biennial Baltic Electronics Conference, Tallinn, Estonia, 2008.
16. U. Pliquett, E. Gersing and F. Pliquett, Biomed. Tech., 2000, 45, 6–13.
17. M. Min, U. Pliquett, T. Nacke, A. Barthel, P. Annus and R. Land, Physiol. Meas., 2008, 29, S185–S192.
18. U. Pliquett, Proceedings of the XV International Conference on Electrical Bio-Impedance (ICEBI) & XIV Conference on Electrical Impedance Tomography (EIT), Heilbad Heiligenstadt, Germany, 2013.
19. C. E. B. Neves and M. N. Souza, Physiol. Meas., 2000, 21, 395–408.
20. T. J. Freeborn, B. Maundy and A. S. Elwakil, Comput. Electron. Agr., 2013, 98, 100–108.
21. F. H. van Heuveln, J. Electrochem. Soc., 1994, 141, 3423–3428.
22. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
23. I. Podlubny, I. Petrás and T. Skovránek, Proceedings of the 13th International Carpathian Control Conference (ICCC 2012), High Tatras, Slovakia, 2012.
24. M. Urquidi-Macdonald, S. Real and D. D. Macdonald, J. Electrochem. Soc., 1986, 133, 2018–2024.

---