A LabVIEW-based electrical bioimpedance spectroscopic data interpreter (LEBISDI) for biological tissue impedance analysis and equivalent circuit modelling

Biological tissues comprise cells and an extracellular matrix. Both have their own electrical behavior and collectively produce a complex, frequency-dependent bioelectrical impedance, Z_b(f) [39]. The impedance Z_b(f) and its frequency-dependent phase angle, θ_b(f) are given by:

where R_b(f) and X_b(f) are the real and imaginary parts of Z_b respectively and j is the complex number √(-1). Both R_b(f) and X_b(f) depend on the signal frequency and tissue composition such that

EIS measures Z_b and its phase angle (θ_b) of a test subject by injecting a constant amplitude, alternating current (AC) signal (I(f)) at different frequency points, f_i (f_i: f₁, f₂, f₃, … f_n), and by measuring the boundary potentials (V(f_i)) through using a linear array of the surface electrodes attached to the subject (Figure 1) though other configurations could also be used such as placing the electrodes like the corners of a rectangle [52].

Fig.1

The impedance of a material is generally measured either by the two-electrode method [8, 20] or four-electrode method [8, 20, 50]. In the two-electrode method, as the name indicates, two surface electrodes are used for current injection and voltage measurement. As shown in Figure 1a, in the two-electrode method the current injection and voltage measurement are conducted through the same electrode pair [8, 20]. Thus, the measurement data contains the effect of contact impedance (Zc). To avoid the contact impedance problem, the impedance measurement is conducted with the four-electrode method as shown in Figure 1b and Figure 1c. Thus, the four-probe method (Figure 1b) of the impedance measurement process is usually preferred to make the measurement insensitive to the impedance of the electrodes and wiring systems. In the four-probe method, a frequency-dependent constant amplitude sinusoidal current (I(f)) is injected through the outermost two electrodes, called the current or driving electrodes (red electrodes in Figure 1b and 1c), and the frequency-dependent AC voltage (V(f)) is measured [13] across the two inner electrodes, called the voltage or sensing electrodes (blue electrodes in Figure 1b and 1c). From Ohm’s law of AC theory, Z_b(f) is the transfer function of the test subject, which is calculated from the voltage-current data by dividing V(f) by I(f):

In bioimpedance spectroscopy, variations in bioelectrical impedance over frequencies are represented as impedance curves and are analyzed to characterize the tissue. As an example, the real part (R_b) and imaginary part (X_b) of Z_b at each frequency point (f_i: i = 1, 2, 3, …, n) are calculated and plotted (along the x-axis and y-axis, respectively) to obtain the Nyquist Plot [6, 13] or the R_b-X_b spectrum.

The lumped electrical impedance value of any material (non-biological and biological) measured at a particular frequency can be represented by the impedance obtained from its equivalent electrical circuit model, which can be a combination of circuit elements such as resistance (R), capacitance (C) and inductance (L). If we define the complex electrical impedance (Z) for any particular material (either non-biological or biological), then

where Re(Z(f)) = R_z(f) represents the magnitude of the real part of complex Z and Im(Z(f)) = X_z(f) represents the magnitude of the imaginary part of the Z(f).

If the material contains both capacitive and inductive elements, then the reactive component (X_z) will have two components called the frequency-dependent inductive reactance component, X_L(f), and the frequency-dependent capacitive reactance component, X_C(f). Thus, we have:

Similarly, in EIS studies on biological material, the measured Z_b of the test subject can also be represented by an equivalent impedance (Z_eq). This Z_eq can be obtained from a combination of some electrical circuit elements (e.g., R and/or C) because the magnetic effect of the biological tissues at low frequency is negligible [53]. Moreover, the equivalent circuit model of biological tissues may also contain some special circuit elements, such as a constant phase element (CPE) [13] or Warburg impedance (W) [13] and/or some complex bioelectrical phenomena occurring inside the biological tissues due to the physiological changes that take place when they are subjected to AC excitation. Warburg impedance represents the electrode’s interfaces and the CPE is often interpreted as a parameter describing the distribution of relaxation times [54] in the system, e.g. due to a distribution of cell sizes.

The Nyquist plot is the plot of imaginary part of impedance (X_z) versus the real part of impedance (R_z). The approximate equivalent circuit model of the material can be found from the X_z-R_z curve (Nyquist plots) which is generally obtained from the EIS data. Electrical impedance spectroscopic analysis followed by the equivalent circuit modeling establishes the relationship between the tissue property and its impedance response. As for example, the Nyquist plots of some basic circuit models have been discussed in the Figure 2.

Fig.2

Figure 2a shows a r-C series circuit and its impedance spectra (Nyquist plot). Figure 2b shows an electrical circuit model containing R_p and C_p connected in parallel and its Nyquist plot. As shown in Figure 2b, the Nyquist plot contains a semi-circular arc that represent the two parallel circuit blocks, (R_p||C_p).

If a model contains a series resistance (r) along with the parallel branch (R_p||C_p) as shown in Figure 2c, the impedance spectra will be shifted toward positive real axis and the amount of the shift will indicate the magnitude of the resistance r. Figure 2d shows an equivalent electrical circuit model of an object containing several R and C components and its impedance spectra. This model contains three sub-circuit blocks or sub-circuit parts connected in series: R_s, (R_p1||C_p1) and (R_p2||C_p2). As shown in Figure 2d, the Nyquist plot contains two semi-circular arcs that represent the two parallel circuit blocks, (R_p1||C_p1) and (R_p2||C_p2). The high-frequency part of the spectra (left side of the graph) cuts the x-axis at a particular distance away from the y-axis, and this distance represents the series resistance (R_s). Similarly, Figure 2e shows the Nyquist plot of a different electrical circuit model, which contains two parts connected in series, (R_p1||C_p1) and ((R_p2+W)||C_p2). Its impedance spectra contain two semi-circular arcs for two parallel circuit blocks, (R_p1||C_p1) and ((R_p2+W)||C_p2). The linear part of the spectra on the low-frequency side (right side of the graph) indicates Warburg impedance (W) [55, 56, 57, 58, 59], resulting from the diffusion effect [58, 59, 60, 61] on electrodes at low frequencies.

In some cases, the Nyquist plots are semi-circular arcs with their centers some distance below the x-axis. These depressed semi-circular Nyquist plots are complex to analyze because of the involvement of the CPE circuit. Figure 3 presents a depressed semi-circular Nyquist plot, where the equivalent circuit contains the CPE in parallel with R. The high-frequency side of the semi-circular arc cuts the x-axis at a particular distance away from the y-axis and the distance represents R_s.

Fig.3

By definition, the impedance of the CPE (Z_CPE) shown in Figure 3 is represented as:

where Q is the magnitude of Z_CPE and Y₀=1/Q, j=√−1 and 0<n<1.

Note that the phase angle (ϕ) of Z_CPE is frequency independent. It thus remains constant throughout the frequency range of EIS. The ϕ of Z_CPE is given by:

For an object with CPE, when n is less than 1, the CPE resembles a capacitor, but not a pure capacitor because ϕ is less than 90°. As the value of n increases, CPE gradually becomes more capacitive. At n = 1, Y₀ becomes a pure capacitor (CCPE) for which

Nyquist plots of the circuits with a CPE are strongly influenced by circuit element combinations. When a material contains CPE and a resistive element, R_p, parallel to each other, the Nyquist plot (R-X plot) of the material will be a semi-circular path depressed by an angle of (1-n)*90°, as shown in Figure 3. But the R-X spectrum of a CPE alone is a straight line, generating an angle of (n*90°) with a positive x-axis (green dotted line in Figure 3).

Biological cells and tissues can be modeled by analysis of their impedance responses obtained from EIS studies conducted within a suitable frequency range. The impedance response of a biological cell is strongly influenced by cell composition [62, 63]. The individual electrical properties of each cell’s composition are thus represented by their equivalent electric circuit elements, which can be used to develop an equivalent circuit model of a single cell [6, 13, 64, 65, 66, 67, 68, 69, 70,], as described in the following sections.

Animal cells can be modeled with an equivalent circuit concept as shown in Figure 4. The resistive path created by the ICF in biological cells to an electric signal (Figure 4a) is represented as a resistive element called ICF resistance (R_ICF). Alternatively, the capacitance [74] offered by the protein-lipid-protein structure [62, 63] of the cell membrane is modeled as cell membrane capacitance (C_CM) [5, 6] (Figure 4a).

Fig.4

The equivalent impedance of the cell membrane and the protoplasm (which comprises cytoplasm and cell organelles) [71, 72] is called Z_PPCM (Figure 4b). Because the ECF is made up of conducting solution, it also creates a resistive path [6, 13, 65] (Figure 4a) that exerts an electrical resistance called ECF resistance (R_ECF). When cells are suspended in or by ECF, R_ECF becomes a resistive element connected as a parallel branch to the Z_PPCM. Therefore, an animal cell suspended in ECFs can be modeled as an equivalent electrical circuit [66, 67, 68, 69, 70] called the equivalent electrical impedance of an animal cell (Z_AC), as shown in Figure 4b.

Plant cells (Figure 5a) have cell walls made up of low-conducting materials, thus creating an additional resistive path that produces an electrical resistance called cell wall resistance (R_CW). Thus, as the envelope surrounding the cell membrane, R_CW is modeled as a resistance connected in parallel with Z_PPCM (Figure 5b) of the plant cell. Moreover, similar to the animal cell, ECFs in plant cells also produce resistive paths (Figure 5b). They are thus represented as R_ECF connected in parallel to the Z_PPCMCW, which is the parallel combination of Z_PPCM and R_CW (i.e., Z_PPCM || R_CW). Thus, as plant cells contain ECFs, a cell wall, a cell membrane and ICFs, they can be modeled as an equivalent electrical circuit called the equivalent electrical impedance of a plant cell (Z_PC), as shown in Figure 5b.

If the plant cell contains a large vacuole, the model can be modified by incorporating the impedance properties exerted by the composition of the vacuole. The vacuole of a plant cell is a cell-like structure made up of vacuole sap surrounded by a vacuole membrane, which is called a tonoplast [73, 74]. Thus, vacuole impedance can be modeled as a separate cell-like structure or a subcell inside the plant cell. The vacuole sap can be considered as the ICF of the vacuole, while the tonoplast can be considered as a membrane. Therefore, the R_ICF in the equivalent circuit model of a plant cell with vacuoles will be replaced by another subcellular impedance block [73, 74] in which the R_ECF will be the R_ICF because the cytoplasm could be considered the extracellular fluid of the vacuole. The equivalent circuit of biological cells can also be modeled with a number of other approaches [1, 75, 76]. The Nyquist plot, Cole-Cole plot [77, 78, 79, 80], impedance equivalent circuit [81, 82], Cole parameter estimation [75, 83, 84] of electrical bioimpedance can be suitably utilized for tissue characterization [34, 85, 86], body composition estimation [84] and disease detection [87]. It is worthwhile to note that the Cole-Cole plot is the Nyquist plot of complex permittivity [77, 78, 79, 80] whereas the Cole plot is for complex impedance or admittance.

Fig.5

A number of impedance data sets were generated and collected to evaluate the performance of LEBISDI. They are described in Section 2.5. Artificial impedance data were simulated and experimental impedance data were collected from various combinations of high-precision resistors and capacitors. Finally, after evaluating and calibrating LEBISDI, we collected impedance data from fruit and vegetable tissues using an impedance analyzer.

The evaluation of the performance of LEBISDI therefore included three types of impedance data: (i) computer-simulated impedance data, (ii) data generated by EIS studies conducted on real electronic circuit components with an impedance analyzer and (iii) the data collected by EIS studies conducted on fruits and vegetables with an impedance analyzer. Data sets (i) and (ii) were used to evaluate LEBISDI, whereas data set (iii) was used to calculate the equivalent circuit parameter for studying the variation in bioimpedance parameters of different tissue compositions.

2.4.1

PC-Based EIS Data Simulation in Matlab

Different electronic circuit combinations were simulated (Fig. 6) and a number of impedance data sets were generated using a specially developed MATLAB-based electrical impedance data-generating program. The PC-based impedance data generator mathematically calculates complex impedance and its phase angles at a number of frequency points within a particular frequency band (10 Hz to 2 MHz) for a number of circuit combinations called the simulated circuit model (SCM). Using the combinations of different circuit elements such as resistance (R), capacitance (C) and inductance (L), we could develop a capacitive circuit model (CCM) and an inductive circuit model (ICM). Figure 6 shows a few examples of the electronic circuit combinations studied; each has different values of resistance and capacitance connected in parallel and is tested to calculate the simulated impedance (Z_S) within a specific frequency range.

Fig.6

Real and imaginary parts of the simulated complex impedance obtained from all SCMs were calculated in the impedance data generator and stored in a .txt file in a desktop computer.

For a C CM made of a resistor and a capacitor connected in parallel, the real part of the impedance (R_z) and the imaginary part of the impedance (X_z) are respectively calculated as

(11)Rz=R(1ωC)2R2+(1ωC)2$${{R}_{z}}=\frac{R{{\left( \frac{1}{\omega C} \right)}^{2}}}{{{R}^{2}}+{{\left( \frac{1}{\omega C} \right)}^{2}}}$$

and

(12)Xz=R2(1ωC)R2+(1ωC)2.$${{X}_{z}}=\frac{{{R}^{2}}\left( \frac{1}{\omega C} \right)}{{{R}^{2}}+{{\left( \frac{1}{\omega C} \right)}^{2}}}.$$

The modulus of the impedance and the phase angle (θ) are respectively calculated as

(13)|Z|=Rz2+Xz2$$\left| Z \right|=\sqrt{R_{z}^{2}+X_{z}^{2}}$$

and

(14)θ=−arctan(XzRz).$$\theta =-\arctan \left( \frac{{{X}_{z}}}{{{R}_{z}}} \right).$$

For the ICM of a resistor and inductor connected in parallel, the real part of the impedance (R_z) and the imaginary part of the impedance (X_z) are respectively calculated as

(15)Rz=R(ωL)2R2+(ωL)2$${{R}_{z}}=\frac{R{{\left( \omega L \right)}^{2}}}{{{R}^{2}}+{{\left( \omega L \right)}^{2}}}$$

and

(16)Xz=R2(ωL)R2+(ωL)2.$${{X}_{z}}=\frac{{{R}^{2}}\left( \omega L \right)}{{{R}^{2}}+{{\left( \omega L \right)}^{2}}}.$$

The modulus of the impedance and the phase angle (θ) are calculated using the formula as shown in the equations (13) and (14).

2.4.3

EIS Data Collection from Fruits and Vegetables with an Impedance Analyzer

The impedance responses of fruits and vegetables were studied by a four-electrode-method-based EIS using a QuadTech7600 impedance analyzer with a 1 mA constant sinusoidal applied current. The impedance variations in cucumber (Cucumis sativus), carrot (Daucus carota), pumpkin (Cucurbita mixta), bottle gourd (Lagenaria vulgaris), banana cortex or pseudo stem of the banana tree (Musa acuminata), potato (Solanum tuberosum), tomato (Lycopersicon Esculentum L), bringle (Solanum melongena), mango (Mangifera indica), apple (Malus domestica) and orange (Citrus sinensis) were studied to measure the Z_b and θ_b at 100 discrete frequency points within a frequency band of 10 Hz to 2 MHz.

Nyquist plots were obtained from R_z and X_z which were calculated from Z and θ. QuadTech 7000-04 alligator clip leads (QuadTech Inc., USA), which are generally used to connect to large multiterminal devices, were used [88] as an interface between the surface electrodes and the impedance analyzer. The QuadTech 7000-04 comprises a BNC-to-BNC cable, four banana plug adapters and four alligator clips, which easily connect to metal conductors and electrodes. Figure 7a presents a schematic of the four-probe-method-based impedance data collection, which shows the arrangement of the current and voltage electrodes used for the EIS studies we conducted on circuit combinations. Figure 7b presents an EIS data collection schematic of the four-electrode-method-based EIS studies conducted on fruit and vegetable samples with separate current and voltage electrodes. As shown in the Figure 7b, the current signal is injected through the outer driving electrodes (red rectangles) and the voltage data are measured across the inner sensing electrodes (blue rectangles).

Fig.7

Circuit element studies were performed by directly connecting 7000-04 alligator clips with the circuit leads. EIS for fruit and vegetable samples required that the lead set was used with an ECG electrode-based four-electrode array (EE-FEA). Four QuadTech solid gel Ag/AgCl ECG electrodes were fixed on sticky insulating tape and the EE-FEA was developed (Figure 8a). The voltage electrodes were placed between current electrodes, with a distance of 50 mm between voltage electrodes (Figure 8a). In addition, the distance between the current-electrode and the voltage-electrode was maintained by separating the voltage electrodes from the current electrodes by 25 mm (Figure 8a). Fruit and vegetable tissue health and temperature were varied in different EIS studies, whereas the amplitude (1mA r.m.s.) remained constant and the sinusoidal current varied from 10 Hz to 2 MHz (Figure 8b).

Fig.8

Prior to EIS studies, all fruits and vegetables were properly cleaned and marked for electrode placement using a measuring tape, and the electrode array was pasted on the skin of the cleaned samples. The QuadTech7000-04 alligator connectors of the impedance analyzer probe were connected to the electrodes by EE-FEA. During EIS studies, Z and θ were set as the primary and secondary parameters, respectively, in the impedance analyzer, and measured at 100 frequency points within 10 Hz to 2 MHz. To study the effect of storage time on vegetable bioimpedance, we conducted EIS studies every 3 days for 22 days on the same samples with the same instrumental setup. To study variation in biological tissue impedance profiles due to boiling, we conducted EIS studies on fruits and vegetables before and after boiling.

The LabVIEW-based electrical bioimpedance spectroscopic data interpreter (LEBISDI) has been developed to study the equivalent circuit parameters of biological tissues using the LabVIEW 2014 software. The NI LabVIEW is a highly productive software development environment that enables the user to obtain a graphical programing and unprecedented hardware integration facility. LabVIEW provides a rapid virtual instrumentation design facility required for the high-speed and highly sophisticated measurement and control system applications. The proposed LEBISDI has been developed with a circle-fitting algorithm and the R-X data are fit to a circular arc from which the center and radius of the circle are calculated. Using the coordinates of the circle’s center, the radius and the other impedance parameters (Z_b, θ_b, R_z and X_z) are calculated. Figures 9a and b show LabVIEW-based graphical user interface (GUI) of the LEBISDI and the Nyquist plot loaded into the LEBISDI respectively^.

Fig.9

The impedance data is fed to the LEBISDI as a matrix [M_z]_100×3 having three columns: f, R_z and X_z respectively. The impedance data matrix [M_z] is saved in a .txt file though the program and can be easily modified to work with .xls file also. All the 100 rows are the f, R_z and X_z data obtained for a particular frequency point f_n. For each frequency data points (f_n), the real (R_z(f_n)) and imaginary (X_z(f_n)) parts of the impedance data are calculated from Z_n and θ_n obtained from the EIS and then are loaded through the GUI of the LEBISDI (Figure 9b) by pressing the “LOAD FILE” button. After that, the “FIT A CIRCLE” button of the LEBISDI GUI is pressed to fit the data along a circular path and the series resistance, parallel resistance, constant phase elements, X_Max (the maximum value of X_z) and X_Max frequency (i.e., the characteristic frequency (f_c)) are calculated.

The real and imaginary parts of the simulated impedance data sets were calculated and Nyquist plots were drawn (Figure 10) in the Matlab-based impedance data generator. The real and imaginary parts of impedance were also stored in a .txt file for use with the LEBISDI assessment. The R-X data corresponding to each frequency point were loaded in LEBISDI (Figure 11) from the .txt file and the Nyquist plots are obtained in the GUI.

Fig. 10

Fig. 11

The Nyquist plots drawn from the real and imaginary parts of the simulated impedance were fitted in LEBISDI to calculate the values of the elements in the circuit combination used in the computer simulation. Impedance parameters calculated by LEBISDI were compared with the actual values of the circuit components used for impedance data simulation to assess the performance of LEBISDI.

Figure 12 shows that the LEBISDI successfully fit the Nyquist plots obtained for all four of the simulated circuit models shown in Figure 6 and accurately calculated the electronic circuit parameters (Table 1). Results from LEBISDI show that the errors found in R_p identification forsimulated circuit models 1, 2, 3 and 4 were 0.0027 %, 0.0187 %, 0.0188 % and 0.0030 %, respectively, and 0.937 %, 0.068 %, 0.068 % and 0.569 %, respectively, for C_p calculations (Table 1). Results also demonstrate that n (power in the CPE model) was equal to 1, indicating that there was no CPE in the designed models, which agrees with results for the simulated models.

Fig. 12

The impedance data obtained from the inductive circuit models are also analyzed with LEBISDI. The Nyquist plots obtained from the data simulator were loaded and fitted in LEBISDI and the impedance parameters used in the computer simulation, such as parallel resistance (R_p) and parallel inductance (L_p), were calculated. The results obtained from the simulated data analysis of the inductive circuit are presented in Figure 13 and Table 2.

Fig. 13

Next, we investigated the performance of LEBISDI with the impedance data generated by the RCM developed with different circuit combinations of real electronic circuit components. Nyquist plots obtained from the real circuit data collected by an impedance analyzer were fitted by LEBISDI (Figure 14) and the impedance parameters were calculated. Figure 14a shows the impedance plots of RCM 1, which is a parallel combination of a 1 kΩ resistor and a 0.1 μF capacitor. Table 3 shows the impedance parameters calculated by LEBISDI for RCM 1 and the error obtained in the fitting. The errors were 0.348 % and 0.01% for resistance and capacitance calculations, respectively (Table 3).

Fig.14

Impedance data were also collected from a parallel combination of a 2 kΩ resistor and a 0.1 μF capacitor called RCM 2. Figure 14b shows the impedance plots for RCM 2 and Table 4 presents the impedance parameters calculated by LEBISDI for this circuit. The errors in the fitting process in LEBISDI were 0.335 % and 5.0 % for resistance and capacitance calculations, respectively (Table 4).

Impedance data were also collected from fruits and vegetables. Figure 15 – 17 show the impedance plots of a cucumber, a carrot and a bottle gourd, respectively. Numerical results for these configurations are reported in Table 5. To investigate the effect of boiling on impedance, data were also collected from the fruits and vegetables before and after boiling. Figures 18a and b show the impedance plots of a green banana before and after boiling, respectively. Numerical results for these configurations are reported in Table 6. After boiling, impedance parameters were lower for both series and parallel resistance, likely because boiling destroyed the structure of the cells; the CPE angle was also reduced. We investigated the role of freshness on impedance variation by collecting data from naturally stored vegetables. Impedance was measured daily and the variation over time was analyzed by LEBISDI. Figure 19 shows the impedance plots of a cucumber collected every 3 days for 22 days, and Table 7 shows the impedance parameters calculated by LEBISDI of impedance in a cucumber throughout the 22-day storage period. Over time, the impedance parameters for parallel resistance gradually increased while the impedance parameters for series resistance and parallel capacitance gradually decreased, likely due to dehydration with storage time; the CPE angle also increased slightly with time.

Fig.15

Fig. 16

Fig. 17

Fig. 18

Fig. 19