Compensation of volatile-distributed current due to variance of the unknown contact impedance in an electrical impedance tomography sensor

In order to compensate for the contact impedance variance, the current pathway length D_e (mm) is considered by calculating the normalized extracellular fluid resistance, called ${{\boldsymbol{\chi }}_{{\rm norm}}}$ (unitless), from all the measurements m as follows:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{\boldsymbol{\chi }}_{{\rm norm}}}=\left(\frac{{{{\bf R}}_{{\rm ecf}}}}{{\rm max}\left[ {{{\bf R}}_{{\rm ecf}}} \right]} \right)\nonumber \end{align} \tag{ 6 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle 0<{{\boldsymbol{\chi }}_{{\rm norm}}}\leqslant 1.\nonumber \end{align} \tag{ 7 }$$

Here, R_ecf (Ω) is the resistance for the measurement at zero frequency f ₀, and ${{{\bf R}}_{{\rm ecf}}}\in {{\mathbb{R}}^{1\times M}}$. The total number of measurements M  =  104 with n_e  =  16 electrodes and m  =  [1, ..., M]. The value of χ could be much higher than R_a  −  R_b. ΔZ, which has a high range order value, makes the reconstruction more ill-posed because the EIT reconstruction method proposed in this study is for cases with a small change in conductivity. Considering the ill-conditioned Jacobian matrix, in the linear regularization of EIT the value of ΔZ should be small compared with the change in conductivity, i.e. ΔZ/Δσ  ⩽  1. In this regard, χ is normalized as we proposed without losing information about the length of the current pathway.

The approximation of equation (6) is based on properties of the Cole impedance model shown in the electrical equivalent circuit in figure 1(a) [28–30]. The equivalent electrical circuit is a model of impedance with multifrequency measurements of biological tissue composed of multicellular cells, as shown in figure 1(b). The equivalent electrical circuit shown in figure 1(a) is widely used for the interpretation of the Nyquist plot of a single curve, as shown in figure 2 for biological tissue impedance measurements which have a dispersion impedance due to the complexity of electrical properties of multicellular systems with multifrequency measurements [30–35]. R_ecf is the resistance when the current flows only in the extracellular fluid for the case of the zero-frequency measurement f ₀. The cell membranes behave as a constant phase element (CPE_m) which has a very high impedance at f ₀. The resistance of intracellular fluid in multicellular cells is represented as R_icf.

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According to the Cole impedance model, R_ecf is calculated using the following equation [28–30]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{{\bf R}}_{{\rm ecf}}}={{{\bf R}}_{0}}-{{{\bf R}}_{\infty }}\nonumber \end{align} \tag{ 8 }$$

where R₀ is the real part of the impedance at zero frequency, ${{{\bf R}}_{0}}\in {{\mathbb{R}}^{1\times M}}$, and ${{{\bf R}}_{\infty }}$ is the real part of the impedance at infinite frequency, ${{{\bf R}}_{\infty }}\in {{\mathbb{R}}^{1\times M}}$. As shown in figure 2, by using the fitting circle curve [36] we can obtain R₀ and ${{{\bf R}}_{\infty }}$ with the Cole impedance model [28, 29].

According to fd-EIT, ΔZ_fd from two sequential frequencies f _a and f _b ( f _a  <  f _b) can be written as follows:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta {{{\bf Z}}_{{\rm fd}}}=\left({{\left[ {\bf R} \right]}_{a}}-{{\left[ {\bf R} \right]}_{b}} \right)\nonumber \end{align} \tag{ 9 }$$

where R_a and R_b are the resistance vectors at f _a and f _b, respectively. R_a and R_b $\in {{\mathbb{R}}^{1\times M}}$. For the volatile-distributed current due to contact impedance variance, the change in impedance ΔZ changes [16]. The changing ΔZ is reflected by the changing of De to ${{\widehat{{\bf D}}}_{{\rm e}}}$. The value of ΔZ_fd is distorted to $\Delta {{\widehat{{\bf Z}}}_{{\rm fd}}}$ (where $\Delta {\bf Z}\ne \Delta \widehat{{\bf Z}}$) as follows:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta {{\widehat{{\bf Z}}}_{{\rm fd}}}{\rm =}\left({{\left[ \widehat{{\bf R}} \right]}_{a}}-{{\left[ \widehat{{\bf R}} \right]}_{b}} \right).\nonumber \end{align} \tag{ 10 }$$

Here ${{\left[ \widehat{{\bf R}} \right]}_{a}}$ and ${{\left[ \widehat{{\bf R}} \right]}_{b}}$ are the resistance vectors at f _a and f _b, respectively, for the volatile-distributed current. Then, the changing ΔZ is compensated by using χ_norm (so-called fd-EIST), and can be written as follows:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta {{{\bf Z}}_{{\rm fds}}}={{\boldsymbol{\chi }}_{{\rm norm}}}\cdot \ast \left({{\widehat{{\bf R}}}_{a}}-{{\widehat{{\bf R}}}_{b}} \right).\nonumber \end{align} \tag{ 11 }$$

The concept of ${{\boldsymbol{\chi }}_{{\rm norm}}}$ works as a weighted parameter vector by considering the changing current pathway length within all the measurements in the EIT sensor. Remember that equation (11) is an element-wise multiplication between two matrix-vectors.

wfd-EIT, as shown in equation (3), modifies fd-EIT by employing a weighted constant ξ in order to increase the accuracy of determination of the source of the measured impedance which can come from a background object, inclusion object or contact impedance variance [13, 27, 37]. fd-EIT attempts to find ΔZ from two sequential frequencies f _a and f _b. The lower frequency measurement f _a is used as the baseline impedance value while the higher frequency measurement f _b is only selected when significant changes in the physiological conditions are detected. Calculation of ξ requires accurate prior information about the distribution of the electrical properties of the background object at different frequencies [38, 39]. For derivation of the equation for fd-EIST, shown in equations (10) and (11), the weighted parameter vector χ_norm is used; to derive a similar equation (3) for wfd-EIT this can be modified to so-called wfd-EIST as shown in equation (4).

The experimental setup comprises an impedance analyzer (HIOKI IM3570, Japan) and a multiplexer (Agilent 34970A, USA) with two modules (34904A, 4  ×  8 two-wire matrix module) as a multifrequency data acquisition system for EIT, a personal computer and a cylindrical tank.

The signal-to-noise ratio of this system is 55 dB for 1 MHz frequency measurement. The impedance analyzer is connected to the multiplexer using four ports—high current (H_c), high potential (H_p), grounded-low current (L_c) and low potential (L_p)—through coaxial cables. The impedance analyzer measures the complex impedance from frequency measurements f   =  500 Hz to 100 kHz with 1 mA constant current. The multiplexer is connected to the contact electrode in a cylindrical tank by coaxial cables.

A cylindrical tank with inner diameter D  =  80 mm and height h  =  40 mm was used as a frame of the sensor. The contact electrode consists of n_e  =  16 stainless steel screw electrodes (one layer) with an angular separation between the electrodes of ϕ  =  22.5°, giving an area contact of each electrode A_el  =  19 mm². The electrodes were attached to the middle point of the tank height h.

Two types of phantom were used, as shown in figures 3(a) and (b). One type of phantom consisted of an inclusion object (potato) in a frequency-dependent background. In this phantom condition, the inclusion object (potato dice) is regarded as the abnormal physiological condition. The background object (saline water and carrot dice) is regarded as the normal physiological condition. Potato dice and saline water and carrot dice were chosen as the inclusion and background objects, respectively, because these materials show frequency-dependent behavior and are often used for research in pre-clinical experimental studies [40]. The electrical properties of the material components inside the phantom are shown in figure 3(c).

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3.2.1. Phantom A: frequency-independent background object.

3.2.2. Phantom B: frequency-dependent background object.

To evaluate the effect of the unknown contact impedance variance, we employed finite element method (FEM) simulations based on COMSOL Multiphysics v4.3a with the electrical circuit and the electric current modules [41] and the free-noise condition. Figure 4 illustrates the equivalent circuit of the EIT sensor including the instrumental effect and the contact impedance variance which is used in the numerical simulation. Here, ρ (=1/σ) (Ω m) is resistivity and A_e is the area of the electrode of height h (mm) and length L_e (mm). The length L_e is the distance between the adjacent ith (transmitter) and j th (receiver) electrodes or the adjacent kth (transmitter) and lth (receiver) electrodes. Figure 4(a) shows the current pathway length D_e (as for example D_e,1 for m  =  1 in figure 4(b)) which is the distance between points f  and g. The values C_i  =  10 pF and R_i  =  5 MΩ represent the instrumental effect obtained according to the study of Zhang et al [42] and their absolute value depends on the data acquisition system. Z_i, Z_j, Z_k and Z_l are the contact impedances whose values are shown in the curve of figure 5. Meanwhile, Z₅, Z_x and Z₆ are the impedances of the domain (the phantom). Using COMSOL Multiphysics, the electric current module is used to calculate the electrical potential distribution ϕ, whilst the electrical circuit module is used to calculate the instrumental effect and the contact impedance variance. The boundary and phantom conditions of the numerical simulation are similar to those used in the experimental studies.

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The reconstructed images of conductivity change Δσ in experiments and numerical simulation studies were calculated using a total variation method, which is a hybrid between l₂-norm- and l₁-norm-based reconstruction methods. The reconstruction based on the total variation method is derived from the use of the following objective function [43]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta \sigma ={\rm arg}\underset{\sigma >0}{\mathop{{\rm min}}}\,\left\{\frac{1}{2}\left\Vert F\left(\Delta \sigma \right)-\Delta \mathbf{Z} \right\Vert_{2}^{2}+\lambda {{\left\Vert L\left(\Delta {{\sigma }_{{\rm true}}} \right) \right\Vert}_{1}} \right\}\nonumber \end{align} \tag{ 12 }$$

where $\Delta \sigma \in {{\mathbb{R}}^{1\times n}}$ is the reconstructed conductivity change, n is the number of pixels, F is the forward solver used to predict the impedance data from Δσ, λ is the regularization parameter, which is calculated using the L-curve method, and L is the regularization matrix, of which L^T · L  =  I, and I is the identity matrix. The L-curve displays a set of points with a representation that minimizes the first term $\left\Vert F\left(\Delta \sigma \right)-\Delta {\bf Z} \right\Vert_{2}^{2}$ versus the second term ${{\left\Vert L\left(\Delta {{\sigma }_{{\rm true}}} \right) \right\Vert}_{1}}$ in the right-hand side of equation (12). The value of λ is selected at the maximum inflection point or the corner of the L-curve [44]. In order to solve equation (12), we use the total variation primal dual-interior point method (TV PD-IPM) [43]. Meanwhile, EIDORS v.3.9 package software is used to calculate the Jacobian matrix S using a homogeneous condition, with conductivity and permittivity values based on the phantom condition [45, 46]. In order to reconstruct the images of conductivity change Δσ, the frequency f _a is fixed at 500 Hz, the lowest frequency measurement from the experiment. After that, f _b was then changed and evaluated based on the reconstructed image.

In this study the image reconstruction performances are evaluated using the values of RMSE, PE, RNG and SD [47]. The lower the value of the evaluation parameter the better the reconstruction performance.

RMSE (unitless) shows the aggregate residual value between the measured impedance and the predicted impedance:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm RMSE}=\frac{\sqrt{{{\left(\Delta {\bf Z}-\Delta {{{\bf Z}}_{{\rm fem}}} \right)}^{T}}\cdot \left(\Delta {\bf Z}-\Delta {{{\bf Z}}_{{\rm fem}}} \right)}}{{{\left(\Delta {\bf Z} \right)}^{T}}\cdot \Delta {\bf Z}}.\nonumber \end{align} \tag{ 13 }$$

Here ΔZ are the measured impedances from the experimental data while ΔZ_fem is the predicted impedance obtained from the FEM model based on the calculated Δσ.

PE (unitless) represents the accuracy of the reconstructed image in detecting the location of the inclusion:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm PE}=r-{{r}_{{\rm true}}}.\nonumber \end{align} \tag{ 14 }$$

r is the predicted center of the inclusion in the reconstructed image, while r_true is the true location of the center of the inclusion in the phantoms.

RNG (unitless) represents the areas of opposite sign surrounding the inclusion of the reconstructed image:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm RNG}=\frac{{{A}_{{\rm out}}}}{{{A}_{{\rm in}}}}.\nonumber \end{align} \tag{ 15 }$$

A_in (mm²) is the area of inclusion in the reconstructed image. A_out (mm²) is the ringing, which is the area outside the inclusion in the reconstructed image that has an opposite sign to A_in (mm²).

SD (unitless) measures the ratio of the inclusion area in the reconstructed image A_in (mm²) which is not covered by the circular area A_c (mm²):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm SD}=\frac{{{A}_{{\rm c}}}}{{{A}_{{\rm in}}}}.\nonumber \end{align} \tag{ 16 }$$

A_c is the circular area whose center is the center of inclusion in the reconstructed image, and the area of A_c is similar to A_in. The circular area is considered in this evaluation because a circle is the typical shape of the reconstructed image.

The correlation coefficient CC (unitless):

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle CC=\frac{\sum\nolimits_{i=1}^{n}{\left(\Delta {{\sigma }_{w}}-\overline{\Delta {{\sigma }_{w}}} \right)\left(\Delta {{\sigma }_{o}}-\overline{\Delta {{\sigma }_{o}}} \right)}}{\sqrt{\sum\nolimits_{i=1}^{n}{{{\left(\Delta {{\sigma }_{w}}-\overline{\Delta {{\sigma }_{w}}} \right)}^{2}}}\sum\limits_{i=1}^{n}{{{\left(\Delta {{\sigma }_{o}}-\overline{\Delta {{\sigma }_{o}}} \right)}^{2}}}}}\nonumber \end{align} \tag{ 17 }$$

is used in the numerical simulation studies to quantify the effect of contact impedance variance. Δσ_w is the distribution of conductivity change when the electrode has contact impedance variance. Meanwhile, Δσ_o is the distribution of conductivity change when the electrode does not have contact impedance variance. The ideal value of CC is 1, which means the two conductivity distributions between Δσ_w and Δσ_o are identical.

As shown in figure 6, the inclusion object (i.e. potato dice area) of phantoms A and B based on conductivity change Δσ in fd-EIST and wfd-EIST is successfully reconstructed just as in the case of fd-EIT and wfd-EIT. The impedance changes ΔZ of fd-EIT, fd-EIST, wfd-EIT and wfd-EIST are acquired at f _a  =  500 Hz and f _b  =  2–6 kHz. The reconstructed images of phantom A produced by fd-EIST and wfd-EIST show a reasonably accurate inclusion shape, while those for fd-EIT and wfd-EIT show a relatively inaccurate inclusion shape (an ellipse-like shape). For phantom B, as shown in figure 7,the inclusion object geometry of fd-EIST and wfd-EIST is more consistent with the true shape of the inclusion compared with other reconstruction methods.

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In figures 6 and 7, fd-EIST and wfd-EIST show the interface boundary between the potato dice and carrot dice relatively consistently compared with fd-EIT and wfd-EIT, regardless of the frequency change, without suffering image artifacts. On the other hand, fd-EIT and wfd-EIT methods could not eliminate image artifacts, which look like an inclusion that appears inconsistently with changing frequency.

4.2.1. RMSE.

For different frequencies f _b, fd-EIST showed the lowest RMSE value (see figures 8(a) and 9(a)) compared with fd-EIT, wfd-EIT and wfd-EIST. As the frequency increased, all reconstruction methods showed an increasing RMSE value.

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4.2.2. PE.

4.2.3. RNG.

4.2.4. SD.

Five different frequencies f _b were used to evaluate the variance in the unknown contact impedance based on the frequency-dependent behavior, i.e. f _b  =  2, 3, 4, 5 and 6 kHz. For the ideal condition, i.e. without contact impedance variance, the curves of frequency-dependent behavior of conductivity change show that the conductivity change Δσ of the reconstructed images increases with increase in frequency [35]. If the frequency-dependent behavior curve shows an irregular change of Δσ for the reconstructed image with increasing frequency, the contact impedance variance is considered to influence the reconstructed image.

For phantom A, as shown in figure 10, fd-EIST and fd-EIT give the best performance. Figure 10 shows the curve of frequency-dependent behavior for four positions in the reconstructed images of phantom A shown in figure 6. Positions 1 and 2 are in the intracellular fluid of potato dice located in a biomaterial having a non-linear frequency-dependent behavior. All reconstruction methods showed an increase in the curve of frequency-dependent behavior with increasing frequency. In positions 1 and 2, fd-EIST and fd-EIT showed an approximate 30% increment in the curve of frequency-dependent behavior. Positions 3 and 4 are in the saline water (a frequency independent-background that is supposed to remain constant as the frequency increases). fd-EIST and fd-EIT showed an increment of less than 5% in the curve of frequency-dependent behavior. Meanwhile, wfd-EIST and wfd-EIT showed an increment of about 10%–20% in the curve of frequency-dependent behavior.

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Meanwhile, for phantom B (figure 11), fd-EIST showed the most consistent frequency-dependent behavior curve in all positions compared with other reconstruction methods. Figure 11 shows the curve of frequency-dependent behavior for four positions in the reconstructed images of phantom B shown in figure 7. Position 1 is located in a biomaterial which has a non-linear frequency-dependent behavior. Positions 2, 3 and 4 are also considered to be in a biomaterial which has a different frequency-dependent behavior from position 1. Position 1 is in the intracellular fluid of potato dice while positions 2, 3 and 4 are in the frequency-dependent background objects (saline water and carrot dice). For position 1, all the reconstruction methods showed an increasing frequency-dependent behavior curve with increasing frequency. However, each reconstruction method showed a different degree of increment. wfd-EIST showed the highest increment while fd-EIT had the lowest increment of the four reconstruction methods. In ideal conditions the curve of frequency-dependent behavior in positions 2, 3 and 4 should show similar degrees of increment as the frequency increases. Among the four reconstruction methods, fd-EIST showed the results closest to ideal, with a 15% increment. fd-EIT produced a frequency-dependent behavior curve which tended to be constant.

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Figure 12(a) shows that the unknown contact impedance variance affects the ideal trend line of the real part of impedance R from all measurements m. The effect of unknown contact impedance variance can be seen in the results of the numerical simulation comparing ΔZ for homogeneous conditions with the carrot dice background condition. Figure 12(a) shows the comparison of ΔZ without and with contact impedance variance evaluated at frequencies f _a  =  500 Hz and f _b  =  2 kHz, and figure 12(b) shows the change in ΔZ due to contact impedance variance.

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5.2.1. Phantom A: frequency-independent background object.

As shown in figure 13 for phantom A, fd-EIST had the ability to overcome the contact impedance variance better than fd-EIT, wfd-EIT and wfd-EIST. The existence of contact impedance variance significantly affects the accuracy of reconstructed images based on fd-EIT and wfd-EIT. As the frequency increases, the separation between potato dice inclusions becomes easier to distinguish. However, qualitative analysis based on reconstructed images is subjective from the user's point of view. Thus, we evaluated the correlation coefficient CC with and without contact impedance variance, as shown in the curve of figure 14. The CC value of fd-EIST is relatively higher than that for fd-EIT, wfd-EIT and wfd-EIST. Even though the CC value increases with frequency f _b there is a trade-off between CC value and accuracy of the reconstructed image in terms of RMSE, PE, RNG and SD as shown in the experimental results.

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5.2.2. Phantom B: frequency-dependent background object.

For phantom B, as shown in figure 15, the existence of contact impedance variance affects the accuracy of reconstructed images in a very similar way to that seen in figure 13. The CC values with fd-EIST and wfd-EIST are relatively higher than with fd-EIT and wfd-EIT, but tend to decrease as frequency f _b increases, as shown in figure 16. Both fd-EIST and wfd-EIST depend on χ_norm, which is calculated based on the resistance of the extracellular fluid R_ecf. According to the Cole impedance model, the resistance of the extracellular fluid R_ecf is acquired when the current flows only in the extracellular fluid. In this case, the current flowing through CPE_m is negligible. We therefore hypothesize that fd-EIST and wfd-EIST are only accurate when the effect of permittivity change Δε_r on impedance Z is relatively low compared with the effect of conductivity change Δσ at two sequential frequencies f _a and f _b.

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We begin our discussion with an explanation of how variance in the unknown contact impedance causes the volatile-distributed current I₁. Unknown contact impedance variance means that the value of contact impedance is not known prior to the experimental measurements. The volatile-distributed current I₁ is analyzed via the impedance change based on a Kirchhoff loop analysis of figure 17, which shows an equivalent circuit of the four-wire measurement method. Z_ie is the capacitive coupling impedance. Z_ie  =  −j /(ωC_i), where C_i is the capacitive coupling between voltmeter wires (H_p and L_p) and the virtual ground that causes impedance artifacts [48]. Z_i, Z_j, Z_k and Z_l are the variance contact impedances, where Z_i  ≠  Z_j  ≠  Z_k  ≠  Z_l. Note that contact impedance has a frequency-dependent behavior, so it is represented as a parallel combination between the resistive and capacitive components. Z_m is the measured impedance of the object of interest, V_x is the measured voltage of the object of interest, I₀ is the injected current and Z₅ and Z₆ are the edge impedances due to the separation between electrodes. In the ideal case, I₃ should be negligible because the input impedance of the voltmeter is very high. Therefore, analysis of the voltage of the object of interest V_x should be considered with I₁:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \begin{array}{@{}lllllllllllllllllllllllll@{}} & {{Z}_{{\rm m}}}=\frac{{{V}_{x}}}{{{I}_{{\rm meas}}}}=\frac{{{V}_{x}}}{{{I}_{1}}} \nonumber \\ & \,\,\quad=\frac{\left({{I}_{3}}\cdot \left({{Z}_{k}}+{{Z}_{x}}+{{Z}_{j}} \right)-{{I}_{2}}\cdot {{Z}_{j}}-{{I}_{1}}\cdot {{Z}_{k}} \right)}{{{I}_{1}}} \end{array}\nonumber \end{align} \tag{ 18 }$$

where ${{Z}_{i}}\neq{{Z}_{j}}\neq{{Z}_{k}}\neq{{Z}_{l}}$. In loop I₁:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{I}_{1}}=\frac{{{I}_{3}}\cdot \left({{Z}_{ie}}+{{Z}_{k}} \right)}{\left(2\cdot {{Z}_{ie}}+{{Z}_{l}}+{{Z}_{6}}+{{Z}_{k}} \right)}.\nonumber \end{align} \tag{ 19 }$$

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Based on equation (19), the contact impedance variances Z_k and Z_l have a greater effect on the intensity of the value of I₁ compared with Z₆. In the ideal case, the value of I₁ is constant for all the measurements. For the EIT measurement, the electrode combination k and l changes in all measurements, and the values of Z_k and Z_l also change among the electrodes. Therefore, the value of I₁ also changes in all measurements, or a volatile-distributed current occurs.

The numerical simulation results shown in figures 12, 13 and 15 provide an understanding of the existence of unknown contact impedance affecting the impedance change ΔZ and the accuracy of the reconstructed images. Even though the numerical simulation could not mimic the experimental conditions perfectly, the equivalent circuit of the four-wire measurement method in a circular EIT sensor (as shown in figure 4) is capable of showing the effect of unknown contact impedance artificially. Generally, a comparison of the experimental and numerical simulation results leads us to conclude that the contact impedance variance causes the volatile-distributed current I₁.

fd-EIST and wfd-EIST show better performances than fd-EIT and wfd-EIT because the reconstructed images have a frequency-dependent behavior that is closer to ideal. The volatile-distributed current I₁ distorts the ideal frequency-dependent behavior of the object of interest. Non-ideal frequency-dependent behavior of the object of interest, as shown in figures 10 and 11, causes inaccurate calculation of the impedance change ΔZ in fd-EIT and wfd-EIT. In order to compensate for the volatile-distributed current I₁ in the object of interest, a weighted parameter vector χ_norm is used in fd-EIST and wfd-EIST to relate the sensor geometry and the current pathway length D_e for all the measurements m, as shown in figure 18. Figure 18 shows χ_norm versus the measurement number list for one rotation of impedance measurement. The value of χ_norm provides not only information about the sensor geometry but also the spatial condition of the inclusion object that is represented as the current pathway length. The green-dashed line gives information about the sensor geometry, which is the normalized linear distance between the transmitter and receiver electrodes $\hat {\rm\bf D}$_e, i.e. the lengths shown as red lines in the circle in figure 18. As shown in figure 18, χ_norm has comparable characteristics for each background condition. χ_norm functions as a baseline impedance value for synchronization and to signify the measured impedance from the inclusion object. This study gives a two-dimensional view of the reconstructed image, but we realize that the current pathway length has a three-dimensional distribution. In this regard, further analysis is needed to describe the behavior of χ_norm for a three-dimensional sensor.

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In the reconstructed image, the color intensity depends on how different the real part of the impedance R is from the impedance at two sequential frequencies f _a and f _b, i.e. R_a  −  R_b. As R_a  −  R_b increases, the intensity of the color of the reconstructed image also increases. Meanwhile, the location of the interface boundary between the inclusion and background object depends on how the measurement number m experiences the change in impedance ΔZ. If the change in ΔZ is measured inaccurately the position of the interface boundary between the inclusion and the background object will be incorrectly recorded as the frequency changes. Figure 19 shows χ_norm calculated from the conditions of phantom B. Even though the change in impedance ΔZ is recognized by using just two sequential frequency impedances, i.e. R_a  −  R_b, it is difficult to distinguish between the sources of the measured impedance because of the volatile-distributed current I₁. A weighted function ξ of wfd-EIT is used to represent the measured impedance from the background object. The value of ξ depends on the selection of f _b as shown in equation (5), and the accuracy of the reconstructed image depends on the accuracy of the prior information about the baseline impedance value of the background object. In a real situation, obtaining prior information about the background object is not a trivial task and can only be achieved by an invasive method, making it difficult to use EIT for functional imaging. In the proposed method χ_norm can overcome the difficulties with wfd-EIT; it behaves as a weighted parameter vector of the real part of the impedance at two sequential frequencies. Compared with wfd-EIT, weighted parameters in fd-EIST and wfd-EIST do not depend on the selection of f _b. Thus, fd-EIST and wfd-EIST do not need prior information about the background object.

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Traditionally, fd-EIT or wfd-EIT reconstructs the interface boundary between an inclusion and background object as a representation of the change in physiological conditions. The unknown contact impedance variance limits the ability of fd-EIT or wfd-EIT to show more features. Taking the frequency-dependent behavior into consideration in the analysis of the reconstructed image improves physiological analysis by increasing the accuracy of interpretation in long-term monitoring applications. In long-term monitoring, the spatial as well as the temporal distribution of the abnormal physiological condition is detected. For the temporal distribution, characterization of the region considered to represent the abnormal physiological condition can be done by analyzing the frequency-dependent behavior.