# Short-Time Impedance Spectroscopy Using a Mode-Switching Nonsinusoidal Oscillator: Applicability to Biological Tissues and Continuous Measurement

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## Abstract

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## 1. Introduction

- (1)
- Design a nonsinusoidal oscillation circuit capable of periodically switching oscillation frequency;
- (2)
- Show that the proposed method can estimate circuit parameters for circuit models close to actual measured objects;
- (3)
- Apply the proposed method to the impedance measurement of biological tissues;
- (4)
- Show that the continuous measurement of impedance is possible.

## 2. Capacitive Coupling Impedance Spectroscopy

#### 2.1. Mode-Switching Nonsinusoidal Oscillator

#### 2.2. Experimental Method

#### 2.2.1. CIS of Parallel RC Circuits and Bioimpedance Models

#### 2.2.2. Application of CIS to Biological Tissues

#### 2.2.3. Continuous Impedance Measurement Using CIS

#### 2.2.4. Data Acquisition and DFT Analysis

## 3. Results and Discussion

#### 3.1. CIS of Parallel RC Circuits and Bioimpedance Models

#### 3.2. Application of CIS to Biological Tissues

#### 3.3. Continuous Impedance Measurement Using CIS

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Mode-switching nonsinusoidal oscillator. The oscillation frequency ${f}_{0}$ and the oscillation period ${T}_{0}$ of the oscillator are switched between four different frequencies by 4-bit binary counters and bilateral analog switches. CLK of 4-bit binary counters denotes a clock signal, and RST represents a reset signal. ${Q}_{0}$, ${Q}_{1}$, ${Q}_{2,}$ and ${Q}_{3}$ of 4-bit binary counters denote zero bits, and the first, second, and third bits of the output signals, respectively. ${C}_{\mathrm{O}}$ represents a bypass capacitor to remove the switching noise in the oscillator, and its value is 0.47 nF. CTR1 and CTR2 are binary counters to control the repeat count and the mode number of the mode switching, respectively.

**Figure 2.**Impedance measurement of porcine myocardium: (

**a**) photograph of the porcine left ventricle coagulated using electrocautery; (

**b**) measurement electrode fixture.

**Figure 3.**Constant current circuit for the LED current control of the photocoupler. The CdS photocell was used as (

**a**) a single resistor, (

**b**) a resistor ${R}_{\mathrm{X}}$ of the bioimpedance model, or (

**c**) ${R}_{\mathrm{PX}}$ of its model.

**Figure 4.**Four-mode oscillation waveforms of ${v}_{1}\left(t\right)$ and ${v}_{12}\left(t\right)$ obtained through the mode-switching nonsinusoidal oscillator. ${R}_{\mathrm{X}}$, ${R}_{\mathrm{PX}}$, ${C}_{\mathrm{PX}}$, and ${C}_{\mathrm{SX}}$ were set to 0.10 k, 0 Ω, 1.016 nF, and 9.76 nF, respectively. The oscillation frequency was switched every two cycles of each ${R}_{\mathrm{F}}$. The DFT analysis was performed on the oscillation waveforms of the second half of two cycles.

**Figure 5.**Frequency spectra of the (

**a**) real part $\mathrm{Re}\left({\dot{Z}}_{\mathrm{X}}\right)$ and (

**b**) imaginary part $\mathrm{Im}\left({\dot{Z}}_{\mathrm{X}}\right)$ of the impedance ${\dot{Z}}_{\mathrm{X}}$ comprising the resistance ${R}_{\mathrm{X}}$ of 0.10 kΩ, resistance ${R}_{\mathrm{PX}}$ of 0 Ω, capacitance ${C}_{\mathrm{PX}}$ of 1.016 nF (parallel RC circuit), and capacitance ${C}_{\mathrm{PX}}$ of 9.76 nF. The dashed lines represent the theoretical curves calculated from Equations (9) and (10).

**Figure 6.**Frequency spectra of the (

**a**) real part $\mathrm{Re}\left({\dot{Z}}_{\mathrm{X}}\right)$ and (

**b**) imaginary part $\mathrm{Im}\left({\dot{Z}}_{\mathrm{X}}\right)$ of the impedance ${\dot{Z}}_{\mathrm{X}}$ comprising the resistance ${R}_{\mathrm{X}}$ of 0.91 kΩ, resistance ${R}_{\mathrm{PX}}$ of 0 Ω, capacitance ${C}_{\mathrm{PX}}$ of 9.85 nF (parallel RC circuit), and capacitance ${C}_{\mathrm{SX}}$ of 9.76 nF. The dashed lines represent the theoretical curves calculated from Equations (9) and (10).

**Figure 7.**Absolute relative estimation errors of the circuit parameters of parallel RC circuits: (

**a**) ${R}_{\mathrm{X}}$ = 0.10 kΩ, ${C}_{\mathrm{PX}}$ = 1.016 nF, $\mathrm{and}{C}_{\mathrm{SX}}$ = 9.76 nF; (

**b**) ${R}_{\mathrm{X}}$ = 0.91 kΩ, ${C}_{\mathrm{PX}}$ = 9.85 nF, and ${C}_{\mathrm{SX}}$ = 9.76 nF.

**Figure 8.**Frequency spectra of the (

**a**) real part $\mathrm{Re}\left({\dot{Z}}_{\mathrm{X}}\right)$ and (

**b**) imaginary part $\mathrm{Im}\left({\dot{Z}}_{\mathrm{X}}\right)$ of the impedance ${\dot{Z}}_{\mathrm{X}}$ comprising the parallel RC circuits for ${C}_{\mathrm{PX}}=$ 1.016 nF and coupling capacitance ${C}_{\mathrm{SX}}=$ 9.76 nF. The DFT data were fitted to Equations (9) and (10) to calculate ${R}_{\mathrm{X}}$, ${C}_{\mathrm{PX}}$, and ${C}_{\mathrm{SX}}$ (dashed lines).

**Figure 9.**${R}_{\mathrm{X}}$, ${C}_{\mathrm{PX}},$ and ${C}_{\mathrm{SX}}$ of the parallel RC circuits estimated using oscillation waveforms and DFT: (

**a**) the estimated ${R}_{\mathrm{X}}$, (

**b**) relative error of the estimated ${C}_{\mathrm{PX}}$, and (

**c**) relative error of the estimated ${C}_{\mathrm{SX}}$. The insets are the circuit model used for fitting, and the estimated parameters are highlighted in pink.

**Figure 10.**Cole–Cole plot of ${\dot{Z}}_{\mathrm{X}1}$ for the parallel RC circuits with ${C}_{\mathrm{PX}}=$1.016 nF and ${C}_{\mathrm{SX}}=$9.76 nF. The dashed lines represent the fitting curves obtained from Equations (9) and (10).

**Figure 11.**Frequency spectra of the (

**a**) real part $\mathrm{Re}\left({\dot{Z}}_{\mathrm{X}}\right)$ and (

**b**) imaginary part $\mathrm{Im}\left({\dot{Z}}_{\mathrm{X}}\right)$ of the impedance ${\dot{Z}}_{\mathrm{X}}$ comprising the bioimpedance models for ${R}_{\mathrm{X}}=$0.30 kΩ, ${C}_{\mathrm{PX}}=$3.22 nF, and coupling capacitance ${C}_{\mathrm{SX}}=$9.76 nF. DFT data were fitted to Equations (9) and (10) to determine ${R}_{\mathrm{X}}$, ${R}_{\mathrm{PX}}$, ${C}_{\mathrm{PX}},$ and ${C}_{\mathrm{SX}}$ (dashed lines).

**Figure 12.**${R}_{\mathrm{X}}$, ${C}_{\mathrm{PX}}$, and ${C}_{\mathrm{SX}}$ of the bioimpedance models estimated using oscillation waveforms and DFT: (

**a**) the estimated ${R}_{\mathrm{X}}$, (

**b**) the estimated ${R}_{\mathrm{PX}}$, (

**c**) relative error of the estimated ${C}_{\mathrm{PX}}$, and (

**d**) relative error of the estimated ${C}_{\mathrm{SX}}$. The insets are the circuit models used for fitting, and the estimated parameters are highlighted in pink.

**Figure 13.**Cole–Cole plot of ${\dot{Z}}_{\mathrm{X}1}$ for the bioimpedance models with ${C}_{\mathrm{PX}}=$3.22 nF and ${C}_{\mathrm{SX}}=$9.76 nF. The dashed lines represent the fitting curves obtained from Equations (9) and (10).

**Figure 14.**Frequency spectrum of the real part $\mathrm{Re}\left({\dot{Z}}_{\mathrm{X}}\right)$ of the impedance ${\dot{Z}}_{\mathrm{X}}$ of the porcine myocardium and coupling capacitance. DFT data were fitted to Equation (9) to determine ${R}_{\mathrm{X}}$, ${R}_{\mathrm{PX}}$, and ${C}_{\mathrm{PX}}$ (dashed lines).

**Figure 15.**Frequency spectra of the impedance ${\dot{Z}}_{\mathrm{X}1}$ of the porcine myocardium: (

**a**) the Cole–Cole plot, (

**b**) the real part $\mathrm{Re}\left({\dot{Z}}_{\mathrm{X}1}\right)$, and (

**c**) the imaginary part $\mathrm{Im}\left({\dot{Z}}_{\mathrm{X}1}\right)$. ${\dot{Z}}_{\mathrm{X}1}$ represents the impedance obtained by subtracting the impedance of the estimated ${C}_{\mathrm{SX}}$ from ${\dot{Z}}_{\mathrm{X}}$. $\mathrm{Re}\left({\dot{Z}}_{\mathrm{X}1}\right)$ equals $\mathrm{Re}\left({\dot{Z}}_{\mathrm{X}}\right)$ in Figure 14.

**Figure 16.**Comparison of equivalent circuit parameters between the untreated and ablated left ventricles of porcine hearts: (

**a**) ${R}_{\mathrm{X}}$, (

**b**) ${R}_{\mathrm{PX}}$, and (

**c**) ${C}_{\mathrm{PX}}$. The sample size is 12.

**Figure 17.**Single-mode oscillation waveforms of ${v}_{12}\left(t\right)$ and ${v}_{1}\left(t\right)$ for the nonsinusoidal oscillator incorporating the CdS photocell (Figure 3a): (

**a**) full waveform and (

**b**) enlarged waveform. The photocurrent ${I}_{\mathrm{Ph}}$ of the photocoupler is changed at 10 and 23 ms. The high and low levels of ${I}_{\mathrm{Ph}}$ are 2.0 and 0.6 mA, respectively. Circuit parameters are estimated for each cycle (blue dashed line). Representative points of time in each cycle are the ends of the cycles such as ${t}_{1}$ and ${t}_{2}$.

**Figure 18.**Resistance ${R}_{\mathrm{X}}$ of the photocell estimated using oscillation waveforms in Figure 17a and DFT: (

**a**) time–resistance curves and (

**b**) saturation values at a high ${I}_{\mathrm{Ph}}$ period. Time of the time–resistance curves corresponds to the representative points of time in Figure 17b.

**Figure 19.**Two-mode oscillation waveforms of ${v}_{12}\left(t\right)$ and ${v}_{1}\left(t\right)$ for the nonsinusoidal oscillator incorporating the CdS photocell and circuit elements of the bioimpedance model (Figure 3b): (

**a**) full waveform and (

**b**) enlarged waveform. The photocurrent ${I}_{\mathrm{Ph}}$ of the photocoupler is changed at 10 and 23 ms. The high and low levels of ${I}_{\mathrm{Ph}}$ are 8.0 and 4.0 mA, respectively. Circuit parameters are estimated for each cycle (blue dashed line). Representative points of time in each cycle are the ends of the cycles such as ${t}_{1}$ and ${t}_{2}$. The frequency f is the oscillation frequency obtained from the second half cycle of the oscillation waveform at ${R}_{\mathrm{F}}$ = 3.3 kΩ.

**Figure 20.**Time–resistance, time–capacitance, and time–frequency curves of the resistance of the photocell and circuit elements of the bioimpedance model estimated using oscillation waveforms and DFT. The photocell was used as (

**a**) ${R}_{\mathrm{X}}$ or (

**b**) ${R}_{\mathrm{PX}}$. The other resistance is 0.20 kΩ; the capacitance ${C}_{\mathrm{PX}}$ is 3.22 nF; and the capacitance ${C}_{\mathrm{SX}}$ is 9.76 nF. The times of these curves correspond to the representative points of time in Figure 19b. The insets are photocell-incorporated circuits.

Mode | ${\mathit{Q}}_{1}$ | ${\mathit{Q}}_{0}$ | SW1 | SW2 | ${\mathit{R}}_{\mathit{F}}$ |
---|---|---|---|---|---|

1 | 0 | 0 | OFF | OFF | ${R}_{\mathrm{F}1}+{R}_{\mathrm{F}2}+{R}_{\mathrm{F}3}$ |

2 | 0 | 1 | OFF | ON | ${R}_{\mathrm{F}1}+{R}_{\mathrm{F}3}$ |

3 | 1 | 0 | ON | OFF | ${R}_{\mathrm{F}1}+{R}_{\mathrm{F}2}$ |

4 | 1 | 1 | ON | ON | ${R}_{\mathrm{F}1}$ |

**Table 2.**Two sets of the oscillation frequencies obtained by combining ${R}_{\mathrm{F}1}$, ${R}_{\mathrm{F}2}$, and ${R}_{\mathrm{F}3}$.

Set | ${\mathit{R}}_{\mathit{F}1}$ (kΩ) | ${\mathit{R}}_{\mathit{F}2}$ (kΩ) | ${\mathit{R}}_{\mathit{F}3}$ (kΩ) | 4 Modes of
${\mathit{R}}_{\mathit{F}}$ (kΩ) | Estimated Oscillation Frequencies (kHz) |
---|---|---|---|---|---|

1 | 3.3 | 1.8 | 3 | 8.1, 6.3, 5.1, 3.3 | 16.9, 22.3, 28.4, 47.9 |

2 | 5.1 | 1 | 2 | 8.1, 7.1, 6.1, 5.1 | 16.9, 19.5, 23.1, 28.4 |

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**MDPI and ACS Style**

Yamaguchi, T.; Ogawa, E.; Ueno, A. Short-Time Impedance Spectroscopy Using a Mode-Switching Nonsinusoidal Oscillator: Applicability to Biological Tissues and Continuous Measurement. *Sensors* **2021**, *21*, 6951.
https://doi.org/10.3390/s21216951

**AMA Style**

Yamaguchi T, Ogawa E, Ueno A. Short-Time Impedance Spectroscopy Using a Mode-Switching Nonsinusoidal Oscillator: Applicability to Biological Tissues and Continuous Measurement. *Sensors*. 2021; 21(21):6951.
https://doi.org/10.3390/s21216951

**Chicago/Turabian Style**

Yamaguchi, Tomiharu, Emiyu Ogawa, and Akinori Ueno. 2021. "Short-Time Impedance Spectroscopy Using a Mode-Switching Nonsinusoidal Oscillator: Applicability to Biological Tissues and Continuous Measurement" *Sensors* 21, no. 21: 6951.
https://doi.org/10.3390/s21216951