# Analog Realization of Fractional-Order Skin-Electrode Model for Tetrapolar Bio-Impedance Measurements

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Skin and Electrode Cole Models

#### 2.1. The Skin Model

#### 2.2. The Electrode Model

## 3. Circuit Realization of the Electrode and Skin Cole Models

#### 3.1. Valsa-Vlach Fractional Order Capacitor RC Network Approximation

#### 3.2. Versatile Active Fractional Capacitor Emulator

#### 3.3. Inverse Follow-the-Leader Feedback Fractional Capacitor Emulator

- ${\alpha}_{5}={b}_{0}=-{\alpha}^{5}-15{\alpha}^{4}-85{\alpha}^{3}-225{\alpha}^{2}-274\alpha -120$,
- ${\alpha}_{4}={b}_{1}=5{\alpha}^{5}+45{\alpha}^{4}+5{\alpha}^{3}-1005{\alpha}^{2}-3250\alpha -3000$,
- ${\alpha}_{3}={b}_{2}=-10{\alpha}^{5}-30{\alpha}^{4}+410{\alpha}^{3}+1230{\alpha}^{2}-4000\alpha -12000$,
- ${\alpha}_{2}={b}_{3}=10{\alpha}^{5}-30{\alpha}^{4}-410{\alpha}^{3}+1230{\alpha}^{2}+4000\alpha -12000$,
- ${\alpha}_{1}={b}_{4}=-5{\alpha}^{5}+45{\alpha}^{4}+5{\alpha}^{3}-1005{\alpha}^{2}+3250\alpha -3000$,
- ${\alpha}_{0}={b}_{5}={\alpha}^{5}-15{\alpha}^{4}+85{\alpha}^{3}-225{\alpha}^{2}+274\alpha -120$,

#### 3.4. Cole Model Tunable Resistor ${R}_{o}$ Realization

#### 3.5. Cole Model Circuit Realization Simulation Results

#### 3.6. Cole Models Parameters Variation and Circuit Emulator Trimming

#### 3.6.1. Electrode Model Parameter Variation

#### 3.6.2. Skin Model Parameter Variation

## 4. Tetrapolar Model Simulation Results

#### 4.1. Case I: Balanced Contact Impedances

#### 4.2. Case II: Imbalanced Contact Impedances

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Brief schematic of a bipolar measurement setup (

**left**) and a tetrapolar measurement setup (

**right**): green rectangles indicate the electrodes placed. Z refers to the impedance under measure, while ${z}_{a}$ and ${z}_{b}$ refer to the bio-impedances between injection and measurement electrodes. The schematics are based on the description in [11], assuming infinite current source output impedance, infinite instrumentation amplifier input impedance, and negligible electrode impedances.

**Figure 4.**Fractional-order capacitor emulator (versatile methodology) [37].

**Figure 5.**Employed CCII [37].

**Figure 7.**Realization of fractional-order capacitor emulator (Inverse Follow-the-Leader Feedback (IFLF) methodology).

**Figure 10.**Layout of the implemented fractional-order skin and electrode models (versatile methodology).

**Figure 11.**Impedance magnitude (

**left**) and phase response (

**right**) of the fractional-order capacitor for the case of electrode model.

**Figure 12.**Impedance magnitude (

**left**) and phase response (

**right**) of the fractional-order capacitor for the case of skin model.

**Figure 13.**Impedance magnitude (

**left**) and phase response (

**right**) of fractional-order electrode model.

**Figure 15.**Impedance magnitude (

**left**) and phase response (

**right**) of the case I fractional-order electrode model.

**Figure 16.**Impedance magnitude (

**left**) and phase response (

**right**) of the case II fractional-order electrode model.

**Figure 17.**Impedance magnitude (

**left**) and phase response (

**right**) of the case III fractional-order electrode model.

**Figure 18.**Impedance magnitude (

**left**) and phase response (

**right**) of the case IV fractional-order electrode model.

**Figure 19.**Impedance magnitude (

**left**) and phase response (

**right**) of the case V fractional-order electrode model.

**Figure 20.**Impedance magnitude (

**left**) and phase response (

**right**) of the case VI fractional-order electrode model.

**Figure 21.**Impedance magnitude (

**left**) and phase response (

**right**) of the case VII fractional-order electrode model.

**Figure 22.**Impedance magnitude (

**left**) and phase response (

**right**) of the case VIII fractional-order electrode model.

**Figure 23.**Impedance magnitude (

**left**) and phase response (

**right**) of the case I fractional-order skin model.

**Figure 24.**Impedance magnitude (

**left**) and phase response (

**right**) of the case II fractional-order skin model.

**Figure 25.**Impedance magnitude (

**left**) and phase response (

**right**) of the case III fractional-order skin model.

**Figure 26.**Impedance magnitude (

**left**) and phase response (

**right**) of the case IV fractional-order skin model.

**Figure 30.**AC magnitude and phase impedance measurements for (

**a**) R

_{b}= 100 Ω, (

**b**) R

_{b}= 1 kΩ, and (

**c**) R

_{b}= 10 kΩ: all electrode and contact impedances are equal (R

_{∞}= 1.5 kΩ). The corresponding RC network approximations are included for comparison.

**Figure 31.**AC magnitude (

**left**) and phase (

**right**) impedance measurements for deviated shunt impedance of the positive voltage electrode (versatile design).

**Figure 32.**AC magnitude (

**left**) and phase (

**right**) impedance measurements for deviated shunt impedance of the positive voltage electrode (IFLF design).

**Figure 33.**Sensitivity performance of magnitude for target impedance ${R}_{b}=100\text{}\mathsf{\Omega}$ using Monte-Carlo analysis (versatile design).

**Figure 34.**Sensitivity performance of magnitude for target impedance ${R}_{b}=100\text{}\mathsf{\Omega}$ using Monte-Carlo analysis (IFLF design).

**Figure 35.**Sensitivity performance of magnitude for target impedance ${R}_{b}=10$ k$\mathsf{\Omega}$ using Monte-Carlo analysis (versatile design).

**Figure 36.**Sensitivity performance of phase for target impedance ${R}_{b}=10$ k$\mathsf{\Omega}$ using Monte-Carlo analysis (versatile design).

Parameter | ${R}_{\infty}$ (k$\mathsf{\Omega}$) | ${R}_{0}$ (M$\mathsf{\Omega}$) | $\tau $ (s) | a | C (nF/sec${}^{1-a}$) |
---|---|---|---|---|---|

Skin | 1.86 | 1.39 | 0.53 | 0.749 | 447 |

Electrode | 0.21 | 1.08 | 1.41 | 0.942 | 1.92 |

Parameter | ${R}_{0}$ (M$\mathsf{\Omega}$) | a | C (nF/sec${}^{1-a}$) |
---|---|---|---|

Skin | 0.64–1.46 | 0.63–0.86 | 61.2–1042.1 |

Electrode | 0.65–2.09 | 0.8–0.99 | 1.39–2.09 |

Electrode Element | Value | Skin Element | Value |
---|---|---|---|

${C}_{1}$ | $232.78$ pF | ${C}_{1}$ | $146.63$ nF |

${C}_{2}$ | $203.19$ pF | ${C}_{2}$ | $81.24$ nF |

${C}_{3}$ | $177.37$ pF | ${C}_{3}$ | $45.01$ nF |

${C}_{4}$ | $154.83$ pF | ${C}_{4}$ | $29.94$ nF |

${C}_{5}$ | $135.15$ pF | ${C}_{5}$ | $13.82$ nF |

${C}_{p}$ | $928.34$ pF | ${C}_{p}$ | $17.16$ nF |

${R}_{1}$ | $683.7$ M$\mathsf{\Omega}$ | ${R}_{1}$ | $1.1$ M$\mathsf{\Omega}$ |

${R}_{2}$ | $75.2$ M$\mathsf{\Omega}$ | ${R}_{2}$ | $188.1$ k$\mathsf{\Omega}$ |

${R}_{3}$ | $8.3$ M$\mathsf{\Omega}$ | ${R}_{3}$ | $32.6$$\mathsf{\Omega}$ |

${R}_{4}$ | $909.4$ k$\mathsf{\Omega}$ | ${R}_{4}$ | $5.6$ k$\mathsf{\Omega}$ |

${R}_{5}$ | 100 k$\mathsf{\Omega}$ | ${R}_{5}$ | $978.5$$\mathsf{\Omega}$ |

${R}_{p}$ | $5.53$ G$\mathsf{\Omega}$ | ${R}_{p}$ | $5.2$ M$\mathsf{\Omega}$ |

**Table 4.**Parameters for the implementation of expression H(s) (12).

Electrode Parameter | Value | Skin Parameter | Value |
---|---|---|---|

${c}_{6}$ | $1.0$ | ${c}_{6}$ | $6.1$ |

${c}_{5}$ | $9.393\times {10}^{4}$ | ${c}_{5}$ | $9.388\times {10}^{5}$ |

${c}_{4}$ | $7.715\times {10}^{8}$ | ${c}_{4}$ | $1.221\times {10}^{10}$ |

${c}_{3}$ | $6.024\times {10}^{11}$ | ${c}_{3}$ | $1.504\times {10}^{13}$ |

${c}_{2}$ | $4.464\times {10}^{13}$ | ${c}_{2}$ | $1.762\times {10}^{15}$ |

${c}_{1}$ | $2.824\times {10}^{14}$ | ${c}_{1}$ | $1.836\times {10}^{16}$ |

${c}_{0}$ | $1.76\times {10}^{14}$ | ${c}_{0}$ | $1.102\times {10}^{16}$ |

${d}_{5}$ | $100.0$ | ${d}_{5}$ | $33.33$ |

${d}_{4}$ | $8.184\times {10}^{6}$ | ${d}_{4}$ | $2.728\times {10}^{6}$ |

${d}_{3}$ | $5.866\times {10}^{10}$ | ${d}_{3}$ | $1.955\times {10}^{10}$ |

${d}_{2}$ | $3.999\times {10}^{13}$ | ${d}_{2}$ | $1.333\times {10}^{13}$ |

${d}_{1}$ | $2.593\times {10}^{15}$ | ${d}_{1}$ | $8.645\times {10}^{14}$ |

${d}_{0}$ | $1.473\times {10}^{16}$ | ${d}_{0}$ | $4.910\times {10}^{15}$ |

OTA | W/L (μm/μm) | CCII | W/L (μm/μm) |
---|---|---|---|

${M}_{n2}$, ${M}_{n3}$ | $2/1$ | ${M}_{p1}$, ${M}_{p2}$, ${M}_{p4}$ | $1.6/0.4$ |

${M}_{n8}$– ${M}_{9}$ | $1/2$ | ${M}_{p5}$, ${M}_{p6}$ | $3.2/0.4$ |

${M}_{n5}$–${M}_{n7}$ | $0.5/4$ | ${M}_{p3}$ | $6.4/0.4$ |

${M}_{n10}$–${M}_{n11}$ | $1/2$ | ${M}_{n1}$-${M}_{n6}$ | $0.8/0.4$ |

${M}_{p1}$–${M}_{p6}$ | $10/5$ | ${M}_{p7}$ | $1.6/0.4$ |

${M}_{n1}$, ${M}_{n4}$ | $0.4/1$ | – | – |

**Table 6.**Values of the capacitors of Figure 4.

Element | Value | Element | Value |
---|---|---|---|

${C}_{1}$ | $1.22$ pF | ${C}_{2}$ | $13.95$ pF |

${C}_{3}$ | $146.67$ pF | ${C}_{4}$ | $1.54$ nF |

${C}_{5}$ | $17.60$ nF | ${C}_{r}$ | $10.0$ nF |

Electrode Model Scaling Factor | Value | Skin Model Scaling Factor | Value |
---|---|---|---|

${G}_{0}$ | $1.707$ | ${G}_{0}$ | $19.188$ |

${G}_{1}$ | $1.705$ | ${G}_{1}$ | $11.080$ |

${G}_{2}$ | $1.505$ | ${G}_{2}$ | $6.167$ |

${G}_{3}$ | $1.315$ | ${G}_{3}$ | $3.418$ |

${G}_{4}$ | $1.148$ | ${G}_{4}$ | $1.881$ |

${G}_{5}$ | $1.000$ | ${G}_{5}$ | $1.000$ |

**Table 8.**Values of the capacitors of Figure 7.

Element | Value | Element | Value |
---|---|---|---|

${C}_{1}$ | $2.58$ pF | ${C}_{2}$ | $141.43$ pF |

${C}_{3}$ | $689.82$ pF | ${C}_{4}$ | $2.75$ nF |

${C}_{5}$ | $13.80$ nF | - | - |

**Table 9.**Values of transcoductance for the skin model of Figure 7.

Parameter | Value | Parameter | Value |
---|---|---|---|

${g}_{ma}$ | $184.84$ nS | ${g}_{mb}$ | $674.02$ nS |

${g}_{mc}$ | $721.35$ nS | ${g}_{md}$ | $729.19$ nS |

${g}_{me}$ | $713.18$ nS | ${g}_{mf}$ | $564.15$ nS |

${g}_{mvi}$ | 55.37 μS | - | - |

Electrode Model Scaling Factor | Value | Skin Model Scaling Factor | Value |
---|---|---|---|

${G}_{0}$ | $0.002$ | ${G}_{0}$ | $0.02$ |

${G}_{1}$ | $0.12$ | ${G}_{1}$ | $0.19$ |

${G}_{2}$ | $0.52$ | ${G}_{2}$ | $0.600$ |

${G}_{3}$ | $1.92$ | ${G}_{3}$ | $1.66$ |

${G}_{4}$ | $8.61$ | ${G}_{4}$ | $5.33$ |

${G}_{5}$ | $422.02$ | ${G}_{5}$ | $50.01$ |

Case/Parameter | ${R}_{o,e}$ (M$\mathsf{\Omega}$) | ${C}_{e}$ (nF/sec${}^{1-a}$) |
---|---|---|

Case I | 1.08 | 1.75 |

Case II | 2.09 | 1.39 |

Case III | 0.65 | 1.92 |

Case IV | 1.51 | 1.92 |

Case V | 0.65 | 1.75 |

Case VI | 1.51 | 1.75 |

Case VII | 0.65 | 2.09 |

Case VIII | 1.51 | 2.09 |

Case/Parameter | ${a}_{s}$ | ${C}_{s}$ (nF/sec${}^{1-a}$) |
---|---|---|

Case I | 0.86 | 65.2 |

Case II | 0.81 | 61.2 |

Case III | 0.82 | 88.9 |

Case IV | 0.78 | 73.1 |

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**MDPI and ACS Style**

Alimisis, V.; Dimas, C.; Pappas, G.; Sotiriadis, P.P.
Analog Realization of Fractional-Order Skin-Electrode Model for Tetrapolar Bio-Impedance Measurements. *Technologies* **2020**, *8*, 61.
https://doi.org/10.3390/technologies8040061

**AMA Style**

Alimisis V, Dimas C, Pappas G, Sotiriadis PP.
Analog Realization of Fractional-Order Skin-Electrode Model for Tetrapolar Bio-Impedance Measurements. *Technologies*. 2020; 8(4):61.
https://doi.org/10.3390/technologies8040061

**Chicago/Turabian Style**

Alimisis, Vassilis, Christos Dimas, Georgios Pappas, and Paul P. Sotiriadis.
2020. "Analog Realization of Fractional-Order Skin-Electrode Model for Tetrapolar Bio-Impedance Measurements" *Technologies* 8, no. 4: 61.
https://doi.org/10.3390/technologies8040061