# Cole-Impedance Model Representations of Right-Side Segmental Arm, Leg, and Full-Body Bioimpedances of Healthy Adults: Comparison of Fractional-Order

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Participants

#### 2.2. Dual-Energy X-ray Absorption Measurements

#### 2.3. Electrical Impedance Measurements

**I+, I−**) and two electrodes measuring the voltage response of the tissue segment (

**V+, V−**). A tetrapolar configuration is utilized to reduce (but cannot remove) the effects of the electrode/tissue interface impedance which is typically much larger than the tissue impedance. For further details regarding tetrapolar measurements, readers are recommended to review the works of Grimnes and Martinsen [25] and Aliau-Bonet and Pallas-Areny [26].

#### 2.4. Outlier Identification/Removal

#### 2.5. Cole-Impedance Model Parameter Identification

**particleswarm**function available in MATLAB with the following options: Swarm Size$=1000$, Social Adjustment Weight $=1$, MinNeighborsFraction $=0.6$, and the hybrid functionality was enabled to apply the

**fmincon**solver after the PSO solver terminated. Hybrid functions can obtain a more accurate solution (to the PSO alone) by starting from the relatively rough solution found by the first solver. This functionality is built into the MATLAB functions to implement the PSO and does not require additional effort on the part of the user to setup. Constraints were also added to the PSO implementation, with lower and upper boundaries for $\left(\right)$ fixed at $\left(\right)$ and $\left(\right)$, respectively.

## 3. Results

#### 3.1. Statistical Testing: Friedman Test

#### 3.2. Statistical Testing: Spearman Correlation

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) ImpediMed SFB7 device to collect participant electrical impedance using (

**b**) referenced electrode locations to measure segmental right arm, segmental right leg, and right full-body of each study participant with tetrapolar configuration.

**Figure 3.**Samples of participant data excluded (solid) from statistical analysis due to (

**a**) high-frequency hook artifact, (

**b**) positive high-frequency reactance, and (

**c**) motion artifacts. Simulations using identified Cole-impedance parameters (dashed) shown to highlight poor fitting for datasets with these characteristics.

**Figure 4.**Process applied to complete set of electrical impedance datasets for outlier identification/removal, resulting in the use of $N=174$ participant datasets (from an original set of $N=185$).

**Figure 5.**Samples of right-side full-body (black), right arm (blue), and right leg (red) bioimpedances collected from participants (solid line) compared to simulations of Cole-impedance model using PSO identified parameters (dashed).

**Figure 6.**Histograms of the Cole-impedance model parameters (${R}_{\infty}$, ${R}_{1}$, C, $\alpha $) identified by the PSO to best-fit the participant right-side full body (blue), right arm (orange), and right leg (yellow) datasets.

**Figure 7.**Scatterplots of (

**a**) ${R}_{\infty}$, (

**b**) ${R}_{1}$, (

**c**) C, and (

**d**) $\alpha $ vs. total segmental tissue, lean tissue, and fat tissue for right-side full body (black), right arm (blue), and right leg (red) impedance datasets.

**Figure 8.**DXA derived contributions of bone mineral content (orange), fat tissue (red), and lean tissue (blue) to overall composition of right arm, right leg, and right body of study participants.

**Table 1.**Median Cole-impedance model parameters (${R}_{\infty}$, ${R}_{1}$, C, $\alpha $) identified by the PSO to best-fit the 174 participant right-side full body, right arm, and right leg bioimpedance datasets.

Body Segment | ${\mathit{R}}_{\mathit{\infty}}$ ($\mathbf{\Omega}$) | ${\mathit{R}}_{1}$ ($\mathbf{\Omega}$) | C$\mathsf{\mu}\mathbf{F}\xb7{\mathbf{sec}}^{\mathit{\alpha}-1}$ | $\mathit{\alpha}$ |
---|---|---|---|---|

Right-side Full Body | $440.28$ | $191.32$ | $0.566$ | $0.7428$ |

Right Arm | $191.39$ | $92.51$ | $2.57$ | $0.6841$ |

Right Leg | $210.56$ | $80.05$ | $0.534$ | $0.8374$ |

**Table 2.**Spearman-rank correlation coefficients (${r}_{s}$) and statistical significant (p) between Cole-impedance parameters and total/lean/fat tissue masses for segmental measurements.

Total Tissue (kg) | Lean Tissue (kg) | Fat Tissue (kg) | |||||||
---|---|---|---|---|---|---|---|---|---|

Full-Body | Arm | Leg | Full-Body | Arm | Leg | Full-Body | Arm | Leg | |

${R}_{\infty}$ ($\Omega $) | |||||||||

${r}_{s}\left(157\right)$ | $-0.731$ | $-0.856$ | $-0.641$ | $-0.862$ | $-0.903$ | $-0.781$ | − | − | − |

p | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 | − | − | − |

${R}_{1}$ ($\Omega $) | |||||||||

${r}_{s}\left(157\right)$ | $-0.375$ | $-0.528$ | $-0.222$ | $-0.267$ | $-0.517$ | − | $-0.323$ | − | $-0.307$ |

p | <0.001 | <0.001 | $0.020$ | $0.003$ | <0.001 | − | <0.001 | − | <0.001 |

C ($\mu \mathrm{F}\xb7{\mathrm{sec}}^{\alpha -1}$) | |||||||||

${r}_{s}\left(157\right)$ | $0.682$ | $0.636$ | $0.542$ | $0.612$ | $0.545$ | $0.611$ | $0.334$ | $0.366$ | − |

p | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 | <0.001 | − |

$\alpha $ | |||||||||

${r}_{s}\left(157\right)$ | $-0.384$ | − | $-0.334$ | $-0.351$ | − | $-0.468$ | − | $-0.386$ | − |

p | <0.001 | − | <0.001 | <0.001 | − | <0.001 | − | <0.001 | − |

**Table 3.**Summary of fractional-order ($\alpha $) reported from studies of skeletal muscle tissue impedance utilizing Cole-impedance model.

Authors | Sample Population | Muscle Location | Frequency Band | Fractional-Order ($\mathit{\alpha}$) |
---|---|---|---|---|

Rigaud et al. [19] | Sheep | Gemellus (Ex Vivo) | $7.2$ kHz–720 kHz | Longitudinal: $0.73\pm 0.05$ Traverse: $0.78\pm 0.02$ |

Arnold et al. [32] | Mice | Gastrocnemius (Surface) | 1 kHz–10 MHz | Longitudinal (Young): $0.682$ Traverse (Young): $0.660$ Longitudinal (Aged): $0.794$ Traverse (Aged): $0.656$ |

Nagy et al. [33] | Mice | Gastrocnemius (Surface) | 1 kHz–10 MHz | Longitudinal: $0.709$–$0.760$ Traverse: $0.713$–$0.748$ |

Clark-Matott et al. [34] | Mice | Gastrocnemius Ex Vivo | 1 kHz–10 MHz | Longitudinal: $0.522$–$0.677$ Traverse: $0.690$–$0.784$ |

Sanchez, Bragos, & Rutkove [35] | Rat | Gastrocnemius (Ex Vivo) Soleus (Ex Vivo) | 1 kHz–1 MHz 1 kHz–1 MHz | Longitudinal: $0.528\pm 0.012$ Traverse: $0.729\pm 0.01$ Longitudinal: $0.695\pm 0.02$ Traverse: $0.736\pm 0.01$ |

Freeborn & Fu [2] | Healthy Adults | Biceps (Surface) | 10 kHz–100 kHz | Pre Exercise: $0.552$–$0.781$ Post Exercise: $0.552$–$0.779$ |

Fu & Freeborn [3] | Healthy Adults | Biceps (Surface) | 10 kHz–100 kHz | Exercised: $0.621$–$0.745$ Unexercised: $0.628$–$0.766$ |

Sato et al. [36] | Healthy Men (Surface) | Lower Extremities | 5 kHz–250 kHz | $0.71\pm 0.03$ |

This work
| Healthy Adults | Right-Body, Segmental Arm/Leg | 3 kHz–200 | Right-Body: 0.687–0.769 Right Arm: 0.606–0.725 Right Leg: 0.778–0.971 |

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**MDPI and ACS Style**

Freeborn, T.J.; Critcher, S.
Cole-Impedance Model Representations of Right-Side Segmental Arm, Leg, and Full-Body Bioimpedances of Healthy Adults: Comparison of Fractional-Order. *Fractal Fract.* **2021**, *5*, 13.
https://doi.org/10.3390/fractalfract5010013

**AMA Style**

Freeborn TJ, Critcher S.
Cole-Impedance Model Representations of Right-Side Segmental Arm, Leg, and Full-Body Bioimpedances of Healthy Adults: Comparison of Fractional-Order. *Fractal and Fractional*. 2021; 5(1):13.
https://doi.org/10.3390/fractalfract5010013

**Chicago/Turabian Style**

Freeborn, Todd J., and Shelby Critcher.
2021. "Cole-Impedance Model Representations of Right-Side Segmental Arm, Leg, and Full-Body Bioimpedances of Healthy Adults: Comparison of Fractional-Order" *Fractal and Fractional* 5, no. 1: 13.
https://doi.org/10.3390/fractalfract5010013