A novel method for estimating the fractional Cole impedance model using single-frequency DC-biased sinusoidal excitation

Abstract

Objective:

The Cole model is a widely used fractional circuit model in electrical bioimpedance applications for evaluating the content and status of biological tissues and fluids. Existing methods for estimating the Cole impedance parameters are often based on multi-frequency data obtained from stepped-sine measurements fitted using a complex non-linear least square (CNLS) algorithm. Newly emerged numerical methods from the magnitude of electrical bio-impedance data-only do not need CNLS fitting, but they still require multi-frequency stepped-sine data. This study proposes a novel approach to estimating the Cole impedance parameters that combines a numerical and time-domain fitting method based on a single-frequency DC-biased sinusoidal current excitation.

Approach:

First, the transient and steady-state voltage response along with the current excitation are acquired in electrical bio-impedance measurement. From the sampled data, a numerical method is applied to provide the initial estimation of the Cole impedance parameters, which are then used in a time-domain iterative fitting algorithm.

Results:

The accuracy of the algorithm proposed is tested with noisy electrical bio-impedance simulations. The maximum relative error of the estimated Cole impedance parameters is 1% considering 2% (34 dB) additive Gaussian noise. Experimental measurements performed on a 2R-1C circuit and some fruit samples show a mean difference less than 1% and 5% respectively compared to the Cole impedance parameters estimated from a commercial electrical bio-impedance analyzer performing stepped-sine measurements and CNLS fitting.

Significance:

This is the first method that allows estimating the Cole impedance parameters from single-frequency electrical bio-impedance data. The approach presented could find broad use in many applications, including single-frequency body impedance analysis.

1. Introduction

Fractional calculus is the branch of mathematics that deals with the derivatives and integrals of functions to non-integer orders [27]. A considerable number of fractional circuit analysis has been found to be better describing real-world phenomena than the traditional integer-order circuit analysis, including in industrial control [19], power electronics [8], and bio-engineering [2], etc.

Electrical bioimpedance is one example where fractional circuit analysis is applied to measure biological tissues [14]. To extract useful information contained in the measured electrical bioimpedance data, an impedance model is usually fitted to the data. The Cole impedance model [4] is one of the widely used fractional circuit model for describing the frequency-dependence electrical bioimpedance data [2].

Today, the main methods for estimating the Cole impedance parameters can be categorized into three groups: (1) frequency-domain complex impedance (using both real and imaginary parts simultaneously) techniques; (2) frequency-domain magnitude-only techniques; (3) time-domain complete-response techniques.

Complex impedance measurement in the frequency-domain is the commonly used method for estimating the Cole impedance parameters. In general, to estimate the Cole parameters, the multi-frequency complex impedance data obtained from stepped-sine measurements are fitted to the Cole model using a gradient-based nonlinear fitting algorithm [32]. However, the accuracy of the estimated results largely depends on the initial values of the fitting process, plus they may converge prematurely to a local minimum due to noisy data or experimental errors [13]. Other stochastic optimization algorithms have been developed to overcome these limitations, including genetic algorithms, particle swarm optimization algorithms, bacterial foraging optimization algorithms, and meta-heuristic optimization algorithms [13,15,34]. These stochastic optimization algorithms can be generally more accurate in estimating the Cole impedance parameters than conventional gradient-based algorithms; however, the former are typically more time-consuming and cumbersome to apply.

Magnitude response-only measurement in the frequency-domain has been recently proposed for estimating the Cole impedance parameters [25]. Although these approaches do not need complex impedance data, they usually require solving nonlinear equation(s) based on multi-frequency data [30]. In these methods, the tissue impedance is usually considered as a basic component of a(n) (i) high-pass filter setup, (ii) integrator setup, (iii) or band-pass filter setup. Then, a stepped-sine voltage excitation is applied as the input signal and the peak output voltage is measured at each applied frequency. Finally, the Cole impedance parameters can be estimated via (i) iterative fitting optimization from the whole measured magnitude response spectrum or (ii) numerical solving nonlinear equations at critical frequencies, such as {ω₀, ω₁, ω_ϕmax, ω_3dB} for the high-pass filter setup [6], {ω₀, ω₁, ω_ϕmin, ω_−3dB} for the integrator setup [?], or {ω₀, ω_c, ω_∞}for the band-pass filter setup [25], where ω₀ and ω_∞ represent the very low frequency when ω → 0 and the very high frequency when ω → ∞, respectively; ω_ϕmax and ω_ϕmin represent the frequency at which the phase of the system’s response reaches a maximum or minimum, respectively; ω_3dB and ω_−3dB are the frequency at which the magnitude of the output voltage increases or drops by 3 dB, respectively; and ω_c is the frequency at which the magnitude response reaches a minimum.

Vastarouchas et al. proposed an improved integrator setup, which only requires the magnitude response at {ω₀, ω_−3dB}. Moreover, the maximum measurement frequency is limited to the half-power frequency (ω_−3dB) of the integrator setup, thus the measurement can be conducted by using a low-frequency unity bandwidth op-amp. However, both the magnitude response and the phase must be recorded in this method [30]. The same group of authors recently proposed a new method by embedding an unknown tissue impedance within a Wien-bridge oscillator for estimating the Cole impedance parameters [29]. This approach doesn’t require stepped-sine based magnitude response measurement in a wide bandwidth. However, it still needs (i) to record two oscillation frequencies and their corresponding oscillation start-up gains and (ii) to solve two complex nonlinear equations.

Another alternative to estimate the Cole impedance parameters is time-domain fitting methods based on complete-response measurement. The time-domain techniques could further simplify hardware design without the need of impedance analyzer or magnitude response analyzer. Moreover, they could significantly save the measurement time since no frequency sweep is required [18].

Freeborn et al. proposed some step excitation-based methods in both potentiostatic mode and galvanostatic mode for estimating the Cole impedance parameters [11,10]. Similarly, Kapoulea et al. proposed a periodic triangle excitation-based method, which only requires a simple fixed-frequency measurement procedure to estimate the Cole impedance parameters. The triangle-wave technique has the advantage on selecting the frequency of the applied excitation compared to the step response technique. For example, a 100 Hz or 1 kHz triangle-wave can be applied to estimate the Cole model parameters over the bandwidth of 1 Hz to 100 kHz [18]. However, these existing time-domain methods usually have unsatisfactory estimation accuracy and heavily rely on some specific types of excitations [36]. We recently proposed a new time-domain fitting approach to estimating the Cole impedance parameters based on fractional operational matrix (FOM), which can be easily applied to an arbitrary excitation by using simple matrix operations [36]. However, FOM method has no optimization on the choice of excitation signal, which may reduce the estimation accuracy and overall practical utility.

This short communication introduces a new FOM-based method for estimating the Cole impedance parameters with initial optimization using single-frequency DC-biased sinusoidal current excitation. To facilitate the comparison of the new estimation method with the former methods, a systemic literature survey of the published articles on this topic is summarized in Table 1. It shows that the method presented has two main advantages: (i) the new technique could expand the use of single-frequency devices in electrical bioimpedance applications restricted to multi-frequency devices. (2) it is the only time-domain technique at present that could estimate the Cole impedance parameters through both numerical computation and iterative optimization, in which the numerical method can be applied to provide the initial estimation of the model parameters, thus reducing the possibility of converge prematurely in the fitting procedure. The step-response technique could also be used for numerical analysis, however, only part of the Cole impedance parameters, such as R₀ and R_∞ can be numerically estimated in advance [10].

Table 1.

Literature survey of the published articles for estimating the Cole impedance parameters, where R, X, |Z|, θ represent impedance real part, imaginary part, magnitude and phase, respectively; |H(jω)| represents the modulus of the frequency response function; H(s) is the transfer function; f_s1,s2, A_s1,s2 represent two oscillation frequencies and the gains, respectively.

Measurement domain	Measurement parameters	Circuit setup	Excitation signal	Numerical(N)/Iterative(I)	Reference
Frequency-domain	R, X, |Z|, θ	Impedance measurement	Multi-frequency	I	[32, 13, 15, 34, 31]
	|H (jω)|	High-pass filter	Stepped-sine	N,I	[6]
	|H (jω)|	Band-pass filter	Stepped-sine	N,I	[25]
	|H (jω)|	Integrator	Stepped-sine	N,I	[30,24]
	fs1,s2, As1,s2	Oscillator	Two sine-waves	N	[29,26]
Time-domain	H (s)	Step response measurement	Step wave	I	[11,10,12]
	H (s)	Triangle-wave response measurement	Triangle wave	I	[18]
	H (s)	Arbitrary-wave response measurement	Multisine	I	[36, 3]
	H (s)	Sine response measurement	DC-biased sine	N,I	Here

2. Approximated time-domain response of the Cole impedance model

Fig. 1 shows the measurement diagram for the measurement method proposed, where u(t) and y(t) represent the single-frequency DC-biased input sinusoidal current excitation and the output voltage response (including the output steady-state y_s(t) and transient responses y_t(t)). The input current excitation is defined as

$$u(t)≔\left(I_0\text{ cos}(2\pi ft+\varphi )+I_1\right)(H(t)-H(t-T)),$$ (1)

where I₀ (A) is the amplitude of the sinusoidal signal; f (Hz) is the excitation frequency; ϕ (rad) is the initial phase; I₁ (A) is the amplitude of the DC-bias signal; T (s) is the measurement time and H(t) is the Heaviside function.

Fig. 1.

Measurement diagram for estimating the Cole impedance parameters. The tissue impedance is represented by the Cole impedance model with three components, R₀, R_∞ and a constant phase element (CPE). The signals u(t) and y(t) = y_s(t) + y_t(t) are the current excitation signal and output voltage response signal, respectively, the latter including the steady-state and transient responses. The measurement time for the initial and ending transient responses is in t = [0, T_t) and t = [T, T + T_t], respectively. The measurement time for the steady-state response is in t = [T_t, T).

The fractional Cole impedance model can be represented as [4]

$$Z(s)≔R_{\infty }+\frac{R_0-R_{\infty }}{1+{\left(s{\tau }_0\right)}^{\alpha }},$$ (2)

where s ≔ jω is Laplace variable, $j≔\sqrt{-1}$ (dimensionless) is the imaginary unit; ω (rad s⁻¹) is the angular frequency; R₀ (Ω) and R_∞ (Ω) represent the resistances when ω → 0 and ω → ∞, respectively; τ₀ (s) is the relaxation time constant; α ∈ (0, 1) (dimensionless) is the fractional exponent representing the distribution of the relaxation time due to the heterogeneous shape of different tissue cells.

Next, the current excitation in (1) can be discretized as follows

$$u≔I_0\text{cos}\left(2\pi nf/f_s+\varphi \right)+I_1,$$ (3)

where f_s (Hz) is the sampling frequency, $N\in \mathbb{N}$ is the number of points, and n = [1, 2, 3, .., N].

The Cole impedance model in (2) can be represented as an N × N matrix using the FOM [36]

$$Z\left(F_{\alpha }\right)≔\left(a{F}_{\alpha }+bA\right){\left(F_{\alpha }+cA\right)}^{-1},$$ (4)

where a ≔ R₀, $b≔R_{\infty }{\tau }_0^{\alpha }$, $c≔{\tau }_0^{\alpha }$, A is the identity matrix with size N × N, and F_α is defined as

$$F_{\alpha }≔{\left(\frac{T}{N}\right)}^{\alpha }\frac{1}{\Gamma (\alpha +2)}\left[\begin{array}{ccccc}{\xi }_1& {\xi }_2& {\xi }_3& \cdots & {\xi }_N\\ 0& {\xi }_1& {\xi }_2& \cdots & {\xi }_{N-1}\\ 0& 0& {\xi }_1& \cdots & {\xi }_{N-2}\\ ⋮& ⋮& ⋮& \ddots & \\ 0& 0& 0& \cdots & {\xi }_1\end{array}\right],$$ (5)

with ξ₁ = 1, ξ_p = p^α+1 − 2(p − 1)^α+1 + (p − 2)^α+1 for p = 2, 3, …, N; Γ(·) is the Gamma function.

Finally, the approximated time-domain Cole model-based response can be calculated using (3) and (4) as [36]

$$\tilde{y}=uZ\left(F_{\alpha }\right)=u\left(a{F}_{\alpha }+bA\right){\left(F_{\alpha }+cA\right)}^{-1}.$$ (6)

3. Proposed algorithm

The pseudo-code for estimating the Cole impedance parameters based on the results from Section 2 is shown in Algorithm 1. The Cole impedance parameters are numerically estimated from Step 1 to 4, and then used as initialization values for the non-linear least square time-domain fitting method in Step 5.

Algorithm 1.

Iterative time-domain fitting algorithm for estimating the Cole impedance parameters using single-frequency DC-biased sinusoidal excitation

1: Input: u(t) and y(t), t ϵ [0, T + Tt]

2: Output: ${\widehat{R}}_0$, ${\widehat{R}}_{\infty }$, $\widehat{\alpha }$, ${\widehat{\tau }}_0$

3: Step 1: required calculations; ΔTk is the steady-state interval of time in k excitation periods.

4:  $I_1\leftarrow \frac{1}{\Delta T_k}{\int }^{\Delta T_k}u(t)dt$

5:  $I_u\leftarrow {\int }^{\Delta T_k}u(t)\text{ cos}(\omega t)dt$

6:  $Q_u\leftarrow {\int }^{\Delta T_k}u(t)\text{ sin}(\omega t)dt$

7:  $I_0\leftarrow \frac{2}{\Delta T_k}\sqrt{I_u^2+Q_u^2}$

8:  $y(t){|}_{t=0^{+}}\leftarrow \underset{t\to 0^{+}}{\text{lim}}y(t)$

9:  $I_y\leftarrow {\int }^{\Delta T_k}y(t)\text{ cos}(\omega t)dt$

10:  $Q_y\leftarrow {\int }^{\Delta T_k}y(t)\text{ sin}(\omega t)dt$

11:  $R\leftarrow \left(I_y{I}_u+Q_y{Q}_u\right)/\left(I_u^2+Q_u^2\right)$

12:  $X\leftarrow \left(I_y{Q}_u-Q_y{I}_u\right)/\left(I_u^2+Q_u^2\right)$

13: Step 2: initial estimation of R0 and R∞

14:  $R_{\infty }^{[0]}\leftarrow \frac{y(t){|}_{t=0+}}{I_0\text{ cos}\varphi +I_1}$

15:  $R_0^{[0]}\leftarrow \frac{1}{\Delta T_k{I}_1}{\int }^{\Delta T_k}y(t)dt$

16: Step 3: initial estimation of α

17:  $A\leftarrow \frac{\left(R_0-R_{\infty }\right)\left(R-R_{\infty }\right)}{{\left(R-R_{\infty }\right)}^2+X^2}$

18:  $B\leftarrow \frac{-\left(R_0-R_{\infty }\right)X}{{\left(R-R_{\infty }\right)}^2+X^2}$

19:  ${\alpha }^{[0]}\leftarrow \frac{2}{\pi }\text{ arctan }\left(\frac{B}{A-1}\right)$

20: Step 4: initial estimation of τ0

21:  ${\tau }_0^{[0]}\leftarrow \frac{1}{\omega }\left[{(A-1)}^2+B^2\right]\frac{1}{2{\alpha }^{\left[0\right]}}$

22: Step 5: iterative non-linear least squares fitting

23:  $a^{[0]}\leftarrow R_0^{[0]},b^{[0]}\leftarrow R_{\infty }^{[0]}{\left({\tau }_0^{[0]}\right)}^{{\alpha }^{[0]}},c^{[0]}\leftarrow {\left({\tau }_0^{[0]}\right)}^{{\alpha }^{[0]}}$

24:  ${\theta }^{[0]}\leftarrow \left\{a^{[0]},b^{[0]},c^{[0]},{\alpha }^{[0]}\right\}\text{ and }m\leftarrow 1$

25: while

26:  None stop criteria is satisfied

27: do

28:  $Z^{[m]}\left(F_{\alpha }\right)=\frac{a^{[m-1]}{F}_{\alpha }^{[m-1]}+b^{[m-1]}A}{F_{\alpha }^{[m-1]}+c^{[m-1]}A}$

29:  $\underset{{}_{\theta }}{\text{min}}\Vert y-u{Z}^{[m]}\left(F_{\alpha }\right){\Vert }_2^2$

30:  $m\leftarrow m+1$

31:  $\theta [m-1]\leftarrow \widehat{\theta }$

32: end while

33:  $\left\{\widehat{a},\widehat{b},\widehat{c},\widehat{\alpha }\right\}\leftarrow \widehat{\theta }$

34:  ${\widehat{R}}_0\leftarrow \widehat{a},{\widehat{R}}_{\infty }\leftarrow \widehat{b}/\widehat{c},{\widehat{\tau }}_0\leftarrow {\widehat{c}}^{\frac{1}{\alpha }}$

4. Materials and methods

4.1. Simulation setup

We study the performance of the method presented conducting noisy simulations in MATLAB (The MathWorks, Natick, MA, USA). The true Cole impedance parameters are R₀ = 6306 Ω, R_∞ = 250 Ω, τ = 36 μs and α = 0.741, which represent a set of typical characteristic parameters of a vegetable impedance [6]. Then, u(t) is generated at f = 50 kHz with T = 0.64 ms, T_t = 0.36 ms, I₀ = I₁ = 0.1 mA, and ϕ = −π/2 rad. The approximated output voltage response $\tilde{y}$ is simulated by using (6) with the sampling rate f_s = 15.625 MHz and the total number of samples N = 15625. Additive white Gaussian noise with 0.5% (46 dB), 1% (40 dB), 2% (34 dB), 5% (26 dB) standard deviation is added to u and $\tilde{y}$, where % (dB) is defined as the (logarithmic) amplitude ratio of the noise and the simulated signals [35]. Finally, the Cole impedance parameters are estimated following Algorithm 1. Here, lsqcurvefit function with the ‘trust-region-reflective’ method is chosen in Step 5 with upper and lower bound half and twice the initial value of the Cole impedance parameters, respectively. The stopping criteria are chosen as follows: (i) Maximum number of model evaluation exceeds 6,000; (ii) Maximum number of iterations reaches 400; (iii) Stop tolerance involved in the model value reaches its default value of 10⁻⁶.

For comparison purposes, a stepped-sine excitation is chosen as the reference measurement method to estimate the Cole impedance parameters consisting of f_k = {4, 5, 7, 10, 13, 17, 23, 31, 41, 55, 73, 98, 131, 175, 234, 313, 418, 559, 748, 1000} kHz, with k = [1, F] and F = 20 the number of frequencies measured [22]. The steady-state output response is acquired by setting the sampling rate and the number of samples as 5 MHz and 5,000 samples respectively for all the measured frequencies giving the total measurement time of 20 ms (transient responses are not considered). Then, additive Gaussian noise with the same characteristics as the former simulation is added to the corresponding input and output signals before processing. Next, the impedance magnitude and phase are calculated from the complex division between voltage and current phasors. Finally, the Cole impedance parameters are estimated by using a least squares curve fitting method proposed in [20].

4.2. Experimental setup

We used an integrated instrument setup STEMlab 125–14 (see Fig. 2) from Redpitaya (Velika pot, Solkan, Slovenia) for testing the performance of the proposed method [36]. The measurement system includes an embedded dual-core processor Zynq 7010 (Xilinx, Inc., San Jose, CA, USA), a high-speed arbitrary waveform generator AD9767 (125 Ms s⁻¹, 14 bits, Analog Devices, Inc., Norwood, MA, USA), a two-channel high-speed digitizer LTC2145 (125 Ms s⁻¹, 14 bits, Analog Devices, Inc.), and a custom-designed analog front end. The system can be controlled by using MATLAB software with standard commands for programmable instrumentation.

Fig. 2.

(a) Photo of the experimental setup presented. (b) Functional block diagram of the system for estimating Cole impedance parameters from an unknown impedance Z using single-frequency DC-biased sinusoidal current excitation. In this figure, r(n) and r(t) are the discrete and the time-continuous reference signals, respectively. u(t) is the input signal measured through a reference resistor R_r = 1 kΩ. y(t) is the output signal measured across the biological tissue under test, u(n) and y(n) are the discretized signals in T = 1 ms.

First, the same input reference DC-biased sinusoidal waveform r(n) used in the simulations is stored in STEMlab. Then, r(t) is generated with the amplitude of 0.1 V for both the sinusoidal signal and the DC-bias signal at the generation rate of 15.625 MHz. Therefore, the fundamental frequency and the duration time of the reference signal are set as f = 50 kHz and t = [0, 0.64] ms, respectively. Next, the corresponding time-continuous current signal u(t) is generated with I₀ = I₁ = 0.1 mA, by using a wideband mirrored current source [37]. The current source flows through the impedance Z in series with a precise reference resistor R_r = 1 kΩ (±0.1%, 10 ppm/°C). The differential voltage across Z and R_r is buffered with a dual operational amplifier (AD8066, Analog Devices, Inc.), followed by a differential amplifier (AD8130, Analog Devices, Inc.). Finally, the input and output signals are acquired synchronously to STEMlab with T + T_t = 1 ms, in which T and T_t are chosen as 0.64 ms and 0.36 ms respectively for the following calculation.

Reference complex impedance data are obtained performing stepped-sine measurements (SFB7, ImpediMed Ltd., Brisbane, Australia) from 3 kHz and 1 MHz (256 frequencies). The device has ±1% measurement error from 50 to 1100 Ω. The Cole impedance parameters are then estimated using Bioimp software provided by the manufacturer.

A 2R-1C circuit and some fruits are chosen as the test samples to demonstrate the performance of the approach presented. The former is soldered on a printed circuit board, which is directly connected to the analog front end using DC power plug connectors. The latter includes 4 types of fruits, namely apricots, pears, bananas and tomatoes (3 samples/type), which are all purchased from the same local market. The experimental measurements are conducted using custom cables with 4-pin header on the end as the 4-electrode configuration (Width: 3.8 mm and Depth: 8.0 mm). To reduce the effect of electrode location on the estimated Cole impedance parameters, both the single-frequency and multi-frequency measurements are conducted on the same location of the fruit samples around their latitudinal positions [9].

5. Simulation and experimental results

The simulation results on estimating the Cole impedance parameters using different excitations are given in Table 2. The relative errors of the estimated model parameters increase with the noise as expected and plateau at 3% and 1% using the single-frequency excitation and the stepped-sine excitation, respectively. Moreover, we also compare the simulation results with other published results. For example, the simulation with 1% noise shows that the maximum relative error of the estimated Cole-parameters is only 0.38% performing DC-biased sinusoidal response measurement in 1 ms, which is much smaller than the simulation results with the same noise level performing step-response measurement in 0.1 μs to 10 s (2.85%) [12], or performing complex impedance spectroscopy measurement in 30 Hz to 32 MHz using 30 logarithmically distributed datapoints (2.09%) [7].

Table 2.

Simulation results using single-frequency (SF) DC-biased sinusoidal excitation and reference stepped-sine (SS) excitation for Cole parameter estimation. Mean ± standard deviation of errors in 50 simulations.

Noise % (dB)	Excitation	${\widehat{R}}_0(\Omega )$	${\widehat{R}}_{\infty }(\Omega )$	$\widehat{\alpha }(-)$	${\widehat{\tau }}_0(\mu \text{s})$
0.5 (46)	SF	0.01 ± 1.2e-4	0.21 ± 2.6e-3	0.03 ± 3.6e-4	0.05 ± 5.8e-4
	SS	0.03 ± 3.2e-4	0.02 ± 3.6e-4	0.02 ± 1.8e-4	0.09 ± 1.1e-3
1 (40)	SF	0.03 ± 3.6e-4	0.38 ± 4.5e-3	0.05 ± 6.9e-4	0.10 ± 1.2e-3
	SS	0.06 ± 6.8e-4	0.06 ± 7.5e-4	0.03 ± 3.9e-4	0.19 ± 2.4e-3
2 (34)	SF	0.06 ± 7.2e-4	0.86 ± 9.9e-3	0.13 ± 1.5e-3	0.24 ± 2.9e-3
	SS	0.10 ± 1.2e-3	0.10 ± 1.2e-3	0.06 ± 6.7e-4	0.37 ± 4.5e-3
5 (26)	SF	0.14 ± 1.7e-3	2.41 ± 2.5e-2	0.33 ± 3.9e-3	0.49 ± 6.3e-3
	SS	0.28 ± 3.4e-3	0.23 ± 2.8e-3	0.13 ± 1.6e-3	0.94 ± 1.1e-2

Before the estimation of the Cole impedance parameters of the fruit samples, we have an assumption that these samples could be well explained by the single-dispersion Cole model according to some published research [1]. Here, an example of the impedance Nyquist plots of the test samples acquired from a stepped-sine measurement is illustrated in Figure 3, which shows that all the electrical bioimpedance of the fruit samples can be fitted well with the single-dispersion Cole model from 3 kHz to 1 MHz (The maximum measurement bandwidth of the bioimpedance analyzer SFB7) as expected.

Fig. 3.

Measured (M) and Fitted (F) electrical bioimpedance data of fruit samples based on single-dispersion Cole model. The marked dots represent the experimental results, while the lines represent the fitted models for (a) Apricots, (b) Bananas, (c) Pears, and (d) Tomatoes.

Experimental results measuring the 2R-1C circuit and the fruit samples are provided in Table 3. Of note, the single-frequency method can be used for accurate estimation of the circuit’s impedance parameters achieving less than 1% relative error, and the estimation of the fruit’s impedance parameters less than 5% mean difference compared to the traditional multi-frequency method. Similar estimation error on the fruit samples (3.0%) is reported conducting two-frequency measurement based on oscillator setup [29]. Moreover, the RMSE of the fitting results for all the test samples is less than 1% in both methods, which is better than the RMSE for apples (2.66%) and bananas (2.82%) presented in [16].

Table 3.

Experimental results using single-frequency (SF) DC-biased sinusoidal excitation and the reference stepped-sine (SS) excitation for Cole parameter estimation of a 2R-1C circuit and some fruit samples. Mean ± standard deviation of impedance parameters in 50 consecutive measurements. The reference (REF) values of the 2R-1C circuit are provided for comparison.

Test Sample	Excitation	${\widehat{R}}_0(\Omega )$	${\widehat{R}}_{\infty }(\Omega )$	$\widehat{\alpha }(-)$	${\widehat{\tau }}_0(\mu \text{s})$	RMSE (%)
2R-1C	SF	3254.3 ± 9.1	98.0 ± 1.7	0.99 ± 3.6e-3	99.9 ± 0.2	0.2 ± 8.9e-2
	SS	3276.6 ± 25.6	97.0 ± 0.1	0.99 ± 2.0e-3	99.1 ± 0.9	0.2 ± 4.7e-2
	REF	3221.0	97.0	1	98.7	-
Apricot-1	SF	1367.8 ± 18.2	46.7 ± 0.5	0.85 ± 6.2e-3	24.7 ± 0.8	0.3 ± 7.7e-2
	SS	1332.1 ± 12.0	47.3 ± 0.1	0.82 ± 6.8e-4	23.4 ± 0.2	0.4 ± 1.7e-2
Apricot-2	SF	1187.5 ± 12.8	50.0 ± 0.7	0.77 ± 2.7e-3	21.1 ± 0.5	0.5 ± 5.2e-2
	SS	1138.1 ± 24.0	48.1 ± 0.1	0.82 ± 1.2e-3	20.9 ± 0.5	0.4 ± 1.8e-2
Apricot-3	SF	1447.6 ± 21.3	45.8 ± 0.8	0.76 ± 1.7e-3	28.1 ± 0.6	0.5 ± 7.8e-2
	SS	1461.0 ± 69.2	47.5 ± 0.1	0.81 ± 8.3e-4	30.4 ± 1.4	0.3 ± 1.8e-2
Banana-1	SF	1719.2 ± 58.2	31.0 ± 1.0	0.76 ± 2.7e-3	26.0 ± 0.2	0.5 ± 6.1e-2
	SS	1691.6 ± 11.6	29.3 ± 0.2	0.73 ± 7.1e-4	24.7 ± 0.2	0.5 ± 1.5e-2
Banana-2	SF	1504.6 ± 28.2	23.5 ± 0.7	0.66 ± 2.3e-3	31.4 ± 0.6	0.4 ± 6.9e-2
	SS	1542.2 ± 48.0	21.4 ± 0.2	0.70 ± 1.2e-3	29.1 ± 0.9	0.6 ± 1.4e-2
Banana-3	SF	1911.9 ± 71.5	24.6 ± 3.7	0.73 ± 1.5e-3	24.4 ± 0.2	0.3 ± 8.1e-2
	SS	1832.3 ± 39.3	22.5 ± 0.2	0.70 ± 6.9e-4	26.1 ± 0.7	0.6 ± 1.8e-2
Pear-1	SF	2205.9 ± 51.2	138.7 ± 1.7	0.75 ± 4.5e-3	34.0 ± 0.4	0.6 ± 6.2e-2
	SS	2157.6 ± 5.8	141.0 ± 0.2	0.80 ± 9.7e-4	35.8 ± 0.2	0.2 ± 2.2e-2
Pear-2	SF	3093.1 ± 46.1	127.5 ± 4.1	0.83 ± 6.8e-3	27.7 ± 1.9	0.3 ± 5.9e-2
	SS	2939.0 ± 19.5	131.5 ± 3.7	0.87 ± 1.8e-3	26.9 ± 2.1	0.2 ± 1.8e-2
Pear-3	SF	2201.7 ± 51.6	120.0 ± 2.1	0.83 ± 1.3e-3	28.0 ± 0.9	0.7 ± 3.1e-2
	SS	2072.8 ± 6.0	125.6 ± 0.3	0.80 ± 7.2e-4	27.4 ± 0.8	0.4 ± 2.2e-2
Tomato-1	SF	1428.0 ± 41.0	49.0 ± 0.7	0.83 ± 5.1e-3	63.9 ± 3.5	0.4 ± 3.1e-2
	SS	1506.8 ± 18.7	49.9 ± 0.7	0.81 ± 1.8e-3	60.9 ± 3.0	0.2 ± 3.1e-2
Tomato-2	SF	1515.1 ± 61.3	58.1 ± 1.7	0.85 ± 9.1e-3	54.2 ± 0.8	0.2 ± 7.8e-2
	SS	1495.5 ± 19.1	56.1 ± 2.0	0.82 ± 4.0e-3	55.8 ± 0.1	0.2 ± 3.2e-2
Tomato-3	SF	1070.1 ± 10.1	61.1 ± 0.7	0.82 ± 5.7e-4	49.3 ± 0.4	0.2 ± 3.1e-2
	SS	1076.9 ± 6.5	59.9 ± 0.1	0.81 ± 1.3e-3	48.8 ± 0.6	0.2 ± 3.3e-2

6. Discussion

This brief communication proposes the first time-domain method capable of estimating the Cole impedance parameters using a single-frequency DC-biased sinusoidal current excitation. Both simulations and experiments are performed and the results compared to the existing reference multi-frequency method based on stepped-sine excitation. The results show the estimated Cole impedance parameters is slightly less accurate than the reference method; however, the measurement time is only 1 ms, which is approximately 20 times less than the stepped-sine measurement time considering only 20 frequencies [22]. In fact, the reduction in time will be greater as increasing the number of measured frequencies, such as 256 frequencies in the case of the commercial impedance analyzer.

As expected, averaging several measurements will result in more accurate estimation of the Cole impedance parameters at the expense of increasing measurement time. Of note, the accuracy of the method proposed also depends on the approximation of output voltage response in (6). As a rule of thumb, the choice of the measurement time T should be done carefully so that the transient response of biological tissues can be recorded completely with sufficient sample data N for the fitting algorithm. Moreover, the performance of the measurement system also plays an important role for accurate estimation of the Cole impedance parameters. For example, the wideband electrical bioimpedance measurement should be accurate for steady-state response analysis [17]. The accuracy of our system is estimated in 1 kHz to 1 MHz with loads from 100 Ω to 2 k, showing the error < 1% and 1° for the amplitude and phase measurement, respectively. Moreover, the slew-rate of the amplifiers should be able to capture short transient responses associated to biological tissues [23]. Here, the slew-rate of the chosen amplifiers is larger than 290 V·μs⁻¹, which could record the maximum change of the voltage response up to·18.56 V at the sampling rate 15.625 Ms·s⁻¹. Besides, neither anti-aliasing filters before signal acquisition nor reconstruction filters after signal generation are required, since these filters could significantly change the shape of the input and output signals in transient response analysis.

The method proposed could find use in many electrical bioimpedance applications, including body composition assessment [33]. Currently, there exist two methods for body composition assessment based on single-frequency and multi-frequency excitation. In the single-frequency method, the body impedance is often measured at 50 kHz with sinusoidal excitation [28]. With a single-frequency impedance value, the multi-parameter relationship between body fluids and impedance is then established with statistical regression analysis. However, the rationale of this approach is unclear and studies have shown a lack of consensus since these equations are empirically derived [21]. On the other hand, the multi-frequency method typically measures body impedance at two or more frequencies performing stepped-sine measurements. As a first approximation, the body impedance measured at low (e.g., 5 kHz) and high (e.g., 100 kHz) frequency is used to approximate Cole impedance parameters R₀ and R_∞, respectively. More accurate methods fit the Cole impedance model using CNLS to calculate the parameters R₀ and R_∞. The estimated Cole impedance parameters are then used to predict body fluids based on combining a physiological model and Hanai’s mixture theory [5]. Our novel method could be implemented in single-frequency devices for estimating body composition from the Cole impedance parameters, an approach analogous to more accurate multi-frequency approaches.

7. Conclusion

A new method for estimating the fractional Cole impedance parameters based on time-domain fitting using single-frequency DC-biased current excitation is proposed. Simulation and experimental results are in good agreement with the reference multi-frequency measurement method. The proposed method could be considered as an attractive alternative to the existing methods based on multi-frequency excitation in electrical impedance analysis.