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.

Keywords: Cole parameters estimation; fractional impedance model; sinusoidal response; time-domain fitting.

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).

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.

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.