A New Technique to Estimate the Cole Model for Bio-impedance Spectroscopy with the High-Frequency Characteristics Estimation

Abstract

Bio-impedance spectroscopy (BIS) is a sophisticated testing technique used to analyze impedance changes at different frequencies. In this study, we investigated the estimation of the Cole Model for BIS measurements without the need for high-frequency resistance and reactance measurements, where they are inaccurate due to leakage capacitences. We employed a Texas Instruments evaluation kit (AFE4300) and compared the Cole plots of two different circuit models of tissue between the proposed configuration and a commercial impedance analyzer used as a reference. To enhance the performance of the AFE4300, we incorporated an external direct digital synthesis (DDS) to generate higher frequencies. The results demonstrated the reliability of the proposed theoretical estimation technique in accurately estimating the resistances and capacitance of the Cole Model.

I. Introduction

Bio-impedance spectroscopy (BIS) aims to assess the impedance of biological tissue at various frequencies. It involves passing weak alternating currents through the tissue and measuring the impedance magnitude and phase difference at each frequency [1]. The fundamental measurement system of BIS involves establishing the relationship between the current and voltage of an object. To measure tissue impedance, a fixed amount of current is injected, and the resulting voltage drop across the biological sample is measured. [1]–[3]. However, the accuracy of BIS measurements at high frequencies (above 200 kHz) is compromised due to distortions caused by parasitic capacitance from the body. This leads to a decrease in the quality of the measurement results [2]. To ensure reliable BIS analysis, it is crucial to process the measurements, as well as the type of error present.

A noteworthy study described in [4], involved the development of a BIS system that utilized a gain phase detector (GPD) chip from Analog Devices. Previous studies have shown that most systems were capable of covering a frequency range of 1 kHz to 100 kHz. Some systems even extended their coverage to higher frequencies, up to 1 MHz, using techniques such as fast Fourier transform (FFT) for impedance calculation or the under-sampling technique. More recent works, such as the one mentioned in paper [2], have observed the use of IQ demodulation in BIS systems, which allows measurements within a frequency range of 4 kHz to 120 kHz. In addition, classification techniques provided by [5] introduced the idea of processing data at high frequencies to achieve greater accuracy. The intention behind employing multiple frequencies of injected current, ranging from 1 kHz to 1 MHz, is to obtain a single Cole plot. This approach enables the measurement of impedance and phase across both very low and very high frequencies, facilitating the observation of bio-impedance values and estimation of tissue characteristics. These measurements can indicate the presence of abnormal fluid accumulation.

In this study, we utilized the AFE4300 model evaluation board from Texas Instruments, USA, which is capable of measuring and performing BIS. The BIS was examined using different circuit models of tissue simultaneously. In each test, an alternating current of 1 mA was passed through the outer pair of electrodes positioned on either side of the model, while the complex impedance was measured from the inner electrode pair. The primary goal of this research is to evaluate the performance of the evaluation board beyond its specified peak frequencies, which correspond to the maximum point of reactance, as stated in its circuit limitations. These limitations are set at 128 kHz in I∕Q mode. To achieve this, we employed an external Direct Digital Synthesis (DDS) system to generate higher frequencies (200 kHz - 250 kHz). By determining the peak frequency of the Cole plot for each type of circuit model of bio-impedance, we identified the maximum point that could be used to estimate the remaining portion of the Cole plot at very high frequencies (up to 1 MHz). This approach was necessary because the results obtained at such high frequencies were deemed unreliable. Eventually, we can evaluate the approach for different environments [6]. This paper is organized as follows: Section II describes the experimental setups and implementation; Section III presents the results obtained from this system; Section IV outlines the main hypothesis; and Section V concludes the study.

II. Methods

A. AFE4300 Circuit Function

The AFE4300 is utilized for the measurement and estimation of the bio-impedance model within the body. The internal analog-to-digital converter (ADC) of this board provides a digital code, which offers a comprehensive real-time view of measurements across different frequencies. The BIS measurement process involves injecting a sinusoidal current into the body, enabling by an internal digital block that consists of a DDS and a low-pass filter, allowing operation in different frequency modes. Ultimately, the direct current (DC) component of the injected current and the resulting voltage across two other terminals are used to determine the body’simpedance [7].

The AFE4300 offers two key features: the ability to connect up to four different load cells and tetra-polar measurements using I/Q measurements. The I/Q demodulator generates two DC values based on the voltage signal. By using a single frequency measurement, these two values are utilized to separate the impedance magnitude from the phase. These equations establish a clear relationship between impedance, phase, and the I/Q codes [8]: $\theta ={\mathit{\text{tan}}}^{-1}\left(\frac{Q_{DC}}{I_{DC}}\right)$ and $Z=\frac{\sqrt{I_{DC}^2+Q_{DC}^2}}{K}$. Also, the device provides additional terminals for driving and current return on one side, and extra terminals on the receive/differential amplifier side. These connections address system non-idealities, compute K, and support up to four external calibration impedances.

B. Bio-impedance Model Analysis

The human body can be represented as a passive parameter model that impedes the flow of electric current, categorizing total body water into two types of fluids: intracellular water (ICW) within the cells and extracellular water (ECW) outside the cells. Cell membranes act as capacitors, allowing an applied sinusoidal current to primarily flow through the ECW at lower frequencies. However, both ICW and ECW are capable of conducting an applied sinusoidal current at higher frequencies [9]. Therefore, as illustrated in Fig. 1(a), the capacitance C_m acts as an open circuit at low frequencies and a short circuit at high frequencies. As a result, the impedance can be approximated as R_e at low frequencies and as R_e ∥ R_i at high frequencies [1].

Fig. 1.

(a) Proposed Circuit Model Representing the Tissue Model (b) Replacement of an External DDS in AFE4300 Evaluation Board

Furthermore, the Cole model demonstrates consistency with measured bio-impedance data. In this model, the capacitor in the three-element model is represented as a constant phase element (CPE) [10], [11]. The phase of the CPE is assumed to be frequency-independent and is determined by the alpha term, as defined in reference [2]. While all the parameters in this model have physiological significance, the alpha term does not [2]. Nonetheless, the following is the definition of the Cole model in question: $Z=R_{\infty }+\frac{R_0-R_{\infty }}{1+{(j\omega \tau )}^{\alpha }}$. Where R_∞ is equal to R_e ∥ R_i, R₀ is equal to R_e, τ is the time constant which is (R_e + R_i)C_m and ω is related to the frequency, and α is the constant term for the phase concept which was discussed about.

C. Estimation Technique

To describe the estimation process in this work, an external DDS system is employed in the AFE4300 circuit, as shown in Fig. 1(b)). This alteration is implemented by disconnecting the external filter that intercepts the internal DDS and substituting it with a function generator (Keysight Technologies 33210A). This configuration enables manual frequency adjustment from outside the circuit, encompassing a broader frequency range. This option was implemented as a replacement when frequency adjustments are needed, specifically for peak frequencies in the curved Cole plots. In other words, these adjustments are necessary to estimate the expected model, as the desired bio-impedance models typically have peak frequencies exceeding 200 kHz. Therefore, the flexibility of the I/Q mode of this evaluation board was increased to accommodate this requirement.

Based on the circuit specifications, an external capacitor is placed in series with the circuit to block DC current after 150 kHz. Additionally, a second-order filter is used to filter the output of the DAC, removing high-frequency images and preventing DC current from entering the body. The output of the filter is then connected to a resistor, which controls the amount of current injected into the body. Although these filters are designed to prevent unwanted interference in various situations, this part of the device can be modified to take advantage of frequency gains. To collect impedance data with adjustable frequencies, we will apply a sinusoidal current signal with a DC offset of 1 mA and a peak-to-peak value of 700 μA, ranging from 1 kHz to beyond 200 kHz. By collecting these data points, we can make a Cole plot for and estimate the resistance value. The demodulation’s I and Q codes correspond to the real and imaginary parts of the impedance, respectively.

One of the major challenges in this study is dealing with the effects of parasitic output capacitors in the circuits used, which have been identified as problematic in previous studies. These capacitors introduce uncertainty and result in inappropriate output behavior when operating at frequencies above 200 kHz. A proposed solution is to derive a model for low frequencies, which would eliminate the need to measure impedance values at high frequencies. In Fig.1(a), the model comprises three elements, specifically resistor and capacitor parameters. By obtaining and estimating these three variables through different equations, we can address this challenge effectively, as follows [2]:

$${\theta }_{\mathit{\text{peak}}}={\mathit{\text{tan}}}^{-1}\left(\frac{R_e}{2\sqrt{R_i\left(R_e+R_i\right)}}\right);f_{\mathit{\text{peak}}}=\frac{1}{2\pi C_m}\sqrt{\frac{1}{R_i\left(R_e+R_i\right)}}$$

The evaluation board was tested using two different circuit models (implemented parameters of model 1 and model 2 in Fig.2. In both tested models, the highest phase value was consistently observed within the frequency range of 200 kHz to 250 kHz. This means that when the peak point of the plot begins to decrease, we can extract the calibrated phase and determine its corresponding frequency. Additionally, with knowledge of the resistance value (R_e) at low frequencies and the provided equations, we can calculate the remaining parameters of the model and make predictions.

Fig. 2.

(a) Cole Plot of Three Different Circuit Models, Recorded by Reference Impedance Analyzer Device (Showing the Variations in the range of 1 kHz - 1 MHz) (b) Variations in the Real Part of the Impedance for Three Circuit Models

The impedance analyzer (Zurich instruments, Switzerland, 2019) provided reference impedance values that matched the I and Q codes from the evaluation board up to a frequency of 275 kHz, as depicted in Fig 2. This figure demonstrates that the obtained Cole plot, based on the reference instrument, exhibits a linear variation at each frequency point. This indicates that with calibration using a highly accurate 1 kΩ resistor available in the instrument’s package, we can easily establish a linear relationship between the obtained reference database and the output of any random model at low frequencies. This allows us to determine the value of R_e. Once we determine the amount of extracellular water (ECW) at very low frequencies by comparing it with the reference Cole plot and obtain the peak frequency and peak phase through repeated measurements of this circuit, we can input them to calculate ICW or R_i. Then, we can determine C_m. Thus, we can estimate the entire model without needing to continue the process to higher frequencies. Finally, this estimated model allows us to refer to our reference instrument and compare the error rate with it. This comparison offers validate the accuracy of our estimation and enables us to provide the best-fitted Cole plot for verifying the accuracy of our estimation.

III. Result

Two different circuit models were used to conduct testing on the evaluation board (particularly model 1 and model 2). Both models included a constant capacitor of 1 nF and had varying extracellular and intracellular resistances of 560 Ω, and 1 KΩ, respectively (as depicted in Fig 1(a)). While our evaluation board covers a frequency range from 1 kHz and 1 MHz, the specific frequency ranges do not align with the reference values, as illustrated in Fig 3(c) & (d). These discrepancies indicate deviations from the original Cole plot, which must be disregarded during calibration. However, through testing two different models, we observed that these unreliable frequency ranges were consistently repeated (as evident from the similarity in error rates observed in Fig 4). Consequently, we can extend these observed discrepancies to all models through analysis.

Fig. 3.

(a) Cole - Cole Plot Based on the Digital Output Codes of AFE4300 Board in frequency range of 1 kHz - 275 kHz (b) Cole - Cole Plot Based on the Digital Output Codes of AFE4300 Board in frequency range of 1 kHz - 485 kHz (c) Unreliable frequency ranges based on the variation of I code of evaluation board (d) Unreliable frequency ranges based on the variation of Q code of evaluation board (e) Calibration of the I code and (f) Calibration of the Q code Based on the available Reference Values

Fig. 4.

(a) Illustration of the error rate in Model 1 (b) Illustration of the error rate in Model 2

From Fig. 3(a) and (b), it was observed that the resistance and reactance values obtained as digital codes aligned well with the reference values up to a frequency of 275 kHz, displaying the expected Cole plot curve. However, as the frequency was increased to approximately 500 kHz, distortion occurred, deviating from the reference values. Therefore, based on Fig. 3(e) and (f), we identified and removed the unreliable frequency range, allowing us to calibrate the evaluation board and showcase the circuit’s accurate performance. The calibration process focused on determining the first and last values of the frequency range, ensuring reliable and precise results.

The evaluation board was calibrated by comparing it to the impedance analyzer, resulting in the derivation of equations for the calibrated resistance and reactance values as follows:

$$\Re (\Omega )=0.1597\times (\mathit{\text{ICode}})-723.42$$ (1)

$$\Im (\Omega )=\frac{{(Q\mathit{\text{ Code}})}^2}{5\times {10}^4}-\frac{Q\mathit{\text{ Code}}}{8}-318.5$$ (2)

where ℜ(Ω) and ℑ(Ω) represent the real and imaginary parts of the impedance, respectively. By substituting the I and Q codes into this equation, the impedance values can be accurately estimated. Subsequently, the obtained codes from the evaluation kit should be inserted into the regression line equations to determine the actual values of impedances and phases. These values can then be displayed in accordance with Fig. 5(a).

Fig. 5.

(a) Variations of the calibrated Impedance and phase in Model 1 (b) Showing the Peak frequency based on the Final Calibrated Cole-Cole Plot

For model 1, the test measurements demonstrated successful estimation of the resistance and reactance values without the need for frequencies above 200 kHz. Analysis revealed that the maximum phase value reached 19.43° at a frequency of 96 kHz, as shown in Fig. 5(b) after the calibration process. Additionally, based on the recorded data, the impedance value at very low frequencies was approximately 1020 Ω, closely matching the extracellular resistance of the cell, which is consistent with the actual model value of 1 KΩ. Subsequently, by inserting the phase and R_e values, the intracellular resistance of the cell was determined, yielding another 1 KΩ, aligning with the model’s specifications. Furthermore, by inputting the obtained peak frequency and the resistance values, the capacitance value was calculated to be approximately 1.1 nF. Consequently, based on these assumptions, it was observed that model 1 was accurately estimated, even without considering high frequencies in this study. This process is repeated in the same manner for models 2 and 3. Based on the results we obtained, which exhibit an error of 4.46% for model 2 and 2.27% for model 3, the resistance values for models 2 and 3 are determined as 585 Ω and 225 Ω, respectively, by substituting the values into the aforementioned formulas.

IV. Discussion

The findings demonstrate that by identifying the frequency point of maximum reactance, typically ranging from 90 kHz to 200 kHz for various models, as recorded by the reference device, the entire model can be estimated. Consequently, it becomes possible to anticipate the remaining Cole plot by utilizing the reference evidence available for a specific model. Fig. 6 illustrates the estimated Cole plot within the frequency range of 1 kHz to 1 MHz, obtained by comparing it with the reference data from the Impedance analyzer. This comparison involves obtaining the passive parameters of the model theoretically based on the described peak frequency process and comparing it with the reference plot of the same model to determine the mean error value. Subsequently, the remaining points can be estimated based on this error, as depicted in Fig. 6. As a result, the need for impedance measurements at high frequencies may no longer be necessary, allowing for a focus on the circuit’s functionality within the 1 kHz to 200 kHz range.

Fig. 6.

Estimated Cole Plot for f Range of 1 kHz to 1 MHz in Model 1

By and large, this study introduces a proposed method for estimating the Cole plot and detecting measurement errors in Bio-impedance Spectroscopy (BIS). The method was developed using a comprehensive database that includes measurements from various models obtained from the reference instrument. Statistical estimation was employed to derive and illustrate Fig. 6, resembling a Cole plot, through a theoretical process. This method involved comparing each test point against the reference graph, and subsequent calibration was performed. During these stages, a minor distortion was identified within the frequency ranges of the property. Notably, the error rate calculated highlighted significant discrepancies at certain high frequencies. This distortion was anticipated due to predictable factors. Then, specific data points responsible for the distortion were omitted from the final results. The ultimate Cole plot was then constructed using the remaining reliable data points. The outcome presented in Fig. 6 exhibited an estimate remarkably close to the ideal model. However, the removal of unreliable points might introduce some constraints, owing to the limitations intrinsic to the evaluation board. Furthermore, the experimentation in this study did not encompass human testing on real-world participants. We intend to validate our technique using human subjects in future research endeavors. Incorporating outcomes from such real-sample testing could facilitate a comparative analysis between the results of circuit models incorporating passive components.

V. Conclusion

This study presents a novel technique for estimating the parameters of the Cole model in Bioimpedance Spectroscopy (BIS) systems without the need for impedance measurements at high frequencies. By modifying an evaluation kit and employing statistical estimation, the proposed technique achieves accurate estimation of the Cole plot characteristics. This advancement enhances the usability and effectiveness of BIS systems for body fluid analysis, offering a more efficient and accessible approach to bio-impedance measurements.