**2. Cole model**

#### **2.1 Introduction**

The frequency dependence of the body impedance, measured for example between the wrist and the ankle, can be understood knowing the behavior of the organic tissue at low frequencies (LF) and at high frequencies (HF). In the LF range (1–70 kHz), the cell membrane capacity has a high impedance value and the electric current flows mainly through ECW (**Figure 1a**). In the HF range (70 kHz–1 MHz), as this impedance decreases, the current flows through both ICW and ECW depending on their relative conductivities and volumes [7] (**Figure 1b**).

At a first glance, this behavior can be modeled with a very simple linear electrical circuit known as the Cole model [2–4] which is shown in **Figure 2**, where:


The AC equivalent resistance of this circuit at zero frequency is *R0* = *Re* and its AC equivalent resistance at infinite frequency is *R*<sup>∞</sup> = (*Ri* � *Re*)/(*Ri* + *Re*).

Approximating *R0* with the measured AC resistance at the minimum angular frequency *ω<sup>m</sup>* and *R*<sup>∞</sup> with the measured AC resistance at the maximum angular frequency *ωM*, the ICW volume *VI* and the ECW volume *VE* are estimated [3] as:

$$\mathbf{V}\_I = \mathbf{k}\_I \cdot \mathbf{W} \mathbf{t} \cdot \left(\mathbf{H} \mathbf{t}^2 / \mathbf{R}\_i\right) \tag{1}$$

**Figure 1.**

*Circuit Models of Bioelectric Impedance DOI: http://dx.doi.org/10.5772/intechopen.91004*

**Figure 2.** *Cole model.*

**83**

*The organic tissue behavior for: (a) LF electrical current flow and (b) HF electrical current flow [7].*

$$V\_E = k\_E \cdot \mathcal{Wt} \cdot \left( Ht^2 / R\_\epsilon \right) \tag{2}$$

where *Ht* is the height, *Wt* is the weight of the subject and *kI* and *kE* are constants that can be determined by the cross validation against other methods [2–3].

#### **2.2 Analysis of measurement results**

Some measurement results for a bioimpedance have been reported in [4] together with a circuit model of the test bench (**Figure 3**). This circuit that includes the Cole model can be used for simulation purposes. In order to estimate the effect

Intracellular water (ICW) can be used to estimate body cell mass (BCM) which is an important indicator of the nutrition status. The evaluation of extracellular water (ECW) is also important to predict changes in fluid distribution for people suffering from wasting diseases, obesity, or patients receiving dialysis [6].

Section 2 describes the well-known Cole model, a linear, first-order *RC* circuit of the human body used for ICW and ECW volume prediction. Section 3 presents a fractional-order impedance whose parameters are identified using the frequency characteristics of the impedance module and can be used for a frequency range of up to three decades. In Section 4, two versions of a ladder *RC* model are presented, one valid for a frequency range of two decades and its extension valid for three decades. Section 5 describes a model consisting of multiple *RC* branches connected in parallel. This model can be viewed as an extension of the Cole model. Finally,

The frequency dependence of the body impedance, measured for example between the wrist and the ankle, can be understood knowing the behavior of the organic tissue at low frequencies (LF) and at high frequencies (HF). In the LF range (1–70 kHz), the cell membrane capacity has a high impedance value and the electric current flows mainly through ECW (**Figure 1a**). In the HF range (70 kHz–1 MHz), as this impedance decreases, the current flows through both ICW and ECW depending on their relative conductivities and volumes [7] (**Figure 1b**). At a first glance, this behavior can be modeled with a very simple linear electrical circuit known as the Cole model [2–4] which is shown in **Figure 2**, where:

The AC equivalent resistance of this circuit at zero frequency is *R0* = *Re* and its

Approximating *R0* with the measured AC resistance at the minimum angular frequency *ω<sup>m</sup>* and *R*<sup>∞</sup> with the measured AC resistance at the maximum angular frequency *ωM*, the ICW volume *VI* and the ECW volume *VE* are estimated [3] as:

*VI* <sup>¼</sup> *kI* � *Wt* � *Ht*<sup>2</sup>

*VE* <sup>¼</sup> *kE* � *Wt* � *Ht*<sup>2</sup>

Some measurement results for a bioimpedance have been reported in [4] together with a circuit model of the test bench (**Figure 3**). This circuit that includes the Cole model can be used for simulation purposes. In order to estimate the effect

that can be determined by the cross validation against other methods [2–3].

where *Ht* is the height, *Wt* is the weight of the subject and *kI* and *kE* are constants

*=Ri*

*=Re*

(1)

(2)

AC equivalent resistance at infinite frequency is *R*<sup>∞</sup> = (*Ri* � *Re*)/(*Ri* + *Re*).

• *Ri* stands for the resistance of the intracellular fluid;

• *Re* stands for the resistance of the extracellular fluid.

• *Cm* is the capacity of the cellular membrane; and

**2.2 Analysis of measurement results**

**82**

some conclusions are presented in Section 6.

*Electrochemical Impedance Spectroscopy*

**2. Cole model**

**2.1 Introduction**

**Figure 1.** *The organic tissue behavior for: (a) LF electrical current flow and (b) HF electrical current flow [7].*

**Figure 2.** *Cole model.*

**Figure 4.** *The Cole model and the simplified circuit (Model 2).*

of the measurement equipment (signal source, cables and connectors), the circuit in **Figure 3** has been simulated. Similar results are obtained by simulating a simpler circuit made only of the Cole model and the signal source (**Figure 4**). The frequency characteristics of the circuit which takes into account the measurement equipment (Model 1) and of the Cole model only (Model 2) are given in **Figure 5**. It follows that the measurement equipment has practically no influence and the result of the measurements is exactly the frequency characteristic of the human body bioimpedance [8].

frequencies. As the resistance values at the minimum and maximum frequencies are used for ECW and ICW volume estimation in Eqs. (1) and (2), we have chosen these three frequencies as *ω<sup>1</sup>* = *ωm*, *ω<sup>2</sup>* = *ωM*, and *ω<sup>3</sup>* corresponding to the intersection points between the measured characteristic and that of the Cole model

**Frequency [kHz] Measured impedance [ohm]**

5.62E+02 5.58E+02 5.54E+02 5.50E+02 5.46E+02 5.29E+02 5.13E+02 5.00E+02 4.88E+02 4.52E+02 4.30E+02 4.18E+02 4.08E+02 4.02E+02 4.00E+02 3.92E+02 3.85E+02 3.66E+02 3.58E+02 3.53E+02

(**Figure 6**). In this case, it is obvious that the results obtained with this Cole model are not fitted to the measured data. This proves that a more accurate model is necessary. Identifying the parameters of this Cole model in a different way, that is, using three frequencies in the middle of the frequency interval, significant errors appear at the minimum and maximum frequencies. It follows that the Cole model is

not suitable for ECW and ICW computation [8].

**Figure 5.**

**Table 1.**

**85**

*The measured human body impedance.*

1.00E+03 2.00E+03 3.00E+03 4.00E+03 5.00E+03 1.00E+04 1.50E+04 2.00E+04 2.50E+04 5.00E+04 7.50E+04 1.00E+05 1.28E+05 1.48E+05 1.60E+05 2.00E+05 2.48E+05 5.00E+05 7.48E+05 1.00E+06

*Simulation results for Model 1 and Model 2.*

*Circuit Models of Bioelectric Impedance DOI: http://dx.doi.org/10.5772/intechopen.91004*

The measured values of the human body impedance as a function of frequency are given in **Table 1** [3].

## **2.3 Parameter identification**

The parameters of the Cole model with the frequency independent values for *Ri, Cm* and *Re* can be identified using the measured impedance values for three

*Circuit Models of Bioelectric Impedance DOI: http://dx.doi.org/10.5772/intechopen.91004*

#### **Figure 5.**

*Simulation results for Model 1 and Model 2.*


#### **Table 1.**

of the measurement equipment (signal source, cables and connectors), the circuit in **Figure 3** has been simulated. Similar results are obtained by simulating a simpler circuit made only of the Cole model and the signal source (**Figure 4**). The frequency characteristics of the circuit which takes into account the measurement equipment (Model 1) and of the Cole model only (Model 2) are given in **Figure 5**. It follows that the measurement equipment has practically no influence and the result of the measurements is exactly the frequency characteristic of the human body

*The Cole model and the equivalent circuit of the measurement equipment (Model 1).*

The measured values of the human body impedance as a function of frequency

The parameters of the Cole model with the frequency independent values for *Ri,*

*Cm* and *Re* can be identified using the measured impedance values for three

bioimpedance [8].

**84**

**Figure 3.**

**Figure 4.**

are given in **Table 1** [3].

**2.3 Parameter identification**

*The Cole model and the simplified circuit (Model 2).*

*Electrochemical Impedance Spectroscopy*

*The measured human body impedance.*

frequencies. As the resistance values at the minimum and maximum frequencies are used for ECW and ICW volume estimation in Eqs. (1) and (2), we have chosen these three frequencies as *ω<sup>1</sup>* = *ωm*, *ω<sup>2</sup>* = *ωM*, and *ω<sup>3</sup>* corresponding to the intersection points between the measured characteristic and that of the Cole model (**Figure 6**). In this case, it is obvious that the results obtained with this Cole model are not fitted to the measured data. This proves that a more accurate model is necessary. Identifying the parameters of this Cole model in a different way, that is, using three frequencies in the middle of the frequency interval, significant errors appear at the minimum and maximum frequencies. It follows that the Cole model is not suitable for ECW and ICW computation [8].

Over 500 kHz, the time delay between the excitation and its response cannot be

As it was shown in the previous section, the frequency characteristic of the Cole

where *ω* is the angular frequency,*Td* is the delay and *α* ∈½ � 0*:*3, 0*:*7 is a coefficient whose value is chosen to fit the values given by Eq. (3) to the |*ZRC*(*jω*)| experimental data. As this formula does not lead to a single valued function *arg*(*ZRC*(*jω*)), the identification of its parameters based on the measured frequency characteristics |*ZRC*(*jω*)| and *arg*(*ZRC*( *jω*)) cannot be made. The measurements of |*ZRC*(*jω*)| and *arg*(*ZRC*(*jω*)) suggest a circuit like that in **Figure 2** having frequency dependent components, rather than this formula that has been obtained starting from the equivalent impedance of the linear circuit in **Figure 2** in which the power *α* < 1 is attached to one term while other terms remain unchanged. Taking into account that the frequency dependence of material parameters is not known for all kinds of tissues, the development of an accurate physical model is very difficult or even impossible [10]. The parameter identification for Eq. (3) is made in [3] (ignoring the uncertainty on phase) starting from measured frequency characteristics |*ZRC*(*jω*)|. A very good fitting of the model characteristics to the experimental data is obtained in this case which describes de Lorenzo model. But the real parts of the body impedance in Eq. (3) at *ω<sup>m</sup>* and *ω<sup>M</sup>* have not the dimension of AC resistances, so the formulae Eq. (1) and Eq. (2) aimed to be employed with the Cole model AC resistances

The parameters of a model with a given structure are extracted or identified using optimization methods. In general, these methods minimize the distance between the measurement results and those obtained by simulation. In the case of semiconductor devices, the parameters of the large signal DC models or those of the small signal AC models are usually extracted using numerical techniques. Some symbolic methods have been used efficiently for parameter identification [11–13]. The circuit functions are generated using a symbolic method, obtaining analytical formulae in terms of *s* and model parameters. These parameters are computed using an optimization method to reach a global minimum of the distance between the measured and simulated values of the circuit functions for a set of test frequencies. The symbolic methods are very efficient for the computation of derivatives which

are usually needed in the optimization procedure. The optimization can be performed using genetic algorithms [14]. Sometimes, hierarchical techniques are

The body impedance analysis method has not yet reached its full potential. Following the trend to improve the method by increasing the level of model accuracy, a new approach to the parameter identification for a linear *RC* model in

employed to obtain combined DC-AC models [15].

*Re* <sup>1</sup> <sup>þ</sup> *<sup>j</sup>ωCm*ð Þ *Re* <sup>þ</sup> *Ri* ½ �*<sup>α</sup> <sup>e</sup>*

�*jωTd* (3)

neglected [3]. In this case, a model with distributed parameters could be more

model with frequency independent values *Ri*, *Cm* and *Re* cannot be fitted to the measured data on a three-decade frequency range. In order to fix this drawback, a modified Cole model has been proposed in [3] in which the body impedance is

accurate.

considered as:

cannot be used properly.

**4.** *RC* **ladder model**

**4.1 Introduction**

**87**

*ZRC*ð Þ¼ *<sup>j</sup><sup>ω</sup> Re*

*Circuit Models of Bioelectric Impedance DOI: http://dx.doi.org/10.5772/intechopen.91004*

> *Re* þ *Ri Ri* <sup>þ</sup>

**Figure 6.** *The simulated Cole model and the measured results.*
