**5.** *RC* **parallel model**

#### **5.1 Introduction**

**4.4 Parameter identification for the three-decade model**

model and the measured results is given in **Figure 12**.

*<sup>s</sup>* <sup>þ</sup> <sup>1</sup> <sup>13</sup>*:*<sup>04</sup> � <sup>10</sup>�<sup>6</sup>*<sup>s</sup>* <sup>þ</sup> <sup>1</sup> <sup>43</sup>*:*<sup>11</sup> � <sup>10</sup>�<sup>7</sup>

*<sup>s</sup>* <sup>þ</sup> <sup>1</sup> <sup>11</sup>*:*<sup>55</sup> � <sup>10</sup>�<sup>6</sup>*<sup>s</sup>* <sup>þ</sup> <sup>1</sup> <sup>38</sup>*:*<sup>08</sup> � <sup>10</sup>�<sup>7</sup>

109*:*02 þ

*YRC*ðÞ¼ *s*

**Figure 10.**

**Figure 9.**

*Cauer synthesis of the* RC *ladder circuit model [8].*

*Electrochemical Impedance Spectroscopy*

*YRC*ðÞ¼ *s* <sup>19</sup>*:*<sup>47</sup> � <sup>10</sup>�<sup>11</sup>*<sup>s</sup>* <sup>þ</sup>

**90**

<sup>17</sup>*:*<sup>81</sup> � <sup>10</sup>�<sup>4</sup> <sup>15</sup>*:*<sup>92</sup> � <sup>10</sup>�<sup>5</sup>

<sup>15</sup>*:*<sup>91</sup> � <sup>10</sup>�<sup>5</sup>

<sup>23</sup>*:*<sup>94</sup> � <sup>10</sup>�<sup>9</sup>*<sup>s</sup>* <sup>þ</sup>

392*:*53 þ

Using the same algorithm presented above applied for the measurement set of

*The frequency characteristic for the Cole model, the two-decade* RC *ladder model and the measured results.*

*<sup>s</sup>* <sup>þ</sup> <sup>1</sup> <sup>18</sup>*:*<sup>59</sup> � <sup>10</sup>�<sup>7</sup>

1

<sup>10</sup>*:*<sup>71</sup> � <sup>10</sup>�<sup>8</sup>*<sup>s</sup>* <sup>þ</sup>

C5 = 16.99 mF, R5 = 106.80 Ω, C4 = 500.71 nF, R4 = 17.39 Ω, C3 = 107.12 nF,

R3 = 42.54 Ω, C2 = 23.94 nF, R2 = 109.02 Ω, C1 = 194.77 pF, R1 = 392.53 Ω.

1

42*:*54 þ

*<sup>s</sup>* <sup>þ</sup> <sup>1</sup> <sup>16</sup>*:*<sup>47</sup> � <sup>10</sup>�<sup>7</sup>

1

<sup>50</sup>*:*<sup>07</sup> � <sup>10</sup>�<sup>8</sup>*<sup>s</sup>* <sup>þ</sup>

1

*<sup>s</sup>* <sup>þ</sup> <sup>1</sup> <sup>75</sup>*:*<sup>82</sup> � <sup>10</sup>�<sup>9</sup>*<sup>s</sup>* <sup>þ</sup> <sup>1</sup>

(6)

*<sup>s</sup>* <sup>þ</sup> <sup>1</sup>

1

17*:*39 þ

1

1

1 <sup>16</sup>*:*<sup>95</sup> � <sup>10</sup>�<sup>3</sup>

*s* þ 106*:*8 (7)

three decades, a more elaborated admittance expression, given in Eq. (6), is obtained [18]. The Cauer synthesis starting from the continued fraction expansion in Eq. (7) gives the circuit presented in **Figure 11**. The parameter values are given above the model. A comparison between the frequency characteristic of the new

A behavioral model, as a linear circuit which can be an extension of the Cole model is the best choice, taking into account that the intracellular and the extracellular water volumes are related to the real part of the model impedance computed at minimum and maximum frequencies [3], this impedance being well defined only for a model of this kind.

An *RC* parallel model, valid for a frequency range of three decades, which can be reduced to the Cole model for a narrow frequency interval, is presented in this section.

### **5.2 Parameter identification for the** *RC* **parallel model**

For the parameter identification of the *RC* parallel model, only the measured frequency characteristic |*YRC*(*jω*)| is used [10]. In order to build this model, the approximation method is employed, followed by the circuit synthesis as it is described in the previous section.

Using the above algorithm, the frequency characteristic |*YRC*(*jω*)| corresponding to the data in [3] has been approximated by the admittance in Eq. (6) with an error *ε* = 0.95 dB using a sweeping step Δ*ω*<sup>m</sup> *=* 8315 Hz. The synthesis of this admittance can be made by the Foster II method which gives the most interesting circuit in **Figure 13**.

The direct employment of the Foster II synthesis algorithm starting from Eq. (6) leads to some negative parameter values. This effect can be avoided performing the Foster II synthesis of |*YLC*(*jω*)|, where *YLC*(*s*) in Eq. (9) is given by the frequency transformation in Eq. (8) [17].

$$Y\_{LC}(\mathbf{s}) = \frac{1}{s} \cdot Y\_{RC}(\mathbf{s}^2) \tag{8}$$

$$\begin{split} Y\_{LC}(s) &= \mathbf{19.47} \cdot \mathbf{10^{-11}s} + \frac{\mathbf{17.81 \cdot 10^4}}{s} + \frac{\mathbf{24.14 \cdot 10^{-10}s}}{\mathbf{11.54 \cdot 10^{-6}s^2} + 1} + \\ &+ \frac{\mathbf{94.53 \cdot 10^{-11}s}}{\mathbf{38.08 \cdot 10^{-7}s^2} + \mathbf{1}} + \frac{\mathbf{50.94 \cdot 10^{-11}s}}{\mathbf{16.47 \cdot 10^{-7}s^2} + \mathbf{1}} + \frac{\mathbf{59.04 \cdot 10^{-13}s}}{\mathbf{15.91 \cdot 10^{-5}s^2} + \mathbf{1}} \end{split} \tag{9}$$

Starting from the partial fraction decomposition in Eq. (9), the parameter values are: C5 = 5.9 pF, R5 = 27 MΩ, C4 = 2.41 nF, R4 = 4.78 kΩ, C3 = 0.945 nF, R3 = 4.03 kΩ, C2 = 0.51 nF, R2 = 3.23 kΩ, C1 = 0.195 nF, R1 = 561.5 Ω.

The resistance corresponding to the volume of the extracellular water can be computed for *fmin* = 1 kHz and has a 560.97 Ω value, which is practically the same with *RE* = 562 Ω given by the Cole model.

The resistance corresponding to the volume of the intracellular water can be computed for *fmax* = 1000 kHz and has a 314.97 Ω value, unlike *Ri* = 352.69 Ω given by the Cole model. Due to the better agreement with experimental data, it is expected that the body water volume prediction will be improved considering these values in Eqs. (1) and (2).

It is very interesting to observe that R1 has a similar value to *Re* in the Cole model, being the equivalent resistance for *f* = 0 Hz. This circuit can be viewed as a generalization of the Cole model. The two branch models contain R1, R2, C2. As the frequency range of interest is extended to higher frequencies, a model with a greater number of branches is needed. The simulated data obtained with models with various numbers of branches, obtained by imposing the same error *ε* on various frequency intervals are given in **Figure 14**.

A similar circuit (**Figure 15**) is given in [3] without pointing out how the resistance and capacitance values can be computed starting from the measured data.

In order to appreciate the agreement between the measured and simulated data, the measuring errors must be known. Unfortunately, no information on these errors is given in [3].

**6. Conclusions**

*The extended Cole model [3].*

**Figure 15.**

**93**

**Figure 14.**

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

decade frequency range.

importance, cannot be used.

volumes have been presented in this chapter.

Four models of the human body bioimpedance used to compute ICW and ECW

*The measured frequency characteristic and some proposed models with 2, 4 and 6 branches [10].*

The first model presented in this chapter is the Cole model. This model is used for body water volume prediction, having frequency independent values for *Ri*, *Re* and *Cm*. It cannot reproduce the measurement results for a three or even for a two-

The second model is based on a fractional exponent formula for the body impedance whose module is fitted to the measured values in [3]. But the real part of this impedance at the minimum and maximum frequencies cannot be computed and, consequently, the ICW and ECW formulae, having an outstanding practical

The next two behavioral models are based on parameter identification. These

models are linear *RC* circuits with frequency independent elements, whose

**Figure 13.** *Foster II synthesis of the new circuit model [10].*

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

The direct employment of the Foster II synthesis algorithm starting from Eq. (6) leads to some negative parameter values. This effect can be avoided performing the Foster II synthesis of |*YLC*(*jω*)|, where *YLC*(*s*) in Eq. (9) is given by the frequency

> 1 *s* � *YRC s*

þ

<sup>50</sup>*:*<sup>94</sup> � <sup>10</sup>�11*<sup>s</sup>* <sup>16</sup>*:*<sup>47</sup> � <sup>10</sup>�<sup>7</sup>

Starting from the partial fraction decomposition in Eq. (9), the parameter values are: C5 = 5.9 pF, R5 = 27 MΩ, C4 = 2.41 nF, R4 = 4.78 kΩ, C3 = 0.945 nF, R3 = 4.03

The resistance corresponding to the volume of the extracellular water can be computed for *fmin* = 1 kHz and has a 560.97 Ω value, which is practically the same

The resistance corresponding to the volume of the intracellular water can be computed for *fmax* = 1000 kHz and has a 314.97 Ω value, unlike *Ri* = 352.69 Ω given by the Cole model. Due to the better agreement with experimental data, it is expected that the body water volume prediction will be improved considering these

It is very interesting to observe that R1 has a similar value to *Re* in the Cole model, being the equivalent resistance for *f* = 0 Hz. This circuit can be viewed as a generalization of the Cole model. The two branch models contain R1, R2, C2. As the frequency range of interest is extended to higher frequencies, a model with a greater number of branches is needed. The simulated data obtained with models with various numbers of branches, obtained by imposing the same error *ε* on

A similar circuit (**Figure 15**) is given in [3] without pointing out how the resistance and capacitance values can be computed starting from the measured data. In order to appreciate the agreement between the measured and simulated data, the measuring errors must be known. Unfortunately, no information on these errors

<sup>24</sup>*:*<sup>14</sup> � <sup>10</sup>�10*<sup>s</sup>* <sup>11</sup>*:*<sup>54</sup> � <sup>10</sup>�6*s*<sup>2</sup> <sup>þ</sup> <sup>1</sup>

> *s*<sup>2</sup> þ 1 þ

<sup>2</sup> (8)

<sup>59</sup>*:*<sup>04</sup> � <sup>10</sup>�13*<sup>s</sup>* <sup>15</sup>*:*<sup>91</sup> � <sup>10</sup>�<sup>5</sup>

*s*<sup>2</sup> þ 1

(9)

þ

*YLC*ðÞ¼ *s*

<sup>17</sup>*:*<sup>81</sup> � 104 *s*

transformation in Eq. (8) [17].

*Electrochemical Impedance Spectroscopy*

*YLC*ðÞ¼ *<sup>s</sup>* <sup>19</sup>*:*<sup>47</sup> � <sup>10</sup>�11*<sup>s</sup>* <sup>þ</sup>

<sup>94</sup>*:*<sup>53</sup> � <sup>10</sup>�11*<sup>s</sup>* <sup>38</sup>*:*<sup>08</sup> � <sup>10</sup>�<sup>7</sup>

with *RE* = 562 Ω given by the Cole model.

various frequency intervals are given in **Figure 14**.

*s*<sup>2</sup> þ 1 þ

kΩ, C2 = 0.51 nF, R2 = 3.23 kΩ, C1 = 0.195 nF, R1 = 561.5 Ω.

þ

values in Eqs. (1) and (2).

is given in [3].

**Figure 13.**

**92**

*Foster II synthesis of the new circuit model [10].*

*The measured frequency characteristic and some proposed models with 2, 4 and 6 branches [10].*

**Figure 15.** *The extended Cole model [3].*

## **6. Conclusions**

Four models of the human body bioimpedance used to compute ICW and ECW volumes have been presented in this chapter.

The first model presented in this chapter is the Cole model. This model is used for body water volume prediction, having frequency independent values for *Ri*, *Re* and *Cm*. It cannot reproduce the measurement results for a three or even for a twodecade frequency range.

The second model is based on a fractional exponent formula for the body impedance whose module is fitted to the measured values in [3]. But the real part of this impedance at the minimum and maximum frequencies cannot be computed and, consequently, the ICW and ECW formulae, having an outstanding practical importance, cannot be used.

The next two behavioral models are based on parameter identification. These models are linear *RC* circuits with frequency independent elements, whose

parameters can be identified starting from the measured values |*ZRC*( *jω*)| reported in [3]. The influence of the measurement equipment including signal source, cables (modeled as transmission lines) and connectors has been shown to be negligible, so |*ZRC*(*jω*)| given in [3] is an accurate representation of the human body impedance modulus [8, 18].

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of bioelectrical-impedance

After the synthesis of the third model, an *RC* ladder, valid for a frequency range between 1 and 100 kHz [8], and its extension to a three-decade frequency interval [17], the fourth model, the *RC* parallel circuit [9], whose validity range is three decades is presented. This model contains some *RC* branches connected in parallel. This model can be simplified, taking into account that the influence of some branches is negligible in a certain frequency range, its ultimate simplification being the linear *RC* Cole model. It follows that this model can be considered as an extension of the linear *RC* Cole model, allowing a good prediction of the intracellular and extracellular water volumes. All these linear lumped *RC* circuits avoid using both intricate frequency dependent elements suggested by the physical interpretation of current conduction in human body and the fractional exponent impedance formula of de Lorenzo model [3].

Even though the modeling of fractional-order circuits is rigorously established [19], a linear *RC* circuit model based on straightforward concepts is more useful for intracellular and extracellular water volume prediction than a fractional-order system. The development of these new models illustrates the actual trend [20] to make noninvasive investigation methods more precise in various areas of medicine [7] as coronary artery disease [21], colorectal cancer [22] and HIV infection [23].
