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

#### **4.1 Introduction**

**3. De Lorenzo model**

*The simulated Cole model and the measured results.*

*Electrochemical Impedance Spectroscopy*

**Figure 6.**

**Figure 7.**

**86**

*Dielectric constant of the muscle tissue vs. frequency [3].*

nonlinear. For example:

The measurement results show that, in a wide frequency range (e.g., for two or

• the electrical permittivity depends on frequency as it is pointed out in **Figure 7** [3];

• the mixture effects have a greater influence on the skeletal muscle resistivity

• due to the complexity of the nonlinear relations between *Ri* and *Re* and ICW and ECW volumes, some heuristic relations as Eqs. (1) and Eq. (2), including the height and weight of the subject are used to compute the body water volumes.

The above properties, including unusual high values of the dielectric constant are discussed in detail in [3, 9]. As our approach is not related to these aspects, we

in the LF range than in the HF range [3]; and

suggest the interested researchers to read these publications.

three decades), the parameters of the circuit are frequency dependent and the relationships between the resistances of this model and the body water volumes are

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

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

bioimpedance spectroscopy is presented in this section. This approach employs the approximation of the measured body admittance modulus |*YRC*(*jω*)| with a physically realizable function followed by the circuit synthesis [16]. This model is a linear *RC* circuit with frequency independent values of resistances and capacitances. As the frequency dependence of the phase angle *arg*(*YRC*(*jω*)) can be computed from |*YRC*(*jω*)| using the Bayard-Bode relationships [17], the measured values of *arg* (*YRC*(*jω*)) are not needed for the parameter identification of this model. Two equivalent circuits of the human body, built using this approach, have been proposed [10–11]. These are ladder circuits which cannot be considered as extensions of the Cole model.

The algorithm for the synthesis of a *RC* one-port in the angular frequency band [*ωm*, *ωM*], where *ω<sup>m</sup>* is the minimum value and *ω<sup>M</sup>* is the maximum value, has the

• set the first zero *z1* corresponding to the minimum angular frequency *ωm*;

• compute the remaining poles and zeros at slope changes, by sweeping the *ω*

Sweeping the frequency axis with a step Δ*ωm*, the algorithm checks the error between the 20 dB/decade asymptote and the given characteristic. This error cannot be greater than an imposed value *ε*. The first pole *p1* is assigned to the last value before that corresponding to an error of *2ε* or greater. If this error occurs after the first angular frequency step Δ*ωm*, then *p1* is placed in the vicinity of *z1*. Afterwards, the first asymptote is translated so that a maximum error of *ε* is obtained. The other asymptotes are determined similarly, in order to fulfill the condition *error* ≤ *ε* for

The following models are build starting from measurement of the body impedance frequency characteristic |*ZRC*(*jω*)| reported in [3, 8]. The simulations presented in Section 2 show that the result of the frequency characteristic measurement is an accurate representation of the human body impedance modulus. Using the above

*<sup>s</sup>* <sup>þ</sup> <sup>1</sup> <sup>22</sup>*:*<sup>70</sup> � <sup>10</sup>�<sup>7</sup>

1

1

<sup>18</sup>*:*<sup>16</sup> � <sup>10</sup>�8*<sup>s</sup>* <sup>þ</sup>

Using a Cauer synthesis, the continued fraction expansion in Eq. (5) is obtained, and the circuit is given in **Figure 9**. The parameter values extracted from Eq. (5) are

106*:*17 þ

A comparison between the frequency characteristic of the *RC* ladder model, the

*<sup>s</sup>* <sup>þ</sup> <sup>1</sup> <sup>28</sup>*:*<sup>82</sup> � <sup>10</sup>�<sup>8</sup>*<sup>s</sup>* <sup>þ</sup> <sup>1</sup>

(4)

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

1

44*:*66 þ

1

1

<sup>15</sup>*:*<sup>69</sup> � <sup>10</sup>�<sup>3</sup>

1

*s* þ 98*:*64 (5)

*<sup>s</sup>* <sup>þ</sup> <sup>1</sup> <sup>10</sup>*:*<sup>23</sup> � <sup>10</sup>�<sup>6</sup>*<sup>s</sup>* <sup>þ</sup> <sup>1</sup> <sup>26</sup>*:*<sup>82</sup> � <sup>10</sup>�<sup>7</sup>

*<sup>s</sup>* <sup>þ</sup> <sup>1</sup> <sup>86</sup>*:*<sup>25</sup> � <sup>10</sup>�<sup>7</sup>

<sup>31</sup>*:*<sup>08</sup> � <sup>10</sup>�9*<sup>s</sup>* <sup>þ</sup>

Cole model and the measured results is given in **Figure 10**.

axis with the step Δ*ω*; a larger Δ*ω* leads to a simpler circuit;

• compute the circuit parameters using a synthesis method.

**4.3 Parameter identification for the two-decade model**

• compute the *RC* admittance expression; and

following steps [16]:

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

each asymptote.

algorithm it follows:

<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>

410*:*65 þ

*YRC*ðÞ¼ *s*

given below.

<sup>71</sup>*:*<sup>97</sup> � <sup>10</sup>�11*<sup>s</sup>* <sup>þ</sup>

*YRC*ðÞ¼ *s*

**89**
