**3. De Lorenzo model**

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


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 suggest the interested researchers to read these publications.

**Figure 7.** *Dielectric constant of the muscle tissue vs. frequency [3].*

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

Over 500 kHz, the time delay between the excitation and its response cannot be neglected [3]. In this case, a model with distributed parameters could be more accurate.

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

$$Z\_{RC}(jo) = \left(\frac{R\_\epsilon}{R\_\epsilon + R\_i}\right) \left(R\_i + \frac{R\_\epsilon}{1 + \left[joC\_m(R\_\epsilon + R\_i)\right]^a}\right) e^{-joT\_d} \tag{3}$$

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 cannot be used properly.
