## 2 Materials and Methods
### Lumped Fractional Order Parametric Models  
Our previous work and original contributions in modeling biological tissues have already long shown the success story of using lumped general order parametric models to capture complex phenomena [9]. Recent works have introduced such models into the control and system theory for use in control applications [31].  
When describing time-varying properties of these static characteristics of stress-strain relationships [10], it is very practical to use generalized order models with electrical equivalent notations for material dynamic properties:  
\[Z_{e}(s)=R_{e}+L_{e}(s)^{\alpha}+\frac{D_{e}}{s^{\beta}} \tag{18}\]  
where \(R_{e}\) is the electrical resistance related to material resistance, \(L_{e}\) the inductance related to inertance or molecular accelerations and \(D_{e}=1/C_{e}\) the compliance value as the inverse of the electrical capacitance. The units depend on the physical units used to derive the impedance from their time-domain measured signals. For electrical circuits, these time signals are in Volts, while for air/liquid measurements the time signals are in pressure (Pa) and flow in liters/s. The elements in this model have been in detail derived from mechanical and electrical equivalent properties analogy in [32]. When interpreting the model at a micro-scale level, the model refers to the diffusion process of glucose molecules in the simulated environment [12]:  
\[Z(s)=\frac{D}{s^{\beta}} \tag{19}\]  
Hence, the complex representation in frequency domain of this generic fractional order model is:  
\[Z_{e}(j\omega)=R_{e}+L(j\omega)^{\alpha}+\frac{D_{e}}{(j\omega)^{\beta}}, \hskip 28.452756 pt-2<\alpha,\beta<2 \tag{20}\]  
When identifying real data samples, the real and imaginary parts can be evaluated in the cost function of a regressor estimator:  
\[\begin{array}{ll}Z_{e}(j\omega)&=R_{e}+L_{e}\omega^{\alpha}\cos\left(\frac{ \alpha\pi}{2}\right)+\frac{D_{e}}{\omega^{\beta}}\cos\left(\frac{\beta\pi}{2 }\right)\\ Z_{e}(j\omega)&=-\left[L_{e}\omega^{\alpha}\sin\left(\frac{\alpha\pi}{2} \right)+\frac{D_{e}}{\omega^{\beta}}\sin\left(\frac{\beta\pi}{2}\right)\right] \end{array} \tag{21}\]  
Figure 4: ModuLab Xm measuring device, by Solartron Analytics.  
In [12], this model was successfully employed to fit the impedance of non-Newtonian solutions like water mixed with glucose and its parameters correlated to different concentrations.  
When the material properties behave as an electrical circuit that has one resistor and two fractional order capacitors, also known as constant phase elements, the number of model parameters can be significantly reduced:  
\[Z_{e}(s)=R_{e}+\frac{D_{e}}{s^{\alpha}} \tag{22}\]  
This was shown to be the case in non-Newtonian fluids with thixotropic properties [9]. In this model, \(\alpha\) can be fixed per tablet geometrical properties, while \(R_{e}\) and \(D_{e}\) are related to dynamic changes in viscosity and density of the material mixture. The drawback of this model structure is that its accuracy is limited over a rather small frequency interval (i.e., 1-2 decade interval).  
An alternative solution for a much broader (i.e., several decades) frequency interval dependence of the polar plot representation is the model:  
\[Z(s)=\frac{Kp}{s^{\beta}(s+p)^{\alpha}} \tag{23}\]  
which is a four parameters model, a lumped version of that proposed in [33]. Several comparisons of estimators on various non-Newtonian fluids have been reported in our prior work in [9]. In this model, all parameters were sensitive to changes in material properties, and the results suggested that in order to increase the detectability of material properties, correlation matrices can be used to calibrate it for particular applications at hand. When parameters \(Kp\), \(\alpha\), \(\beta\) are fixed to the material specifications, the parameter \(p\) has an excellent sensitivity to capture changes in viscosity.