## 2 Material and Methods
### Impedance Model Selection  
It has been already proven that fractional-order impedance models are natural solutions for characterizing frequency response of the anatomical and structural composition of the respiratory system [13]. The fractional-order model for human respiratory system provides an expression for the input impedance (measured at the mouth of the patient) by considering a series arrangement of a resistance in series with two fractance elements, as follows:  
\[Z_{r}(s)=R_{r}+L_{r}s^{a_{r}}+\frac{1}{C_{r}s^{b_{r}}} \tag{1}\]  
where \(Z_{r}(s)\) is the respiratory impedance; \(R_{r}\) is airway resistance in kPa/(l/s); \(L_{r}\) is inductance kPa/(l/s^{2}); \(C_{r}\) is capacitance in L/kPa; \(\beta_{r}\) is the fractional order; \(s\) is the Laplace  
Figure 2: Schematic of the Equivital monitoring system.  
operator; and the subscript \(r\) denotes respiratory model. The basic concept of a fractional order in the frequency domain is given in Appendix A.  
From the identified model parameters, one can derive the tissue damping \(G_{r}\) and elastance \(H_{r}\), defined as [13]:  
\[\begin{array}{l}G_{r}=\frac{1}{C\epsilon^{\alpha\beta r}}\cos(\beta_{r}\frac{ \pi}{2})\\ H_{r}=\frac{1}{C\epsilon^{\alpha\beta r}}\sin(\beta_{r}\frac{\pi}{2})\end{array} \tag{2}\]  
The hysteresivity coefficient \(\eta_{r}\) (dimensionless) is defined as [24]  
\[\eta_{r}=\frac{G_{r}}{H_{r}} \tag{3}\]  
This parameter characterizes the heterogeneity of the lung tissue and has been shown to vary and correlate significantly with pathology [13].  
In this study, two other fractional-order models, i.e., a Cole model and a Warburg model, were used to fit the measured data. A fractional-order Cole model has shown to be successful in characterizing biological tissues [25]:  
\[Z_{c}(s)=R_{c}+\frac{K_{c}}{1+(\frac{1}{p})^{\beta_{c}}} \tag{4}\]  
where \(R_{c}\) denotes the gain at infinite frequencies, \(K_{c}\) denotes the relative gain difference between low and high frequencies, \((\frac{1}{p})^{\beta_{c}}\) denotes pseudo-capacitance of the constant phase element, and the subscript \(c\) denotes the Cole model. It follows that \(\tau=\frac{1}{p}\) is the relaxation time constant as per analogy to Debye material characteristics. Lastly, the Warburg diffusion element that models the diffusion process in dielectric spectroscopy is considered as  
\[Z_{w}(s)=\frac{K_{w}}{\sqrt{s}} \tag{5}\]  
where \(K_{w}\) is the Warburg coefficient (or the Warburg constant).  
The best-fitting fractional-order impedance model is selected using the total error between the real impedance and the model estimated impedance, which is calculated as  
\[E_{T}=\sqrt{E_{R}^{2}+E_{I}^{2}} \tag{6}\]  
where \(E_{R}\) and \(E_{I}\) denote the errors on the real and imaginary parts, respectively. They are found using following formulas:  
\[E_{R}=\frac{1}{N_{s}}\sqrt{\sum_{k=1}^{N_{s}}(Z_{Rk}-\hat{Z}_{Rk})^{2}}\quad E _{I}=\frac{1}{N_{s}}\sqrt{\sum_{k=1}^{N_{s}}(Z_{Ik}-\hat{Z}_{Ik})^{2}} \tag{7}\]  
where \(Z_{Rk}\) and \(Z_{Ik}\) represent the real and imaginary parts of the real impedance, respectively, while \(\hat{Z}_{R,k}\) and \(\hat{Z}_{I,k}\) are their model estimated impedance values. \(N_{s}\) is the total number of measured points in the frequency interval of interest.