# In Vitro Estimation of Relative Compliance during High-Frequency Oscillatory Ventilation
## 2 Materials and Methods  
The experiment setup is presented in Figure 1. In the experiment, we used a 3100B high-frequency oscillatory ventilator with a standard patient circuit for an adult patient. The patient circuit was connected to a glass container that represented the model of the respiratory system. The constant volume of the container determined its compliance. We gradually ventilated three different glass containers with volumes of 54, 35, and 25 L, which represented respiratory systems with the values of compliance 37, 24, and 17 mL/cmH2O, respectively. The measurements were realized in two phases: with and without additional parabolic resistor Rp5 (Michigan Instruments, Grand Rapids, MI, USA), which simulated an increase in the flow resistance of the airways. In the laboratory experiment, we used the following ventilatory parameters: bias flow = 30 L/min, inspiration to expiration time ratio \(I\):\(E\) = 1:1, ventilatory frequency \(f\) = 5 Hz, \(CDP\) = 12 cmH2O, and amplitude of the pressure oscillations \(\Delta P\) = 20 cmH2O. The ventilatory parameters were determined according to [37]. In clinical practice, the \(CDP\) and \(\Delta P\) are titrated at the start of HFOV to reach physiological levels of P\({}_{\text{a}}\)O\({}_{\text{2}}\) and P\({}_{\text{a}}\)CO\({}_{\text{2}}\). We did not titrate \(CDP\) and \(\Delta P\) in the ventilation because there was no gas exchange in our in vitro laboratory experiment. Pressure \(p_{\text{aw}}\) and flow \(q_{\text{aw}}\) of the air were recorded at the entry of the model of the respiratory system using a measuring system designed for HFOV monitoring [38]. The flow was measured as a pressure difference across the orifice. Both \(p_{\text{aw}}\) and \(q_{\text{aw}}\) signals were recorded using a sampling frequency of 1000 Hz. The relative compliance \(X_{\text{RS}}\) was calculated based on the power-spectral density \(S_{\text{xx}}\) of the pressure and the cross-spectral density \(S_{\text{yx}}\) of flow and pressure. The spectral densities were calculated from \(p_{\text{aw}}\) and \(q_{\text{aw}}\) using windows with a width of 2 s that moved along the signals with a step of 1 s. In the windows, the average value of a signal in the time domain was subtracted from the signal to remove the DC component in the frequency domain. Both spectral densities \(S_{\text{xx}}\) and \(S_{\text{yx}}\) were calculated using [PERSON]'s averaged modified periodogram method of spectral estimation [39]. When calculating the spectral densities, sections of both signals within the same window were split into eight segments, with a 50%  
Figure 1: The arrangement of the experiment. overlap between the neighboring segments. In a particular 2-s window, an \(i\)-th sample \(y_{K}(i)\) of the \(K\)-th segment of the \(p_{\rm aw}\) or \(q_{\rm aw}\) signal was thus acquired as:  
\[y_{K}=y(i+(K-1)\cdot 222), \tag{2}\]  
where \(y\) represents the \(p_{\rm aw}\) or \(q_{\rm aw}\) signal in the window, \(i\) has a range of 1 to 444, and \(K\) has a range of 1 to 8. Each segment \(y_{K}\) was filtered in the time domain by a low-pass FIR filter using the Hamming window. Fourier transforms \(P_{K}\) and \(Q_{K}\) of \(y_{K}\) were obtained for the pressure and flow signal segments, respectively. Modified periodograms \(I_{K}\) of \(K\)-th segment were calculated from \(P_{K}\) and \(Q_{K}\) as:  
\[I_{K}=\frac{1}{w^{T}\cdot w}\cdot P_{K}\cdot P_{K}{}^{*} \tag{3}\]  
for \(S_{\rm xx}\) and as:  
\[I_{K}=\frac{1}{w^{T}\cdot w}\cdot Q_{K}\cdot P_{K}{}^{*} \tag{4}\]  
for \(S_{\rm yx}\). In equations (3) and (4), \(P_{K}{}^{*}\) represents a complex conjugate of \(P_{K}\), \(w\) is a column vector of coefficients of the previously used Hamming window, and \(w^{T}\) represents the transpose of \(w\). Finally, the spectral densities \(S_{\rm xx}\) and \(S_{\rm yx}\) were calculated as an average value of \(I_{K}\) in a given window. In the \(S_{\rm xx}\) and \(S_{\rm yx}\), the frequency \(f_{\rm max}\) corresponding to the maximal power was found.