Detailed Respiratory System Model

Various models of the respiratory system are available. By far the most used model is the linear one-compartment model from which the equation of motion is derived. We have chosen to use a more elaborate version of this model, which is a modified variant of the model in Liu et al. [17] (link) and is previously described and validated in [20] , [19] . The model is shown on the right in Fig. 1.
Instead of one constant resistance in the linear model, the chosen respiratory model consists of three variable resistances that model: 1) the upper airways ( $R_u$ ), modeled by a so-called Rohrer resistance to account for turbulence, 2) collapsible airways ( $R_c$ ) which is inversely proportional to the air volume in the collapsible airway, and 3) small airways ( $R_s$ ), which depends on the volume of air in the lung.
The volume of the chest wall ( $V_{\mathrm{cw}}$ ), lung volume ( $V_a$ ) and collapsible airway segment ( $V_c$ ) are modeled with a sigmoidal relation, depending on the transmural pressures over the element (see Table 1). This corresponds well with the pressure-volume relationships that are described in the literature for the lung and chest wall [15] (link).Table 1

The model equations of the respiratory system.

Table 1

	Equation	Model parameters
Ru	Au + Kuflow	Au, Ku
Rc	Kc(Vcmax/Vc)2	Kc, Vcmax
Rs	AseKs(VA−RV)/(V⁎−RV)+Bs	As, Ks, RV, Bs, V⁎
Vcw	TLC−RV0.99+exp⁡−(Pcw−Acw)Bcw+RV†	TLC, RV, Acw, Bcw
Vc	Vcmax(1+e−Ac(Pc−Bc))Dc†	Vcmax, Ac, Bc, Dc
VA	Al(1+e−Bl(Pa−Dl))†	Al, Bl, Dl
Cg	FRC/970	–

†

Pa, Pc, Pcw are the pressures over the lung compliance, collapsible airway compliance, and chest wall compliance, respectively.

The viscoelastic properties of the respiratory system are modeled by a linear-solid model. It is formed by a constant resistance, $R_{\mathrm{ve}}$ , and a constant compliance, $C_{\mathrm{ve}}$ , in series with the lung-chest wall compliances (Fig. 1). This is a common method to model the viscoelastic behavior of tissue.
Airflow needs pressure generation, and according to Boyle's law, pressure generation means that a mass of air is compressed or decompressed relative to its equilibrium volume at atmospheric pressure. This leads to Equation (1). $P_{\mathrm{atm}}{V}_{\mathrm{atm}}=(P_{\mathrm{atm}}+\mathrm{\Delta }P)(V_{\mathrm{atm}}+\mathrm{\Delta }V),$ where $P_{\mathrm{atm}}$ is the absolute atmospheric pressure (970 cmH₂O is the absolute atmospheric pressure minus water vapor pressure), $V_{\mathrm{atm}}$ is the volume of air in the lung at atmospheric pressure, $P_{\mathrm{atm}}+\mathrm{\Delta }P$ is the absolute pressure in the lung, and $V_{\mathrm{atm}}+\mathrm{\Delta }V$ is the volume of the same mass of air in the lung. Assuming $P_{\mathrm{atm}}$ is much larger than ΔP (970 cmH₂O vs +/- 0-50 cmH₂O), the equation leads to the solution given by Equation (2). $\mathrm{\Delta }V=\frac{-\mathrm{\Delta }P{V}_{\mathrm{atm}}}{P_{\mathrm{atm}}}.$ ΔV is often referred to as ‘shift volume’ [21] (link), [16] (link), which is one of the driving forces during spontaneous breathing and used during full body plethysmography to measure the total lung volume. In the electrical equivalent model, this can be accounted for by adding the compliance ( $C_g$ ) between the alveolar space and atmospheric pressure (Fig. 1). $C_g$ is then equal to $V_{\mathrm{atm}}/P_{\mathrm{atm}}$ , where we assume that $V_{\mathrm{atm}}$ is equal to the functional residual capacity (FRC) and $P_{\mathrm{atm}}$ is 970 cmH₂O. The charge on $C_g$ is then equal to the shift volume ΔV.
The model equations are summarized in Table 1. The values of the model parameters in Table 1 are given in the supplementary material. Normal, obese, and ARDS parameter sets are available and include inter-patient variability by the availability of four different parameter sets per pathology.
In the model, a pressure source, $P_{\mathrm{mus}}$ , simulates the total pressure generated by the respiratory muscles and serves as a ground truth for our comparison. Various shapes of this muscle effort source are proposed in the literature to simulate spontaneous breathing patients [22] . A rounded trapezoid with a different slope for the rising edge and the falling edge is used, which is common in the literature. For the purpose of this study, we keep the rise and fall time equal while varying the depth of the muscle waveform (see Section 2.4 for the precise values).