# In-Vitro Experimental Modeling of Oscillatory Respiratory Flow in a CT-Scanned OSAHS Tract
## 2 In-Vitro Experimental OSAHS Flow Model
### Oscillatory Respiratory Flow Rate  
In the present unsteady measurement, the realistic normal male breathing flow rate transients at rest were used to determine the modeling flow rate transients. Statistical data of the breathing flow rate transients compiled by a Beijing medical institution from hundreds of supine male adults were used. Similar flow rate transient curves were also confirmed in [PERSON] and [PERSON] [21] and [PERSON] and [PERSON] [33]. The tide volume (\(V_{\mathrm{T}}\)) here was chosen to be 500 mL. The case where the inspiratory air volume is a little larger than the expiratory one [34] was not considered in the present test rig. Figure 3: Optical index matching between ETA model and working fluids: (**a**) air; (**b**) water; (**c**) mixture of water and glycerol. The typical breathing cycle period at rest is within 3-5 s; 5 s were chosen due to the severe status of the OSAHS patient. To mimic the realistic flow physics and respiratory physiology, the similarities of Womersley number (\(\alpha=L/2\cdot\sqrt{2\pi\cdot f_{\mathrm{b}}/\
u}\)) and Reynolds number (\(Re=V\cdot L/\
u\)) were kept between the breathing air and the modeling water-glycerol liquid [23; 29; 32; 35]. The two similarity criteria ensured flow similarity such as the same relationship of pressure drop coefficient vs. flow coefficient. The Womersley number is a dimensionless number in biofluid mechanics representing the ratio of transient inertial force to viscous force and is an important similarity criterion for the oscillatory respiratory flows. Here, the \(Re\) number can be defined as \(Re=4f_{\mathrm{b}}\cdot V_{\mathrm{T}}/(L\cdot\
u)\), as mentioned by [PERSON] et al. [23; 24]. In these definitions, \(L\), \(V_{\mathrm{T}}\), \(f_{\mathrm{b}}\) and \(\
u\) refer to the inlet hydraulic diameter of the tract, tide volume, frequency of the oscillatory respiration, and kinematic viscosity of the working fluid, respectively. The kinematic viscosity (\(\
u\)) of the water-glycerol mixture and breathing air was \(8.933\times 10^{-6}\) m\({}^{2}\)/s and \(15.03\times 10^{-6}\) m\({}^{2}\)/s, respectively, at room temperature during the experiments. In this oscillatory respiratory flow, the maximum Reynolds numbers during inspiration and expiration phases were 1442 and 1149, respectively, and the Womersley number was 2.89. To maintain the similarity of Reynolds number and Womersley number, the breathing cycle period (1/\(f_{\mathrm{b}}\)) of 5 s was tuned to be 8.5 s in the present in-vitro experimental model where the working fluid was taken as the water-glycerol liquid. The breathing flow rate transients as well as those employed in the experiment are shown in Figure 4. Under such oscillatory boundary condition, roles of inlet and outlet of this model would switch periodically. At the inspiration stage, the boundary at the mouth side can be considered as the inlet, and the boundary at the trachea side was the outlet. While at the expiration stage, the contrary would be the truth. To mimic the modeling flow rate transients as shown in Figure 4, several apparatuses and procedures were carefully designed in this research. The schematic and photo of the experimental setup are presented in Figure 5. A programmable logic controller (PLC), servo-motor, servo-motor driver, gear box, bent axle, piston, air exhausting device, reservoir and \(V_{\mathrm{T}}\) monitor (Figure 5a) were employed to provide precise control over piston movements and thus experimental flow rate transients. Figure 5b,c presents a photo of the main apparatus and experimental rig, respectively. First, a PLC program was developed and run on a personal computer to transfer the PLC input digital signal into the PLC output analog voltage signal. Second, step by step, the driver of the servo-motor drove the movements of the servo-motor. The duration of the movement in each step was 20 ms. Thus, a total of 425 step movements were precisely controlled over the present single respiratory cycle (8.5 s).