# Estimating Airway Resistance from Forced Expiration in Spirometry
## 2 Materials and Methods
### Airway Resistance from Spirometry  
#### 2.4.1 Deflating Balloon Model of Forced Expiration  
During forced expiration, a person empties his lungs forcefully from total lung capacity (TLC) until a residual volume (RV) is reached [4]. As such, one can draw an analogy with an elastic balloon that deflates from a volume of TLC until RV (Figure 1) or from a volume of FVC until 0 (since FVC is the difference between TLC and RV), under the influence of a driving pressure. At a given time t, we denote the volume of the balloon as x, and the velocity of deflation as \(\dot{x}\). If we ignore the effect of intra-thoracic gas compression [19], then the volume of air coming out of the balloon is FVC- x and the magnitude of airflow is \(\dot{x}\), which we assume to be laminar. Furthermore, we considered a coordinate system such that all vectors in the direction of deflation are negative. First, we write the equation of motion of the deflating balloon as:  
\[\dot{I}\ddot{x}\ =\ P_{alv}+P_{res} \tag{1}\]  
where \(\dot{I}\ddot{x}\) is the inertial pressure, \(\mathrm{P_{alv}}\) is the alveolar pressure, and \(\mathrm{P_{res}}\) is the resistive pressure. Equation (1) implies that the inertia of a deflating balloon is the result of \(\mathrm{P_{alv}}\), which is a driving pressure and \(\mathrm{P_{res}}\), which opposes deflation [9; 20]. The initial conditions of Equation (1) at \(\mathrm{t}=0\) are:  
\[x(0)\ =\ FVC\ and\ \dot{x}(0)\ =\ 0 \tag{2}\]  
and final conditions at \(\mathrm{t}=\mathrm{t_{n}}\) (time at the end of forced expiration) are:  
\[x(t_{n})\ =\ 0\ and\ \dot{x}(t_{n})\ =\ 0 \tag{3}\]  
At the beginning of the forced expiration, flow increases from zero to peak expiratory flow (PEF) within a very short interval \(\mathrm{t_{1}}\) (\(0<\mathrm{t_{1}}<<\mathrm{t_{n}}\)). While the airflows in this phase are greatly influenced by expiratory muscle effort [21], the airflows during \(\mathrm{t_{1}}\) until \(\mathrm{t_{n}}\), generally after 50% of expired volume are known to be effort-independent [22]. To avoid effort-dependent biased estimates, we drop the former phase and consider only the later phase (\(\mathrm{t_{1}}\leq\mathrm{t}\leq\mathrm{t_{n}}\)) by modifying the initial conditions as:  
\[x(t_{1})\ =\ \mathrm{FVC}-\Delta v\ \mathrm{and}\ \dot{x}(t_{1})\ =\ - \mathrm{PEF} \tag{4}\]  
where \(\Delta v\) is the volume of air expired between 0 and \(\mathrm{t_{1}}\), and can be written in terms of measured airflow F(t) as:  
\[\Delta\ =\ \int_{0}^{t_{1}}F(t)\ dt \tag{5}\]  
In Equation (1), the driving pressure \(\mathrm{P_{alv}}\) is a sum total of pleural pressure (\(\mathrm{P_{pl}}\)), which arises purely out of expiratory muscle effort, and elastic recoil pressure (\(\mathrm{P_{st}}\)), which is a restoring pressure generated by the inherent elasticity in the lungs [9]. According to the data produced by [PERSON] et al. [23], \(\mathrm{P_{alv}}\) decreases non-linearly with lung volume from a peak until zero during forced expiration. However, we make a simplifying assumption by representing \(\mathrm{P_{alv}}\) as a linear function of \(\mathrm{x}\) with a coefficient \(\mathrm{k}\). Thus,  
Figure 1: Model of a balloon that deflates from total lung capacity (or a volume of FVC) at time time (\(\mathrm{t}\)) = 0 until residual volume (or a volume of 0) at \(\mathrm{t}=\mathrm{t_{n}}\) with a peak airflow resulting at \(\mathrm{t}=\mathrm{t_{1}}\). The deflation process is driven by alveolar pressure, which arises due to a combination of expiratory muscle effort and elastic recoil of the lungs; while it is opposed by a resistive pressure, which depends on the width of the outlet. \[P_{\mathit{adv}}\ =\ -\ \mathit{k}x \tag{6}\]  
The resistive pressure \(\mathrm{P_{res}}\) in Equation (1) is determined by the width of the outlet of the balloon (Figure 1a). \(\mathrm{P_{res}}\) decreases as the width of the outlet increases.