= Unequal filling times in apparent lung compliance
\b[Unequal filling times in apparent lung compliance] occur when different communicating lung regions have different resistance-compliance <time constants>. An observation made before all regions equilibrate can measure less volume response than the equilibrium elastic relation predicts. This provides a concrete mechanism for an <observation-time dependence of apparent compliance> without changing the underlying component compliances.
Consider an original passive model with parallel compartments exposed to one imposed pressure step $\Delta P$. In compartment $i$, let added volume be $v_i$, <airway resistance> $R_i$ and elastic <lung compliance> $C_i$. The equation and initially unfilled solution are
$$
R_i\frac{dv_i}{dt}+\frac{v_i}{C_i}=\Delta P,\qquad
v_i(t)=C_i\Delta P\left(1-e^{-t/\tau_i}\right),\qquad \tau_i=R_iC_i.
$$
The apparent finite-time response per unit pressure is therefore
$$
C_{\rm app}(T)=\frac{\sum_i v_i(T)}{\Delta P}
=\sum_i C_i\left(1-e^{-T/\tau_i}\right).
$$
For positive component values it rises toward $C_\infty=\sum_iC_i$. This is a pressure-step response, not a definition that replaces every breathing-cycle compliance convention.
Take $C_1=C_2=0.10\,\mathrm{L\,(cm\,H_2O)^{-1}}$, $\tau_1=0.2$ s and $\tau_2=2$ s. At $T=0.5$ s, the fast region has filled about 91.8% of its eventual increment, while the slow region has filled only 22.1%. The summed apparent compliance is about 0.114, compared with the equilibrium value 0.20 in the same units. After 2 s it is about 0.163. If both time constants were 0.2 s, the 0.5-s value would instead be about 0.184, despite the same equilibrium compliance.
Early and late observations constrain different combinations. The initial volume slope is $\Delta P\sum_i1/R_i$, whereas the final volume change is $\Delta P\sum_iC_i$. A sum of exponentials also need not have one fixed decay time: a single fitted <time constant> depends on the fitted interval.
A primary pilot study comparing static and dynamic respiratory-system measurements discusses expiration-based estimates and the possibility of multiple time constants.<Static and Dynamic Measurements of Compliance and Driving Pressure: A Pilot Study by Pierre Tawfik and others>[Introduction, Method D; Study Procedures, two measured expiratory time constants]{ref} The constructed compartment example explains why measurement duration and model form belong with the reported value. Distinguishing lung-only from total respiratory-system mechanics still requires the appropriate internal pressure reference.
\Image[/lung-compartment-filling-times.png]
{title=Original two-compartment pressure-step model: unequal <time constants> produce an observation-time dependence of apparent <lung compliance>}
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