= Early-time resistance dominance of a filling response
\b[Early-time resistance dominance of a filling response] means that the initial <volume> admitted by a <pressure> step is governed mainly by <airway resistance>, even when the eventual <volume> depends strongly on <lung compliance>. For a passive compartment with constant <airway resistance> $R$, <lung compliance> $C$, imposed <pressure> increment $\Delta P$ and initial added <volume> zero,
$$
R\frac{dv}{dt}+\frac vC=\Delta P,\qquad
v(t)=C\Delta P\left(1-e^{-t/(RC)}\right).
$$
Initially the elastic recoil increment is zero, so the initial <volumetric flow rate> is $\Delta P/R$. The initial slope identifies $R$ within this model, not $C$.
A <Taylor series> makes the separation explicit:
$$
\frac{v(t)}{\Delta P}=\frac tR-\frac{t^2}{2R^2C}+O(t^3),\qquad
v'(0)=\frac{\Delta P}{R},\qquad
v''(0)=-\frac{\Delta P}{R^2C}.
$$
Information about <lung compliance> first enters the curvature. Recovering that curvature from noisy early measurements can be much harder than measuring the initial slope. The dimensionless ratio $x=t/(RC)$ controls the approximation: the exact response divided by the resistance-only prediction is $(1-e^{-x})/x$.
For an original calculation, take $R=10\,\mathrm{cm\,H_2O\,s\,L^{-1}}$, $\Delta P=10\,\mathrm{cm\,H_2O}$ and compare $C=0.10$ with $C=0.50\,\mathrm{L\,(cm\,H_2O)^{-1}}$. Both predict initial <volumetric flow rate> 1 L/s. At 0.05 s their admitted <volumes> are approximately 0.04877 and 0.04975 L, although their equilibrium added <volumes> are 1 and 5 L. An apparently precise short filling trace can therefore provide little discrimination between very different equilibrium <lung compliances>.
This is an ideal pressure-step calculation. A finite instrument response, an unknown driving <pressure>, nonlinear <airway resistance> or multiple compartments changes what can be inferred. A primary ventilation study explicitly measured multiple expiratory <time constants> because deflation need not follow one exponential.<Static and Dynamic Measurements of Compliance and Driving Pressure: A Pilot Study by Pierre Tawfik and others>[Materials and Methods, Study Procedures: two expiratory time constants rather than an assumption of uniexponential deflation]{ref} An early filling slope and an equilibrium <lung compliance> answer different experimental questions.
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