= Operating-point time constants with nonlinear airway resistance
\b[Operating-point time constants with nonlinear airway resistance] describe small deviations about a specified flow and volume trajectory. The finite ratio of pressure to flow does not generally give the resistance governing those deviations. Instead, local response uses the derivative of the pressure-flow curve at the operating flow.
In an original one-compartment model, let elastic <lung compliance> $C$ be locally constant and let resistive pressure be $F(Q)$. With volume $v$ measured relative to an elastic reference,
$$
P(t)=\frac{v(t)}C+F(\dot v(t)).
$$
A baseline trajectory $v_0(t)$ has flow $Q_0=\dot v_0$. Linearizing for small perturbations gives
$$
\delta P=\frac{\delta v}C+F'(Q_0)\,\delta\dot v.
$$
When the local coefficients are constant, the <time constant> is $\tau_{\rm local}=CF'(Q_0)$. It is the differential <airway resistance> times compliance, rather than the secant ratio $CF(Q_0)/Q_0$.
For the stipulated curve $F(Q)=2Q+4Q|Q|$, with numerical pressure in centimetres of water and flow in litres per second, $F'(0.5)=6$ while $F(0.5)/0.5=4$. With $C=0.20\,\mathrm{L\,(cm\,H_2O)^{-1}}$, the local time constant is 1.2 s, not 0.8 s. A small added pressure step has volume response
$$
\delta v(t)=C\,\delta P\left(1-e^{-t/(1.2\,\mathrm s)}\right).
$$
After 0.8 s it has reached about 48.7% of its final increment; using the secant-based time constant would incorrectly predict 63.2%. The numbers describe the local model, not measured airway coefficients.
<Baseline-trajectory dependence of local respiratory time constants> is essential. An isolated filling compartment cannot maintain nonzero flow at a truly stationary volume. The $Q_0=0.5$ example therefore refers to a driven baseline trajectory, or another arrangement with balancing flow. For an undriven stationary compartment, the baseline flow is zero. A large transient that substantially changes $Q$ also changes $F'(Q)$, so one constant time scale need not describe its whole course.
A primary ventilator study compared mechanics measurements across inspiratory and expiratory methods and varied flow-related settings.<Static and Dynamic Measurements of Compliance and Driving Pressure: A Pilot Study by Pierre Tawfik and others>[Introduction, methods C and D; Study Procedures]{ref} Report the operating state, perturbation size and pressure references with any inferred time constant. A single finite pressure-flow ratio does not establish a universal relaxation property.
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