 Operating-point time constants with nonlinear airway resistance

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,A baseline trajectory $v_0(t)$ has flow $Q_0={\dot{v}}_0$. Linearizing for small perturbations givesWhen the local coefficients are constant, the time constant is ${\tau }_{\mathrm{local}}=C{F}^{\prime }(Q_0)$. 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, $F^{\prime }(0.5)=6$ while $F(0.5)/0.5=4$. With $C=0.20\mathrm{L}(\mathrm{cm}{\mathrm{H}}_2\mathrm{O}{)}^{-1}$, the local time constant is 1.2 s, not 0.8 s. A small added pressure step has volume responseAfter 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 $F^{\prime }(Q)$, 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.^[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.