Baseline-trajectory dependence of local respiratory time constants

Baseline-trajectory dependence of local respiratory time constants means that the linearized response can change as volumetric flow rate and elastic state change during a breath. A time constant measured at one operating point need not describe the entire trajectory, even when the underlying constitutive laws stay fixed.

In an original one-compartment model, let pressure satisfy $P=G(v)+F(\dot{v})$. Around a specified baseline $v_0(t)$, a small perturbation obeysThe frozen-coefficient local time constant is $\tau (t)=R(t)/E(t)$ when both coefficients are positive. For zero perturbing drive, the actual response is insteadThis follows by an integrating factor for the linear differential equation. The accumulated relaxation rate, rather than one selected endpoint value, controls the decay.

Choose a stipulated driven baseline volumetric flow rate $Q_0(t)=0.5e^{-t/t_b}$ L/s with $t_b=1$ s. Let elastic compliance be 0.20 L per cm water and resistive pressure be $F(Q)=2Q+4Q|Q|$ in the corresponding numerical units. Then $E=5$ and $R(t)=2+4e^{-t/t_b}$. The local time constant decreases from 1.2 s toward 0.4 s as the baseline volumetric flow rate falls.

Writing numerical time $u=t/t_b$, the homogeneous perturbation has the exact relative amplitudeAt one second it is about 0.322. Freezing the initial time constant would instead give $e^{-1/1.2}\approx 0.435$; freezing the limiting value would give $e^{-1/0.4}\approx 0.082$. These are distinct approximations to a changing-coefficient problem, not measured patient responses.

The baseline here is explicitly driven: its external pressure must sustain the chosen trajectory. It is not claimed to be the natural unforced expiration law of the nonlinear compartment. The primary respiratory pilot study compares methods that depend on volumetric flow rate, pressure and expiratory time scales, and discusses why expiration need not have one exponential time constant.^[ref] A local fitted value belongs to its operating trajectory and observation interval.