= Baseline-trajectory dependence of local respiratory time constants
\b[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 obeys
$$
R(t)\,\delta\dot v+E(t)\,\delta v=\delta P,
\qquad R(t)=F'(\dot v_0(t)),\quad E(t)=G'(v_0(t)).
$$
The 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 instead
$$
\delta v(t)=\delta v(t_0)\exp\left[-\int_{t_0}^t\frac{E(s)}{R(s)}\,ds\right].
$$
This 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 amplitude
$$
\frac{\delta v(t)}{\delta v(0)}=\left(\frac{2e^u+4}{6}\right)^{-5/2}.
$$
At one second it is about 0.322. Freezing the initial <time constant> would instead give $e^{-1/1.2}\approx0.435$; freezing the limiting value would give $e^{-1/0.4}\approx0.082$. 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>.<Static and Dynamic Measurements of Compliance and Driving Pressure: A Pilot Study by Pierre Tawfik and others>[Methods C and D; Study Procedures; Discussion]{ref} A local fitted value belongs to its operating trajectory and observation interval.
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