= Finite-time ambiguity between stiffness and slow filling
\b[Finite-time ambiguity between stiffness and slow filling] means that a small <volume> response at one time can come from a stiff compartment or from a more compliant compartment that has not finished filling. In a passive single-compartment pressure-step model,
$$
A(T)=\frac{\Delta V(T)}{\Delta P}=C\left(1-e^{-T/\tau}\right),\qquad \tau=RC.
$$
One observation supplies one equation for equilibrium <lung compliance> $C$ and <time constant> $\tau$. It does not generally identify both.
An original family makes the ambiguity exact. For an observed $A(T)=A>0$, choose any $C>A$ and set
$$
\tau(C)=-\frac{T}{\log(1-A/C)}.
$$
Every choice reproduces the observation. At $T=1$ s and stipulated $A=0.10\,\mathrm{L\,(cm\,H_2O)^{-1}}$, equilibrium compliances 0.12, 0.20 and 1.00 correspond to <time constants> about 0.558, 1.443 and 9.491 s. Their agreement at one time does not imply agreement about equilibrium <volume> response.
Later measurements distinguish these particular models. At $2T$, the predicted response is
$$
A(2T)=C\left[1-\left(1-\frac AC\right)^2\right]=2A-\frac{A^2}{C}.
$$
The three examples predict approximately 0.1167, 0.1500 and 0.1900 in compliance units. Those differences motivate observing more than one stage of the transient. They do not prove that a one-compartment model is adequate for an actual lung.
<Early-time resistance dominance of a filling response> explains the limiting behavior. Expanding the exponential for $T\ll RC$ gives
$$
A(T)=\frac TR-\frac{T^2}{2R^2C}+\cdots.
$$
The leading response depends on resistance rather than equilibrium compliance. As $C$ grows in the indistinguishable family, $\tau\sim TC/A$ and $R=\tau/C$ tends to $T/A$. Much larger equilibrium compliance can therefore be hidden behind similar early filling.
The primary respiratory pilot study discusses estimates based on expiratory <time constants> and why deflation need not be uniexponential.<Static and Dynamic Measurements of Compliance and Driving Pressure: A Pilot Study by Pierre Tawfik and others>[Introduction, Method D; Study Procedures; Discussion of multiple time constants]{ref} An experiment should constrain its <pressure> step, observation duration and model class. A fitted finite-time response is not automatically an identified equilibrium material property.
\Image[/same-early-response-different-lung-compliances.png]
{title=Original passive models sharing one early response while predicting different equilibrium <lung compliances> and <time constants>}
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