= Observation-time dependence of apparent compliance
\b[Observation-time dependence of apparent compliance] is the change in an inferred volume-per-pressure response when the measurement is made at different stages of filling or relaxation. For an original single passive compartment exposed to a pressure step,
$$
A(T)=\frac{\Delta V(T)}{\Delta P}=C\left(1-e^{-T/\tau}\right).
$$
The equilibrium compliance is $C$, but a finite-time observation is smaller whenever $T$ and $\tau$ are positive and finite. A short observation can therefore confound stiffness with slow filling.
Two observation times identify the parameters only within the assumed model. Taking measurements at $T$ and $2T$ and defining $r=A(2T)/A(T)$ gives
$$
r=1+e^{-T/\tau},\qquad
\tau=-\frac{T}{\log(r-1)},\qquad
C=\frac{A(T)}{2-r},\qquad 1<r<2.
$$
For stipulated $A(1\,\mathrm s)=0.10$ and $A(2\,\mathrm s)=0.15\,\mathrm{L\,(cm\,H_2O)^{-1}}$, this model gives $C=0.20$ and $\tau=1/\log2\approx1.44$ s. The parameter values follow from the model as well as the data.
A second original example exposes <finite-time ambiguity between stiffness and slow filling>. Suppose the real response is the sum of two compartments, each with compliance 0.10, but time constants 0.2 and 2 s. Then $A(0.5\,\mathrm s)\approx0.11391$ and $A(1\,\mathrm s)\approx0.13867$. Fitting the single-compartment formulas to just those two values gives $\tau\approx0.328$ s and equilibrium compliance about 0.146, although the actual equilibrium sum is 0.20. Two exact measurements can fit the wrong model exactly.
Even one perfect finite-time value supplies no finite upper bound on equilibrium compliance without further restrictions. For an observed $A>0$ at time $T$, every choice $C>A$ can reproduce it with
$$
\tau=-\frac{T}{\log(1-A/C)}.
$$
Increasing compliance while increasing the filling time can preserve the measured response. Additional times, an independently bounded time scale or a validated compartment model add information.
The primary pilot study's discussion of non-uniexponential deflation and multiple time constants illustrates the experimental importance of this distinction.<Static and Dynamic Measurements of Compliance and Driving Pressure: A Pilot Study by Pierre Tawfik and others>[Study Procedures and Discussion of method D]{ref} The constructed calculations here are pressure-step examples, not patient measurements or a replacement for the study's actual compliance definitions.
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