= Bounding uncollected expiratory volume from terminal flow
\b[Bounding uncollected expiratory volume from terminal flow] uses an explicit restriction on future <volumetric flow rate> to limit the air missed when a recorded expiration stops. A small final <volumetric flow rate> alone gives no such guarantee: an arbitrarily long, slowly decaying tail can have substantial remaining <volume>.
Let $q(s)$ be the nonnegative future <volumetric flow rate>, with $s=0$ at the recording endpoint. Suppose an independently justified bound on its decay holds throughout the tail:
$$
q'(s)\leq-\frac{q(s)}{\tau_{\max}},\qquad q(0)=q_T.
$$
Differentiating $e^{s/\tau_{\max}}q(s)$ shows that it cannot increase. Hence
$$
q(s)\leq q_Te^{-s/\tau_{\max}},\qquad
L=\int_0^\infty q(s)\,ds\leq q_T\tau_{\max}.
$$
This is an upper bound from a specified <differential inequality>, not a fitted exponential silently treated as established physiology.
For original values $q_T=0.04\,\mathrm{L\,s^{-1}}$ and $\tau_{\max}=3$ s, the missed <volume> is at most 0.12 L. If the collected <volume> is $M=3.00$ L and the correctly captured first-second <volume> is $A=2.40$ L, then
$$
\frac{A}{M+0.12}\leq\frac{A}{M+L}\leq\frac AM,
\qquad 0.76923\leq\frac{A}{M+L}\leq0.80000.
$$
If the terminal instrument reading has a deterministic error bound $u$, use $q_T\leq q_{\rm measured}+u$ in the same argument. An uncertainty statement about the final reading does not itself establish the future decay assumption.
The assumption is essential. Exponential tails $q(s)=q_Te^{-s/\tau}$ with the same terminal <volumetric flow rate> have missed <volumes> $q_T\tau$, unbounded as $\tau$ increases. This explains why the endpoint of a trace cannot by itself certify complete expiration. The primary <spirometry> statement separately specifies end-of-forced-expiration and measurement acceptability criteria.<Standardization of Spirometry 2019 Update by Brian L. Graham and others>[FVC Maneuver, End of Forced Expiration, and Table 7]{ref} The inequality is an illustrative model bound, not an additional clinical stopping rule.
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