Bounding uncollected expiratory volume from terminal flow

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:Differentiating $e^{s/{\tau }_{\max }}q(s)$ shows that it cannot increase. HenceThis is an upper bound from a specified differential inequality, not a fitted exponential silently treated as established physiology.

For original values $q_T=0.04\mathrm{L}{\mathrm{s}}^{-1}$ 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, thenIf the terminal instrument reading has a deterministic error bound $u$, use $q_T\le q_{\mathrm{measured}}+u$ 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 $q(s)=q_T{e}^{-s/\tau }$ 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.^[ref] The inequality is an illustrative model bound, not an additional clinical stopping rule.