Tail-volume loss and ratio inflation in truncated expiration

Tail-volume loss and ratio inflation in truncated expiration connects unrecorded late gas to a biased FEV1 to FVC ratio. Suppose a valid first-second volume $A$ is retained, the recorded total is $M$, and additional uncollected volume is $L\ge 0$. The complete ratio is $A/(M+L)$, while using $M$ as FVC gives $A/M$.

If independent information bounds $L\le L_{\max }$, thenFor an original example with $A=2.4$ L, $M=3$ L and $L_{\max }=1$ L, the complete ratio lies between 0.60 and 0.80 under this bound. This is a deterministic interval conditional on the tail bound, not automatically a statistical confidence interval.

Bounding uncollected expiratory volume from terminal flow adds a model. Let volumetric flow rate after the stop time be $Q(T+s)=q_T{e}^{-s/\tau }$. The missing volume isIf $\tau \le {\tau }_{\max }$, this gives $L\le q_T{\tau }_{\max }$. For stipulated terminal volumetric flow rate $q_T=0.050\mathrm{L}{\mathrm{s}}^{-1}$ and ${\tau }_{\max }=4$ s, the bound is 0.20 L. With the same $A$ and $M$, the ratio is then between 0.75 and 0.80.

The bound extends to a positive sum of exponential tails: if each component has terminal volumetric flow rate $q_i$ and time constant at most ${\tau }_{\max }$, then $L={\sum }_i{q}_i{\tau }_i\le {\tau }_{\max }{\sum }_i{q}_i$. Without a justified upper time scale, however, very small terminal volumetric flow rate can coexist with a large slowly emptying volume. A visually flat trace alone does not supply that missing assumption.

The relative inflation from using the truncated ratio is especially simple:for positive $A$. In the bounded 0.20 L example it is at most $0.20/3\approx 6.67\%$. The result assumes termination after a valid first second; an invalid numerator needs another error model.

The primary spirometry standard evaluates end of forced expiration and distinguishes it from the validity of FEV1.^[ref] The calculations explain sensitivity to lost tail volume rather than replace the standard's actual maneuver criteria.

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.