= Flow-history ambiguity of a timed expiratory volume
\b[Flow-history ambiguity of a timed expiratory volume] arises because integrating <volumetric flow rate> retains the total gas moved but discards its detailed time ordering. If a timed measurement is $A=\int_0^TQ(t)\,dt$, then every admissible perturbation $h$ satisfying
$$
\int_0^Th(t)\,dt=0
$$
produces the same $A$ from $Q+h$. Nonnegative flow restricts the possibilities, but generally does not leave a unique waveform. This is an <integral null-space ambiguity in expiratory measurements>.
An original example preserves both <FEV1> and <FVC>. Let a constant baseline flow $Q_0=1\,\mathrm{L\,s^{-1}}$ continue for 4 s. Modify only the first second, with $T=1$ s, by
$$
h(t)=a\sin(2\pi t/T),\qquad a=0.5\,\mathrm{L\,s^{-1}},\qquad 0\le t\le T,
$$
and set $h=0$ afterward. The modified flow stays between 0.5 and 1.5 L/s during the first second. Its cumulative difference from the baseline is
$$
H(t)=\frac{aT}{2\pi}\left[1-\cos(2\pi t/T)\right],\qquad 0\le t\le T.
$$
At one second the difference returns to zero. Both traces therefore have <FEV1> of 1 L, <FVC> of 4 L and <FEV1 to FVC ratio> of 0.25. Their <peak expiratory flows> differ: 1 versus 1.5 L/s. At half a second, the modified trace has expelled approximately 0.159 L more than the baseline, before the later flow difference cancels it.
These artificial waveforms isolate information loss; they are not physiological reference curves. A timed volume is not interchangeable with a peak value or an entire flow-volume history. The 2019 technical statement requires inspection of both volume-time and flow-volume displays because the displays expose different features of a maneuver.<Standardization of Spirometry 2019 Update by Brian L. Graham and others>[Equipment, Display: complementary volume–time and flow–volume graphs]{ref}
Adding more timed integrals can constrain the waveform more strongly, but finitely many such summaries still leave many possible continuous histories unless a restrictive model is imposed. Report which waveform features were actually measured rather than inferring all of them from one scalar volume.
\Image[/equal-expiratory-volumes-different-flows.png]
{title=Original flow histories with identical first-second expired <volume>; equal integrals conceal different <volumetric flow rates>}
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