 Integral null-space ambiguity in expiratory measurements

Integral null-space ambiguity in expiratory measurements is the existence of different volumetric flow rate functions producing the same finite collection of cumulative expired volumes. Let $Q$ belong to an appropriate function space and let the measurement map beThis is a linear map. Every perturbation $h$ in its null space leaves all the measurements unchanged: $L(Q+h)=LQ$.

The null space is large even when observations occur at several times. Put $T_0=0$. On any interval $[T_{j-1},T_j]$, choose a sine wave completing an integer number of cycles and set it to zero outside that interval. Its integral across that interval is zero, so every recorded cumulative volume is unchanged. Different cycle counts give infinitely many such functions. Sufficiently small amplitudes preserve positive volumetric flow rate wherever the baseline stays bounded away from zero.

An original example records volumes at 0.5, 1 and 2 s from a baseline volumetric flow rate of 1 L/s. Add $h(t)=0.4\sin (4\pi t)$ L/s only over the first 0.5 s. All three recorded volumes remain 0.5, 1 and 2 L, but the instantaneous volumetric flow rate ranges from 0.6 to 1.4 L/s during that interval. The added shape is invisible to these cumulative measurements.

A finite-dimensional model changes what can be identified. If $Q(t)={\sum }_{j=1}^k{a}_j{\varphi }_j(t)$ is restricted to known basis functions, the observations become $y=Ma$, where $M_{ij}={\int }_0^{T_i}{\varphi }_j(t)dt$. Unique coefficients require matrix rank $k$. For the stipulated linear-flow model $Q=a+bt$, two observations have rows $(T_i,T_i^2/2)$ and determinantDistinct positive times therefore identify $a,b$ in that model. Volumes 1.5 L at 1 s and 4 L at 2 s yield $a=1$ L/s and $b=1\mathrm{L}{\mathrm{s}}^{-2}$.

That model-based uniqueness is not recovery of an arbitrary waveform. The primary spirometry statement's complementary volume-time and flow-volume displays preserve information beyond a few timed volumes.^[ref] State the allowed model class before treating a small set of summaries as a complete measurement.