Spirometry

Spirometry measures the volume of air moved into or out of the respiratory system as a function of time. A spirometer records displaced gas directly or derives volume by integrating volumetric flow rate. The resulting trace describes gas movement at the mouth; it does not independently determine alveolar pressure, pleural pressure or lung compliance.^[ref]

Two important outputs of a forced expiration following maximal inspiration are forced vital capacity, the total volume expelled in that maneuver, and forced expiratory volume in one second, the volume expelled during its first second. Their ratio compares early emptying with the eventual expelled volume.^[ref] Equal final volumes can accompany substantially different emptying rates.

An original mathematical example sets expelled volumewhere the numerical time $t$ and time constant $\tau$ are in seconds. Both $\tau =1$ and $\tau =2$ give an asymptotic expelled volume of 4 L. Their first-second volumes are instead $4(1-e^{-1})\approx 2.53$ L and $4(1-e^{-1/2})\approx 1.57$ L. The corresponding ratios to the limiting volume are approximately 0.632 and 0.393. These smooth toy curves explain the measurements; they are not measured traces or diagnostic reference limits.

Spirometry measures volume differences. The gas left after maximal expiration is residual lung volume, which has not crossed the instrument. Consequently a volume trace alone does not determine total lung capacity: different absolute starting volumes can produce the same expelled-volume trace. This is the absolute-volume ambiguity in spirometry.

Timing, effort, leaks and early termination can alter what a trace represents. Distinguish an instrumentally repeatable signal from a maneuver that answers the intended physiological question. The primary technical statement treats equipment quality assurance and maneuver acceptability separately.^[ref]

A peak flow meter is an instrument that measures peak expiratory flow, the greatest outward volumetric flow rate reached during a specified forced expiration. A portable instrument can retain that maximum without recording the whole volume versus time history. A spirometer that records a suitable flow trace can also determine the maximum, but the two outputs contain different information.

Basil Martin Wright and C. B. McKerrow's 1959 primary paper described a portable instrument for maximum forced expiratory flow measurement.^[ref] The Wood Library-Museum preserves an Airmed example: its collection record places the device in the practical transition from bulky laboratory measurements to portable measurements.^[ref]

A peak alone cannot determine expelled volume. For two original triangular flow pulses, stipulate the same maximum $Q_p=8\mathrm{L}{\mathrm{s}}^{-1}$ but base durations 0.5 and 1.0 s. Their peaks both correspond to 480 L/min, while integrating their traces gives $Q_{p}D/2=2$ and 4 L. The second pulse expels twice as much air without a greater peak.

Finite instrument response introduces a separate issue. As an illustrative first-order low-pass filter model, suppose the indicated volumetric flow rate obeysFor a rectangular flow pulse of height $Q$ lasting $D$, the indicated peak is $Q(1-e^{-D/{\tau }_m})$. Durations $D={\tau }_m$ and $D=3{\tau }_m$ produce only about 63.2% and 95.0% of the input peak. This is a response model, not an assertion that the historical instrument has that exact mechanism or time constant.

A useful measurement therefore specifies the maneuver, units and instrument response as well as the displayed maximum. A retained peak does not establish forced vital capacity, forced expiratory volume in one second or lung compliance. Those require different information; spirometry standards distinguish equipment performance from maneuver quality.^[ref]

Peak expiratory flow is the greatest outward volumetric flow rate reached during a specified forced expiration. In an ideal continuous record,It has units of volume per time, commonly litres per second or litres per minute, whereas FEV1 and FVC are integrated volumes. The 2019 technical statement defines the peak from a maximal forced expiratory maneuver and specifies its reporting units.^[ref]

An original pair of triangular volumetric flow rate histories separates the quantities. Let numerical time be in seconds. One curve is $Q_1(t)=2(1-t)$ L/s over $0\le t\le 1$; the other is $Q_2(t)=4(1-2t)$ L/s over $0\le t\le 0.5$, followed by zero. Both expel 1 L in total and in the first second. Their peaks are nevertheless 2 and 4 L/s, or 120 and 240 L/min. Equal expired volume does not identify the highest volumetric flow rate.

The instrument history reflects the value of measuring that maximum directly. In 1959, Basil Martin Wright and C. B. McKerrow published a description of a portable instrument for maximum forced expiratory volumetric flow rate.^[ref] The Wood Library-Museum's original-object catalogue identifies its Wright peak flow meter, records the author as B. M. Wright (Basil Martin), and describes the instrument's portability.^[ref]

A peak flow meter and a full spirometer record different amounts of information. A retained peak does not reconstruct the preceding waveform or total volume. Conversely, numerical differentiation and finite sensor response can affect a peak inferred from a recorded volume trace. Adequate temporal response, effort and the starting lung volume belong to the measurement definition. The primary standard connects early volumetric flow rate and the peak with maneuver effort and requires complementary waveform displays.^[ref]

The triangular examples are mathematical constructions, not reference values or a diagnostic rule. Comparing peak measurements requires comparable units, maneuvers and instrument response rather than treating a maximum as a complete description of lung function.

The FEV1 to FVC ratio is the dimensionless fraction obtained by dividing forced expiratory volume in one second by forced vital capacity:Under an ideal complete-expiration model with nonnegative outward flow, the first-second expelled volume cannot exceed the total, so $0\le R\le 1$. The ratio describes early emptying relative to the measured total; it is not itself a volume or an instantaneous volumetric flow rate.^[ref]

An original comparison makes the information loss clear. FEV1 of 3 L with FVC of 4 L gives the same ratio, 0.75, as 1.5 L with 2 L. Equal ratios do not imply equal absolute lung size, equal total expired volume or equal complete flow histories. Expressing 0.75 as 75% changes the display, not the quantity.

Common-gain cancellation and flow-offset bias in expiratory ratios distinguishes two instrument errors. If both measured volumes are multiplied by the same positive gain $g$, their ratio is unchanged. A constant flow offset $b$ instead accumulates for different durations. With true first-second volume $A$, total volume $B$, first-second duration $t_1$ and complete observation duration $T$,The cancellation of a common gain does not cancel the duration-dependent offset.

For stipulated $A=2.4$ L, $B=4$ L, $t_1=1$ s and $T=6$ s, the true ratio is 0.60. A common gain of 1.05 still gives 0.60. With unit gain and $b=0.020\mathrm{L}{\mathrm{s}}^{-1}$, however, the measured ratio becomes $2.42/4.12\approx 0.587$. These are constructed measurement errors, not observed patient data.

Early termination bias in forced vital capacity acts differently again: losing late expired volume can lower the denominator while leaving a valid first second intact, increasing the ratio. This is why a ratio needs valid constituent measurements. The technical statement treats FEV1 and FVC acceptability separately.^[ref] A clinical interpretation additionally requires the appropriate reference framework; the arithmetic alone does not identify a disease mechanism.

Common-gain cancellation and flow-offset bias in expiratory ratios distinguishes an error that rescales both volumes equally from one that accumulates differently over time. Let true FEV1 be $A$, true FVC be $B$, and their ratio be $R=A/B$. A shared positive volume gain $g$ cancels exactly: $(gA)/(gB)=R$. The result requires the same gain for both quantities, not merely the same instrument name.

A constant error $b$ in measured volumetric flow rate has another effect. If the first-second interval lasts $t_1$ and the full recording lasts $T$, integration givesThe ratio alone identifies neither $b$ nor $g$ separately; only their relative combination enters this model. Nor does the positive duration-dependent error necessarily raise the ratio.

The exact difference isFor small $b$, its sensitivity coefficient at zero offset is $(t_{1}B-TA)/(g{B}^2)$. If average early volumetric flow rate $A/t_1$ exceeds average whole-record volumetric flow rate $B/T$, a positive volumetric flow rate offset lowers the ratio. If the averages agree, that offset leaves the ratio unchanged in this model.

An original example uses $A=2.4$ L, $B=4$ L, $t_1=1$ s and $T=6$ s. The true ratio is 0.60. At $g=1$, the zero-offset sensitivity is $-0.65\mathrm{s}{\mathrm{L}}^{-1}$. An offset $b=0.020\mathrm{L}{\mathrm{s}}^{-1}$ gives a first-order shift of $-0.013$; the exact ratio is $2.42/4.12\approx 0.58738$. A common 5% gain error without the offset would leave 0.60 unchanged.

Independent numerator and denominator gains would instead multiply $R$ by $g_A/g_B$. Clipping, early termination and different timing conventions are additional measurement effects, so gain cancellation is not a general guarantee of ratio accuracy. The GUM requires the full measurement model and shared-input effects to be retained.^[ref] The spirometry statement separately checks the acceptability of the constituent volumes.^[ref]

Absolute-volume ambiguity in spirometry is the undetermined volume offset left when spirometry measures only air entering or leaving the mouth. If $Q(t)$ is outward volumetric flow rate, expelled volume is $E(t)={\int }_0^{t}Q(s)ds$, while absolute lung volume isThe same measured trace is compatible with every admissible initial volume $K$. Differentiation removes the constant: neither integrating flow nor improving its numerical precision recovers that missing offset.

An original example has 4 L expelled between maximal inspiration and maximal expiration. An initial volume of 6 L leaves 2 L, while an initial volume of 8 L leaves 4 L. Both give the same forced vital capacity and can have exactly the same flow-time curve. They differ in total lung capacity and residual lung volume, which the expelled-volume trace alone does not identify. The pulmonary-function text distinguishes spirometric volume changes from absolute lung-volume measurements.^[ref]

Helium dilution measurement of lung volume adds independent information. In an ideal closed circuit, an external reservoir has known volume $V_A$ and initial helium fraction $c_0$, while the communicating lung gas initially contains no helium. After complete mixing to fraction $c_f$, conservation givesFor stipulated $V_A=3$ L, $c_0=0.10$ and $c_f=0.04$, the communicating lung volume at connection is 4.5 L. This original calculation assumes no leaks, complete mixing, consistent gas conditions and appropriate accounting for the apparatus volume. The primary textbook describes the closed-circuit mixing apparatus and helium concentration measurement.^[ref]

Communicating gas volume versus trapped gas volume limits this inference: gas that does not communicate with the tracer during the measurement is not counted by the ideal mixing balance. Adding a tracer or another independent measurement breaks one ambiguity only under its own experimental assumptions. A well-resolved spirometric trace and a validated absolute-volume measurement answer complementary questions.

Communicating gas volume versus trapped gas volume distinguishes gas that exchanges with a measurement circuit during a specified experiment from gas that is physically present but not reached by that experiment. The distinction depends on the pathway, mixing time and protocol. A region with no exchange during a test contributes no tracer dilution even though its gas occupies space in the lung.

An original ideal closed-circuit model separates these volumes. An external reservoir of volume $V_A$ initially contains helium fraction $c_0$. Communicating lung gas has volume $V_c$ and initially no helium; an additional volume $V_t$ is isolated throughout the test. After complete mixing of the accessible part,The actual lung gas volume in this model is $V_c+V_t$. The final tracer concentration does not identify $V_t$ because that volume does not enter the conservation equation.

Take stipulated $V_A=3$ L, $c_0=0.10$, $V_c=3$ L and $V_t=2$ L. The equilibrium concentration is $0.30/6=0.05$, and the dilution calculation correctly returns 3 L of communicating gas. The total gas volume is nevertheless 5 L. If all 5 L instead communicate and equilibrate, the final fraction becomes $0.30/8=0.0375$. These are constructed gas balances, not measurements from patients.

Helium dilution measurement of lung volume implements the accessible-volume idea using a known gas mixture and concentration measurement. The primary textbook describes the closed circuit and contrasts dilution methods with body plethysmography.^[ref] Plethysmographic inference uses pressure-volume behavior, so it has a different measurement model and its own assumptions.

Protocol-dependent accessibility of lung gas also covers slow exchange. A poorly ventilated region may contribute little during a short measurement yet exchange appreciably over a longer interval. Calling its gas permanently trapped solely because one tracer trace has not equilibrated would exceed the observation. Conversely, an apparently stable accessible compartment need not demonstrate access to every region.

Comparison between methods is therefore useful only with their pressure, mixing and equilibration conditions specified. A difference can reveal different sampled gas compartments, but leaks, technical errors and violated model assumptions must also be considered before assigning the difference to one anatomical cause.

Protocol-dependent accessibility of lung gas means that the volume reached by a tracer gas depends on how long and how the tracer gas is allowed to mix. Gas that communicates slowly can be partly represented during one observation window and more fully represented during another. This differs from assuming that one fixed volume is permanently isolated.

An original compartmental model makes the dependence quantitative. A reservoir of volume $V_A$ initially contains helium fraction $c_0$ and rapidly mixes with a fast lung region $V_f$. Their combined volume is $V_0=V_A+V_f$, with initial fraction $c_f(0)=V_A{c}_0/V_0$. A slow region of volume $V_s$ initially contains no tracer gas and exchanges equal gas volumes in each direction at rate $k$. Assuming constant volumes and well-mixed compartments,Total tracer gas is conserved, while the concentration difference decays withApplying an equilibrium dilution formula before equilibration gives an apparent lung volume ${\hat{V}}_L(t)=V_A[c_0/c_f(t)-1]$. It starts at $V_f$ and tends toward $V_f+V_s$ in this ideal model.

For stipulated $V_A=3$ L, $V_f=3$ L, $V_s=2$ L, $c_0=0.10$ and $\tau =10$ s, the recorded concentration is $0.0375+0.0125e^{-t/(10\mathrm{s})}$. After 10 s, the apparent volume is about 4.13 L; after full equilibration it is 5 L. The change occurs without adding lung gas: the tracer gas is reaching more of the existing volume. These are constructed mixing calculations, not a clinical waiting-time recommendation.

The primary respiratory text describes a closed-circuit helium apparatus with continuous concentration measurement using a thermal-conductivity meter, together with gas-volume corrections. It also states that poor communication and short dilution time can limit the represented gas volume.^[ref] Leaks, changing gas conditions and unequal starting volumes can produce additional differences. A protocol-defined apparent volume therefore needs its mixing and measurement conditions attached to the result.

Forced expiratory volume in one second, abbreviated FEV1, is the gas volume expelled during the first second of a forced expiration timed according to the specified start-of-test convention. It is a volume, normally reported in litres, rather than a volumetric flow rate. FEV1 measures early emptying, while FVC describes the total volume expelled in the maneuver.^[ref]

For an ideal start with no timing offset,An original exponential model $E(t)=V_{\infty }(1-e^{-t/\tau })$ therefore gives ${\mathrm{FEV}}_1=V_{\infty }(1-e^{-1\mathrm{s}/\tau })$. At stipulated $V_{\infty }=4$ L and $\tau =2$ s, the first-second volume is about 1.57 L. The integral's units are litres even though the integrand is expressed in litres per second.

Flow-history ambiguity of a timed expiratory volume explains why FEV1 is not a complete flow trace. Over the first second, constant flow $Q_1=2\mathrm{L}{\mathrm{s}}^{-1}$ and the decreasing flow $Q_2=(3-2t)\mathrm{L}{\mathrm{s}}^{-1}$, with numerical $t$ in seconds, both integrate to 2 L. Their initial flows differ, and their instantaneous flows agree only at 0.5 s. These constructed histories show that equal FEV1 need not imply equal peak flow or equal waveform.

Real timing does not simply begin whenever a recorder is switched on. The 2019 statement uses back-extrapolation to define Time 0 and includes the volume expired before that time in both FEV1 and FVC.^[ref] State the convention rather than silently replacing it with an arbitrary one-second window.

The FEV1 to FVC ratio divides early emptying by total expelled volume. It is dimensionless and depends on both measurements. A reproducible FEV1 alone does not establish complete expiration, absolute lung size or the pressure-flow relation; these require additional observations or a different measurement model.

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 $A={\int }_0^{T}Q(t)dt$, then every admissible perturbation $h$ satisfyingproduces 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 $Q_0=1\mathrm{L}{\mathrm{s}}^{-1}$ continue for 4 s. Modify only the first second, with $T=1$ s, byand 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 isAt 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.^[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.

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.

Forced vital capacity, abbreviated FVC, is the total gas volume expelled in a forced expiration following maximal inspiration, under the stated maneuver and acceptance criteria. It is a difference between lung volumes, not the total lung capacity: residual lung volume remains after expiration. The 2019 technical statement defines the maneuver and separately evaluates whether its start and end support the reported quantities.^[ref]

An original compartment illustration starts with 6 L of lung gas and ends with 2 L. The expelled volume is 4 L. Starting with 8 L and ending with 4 L gives the same FVC, despite different absolute volumes. Thus absolute-volume ambiguity in spirometry applies even to a perfectly measured, fully completed expiration.

Duration also affects what is collected. For a constructed exponential emptying curve,At $V_{\infty }=4$ L, $\tau =2$ s and $T=6$ s, about 3.80 L has been expelled and about 0.20 L of this model's eventual expelled volume remains. This is an explanation of finite-time capture, not a universal breathing time requirement or a patient model.

Early termination bias in forced vital capacity can also change a ratio. Suppose a complete hypothetical maneuver has FEV1 of 2.4 L and FVC of 4.0 L, giving a FEV1 to FVC ratio of 0.60. If a recording ends later than the first second but captures only 3.0 L, treating that incomplete total as FVC would yield 0.80 with the same numerator. An apparently larger ratio can therefore result from losing late expired volume, rather than from an improvement in early emptying.

The technical statement distinguishes FEV1 and FVC acceptability because errors can affect the two quantities differently.^[ref] Reported volume, repeatability and maneuver completeness are distinct pieces of evidence. Agreement between repeated incomplete traces does not independently establish that their common endpoint is valid.

Early termination bias in forced vital capacity occurs when a recording misses late expired gas but its incomplete total is treated as FVC. If the first second was validly measured, FEV1 can remain unchanged while the denominator of the FEV1 to FVC ratio becomes too small. This can make the ratio appear larger without any change in early emptying.

An original algebraic model writes true total expired volume as $B$, measured first-second volume as $A$, and lost tail volume as $L$, with $0\le L<B$. ThenFor positive $A$, the relative ratio inflation is $\epsilon /(1-\epsilon )$. Losing 10% of the total volume raises the ratio by about 11.1%; losing 25% raises it by one third. For stipulated $A=2.4$ L and $B=4$ L, losing 1 L changes the ratio from 0.60 to 0.80.

Tail-volume loss and ratio inflation in truncated expiration can also be studied as a function of time. In a constructed exponential model $E(t)=B(1-e^{-t/\tau })$, a stop at $T\ge 1$ s loses fraction $\epsilon =e^{-T/\tau }$. Keeping the ratio inflation below a chosen positive fraction $\delta$ requiresAt stipulated $\tau =2$ s and $\delta =0.01$, this model needs about 9.23 s. This is a mathematical illustration of tail capture, not a prescribed clinical expiration duration. Different time constants give different uncollected tails at the same clock time.

The primary technical statement assesses end of forced expiration and distinguishes FEV1 from FVC acceptability rather than treating one fixed duration as universally sufficient.^[ref] If the stop also invalidates the first second, the assumption that $A$ is unchanged no longer holds, and the ratio's error cannot be inferred from denominator loss alone.

Repeatability does not remove this bias. Repeated equally incomplete traces can agree closely while missing the same late gas. Timing quality, maneuver completeness and reproducibility are separate observations.

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.

A spirometer is an instrument for measuring gas volume changes associated with breathing. Its output supports spirometry. A displacement design follows the movement of a gas-containing chamber; a flow-measuring design records volumetric flow rate and integrates it. In a pneumotachograph, a measured pressure difference across a flow-resistive element supplies the signal used to estimate volumetric flow rate.

For an ideal piston with cross-sectional area $A$ and displacement $x$, the transferred volume is $V=Ax$. For a flow-measuring instrument,These are different measurement chains for the same volume change. The 2019 primary technical statement explicitly discusses volume and flow instruments and their quality assurance.^[ref]

An original error model makes integration drift visible. If a flow sensor reports $Q_m(t)=gQ(t)+b$, with constant gain $g$ and zero offset $b$, its integrated result isAn offset of $0.020\mathrm{L}{\mathrm{s}}^{-1}$ contributes 0.020 L during one second but 0.20 L during ten seconds. It can therefore change a long-expiration total more than an early timed volume, even though the flow offset is constant. A gain error instead scales both in proportion when no offset is present.

Spirometer calibration verification uses a known gas transfer to test the assembled measurement chain. The 2019 statement specifies at least daily verification with a 3-L syringe at several flows and a measured-volume accuracy requirement of ±3%.^[ref] For a stipulated ideal 3.000 L transfer, a report of 3.090 L differs by 3%; identical total volumes at different transfer speeds can also expose a flow-dependent response.

Measurement calibration, leak checks and correct timing address different errors. Agreement with a syringe transfer establishes a specific instrument response; it does not by itself show that a person's breathing maneuver was complete or physiologically comparable to another attempt.

Spirometer calibration verification checks whether a spirometer measures a known input within specified limits. It differs from measurement calibration, which establishes the relation between input and indication, and from an adjustment that changes that relation. A satisfactory verification applies to the tested equipment, assembly and conditions; it does not validate a person's expiration maneuver.

The primary 2019 technical statement specifies daily verification with a 3-L syringe cycled over multiple flows. Its stated measured-volume accuracy requirement is ±3%, and a filter used for testing is also used during verification.^[ref] The known volume is useful precisely because it can be delivered at different speeds.

An original gain-and-offset model explains this design. If measured flow is $Q_m=gQ+b$, integrating one transfer of known volume $V_0$ over duration $T$ givesFor the same $V_0$ at two durations,A single transfer cannot distinguish gain from zero offset in this model; two different durations can.

Take stipulated 3.000 L transfers lasting 1 s and 6 s, reported as 3.020 and 3.120 L. The inferred offset is $0.100/5=0.020\mathrm{L}{\mathrm{s}}^{-1}$, while $g=1.000$. The slow transfer is 4% high, exceeding the cited 3% limit. If both transfers instead read 3.060 L, this model gives $g=1.020$ and $b=0$: a 2% gain error. These are constructed instrument responses, not a calibration record for an actual device.

Identifying flow-sensor gain and offset with timed transfers relies on the model assumptions. Nonlinear response, leaks or an inaccurately known syringe volume can produce other patterns. Reference-volume uncertainty belongs in the uncertainty budget, and repeated transfers share it rather than independently re-estimating it. A test at several flows is therefore more informative than repeating one convenient delivery speed.

Identifying flow-sensor gain and offset with timed transfers uses a known gas volume delivered over different durations to distinguish multiplicative response from accumulated zero error. In the model $Q_m=gQ+b$, a transfer of volume $V_0$ lasting $T_i$ producesThe observations form a linear regression with design matrix rows $(V_0,T_i)$. If $V_0\ne 0$, gain and offset are identifiable only when the durations are not all equal. Repeating one duration measures the same combination of $g$ and $b$ more precisely, without separating them.

For two noise-free transfers,Stipulated 3 L transfers lasting 1 and 6 s, measured as 3.020 and 3.120 L, give $\hat{b}=0.020\mathrm{L}{\mathrm{s}}^{-1}$ and $\hat{g}=1.000$. The different durations expose an offset that one transfer could misinterpret as a gain error.

Timing spread and gain-offset estimator covariance quantifies the design. Suppose $n$ observations have independent equal volume-error variance $s^2$, and define $S_{TT}={\sum }_i(T_i-\overline{T}{)}^2$. Ordinary least squares givesUnder the stipulated model, wider timing separation improves offset precision. For the two durations above and $s=0.010$ L, the offset's standard uncertainty is about $0.00283\mathrm{L}{\mathrm{s}}^{-1}$ and the gain's about 0.00406.

The reference syringe volume was treated as known in those formulas. If it is uncertain, repeated sensor tests share that reference uncertainty. Indeed the observations identify only the product $g{V}_0$ when both factors are unknown. Additional repetitions cannot establish an absolute gain without an independently characterized volume. The GUM treats fit uncertainties and common-reference correlations as distinct contributions.^[ref]

The primary spirometry statement's multiple-flow syringe checks test the practical response over different delivery rates.^[ref] Actual leaks, flow-dependent nonlinearities and uncertain timing require a richer model than a constant gain and offset.

Timing spread and gain-offset estimator covariance describes how delivery durations affect both precision and correlation when a known volume calibrates an integrated volumetric flow rate sensor. In the model $y_i=g{V}_0+b{T}_i+{ϵ}_i$, assume $V_0$ and durations $T_i$ are known and independent volume errors have common variance $s^2$. Define $S_{TT}={\sum }_i(T_i-\overline{T}{)}^2$.

The least squares slope estimates offset $b$, while the intercept divided by $V_0$ estimates gain $g$. Their covariance isThe negative covariance reflects a tradeoff: a larger fitted offset can be compensated by a smaller gain while preserving responses near the middle of the observed durations.

For an original design with $V_0=3$ L, durations 1 and 6 s, and $s=0.010$ L, the standard uncertainties are approximately 0.00283 L/s for offset and 0.00406 for gain. Their correlation is about $-0.814$. Moving the second duration to 2 s, with everything else stipulated unchanged, makes the offset uncertainty five times larger. Wider duration spread supplies more leverage for separating slope from intercept within this model.

Prediction shows why the covariance matters. For a new duration $T$, the fitted mean response is $\hat{y}(T)=V_0\hat{g}+T\hat{b}$, withAt the midpoint 3.5 s in the example, its standard uncertainty is $s/\sqrt{2}\approx 0.00707$ L. Incorrectly dropping the gain-offset covariance would give about 0.0157 L. This is uncertainty of the fitted mean; an independent future observation adds its own error variance.

Centering time at $\overline{T}$ makes the fitted midpoint response and slope uncorrelated under the stated equal-variance model. Transforming back to physical gain restores the covariance, so a change of coordinates does not remove shared uncertainty. The GUM explicitly retains such covariance terms in propagation.^[ref] Reference-volume uncertainty, uncertain timing and nonlinear volumetric flow rate response require additional terms beyond this original regression calculation.