= Secant versus differential airway resistance
\b[Secant versus differential airway resistance] distinguishes a finite pressure-to-flow ratio from the local slope of a pressure-flow relation. If $Q$ is <volumetric flow rate> and $\Delta P$ the corresponding airway <pressure difference>, define
$$
R_{\rm sec}(Q)=\frac{\Delta P(Q)}Q,\qquad
R_{\rm diff}(Q_0)=\left.\frac{d\Delta P}{dQ}\right|_{Q_0}.
$$
The secant joins the origin to the operating point. The differential value describes a sufficiently small change about that operating point. Both have units of pressure divided by flow, but they agree at every flow only for a linear relation through the origin.
An original nonlinear model is $\Delta P=aQ+bQ|Q|$, with $a,b>0$. For nonzero flow,
$$
R_{\rm sec}=a+b|Q|,\qquad R_{\rm diff}=a+2b|Q|.
$$
Choose coefficients $a=2$ and $b=4$ when pressure is expressed in centimetres of water and flow in litres per second. At $Q_0=0.5\,\mathrm{L\,s^{-1}}$, pressure is 2 cm water, so the secant resistance is 4 and differential resistance is 6 in $\mathrm{cm\,H_2O\,s\,L^{-1}}$. At $Q=0.6$, pressure is 2.64 cm water. The tangent predicts an increase $6(0.1)=0.60$ cm water; the exact increase is 0.64. Their difference is the quadratic remainder $b(\Delta Q)^2$.
At negative flow the pressure reverses sign in this odd model, but both resistance measures remain positive. At zero flow the ratio needs its limiting definition, which here equals $a$. These are properties of the stipulated curve, not measured airway coefficients.
<Operating-point time constants with nonlinear airway resistance> use the differential value. Linearizing a flow equation around a fixed operating point gives $\delta P\simeq R_{\rm diff}\delta Q$. If that flow charges a local elastic compartment of <lung compliance> $C$, the small-perturbation <time constant> is $R_{\rm diff}C$, rather than an arbitrary finite secant ratio multiplied by $C$.
An experiment must identify the pressure drop belonging to the <airways>. Mouth-to-pleural pressure also contains elastic loading, so its slope is not automatically airway resistance. The pulmonary-function text treats airway resistance and lung compliance separately, and the 1959 study's differential pressure and balloon apparatus illustrates the need for multiple mechanical measurements.<Pulmonary Function Testing by Warren M. Gold and Laura L. Koth>[Airway Resistance and Static and Dynamic Compliance]{ref}<Airway Resistance and Lung Compliance by Ole Storstein and others>[Methods, mouthpiece pressure and intra-esophageal balloon]{ref}
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