 Secant versus differential airway resistance

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, defineThe 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,Choose coefficients $a=2$ and $b=4$ when pressure is expressed in centimetres of water and flow in litres per second. At $Q_0=0.5\mathrm{L}{\mathrm{s}}^{-1}$, pressure is 2 cm water, so the secant resistance is 4 and differential resistance is 6 in $\mathrm{cm}{\mathrm{H}}_2\mathrm{O}\mathrm{s}{\mathrm{L}}^{-1}$. 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 $\delta P\simeq R_{\mathrm{diff}}\delta Q$. If that flow charges a local elastic compartment of lung compliance $C$, the small-perturbation time constant is $R_{\mathrm{diff}}C$, 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.^[ref][ref]