= Elastic work in a pressure-volume model
\b[Elastic work in a pressure-volume model] is the <work> required to move along the reversible elastic <pressure> response as <volume> changes. With added <volume> $v$ measured from an equilibrium state and recoil <pressure> $P_e(v)$ measured relative to that state,
$$
W_e(v)=\int_0^v P_e(x)\,dx.
$$
It is the <area> under the elastic <pressure> versus <volume> curve, not the final <pressure> multiplied by the final added <volume>. Constant <respiratory-system elastance> $E$ gives $P_e=Ev$ and $W_e=Ev^2/2$.
Nonlinearity matters even when the final <pressure> agrees. In an original SI example let
$$
P_e(v)=av+bv^3,\qquad
a=8\times10^5\,\mathrm{Pa\,m^{-3}},\qquad
b=8\times10^{11}\,\mathrm{Pa\,m^{-9}}.
$$
At $v=5\times10^{-4}\,\mathrm{m^3}$, the linear and cubic terms contribute 400 and 100 Pa, respectively. Integrating gives
$$
W_e=\frac12av^2+\frac14bv^4=0.100+0.0125=0.1125\,\mathrm J.
$$
A straight line from the origin to the same endpoint would instead give 0.125 J. Equal endpoint <pressure> and <volume> do not specify the entire loading <work>.
The local <respiratory-system elastance> is $dP_e/dv=a+3bv^2$, whereas the endpoint secant is $P_e/v=a+bv^2$. The integral uses the whole elastic curve. A constant baseline <pressure> $P_0$ would add $P_0v$ to boundary <work> computed from the absolute driving <pressure>; that term is absent from the baseline-relative elastic <energy> defined here.
Actual inflation and deflation can follow different paths. In a passive cyclic model, the enclosed <hysteresis> <area> $\oint P\,dV$ represents net mechanical <energy> dissipated per cycle rather than reversible <energy> stored at one endpoint. Resistive <pressure> and muscle activity must therefore be distinguished from elastic recoil. The primary pulmonary-function chapter explains why static and dynamic <lung compliance> use different measurement conditions.<Pulmonary Function Testing by Warren M. Gold and Laura L. Koth>[Lung Elasticity, Static and Dynamic Compliance: pressure-volume measurements and zero-flow conditions]{ref} The constructed curve above illustrates integration; it is not an observed patient curve.
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