Elastic work in a pressure-volume model

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,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 $W_e=E{v}^2/2$.

Nonlinearity matters even when the final pressure agrees. In an original SI example letAt $v=5\times 10^{-4}{\mathrm{m}}^3$, the linear and cubic terms contribute 400 and 100 Pa, respectively. Integrating givesA 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 $d{P}_e/dv=a+3b{v}^2$, whereas the endpoint secant is $P_e/v=a+b{v}^2$. The integral uses the whole elastic curve. A constant baseline pressure $P_0$ would add $P_{0}v$ 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 $\oint PdV$ 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.^[ref] The constructed curve above illustrates integration; it is not an observed patient curve.