Respiratory-system compliance ($C_{\mathrm{rs}}$)

Respiratory-system compliance is the change in volume per change in elastic pressure across the passive lung and chest wall together. For a local static relation,Lung compliance alone instead uses transpulmonary pressure, $P_L=P_{\mathrm{alv}}-P_{\mathrm{pl}}$. The whole-system and lung-only quantities are different even when they refer to the same volume change.

At that shared volume, the elastic pressure increments add. With component compliances $C_L$ and $C_W$,The reciprocals are component stiffnesses; respiratory-system elastance is $E_{\mathrm{rs}}=1/C_{\mathrm{rs}}$. For positive component compliances, the combined value is smaller than either component. OpenStax describes opposing lung and wall recoil and their pressure references.^[ref]

An original example uses $C_L=0.20$ and $C_W=0.10\mathrm{L}(\mathrm{cm}{\mathrm{H}}_2\mathrm{O}{)}^{-1}$. The combined compliance is about 0.0667 in the same units. An added 0.50 L requires 2.5 cm water of lung elastic pressure and 5 cm water of wall pressure, totaling 7.5 cm water. Attributing that total entirely to the lung would underestimate lung-only compliance threefold.

In a passive ventilated-breath measurement, a zero-flow plateau airway pressure can be compared with the appropriately measured end-expiratory pressure. Under the stated equilibrium and no-effort assumptions, an interval estimate isFor stipulated $V_T=0.5$ L, $P_{\mathrm{plat}}=12.5$ and $P_{\mathrm{end}}=5$ cm water, this is again about 0.0667. The baseline must account for the relevant positive end-expiratory pressure; a pressure measured during nonzero flow also contains a resistive contribution. The primary pilot study explicitly compares pressure-based respiratory-system estimates under passive ventilation.^[ref]

State whether the number is a local slope or an interval ratio, which pressure references were used, and whether muscle activity and flow were excluded. Separating lung and wall contributions requires internal pressure information beyond the whole-system volume response.

Respiratory-system elastance is the local increase in passive elastic pressure per increase in respiratory volume. It is the reciprocal of local respiratory-system compliance:Its units are pressure per volume, such as $\mathrm{Pa}{\mathrm{m}}^{-3}$ or $\mathrm{cm}{\mathrm{H}}_2\mathrm{O}{\mathrm{L}}^{-1}$. A larger elastance means that the same small volume increment needs a larger elastic pressure increment.

The lung and chest wall share a volume increment while their pressure increments add. Thus their elastances add,whereas their compliances do not. For stipulated $C_L=0.20$ and $C_W=0.10\mathrm{L}(\mathrm{cm}{\mathrm{H}}_2\mathrm{O}{)}^{-1}$, the elastances are 5 and 10, totaling 15 in reciprocal units. Combined compliance is $1/15\approx 0.0667$. The underlying pressure partition is distinct from attributing all measured stiffness to the lung.^[ref]

Elastic work in a pressure-volume model gives another meaning to stiffness. Around a passive zero-recoil equilibrium, let added volume be $v$ and pressure $P=Ev$ with constant elastance. The reversible work needed to move from zero to $v$ isFor an original SI example, $E=10^6\mathrm{Pa}{\mathrm{m}}^{-3}$ and $v=5\times 10^{-4}{\mathrm{m}}^3$ give final pressure 500 Pa and elastic work 0.125 J. Doubling elastance at the same volume doubles this work; doubling volume at fixed elastance quadruples it. Resistive losses and active muscle work are outside this ideal elastic calculation.

If the pressure-volume curve is nonlinear, local elastance $dP/dV$ can differ from the interval ratio $\Delta P/\Delta V$. Its reciprocal corresponds to the local compliance at that state, not an average across an arbitrary range. A primary ventilator study compared different pressure-volume methods under passive conditions, making this measurement definition operational.^[ref] Report volume range, passive conditions and whether the value is a derivative or a secant.

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.