= Respiratory-system compliance
{title2=$C_{\rm rs}$}
\b[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,
$$
C_{\rm rs}=\frac{dV}{dP_{\rm rs}},\qquad
P_{\rm rs}=P_L+P_W=P_{\rm alv}-P_{\rm body\ surface}.
$$
<Lung compliance> alone instead uses <transpulmonary pressure>, $P_L=P_{\rm alv}-P_{\rm pl}$. 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 $C_W$,
$$
\frac1{C_{\rm rs}}=\frac1{C_L}+\frac1{C_W},\qquad
C_{\rm rs}=\frac{C_LC_W}{C_L+C_W}.
$$
The reciprocals are component stiffnesses; <respiratory-system elastance> is $E_{\rm rs}=1/C_{\rm rs}$. For positive component compliances, the combined value is smaller than either component. OpenStax describes opposing lung and wall recoil and their pressure references.<Anatomy and Physiology 2e by J. Gordon Betts and others>[§22.3, pleural pressure and thoracic wall compliance]{ref}
An original example uses $C_L=0.20$ and $C_W=0.10\,\mathrm{L\,(cm\,H_2O)^{-1}}$. 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 is
$$
C_{\rm rs,interval}=\frac{V_T}{P_{\rm plat}-P_{\rm end}}.
$$
For stipulated $V_T=0.5$ L, $P_{\rm plat}=12.5$ and $P_{\rm end}=5$ 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.<Static and Dynamic Measurements of Compliance and Driving Pressure: A Pilot Study by Pierre Tawfik and others>[Introduction, Methods A–D; Study Procedures]{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.
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