= Protocol-dependent accessibility of lung gas
\b[Protocol-dependent accessibility of lung gas] means that the <volume> reached by a <tracer gas> depends on how long and how the <tracer gas> is allowed to mix. Gas that communicates slowly can be partly represented during one observation window and more fully represented during another. This differs from assuming that one fixed <volume> is permanently isolated.
An original <compartmental model> makes the dependence quantitative. A reservoir of <volume> $V_A$ initially contains <helium> fraction $c_0$ and rapidly mixes with a fast lung region $V_f$. Their combined <volume> is $V_0=V_A+V_f$, with initial fraction $c_f(0)=V_Ac_0/V_0$. A slow region of <volume> $V_s$ initially contains no <tracer gas> and exchanges equal gas <volumes> in each direction at rate $k$. Assuming constant <volumes> and well-mixed compartments,
$$
V_0\dot c_f=-k(c_f-c_s),\qquad V_s\dot c_s=k(c_f-c_s).
$$
Total <tracer gas> is conserved, while the concentration difference decays with
$$
\tau=\frac{V_0V_s}{k(V_0+V_s)},\qquad
c_\infty=\frac{V_Ac_0}{V_0+V_s},\qquad
c_f(t)=c_\infty+\frac{V_s}{V_0+V_s}c_f(0)e^{-t/\tau}.
$$
Applying an equilibrium dilution formula before equilibration gives an apparent lung <volume> $\widehat V_L(t)=V_A[c_0/c_f(t)-1]$. It starts at $V_f$ and tends toward $V_f+V_s$ in this ideal model.
For stipulated $V_A=3$ L, $V_f=3$ L, $V_s=2$ L, $c_0=0.10$ and $\tau=10$ s, the recorded concentration is $0.0375+0.0125e^{-t/(10\,\mathrm s)}$. After 10 s, the apparent <volume> is about 4.13 L; after full equilibration it is 5 L. The change occurs without adding lung gas: the <tracer gas> is reaching more of the existing <volume>. These are constructed mixing calculations, not a clinical waiting-time recommendation.
The primary respiratory text describes a closed-circuit <helium> apparatus with continuous concentration measurement using a thermal-conductivity meter, together with gas-volume corrections. It also states that poor communication and short dilution time can limit the represented gas <volume>.<Pulmonary Function Testing by Warren M. Gold and Laura L. Koth>[Closed Circuit Methods, Figure 25-10, and discussion of poorly communicating gas]{ref} Leaks, changing gas conditions and unequal starting <volumes> can produce additional differences. A protocol-defined apparent <volume> therefore needs its mixing and measurement conditions attached to the result.
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