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Unequal filling times in apparent lung compliance 2026-11-05

Unequal filling times in apparent lung compliance occur when different communicating lung regions have different resistance-compliance time constants. An observation made before all regions equilibrate can measure less volume response than the equilibrium elastic relation predicts. This provides a concrete mechanism for an observation-time dependence of apparent compliance without changing the underlying component compliances.

Consider an original passive model with parallel compartments exposed to one imposed pressure step

Δ P

. In compartment

i

, let added volume be

v_{i}

, airway resistance

R_{i}

and elastic lung compliance

C_{i}

. The equation and initially unfilled solution are

R_{i} \frac{d v _{i}}{d t} + \frac{v _{i}}{C _{i}} = Δ P, v_{i} (t) = C_{i} Δ P (1 - e^{- t / τ_{i}}), τ_{i} = R_{i} C_{i} .

(1)

The apparent finite-time response per unit pressure is therefore

C_{app} (T) = \frac{\sum _{i} v _{i} ( T )}{Δ P} = \sum_{i} C_{i} (1 - e^{- T / τ_{i}}) .

(2)

For positive component values it rises toward

C_{\infty} = \sum_{i} C_{i}

. This is a pressure-step response, not a definition that replaces every breathing-cycle compliance convention.

Take

C_{1} = C_{2} = 0.10 L (cm H_{2} O)^{- 1}

τ_{1} = 0.2

s and

τ_{2} = 2

s. At

T = 0.5

s, the fast region has filled about 91.8% of its eventual increment, while the slow region has filled only 22.1%. The summed apparent compliance is about 0.114, compared with the equilibrium value 0.20 in the same units. After 2 s it is about 0.163. If both time constants were 0.2 s, the 0.5-s value would instead be about 0.184, despite the same equilibrium compliance.

Early and late observations constrain different combinations. The initial volume slope is

Δ P \sum_{i} 1/ R_{i}

, whereas the final volume change is

Δ P \sum_{i} C_{i}

. A sum of exponentials also need not have one fixed decay time: a single fitted time constant depends on the fitted interval.

A primary pilot study comparing static and dynamic respiratory-system measurements discusses expiration-based estimates and the possibility of multiple time constants.^[ref] The constructed compartment example explains why measurement duration and model form belong with the reported value. Distinguishing lung-only from total respiratory-system mechanics still requires the appropriate internal pressure reference.

Figure 1.
Original two-compartment pressure-step model: unequal time constants produce an observation-time dependence of apparent lung compliance
. **Details**.

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