Steady and unsteady pressure-flow relationships in central airways

We measured pressure-flow relationships in a noncompliant five-generation cast of human central airways using air, HeO2, and SF6O2 at 0, 0.25, 0.50, 1.0, and 2.0 Hz with tidal volumes of 0.25, 0.5, and 1.0 liter. When dimensionless pressure drops for steady inspiratory and expiratory flows of the various gas mixture were plotted against Reynolds' number on a log-log scale (Moody diagram), they formed two curves as fluid mechanical theory predicts. At frequencies higher than 0.25 Hz, data obtained from 1) the same gas and same stroke volume, 2) the same frequency and same stroke volume but different gases, and 3) the same gas at the same frequency but with different stroke volumes, all deviated from the steady flow curves in the Moody diagram, always tending to increase the dimensionless pressure drop. The increase was largest when instantaneous flow was near zero and was minimal at the peak flow in a given inspiration or expiration. These data led to the identification of a dimensionless parameter (epsilon) that reflects the relative importance of local acceleration (unsteadiness) to convective acceleration at any given instant during a flow cycle. A dimensional analysis then reveals that the pressure-flow relationship in a given airway system is uniquely and completely determined by a combination of three dimensionless parameters: Reynolds number (Re), Womersley number (alpha), and the new parameter (epsilon). With this set of parameters we can explain all reported apparent paradoxes as well as the present findings.