Relaxation behaviour in ultrametric spaces

Trapping of random walks on small-world networks

We investigate the trapping of random walkers on small-world networks (SWN's), irregular graphs. We derive bounds for the survival probability Phi(SWN)(n) and display its analysis through cumulant expansions. Computer simulations are performed for large SWNs. We show that in the limit of infinite sizes, trapping on SWNs is equivalent to trapping on a certain class of random trees, which are grown during the random walk.

Effect of airway closure on ventilation distribution

We examined the effect of airway closure on ventilation distribution during tidal breathing in six normal subjects. Each subject performed multiple-breath N2 washouts (MBNW) at tidal volumes of 1 liter over a range of preinspiratory lung volumes (PILV) from functional residual capacity (FRC) to just above residual volume. All subjects performed washouts at PILV below their measured closing capacity. In addition five of the subjects performed MBNW at PILV below closing capacity with end-inspiratory breath holds of 2 or 5 s. We measured the following two independent indexes of ventilation maldistribution: 1) the normalized phase III slope of the final breaths of the washout (Snf) and 2) the alveolar mixing efficiency of those breaths of the washout where 80-90% of the initial N2 had been cleared. Between a mean PILV of 0.28 liter above closing capacity and that 0.31 liter below closing capacity, mean Snf increased by 132% (P less than 0.005). Over the same volume range, mean alveolar mixing efficiency decreased by 3.3% (P less than 0.05). Breath holding at PILV below closing capacity resulted in marked and consistent decreases in Snf and increases in alveolar mixing efficiency. Whereas inhomogeneity of ventilation decreases with lung volume when all airways are patent (J. Appl. Physiol. 66: 2502-2510, 1989), airway closure increases ventilation inequality, and this is substantially reduced by short end-inspiratory breath holds. These findings suggest that the predominant determinant of ventilation distribution below closing capacity is the inhomogeneous closure of airways subtending regions in the lung periphery that are close together.