Volume distributions of avalanches in lung inflation: A statistical mechanical approach

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Abstract
To study the dynamics of lung inflation, we introduce a statistical mechanical model that incorporates experimental observations that, during lung inflation from low volumes, (i) each individual airway segment opens when the external inflation pressure reaches a critical opening threshold corresponding to that segment and (ii) airway opening in the lung occurs in cascades or by avalanches. The model includes realistic asymmetry of the bronchial tree, tissue elasticity, and airway and alveolar dimensions. We perform numerical simulations of lung inflation to study the effects of these attributes on the volume distributions of both the first and all avalanches for three different distributions of critical opening threshold pressures: (a) a generation-independent, (b) a slightly generation-dependent, and (c) a highly generation-dependent distribution. For both the first and all avalanches we find that the volume distribution is a power law, except for the highly generation-dependent threshold distribution. Asymmetry and realistic airway and alveolar dimensions slightly modify the scaling region, but retain a power-law behavior as long as the distribution of threshold pressures is generation independent or slightly generation dependent. Also, for such a distribution of threshold pressures, the scaling exponent of the most realistic model (the asymmetric tree with realistic airway and alveolar dimensions and tissue elasticity) is 2, which is the value obtained both analytically using percolation theory and from simulations on a Cayley tree. Thus the power-law behavior and the scaling exponents are a consequence of finite-size effects and a distribution of threshold pressures that is generation independent or slightly generation dependent. We also predict the pressure-volume relationship of the model, which is easily and noninvasively accessible in clinical settings. The results of the avalanche size distributions and pressure-volume curves support the notion that at low lung volumes, the distribution of the critical opening threshold pressures in the normal lung is most likely wide with negligible generational dependence.