The inhalation of micron-sized aerosols into the lung's acinar region may be recognized as a possible health risk or a therapeutic tool. In an effort to develop a deeper understanding of the mechanisms responsible for acinar deposition, we have numerically simulated the transport of nondiffusing fine inhaled particles (1 mum and 3 microm in diameter) in two acinar models of varying complexity: (i) a simple alveolated duct and (ii) a space-filling asymmetrical acinar branching tree following the description of lung structure by Fung (1988, "A Model of the Lung Structure and Its Validation," J. Appl. Physiol., 64, pp. 2132-2141). Detailed particle trajectories and deposition efficiencies, as well as acinar flow structures, were investigated under different orientations of gravity, for tidal breathing motion in an average human adult. Trajectories and deposition efficiencies inside the alveolated duct are strongly related to gravity orientation. While the motion of larger particles (3 microm) is relatively insensitive to convective flows compared with the role of gravitational sedimentation, finer 1 microm aerosols may exhibit, in contrast, complex kinematics influenced by the coupling between (i) flow reversal due to oscillatory breathing, (ii) local alveolar flow structure, and (iii) streamline crossing due to gravity. These combined mechanisms may lead to twisting and undulating trajectories in the alveolus over multiple breathing cycles. The extension of our study to a space-filling acinar tree was well suited to investigate the influence of bulk kinematic interaction on aerosol transport between ductal and alveolar flows. We found the existence of intricate trajectories of fine 1 microm aerosols spanning over the entire acinar airway network, which cannot be captured by simple alveolar models. In contrast, heavier 3 microm aerosols yield trajectories characteristic of gravitational sedimentation, analogous to those observed in the simple alveolated duct. For both particle sizes, however, particle inhalation yields highly nonuniform deposition. While larger particles deposit within a single inhalation phase, finer 1 microm particles exhibit much longer residence times spanning multiple breathing cycles. With the ongoing development of more realistic models of the pulmonary acinus, we aim to capture some of the complex mechanisms leading to deposition of inhaled aerosols. Such models may lead to a better understanding toward the optimization of pulmonary drug delivery to target specific regions of the lung.

Low Reynolds number flows (Re<1) in the human pulmonary acinus are often difficult to assess due to the submillimeter dimensions and accessibility of the region. In the present computational study, we simulated three-dimensional alveolar flows in an alveolated duct at each generation of the pulmonary acinar tree using recent morphometric data. Rhythmic lung expansion and contraction motion was modeled using moving wall boundary conditions to simulate realistic sedentary tidal breathing. The resulting alveolar flow patterns are largely time independent and governed by the ratio of the alveolar to ductal flow rates, Qa/Qd. This ratio depends uniquely on geometrical configuration such that alveolar flow patterns may be entirely determined by the location of the alveoli along the acinar tree. Although flows within alveoli travel very slowly relative to those in acinar ducts, 0.021%

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Global attractive stability conditions for the equilibrium of the bouncing ball system have been proven in this paper using an extension of Lyapunov’s direct method to non-autonomous systems. The bouncing ball system is a non-smooth nonautonomous system with a unilateral frictionless constraint and has been extensively studied in the literature as a standard example of chaotic dynamics. The system consists of a point mass in a constant gravitational field, which bounces inelastically on a flat vibrating table. Furthermore, it is proven that the attractivity of the equilibrium is symptotic, i.e. there exists a finite time for which the solution has converged exactly to the equilibrium. For this attraction time, an upper bound is given in this paper.

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