Perimeter growth of a branched structure: Application to crackle sounds in the lung

We study an invasion percolation process on Cayley trees and find that the dynamics of perimeter growth is strongly dependent on the nature of the invasion process, as well as on the underlying tree structure. We apply this process to model the inflation of the lung in the airway tree, where crackling sounds are generated when airways open. We define the perimeter as the interface between the closed and opened regions of the lung. In this context we find that the distribution of time intervals between consecutive openings is a power law with an exponent $\beta \approx 2.$ We generalize the binary structure of the lung to a Cayley tree with a coordination number Z between 2 and 4. For $Z=4,$ $\beta$ remains close to 2, while for a chain, $Z=2\mathrm{}$ and $\beta =1,$ exactly. We also find a mean field solution of the model.