 
Summary: Perimeter growth of a branched structure: Application to crackle sounds in the lung
Adriano M. Alencar,1,2,
* Sergey V. Buldyrev,2
Arnab Majumdar,2
H. Eugene Stanley,2
and Be´la Suki1
1
Department of Biomedical Engineering, Boston University, Boston, Massachusetts 02215, USA
2
Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA
Received 25 February 2003; published 21 July 2003
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 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, remains close to 2, while for a chain, Z 2 and 1, exactly. We also find a
mean field solution of the model.
DOI: 10.1103/PhysRevE.68.011909 PACS number s : 87.19. j, 43.25. y
