Use of fractal geometry to propose a new mechanism of airway-parenchymal interdependence

The topic of airway-parenchymal interdependence (API) is of great importance to those interested in identifying factors that influence airway patency. A carefully designed experiment has raised questions about the classical concept of API. This paper proposes a new mechanism of API. The pulmonary lobe is an aggregated body consisting of many Miller’s lobular polyhedrons and a fractal bronchial tree. The fractal cartilaginous bronchial tree was assumed to be characterized by both Horton’s ratio ( 1 2    j j L L , where 1  j L and j L denote the mean lengths of branches at Horsfield’ order of j + 1 and j) and power laws between diameters and lengths of branches. Fluid dynamic parameters of fractal trees were assumed to be interrelated among powers and λ. A noncartilaginous lobular bronchiole is adjoined to the edge of a lobular polyhedron, and is encircled by an inextensible basement membrane to reflect a reversible relationship of rlLl = constant(c), where rl and Ll denote the diameter and the length of a lobular bronchiole, respectively. API at the level of the lobular bronchiole was described by        log 1 1 5 log       l l r h  c , where rl and hl denote the diameter of the lobular bronchiole and the parenchymal parameter relating the size of the lobular polyhedron, respectively. If the distribution in sizes of the lobular polyhedrons was described by a Weibull’s probability density function characterized by the shape parameter m as well as the fractal parameter λ = 0.5, the diameter R of a cartilaginous bronchial branch was determined by      log 3 7 log   R F m h c , where F(m) denotes a function of m, and h denotes the mean size of the polyhedrons in the lobe. As a conclusion, API can be described by a combination of both lobular API and corresponding adaptive changes in the degree of contraction of airway smooth muscles.