A simple geometrical pattern for the branching distribution of the bronchial tree, useful to estimate optimality departures.

The design of the bronchial tree has largely been proposed as a model of optimal design from a physical-functional perspective. However, the distributive function of the airway may be more related to a geometrical than a physical problem. The bronchial tree must distribute a three dimensional volume of inspired air on a two dimensional alveolar surface, included in a limited volume. It is thus valid to ask whether an optimal bronchial tree from a physical perspective is also optimum from a geometrical point of view. In this paper we generate a simple geometric model for the branching pattern of the bronchial tree, deducing relationships that permit estimation of the departures from the geometrical optimum of each bifurcation. We also, for comparative purposes, estimate the departures from the physical optimum. From the geometrical assumptions: i) a symmetrical dichotomic fractal design, ii) with minimum volume and iii) maximum dispersion of the terminal points; and several simulations we suggest that the optimality is characterized by a bifurcation angle theta approximately 60 degrees and a length reduction scale gamma = (1/2)(1/3) = 0.7937. We propose distances from the physical and geometrical optimality defined as Euclidean distances from the expected optima. We show how the advanced relationships and the distances can be used to estimate departures from the optimality in bronchographs of four species. We found lower physical and geometrical departures in the distal zone than those of the proximal zones, as well as lower physical than geometrical departures from optimality.

Departures from the physical optimality in the bronchial tree of rats (Rattus norvegicus).

We studied the departure from the physical optimality of the bronchial tree of rats using both i) the minimum volume and power and ii) the minimum surface and drag criteria, considering the bronchial junction as the unit study based on Zamir's model for vascular trees. Our results show deviations of the junctions of the bronchial tree from the expected optimums in the proximal airway that can be explained by both, the turbulent or transitional flow regime, and the airway's necessity to distribute its terminal branches in the alveolar surface filling the thoracic volume. The departures of the observed values at the optimum for the minimum volume and power were significantly different than the obtained departure values for the minimum surface and drag criteria. The departure from the optimum was directly related to the diameter of the smallest branch. The slopes of the regressions for the two criteria were different. The regression lines intercept at a bronchial diameter d2 = 0.129 mm. This result agreed with the idea that the tube diameter is limited at small values by the increasing flow resistance with decreasing tube diameter while at large values is limited by the increasing tube volume and dead space with increasing tube diameter.

Departures from the physical optimality in the bronchial tree of rats (Rattus norvegicus)

We studied the departure from the physical optimality of the bronchial tree of rats using both i) the minimum volume and power and ii) the minimum surface and drag criteria, considering the bronchial junction as the unit study based on Zamir's model for vascular trees. Our results show deviations of the junctions of the bronchial tree from the expected optimums in the proximal airway that can be explained by both, the turbulent or transitional flow regime, and the airway's necessity to distribute its terminal branches in the alveolar surface filling the thoracic volume. The departures of the observed values at the optimum for the minimum volume and power were significantly different than the obtained departure values for the minimum surface and drag criteria. The departure from the optimum was directly related to the diameter of the smallest branch. The slopes of the regressions for the two criteria were different. The regression lines intercept at a bronchial diameter d2 = 0.129 mm. This result agreed with the idea that the tube diameter is limited at small values by the increasing flow resistance with decreasing tube diameter while at large values is limited by the increasing tube volume and dead space with increasing tube diameter

Ontogenic changes in the fractal geometry of the bronchial tree in Rattus norvegicus

Respiration and metabolism change dramatically over the course of the development of vertebrates. In mammals these changes may be ascribed to organogenesis and differentiation of structures involved in gas exchange and transport and the increase in size. Since young as well as mature individuals must be well-designed if the species is to survive, the physiological changes during the development should be matched with geometrical or structural adjustments of the respiratory system. The aim of this study was to evaluate changes in the fractal geometry of the bronchial tree during the postnatal development of the rat. The average fractal dimension of the bronchial tree of the rats was 1.587, but that of juveniles was larger than that of the adults. We found a significant negative correlation between age and fractal dimension. This correlation could be considered be misleading because of the difficulty of separating age/body size effects. Nevertheless, because fractal dimensions of the bronchial tree of rabbits and humans are known to be similar, 1.58 and 1.57 respectively, the body size effect may be nil. To our knowledge, this is the first report of ontogenetic changes in the fractal dimension of the bronchial tree in mammals.