The influence of variability on the optimal shape of an airway tree branching asymmetrically

The asymmetry of the bronchial tree has been reported on numerous occasions, and bronchi in the lung bifurcate most of the time into a major and a minor daughter. Asymmetry is most probably bound to play a role on the hydrodynamic resistance and volume occupation of the bronchial tree. Thus, in this work, we search for an optimal asymmetric airway tree crossed by Poiseuille flow that would be a good candidate to model the distal conductive part of the lung. The geometry is controlled by major and minor diameter reduction factors that depend on the generation. We show that the optimal asymmetric tree has diameter reduction factors that are adimensional from the second level of bifurcation and that they are highly dependent on the asymmetric ratio that defines the relative sizes of the major and minor branches in a bifurcation. This optimization also gives access to a cost function whose particularity is to be asymmetric around its minimum. Thus, the cliff-edge hypothesis predicts that if the system suffers variability, then the best tree is shifted from the optimal. We apply a recent theoretical model of cliff-edge in order to measure the role of variability on the determination of the best asymmetric tree. Then, we compare our results with lung data of the literature. In particular, we are able to quantify the variability needed to fit the data and to give hypothesis that could explain, at least partially, the shift found between the optimal tree and the measures in the case of asymmetric bronchial trees. Finally, our model predicts that, even if the population is adapted at best, there always exist individuals whose bronchial trees are associated with larger costs comparatively to the average and who ought to be more sensitive to geometrical remodeling.