It is shown in this paper that the nearly constant length-to-diameter ratio observed with conducting airways of human bronchial tree can be explained based on the fluid dynamic optimality principle. In any branched tube there are two pressure loss mechanisms, one for wall friction in the tube section and the other for flow division in the branching section, and there exists an optimal length-to-diameter ratio which minimizes the total pressure loss for a branched tube in laminar flow condition. The optimal length-to-diameter ratio predicted by the pressure loss minimization shows an excellent agreement with the length-to-diameter ratios found in the human conducting airways.

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In the human bronchial tree the branching angle becomes larger with generation or for the smaller branches. Previous theories based on single parameter optimization have not been successful at all in predicting the consistent increasing trend of branching angle with continued bifurcation. In this study a new theory for the optimality of the branching angle is proposed, which is based on the optimization between dual competing performances, the maximum space-filling capability at the expense of minimum energy loss. A large-angle branching gives an effect of delivering air into a new direction away from the preceding airways. It then has an effect of utilizing the lung volume with better uniformity, but at the same time inevitably requires a high pressure loss. It is shown in this paper that the ever increasing branching angle with generation can be well explained as the optimum branching structure where the dual opposing performance of space filling and pressure loss is optimized. In estimating the pressure loss, branching loss is considered in addition to the Poiseuille loss. Change of predicted optimum branching angle with generation shows an excellent agreement with the observed data found in the human conducting airways.

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