Branching design of the bronchial tree based on a diameter-flow relationship

Kitaoka, Hiroko, and Béla Suki. Branching designof the bronchial tree based on a diameter-flow relationship.J. Appl. Physiol. 82(3): 968-976, 1997.We propose a method for designing the bronchial tree where thebranching process is stochastic and the diameter(d) of a branch is determined by itsflow rate (Q). We use two principles: the continuumequation for flow division and a power-law relationship betweend and Q, given by Q ~ dⁿ,where n is the diameter exponent. The value ofn has been suggested to be ~3. Weassume that flow is divided iteratively with a random variable for theflow-division ratio, defined as the ratio of flow in the branch to thatin its parent branch. We show that the cumulative probabilitydistribution function of Q, P(>Q) is proportional to Q¹. Weanalyzed prior morphometric airway data (O. G. Raabe, H. C. Yeh, H. M. Schum, and R. F. Phalen, Report No.LF-53, 1976) and found that the cumulative probabilitydistribution function of diameters, P(>d), isproportional to dⁿ, which supportsthe validity of Q ~ dⁿ sinceP(>Q) ~ Q¹. This allowed us toassign diameters to the segments of the flow-branching pattern. Wemodeled the bronchial trees of four mammals and found that theirstatistical features were in good accordance with the morphometricdata. We conclude that our design method is appropriate for robustgeneration of bronchial tree models.