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

We propose a method for designing the bronchial tree where the branching process is stochastic and the diameter (d) of a branch is determined by its flow rate (Q). We use two principles: the continuum equation for flow division and a power-law relationship between d and Q, given by Q similar to d(n), where n is the diameter exponent. The value of n has been suggested to be similar to 3. We assume that flow is divided iteratively with a random variable for the flow-division ratio, defined as the ratio of flow in the branch to that in its parent branch. We show that the cumulative probability distribution function of Q, P(>Q) is proportional to Q(-1). We analyzed prior morphometric airway data (O. G. Raabe, H. C. Yeh, K. M. Schum, and R. F. Phalen, Report No. LF-53, 1976) and found that the cumulative probability distribution function of diameters, P(>d), is proportional to d(-n), which supports the validity of Q similar to d(n) since P(>Q) similar to Q(-1). This allowed us to assign diameters to the segments of the flow-branching pattern. We modeled the bronchial trees of four mammals and found that their statistical features were in good accordance with the morphometric data. We conclude that our design method is appropriate for robust generation of bronchial tree models.