Approximation of a series with zero values by an invariant cumulative probability distribution curve

1996 ◽  
Vol 30 (12) ◽  
pp. 749-751
Author(s):  
V. F. Zalesskii
Keyword(s):  
Probability Distribution ◽  
Distribution Curve ◽  
Cumulative Probability ◽  
Cumulative Probability Distribution ◽  
Zero Values

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

1997 ◽  
Vol 82 (3) ◽  
pp. 968-976 ◽  
Author(s):  
Hiroko Kitaoka ◽  
Béla Suki
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Probability Distribution Function ◽  
Design Method ◽  
Random Variable ◽  
Bronchial Tree ◽  
Cumulative Probability ◽  
Cumulative Probability Distribution ◽  
Tree Models ◽  
Flow Division

Kitaoka, Hiroko, and Béla Suki. Branching design of 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 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 ∼ d n, where n is the diameter exponent. The value of n has been suggested to be ∼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, H. 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 ∼ d n since P(>Q) ∼ 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.

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