Analytical Design of the Human Bronchial Tree for Healthy Patients and Patients with Obstructive Pulmonary Diseases

Abstract

The study is aimed at the analytical design of the full human bronchial tree for healthy patients and patients with obstructive pulmonary diseases. Analytical formulas for design of the full bronchial tree are derived. All surfaces of the bronchial tree are matched with the second-order smoothness (there are no acute angles or ribs). The geometric characteristics of the human bronchial tree in the pathological case are modeled by a "starry" shape of the inner structure of the bronchus; the pathology degree is defined by two parameters: bronchus constriction level and degree of distortion of the cylindrical shape of the bronchus. Closed analytical formulas allow the human bronchial tree of an arbitrary complexity (up to alveoli) to be designed; moreover, the parametric dependences make it possible to specify any desirable degree of airway obstruction.

Method of Constructing an Asymmetric Human Bronchial Tree in Normal and Pathological Cases

“…The internal cross section of the bronchi is not rigorously circular even in the nonpathological case [16,17]. A brief review of the histological structure of the bronchi can be found in [2]. It should be only noted that the internal cross section of the bronchi has a "starry" shape.…”

“…The function that describes the cross-sectional shape of the bronchi of healthy patients and patients with obstructive pulmonary diseases can be written in the following form [2]:…”

“…Regular dichotomy is a symmetric model of the bronchial tree where all elements (branches) in one generation of bifurcations have identical sizes and bronchus deflection angles. Such a method of symmetric bronchial tree design was proposed in [2].…”

“…The upper part of the bronchial tree with well-described morphometric data is reviewed in several papers [3][4][5][6][7][8][9][10][11][12][13][14][15]. An advanced review of these investigations can be found in [2]. It should be only noted here that it is difficult to replicate the bronchial tree design of these researchers because: (1) not all morphometric data are provided in publications, (2) the algorithm of bifurcation design in the 3D simulation code used by researchers is not described in sufficient detail, and (3) the algorithm of construction of an individual bifurcation is too complicated (a detailed description of construction of an individual bifurcation in [5] takes eight pages).…”

“…This method does not require manual operations to design a bronchial tree and can be readily reproduced by other researchers. An algorithmic method of constructing regular dichotomy on the basis of simple analytical formulas was proposed in [2]. This method is extended below to the case of irregular dichotomy for an asymmetric bronchial tree.…”