Stochastic model of the pulmonary airway tree and its implications for bronchial responsiveness

Abstract

The resistance of the pulmonary conducting airway tree (Raw) is a consequence of the resistances of its component airways and how they are connected together. To date, theoretical calculations of Raw have been performed with the aid of mathematical models of the airway tree that are purely deterministic. That is, the mechanical properties of the component airways in these models are precisely defined functions of generation number. Such models take no account of the fact that the airways of a given generation are not all exactly the same but rather exhibit a spectrum of wall thicknesses, amounts of smooth muscle, and number of parenchymal attachments. In the present study, the properties of a 10-generation stochastic airway tree model are investigated. The lengths and radii of the airways in the tree are drawn randomly from probability distribution functions (PDFs), the means of which are deterministic functions of generation number and the standard deviations are assigned various values. Monte Carlo simulation is used to estimate the PDF of Raw itself in various conditions. We show that the relative width of the PDF of Raw may be comparable to that of the PDFs from which the individual airway radii were drawn. It is also shown that when bronchoconstriction is simulated by narrowing each airway by a random amount the resulting PDF for Raw may increase in width many times. We conclude that the variations in airway responsiveness seen in nature can only be properly understood when the distribution of airway properties within the lung are taken into account.