Analysis of a continuous, elastic, single-chambered model of the mammalian lung

A continuous, elastic, single-chambered model of the mammalian lung, suitable for the investigation of regional ventilation and distributional defects, has been described and theoretically analyzed. The model consists, mathematically, of a sponge-like tissue matrix contained within a spherical cone which 'breathes' through a cylindrical 'trachea' via motion of the radial extremity of the cone. The tissue matrix has a radially varying permeability which gives it regional laminar airway resistances consistent with those of an actual bronchial tree. The 'trachea' has a volume and laminar airway resistance equal to that of the trachea and first three bronchial generations, inclusively. The final equation describing the internal state of the model is obtained from the equations governing the state, conservation of linear momentum, and conservation of mass for both the respiratory gas and the tissue matrix. These include a term for viscous interactions between tissue and air. The resulting equation is an inhomogeneous, non-linear, second-order partial differential equation, the solution of which is further complicated by the motion of the radial extremity of the cone. A method of describing the moving boundary is formulated and a brief description of a numerical method currently being used to attempt a one-dimensional solution for the case of a normal lung undergoing quiet breathing is presented. The value of this model lies in its utility for investigating distributional defects in the pulmonary system. By changing the distribution of permeability, porosity, and tissue compressibility, in the model it is possible to simulate the ventilatory effects of both bronchial constriction and occlusion at various sites and the effects of tumors and emphasematous buli. The ease with which this model can 'perform' routine pulmonary function tests makes it ideally suited for studies of the correlation of various pulmonary disorders with the results generally obtained from clinical pulmonary function studies.