Piotr Kowalczyk

This article presents a numerical model which enables the approximate determination of the distribution of strains and stresses in lung tissue during respiration. It includes the formulation of a basic system of differential equations for the problem, time and space sampling of the system and the development of an algorithm for solving the system, and implementation of the system in a computer program. The computer model accounts for the non-linear mechanical properties of the tissue, their extensive deformation, and the effect of variations in the distribution of gas pressure and flow in the air spaces of the lungs. Part 1 explains the basic approach to the subject and briefly discusses extant research on the mechanics of the respiration process. Part 2 presents a physical model of the problem. It also contains a brief discussion of the anatomy and physiology of the lungs. Then follows a discussion of extant knowledge of the mechanical properties of lung tissue and a critical overview of previously used constitutional models of lung tissue. The following subsection contains an analysis of air flow in respiratory tracts and presents a model of the mechanical relationships between this flow and the deformation of the tissue skeleton. It consists of a system of discretely modelled air ducts embedded in a two-phase porous medium which fills the spaces of the lungs. Next the author writes a system of differential equations for the problem which includes an equation for flow in the respiratory tree, an equation for equilibrium in the lung tissue, and an equation for gas filtration in the tissue, along with their respective boundary and initial conditions. The author introduces the concept of a contact layer along with an additional equilibrium equation for the purpose of simplifying the numerical realization of a mixed boundary condition for displacements and stresses. In Part 3 the author presents a numerical algorithm which provides an approximate solution of the equations of the problem by means of an increment version of the finite element method. The author gives a matrix formulation of a conjugate problem in which the unknown variables at each step are the increments of flows in the components of the respiratory tree and the increments of displacements and pressures at the nodes of the finite element grid. The author describes an iterative procedure for solving the problem with an implicit numerical time integration scheme. Examples of numerical computations performed by means of the aforementioned algorithm implemented in a computer program are discussed in Part 4. Among other subjects, the author studied the static characteristics of a model of the human lungs under the action of the tissue's own weight and the effect of the inhalation rate on local ventilation and stresses in different regions of the lungs. Part 5 contains a critical analysis of the model and its results and recommendations for further research in the field. It also summarizes the original work embodied in the thesis. (Author (revised))