The objective of this thesis is the development and evaluation of a mathematical model that captures the deformation behavior of the human airway under smooth muscle contraction. The problem consists of a multilayered circular cylindrical tube subjected to an active contractile stress generated by the smooth muscle layer. The problem formulation assumes axisymmetric deformation for an incompressible isotropic neo-Hookean material. The model is formulated as fully nonlinear and accommodates large deformation, since these are expected in the human airway and other soft biological materials. The equilibrium equations are obtained, as well as the strain-displacement relationship, boundary conditions, and constitutive equations that describe the contractile stress for neo-Hookean hyperelastic solids. The main focus of the project is to obtain the change in airway caliber after a prescribed contractile stress is introduced in the smooth muscle layer. A MATLAB code was written to numerically solve the resulting nonlinear algebraic equation, which obtains the change in airway caliber. The code allows the user to analyze and change the material parameters that govern the solution, as well as input different dimensions for the multilayer cylinder. In the deformation of the airway, the lung parenchyma plays a very important role in preventing full closure of the airway. Here, the lung parenchyma is treated as an infinitely large continuous solid, in which the airway is embedded. The airway generations selected for further investigation are generations 0, 4, 8, 12 and 16. For the lung parenchyma infinite domain assumption, the percent airway caliber change for the critical contractile stresses of the airway generations were 23.86%, 23.68%, 23.95%, 24.56%, and 25.51% for generations 0,4,8,12, and 16, respectively.;The mathematical model does not account for the buckling that is observed in the deformation of the human airway, since the deformation is assumed to be axisymmetric. When the contractile stress in the smooth muscle layer exceeds a certain value, the airway buckles and folds. Therefore, finite element simulations were conducted using ABAQUS software to determine the critical contractile stress at which the airway will buckle. Buckling analyses were performed in order to obtain the buckling modes of the airway, as well as the critical load that will cause the it to buckle, for airway generations 0, 4, 8, 12, and 16. The critical load in which the airway buckles and folds for these generations of the airway are 1.755 kPa, 1.572 kPa, 1.493 kPa, 1.426 kPa, and 1.374 kPa respectively. Contractile stresses used in the analytical model that are larger than the critical buckling loads will provide invalid results since the assumptions made to develop the model will no longer be valid. After the buckling analysis, the first buckling mode is introduced in a post buckling analysis in order to obtain the average change in caliber and the active contractile stresses.
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