We have developed a mathematical model for a tracheal ring that consists of
a "horseshoe" of cartilage with its tips joined by a membrane. The ring is
subjected to a uniform transmural pressure (Ptm) difference. The model was
used to calculate the cross-sectional area (A) of the trachea. Whereas the
mechanics of the deformation of the cartilage were analyzed using elastica
theory, the posterior membrane was treated as a simple membrane that is in
extensible under changes in Ptm. The membrane can be specified to be of any
length less than baseline and thus can represent a posterior membrane unde
r tension. The cartilage can have specifiable nonuniform unstressed curvatu
re as well as nonuniform bending stiffness. We have investigated the effect
on the tracheal A-Ptm curve of posterior membrane length and tensile force
in the membrane, cartilage shape and elasticity, and localized weakening o
f the cartilage. The model predictions are in good agreement with magnetic
resonance imaging data from rabbit tracheas and show that the shape of the
horseshoe as well as the posterior membrane force are important determinant
s of tracheal compliance.