Tension and stress calculations in a 3-D Fourier model of gall bladder geometry obtained from MR images

Biliary tract obstruction results in dilatation of the biliary tract including the gall bladder and induction of symptoms such as abdominal pain. Since the pain receptors are likely mechano-sensitive receptors, it is important to develop tools for studying the distributions of tension and stress in the wall of the gall bladder. Wall tension and stress can be determined using Laplace's equation and the three-dimensional (3-D) geometry of a thin walled organ under equilibrium conditions. The objective of this study was to develop an analytical method to describe the 3-D geometry of the porcine gall bladder. The Fourier series method was used to describe the organ surface geometry obtained from magnetic resonance (MR) images. MR images of nine normal and three obstructed porcine gall bladders were analyzed. The curvature was computed throughout the gall bladder surface and the wall tension was computed using Laplace's equation. The spatial distributions of principal curvatures, tensions, and stresses were nonhomogeneous in the gall bladder because of its complex geometry. The extremum values of curvatures did not differ between normal and 2-day obstructed gall bladders. The pressure, tension, and stress were significantly higher, however, in the obstructed gall bladders (p < 0.05). This study provides an analytical tool for characterizing the complex 3-D geometry of an organ obtained from a clinical imaging modality.