On the mechanical behavior of the human biliary system

Figure 1 The geometry of the biliary tract, where the valves of Heister in the cystic duct are shown.

Figure 2 Two types of the cystic duct models used by Ooi et al[43] (modified from Figure 2 in the Journal of Biomechanics, vol 37, page 1913-1922) .

Figure 3 Computational models built from two patients' biliary system images used by Ooi et a[[43] (modified from Journal of Biomechanics, vol. 37, page 1913-1922, Figure 2). Patient A is reported to have gallstones, and patient B is the normal control.

Figure 4 The resistance, Rd, plotted with the Reynolds number, Re (which is defined as the ratio of inertia over viscous forces, and is proportional to the flow rate for a given duct and bile). n is the number of the baffles in the duct[43] (modified from Journal of Biomechanics, vol. 37, page 1913-1922, Figure 6).

Figure 5 One-dimensional model of human billiary system and bile flow directions in the (a) emptying and (b) refill phases, (c) details of baffles at one section of the cystic duct[44]. p is the pressure inside the gallbladder, and Q is the flow rate of bile fluid. (Modified from figure 2 to be published in ASME Journal of Biomechanical Engineering).

Figure 6 Pressure loss variations with baffle height ratio (A) and baffle number (B) for both rigid and elastic models[44]. (Modified from figure 8 to be published in ASME Journal of Biomechanical Engineering).

Figure 7 Gallbladder body shape during emptying is assumed to be ellipsoidal with major and minor axis lengths D₁, D₂ and D₃ (D₁ > D₂≥ D₃); the gallbladder is subjected to a uniform internal pressure. The stress due to the pressure at any point P has three components, σ_θ (meridian), σ_φ (latitude) and τ_θφ (in surface)[85].

Math 1 Math(A1).

Figure 8 Li et al[85] calculated the stress distributions for two different subjects: R (left) and G (right). The top frame is the principal meridian stress, σ_θ, the middle frame is the principal latitude stress, σ_φ, and the bottom one is the in-plane shear stress τ_θφ. It is seen that the location and the maximum stresses change as the ratio of the lengths of the three axes change.