Mathematical modelling of shape memory stent placing at endobiliary interventions

Gallbladder and biliary tract diseases ranks third in Russia by number of patients. The most common complication of the gallbladder disease is the papillary stenosis. The papillary stenosis results in narrowing of the confluence of the duodenum, common bile duct and pancreatic duct in common. To eliminate papillary stenosis the surgeons apply transpapillary endobiliary surgical treatment methods aimed at providing normal bile outflow and pancreatic juice into the duodenum. Endoscopic papillotomy, ballon dilatation, and stanting of the common bile duct are widely performed surgical operations. However, the last endobiliary intervention is not always successful because physicians consider only their subjective experience, and there are not complete biomechanical models of stent-bile duct interaction that could quantify the surgical procedure effectiveness. Thus, the aim is to develop and improve the installation procedure of the shape memory biliary stent in terminal department of the common bile duct with papillary stenosis on the basis of mathematical modeling for the objectification of treatment and improve patient the quality of life during endobiliary surgery. The presented aim is achieved by the solving of two interconnected problems. Problem 1 is devoted to the procedure of pre-operative stent treatment and estimation of the link between shape memory stent deformation and the pressure, applied to its external surface. Problem 2 is related to determination of stress-strain state of the external cylinder (duct) after stent expansion. To solve the problems, the theory of elasticity relations, constitutive equations for mechanical behavior of shape memory alloy and eigenstrain control theory were adopted. According to computations, shape memory stent can be inserted to achieve normal duct diameter (8 мм) in case when stenosed duct diameter is not less than 2.4 mm.