Mathematical modelling of flow through shunts : application to patent ductus arteriosus and side-to-side anastomosis

• Main Author: Setchi, Adriana
• Other Authors: Siggers, Jennifer ; Mestel, Jonathan ; Parker, Kim
• Published: Imperial College London 2012
• Subjects: 612.133
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.560698

This thesis develops mathematical models for the flow through shunts in the human body in two particular parameter regimes: flows of large Womersley number, i.e. ones dominated by high-frequency terms, and flows of low Reynolds number, e.g. of high viscosity. The first regime can be extended to flows with the properties that the Womersley number is of order bigger than the square root of the Reynolds number. The geometries that are considered in this work are idealised and therefore enable for the mathematical understanding of some fascinating, complicated, and not-very- well-understood problems in fluid dynamics. Analytical solutions are derived for high-frequency flow in three different idealised geometries. These rely on solving Laplaces equation for all linearly independent steady solutions in each particular geometry, and providing a suitable time-dependent behaviour through the choice of boundary conditions. The analysis uses complex potential theory, Schwarz- Christoffel transformations, conformal mappings and Fourier series. The solutions are applied to study the hemodynamics in the vicinity of a patent ductus arteriosus (PDA): a shunt between the aorta and pulmonary artery in some adults. Of particular interest are the distributions of velocity and pressure in the two arteries, as well as the shear stress at the cardiovascular walls, during a cardiac cycle in asymptomatic patients. Different hypotheses are tested by introducing different boundary conditions. The main results are based on the assumption that the flow in asymptomatic PDA patients is similar to the flow in healthy adults. The thesis also considers the low-Reynolds-number flow in a two-dimensional geometry of a shunt between two vessels. An analytical solution is derived by constructing piece-wise continuous functions that solve the biharmonic equation. The method uses the orthogonality properties of Papkovich- Fadle eigenfunctions. This work is applied to model the flow distribution in a side-to-side anastomosis.