Mathematical model of flow through the patent ductus arteriosus
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Mathematical Biology
Mathematical model of flow through the patent ductus arteriosus Adriana Setchi · A. Jonathan Mestel · Jennifer H. Siggers · Kim H. Parker · Ming Wang Tan · Kangwen Wong
Received: 12 March 2012 / Revised: 21 August 2012 © Springer-Verlag Berlin Heidelberg 2012
Abstract The ductus arteriosus is one of several shunts in the cardiovascular system. It is a small vessel connecting the aortic arch and pulmonary artery that allows blood to bypass the pulmonary circulation. It is open during foetal development because the foetal lungs cannot function and oxygenation of the blood occurs by exchange with the maternal blood in the placenta. Normally it closes a few days after birth; however, in a small number of people closure does not occur, leading to a condition known as patent ductus arteriosus. In this paper our aim is to investigate the resulting cardiovascular effects. We develop a mathematical model of the haemodynamics in three different idealised geometries by assuming that the entry flow is irrotational and remains so in the core until at least the shunt position. We argue that separation or diffusion of vorticity into the core flow is delayed due to the high frequency associated with the pulsatile component of the flow profile. The analysis uses complex potential theory, Schwarz–Christoffel transformations, conformal mappings and Fourier series. The main results are based on the assumption that the flow in patients with patent ductus arteriosus is similar to the flow in healthy adults, and we apply this assumption using boundary conditions that are representative of physiological values in healthy adults. The model suggests that the pressures in the aorta and pulmonary artery are likely to equalise, that the shear stress increases near the edges of the shunt and that backflow A. Setchi (B)· J. H. Siggers · K. H. Parker · M. W. Tan · K. Wong Department of Bioengineering, Imperial College London, London SW7 2AZ, UK e-mail: [email protected] J. H. Siggers e-mail: [email protected] K. H. Parker e-mail: [email protected] A. J. Mestel Department of Mathematics, Imperial College London, London SW7 2AZ, UK e-mail: [email protected]
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of large volumes may occur from the pulmonary artery into the aorta or towards the ventricles due to the presence of the patent shunt. Our results strongly suggest that an abnormal compensatory physiology develops in patients with patent ductus arteriosus. Keywords Patent ductus arteriosus · Cardiovascular shunt · Potential flow · High Womersley number Mathematics Subject Classification
92C35 · 76Z05
1 Introduction The ductus arteriosus is a vascular connection between the pulmonary artery and the aortic arch in the developing foetus that ensures oxygen is efficiently transported around the foetal circulation. It protects the right ventricle from pumping against the high resistance in the fluid-filled lungs by shunting most of the flow from the right ventricle into the aorta. Usually, it is functionally closed within days
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