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 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.
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Notes
These rates in healthy adults are \(1, 5\) and \(12~\%\) as estimated by insurance companies in 2012.
References
Abbott JA, Shively HH (1973) Auscultatorily silent patent ductus arteriosus: report of two cases with normal pulmonary presures. Chest 63:371–375
Aoyagi S, Chihara S, Fukunaga S, Mori R, Suda K (2009) Transcatheter coil embolization for patent ductus arteriosus in the elderly: report of a case and review of published work. Geriatr Gerontol Int 9(3):329–332
Aziz K, Tasneem H (1990) Evaluation of pulmonary arterial pressure by Doppler colour flow mapping in patients with a ductus arteriosus. Br Heart J 63:295–299
Baer D (2005) Respiratory physiology–the essentials, 4th edn. Williams & Wilkins, Baltimore
Batchelor GK (2009) An introduction to fluid dynamics. Cambridge University Press, Cambridge
Becker TE, Ensing GJ, Darragh RK, Caldwell RL (1996) Doppler derivation of complete pulmonary artery pressure curves in patent ductus arteriosus. Am J Cardiol 78:1066–1069
Blyth MG, Mestel AJ (2002) The influence of geometry on inviscid decay rates in haemodynamic flows. J Fluid Mech 462:185–207
Campbell M (1968) Natural history of persistent ductus arteriosus. Br Heart J 30:4–13
Caro CG, Pedley TJ, Schroter RC, Seed WA (1978) The mechanics of the circulation. Oxford Medical Publications, Oxford
Cassidy HD, Cassidy LA, Blackshear JL (2009) Incidental discovery of a patent ductus arteriosus in adults. J Am Board Fam Med 22(2):214–218
Chandran KB (1993) Flow dynamics in the human aorta. J Biomech Eng 115:611–616
Crossley KJ, Allison BJ, Polglase GR, Morley CJ, Davis PG, Hooper SB (2009) Dynamic changes in the direction of blood flow through the ductus arteriosus at birth. J Physiol 587(19):4695–4703
Fechtrup C, Kerber S, Karbenn U, Borggrefe M, Breithardt G (1993) Value of intravascular ultrasound in the diagnosis and characterization of patent ductus arteriosus in an adult patient. Eur Heart J 14:1148–1149
Firmin DN, Mohiaddin RH (1996) In: How TV (ed) Advances in hemodynamics and hemorheology. JAI Press, San Jose
Friedlander E (2012) Pathology lectures. An image found online at http://www.pathguy.com/sol/08569.jpg. Accessed August 2012
Gray H (1918) Anatomy of the human body. Lea & Febiger, Twentieth edition
Grossman W, Baim D (2005) Grossman’s cardiac catheterization, angiography, and intervention, 7th edn. Williams & Wilkins, Baltimore
Guettouche A, Challier JC, Ito Y, Papapanayotou C, Cherruault Y, Azancot-Benistry A (1992) Mathematical modeling of the human fetal arterial blood circulation. Int J Biomed Comput 31:127–139
Huikeshoven F, Coleman T, Jongsma H (1980) Mathematical model of the fetal cardiovascular system: the uncontrolled case. Am J Physiol 239:317–325
Levick JR (1991) An introduction to cardiovascular physiology, 1st edn. Butterworth-Heinemann Ltd., Oxford
MacDonald DA (1974) Blood flow in arteries. Edward Arnold, London
McArdle WD, Katch FL, Katch VL (2009) Exercise physiology: nutrition, energy, and human performance, 7th edn. Williams & Wilkins, Baltimore
Menigault E, Berson M, Vieyres P, Lepoivre B, Pourcelot D, Pourcelot L (1998) Feto-maternal circulation: mathematical model and comparison with Doppler measurement. Eur J Ultrasound 7:129–143
Milnor WR (1989) Hemodynamics. Williams & Wilkins, Baltimore
Morgan-Hughes GJ, Marshall AJ, Roobottom C (2003) Morphologic assessment of patent ductus arteriosus in adults using retrospectively ECG-gated multidetector CT. Am J Roentgenol 181:749–754
Morrison LW, Bekey G, Brinkman CR, Assali NS (1971) Computer simulation of fetal baroreceptor function. Comput Biomed Res 3:561–574
Myers LJ, Capper WL (2002) A transmission line model of the human foetal circulatory system. Med Eng Phys 24:285–294
Ozmen J, Granger EK, Robinson D, White GH, Wilson M (2005) Operation for adult patent ductus arteriosus using an aortic stent-graft technique. Heart Lung Circ 14(1):54–57
Pennati G, Bellotti M, Fumero R (1997) Mathematical modelling of the human fetal cardiovascular system based on Doppler ultrasound data. Med Eng Phys 19(4):327–335
Pennati G, Fumero R (2000) Scaling approach to study the changes through the gestation of human fetal cardiac and circulatory behaviors. Ann Biomed Eng 28:442–452
Reel S, Alexander I, Appling S, Arnstein P (2006) Straight A’s in medical-surgical nursing: a review series, 3rd edn. Williams & Wilkins, Baltimore
Rerych SK, Scholz PM, Newman GE, Sabiston DC Jr, Jones RH (1978) Cardiac function at rest and during exercise in normals and in patients with coronary heart disease: evaluation by radionuclide angiocardiography. Ann Surg 187(5):449–458
Riley N (1998) Unsteady fully-developed flow in a curved pipe. J Eng Math 34:131–141
Setchi A (2012) Mathematical modelling of flow through shunts: application to patent ductus arteriosus and side-to-side anastomosis. PhD thesis. Imperial College, London
Smiesko V, Johnson PC (1993) The arterial lumen is controlled by flow-related shear stress. News Phys Sci 8:34–38
Smolich JJ, Mynard JP, Penny DJ (2011) Pulmonary trunk, ductus arteriosus and pulmonary arterial phasic blood flow interactions during systole and diastole in the fetus. J Appl Physiol 110:1362–1373
Spach MS, Serwer GA, Anderson PA, Canent RV, Levin AR (1980) Pulsatile aortopulmonary pressure-flow dynamics of patent ductus arteriosus in patients with various hemodynamic states. Circulation 61:110–122
Waite L, Fine J (2007) Applied biofluid mechanics. McGraw-Hill, Maidenheach
Wijngaard JPHM, Westerhof BE, Faber DJ, Ramsay MM, Westerhof N, van Gemert MJC (2006) Abnormal arterial flows by a distributed model of the fetal circulation. Am J Physiol Regul Integr Comp Physiol 291:1222–1233
Wilson N, Dickinson DF, Goldberg SJ, Scott O (1984) Pulmonary artery velocity patterns in ductus arteriosus. Br Heart J 52:462–464
Wiyono SA, Witsenburg M, de Jaegere PPT, Roos-Hesselink JW (2008) Patent ductus arteriosus in adults: case report and review illustrating the spectrum of the disease. Neth Heart J 16(7/8):255–259
Zabielski L, Mestel AJ (1998) Unsteady blood flow in a helically symmetric pipe. J Fluid Mech 370:321–345
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Setchi, A., Mestel, A.J., Siggers, J.H. et al. Mathematical model of flow through the patent ductus arteriosus. J. Math. Biol. 67, 1487–1506 (2013). https://doi.org/10.1007/s00285-012-0596-8
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DOI: https://doi.org/10.1007/s00285-012-0596-8