Simplified blood flow model with discontinuous vessel properties: Analysis and exact solutions
We formulate a simplified one-dimensional time-dependent non-linear mathematical model for blood flow in vessels with discontinuous material properties. The resulting 3 × 3 hyperbolic system is analysed and the associated Riemann problem is solved exactly
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Simplified blood flow model with discontinuous vessel properties: analysis and exact solutions Eleuterio F. Toro, and Annunziato Siviglia
Abstract. We formulate a simplified one-dimensional time-dependent non-linear mathematical model for blood flow in vessels with discontinuous material properties. The resulting 3 × 3 hyperbolic system is analysed and the associated Riemann problem is solved exactly, including tube collapse. Our exact solutions constitute useful reference solutions for assessing the performance of numerical methods intended for simulating more general situations. In addition the presented model may be a useful starting point for numerical calculations involving rapid and discontinuous material properties variations.
2.1 Introduction The theoretical study of blood flow phenomena in humans through mathematical models is closely related to the study of flow of an incompressible liquid in thinwalled collapsible tubes. In fact the applicability of theoretical models for thinwalled collapsible tubes covers a wider variety of physiological phenomena as well the design of clinical devises for practical medical applications. In this paper we are interested in theoretical models for blood flow in medium to large arteries and veins regarded as thin-walled collapsible tubes. We centre our attention on onedimensional, time-dependent non-linear models. Classical works on this subject are, for example, [12, 17] and the many references therein. For more recent works see [1, 3, 4, 7, 8, 18, 20] to name but a few. This paper is motivated by physical situations of medical interest in which certain properties that characterize blood vessels, external pressures and body forces Eleuterio F. Toro ( ) Laboratory of Applied Mathematics, Faculty of Engineering, University of Trento, Italy e-mail: [email protected] Annunziato Siviglia Laboratory of Applied Mathematics, Faculty of Engineering, University of Trento, Italy e-mail: [email protected]
Ambrosi D., Quarteroni A., Rozza G. (Eds.): Modeling of Physiological Flows. DOI 10.1007/978-88-470-1935-5 2, © Springer-Verlag Italia 2012
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E.F. Toro, and A. Siviglia
change rapidly, or even discontinuously. Physical quantities of interest are vessel wall thickness, equilibrium cross sectional area and Young’s modulus. A prominent example arises in the surgical treatment of Abdominal Aortic Aneurysms (AAA) [23] that includes the insertion of stents. These devises do not always match the compliance properties of natural vessels and discontinuous jumps of physical properties may arise, influencing significantly the wave propagation phenomena associated with the hemodynamics. External pressures and body forces are another source of potentially rapid or even discontinuous variations, which again will influence the wave phenomenon [12]. Here we formulate a mathematical model that allows for discontinuous variation of certain vessel properties, all in the context of simplified one-dimensional flow. In spite of the very strong assumptions, we still expect the one-dimensional model
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