Biomedical Engineering Reference
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2
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 proper-
ties. 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 in-
tended for simulating more general situations. In addition the presented model may
be a useful starting point for numerical calculations involving rapid and discontinu-
ous material properties variations.
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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 thin-
walled collapsible tubes. In fact the applicability of theoretical models for thin-
walled 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 one-
dimensional, 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 cer-
tain 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: toro@ing.unitn.it
Annunziato Siviglia
Laboratory of Applied Mathematics, Faculty of Engineering, University of Trento, Italy
e-mail: nunzio.siviglia@ing.unitn.it
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