A One-Dimensional Model of the Cardiovascular
System
J. Alastruey
[email protected]
Department of Aeronautics
Imperial College
South Kensington Campus
London, SW7 2AZ
United Kingdom
K.H. Parker
[email protected]
Department of Bioengineering
Imperial College
South Kensington Campus
London, SW7 2AZ
United Kingdom
Dr. J. Peiró
[email protected]
Department of Aeronautics
Imperial College
South Kensington Campus
London, SW7 2AZ
United Kingdom
Dr. S.J. Sherwin
[email protected]
Department of Aeronautics
Imperial College
South Kensington Campus
London, SW7 2AZ
United Kingdom
Abstract
A three-dimensional simulation of the cardiovascular system is computationally very expensive and nowadays applied only to the study of
localised parts in the system. Due to the large wave-length of the arterial pulse waves compared to the vessel diameters, one-dimensional (1D)
models permit an efficient simulation of the wave propagation in the larger
arteries, which allows us to study the effects of local changes on the global
system. However, an accurate quantitative validation of 1D models is very
complicated because some of the elastic and geometric properties of the
system are very difficult to measure in-vivo.
Figure 1: The topology of the arterial network used in the validation of our 1D
model against the experimental model by Segers et al. [1] (left) and in the study
of the cerebral circulation (right).
We have tested the accuracy of our numerical scheme against a welldefined laboratory model, developed by Segers et al. [1], in which most
of the parameters required by the numerical algorithm can be easily measured and the results produced are clinically relevant, because the geometric and elastic properties of the 55 human arteries shown in Figure 1
(left) are well approximated by the latex tubes of the model. The system
also contains a pump capable of producing flow waves similar to the waves
produced by the human heart and a model of the microcirculation at each
terminal branch, which is connected to a conduit that returns the liquid
to the heart to close the circuit.
The governing equations of the 1D model are obtained by applying conservation of mass and momentum to a 1D impermeable tubular control
volume of inviscid, incompressible, and Newtonian fluid, and by considering a tube law that relates changes in pressure to changes in cross-sectional
area. For further details refer to [2]. The system of equations is solved by
Figure 2: Pressure (top) and flow rate (bottom) time histories along the aorta
(ascending, thoracic and abdominal) in the experimental model by Segers et al.
[1] (left) and the numerical model (right).
means of a high-order discontinuous Galerkin numerical scheme [3].
In our presentation we will show that our 1D model is able to reproduce
the experimental pressure and flow pulse waves, including the diacrotic
notch and a period of flow reversal at the end of systole in the aorta, the
peak pressure increase and mean flow rate decrease along the aorta as we
move away from the heart and the diastolic decay (Figure 2). We will
also present a study made using the 1D formulation on the role played
by the communicating arteries of the circle of Willis in distributing blood
efficiently to the main arteries that perfuse the brain, and how the circle
compensates the occlusion of one common carotid artery when one or two
communicating vessels are absent. The network considered in this study
is shown in Figure 1 (right).
References
[1] Segers, P., Dubois, F., De Wachter, D., and Verdonck, P., Role
and relevancy of a cardiovascular simulator, Cardiovascular Engrf. 3
(1998), 48–56.
[2] Sherwin, S.J., Franke, V., Peiró, J., and Parker, K.H., One-dimensional
modelling of a vascular network in space-time variables, J. Engrg. Math.
47 (2003), 217–250.
[3] Karniadakis, G.E. and Sherwin, S.J., Spectral/hp element methods
for CFD, Oxford University Press, UK, 1999.