Biomedical Engineering Reference
In-Depth Information
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 prop-
erties may arise, influencing significantly the wave propagation phenomena associ-
ated 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 to provide by itself useful information for practical purposes.
Moreover, one-dimensional models are an integral part of large models in multiscale
approaches [18] and thus the present work may also be useful in the construction of
more realistic models.
In current models used for numerical simulation of blood flow phenomena the
effect of the variation of the above mentioned quantities enters the equations in the
form of source terms; see [20], for example. In particular, for external forces such
as muscle forces, the corresponding source term involves a pressure gradient source
term, analogous to the geometric source term given by bottom variation in shallow
water models [21]. In the numerical analysis literature it is well known that such
source terms are likely to cause serious numerical difficulties. An important issue of
that of constructing
well balanced schemes
that achieve equilibrium between advec-
tive and source terms in the equations near the steady state [9, 13, 16]. The severity
of the numerical difficulties increases as spatial gradients of the physical quantities
of interest increase.
In this paper we formulate and study a simplified model in which discontinuities
of two parameters are permitted, namely wall thickness and Youngs modulus. We
study the mathematical properties of the resulting 3
×
3 hyperbolic system and ob-
tain the exact solution for the Riemann problem. Exact solutions include the case
of tube collapse and constitute reference solutions for assessing the performance of
numerical methods intended for general use. Potentially, the proposed formulation
would facilitate the numerical treatment of source terms due to spatial variation of
material properties and external forces. There are, however, two major difficulties
with our formulation. One is the potential occurrence of
resonance
[10] and [14],
and the possibility of non-uniqueness. The second problem is the non-conservative
character of the model, with all the attendant mathematical and numerical implica-
tions [5]. The issue of resonance is currently the subject of further studies by the
authors and results will be published elsewhere.
The rest of the paper is structured as follows. In Sect. 1 we review the governing
equations and the tube law to be used. In Sect. 2 we introduce and study a 3
3
hyperbolic model with discontinous property variations. In Sect. 4 we formulate and
solve exactly the Riemann problem. In Sect. 5 we show sample exact solutions. In
Sect. 6 we derive exact solutions for tube collapse and give some examples. Con-
clusions are drawn in Sect. 7.
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