Biomedical Engineering Reference
In-Depth Information
and/or the exits of the 3D domain. This approach is embraced by formulation (9.24)
when taking
0, for which we would obtain equations of the form of (9.52a)-
(9.52b) with a given function
γ
=
characterizing the velocity profile, as in (9.53).
Note that while the velocity profiles can exhibit large variations over the cross-
sectional areas of the vessels, the pressure remains, in most cases, almost uniform
(as long as the coupling interfaces are somewhat normal to the axial direction of the
vessel). In this sense the choice
ϕ
(
x
,
y
)
1 considered here appears more adequate for the
type of problems faced in the application examples. Moreover, instead of the sim-
ple 1D model adopted here (derived from a flat velocity profile), more sophisticated
1D models could be considered. For all these models, the proposed non-classical
extended variational formulation will yield automatically the appropriate coupling
conditions satisfied by the primal (kinematic) and dual (traction) variables over the
coupling interfaces.
Specifically, in the construction of our 1D model we neglect the viscous effects
associated to the axial gradients of the velocity field, that is, we neglect terms of the
form
∂
u
1
,
z
γ
=
∂
z
. A simple dimensional analysis justifies this consideration [25]. Therefore,
in the coupling conditions no viscous terms will emerge from the 1D domain to the
coupling interface (left-hand side of (9.51b)). In such a case, the traction over the 3D
domain is exclusively due to the pressure in the 1D model (see Sect. 9.4.8 below).
Finally, the reader interested in more details and applications of this extended
variational formulation to others physical problems, like heat conduction and solid
mechanics, should refer to [3, 4, 5, 7].
9.4.2 Open/closed-loop representations of the cardiovascular system
As mentioned previously, the dimensionally-heterogeneous approach to model the
systemic circulation is able to couple several elements of interest in the analysis: (i)
the complexity of blood flow circulation in specific arterial districts such as bifurca-
tions, aneurisms and other geometrical singularities in general (by using 3D models
with geometry coming from patient-specific medical image data); (ii) the complex
systemic behaviour that leads to the conformation of the cardiac pulse (by using 1D
models) and (iii) the influence of the remaining (complementary) part of the circula-
tory system like arterioles, capillaries, venules, veins, pulmonary circulation, heart
chambers and valves, among others (by using 0D models).
In the literature several approaches have been adopted that integrate different lev-
els of circulation in the sense introduced in the previous paragraphs. Mostly, models
based on lumped representations were employed to accomplish this task, incorporat-
ing 0D models to simulate flow in the larger arteries, veins and cardiac circulation
[31, 35, 38, 52, 54, 55]. Distributed models to simulate the blood flow in compli-
ant vessels have represented an exhaustive area of research over the last decades
[1, 33, 46, 53, 56, 57, 64]. More recently, 1D models of the arterial circulation have
been coupled to 0D models of the venous-cardiac-pulmonary circulation to study
the influence of arterial stenoses on wave propagation [40]. In particular, the 1D
model employed in Liang and Takagi, [40], which was taken from Stergiopulos et
al., [56], comprises 55 arterial segments and a 0D lumped representation for the pe-
