Biomedical Engineering Reference
In-Depth Information
problem is solved monolithically. The non-linearities are treated using Picard itera-
tions. For the 1D model the finite element is applied to the equations written along
the characteristic lines with stabilization provided by least-square terms. The inter-
ested reader is referred to [6, 60] for a more detailed account of these well-known
numerical techniques that, not being the main focus of our investigation, will not
be discussed any further. To deal with the heterogeneous coupling we perform a
partition of the problem into two subproblems: (i) the stand-alone 3D model under
analysis and (ii) the complementary 1D-0D model (open loop or closed loop, de-
pending on the case). For the coupling between these two submodels we employ
different techniques that will be commented in due course.
9.5.1 Blood flow in an abdominal aneurism
In this case we restrict the model to the open-loop representation of the CVS (see
Sect. 9.4.2). The geometry was obtained from medical images through standard seg-
mentation procedures. Fig. 9.8 displays the final geometry and its posititiong within
the arterial network. Also in that figure the flow rate boundary condition at the aortic
root is shown.
The finite-element mesh of the 3D model consists of 0.8M nodes and 4.7M tetra-
hedra. The time step used in the simulation was 6
10
−
4
s. Two cardiac cycles,
.
25
·
of period
T
8 s, were run from at-rest initial conditions. In this case the cou-
pling scheme for the interactions between the 3D and the open-loop 1D-0D models
is explicit, that is, we solved at each time step the 1D model and the 3D model just
once. Based on past experience, in fact, we have found that for the flow regimes
encountered in the abdominal aorta, together with the geometrical features of the
1D segment that is being substituted by a 3D geometry, makes the explicit approach
feasible, reliable and, evidently, computationally cheaper. The results presented cor-
respond to the second heart beat.
In Fig. 9.8 the flow rate and pressure at the coupling interfaces are presented. No-
tice the increased pressure drop in the left common iliac as a result of the reduction
in the arterial lumen right after the aneurism in that branch. Another remarkable fact
is the change in the flow rate at proximal and distal coupling interfaces. Indeed, the
aneurism acts as a large capacitor, changing the way in which the blood flow is re-
leased to the iliacs. Specifically, notice that the solutions obtained at these coupling
interfaces differ significantly from what would be expected in an aneurysm-free ge-
ometry like the one analyzed in [8].
Concerning the 3D blood flow, the sequence shown in Fig. 9.9 presents the blood
flow patterns in the aneurismal region for several time instants throughout the car-
diac cycle. Note that the blood flow is actually a jet flow impiging on the posterior
arterial wall and producing a quite complex secondary circulation in the anterior
region of the aneurism.
We also carried out the calculation of the WSS and OSI hemodynamics indices.
Fig. 9.10 presents these two fields. Notice that the complex flow observed in Fig. 9.9
is manifest in the structure of the OSI field, which features larger values on the lat-
eral zones of the arterial wall as a result of the vortical pattern induced by the jet
=
0
.
