Biomedical Engineering Reference
In-Depth Information
a reasonable first approximation to the actual behavior of blood at shear rates
observed in large arteries [
16
].
An important advantage of the framework we use to obtain the CFD results for
the aortic blood flow is its fully automatic nature. This is achieved using an Eulerian
framework and embedding the boundary using a level set function, which implicitly
defines the computational domain. The level set is computed as the signed distance
from the aortic triangular mesh. This is done first in the cells located near the mesh
itself, then the signs are extended to the rest of the rectangular domain by solving an
extrapolation equation in fictitious time similarly to [
17
,
18
]. Note that the exact
values of the level set are important for the various discretizations only near the
boundary, and only its sign matters away from the mesh. This is visible in Fig.
3
a.
To reduce the sensitivity of the simulation results with respect to the outlet
boundary conditions, and thus improve the simulation results, we extend the outlet
before applying the pressure outlet boundary condition. The extension is approxi-
mately 6-8 times the radius of the outlet.
The boundary conditions used in the simulations are as follows: at the aortic
walls we use no-slip for the velocity and the appropriate normal balance (translating
into a Neumann boundary condition) for the pressure. The inflow velocity is
extrapolated from the MRI, using smooth kernels, to all the inlet nodes, while
pressure proportional to the flow is set as a Dirichlet boundary condition on the
outlet faces. The inflow velocities are also interpolated in time using second order
accurate interpolation.
Our computational algorithm starts at a given time step
n
from the velocity and
pressure information at the previous time step
u
n
1
p
n
1
, and computes
u
n
,
p
n
following a fractional step projection method. The geometry (hence the level set)
is considered static in this paper, but the level set formulation allows for an easy and
robust extension of the algorithm to moving boundaries. Our algorithm computes
the solution to the unsteady 3D Navier-Stokes equations in the following steps:
(1) convective update for the velocity
u
, (2) semi-implicit update for the velocity
(viscous force contribution), (3) pressure update by solving the pressure Poisson
equation with mixed boundary conditions, (4) new velocity update. The algorithm
uses subcycling to enforce the Courant-Friendrichs-Levy (CFL) condition at every
time step. Several results obtained with our system are described and discussed in
Experiments and validation section.
;
2.5 Discussion of Limitations
The current simulations use several simplified assumptions involving the boundary
conditions and the constitutive equations, which may benefit from further
extensions. The geometric model is temporally fixed, which may be improved by
using a moving boundaries formulation, as done for example in [
17
]. As underlined
by [
19
] specifying the correct boundary conditions is of utmost importance for the
realism of the computations, e.g., for simulating the phase delay between flow and
