Biomedical Engineering Reference
In-Depth Information
In addition, at the high mesh resolution required to sample low-noise ESS data, the
LB method requires rather small time steps (of the order of 10 6 sforaresolution
of 20
m).
The wall shear stress, which is central to hemodynamic applications, can be
computed via the shear tensor
m
u T
x
;
t
Þnr r
ð
u
þr
Þ
evaluated via its kinetic
representation
X p c p c p ð
Þ¼
3 no
c S
f e p Þð
x
;
t
f p
x
;
t
Þ:
(10.4)
The tensor second invariant is the endothelial shear stress or ESS:
r
1
2 ðs : sÞð
x w ;
t
Þ¼
x w ;
t
Þ
;
(10.5)
where x w represents the position of sampling points in close proximity to the mesh
wall nodes.
provides a direct measure of the strength of the near-wall shear
stress [ 5 ]. It is worth mentioning that the ESS evaluation via ( 10.4 ) is completely
local and does not require any finite-differencing procedure. Thus it is particularly
advantageous near boundaries where the computation of gradients is very sensitive
to morphological details and accuracy. In order to sample high signal/noise ESS
data, the LB mesh needs high spatial resolution, with mesh spacing being as
small as Dx
x w ;
t
Þ
'
50
m
m for standard fluid-dynamic simulations, or being as small
as Dx
m in order to account for the presence of RBCs.
For the multiscale simulations of blood flows, we have developed the MUPHY
software [ 3 ]. Such simulations in extended arterial systems, are based on the
acquisition of MDCT data which are segmented into a stack of slices, followed
by a mesh generation from the segmented slices. For a typical coronary artery
system, the procedure to build the LB mesh from the MDCT raw data starts from a
single vessel, formatted as stacked bi-dimensional contours (slices), with a nominal
resolution of 100
'
10
m
m. In spite of recent technological progress, this resolution is
still insufficient, and the inherently noisy geometrical data pose a problem in the
evaluation of ESS, a quantity that proves extremely sensitive to the details of the
wall morphology. Raw MDCT data present a mild level of geometric irregularities,
as shown in Fig. 10.1 , that can affect the quality of the LB simulations. For the
simulation, we resort to regularize the initial geometry by smoothing the sequence
of surface points via a linear filter along the longitudinal direction. Similarly,
one could filter out surface points along the azimuthal contour. We have shown
that such smoothing is necessary in order to avoid strong artifacts in the simu-
lation results [ 28 ]. Even if the precise shape of the vessel is unknown, as it falls
within the instrumental indeterminacy, the numerical results converge to a common
fluid-dynamic pattern as the smoothing procedure reaches a given level. The
regularized geometries are still of great interest because they obey the clinical
perception of a smooth arterial system, and, moreover, the smoothing procedure
falls within the intrinsic flexibility of the arterial system.
m
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