Biomedical Engineering Reference
In-Depth Information
c
s
, where the speed of sound is given by
=
ρ
The equation of state is
given
by
p
1
=
/
the advection as
c
s
3 lattice units per time step. The equilibrium part is
unchanged by the collision. Further, in the simulation scheme only the shear modes
survive the collision process. Therefore, only the nonequilibrium momentum flux
changes. The post collision nonequilibrium momentum flux
neq
∗
is given by
Π
1
3
(
neq
∗
¯
neq
:
1
Π
=(
1
+
λ
)
Π
neq
+
1
+
λ
v
)(
Π
)
1
,
(22)
where the over bar indicates the traceless part of the momentum flux. The star
indicates that a value is post collision. The parameter
and
controls the rate of relaxation of the stress tensor. By this it as well regulates the
shear viscosity
λ
is an eigenvalue of
Λ
1
6
ρ
(
μ
=
−
2
/
λ
+
1
)
.
(23)
For stability reasons
0. The kinetic modes can also
contribute to the post-collision distribution, but the eigenvalues of these modes were
set to be 0, so they have no effect on the post-collision distribution
f
i
. The post-
collision distribution is given in terms of the mass-density
λ
has to be in the range
−
2
<
λ
<
and the momentum-
density
j
(or the velocity
u
and the updated post-collision momentum flux
ρ
Π
∗
)
f
i
=
w
2
Π
αβ
w
3
(
Π
αα
−
c
s
)
.
f
i
+
Ω
i
(
f
)=
w
0
ρ
+
w
1
j
α
c
i
α
+
c
i
α
c
i
β
+
¯
3
ρ
(24)
The coefficients
w
i
have their origin in the different lengths of the vectors
,i.e.,
the absolute velocity needed to reach the neighbor cell in one time step, depending
whether it is a nearest neighbor or a diagonal neighbor.
|
c
i
|
B
Vessel Tracing
For completeness, this section presents methods to model complex vasculature in
3-D images using
cylindroidal superellipsoids
, along with robust estimation and
detection algorithms that are detailed in [
90
,
91
].
B.1
Superellipsoids
Superellipsoids are geometric modeling primitives that have been used in early
computer vision literature [
6
]. A cylindroidal superellipsoid is given in implicit
form by
2
1
ε
2
ε
2
S
(
x
;
ε
)=
|
x
|
+
|
y
|
+
|
z
|
(25)
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