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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_3779_Библиотеки_им_академика_М_И_Перельмана
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90 Chapter 5
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Z
U
J dx þ
Z
vu
vt
^
U
s
JSsFT: VJ dx
^
s
Z
rfJðVvÞF1ðuÞv
^
U
f
Z
pJFTðuÞ: VJ dx ¼ 0
^
U
f
WG
s0
,
f 0
Z
$f dx
s0
v$f dx ¼ 0 (5.18)
^
U
s
WG
f 0WGsf
and
for all J ˛ H
for all f ˛ H
Z
vv
J dx þ
r
s
vt
^
U
s
Z
þ
vv
rfJ
vt
^
U
f
Z
2mfJDuv: DuJ dx þ
þ
^
U
1
0
1
0
f
b
U, such that J ¼ 0onG
b
U, such that f ¼ 0 on G
Z
JðVvÞ: FTðuÞq dx ¼ 0 (5.19)
^
U
f
for all q ˛ L
2
b
U. Note that integrals over the interface in Eq. (5.17) cancel out due to the
interface condition (4.58). Moreover, we have u ¼ 0 on G
.
G
s0
s0
WG
vu
vt
f 0
J dx
; v ¼ vDon G
; v ¼ 0 on
f 0
(5.17)
Equalities (5.17)e(5.20) serve to define the FEM, which consists of finding
0
u
; vh; p
f
h
Eqs. (5.17)e(5.20) (in place of u, v, and p) and for all J ˛ V
The coupling condition on G
˛ V
g
h
Vh ℚhsuch that vh¼ v
h
is enforced strongly for FE functions:
sf
vu
h
vt
¼v
on G
D;h
on Gsf. (5.20)
h
; vh¼ 0 on Gs0and satisfying
f 0
0
; f˛ V
h
00
, and q ˛ ℚh.
h
We note that the strong enforcement of the interface condition (5.20) together with
condition (5.18) implies that the equality
h
vt
is satisfied in the usual sense inbUs.
¼ v
h
vu
Eqs. (5.17)e(5.20) subject to initial conditions and a choice of continuous extension of u
frombUsontobUf, ensuring uh˛ V
0
defines the system of ODEs. Of course, for the fully
h
discrete method, we still have to discretize in time. This, however, decouples from space
discretization (in the spirit of the method of lines), since the spacial discretization is done
in the time-independent reference domain. Time discretizations will be discussed later in
Section 5.5.1.
As for the extension equation, the user is free to choose a smooth extension of the
displacement field frombU
(Eq. 4.60). However, this freedom is troublesome: in certain
s
h
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cases, a chosen extension can fail to produce physically meaningful displacements, and the
computational mesh becomes tangled inbU
. The choice is often ad hoc, as we will see in
f
the next subsection.
If one considers the case of the compressible Saint VenanteKirchhoff material, the second
term in Eq. (5.17) becomes
Z
FðuÞSðu; uÞ: VJ dx
^
U
s
where
Sðu
; u2Þ¼lstrðEðu1; u2ÞÞI þ 2msEðu1; u2Þ; Eðu1; u2Þ¼
1
Note that Sðu
; u2Þ¼STðu1; u2Þ¼Sðu2; u1Þ.
1
1
2
ÞTFðu2ÞI
Fðu
1
.
s
In the case of the incompressible neo-Hookean material, the following modifications to the
FE formulation should be made:
(i) Change the domain of integration in the pressure-dependent term (sixth term in
Eq. (5.17)) to the wholebU, so it now reads:
Z
phJFTðuhÞ: VJhdx; (5.21)
^
U
W^U
s
f
(ii) Replace the second term in Eq. (5.17) with
Z
m
FðuhÞ: VJhdx;
s
^
U
s
(iii) Consider the incompressibility condition in the form of identity (4.69), and add the
following constraint to the FE formulation:
Z
JV vh: FTðuhÞqhdx ¼ 0 cqh˛ ℚh:
^
U
s
Hence instead of Eq. (5.19), we enforce the constraint in the whole reference domainbU:
Z
JðVvhÞ: FTðuhÞqhdx ¼ 0 c qh˛ ℚh: (5.22)
^
U
W^U
s
f
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5.3.3 Examples
We illustrate the material of Section 5.3 with several numerical examples. For these
examples, we use P2eP1 (TayloreHood) elements for blood velocity and pressure
variables and P2 elements for the vessel wall displacements. FE discretizations are
implemented on the basis of the open source packages Ani2D [170] and Ani3D [171]. The
time stepping and linearization of nonlinear systems are such as explained in Section 5.5.
For the first experiment, we consider propagation of a pressure impulse in a flexible tube
with the circular cross section filled by incompressible viscous fluid [139]. The tube is
fixed at both ends. Initially, the tube is nondeformed, and the fluid is at rest. This is the
simplest model for the 3D blood flow through a compliant artery, which serves for
validating of FSI solvers [128,134,142,143,164,175].
The length and inner diameter of the tube are 50 and 10 mm, respectively, and the tube
wall is 1 mm thick. The fluid density is 10
The wall has density r
¼ 1.2$103g/mm3. The Saint VenanteKirchhoff hyperelastic
s
model is used with elastic modulus E ¼ 3$10
left open boundary of the tube, an impulse of external pressure p
applied during first 3 ms, followed by zero pressure p
boundary, the external pressure p
ext
3
g/mm3, and kinematic viscosity is 3 mm2/s.
5
g/mm/s2and Poisson’s ratio n ¼ 0.3. On the
¼ 1.333$103Pa is
ext
, whereas on the right open
ext
is always zero. These conditions generate a pressure
impulse that travels along the tube. Fig. 5.5 demonstrates the velocity field in the middle
cross section and wall displacement exaggerated by a factor of 10 for clarity, and the
radial and axial components of the displacement of the inner tube wall at half the length
of the pipe, as functions of time. The extension Eq. (5.16) was used in the simulation.
For the second experiment, we reconsider the idealized model of the aortic bifurcation,
which generates the solution of the incompressible NaviereStokes equations shown in
Fig. 5.3. Although the waveforms in vessels with compliant and rigid walls are different
under the same boundary conditions, the aortic bifurcation reflecting severely the pressure
wave [120] can be interpreted as if it has stiffer walls and even rigid walls. Fig. 5.6 shows
cross-sectionally averaged flow rate waveform and pressure waveform at the bifurcation
junction computed from the solutions to Eqs. (5.1) and (5.17) with time-dependent
boundary conditions [132] and linear elastic model with Young’s modulii for inlet and
outlet cylinders E
m
¼ 4 mPa s, density rb¼ 1060 kg/m3, and mean flow rate Qa¼ 0.4791 1/min.
b
¼ 500 kPa, Ei¼ 700 kPa, density rw¼ 1 g/cm3, blood viscosity
a
In the third experiment, we consider a 2D vessel model with an aneurysm [202]. The
computational domain Uð0Þ3 [e8, 0] [0, 8] mm
wall of the vessel is presented by the shaded part, and the rest is the fluid domain. The
upper open part of the boundary is the inflow with inlet velocity
v
ð0; y; tÞ¼50ð8 yÞðy 6Þð1 þ0:75 sinð2ptÞÞ; 6 y 8. The bottom open part of
1
2
is shown in Fig. 5.7. The compliant
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Figure 5.5
Pressure wave test: velocity field, press ure distribution, velocity vectors, and 10-fold enlarged
structure displacement for t ¼ 3ms (top, left) t = 9ms (top, right), and the axial and radial com-
ponents of displacement of the inner wall at half the length of the pipe (bottom).
Figure 5.6
Flow rate waveform (left) and pressure waveform (right) at the junction computed from the
solutions to Eqs. (5.1) and (5.17).
94 Chapter 5
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Figure 5.7
The computational domain for the 2D model aneurysm and the absolute value of the wall shear
stress on the inside of the aneurysm wall.
the boundary serves as the outflow with natural boundary condition. The upper and lower
ends of the artery walls are fixed. The wall material is assumed to be incompressible neoHookean, although compressible neo-Hookean and Saint VenanteKirchhoff models are
applicable as well [173]. We take r
m
¼ 3.4983$103Pa$s from Ref. [202], the shear modulus ms¼ 270 kPa from Ref. [120],
f
¼ 1.12$103kg/m3, rf¼ 1.035$103kg/m3, and
s
where it was experimentally measured for dog’s artery. The extension Eq. (4.63) uses
¼ msand lm¼ 4ls.InFig. 5.7, we present the maximum of the absolute values of the
m
m
wall shear stress (WSS) evaluated along the dilatation wall. WSS peak values along the
vessel wall are crucial in estimating the risk of both aneurysm formation in the initial
stages and the eventual rupture.
For the final experiment in this section, we consider the interaction of a three-dimensional
clamped beam with a fluid flowing in a pipe [149]. The computational domain is obtained
from computer-aided design but is also motivated by biomedical applications. We address
transient laminar flow at maximal Reynolds number 1283, which is realistic for
cardiovascular flows. The benchmark has been used to validate FSI solvers in
Refs. [150,165,172].
Two inlets of diameter 21.9 mm and length 29.5 mm merge smoothly into a cylindrical
domain of length 173.55 mm, which terminates with an outlet of diameter 76.2 mm.
A silicon filament 2 mm 11 mm 65 mm is attached to the wall of the flow chamber in
the merging section z ¼ 0 (see Fig. 5.8). The parameters of Eqs. (4.64)e(4.66) are
r
¼ 1063 kg/m3, rf¼ 1164 kg/m3, and mf¼ 11.49 mm2/s. Saint VenanteKirchhoff
s
material of the filament has the Young modulus E ¼ 2.1626$10
n ¼ 0.3151 [165]. The gravity volumetric force acts along the y-direction. The inflow
velocities are parabolic and periodic with the frequency 1/6 s
5
Pa and Poisson ratio
1
, peak velocities recovered
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Figure 5.8
Left: flow chamber. Right: comparison of y-displacement of the point of filament with coordinate
zz53; x ¼ 0 for t ˛ [0, 6] and recorded experimental data (bottom).
Figure 5.9
Flow streamlines colored by the velocity magnitude at t ¼ 1.153 s (left) and t ¼ 2.449 s (right).
from the experimental data. On the outflow boundary, we prescribe directional outflow
condition as in Eq. (5.6) with a ¼ 1, but nonzero right-hand side g
2
g ¼ 9.81 m/s
is the acceleration of gravity.
¼ gy n, where
N
The dynamics of the filament is driven by normal stresses exerted on the beam by
upcoming flow jet and the buoyancy force. Generated unsteady vortical structures interact
with the filament and may influence its motion [172]. The swing of the filament computed
numerically matches the experimental data (see Fig. 5.8). Fig. 5.9 illustrates the predicted
flow at a couple of time instances. Extension Eq. (5.16) was found capable to provide the
sequence of untangled meshes.
5.4 Simulation of blood flow in the heart
This section discusses the numerical modeling of blood flow in the heart of a patient. Two
major stages are the reconstruction of the flow domain from a sequence of medical images
and the numerical solution of flow equations in the reconstructed time-dependent domain.
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5.4.1 Reconstruction of the heart beat
Our input data are a sequence of 100 contrast-enhanced CT images of beating heart within
one cardiac cycle. As discussed in Chapter 3 (Section 3.5.2), these data are processed to a
sequence of 100 tetrahedral meshes for the left ventricle cavity. The meshes with 14,033
nodes, 88,150 edges, and 69,257 tetrahedra are topologically invariant in the sense that
they have the same nodes, edges, faces, and cells, only node positions are different (see
Figs. 3.10, 3.11). The time interval between two consecutive shapes in the sequence is
12.7 ms; this is the best time resolution for the currently available CT measurements. When
the deformation is fast, large boundary displacements happen over one time interval, which
is the potential cause of numerical error. To cope with this error, we generate a new series of
1981 meshes with 20 times smaller time step Dt ¼ 0.635 ms. The mesh 20(i e 1) þ1 from
the refined series coincides with the mesh i from the coarse series. The nodal coordinates of
intermediate meshes are interpolated by cubic splines from nodal coordinates of available
100 meshes. The meshes from the refined series have good quality. In particular, no cells
violate the condition for the Jacobian of the recovered deformation J > 0, although the
ventricle volume varies considerably within the cardiac cycle (see Fig. 5.10).
5.4.2 Mathematical model and finite element method
Consider a time-dependent domain UðtÞ3 R3occupied by fluid. To formulate a flow
problem, we introduce the reference domainbU ¼ Uð0Þ and a mapping from the
spaceetime cylinder Q :¼bU ½0; T to the physical domain,
phys
x: Q / Q
:¼ W
The mapping is assumed to be level preserving, i.e., x
We assume also that the evolution of UðtÞ is sufficiently smooth x ˛ C
Figure 5.10
Time dependence of ventricle volume.
t ˛ ½0;T
UðtÞftg:
b
U ft
g
¼ UðtÞ for all t ˛ [0, T].
3
ðQÞ3and that there
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exist such positive reals CF; cJthat, for the spatial gradient matrix of x by F ¼ Vxx and
J ¼ detðFÞ, it holds
J cJ> 0; sup
inf
Q
kFk
Q
þ
F
1
F
C
: (5.23)
F
F
Given the mapping, the fluid equations can be written in the quasi-Lagrangian form (4.27)
e(4.29) for the velocity vector fieldbv(x, t) and the pressure functionbp(x, t) defined inbU
for all times t ˛ [0, T]. To set up boundary conditions, we first distinguish between
D
0
; vU
D
0
ns
N
0
ðtÞ,
¼
; vU
ns
ns
are
0
ðtÞ is
different types of the boundaries in the physical flow domain: the no-slip vU
D
Dirichlet vU
vUðtÞ¼vU
1
vUDðtÞ; vU
x
independent of t so that we can use notations vbU
ðtÞ, and outflow vUNðtÞ parts of the boundary, and
ns
ðtÞWvUDðtÞWvUNðtÞ. In the reference domain, we define vU
N
¼ x1vUNðtÞ; vU
0
ns
¼ x1vUnsðtÞ and assume that vU
0
D
; vbUN; vbUnsfor them. On vUns,we
impose no-penetration no-slip boundary condition, i.e., the fluid velocity on vU
equal to the material velocity of the boundary (see the discussion in the following):
b
v ¼x
on vbUns; (5.24)
t
while onbU
D
andbUN, we prescribe Dirichlet and Neumann conditions,
Here,bv
b
v ¼bv
is a given velocity, and n is the exterior unit normal vector on vbU.IfvbUN¼ B,
D
on vbUD;bsbn ¼bg on vbUN: (5.25)
D
we assume
Z
b
n$x
ns
v^U
The FEM builds on the weak formulation of Eqs. (4.27)e(4.29). It consists of finding
v(t, x) and p(t, x) such that for all t ˛ [0, T] v(t, $) ˛ H
satisfying v ¼ x
on vbUns, v ¼ vDon vbUDand
t
Z
JvtJhdx þ
^
U
Z
pJFT: VJ dx þ
^
U
Z
Jf $J dx þ
¼
^
U
Z
JðVvÞF1v x
^
U
Z
Z
Jg$J ds
N
v^U
^
U
Z
b
v^U
n$bv
D
ds þ
t
$J dx þ
t
qJFT:Vv dx
ds ¼ 0: (5.26)
D
1
b
Uand p(t, $) ˛ L
2
b
U,
Z
2mJDxv: DxJ dx
^
U
(5.27)
98 Chapter 5
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for all J ˛ H
looking forfv
Z
vv
J
vt
^
U
Z
phJFT: VJhdx þ
^
U
Z
Jf $Jhdx þ
¼
^
U
1
b
U; J ¼ 0onvbU
ðtÞ; phðtÞg˛ Vh ℚh, satisfying (5.28) for all Jh˛ V
h
ns
WvbUD, q ˛ L
Z
h
$J
h
dx þ
JðVvhÞF1vh x
^
U
Z
qhJFT: Vvhdx
^
Jg$Jhds:
N
v^U
U
Z
2
b
U. The FEM, naturally, consists of
t
$J
h
dx þ
Z
2mJDxvh: DxJhdx
^
U
0
; qh˛ ℚh:
h
(5.28)
5.4.3 Numerical challenges of patient-specific ventricle simulations
The normal velocity of the boundary vUðtÞ is yG¼bn$xt+x
be recovered from the CT data, as discussed in Chapter 3. However, the material tangential
velocity of the boundary is defined by the tangential part of x
mapping, i.e., xðx; tÞ; t ˛ [0, T], which defines the material trajectory for x ˛bU (or at least
for x ˛ vbU). In some applications, such Lagrangian mapping is not available, and in this
case, Eq. (5.24) may produce spurious or zero tangential velocities on the boundary. For
example, this may happen if x is reconstructed from medical images. Thus, in practice, one
may or may not amend Eq. (5.24) based on any additional information (e.g., tagged MRI,
speckle tracking echocardiography) about the tangential motions for a better model.
Besides the ambiguity of tangential motion, the boundary velocities v
compatible with the divergence constraint, i.e., satisfy Eq. (5.26). This additional
compatibility is dictated by the integration by parts formula (Gauss’s theorem) in the
absence of inflow/outflow boundary (vbU
N
¼ B). Displacements of the ventricle wall
contain an error due to roughness of the CT data, its noise, and postprocessing. There is a
little chance to satisfy Eq. (5.26) accurately. Therefore, in this case, we let
ðtÞ : ¼
c
┴
R
v^U
JFTv
ns;D
ðtÞ$nds and update the boundary data v
h
1
. Both normal n and y
only if x is the Lagrangian
t
ns;D
should be
h
ns;D
h
ðtÞ : ¼ v
ns;D
h
ðtÞc┴.
can
G
For the ventricle blood flow, the inflow and outflow parts of the boundary are defined as
those representing the mitral and aortic valves, respectively. In the simplest case, the
valves function is modeled by the Dirichlet condition for the mitral and aortic valves in
their closed state and the homogeneous Neumann condition for the valves in their open
state. The latter, however, do not account for the stresses produced by valves on the
passing blood flow. To model the resistance of aortic valves to the outflow, the Neumann
boundary conditions can be modified as suggested, for example, in Ref. [195] to include
the “momentum” flux,
ðv 5 ðv x
ÞbsÞn ¼bg;
t
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wherebg represents the external normal stress, which increases linearly with respect to the
flow rate through the aortic orifice. Yet, the most adequate boundary conditions for valves
are derived using the geometrical multiscale approach with an FSI model for the
bloodevalves interaction and a 1D network hemodynamic model to represent the
remaining downstream vasculature (see, e.g., Ref. [108]).
It is well known that the systolic contraction and twist of left ventricle produce helical
outflow through the aortic valve. The outflow helicity is additionally enforced by the
presence of carneal trabecules. The segmentation of the medical images may fail to
recover this anatomical detail. If the geometrical model does not contain the network of
carneal trabecules, it may produce less helical flow compared with the actual one.
Another challenge of the cardiac hemodynamics is its transitional and even turbulent flow
regime [123,136,182]. If the computational mesh is not sufficiently fine to resolve all
scales in the flow, then stabilization has to be adopted (see Section 5.2.3).
5.4.4 Patient-specific simulation of blood flow in the left ventricle
We illustrate the topic of this section by performing numerical simulations of
hemodynamics in a simplified model of the human left ventricle. Our main tool is the finite
element method (Eq. 5.28) for solving the NaviereStokes equations in a moving domain.
Simplifications are made by omitting some fine anatomical structures, using simple
boundary conditions on the aortic and mitral orifices, and neglecting the ventricle twisting.
The motion of the ventricle UðtÞwithin one cardiac cycle is given by a sequence of 1981
k
topologically invariant tetrahedral meshes U
reference grid the first mesh in the sequence and identify the coordinate of each node x
k
U
with x
h
k
1
x
. The mapping x
k
is the continuous piecewise linear vector function with
; k ¼ 1; .; 1981. We associate with the
h
k
in
prescribed values at the reference grid nodes. The stability and accuracy of the FEM under
some smoothness assumptions on the mapping x is analyzed in Ref. [174].
The ventricle boundary is split into aortic valve and mitral valve patches and the
remaining part of the boundary. In the systole phase, t ˛ [0, 355] ms, we set the
“do-nothing” boundary condition (5.25) withbg ¼ 0 (normal stress vanishes) on the patch
associated with the aortic valve. In the diastole phase, t ˛ [355, 1257] ms, the same
boundary condition on the normal stress is imposed on the patch associated with the mitral
valve. The zero normal stress condition imitates valve’s opening. On the remaining part of
the boundary, including the aortic valve during the diastole phase and the mitral valve
during the systole phase, we impose the no-penetration no-slip condition (5.24), v ¼ x
The viscosity of blood was set m ¼ 4mm
s, which corresponds to the Reynolds number of order 10
2
/s, and the maximum velocities attain 1000 mm/
3
. Thus, we stabilize the flow
.
t
using the Smagorinsky turbulent viscosity (5.15).
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