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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_3592_Библиотеки_им_академика_М_И_Перельмана
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Vascular and Intravascular Imaging Trends, Analysis, and Challenges, Volume 1
https://t.me/medicina_free
concentration is a function of the wall shear stress, while the adhesion of monocytes
is a function of shear stress and VCAM. Finally, the alterations of the arterial wall
are simulated. A finite element solver is used to solve the system of the equations.
The lumen is defined as a 2D domain while the intima is simplified as 1D model due
to its thin geometry. First, the LDL penetration to the arterial wall as well as the
wall shear stress is calculated. Then the concentration of the various components of
the model is calculated in order to simulate the intima fattening in the final step.
The LDL penetration is defined by the convection–diffusion equation, while the
endothelial permeability is shear stress dependent. This model produces results
about the initial stages of the atherosclerotic plaque formation. More specifically,
concentration of LDL is calculated on the artery wall and in the next step the
oxidized LDL. Furthermore, monocytes and their modified form (macrophages) are
also counted. The solution to the system provides the user with the concentration of
foam cells created when a threshold LDL concentration is reached.
The previous model describes the initial stages of atherosclerosis. However,
atherosclerosis is characterized by the proliferation of smooth muscle cells (SMCs).
A medical user needs a prediction for plaque formation, which is based on the
concentration of SMCs, the necrotic core and the extracellular matrix. In this respect
a new approach to count the concentration of SMCs is being developed. The user is
also provided with results regarding the formation of plaque in an overall manner.
The fluid is assumed to be steady, incompressible and laminar. For modeling fluid
dynamics in the lumen, the following Navier–Stokes equations are used:
2
μρ−∇ + ·∇ +∇ =uuup() 0
lll
∇=u 0,
l
l
(3.12)
(3.13)
where ulis blood velocity, plis pressure, μ is blood dynamic viscosity and ρ is blood
density.
Darcy’s law were used to model mass transfer across the wall (transmural flow) of
the blood vessel:
⎞
⎛
k
⎟
⎜
−∇ =u
w
p 0 (3.14)
w
⎟
⎜
μ
⎠
⎝
p
∇=u 0,
w
(3.15)
where uwis transmural velocity, pwpressure in the arterial wall, μpis the viscosity of
the blood plasma and k is the Darcian permeability coefficient of the arterial wall
(3.14) and (3.15). Convective diffusion equations are used for modeling mass
transfer in the lumen:
∇· − ∇ + =Dc cu( ) 0, (3.16)
ll ll
where clrepresents the blood concentration in the lumen and Dlis the diffusion
coefficient of the lumen.
3-12
v
v
F
F
F
F
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The following convective diffusion reactive equation is used for modeling mass
transfer in the wall, related to transmural flow:
∇· − ∇ + =Dc Kcu rc( ) , (3.17)
ww ww ww
where cwis the solute concentration in the arterial wall, Dwis the diffusive coefficient
of solution in the wall, K is the solute lag coefficient and r
is the consumption rate
w
constant.
The coupling of fluid dynamics and solute dynamics at the endothelium was
achieved by the Kedem–Katchalsky equations:
v
pd
sf
δπ=Δ−ΔJLp()
δ=Δ+ −JPc Jc(1 ) ,
v
(3.18)
(3.19)
where Lpis the hydraulic conductivity of the endothelium, Δc is the solute
concentration difference across the endothelium, Δp is the pressure drop across
the endothelium, Δπ is the oncotic pressure difference across the endothelium, σ
the osmotic reflection coefficient, σ
endothelial permeability and
is the solvent reflection coefficient, P is the solute
f
is the mean endothelial concentration [54].
c
d
The inflammatory process is modeled using three additional reaction–diffusion
partial differential equations [55]:
∂=Δ− ·
OdOkOM
11
t
∂+ =Δ− · + +
MvMdMkOMSS
div( ) /(1 )
tw
∂=Δ− + · + −
SdS SkOM OO
31
t
21
λγ
(),
thr
(3.20)
where O is the oxidized LDL in the wall, M and S are concentrations in the intima of
macrophages and cytokines, respectively; d
, d2, d3are the corresponding diffusion
1
coefficients; λ and γ are the degradation and LDL oxidized detection coefficients;
and
is the inflammatory velocity of plaque growth [55, 56].
w
is
3.2.5 Discrete approach
Blood flow in the small coupled domain within an FE mesh is viewed as a motion of
the collection of DPD particles. The motion of each DPD particle is described by the
following Newton’s law equation:
where
of velocity;
is the mass of particle i;
m
i
C
,
ij
(Brownian) interaction forces that particle j exerts on particle i, respectively,
provided particle j is within the radius of influence
external force exerted on particle i, which usually represents gradient of pressure or
C
=+++
m vFFFF,
∑
ii
D
ij
and
R
ij
()
j
are the conservative (repulsive), dissipative and random
D
ij
ij
is the particle acceleration as the time derivative
i
3-13
R
ext
ij
i
of particle i; and
r
c
i
ext
(3.21)
is the
F
F
L
L
Vascular and Intravascular Imaging Trends, Analysis, and Challenges, Volume 1
https://t.me/medicina_free
Figure 3.3. Interaction forces in the DPD approach.
gravity force as a driving force for the fluid domain [57]. Hence, the total interaction
force
(figure 3.3) between the two particles is
ij
C
D
=++FF F F. (3.22)
ij
ij
R
ij
ij
3.2.6 DPD modeling of oxidized LDL particle adhesion to the wall
When an oxidized LDL comes close to the wall and if the shear rate allows, it binds
to the wall. However, when adhered LDL particles are exposed simultaneously to
other forces stronger than the binding forces, the bonds break. To incorporate LDL
a
adhesion to vessel walls, we introduce an attractive (bonding) force (
) in addition
ij
to the conservative, viscous and random interaction forces. As an approximation, we
model the attractive force with a linear spring attached to the LDL’s surface. The
spring is attached to the vessel wall or to an already adhered LDL particle. The
effective spring constant for LDL adhesion on the vessel wall, or to another
stationary LDL particle, is denoted by k
bw
.
The additional parameter involved in the model is the size of the domain from the
wall
LDL coated wall (
) for which the action of attractive force needs to be
max
considered. We take the attractive force as
where
a
=−Fk LL
1/
bw w
w
is the distance of the oxidized LDL from the wall.
w
(
wall
max
)
(3.23)
3.3 Results
3.3.1 Coupled method for modeling of atherosclerosis
Atherosclerosis development for two patients specific in the left and right coronary
artery was simulated. Upstream of the bifurcation level there was plaque progression
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Figure 3.4. The right coronary artery for a specific patient. Coupled simulation FE and DPD method.
(figure 3.4). The biomolecular parameters ICAM1, LDL, high-density lipoprotein
(HDL) and triglycerides are used for the computer simulation. The coupled
continuum and discrete method were implemented.
3.3.2 Stent deployment modeling
A system of stent deployment is modeled which consists of three parts: the balloon,
stent and blood vessel. The first part, the elastic balloon, should be inflated in order
to open the stent and blood vessel with stenosis. The second part is the stent, a wired
structure that should open and hold the narrowed blood vessel. The third part is the
blood vessel with a narrowing caused by plaque progression.
The model consists of eight-node brick linear finite elements. The materials of all
three structures are linearly elastic but with large deformations. The boundary
conditions applied in this model are fixed nodes at the beginning and at the end of
the blood vessel with stenosis, a prescribed pressure in the balloon and symmetry
boundary conditions at all three parts of the FE mesh (because we model half of the
model based on symmetry assumptions). The value of the pressure is not significant in
this case, because it is fitted only to open stenosis. The pressure increases linearly over
time. The simulation has 200 time steps of 5 ms. In addition to the mentioned number
of 3D finite elements, when the balloon and stent are in contact, or when the stent and
blood vessel are in contact, there is a variable number of elastic support elements,
which depends on the contact area size between the 3D elements in contact.
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Figure 3.5. FE model of (a) the elastic balloon, (b) the stent and (c) the blood vessel with stenosis.
The three parts of the model are shown separately in figure 3.5. There are (a) the
FE mesh of the balloon, (b) the FE mesh of the stent and (c) the FE mesh of the
blood vessel with stenosis obtained using parametrized geometry.
The initial results for the stent deployment model obtained with the solver
developed using the PAK software package are presented [51]. The PAK software is
upgraded with a contact algorithm developed during this study. The initial model is
parametric, as explained in the previous section.
The results for a stent opened by an inflated balloon are given in figure 3.6. There
are two time steps: time step 10 and 160. These two steps are characteristic because
in time step 10 contact appears between the elastic balloon and stent, and in time
step 160 the narrowed blood vessel is completely opened by the balloon and stent. As
can be seen from the images, at the beginning of the simulation, there is no contact
between the elastic balloon and stent, and only the balloon has deformation. At time
step 10 the contact appears, and the balloon starts opening the stent and also the
blood vessel. At time step 160 the diameter of the narrowed blood vessel is restored
to its original dimensions.
3.3.3 Deformable artery wall
In the case of arteries with deformable walls, additional input data are required for
blood flow simulations compared to the case of arteries with rigid walls. First, the
3D geometry model of the arterial wall is required (figure 3.7). In addition to the
boundary conditions for the fluid domain, the user must also specify the appropriate
boundary conditions for the solid domain. The boundary condition for the solid
domain is the area that is considered to be fixed, in order for the model to be
constrained. This is an issue of great importance since it greatly alters the generated
results regarding the displacement of the arterial wall. Furthermore, the interface
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Figure 3.6. Results obtained using the developed parametric model. The displacement fields for: the stent with
balloon, (a) time step 10 and (b) time step 160; the stent, (c) time step 10 and (d) time step 160; and the whole
model, (e) time step 10 and (f) time step 160.
Figure 3.7. 3D reconstructed model for the arterial wall (right) and the lumen (left).
between the arterial wall and the lumen is specified as fluid–structure interface where
interaction occurs between the fluid (blood) and the solid (wall). Furthermore, the
user must also specify the material of the arterial wall (i.e. linear elastic, hyperelastic,
etc). The choice of the wall material can also influence the obtained results on WSS
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Figure 3.8. Wall deformation (right) and WSS distribution (left).
distribution as well as the wall deformation. However, there is also a great difference
regarding computational time. Moreover, the definition of these parameters is of
great significance since it is very difficult to acquire exact patient-specific data. The
final output data include the areas of low WSS, the WSS distribution within the
arterial wall and the wall deformation (figure 3.8), which can provide useful
information on plaque generation and progression. The user has the capability to
visualize the results in an appropriate environment.
3.3.4 Nitinol material model
In order to perform computer modeling of the combined effects of the surrounding
arterial wall and inner forces of blood and stent deployment against the arterial wall,
a 3D reconstruction from IVUS and angiography is derived.
The FE model consists of the solid domain and the fluid domain. The solid
domain consists of the stent and arterial wall. The fluid and solid domains are
modeled using 3D-eight-node FEs.
The boundary conditions for the solid surrounding the artery are as follows. It is
assumed that the first and last cross-sections do not move axially, hence all FE
element nodes in these cross-sections are axially restrained.
It is also assumed that the wall material is orthotropic nonlinear elastic, and the
Fung material model is adopted [25]. The strain energy function is defined. The
material parameters
ca a a,, ,
12 4
are determined using the data fitting procedure
from [19 ]. Material parameters obtained from the fitting procedure are
====caaa0.7565[MPa], 0.166, 0.084, 0.045. (3.24)
12 4
For the stent material, the alloy of Nitinol is adopted for the definition of the
material. The material parameters characterizing this alloy are [19]
f
AS
ν
SA
s
L
f
SA
(3.25)
1
−
==
EC60 000 [MPa] 0.3
AS
σσσσ
====
520 750 550 200
s
AS SA
ββ ε
====
250 20 7.5% 0 [MPa K ],
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μ
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where allσandβparameters are in MPa. The material parameters of blood are the
density
ρ =×
1.05 10 [g mm ]
33
and dynamic viscosity
=×
3.675 10 [Pa s]
−
3
.
−−
3.3.5 Stress analysis for stent deployment
According to the boundary conditions and loads mentioned above, the numerical
analysis of the material behavior of this complex model is performed. To examine
different loading conditions, we apply the hemodynamic flow as well as stent
deployment procedure at the arterial wall.
The stent is loaded by an internal uniform radial pressure which varies linearly
from 0 to 1 MPa. Due to the artery incompressibility requirement and to avoid
locking problems, eight-node brick elements are used in all the analyses [19]. In
particular, in the simulations we use up to 232 214 elements and 257 532 nodes,
resulting in 666 354 variables. The interaction between the expanding stent and the
artery is described as contact between deformable surfaces. As contact conditions,
we set finite sliding, no friction, with the constraint enforced by a Lagrange
multiplier method. The stenotic segment of the artery which was examined before
and after stent deployment is presented in figure 3.9.
Blood flow analysis was performed using the FE method described in the methods
section. The shear stress distribution before and after stent deployment is shown in
figure 3.10. It can be seen that the stent reduced wall shear stress significantly after
deployment, which is caused by opening the artery and reducing the narrowing.
The effective von Mises stress distribution in the stent is presented in figure 3.11.
It can be observed that highest stresses are located near the connectors between the
stent struts. These parts are subjected to plastic deformation with maximal stress
around 180 MPa.
The effective stress distribution in the arterial wall at the two different crosssection locations at the end of stent deployment with maximum deployment pressure
is shown in figure 3.12. It can be observed that higher stress exists when wall
thickness is reduced during the deployment procedure.
Effective stress distribution for the inflation pressure 1 kPa for stent deployment
in the carotid artery is presented in figure 3.13.
Figure 3.9. Stent positioning before (a) and after (b) stent deployment.
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Figure 3.10. Effective stress distribution inside the arterial wall after stent deployment. The units are in MPa.
Figure 3.11. Effective von Mises stress distribution for inflation pressure of 1 MPa. The units are in MPa.
3.3.6 Plaque concentration for stented arteries
Plaque progression for a specific patient in the left coronary artery was detected
using CTA image analysis at baseline and after 12 months. Volume progression
from 50% to 70% was observed with segmentation and registration of CT images.
We first examined the baseline and follow-up shear stress distribution (figure 3.14).
The boundary conditions for the blood inflow were the same for both the baseline
and follow-up studies because we consider there was not significant change.
Downstream, the bifurcation level in the second marginal branch showed predominantly low WSS values occurring at baseline (figure 3.15, left panel). A similar
situation with the same patient was observed in the right coronary artery in the distal
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Figure 3.12. Effective stress distribution in the two different cross-section locations inside the arterial wall at
the end of stent deployment.
Figure 3.13. Stent deployment in the carotid artery. Stress distribution for the inflation pressure 1 kPa.
portion of the artery (figure 3.15), that showed a progression of the baseline stenosis.
The location of the lowest WSS in the distal portion of the vessel corresponded to the
site of plaque growth after 12 months. The biomolecular parameters cholesterol,
LDL, HDL and triglycerides for the patient at baseline are listed in table 3.1. These
parameters are used for the computer simulation. It can be seen that intra-plaque
WSS values were lower at baseline compared to the follow-up situation. The red in
figures 3.14 and 3.15 denotes the maximal plaque concentration, which directly gives
the plaque volume for the left and right specific patient coronary arteries.
3.4 Discussion and conclusions
We analyze stent modeling with plaque formation and progression for specific
patients in the coronary and carotid arteries. A coupled 3D artery reconstruction
from IVUS and angiography imaging modalities is described, with the ability to
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