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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_3774_Библиотеки_им_академика_М_И_Перельмана
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control volume control volume
volume centroid
ab
Computational Fluid Dynamics intheArterial System: Implications forVascular…
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volume centroid
P
P
Fig. 8.11 Control volume of (a) cell-centred and (b) cell-vertex scheme
dependent values are stored at the nodes, but in nite volume method, the dependent values are stored in the centre of the nite volume.
8.4.5 Fluid: Solid Interaction
Modelling Fluid-Solid Interaction (FSI) as a movable or deformable structure, combines the laws of structural mechanics and uid ow [21]. The interface between the
solid wall and the uid is a means of transferring force and heat from one to the
other. FSI allows intersection of two separate meshes to be coupled and transfer heat
and force between them. This interaction could be one-way or two-way (fully) coupled. In one-way coupled interaction, the solver simulates one media and then the
other, whereas in the fully-coupled method, both media are solved in the same system simultaneously. One of the main advantages of the fully-coupled method is that
it converges to the results in fewer iterations (in a shorter time), although each iteration needs more computing memory (Fig.8.12).
The results of a CFD solution provide pressure data from blood ow at the interface, and the structural FEA solver uses the CFD results (blood pressure and forces)
as an input to calculate the loads and deformations on the blood vessel. The deformed
solid structure (blood vessel) makes a new FSI surface which in turn changes the
boundary conditions of the blood ow for the next CFD calculation. This is a coupled approach where the interaction between uid ow and a solid structure subject
to moving boundaries are solved through multiple cycles. As the below gure
shows, two-way coupling uses an iteration procedure in time-dependent calculations. FSI solution can help us calculate load transfer and get a better surface map,
and consequently, a better estimate of the structural response [22].
The FSI modelling procedure is based on the generation of meshes. Through
using a conformed mesh, FSI considers the interface characteristics as the physical
boundary conditions. Therefore the interface becomes a part of the solution. Due to
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S. Mishani et al.
CFD meshing CFD meshing
CFD meshing
update
Fig. 8.12 Workow of two-way FSI
Pressure
transferring-
fluid to solid
Deformed
geometry
Ye s
CFD meshing
Geometry
shape
Changed?
No
Final results
the solid structure deformation, mesh updating is needed as a part of the solution in
every iteration. FSI models need extensive computation and are therefore time
consuming.
8.5 Material Properties
8.5.1 Blood Flow Properties
Viscosity is an important variable to describe uid behaviour and its resistance
to blood ow in the vasculature. Internal friction of the moving uid generates
uid shear stress that determines uid shear strain. If uid shear stress is directly
proportional to the uid shear strain, the uid is called Newtonian and if not, it
is non- Newtonian. Blood is an incompressible suspension and it behaves as a
non-Newtonian (shear-thinning) uid. It is much easier to numerically model a
Newtonian uid than a non-Newtonian uid, and for some calculations, the assumption of Newtonian behaviour is acceptable (in straight and xed cross-section area
vessels with an arterial diameter greater than 10mm) [23–25].
Blood ow has a complex rheology, so a wide range of constitutive equations
have been proposed to model its non-Newtonian behaviour and dene the relationship between shear stress and shear strain rates. Numerical simulation of blood ow
requires an appropriate constitutive model to reect its shear thinning properties
[26]. The most prevalent non-Newtonian models used for blood viscosity computation, with different degrees of accuracy, are the power law, Casson, Carreau and
Carreau/Yasuda models [23].
The power-law model describes the shear thinning behaviour of blood that
depends mainly on the haematocrit, which varies from one person to another.
Equation 8.3 expresses the relationship between the two; the higher the haematocrit
the greater the blood viscosity [27];
m
=+ -+1 4175 5 878 15 98 31 964
..
,HH H
(8.3)
se
=´E ,
Fluid Shear rate
w viscosity model
Fluid Viscosity
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Fig. 8.13 Blood ow
viscosity behaviour by
Newtonian, power-law and
Carreau models [
28]
Power-la
Carreau viscosity model
Newtonian Fluid
where ‘μ’ is the viscosity of whole blood in poise and H is the haematocrit (%)
divided by 100.
The Casson model is specically used for low shear stress blood ow in narrow
arteries. The Carreau and Carreau-Yasuda models, as shown in Fig.8.13, are similar
to the power-law model, but they t both Newtonian and non-Newtonian blood ow.
Although the instantaneous shear rate of blood ow varies from zero to approximately 1000 s−1 in a cardiac cycle, blood ow in a shear rate range greater than 100
s−1 is almost constant (thus it is behaving as a Newtonian uid), with an approxi-
mate viscosity of 0.035 poise (Pa.s) [23, 28, 29] Blood viscosity also depends on
factors such as smoking status and cholesterol levels, etc. [30].
8.5.2 Blood Vessel Properties
In addition to the geometry of the arterial wall, the mechanical properties of the
artery should be considered. Hooke’s law describes the relationship between the
stress and strain rates for most homogenous materials (Eq. 8.4) within a certain
range of stresses, under simple uniaxial loading conditions [31]. The stress versus
strain curve of a multi-linear homogeneous material is shown in Fig.8.14. The
modulus of elasticity, E1, is the slope of the linearly elastic range in the stress–strain
curve. However, the linear range of plastic properties of the material can be shown
by Young’s modulus (e.g.E2 andE3).
where:
• E: Modulus of elasticity or Young’s modulus (N/m2),
• σ: Normal stress (N/m2), and.
• ε: Strain (unitless).
(8.4)
186
e
d
dt
12
Plastic zone
t
Stress (s)
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Elastic zone
Ultimate point
E
Yield point
E
2
E
1
Fig. 8.14 Stress–strain curve for a homogenous material
3
Hooke’s Law E =
Strain (e)
s
e
Failure poin
Arteries are not linearly elastic and so not homogenous. The slope of the uniaxial
loading curve varies with stress and strain, so, as strain rate increases in an artery,
the artery becomes stiffer. Arteries are made of three different types of material,
elastin, collagen and smooth muscle. Elastin is easily stretched while Collagen is
not and resists arterial stretch. Moreover, the contraction and expansion of smooth
muscle in the artery can resist strain. Collagen bres have a wavy pattern in stressfree conditions. They only stretch when the artery has been already stretched and
the collagen bres straightened. There are three major zones in the stress-strain
curve in the arterial wall including (Fig.8.15):
• Zone I: the toe region,
• Zone II: the linear region, and
• Zone III: the yield and nally failure region.
Arterial walls have viscoelastic properties. The stress in viscoelastic material
relates not only to load variation, but also on the rate of change of strain which is
dependent on time, as shown in eq. 8.5 [32].
A large artery consists of three layers; the intima, media and adventitia [33], which
are made from different constituent materials (Collagen, Elastin, and Smooth
se
=´+´EE
,
(8.5)
Strain (e)
Stress (s)
Zone I Zone II Zone III
Computational Fluid Dynamics intheArterial System: Implications forVascular…
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E
Hooke’s Law E =
Fig. 8.15 Stress–strain curve for an arterial wall
s
e
E
1
E
3
2
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Table 8.1 Mechanical properties of constituents of a
large artery
muscle) with different mechanical properties (as shown in Table8.1). Hence, arter-
Arterial tissue E (MPa)
Collagen 0.3–10
Elastin 0.6
Smooth muscle 0.01
0.01
0.25
E modulus of elasticity, MPa
megapascals
ies are nonhomogeneous and anisotropic. The intima is made up of a monolayer
endothelial lining, which is supported by loose connective tissue and allows the
intima to move relative to the media. The media is made up of elastin bres, smooth
muscle cells and collagen bres. Elastin bres are able to stretch up to 2–3times
their original length without rupture. Collagen bres are up to 5000 times stiffer
than elastin. Smooth muscle cells contribute to arterial wall stiffness. The adventitia, is made of tough collagen bres and connective tissue.
3
The average density of the artery is 1080kg/m
[34–37], and the average Modulus of elasticity is 1MPa (Mega-pascal) [38,
and the Poisson’s ratio 0.5
39]. The ultimate tensile stresses in the adventitia is almost three times more
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than the ultimate tensile stress in the media and intima. The mechanical parameters of the arterial wall play a crucial role in stress and strain analysis, as
atherosclerosis can dramatically reduce the modulus of elasticity of the arterial
wall [40].
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8.5.3 Initial andBoundary Conditions
Every biomedical numerical simulation is dened under the limits of boundary conditions which describe the physiological conditions. The results of CFD modelling
are valid, if boundary condition parameters are incorporated in the governing equations [5]. Some of the common initial and boundary conditions in CFD are the
following;
• Inlet and outlet ow conditions;
• Loads and displacements in uid solid interactions or contact nodes;
• Velocity and pressure conditions;
• Axisymmetric/symmetric boundary conditions;
• Periodic/cyclic initial conditions.
8.6 Pulsatile Flow inArteries
A uid with periodic ow variation is known as pulsatile ow (Womersley ow).
The oscillatory nature of pulsatile blood ow imposes other forces, (apart from uid
driving forces in the case of steady ow), through endothelial cells to the arterial
wall when viscoelastic (non-Newtonian) properties of blood have been taken into
account [41]. Blood ow through curvatures and bifurcations of large and medium
sized arteries can induce secondary ows (ow separation and recirculation). The
secondary ows can be expected to have signicant effects on arterial structure in
the presence of atherosclerosis [42].
A numerical model can simulate pulsatile blood ow in an artery to investigate
the effect of blood ow on the arterial wall to determine the resultant stresses. The
numerical model develops a shear-thinning rheological behaviour and uses the
time-dependent, three-dimensional, incompressible Navier-Stokes equation for
non-Newtonian uid. The main concern of numerical modelling is to create an
interaction surface between periodically varying blood ow pressure and the elastoplastic blood vessel. The stress analysis of a typical bifurcating artery, as shown in
Fig.8.16, under pulsatile blood ow pressure shows the distribution of maximum
stresses at the innermost layer (intima) of an arterial wall. The shear stress contour
plot shows the variation of the stress from 0.017–2.459MPa (at the apex) which has
been observed in the literature [43].
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Fig. 8.16 Maximum stress contours at the inner surface of an arterial bifurcation
8.7 What Is Convergence?
CFD analysis deals with the governing Navier-Stokes equations which have inherent complexity and are highly non-linear in nature. As a result, CFD models cannot
be solved easily. The convergence criteria in CFD analysis dene how close to the
exact solution is acceptable. In other words, the convergence criterion is the allowable error in the calculation. If the CFD model does not converge, then mesh renement and changing the boundary conditions can be considered.
8.7.1 Result Analysis (Post-processing)
Nowadays, nite element software has been able to analyse and interpret complex
structural and uid ow systems using powerful computers. This capacity generates
more nite element results, so the outcome of the design and analysis calculations
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Fig. 8.17 Velocity ow vectors
can be an enormous le. Therefore, post-processing software helps the user in making design/analysis decisions by interpreting the FEA results by displaying them in
graphical form (Figs.8.17, 8.18, and 8.19) [44].
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8.7.2 Validation andVerication
Verication and validation are the methods used to assess and estimate the error and
uncertainty of CFD modelling and its results. Verication and validation generate
condence in the accuracy and reliability of CFD simulations [45, 46]. To verify
and validate a CFD model, one or more of the following can be used:
1. Analytical methods;
2. Experimental methods;
3. Results from similar literature;
4. Benchmark simulation similar to the study.
8.8 Clinical Applications
CFD analysis allows medical researchers to use medical imaging of arteries to study
haemodynamics, predict blood-ow behaviour and assess blood pressure distribution [47]. Computational simulation can obtain valuable information from invivo
measurements to quantify haemodynamic parameters of patient-specic anatomy
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Fig. 8.18 Velocity ow pathway
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Fig. 8.19 Total deformation of Aorta
and pathology and consequently inform clinical decision-making. Furthermore,
CFD techniques can be used to investigate pathological change in vessels but also
to support preventative initiatives which lower clinical events. Finite element analysis utilises a multi-layer arterial wall and elastoplastic mechanical properties which
are representative of invivo situations and hard to reproduce experimentally.
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8.8.1 Atherosclerosis
Blood ow velocity, oscillating pressure, and resultant wall shear stress (WSS) have
been suggested as playing key roles in the development of early atherosclerosis
[48]. Atherosclerosis affects specic arterial geography with certain biomechanical
properties. The link between luminal haemodynamic and vascular geometries can
be easily explored with CFD and explains why atherosclerotic plaque develops at
bifurcations, branch origins and areas of external constraint such as the adductor canal.
CFD analysis of the stress contours and ow velocity proles at the carotid artery
bifurcation illustrate that arterial WSS is highest at the bifurcation apex, and lowest
at the lateral side of the interior carotid artery origin [49]. From the ow point of
view, the WSS depends on the magnitude and direction of the blood ow velocity
vector at the arterial wall; i.e. WSS is highest in regions where blood ows parallel
to the wall and lowest in regions where the ow is not parallel (secondary ow)
[50]. Low WSS is traditionally associated with increase particle transit times and
therefore plaque formation [3, 24].
8.8.2 Stenosis
Arterial stenosis from plaque burden results in reduced luminal ow and a pressure
drop as well as turbulence and vortices ow post stenosis, which may cause intimal
damage and dilatation. Compression of the subclavian artery as it crosses the rst
rib can cause stenosis if the artery is constrained by other neighbouring structures
[51]. Blood ow is highest within the stenosis, becoming turbulent with lower pressure after the stenosis [52].
CFD is a reliable tool which can quantitatively investigate the haemodynamic
characteristics of blood ow and consequently calculate the WSS distribution along
the length of diseased artery and nd the separation point post-stenosis for different
degrees of narrowing. The turbulence precipitated by a stenosis results in forces on
the wall of the artery beyond the stenosis at angles away from the direction of ow
resulting in post-stenotic dilatation i.e. WSS forces work within the wall at the stenosis and dilating radial forces work on the wall beyond the stenosis.
8.8.3 Aneurysm
Aneurysm is a dilation of the arterial wall and the result of a complex biomechanical
and biological predisposition which is created by pulsatile blood pressure creating radial forces on the wall of the artery. The interaction between arterial WSS
and blood ow parameters play a critical role in the mechanisms of aneurysm
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