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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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8 Computational Fluid Dynamics intheArterial System: Implications forVascular…
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• Secondary Flow: In non-uniform and non-circular conduits, there exists the
phenomenon that relatively minor uid particles ow away from main uid
stream, which ows parallel to intramural surface of the conduit, toward the
surface of conduit at any ow section, this minor ow is called second-
ary ow.
• Separation point: In a conduit, when a uid ows over a curved surface, such
as a bifurcation or stenosis, the uid may separate from the inner surface of the
conduit (wall boundary layer). This point, at which the wall shear stress van-
ishes, is called the ow separation point. The characteristics of the ow after the
separation point will be turbulent with eddies and vortices.
• Poisson’s ratio: The ratio of the change in lateral strain to the axial strain of a
material under uniaxial elastic stretching.
• Control volume: Creating a mathematical model of a volume in space which is
xed or moving with constant velocity through uid ow. The surface enclosing
the control volume is referred as the volume boundaries.
• Boundary conditions (BC): The physical conditions that are required in a
model to set specic parameters for displacements and forces.
• Discretisation (Meshing): The process of subdividing the domain (body) of a
nite element system with associated nodes and elements.
• Mesh renement: Increasing the number of elements in a domain to improve
the accuracy of a solution, but requiring longer processing time.
• Element (Cell): The small component of domain in a structure of uid which is
interconnected to other components within the domain.
• Nodes: The points on the corners of each element.
• Edge: Boundary of a face of an element.
• Face: Boundary of an element.
• Domain: A grouping of nodes, faces, and elements of a structure or uid.
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8.3 Finite Element Analysis (FEA)
Finite element analysis (FEA) is a numerical technique used to obtain an approximate answer to a complex engineering and physical problem in a specic domain.
In the early 1940s, FEA was developed to solve complex elasticity and structural
analysis problems such as buildings and bridges etc. [5]. FEA is based on building
a complicated object by assembling small and manageable pieces, like a child’s
Lego® (Fig.8.1). These small parts are rods, plates and blocks representing 1-D,
2-D and 3-D elements, respectively, whose material properties and behaviour are
readily understood. The pieces or units are called elements. The domain is composed of a nite number of elements connected to each other in the end nodes.
These nite elements are connected to neighbouring elements at ‘nodes’ [6]. Some
of these nodes may be given limitations in degrees of freedom (DOF) to constrain
them in specic positions and directions. Relevant force and pressure set in the
boundary conditions are then imposed on the elements.
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face
nodes
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Fig. 8.1 The nite
element grid
cell
S. Mishani et al.
edge
A simple mathematical and physical model can be solved through a number of
equations using calculus or trigonometry analysis techniques. This is an analytical
solution and there will be an exact answer. In complex equations, the analytical
techniques become too complicated so numerical methods are used instead.
However the result is an approximation, and consequently, engineers and scientists
use FEA methods to build computational models and enhance the accuracy of the
solution with the aim of converging around the exact value.
Finite element analysis starts with understanding the geometry and physics of the
system. The geometry is represented using a 3D-model. A typical 3D-model will accurately describe the shape and structure. The FEA also considers the physics that is
related to the modelling, such as the material properties, the forces and constraints, and
other parameters which can affect the results. The physics of the model is not always
precisely known, so the accuracy can vary, however it is easier to develop a simple
model and then add complexity rather than use a complex model then simplify it.
The mesh is the partition of a given numerical model into elements, such that
every point of the model is found in one of those nite elements. The size and shape
of the elements is the challenging part of FEA.For example, smaller sized elements
give more accurate results but increase the processing time. Apart from the appearance of the elements, the model must represent the material properties of the object,
and imposed forces and constraints. Thus the nite element method analyses complex structures based on geometry and their physical properties [7].
There are three main steps to FEA (Fig.8.2);
1. Build the model (Pre-processing)
2. Solve the model (Solver)
3. Display the results (Post-processing)
The nite element method used in engineering and research utilizes commercial
software which comes with pre- and post-processing abilities e.g. ANSYS
pre-processor is used to feed modelling and design data into the software and the
post-processor is used to present the results in the form of a graph, table and contour
map. However the ability to use nite element software (pre- and post-processing)
depends on the user having the core knowledge and understanding of FEA needed
to obtain accurate results [8].
®
[5]. The
r
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Computational Fluid Dynamics intheArterial System: Implications forVascular…
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Fig. 8.2 Three steps of
model construction
No
Start
Geometry
(Clinical Imaging)
Segmentation
(Reconstruction)
Meshing
(Discretisation)
Initial and Boundary
Condition
Simulation
Ye s
Convergence
Pre-processor
Solver
8.4 Computational Fluid Dynamics (CFD)
In the past, CFD has provided a powerful and popular tool for the study of haemodynamic and image-based modelling of blood ow in the development, diagnosis,
and also treatment of cardiovascular disease. CFD is a ow measurement technique helping to make links between haemodynamic arterial wall shear stress
(WSS) and the distribution of atherosclerotic plaque to explain why the plaque
Validation
Displaying Result
Stop
Post-processo
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develops at arterial junctions, for example [9]. Moreover, CFD can also be linked
to medical images. These images can be extracted from invivo vascular geometry
and, along with ow data from ultrasound, can provide adequate inputs for CFD
modelling [10].
The essential part of producing a CFD model is deriving volumetric (3D) images
from lumen geometry, often derived from Magnetic Resonance Imaging (MRI) or
Computed Tomographic (CT) imaging, with the inlet and outlet blood ow parameters taken from medical ultrasound. CFD analyses the pressure and WSS for every
element. Therefore, the results can depict the time and position (temporal and spatial) distribution of the forces on the innermost surface of the arterial wall.
One of the main advantages of using CFD over experimental methods is that all
blood ow properties (such as velocities, pressures, shear force) can be calculated
throughout the simulation process. The other advantage is blood ow visualization
which provides valuable metrics for quantitative analysis.
The processes of deriving the CFD model from the MRA or CT scan of patients
include:
• Specifying the innermost part of the blood vessel (lumen) geometry of the
patient;
• Reconstruction of 3D anatomic geometry from 3D digital imaging in medicine
DICOM format using commercial imaging software;
• Extracting the specic 3D geometry in a numerical domain (‘.step les’ or ‘.stl
les’ formats);
• Meshing and simulation of the numerical domain on the CFD.
S. Mishani et al.
8.4.1 Segmentation andReconstruction
Segmentation has a crucial role in the image analysis process. It is the process of
partitioning a digital image into multiple segments of non-overlapping and constituent regions with homogeneous intensity or texture. Image segmentation through an
automatic or semi-automatic process extracts the desired features of an object or
other relevant information from a digital image [11].
Image segmentation is the removal of all parts of the image that are not required,
leaving only the area of direct interest. The resultant 3D image is then easier to
analyse. Blood vessel segmentation is needed to identify the vessel structures in the
vicinity of organs, bones and other structures. Segmentation methods convert medical images to digital geometries which show the region of interest of the model [12].
The structure of invivo organs cannot be measured easily (Fig.8.3) so, by using
reconstruction, relevant structures in medical images are captured (Fig.8.4) and
converted into digital 3D images (Fig.8.5). Development of 3D artery reconstruction algorithms can be used in the assessment of disease such as the evaluation of
the role of local hemodynamic forces on plaque progression [13].
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Fig. 8.3 3D image of chest (mrml format)
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Fig. 8.4 The aorta is
highlighted as separate
from the rest of the chest
elements
CTA and MRA imaging provide images in DICOM le format (‘.dcm les’). 3D
Data Loading and visualization of DICOM can covert to:
• Volume rendering of 3D images, in ‘.step les’ format
• Surface rendering of 3D images, in ‘.stl les’ format
P
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Fig. 8.5 3D digital image,
post segmentation
S. Mishani et al.
26.8 mm
R
16.5 mm
160 mm
8.4.2 Numerical Simulation
CFD is a computer simulation method for prediction of blood ow behaviour using
hemodynamic parameters. The objective of CFD is to quantify ow characteristics
at desired locations in a specic geometry through developing the governing equations (the Naiver–Stokes equations). The additional inputs to the geometry of CFD
model are:
• Fluid properties (blood density and viscosity),
• Boundary conditions (rigid or exible blood vessel),
• Initial conditions (pulsatile blood ow data),
• Discretis
• Blood ow pattern (steady/unsteady and laminar/turbulent).
This information allows CFD to solve the Navier-Stokes (Eq.8.1) and continuity
(Eq.8.2) equations to converge towards the nal solution [14].
ation information (mesh size/quality and time steps) and,
r
Inertia (per volume)
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æ
ç
¶
v
ç
çç
ç
Unsteady
ç
acceleration
è
¶
{
+Ñ
t
{
change in
density
over time
vv
{
convective
acceleration
¶
r
¶
t
.
+Ñ
ö
Divergence of stress
÷
6744844
÷
=-Ñ
÷
{
Pressure
÷
gradie
÷
ø
r
v
()
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Rate of
mass
flux
+Ñ +
m
p
{
Viscosity
effect
nnt
=
0,
Othe
rr
body
force
}
2
v
f
,
(8.1)
(8.2)
t
8 Computational Fluid Dynamics intheArterial System: Implications forVascular…
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where:
ρ: Fluid density
t: Time
v: Flow velocity
p: Pressure
μ: Dynamic viscosity
¶¶v
: Change in velocity over time
∇: Gradient of vector parameter
CFD methods for a solution of uid ow are classied into two distinct categories [15];
• Density based: suitable for high speed and compressible uids,
• Pressure based: suitable for slow speed and incompressible uids.
There are three different points of view in analysing problems in uid
mechanics [15];
1. Eulerian approach: Concerned with the uid properties (pressure, velocity, den-
sity, etc.) in a specic point in space e.g. sitting on a riverbank and watching the
water pass by a xed location.
2. Lagrangian approach: Concerned with a specic particle of uid as it moves
through space, e.g. sitting in a boat and drifting down a river.
3. Arbitrary-Lagrangian-Eulerian approach: Concerned with a specic particle of
uid which can move, while the boundaries and interfaces move too, e.g. sitting
in a moving vehicle on the riverbank and watching the water ow by.
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In the context of blood vessels, the type of discretisation of the uid domain can
be described with either Arbitrary Lagrangian-Eulerian or Eulerian formulation, but
the former will give more accurate results at the interface between the solid and the
uid [16].
8.4.3 Meshing
FEA software uses a Computer-Aided Design (CAD) model which represents the
shape and structure of the simulating system as well as the material properties, and
the applied forces and constraints (initial and boundary conditions). The process of
dividing the CAD model into smaller domains called elements is referred to as
meshing. The accuracy and robustness of the nite element computations are
strongly related to the meshing size and quality (shape of the elements formed).
Starting with a bigger size and simple shape mesh and adding complexity is easier
than starting with small size and complex ones. In CFD model analyses with extrarened meshes, the computation time can take days and require supercomputers.
There are four common 3D-element (volume) shapes, as shown in Fig.8.6;
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Tr iangular PrismHexahedron
S. Mishani et al.
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Fig. 8.6 Basic 3Delement shapes
Tetrahedron
Pyramid
1. Tetrahedron, with 4 faces and 6 edges. Tetrahedral meshes are relatively simple
and can t complex geometry better, but suffer from wall boundaries in CFD
modelling. When modelling the interface between a wall and a uid, the varia-
tions in uid properties (velocity, temperature, density) will be much lower along
the direction the uid is travelling, than those in the cross-stream direction.
Therefore, a thin brick- type mesh will be more suitable than other shapes.
2. Pyramid with 5 faces and 8 edges. The pyramid is used in transition areas
between square and triangular—faced elements.
3. Hexahedron (brick), with 6 faces and 8 edges. For the same mesh quantity, hexa-
hedral meshing has the highest accuracy and more computational efciency with
uid ow direction.
4. Prism (wedge), with 5 faces and 9 edges. Prism meshing can efciently resolve
boundary layer problems.
The length of arteries is greater than their diameter, therefore the gradient (variation) of blood ow velocity in an artery is much greater in the radial than the longitudinal direction. An accurate resolution discretization with uniform mesh size of
lumen geometry can impose excess computation. Therefore, each component (e.g.
the blood ow (Fig.8.7) and the arterial wall (Fig.8.8)) needs to be discretised with
a different resolution in different directions (anisotropic meshing) [17, 18]. The
blood ow, as a non-Newtonian uid, in a time-varying velocity prole imposes
shear stress to the innermost aspect of the arterial wall. So, the near—wall regions
must be the nest mesh possible to create a suitable boundary layer accurately [19].
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Computational Fluid Dynamics intheArterial System: Implications forVascular…
Fig. 8.7 Blood ow and artery vessel
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Fig. 8.8 Aorta shell meshing (STL format)
There are two basic ways to enhance the accuracy of the nite element model [20];
1. Increasing the number of elements in the model (Fig.8.9).
2. Using a higher-order of the elements (Fig.8.10).
The accuracy of the nite elements model is inuenced by both spatial and temporal (mesh size and time step) renement factors, so the more renement produces
a more accurate result. Spatial discretisation (meshing) is the process of subdividing
the geometry of the model into a number of discrete volume elements (cells).
Temporal discretisation (time-step) is a time-dependent solution of the FE model.
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Fig. 8.9 Volume meshing (step format)
Fig. 8.10 Volume
sub-layers meshing
S. Mishani et al.
8.4.4 Element-Based vs. Volume-Based Meshing
ANSYS-FLUENT and ANSYS-CFX are two independent solvers developed for
performing CFD analysis. The main difference is in their discretization (meshing)
methods, called nite difference and nite volume methods respectively. In the
nite difference method, the number of control volumes are equal to the number of
elements (as cell-centred or element-centred), whereas in the nite volume method,
the control volume is assembled around the nodes (as vertex-centred or nodecentred), so CFX solvers divide each element into sub-elements. In another words,
in the nite difference method, the dependent values are stored at the node, but in
the nite volume method, the dependent values are stored in the centre of the volume (Fig.8.11).
Although, CFX uses only one solving approach for the governing equation of
motion and FLUENT offers several solving approaches, Fluid-Solid Interaction
(FSI) can perform better in CFX than FLUENT because there is a stronger connection between the volumes in CFX.In the case of a geometry with irregular
and complex shape and structure, high quality meshes are produced in a CFX
solver, saving a lot of pre-processing time. In the nite difference method, the
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