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1. Introduction
Blood flow is fundamental to the functioning of the human body.
It is involved in several physiological processes that are neces-
sary for life:
1. Transport of oxygen: Blood is responsible for the transport
of oxygen from the lungs to all tissues of the body.
2. Transfer of nutrients and removal of waste products: It also
transports nutrients produced as a result of the digestion of
food to cells throughout the body and helps to remove met-
abolic byproducts, including carbon dioxide from cells.
3. Pressure regulation: The heart and blood vessels regulate
blood flow and pressure. This is necessary to ensure that
blood reaches all parts of the body.
4. Temperature regulation: Blood intensifies the heat ex-
change between the body and the surroundings.
5. Immune system support: The leukocytes contained in the
blood defend the body against infections, viruses, and
other pathogens.
6. Distribution of hormones: Hormones, produced by various
glands that regulate physiological processes, are trans-
ported in the bloodstream to the target organs and tissues.
7. Clotting and wound healing: The healing process is initi-
ated by platelets that initiate clot formation to prevent ex-
cessive bleeding.
8. Maintaining the pH and electrolyte balance: Blood con-
tains buffers that can handle changes in pH and electrolytes
(such as sodium, potassium, and calcium) that are crucial
for different physiological functions.
The paper describes some aspects of modelling blood flow
in the arteries that have been developed by the authors of this
paper within recent research projects. The article does not de-
Selected aspects of blood flow simulations in arteries
Ryszard Białecki*a, Wojciech Adamczyka, Ziemowit Ostrowskia
a Silesian University of Technology, Department of Thermal Technology, Faculty of Energy and Environmental Engineering, Konarskiego
22, 44-100 Gliwice, Poland
*Corresponding author email: ryszard.bialecki@polsl.pl
Received: 17.12.2023; revised: 10.02.2024; accepted: 10.03.2024
Abstract
This article discusses selected aspects of modelling blood flow in the arteries. The method of reproducing the variable-in-
time geometry of coronary arteries is given based on a sequence of medical images of different resolutions. Within the
defined shapes of the arteries, a technique of generation of numerical meshes of the same topology is described. The
boundary conditions and non-Newtonian rheological models used in blood flow are discussed, as well as the description
of blood as a multiphase medium. The work also includes a discussion of tests on the phantom of the carotid artery for the
accuracy of measurements made using ultrasonography.
Keywords: Blood flow; CFD; Coronary arteries; Boundary conditions; Blood rheology; Blood as a multiphase medium
Vol. 45(2024), No. 1, 145‒153; doi: 10.24425/ather.2024.150447
Cite this manuscript as: Białecki, R., Adamczyk, W., & Ostrowski, Z. (2024). Selected aspects of blood flow simulations in arteries.
Archives of Thermodynamics, 45(1), 145‒153.
Co-published by
Institute of Fluid-Flow Machinery
Polish Academy of Sciences
Committee on Thermodynamics and Combustion
Polish Academy of Sciences
Copyright©2024 by the Authors under license CC BY 4.0
http://www.imp.gda.pl/archives-of-thermodynamics/
Białecki R., Adamczyk W., Ostrowski Z.
146
Nomenclature
A ‒ cross-sectional area, m2
D ‒ vessel diameter, mm
E ‒ Young modulus, GPa
f ‒ frictional force
h ‒ wall thickness, mm
k ‒ constant in Quemada viscosity
ΔL ‒ distance between the measurement points, mm
p ‒ pressure, mmHg
Q ‒ volumetric flow, m3/s
R ‒ tube radius, mm
S ‒ cross-sectional area, m2
t ‒ time, s
U cross-sectional averaged velocity, m/s
u ‒ local velocity, m/s
Wo ‒ Womersley number
Greek symbols
‒ shear rate
– dynamic viscosity, Pas
‒ kinematic viscosity, m2/s
ρ – blood density, kg/m3
‒ wall shear stress, Pa
ω ‒ circular frequency of the flow changes
Subscripts and Superscripts
c – critical shear rate
p – plasma
w – wall
Abbreviations and Acronyms
CTA – Computed Tomography Angiography
FSI – Fluid-Structure Interaction
ICA – Invasive Coronary Angiography
MB – Myocardial Bridge
LV – Left heart Ventricle
USG – Ultrasonography
RBC – Red Blood Cells
WSS Wall Shear Stress
scribe a complete research program but focuses on some im-
portant aspects of blood flow that the team encountered during
ongoing research projects. Emphasis was placed on the issues
that caused the team special difficulties. The article presents
ready-made solutions to some problems but also shows possible
approaches not yet implemented in the practice of the team. A
comprehensive overview of the applications of CFD in hemody-
namics can be found in references [1, 2].
The methods and examples discussed in the article concern
the arteries of the systemic circulatory system. The walls of the
arteries, unlike the walls of the veins, experience significant dif-
ferences in cyclic pressure caused by heart contractions.
The characteristic features to be considered in flow simula-
tions in the arteries are:
complex geometry and numerous arterial branches,
pulsatile, cyclic nature of pressure and flow variability,
deformation of the vessel walls,
non-Newtonian rheology,
nonstandard boundary conditions.
2. Dimensionality of the model
The total length of the arteries in the human body is estimated
to be between 100 000 and 160 000 km. The development of a
model of geometry of the entire circulatory system is not realis-
tic. There are models of blood circulation in the human body
based on one-dimensional models and the restriction of geome-
try to larger-diameter vessels. An example of such an approach
is the open code STARFiSh developed at NTNU Trondheim [3].
The code is based on a solution of a system of first-order differ-
ential equations with partial derivatives:
, (1)
. (2)
where A is the cross-sectional area, Q is the volumetric flow, p
is the pressure assumed constant over the cross-section, ρ is the
density of blood, f is the frictional force,
accounts for nonlin-
earity in the cross-sectional integration of the local velocity u:
, (3)
where U is the cross-sectional averaged velocity.
It has been recognized that the distribution of the shear
stresses on the wall of the vessel determines the tendency of dep-
osition of atherosclerotic plaques at the internal layer (endothe-
lium) of the vessel [4]. At the level of histopathology, endothe-
lial cells change their shape from elongated to nearly round and
their state from atheroprotective to atherogenic [5].
Equations (12) are very useful to simulate the blood flow
within the whole body. However, 1D models do not provide de-
tails of the blood flow pattern in the vessel, specifically the shear
stress at the wall. The latter can be determined only using 2D or
3D models, preferably in time-dependent geometry. 2D models
are applied to reduce computational time; in most cases, these
papers assume axisymmetry of the geometry [6]. Both 2D and
3D models are based on solutions of the Navier-Stokes equa-
tions (mass and momentum conservation). The simplest cases
do not account for pulsatile flow using average or extremum
flow conditions [7].
More sophisticated approaches use time-dependent bound-
ary conditions (pressure of flow) but neglect the deformation of
the walls of the vessels during the cardiac cycle. For most of the
major healthy arteries, the vessel diameter changes during the
cardiac cycle are approximately 5–10% [8]. Therefore, there are
numerous articles in which the walls of the vessels are treated as
rigid [9].
The next complexity involves accounting for the periodic de-
formation of the vessels. 3D simulations where the flexibility of
the walls is accounted for are computationally demanding com-
pared to rigid wall modelling. In this case, two techniques are
available when:
Selected Aspects of Blood Flow Simulations in Arteries
147
the change in geometry is known from medical imaging
techniques such as noninvasive coronary CT (CCTA), in-
vasive coronary angiography (ICA), and intravascular ul-
trasound (IVUS) [10];
the deformation of the vessels is obtained by applying the
fluid-structure interaction where the stress/stress field in
the walls is coupled with the blood flow pattern [11].
The remaining portion of the paper is devoted to 3D models.
3. Retrieving the time-dependent geometry. Case
of coronary arteries
Coronary arteries are located on the surface of the heart muscle
(myocardium). As a result, these arteries are not squeezed at sys-
tole and can transport oxygen and nutrients to the heart muscle.
Arteries are permanently attached to the myocardium whose
volume and shape change during the cardiac cycle causing var-
iations in artery length and curvature. In addition, pressure pul-
sations generated by the contraction of the left ventricle generate
changes in the lumen of the vessel.
The geometry of the coronary artery can even be more com-
plex in the presence of the so-called myocardial bridge (MB)
when muscle bands overlay a segment of the coronary artery.
This condition is mostly asymptomatic, however, in some cases,
it may result in myocardial ischemia, angina, acute coronary
syndrome, etc. The presence of MB causes an oscillation in
blood flow that results in a deposition of the atherosclerotic
plaque proximal (before the inflow) on MB.
The geometry of the vessel can be retrieved by several mo-
dalities. The medical images that were used in the paper were
acquired at the Silesian Centre of Heart Diseases using angio-
computed tomography.
The raw data used to retrieve the geometry of the coronary
arteries consisted of a set of ECG-gated angio-CT images rec-
orded every 10% of the time of the cardiac cycle. To reduce the
harmful effect of the X-rays on the patient, the radiation dose
was not only low but also modulated within the cycle. As a re-
sult, the resolution of these packages of images was low. Addi-
tional high-resolution images have been taken separately for di-
astole and systole. The total set consisted thus of images of var-
ious resolutions and quality, specifically 10 images of 256 × 256
resolution and 2 images (diastole/systole) of 512 × 512 resolu-
tion. The images have been recorded by a 128-slice dual-source
computed tomography scanner (SOMATOM Definition Flash,
Siemens Healthineers, Forchheim, Germany) using beam colli-
mation 2 × 64 mm × 0.6 mm, slice thickness of 1.5 mm and re-
construction interval of 0.5 mm. Figure 1 shows an example of
a raw image.
The set of images requires intensive processing consisting
of:
1. segmentation of low-resolution images (extraction of cor-
onary artery shapes from 3D image data);
2. co-registration of the low-resolution images corresponding
to subsequent time instants of the cardiac cycle (the pro-
cess of transforming images into one coordinate system).
The result is a set of nonlinear transformations showing the
transition between subsequent images;
3. segmentation of the high-resolution data and generation of
a 3D surface in STL format;
4. smoothing the obtained in step 3 STL surface;
5. generation of CFD mesh in the smoothed object obtained
in step 4;
6. transformation of the mesh obtained in step 5 using the
nonlinear transformation resulting from step 2 (co-registra-
tion).
Segmentation of medical images (steps 1 and 3) was carried
out using the active contour method with threshold-based pre-
segmentation implemented in the ITK SNAP package [12]. The
theory behind this method is to find the close contours by solv-
ing the equation:
, (4)
where: C ‒ closed contour parametrized by u, υ, t with u and υ
denoting the local coordinates and t – time,
– the sum of
internal and external forces acting on the contour in the normal
direction: internal forces associated with the curvature, external
with the gradient of the imaged intensity.
The co-registration process consists of three steps: rigid
body movement, affine transformation, and nonlinear transfor-
mation based on diffeomorphism. These transformations are ex-
ecuted using the ANTs package [13].
The raw 3D surface retrieved from the high resolution data
was smoothed using the GeoMagic software [14]. The CFD
mesh generation is performed using a standard Fluent Ansys
mesher [15]. The nodes of the CFD mesh generated in step 5 are
projected using the transformation matrix obtained in step 5 onto
every low-resolution object. Based on this surface mesh, the vol-
umetric mesh at each step is generated. The advantage of the
proposed method is the generation of a CFD mesh of the same
topology at each instant of time. Due to this, the interpolation
errors between grids are avoided, the mesh morphing utility can
be used directly, and the CFD calculation time is reduced.
The scheme of generation of the geometry of the coronary
arteries is shown in Fig. 2. The resulting geometries at 30, 50,
70 and 90% of the cardiac cycle are shown in Fig. 3.
Fig. 1. An example of a raw high-resolution image.
Białecki R., Adamczyk W., Ostrowski Z.
148
4. Measurement of the phantom deformation of
the carotid artery
The stiffness of the arteries is a measure of the rigidity of the
arterial walls. This property changes with age due to remodeling
of the artery wall structure where elastic elastin fibers are re-
placed by stiffer collagen ones. Increased stiffness is a condition
that can have several serious implications for overall well-being.
Increased stiffness may result in:
Hypertension that results from the superposition of the
main pressure pulse generated by the systole of the left
heart ventricle (LV) and the wave reflected from the bifur-
cations of the arteries;
Thickening of the heart muscle (LV hypertrophy) due to
the increased workload of the left ventricle resulting from
its increased workload contributing to the increased risk of
heart failure;
Reduced blood flow to organs due to the reduced ability of
the blood vessels to constrict and dilate;
Damage of end organs such as kidneys, eyeballs and brain
caused by the reduced ability to dump the pulsation of the
blood pressure.
The stiffness of the blood vessels is assessed by measuring
the pulse wave velocity PWV obtained from the time that elapses
between pressure peaks in two arteries (usually carotid and fem-
oral). This value is obtained using the Moens-Kortweg equation
[16]:
, (5)
where: E ‒ Young modulus of the wall, h ‒ wall thickness,
D ‒ vessel diameter, ρ ‒ blood density, ΔL ‒ distance between
the measurement points along the blood vessel, Δt ‒ time that
elapsed between the pressure peaks at measurement points.
Equation (5) gives the mean values of the stiffness of the
vessel wall between the pressure measurement points. The pro-
ject aimed to determine the local stiffness of the carotid artery
wall by solving an inverse problem of blood flow in a deforming
conduit (solving the fluid-structure interaction (FSI) problem)
based on the measured deformation of the walls. The latter was
measured using the ultrasound scanner.
The key issue was to determine the precision of the ultraso-
nography (USG) measurements and the validation of the FSI
model based on measurements performed on a phantom under-
going a load similar to the carotid artery. For this purpose,
a phantom was built in which pressure pulsations mimicking the
work of the heart are produced by a pump, and the deformations
of the flexible tube are measured in two perpendicular planes by
high-resolution digital cameras. Parallel to these deformation
measurements, deformation is measured with a USG scanner.
Figure 4 shows the scheme of the experimental rig. The photo
of the rig is shown in Fig. 5. The comparison of the camera and
USG scanner uncertainties in measured displacements is shown
in Fig. 6. As can be seen, the ultrasound-derived displacements
exhibit good agreement (mean difference of 0.0113 mm) in
comparison with the camera-generated data.
Fig. 2. Scheme of retrieving the time-dependent geometry
of the coronary arteries.
Fig. 4. Scheme of the phantom (a). EH is the Endress Hauser electromagnetic flow meter.
Side view (b). The pressure gauges are installed at the inlet and outlet of the installed elastic tube [17].
Fig. 3. Geometries of the coronary arteries at 30, 50, 70 and 90%
of the cardiac cycle.
Selected Aspects of Blood Flow Simulations in Arteries
149
5. Boundary conditions
In the case of 3D simulations, simulation of blood flow in the
entire cardiovascular system would lead to a prohibitively long
calculation time. Therefore, the computational area is limited to
a small portion of the circulatory system with at least one inlet
and one outlet defined at fictitious surfaces closing the truncated
vessel. To obtain the blood flow solution of the Navier-Stokes
equations, unambiguous boundary conditions should be speci-
fied at these fictitious surfaces. These boundary conditions are
formulated in terms of velocity (mass flow), pressure, or a rela-
tionship between these two parameters. It should be stressed that
the fluid mechanics outflow conditions commonly used cannot
be used in blood flow simulations. Such boundary conditions
can be applied only in the case of fully developed velocity pro-
files that correspond to flow in a long, constant cross-section.
Moreover, although the flow in the arteries is, with very few
exceptions, laminar, the velocity profile in the arteries is not
parabolic. It is the pulsating blood flow generated by the con-
traction of the left ventricle that distorts the velocity profile. The
resulting distribution of the velocity is, for the idealized case of
flow in a stiff, constant cross-section tube, governed by the
Womersley number defined as:
, (6)
where: R ‒ tube radius, ω ‒ circular frequency of the flow
changes,
‒ kinematic viscosity. Figure 7 shows the speed pro-
files for different Womersley numbers.
The boundary conditions can be defined using three ap-
proaches:
measurements (patient-specific data),
statistics (population data),
estimates based on physiological assumptions and models.
It is recommended to use, whenever applicable, the first op-
tion and define the velocity or pressure. Statistical data cannot
be patient-specific and only represent the general population. In-
flow and outflow conditions can also be assessed using theories
such as Murray's law [19] which defines the link between mass
flow rate and vessel diameter, or the theory of constant wall
shear stress [20]. Alternatively, the interaction of the computa-
tional domain with the remaining part of the cardiovascular sys-
tem can be addressed using the multi-scale approach. The idea
is to couple the 3D model with the 1D [21] or 0D [22] model
describing the behaviour of the peripheral vascular system. The
latter is usually based on the hydraulic/electric analogy with
flow resistance resulting in pressure drop and capacitance de-
scribing vascular compliance. The lower-order model is de-
scribed by an ordinary differential equation whose parameters,
describing the behaviour of the vascular system outside of the
3D model, are not easy to determine. Implementation of this
type of nonlinear boundary condition leads to an iterative proce-
dure and often requires stabilization.
5.1. Velocity boundary conditions
In some cases, blood speeds can be measured using non-invasive
methods such as the Doppler technique associated with ultra-
sound, magnetic resonance imaging and computed tomography
angiography (CTA) (which requires contrast). Invasive methods
Fig. 6. Comparison of displacements recorded by both cameras with uncertainty
represented by overlaid shaded areas. In both cases, uncertainty is characterized as
dr el,cam ± σres,cam [17].
Fig. 5. Photo of the test rig: A ‒ periodic pump, B ‒ flowmeter,
C ‒ backlight, D ‒ arterial phantom, E ‒ top camera (MIRO),
F ‒ reservoir tank, G ‒ pressure transducers, H ‒ side camera
(VEO) [17].
Fig. 7. Four pulsatile flow profiles in a straight tube. The first graph
(in blue) shows the pressure gradient as a cosine function,
and the other graphs (in red) show dimensionless velocity profiles
for different Womersley numbers [18].
Białecki R., Adamczyk W., Ostrowski Z.
150
include arterial catheterization, a method that requires the intro-
duction of a catheter into an artery, and rather rarely used elec-
tromagnetic flowmetry. The latter consists of measuring the
generated voltage induced in coils placed near a blood vessel
immersed in a variable. magnetic field.
5.2. Pressure Boundary Conditions
Non-invasive pressure measurement can be carried out at arter-
ies located close to the surface of the body (carotid, radial, fem-
oral) using the applanation tonometer technique [23, 24]. Pres-
sure can be measured by invasive arterial catheterization.
6. Wall shear stress and oscillatory shear stress in-
dex
The deposition of atherosclerotic plaque on the walls of arteries
depends on the wall shear stress (
w, WSS). Typically, the onset
of plaque occurs at low
w locations while a high WSS prevents
plaque deposition [4]. Another parameter that indicates plaque
deposition is the oscillatory shear stress index OSI defined as
[25]:
(6)
Figure 8 shows the distribution of both indices in the left as-
cending coronary artery. The results have been obtained for the
inflow condition: pulsatile pressure, and the outflow condition:
mass distribution between outlets based on the Murray law (in-
cluding accumulation in the volume of the arteries).
7. Rheology of blood
Blood is a non-Newtonian, multiphase fluid with cellular ele-
ments: red blood cells (RBC), white blood cells and platelets
suspended in plasma. The latter is an aqueous solution contain-
ing organic molecules, proteins and salts. Plasma and red blood
cells (erythrocytes) play a dominant role in the definition of the
viscosity of blood. At low shear rates, the aggregation of RBC
occurs accompanied by increased blood viscosity [27].
There are two possible approaches to account for the non-
Newtonian properties of blood. The first is the usage of nonlin-
ear constitutive laws for viscosity where the blood is treated as
a one phase fluid. The second is to treat blood as a multiphase
medium.
The most popular are [28]:
Carreau:
), (7)
Casson:
, (8)
Quemada:
, (9)
where
is the shear rate and the remaining variables are con-
stants, whose values can be found in [29].
Figure 9 shows the velocity contours corresponding to dif-
ferent instants of the cardiac cycle obtained using various rheo-
logical models [30] obtained for one velocity inlet and four pres-
sure outlets, the numerical mesh consisting of 1.3 million cells.
8. Blood as a multiphase medium
When using non-Newtonian constitutive laws, blood is treated
as a homogeneous liquid of viscosity that depends on the shear
rate. An alternative approach is to take into account the presence
of granular phases (blood cells) immersed in a liquid (plasma).
The two most popular approaches are known as Euler-Euler and
Euler-Lagrange. In the former, the Navier-Stokes equations for
plasma are solved in the Eulerian coordinate frame. The granular
phases are also modelled as fluids with appropriately defined
properties and solved in the Euler coordinate frame. Both
plasma and granular phases are then as interpenetrating con-
tinua.
In the Euler-Lagrange approach, first the plasma is solved
separately in Eulerian coordinates, and then the fate of the gran-
ular phase is traced in the Lagrangian coordinates frame. As the
volume fraction of the blood cells is high, the interactions (col-
lisions) between the tranced cells should be accounted for. This
requires a solution of an additional set of equations based on the
kinetic theory of granular flow [31].
Figure 10 shows the results of simulations of blood flow in
the aorta with coarctation (congenital narrowing) obtained by
the Euler–Euler technique for the inlet pressure conditions and
the boundary conditions of the 3-element Windkessel outlet
[32].
Fig. 8. Oscillatory shear stress index (top) and wall shear stress
(bottom) for the maximum flow [26].
Selected Aspects of Blood Flow Simulations in Arteries
151
The results (see Fig. 9) confirm the presence of the Fhraeus-
Lindqvist effect resulting from the movement of the RBC to the
centre of the vessel leaving plasma near the wall [33].
9. Conclusions
CFD can be used successfully to simulate blood flow in the ar-
teries. Reliable simulation results are based on the restoration of
vessel geometry, which requires the processing of medical im-
ages obtained using tomographic modalities. The difficulties en-
countered in these activities result from the variability of images
over time. The article shows a method that allows not only to
reproduce such shapes but also to generate numerical grids with
the same topology in subsequent time steps. Difficulties in blood
flow simulations are also caused by the proper definition of
boundary conditions and non-Newtonian blood rheology. An
overview of the types of boundary conditions and rheological
models encountered and examples of their application are given.
Acknowledgements
The research leading to these results is funded by the Norwegian
Financial Mechanism 2014-2021 operated by the National Sci-
ence Center, PL (NCN) within GRIEG program under grant
Fig. 10. Pressure distribution in the aorta with systole (left). Velocity of the RBC (middle). Volume fraction of RBC in left subclavian artery [32].
Fig. 9. Times of the cardiac cycle (red arrow) (left). Velocity profiles evaluated using various rheological models (middle). Geometry of the computa-
tional domain (bottom). Circle denotes the location of the cross section where the velocity is shown [30].
Białecki R., Adamczyk W., Ostrowski Z.
152
#UMO2019/34/H/ST8/00624, project (ENTHRAL), and by the
National Science Center within OPUS scheme under contract
2017/27/B/ST8/01046, and by Ministry of Education and Sci-
ence (Poland) under statutory research funds of the Faculty of
Energy and Environmental Engineering of SUT under contract
BK-252/RIE6/2023 08/060/BK_23/1096.
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