Nabila Zaman, M Ferdows, MA Xenos*, KE Hoque and EE Tzirtzilakis 
Corresponding Author: Michalis Xenos, Section of Applied Mathematics and Engineering Research, University of Ioannina, Ioannina 45110, Greece 
Received: February 05, 2020; Accepted: February 11, 2020; Published: August 09, 2020 
Citation: Zaman N, Ferdows M, Xenos MA, Hoque KE & Tzirtzilakis EE. (2021) Effect of Angle Bifurcation and Stenosis in Coronary Arteries: An Idealized Model Study. BioMed Res J, 4(2): 220234. 
Copyrights: ©2021 Zaman N, Ferdows M, Xenos MA, Hoque KE & Tzirtzilakis EE. This is an openaccess article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. 
In this study, we have used
patientspecific computed tomography angiography data to produce idealized
coronary artery models with different left anterior descending (LAD) and left
circumflex (LCx) arteries with various bifurcation angles and degrees of
stenosis. Since coronary artery stenosis and bifurcation angles are closely
associated to myocardial infarction (heart attack), the main goal of this study
is to analyze whether the bifurcation angles have an impact in the coronary
blood flow.
Six idealized models have constructed with
two arterial bifurcation angles, 93° and 75°. Those models are; one healthy
model and two diseased (1 stenosis model and 2 stenosis model) models, for each
arterial bifurcation angles 93°
and 75°
respectively. After reconstructing the computational
models, numerical studies have been performed to obtain the hemodynamics
parameters, such as velocity magnitudes, pressure gradients and fractional flow
reserve (FFR). FFR measures the severity of stenosis in the idealized models.
We used the finite element method to
provide concise and comprehensive solution over the curved surfaces. From the
detailed simulation results, the information of the hemodynamic parameters is
compared with the clinical data for validating the obtained results. From the
simulated idealized models, we notice that due to the stenosis in the main LAD artery
the blood flow is much higher in the bifurcated stenosis region than other
locations in the 3D stenosis model. From the FFR value which is measured as the
ratio of the downstream and upstream pressure, we observe that the severity of
stenosis is much higher for the arterial bifurcation with larger angle 93°.
Severity of stenosis differs for different
angles in the diseased (stenosed) models. So, the computational hemodynamics
(CHD) parameters provide information about the condition of stenosis in the idealized
models and can give a clearer insight about the blood circulation in the human
body.
So, this is worthy of investigation as it
has influence on the propagation of coronary artery disease (CAD). Thus, more
accurate simulation results may predict the patient conditions and it might be
replaced with the invasive test catheterization for the diagnosis of the
disease for evaluating FFR in the future.
Keywords: Fractional flow reserve (FFR), Left
anterior descending (LAD), Coronary artery, Myocardial infarction
INTRODUCTION
Cardiovascular
disease (CAD) is a serious heart condition affecting many developed countries
in the world and resulting in high morbidity rates [1]. It is usually caused by
breakdown in the normal cardiovascular functions and including abnormal flow
situations. One of the most common causes of CAD is atherosclerosis, which is
the disease of the vascular system [2,3].
It is a
cardiovascular condition that can occur in any part of the vascular system.
Especially, it can exist in bifurcated arteries such as the left and right
coronary arteries, abdominal aortic bifurcation or carotid artery bifurcation.
In this study, the left coronary artery is examined as an exemplification using
wall shear stress (WSS) and pressure gradient (PG). An attempt has been made to
find the relationship between bifurcated arterial geometry and hemodynamics.
The result demonstrated that the region of low WSS area and magnitudes of
maximum PG increases with the bifurcation angles [4].
The most
common type of CAD is coronary artery stenosis which is initiated by the
buildup of plaques on the endothelial walls of coronary arteries. It leads to
a reduction in arteries crosssectional area and blood supply to the myocardium
[5]. The rapid development of noninvasive imaging technologies, such as
computed tomography angiography (CTA) and magnetic resonance imaging (MRI), has
proven valuable to characterize the anatomic severity of CAD with fair cost and
less complication [6].
This study
proposes a noninvasive method to determine fractional flow reserve (FFR) by
combining computed tomography angiographic (CTA) data and computational fluid
dynamics (CFD) technique. Hence in this study the effects of diameter stenosis
(DS), stenosis length and location on FFR has been explored. Diameter stenosis
(DS) is commonly applied to quantify the anatomic severity of CAD. It expresses
the ratio of the lumen diameter at a stenotic region over that of a “normal”
segment [7].
Recent
coronary computed tomography angiography (CCTA) studies have noted higher
transluminal contrast agent gradients in arteries with stenotic lesions, but
the physical mechanism responsible for these gradients is not clear. In its
current form, CCTA is used to identify arterial stenosis as well as to evaluate
the size, shape and area reduction due to the lesion. This information is
subsequently used to make decisions on surgical intervention as well as to plan
the surgery. However, this approach does not provide any information about the
coronary hemodynamics, such as velocity or pressure drop across the lesion,
which is a more direct measure of the functional significance of the arterial
lesion and the resulting ischemia. CFD modeling coupled with contrast agent
dispersion has been used to investigate the mechanism for contrast agent
gradients [8].
Currently, there are no reliable ways to predict an aneurysm.
Multiple studies have been performed to determine the etiology of aneurysms,
but all of them have been inconclusive. In this study the occurrence of
aneurysms in the main middle cerebral artery (MCA) to asymmetry between the
bifurcation angles has been analyzed. Hence using MCA models, the morphological
differences have been found examining the changes in the fluid flow and wall
shear stress [9].
The causes of atherosclerosis are multifactorial and identification
of these factors could allow earlier detection and prevention of the disease.
Hemodynamics and vessel geometry may play an important role in the cause of
plaque formation since atherosclerotic plaques occur frequently in
wellrecognized arterial regions of curvature, bifurcated area and vessel
branches [10].
Blood flow variations, particularly those related to the rate of
change of streamwise velocity perpendicular to the blood vessel wall (known as
the wall shear stress), have been reported to be related to the pathogenesis of
atherosclerosis [11]. The purpose of this study was to investigate the
hemodynamic effect of variations in the angulations of the left coronary
artery, based on simulated and realistic coronary artery models. Twelve models
consisting of four realistic and eight simulated coronary artery geometries
were generated with the inclusion of left main stem, left anterior descending
(LAD) and left circumflex (LCx) branches. The simulated models included
coronary artery angulations and CFD analysis were performed to simulate realistic
physiological conditions that reflect in the vivo cardiac hemodynamics. The
WSS, velocity flow patterns and wall pressure were measured in simulated and
realistic models during the cardiac cycle. The results show that a disturbed
flow pattern is observed in models with wider angulations, and wall pressure is
found to reduce when the flow changed from the left main stem to the bifurcated
regions, based on simulated and realistic models. There is a direct correlation
between coronary angulations and subsequent hemodynamic changes, based on
realistic and simulated models [12].
Development of many conditions and disorders, such as
atherosclerosis and stroke, are dependent upon hemodynamic forces. To
accurately predict and prevent these conditions and disorders hemodynamic
forces must be properly mapped. In this study, a comparative shearrate
dependent fluid (SDF) constitutive model, based on the works by Yasuda et al.
[13] in 1981, against a Newtonian model of blood is analyzed. Numerical simulations
show that the Newtonian model gives similar velocity profiles in the 2D cavity
given different height and width dimensions, given the same Reynolds number.
Conversely, the SDF model gave dissimilar velocity profiles, differing from the
Newtonian velocity profiles by up to 25% in velocity magnitudes. This
difference affect estimation in platelet distribution within blood vessels and
magnetic nanoparticle delivery.
Arteries are normally straight to efficiently transport blood to
distal organs. However, arteries may become tortuosity as a result of arterial
remodeling [14]. Tortuosity is widely observed in coronary arteries [15,16].
Tortuous coronary arteries may hamper the ventricular function and have been
proposed as an indicator of the ventricular dysfunction [17]. Coronary tortuosity
(CT) is associated with reversible myocardial perfusion defects and chronic
stable angina [18,19].
Tortuous coronary arteries are
commonly observed in clinical screenings and it may cause a reduction of the
coronary pressure. However, whether this reduction leads to significant
decreasing in the coronary blood supply is still unknown. The purpose of this
study was to investigate the effect of the coronary tortuosity (CT) on the
coronary blood supply. CFD is conducted to evaluate the impact of tortuosity on
the coronary blood supply. Two patientspecific LAD models and the
corresponding nontortuous models are reconstructed to perform three
dimensional CFD analysis. The lumped parameter model is coupled to the outlet of
the simulated branches to represent the absent downstream vasculatures. The
rest and exercise conditions are modeled by specifying proper boundary
conditions. Under resting condition, the mean flow rate could be maintained by
decreasing less than 8% of the downstream vascular bed’s resistance for
tortuous models. While during exercise (maximal dilation condition), the
maximal coronary blood supply would reduce up to 14.9% due to tortuosity.
Assuming that the flow rate can be maintained by the autoregulation effect
under the maximal dilation condition, the distal resistances for CT models
still have to reduce more than 23% to maintain blood perfusion [20].
CAD is responsible for 7.3 million
deaths and 58 million disabilityadjusted life years lost worldwide [21]. It is
generally accepted that CAD is an inflammatory disease with lipid deposition in
the arterial wall as an initial stage of atherosclerosis [2225]. Although the
risk factors for atherosclerotic coronary plaque formation, including high
cholesterol, diabetes and hypertension are systemic in nature, plaques are
located at specific sites in the coronary artery where distributed flow and low
endothelial shear stress occur [26, 27]. This study provides an overview of the
CFD applications in CAD, including biomechanics of atherosclerotic plaques,
plaque progression and rupture; regional hemodynamics relative to plaque
location and composition [28].
Blood flow in arteries is
dominated by unsteady flow phenomena. The cardiovascular system is an internal
flow loop with multiple branches in which a complex liquid circulates. A
nondimensional frequency parameter, the Womersley number, governs the
relationship between the convective and viscous forces. The arteries are living
organs that can adapt to and change with the varying hemodynamic conditions. In
certain circumstances, unusual hemodynamic conditions create an abnormal
biological response. Velocity profiles skewing can create pockets in which the
direction of the wall shear stress oscillates. Atherosclerotic disease tends to
be localized in these sites and results in a narrowing of the artery lumena
stenosis. The stenosis can cause turbulence and reduce flow by means of viscous
head losses and flow choking. Very high shear stresses near the throat of the
stenosis can activate platelets and thereby induce thrombosis, which can
totally block blood flow to the heart or brain [29].
The CFD enabled the simulation of coronary
blood flow conditions in idealized and patientbased coronary models. The simulation
allows to produce a noninvasive assessment of the hemodynamic parameters, such
as blood velocity magnitude, relative pressure difference and FFR. Hence these
hemodynamic mechanisms provide the information on coronary stenosis conditions
and predict the severity of coronary arterial lumen area which is responsible
for the heart attack of the patients. In the results, the expected outcomes in
both cases, for instance, a higher blood velocity in the coronary vessels tends
to stretch the contrast agent gradient and a lower blood velocity magnitude
tends to steepen the contrast agent gradient has been reflected. The pressure
difference and FFR results allow to distinguish the unstenosed and stenosis
arterial models [30].
MODEL
GENERATIONS
Construction of idealized 3D models
It is quite clear that most of the
coronary main arteries are bifurcated everywhere. In the most cases stenosis
occurs near the bifurcations or artery bending regions. To describe these
characteristics, we construct an idealized model imitating patientspecific
computed tomography coronary artery angiography model with different left
anterior descending (LAD) and left circumflex (LCx) arteries. In this study,
the 3D idealized models are generated by computeraided design software FreeCAD
and then the models are imported into COMSOL Multiphysics software, for grid
generation and simulation. The models are made to match images of the coronary
artery bifurcation during operations or imaging since no digital scans are
available to import into COMSOL. The 3D idealized models are built using
FreeCAD software. The idealized 3D model consists of two main arteries which
are LAD and LCx, each with one side branch. The length of LAD is 20 cm and the
length of LCx is 14 cm. The branch from LAD artery has length 5 cm and the
branch from LCx artery has length 7 cm. The structure and dimension of the
models are based generally on a typical left coronary artery that follows the
anatomically close information.
To curve the models, we use Bsplines which are
easy to represent the bending nature of the LAD, LCx arteries and the side
branches. For building the arterial tube we use again Bsplines with the same
center but different radius. Taking the difference of radii, we build the
arterial wall that is rigid inside for the blood flow. Using Boolean functions,
we can get accurate shapes of the 3D idealized models. In the diseased model
the stenoses are generated using the software.
This computational idealized model
contains one inlet and four outlets that are represented by outlet 1, outlet 2,
outlet 3 and outlet 4, respectively. The radius of inlet is 2.5 mm and the
radius of outlets 1 and 2 is 1.8 mm and 1.7 mm respectively. Again, the radius
of outlets 3 and 4 is 1 mm each.
We construct six computational idealized
models. The first 3 models are constructed where we choose that the arterial
bifurcation is 93°.These 3 models are: healthy (unstenosed)
model and two diseased models (one stenosis model and a model with two
stenoses). In the healthy (unstenosed) idealized model the measured angle
between LAD artery and its side branch is 58° (Figure 1) and the measured angle between LCx artery and its side
branch is 35°. We have measured these angles using
FreeCAD software.
Grid
generation in 3D idealized models
Once all the models are prepared, a mesh
analysis is performed to find the maximum and minimum mesh required to get
accurate results. The models are simulated for one cardiac cycle at various
mesh settings until required convergence is achieved (Figure 7, 8).
The domain is set to the predefined finer
setting. This setting created a gradient in element size with smaller elements
being on the boundaries of the model and larger elements in the middle of the
domain. Meshing gets finer closer to the boundary because the fluid velocity
reduces rapidly as it approaches to the wall, no slip condition. The predefined
mesh fluid dynamic settings also added a layer of elements between the boundary
and the domain. In all models we have used tetrahedral elements. Tetrahedral
elements are used for the domain since they model accurately complex 3D
structures. An internal volume of boundaries is covered by unstructured
tetrahedral domain cells where the largest cell size is 1.2 mm and the smallest
cells are 0.7 mm. The finer meshing gives highquality elements that improve
numerical stability and increases the likelihood of attaining a reliable
solution. In the whole domain, the amount of mesh is 6.67877×10^{5} elements,
where 75164 in the fluid zone, 53966 in boundary elements and 1825 are edge
elements. The other parameter settings of the meshing are the growth rate,
defined as 1.2, the curvature factor 0.6 and resolution in narrow regions is
zero.
Arterial input function
and numerical simulation
Quantification of
tissue perfusion relies on the determination of arterial input function (AIF).
The choice of AIF can have a profound effect on the blood flow maps generated
on perfusionweighted MR imaging. Automation of this process could
substantially reduce operator dependency and increase consistency. AIF plays an
important role to measure the concentration of blood contrast agent under the
timeindependent condition at the outset position of the coronary artery. The
erroneous determination of the AIF could result in miscalculation of perfusion
parameters.
We use the following
equation as AIF:
f(t) = {
(1.75α) sin (πt), 0 ≤ t ≤ 0.20 (1)
1α
cos(2π(t0.5)),
0.20≤ t ≤ 1.5
Equation (1) is a simple trigonometric
function,
In a clinical study on
coronary artery blood flow, the average blood velocity of LAD is measured as
17.98 ± 5.66 cm/s using digital tracing coronary angiography [31]. We compare
the simulation results with the clinical data [31] for angles 93°and 73° with the flow
velocity in cm/s. For the arterial bifurcation with angles 93° and 75° we calculate the
flow velocity using the same approach for materials, boundary conditions and
mesh generation.
RESULTS AND DISCUSSION
We present a comparative study between
healthy (unstenosed) and diseased (one stenosis model and a model with two
stenoses) models for describing coronary blood flow. We display our simulation
results of hemodynamics parameters (velocity magnitudes, pressure difference).
For validation purposes, we also compared our data with literature [30].
Additionally, we calculate FFR value from the pressure difference. Hence, from
FFR value we measure the severity of stenosis of the idealized coronary artery
models. We finally discuss the effect of bifurcation angles of stenosis in
coronary blood flow of the models.
Post processing, visualization
and data comparison
The CFD solver provides the simulation
results, with the appropriate physical properties and boundary conditions as
described before [30]. The presented solutions in this section rely on
information such as which solver is selected and which component the solution
applies to (for models with multiple components). Figure 10 shows the velocity magnitudes for the arterial
bifurcation with angle 93° of the healthy, one stenosis model and
two stenoses idealized model.
Figures 14 and 15 present the simulation result of the pressure field in the arteries under consideration. Figures 14 and 15 present the
pressure gradient in the six idealized models. The relative pressure also
represents the physically viable conditions, such as in the healthy
(unstenosed) model, the relative pressure in different locations of the model
are uniform. The one
and two stenoses models refers the pressure differences are very distinguishable
between the upstream and
downstream flow of the lesion
areas [30]. For the arterial bifurcation with
angle 95° and angle 75°, we measure the FFR and stenosis percentage
that is obtained from the simulation results and the results are shown in Table 2.
From Tables 1 and 2, we can see the
hemodynamic results for the bifurcation angles, 93° and 75°. The FFR value is
calculated from the ratio of the downstream and upstream relative pressure of
the stenosis model and the percentage of stenosis is calculated from equation
(2) [32].
Percentage of stenosis = [1 (D
_{stenosis} ÷
For the arterial
bifurcation with the larger angle 93°, we can see from Table 2 that the measured FFR for the one stenosis model is 0.71,
which is less than 0.80 and the stenosis percentage is 29%. So, it is not a
severe problem. Again, from Table 2
we see that measured FFR for the two stenoses model is 0.40, which is also
Hence, we made a
comparison between models of two different bifurcation angles using numerical
simulation based on velocity magnitudes and pressure gradient. The simulated
values indicate how the angles affect hemodynamics of bifurcated coronary
arteries. Certainly we can observe that the severity of stenosis differ for
different angles and stenosis in the various models.
For the arterial
bifurcation with larger angle 93° the severity of stenosis is much higher than
the severity of stenosis for the smaller angle 75° for the two stenoses model.
The observed results prove that the bifurcation angle has an effect on the
pressure gradient as well as velocity magnitudes when a stenosis exists in the
model. From the simulation result we also analyze geometrical variations, such
as degrees of arterial occlusions at the proximal sections of the bifurcated
arteries, curvature of the arteries and the combined effect of these
geometrical variations. This can provide a clearer insight in how irregular
geometries of the coronary arteries can affect coronary blood flow. Thus we
suggest that the performed blood flow simulations in different physiological
conditions provide feasible results in the idealized 3D models.
CONCLUSION
The purpose of this study was to
noninvasively assess hemodynamic parameters with CFD in coronary
atherosclerosis for idealized coronary artery model from patientspecific
computed tomography angiography data. The hemodynamic parameters allow to
describe the intrinsic blood flow patterns. This has been accomplished with the
development of a set of techniques which includes idealized model
reconstruction, analyze and discretize 3D model from computeraided design
model and numerical simulation.
Computational hemodynamics that
relies on a numerical simulation framework has been implemented to verify the
effect of bifurcation angles of stenosis in coronary blood flow using idealized
coronary artery model. For achieving this we first construct the geometry of
the idealized model imitating patient specific coronary artery data using
FreeCAD software. Since the geometry of the 3D model has uniform shape the
finite element method is used to produce computational meshes, so that it can
provide a concise and comprehensive solution over curved surfaces and 3D volume
cells. CFD solver is employed for generating the hemodynamic parameters
(velocity magnitude, pressure gradient). We calculate the severity of stenosis
from FFR which is measured as the ratio of downstream and upstream pressure.
The simulation results are compared with the clinical data. We obtain the
following findings: Both Figures 10 and 11
show that, due to the stenosis in the main LAD artery, the blood flow is much
higher in the stenosis region than other locations in the 3D model.
The pressure difference and FFR
results allow to distinguish between healthy (unstenosed) and diseased
(stenosed) models with arterial bifurcation of two different angles 93° and 75°.
From the FFR values it is observed
that the severity of stenosis is much higher in the two stenoses model with
arterial bifurcation of angle
93° and 75°. So, the bifurcation angle with the existence of a stenosis has
significant impact on the hemodynamic parameters of the idealized model.
The techniques developed and presented in
this study are very promising for the future. If it is possible to reduce the
limitations of this work, then this noninvasive method might replace the
expensive and invasive diagnosis test catheterization for evaluating FFR in
coronary arteries. Utilizations of hemodynamic indicators, such as blood
velocity, pressure difference, FFR value can give a clear insight of blood
circulation in the human body. Hence, the combined application of CFD with CCTA
procedure is not only saving direct healthcare costs but also could reduce indirect
costs and patient risks, such as heart attack, stroke and death. So, this
procedure can provide medical experts support in diagnosing cardiovascular
disease and for better treatments of diseased arteries.
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