Ain Shams Engineering Journal 14 (2023) 102535

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Ain Shams Engineering Journal

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Numerical simulation of thermal behavior of cerebral blood vessels using

computational hemodynamic method
Yutao Li a, *, Shahab Naghdi Sedeh b, As’ad Alizadeh c, Maytham N. Meqdad d,
Ahmed Hussien Alawadi e, f, g, Navid Nasajpour-Esfahani h, Davood Toghraie b,
Maboud Hekmatifar b
a
Wuhan Third Hospital, Hubei, Wuhan 430070, China
b
Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran
c
Department of Civil Engineering, College of Engineering, Cihan University-Erbil, Erbil, Iraq
d
Intelligent Medical Systems Department, Al-Mustaqbal University, 51001 Babil, Iraq
e
College of Technical Engineering, The Islamic University, Najaf, Iraq
f
College of Technical Engineering, The Islamic University of Al Diwaniyah, Iraq
g
College of Technical Engineering, The Islamic University of Babylon, Iraq
h
Department of Material Science and Engineering, Georgia Institute of Technology, Atlanta 30332, USA

A R T I C L E I N F O A B S T R A C T

Keywords: Nowadays, cardiovascular illnesses are among the leading causes of death in the world. Thus, many studies have
Viscosity model been performed to diagnose and prevention of these diseases. Studies show that the computational hemodynamic
Thermal effect method (CHD) is a very effective method to control and prevent the progression of this type of disease. In this
Cerebral blood vessel
computational paper, the impression of five non-Newtonian viscosity models (nNVMs) on cerebral blood vessels
Non-Newtonian blood flow
Dimensionless pressure
(CBV) is investigated by CHD. In this simulation, blood flow is supposed steady, laminar, incompressible, and
Nusselt number non-Newtonian. The parameters of Nusselt number (Nu), dimensionless temperature (θ), pressure drop (Δp), and
dimensionless average wall shear stress (DAWSS) are also investigated by considering the effects of heat
generated by the body. Utilizing the FVM and SIMPLE scheme for pressure–velocity coupling is a good approach
to investigating CBVs for five different viscosity models. In the results, it is shown that the θ and Δp+ increase
with increasing Reynolds number (Re) in the CBVs. By enhancing the Re from 90 to 120 in the Cross viscosity
model, the Δp+ changes about 1.391 times. The DAWSS grows by increasing the Re in all viscosity models. This
increase in DAWSS leads to an increasing velocity gradient close to the cerebral vessel wall.

1. Introduction Oxygen consumption is relatively high in the brain and neurons, but
their storage is very low [5]. Therefore, the amount of blood vessels and
Blood flow is made up of a combination of cells and fluid [1]. The cell capillaries in the brain tissue is very high because it can meet the need
part, which contains red and white blood cells and platelets, makes up for about 55 ml of blood per minute for every 100 g of brain tissue [6–8].
45 % of the blood volume. The fluid (plasma) part, which contains The human brain needs about 15 percent of the heart’s output to get the
water, salts, hormones, coagulants, organic and fatty substances, pro­ oxygen and glucose it needs. In other words, the brain needs a lot of
teins, and sugars, makes up 55 % of the blood volume. It is also blood circulation to maintain its health. When this circulation is
responsible for transporting the absorbed food from the gastrointestinal impaired, the brain may be damaged, and many complications and
tract to the tissues and body cells and the excretion of waste products to disabilities can occur as a result. Today, one of the most significant
the kidneys and liver [2,3]. Cerebral blood flow provides the oxygen and causes of fatality, especially in developed countries, is cardiovascular
nutrients needed for the brain to function properly [4]. It carries blood, disease [9,10]. Dynamic properties of blood flow play an important role
oxygen, and glucose to the brain. Although the brain makes up a small in understanding and treating many cardiovascular diseases. Therefore,
portion of the total body weight, it needs a lot of energy to function. it is necessary to study and analyze the characteristics of blood flow in

* Corresponding author.
E-mail addresses: Liyutao7526@126.com (Y. Li), Toghraee@iaukhsh.ac.ir (D. Toghraie).

https://doi.org/10.1016/j.asej.2023.102535
Received 25 December 2021; Received in revised form 27 June 2023; Accepted 16 October 2023
Available online 27 October 2023
2090-4479/© 2023 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Ain Shams University. This is an open access article under the CC
BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Y. Li et al. Ain Shams Engineering Journal 14 (2023) 102535

Fig. 1. The Schematic of the simulated CBV.

magnetic drug targeting in a penetrable microvessel. The results showed

Table 1
that the permeability of the carrier particle increases the tendency of the
Thermophysical characteristics of blood and vessels [26,27].
carrier particles to capture close by tumor position.
Parameter Blood Vessel Maiti et al. [20] investigated the fractional order model for the
Density 1060 kg.m− 3 1190 kg.m− 3 thermochemical flow of blood. The results showed that blood velocity
Thermal conductivity 0.556 W.m− 1.K− 1 0.235 W.m− 1.K− 1 and temperature both lessen in arising values of the fractional-order
Heat capacity 3770 J.kg− 1.K− 1 3600 J.kg− 1.K− 1 parameter as the memory effect. The penetrability of the blood flow
medium withstands to drive the fluid fast. Sharma et al. [21] investi­
the vascular sections. In many studies on the dynamic properties of gated the hemodynamical analysis of MHD two-phase blood flow. The
blood flow, blood flow has been considered to be a Newtonian or non- results showed that the curvature and penetrability of the arterial wall
Newtonian monolayer fluid. In a numerical study, Liu et al. [11] increase the risk of atherosclerosis formation, while the implication of
analyzed the pulsed blood flow along a sampled vessel. In their study, heat source on the blood flow lower this risk. Daset al. [22] investigated
blood flow was considered to be Newtonian fluid, and the given vessel the effect of hall and ion slip currents on electromagnetic blood flow
was assumed to be inelastic. The Navier-Stokes equations governing conveying hybrid nanoparticles. The results show that hybrid nano­
blood flow are solved in their study using the finite difference method. particle concentration has an important role in the heat-conducting
The hypothesis that blood flow is Newtonian for flow with high shear nature of blood which is cardinal to life support. Prakash et al. [23]
strain is agreeable, which is true for flow along vessels with an interior investigated the effects of stenoses on the non-Newtonian flow of blood
diameter larger than one millimeter [12–14]. On the other hand, in blood vessels. The results show that stenoses size lessens the flow rate
decreased cardiovascular function leads to impaired regulation of ce­ and enhance the wall shear stress as well as resistance to flow.
rebral blood flow. Therefore many studies have been done on cardio­ In this paper, the effects of the five distinct nNVMs of cerebral blood
vascular disease and cerebral blood flow. Tarumi et al. [15] investigated flow on Δp, DAWSS, θ, and Nu are studied. An open-source software
cardiovascular factors and clinical implications for cerebral blood flow [24] based on MRI and DICOM is used to construct a 3D computational
in adults. The results show that regular aerobic exercise improves car­ model of the CBVs. The blood flow is treated as a single-phase fluid. The
diovascular function and better regulation of cerebral blood flow. Re ranges from 30 to 120. The vessel wall is also assumed solid.
Therefore, it reduces the risk of dementia. Matthew et al. [16] investi­
gated the risk of cardiovascular disease and the velocity of cerebral 2. Numerical methods
blood flow. The results show that cardiovascular disease has different
effects on moderate blood flow velocity. Also, the study of cerebrovas­ In this numerical analysis, the effects of five different nNVMs such as
cular reactions and arterial stiffness can contribute to brain pathology Power-law, Cross, Quemada, Carreau, and Carreau-Yasuda on the pa­
and cognitive impairment. Jennings et al. [17] investigated cardiovas­ rameters of Δp+, DAWSS, θ, and Nu are investigated. Several approaches
cular disease using cerebral blood flow. The results show that aging and have been proposed so far to create 3D biological geometries. This study
cardiovascular disease lead to decreased cerebral blood flow. Moitoi employs the DICOM and MRI approaches for creating 3D geometries of
et al. [18] investigated the impression of using the magnetic drug tar­ blood vessels. Open-source software like SimVascular [24] is utilized to
geting at the same time as Caputo-Fabrizio fractionalized blood flow make geometry from images into a 3D CAD file. The Schematic of the
through a permeable vessel. The results showed that enhancing Re re­ simulated CBV is represented in Fig. 1. The geometry creation process is
sults in lessening the tendency of the drug to trap near the tumor site, briefly expressed as follows:
whereas the pulsatile frequency presents a conflicting phenomenon.
Shaw et al. [19] investigated the permeability and stress-jump effects on • Draw a path line along the length of the vessel (using xy-plane)

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Y. Li et al. Ain Shams Engineering Journal 14 (2023) 102535

Fig. 2. A) grid independency of the 3d model of the cbvs for numerical computation, b) laminar and turbulent blood flow patterns.

• Draw a vessel wall line using images (perpendicular to xy-plane) The range of velocity for this situation of the thoracic aorta and the
• Create point clouds blood temperature is considered between 0.15 and 0.45 m.s− 1 and
• Create a shell using point clouds (vessel wall) 309.55 K based on clinical data, respectively. The heat flux on the
• Create a volume of the vessel thoracic aorta wall is considered constant heat flux at normal daily ac­
tivity. The DICOM and MRI method and SimVasular software were used

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Y. Li et al. Ain Shams Engineering Journal 14 (2023) 102535

Fig. 3. Nu in terms of Re amount in the CBVs for all five viscosity.

to construct the 3D geometry of blood vessels with excellent accuracy [

∂v ∂v ∂v
]
∂p
[[
∂
(
∂v ∂
)
∂v ∂
(
∂v
) ( )]]
[25]. Heat flux is also intended for three types of daily activities. The ρ u +v +w = − + η + η + η (y
∂x ∂y ∂z ∂y ∂x ∂x ∂y ∂y ∂z ∂z
body produces this heat flux on the vessel wall. The thermophysical
characteristics of blood and vessels are reported in Table 1. − direction)
In this numerical analysis of non-Newtonian blood flow for the (2b)
abdominal aorta, the biological fluid is assumed a time-dependent [ ] [[ ( ) ( )
laminar incompressible flow. The FVM and SIMPLE schemes are used ∂w ∂w ∂w ∂p ∂ ∂w ∂ ∂w
ρ u +v +w = − + η + η
to couple velocity and pressure [28]. The conversion equations, ∂x ∂y ∂z ∂z ∂x ∂x ∂y ∂y
( )]]
including mass, momentum, and energy equations solved by the ap­ +
∂
η
∂w
(z − direction) (2c)
proaches above, are represented as follows: ∂z ∂z
Continuity equation:
Energy equation:
→
∇.(ρ. V ) = 0 (1) [
∂T ∂T ∂T
] [ 2
∂ T ∂2 T ∂2 T
]
ρC p u + v + w =k + + (3)
Momentum equation: ∂x ∂y ∂z ∂x2 ∂y2 ∂z2
[
∂u ∂u ∂u
]
∂p
[ (
∂ ∂u
)
∂
(
∂u
)
∂
(
∂u
)] The Quemada viscosity model can be defined as follows [29–31]:
ρ u +v +w = − + η + η + η (x
∂x ∂y ∂z ∂x ∂x ∂x ∂y ∂y ∂z ∂z η = K γ̇n (Power − lawmodel) (4)
− direction)
(2a)

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Y. Li et al. Ain Shams Engineering Journal 14 (2023) 102535

Fig. 4. The θ at various Re in a CBV for all five viscosity.

η = η∞ +
η0 − η∞
(Crossmodel) (5) the heat transfer coefficient (HTC). Moreover, the average Nu is calcu­
1 + λγ̇n lated by volume integration of the entire calculation zone.
[ The local heat transfer coefficient (LHTC) is shown as follows [29]:
√̅̅̅̅̅̅̅̅̅ ]− 2
1 k0 + k∞ γ̇/γc
η = η∞ 1− √̅̅̅̅̅̅̅̅̅ Ht (Quemadamodel) (6) q″(x)
2 1 + γ̇/γc h(x) = (10)
(Tw (x) − Tb )
[ ]b ∫
(7)
2 1
η = η∞ + [η0 − η∞ ] − 1 + A|γ̇| (Carreaumodel) Tw (x) = TdA (11)
A
[ ]a−b 1 ⃒ ⃒
η = η∞ + [η0 − η∞ ] − 1 + [λγ̇]
b
(Carreau − Yasudamodel) (8) ∫ ⃒→ ⃒
T ρ⃒ V dA⃒
Tb (x) = ∫ ⃒⃒→ ⃒⃒ (12)
where constant parameters are defined as K = 0.035, n = 0.6, η∞ = ρ⃒ V dA⃒
0.0033, η0 = 0.056, A = 10.976, b = 1.23, λ = 8.2 and a = 0.64.
Besides, the average Nu is signified as follows:
The dimensionless parameters are illustrated as follows:
h(x)Dh
x y z Tb − Tin Nu(x) = (13)
X = -Y = -Z = -θ= - Δp+ k
D D D Tw − Tin
Δp τ ∫ L
= - DAWSS = (9) Nu =
1
Nu(x)dx (14)
Δps τs L 0

The surface integration of each segment calculates the Nu following

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Y. Li et al. Ain Shams Engineering Journal 14 (2023) 102535

Fig. 5. Change in Δp+ at various Re in the CBVs for all five viscosity models.

Fig. 6. Changes of DAWSS versus Re in the CBVs for all five viscosity models.

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Y. Li et al. Ain Shams Engineering Journal 14 (2023) 102535

hDh CBVs (Fig. 4). This can be attributed to two reasons: 1) a faster
Nu = (15)
k expanding thermal border layer in the CBVs that reduced the tempera­
Δp can be measured as follows: ture difference between blood flow and the blood vessel wall. This is
because some viscosity models represent a higher viscosity than other
2fLρu2in models, and 2) increasing velocity by enhancing the Re at the entrance.
ΔP = (16)
Dh For example, the growth in Re from 60 to 90 in the Carreau-Yasuda
viscosity model in the running position of the body leads to a growth
Finally, the grid independency of the simulated structure is exam­
in the θ of about 1.1446 times.
ined. Checking grid independency is one of the most significant parts of
Fig. 5 depicts the diagram of Δp+ in terms of Re in the CBVs for all
any simulation. Generally, it can be expressed that if the simulation
five nNVMs. As shown, the Δp+ enhances by increasing the Re. This can
study of grid independency is not done in the simulation, the simulation
be attributed to two main reasons: 1) In some models, with increasing
answers are not trustworthy and cannot be mentioned. The grid inde­
the Re and blood viscosity, the amount of velocity and viscosity of ce­
pendence of the 3D model of the simulated CBVs is represented in
rebral blood flow increases more than in other models, respectively. And
Fig.2a. The relationship between Re and vessel diameter is about natu­
2) the increasing blood flow viscosity by increasing Δp+. For example,
rally smaller vessels and how smaller diameter affects hemodynamics.
by enhancing the Re from 90 to 120 in the Cross viscosity model, the
Considering that the blood flow is assumed to be laminar. In laminar
Δp+ changes about 1.391 times.
flow, the pressure changes have a direct relationship with the flow rate
Fig. 6 shows the DAWSS at different Re in the CBVs, along with the
and it enhances the growth of the flow speed, but it reaches a constant
effects of the five nNVMs. The results express that the DAWSS enhances
value in the turbulence region. As a result, it is better to check the Δp in
by increasing the Re in all viscosity models. This increase in DAWSS
the laminar areas (Fig. 2b).
leads to an increasing velocity gradient near the cerebral vessel wall.
The increased velocity gradient can increase blood flow viscosity with
3. Results and discussion
an increase in some viscosity models than other models and increase Re
by changing cerebral blood flow velocity at the entrance.
The impression of the Nu on different Re in the CBVs for three types of
Figs. 7a and 7b show pressure contours at various Re and velocity
heat fluxes produced by the body, consisting of sleeping, standing, and
contours, respectively. These figures indicate an increase in CBV pres­
running, are represented in Fig. 3 (from Ref. [27]). This figure also re­
sure and velocity with increasing the Re. A rise is observed in Δp at
ported the effect of five nNVMs on the Nu. The Nu increases by increasing
branching parts of the 3D cerebrovascular model. Fig. 7c depicts the
the entrance velocity because of an extended thermal border layer by
velocity streamline contours at various Re and velocity behavior in the
increasing the heat produced by the body in various positions. Further­
geometry. Figs. 7d, 7e, and 7f show the velocity component in each
more, the expansion of the thermal boundary layer is reduced by
direction.
increasing velocity.
Fig. 4 indicates the θ at different Re amounts in the CBVs for five
nNVMs. As mentioned above, the θ enhances with increasing Re in the

Fig. 7a. The 3D velocity contour at different Re.

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Y. Li et al. Ain Shams Engineering Journal 14 (2023) 102535

Fig. 7b. The 3D pressure contour at different Re.

Fig. 7c. The 3D velocity streamlines at different Re.

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Fig. 7d. The 3D velocity u at different Re.

Fig. 7e. The 3D velocity v at different Re.

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Fig. 7f. The 3D velocity w at different Re.

4. Conclusion References

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