# Non-Newtonian and Newtonian blood flow in human aorta: A transient analysis

**Vinoth R ^{1,2*}, Kumar D^{2}, Raviraj Adhikari^{3} and Vijay Shankar CS4**

^{1}Department of Electronics and Communication Engineering, Manipal Institute of Technology, Manipal University, Manipal, Udupi, Karnataka, India

^{2}Department of Electronics and Communication Engineering, Periyar Maniammai University, Vallam, Thanjavur, Tamil Nadu, India

^{3}Department of Mechanical and Manufacturing Engineering, Manipal Institute of Technology, Manipal University, Manipal, Udupi, Karnataka, India

^{4}Department of Cardio Vascular and Thoracic Surgery, Apollo Hospitals, Greams Road, Chennai, Tamil Nadu, India

- *Corresponding Author:
- Vinoth R

Department of Electronics and Communication Engineering

Periyar Maniammai University, Tamil Nadu, India

**Accepted on **December 15, 2016

**Visit for more related articles at**Biomedical Research

## Abstract

Pulsatile blood flow in an aorta of normal subject is studied by Computational Fluid Dynamics (CFD) simulations. The main intention of this study is to determine the influence of the non-Newtonian nature of blood on a pulsatile flow through an aorta. The usual Newtonian model of blood viscosity and a non- Newtonian blood model are used to study the velocity distributions, wall pressure and wall shear stress in the aorta over the entire cardiac cycle. Realistic boundary conditions are applied at various branches of the aorta model. The difference between non-Newtonian and Newtonian blood flow models is investigated at four different time instants in the fifth cardiac cycle. This study revealed that, the overall velocity distributions and wall pressure distributions of the aorta for a non-Newtonian fluid model are similar to the same obtained from Newtonian fluid model but the non-Newtonian nature of blood caused a considerable increase in Wall Shear Stress (WSS) value. The maximum wall shear stress value in the aorta for Newtonian fluid model was 241.706 Pa and for non-Newtonian fluid model was 249.827 Pa. Based on the results; it is observed that the non-Newtonian nature of blood affects WSS value. Therefore, it is concluded that the non-Newtonian flow model for blood has to be considered for the flow simulation in aorta of normal subject.

## Keywords

Computational fluid dynamics, Fluid-structure interaction, Aorta, Newtonian model, Non-Newtonian model, Wall shear stress, Wall pressure.

## Introduction

The cardiovascular system maintains an adequate blood flow to all cells in the body. The flow of blood in the cardiovascular system depends upon the pumping mechanism of the heart. This mechanism induces the blood to flow in pulsatile nature. The aorta is the largest and most important artery in cardiovascular system; it carries blood from the heart to other organs in a body. The occurrence of many diseases in cardiovascular system has been associated with blood flow behaviour in the blood vessels [1]. The investigation on blood flow in aorta is crucial because the flow of blood changes hemodynamic stresses act upon the aortic wall. The relationship between hemodynamic stresses and corresponding changes in the layer of blood vessel wall is the major cause of aneurysm [2], other lesions. Therefore, the interaction between hemodynamic stresses and physiological behavior of blood vessel wall plays an important role in aneurysm formation, progression and rupture. Computational Fluid Dynamics (CFD) and Fluid-Structure Interaction (FSI) techniques have been widely used for simulating the blood flow in idealized and patient-specific aorta models.

The blood is a non-Newtonian fluid and it follows Newtonian
nature when the shear rate is above 100 s^{-1} [3,4]. The effect of
non-Newtonian behavior of flow is not significant in large
blood vessels like aorta, where the shear rate is high.
Considering the blood as a Newtonian fluid is a satisfactory
assumption for large arteries such as aorta [3,5,6]. In transient
analysis, the non-Newtonian flow effects could become
significant when the shear rate is below 100 s^{-1} [3]. Some
authors concluded that the non-Newtonian fluid approximation
for flow in large arteries is crucial [7-9] while others found it is
an unimportant assumption [10,11]. Gijsen et al. [9]
highlighted various differences between non-Newtonian and Newtonian flow patterns when they studied flow through 90°
curved tube. Li et al. [12] simulated blood flow through an
Abdominal Aortic Aneurysm (AAA) model with Stent-Graft
(SG) using non-Newtonian fluid assumption. They discussed
about key biomechanical factors which are causing SG
migration. The authors concluded that the blood flow
conditions, aneurysm and SG geometries were major reasons
for SG migration. Amblard et al. [13] developed a
methodology using non-Newtonian fluid approximation to
observe the relation between the aorta’s wall and endocraft to
find when type I endoleaks could occur. They evaluated the
stresses on the aorta’s wall generated by the blood flow.

In the present study, CFD approach is used to simulate the blood flow through an aorta with Newtonian fluid assumption for assessing the hemodynamic parameters. This study also depicts the influence of non-Newtonian nature of blood on various physiologically important flow parameters like velocity, wall pressure and wall shear stress (WSS). The Casson non-Newtonian model is used in this study because of its capability in representing the non-Newtonian blood rheology [14]. The blood flow through a bifurcation with an aneurysm in the cerebrovascular system was simulated with the Casson non-Newtonian model by Perktold et al. [10]. Perktold et al. [6] studied pulsatile flow characteristics through a human carotid bifurcation with the Casson model approximation. The blood assumed as power-law non-Newtonian fluid by Liepsch et al. for their flow study through renal artery and T-shaped bifurcations [15-19]. The complex flow regions could happen in the abdominal aorta segment due to bifurcations, branches and curvature of the arteries [20]. Wiwatanapataphee et al. [21] studied the effect of branching vessel on the pulsatile blood flow in the human coronary artery with non-Newtonian fluid assumption. The authors concluded that the branching of artery influences the flow in the artery substantially. Shahcheraghi et al. [22] described the influence of aortic branches on flow of blood. In this work, those aortic branches recommended by Shahcheraghi et al. are considered. The aorta model is simplified by excluding coronary arteries, intercostal arteries, gonadal artery, arteries branching from the brachiocephalic trunk, subclavian artery and celiac trunk. The inflow data to the aorta used in this study was measured at the ascending aorta past the coronary arteries and the coronary arteries carry approximately 4-5 percent of the whole cardiac output [23]. Hence, the coronary arteries are not considered in the present work. The intercostal arteries carry less than 1percent of the whole cardiac output [23] and hence these arteries are not considered in the present work. The brachiocephalic trunk, common carotid artery, left subclavian artery, celiac trunk, renal arteries, superior mesenteric artery, inferior mesenteric artery and common iliac arteries are included in the aorta model. In order to reduce the computational time the other arteries and branches are not considered in the aorta model.

There are substantial studies reported on the CFD simulation of blood flow which investigated the influence of the non- Newtonian nature of blood on a pulsatile flow in idealized aorta, coronary artery and carotid artery. The CFD analyses have been carried out by a few authors with symmetric aortic bifurcation geometry. The aortic bifurcation is not a symmetric geometry and the influence of asymmetry in geometry on flow analysis is crucial [24]. To the best of knowledge of authors, the CFD analysis of blood flow in an aorta model with Newtonian, non-Newtonian fluid assumptions had not been attempted. There are a few studies which have been conducted with realistic geometry of aorta. The purpose of this study is to find the influence of the non-Newtonian nature of blood on a pulsatile flow through an aorta. The physiologically important flow parameters such as velocity distribution, wall pressure and wall shear stress are estimated in the aorta through CFD simulation with both non-Newtonian and Newtonian fluid models.

## Methodology

*Aorta geometry and mesh generation*

The subject was a 36 year old female without any known heart
disease. The heart rate of this subject was 78 bpm and her
blood pressure range was (120-160) mmHg/(62-80) mmHg.
The cardiac output and blood flow can be affected by abnormal
heart rate. The elevated heart rate can initiate and promote the
atherosclerotic process [25] by several mechanisms and WSS
is one of the mechanisms [26]. In the present study, heart rate
of the subject considered was normal and there would not be
any effect on blood flow. All medical data were obtained from
the subject for diagnostic purposes. This study has been
approved by the Institutional Ethics Committee-Clinical
Studies (Apollo Hospitals, Chennai, India) and adapted to the
declaration of Helsinki. The sectional image of aorta of the
representative case was obtained with the aid of a 320-slice
Computed Tomographic (CT) scanner (Aquilion one, Toshiba
Medical System). The intravenous contrast medium was
injected during imaging process to acquire good quality image.
The slice thickness was 0.5 mm and pixel size was 0.5 mm.
The 3D geometry of aorta was reconstructed from the dicom
format files of CT image using Mimics v17.0 (Materialise,Belgium). The reconstructed 3D geometry of aorta of normal
subject is shown in **Figure 1**.

The fluid domain geometry data of the aorta model was imported to ANSYS Workbench (v15, ANSYS, Inc.) in IGES format. The model was discretized by using tetrahedral fluid elements.

**Mesh sensitivity test: **Three tetrahedral meshes were created
for the fluid domain geometry of the aorta model with element
numbers 625456, 1037252 and 1382806. Mesh sensitivity test
was performed with steady state boundary conditions using the
laminar flow model by monitoring maximum value of WSS of
the model.

The Navier-Stokes and continuity equation are the governing
equations for the motion of three-dimensional fluid (blood). A
finite volume based solver ANSYS CFX was employed to
solve these equations. Laminar flow model was used for the
simulation of steady flow in aorta. The RMS residual value of
10^{-4} was taken as maximum value to obtain the accuracy of the
solution. The maximum iteration was considered as 500 and
minimum iteration was considered as 100 in convergence
control to achieve convergence in each simulation. The
Reynolds number for blood flow was fixed at 1350 [27]. The
average velocity of 0.225 m/s was obtained from the Reynolds
number and imposed at the inlet of ascending aorta as
boundary condition. Due to lack of pressure data, opening
boundary condition was assumed at the outlet of all the
branches of aorta with relative static pressure of 120 mmHg.
The wall of aorta was assumed as rigid and no-slip condition
was employed at the wall. Blood flowing through the aorta was
approximated to be a Newtonian, homogeneous and an
incompressible fluid. The blood density was taken as 1050 kg
m^{-3} and dynamic viscosity of blood as 0.0035 Ns m^{-2} [3,5,28].

Particulars of mesh sensitivity test for the aorta model are
shown in **Table 1**. The difference in maximum WSS value
between fine and medium mesh was 6.23%, but between
medium and coarse mesh was 15.84%. The 1382806 elements
mesh model was considered for further analysis.

Mesh | Number of elements | WSS (Pa) | Difference (%) |
---|---|---|---|

Coarse mesh | 625456 | 24.778 | 15.84 |

Medium mesh | 1037252 | 28.704 | |

6.23 | |||

Fine mesh | 1382806 | 30.495 |

**Table 1.** Mesh sensitivity test result shows maximum wall shear stress
value for different mesh element numbers of aorta model.

*Transient state simulation and boundary conditions*

**Newtonian model: **The pulsatile blood flow in aorta was
investigated by using transient analysis. A time varying
pulsatile velocity profile at the ascending aorta inlet and
pressure waveforms at the outlet of iliac arteries were imposed
in simulations based on data from Olufsen et al. [23]. These
waveforms are shown in **Figure 2**. (**Figures 2a** and **2c**) and
these conditions had been verified by several experimental data [5,29]. The cardiac cycle period was 1 s with peak blood flow
occurred at 0.14 s. A fifteen percent of the inlet flow volume
was assumed for brachiocephalic artery; five percent of the
inlet flow volume was prescribed at common carotid artery and
left subclavian artery [22]. A total proportion of ten percent of
the thoracic mass flow was considered for each renal artery
[30] and pulsatile pressure was fixed at other branches. Blood
was treated as a homogeneous, Newtonian and an
incompressible fluid. The blood flow can be considered as
laminar in large blood vessels like aorta and it was found to be
laminar in Abdominal Aortic Aneurysms (AAAs) during
exercise [31]. In this study, the maximum Reynolds number
based on the flow velocity was 3971. Since the maximum
Reynolds number was lower than the threshold Reynolds
number [32,33] the flow was assumed as laminar. Various
cardiac cycles are required for achieving convergence for the
transient analysis [22,34]. In CFD simulation, five cardiac
cycles (**Figure 2b**) were used with time step of 0.002 s. The
fifth cardiac cycle was used as the final periodic solution to
obtain the hemodynamic parameters from the model. The other
properties and boundary conditions were same as that of steady
state analysis.

**Non-Newtonian model:** Shear thinning or pseudoplastic fluids
are the fluids whose effective viscosity decreases with increase
in shear rate. This fluid structure is time-independent. The
Casson model was recommended for shear thinning liquids
[35]. In the present work, the Casson model was used in
transient simulation to approximate the non-Newtonian flow.
The dynamic viscosity for the Casson model is given by
Equation 1 [36].

√u=√τY/γ+√K → (1)

Where is the dynamic viscosity, γ the shear strain rate, yield stress and the viscosity consistency. The yield stress of human blood in normal condition is between 0.0003 Pascal and 0.02 Pascal [37]. The yield stress was taken as 0.004 Pascal. The other properties and boundary conditions were assumed same as Newtonian model simulation condition.

## Results

The hemodynamic parameters were measured at four different
time instants during the cardiac cycle. These four time instants
(t=4.08 s (maximum acceleration), 4.14 s (peak systole), 4.28 s
(maximum deceleration) and 4.65 s (mid-diastole)) represent
critical flow rate phases in the cardiac cycle. **Figure 3** depicts
velocity vectors of the aorta with a Newtonian and a non-
Newtonian model at four time instants in the cardiac cycle. The
velocity vector plot shows region of high velocity in aortic
arch, branches of the distal portion of the aorta with Newtonian
and non-Newtonian models. The region of low velocity
appears in the ascending and descending aorta. The maximum
velocity value at peak systole is increased insignificantly in the
non-Newtonian model. The high velocity regions at middiastole
are reduced in the non-Newtonian model. The overall
velocity vector patterns of the aorta for a non-Newtonian fluid
model are similar to velocity vector patterns of a Newtonian fluid model. **Figure 4** presents the variation of wall pressure of
the aorta at peak systole in the cardiac cycle. Since wall
pressure contours of the aorta at all four time instants of the
cardiac cycle for a non-Newtonian fluid model are similar to
wall pressure contours of Newtonian model, only the wall
pressure contour at peak systole is presented here. In both
Newtonian and non-Newtonian models, the wall pressure
diminishes in flow direction. The maximum wall pressure
value of the aorta is increased in non-Newtonian model. **Figure
5** shows wall shear stress pattern of the aorta for the non-
Newtonian model at different time instants in the cardiac cycle.
The range of wall shear stress is between 0.003 and 249.8 Pa.
At time t=4.08 s of the cardiac cycle very low wall shear stress
(0.046-53.142 Pa) is distributed uniformly in the ascending
aorta, arch of aorta and descending aorta. Quite marked regions
of low shear stress (53.142-106.239 Pa) are appeared at all the
branches of abdominal aorta. At peak systole, the wall shear
stress reaches its maximum value and a few patches of low
wall shear stress (62.476-124.926 Pa) are appeared in the
abdominal aorta region. At time t=4.28 s of the cardiac cycle
few patches of low wall shear stress (16.495-32.978 Pa) are
distributed in aortic arch, number of this patches are increased
in the posterior view of the aortic arch and abdominal aorta. At
mid-diastole, the intensity of low wall shear stress
(1.235-2.468 Pa) patches has increased in the region of aortic
arch and abdominal aorta (anterior and posterior view). Five
intense regions of low shear stress appear in posterior view of
descending aorta.

At all-time instants, the region of high wall shear stress appears at the proximal end of the branches and aortic bifurcation. This high wall shear stress exists due to sudden contraction and At all-time instants, the region of high wall shear stress appears at the proximal end of the branches and aortic bifurcation. This high wall shear stress exists due to sudden contraction and bifurcation of the artery. Since at mid-diastole the flow has almost reduced to zero, the maximum value of shear stress was only 4.933 Pa. The peak velocity of reverse flow is insignificant; hence low wall shear stress is uniformly distributed over the entire artery. During flow acceleration phases the value of wall shear stress begins to increase and patches of low wall shear stress are distributed in the aorta. As the blood flow decelerates, the value of wall shear stress begins to decrease and intensity of low wall shear stress patches are increased.

In general, the high wall shear stress regions are dominant at high flow velocities close to the distal portions of the artery and at all-time instants of the cycle the low wall shear stress regions are present close to the proximal portions of the artery [38].

In this study, at peak systole the high shear stress occurred in abdominal region and low shear stress distributed in descending aorta region. The low wall shear regions are more in descending aorta due to its larger diameter than abdominal aorta. The high wall shear stress regions are observed where there is a narrowing in the aorta geometry profile.

**Figures 6-8** indicate the wall shear stress distribution of aorta at
four time instants for a non-Newtonian and a Newtonian fluid
model. It is interesting to see from **Figures 6** and **7** that the wall
shear stress patterns are similar in non-Newtonian and Newtonian models at time t=4.08 s of the cardiac cycle. At
peak systole the shear stress pattern of non-Newtonian model
is similar to the same of Newtonian model but few patches of
low shear stress are distributed in aortic arch and its branches
in only non-Newtonian model. It can be observed from** Figures
6** and **7** that the intensity of low WSS patches in aortic arch,
descending region and abdominal region is more in non-
Newtonian model than Newtonian model.

**Figure 8** depicts that the intensity of WSS patches at ascending
aorta inlet, bifurcation outlet and branches of aorta in non-
Newtonian model is greater than in Newtonian model. Hence,
the non-Newtonian nature of blood affects the inlet, outlets and
branches more than inner and outer regions of aorta. This effect
exists because the shear rate is lower in inlets, outlets and
branches than inner and outer regions of aorta. In non-
Newtonian model, the variation in viscosity is inversely
proportional to variation in shear rate.

The results from **Figures 6** and **7** indicate that the WSS values
in non-Newtonian model are greater than WSS values in
Newtonian model. It can be observed from the results that the
non-Newtonian model is more significant during flow
decelerating phase and also when the flow velocity is close to
zero. This observation agrees to those observed by Caballero et
al. [39].

## Discussion

Hemodynamic parameters of blood flow in arteries are very
crucial. It plays a key role in the initiation and occurrence of
many diseases in arterial system. Flow recirculation,
separation, low and oscillating wall shear stresses are to be
considered as atheromatic factors to develop diseases in arterial
system [40,41]. In many studies the blood has been assumed as
Newtonian fluid but arterial flow and stress pattern can be
affected by rheological properties of the blood. Gijsen et al.
[42] found from their study on blood flow in a carotid
bifurcation model the velocity distribution of flow could be
affected by shear thinning non-Newtonian property of blood.
The idea of this study was to investigate the difference between
Newtonian and non-Newtonian blood flow models of aorta.
Results for transient analysis of blood flow in aorta have been
presented in the preceding section. Results of velocity
distribution, wall pressure and WSS distribution of aorta at
four different time instants in the cardiac cycle have been
presented for Newtonian and non-Newtonian fluid models.
This study provides an idea about whether to include non-
Newtonian blood models during modelling of blood flow in
aorta or not. The velocity vector plots of the aorta for a
Newtonian and a non-Newtonian model are shown in **Figure 3**.
This vector plot shows region of high velocity in distal part and
branches of the distal portion of the aorta with Newtonian and
non-Newtonian models. These results agree to those observed
by Sheidaei et al. [43]. The overall velocity vector patterns of
the aorta for a non-Newtonian fluid model are similar to
velocity vector patterns of a Newtonian fluid model. The
variation in yield stress ( ) within a physiological value
fundamentally affects the velocity and WSS profiles. Some
yield stress values for normal human blood available in the
literature are taken in simulating the non-Newtonian flow
through aorta. The results obtained from the simulation
indicate that the velocity value decreases as the increases as
shown in **Figure 9**. Sankar et al. [44] had obtained similar
pattern of variation of velocity with respect to yield stress from
their steady flow analysis on blood flow through a catheterized
artery. This is because the shear thinning non-Newtonian
property of blood decreases the peak velocity of the blood
flow. The wall pressure distribution of aorta at peak systole is
shown in **Figure 4**. The peak value of wall pressure for non-
Newtonian model is greater than the value of Newtonian
model. **Figures 5-8** present the WSS distribution of aorta. At
all-time instants, regions of low WSS appeared at the proximal
end (ascending aorta, aortic arch and descending aorta) of the
aorta and regions of high WSS predominate at the distal end
(abdominal aorta) of the aorta. This type of WSS pattern
occurred because of tapering of aorta towards the distal end.
This result is consistent with results obtained by Kirpalani et
al. [45] and Myers et al. [46]. The maximum shear stress value
for Newtonian fluid model is 241.706 Pa. Zhonghua et al. [47]
obtained considerably higher value than this, but their WSS
distribution was almost similar to those obtained from this
study. It is noticed from the result analysis, that the wall shear
stress value for the non-Newtonian flow is higher than that of
the Newtonian flow. This is in good agreement with results obtained by Lou et al. [48] and Chen et al. [49]. It is found that
the WSS value increases marginally as the yield stress
increases and it is shown in **Figure 10**. This effect occurred
because the variation of dynamic viscosity is directly
proportional to variation of yield stress. Sankar et al. [44] and
Lou et al. [48] had found similar pattern of variation of WSS
with respect to yield stress from their study. It is considered
that the hemodynamic stresses are to have significant effects
on the development of lesions in the arterial system. The WSS
acts directly on the aortic wall. The mechanoreceptor converts
wall shear stress into its equivalent biological signal and this
signal alters the cellular function of the endothelial cells
[50,51]. The wall shear stress affects the rate of mass transport
throughout the arterial walls and it influences the occurrence of
atherogenesis [44]. The low shear stress has been linked with
thrombus formation [52], aneurysm progression [53] and
aneurysm ruptures [54,55] whereas high wall shear stress has
been associated with cerebral aneurysms [54,56]. The variation
in wall shear stress is one of the factors to cause intimal
thickening [57-59]. Fry [60], Caro et al. [61], Giddens et al.
[62] and Ku et al. [63] had investigated and reported that the
WSS could be a substantial factor to cause coronary artery
disease. In clinical applications, the assessed WSS could assist
to understand about the pathological conditions of
cardiovascular system and factors affecting endothelial cells.
The non-Newtonian nature of the human blood makes
differences in the flow behavior according to people’s age. The
shear rate in the blood circulation of elderly people is lower
[64] than those of younger people. The blood acts like a
Newtonian fluid in younger people due to this higher shear
rate. The non-Newtonian nature of blood affects velocity, WSS
values and distributions of pulsatile flow. It is very important
to consider the non-Newtonian nature of blood in simulations
during the assessment of relationship between arterial diseases
and hemodynamic parameters of the blood flow. A Newtonian
model for blood could render inappropriate interpretation of
results of analysis. Based on the above discussions, it could be
concluded that using a non-Newtonian model for blood is an
appropriate assumption.

## Limitations

In the current study, the aorta wall was assumed as rigid. In reality the aorta walls are hyper elastic. Due to this, the results of this investigation may not be generalized to realistic conditions. The outcome of this study is based on the analysis of one subject. Liu et al. [65] investigated the influence of both non-Newtonian nature and pulsatile nature of blood flow on the distributions of luminal surface Low-Density Lipoprotein (LDL) concentration and oxygen flux within luminal wall of an aorta. In this study, the aorta model constructed from Magnetic Resonance Imaging (MRI) technique. The authors compared the results of Newtonian model with non-Newtonian one under steady and transient state conditions. They reported that, WSS value could be increased in aorta by the non-Newtonian nature of blood under steady state condition. This result was obtained from the analysis of one subject. The outcome of the present investigation agrees to those observed by Liu et al. The reproducibility of results may be negatively influenced by reduced number of subjects. This may be overcome by considering more number of subjects which make the results statistically significant.

## Conclusion

In this study, the CFD models of aorta are constructed to investigate the non-Newtonian nature of blood on a pulsatile flow. Results of velocity distribution, wall pressure and WSS distribution of aorta at four different time instants in the cardiac cycle have been presented for Newtonian and non- Newtonian fluid models. It is concluded that the flow patterns of Newtonian and non-Newtonian blood models are similar, but the non-Newtonian nature of blood caused a significant increase in wall Shear Stress (WSS) patterns. It is very difficult to observe the quantitative information of hemodynamic profiles like flow parameters, wall pressure and WSS in vivo. Computed profiles from the aorta model could be used as a diagnostic tool in clinical applications. For example, the measured flow profiles from a diseased subject could be compared with flow profiles from a healthy subject. The information generated from this comparative study plays an important role in understanding of the pathologic condition of diseased subject. The computed hemodynamic profiles from the CFD analysis could also be used with surgery and anesthesia simulators to train the medical professionals.

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