J. Biomedical Science and Engineering, 2013, 6, 14-20 JBiSE
http://dx.doi.org/10.4236/jbise.2013.612A003 Published Online December 2013 (http://www.scirp.org/journal/jbise/)
Effect of tapered angles in an artery on distribution of
blood flow pressure with gravity considered*
Wenying Mu1, Shanguang Chen2, Changsheng Ma1, Jianz e ng Dong1
1Department of Cardiology, Beijing Anzhen Hospital, Capital Medical University, Beijing, China
2National Key Laboratory of Human Factors Engineering, China Astronaut Research and Training Center, Beijing, China
Received 10 October 2013; revised 15 November 2013; accepted 29 November 2013
Copyright © 2013 Wenying Mu et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
The tapered angles of an artery significantly influence
the local hemodynamics. However, as gravity is con-
sidered, little is known about the effect of tapered
angles on the hemodynamics. In this study, we ex-
plored whether the effect of tapered angles on the
distribution of blood flow pressure (DBFP) differed
with gravity considered or not. Numerical simulations
of the DBFP in a single vessel were performed based
on such tapered angles as 0˚, 0.5˚ and 1˚. In the model
used for simulation, gravity was introduced as a body
force. We obtained the following simulations: i) The
larger the tapered angles were, the better distributed
the blood flow pressure; ii) The tapered effect was an
important factor leading to nonlinearity in blood flow
pressure; iii) Gravity affected DBFP coupling with
the tapered angles, yet independently influenced the
dimension of the DBFP. At the same time, the effec-
tive intensity of gravity decreased with the increase of
tapered angles.
Keywords: Tapered Angles; Distribution of Blood Flow
Pressure (DBFP); Gravity; Numerical Simulation
1. INTRODUCTION
As human enters into space and exposes to microgravity,
a series of adaptations, such as blood redistribution, oc-
cur [1]. The alteration is simulated by tilt table test on the
ground, which is adopted widely during the selection and
training of astronauts [2-4]. The essential difference be-
tween the simulation test and real situation in space lies
in the change of gravity.
Therefore, it is the first step to determine the effect of
gravity on the hemodynamics.
The distribution of blood flow pressure characterizes
the cardiovascular function of human body and repre-
sents the space state of blood flow. The arterial tapered
angles affected greatly the hemodynamics of the arterial
system in mammals [5]. For human arterial system, ta-
pered angles at different levels are different and have an
important impact on the hemodynamics. So in many
cases, the tapered angle may be an important factor
leading to cardiovascular dysfunction or diseases [6,7].
In recent years, many researchers have investigated
the tapered angles in the clinical study of human arterial
blood flow. Some of them have divided the stenosed ar-
teries into three types of non-tapered angle, divergent
tapered angle and convergent tapered angle to explore
the relationship between the arterial stenosis and tapered
angles. Among these studies, the effect of tapered angles
on hemodynamics is an important part discussed [6,8-10].
In addition, some researchers have concluded that the
tapered angles in different arterial sections influence the
fatty streaks and injury distribution of the artery on the
basis of the measurement of the medical images, and
have suggested that the tapered angle is an important
factor leading to the development of atherosclerosis [11].
In the above study, gravity was not considered as a con-
tributor.
In contrast, in the aerospace field, the variation of
gravity is a key issue that needs to be taken into account.
However, the existed mathematical models in cardio-
vascular hemodynamics are mainly lumped parameter
models, which are founded on the thought of analog cir-
cuit. So, gravity could only be considered in scalar quan-
tity, while the effect of tapered angle could not be in-
cluded at all in the lumped-parameter model [12,13]. At
present, considering as many factors as possible to obtain
the precise results is bringing more and more attention.
The distribution of blood flow pressure (DBFP) can
quantify the temporal and spatial information of cardio-
vascular system more accurately. Yet up to now, under
different tapered angles and with gravity involved as a
*There are no conflicts of interest for the authors of this study.

W. Y. Mu et al. / J. Biomedical Science and Engineering 6 (2013) 14-20 15
body force, DBFP has been studied inadequately.
In this study, by the aid of the postural model in which
gravity was considered as a body force [14], DBFP in a
single vessel was simulated numerically under different
tapered angles with or without gravity considered as a
contributor. Then it was made clear if the effect of ta-
pered angles on DBFP depended on gravity or not.
2. METHODS
2.1. Mathematical Model
The mathematical model with posture was adopted [14].
In this model, gravity was introduced as body force.
Therefore, the native of gravity as a vector was reflected.
Equation of Continuity

0A 

U (1)
Equation of Motion

b
  


UUF P
 

(2)
where is the inertial force,

 

UU 




P is the
surface force, b


F is the body force, which was iden-
tified as the key term to reflect the postural change,

represents the blood density, represents the body
force per unit mass of blood,


represen the second
order stress tensor of the blood pressure,

represents
the blood velocity, and A represents the cross-sectional
area of the vessel.
b

F
Pts

U
2.2. Finite Element Model
The carotid artery was taken as a computational case.
The carotid artery is one of the systemic arteries. The
blood flow in it is assumed to be homogeneous, incom-
pressible, viscous and Newtonian fluid [15,16], the ves-
sel wall can be assumed to be rigid [17,18]. The geomet-
ric model of the single vessel is shown in Figur e 1.
The geometric dimensions of the vessel [19] and the
material parameters of blood [15,16] are listed in Table1.
The value of D2 was determined by the tapered angle
α, and α was among [0˚,1˚], h represents the wall thick-
ness of the vessel. The inlet-outlet pressures were set to
be constant respectively, and the pressure difference was
set 199.98 Pa (1.5 mmHg). The solid-wall boundary con-
dition was applied to the surrounding of the vessel wall.
The tool used in numerical computation was ADINA
(Automatic Dynamic Incremental Nonlinear Analysis,
ADINA R&D, Inc., Watertown, MA, USA), a comer-
cial software package for finite element analysis. Some
researchers before [20,21] had used the software in bio-
logical system. Owing to the consideration of the effect
of gravity, the model was meshed by three-dimensional
fluid element. Then the finite element model was built up
(shown in Figure 2).
Figure 1. Geometric model of the single vessel.
Figure 2. Finite element model of a single vessel.
Table 1. Geometric and material parameters of the single ves-
sel.
L
(mm)
D1
(mm)
h
(mm)
α
(˚)

(Pa·s)
ρ
(kg·m−3)
145.00 10.74 0.35 [0˚,1˚] 0.0035 1050.00
(The viscosity  and the density ρ equal to the normal values of human
blood at 37˚C).
For the eleven tapered angles among [0˚,1˚], with
gravity considered or not, the corresponding finite ele-
ment models were presented respectively. Then by com-
paring the typical numerical results at 0˚, 0.5˚ and 1˚
tapered angles, it was found out whether there existed
some difference in the effect of tapered angles on the
blood flow pressure distribution with gravity considered
or not.
3. RESULTS AND DISCUSSIONS
With gravity considered or not, in fluid-only model, as
tapered angle α was equal to 0˚, 0.5˚, 1˚ respectively, the
contour charts of the blood flow pressure distribution in
the transverse and longitudinal sections, were shown
respectively in Figures 3(a), (b) and 4.
In the contour chart of blood flow pressure distribution
in longitudinal section (Figures 3a(i)-3a(iii)), with grav-
ity considered (0G

) and taking 0˚ tapered angle as the
reference one (namely cylindrical vessel commonly
adopted), the pressure distribution of blood flow was
typically three-dimensional asymmetrical distribution.
And with tapered angles increasing, the pressure differ-
ence in the vessel decreased, and the distribution non-
uniformity of blood flow pressure weakened, the pres-
sure gradient produced from the effect of gravity reduced

W. Y. Mu et al. / J. Biomedical Science and Engineering 6 (2013) 14-20 16
gradually, yet the pressure distribution was invariably
three-dimensional asymmetrical one.
Figures 3b(i)-3b(iii) shows the pressure distribution
of transverse section under different tapered angles with
gravity considered (). With tapered angles in-
creasing from 0˚ to 1˚, the distribution of blood flow
pressure tended gradually to be uniform and the effect
intensity of gravity decreased gradually. This further
demonstrated the conclusion drawn from the contour
chart of the pressure distribution in the longitudinal sec-
tion. In order to further identify the pure effect of gravity
and the combined effect of gravity and tapered angles on
DBFP, the magnitude of the blood flow pressure along
the symmetrical centerline and DBFP in the vessel were
compared respectively under different tapered angles
with or without gravity considered.
0GFigure 4 shows the comparison of the blood flow
pressure distributions of longitudinal and transverse sec-
tions under different tapered angles with gravity consid-
ered or not. Whether gravity was considered or not, the
distribution and magnitude of blood flow pressure in the
vessel varied with tapered angles. With gravity consid-
ered, DBFP was always three-dimensional asymmetrical
one at all three tapered angles.
(i) (ii)
(iii)
(a)
(i) (ii)
(iii)
(b)
Figure 3. Longitudinal and transverse section contour charts of distribution of blood flow pressure with different tapered angles
(ΔP = 199.98 Pa, , supine posture, θ is the tapered angle): (a) Longitudinal pressure distribution; (b) Transverse pressure
distribution; (i) θ = 0˚, (ii) θ = 0.5˚, (iii) θ = 1˚.
0G

W. Y. Mu et al. / J. Biomedical Science and Engineering 6 (2013) 14-20 17
(i)
(ii)
(a)
(i)

W. Y. Mu et al. / J. Biomedical Science and Engineering 6 (2013) 14-20 18
(ii)
(b)
Figure 4. Longitudinal and transverse section contour chart of distribution of blood flow pressure with different tapered angles
(ΔP = 199.98 Pa, and G = 0, supine posture, the tapered angle θ is 0˚, 0.5˚, 1˚.): (a) Longitudinal pressure distribution;
(i) G = 0, (ii) . (b) Transverse pressure distribution; (i) G = 0, (ii)
0G
0G0G

.
At 1˚ tapered angle, the effect intensity of gravity was
18.4% less than that at 0˚ tapered angle as reference state,
and 10.5% less at 0.5˚ than that at 0˚. With gravity ne-
glected (), the blood flow pressure distribution was
typically two-dimensional axisymmetrical one, and it
was not affected by tapered angles. Accordingly, we con-
cluded that the tapered angles affected only the magni-
tude of blood flow pressure and its position occurred, but
not the dimension of DBFP. This was similar to Liu et
al.’s results [22] which have reported that the tapered
angles do not affect the flow type and distribution, and
only change the value of the wall shear stress in the nu-
merical simulation of the effect of the tapered angles and
stenosis on the pulsating flow in an artery.
0G
Figure 5 shows the blood flow pressure under differ-
ent tapered angles along the axisymmetrical centerline of
the vessel with gravity considered (). From the
inlet to outlet of the vessel, at 0˚ tapered angle, the rela-
tionship between the blood flow pressure along the axi-
symmetrical centerline of the vessel and the distance
from the vessel inlet visualized as a straight line, at 1˚
tapered angle, the relationship figure became a curve,
and at 0.5˚ tapered angle, the curve was between the
above two figure lines.
0G
To further analyze the effect of gravity, the blood flow
pressure figures at different tapered angles along the ax-
isymmetrical centerline in the vessel with or without
gravity considered were compared (shown in Figure 6).
The simulation indicated that the blood flow pressure
change at different tapered angles along the axisymmet-
rical centerline of the vessel with gravity neglected
() was similar to that with gravity considered
(
0G
0G

). With tapered angles increasing, the relationship
between the blood flow pressure along axisymmetrical
centerline of the vessel and the distance from the vessel
inlet, developed from linear relation at 0˚ tapered angle
to non-linear relation at 1˚ tapered angle. Also with ta-
pered angles increasing, the nonlinearity was more evi-
dent. This means that the tapered effect of the vessel was
one of the import factors leading to the nonlinearity of
hemodynamic parameters and it was independent of
gravity.
In addition, for the same tapered angle, the curves of
blood flow pressure with gravity considered were all
above those with gravity neglected. This indicated that
the effect of gravity increased the magnitude of blood
flow pressure, and the extent of increase in it varied with
tapered angles. Ignoring the inlet-outlet effect, the blood
flow pressure at 0˚ tapered angle increased with the same
amplitude from inlet to outlet. Whereas at 0.5˚ and 1˚
tapered angles, the increase rate of blood flow pressure
gradually decreased. At the same position, the increase
magnitude was the most at 0˚ tapered angle, and at 0.5˚,
it took the second place, and the least at 1˚ tapered angle.
This suggested that gravity coupling with the tapered
angles, affected the value of the blood flow pressure and
its effect intensity. With tapered angles increasing, the
effect intensity of gravity decreased. This was consistent
with the conclusion drawn from the contour diagram of
blood flow pressure distribution in longitudinal and
transverse section.
This study suggest that without considering the ta-
pered effect, taking cylindrical tube as the geometrical
model in vessel study, there may be a big difference in

W. Y. Mu et al. / J. Biomedical Science and Engineering 6 (2013) 14-20 19
Figure 5. Blood flow pressure along the symmetrical centerline of the vessel with different tapered angles (ΔP =
199.98 Pa, , supine posture, θ is the tapered angle): (a) θ = 0˚; (b) θ = 0.5˚; (c) θ = 1˚. 0G
Figure 6. Curve of blood flow pressure at symmetrical centerline of the vessel with different tapered angles (ΔP =
199.98 Pa, and G = 0, supine posture, θ is the tapered angle): (a) θ = 0˚; (b) θ = 0.5˚; (c) θ = 1˚. 0G
the distribution and value of blood flow pressure be-
tween the obtained results and the real physiological
situation. Therefore, considering the combined action of
gravity and tapered angles will make the results in this
study more close to the real physiology.
However, there are still some important factors not
considered in the current study and could be added to
improve the results: a) non-Newtonian flow properties
and turbulent property of blood; b) patient-specific data;
c) effect of vessel wall. Simple and ideal geometrical and
physical model aids in reducing the computational cost
and obtaining the results rapidly. For the present study,
this assumption would not alter the total trend of DBFP
with gravity and tapered angles, so it is easier to get the
pure results about the effect of different tapered angles
with gravity considered or not, which excludes the other
action.
4. CONCLUSION
In conclusion, whether gravity is considered or not, the
tapered effect is one of the important factors leading to
nonlinearity in such hemodynamic parameters as blood
flow pressure. The tapered angles, together with gravity,
affected the magnitude of the blood flow pressure. Grav-
ity, however, independently influenced the dimension of
the DBFP, which had nothing to do with the tapered an-
gles.
5. ACKNOWLEDGEMENTS
The authors would like to thank Prof. LIU Da’an of Institute of Geol-
-ogy and Geophysics, Chinese Academy of Sciences for providing us
with the computational tool freely. We also thank Prof. Yang C. and Dr.
Bai Rong for their helpful advice.
Also, this paper was supported by National Natural Scientific Foun-

W. Y. Mu et al. / J. Biomedical Science and Engineering 6 (2013) 14-20 20
dation of China (No.81227001) and the National Science and Technol-
ogy Support Program (No.2012BAI14B04).
REFERENCES
[1] Grigoriev, A.I., Kotovskaya, A.R. and Fomina, G.A.
(2011) The human cardiovascular system during space
flight. Acta Astronautica, 68, 1495-1500.
http://dx.doi.org/10.1016/j.actaastro.2009.11.013
[2] Hughson, R.L., Yamamoto, Y., Maillet, A., Fortrat, J.O.,
Traon, A., Butler, G.C., Giiellr, A. and Gharib, C. (1994)
Altered autonomic regulation of cardiac function during
head-up tilt after 28-day head-down bed-rest with counter-
measures. Clinical Physiology, 14, 291-304.
http://dx.doi.org/10.1111/j.1475-097X.1994.tb00386.x
[3] Arbeille, P., Sigaudo, D., LeTraon, A.P., Herault, S.,
Porcher, M. and Gharib, C. (1998) Femoral to cerebral
arterial blood flow redistribution and femoral vein disten-
sion during orthostatic tests after 4 days in the head-down
tilt position or confinement. European Journal of Applied
Physiology and Occupational Physiology, 78, 208-218.
http://dx.doi.org/10.1007/s004210050409
[4] Hamilton, D.R., Sargsyan, A.E., Garcia, K., Ebert, D.J.,
Whitson, P.A., Feiveson, A.H., et al. (2012) Cardiac and
vascular responses to thigh cuffs and respiratory maneu-
vers on crewmembers of the International Space Station.
Journal of Applied Physiology, 112, 454-462.
http://dx.doi.org/10.1152/japplphysiol.00557.2011
[5] Hall, P. (1974) Unsteady viscous flow in a pipe of slowly
varying cross-section. Journal of Fluid Mechanics, 64,
209-226. http://dx.doi.org/10.1017/S0022112074002369
[6] Bakırtaş, I. and Demiray, H. (2005) Weakly non-linear
waves in a tapered elastic tube filled with an inviscid
fluid. International Journal of Non-Linear Mechanics, 40,
785-793.
http://dx.doi.org/10.1016/j.ijnonlinmec.2004.03.003
[7] Sankar, D.S. and Hemalatha, K. (2007) Non-linear ma-
thematical models for blood flow through tapered tubes.
Applied Mathematics and Computation, 188, 567-582.
http://dx.doi.org/10.1016/j.amc.2006.10.013
[8] Mandal, P.K. (2005) An unsteady analysis of non-New-
tonian blood flow through tapered arteries with a stenosis.
International Journal of Non-Linear Mechanics, 40, 151-
164. http://dx.doi.org/10.1016/j.ijnonlinmec.2004.07.007
[9] Nadeem, S. and Akbar, N.S. (2010) Simulation of the
second grade fluid model for blood flow through a ta-
pered artery with a stenosis. Chinese Physics Letters, 27,
068701-1-068701-4.
http://dx.doi.org/10.1088/0256-307X/27/6/068701
[10] Nadeem, S. and Akbar, N.S. (2011) Power law fluid mo-
del for blood flow through a tapered artery with a stenosis.
Applied Mathematics and Computation, 217, 7108-7116.
http://dx.doi.org/10.1016/j.amc.2011.01.026
[11] Sundell, P.M. and Roach, M.R. (1998) The role of taper
on the distribution of atherosclerosis in the human infra-
renal aorta. Atherosclerosis, 139, 123-129.
http://dx.doi.org/10.1016/S0021-9150(98)00068-9
[12] Melchior, F.M., Srinivasan, R.S., Thullier, P.H. and Clere,
J.M. (1994) Simulation of cardiovascular response to
lower body negative pressure from 0 to 40 mmHg. Jour-
nal of Applied Physiology, 77, 630-640.
[13] Olufsen, M.S., Ottesen, J.T., Tran, H.T., Ellwein, L.M.
and Novak, V. (2005) Blood pressure and blood flow
variation during postural change from sitting to standing:
Model development and validation. Journal of Applied
Physics, 99, 1523-1537.
http://dx.doi.org/10.1152/japplphysiol.00177.2005
[14] Mu, W.Y., Yu, G., Zhuang, F.Y. and An, G.M. (2009)
Mathematical model of hemodynamics with posture.
Journal of Medical Biomechanics, 23, 441-446.
[15] Caro, C.G., Pedley, T.J., Schroter, R.C. and Seed, W.A.
(1978) “Blood” and “The systemic arteries”. In: The me-
chanics of the circulation, Oxford, New York, 4-30.
[16] Fung, Y.C. (c1981) The flow properties of blood. In: Bio-
mechanics: Mechanical properties of living tissues, Sprin-
ger-Verlag, New York, 62-98.
[17] Taylor, C.A. and Draney, M.T. (2004) Experimental and
computational methods in cardiovascular fluid mechanics.
Annual Review of Fluid Mechanics, 36, 197-231.
http://dx.doi.org/10.1146/annurev.fluid.36.050802.12194
4
[18] Kwak, D., Kiris, C. and Kim, C.S. (2005) Computational
challenges of viscous incompressible flows. Computers &
Fluids, 34, 283-299.
http://dx.doi.org/10.1016/j.compfluid.2004.05.008
[19] Matthys, K.S., Alastruey, J., Peiró J., Khir A.W., Segers
P., Verdonck P.R., Parker, K.H. and Sherwin, S.J. (2007)
Pulse wave propagation in a model human arterial net-
work: Assessment of 1-D numerical simulations against
in vitro measurements. Journal of Biomechanics, 40,
3476-3486.
http://dx.doi.org/10.1016/j.jbiomech.2007.05.027
[20] Kaazempur-Mofrad, M.R., Bathe, M., Karcher, H.,
Younis, H.F., Seong, H.C., Shim, E.B., et al. (2003) Role
of simulation in understanding biological systems. Com-
puters & Structures, 81, 715-726.
http://dx.doi.org/10.1016/S0045-7949(02)00481-9
[21] Tang, D., Yang, C., Kobayashi, S., Zheng, J. and Vito,
R.P. (2003) Effect of stenosis asymmetry on blood flow
and artery compression: a three-dimensional fluid-struc-
ture interaction model. Annals of Biomedical Engineering,
31, 1182-1193. http://dx.doi.org/10.1114/1.1615577
[22] Liu, G.T., Wang, X.J., Ai, B.Q. and Liu, L.G. (2004)
Numerical study of pulsating flow through a tapered ar-
tery with stenosis. Chinese Journal of Physics, 42, 401-
409.