A Patient-Speciﬁc Three-Dimensional Hemodynamic Model
of the Circle of Willis
, and A
Department of Mechanical Engineering, K. N. Toosi University of Technology, 15 Pardis St., Tehran 1999143344, Iran
(Received 16 January 2017; accepted 4 September 2017; published online 14 September 2017)
Associate Editor Stephen B. Knisley oversaw the review of this article.
Abstract—Circle of Willis (CoW) is one of the most
important cerebral arteries in the human body and various
attempts have been made to study the hemodynamic of blood
ﬂow in this vital part of the brain. In the present study, blood
ﬂow in a patient speciﬁc CoW is numerically modeled to
predict disease-prone regions of the CoW. Medical images
and computer aided design software are used to construct a
realistic three-dimensional model of the CoW for this
particular case. The arteries are considered as elastic conduits
and the interactions between arterial walls and the blood ﬂow
are taken into account. Mooney-Rivlin hyperelastic model is
used to describe the behavior of arterial walls and blood is
considered as a non-Newtonian ﬂuid obeying the Carreau
model. An available experimental-based pulsatile velocity
proﬁle is used at the entrance of the CoW. The ﬁnite element-
based commercial software, ADINA, is used to solve the
governing equations. Blood pressure and velocity and arterial
wall shear stress are calculated in different regions of the
CoW. A simpliﬁed form of the model is also compared with
the available published data. Results afﬁrmed that the
proposed computational model has the potential to capture
the hemodynamic characteristics of the CoW. The compu-
tational results can be used to determine disease-prone
locations for a given CoW.
Keywords—Circle of Willis, Hemodynamic, Pulsatile ﬂow,
Patient-speciﬁc model, Finite element method.
Blood ﬂows throughout the body, feeds the cells and
removes the wastes and disposables. It has been known
from a long time ago that a healthy blood ﬂow in the
body is vitally important. However, attempts to
understand and explain the blood circulation in a
systematic way are not that old. Some noticeable
works regarding the blood circulation return back to
the seventeenth century. In 1628, Harvey discovered
the blood pumping mechanism and the important role
of the heart.
Later, around 1735, Stephen Hale mea-
sured the ﬂuctuating blood pressure and noticed the
elastic properties of the arteries.
The famous Euler
equations were developed around 1775 to explain the
hemodynamic in the arteries.
More recently and during the past few decades the
progress in numerical methods and computer software
and hardware has paved the way for the simulation of
the blood ﬂow. Currently, powerful commercial com-
putational ﬂuid dynamics (CFD) tools are available
and can be used to obtain a rather realistic picture of
the details of the blood ﬂow in many important parts
of the body.
The interest in scientiﬁc explorations in
the general area of blood ﬂow has been boosted up due
to the sheer fact that the modern life styles seem to
somehow enhance the blood-related diseases. It has
been reported that the cardiovascular disease (CVD) is
the leading cause of death in the United States and is
responsible for 17% of national health expenditures.
Total direct medical costs of CVD are projected to
triple from $273 billion to $818 billion Between 2010
and 2030. Table 1 provides some relevant information
in this regard.
Detailed knowledge of the blood ﬂow is very
important in detecting arterial diseases. By including
realistic features of the blood ﬂow in the simulations, it
is expected that abnormalities and risks associated with
the cardiovascular system are better predicted. Such
knowledge and studies have many applications in
surgical planning and design of medical devices as
To simulate the blood ﬂow, 0D, 1D, 2D and 3D
models have been proposed.
models, in general, are capable of capturing more de-
Address correspondence to Hamed Rezaie, Department of
Mechanical Engineering, K. N. Toosi University of Technology,
15 Pardis St., Tehran 1999143344, Iran. Electronic mail:
Cardiovascular Engineering and Technology, Vol. 8, No. 4, December 2017 (
2017) pp. 495–504
2017 Biomedical Engineering Society