15th International Conference on Experimental Mechanics
PAPER REF: 2693
PULSATILE BLOOD FLOW IN HUMAN CAROTID BIFURCATION
Luísa C. Sousa *1,2, Catarina F. Castro1,2, Carlos C. António1,2, Rui Chaves2, Rosa Santos3,4, Pedro Castro3,4
and Elsa Azevedo3,4
1
Department of Mechanical Engineering, FEUP, University of Porto, Porto, Portugal
2
Institute of Mechanical Engineering (IDMEC), University of Porto, Porto, Portugal
3
Hospital São João, Porto, Portugal
4
Faculty of Medicine (FMUP), University of Porto, Porto, Portugal
(*)
Email: [email protected]
ABSTRACT
The purpose of this paper is to present a numerical simulation system for blood flow in the
carotid artery bifurcation and its validation using experimental data collected in clinical
practice. Blood flow is described by the incompressible Navier-Stokes equations and the
simulation is carried out under pulsatile conditions using inlet velocities given by a flow wave
form obtained by ultrasound measurements.
Keywords: carotid artery bifurcation, computational fluid dynamics, finite element method,
image-based analysis
INTRODUCTION
The accumulation of plaques on an artery wall is a progressive disease facilitated by local
irregular flow fields. Although clinical symptoms of this disease become evident when the
stenosis percentage is around 70%, modifications of the flow characteristics such as
separation and flow-reversal zones, occur at milder degrees of stenosis. It is well established
that once a mild stenosis is formed in the artery, altered blood flow and stress distribution in
the arterial wall contribute to further progression of the disease.
Recent non-invasive medical imaging data acquisition made feasible to construct three
dimensional models of blood vessels (Schumann et al., 2008). Colour Doppler ultrasound is
inexpensive, widely accessible, fast and safe, and provides real-time images of endovascular
structure; also measuring techniques have improved to provide accurate information on the
flow fields. Validated computational fluid dynamics models using data obtained by these
currently available measurement techniques can be very valuable in the early detection of
vessels at risk and prediction of future disease progression.
The objective of this paper is to present a three dimensional numerical simulation system for
the study of arterial blood flow under pulsatile conditions by the Finite Element Method
(FEM) (Sousa et al., 2011) and compare numerical results with experimental ones obtained
from Doppler ultrasound blood flow measurements in patients. The developed software is
applied to the study of a carotid arterial bifurcation.
CAROTID ARTERY BIFURCATION DATA
Extra-cranial carotid arteries of symptomatic patients were scanned by medical experts using
a digital ultrasound acquisition system, consisting of a standard commercial ultrasound
scanner (General Electric vivid e). Then, a developed semi-automatic procedure determines
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the 3D geometry of the carotid arterial bifurcation to be reconstructed from 2D longitudinal
images and complemented by a serial of 2D transverse images. Since vessel geometry
determines important features of the flow it is important to accomplish the ability to achieve
anatomically correct geometries. The arterial geometry was based on in vivo image data with
minimum volume at diastolic pressure according to the approved protocol. Details of the 3D
geometry reconstruction procedures have been reported elsewhere (Chaves et al., 2012). In
summary, three main steps were involved in developing the carotid bifurcation artery model:
acquiring the in vivo anatomical data via ultrasound imaging, image surface reconstruction
and 3D Finite element mesh definition.
According to the protocol, flow velocity was measured at different central locations in the
common carotid and also at internal and external carotid arteries. Ideally instead of centreline
velocity, velocity profiles should be acquired and used as boundary conditions for the flow
simulations. However, in clinical studies, the prolonged scan time of MR velocity
measurement could be rather prohibitive whereas the well subject tolerated and easy access of
Doppler ultrasound render it a more practical choice.
NUMERICAL SIMULATION MODEL
Navier-Stokes equations describe the mechanics of fluid flow. They state the dynamical
balance between the internal forces due to pressure and viscosity of the fluid and the
externally applied forces. Considering isothermal conditions the time dependent
incompressible blood flow is governed by the momentum and mass conservation equations,
the Navier-Stokes equations given as:
 ∂u

+ u.∇u  = ∇σ + f
 ∂t

∇.u = 0
ρ
(1)
where u and σ are the velocity and the stress fields, ρ the blood density and f the volume
force per unit mass of fluid.
The equation system Eq. (1) can be solved for the velocity and the pressure given appropriate
boundary and initial conditions. In this study the biochemical and mechanical interactions
between blood and vascular tissue are neglected. In an elderly person the walls of the arteries
thicken, loose their elasticity, become stiffer and the effects of wall compliance can be
neglected. Then no slip at the interface with the rigid vessel wall can be assumed. At the flow
entrance Dirichelet boundary conditions are considered prescribing a parabolic distribution
for the time dependent value u D of the velocity on the portion ΓD of the boundary
u ( x, t ) = u D ( x, t ) , x ∈ ΓD .
At an outflow boundary ΓN the condition describing surface traction force h is assumed. This
can be described mathematically by the condition:
−δ + +
= ℎ , = 1,2,3onΓ (2)
where n j are the components of the outward pointing unit vector at the outflow boundary.
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Editors: J.F. Silva Gomes and Mário A.P. Vaz
15th International Conference on Experimental Mechanics
FINITE ELEMENT METHOD (FEM)
A finite element approach for the solution of the time-dependent incompressible NavierStokes equations is presented. For a 3D analysis hexahedral meshes often provide the best
quality solution minimizing errors due to numerical diffusion whenever a good alignment
between mesh edges and flow exists (De Santis et al., 2010). In this work a spatial
discretization with isoparametric brick elements of low order with trilinear approximation for
the velocity components and element constant pressure is adopted.
The numerical procedure for the discretization in space uses the Galerkin-finite element
interpolation. The discretization in time is obtained by an implicit fractional step-method in
which the time advancement is decomposed into a sequence of two steps (Blasco et al., 1998;
Quarteroni et al., 2003):
•
Calculation of an auxiliary velocity field un +1 / 2 from the equations of motion, which include
the viscous and convective effects; known pressure values from the previous time step n or
previous iteration step m are used;
•
Calculation of the end-of-step velocity field un +1 and pressure p n +1 by the solution of the
Stokes problem using lumped mass matrix. The method includes a diffusion term in the
incompressibility step, which allows the imposition of the full boundary conditions for the
velocity while needing no pressure boundary conditions (Blasco et al., 1998).
Due to high Reynolds numbers, the treatment of the convective term needs a numerical
scheme with a stabilization technique like the Streamline upwind/Petrov Galerkin method,
(SUPG-method) in order to avoid oscillations in the numerical solution (Hughes et al., 1986;
Hughes et al., 1989). The SUPG-method produces a substantial increase in accuracy as
stabilizing artificial diffusivity is added only in the direction of the streamlines and crosswind
diffusion effects are avoided.
RESULTS
Doppler ultrasound measurements of the left carotid arteries of one particular patient were
obtained at the hospital practice. The medical doctor was able to make a rough outline of the
intima-media region boundaries in the B-mode image shown in Fig. 1. The reconstructed
geometry and meshing of the carotid bifurcation based on in vivo image data with minimum
volume at diastolic pressure according to the approved protocol is shown in Fig. 1. The flow
wave form for one cardiac cycle has been calculated from ultrasound measurements at the
distal common carotid artery section (DCCA), at 1.5 cm before bifurcation. Numerical
simulation is performed considering pulsatile blood flow with time dependent parabolic inlet
velocity given by this wave form.
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Fig.1 Doppler image and finite element mesh of the carotid bifurcation
Convergence has been carefully tested. Refinement of both temporal and spatial resolutions
was performed until changes in predicted velocities and wall shear stress became negligible.
A mesh of 40 thousand tetrahedrons with trilinear approximation for the velocity components
and element constant pressure is adopted.
Velocity results are presented and compared with experimental data flow wave forms at three
locations. In Figs. 2-4, the measured and calculated flow wave forms, during one cardiac
cycle are shown at the common carotid artery (APPEX), PICA and PECA. APEX represents
the bifurcation entrance; the proximal external carotid artery (PECA) corresponds to the most
proximal point at the external carotid artery without flow disturbance from bifurcation (1.68
cm from DCCA); PICA, the proximal internal carotid artery is situated 1.65 cm after DCCA,
at the internal carotid artery.
60
v (cm/s)
40
20
0
0,0
0,3
0,5
t/tp
0,8
1,0
Fig.2 Obtained flow wave form and measured diastolic velocity at the APPEX
60
v (cm/s)
40
20
0
0,0
0,3
0,5
t/tp
0,8
1,0
Fig.3 Obtained flow wave form and measured diastolic velocity at PICA
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Editors: J.F. Silva Gomes and Mário A.P. Vaz
15th International Conference on Experimental Mechanics
This study shows the accuracy and efficiency of the carotid bifurcation blood flow simulation
method. At all three positions in the carotid bifurcation there is a good agreement in shape
between the obtained flow wave forms and those obtained experimentally by Doppler
ultrasound measurements in clinical practice. At all three positions the direct influence of the
inflow pulse, which reflects the mechanical heart action, is largely preserved.
60
v (cm/s)
40
20
0
0,0
0,3
0,5
t/tp
0,8
1,0
Fig.4 Obtained flow wave form and measured diastolic velocity at PECA
Fig.5 Velocity field in the symmetric plane of the carotid bifurcation t = 0.37s
Velocity field in the symmetric plane of the carotid bifurcation is shown in Fig. 5 for the
cardiac cycle instant t = 0.37. It can be observed a strongly skewed axial velocity in the
carotid sinus with high velocity gradients at the internal divider wall. Near the outer sinus
wall (the wall opposite the divider wall) stagnation corresponding to a recirculation region is
detected by the developed code. At the external carotid high velocity gradients are also
detected, due to the narrow section.
CONCLUSIONS
A computational finite element model for simulating blood flow in arteries is presented.
Blood flow is described by the incompressible Navier-Stokes equations and the simulation is
carried out under steady and pulsatile conditions. The accuracy and efficiency of the blood
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simulation is validated comparing with experimental data obtained by in vivo measurements.
Velocity fields and flow wave forms are in good agreement with those obtained by ultrasound
measurements.
The presented research is helpful for preventing cerebral vascular accidents once the
developed code is able to identify stagnation and reversal flow regions in vessels with
complex geometry like in stenosed arteries or carotid bifurcation.
ACKNOWLEDGMENTS This work was partially done in the scope of project PTDC/SAUBEB/102547/2008, “Blood flow simulation in arterial networks towards application at
hospital”, financially supported by FCT – Fundação para a Ciência e a Tecnologia from
Portugal.
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