25
Part 14 Bio-Fluid Mechanics
Part 14 Bio-Fluid Mechanics
Flow Structures in Physiological Conduits
A.M. Gambaruto and A. Sequeira
Abstract The study of topological structures in the case of a peripheral planar by-
pass graft and a cerebral aneurysm are presented. The Taylor series expansion of the
velocity field to first order terms leads to a system of ODEs, the solution to which
locally describes the motion of the flow. If the expansion is performed on the wall
shear stress, critical points can be identified and the near-wall flow field parallel to
the wall concisely described. Furthermore the expansion can be expressed in terms
of relative motion and the near-wall convective transport normal and parallel to the
wall can be accurately derived on the no-slip domain.
1 Introduction
It is known that the haemodynamics in arteries is linked to disease formation such
as atheroma and aneurysms. While the relationship between the flow field and dis-
ease are not fully understood, fluid mechanics parameters on and near the artery
wall, such as wall shear stress and derived measures, are among the most commonly
sought correlators to disease. Furthermore the non-planarity and tortuousity of ves-
sels play a determining role in the arterial system, resulting in a strong influence
of the local vessel topology on the flow field. In this work the flow structures for a
peripheral bypass graft and a cerebral aneurysm, shown in Figure 1, are studied for
steady state simulations.
In specific, the proposed methods are based on using existing theory of critical
points [1, 2] as well as novel methods to look at the near-wall flow that is important
A.M. GambarutoDepartment of Mathematics and CEMAT, Instituto Superior Tecnico, Lisbon, Portugal, e-mail:[email protected]
A. SequeiraDepartment of Mathematics and CEMAT, Instituto Superior Tecnico, Lisbon, Portugal e-mail:[email protected]
1
2 A.M. Gambaruto and A. Sequeira
(a) (b)
Fig. 1 (a) Cerebral vasculature and aneurysm (flow is from bottom to top) and (b) planar peripheralbypass graft (flow enters from top-left vessel and exits through top-right and bottom vessels).
in medical applications [3]. These will be used to observe the different topological
structures in a detailed manner, and indicate a means to accurately formulate the
near-wall transport from the no-slip domain.
2 Taylor series expansion of the velocity field
Performing a Tarlor expansion of the velocity ui, i = 1, . . . ,3, in terms of the spatial
coordinates xi we obtain
ui = xi = Ai +Ai jx j +Ai jkx jxk + . . . (1)
Let the coordinate system be such that the origin follows a fluid particle by trans-
lation and no rotation. In such a reference frame Ai = 0, while higher order terms
are finite, and the origin is a critical point. By truncating the series to keep only the
first-order term (the velocity gradient tensor), hence x = (∇u) ·x, or explicitly
⎛⎝
x1
x2
x3
⎞⎠=
⎛⎜⎝
∂ x1∂x1
∂ x1∂x2
∂ x1∂x3
∂ x2∂x1
∂ x2∂x2
∂ x2∂x3
∂ x3∂x1
∂ x3∂x2
∂ x3∂x3
⎞⎟⎠⎛⎝
x1
x2
x3
⎞⎠ (2)
From this set of three first-order ODEs the eigenvalues (λi) are obtained, which can
be either three real values or one real and a complex-conjugate pair, the sum of
which equals zero for incompressible fluids. The corresponding eigenvectors (ζi)
form planes to which the solution trajectories osculate, being either a node-saddle-
saddle arrangement if the eigenvalues are real or a focus-stretching if a complex-
Flow Structures in Physiological Conduits 3
R
Q
R +Q = 03__ 2
427
Stable-focus-stretching
Unstable-focus-compressing
Stable-node-saddle-saddle
Unstable-node-saddle-saddle
X
Y
Z
X
Y
Z
(a) (b) (c)
Fig. 2 (a) Solution trajectories can be either node-saddle-saddle or focus-stretching, as well aseither stable or unstable. The axes are the invariants of ∇u. (b) Sample trajectory and (c) lowerregion detail of trajectory in a vortical structure showing the plane of swirling and the axis ofstretching, given by the eigenvectors of the velocity gradient tensor. (It should be noted that forthis detail the foci are stable and λ1 > 0.)
conjugate pair exist. An example of the solution planes is shown in Figure 2 for a
passive particle path in a vortical structure.
The eigenvalues of ∇u satisfy the characteristic equation.
λ 3 +Pλ 2 +Qλ +R = 0 (3)
where the three invariants are given by
P =−tr[A]Q = 1
2 (P2 − tr[A2])
R =−det[A](4)
Furthermore, invariants of the rate-of-shear tensor and rate-of-rotation tensor are
obtained from the above equations by setting in turn symmetric and anti-symmetric
parts of the velocity gradient tensor to zero. The full set of invariants can be used to
describe the local, instantaneous dynamics of the flow. A cross section example is
shown in Figure 3, where the in plane particle paths are used to explain the flow dy-
namics. It is clear that the eigenvalues and invariants can yield insight into studying
and comparing the flow field locally in physiological situations.
If the wall shear stress magnitude and direction are projected onto the piecewise-
linear planar triangulated mesh that defines bounding geometry, then from a similar
analysis the critical points of the traction force can be described. These can be either
foci or saddle configurations. In doing so the near-wall flow parallel to the wall can
be described elegantly as shown in Figure 4.
4 A.M. Gambaruto and A. Sequeira
(a) (b)
(c) (d)
Fig. 3 Cross section of anastomosis with: (a) λ1; (b) |λ3/λ2| (to indicate the spiralling compactnessfor a focus configuration); (c) Q (the second invariant of the velocity gradient tensor); (d) velocitymagnitude (m s−1).
Fig. 4 Two views of the aneurysm surface with surface shear lines and locations of critial pointsof the wall shear stress, both foci and saddles. These are used to succinctly and clearly describe thenear-wall fluid motion parallel to the wall
3 Taylor series expansion of the relative position
In a similar manner, but performing the expansion on the relative position instead,
the near-wall convective transport is derived [3]. The leading terms relate the wall
shear stress as the components parallel to the wall while the normal component is
Flow Structures in Physiological Conduits 5
given by the sum of the spatial gradients of the wall shear stress, as shown in Figure
5. It is apparent that the surface shear lines also shown in Figure 5 indicate move-
ment towards or away by their separating or coalescing, respectively (for constant
wall shear stress). The wall shear stress, its spatial gradient sum [3] and its spatial
gradient magnitude are commonly correlated to disease initiation and progression.
(a) (b)
Fig. 5 (a) Graft with spatial wall shear stress gradient sum and surface shear lines; (b) aneurysmwith spatial wall shear stress gradient sum. Blue indicated flow to the wall while red indicates flowfrom the wall. A large motion of flow perpendicular to the wall is seen in the graft geometry whilelittle is seen for the aneurysm.
4 Conclusion and future work
The Taylor expansion, while performed locally, acts as a perturbation analysis and
is able to perceive the neighbouring flow field. As seen in [3] this permits measures
from the flow field core to be extrapolated from the no-slip domain, allowing for a
means to describe the flow field and hence the near-wall transport directly.
In this work we have shown that the local dynamics for both the core flow and
the near-wall flow (described using the traction force, hence wall shear stress) can
be studied using the Talyor series expansion of the velocity or relative position. This
leads to a local information of the flow field that can aid in discussing phenomena of
human physiology in normal and diseased states. Future work includes using higher
order terms to allow for a more detailed information of the flow field and a greater
range in the locality of the analysis.
6 A.M. Gambaruto and A. Sequeira
References
1. Chong, M.S., Perry, A.E.: A general classification of three-dimensional flow fields. Phys.Fluids A, 2(5):765–777 (1990)
2. Gambaruto, A.M., Moura, A., Sequeira, A.: Topological flow structures and stir mixing forsteady flow in a peripheral bypass graft with uncertainty. Int. J. Num. Meth. Biomed. Eng.,26(7): 926–953 (2010)
3. Gambaruto, A.M., Doorly, D.J., Yamaguchi, T.: Wall Shear Stress and Near-Wall ConvectiveTransport: Comparisons with Vascular Remodelling in a Peripheral Graft Anastomosis. J.Comp. Phys., 229(14):5339–5356 (2010)
Fluid mechanics in aortic prostheses after aBentall procedure
M.D. de Tullio∗, L. Afferrante∗, M. Napolitano∗, G. Pascazio∗ and R. Verzicco+
∗ CEMeC & DIMeG, Politecnico di Bari, Via Re David 200, 70125 Bari, Italy+DIM, Universita di Roma Tor Vergata, Via del Politecnico 1, 00133 Roma, Italy
E-mail: [email protected], [email protected], [email protected], [email protected]@uniroma2.it
Abstract The simultaneous replacement of a diseased aortic valve, aortic root and
ascending aorta with a composite graft equipped with a prosthetic valve is a nowa-
days standard surgical approach, known as the Bentall procedure: the Valsalva si-
nuses of the aortic root are sacrificed and the coronary arteries are reconnected
directly to the graft. In practice, two different composite–material prostheses are
largely used by surgeons: a standard straight graft and the Valsalva graft with a
bulged portion that better reproduces the aortic root anatomy. The aim of the present
investigation is to study the effect of the graft geometry on the the flowfield as well
as on the stress concentration at the level of coronary–root anastomoses during the
cardiac cycle. An accurate three-dimensional numerical method, based on the im-
mersed boundary technique, is proposed to study the flow inside moving and de-
formable geometries. Direct numerical simulations of the flow inside the two pros-
theses, equipped with a bileaflet mechanical valve with curved leaflets, under phys-
iological pulsatile inflow conditions show that, when using the Valsalva graft, the
stress level near the coronary–root anastomoses is about half that obtained using the
standard straight graft.
1 Geometries and materials
Two prostheses can be used to perform the Bentall procedure [2]. One is the stan-
dard straight graft, with a constant orientation of the textile for the whole length. The
other one is the Valsalva graft that exhibits three main portions, namely, the collar,
the skirt and the body: while the collar and the body have the same properties of the
straight tube prosthesis, the skirt is created by taking a section of graft and sewing
it to the body with crimps in the vertical rather than in the horizontal direction. Fig-
ure 1 shows the geometrical model used in the numerical simulations. The model is
discretized by triangular elements, as requested by the ray–tracing technique used in
the geometrical preprocessor [6]. The inflow tube and the valve housing are consid-
1
2 M.D. de Tullio et al.
(a)
Fig. 1 Model of the graft.
ered to be rigid, whereas the skirt, the body and the coronaries are discretized by 0.3
mm-thick linear elastic shell elements having large deflection capability. The same
geometrical model is considered for both grafts, while the different behavior of the
two ducts in the skirt region is taken into account by a different material model. The
valve considered is a bileaflet 25 mm Bicarbon mechanical valve by Sorin Biomed-
ica. The leaflets have a curved profile and exhibit a rotation range equal to 60◦, with
a fully open position of 10◦ with respect to the streamwise direction.
As shown by Lee and Wilson [7], woven Dacron grafts can be characterized by
orthotropic material properties, the stress–strain relationship being correctly mod-
eled with a linear model in both the longitudinal and the circumferential directions.
Here, the Dacron graft is considered as transversely isotropic, with the following
elastic constants (z is the streamwise direction): Young’s moduli Ex = Ey = 12MPa,
Ez = 1.2MPa; Poisson’s ratios νxz = νyz = 0.15, νxy = 0.1; Shear moduli Gxz =Gyz = 5.2MPa, Gxy = 0.55MPa. It is noteworthy that for the Valsalva graft the or-
thotropic directions of the material are inverted at the skirt region, in order to take
into account the different behavior of the prosthesis. A linear elastic material with
Young’s modulus equal to 2 MPa is used for the two coronaries. The leaflets are
made of pyrolytic carbon, with density of ρl = 2000 kg/m3, and are considered to
be rigid.
2 Numerical method and simulation details
At every point of the time–dependent fluid domain, the Navier–Stokes equations for
an incompressible viscous Newtonian fluid are solved, coupled with the ordinary
differential equations governing the motions of the two leaflets. The fluid and the
rigid leaflets are treated as elements of a single dynamical system, all governing
equations being integrated simultaneously in the time–domain by a strong coupling
scheme. Details on the basic fluid solver are given in Verzicco and Orlandi [9] and
Fluid mechanics in aortic prostheses after a Bentall procedure 3
Fadlun et al [3], whereas the complete method and several numerical validations can
be found in de Tullio et al [8]. Then, the Navier equations governing the dynam-
ics of the deformable solid region are solved using the finite-element commercial
software ANSYSR©
[1]. Such a weak-coupling is employed in order to reduce the
computational cost of the overall procedure and to use optimized solvers for both
the fluid and the elastic–body problems.
In more details, at each time step, the following procedure is employed, where
n and n + 1 indicate the old and new time levels, i = 1,2 indicates the leaflets,
subscripts f and s indicate fluid and structure quantities respectively:
1. iteratively solve the system of the fluid and leaflets’ equations using the positions
and velocities of the structural nodes xns and un
s as boundary conditions for the
fluid domain, finding the fluid velocities and pressures, un+1f , pn+1
f , as well as the
leaflets angular positions and velocities, θ n+1i and θ n+1
i ;
2. find the loads exerted on the structure by the fluid Φn+1f =Φn+1
f (xns ,u
n+1f , pn+1
f ,θ n+1i
and θ n+1i );
3. solve the structural equations with the computed loads, Φn+1s = Φn+1
f , to obtain
the new positions and velocities of the structural nodes, xn+1s and un+1
s , to be used
as boundary conditions for the successive time step.
In this way, the solution of the structural solver is needed only once per time step.
Stability is ensured by the very small time steps required by the flow solver to cap-
ture the time-history of the smallest turbulent scales. The coupling of the flow solver
with the deformable structure has been validated considering a non-constrained pipe
conveying a stationary flow: the present results, non reported here for brevity, agree
well with the analytical predictions of the dimensionless pipe radius variation with
dimensionless streamwise position at different Reynolds numbers, reported in [4].
3 Results and discussion
Typical physiological conditions for an adult male are considered. The cycle dura-
tion is set at 866 ms, corresponding to about 70 beats/min. The mean flow rate was
adjusted to about 5 l/min with a peak flow rate of about 28 l/min. The blood kine-
matic viscosity and density are set to ν = 3.04×10−6 m2/s and ρb = 1060 kg/m3,
respectively. The peak Reynolds number, based upon the bulk velocity at the
peak inflow, U = 0.95 m/s, and the inflow tube diameter is about 7800. After a
grid convergence study, a background cylindrical structured grid is used, having
217×165×250 nodes in the azimuthal, radial and axial directions respectively. For
the fluid domain inflow section, the pressure and velocity profiles are imposed in
order to mimic the physiological conditions produced by the heart in the left ventri-
cle, as shown in Fig. 2a. Concerning the deformable structure, it is fully constrained
at the lowest nodes (where the mechanical valve is fixed to the prosthesis), while a
longitudinal displacement is applied to the highest nodes to simulate the longitudi-
4 M.D. de Tullio et al.
time [ms]
aort
icflo
wra
te[l
min
-1]
aort
icpr
essu
re[k
Pa]
0 200 400 600 800-5
0
5
10
15
20
25
30
8
10
12
14
16
18
(a)
time [ms]
Λ
aort
icflo
wra
te[l
min
-1]
0 200 400 600 800-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-5
0
5
10
15
20
25
30
valsalvastraight tube
(b)
Fig. 2 (a) Inflow aortic flowrate and pressure. (b) Time variation of the phase-averaged leaflets’
angular position Λ =(αopen−α)
(αopen−αclosed ). The aortic flow rate is also reported for reference.
nal stresses reported in the natural aorta, according to the results of Han and Fung
[5]. Here, the maximum displacement of about 5 mm is modulated in time by the
pressure curve depicted in Fig. 2a. Five complete cardiac cycles are considered.
The phase-averaged angular displacements of the two leaflets during the cardiac
cycle are shown in Fig. 2b for both prostheses: very little sensitivity to the aortic
root geometry is observed. The leaflet dynamics is greatly influenced by the pres-
sure gradient through the valve, and this is evident during the opening phase, where
the flow is accelerated. Small differences are noticeable during the closing phase,
where the deceleration of the flow induces high turbulence levels, and therefore the
forces on the leaflets are influenced by the fluctuations in time of the pressure and
velocity fields. The out-of-plane vorticity contours in the symmetry plane are shown
in Fig. 3, at three significant times. The typical configuration of the bileaflet valves,
forming three jets, with strong shear layers shed from the valve housing and the tips
of the leaflets is noticeable. At the peak flow rate, the shear layers become unstable
and a strong small-scale turbulence production is observed in the wake and in the si-
nuses region. The Valsalva prosthesis exhibits an axisymmetric recirculation region,
due to the sudden expansion flow past the valve, larger than that occurring for the
case of the straight prosthesis and more similar to that occurring in the physiolog-
ical situation. During the decelerating phase, the flow becomes turbulent with high
production of small scale structures downstream the valve. The leaflets start closing
as soon as the pressure starts decreasing and quickly reach a complete closure: the
flow stops and viscosity dissipates the small scale structures until the beginning of
the new cycle. Fig. 4a provides the time histories of the Von Mises stresses near
the coronary–root anastomoses for both the prostheses, while contour maps of the
Von Mises stress distributions are plotted in Fig. 4b, at the minimum and maximum
pressures registered during the cardiac cycle. The maps clearly show that the stress
values, in the region corresponding to the sinuses, are smaller for the Valsalva pros-
thesis than for the straight one. In particular, the maximum stress for the Valsalva
Fluid mechanics in aortic prostheses after a Bentall procedure 5
(a) (b) (c)
(d) (e) (f)
Fig. 3 Out of plane vorticity in the symmetry plane for the Valsalva (top) and straight (bottom)grafts: (a),(d) end opening of the leaflets; (b),(e) flowrate peak; (c),(f) start closure of the leaflets.
time [ms]
σ eqv
0 200 400 600 8002.00E+05
4.00E+05
6.00E+05
8.00E+05
1.00E+06
1.20E+06
1.40E+06
valsalvastraight tube
(a) (b)
Fig. 4 (a) Von Mises stress versus time in a region near the coronary–root anastomosis. (b) VonMises stress contours at minimum and maximum levels of pressure during a cardiac cycle.
graft is about one half that for the straight one. The highest stress values are ob-
tained for the Valsalva graft near the sinotubular junctions but they raise no concern
because, in contrast with the hand–sewn coronary–root anastomoses, this zone is an
all-Dacron component, reinforced by machine suturing during manufacturing, and
thus it is not prone to rupture.
6 M.D. de Tullio et al.
4 Conclusions
The numerical tool developed by the authors provides a very detailed description of
the unsteady flowfield inside two deformable Dacron prostheses carrying a bileaflet
mechanical heart valve, under physiological pulsatile conditions. The code allows
one to integrate the loads and thus to evaluate the stresses during the cardiac cycle.
Results show that the dynamics of the leaflets is influenced only slightly by the
prosthesis geometry, whereas near the coronary–root anastomoses the stress level
for the Valsalva prosthesis having a bulged portion is about half that obtained using
the standard straight tube. Therefore, the Valsalva graft could reduce complications
like bleeding and pseudo–aneurysm formation, more likely to occur when using the
straight one. This is the main and very interesting result of this work.
Acknowledgements This research was funded by by MIUR and Politecnico di Bari under contractCofinLab 2001.
References
[1] ANSYSR©
(2009) ANSYS, Inc. http://www.ansys.com
[2] Bentall H, De Bono A (1968) A technique for complete replacement of the
ascending aorta. Thorax 23:338–339
[3] Fadlun EA, Verzicco R, Orlandi P, Mohd-Yosuf J (2000) Combined immersed-
boundary finite-difference methods for three-dimensional complex flow simu-
lations. J Comput Phys 161:35
[4] Fung Y (1984) Biodynamics. Circulation. Springer-Verlag, New York Berlin
Heidelberg Tokyo
[5] Han H, Fung Y (1995) Longitudinal strain of canine and porcine aortas. J of
Biomechanics 28:637–641
[6] Iaccarino G, Verzicco R (2003) Immersed boundary technique for turbulent flow
simulations. Appl Mech Rev 56:331
[7] Lee J, Wilson G (1986) Anisotropic tensile viscoelastic properties of vascular
graft materials tested at low strain rates. Biomaterials 7:423–431
[8] de Tullio M, Cristallo A, Balaras E, Verzicco R (2009) Direct numerical sim-
ulation of the pulsatile flow through an aortic bileaflet mechanical heart valve.
Journal of Fluid Mechanics 622:259–290
[9] Verzicco R, Orlandi P (1996) A finite difference scheme for threedimen-
sional incompressible flows in cylindrical coordinates. Journal of Computa-
tional Physics 123:402–413
Optimisation of Stents for Cerebral Aneurysm
C.J. Lee1, S. Townsend 2 and K. Srinivas3
Abstract Stents are used to effect a flow diversion in an aneurysm to reduce the risk
of its rupture. Following the technique of exploration of design space, the present
work attempts to optimize the design of stents. In this study both two-dimensional
simplification of stents and the three-dimensional ones are considered. Optimiza-
tion determines the most effective arrangement of struts and gaps within the stents.
Velocity and vorticity reduction within the aneurysm form the objective functions.
Optimisation is performed by considering random design of stents and arriving at
what are called Non-Dominated solutions. The designer identifies the one that suites
the requirements best. It is shown that for best flow diversion, a strut at the proximal
end is a necessity.
1 Introduction
Aneurysms are caused by pathological dilation of the arterial wall. They can grow
in size, which can cause internal hemorrhage leading to stroke or death [1]. To avoid
rupture, stents are used to cause a flow diversion, i.e., reduce flow activity within. A
number of experimental and computational studies of the flow in a stented aneurysm
are available in the literature. In particular, detailed studies on a two-dimensional
simplified model of stented aneurysms carried out by Hirabayahsi et. al. [2] show
C.J. LeeSchool of Aerospace, Mechanical and Mechatronic Engineering, University of Sydney, NSW 2006,Australia e-mail: [email protected]
S. TownsendSchool of Aerospace, Mechanical and Mechatronic Engineering, University of Sydney, NSW 2006,Australia e-mail: [email protected]
K. SrinivasSchool of Aerospace, Mechanical and Mechatronic Engineering, University of Sydney, NSW 2006,Australia e-mail: [email protected]
1
2 C.J. Lee1, S. Townsend 2 and K. Srinivas3
that the position of gaps and struts, and their size has a considerable effect on the
flow diversion caused. One of our objectives in the present research is to take the
next step, and optimize the design of the stents for an effective flow diversion. This
is similar to the work reported by Srinivas et. al. [3], which considered optimization
of stents for coronary arteries. In this work we present two-dimensional results for
aneurysms and some preliminary results for three-dimensional case.
2 Methodology
Optimisation follows the Exploration of Design Space [3,4,5] approach. To start
with a large number of samples called individuals are selected within the range of
design variables using the Latin Hypercube technique [5]. Objective functions are
computed for each of these samples. An optimization procedure based on Kriging
[5] is then carried out. What results will be a set of optimum or non-dominated
solutions. The designer selects the one that suites best for a given situation.
2.1 Stents considered
Both two- and three-dimensional aneurysm geometries are considered in this study,
as shown in Fig. 1.
For a two-dimensional simplification, the stent placed at the neck of the aneurysm
cavity appears as an arrangement of struts and gaps, whereas for a three-dimensional
model the stent is created using Pro Engineer to be a close resemblance to a real
stent.
Many different cases of two-dimensional stents were considered in the study;
only a typical one is presented here. This consists of five struts of uniform width,
each being 0.4 mm with varying gap sizes. The porosity in this case is 80% while
the thickness of each strut is 0.1 mm.
Fig. 1 Geometry of the aneurysm considered: 2D model (left) and 3D model (right)
Optimisation of Stents for Cerebral Aneurysm 3
The other case presented is a three-dimensional one. At the time of writing this
paper, only one three-dimensional case has been completed. Similar to the two-
dimensional case there are five struts of uniform width and thickness, 0.4 mm and
0.1 mm, respectively. Only the gap sizes are varied, with the mid-cross section
porosity fixed to 80%. A typical three-dimensional stent model is shown in Figure
2. The results for this case will be presented as Case 2 in the Results section.
Fig. 2 A typical three-dimensional stent modelcreated using Pro-Engineer
2.2 Objective Functions
As mentioned before, the aim of stent design is to reduce the flow velocity within
the cavity. In addition, vorticity is a good indicator of viscous activity. Hence, the
following objective functions are chosen.
1. Velocity Reduction in the cavity, expressed as a percentage, ΔV = (V NS−V S)VNS
×100, where V NS, and V S, are the area weighted average of velocity magnitude
within the cavity without the stent and in presence of the stent, respectively.
2. Vorticity Reduction in the cavity, expressed as a percentage, Δω =∫
d ωNS−∫
d ωS∫d ωNS
×100, where ω is vorticity, NS and S, represent within the cavity without the stent
and in the presence of the stent, respectively, and d denotes the interior of the
cavity.
Both of the above objective functions are to be maximized.
2.3 Computational Details
All aneurysm and stent models were modeled and meshed using GAMBIT 2.3, with
approximately 100,000 grid nodes within the region and approximately 350 points
specified on the cavity surface, using triangular meshing scheme.
The fluid considered is blood with a density of 1060 kg/m3, a viscosity of 0.0035
kg/m·s, and the flow velocity at the entry is 0.3 m/s. The Reynolds Number based on
4 C.J. Lee1, S. Townsend 2 and K. Srinivas3
entry conditions is 363. The flow in the vessel is considered as steady and Newto-
nian. The governing equations are those for a steady, two-dimensional laminar flow.
The boundary conditions applied are the no flow conditions on all solid boundaries,
constant velocity condition at the flow entry and outflow conditions at the exit.
All computations were performed using FLUENT 6.3, on personal computers
available at the University of Sydney, with the CPU time of between two minutes to
one hour depending upon the case considered. Computations for three-dimensional
models required CPU times around five hours. A number of grid dependence studies
were performed for the no-stent case and for a few of the stented cases. The solution
was deemed to have converged when the residual for each of the equations was
reduced to a level below 10-6. The next step is to generate what are known as non-
dominated solutions using Kriging explained in Srinivas et al. [3]. From these a
compromise solution is identified and forms the optimums solution or stent.
3 Results & Discussion
Fig. 3 shows the velocity vectors and vorticity contours for the no-stent case.
Fig. 3 Velocity vectors (left) and vorticity contours (right) for no-stent case
Case 1
Forty random samples were considered for this case. The objective functions are
plotted in Fig. 4, which also shows the non-dominated solution for this case and con-
sists of as many as 30 solutions lying close to each other. Figure 5 shows the results
for the best, the worst and the compromise stent identified in the non-dominated
solutions for this case. The results for this case clearly indicate that for a good flow
diversion, one should place struts at the proximal end of the neck.
Optimisation of Stents for Cerebral Aneurysm 5
Fig. 4 Non-dominated solutions for case 1
Fig. 5 Velocity (top) and vorticity contours and strut arrangements for case 1, (a) best stent and(b) worst stent and (c) compromise stent
Case 2
Fig. 6 shows the results for the best and the worst stent for this case. The three-
dimensional results are in agreement with the results from two-dimensional case that
for a good flow diversion, one should place struts at the proximal end of the neck.
Velocity reduction in the dome ΔV had a minimum of 20.55% while its maximum
6 C.J. Lee1, S. Townsend 2 and K. Srinivas3
Fig. 6 Velocity vectors, vorticity contours and strut arrangements for case 2, (a) no-stent, (b) beststent and (c) worst stent
was 36.66%, reduction in vorticity Δω varied between 8.9% and 23.38%. Rigorous
optimization is being carried out now.
4 Conclusion
Two-dimensional and three dimensional stents have been optimized for an effective
flow diversion for an aneurysm. The stent consists of five struts of uniform width
and thickness; the design variable being the gap sizes. It is revealed that a strut at
the proximal end of the aneurysm is essential for a good flow diversion.
References
1. Sforza, D.M., Putman, C.M., Cebral, J.R.: Hemodynamics of Cerebral Aneurysms. Ann. Rev.Fluid Mech., 41, pp. 91-107 (2009)
2. Hirabayashi, M., Ohta, M., Barath, K., Rufenacht, D.A., Chopard, B.: Numerical Analysisof the Flow Pattern in Stented Aneurysms and its Relation to Velocity Reduction and StentEfficiency. Math. Comput. Simulat., 72, pp. 128-133 (2006)
3. Srinivas, K., Ohta, M., Nakayama, T., Obayashi, S., Yamaguchi, T.: Studies on Design Opti-mization of Coronary Stents. J Medical Devices, 2, pp. 011004-1-011004-7 (2008)
4. Jeong, S., Obayahsi, S.: Multi-Objective Optimization using Kriging Model and Data Mining.KSAS International Journal, 7, pp. 1-12 (2006)
5. Myers, R.H., Montgomery, D.C.: Response Surface Methodology: Process and Product Op-timization Using Designed Experiments, Wiley, New York, pp. 1-84 (1995)
Micron Particle Deposition in the Nasal CavityUsing the v2-f Model
Kiao Inthavong, Jiyuan Tu, Christian Heschl
Abstract From a health perspective inhaled particles can lead to many respiratory
ailments. In terms of modelling, the introduction of particles involves a secondary
phase (usually solid or liquid) to be present within the primary phase (usually gas
or liquid). The influence of the fluid flow regime, whether it is laminar or turbu-
lent plays a significant role on micron particle dispersion. RANS (Reynolds Av-
eraged Navier-Stokes) based turbulence models provide simpler and quicker mod-
elling over the more computationally expensive Large Eddy Simulations. However
this comes at an expense in that the RANS models fails to resolve the near wall tur-
bulence fluctuating quantities due to the turbulent isotropic assumption. This error
further propagates to the Lagrangian particle dispersion. Using the v2-f the normal
to the wall turbulent fluctuation, can be solved and used on the particle dispersion
model directly in order to overcome the isotropic properties of RANS turbulence
models. This technique is first validated against experimental pipe flow for a 90o-
bend and then applied to particle dispersion in a human nasal cavity using Ansys-
Fluent. The results arising from the nasal cavity application will increase the un-
derstanding of particle deposition in the respiratory airway. Greater knowledge of
particle dynamics may lead to safer guidelines in the context of exposure limits to
toxic and polluted air.
Kiao InthavongRMIT University, Bundoora, Australia e-mail: [email protected]
Jiyuan TuRMIT University, Bundoora, Australia e-mail: [email protected]
Christian HeschlFachhochschulstudiengange Burgenland - University of Applied Science, Pinkafeld, Austria e-mail: [email protected]
1
2 Kiao Inthavong, Jiyuan Tu, Christian Heschl
1 Introduction
Studies of gas-particle flows in the human nasal cavity have generated a lot of in-
terest recently as computational modelling offers a complementary alternative to
experimental methods. In particular the inhalability of particles has been studied
which showed aspiration efficiencies of 60-80% [3] and 50-95% [2] for micron par-
ticles between 1-40μm. In terms of modelling, the introduction of particles involves
a secondary phase (usually solid or liquid) to be present in conjunction with the pri-
mary phase (usually gas or liquid), leading into the field of multiphase flows. The
dispersed phase can be modelled under two different approaches, i.e. Lagrangian or
Eulerian. Both approaches have their own advantages in computational modelling,
however this paper is limited to the Lagrangian approach only. Micron particles
are dominated by its inertial property which lead to inertial impaction upon sud-
den changes in the airflow streamlines. When the flow field is turbulent, turbulent
dispersion of the particles has to be addressed. Reynolds Averaged Navier Stokes
(RANS) based turbulence models are often used to resolve the flow field, as it pro-
vides simpler and quicker modelling over the more computationally intensive Large
Eddy Simulations. However this comes at an expense in that the RANS models fails
to resolve the turbulence dissipation and anisotropy in the near wall regions (i.e.
fixed stationary surfaces or boundaries). The turbulence fluctuating quantities are
overpredicted by RANS models and this error propagates to the particle dispersion.
In this paper, the v2-f and the k-ε turbulence model are used to solve the fluid flow,
while a dispersed phase model (Lagrangian reference) is used to track the individ-
ual particles. Turbulent dispersion of particles is modelled by the so-called Discrete
Random Walk found in Ansys Fluent. The turbulent particle tracking scheme is
evaluated and the requirements for the models to account for the anisotropic flow
behaviour in the near wall region is discussed. Successful modelling of micron par-
ticles will allow more flexibility in simulations of gas-particle flows for inhalation
toxicology, and drug delivery studies through the human respiratory system.
2 Model Description
The 90o curvatures in the nasal cavity are located just after the nostril entrance, and
at the nasopharynx region (Fig. 1). These two bends act as a naturally occurring filter
system that traps high inertial particles as they travel through the airway. A simpler
test case for evaluation of the CFD modelling is to use a 90o-bend pipe based on
experimental [5] and Large Eddy Simulation [1] data (Fig. 1). The computational
pipe (0.6million cells) has a diameter of 0.02, radius of curvature Rb = 0.056, a
curvature ratio of Ro = 5.6, Re=10,000 and De=4225. The nasal cavity (3.5million
cells) is subjected to a Re=2498 at the outlet which corresponds to a flow rate of
20L/min.
The realizable k-ε turbulence model with enhanced wall function is applied
through Ansys-Fluent and its model equations are provided in Ansys-Fluent (Ansys
Micron Particle Deposition in the Nasal Cavity Using the v2-f Model 3
Fig. 1 (a)Lateral view of the nasal cavity model showing the left cavity side. Two 90o curvaturebends are present; one at the nostril inlet and the other at the nasopharynx region. (b)Lateral viewof the 90o bend pipe.
2009). The v2-f model was implemented in Fluent via user-defined scalar inter-
face (UDS). The segregated solver in Fluent is used to solve the additional trans-
port equations for v2 and f, and therefore the code friendly v2-f version by Lien
and Kalitzin [4] is applied to improve convergence. For the Discrete Phase Model
(DPM) the Lagrangian approach is used. The normal (perpendicular to the wall)
fluctuation component, (v′) in the near wall region (y+ ≈ 0-30) is significantly
damped in comparison to the corresponding fluctuating components, u′ and w′. With
the v2-f model, the turbulent fluctuation component perpendicular to the wall is re-
solved which can then be applied to the DRW model in the near wall region. The
modified DRW model applied to regions where y+ < 30 is then reformulated as
v′ = ζ(v2)0.5
. The number of droplets tracked was checked for statistical indepen-
dence since the turbulent dispersion is modelled based on a stochastic process. Inde-
pendence was achieved for 40,000 droplets since an increase of droplets to 60,000
droplets yielded a difference of 0.1% in the inhalation efficiency. To achieve the uni-
form droplet concentration assumption, droplets were released at the same velocity
as the freestream.
3 Results and Discussion
Velocity contours at the 45o and 90o deflection are shown in Fig. 2. It can be seen
that the high velocity region moves from the inner wall to the outer wall (from the
45o deflection to the 90o deflection). A larger region of slower velocity is found at
the inner wall region as the secondary flow effects progress with the flow moving
through the bend from 45o to 90o. In addition the streamlines highlight the move-
ment of the fluid from the core towards the outer pipe wall with two resultant vor-
tices near the inner wall region. The secondary flow features are captured well with
the LES model by [1]. The v2-f model was also able to capture some of the sec-
ondary flow effects although at a reduced level. The region of slow moving fluid is
much smaller than that of the LES data especially at the 45o deflection. Even worse
4 Kiao Inthavong, Jiyuan Tu, Christian Heschl
Fig. 2 v2-f and k-ε simulationresults compared with LESdata by Berrouk and Laurence(2008). The angled deflectionsindicate the pipe curvature lo-cation. The inside wall of thepipe is on the left side and theouter wall of the pipe is onthe right indicated by I and Orespectively.
performing is the k-ε model which does not capture the slow moving region at all
at the 45o deflection. Furthermore, the streamlines in the core flow, moving towards
the outer wall are distorted at the 90o deflection which is not reproduced by the two
RANS turbulence models.
Fig. 3 a) RMS of velocity fluctuations for v’. The CFD simulations for the v2 − f and k− εsimulations had a Reτ =267 while the DNS data from Abe et al. (2001) is shown for Reτ =395. b)Particle deposition in a pipe bend for Re=10000.
The normal fluctuating velocity component (v’) taken at the pipe bend entrance
is chosen for comparison with DNS data because this is the component that is
overpredicted by the DRW models when a RANS-based turbulence model is ap-
plied. In the DRW model, the particle takes the fluctuating velocity component as
v′ = u′ = w′ =√
2k/3. Fig. 3 shows that when this occurs the v’ component is
overpredicted in both the v2-f and the k-ε model. Because the v2-f model provides
directly the v2 profile near the wall, the v2 can be defined directly into the DRW
Micron Particle Deposition in the Nasal Cavity Using the v2-f Model 5
model. Its profile is shown in Fig. 3 by the dashed line and denoted as v2-f(v2)
which shows a better improvement in the near wall region. Therefore the proposed
modification of the DRW model is expected to improve the turbulent particle dis-
persion. The predicted deposition of 1-30 μm particles in a 90o bend pipe compared
with the experimental data of Pui et al. [5] is shown in Fig. 3b, where the particle
Stokes number is based on the pipe inlet conditions. The deposition efficiency for
the DRW model taking the default isotropic fluctuating component from k (turbulent
kinetic energy) shows an overprediction in the deposition for St < 0.1 (square sym-
bols in Fig. 3b). The overprediction becomes greater, away from the experimental
data as the St decreases further. When the DRW model is modified and the normal
fluctuating component takes on the v2 profile, the deposition efficiency for St < 0.1
is improved.
Fig. 4 Flow field and parti-cle deposition patterns takenat cross-sectional slices A-Das depicted in Fig. 1. Thecontours show axial velocitymagnitudes while the surfacestreamlines represent the sec-ondary flow. Particles passingthrough the slices are colouredby velocity magnitude. Thepercentage of 1μm and 15μmparticles passing through eachcross-section is given beloweach slice.
Particle tracking analysis using the DRW-mod-v2-f model is performed and the
coordinates of the particles as they move through specified slice planes in the ge-
ometry are recorded. This can give an indication of how the particles are moving
through the nasal cavity, and also how many are passing through. Three cross-
sectional slices are created (Fig.1) which are viewed from the front of the nose. Slice
A exhibits the highest maximum velocity, with streamlines directed toward the inner
nasal septum wall. At this slice, only 77% of 15μm particles pass through, mean-
ing that ≈23% has deposited already in the anterior nasal cavity (nasal vestibule)
region. The particles are concentrated close to the ceiling of the passageways with
high velocities. The streamlines in Slice A tell us that the secondary flow will push
the particles both upwards and downwards. This results in the 1μm particle dispers-
ing more evenly through Slice B whereas the 15μm particles remain close to the
6 Kiao Inthavong, Jiyuan Tu, Christian Heschl
top of the slice because of its high inertial property. All particles are mainly passing
through the inner side of the passageway (the nasal septum wall side). For the 15μm
particles there is a large drop in the number of particles passing through from Slice
A to Slice B ( 61%), meaning that this is the main section of deposition for the parti-
cles. Interestingly this is also the main deposition region for the 1μm particles with
a percentage deposition of 3.3%. At Slice C, the airflow between the left and right
nasal cavity chambers have merged. Here we see complex secondary flow patterns,
exhibiting two vortices and two peak axial velocity regions each of which are almost
symmetrical to each other. The streamlines from the left and right sides converge in
the middle of the slice and are directed towards the inner curvature wall side of the
slice. Nearly all the particles moving from Slice B to Slice C have passed through
the passageway which has now expanded in cross-sectional area. The main cluster
of particles for the 15μm particles is still found in the superior regions of the slice,
which is now the outer curvature wall side. These flow patterns provide a predictive
tool as to where the particles may travel. The particle tracking model can be used to
determine the localised regions of high particle deposition.
4 Conclusion
The Discrete Random Walk (DRW) model used in Ansys-Fluent was used to simu-
late dispersed particles through a wall-bounded geometry such as the human nasal
cavity. A breathing rate of 20L/min was used and a RANS based turbulence models
in the form of the k-ε and the v2-f model was applied. It was shown that by applying
the small modification of v′=
√v2 directly the DRW can take on a more realistic
turbulent dispersion in the near wall region. A 90o pipe bend was used to validate the
model which showed an improvements to the particle deposition. The nasal cavity
was then used as an application which also showed that the modified DRW model
improved the deposition efficiency for smaller inertial particles.
References
1. Abdallah S. Berrouk and Dominique Laurence. Stochastic modelling of aerosol deposition forles of 90◦ bend turbulent flow. International Journal of Heat and Fluid Flow, 29(4):1010–1028,2008.
2. Nola J. Kennedy and William C. Hinds. Inhalability of large solid particles. Journal of AerosolScience, 33:237–255, 2002.
3. Camby Mei King Se, Kiao Inthavong, and Jiyuan Tu. Inhalability of micron particles throughthe nose and mouth. Inhalation Toxicology, 22(4):287–300, NaN.
4. F. Lien and G. Kalitzin. Computations of transonic flow with the v2-f turbulence model. Inter-national Journal of Heat and Fluid Flow, 22:53–56, 2001.
5. David Y. H. Pui, Francisc Romay-Novas, and Benjamin Y. H. Liu. Experimental study ofparticle deposition in bends of circular cross section. Aerosol Science and Technology, 7(3):301– 315, 1987.
