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Numerical investigation of fluid–particle interactions for embolic stroke

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Numerical investigation of fluid–particle interactions for embolic stroke

Abstract

Roughly one-third of all strokes are caused by an embolus traveling to a cerebral artery and blocking blood flow in the brain. The objective of this study is to gain a detailed understanding of the dynamics of embolic particles within arteries. Patient computed tomography image is used to construct a three-dimensional model of the carotid bifurcation. An idealized carotid bifurcation model of same vessel diameters was also constructed for comparison. Blood flow velocities and embolic particle trajectories are resolved using a coupled Euler– Lagrange approach. Blood is modeled as a Newtonian fluid, discretized using the finite volume method, with physiologically appropriate inflow and outflow boundary conditions. The embolus trajectory is modeled using Lagrangian particle equations accounting for embolus interaction with blood as well as vessel wall. Both one-and two-way fluid–particle coupling are considered, the latter being implemented using momentum sources augmented to the discretized flow equations. It was observed that for small-to-moderate particle sizes (relative to vessel diameters), the estimated particle distribution ratio—with and without the inclusion of two-way fluid– particle momentum exchange—were found to be similar. The maximum observed differences in distribution ratio with and without the coupling were found to be higher for the idealized bifurcation model. Additionally, the distribution was found to be reasonably matching the volumetric flow distribution for the idealized model, while a notable deviation from volumetric flow was observed in the anatomical model. It was also observed from an analysis of particle path lines that particle interaction with helical flow, characteristic of anatomical vasculature models, could play a prominent role in transport of embolic particle. The results indicate therefore that flow helicity could be an important hemodynamic indicator for analysis of embolus particle transport. Additionally, in the presence of helical flow, and vessel curvature, inclusion of two-way momentum exchange was found to have a secondary effect for transporting small to moderate embolus particles—and one-way coupling could be used as a reasonable approximation, thereby causing substantial savings in computational resources.
Theor. Comput. Fluid Dyn.
DOI 10.1007/s00162-015-0359-4
ORIGINAL ARTICLE
Debanjan Mukherjee ·Jose Padilla ·Shawn C. Shadden
Numerical investigation of fluid–particle interactions for
embolic stroke
Received: 2 February 2015 / Accepted: 27 June 2015
© Springer-Verlag Berlin Heidelberg 2015
Abstract Roughly one-third of all strokes are caused by an embolus traveling to a cerebral artery and blocking
blood flow in the brain. The objective of this study is to gain a detailed understanding of the dynamics of embolic
particles within arteries. Patient computed tomography image is used to construct a three-dimensional model of
the carotid bifurcation. An idealized carotid bifurcation model of same vessel diameters was also constructed
for comparison. Blood flow velocities and embolic particle trajectories are resolved using a coupled Euler–
Lagrange approach. Blood is modeled as a Newtonian fluid, discretized using the finite volume method, with
physiologically appropriate inflow and outflow boundary conditions. The embolus trajectory is modeled using
Lagrangian particle equations accounting for embolus interaction with blood as well as vessel wall. Both one-
and two-way fluid–particle coupling are considered, the latter being implemented using momentum sources
augmented to the discretized flow equations. It was observed that for small-to-moderate particle sizes (relative
to vessel diameters), the estimated particle distribution ratio—with and without the inclusion of two-way fluid–
particle momentum exchange—were found to be similar. The maximum observed differences in distribution
ratio with and without the coupling were found to be higher for the idealized bifurcation model. Additionally,
the distribution was found to be reasonably matching the volumetric flow distribution for the idealized model,
while a notable deviation from volumetric flow was observed in the anatomical model. It was also observed
from an analysis of particle path lines that particle interaction with helical flow, characteristic of anatomical
vasculature models, could play a prominent role in transport of embolic particle. The results indicate therefore
that flow helicity could be an important hemodynamic indicator for analysis of embolus particle transport.
Additionally, in the presence of helical flow, and vessel curvature, inclusion of two-way momentum exchange
was found to have a secondary effect for transporting small to moderate embolus particles—and one-way
coupling could be used as a reasonable approximation, thereby causing substantial savings in computational
resources.
Keywords Hemodynamics ·Embolic stroke ·Fluid–particle coupling ·Helicity ·Collision
Communicated by Rajat Mittal.
D. Mukherjee (B
)·S. C. Shadden
Department of Mechanical Engineering, University of California, Berkeley, Berkeley, CA, USA
E-mail: debanjan@berkeley.edu
S. C. Shadden
E-mail: shadden@berkeley.edu
J. Padilla
Los Angeles City College, Los Angeles, CA, USA
D. Mukherjee et al.
1 Introduction
Stroke is among the most severe and common forms of cardiovascular disease, which involves disruption in
blood supply to the brain. Over 85 % of strokes are ischemic, and about 3540 % of these are due to embolism
[3]—that is, extracranial particles, often originating from the heart, travel to the supplying arteries in the brain
to cause an occlusion. These emboli are typically composed of thrombotic or fatty material. The mechanics
of the transport of an embolic particle under the combined action of blood–embolus interaction, unsteady and
pulsatile flow in the blood vessels, and embolus collisions with the vessel walls is a complex issue. Deeper
insights into these phenomena may lead to a better understanding of the mechanisms underlying embolic stroke
risk, as well as other particle transport scenarios such as drug delivery [32].
Previous studies have attempted to quantify the distribution fraction of emboli across arterial branches.
Macdonald and Kowalczuk [22] performed an experimental study by injecting agarose particles into the
internal carotid artery of monkeys and concluded that the particles entered the penetrating arteries of the brain
in proportion of their diameters. Rapp et al. [31] in a recent study on rats demonstrated that emboli shape and
composition are important factors in governing the extent of brain injury once they move up to the cerebral
vasculature. Pollanen [30], and more recently Bushi et al. [6] performed experimental studies on embolic
particle transport across idealized Y-bifurcation geometries. Their studies indicated a bias in the distribution
of larger particles into the wider branching vessels, over and above what volumetric flow distributions may
suggest. Chung et al. [9] used a more realistic anatomical model of the Circle of Willis for their experiments
on embolus transport across the cerebral arteries and suggested a similar bias of larger particle sizes migrating
toward the wider branching vessels.
While experimental studies have characterized overall embolus distribution fractions in relation to volumet-
ric flow, it is difficult to probe deeper into the underlying fluid–particle–vessel interactions from such studies.
CFD models, on the other hand, are effective to study detailed flow information and performing paramet-
ric variations. Simulations of embolic particle trajectories using flow fields obtained from image-based CFD
present a powerful alternative for investigating transport of emboli to the brain. Fabbri et al. [12] presented a
simulation study on embolus transport within the cerebral vasculature, and through their simulation data, they
report a size-dependent preference of embolus transport into the wider arteries, as was reported by Chung et al.
[9]. Carr et al. [8] performed a series of three-dimensional CFD simulations to study the transport of embolic
particles originating from the heart, across the aortic arch, into the supplying vessels to the head. Their study
showed a strong size-dependent preference between the right and the left branches of the carotid and vertebral
arteries for the distribution of the embolic particles being transported across the aortic arch.
Accurate computational modeling of the particle trajectories requires consistent numerical resolution of
the interaction between the fluid and the particle. Of greatest consideration are the fluid forces on the particle,
but the presence of the particle also disturbs the flow and leads to an exchange of momentum locally back
into the fluid. The mathematical resolution of this two-way momentum exchange is often referred to as “two-
way coupled” or “fully coupled” fluid–particle interaction. While modeling fully coupled interactions is more
accurate and rigorous, it is far more computationally expensive than employing a one-way coupling, whereby
the fluid forcing on the particle is considered without the reciprocal momentum exchange on the fluid. A major
advantage of a one-way coupling is that the fluid dynamics can be solved independently of particle dynamics,
and a host of particle dynamics simulations can be carried out as a post-processing step on the CFD flow data.
The two-way coupled interactions are generally dependent upon the size and the volume fraction of the
particles considered, and the nature of the background flow structures [4,10]. The effect of the two-way coupling
on the flow, and on the particle trajectories is still a subject of active research, and as for their role in simulations
of embolus laden hemodynamics, not much attention has been given. Carr et al. [8] employed one-way coupled
interactions for their simulations, but hint at the considerations of coupling possibly being of importance in
estimating embolus trajectories and their distribution especially for the larger particles considered. The study
by Fabbri et al. [12] is the only work thus far, that does consider both one and two-way coupled models in their
simulations. However, their treatment is limited, and they do not outline the role that is played by coupling
in altering distribution statistics. Intuitively, it is expected that for very small particle sizes (relative to vessel
diameter), the particles will follow the background flow, and their distribution across bifurcations will mirror
volumetric flow division. For large particle sizes, their influence on the flow features surrounding them will
become a necessity in determining their transport. In the intermediate size ranges, it is difficult to state a priori
whether one-way coupling is a reasonable approximation, or fully coupled simulations are required. These
considerations led us to present two guiding questions for the work presented here:
Numerical investigation of fluid–particle interactions
Fig. 1 A schematic description of the overall image-based CFD simulation framework
How does fully coupled fluid–particle interaction alter small- to intermediate-sized embolus particle dis-
tribution across a bifurcation (when compared with one-way coupled interaction models)?
What role does the mechanics of particle interaction with fully three-dimensional flow structures play as
the particle is distributed across a bifurcation into the branching vessels?
To address these questions, a coupled Euler–Lagrange CFD framework is used to study the transport of embolic
particles across the human carotid bifurcation. The carotid bifurcation is chosen because it is a critical junction
in guiding emboli to the supplying vessels of the brain. The entire framework was implemented using the
finite volume CFD solver package Fluent by extensive customization through user defined function libraries.
The organization of this paper is as follows: Sect. 2outlines the detailed mathematical model, and the CFD
framework and its implementation; Sect. 3outlines the observations from the numerical experiments performed,
in the context of addressing the two questions mentioned above; and Sect. 4presents our concluding remarks
and few comments about future research directions.
2 Modeling
2.1 Image-based CFD framework
Vessel lumen for the chosen vasculature segment was reconstructed from patient computed tomography (CT)
scan data and was translated into a three-dimensional computer model using ITK-SNAP—an image process-
ing package based on active contour segmentation techniques [33]. The parent vessel diameter for the patient
model chosen was approximately 5.2 mm, with the daughter vessels having diameters approximately 3.2 and
4.2 . The domain was discretized into a computational mesh comprising 123,293 tetrahedral elements (average
element volume 0.022 mm3, average element face area 0.157 mm2). The vessel geometry and diameter infor-
mation was used to create an additional model of an idealized Y-bifurcation with same vessel diameter as the
anatomical bifurcation (model not shown in Fig. 1), which was discretized into a computational mesh compris-
ing 90,925 tetrahedral elements (average element volume 0.018 mm3, average element face area 0.115 mm2).
This idealized model was used to create a corresponding set of simulations, such as to enable investigations
into how embolic particles react with vortical flow typically observed in anatomical geometries, as compared
with idealized geometries (under same volume flow rates, and same cardiac periods). Blood was assumed to
be a Newtonian fluid with constant density 1025 kg/m3, and viscosity 0.004 Pa s [28]. A finite volume-based
D. Mukherjee et al.
computational scheme was employed to resolve the hemodynamics within the vasculature segment, wherein
the general form of the continuity and the Navier–Stokes equations were represented as follows:
∂(ρfφ)
t+∇·fφu)=∇·φφ+Sφ(1)
with ρfbeing the fluid (blood) density, φ=1 for the continuity equation, and φ=ufor the Navier–Stokes
equation. The terms φand Sφare 0 for the continuity equation, while for the Navier–Stokes equation,
∇·φφ=−p+∇·τ(pbeing the hydrostatic pressure, and τbeing the viscous stresses), and
Sφincorporates the gravity body forces and other momentum source terms. The conservation equations are
discretized spatially by integrating the equations over each computational cell, to obtain the following numerical
form (refer to Ferziger and Peric [13] for details)
V
∂(ρfφ)
tdV=−ρfφu·ndA+φφ·ndA+V
SφdV(2)
∂(ρφ)
tV=−
1,Nfaces
k
ρfφkuk·nkAk+
1,Nfaces
k
φ,kφkuk·nkAk+SφV(3)
The temporal component is discretized using a one-step implicit Euler scheme, as follows:
∂(ρφf)
tV=F(t), t)φN+1φN
t=1
ρfVFφN+1etc. (4)
A pulsatile, analytical, volumetric flow profile of the form as defined by Olufsen [29] was prescribed at
the inlet boundary (common carotid artery) (see Fig. 1). The parameters for the profile are chosen such that
the total volumetric inlet flow is consistent with available flow data in the literature for the common carotid
artery [20]. The inlet flow at each instant in time is mapped to a parabolic profile across the cross-section of the
inlet. The outlet boundary conditions are set using single element Windkessel resistors. In order to estimate the
resistance value at each outlet, the total flow is assumed to be distributed to each outlet, on an average, based
on the cross-sectional areas of the outlets, i.e., QiAi[34]. The mean pressure at each outlet is assumed
to be 93 mm/Hg, which when divided by the average outlet flow as computed above, gives the resistance
magnitudes.
2.2 Embolus particle motion
The embolus is assumed, to a first-order approximation, to be a spherical particle, with a nominal density of
1100.0 kg/m3. The motion of this particle within the vasculature is now modeled using a modified form of the
Maxey–Riley equation [23], derived from a superposition of the steady and the unsteady forces on the particle
arising from the background flow as well as the disturbed flow around the particle. In its most generalized
form this equation is (see also [5,18] for mathematical details and alternative forms of this equation)
mp
dvp
dt=1
2ρfCDπD2
p
4|uvp|uvp

steady drag force
+μf
πD3
p
82u

Faxen corr. drag
+mpg

gravity
+Vp(−∇p+∇·τ)

fluid stresses, undisturbed flow
+CaρfVp
2Du
Dt dvp
dt

added mass, unsteady
+CaρfVp
2
d
dtD2
p2u
40

Faxen corr. added mass
+3
2D2
pπμfρft
0
K(tt)g(t)dt+u(0)vp(0)
t

history force, and correction for initial velocities
+I(P,W)Fcontact

particle-wall contact
+
i=j
I(Pi,Pj)Fcontact

particle-particle contact
,(5)
Numerical investigation of fluid–particle interactions
where ρp,mp,andDpare the particle density, particle mass, and particle diameter, respectively, Vpdenote the
overall particle volume, CDdenotes the drag coefficient, and Cadenotes the added mass coefficient. The drag
coefficient is estimated using the correlation presented by Haider and Levenspiel [16]—which was chosen
for its applicability over a broad range of Reynolds numbers, and the reportedly low mean square error when
compared to values of drag coefficients obtained from a broad range of available experimental data. The
correlation is
CD(Rep)=24
Rep1+0.1806Re0.6459
p+0.4251
1+6880.95
Rep
,(6)
where Repdenotes the particle slip-velocity-based Reynolds number defined as Rep=uvpρfDpf.
The added mass coefficient is assumed to be Ca=0.5 for the spherical embolus geometry. The full form
of the particle motion equation as presented in Eq. (5) can be computationally intensive, especially for fully
coupled fluid–particle interaction calculations—specifically, the evaluation of the history terms can add on
considerable computational complexity. This motivates further investigations into the individual force terms
in Eq. (5) so as to be able to formulate a simpler, and computationally less intensive, first-order approximation
of the particle motion equations. Considering a characteristic free-stream flow velocity Uand characteristic
length scale to be that of the particle diameter Dp, a non-dimensionalized form of the particle motion equation
in Eq. (5) can be obtained as follows (only the fluid forces have been presented, since the contact interactions
will necessarily need to be retained for a correct description of particle motion):
dv
p
dt=CDRep
24 ·Dp
U·1
τpuv
p+1
24 ·1
τp·Dp
U2u+g
+ρf
ρpDu
Dtg+Ca
2·ρf
ρpDu
Dtdv
p
dt+1
40
d
dt2u
+3
2π·ρf
ρp·1
τp·Dp
UK·gdt+u(0)v
p(0)
t,(7)
where all terms with a superscript denote non-dimensionalized terms, and the parameter τp=ρpD2
p/18μfis
the individual embolus particle response time to the background flow. It is evident from the non-dimensional
form that for density ratios ρfpclose to unity (which is the case for embolus and blood, ρfp=0.932), the
effect of the forces due to undisturbed flow and added mass effects become significant. The Faxen corrections
become significant for cases where the flow field exhibits substantial spatial non-uniformity and curvature—
and to a first-order approximation, they can be neglected for our problem. Additionally, the relative contribution
of the history forces as compared to the drag forces (which are typically the most dominant fluid forces on
the particle) scales as the square-root of the particle Stokes number. Hence, for cases where the particles
are considered to be small to moderately small in size—the Stokes numbers are small, and to a first-order
approximation the history forces can also be neglected. This leads us now to a simpler form of the particle
motion equation
mp
dvp
dt=Fp=1
2ρfCDπD2
p
4|uvp|uvp+Vp(−∇p+∇·τ)
+CaρfVp
2Du
Dt dvp
dt+I(P,W)Fcontact.(8)
The equations are integrated in time using a generalized, one-step θmethod as follows:
vN+1
pvN
p
t=θ
mp
FN+1
p+(1θ)
mp
FN
p(9)
where typically θ∈[0,1], and the subscript Nindicates the time-step index. For the simulations presented
here we have used θ=0.5, which corresponds to the trapezoidal method.
D. Mukherjee et al.
2.3 Embolus contact with vessel wall
The embolus interaction with vessel walls is incorporated in Eq. (8) above through the last term on the right-hand
side, wherein a contact force is applied to the particle based upon an indicator function I(P,W). This indicator
function is calculated from a triangulation of the vessel wall surface, represented as W=Ntriangles
w=1Tw, with
each triangle having a unique normal nw. The contact of the spherical embolus is checked against the wall
triangulation (see Ericson [11] for details), and I(P,T)=1 if the embolus particle Pis in contact with the
triangle T, and 0 otherwise. A direct balance of linear and angular momenta of the particle can now be used
to obtain a contact force estimate. Considering a spherical particle approaching the vessel wall surface, and
coming in contact with a triangle Twthe particle velocity at the point of contact (denoted by subscript pc) is
vpc =vp+ωp×Dp
2nw.(10)
This helps define a unique tangent slip vector for the contacting triangle Tw(assuming that wall velocity at the
point of contact is determinable to be vwall)
tw=vpc vwallvpc vwall·nwnw
vpc vwallvpc vwall·nwnw.(11)
The overall embolus-wall contact force can now be represented as
Fcontact =
Ncontact
w=1
fN,wnw+fT,w tw,(12)
where Ncontact represents the discrete triangle elements on the wall that the embolus particle is in contact with.
A direct linear momentum balance of the particle during the contact duration gives
mpvp(t+δt)mpvp(t)=t+δt
t
[Fcontact +Ffluid]dt,
=Fcontact δt+Ffluid δt,(13)
where the impulses of the contact and the fluid forces during the contact duration are replaced by an average
impulse over the contact duration (denoted by ·). The effect of inelastic deformations and energy loss during
contact can be now incorporated by means of a restitution coefficient that is defined by decomposing the
overall collision process into a “compression” phase and a “recovery” phase and taking the ratio of the contact
impulses during these two phases as
e=Recovery impulse
Compression impulse =t+δt
t+δtfN,wdt
t+δt
tfN,wdt=fN,w Rtδt)
fN,wCδt.(14)
Decomposing the normal component of the linear momentum balance for the particle into the compression and
recovery phase, replacing the impulse integrals with average impulses, and using the definition of restitution
coefficient as above, the total average contact impulse in the direction normal to the triangle contacting the
particle can be given as
fN,wδt=(1+e)mpvwall,nmpvp,n(t)−Ffluid,nδt,(15)
where vwall,n=vwall ·nw,v
p,n=vp·nw,andFfluid,n=Ffluid ·nw. For a numerically discretized solution of
the particle motion equation, the contact force for the time step when contact event occurs is given as
fN,w =fN,wδt
t,(16)
where tis the numerical discretization time step. The impulse-based estimate of contact forces is chosen
because it is less restrictive in terms of time-step requirement for resolution of the contact (as opposed to
models based on contact deformation, where explicit resolution of the contact durations is required), and also
because it ensures momentum conservation for the contact explicitly. The collision durations can be estimated
based on the approach velocity and material properties of the contacting bodies (see Johnson [17] for detailed
theoretical discussion on the same). However, for the purpose of the present simulations, we chose the duration
to be a very small number in comparison with the numerical time step: δtCtand C1.
Numerical investigation of fluid–particle interactions
2.4 Particle–fluid two-way coupling
Modeling of the particle–fluid two-way coupling continues to be an active area of research, and various
formulations for the coupling interactions have been proposed in the existing literature, see [4,10] for detailed
reviews. Broadly, such formulations can be categorized into two types—methods which directly aim to resolve
the flow around the particle, and methods that consider the particles to interact with the fluid via some form
of momentum sources. The former, while offering more numerical resolution and accuracy, usually bear
substantial computational expense when compared to the latter. This is an important factor for consideration
for the present study, since for elucidating meaningful statistics of particle distribution, we need a framework
that can repeat simulations for various particle sizes, configurations, and for large populations of particles. In
this context, we employ here a particle-source-type coupling strategy where the coupling between the particle-
phase and the fluid is achieved by augmenting the Navier–Stokes equations with momentum source terms that
incorporate the transfer of momentum onto the carrier fluid by the particle. Specifically, denoting the sum of
all external forces on the particle as Fpsuch that mpdvp/dt=Fp(see Eqs. 5,8), the source term for each
computational cell can be represented as follows:
Sp=−1
V
p
NpFp(17)
In order to ensure that the effect of the momentum transfer is smoothly distributed over mesh nodes neighboring
the cell where the particle centroid lies, a kernel averaging of the source terms across the nodes is employed,
where the source terms at a node n with nodal coordinates xnare obtained from
Snp =GxpxnSp,(18)
where Gis an appropriately chosen weighting function. For our simulations we chose Gto be a Gaussian kernel
G(x;a,h)=a
π3/2
exp ax·x
h2.(19)
The parameter ais a bandwidth parameter for the Gaussian kernel and has been chosen for our simulations
to be unity, while his a cell size parameter (see [2] for additional details). This allows for simple modeling
of transfer of momentum to cells occupied by the particle, for particle sizes that are bigger than a single cell,
but not multiple order larger. While this coupled particle source finite volume method provides a numerically
lean, first-order approximation for the coupling—the applicability of this method will fail for particle sizes
that are multiple times larger than the element size used in the mesh (which would be the case, for example,
for particles equivalent to the size of the vessel diameter). Thus the framework cannot be used for these larger
particles. However, in consonance with our research questions stated in Sect. 1—our objective is to clarify
the role of coupling in small to moderate size particles when compared to the vessel diameter (knowing that
considerations of the coupling will have a prominent role for large particles).
2.5 Numerical implementation
The fully discretized continuity and Navier–Stokes equations were solved using the well-benchmarked finite
volume solver Fluent (part of the Ansys software suite). The Fluent solver was extensively customized with
user defined function libraries for handling physiological boundary conditions, for handling embolus particle
contact forces, and for post-processing of embolus trajectory data using the Visualization Toolkit (VTK)
library. The Navier–Stokes pressure velocity coupling for the finite volume solver was implemented using the
PISO scheme, with second-order discretization for pressure, and second-order upwinding used for momentum
equation. The mesh sizes chosen for the discretization was guided by a mesh convergence study to ensure the
spatial resolution is sufficient. This was ensured by starting from a baseline mesh size choice guided by existing
literature on computational hemodynamics, and choosing 3–4 levels of refinement for both the idealized and
anatomical models. The convergence for the flow variables were determined based on globally scaled residuals
of the momentum and continuity equations, and the limits for these were set to 0.001. Additionally, volumetric
flow, and pressure, integrated across the inlet and outlet faces were also tracked as monitors—such that the
simulations were run for sufficient number of cardiac cycles until the difference between these monitors
between subsequent cardiac cycles became negligible. In order to ensure a consistent initial condition for the
D. Mukherjee et al.
0 100 200 300 400 500 600 700 800
Particle sizes (microns)
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
Distribution fractions
ext.car.one-way
ext.car.two-way
ext.car.mean-flow
int.car.one-way
int.car.two-way
int.car.mean-flow
Fig. 2 A comparison of estimated embolic particle distribution fractions amongst the external and internal carotid arteries for an
idealized Y-branch model of the human carotid bifurcation
particle laden flow simulation, both coupled and one-way, the fluid velocity field is first initialized with an
initial flow field that is obtained by solving a Euler problem over the entire domain with appropriate boundary
conditions. This velocity field is then allowed to evolve for one full cardiac cycle using the inlet flow profile as
described in Sect. 2.1 and in Fig. 1, and the particles are released from the inlet either at the start of the second
cardiac cycle, or later, after having allowed the simulations to evolve for a few cycles as mentioned above. The
discrete particle equations themselves are integrated with a time step of 0.0001s, ensuring that the particle is
sub-stepped ten times for each fluid time integration step. The iteration tolerance for the iterative solution of
Eq. (9)wassetto10
5.
3 Simulations of embolus transport: results and discussion
3.1 Embolus distribution and fluid–particle two-way coupling
In order to demonstrate the difference in embolus particle distribution with and without the inclusion of fully
coupled particle fluid interactions, the trajectories for a sample of 5000 embolic particles were computed for
both the anatomical and idealized bifurcation models (as mentioned in Sect. 2.1). The particle sizes were
chosen to be in the range of 10–750 µm. These sizes were chosen such that they were in the small-to-moderate
regime when compared to vessel diameters—up to about 17% of the ICA diameter for the anatomical model,
and about 18.8 % of the ICA diameter for the idealized geometry. This choice is consistent with the limitations
of the coupling framework chosen, as detailed in Section 2.4. The particles were released from the common
carotid artery inlet, at the end of one cardiac cycle, and then allowed to flush out of the bifurcation model over
three more cardiac cycles. For the one-way coupled simulations, all 5000 particles could be released at once,
as they are assumed to not interact with the flow or interact with each other via collisions. On the other hand,
for the two-way coupled simulations, each embolus trajectory computation was performed individually, each
of which requiring an independent solution of the flow field. It is noted here that, for the flow simulations, the
peak Reynolds number based on inlet flow was 252.00, and the peak Womersley number, based on common
carotid inlet diameter, was 43.54.
For all the cases, the number of particles being transported to the external and internal common carotid
was explicitly tracked. The fraction of particles distributed to the branches for the idealized bifurcation model
has been presented in Fig. 2and compared with the volumetric flow division to each branch. Figure 3shows
the corresponding distribution for the anatomical bifurcation model. For each of the geometries, for both
the external and internal carotid arteries, the estimated particle distribution fraction is compared between
Numerical investigation of fluid–particle interactions
0 100 200 300 400 500 600 700 800
Particle sizes (microns)
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
Distribution fractions
ext.car.one-way
ext.car.two-way
ext.car.mean-flow
int.car.one-way
int.car.two-way
int.car.mean-flow
Fig. 3 A comparison of estimated embolic particle distribution fractions amongst the external and internal carotid arteries for an
anatomical model of the human carotid bifurcation
simulations that accounted for fully coupled fluid–particle interactions and those that included only one-way
interactions. It was found that for the idealized geometry, the maximum deviations between the fully coupled
and one-way estimates was 15 %. Additionally, it is observed that for the idealized bifurcation results shown
in Fig. 2that the distribution fractions matched reasonably with volumetric flow distributions for all the particle
sizes considered. On the other hand, for the anatomical model results presented in Fig. 3, it is evident that
the estimated distribution fractions show a marked size-dependent bias when compared to volumetric flow
distributions. It was also found that for the anatomical model the maximum deviations between the fully coupled
and one-way coupled estimates for particle distribution fractions was 5.3 %—notably lower as compared to
the idealized geometry.
An important aspect that is worth a mention here is that the particle sizes considered (in comparison with
vessel diameter) are well below the regimes where geometric obstruction would become a major factor in
reducing flow of particles into the narrower branch, and biasing them toward the wider branch. The differences
in the size-dependent distribution fraction between the anatomical and the idealized geometries therefore
motivate further discussion of the mechanics of the flow. The embolus particle interaction with the local flow
structures plays a significant role in the distribution of the particles across a bifurcation, which is an aspect
that cannot be entirely understood using distribution fractions alone.
3.2 Interaction of emboli with vortical flow
A comparison of the emboli trajectories between the idealized and the anatomical models as presented in
Fig. 4(for 750 µm emboli) reveals an important factor. For the idealized bifurcation (Fig. 4, left panel), the
emboli travel in rectilinear paths up to the bifurcation. The local flow structures at the bifurcation alone, in
combination with particle inertia, will thereby determine distribution behavior. On the other hand, the particle
pathlines for the anatomical model (Fig. 4, right panel) exhibit substantial helical motion prior to approaching
the bifurcation. Realistic arterial geometries, such as the one considered here, are in general curved, and non-
planar, resulting in secondary vortical and swirling flow [7,19,20,26]. Specifically, for the anatomical carotid
bifurcation geometry, Fig. 5displays snapshots of velocity vectors decomposed along sections of the common
carotid artery, and at multiple instants during the cardiac cycle. The decomposed velocity vectors clearly
indicate the presence of a circulating flow pattern evolving with time and length along the arterial segment.
Such flow patterns are not observed in idealized, planar, arterial geometries. Thus significant differences in
how particles are distributed across idealized bifurcation geometries as compared to anatomically realistic
bifurcations may be in large part due to helical flow structures.
D. Mukherjee et al.
Fig. 4 Particle path lines for the 750µm emboli traveling across the idealized (left), and the anatomical (right) bifurcations—
colored by the branch vessel they migrate to across the bifurcation
In order to directly correlate the role that this swirling flow plays in transporting embolic particles, an
appropriate descriptor of the flow that can be employed is the flow helicity. Helicity, is commonly described
in terms of the helicity density, which is defined as the dot product of flow velocity and flow vorticity Hk=
u·[∇×u]. In its purely physical interpretation, helicity is an indicator of the topology of vortex-lines in a
flow [24,25] and is representative of the spiraling flow in a specified volume of fluid. A modified form of this
variable, referred to as the Local Normalized Helicity (or Relative Helicity [21]) has recently seen substantial
interest owing to its utility in indicating the swirling flow structures in a volume of fluid [14,27]. This parameter
was introduced (and modified) by Grigioni et.al. [15], and is defined as
Hn(x,t)=u(x,t)·[∇×u(x,t)]
u(x,t)·[∇×u(x,t)].(20)
The variable Hnvaries spatio-temporally and is a signed variable such that Hn∈[1,1].Itisameasureof
the angle that the local flow velocity makes with the axis of local vorticity and, thereby, provides a normalized
indicator of the swirling structure of the flow. A purely helical flow would lead to Hn1, with the sign
being negative for counterclockwise swirling flow, and positive for clockwise swirling flow. For the anatomical
bifurcation model, a clear distinction of the regions of the flow where strong swirling motion is dominant, can
now be provided as in Fig. 6using the local normalized helicity (Hn). The figure shows streamlines, colored
by Hn, for successive instants across systole and diastole during one cardiac cycle. A strong forward spiraling
flow is observed during the systole, which starts spiraling downstream into the internal carotid artery ([14,20]),
and reverses direction in the common carotid during the diastole where slow retrograde flow is common.
In order to characterize the swirling kinematics of the individual embolic particles and identify regions
where particles are experiencing a higher extent of helical motion, a simple velocity-based indicator can be
devised. The individual vessel centerlines were extracted using the Vasculature Modeling Tool Kit (VMTK)
open source library [1]. Centerline coordinates in VMTK are computed from a minimization of the radius of the
maximal inscribed sphere within the boundary mesh of the vessel. The obtained collection of discrete points,
representing the centerline, may therefore contain some irregular fluctuations at some locations in the vessel.
In order to ensure this does not lead to any uncertainties that may further propagate into our analysis, local
polynomial spline interpolation was used to obtain a smooth representation of the centerline.Mathematically, we
now represent this smoothed centerline as a collection of ordered points C={xk|k=1,...,n}. Assuming that
the centerline defines the preferred axial flow direction at any section of the vessel, we now drop a perpendicular
from the current embolus location xpto the centerline Cand represent this point on the centerline as x.We
define a radial vector eras
Numerical investigation of fluid–particle interactions
Fig. 5 Snapshots of velocity vectors decomposed along multiple section planes across the length of the common carotid up to
the bifurcation, with snapshots IV and V having vectors scaled by a factor of 15 and 75 respectively, owing to the low velocity
magnitudes during diastole. Snapshots taken at 0.05 s (I), 0.15s (II), 0.25 s (III), 0.45 s (IV), 0.85 s (V) from the start of the cardiac
cycle
D. Mukherjee et al.
Fig. 6 Snapshots depicting the flow streamlines at successive instants during the cardiac cycle (shown on left inset). Snapshots
are taken at 0.05 s (1), 0.10 s (2), 0.15s (3), 0.25s (4), 0.45s (5), and 0.70 s (6) from the start of the cardiac cycle. The streamlines
are colored by normalized flow helicity, to indicate the prominent swirling flow in the common carotid
er=xxp
xxp.(21)
Additionally, a tangential flow direction can be specified at the location of the embolus as
ez=xk+1xk
xk+1xk,(22)
where kand k+1 denote the points immediately preceding and following xon the centerline. The vectors
erand ezcan now be used to define an azimuthal direction
ea=ez×er.(23)
The individual embolus particle velocity at every instant of time can now be decomposed into an azimuthal
and axial component, and a ratio of these components is defined to be the particle swirl ratio
sw=va
vz=vp·ea
vp·ez
.(24)
The swirl ratio va/vzcan be now used to characterize the instantaneous tendency of the particle to undergo
helical or swirling motion. It is noted that this ratio is a signed quantity—meaning that it can distinguish
between counter-clockwise and clockwise helical motion of the particles. While the absolute magnitude of
this variable will not be of relevance by itself, the relative levels of this ratio among particles distributed at
various locations, and during various instants of time, will provide an indicator of the tendency of a particle to
move straight as compared to following a helical trajectory (since particles moving straight will end up with
the swirl ratio being 0).
A comparison of the relative magnitudes of the swirl ratio has been compiled for successive instants in the
cardiac cycle for each particle size and presented in Fig. 7. The particles marked red possess a high, positive
value for sw—thereby possessing a higher tendency to undergo counterclockwise helical motion (and similarly
particles marked blue will possess a higher tendency to undergo clockwise helical motion). The locations of the
high magnitude of the swirl ratio can be directly compared with the locations of the high and low normalized
helicity as presented in Fig. 6. By definition, a high negative Hnindicates strong counter-clockwise helical
flow—and its spatial extent as observed from this figure correlates well with the spatial extent of the high
positive particle swirl number. It is evident that upon being introduced into the swirling flow as described here,
the particle trajectory will be determined by the strength and spatial extent of the swirling flow, the momentum
with which the particle approaches the swirling flow region, and the particles own inertia. The particle response
time to the local fluid velocity field is typically defined as τp=ρpD2
p/18μf. While the local flow helicity
will tend to induce a helical, swirling motion to the particles, larger particles are slow to respond, and deviate
Numerical investigation of fluid–particle interactions
Fig. 7 An illustration of the swirling motion of the particles based on the kinematic swirl ratio (referred to in the figure by ‘swirl’).
From left to right, successive snapshots during the cardiac cycle have been presented at 0.04, 0.05, 0.06, 0.07, 0.08, and 0.09 s
respectively, with particle size increasing from top to bottom. All the results presented here are from one-way coupled simulations
from the circulating flow entering the bifurcation. This leads to particles being preferentially directed toward
the external carotid branch as presented in Fig. 3. The spatial extents of high Hn, and high swwill correlate
most closely for smaller particles but in contrast to quantification of helicity based on tracer particles [15],
the helical flow of inertial particles will be size dependent, and particle response times will be an additional
parameter that needs to be considered.
D. Mukherjee et al.
0.0 0.2 0.4 0.6 0.8 1.0
Release instants in a cardiac cycle
0.0
0.2
0.4
0.6
0.8
1.0
Particle distribution ratio
ext-car
int-car
cardiac flow
Fig. 8 A comparison of the fraction of particles distributed to the internal and external carotid arteries, for simulations where
particle release instants during the cardiac cycle was varied (all release instants denoted on the black curve)
An additional set of numerical experiments were performed wherein particles of size 750 µm(thelargest
particles considered) were released into the anatomical bifurcation model from the common carotid inlet,
at multiple instants during the cardiac cycle. For each release instant, the particles were assigned the same
initial velocity as the inflow velocity at that location in space and time. The corresponding particle distribution
fractions—estimated in a similar manner as discussed in Sect. 3.1—have been presented in Fig. 8for each
release. The resulting distribution fractions show notably different trends for particles released during different
stages of the cycle. The preference of particles to migrate toward the internal carotid artery is higher for
particles released during systole and opposite trend is observed during diastole. During transition from systole
to diastole, the behavior becomes unstable and less predicable. Figure 9shows three illustrations of the particle
pathlines for releases during peak systole, transition from systole to diastole, and diastole. This provides a
visual explanation of the distribution fractions observed in Fig. 8. For the particles released during systole
and diastole, the characteristic swirling motion is observed from the particle pathlines. For particles released
during the diastole, they remain nearly statically suspended until the next cardiac cycle begins. Once systole
starts, these particles with nearly zero velocities are accelerated toward the regions of high local helicity.
This is in contrast with the particles released during the rest of the cardiac cycle where the initial velocity
with which they are released is more substantial. Upon entering the proximal region of high helicity, these
particles possess higher momentum as compared to the ones released during diastole. This leads to contrasting
level of interactions with the dominant swirling flow structures prior to approaching the bifurcation. This also
explains the extremal behavior during the transition phase—where the particles are released with a slower
initial velocity into a decelerating flow. The centrifugal accelerations on the particles become the dominant
forcing as opposed to the fluid–particle interaction, leading to a majority of the particles migrating toward the
outer extent of the curved vessel, until the next cardiac cycle begins. From Fig. 6, we see that these particles are
drawn into regions where helicity is low, and the flow is relatively straight—and hence they become strongly
preferentially directed toward the internal carotid. These observations provide additional insights into the
potential significance that helical flow may play in transporting embolic particles.
4 Concluding remarks and outlook
Coupled Euler–Lagrange CFD simulations have been presented to understand the mechanics of embolic particle
transport near the carotid bifurcation. Particle motion was modeled using a discrete particle dynamics technique.
The difference in embolus distributions with and without considerations of fully coupled, two-way fluid
particle interactions was approximately 15 and 5.3 %, respectively, for the idealized and the anatomical models.
Additionally, observed flow patterns and particle trajectories indicate that helical flow plays an important role
Numerical investigation of fluid–particle interactions
Fig. 9 Particle path lines for embolus particles released during peak systole, systole–diastole transition, and during diastole
(release times shown on left inset). Pathlines have been colored based on the branch vessel that the particle migrates to. Snapshot
left corresponds to 0.15 s, center corresponds to 0.45 s, and right corresponds to 0.70 s
in preferentially directing particles toward one branch as the particles approach a bifurcation. Specifically for
the observations from the numerical experiments presented here, the curvature and geometric irregularities, and
dominant swirling flow played a leading order effect when compared to two-way fluid–particle momentum
exchange. The distribution statistics therefore were seen to be more sensitive to coupling for the idealized
geometry—where geometry and flow helicity factors are secondary. The role of the flow helicity is further
elucidated by devising a kinematic swirl ratio for the individual, inertial embolus particle—and regions of
high swirl ratio with regions of high local normalized flow helicity. Since the particle trajectories result from
their interaction with such helical flow, for varying particle release times during the cardiac cycle, and varying
initial velocities, the particle trajectories show substantial variations. The resultant particle distribution ratios
are found to be notably different for particles released during systole and diastole.
It is noted that while the consideration of two-way coupling is theoretically more accurate and rigorous, it
adds significant computational costs to simulations of particle trajectories—especially since multiple embolic
particles cannot be simulated simultaneously owing to the alterations in flow by the momentum exchange from
the particle to the fluid. For the purpose of quantifying distribution statistics of emboli, typically samples of
many thousands of particles need to be simulated. Thus, the added computational costs could be detrimental to
the utility that numerical simulations can play in understanding general trends. It is therefore beneficial to have
observed that for small- to moderate-sized particles, in the presence of dominant, circulating flow structures,
the consideration of one-way fluid particle coupling is a reasonable first-order approximation. This statement,
however, must be dealt with a little more caution. For example, at the aortic arch where strong, pulsatile,
vortical flow structures are found, the motion of small to moderate particles can be estimated to a first-order
using one-way coupling models. However, it is not clear from the studies presented here as to whether the same
can be claimed about embolus transport in the arteries of the Circle of Willis—where the flow is relatively
slower and more unidirectional. Additionally, in order to understand the mechanics of embolic occlusion of
an artery—which ultimately leads to stroke—the role of fully coupled fluid–particle interactions, and also
detailed particle interactions with the vessel wall, cannot be ignored.
D. Mukherjee et al.
An important factor is the choice of an idealized inflow profile at the common carotid artery inlet. The
in vivo flow profile may not correspond to this idealized profile, and this will affect the type of helical flow
generated in the carotid artery. Instead, an inflow profile could be obtained by mapping measured flow data onto
the carotid inlet face, or extending the arterial domain proximally so that idealizations introduced at the inflow
boundary have minimal effect on the carotid artery flow structure. Nonetheless, the conclusions regarding the
relevance of vessel geometry in inducing complex helical flow, and the relative importance of coupling in the
presence of these flow structures, can be considered valid regardless of the choice of inflow profile. Namely,
the primary factors governing the comparisons presented here were relative particle diameter and curvature
differences between the anatomical geometry versus the idealized geometry. Regardless of inflow profile, the
vessel curvature induces marked differences in flow between the anatomical and idealized geometries, and
because of this, the two-way particle–fluid momentum exchange is a more prominent factor in the idealized
model where such flow structures are either negligible or non-existent.
A limitation of the numerical resolution of the two-way coupled interactions is that the particles are assumed
to interact with the fluid as distributed momentum sources. This may be reasonable for smaller particle sizes,
but cannot be extended to particle sizes that are significantly greater than the computational cell size. A rigorous
treatment of the coupling will then necessitate a better resolved treatment of the particle fluid boundary, and
finer mesh for capturing the flow around the boundary. The development of an appropriate numerical method
for this is currently being investigated.
We also remark that the investigations on the particle interactions with the flow structures were only intended
to provide insights into an appropriate hemodynamic descriptor for embolic particle transport. In that regard,
the observations from this study were found to be indicative of the notable effect of local flow helicity. The
study presented here does not span multiple patient geometries to be able to establish statistical correlations
regarding particle interactions with helical flow. Such investigations will further enhance the role of helicity
in the mechanics of embolus transport—and are currently being investigated by the authors. Additionally,
helical nature of flow could be either affected by, or correlated with, other physiological factors, and thereby
could also alter embolus distribution across an arterial segment. Detailed insights into how the effect of such
physiological factors can be quantified and used in stroke diagnostics to characterize stroke risks, requires
further investigations.
Swirling, helical flow is a common occurrence in various locations along the major arteries of the cir-
culation pathway. The observation that flow helicity has a prominent role to play, and that volumetric flow
distribution, and particle distribution fractions alone are insufficient to represent this effect is of great rele-
vance for understanding the mechanisms underlying cardiogenic embolic stroke, and hemodynamic factors
affecting stroke risks. Existing studies in the literature oftentimes describe experimental results on particle
distribution fractions in terms of volumetric flow ratios alone—thereby presenting an empirical correlation for
the data. In the context of the insights gained from this paper, this may be an oversimplification in the context
of representing the fluid–particle interaction. Additionally, idealized bifurcation geometries have often been
used as experimental models for performing studies on particle transport across bifurcation, and quantifying
their ratios. Such experimental setup will lead to inaccuracies, and will not be able to capture the effect of the
swirling flow and vessel curvature—phenomena that were found to have substantial role in particle transport
within the arteries supplying the brain.
Acknowledgments This research was supported by the American Heart Association, Award No. 13GRNT17070095. JP was
supported through the “Transfer To Excellence Summer Research Internship” program organized by U.C. Berkeley, and supported
by the National Science Foundation.
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... Several studies have applied particle tracking to identify flow disturbances in, e.g., carotid bifurcation models, contributing to providing a deeper understanding of the hemodynamics-driven processes underlying atherosclerosis onset/progression [39][40][41][42]. Moreover, particle tracking has been used to study the hepatic perfusion in the Fontan circulation [43,44], identify the ∇Φ 0 ( ) ≈ [ optimal left ventricular assist device cannula outflow configurations [45], obtain a deeper understanding of the dynamics of embolic particles within arteries [46], and detect peculiar intravascular helical flow patterns in the aorta from in vivo, 4D-flow MRI data [47,48]. ...
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