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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 35−40 % 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+∇·(ρfφu)=∇·φ∇φ+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,

∇·φ∇φ=−∇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+φ∇φ·ndA+V

SφdV(2)

∂(ρφ)

∂tV=−

1,Nfaces

k

ρfφkuk·nkAk+

1,Nfaces

k

φ,k∇φkuk·nkAk+SφV(3)

The temporal component is discretized using a one-step implicit Euler scheme, as follows:

∂(ρφf)

∂tV=F(φ(t), t)⇒φN+1−φN

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., Qi∝Ai[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|u−vp|u−vp

steady drag force

+μf

πD3

p

8∇2u

Faxen corr. drag

+mpg

gravity

+Vp(−∇p+∇·τ)

ﬂuid stresses, undisturbed ﬂow

+CaρfVp

2Du

Dt −dvp

dt

added mass, unsteady

+CaρfVp

2

d

dtD2

p∇2u

40

Faxen corr. added mass

+3

2D2

p√πμfρft

0

K(t−t∗)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=u−vpρfDp/μf.

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

τpu∗−v∗

p+1

24 ·1

τp·Dp

U∇∗2u∗+g∗

+ρf

ρpDu∗

Dt∗−g∗+Ca

2·ρf

ρpDu∗

Dt∗−dv∗

p

dt∗+1

40

d

dt∗∇∗2u∗

+3

√2π·ρf

ρp·1

√τp·Dp

UK∗·g∗dt∗+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 ρf/ρpclose to unity (which is the case for embolus and blood, ρf/ρp=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|u−vp|u−vp+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

p−vN

p

t=θ

mp

FN+1

p+(1−θ)

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+δt∗fN,wdt

t+δt∗

tfN,wdt=fN,w R(δt−δt∗)

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,n−mpvp,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: δt≈Ctand 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 =Gxp−xnSp,(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 Hn=±1, 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=x⊥−xp

x⊥−xp.(21)

Additionally, a tangential ﬂow direction can be speciﬁed at the location of the embolus as

ez=xk+1−xk

xk+1−xk,(22)

where kand k+1 denote the points immediately preceding and following x⊥on 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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