5th International Conference on Computational and Mathematical Biomedical Engineering - CMBE2017
10–12 April 2017, United States
P. Nithiarasu, A.M. Robertson (Eds.)
DISCRETE PARTICLE MODELING FOR THROMBOTIC AND
EMBOLIC PHENOMENA IN ARTERIES
Debanjan Mukherjee, Neel D. Jani, and Shawn C. Shadden
University of California, Berkeley, Department of Mechanical Engineering, Berkeley, CA, USA
Thrombosis and embolisms are strongly inﬂuenced by blood ﬂow and hemodynamic loading, and
resolving this connection is a challenge. Discrete particle methods, in conjunction with image-based
modeling and computational ﬂuid dynamics, provide viable avenues to address this. We present
two speciﬁc case-studies for elucidating particle based models for thrombosis and embolus transport.
We ﬁrst illustrate a one-way coupled ﬂuid-particle framework for investigating the role of cerebral
artery anatomy on embolus transport. Thereafter, we describe a ﬁctitious domain, discrete particle
framework for modeling ﬂow around and within a clot. Using these examples, we demonstrate the
utility and efﬁcacy of particle-based modeling techniques for thrombotic and embolic phenomena.
Key words: thrombosis, embolism, stroke, particles
Thrombosis and embolisms comprise the primary cause of several major cardiovascular diseases in-
cluding stroke and heart attack. In addition, embolic events during surgery, and medical device in-
duced thrombosis, often cause serious complications and adversely affect patient health and recovery.
These phenomena are intimately related to blood ﬂow and hemodynamic forces. However, predictive
understanding of the interaction of an arbitrarily shaped clot (thrombus) or an embolus with complex
hemodynamics within real human anatomy poses challenges. The interaction of a ﬂoating embolus
with unsteady ﬂow structures characteristic of large arteries is complex, and often chaotic. This ren-
ders difﬁculty in predictively identifying locations where the embolus is transported to. Correspond-
ingly, realistic thrombi in large arteries have arbitrary aggregate morphology and microstructure,
often varying with time. Evaluation of ﬂow structures around, and hemodynamic loading on, such
an aggregate geometry comprises a challenging task. Computational tools devised using a combina-
tion of image-based modeling, computational ﬂuid dynamics, and discrete particle methods provide
a suitable alternative to address these challenges. In prior works, we have employed such methods
for understanding the transport of emboli along arteries , and embolus distribution to the cere-
bral arteries for stroke . Here we provide two speciﬁc computational case-studies, that further
establish the utility and efﬁcacy of these models in addressing the aforementioned challenges, and
answering key questions pertaining to stroke and thrombosis. First, using a combined ﬂuid-particle
simulation framework, we illustrate evidence on the role of cerebral artery anatomy and topology in
affecting embolic stroke risks. Second, we evaluate ﬂow around and within a thrombus with varying
microstructure using a ﬁctitious domain discrete-particle method, which circumvents the difﬁculties
in resolving and meshing arbitrary geometries and microstructures typical of realistic thrombi.
2.1 Computational ﬂuid dynamics
Image data was used to generate the model for a computational domain. For the embolus transport
case-study, this was achieved through image segmentation and lofting operations based on patient
computed tomography (CT) scans. For the thrombus case-study, ﬂuorescent microscopy data was
used to identify the thrombus boundary and potential locations of platelets within the thrombus ,
and a 2D thrombus was reconstructed within a channel. Once created, the respective computational
domains were discretized into meshes comprising linear triangular (for 2D) and tetrahedral (for 3D)
elements. The incompressible Navier-Stokes equation for momentum, and the continuity equation,
were then solved using a Petrov-Galerkin stabilized ﬁnite element formulation . For the patient
arterial hemodynamics simulations, inlet ﬂow boundary conditions were assigned based on measured
inﬂow proﬁle at the aortic inlet presented in literature. For all arterial outlets, lumped resistor outﬂow
boundary conditions were employed. Blood density was assumed to be 1.06 g/cc, and viscosity 4.0
cP. The resistance values for the six major cerebral arteries were tuned to achieve average ﬂow rates
obtained from measured MR data. Outﬂow resistor values at all other arterial outlets were chosen by
dividing the remainder ﬂow in proportion to their cross-sectional areas. The patient hemodynamics
simulations were run for three successive cardiac cycles for convergence, and the ﬁnal cardiac cycle
ﬂow-data was assumed periodic thereafter for subsequent particle transport calculations . For
the thrombus case-study hemodynamics simulations, the inﬂow boundary condition consisted of a
speciﬁed parabolic velocity proﬁle, and a standard constant pressure outﬂow boundary condition was
employed. The ﬂuid was assumed to be plasma with density 1.025 g/cc, and viscosity 1.7 cP. The ﬂow
around the thrombus was computed for a few time-steps for velocity and pressure ﬁelds to achieve
convergence, and the converged velocity was compared across various thrombus microstructures.
2.2 Dynamics of embolic particles
For the embolus transport case-study, embolic particles were assumed to be spherical, and their dy-
namics was modeled using a one-way coupling scheme based on a modiﬁed form of the Maxey-Riley
equation [4, 2]. An ensemble of 5,684 emboli were released from locations sampled on the inlet plane
at the ascending aorta - each representative of an individual sample of a cardiogenic thrombo-emboli
(density: 1.10 g/cc) of diameter 1.0 mm. These sampled emboli trajectories were simultaneously
integrated, without any collisional interactions between them (independent samples). Collisional in-
teraction between each embolus and artery wall was handled using a modiﬁed elastohydrodynamic
collision model. In this model, collisional velocities were obtained as a function of normal and tan-
gential restitution coefﬁcients, which were derived assuming pairwise viscoelastic collisions, and
modiﬁed further to account for additional energy dissipation due to increased pressure loading from
the lubrication layer of ﬂuid in the contact region. The embolus dynamics was computed for 10
cardiac cycles, based on looped ﬂow-data from the ﬁnal cardiac cycle of the patient-speciﬁc arte-
rial ﬂow simulations as described above. The ﬁnal particle distribution to each outlet was extracted,
and divided by the ensemble size to get distribution fractions (which correspond to probability of an
embolic event at locations following the outlet).
2.3 Fictitious domain discrete element method for thrombus
Evaluation of hemodynamics around a thrombus requires resolving the inﬂuence of the thrombus
on the ﬂow. Efﬁcient mesh or lattice based descriptions of the arbitrary thrombus geometry and
blood-thrombus interface, that can grow or shrink with time, is difﬁcult. Resolution of thrombus
internal microstructure for understanding intra-thrombus transport renders added complications. Here
we devised an alternative strategy where the arbitrary shape and micro-structure of the thrombus is
handled by representing it using a collection of mesh-free, off-lattice, discrete elements or particles.
Each discrete element was modeled as a superquadric geometry object, whose shape and size can
be parametrically varied, and which are characterized by an analytical level-set function classifying
the mesh or quadrature points inside/outside of an element. The inﬂuence of this discrete element
aggregate was then included within the stabilized ﬁnite element formulation mentioned in Section
2.1 by (i) replacing the separate blood and thrombus domains with a single, continuous, and simpler
‘ﬁctitious domain’ computational mesh, (ii) embedding the discrete element representation of the
thrombus within this ﬁctitious domain, and (iii) incorporating the interaction of the thrombus with the
ﬂow using a penalty term added to the stabilized variational form within the thrombus domain. The
penalty formulation imposes the ﬂuid ﬂow velocity at any point to take up the corresponding local
velocity of the thrombus subdomain, thereby coupling the two domains.
3 RESULTS AND CONCLUSIONS
3.1 Inﬂuence of Circle of Willis anatomical variations on embolic occlusion risks
Figure 1: Illustration of variations in embolus transport to the six major cerebral vasculature regions across
six topological variations in Circle of Willis anatomy. Panel a denotes the topologies obtained by removing
anterior communicating artery (AcoA), left/right communicating arteries (L/R.Comm), and left/right posterior
connecting segment (L/R.P1). Corresponding embolus distribution fractions are denoted in panel c (within
square brackets), and variations in embolus distribution are compared with ﬂow distribution in panel b.
Data from a collection of 25 research articles reporting the frequency of the various observed anatom-
ical variations of the Circle of Willis (CoW) were compiled. By using ranking statistics to categorize
and order this data, the ﬁve most common anatomical variations were selected (see Fig:1, panel a).
Embolus distribution fractions to the six major cerebral arteries for the complete CoW topology, and
the selected ﬁve incomplete topologies created by iteratively detracting vessels from the complete
topology, were then compared and contrasted. The distribution fractions (in terms of percentage of
total emboli reaching the CoW) have been presented in Fig:1, panel c. Panel b compares the coefﬁ-
cient of variation (ratio between standard deviation and mean) for the ﬂow distribution and embolus
distribution to each of the six cerebral arteries, across all the six CoW topologies considered. Since
the boundary conditions for all the anatomical models were tuned to the same cerebral artery outﬂow
data, and all other factors barring the CoW anatomy were held ﬁxed, we observe minimal variations
in ﬂow distribution. However, signiﬁcant variations in embolus distribution are observed, being about
5 times or higher compared to ﬂow distribution for most cases. With all other factors being controlled
for, these differences can be attributed to the variations in CoW anatomy. Embolic particles, owing
to their inertia, do not follow ﬂow distribution exactly, and are likely to be rerouted differently across
the different arterial topologies of the CoW - thus affecting embolus transport and embolic stroke
risks. Further analysis of ﬂow through the communicating arteries of the CoW may elucidate how
blood ﬂow reroutes itself to maintain speciﬁc ﬂow to each cerebral vascular bed, and explain how
they correspondingly inﬂuence inertial embolic particles to reroute and travel to these vascular beds.
3.2 Extra-thrombus and intra-thrombus ﬂow
For this second case-study, the system described in  was employed as the model system, with a
thrombus obtained from an injured mouse cremaster muscle, and reconstructed using discrete ele-
ments with planar aspect ratio 1:0.6. For an inlet ﬂow with peak centerline velocity of 2 mm/s, the
ﬂow ﬁeld obtained using the devised ﬁctitious domain approach (Section 2.3) has been presented in
Figure 2: Results for ﬂow around and within a clot with platelets represented as discrete elements. Base
clot model, created based on experimental data, is modiﬁed to generate clots with varying microstructure.
Intra-thrombus ﬂow is orders of magnitude less than extra-thrombus ﬂow, and governed by morphology and
microstructure of the clot.
Fig: 2 (left). The observed peak extra-thrombus ﬂow velocity is 3.2 mm/s, while peak intra-thrombus
velocity is around 3.29 µm/s. These observations, along with the spatial ﬂow pattern around the
thrombus and within the thrombus interstices, are in excellent agreement with the reported results
in , obtained from numerical simulations performed by explicitly meshing the clot (no embedded
domains). Thrombi with varying microstructures were then generated by using the parametric discrete
element shape deﬁnitions to methodically vary the shape and/or size parameters for each individual
element representative of a platelet. This has been illustrated for three different variations in clot
morphology (relative to the original morphology) in Fig: 2 (M1-M3, right). Intra-thrombus velocity
is observably inﬂuenced by pore space size as well as pore connectivity. In particular, morphology
M2 is more porous as compared to the remaining three (including the original), enabling higher ﬂow.
Variations in microstructure and porosity were also observed to have small but noticeable inﬂuence on
extra-thrombus ﬂow. This was observed, for example, by comparing peak extra-thrombus velocities,
which is highest for morphology M3 (3.33 mm/s), and lowest for M2 (3.26 mm/s) with a velocity dif-
ference of about 70 µm/s, that is likely to increase for higher incoming ﬂow at the inlet. These results
and observations clearly indicate that not only does the devised ﬁctitious domain discrete-element
method resolve the appropriate ﬂow characteristics around a thrombus, but it also enables ﬂexible
representation of thrombus morphology information and rapid analysis of inﬂuence of microstructure
variations on ﬂow and intra-thrombus transport.
The Authors acknowledge the support of the American Heart Association through two grants: 13GRNT-
17070095 and 16POST-27500023. Computational resources for large parts of this research is pro-
vided by the Berkeley Research Computing program through the Savio compute cluster.
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