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Discrete Particle Modeling For Thrombotic And Embolic Phenomena In Arteries

5th International Conference on Computational and Mathematical Biomedical Engineering - CMBE2017
10–12 April 2017, United States
P. Nithiarasu, A.M. Robertson (Eds.)
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 [1], and embolus distribution to the cere-
bral arteries for stroke [2]. 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 [3],
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 [2]. 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 [2]. 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.
3.1 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 [3] 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 [3], 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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sessment of the relation between embolism source and embolus distribution to the circle of
willis for improved understanding of stroke etiology. Journal of Biomechanical Engineering,
138:081008-081008-13, 2016.
[3] M. Tomaiuolo, T.J. Stalker, J.D. Welsh, S.L. Diamond, T. Sinno, and L.F. Brass. A systems
approach to hemostasis: 2. Computational analysis of molecular transport in the thrombus mi-
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[4] M. Maxey, and J. Riley. Equation of motion for a small rigid sphere in a nonuniform flow.
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ResearchGate has not been able to resolve any citations for this publication.
Full-text available
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.
Stroke caused by an embolism accounts for about a third of all stroke cases. Understanding the source and cause of the embolism is critical for diagnosis and long-term treatment of such stroke cases. The complex nature of the transport of an embolus within large arteries is a primary hindrance to a clear understanding of embolic stroke etiology. Recent advances in medical image-based computational hemodynamics modeling have rendered increasing utility to such techniques as a probe into the complex flow and transport phenomena in large arteries. In this work we present a novel, patient-specific, computational framework for understanding embolic stroke etiology, by combining image-based hemodynamics with discrete particle dynamics and a sampling-based analysis. The framework allows us to explore the important question of how embolism source manifests itself in embolus distribution across the various major cerebral arteries. Our investigations illustrate prominent numerical evidence regarding (i) the size/inertia dependent trends in embolus distribution to the brain, (ii) the relative distribution of cardiogenic versus aortogenic emboli amongst the anterior, middle, and posterior cerebral arteries, (iii) the left versus right brain preference in cardio-emboli and aortic-emboli transport, and (iv) the source-destination relationship for embolisms affecting the brain.
Key Points Hindered diffusion becomes the dominant force of molecular movement in a thrombus. The thrombus core acts as a selective molecular prison.