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Fictitious Domain Particle-Based Modeling For Thrombosis

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Flow around and within a micro-scale thrombotic platelet plug, alongwith illustration of variations in intrathrombus flow for varying internal thrombus microstructure. For the micro-scale flow-thrombus interaction simulation, we chose the model system described in [4]. Specifically, a thrombus formed within an injured vessel in a mouse cremaster muscle was imaged, and reconstructed using discrete particles. Each particle was a superquadric with planar aspect ratio 1.0:0.6. The reconstructed thrombus was embedded within a rectangular channel with width representative of a small arteriole, and a parabolic flow profile with peak centerline velocity 2 mm/s was imposed at the vessel inlet. The background fluid was plasma (density: 1.025 g/cc, viscosity: 1.7 cP), and the corresponding flow velocity fields around and within the reconstructed thrombus is presented in Fig: 2. The observed maximum flow velocity within the channel was 3.2 mm/s, while peak intrathrombus velocity was 3.29 µm/s. These values, alongwith the spatial variation of flow, is in excellent agreement with the results obtained from simulations with explicitly meshed clots (i.e. no embedded domains) as reported in [4]. The discrete particle approach enables modeling varying clot microstructures by further parametric modifications of individual discrete element shape/size. This has been illustrated in Fig: 2 for three different variations (M1-M3) in thrombus morphology with respect to the original thrombus. Corresponding intra-thrombus flow velocities are observed to be influenced by thrombus interstitial space and connectivity. Microstructural variations are also observed to have small but noticeable influence on extra-thrombus flow. For example, when the peak extra-thrombus velocities are compared, the highest is observed for model M3 (3.33mm/s) and the lowest for model M2 (3.26 mm/s). The difference between the two is 70 µm/s, likely to increase with higher incoming flow at the inlet, and caused primarily by variations in interstitial space morphology.
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SB3C2017
Summer Biomechanics, Bioengineering and Biotransport Conference
June 21–24, 2016, Tucson, AZ, USA
FICTITIOUS DOMAIN PARTICLE-BASED MODELING FOR THROMBOSIS
Debanjan Mukherjee and Shawn C. Shadden
Department of Mechanical Engineering
University of California, Berkeley
Berkeley, CA, USA
INTRODUCTION
Thrombosis is a key cause of severe cardiovascular diseases like heart
attack and stroke [1]. In addition, it leads to major complications dur-
ing surgical procedures, and medical device deployment. Blood flow and
hemodynamic loading plays a crucial role in thrombosis. The evaluation
of blood flow patterns and flow-induced loading on a realistic blood clot
within an anatomically realistic vasculature is a complicated task. The
complication arises primarily due to the complexity of interactions be-
tween unsteady, pulsatile flow of blood with an arbitrarily shaped aggre-
gated blood clot (thrombus). Realistic thrombi in large arteries possess
arbitrary aggregate morphology and microstructure varying with time as
the thrombi grow (or embolise). Efficient mesh or lattice-based descrip-
tion of the blood-thrombus interface, that can grow or shrink with time,
is difficult. Furthermore, understanding intra-thrombus flow and transport
requires resolution of thrombus internal microstructure, which is hetero-
geneous and non-uniform. Resolving these characteristic microstructures
using conventional mesh based methods is difficult, and often a homoge-
nized porous medium approach is employed which does not use an explicit
description of the internal microstructure.
Here, we address this challenge by proposing a modeling framework using
a combination of fictitious domain finite element method and discrete el-
ement method. The central premise here is to replace the complex blood-
thrombus domain by a simpler, regular, “fictitious” domain that consti-
tutes a background mesh, and numerically embed the blood (fluid), and
thrombus (solid) domains. The thrombus domain is further described us-
ing a mesh-free off-lattice discrete element method which renders substan-
tial flexibility in representation of complex thrombus geometry and mi-
crostructure. Two illustrative numerical examples are provided to demon-
strate the efficacy of the proposed approach.
METHODS
We consider a single computational domain, referred to as fictitious do-
main as above, and assumed blood to be an incompressible, Newtonian
fluid whose motion across this domain was modeled using a Petrov-
Galerkin stabilized finite element formulation. The velocity and pressure
boundary conditions on the original blood domain were imposed on the
corresponding boundaries in the fictitious domain. The influence of the
clot sub-domain on blood flow was then accounted for by imposing a
source function within the Petrov-Galerkin variational formulation for the
flow. For the present study, this function was chosen in form of a penalty
term that enforces the flow velocity to match the local velocity within the
thrombus sub-domain. The penalty constant for this added penalty term
was scaled with element size to ensure the additional penalization does not
induce ill-conditioning of the resultant matrix system of equations gener-
ated from the finite element formulation. The overall discretized system
was integrated over time using a one-step midpoint integration scheme.
The implementation of this penalty coupling implicitly assumed a defi-
nition of the thrombus sub-domain. The arbitrary shape and microstruc-
ture of the thrombus sub-domain was handled here by representing the
thrombus as a collection of discrete particles. This involved identifying
the thrombus manifold geometry information from medical images (for
macro-scale thrombi) or platelet centroidal locations from fluorescent mi-
croscopy images (for micro-scale thrombus geometry), and generating an
ensemble of particles based on this information. Illustrative examples of
this particle based representation procedure has been presented in Fig: 1.
Within the ensemble, each discrete particle was modeled as a superquadric
FIGURE 1: Illustration of the reconstruction of a thrombus us-
ing discrete particles for micro- (a) and macro-scale (b) clots.
object, which can be uniquely identified using three size parameters and
two shape parameters (in 3D) [2]. Each superquadric discrete particle has
a uniquely defined in-out function that helps identify computational nodes
that lie inside or outside of a particle. This was used to identify the inte-
rior locations of the overall thrombus sub-domain for imposing the penalty
term. Additionally, the flexibility of superquadric discrete particle repre-
sentation enabled easy manipulation of individual particle geometries by
modifying the shape and size parameters. This enabled rapid variations
in microstructure and morphology for a given thrombus sub-domain con-
figuration. The complete formulation was implemented as a solver using
the open-source finite element library FEniCS [3]. Here we present results
from two representative numerical case-studies that demonstrate the effi-
cacy of the proposed fictitious domain discrete particle method in resolv-
ing macro-scale flow around a clot in a large artery, as well as micro-scale
intra-thrombus flow.
1
Poster Presentation #P252
Copyright 2017 The Organizing Committee for the 2017 Summer Biomechanics, Bioengineering and Biotransport Conference
RESULTS
Microscale interactions
FIGURE 2: Flow around and within a micro-scale throm-
botic platelet plug, alongwith illustration of variations in intra-
thrombus flow for varying internal thrombus microstructure.
For the micro-scale flow-thrombus interaction simulation, we chose the
model system described in [4]. Specifically, a thrombus formed within an
injured vessel in a mouse cremaster muscle was imaged, and reconstructed
using discrete particles. Each particle was a superquadric with planar as-
pect ratio 1.0:0.6. The reconstructed thrombus was embedded within a
rectangular channel with width representative of a small arteriole, and a
parabolic flow profile with peak centerline velocity 2 mm/s was imposed
at the vessel inlet. The background fluid was plasma (density: 1.025 g/cc,
viscosity: 1.7 cP), and the corresponding flow velocity fields around and
within the reconstructed thrombus is presented in Fig: 2. The observed
maximum flow velocity within the channel was 3.2 mm/s, while peak intra-
thrombus velocity was 3.29 µm/s. These values, alongwith the spatial
variation of flow, is in excellent agreement with the results obtained from
simulations with explicitly meshed clots (i.e. no embedded domains) as re-
ported in [4]. The discrete particle approach enables modeling varying clot
microstructures by further parametric modifications of individual discrete
element shape/size. This has been illustrated in Fig: 2 for three different
variations (M1-M3) in thrombus morphology with respect to the original
thrombus. Corresponding intra-thrombus flow velocities are observed to
be influenced by thrombus interstitial space and connectivity. Microstruc-
tural variations are also observed to have small but noticeable influence on
extra-thrombus flow. For example, when the peak extra-thrombus veloci-
ties are compared, the highest is observed for model M3 (3.33mm/s) and
the lowest for model M2 (3.26 mm/s). The difference between the two
is 70 µm/s, likely to increase with higher incoming flow at the inlet, and
caused primarily by variations in interstitial space morphology.
Macroscale interactions
For demonstrating the efficacy of the proposed fictitious domain method
in resolving macro-scale thrombus-hemodynamics interactions, a model
system comprising an occlusion within an idealized channel was devised.
As described in Fig: 3.a., the channel width (dv) was assumed to be that
of the common carotid artery (6.0 mm), with a measured inflow profile
imposed at the channel inlet to drive the flow (Fig: 3.b.). An idealized
hemispherical occlusion was embedded within the channel, with a radius
dc. The flow was started from rest, and computed over three cardiac cy-
cles, and the flow field obtained from the final cardiac cycle was compared
across varying occlusion sizes (in comparison with channel width). The
corresponding flow fields obtained for three different occlusions (dc/dv=
0.25, 0.33, 0.42) have been presented across five successive instants dur-
ing a cardiac cycle in panels c1-c3 in Fig: 3. All velocity magnitudes have
been scaled to 700.0 cm/s. The results capture the temporally varying re-
circulation region and flow reattachment distal to the occlusion, and the
resultant vortical structures in the vicinity of the clot. This is relevant
for progression of thrombotic activity in large artery thrombosis, since
the resultant temporally varying, oscillating, shear loading on platelets,
and the extent of recirculatory flow have an intimate role to play in me-
chanical platelet activation and subsequent aggregation of platelets. The
results from the simulations presented here clearly indicate that the pro-
posed numerical method is capable of capturing such spatiotemporally
varying flow in the proximity of a large-artery thrombus. We remark here,
in addition, that while the model system here involved an idealized cir-
cular/hemispherical occlusion, following the flexibility of shape represen-
tation as described in the previous sections, this can now be extended to
model macroscopic occlusions of any shape.
FIGURE 3: Macroscale interactions between an idealized
thrombotic occlusion and pulsatile hemodynamics, in a chan-
nel of width equivalent to common carotid artery.
DISCUSSION
The results presented here establish the ability of the proposed fictitious
domain discrete particle framework to resolve thrombus-hemodynamics
interactions and potentially enable modeling of both intra-thrombus and
extra-thrombus flow. While the former has broad implications in under-
standing transport of coagulation agonists and thrombolytic drugs within
an existing thrombus, the latter is of key interest in the biomechanics of
thrombotic disease progression and embolization. The premise of para-
metrically defined discrete particle aggregates embedded within a simpli-
fied mesh addresses the challenge of modeling realistic thrombus mor-
phologies. However, when compared with an explicitly meshed blood-
thrombus interface, in the proposed fictitious domain approach, we recover
only a discretized approximation of that interface. This may be associ-
ated with numerical errors, and a closer inspection of the bounds of this
approximation error is underway. The imposition of a single simplified
background mesh, also enables this framework to directly use the native
parallelization of existing parallel flow solvers, thus allowing for poten-
tially large-scale patient-specific simulations. This is also currently an area
of active research interest.
ACKNOWLEDGMENTS
This research was supported by the American Heart Association, Award
No. 16POST27500023.
REFERENCES
[1] Raskob, G.E., et.al. Arterioscler. Throm. Vas., 34(11):2363-71, 2014.
[2] Barr, A.H. IEEE Comput. Graph., 1.1:11-23, 1981.
[3] Alnaes, M.S., et.al. Arch. Numer. Software, 3(100), 2015.
[4] Tomaiuolo, M., et.al. Blood, 124(11):1816-1823, 2014.
2
Poster Presentation #P252
Copyright 2017 The Organizing Committee for the 2017 Summer Biomechanics, Bioengineering and Biotransport Conference
ResearchGate has not been able to resolve any citations for this publication.
  • G E Raskob
Raskob, G.E., et.al. Arterioscler. Throm. Vas., 34(11):2363-71, 2014.
  • A H Barr
Barr, A.H. IEEE Comput. Graph., 1.1:11-23, 1981.
  • M Tomaiuolo
Tomaiuolo, M., et.al. Blood, 124(11):1816-1823, 2014.
  • M S Alnaes
Alnaes, M.S., et.al. Arch. Numer. Software, 3(100), 2015.