Study of blood flow impact on growth of blood clot using a multiscale model

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Study of blood flow impact on growth of blood
clot using a multi-scale model

• Mark Alber
• malber_at_nd.edu
• Departments of Mathematics and Physics and
• Interdisciplinary Center for the Study of
  Biocomplexity
• http//www.nd.edu/icsb/
• University of Notre Dame, Notre Dame, IN 46556,
  USA
• July 22, 2009

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Blood Clot Formation

• Following a rupture in blood vessels, components
  in blood and the vessel wall interact rapidly to
  form a clot to limit hemorrhage. This hemostatic
  response is not only rapid but regulated, since
  excessive and inappropriate clotting in a damaged
  blood vessel can block blood flow. The biomedical
  importance of these reactions is underscored by
  the severe and lethal consequences resulting from
  failure and disregulation of the hemostatic
  system.
• While significant advances have been made in
  understanding the processes involved in the
  development of a thrombus, many important
  questions remain unresolved. In particular, it is
  not clear why a thrombus initiated after vascular
  injury stops growing. Current models suggest that
  thrombus development involves incorporation of
  activated platelets and procoagulant
  microparticles on the thrombus surface generating
  a prothrombotic environment for continued growth.
  While many anti-thrombotic processes might limit
  the growth of a developing thrombus, there is
  limited experimental evidence demonstrating the
  role of any particular mechanism.

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Thrombus Development
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Interface
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Cell-State Transition Map
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Cellular Dynamics extended CPM

• Energy minimization formalism
• extended by Graner and Glazier, 1992
• DAH Contact energy depending on cell types
  (differentiated cells)
• Extensions
• J_cell_cell is type dependent
• Other terms Cell volume, Chemotaxis/Haptotaxis
• Metropolis algorithm probability of
  configuration change

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Cellular Potts Model
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Additional Terms in the Hamiltonian
The flow force applied to a cell i with high
level of fibril (platelet or blood cell) is
calculated as an integral of blood pressure along
a cell membrane. Pk is the pressure applied to
the bloodcell interface segment k nk is the
inward unit normal of the bloodcell interface
segment k and Sk is the membrane length of the
bloodcell interface segment k.
The flow energy change for the cell i for a given
state change. Ddi is the change in the position
of the center of mass of cell i caused by the
state change and Ke1 is a flow energy constant.
Cells (platelets and blood cells) inside of a
cell cluster do not have direct blood contact and
flow energy for those cells is zero.
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Multiscale model of thrombus formation
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Chen et al. Journal of the Royal Society
Interface 2008
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Different Pulse Rates
The effect of blood flow rate on the clot growth
non-Newtonian flow
Xu, Z., Chen, N., Shadden, S., Marsden, J.E.,
Kamocka, M.M., Rosen, E.D., and M.S. Alber, Study
of Blood Flow Impact on Growth of Thrombi Using a
Multiscale Model, Soft Matter 5, 769779 (2009)

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Analysis of the flow near the clot using
Lagrangian Coherent Structures (LCS)
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Sub-cellular element modeling

• In contrast to the Cellular Potts models (CPM), a
  dynamical system is developed that evolves in
  time. Cell adhesion, elasticity and flow coupling
  can be directly related to real-world
  measurements. No empirically derived parameters,
  as in CPM.
• Harmonic restraints are introduced between the
  sub-cellular elements (SCE) to give the correct
  cell shape and elasticity. The resolution is
  determined by the number of elements.
• Inter-Cellular Adhesion is modeled by using the
  Morse or Lennard-Jones potentials.
• Navier-Stokes Flow can interact with the cells
  via velocity/surface accessible area of the SCE,
  or an Immersed Element Method.

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Sub-cellular element interactions
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As an alternative submerged particles, we can
couple the SCE dynamics (SCED) to flow in a
similar manner to the current CPM coupling of
                . The CPM
coupling has no steady state solution (a better
method would be                        ), so
must
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Sub-cellular element forces

• Elasticity harmonic restraint energy.
• Cell adhesion Lennard-Jones potential energy.

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The Effects of FVII on the Structure of Venous
Thrombi
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Thrombi formed in wild type and low FVII animals
(R10, about 13 min).
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Diagram shows complex network of reactions
included in the coagulation (black),
anticoagulant Protein C (red) and fibrinolytic
pathways (green). This includes both
solution-phase and membrane-phase reactions in
which the concentrations of membrane
binding-sites are limited and treated as control
variables.
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Delayed thrombin production in low factor VII
case. The delayed thrombin generation is
consistent with the experimental results. Because
of low thrombin level at the initial stage, less
platelets are activated and a very small amount
of fibrinogen is converted to fibrin and
formation of fibrin network occurs at a much
later time. Simulations suggest that FVII/TF is
required to stabilize the clot following initial
aggregation of platelets. Interestingly, thrombi
in fibrinogen deficient mice, keep rowing after
injury, so we hypothesize that the stabilization
of the clot is not solely dependent on fibrin.
FVII/TF is doing something other than just
generating fibrin. Most likely, FVII/TF thrombin
generation might be doing something to increase
'adhesiveness' or stability of platelet
aggregates in the absence of fibrinogen.
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(a) Scanning electron micrographs (SEMs) of
fibrin networks obtained by Dr. Wolberg lab at
the University of North Carolina at Chapel Hill,
(b) A brin ber network representation, (c) A
2-dimensional microscopic domain of brin network
model 500 branch points in a 100 m square
domain.
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• R. A. Campbell, K. A. Overmyer, C. R. Bagnell,
  and A. S.Wolberg. Cellular procoagulant
• activity dictates clot structure and stability as
  a function of distance from the cell surface.
  Arterioscler Thromb Vasc Biol, 28(12)22472254,
  2008.
• P. L. Chandran and V. H. Barocas. Ane versus
  non-ane bril kinematics in collagen
• networks theoretical studies of network
  behavior. J Biomech Eng, 128(2)259270, 2006.
• P. L. Chandran and V. H. Barocas. Deterministic
  material-based averaging theory model
• of collagen gel micromechanics. J Biomech Eng,
  129(2)137147, 2007.
• .

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Future Plans

• Model validation
• Extension to 3D and Parallelization
• Incorporation of various coagulation (anti
  coagulation) pathway models
• Fibril network submodel
• Use multiscale models to run predictive
  simulations

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Zhiliang Xu, Chris Sweet, EunJung Kim, Joshua
Lioi, Department of Mathematics, University of
Notre Dame Malgorzata Kamocka, Elliot Rosen,
Department of Medical and Molecular Genetics,
Indiana University School of Medicine Danny
Chen, Computer Science and Engineering
Department, University of Notre Dame Pavel
Lushnikov, Department of Mathematics, University
of New Mexico NSF DMS-0800612 NIH
1R0-GM076692-01 Interagency Opportunities in
Multiscale Modeling in Biomedical, Biological and
Behavioral Systems NSF 04.6071
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Continuous Limit of the CPM
diffusive time-space scaling
Master equation. Tl(x,L x,L) and Tr(x,L
x,L) correspond to transitional probabilities
for a cell of length L and center of mass at x to
change into a cell of length L and center of mass
at x. Subscripts l and r correspond to a
transition due to the addition removal of a pixel
from the left right side of a cell, respectively.
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Fokker-Planck Equation and Keller-Segel Model
continuous time variable
Emin is a minimum of energy E(L) as a function of
L for a fixed x.
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Continuous Limit of the 2D CPM with excluded
volume constraint
Alber et al. Phys.Rev.Lett. 2007 19
99(16)168102
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Comparison between mesoscopic CPM and macroscopic
continuous model

• Plot of a two-dimensional probability density
  distributions for a CPM simulation of 12 cells
  and numerical solution p(x y t) of the
  continuous.
• Cross sections of pcpm(x0 y t) and pcont(x0 y
  t) at x0 530 as functions of y.

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Equation with nonlinear diffusion term and
without blow up in finite time
Lushnikov, P.P., Chen, N., and M.S. Alber 2008,
Macroscopic dynamics of biologicalcells
interacting via chemotaxis and direct contact,
Phys. Rev. E. 78, 061904.
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Monte Carlo
3.0
Chemical 0.5 Production rate
1.5
Continuous Model
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Measure of the average distance between two
neighboring branches. The left picture is the
two dimensional cell density distribution
calculated from the CPM. The right picture is the
Fourier transform result (200 x 200 modes).
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Parallel Implementation of the Cellular Potts
Model
Each computer node consists of four subgrids. At
any given time, calculations are performed on
only one subgrid of each node indicated in
different shading in the figure. Each node
includes a set of buffers which duplicate the
border areas of neighboring subgrids. During
simulations pixel information in neighboring
nodes is retrieved from these buffers. Chen, N.
et al., 2007, Computer Physics Communications 76,
670-681 Christley, S. et al., 2007, PLoS
Computational Biology 3, 4, e76.