A Diffusion Model of Platelet Derived Growth Factor PDGF for Hybrid Modeling of Intimal Hyperplasia

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A Diffusion Model of Platelet Derived Growth
Factor (PDGF) for Hybrid Modeling of Intimal
Hyperplasia
Presented by Katelyn Swift-SpongResearch
Alliance in Math and ScienceComputational
Sciences and Engineering Division Mentor Dr.
Richard Ward August 13, 2008 Oak Ridge, Tennessee
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Intimal Hyperplasia

• Occurs in arterial wall as response to injury
  such as balloon angioplasty
• Smooth muscle cells (SMCs) migrate from media to
  intima layer of the arterial wall
• Migration due to presence of the chemoattractant
  PDGF and breakdown of collagen by matrix
  metalloproteinase (MMP) enzyme kinetics
• Results in restenosis (re-narrowing) of the
  artery

Diagram of Artery
Intima
Lumen
Endothelial cell layer
Media
Internal elastic lamina
Adventitia
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Intimal Hyperplasia
Lumen
Lumen
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Hybrid Intimal Hyperplasia Model

• Needed for a greater understanding of the process
• Use for prediction of restenosis
• Cell movement modeled with discrete event
  simulation
• Chemical diffusion of Platelet Derived Growth
  Factor (PDGF) is continuous

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Three Dimensional Diffusion Model of PDGF

• Developed using C
• Visualization with Matlab and VisIT
• Used finite difference method to solve the
  diffusion equation
• Applied no flux boundary conditions

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Finite Difference Method

• An explicit method used to solve the diffusion
  equation in three dimensions
• Continuous partial differential equation solved
  using discrete space and time steps
• Forward time and central space discretization
• Concentration at previous time step used to
  compute concentration at next time step

Diffusion equation
Finite difference approximation
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Stability

• Time step must be small enough to satisfy this
  condition
• Diffusion coefficient for PDGF in a solution is
  around 10-6 cm2/s

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Model Results

• Began with one dimensional model and expanded it
  to two and three dimensions
• Used visualization for model validation
• Used initial conditions of constant concentration
  of one in top half and zero in bottom
• Applied no flux boundary conditions

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Integration of 3D Diffusion Model

• Previously developed cell migration model was
  recently extended to 3D
• Needed 3D diffusion model
• Incorporate into discrete event simulation of
  cell migration
• Leads to hybrid modeling environment

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Trilinear Interpolation

• Concentration within grid needed for cell
  migration model
• Computes PDGF concentration, C(x,y,z), at
  specified time
• Uses linear interpolation method seven times

Image from Wikipedia
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Lagrange Interpolating Polynomial

• Used to determine partial derivative of
  concentration at point within grid
• Computes
• Interpolates a polynomial of degree n-1 that
  passes through n points
• Partials used by cell migration model

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Future Work

• Incorporate matrix metalloproteinase (MMP) 2
  enzyme kinetics model developed with JSim and
  Systems Biology Workbench (SBW) into IH model
• Develop collagen degradation model based on MMP-2
  kinetics
• Include effects of hormones from hormone
  replacement therapy

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Model Interoperability

• Tested capabilities of extensible markup language
  (XML) based programs
• Cellular Open Resource (COR), SBW, and JSim do
  not support three dimensional partial
  differential equations
• Not all CellML and Systems Biology Markup
  Language (SBML) files could be used with these
  programs
• Easy to use without programming background

Screen shots of JSim, SBW, and COR user interfaces
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Conclusions

• Achieved successful integration of diffusion
  model into cell migration model
• Need an integrated modeling language that extends
  beyond the present capability
• Capable of integrating the kinetics models
  described using the XML formats, such as CellML
  and SBML, with the diffusion models for
  biochemicals and cells
• Still needed in hybrid IH model
• Collagen breakdown by MMP2
• Effect of hormones from hormone replacement
  therapy on cell migration

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Acknowledgments

• Special thanks to Richard Ward, Kara Kruse, and
  Jim Nutaro for their help and guidance.
• The Research Alliance in Math and Science program
  is sponsored by the Office of Advanced Scientific
  Computing Research, U.S. Department of Energy.
• The work was performed at the Oak Ridge National
  Laboratory, which is managed by UT-Battelle, LLC
  under Contract No. De-AC05-00OR22725. This work
  has been authored by a contractor of the U.S.
  Government, accordingly, the U.S. Government
  retains a non-exclusive, royalty-free license to
  publish or reproduce the published form of this
  contribution, or allow others to do so, for U.S.
  Government purposes.

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Questions
16 Managed by UT-Battellefor the Department of
Energy
UTBOG_Computing_0801