Numerical modelling of epidermal wound healing

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Numerical modelling of epidermal wound healing
E. Javierre, S. van der Zwaag Fundamentals of
Advanced Materials
F.J. Vermolen, C. Vuik Delft Institute of
Applied Mathematics
Delft University of Technology
EUNUMATH 2007 Graz, Austria September 10-14, 2007
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Outline

• Introduction
• Motivation and objectives
• Biological background
• The mathematical model
• Wound closure
• Wound edge tracking
• Coupling with angiogenesis
• The numerical method
• Numerical results
• When a wound will heal?
• What is the role of wound geometry?
• How does angiogenesis affect the healing process?
• Conclusions References

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Introduction

• Self healing materials
• Man-made materials with the property of
  undergoing autonomous repair whenever a defect
  occurs
• Aerospace (aircrafts, space shuttles), civil
  structures (roads, bridges), microelectronic
  circuits (cell phones, laptops)
• Challenging and very attractive topic due to
  economical and safety reasons
• Healing processes should mimic those observed in
  nature
• Aims
• Develop a simple model for wound healing
  (closure angiogenesis)
• Investigate the influence of key physiological
  parameters
• Export insights to the development of
  self-healing materials

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Biological background
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The model for wound closure

• Wound closure entirely due to cell mitosis and
  migration (epidermal wounds)
• Increased cellular activity in an active layer
  surrounding the wound
• Generic epidermic growth factor responsible of
  cellular response
• Cell migration is dose-dependent
• Wound edge advancing front of cells closing
  the wound
• Closure rate linearly dependent on wound edge
  curvature

Adam et al. 99
Wound edge must be followed as a moving interface
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Wound edge as a moving interface
Level Set Method for tracking the wound edge

• Level Set Function continuous such that

• Interface advection
• where is any continuous extension of the
  front velocity onto

• To prevent steep gradients and provide a simple
  identification of
• is a distance function

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The model for angiogenesis

• At injury, vascular network is removed from the
  wounded area, and oxygen levels are decreased
• Lack of oxygen at the wound area triggers
  release of macrophage-derived growth factors
• MDGFs stimulate vascular grow
• Capillaries supply oxygen and nutrients
  essential in the healing process

Maggelakis 03
Oxygen concentration
MDGF concentration
Capillary density
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Coupled clousure and angiogenesis

• Oxygen acts as activator/inhibitor of epithelial
  cell proliferation

• An increased capillary density is permitted near
  the wound edge

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Numerical solution

• Finite Element Method
• Galerkins method with piecewise linear elements
  for the diffusion-reaction equation(s)
• Newton-Cotes integration for element vectors and
  matrices
• IMEX time integration (explicit on the coupled
  reaction terms)

• Local grid refinement
• Elements within a certain distance to the front
  are refined
• Mesh consistency
• division of edges division of element
• 1 edge refnratio equally-sized
  sub-edges
• To prevent ill-shaped elements refnratio 2,3

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Numerical solution

• Moving interface
• Pure advection equations due to level set
  formulation
• Two nested Cartesian grids due to refinement

• Fine Cartesian band
• Velocity extension (linear interpolation at the
  interface, few pseudo-time iterations)
• Interface advection (Explicit Euler, 1st order
  upwind)
• Re-initialization (2nd order Fast Marching
  Method)

Coarse Cartesian grid - Level Set extension
(2nd order Fast Marching Method)
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Numerical results wound closure
Adam et al. 99, steady-state solution Healing
can not always be initiated
Once initiated would healing continue
successfully?
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Numerical results wound closure
t10min, 0 healed
t1h20min, 5 healed
t10h, 50 healed
t22h15min, 95 healed
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Numerical results closure angiogenesis
5 healed, t8h55min
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Numerical results closure angiogenesis
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Numerical results closure angiogenesis
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Numerical results closure angiogenesis
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Conclusions

• We have presented a simple model for epidermal
  wound healing, in which the advancing front of
  cells is dealt as a moving interface
• This basic model has been coupled with
  angiogenesis to investigate the role of oxygen in
  the healing process
• The numerical solution is based on an adaptive
  mesh strategy that facilitates the combination of
  Finite Element and Finite Difference/Finite
  Volume methods
• The numerical results evidence that
• the healing process can be unsuccessful if the
  active layer becomes insufficient
• wound geometry plays an important role in the
  incubation period as well as in the healing
  progress
• an increased production of EGF given by the
  oxygen levels may become counterproductive for
  fast recoveries of the vascular system

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References Contact
Selected publications F.J. Vermolen, W.G. van
Rossum, E. Javierre and J.A. Adam, Modeling of
self-healing of skin tissue, In Self Healing
Materials An Alternative Approach to 20
Centuries of Materials Science, Springer Series
in Materials Science F.J. Vermolen, W.G. van
Rossum, E. Javierre and J.A. Adam, A numerical
model for epidermal wound healing, In
Proceedings of Int. Conf. on Computational
Methods for Coupled Problems in Science and
Engineering COUPLED PROBLEMS 2007 Further
information E-mail e.javierre_at_ewi.tudelf
t.nl Web site http//ta.twi.tudelft.nl/nw/use
rs/perez/
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