Mathematical modelling of angiogenesis in wound healing

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Mathematical modelling of angiogenesis in wound
healing

• J. Arnold1, A. Anderson1,
• M. Chaplain1, S. Schor2
• 1 Mathematical Biology Group, Division of
  Mathematics, University of Dundee
• 2 Cell Molecular Biology Unit, Dundee Dental
  School, University of Dundee

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Contents

• Biological Motivation
• Biological Mechanisms
• Mathematical Background
• Continuous Model
• Discrete Model
• Conclusions Future Work

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Biological Motivation

• Wound healing is a vital process
• Wounds give rise to both medical and cosmetic
  challenges
• Ultimate aim is a speedy recovery to (ideally)
  the unwounded state
• More realistic goals are to reduce scarring in
  healed wounds, and to minimise time in the
  wounded state

Source http//www.nursing.iuowa.edu/sites/chronic
wound/!def.htm
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Biological Mechanisms

• Differing severities of wounds, but process
  remains more or less the same
• Once a wound occurs, a multitude of biological
  and chemical processes take place
• 3 overlapping phases inflammation,
  proliferation, maturation

Source http//www.orthoteers.co.uk/Nrujpij33lm/O
rthwound.htm
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Biological Mechanisms

• Inflammation
• Wound fills with blood and many substances
  including macrophages and leucocytes
• Essentially this phase cleans the wound of dead
  cellular material and bacterial infection and
  sets up chemical gradients in the wound space
• Proliferation
• Cells move into the wound and begin to create new
  tissue in the wound space
• It is in this phase that angiogenesis takes
  place, and upon which our model is based
• Maturation
• ECM is remodelled during this phase and collagen
  continues to be deposited, increasing the
  strength of the scar tissue
• Lasts indefinitely, but most activity in 1st year

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Inflammation

• During this phase, blood enters the wound and
  signalling processes are enabled that give rise
  to the activation of clotting cascades and
  aggregation of macrophages and leucocytes,
  amongst other substances
• Essentially this phase prepares the wound for
  repair by removing any dead cellular material and
  foreign (bacterial) substances and also by
  creating some chemical gradients in the wound
  space

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Proliferation

• Once the wound has been cleaned and prepared by
  the inflammatory phase, the process of healing
  can begin
• Characterised by the formation of granulation
  tissue fibroblasts, inflammatory cells, new
  capillaries embedded in a loose extra cellular
  matrix (ECM) of collagen, fibronectin and
  hyaluronic acid
• Fibronectin is an ECM macromolecule, and is used
  to model the ECM
• Re-epithelialization begins during this phase to
  form a new layer of skin over the wound
• Fibroblasts are involved in wound contraction,
  fibronectin production and collagen accumulation
• The exact processes of fibroblast migration
  proliferation are unknown

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Proliferation
Granulation tissue (during proliferation phase)
Fibroblasts and endothelial cells
Source http//medic.med.uth.tmc.edu/edprog/Path/I
nflamm/Inflammshort.htm
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Maturation

• The ECM is formed in a loose manner and is
  relatively weak (20 of final strength after 3
  weeks)
• Maturation involves remodelling of the ECM and
  continuing collagen deposition, which adds to the
  strength of the resulting scar tissue
• Gradual reduction in vasculature and cellular
  structure ensues
• This phase lasts in a highly active state for
  around 1 year, but remodelling continues
  indefinitely
• Final strength of wound is around 70-80 of that
  of uninjured tissue

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Mathematical Background

• Previous work on wound healing has been mainly
  continuous models in 1-D
• Usual advantages and disadvantages of continuous
  modelling of biological systems apply
• Work by Anderson and Chaplain (1998) in 2-D
  Cartesian coordinates introduced a novel way of
  obtaining a discrete system from the continuous
  model. We will use a modification of this
  technique
• We will develop initially a continuous model, and
  from that the discrete model

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Continuous Model

• Considers the movement of endothelial cell (EC)
  densities under the influence of chemotaxis and
  haptotaxis
• Working in 2D polar coordinates representing a
  punch wound
• Variablesn(r,q,t) EC densityc(r,q,t)
  angiogenic factor concentrationf(r,q,t)
  fibronectin concentration

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Continuous Model
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Continuous Model

• Boundary conditionsZero flux on the r
  boundaries, except for the inner c boundary where
  c(a,q,t)1.q boundaries are matched, i.e.
  n(r,0,t) n(r,2p,t), etc.
• Initial conditionsPatches of endothelial cells
  lie in thin areas around the wound edge.A
  straight line gradient of angiogenic factor
  exists.Fibronectin is constant to begin with.

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Continuous Model - Results
Endothelial Cell Densities
Chemotaxis only
Chemotaxis and haptotaxis
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Discrete Model

• The discrete model is formed by considering the
  numerical scheme discretisation of the cell PDE
• A biased random walk model is obtained
• Allows explicit functions of cells to be
  considered, and also shows the dendritic
  structure of the formed blood vessels

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Discrete Model

• Consider the movement of tip-cells, which are
  active areas at the end of blood vessels
• Tip-cells can fuse if they come into contact with
  vessels that arent from the same parent to form
  loops (anastomosis)
• They can branch if conditions are favourable,
  thereby creating a new tip-cell that lays a new
  vessel
• Tip-cells move by growing the vessel in their
  wake. The movement direction is chosen by the
  random walk, and whether or not space is available

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Discrete Model
Coefficients of Euler finite difference scheme
elements provide probabilities of movement. P0
stay in same position P1 move ni,j in
decreasing r direction P2 move ni,j in
increasing r direction P3 move ni,j in
decreasing q direction P4 move ni,j in
increasing q direction
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Discrete Model
(matrix degrading enzyme)
(oxygen)
(fibroblasts)
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Discrete Model - Results
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Good Heal
Endothelial Cells
Fibronectin
Fibroblasts
Oxygen
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Bad Heal
Endothelial Cells
Fibronectin
Fibroblasts
Oxygen
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Conclusions Future Work

• Haptotaxis aids movement of cells perpendicular
  to the wave front movement
• If angiogenesis network grows too quickly,
  healing can fail
• Use biological data to validate the model
• Apply network flow techniques to determine good
  structures and improve the model

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References

• ANDERSON A.R.A. CHAPLAIN M.A.J., 1998.
  Continuous and Discrete Mathematical Models of
  Tumor-induced Angiogenesis. Bulletin of
  Mathematical Biology 60, 857-900.
• CLARK, R.A.F., 1988. The Molecular and Cellular
  Biology of Wound Repair. New York Plenum Press.