# Modelling Collagen Alignment in Dermal Wounds

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Modelling Collagen Alignment in
Dermal Wounds
John Dallon, Jonathan Sherratt, Mark Ferguson and
Philip K. Maini
Key Players
secrete collagen

Collagen slows down fibroblasts

Fibroblasts < ----------- >   collagen
alignment
Alignment problems occur in a wide variety of applications:

crystals, ecology, developmental biology etc

Mathematical approaches: integro-partial-differential equations,

discrete orientation …
The Orientation Model

Variables

Cells – discrete objects
paths given by f

Collagen – continuous vector field denoted by c (x, t)
df ( ) 
i
v( )
 f ( )  s
i

d                v( )
f (  )
   i

v( )  (1  )c( f ( ), )  
i

f (  t )
  i

Where ρ and s are positive constants with s
representating the cell speed and  a time lag

f (  )
       i

f ( x, )   w ( x, )
N

f (  )
i
i 0            i

where  is again a time lag and N is the total
number of cells
d ( x, t )
 f sin(  0)
dt

Here ө is the angle of c, the vector representing the
collagen direction and α (x, t) is the angle of f (x,t).
Fig. 3. The collagen orientations and cell positions for a typical simulation. In (a)
the initial random collagen orientation is shown and in (b) the collagen orientation
is shown after 100 hr of remodelling by the fibroblasts on a domain of 0.5 mm x
1.0 mm. The cells have a speed of 15 m hr 1,  5,   0,  0.15 and the
numerical grid for the vector field is 51 x 101.
Fig.4. The effect of altering the rate at which the fibroblasts change the fibre direction. It
is seen that as the influence of the cells on the collagen orientation increases the pattern
has more structure in (a) where κ = 20. The simulations shown here have the same
parameters and set-up as that shown in Fig. 3.
The Tissue Regeneration Model
fibrin network is represented by b (x,t)

 (t )  s ( c , b ) v (t )
f  i

v (t )
i
u ( f (t ), t )        f (t  )
   i

v(t )  (1  )                 
i
u ( f (t ), t )         f (t 
      i

u( x, t )  (1 )c( x, t )  b( x, t )
d c ( x, t )                         N
 ( p  d c ( x , t ) )  w ( x, t )
c     c                    i
dt                               i 1

d b ( x, t )                             N
  d b ( x, t )  w ( x, t )
b                         i
dt                                 i 1
• Altering the speed of fibroblasts:
increasing the speed leads to greater
alignment. Can be done using a
chemoattractant (Knapp et al, 1999 – can
increase speed 3-fold (Ware et al, 1998))
or altering the integrin expression levels of
the fibroblasts (Palecek et al, 1997)
• Reducing contact guidance: inhibit the
formation of microtubules with colcemid
(Oakley et al, 1997); treatment with
colchicine (causes rounder morphology
(Mercier et al, 1996)
• Effects of initial collagen orientation:
transplant pieces of tendon (Matsumoto et
al, 1998); place pieces of oriented gel
(Guido and Tranquillo, 1993)
• Altering the profile of transforming growth
factor beta can have profound effects on
the healing process, including significantly
increasing or decreasing the degree of
scarring (Shah et al, 1992, 1994, 1995,
1999)
Effects of TGF-beta
• Cell proliferation – biphasic (depends on age)
• Cell motility – biphasic effect on directed cell
movement (chemotaxis)
• Collagen production – increase collagen
production and decrease collagenase production
• Cell reorientation – development of lamellipodia
and filopodia depends on concentration levels
Model results
• Effects on cell proliferation, migration and
extracellular matrix production influence
collagen alignment in only a MINOR way

• Regulation of filopodial extensions by
TGF-beta could be the CRUCIAL property
Interpretation
• Adding TGF-beta-3 causes more cell
reorientation, leads to less alignment and
scarring is reduced

• Antibodies to TGF-beta-1 and 2 would, in this
interpretation, lead to more alignment and hence

• Both these isoforms bind to cells competitively
(Altomonte et al, 1996, Piek et al, 1999)
Model considered wound is isolation.

If we embed it in tissue we find that the
time taken for the cells to enter and “heal”
the wound is too long.
McDougall and Sherratt
• Add a chemoattractant produced in the
wound (PDGF, IL-Ibeta, TNF-alpha)

• Reaction-diffusion equation at steady state

• Cells velocity now depends on size of
chemical gradient and is in the direction
• Fibroblast density is low at top and high
at bottom (staining experiments)
RESULTS
• Wound heals in reasonable time
• Widely dispersed chemoattractant prolife
leads to greater degree of interdigitation
• Uniform cell distribution in the unwounded
skin leads to parallel alignment rather
than perpendicular alignment (w.r.t.
bottom of wound)
• Switching off the speed cue leads to fewer
cells entering the wound. Orientation not
altered

Switching off the directional cue (but not speed)
is worse

Pattern of alignment depends crucially on
the form taken for velocity dependence
Therapeutic aspects
• Decrease the sensitivity of fibroblast
(add agent that binds competitively to
receptors – mannose 6 phosphate acts in
this way [Ferguson and O’Kane, 2004]
have shown this reduces scarring)
References

J.C. Dallon, J.A. Sherratt, P.K. Maini, J.theor.Biol., 199, 449-471 (1999)

J.C. Dallon, J.A. Sherratt, P.K. Maini, M. Ferguson,
IMA J.Math.Appl.Biol.Med, 17, 379-393 (2000)

J.C. Dallon, J.A. Sherratt, P.K. Maini, Wound Repair and Regeneration, 9,
278-286 (2001)

S. McDougall, J. Dallon, J. Sherratt, PKM, (submitted)

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