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
Fibroblasts degrade fibrin
             secrete collagen

Collagen slows down fibroblasts

Fibroblasts < ----------- >   collagen
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


Cells – discrete objects
        paths given by f

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

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

v( )  (1  )c( f ( ), )  

                                 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, )

                        f (  )
            i 0            i

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

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

v(t )  (1  )                 
                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,
         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
• 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
  more scarring. CONTRADICTION

• 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
  of the gradient
• Fibroblast density is low at top and high
  at bottom (staining experiments)
• Wound heals in reasonable time
• Widely dispersed chemoattractant prolife
  leads to greater degree of interdigitation
  (better linked)
• 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

 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
  reorientation to chemoattractant gradients
  (add agent that binds competitively to
  receptors – mannose 6 phosphate acts in
  this way [Ferguson and O’Kane, 2004]
  have shown this reduces scarring)

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)