Clark

Jonathan Sherratt

Luke Olsen

Murray

Olsen

Philip Maini

English

Olsen, L.

Maini, P. K.

Sherratt, J. A.

Oxford University Research Archive (ORA)

A mechanochemical model for normal and abnormal dermal wound repair

Oxford University Research Archive

Mathematical Institute Eprints Archive

Nonlinear Analysis, Theory, Methods & Applicarions, Vol. 30, No. 6, pp. 3333-3338, 1997 
Proc. 2nd World Congress of Nonlinear Analysrs 
Q 1997 Elsevier Science Ltd 
PII: SO362-546X(96)00343-4 
Printed in Great Britain. All rights reserved 
0362-546X/97 $17.00 + 0.00 
A MECHANOCHEMICAL MODEL FOR NORMAL AND ABNORMAL DERMAL 
WOUND REPAIR 
LUKE OLSEN, t PHILIP MAINIt and JONATHAN SHERRATTI 
tCentre for Mathematical Biology, Mathematical Institute, 24-29 St Giles’, Oxford, OX1 3LB, UK; and 
$Nonlinear Systems Laboratory, Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK 
Key words and phrases: Wound contraction, fibroproliferative disorders, growth factors 
1. INTRODUCTION 
The healing of full-thickness excisional skin wounds in adult mammals involves a complex sequence 
of inter-regulatory biological processes (see [l] f  or review). A key process involves the movement of 
cells into the wound in response to mechanical and chemical cues. These cells exert traction forces 
which pull the wound together, leading to normal wound healing, in which the skin is in a contracted 
state. Abnormal regulation of this process may result in a number of healing disorders. One such 
disorder results in increased tissue in the wound leading to fibroproliferative diseases such as keloid 
scarring. In this paper we consider the general mathematical model framework for dermal wound 
healing proposed in [a]. We briefly describe the model in Section 2 and review its properties. In 
Section 3 we show, by looking at a simpler version of the model, that it can generate contracted 
steady states. In Section 4, we investigate another version of the model to focus on fibroproliferative 
diseases. 
2. MECHANOCHEMICAL MODEL FRAMEWORK 
Here, we briefly present a mechanochemical model for dermal wound healing based on the frame- 
work developed by Murray and coworkers (see [3] f  or review). For full details and experimental 
justification for each term, see [2]. The full model consists of five field variables and, for simplicity, 
we present here only the one-dimensional version of the model. The model considers two distinct 
cell types - fibroblasts and myofibroblasts, denoted by n(z,t) and m(z, t), respectively, where zr is 
space and t is time. It is assumed that fibroblasts secrete a growth factor, c(z, t), and move into the 
wound in response to gradients in c. The fibroblast equation takes the form: 
g = Dn$ - T&,n); + n$] + R(c)n(l - ;, - = 
ck + c 
+ t&m - d,n. (2.1) 
The first three terms on the right hand side model the following contributions to cell flux: cell 
diffusion with constant diffusion coefficient, D,, cell chemotaxis with chemotactic sensitivity x(c, n), 
and convection in response to the displacement, ~(2, t), of the extracellular matrix (ECM) substratum 
on which cells move. The next four terms describe, respectively, logistic cell growth with linear 
rate enhanced by growth factor, conversion to myofibroblast phenotype mediated by growth factor, 
conversion from myofibroblast to fibroblast cell type, and cell death. 
The myofibroblasts satisfy the equation 
dm 
at = ax 
AI-m:] + c,R(c)m(l - E) + 5 - &n - d,m, 
k c 
(2.2) 
where it is assumed that the dominant contribution to myofibroblast flux is convection, and that 
mitosis takes the same form as that for fibroblasts to within a scale factor cr. 
The growth factor, c, diffuses, is convected by the ECM, and is secreted and degraded by both 
cell types. Therefore, it satisfies the equation 
3333 
3334 Second World Congress of Nonlinear Analysts 
g = Dcg + &-~$1 + S(n, m, c) - d,c. 
The ECM: density p(z, t), moves primarily by convection and satisfies the equation 
(2.4) 
where B(n, m, c, p) represents ECM biosynthesis and degradation. 
Finally, modelling the ECM as a linear, isotropic, viscoelastic material, the displacement 21 satisfies 
the force balance equation 
p&+EZ+ wh m, P) ax = Fb, u), (2.5) 
where the first two terms on the left hand side model viscous and elastic forces, respectively, and the 
third term models cell traction forces. These forces are balanced by the body forces F(p, u). 
Equations (2.1)-(2.5), with appropriate initial and boundary conditions (see below), consti- 
tute the mechanochemical model framework. Using biologically realistic forms for the functions 
x(c,n),R(c),S(n,m,c),B(n,m.c,p),7(n,p),F(p,v), and estimates derived from experimental data 
for the parameters D,, DC, K,kl,k2,Ck,d,,d,,d,,~~,~ and E, it can be shown that this model ex- 
hibits solutions for the decay of growth factor and rate of wound closure that closely agree with 
experimental results (see [2] for full details). Using a caricature model, we now proceed to analyse 
the possible contracted steady states exhibited by the model. 
3. SPATIALLY VARYING CONTRACTED STEADY STATES 
To consider the potential of the model framework (2.1))(2.5) to exhibit spatially varying con- 
tracted steady states we consider a simpler version of the model of the form 
2 = Dn$ + &n$] + n(l - n) (3.1) 
(3.2) 
p&+*2+ ah P) ~ = F(p,u). ax 
Here, we have chosen to consider the purely mechanical aspects of the dynamics and have sim- 
plified the model by considering only one cell type, have non-dimensionalised the model so that the 
logistic growth term takes an algebraically simpler form, and assumed that there is negligible syn- 
thesis and degradation of ECM on the timescale of wound closure. This is a reasonable assumption 
to make in the stages prior to tissue remodelling. 
By defining the initial wound space as -1 5 z 5 1 and using symmetry at x = 0 (the wound 
centre), we may restrict attention to the semi-infinite domain 0 5 z < 03. The initial half-wound is 
set to unity by the scaling for z. The boundary conditions are thus 
g(O, t) = $(O, t) = ~(0, t) = 0 and n(co, t) = p(co, t) = 1, u(oo, t) = 0. 
Second World Congress of Nonlinear Analysts 3335 
The initial conditions are 
n(x, 0) = H(x - l), p(z,O) = pi + (1 - pi) H(x - l), U(Z,O) = 0, 
where the initial ECM density pi inside the wound is due to the early, provisional wound matrix 
which is low in collagen and satisfies 0 < pi < 1, and H(.) is the Heaviside step function. 
Consider now the healed steady state, n = 1. Linearising (3.2) about the init,ial profile, we have 
{ 
Pi (1 - au/ax) ) 0 5 x < 1 
P= 
I -au/ax, a>1 
as suggested by the small-strain restriction (since the convective flux should be small). Substituting 
this into the steady state equation for u, we have a second order ordinary differential equation for u, 
which we can write in the (resealed) form 
21’ = v (2) 
1 
S&U(l - w) 
1 - pi7[p,(l - v)J’ 
O<x<l 
v’ = 
su(1 - U) 
1 - 7(1 - V) ’ 
x>l 
(3) 
where 7(p) G vlnzl, and u satisfies the boundary conditions u(O) = U(CCI) = 0. He we have 
assumed that the body force, F(p,u), is due to external tethering to the basement membrane and 
have modelled it by a linear spring, that is, F(p,u) = sup, where s is a constant. 
Standard phase plane analysis of (3.4)-(3.5) s h ows that for x > 1, the origin is a saddle (centre) 
i f f  7(l) < 1 (> 1). However, linear stability analysis of the partial differential equation system (3.1)- 
(3.3) shows that the healed steady state is stable i f f  7(l) < 1. As this must be the case, we have 
that the origin of the ordinary differential equation system (3.4)-(3.5) is a saddle, and the boundary 
condition I = 0 implies that the solution must converge towards the origin along the stable 
manifold as x tends to co. By tracing backwards in x from infinity along the stable manifold, the 
solution reaches a point in the (u, u/)-phase plane corresponding to x = 1 where u = ~1, say. This 
must match the solution for 0 5 x < 1. 
Now, at the wound centre, u(O) = 0, but w(0) . IS unspecified. Rather, it is determined by matching 
to the “outer” solution at x = 1. For 0 5 1 < 1, it can be shown that the origin can either be a 
saddle or a centre, depending on the form of the function ~(n, p) and the values of the parameters. 
If  the origin is a saddle point, then the solution in 0 5 x < 1 is expected to be either monotonic 
increasing with increasing gradient or monotonic decreasing with decreasing gradient. I f  the origin 
is a centre, then the solution in 0 < z < 1 may be oscillatory. Figure 1 illustrates the qualitative 
construction of such a solution and Figure 2 illustrates various theoretically-possible forms of steady 
state solutions. Modelling the traction term, 7(n,p), by ~(n,p) = ~np/(T~ + p’), to account for 
the fact that traction forces depend on adhesion between cell surface receptors and binding sites on 
collagen fibres, but the ability of a cell to extend and retract protrusions within a collagen substrate 
is inhibited at relatively high collagen densities, where TO and T are constant parameters, we have 
found, by numerical simulation, steady states for (3.1)-(3.3) of the form illustrated in Figure 2(a)-(e). 
3336 Second World Congress of Nonlinear Analysts 
Fig. 1. Qualitative illustration of a possible solution trajectory as described in the text, showing a 
case in which the origin is a centre for 0 5 z < 1 and a saddle point for x > 1 with U(Z) + 0 from 
below as 2 -+ 00. See also Figure 2(b) below. Dashed curves denote phase trajectories, with the 
contracted solution curve highlighted by solid arrows. 
(4 
Fig. 2. Possible qualitative forms of the solution u(z) of the boundary value problem (3.4)-(3.5), 
representing contracted tissue displacement profiles. The point (u, u’) = (0,O) must be a saddle point 
for z > 1 in the (u, u/)-phase plane, with u increasing to zero and U’ decreasing to zero monotonically 
along the stable manifold in the top-left quadrant as z -S co. For 0 5 CC < 1, the origin may be either 
a saddle point, in which case the profiles for u and u’ are monotonic decreasing as shown in (a), or 
a centre, in which case u and 2~’ oscillate about the origin as shown in (b-f); within this region, any 
number of oscillations is possible-for example, (f) is equivalent to (b) modulo one period. Note that 
the above steady-state profiles but with reversed signs of u and U’ are also admissible solutions of 
(3.4)-(3.5), representing expanded tissue displacement profiles since ~(1) would be positive. Recall 
that x = 1 is the initial wound boundary. 
Second World Congress of Nonlinear Analysts 
4. FIBROPROLIFERATIVE WOUND HEALING DISORDERS 
3337 
We now consider the application of the mechanochemical model framework to fibroproliferative 
wound healing disorders. Such disorders are characterised by abnormally large amounts of tissue. The 
full model can exhibit solutions in which an excess of cells is observed, corresponding to a pathological 
state. To understand this more fully, we focus purely on the chemical aspects of (2.1)-(2.5). Consider 
now the cell-growth factor sub-model of (2.1)-(2.5), namely, 
(4.1) 
where x(c, n) = a/(/3 + c)~, and cr,p, I’, &, IE, and y  are positive constants (see [4] for full details). 
This caricature model has two uniform steady states, (n, c) = (0, 0), (K, 0) corresponding, respec- 
tively, to the trivial, or non-healing, state, and the normal dermal state. For appropriate parameter 
values, two other steady states exist which have both n and c non-zero, with n > K. These are the 
pathological, or diseased, steady states. A bifurcation analysis can be carried out for (4.1)-(4.2), in 
the absence of diffusion, and it can be shown that for ~~ = K: (calculated in terms of the other pa- 
rameters) the dermal steady state remains locally stable but loses global stability as the pathological 
steady states appear. At ~~ = & the dermal steady state loses stability and the pathological state 
with higher cell density level becomes globally stable. 
The spatiotemporal dynamics of the model can be analysed by travelling wave analysis, which 
shows that travelling wave trajectories from the dermal state to the pathological state are possible 
and a minimum wavespeed can be determined. Numerical simulations of the system show that such 
travelling waves do exist, but that reducing K~ can cause the waves to stop and to regress (see Figure 
3). 
5. DISCUSSION 
The mechanochemical framework presented in Section 2 has been proposed to model dermal 
wound healing. Numerical simulations of the full model have shown that it can capture key aspects 
of normal and abnormal wound healing. The complexity of the model makes it difficult to fully un- 
derstand what roles the various mechanisms play during these processes. To investigate the model in 
more detail, we have chosen to analyse caricature models which focus on certain model mechanisms. 
In Section 3, we considered a purely mechanical model and showed how it could exhibit contracted 
steady states, characteristic in normal wound healing. In Section 4, we focussed on the chemical as- 
pects of the full model and showed how the system could evolve to a pathological state corresponding 
to a fibroproliferative state. In this case, our analysis shows that reducing K~, the linear rate at which 
cells secrete growth factor, can cause the disease to regress back to the normal dermal state. More 
detailed analysis of this model ([4]) d e t ermines analytically how the bifurcation values of q depend 
on the other parameters in the model. This provides a clinically-testable method to help reduce this 
type of fibroproliferative disorder. 
3338 
120 
100 
80 
60 
40 
20 
Second World Congress of Nonlinear Analysts 
cells, n(x,t) chemical, c(x.t) 
I I 
X X 
n(x.t) c(x,t) 
120 
100 
80 
60 
40 
20 
0 
0 5 10 0 5 
Fig. 3. Numerical simulations of (4.1)-(4.2) showing progression to pathological steady state (a), 
and cessation and regression for the case where IE, in reduced to zero after a certain time (b). See 
[4] for parameter values. 
Acknowledgement: L. 0. expresses gratitude to the Engineering and Physical Sciences Research 
Council of Great Britain for a Research Studentship award to support this work. We also thank the 
London Mathematical Society for support. 
REFERENCES 
[l] Clark, R. A. F., Overview and general considerations of wound repair, in The Molecular and 
Cellular Biology of Wound Repair (Edited by R. A. F. Clark and P. M. Henson), pp. 3-34. Plenum, 
New York (1988). 
[2] Olsen, L., Sherratt, J. A. & Maini, P.K., A mechanochemical model for adult dermal wound 
contraction and the permanence of the contracted tissue displacement profile, J. theor. Eiol. 17’7, 
113-128 (1995). 
[3] Murray, J.D., Mathematical Biology, Springer-Verlag (1989). 
[4] Olsen, L., Sherratt, J. A. & Maini, P.K., A mathematical model for fibroproliferative wound 
healing disorders, Bull. Math. Biol. 58, 787-808 (1996). 


A mathematical model for fibroproliferative wound healing disorders,

A mechanochemical model for adult dermal wound contraction and the permanence of the contracted tissue displacement profile,

Overview and general considerations of wound repair, in The Molecular and Cellular Biology of Wound Repair (Edited by

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A mechanochemical model for normal and abnormal dermal wound repair

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