Several authors have predicted that cell propagation in a number of biological contexts, for example, wound healing, tumour cell invasion, angiogenesis etc., occurs due to a constant speed travelling wave of invasion. The analyses of these models to arrive at this prediction is, in many cases, essentially an extension of the classical analysis of Fisher's equation. Here, we show that a very simple wound healing assay does indeed give rise to constant speed travelling waves. To our knowledge, this is the first verification of Fisher's equation in a medical context. © 2004 Elsevier Ltd. All rights reserved

Leavesley, D

Maini, PK

McElwain, DLS

Applied Mathematics Letters

Several authors have predicted that cell propagation in a number of biological contexts, for example, wound healing, tumour cell invasion, angiogenesis etc., occurs due to a constant speed travelling wave of invasion. The analyses of these models to arrive at this prediction is, in many cases, essentially an extension of the classical analysis of Fisher's equation. Here, we show that a very simple wound healing assay does indeed give rise to constant speed travelling waves. To our knowledge, this is the first verification of Fisher's equation in a medical context

Maini, P. K.

McElwain, S.

Leavesley, D.

Oxford University Research Archive

Available online at www.sciencedirect.com Applied . c , . . c .  Mathematics Letters ELSEVIER Applied Mathematics Letters 17 (2004) 575-580 www.elsevier.com/locate/aml Travell ing Waves  in a Wound Heal ing  Assay P. K. MAINI Centre  for Mathemat ica l  Biology, Mathemat ica l  Ins t i tu te  24-29 St Giles', Oxford, OX1 3LB, U.K. D. L. S. MCELWAIN School of Mathematical Sciences Queensland University of Technology GPO Box 2434, Brisbane, Australia Q4001 D. LEAVESLEY School of Life Sciences Queensland University of Technology GPO Box 2434, Brisbane, Australia Q4001 (Received and accepted January 2003) Communica ted  by G. Wake A b s t r a c t - -  Several authors have predicted that  cell propagation in a number of biological con- texts, for example, wound healing, turnout cell invasion, angiogenesis etc., occurs due to a constant speed travelling wave of invasion. The analyses of these models to arrive at this prediction is, in many cases, essentially an extension of the classical analysis of Fisher's equation. Here, we show that  a very simple wound healing assay does indeed give rise to constant speed travelling waves. To our knowledge, this is the first verification of Fisher's equation in a medical context. (~) 2004 Elsevier Ltd. All rights reserved. K e y w o r d s - - F i s h e r ' s  equation, Human peritoneal mesothelial cells, Invasion waves. I N T R O D U C T I O N  Severa l  a u t h o r s  have  p r o p o s e d  t h a t  c o n s t a n t  profile, c o n s t a n t  speed  t r ave l l i ng  waves  p lay  an  i m p o r t a n t  role in  a n u m b e r  of m ed i ca l  app l ica t ions .  For  example ,  in  e p i d e r m a l  w o u n d  heal ing ,  mode l s  h a v e  b e e n  p r o p o s e d  in  w h i c h  cells move  a n d  pro l i fe ra te ,  s o m e t i m e s  in r e s p o n s e  to  chemica l  s ignals ,  f o r m i n g  a wave  w h i c h  moves  in to  t h e  space  to  close t h e  w o u n d  (see, for example ,  [1-3]). M e c h a n o c h e m i c a l  m o d e l s  for d e r m a l  w o u n d  hea l ing  h y p o t h e s i z e  t h a t  cells exe r t  t r a c t i o n  forces on  e x t r a c e l l u l a r  m a t r i x ,  s e t t i n g  u p  m o t i o n  a n d  g u i d a n c e  cues to  w h i c h  t h e y  r e s p o n d  as t h e y  invade  t h e  w o u n d  space  a n d  e v e n t u a l l y  close t h e  w o u n d  [4,5]. This work was carried out at Queensland University of Technology during a visit by P. K. Maini funded under the Visiting Fellowship Scheme and finished during a Royal Society Leverhulme Trust Senior Research Fellowship (PKM). We thank F. Stoop for assistance with the analysis and B. Addison-Smith for numerical work. P .K.  Maini also thanks W. van Saarloos for constructive comments. 0893-9659/04/$ - see front matter (~) 2004 Elsevier Ltd. All rights reserved. doi: 10.1016/j.aml.2003.01.011 Typeset by .42kdS-TEX 576 P.K. MAINI et al. ~mour cancer cells are hypothesized to outcompete normal cells setting up waves of inva- sion [6], or to invade normal tissue by degrading extracellular matrix with proteases [7]. Angio- genesis, the process by which a tumour recruits endothelial cells and establishes its own vascular system, is thought to occur as a directed wave propagation of cells from the normal host vascu- lature to the tumour under the influence of tumour angiogenesis factors secreted by tumour cells (see, for example, [8]). The above processes are modelled, in the main, by systems of coupled nonlinear partial- differential equations which describe the spatiotemoporal evolution of cells and signalling cues. Cells are assumed to diffuse (usually modelled by Fickian diffusion) and respond to various chemical signals via chemokinesis and/or chemotaxis, and to mechanical signals via haptotaxis (movement up adhesive gradients), and contact guidance. The models are solved using a mixture of analytical and numerical techniques and have been shown to exhibit constant profile constant speed travelling waves in cell density. Although there is good evidence that such propagating travelling waves exist in ecology (see, for example, [9]), we are unaware of this being shown conclusively in the aforementioned medical applications. In this paper, we consider a much simpler system and show that a wound healing assay does indeed appear to admit travelling waves of constant speed. Moreover, we show that the data is well fitted by Fisher's equation, a single reaction-diffusion equation that was initially proposed as a model for the spread of an advantageous gene in a population [10,11]. In the next section, we describe briefly the experimental set-up and results. Then, we consider the Fisher equation and compare its behaviour to the experimental results. Finally, we discuss the implications of this work. W O U N D  H E A L I N G  A S S A Y  A N D  R E S U L T S  We refer the reader to the paper [12] for full experimental details. Briefly, human peritoneal mesothelial cells (HPMCs) were grown as monolayers on different substrates to confluence. In- tracellular matrices used were collagen I, collagen IV, laminin, fibronectin, vitronectin, and hyaluronic acid. A 4 mm scrape wound was then made and the displaced cells were removed. The remaining cells were bathed in fresh culture medium and the position of the invading cell front was noted against a reference grid. Figure 1 shows the typically observed sharp front of cells invading the space left by the scrape. The results for all substrates and all experiments were remarkably consistent. After a short period of time, the cell fronts progress with what  appears to be a constant wavespeed (see Figure 2). T H E  M O D E L  We propose that  the waves observed are Fisher waves. Tha t  is, we assume that  the cell movement can be modelled by Fickian diffusion and that  proliferation satisfies a logistic growth curve. Furthermore, we assume from the experimental set up that  the process is occurring in one space dimension. Hence, the model is an  02n 0-7 = D-o~x2 + r n ( g  - n),  (1) where n(x ,  t) denotes cell number density at position x and time t, D is the diffusion coefficient (assumed constant), r N  is linear growth rate of cells, and N is the carrying capacity. For the experimental set up, the initial conditions would be of Heaviside type. However, we actually take as our starting point the first measurement, which is after 9.5 hours. At this stage, the initial condition will be a "smoothed" Heaviside. However, the exact profile of the initial condition is not as important  as the observation that  it has compact support.  Under these Travelling Waves 577 Figure 1. A typical human peritoneal mesothelial cell front ten hours after wounding. conditions, it is well known that the model exhibits a constant profile travelling wave moving with constant speed (see, for example, [13] for references) v = 247- . (2) Thus, we see that  this travelling wave solution has a nontrivial set of initial conditions for which it is asymptotically attracting. Figure 3 illustrates a typical plot of front distance from origin with time, together with a least-squares fit assuming a front moving with constant speed. As can be seen, the actual wave initially moves at a speed greater than that  of the fit, then it moves more slowly, then the speed increases almost monotonically to the asymptotic wavespeed. In all the experiments we analysed, the wavefront initially moves at a speed greater than the speed predicted from the least-squares fit. Then, the speed decreases below that  of the fitted speed. After this, in some cases, we see a monotonic increase to the fitted speed, in other cases we see a slight overshoot (as in Figure 3) before it eventually settles down. Figure 4 illustrates this more clearly. Note that  the prediction from the Fisher equation is that  initially the wavespeed will be greater than the asymptotically predicted wavespeed, then the speed will decrease and approach the asymptotic wavespeed from below [14]. This is very close to the behaviour that we observe experimentally. To make a quantitative comparison between the predicted wavespeed and the actual wavespeed, we need to know the cell doubling time and diffusion coefficient. Unfortunately, these quantities were not measured in the original experiments. However, we know that  typical cell doubling times 578 P . K .  MAINI et al. 4 E ~ 3  2 20 40 60 80 100 120 Time (hours) Figure 2. Typical plots from experimental data  for the position of the invading cell front (taking the origin as the position at 9.5 hours and measuring t ime from that  point) over different substrates. Distance units are 0.25 mm: control (•), colla- gen I (o), collagen IV (ZX), laminin (x) ,  fibronectin (*), vitronectin (e), hyaluranic acid (+). 140 3.5 2.5 E 2 t, ~1.5 0.5 0 ~= 0 20 40 60 80 1 O0 120 140 Time (hours) Figure 3. A least-squares straight-line fit to a typical data  set. Also plotted are the positions of the  computed wavefront assuming different threshold values defining the front: experimental data  (x) ,  least-squares fit (*), u ---- 0.5 (o), u ---- 0.1 (o), = o.ol (A). 160 Travelling Waves 579 0 . 0 4  0.035 0.03 -~ 0 .  025 0 . 0 2  o .  u) O. 015 0.01 1 0.005 0 K c. o o o o = o : ~ X J 0 20 40 60 so 100 120 Time (hours) Figure 4. Velocity plots corresponding to Figure 3. Velocity estimates from the experimental data  (×), least squares fit (*), u = 0.5 (o), u ---- 0.1 (o), u = 0.01 (A). 140 range between one and seven days (see Table I in [15]), and typical diffusion coefficients are of the order of 10 - 6  mm2s -1. So, assuming a cell doubling time of five days and a diffusion coefficient of say 10 -6 mm2s -1, we predict a wavespeed of 9.1 × 10 - 3  mmhr -1, which compares well with the actual measured wavespeeds that lie within the range 5.4 × 10 - 3  m m h r - l - - 9 . 4  × 10 -3 mmhr -1 calculated from the data (variation arising due to different patients and different substrates). CONCLUSIONS Several authors have predicted constant speed travelling waves for various biological processes. This prediction has been based on models that  are, essentially, extensions of the basic Fisher equation in which cell movement is affected by mechanochemical cues and cell division is affected by chemical signals. In this paper, we have considered a very simple wound healing migration assay and have shown that it exhibits constant speed travelling waves with a speed consistent with that  predicted by Fisher's equation. Moreover, we have shown that  the way in which the wavespeed approaches this asymptotic wavespeed is very close to that  predicted by the model. To our knowledge, this is the first verification of Fisher's equation in a medical context. R E F E R E N C E S  1. J.A. Sherratt  and J.D. Murray, Models of epidermal wound healing, Proc. Roy. Soc. Lend. B 241, 29-36, (1990). 2. H. Sheardown and Y.-L. Cheng, Mechanisms of corneal epithelial wound healing, Chem. Eng. Sci. 19, 4517- 4527, (1996). 3. P.D. Dale, J.A. Sherratt  and P.K. Maini, The speed of corneal epithelial wound healing, Appl. Math. Lett. 7 (2), 11-14, (1994). 4. J.D. Murray, P.K. Maini and I~.T. Tranquillo, Mechanochemical models for generating biological pat tern  and form in development, Phys. Reports 171, 59-84, (1988). 5. L. Olsen, J.A. Sherratt  and P.K. Maini~ A mechanochemical model for adult dermal wound contraction: On the permanence of the contracted tissue displacement profile, J. Theor. Biol. 177, 113-128, (1995). 6. P~.A. Gatenby and E.T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res. 56, 4740-4743, (1996). 580 P . K .  MAINI et al. 7. A.J. Perumpanani, J.A. Sherratt, J. Norbury and H. Byrne, A two parameter family of travelling waves with a singular barrier arising from the modelling of matrix mediated malignant invasion, Physica D 126,145-159, (1999). 8. M.A.J. Chaplain and H.M. Byrne, Mathematical models for tumour angiogenesis: Numerical simulations and nonlinear wave solutions, Bull. Math. Biol. 57, 461-486, (1995). 9. A. Okubo, P.K. Maini, M.H. Williamson and J.D. Murray, On the spatial spread of the grey squirrel in Britain, Proc. Roy. Soc. Lond. B 238, 113-125, (1989). 10. R.A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7, 353-369, (1937). 11. A. Kolmogoroff, I. Petrovsky and N. Piscounoff, ]~tude de l'~quation de la diffusion avec croissance de la quantit~ de mati~re et son application ~ un probl~me biologique, Moscow Univ. Bull. Math 1, 1-25, (1937). 12. R.J. Faull, J.M. Stanley, S. Fraser, D.A. Power and D.I. Leavesley, HB-EGF is produced in the peritoneal cavity and enhances mesothelial cell adhesion and migration, Kidney Int. 59, 614-624, (2001). 13. J.D. Murray, Mathematical Biology, Second Edition, Springer, Berlin, (1993). 14. U. Ebert and W. vanSaarloos, Front propagation into unstable states: Universal algebraic convergence towards uniformly translating pulled fronts, Physica D 146, 1-99, (2000). 15. P.K. Maini, S. McElwain and D. Leavesley, A travelling wave model to interpret a wound healing cell migration assay for human peritoneal mesothelial cells (submitted). 

English

Travelling waves in a wound healing assay

Oxford University Research Archive (ORA)

AbstractSeveral authors have predicted that cell propagation in a number of biological contexts, for example, wound healing, tumour cell invasion, angiogenesis etc., occurs due to a constant speed travelling wave of invasion. The analyses of these models to arrive at this prediction is, in many cases, essentially an extension of the classical analysis of Fisher's equation. Here, we show that a very simple wound healing assay does indeed give rise to constant speed travelling waves. To our knowledge, this is the first verification of Fisher's equation in a medical context

Maini, P.K

McElwain, D.L.S

Elsevier - Publisher Connector 

Available online at www.sciencedirect.com Applied 
. c , . . c .  Mathematics 
Letters 
ELSEVIER Applied Mathematics Letters 17 (2004) 575-580 
www.elsevier.com/locate/aml 
Travell ing Waves in a Wound Heal ing Assay 
P. K. MAINI 
Centre for Mathematical  Biology, Mathematical  Inst itute 
24-29 St Giles', Oxford, OX1 3LB, U.K. 
D. L. S. MCELWAIN 
School of Mathematical Sciences 
Queensland University of Technology 
GPO Box 2434, Brisbane, Australia Q4001 
D.  LEAVESLEY  
School of Life Sciences 
Queensland University of Technology 
GPO Box 2434, Brisbane, Australia Q4001 
(Received and accepted January 2003) 
Communicated by G. Wake 
Abst rac t - -  Several authors have predicted that cell propagation i  a number of biological con- 
texts, for example, wound healing, turnout cell invasion, angiogenesis etc., occurs due to a constant 
speed travelling wave of invasion. The analyses of these models to arrive at this prediction is, in 
many cases, essentially an extension of the classical analysis of Fisher's equation. Here, we show that 
a very simple wound healing assay does indeed give rise to constant speed travelling waves. To our 
knowledge, this is the first verification of Fisher's equation in a medical context. (~) 2004 Elsevier 
Ltd. All rights reserved. 
Keywords - -F i sher ' s  equation, Human peritoneal mesothelial cells, Invasion waves. 
INTRODUCTION 
Several authors  have proposed that  constant  profile, constant  speed travel l ing waves play an 
impor tant  role in a number  of medical  appl icat ions.  For example,  in ep idermal  wound heal ing, 
models  have been proposed in which cells move and  prol i ferate, somet imes in response to chemical  
signals, forming a wave which moves into the  space to close the  wound (see, for example,  [1-3]). 
Mechanochemica l  models  for dermal  wound heal ing hypothes ize  that  cells exert  t rac t ion  forces on 
extracel lu lar  matr ix ,  set t ing  up mot ion  and  guidance cues to which they  respond as they  invade 
the  wound space and  eventua l ly  close the  wound [4,5]. 
This work was carried out at Queensland University of Technology during a visit by P. K. Maini funded under 
the Visiting Fellowship Scheme and finished during a Royal Society Leverhulme Trust Senior Research Fellowship 
(PKM). We thank F. Stoop for assistance with the analysis and B. Addison-Smith for numerical work. P.K. 
Maini also thanks W. van Saarloos for constructive comments. 
0893-9659/04/$ - see front matter (~) 2004 Elsevier Ltd. All rights reserved. 
doi: 10.1016/j.aml.2003.01.011 
Typeset by .42kdS-TEX 
576 P.K. MAINI et al. 
~mour  cancer cells are hypothesized to outcompete normal  cells setting up waves of inva- 
sion [6], or to invade normal  tissue by degrading extracellular matrix with proteases [7]. Angio- 
genesis, the process by which a tumour  recruits endothelial cells and establishes its own vascular 
system, is thought to occur as a directed wave propagation of cells from the normal  host vascu- 
lature to the tumour  under the influence of tumour  angiogenesis factors secreted by tumour  cells 
(see, for example, [8]). 
The  above processes are modelled, in the main, by systems of coupled nonlinear partial- 
differential equations which describe the spatiotemoporal evolution of cells and signalling cues. 
Cells are assumed to diffuse (usually modelled by Fickian diffusion) and respond to various 
chemical signals via chemokinesis and/or chemotaxis, and to mechanical signals via haptotaxis 
(movement  up adhesive gradients), and contact guidance. The  models are solved using a mixture 
of analytical and numerical techniques and have been shown to exhibit constant profile constant 
speed travelling waves in cell density. 
Al though there is good evidence that such propagating travelling waves exist in ecology (see, 
for example, [9]), we  are unaware of this being shown conclusively in the aforementioned medical 
applications. 
In this paper, we  consider a much simpler system and show that a wound healing assay does 
indeed appear to admit travelling waves of constant speed. Moreover, we  show that the data is 
well fitted by Fisher's equation, a single reaction-diffusion equation that was initially proposed 
as a model  for the spread of an advantageous gene in a population [10,11]. 
In the next section, we describe briefly the experimental set-up and results. Then, we  consider 
the Fisher equation and compare its behaviour to the experimental results. Finally, we  discuss 
the implications of this work. 
WOUND HEAL ING ASSAY AND RESULTS 
We refer the reader to the paper [12] for full experimental details. Briefly, human peritoneal 
mesothelial cells (HPMCs) were grown as monolayers on different substrates to confluence. In- 
tracellular matrices used were collagen I, collagen IV, laminin, fibronectin, vitronectin, and 
hyaluronic acid. A 4 mm scrape wound was then made and the displaced cells were removed. 
The remaining cells were bathed in fresh culture medium and the position of the invading cell 
front was noted against a reference grid. Figure 1 shows the typically observed sharp front of 
cells invading the space left by the scrape. 
The results for all substrates and all experiments were remarkably consistent. After a short 
period of time, the cell fronts progress with what appears to be a constant wavespeed (see 
Figure 2). 
THE MODEL 
We propose that the waves observed are Fisher waves. That is, we assume that the cell 
movement can be modelled by Fickian diffusion and that proliferation satisfies a logistic growth 
curve. Furthermore, we assume from the experimental set up that the process is occurring in one 
space dimension. Hence, the model is 
an 02n 
0-7 = D-o~x2 + rn(g  - n), (1) 
where n(x,  t) denotes cell number density at position x and time t, D is the diffusion coefficient 
(assumed constant), rN  is linear growth rate of cells, and N is the carrying capacity. 
For the experimental set up, the initial conditions would be of Heaviside type. However, we 
actually take as our starting point the first measurement, which is after 9.5 hours. At this stage, 
the initial condition will be a "smoothed" Heaviside. However, the exact profile of the initial 
condition is not as important as the observation that it has compact support. Under these 
Travelling Waves 577 
Figure 1. A typical human peritoneal mesothelial cell front ten hours after wounding. 
conditions, it is well known that the model exhibits a constant profile travelling wave moving 
with constant speed (see, for example, [13] for references) 
v = 247- . (2) 
Thus, we see that this travelling wave solution has a nontrivial set of initial conditions for 
which it is asymptotically attracting. Figure 3 illustrates a typical plot of front distance from 
origin with time, together with a least-squares fit assuming a front moving with constant speed. 
As can be seen, the actual wave initially moves at a speed greater than that of the fit, then it 
moves more slowly, then the speed increases almost monotonically to the asymptotic wavespeed. 
In all the experiments we analysed, the wavefront initially moves at a speed greater than the 
speed predicted from the least-squares fit. Then, the speed decreases below that of the fitted 
speed. After this, in some cases, we see a monotonic increase to the fitted speed, in other cases we 
see a slight overshoot (as in Figure 3) before it eventually settles down. Figure 4 illustrates this 
more clearly. Note that the prediction from the Fisher equation is that initially the wavespeed 
will be greater than the asymptotically predicted wavespeed, then the speed will decrease and 
approach the asymptotic wavespeed from below [14]. This is very close to the behaviour that we 
observe xperimentally. 
To make a quantitative comparison between the predicted wavespeed and the actual wavespeed, 
we need to know the cell doubling time and diffusion coefficient. Unfortunately, these quantities 
were not measured in the original experiments. However, we know that typical cell doubling times 
578 P .K .  MAINI et al. 
4 
E 
~3 
2 
20 40 60 80 100 120 
Time (hours) 
Figure 2. Typical plots from experimental data for the position of the invading 
cell front (taking the origin as the position at 9.5 hours and measuring time from 
that point) over different substrates. Distance units are 0.25 mm: control (•), colla- 
gen I (o), collagen IV (ZX), laminin (x), fibronectin (*), vitronectin (e), hyaluranic 
acid (+). 
140 
3.5 
2.5 
E 2 
t, 
~1.5 
0.5 
0 ~= 
0 20 40 60 80 1 O0 120 140 
Time (hours) 
Figure 3. A least-squares straight-line fit to a typical data set. Also plotted are 
the positions of the computed wavefront assuming different hreshold values defining 
the front: experimental data (x), least-squares fit (*), u ---- 0.5 (o), u ---- 0.1 (o), 
= o.ol (A). 
160 
Travelling Waves 579 
0.04  
0.035 
0.03 
-~ 0.  025 
0.02  
o. 
u) O. 015 
0.01 1 
0 .005 
0 
K 
c. o o o o = o : ~ X 
J 
0 20 40 60 so 100 120 
Time (hours) 
Figure 4. Velocity plots corresponding to Figure 3. Velocity estimates from the 
experimental data (×), least squares fit (*), u = 0.5 (o), u ---- 0.1 (o), u = 0.01 (A). 
140 
range between one and seven days (see Table I in [15]), and typical diffusion coefficients are of the 
order of 10 -6  mm2s -1. So, assuming a cell doubling time of five days and a diffusion coefficient of 
say 10 -6 mm2s -1, we predict a wavespeed of 9.1 × 10 -3  mmhr -1, which compares well with the 
actual measured wavespeeds that lie within the range 5.4 × 10 -3  mmhr- l - -9 .4  × 10 -3 mmhr -1 
calculated from the data (variation arising due to different patients and different substrates). 
CONCLUSIONS 
Several authors have predicted constant speed travelling waves for various biological processes. 
This prediction has been based on models that are, essentially, extensions of the basic Fisher 
equation in which cell movement is affected by mechanochemical ues and cell division is affected 
by chemical signals. In this paper, we have considered a very simple wound healing migration 
assay and have shown that it exhibits constant speed travelling waves with a speed consistent 
with that predicted by Fisher's equation. Moreover, we have shown that the way in which the 
wavespeed approaches this asymptotic wavespeed is very close to that predicted by the model. 
To our knowledge, this is the first verification of Fisher's equation in a medical context. 
REFERENCES 
1. J.A. Sherratt and J.D. Murray, Models of epidermal wound healing, Proc. Roy. Soc. Lend. B 241, 29-36, 
(1990). 
2. H. Sheardown and Y.-L. Cheng, Mechanisms of corneal epithelial wound healing, Chem. Eng. Sci. 19, 4517- 
4527, (1996). 
3. P.D. Dale, J.A. Sherratt and P.K. Maini, The speed of corneal epithelial wound healing, Appl. Math. Lett. 7 
(2), 11-14, (1994). 
4. J.D. Murray, P.K. Maini and I~.T. Tranquillo, Mechanochemical models for generating biological pattern and 
form in development, Phys. Reports 171, 59-84, (1988). 
5. L. Olsen, J.A. Sherratt and P.K. Maini~ A mechanochemical model for adult dermal wound contraction: On 
the permanence of the contracted tissue displacement profile, J. Theor. Biol. 177, 113-128, (1995). 
6. P~.A. Gatenby and E.T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res. 56, 4740-4743, 
(1996). 
580 P .K .  MAINI et al. 
7. A.J. Perumpanani, J.A. Sherratt, J. Norbury and H. Byrne, A two parameter family of travelling waves with 
a singular barrier arising from the modelling of matrix mediated malignant invasion, Physica D 126,145-159, 
(1999). 
8. M.A.J. Chaplain and H.M. Byrne, Mathematical models for tumour angiogenesis: Numerical simulations and 
nonlinear wave solutions, Bull. Math. Biol. 57, 461-486, (1995). 
9. A. Okubo, P.K. Maini, M.H. Williamson and J.D. Murray, On the spatial spread of the grey squirrel in 
Britain, Proc. Roy. Soc. Lond. B 238, 113-125, (1989). 
10. R.A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7, 353-369, (1937). 
11. A. Kolmogoroff, I. Petrovsky and N. Piscounoff, ]~tude de l'~quation de la diffusion avec croissance de la 
quantit~ de mati~re t son application ~ un probl~me biologique, Moscow Univ. Bull. Math 1, 1-25, (1937). 
12. R.J. Faull, J.M. Stanley, S. Fraser, D.A. Power and D.I. Leavesley, HB-EGF is produced in the peritoneal 
cavity and enhances mesothelial cell adhesion and migration, Kidney Int. 59, 614-624, (2001). 
13. J.D. Murray, Mathematical Biology, Second Edition, Springer, Berlin, (1993). 
14. U. Ebert and W. vanSaarloos, Front propagation into unstable states: Universal algebraic onvergence 
towards uniformly translating pulled fronts, Physica D 146, 1-99, (2000). 
15. P.K. Maini, S. McElwain and D. Leavesley, A travelling wave model to interpret a wound healing cell migration 
assay for human peritoneal mesothelial cells (submitted). 


Travelling waves in a wound healing assay 

Maini, Philip

ORA - Oxford University Research Archive

GÖRKEMLİ, Burcu

Erciyes University - AVESIS