Mechanistic
Model of Wound Contraction
ROBERTT. TRANQUILLO, PH.D.,*,l ANDJ. D. MURRAy,t PH.D., D.Sc.
.Department o{ Chemical Engineering and Materials Science, Universit), o{ MinnRsota. .Yinneapolis, MinnRsota 55455;
and tApplied Mathematics FS-20, Universityo{ Washington, Seattle. Washington 98195
Submitted for publication July 8, 1991
The
interplay
mechanical
tion
bet~~.een
phenomena
is examined
The
model
wound
ing
,~'hich
with
accounts
extracellular
migration
and
traction
character.
tissue
with
respect
sistent
with
model
which
for
fibroblasts
and,
local
to the
wound
wound
for
center,
is found
of an inflammation-derived
growth,
contracting
tion
the
wound.
wound
center
exponential
largely
consistent
mon
(Plast.
extent
geometry.
is shown
rate
with
of contraction
the
Burg.
features
case
to
obey
the
findings
72,
from
commonly
and
of McGrath
66,
1983)
rate
initial
trac-
approach
law
contraction
of the locus of cell force during wound contraction
for example,the "picture frame
blastic cells pull
.'
in from the wound edge, and the "pull"
where fibroblastic cells already within
ate a tension which causesthe inward movement
wound boundary. Even the nature of the cell force
been debated [5]: the "contraction force"
with the hypothesized shortening of the entire cell
of a
where
as fibroblasts
contraction
the
of the
mediaproperties
the qualitative
increase
Reconstr.
independence
final
to
base
of a con-
or chemotactic
In particular,
is prescribed
ser ved
the
do predict
of the
influence
gradient
and
position
to be incon-
Extensions
the possible
cell
of
traction,
fibroblasts
their
that
tor
of the
of
of exert-
motility,
centration
on the
the
has a viscoelastic
irrespective
contraction.
account
their
assumes
same
modelo
in the course
which
~vhich
the
contrac-
by a combination
~vith
ECM,
model,
are
in wound
repopulating
(ECM)
ing and inflammation, then a rational basis for the design and application ofpharmacologic modulators of cell
behavior to augment or mitigate wound contraction (as
appropriate for the nature of the wound being managed)
would be available. The model proposed here is a first
attempt in that direction.
A preponderance of evidence implicates fibroblastic
cells as the primary force driving wound contraction [3J.
However, the signals which determine the spatial and
temporal distribution of fibroblastic cells in and around
a wound, as well the factors which govern the magnitude
of force transmitted by the cells to local extracellular
matrix (ECM) fibers, are poorly understood. These .
and bio-
of a mathematical
associated
A base
properties
result
aid
proliferation
of the
biochemical,
the
matrix
Carees
deformation
cellular,
obto
be
and
Si-
concerning
constant
wound
size
and
and
!& 1993 Academic Press,lnc.
l. INTRODUCTION
Wound contraction, the centripetal movement of the
wound boundary and adjacent uninjured skin toward the
wound center, is a hallmark ofhealing full-thickness cutaneous wounds [1]. There is strong motivation for understanding the interplay between the complex cellular,
biochemical, and biomechanical phenomena which result in wound contraction, since it can be a beneficial or
deleterious consequence of wound healing [2]. If a validated predictive model of wound healing were developed,
one which accounted for the salient phenomena and
provided a framework for accommodating the emerging
understanding of the relationship between wound heal-
To whom correspondence should be addressed.
233
"traction force"
podal activity of a cultured fibroblast [7]. The
mechanism by which fibroblasts and/or
causewound contraction is still an open question. Fortunately, the modeling approach described here is sufficiently abstract to accommodate both mechanisms (although here we refer to the operative force solely as traction force, reflecting our bias, and use "fibroblast" to
mean any fibroblastic cell exerting traction). Nonetheless,the traction force which presumably allows a fibroblast to migrate from the surrounding dermis into the
developing granulation tissue at the same time causes
some local reorganization ofECM fibers. This fiber reorganization, in turn, causes a local deformation of the
ECM, and the macroscopic manifestation of all of the
local deformations due to the traction forces exerted by
all the fibroblasts involved in wound healing is wound
contraction. Of course, other forces, such as viscous,
elastic, and osmotic forces, also contribute to the actual
rate and extent of wound contraction.
0022-4804/93 $4.00
Copyright @ 1993 by Academic Press, lnc.
All rights of reproduction in any form reserved.
1
234
JOURNAL
2, AUGUST 1993
tially independent of both the excised wound afea and
the wound geometry for the limited casesstudied (Table
1). The majar difference in the two wound geometries
~
studied is associated with Ao, with the immediate expanti
1
sion of square wounds being considerably greater than
~ 1
that of circular wounds ofthe same excised afea, leading
05
Irlc
to a larger Ao. Wound contraction in humans is algo
characterized by the lag and exponential phases [9] al0.6
though there are no data to prove the same conclusions
0.4
; ~
regarding the independence of Af and kc from initial
e
wound afea and geometry. These characteristic phases
o.~
[
provide a basis for validation of any mathematical model
00'
_L
I
I
I
I
a
10
20
30
40
50
of the dynamics of wound contraction.
It is appropriate that the mathematical model be abplatea u
exponential
stract in the sense of omitting uncertain details as \Ve11
phase
phase
as flexible in the sense of being able to accommodate
Time (days)
fundamentally different cell biology and more compliFIG.1.
A plot ofthe wound afea normalized to the excised afea cated biomechanics when they are postulated to be of
for a full-thickness rat wound (Fig. 1 of McGrath and Simon [10]). Ao
significance. The model used in this work, based onthe
is the wound afea when contraction begins, Af is the afea remaining
continuum modelproposedby G. F. Oster, J. D. Murray,
aftercontraction is completed, andA. = Ao -Afis the contracted afea.
and their colleagues [11, 12], possessesboth ofthese feaSee text for details.
tures. It accounts for known cell functions, such as migration and mitosis, and the coupling of cell traction
forces to the mechanical state of the ECM. This couWounds are not amenable to detailed quantitative
pling leads to a predicted traction-induced deformation
study', and almost all such studies conducted to date
of the ECM. In this paper, we examine predictions of
have been performed in animal models rather than hucontraction for an idealized dermal wound using extenman subjects. Whereas up to 100% of an animal wound
sions of the Oster-Murray model which incorporate saafea may be closed by contraction owing to the presence
lient features of wound healing suggestedabove and exof the panniculus carnosus, this value is considerably
panded upon below. Our main objective here is to proless in humans, for example, 20-40% of postexcisional
vide a framework for understanding how known and
wound afeas ofthe forearm [8,9] (the balance ofwound
speculated fibroblast functions are orchestrated with
closure is attributable to reepithelialization over the
other cellular, biochemical, and biomechanical events in
granulation tissue). Not~'ithstanding this anatomical
wound healing to give rise to contraction. Our primar)'
difference, it is generally believed that the underlying
criterion of whether a particular set of fibroblast funcmechanisms are similar, and there are several quantitations, ECM properties, and wound healing events is suftive studies based on animal wounds addressing the
ficient is qualitative consistency with the contraction
long-standing controversy of what type of wound clases
curve typified in Fig. 1. The number of sets which could
the fastest. This is typically accomplished by measuring be considered is quite large given the complexity of the
wound afea following the creation of an excisional
wound healing response;therefore our survey cannot be
wound, using tattoo marks to delineate the wound
comprehensive. Our strategy has been to examine a
boundary and distinguish contraction from reepithelial"base model" consisting of a minimal set upon which
ization. McGrath and Simon [10] definitively showed
there seemsto be general agreement (although not necesthat the contracting phase of excised dermal wounds on
sarily agreement on how they are modeled). Given that.
rats could be described with the simple exponential dethe base model does not yield a contraction curve qualitapendence on time
tively consistent with Fig. 1, extensions of the base
model are made by incorporating additional fibroblast
A(t) = Af + (Ao -Af) exp( -ket),
(1) functions, ECM properties, and wound healing events
and then checking for consistency with Fig. 1. In addiwhere Ao is the wound area when contraction begins, Af tion to identifying those sets which meet the criterion,
is the area remaining after contraction is completed, the predictive use of the model is demonst:rated by sugboth areas being scaled to the excised area, and heis the gesting how modification of fibroblast functions encontraction rate constant. Excisional wounds undergo hances or diminishes the rate and extent of contraction,
whichever is desired for managing a particular dermal
an immediate expansion, exhibit a "plateau" or "lag"
phase of several days, and then exhibit the exponential wound. A full description of this work, including details
contracting phase described by (1), which occurs on the of the mathematical model and its numerical analysis, is
arder ofweeks (Fig. 1). Af and hewere found to be essen- found in Tranquillo and Murray [13].
I
I
11
Q)
~
"
1':ti
OF SURGICAL RESEARCH: VOL. 55, :\0.
I
TRANQUILLO
AND MURRA Y: MECHANISTIC
MODEL OF WOUND
CONTRACTION
235
TABLEI
Contraction
Data ol McGrath and Simon [1O) lar Full-thickness
11. BASE MODEL
The general features ofthe Oster-Murray model from
which the base model for dermal wound contraction is
formulated have been described at length elsewhere [11,
12]. Only a brief summary is given here with emphasis
on considerations particular to wound healing and contraction. General modeling and analysis issues which
apply to all model extensions presented in Section III
are also discussed in this section. Since we are concerned
only with wound contraction here, we do not. consider
the subsequent phases of wound healing, and assuming
wound contraction is not significantly influenced by reepithelialization, we neglect the latter as well [14].
Model Formulation
In the formulation of the base model and all of its
extensions, there are only three variables: the fibroblast
concentration, n, the ECM concentration, p, and the displacement vector for the fibroblastjECM composite, U,2
all ofwhich are functions oftime, t, and spatial position,
x. The current formulation reflects several significant
simplifications which are consistent with the chosendegree of abstraction for the model at present: dynamics
associated with migration and mitosis of fibroblasts
alone are described explicitly (neglecting other cell types
at this point), dynamics associated with movement and
synthesis of the collective ECM are described (rather
than for its constitutive components), and macroscopic
stresses developed in the ECM associated with fibroblasts distributed throughout exerting traction forces
are described ("coarse graining" of the point forces exerted by individual cells on individual ECM fibers). We
briefly discuss these simplifications next.
The first simplification may be the most severe.Fibroblasts appear to transform between dermal fibroblast
2 U is the vector connecting a material point in the composite 10cated at position x at time t with its initial position, Xo. It measures
the magnitude and direction of displacement of a material point due to
acting forces. Mathematically, Xo(x, t) + u(x, t) = x.
Wounds on Rats
and myofibroblast phenot)rpes during the course of
wound healing, which vary with respect to their migration, biosynthetic activities, and traction force capacity
[3). In this base model, however, we consider only a single population of fibroblasts exerting a constant traction
force. Inflammatory, immune cells, and endothelial cells
algo exhibit transient appearances in the granulation
tissue and adjacent dermis. In several extensions of the
base model, an implicit account is made for the speculated role of wound macrophages in releasing diffusible
bioactive mediators which modulate fibroblast behavior
(migration, mitosis, and traction force) at the wound site
by postulating that a concentration gradient of such a
mediator exists during the course of wound contraction.
However, no account is made of the other majar cell type
believed to significantly influence the wound healing process, the endothelial cell, which through the process of
angiogenesis causes granul.ation tissue to be vascularized, thus influencing the concentrations of metabolic
nutrients for fibroblasts. Instead of attempting to include angiogenesis in this first-generation model, we assume that wound contraction can be described to a first
approximation without its inclusion; i.e., angiogenesis is
assumedto proceed such that there are no limitations of
oxygen and other blood-derived nutrients which would
modulate the fibroblast responses.Note that any simplifying assumption can be appropriately modified where
reliable modeling can be made.
The second simplification means that the variable p
represents the collective fibrous network of both the
wound ECM and the surrounding dermis (i.e., the total
concentration of the ECM irrespective of composition
and, therefore, of location). In reality, extensive modification of the wound ECM composition occurs during
wound healing, the initial fibrin clot transforming into
dynamic granulation tissue with time-varying percentagesof collagen, fibronectin, hyaluronic acid, and other
proteoglycans and glycosaminoglycans [15, 16), all of
which are synthesized by infiltrating cells and in turn
likely influence their behavior in a concentration-dependent manner. Consistent with the abstraction embodied in the definition of p is the assumption that the
236
JOURNAL
OF SURGICAL RESEARCH: VOL. 55, NO. 2, AUGUST 1993
rheological properties of the ,..'ound ECM and surrounding dermis are the same. This is algo a gross, but appropriate, assumption, given that kno\\'ledge of granulation
tissue rheology is virtually nonexistent. Also consistent
is the notion that "direct synthesis" of fibrous network
is meaningful in the context of the model; i.e., no explicit
account is made for the complex transport and assembly
steps of collagen and other ECM molecules.
The final simplification which makes account of traction forres exerted by cells on the ECM in which they
reside as macroscopic "active stress" is the crux of the
Oster-Murray modeling approach. As indicated earlier,
details of how the cells generate and transmit motility
forces which create this traction stress are unnecessary
in this continuum formulation. The modeling of this
traction stress is discussed further below.
A complete discussion of the base model assumptions
and associated model parameters and equations is given
in the Appendix. In summary, the base model assumes
the following cell and ECM properties:
Wound
- ---
~
---
Steady Concentration
Gradient
of Inflammatory
Mediator
(fom1edby leukocytes prior:o fibroplasia)
--
--
-
ciot)
-
i'~
x=o
f1-:"--.:-""
/
(rapidly ¡Oled-rKlrin
fA'
/"
I
-
Fibroblasts:
migration
logis.itic gro\'o1h
tractlon
force
Space
"ECM/cell
composite"
¡
homogeneous -wound matr.. and de,",. a'e not distjnguisheCS
linear viscoelastlc
stresses
traction stress
proportional to local ECM and fibrOb:as:concent'a1ionswrth
CO?!aCI
inhibilion at high librob'as! conoernrations
elastlc restonng force
j ~
FIG. 2. Schematic of idealized initial state of a dermal '\"ound
sho,,-ing majar assumptions about fibroblast and ECM (dermis and
wound) properties. See text for details.
concentrations which may act to synchronize and possibly enhance rather than inhibit traction force [18]. AIternative functional forms for T can easily be assessed
within the framework of this modelo
To compare predictions of this base model and the
extensions
that follow with the results of McGrath and
fibroblasts
random migration, passive conSimon
[10]
for contraction of full-thickness excisional
vection (with ECM), logistic
wounds, the initial state of the model wound is taken to
growth
coincide with the state ofthe experimental wound followECM
passive convection
ing the immediate expansion phase (e.g., A = 1.35 at t =
cell/ECM composite simple viscoelastic material con- O in Fig. 1 is normalized to A = 1 at t = O in the model).
taining traction-exerting cells ECM, devoid of fibroblasts, is assumed to be present in
and with elastic subdermal at- the wound initially; an accurate assumption gi,en the
tachments.
rapid formation of the fibrin clot. Furthermore, the
ECM concentration in the wound is assumed to be the
The equation which defines how u varíes with time same initially as the dermis outside the wound boundand position is based on a balance of the forces assumed ary, defined as x = L, with value Po (x = O defines the
to be acting on the cell/ECM composite, which we state wound center and L is the characteristic linear dimenabove. The key term in the force balance is that which sion of the wound). The wound is considered to be suitaccounts for traction-exerting cells distributed in the ably isolated so that the surrounding dermis can be apECM, since in this model an imbalance of traction force proximated as an "infinite medium" (equivalently, there
is the driving force for contraction. We assume that the are no "boundary effects"). This implies that variables
traction force acts isotropically, like a pressure, with should assume values characteristic of uninjured dermis
magnitude proportional to the product n' p (implying a sufficiently far from the wound: p = Po,n = N (the maxirapid equilibrium of cell pseudopod-ECM fiber anchor- mum cell concentration in logistic growth), u = O (no
age interactions or receptor-mediated adhesions), where displacement). These conditions for a "one-space-dithe proportionality factor, 7, reftects the traction stress mension" model wound (seebelow) are illustrated in Fig.
generated per unit cell and ECM concentrations. To ac- 2 along with a summary of the model assumptions.
commodate the in vitro observation of contact inhibition
Two wound geometries are considered here, a linear
of motility [17] and, therefore, of traction force genera- wound, which is consistent with long parallel wound
tion, 7 is taken to be a monotonic decreasing function of boundaries, and a circular wound. We assume that one
cell concentration, specifically 7 = 70/(1 + Xn2). This spacedimension applies, where variables depend only on
states that if the local cell concentration increases, all one spatial coordinate, linear or radial (i.e., no depenelse being equal, the traction exerted by each cell in the denceofvariables on vertical position). This ássumption
locale decreases.The traction parameter 70is a measure is generally not true; however, our modeling of subderof the traction exerted by a cell on the local ECM in the mal attachment of the ECM (see the Appendix) makes
absence of contact inhibition, which is measured by X. an approximate account for the vertical dimension. Note
The functional forro of 7 is an open question, as is its that for a one-space-dimension model, the displacement
validity for fibroblasts in vivo. lndeed, Gabbiani and co- vector u becomes simply a scalar variable, u (here, linear
workers noted that myofibroblasts possessnumerous in- or radial displacement depending on linear or radial distercellular gap junctions and desmosomes at high cell tance from the wound center, respectively).
Alodel
TRANQUILLO
AND MURRA Y: MECHANISTIC
Predictiol1S
Details of the numerical solution of the modeling
equations are found in Tranquillo and Murray [13]. A
nonlinear least squares regression fit of predicted values
of the normalized wound afea ayer a series of times,
A(tj), was performed to determine optimal values for the
parameters defining (1): Ao, Af, and k, the dimensionless
contraction late constant (k = k"L2j:IJ, where k" is the
dimensional contraction late constant and :IJ is the cell
"diffusion coefficient" measuring random migration).
There is variability in the goodnessof fit among different parameter sets, indicating that the exponential fit is
more or less satisfactory depending on the parameter
values. AIso, the regression value for k can be quite sensitive to the number of included A(tj) points at long
times when the steady state is approached (i.e., when
variables no longer change with time). Most of the fits
underlying the parametric sensitivity plots (e.g., Fig. 4)
are at least as satisfactory as the one presented in Fig.
3d; however, variations in k of up to :tO.1 can be associated with different time cutoffs.
The model variables are made dimensionless by
choosing appropriate scaling factors, as shown below:
n
ñ=-,
No
p
p=~
-u
u=-
L'
-x
x=-
L'
~
t
t = L27:15 (2)
MODEL OF WOUND CONTRACTION
237
the unperturbed state (A -1) rather than any contraction of the wound boundary inward (A < 1). It thus
seems that the base model set of fibroblast and ECM
properties, which we consider to be a minimal set ofproperties applicable for dermal wound healing, is inadequate for simulating wound contraction.
III. MODEL EXTENSIONS
AII extensions of the base model add to the fibroblast
functions, ECM properties, and wound healing events
assumed for the base modelo Except for the "ECM biosynthesis" extension, they are motivated by the widely
recognized infiuence of infiammation-derived biochemical mediators on fibroblast functions. Although the prevailing consensus is that wound contraction is predominantly regulated by the infiammatory phase which precedes and overlaps it, positive feedback intrinsic to the
fibroblast response (to the initial infiammatory stimulus) may eventually be the regulatory determinant. Such
feedback, leading to enhanced fibroblast accumulation
in the wound, might be both biochemical and biomechanical (see Discussion). We have chosen here to circumvent the added complexity of a variable concentration field oí the mediator and simply assume that a
steady-state concentration exponentially decreasing
away from the wound center exists ayer the time-course
of contraction.
(all variables and parameters are dimensionless hereafter, although tildes are omitted). There are several ad- A. Traction Variation
vantages to doing this: the dimensionless variables do
Among the various ways that an infiammatory medianot have units and take on values of arder one. Also,
tor
could infiuence the behavioral properties of fibrowhen the variables in the governing model equations are
blasts, it has been suggestedthat fibroblasts may transmade dimensionless, five dimensionless parameter
groupings arise (seethe Appendix). Since there is a scar- form into traction-enhanced myofibroblasts around the
wound due to some infiammation-derived growth factor
city of data currently available from which to define values for the 10 model parameters appearing in the base [3]. A simple alternative to modeling the population dymodel, leaving a very large number ofparameter combi- namics of the myofibroblasts explicitly is to make TO'
nations for characterization, another advantage of sca- measuring fibroblast traction, an increasing function of
the mediator concentration, c. Thus, as cells approach
ling the variables is that only five parameter groups
the wound center, they encounter a greater mediator
must be assignedvalues. Furthermore, four ofthe groups
concentration (by assumption, above) and exert a
measure the relative importance of competing phenomgreater traction. We assume for illustration that Tohas a
ena: ro represents the ratio of characteristic times for
saturable
dependence on c,
fibroblast repopulation of the wound by random migration across the wound spaceto that by proliferation, JLa
characteristic ratio of viscous force to elastic force acting
'To = 'TO[1 + 'Tf~}.
(3)
in the wound, TOthe ratio oftraction force to ECM elastic force, and s the ratio of subdermal elastic force to
ECM elastic force. The values used for the dimension- where Tf is a traction force enhancement coefficient. By
less groups in the results presented are mostly of arder prescribing c to be 1 at the wound center, TOincreases
from a value characteristic of the uninjured dermis, TO'
one, which means that one phenomena is not dominatfar from the wound to a maximum value 50. Tf % greater
ing the other.
We did not observe predictions of the base model con- at the wound center.
An example of the model predictions is given in Fig. 3
sistent with features of a contracting wound (Fig. 1), at
for
a linear wound. The steady profile for TObased on the
least for the parameter combinations examined. Typiabove
assumptions is indicated. The repopulation of the
cally, there is a slight expansion of the wound boundary
wound
by fibroblasts resulting from random migration
outward (A > 1) followed by relaxation inward back to
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TRANQUILLO
AND MURRA Y: MECHANISTIC
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CONTRACTION
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MODEL OF WOUND
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Force Enhancement
Coefficient.
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ti
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TI
FIG. 4. Dependence for the "variable traction" extension of (a) the final wound area at steady state {"uncontracted" area), Ac, and (b) the
dimensionless contraction late constan!, k, on the traction force enhancement coefficient, TC.Other parameter values are the same as those in
the legend to Fig. 3. See text for discussion.
and proliferation is seen in Fig. 3a. The evolution of the
cell/ECM composite to a contracted steady state (cf. a
contracture) is apparent from the simulated time-course
of displacing tattoo marks in Fig. 3e derived from the
displacement profiles in Fig. 3b (note that u(x, t) < O
means the material point located at position x at time t
has displaced toward the wound center from its initial
position, and conversely for u(x, t) > O). Associated with
the contraction is an increase in the ECM concentration
(p> 1) in the wound (Fig. 3c). The plot ofthe wound area
(based on the location of the wound margin) over time in
Fig. 3d is qualitatively consistent with the data in Fig. 1,
suggesting that the fibroblast-ECM-inflammatory mediator mechanisms of this scenario can interact to yield
the empirical exponential fit of (1).
Given that these results are in accord with the simple
criterion of contraction, it is of interest to determine
how other parameters of the model affect the qualitative
and quantitative features ofthe predictions for this case.
The dependence of the dimensionless contraction late
constant, k, and the wound afea remaining after contraction is completed, Af, on the traction force enhancement
factor, Tf, is presented in Fig. 4. Af decreasesfor increasing Tf(Fig. 4a), as expected, since a greater traction exerted per cell suggests an increased final extent of contraction. Figure 4b indicates only a weak dependenceof
k on Tfover the range investigated.
It is algo oí interest to compare predictions írom this
case with the findings oí McGrath and Simon [10] that
kc and Ar are essentially independent oí initial wound
size and geometry. Ií parameters íor the illustrative case
oí Tr = 2 in Fig. 3 (k = 0.910, Ar = 0.906) are accordingly
adjusted to consider a linear wound of one-halí the initi al area, Ao/2 (i.e., one-half the initial width, L/2), the
model predictions are k = 0.196 andAr = 0.826 (Table 2).
Accounting íor the scaling dependence of k on L (since
(1) in terms oí dimensionless time is used to fit the predicted A(t;) values), the expected value oí k, ií kc is indeed independent oí geometry, is k = 0.222. Thus, in the
case oí a linear wound, the model predicts a slight dependence of kc on initial wound area (relative change in
kc, ~kc = -12%), atleastíorthiscombinationofparameters (a change oí this magnitude may not be significant
given the variability in k associated with an imperfect
exponential fit which was noted earlier). This is consistent with the lack oí dependence found by McGrath and
giman for a square wound with one-halí the initial area
(Table 1; ~kc = +18% was determined not statistically
significant, nor were any oí their values íor ~kc or Mr).
For a circular wound with the same linear dimension
as the linear wound (i.e., same diameter as width) the
model predicts k = 0.952, quite clase to the values predicted for the linear wound (Table 2: ~kc = -4.6%), in
accord with McGrath and Simon's finding concerning
FIG.3.
Representative predictions of tbe "traction variation" extension for Tr = 2 and a linear wound: (a) fibroblast concentration, n, (b)
displacement of the cell/ECM composite, u, and (c) ECM concentration, p, are plotted versus distance from the wound center for a series of
times after wounding. x = Ois the wound center and x = 1 is the initiallocation ofthe wound boundary. Since the profiles are symmetrical about
x = O, only the results for x ~ O are shown. The traction profile associated with the inflammatory mediator gradient is indicated (not to scale;
actual range is 0.5-1.0; see[3]). In (d) the relative wound area (actual area normalized to the initial area following immediate expansion) plotted
as a function of time is described by the continuous curve (A > 1 is associated with wound expansion; A < 1 is associated with wound
contraction). The dashed line is the nonlinear least squares regression fit of the numerical data A(l, tJ to tbe exponential forro of (1)
investigated by McGrath and Simon [10]. In (e) the movement oftattoo marks located across the wound and adjacent dermis is simulated for a
series of times over the course of contraction. Parameter values are r = 1, Jl.= 1, s = 1, TO= 0.5, ~ = 1. See text for discussion.
~"~;
~
JOURNAL
240
OF SURGICAL RESEARCH: VOL. 55, NO. 2, AUGUST 1993
TABLE2
"Traction
Variation"
Model Predictions
.lk.
Wound
Large circular (A ~ Ao)
Large linear (A = Aod)
Smalllinear (A = Ao/2)
Small circularc (A = Ao/2)
V. large circularc (A = Uo)
k"
0.952
0.910
0.196
0.422
2.12
kp..d°
-
(%)
-4.6
0.222
0.455
1.82
-12
-7.2
+16
.
Arb
Ab
0.858
0.906
0.826
0.814
0.900
0.142
0.094
0.174
0.186
0.100
.l.4(b
~b t
(%)
(?f,)
-
-
+5.6
-8.8
-5.1
+4.9
-34
+85
+31
-30
.k is a dimensionless form of kc; kpM is the predicted value oí k assuming k" is const~nt.
b These values are relative to the maximum area precontraction.
c~ values are relative to the large circular wound.
d Based on width equal to diameter of the large circular wound.
.~ values are relative to the large linear wound.
the independence of kc from wound geometry (Table 1:
Ákc = -12%). Although comparison data are not available, model predictions were obtained for a circular
wound with initial afea Ao/2 (i.e., L/21/2), where k =
0.422 (vs prediction ofO.455, Ákc = -7.2%), and for initi al afea 2Ao (i.e., L. 21/2),where k = 2.12 (vs prediction
of 1.82, .:lkc= + 16%). Thus, the model is consistent with
their conclusion concerning independence of kc from Ao
for circular as well as linear geometry.
Comparisons of the wound afea are more difficult to
interpret since the values of Ar observed by McGrath
and giman are much smaller than those predicted by our
model under the small strain (therefore, small displacement, small contraction) assumption implicit in our
model equations (see the Appendix). Comparing values
of LlAr, the relative change in final wound afea, the
model predictions agree with McGrath and Simon's
conclusion that the extent of contraction is independent
of initial wound size; agreement does not exist for LlAc
values, the relative change in contracted afea, the model
predictions showing a strong dependence on initial
wound size.
It is reasonable to question whether their finding that
kcand Ar are independent of initial wound afea, based on
square and circular wounds, should, in fact, apply to linear wounds as well, since now the geometries are fundamentally different. Our model predictions suggest so,
but this remains to be verified.
B. Growth Variation
knowledge, in vitro data from cultured fibroblasts in tissue-equivalent collagen gels. Here we assume for illustration that fibroblasts will proliferate to a higher maximuro cell concentration in the presence of a growth factor and take the functional dependence of N on c again
to be a simple saturable dependenceas was done for T in
(3) (defining Nf analogously to Tf).
To meet the A < 1 contraction criterion, it was necessary to reduce X,the parameter measuring the extent of
contact inhibition of traction, from the value 1 assumed
until now to 0.1 (inhibition now does not occur until
higher cell concentrations). To facilitate the computations, .ILwas increased from 1 to 10 (ECM viscous forces
are now an arder of magnitude greater than ECM elastic
forces). Otherwise, the parameter values are as for the
previous cases(note that Tf= Oapplies here, i.e., no variable traction). As seenin Fig. 5 the profiles are distinctl:-'
different from the "traction variation" extension,.notably the increased cell concentration in the wound. AIso,
at least for these parameter values, the point of maximuro displacement toward the wound center occurs outside the wound (at x ~ 2.5) rather than near the initial
location ofthe wound margin (x = 1) (cf. Fig. 3b). Unlike
the traction variation results, the dependence of Af on
the varied parameter (here, N J is not monotonic; rather
an optimal Nf (~5) exists for minimum Af (Fig. 6a). Thi~
is due to inhibition of traction force in the wound as tht:'
cell concentration increases (relative to X) with increasing Nf. AIso, k now shows a significant dependence 011
the varied parameter, N f. The increase in k with increasing Nf reflects the more rapid fibroblast repopulation 01'
the wound (cf. Figs. 3a and 5a). This is due to a greater
cell growth late in the direction of the wound associateo
with increasing N in the lo gistic growth late law.
Given the feasibility of topical application of patientderived growth factors to cutaneous wounds [19], it is of
interest to understand the consequencesof a concentration gradient of a single growth factor around the wound
site which affects the mitotic characteristics of infiltratC. ChemotaxÍ8
ing fibroblasts. Unfortunately, there are no in vivo data
Fibroblasts are considered to be chemotactic to a numfrom which to deduce how the parameters of the assumed logistic growth rate (see the Appendix), r and N, ber offactors which are generated atthe wound site (i.e..
may depend on growth factor concentration, nor, to our cell-derived chemotactic factors, including many growt 11
TRANQUILLO AND MURRAY: MECHANISTIC MODEL OF WOUND CONTRACTION
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FIG.5.
Representative predictions ofthe "growth variation" extension for Nf = 10 and a linear wound: seethe legend to Fig. 3. The profile
in maximum fibroblast concentration, N, associated with the inflammatory mediator gradient is indicated (not to scale; actual range is 1.0-6.0;
see [3]). Other parameter values are r = 1, ¡L = 10, s = 1, TO= 0.5, X = 0.1. See text for discussion.
factors, and proteolytic ECM fragments [20,21]), and it
is typically assumed that chemotaxis is an important
mechanism by which fibroblasts, as well as most other
cells involved in wound healing, localize in the wound.
Here we assume the simplest description of chemotaxis,
based on adding a chemotactic term to the random migration term in the cell conservation equation defining n
(see the Appendix). The chemotactic term is proportional to the product of the cell concentration, the spati al gradient of the chemotactic factor concentration,
and the chemotaxis coefficient, x, which measures the
directional bias exhibited by a cell in a chemotactic gradient (the steady chemotactic factor concentration profile is taken to be the same as that for c previously used
for the traction variation and "growth variation" extensions). Recognizing that chemotaxis serves as a mechanism oí fibroblast accumulation, it is expected that these
results would share similarities with the growth variation extension, for the same reasons (Fig. 7). Over the
range oí X examined (x = 0-1.2), the dependencies oí k
and Af on x are qualitatively similar to those for Tf (Fig.
4). Although dependencies similar to those for Nf (Fig. 6)
appear for larger values of x (greater directed migration), the small strain assumption becomes more violated and the results less valido
In passing, we note that fibroblasts resident in the
dermis are believed to be essentially nonmigrating (zero
speed)in the absenceof any stimulation, so chemotactic
factors typically increase the speedof movement (orthokinesis) as well as bias the direction of movement (chemotaxis). An account for orthokinesis requires yet another migration term, but the theoretical consequences
are well known: if speedincreases only as a cell migrates
toward the wound, orthokinesis will counteract the accumulation of fibroblasts in the wound associated with
chemotaxis; if speedincreases then decreases,it will enhance the accumulation where the speedpassesthrough
its maximum value.
D. ECM Biosynthesis
Here, the base model is extended to include an ECM
generation term to account for the extensive deposition
of collagen and other ECM macromolecules by fibro-
.
~
JOURNAL
242
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OF SURGICAL RESEARCH: VOL. 55, NO. 2, AUGUST 1993
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O
2
Maximum
.¡
Cell Concentration
6
,:
8
Enhancement
10
Coefficient,
12
NI
FIG. 6. Dependence for the growth variation extension of (a) the final wound ares at steady state (uncontracted afea), Af, and (b) the
dimensionless contraction late constant, k, on the maximum cell concentration enhancement coefficient, Nf. Other parameter values are the
same as those in the legend to Fig. 5. See text for discussion.
an expansion of the wound boundary followed by relaxation to its initial position as fibroblasts repopulate the
wound. Thus, the base model, which takes no account of
the infiuence of bioactive factors generated from the infiammatory processes, appears inadequate to explain
wound contraction. Interestingly, infiammation and
wound contraction are both notably absent in early gestation fetal wound healing ¡'22].
Several scenarios suggestedby the possible role of infiammation-derived mediators of fibroblast behavior in
the wound healing response were examined by appropriate extension of the base modelo Upon postulating a
steady concentration gradient of an infiammatory factor
emanating from the wound during the time-course of
contraction, model predictions are qualitatively consistent with wound contraction for the caseswhere the factor enhances any of the following fibroblast properties:
traction force, maximum concentration in the ECM, and
chemotaxis. As a quantitative test of the model, the predicted variation of uncontracted wound afea with time
IV. DISCUSSION
was fit with the empirical, exponential rate contraction
This work marks an initial effort aimed at elucidating law (1) for characterization in terms of the dimensionhow the cellular, biochemical, and biomechanical phe- less contraction rate constant, k, and the final unconnomena known or speculated in the dermal wound heal- tracted area relative to the initial wound afea, Af. In
ing process are orchestrated to give rige to wound con- accord with the experimental findings of McGrath and
traction. Given the complexity of the phenomena, com- Simon [10], k and Af are only weakly dependent on both
pounded by the uncertainty of many regulatory points, wound geometry (based, however, only on a comparison
the strategy that we have adopted is to establish a con- of linear and circular wounds of the same characteristic
tinuum model framework based on a set of cell functions dimension) and initial afea (based on a comparison of
and ECM properties which seem intrinsic to any sce- wounds with half and double the initial afea) for the
nario of wound contraction: fibroblasts repopulate the traction variation extension of the base model, at least
for the parameter values tested. In this extension, the
wound ECM by a combination of migration and proliferation and, in the course of exerting traction forces asso- traction force parameter is a saturating function of the
ciated with their motility, deform the local ECM, which local concentration of the inftammatory mediator.
The model predictions concerning the occurrence and
has a viscoelastic character. Numerical solutions of the
extent
ofwound contraction (i.e., the parametric depenequations describing a base model where cell and ECM
dencies
of AJ are intuitively obvious based on the fact
properties do not vary with time or position show only
blasts in the granulation tissue. Since the regulation of
this process is poorly understood, a particular functional
dependence of the net ECM generation late on the
model variables (n, u, p, and c) cannot be independently
justified. We assume, solely for the purpose of illustration here, that this late is proportional to that of fibroblast proliferation (i.e., a logistic growth-type late: bn(l
-n», so that there is ECM accumulation until cells
repopulate the wound (n -1). Although this late forro
yields a contracted steady state (Fig. 8) without having
to invoke the influence of an inflammatory mediator,
the fact that p attains a steady state on the time scale of
wound contraction is not consistent with the observation that collagen continues to accumulate for extended
periods after completion of wound contraction. A
greater understanding of ECM regulation in wound healing is needed to guide any future proposition of realistic
late forros for ECM generation.
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FIG. 7. Representative predictions oCthe "chernotaxis" extension Corx = 1 and a linear wound: seethe legend to Fig. 3. Here, X (=xco/.'D) is
a dirnensionless group rneasuring chernotactic migration relati,'e tO randorn rnigration. The inflarnrnatory mediator concentration gradient is
indicated (not to scale). Other pararneter values are r = 1, J1= 1, s = 1, TO= 0.5, A = 1 (sarneas Fig. 3, except Tf = O). See text Cordiscussion.
that in this model, wound contraction is driven by an
imbalance of traction force. Within the context of the
current model, that is possible if (i) the fibroblast concentration in the wound is greater than that outside the
wound (but not so great that contact inhibition, if any, is
significant), each cell generating the same traction (e.g.,
the growth variation and "chemotaxis" extensions), or
(ii) the fibroblast concentration in the wound is the
same as that outside the wound, but cells closer to the
wound center generate more traction (e.g., the traction
variation and ECM biosynthesis extensions). These assertions are consistent with the classical "pull" hypothesis of wound contraction and follow directly from the
form of the traction force term in the mechanical force
balance, traction force ocn' p at low values of n (n and p
are the fibroblast and ECM concentrations, respectively), which drives the displacement of the cell/ECM
composite. They are algo supported by the parametric
dependencies presented for Ar. Clearly, the functional
form of the traction force term is critical to the model
predictions and needs to be justified theoretically (e.g.,
from a microstructural model of cell-fiber mechanical
interactions) and experimentally (e.g., from force measurements of fibroblasts cultured in tissue-equivalent
collagen gels [23, 24]).
A number of unsatisfactory aspects of the base model
and its extensions are discussed in their respective sections. The prescription of the mediator concentration
gradient in lieu of it being autonomously determined
from a mathematical model of infiammation affords
great simplicity but has undesirable consequences: it
creates some ambiguity in comparing the results for the
linear and circular wounds, because identical infiammatory mechanisms will not generally give rige to the same
gradient in different geometries. More seriously, besides
circumventing the question oí the causal factors and ultimate regulatory control of wound contraction, it does
not allow for the investigation oí potentially important
biochemical feedback effects due to autocrine íactors
which infiltrating fibroblasts probably release. AIso, it
imposes constraints on the predicted rate of contraction,
if in reality the gradient is dissipating beíore contraction
244
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JOURNAL OF SURGICAL RESEARCH: VOL. 55, NO. 2, AUGUST 1993
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FIG. 8- Representative predictions for the "ECM biosynthesis" extension with b = 5 and a linear wound. See the legend to Fig. 3. Here, b
(=bNL2/.'D) is a dimensionless ECM biosynthesis late constant. Parameter values are ro = 1,Jl = 1, s = 1, TO= 0.5, ~ = 1 (same asFig. 3, except TC
= O). See text for discussion.
is complete. To investigate ibis consequence further, we
prescribed dissipation ofthe mediator gradient to simulate the resolution of inftammation by prescribing that
the mediator concentration at each point in space decay
exponentially with time (c oce t). Results for the illustrative traction variation case of Fig. 3 for different values ofthe temporal decayrate constant, "', are presented
in Fig. 9 (the time required for c to decay 95% of its
initial value at any point in the gradient is 30, 3, 0.3 for '"
equal to 0.1, 1, 10, respectively). Note that in eachcase,a
contracted steady state (cf. a contracture) results even
after the mediator gradient which elicited the contraction has dissipated. Obviously, both the kinetics (k) and
the final extent of contraction (Ar) can vary with the rate
oí gradient dissipation ("').
We have concentrated here on some scenarios by
which wound contraction could result from the modulation oí a fibroblast function by an inftammatory mediator. However, that is only part of a much broader picture.
There are a number of biomechanical as opposed to biochemical regulatory mechanisms which are worth considering, necessarily so, since a variety of fibrotic patho-
logies do not involve a prominent inflammatory component. For example, alignment of ECM fibers in and
around the wound, due initially to platelet contraction of
the fibrin clot and subsequently due to wound contraction, may be a mechanism leadingto localizationoffibroblasts in the wound via contact guidance and haptotaxis,
as we investigate elsewhere [24]. Of course, a more realistic constitutive stress-strain relation for the dermis and
granulation tissue, in particular one that admits large
strains and accounts for the dependence of rheological
parameters on the composition and orientation state of
the ECM fiber network, is a necessary refinement for
clinical applicability. The complex evolution ofthe composition and structure of granulation tissue has implications not only for the forro of the stress-stram relation
and parameter values defined therein, but algo for the
role of the traction stress termo We have illustrated this
in a contrived way (ECM biosynthesis extension) by allowing for the generation of ECM during the process of
wound repopulation by fibroblasts, resulting in increased p, which by virtue of the assumed functional
forro of the traction force term drives wound contrac-
~
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vation, yielding the equation describing u. The specific
terms of these equations depend, of course, on both the
phenomena presumed operative and their mathematical
modeling. The general speciesconservation equation expressesthe requirement that the rate of accumulation of
a species in a control volume equal the net flux of the
species through the boundaries of the control volume
plus the net rate of generation within the control volume, where the control volume is any arbitrary region of
space where the speciesexists [e.g., 27]:
0.98
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0.92
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= {net inward flux} + {net generation}.
(Al)
Time,T
FIG.9.
Relative wound afea plotted as a Cunction oCtime Corthe
variable traction extension including dissipation oCthe biochemical
mediator gradient. '" is the exponential decay late constant Cor the
mediator concentration. Parameter values are the same as those in the
legend to Fig. 3. See text Cordiscussion.
For the fibroblast conservation equation, this translates
into
{rate of cell accumulation}
= {net inward migration flux}
+ {net inward convective flux} + {mitosis rate}
tion. Inclusion of mechanisms which model the evolution of granulation tissue composition and structure
more realistically would allow the contribution of this
effect to be understood better and possibly provide a
basis for using the model to consider the occurrence of
hypertrophic scars and keloids and their relationship to
wound contraction. Another class of biomechanical
mechanisms relates to the probable modulation of fibroblast metabolism by stress and strain in the deforming
ECM. There are many examples recently documented
for altered metabolism of cells cultured on substrata
which are externally deformed or subject to fluid shear,
which suggests that fibroblast functions may be modulated during wound contraction. Finally, electrical fields
generated intrinsically within the wound and applied externally may affect various aspects of the cell and tissue
response [25].
In summary, there are many scenarios for wound contraction which can be considered given the complexity of
the wound healing process. This "mechanocellular"
modeling approach offers a possibility of considering the
effects of known and speculated cell proliferation, migration, and traction responses and ECM rheological
properties individually as well as collectively. An improved uriderstanding of how these effects are interrelated through modeling, which will be promoted by applying this modeling approach to a novel in vitro wound
healing and contraction assay [24, 26], should lead to
new therapies for controlling wound contraction and optimizing the wound healing response.
APPENDIX
The three variables of the model satisfy fundamental
conservation laws: species conservation, yielding equations describing n and p, and linear momentum conser-
-{death
rate}.
(A2)
The convective term is a passive contribution to the
net cell flux due to any time-dependent displacement of
the cell/ECM composite. This contribution occurs irrespective of the assumed modes of active cell migration
and simply states that if a force imbalance causes the
ECM to deform, resident cells will be carried along with
the deforming ECM. For the base model we assume only
that fibroblasts migrate randomly ("cell" will mean fibroblast hereafter). We take the cellular equivalent of
Fick's Law for modeling random migration, as demonstrated for fibroblasts on substrata [28] and other cell
types in isotropic collagen gels in vitro [29], involving a
cell diffusion coefficient, :IJ (\\'hen migration is random,
:IJ is proportional to the mean square cell displacement
divided by time, for times much greater than the cell
directional persistence time). A reasonable choice for
modeling the net rate of cell proliferation (mitosis plus
death) is the logistic gro\vth rate, which is consistent
with data for cultured dermal fibroblasts in collagen gels
[30] and myofibroblasts on substrata [31], involving
first-order growth rate constant r and maximum cell
concentration N (In 2/r is the mínimum "doubling time"
as the cells proliferate from an initial concentration
much less than N' , N is the concentration to which cells
will proliferate characteristic of the extracellular environment).
Given these assumptions and definitions, the mathematical forro of (A2) is
(A3)
In the chemotaxis case oí Section IIIC, the chemotactic
flux term \7. xn\7c is added to the right side oí (A3) [32].
246
JOURNAL
OF SURGICAL RESEARCH: VOL. 55, NO. 2, AUGUST 1993
viscosities, JIl and JI2,Young's modulus, E, and Poisson's
ratio, v.
This stress-strain relation is modified heuristically to
account for traction-exerting cells distributed in the
ECM by adding a term which accounts for a resultant
isotropic (traction) stress, as discussed under Model
Formulation. Another force to be considered is the reaction force on the celljECM composite due to forces transmitted through attachments of the ECM to subdermal
structures (e.g., muscle fasciae). For simplicity, we
model this force as alinear, elastic, restoring body force
measured by the elastic constant, s.
The linear stress-strain assumption applies only for
sufficiently
small strains. Furthermore, anisotropy asop
OU
(A4) sociated with stress-induced fiber alignment is not ad-+\7op-=O;
rlt
ot
missible (this is to be distinguished from fiber alignment
occurring local to a cell due to its traction structuring
In the ECM biosynthesis case of Section IIID, the net activity [36], which is not relevant for this macroscopic
generation rate bn(l -n) is added to the right sirle of description). Ho\\'ever, in wounds which exhibit signifi(A4).
cant contraction, strains are considerable andalignment
The momentum conservation equation expressesthe of ECM fibers directed toward the wound center probarequirement that the rate of accumulation of momentum bly occurs. Thus, a strictly valid comparison of the
in the control volume equal the net momentum flux model predictions can only be made with wounds exhibitthrough the boundaries ofthe control ,'olume associated ing minimal contraction. The occurrence of osmotic
with material flow and surface stressesplus the net rate forces which may develop in the wound space,particuof momentum generation within the control volume due larly in the early stages of wound healing when hyalto bodyforces. When inertial forces associated with ac- uronic acid is present in elevated concentration [16], are
celeration of mass in the control volume are negligible, neglected here, although they can be readily incorpoas is the case for celljECM mechanics, then the momen- rated into the mechanical force balance.
tum conservation equation expressest.hat the sum of all
Given these assumptions and definitions, the matheforces acting on the control volume in a specified direc- matical forro of (A5) is
tion must sum to zero. If the control ,'olume is taken to
be a box with sides of area A, t.hen the following must
hold in each of the three orthogonal directions:
For the ECM conservation equation, (Al) states that
terms reflecting net flux and net generation of ECM
must be specified. Since the ECM is modeled as an aggregate of networked fibers, there is only a convective
component to its flux (i.e., the fiber network does not
diffuse). Fibroblasts playa key, but incompletely understood, role in the regulated levels of collagen, fibronectin, and other constituents of the dermis and granulation tissue. However, to keep the base model as simple as
possible, no generation term is included (see ECM Biosynthesis). In this case, the mathematical forro of (Al)is
simply
{net body force} + A .{net surface stress} = O. (A5)
(A6)
In our case,this represents a mechanical force balance
between traction forces exerted by the cells and forces where ~ =%<vu+ VUT) = strain tensar, u = x -Xo =
associated with the physicochemical properties of the displacement vector, 8 = v. u = dilation.
The specific forms of the above equations for linear
ECM. Theconstitutive stress-strain relation for dermis,
if one were available, would be quite complex [33], and and circular wound geometries can be found in [13]. The
this must certainly also be true for granulation tissue. five dimensionless parameter groupings which alise
An accurate model of the dermis would also entail at when the governing equations are made dimensionless
least a "biphasic" description (aka poroviscoelastic), using [2] are
where mechanical force balances are applied to distinct
fiber and "solution" (i.e., nonfiber) phases [34, 35]. Con#L
~ = E(L2j
1+#L" ) v
¡: = rN-L2 ,\ =)"N'"
sistent with our modeling philosophy but with the loss of
:IJ
considerable generality, we describe the cell-free ECM
~
--v ToPoN"
S = :!'-pob2~
in the base model and in all extensions as a homoge(A7)
TO -E
E-u,
.neous solid in which elastic and viscous forces, varying
linearly with strain and strain rate, respectively, can
arise (recall that we make no rheological distinction be- where ti = (1 + v)(1 -2v)/(1 -v).
tween the dermis and granulation tissue at this stage).
ACKNOWLEDGMENTS
Even these simplifying assumptions do not uniquely define the constitutive stress-strain relation, and we
R.T.T. acknowledges the support of a NATO Postdoctoral Fellowchoose a standard form which entails the shear and bulk ship in Science administered by the National Science Foundation
~
TRANQUILLO
AND MURRA Y: MECHANISTIC
(Grant RCD-8651í33), the Centre for Mathematical Biology, Oxford
Uni\'ersity, for its hospitality during the tenure of his NATO fellowship ",hen this ,,'ork was initiated, and a grant from the Minnesota
Supercomputer Int\titute, This work (J,D,M,) was supported in part
by Grant DMS-9003339 from the National Science Foundation.
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