A morphoelastic model for dermal wound...

Biomech Model Mechanobiol manuscript No.(will be inserted by the editor)

A morphoelastic model for dermal wound closure

L. G. Bowden · H. M. Byrne · P. K. Maini · D. E. Moulton

Received: date / Accepted: date

Abstract We develop a model of wound healing in theframework of finite elasticity, focussing our attention onthe processes of growth and contraction in the dermal layerof the skin. The dermal tissue is treated as a hyperelasticcylinder that surrounds the wound and is subject tosymmetric deformations. By considering the initial recoilthat is observed upon the application of a circular wound,we estimate the degree of residual tension in the skin,and build an evolution law for mechanosensitive growth ofthe dermal tissue. Contraction of the wound is governedby a phenomenological law in which radial pressure isprescribed at the wound edge. The model reproduces threemain phases of the healing process. Initially the woundrecoils due to residual stress in the surrounding tissue; thewound then heals as a result of contraction and growth; andfinally, healing slows as contraction and growth decrease.Over a longer time period the surrounding tissue remodels,returning to the residually stressed state. We identify thesteady state growth profile associated with this remodelledstate. The model is then used to predict the outcome ofrewounding experiments designed to quantify the amount ofstress in the tissue, and also to simulate the application ofpressure treatments.

Keywords finite elasticity · wound healing · dermis ·volumetric growth · contraction

L. G. Bowden · H. M. Byrne · P. K. Maini · D. E. MoultonWolfson Centre for Mathematical Biology, Mathematical Institute,University of Oxford, Andrew Wiles Building, Radcliffe ObservatoryQuarter, Woodstock Road, Oxford, OX2 6GG, UKTel.: +44 1865 283881E-mail: [email protected]

1 Introduction

Wound healing is the physiological process by whichdamaged tissue repairs and regenerates. Most commonlythis tissue is the skin and the damage is caused by controlledsurgical procedures or traumatic accident. In either case itis desirable for the wound to heal quickly and efficiently,restoring the skin’s mechanical, protective and regulatoryfunctions. The estimated cost of pathological wound relatedsurgical procedures and subsequent treatment is a staggering£1 billion a year in the UK alone (Hex et al, 2012).

Surface wounds break only the outermost layer of theskin, the epidermis. Epidermal wounds usually heal withoutcomplications by proliferation and migration of epithelialcells across the defect. More problematic are wounds thatalso damage the underlying dermis, a thicker layer ofcollagenous elastic tissue.

Wound contraction is responsible for up to 80% ofdermal healing (McGrath and Simon, 1983). The mainprocess by which this occurs is fibroblasts pulling the woundedges inwards. Additionally, growth of new tissue within thesurrounding healthy dermis, also regulated by fibroblasts,may contribute to healing. The hole that remains is initiallyfilled with extra-cellular matrix and over a longer period oftime is remodelled into scar tissue.

Immediately after injury, the residual tension in theskin is released at the wound edge, causing retractionof the wound edge (McGrath and Simon, 1983). Healingthen proceeds through four main overlapping phases:haemostasis, inflammation, proliferation and remodelling.Firstly, haemostasis is the formation of a blood clot inthe wound space. This forms within hours of wounding,is made primarily of fibrin, and prevents blood loss. Theclot acts as a source of blood-derived chemotactic factorswhich initiate the migration of inflammatory cells into thewound space (Wahl et al, 1989). The inflammatory phase

2 L. G. Bowden et al.

lasts for up to a week (Jeffcoate et al, 2004) during whichinflammatory cells clear the wound site of bacteria, deadtissue and other foreign bodies (Singer and Clark, 1999).Fibroblasts naturally present in the surrounding dermaltissue proliferate in response to growth factors secretedby inflammatory cells (Clark, 1988) and migrate up thechemotactic gradient created by chemoattractants from thefibrin clot. Fibroblasts initiate the formation of collagenfibres that increase the mass of the dermal tissue. Epithelialcells migrate and proliferate into the wound in responseto growth factors. This process, termed re-epithelialisation,ensures that an epidermal covering is restored. Contractionof the wound edge begins approximately 2 to 5 days postwounding (Monaco and Lawrence, 2003) as fibroblastsarranged at the wound edge crawl across the substratumtowards the wound centre resulting in an inward movementof the dermal edges. Many of the fibroblasts infiltrate andbreak down the fibrin clot, replacing it with a collagen-richmatrix or scar tissue. Even after the wound has healed thescar and surrounding tissue remodel over several months.This process is important aesthetically and further allows thetissue to regain mechanical integrity.

Wound healing has been studied in a mathematicaland computational context for over 20 years. Developingaccurate mathematical models of wound healing thatcan be validated experimentally can play a crucial rolein understanding and exploring the mechanisms drivinghealing. Sherratt and Murray (1990) were the first totranslate this biological phenomenon into mathematicalterms, formulating reaction-diffusion equations to describeepidermal wound healing. Extensions to this model sawan investigation into healing rates and patterns for variouswound shapes (Sherratt and Murray, 1992). For fullthickness wounds, mathematical models typically focus onone of the processes contributing to dermal healing, forexample, wound contraction (Tranquillo and Murray, 1992;Tracqui et al, 1995; Olsen et al, 1999; Yang et al, 2013),or tissue synthesis (Segal et al, 2012). There are two mainapproaches to modelling wound contraction. One is throughpartial differential equation models (PDEs) based on theprinciples of mass and momentum balance. Typically thesecouple mass balances for the density of fibroblasts andextra-cellular matrix (ECM) with a momentum balancefor the displacement of the cell-ECM continuum andconstitutive laws that define the mechanical properties ofthe tissue (Tranquillo and Murray, 1992; Olsen et al, 1999).The second approach is based on a hybrid framework.Models of this kind focus on the interactions betweenfibroblasts, considered as discrete entities, and the ECM,considered as a continuous variable. The direction offibroblast migration is governed by ECM fibre orientation,which cells can change (Dallon et al, 1999; Dallon, 2000;McDougall et al, 2006). Although most models of wound

healing are formulated either as systems of PDEs orhybrid models, other less widely used approaches meritdiscussion. For example, Segal et al (2012) studied thecontribution of collagen accumulation in a wound to healingof the tissue. Their spatially-averaged model consisted ofa system of time-dependent, ordinary differential equations(ODEs). Although the complexity was reduced by adoptinga spatially-averaged framework, the model still had alarge number of parameters that needed to be determinedexperimentally.

The various processes contributing to healing overlap,therefore models that combine these processes may offer amore in-depth understanding of wound healing. Vermolenand Javierre (2012) adapted and coupled previous modelsof wound contraction (Tranquillo and Murray, 1992), angio-genesis (Maggelakis, 2004) and epidermal wound closure(Sherratt and Murray, 1991) to provide a descriptive modelof dermal regeneration. The authors used their model to in-vestigate the effects of the various contributing processes tohealing of the dermis. Although the model was more com-plete in that many processes were included, this came at thecost of an increased number of undetermined model param-eters. We have previously adopted an ODE framework, fo-cussing on the dominant processes contributing to healingof a full thickness wound (Bowden et al, 2014). A system ofthree ODEs was derived to track changes in the epidermaland dermal wound areas over time, coupled to a phenomeno-logical force balance. Growth of new tissue was governed bya modified logistic growth law with dermal growth enhancedby mechanical stretch caused by contraction of the tissue.Although the model combined the effects of growth and con-traction, and contained few parameters, it neglected spatialeffects, which could play an important role in mechanosen-sitive growth.

In the models discussed above, mechanical effectsare typically included via linear elasticity, if at all.Several factors highlight the importance of the mechanicalenvironment in healing and the value of a nonlinear elasticmodelling framework. When a circular wound is formedin mice, following injury the skin retracts to 120% of thewounded area (McGrath and Simon, 1983). The retractionimplies the presence of residual tension in the skin that isreleased when the skin is cut, while the large degree ofretraction points to a nonlinear regime. It is also known thatgrowth can be stimulated by local changes in mechanicalstress. Such stress is generated in healing tissue by woundcontraction, largely driven by fibroblast activity at thewound edge. While models of wound contraction typicallyfocus on the interior of the wound: e.g. the formation andremodelling of the central granulation tissue into scar tissue(Cumming et al, 2010; Yang et al, 2013), the “pulling” of thefibroblasts on the surrounding dermal tissue directly impacts

A morphoelastic model for dermal wound closure 3

the stress and growth in the tissue, which is crucial for thefinal size of the defect and the potential for scarring.

Morphoelasticity provides a theoretical framework formodelling growth in elastic tissues. It enables for amathematically tractable approach for studying the growthand remodelling of elastic tissue, while allowing for largedeformations, anisotropy and heterogeneity. It has beenapplied to various biological tissues including arteries(Taber and Eggers, 1996; Rachev et al, 1998; Taber, 1998,2001; Goriely and Vandiver, 2010), the heart (Lin andTaber, 1995), the trachea (Moulton and Goriely, 2011),plant stems (Goriely et al, 2010), tumours (Ambrosiand Mollica, 2004; Ciarletta et al, 2011) and the skin(Ciarletta and Ben Amar, 2012). Despite this wide rangeof applications, few mathematical models of wound healingincorporate nonlinear elasticity. Yang et al (2013) developeda biomechanical model of wound healing that focussed onthe formation and remodelling of the scar tissue, whichhas nonlinear elastic properties. Wu and Ben Amar (2015)used the decomposition of the deformation tensor to studythe closure of a circular epidermal wound and investigatedstress induced instabilities of the regular wound geometry.Contraction of the nonlinear elastic tissue was assumedto be generated by circumferential resorption analogousto a model of embryonic wound healing (Taber, 2009)in which the author formulates mechanosensitive growthlaws with an evolving target stress. The main focus ofthis paper is the development of a morphoelastic model ofdermal closure, that accounts for growth and contraction.Although wounding also affects the epidermal layer we donot explicitly model re-epithelialisation but incorporate onlythe effects of the epidermis on the dermis. By postulatingmechanosensitive growth laws, we aim to understandhow the mechanical environment in the dermal tissuesurrounding the wound impacts the healing process. Weuse the model to gain mechanistic insights, investigate thefeedback between growth and contraction in dermal closure,and predict behaviours that can be tested experimentally.

The paper is organised as follows. In Section 2we develop the governing equations for the growth andmechanical equilibrium of the tissue, using the frameworkof morphoelasticity. The presence of volumetric growthis demonstrated by considerations of residual tension inthe skin and the form of mechanosensitive growth laws ismotivated by a simple analysis of homogeneous growth.In Section 3 we present typical results of wound healingsimulations. We show that the model can exhibit normalhealing behaviour and show the effect of key parameterson healing. We also consider the possibility of tissueremodelling to a steady state of residual stress over a longtime scale. We then use the model to predict the outcomeof hypothetical rewounding experiments as a method ofdetermining the stress in the tissue and the outcome of

applying pressure to control wound closure. Conclusionsand discussion are provided in Section 4.

2 Model description

In this section we develop a mechanical model of dermaltissue subjected to a circular wound. We treat the dermaltissue as a cylindrical elastic annulus surrounding the woundsuch that the outer radius is far from the wound edge andthe effects of dermal tissue external to the cylinder canbe approximated by imposing a boundary condition at theouter edge. Our model is developed by considering thewound geometry, the tissue mechanics (balance of linear andangular momentum) and by prescribing constitutive lawsrelating stress to strain, and for growth.

2.1 Geometry

The geometry is pictured schematically in Fig. 1. In order toincorporate the presence of residual tension in the skin, weidentify four states. The pre-wounded tissue is denoted bystate S1 – this tissue is in a state of residual tension, whichfor a finite cylinder can be assumed to arise from a pressureTres applied by the external skin. If excised, the residualtension is relieved and the tissue relaxes to a stress-freeconfiguration, denoted S0. We assume a wound of radius Ais formed in the tissue at time t = 0, at which point the stresson the inner surface is relieved and the wound fully recoilsto a radius aR by time tR – we refer to this configuration asstate S2. Fibroblast activity then creates a contraction forcefc at the inner edge, so that at times t > tR the wound radiusis a; this current state is referred to as S3. Growth does notoccur until contraction begins, thus there is assumed to beno growth in states S0, S1, S2.

We restrict attention to symmetric deformations, thusall configurations are described as cylindrical annuli. Inparticular, the radial and axial coordinates R0 and Z0 in thestress-free reference state (S0) are deformed, at time t, tor and z in the current state (S3). The geometrical regionassociated with the reference state is

A0 ≤ R0 ≤ B0, 0≤Θ ≤ 2π, 0≤ Z0 ≤ L0,

where A0, B0 and L0 denote the wound radius, the outerradius of the modelling domain and the initial thickness ofthe tissue in the stress-free state, respectively. This region isdeformed in the current state to

a≤ r ≤ b, 0≤ θ ≤ 2π, 0≤ z≤ l,

by the maps

r = r(R0, t), θ =Θ , z = λ (t)Z0, (1)

where λ is the axial stretch.


S2: recoiled state

S3: current state

b

al

Nz

Tresfc

lR

Tres0

Nz

A0 � R0 � B0 and 0 � Z0 � L0

L0 A0

B0

S1: residually stressed state L A

BTres

Nz

A � R � B and 0 � Z � L

S0: stress-free reference state

excised

a � r � b and 0 � z � l

aR

bR

in vivo pre-wound (t=0)

post-wound (t=tR)

post-wound (t>tR)

aR � r � bR and 0 �z � lR

Fig. 1 Schematic representation of the mathematical model. Distancefrom the wound centre in the stress-free state can be described by theindependent variable R0, where A0 ≤ R0 ≤ B0. The cylinder representsthe dermal tissue which, under some deformation F, the reference stateis mapped to the current state. Distance from the wound centre in thecurrent state can be described by the dependent variable r = r(R0, t),where a≤ r ≤ b, at time t. In vivo the tissue is residually stressed withexternal radial pressure Tres MPa and axial load Nz N. After wounding,tension is released on the wound edge causing the wound to recoil,resulting in the recoiled state, where aR ≤ r ≤ bR. As the wound healsit is subject to contraction modelled as a radial pressure, fc MPa, on thewound edge. Far from the wound the tissue remains residually stressed

2.2 Morphoelastic framework

Consider an elastic body which, in its reference state,is described by the material coordinates X. After somedeformation, the body in the current state is described byx = x(X, t). The map from the reference to the current stateis described by the geometric deformation tensor (Rivlin,1948b), F= ∂x

∂X , which in a cylindrical geometry is given by

F= diag(

∂ r∂R0

,r

R0,λ

). (2)

Following the fundamental assumption of morphoelas-ticity, the deformation F is viewed as a combination of twoprocesses (Rodriguez et al, 1994). The local addition (or re-moval) of material to the stress-free state, described by thegrowth tensor G, changes the mass of the body. To accom-modate any growth incompatibilities that the body may un-

dergo we introduce an elastic deformation, described by theelastic tensor A. Thus, the geometric deformation tensor canbe decomposed as F = A ·G, represented schematically inFig. 2.

stress-free reference state

(S0)

currentstate

(S1,2,3)

F

AG

stress-freegrown state

Fig. 2 A schematic representation of the deformation of an elasticbody. The deformation tensor F can be decomposed into a growthtensor G, describing the local addition of material, and an elastic tensorA, describing the elastic response of the body. The deformation isapplied to a stress-free state, corresponding to S0 in Fig. 1, resultingin the current state. Depending on the deformation, the current statecorresponds to S1, S2 or S3 in Fig. 1

For symmetric deformations we write the growth tensoras G = diag(γr,γθ ,γz) where γr, γθ and γz, representradial, circumferential and axial growth (or resorption)respectively. Note that each γi can be a function of position,such that γi > 1 (< 1) signifies a local increase (decrease)in mass in the direction i. The elastic tensor is given byA = diag(αr,αθ ,αz) where αr, αθ and αz are the principlestretches in the radial, circumferential and axial directions,respectively.

Mechanical testing has shown that the skin is almostincompressible (North and Gibson, 1978), and so we assumedet(A) = 1. The deformation then satisfies

∂ r∂R0

=γrγθ γzR0

λ r(3)

from which we ascertain

r2(R0) = a2 +2∫ R0

A0

γr(R)γθ (R)γz(R)Rλ

dR. (4)

For given growth functions γi(R0), the deformation is fullydetermined once a and λ are known. The correspondingprinciple stretches of the elastic deformation tensor can bewritten

A= diag(

γz

αλ,α,

λ

γz

), where α =

rγθ R0

. (5)

The system mechanics are described by the balanceof linear and angular momentum and a constitutive stress-strain relationship. If the tissue has density ρ and is subject


to body forces fb, the balance of linear momentum reads

ρ x−∇ ·T−ρfb = 0, (6)

where a dot represents a derivative with respect to time. InEq. (6), T is the Cauchy stress tensor. The balance of angularmomentum gives that T is symmetric; in the cylindricalgeometry it is diagonal with radial, circumferential and axialcomponents denoted Trr, Tθθ and Tzz respectively.

The stress-strain constitutive relationship for an incom-pressible hyperelastic material (Eringen, 1962) is given by

T= A · ∂W∂A− pI, (7)

where the hydrostatic pressure p ensures incompressibilityand W = W (αr,αθ ,αz) is the strain-energy function. Wemodel the dermal tissue as a neo-Hookean material (Rivlin,1948a) with strain-energy function

W (αr,αθ ,αz) =µ

2(α

2r +α

2θ +α

2z −3

). (8)

Expressing Eq. (7) in component form we have

Trr = αrWr− p, Tθθ = αθWθ − p, Tzz = αzWz− p, (9)

where Wi = ∂W/∂αi.The mechanical description is completed by prescribing

appropriate boundary conditions. For example, we mayspecify the radial stress Trr|r=a,b and/or the axial load, Nz =∫

Tzz dA, with integration over the top of the annulus.

2.3 Residual stress

Implicit in the constitutive law in Eq. (7) is the assumptionthat F describes a deformation from a stress-free referencestate. Since the skin is naturally residually stressed in vivo,we require a map from the observed residually stressedstate to the unknown stress-free state. In other words, S1is the typical observed pre-wound state, but the mechanicaldescription entails a map from S0 to S3, hence we first mustdetermine the state S0. In this section we determine thestate S0 and estimate the value of Tres by considering theinitial recoil upon wounding. Typical wound healing dataare given as a time series of averaged wound areas (Bowdenet al, 2014) where initial measured quantities include theinitial wound radius A, the recoiled wound radius aR, and thetissue thickness L. Given these measurements, the referencegeometry and the residual stress can be calculated in thefollowing way.

Let F1 and F2 denote the deformations from the stress-free state S0 to the residually stressed (S1) and recoiled (S2)states, respectively; i.e. F1 =

∂X∂X0

and F2 =∂xR∂X0

.

Assuming no body forces and that mechanical equilib-rium is reached after the deformations F1 and F2, Eq. (6)reads

∇ ·T= 0, (10)

For plane stress, the only nontrivial component is

∂Trr

∂ r+

1r(Trr−Tθθ ) = 0. (11)

In state S1, the residual tension is assumed to be due topressure at the external boundary, so that Trr(B0) = Tres.1

Upon injury the wound edge is relieved of tension, causingit to retract to a size bigger than that of the initial wound.Therefore, in the recoiled state S2, we have Trr(A0) = 0.Far from the wound we expect the healthy dermis to remainresidually stressed so that Trr(B0) = Tres in S2.

The deformation F1 is found by solving Eq. (3) withG= I (no growth has occurred yet)

R =R0√λ1

, (12)

where λ1 is determined by imposing R(B0) = B in Eq. (12)resulting in

R =BB0

R0. (13)

The components of the elastic tensor are

αr = αθ =BB0

, and αz =B2

0B2 . (14)

Since αr = αθ , Eqs. (9) give Trr = Tθθ and imposingTrr(B0) = Tres in Eq. (11) we deduce that Trr ≡ Tres isconstant. From Eq. (9), the axial stress is also constant andcan be determined from the boundary condition

Nz = 2π

∫ B

0TzzR dR = πTzzB2. (15)

Assuming no axial load in state S1 (Nz = 0), then Tzz = 0and the residually stressed state is one of constant planarisotropic stress with T∗ = diag(Tres,Tres,0).

For a neo-Hookean material with Nz = 0, Eqs. (8), (9),and (14) imply

Tres +µ

(B4

0B4 −

B2

B20

)= 0. (16)

For the deformation F2, Eq. (4) with G= I and r(A0) =

aR gives

r2 =R2

0−A20

λ2+a2

R. (17)

1 We note that there is a one-to-one map between all states. Ourconvention is generally to view all spatially dependent variables asfunctions of the independent variable R0. So, for example, we writeTrr(R) as Trr(R0) = Trr(R(R0)).


The radial stress is determined by substituting Eqs. (8)and (9) into Eq. (11) and integrating subject to Trr|r=aR = 0:

Trr =µ

2λ2

[log(

λ2(r2−a2R)+A2

0

A20

)− log

(r2

a2R

)+

(A2

0λ2−a2

R

)(1r2 −

1a2

R

)](18)

We also have the axial boundary condition

Nz = 2π

∫ b

aTzzr dr = 0 (19)

with Tzz = Trr + αzWz − αrWr from Eq. (9). ImposingTrr|r=bR = Tres, Eqs. (16) and (18) can then be equated toeliminate Tres, giving

12λ2

[log(

B2

A2

)− log

(b2

R

a2R

)+

(A2

0λ2−a2

R

)(1

b2R− 1

a2R

)]+

(B4

0B4 −

B2

B20

)= 0. (20)

Given measurements of the initial and recoiled woundradii, A and aR, the thickness of the tissue, L, in vivo, thetissue stiffness µ , and the outer radius B, Eqs. (19) and (20)can be solved simultaneously for B0 and λ . The stress-freereference geometry, {A0,B0,L0}, and the residual stress,Tres, are then fully determined.

2.4 Domain size

From Eq. (16), we note that the residual stress dependson the outer radius B. However, we expect this to saturatewith large B, thus we seek a domain size large enough thatTres in Eq. (16) is not affected by small changes in thelocation of the outer radius. Before continuing, we note thatthere exists a wide range of parameter values and woundsizes in the literature. This is partly due to measurementsbeing taken from different species. Since most data availableare for wound healing in mice we choose our referencevariables and parameter values accordingly. Following theexperiments of McGrath and Simon (1983) (in whichcircular wounds in mice retract to a radius approximately110% of the initial cut), we take an initial wound radiusA = 4 mm that recoils to aR = 4.4 mm. For these valueswe compute Tres in Eq. (16) while varying B – the result isplotted in Fig. 3 for three different choices of µ . We observethat Tres asymptotes as B increases; and thus assign the outerradius of the domain to B = 20 mm.

0 10 20 30 40 500

0.02

0.04

0.06

0.08

0.1

0.12

B (mm)

Tre

s (M

Pa)

µ=0.1 MPa

µ=0.3 MPa

µ=0.5 MPa

Fig. 3 Residual stress as a function of the outer radius. The residualstress is calculated according to Eq. (16) for a wound with A = 4mm, aR = 4.4 mm and L = 0.1 mm for increasing B. The shearstress is µ = 0.1 (−), µ = 0.3 (- -) and µ = 0.5 MPa (· -). Thecalculated residual stress increases with shear stress and for largeenough modelling domain Tres plateaus

2.5 Shear elastic modulus

Measured values for the shear elastic modulus µ for dermaltissue may vary by a factor of 3000 depending on the modeland experimental apparatus used (Diridollou et al, 2000).A large range of residual stress in the skin has also beenreported, between 0.005 and 0.1 MPa (Diridollou et al,2000; Jacquet et al, 2008; Flynn et al, 2011). As the stresshas linear dependence on µ , we can use the residual stresscalculation to estimate a physiological range for µ .

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

µ (MPa)

Tre

s (M

Pa)

range for Tres

Fig. 4 Residual stress as a function of the shear elastic modulus. Theresidual stress is calculated according to Eq. (16) for a wound with A =4 mm, aR = 4.4 mm and L = 0.1 mm for varying µ . For biologicallyrealistic values of the residual stress as stated in the literature, the shearelastic modulus must lie in the range 0.02 < µ < 0.52 MPa

In Fig. 4 we plot Tres against µ . For 0.02 < µ < 0.52MPa, the residual stress lies within the previously reportedrange 0.005 < Tres < 0.1. Although the biological literature


estimates that the shear stress can take values between 0.006and 20 MPa, our model suggests that µ > 0.52 MPa yieldsunphysiological values of the residual stress.

2.6 Contraction alone

The calculations of Sect. 2.3 have established the stress-free state and the residual tension. We can now turn to thehealing of a wound. Our primary modelling aim is to exploremechanosensitive growth in the healing dermal tissue. Thisrequires the definition of an evolution law for the growthtensor G. To motivate this, as a starting point we considerthe possibility that no growth occurs. That is, we supposethat wound closure is driven by contraction only, and we usethe mechanical framework to quantify the degree of stressthat would be generated in the tissue in this scenario.

Wound contraction in adult dermal wounds is drivenby fibroblasts that localise on the wound edge and pullthe tissue inwards (Ehrlich, 1988; Ehrlich and Rajaratnam,1990). Dermal closure can reduce a wound by up to 80%of its original area. If there is no growth, we can simplyimpose a deformation from the post-wound state S2 to anequilibrium configuration in which the wound radius hasdecreased by the appropriate amount. From Eqs. (8), (9)and (11) we deduce that the radial stress at the innerboundary required to deform the wound radius from itsoriginal size to its final contracted size ac, is given by

Trr(A0) = Tres−µ

2λ

[log(

B20

A20

)− log

(B2

0−A20 +λa2

c

λa2c

)+(A2

0−λa2c)( 1

B20−A2

0 +λa2c− 1

λa2c

)], (21)

where λ is determined by solving Eq. (19) and the referencevariables A0 and B0 are as calculated in Sect. 2.3.

Mechanical testing reveals that, on average, mouse skinfails to withstand stresses higher than 2 MPa (Bermudezet al, 2011) and samples of dermis from diabetic mice tearat stresses as low as 0.5 MPa. In Fig. 5 we plot the stress atthe inner edge as a a function of µ . We find that, for a widerange of µ , the radial stress on the wound edge is higher than0.5 MPa. This suggests that if healing occurs by contractionalone then stresses higher than the tissue can withstand willbe generated. Thus for dermal closure to reduce a woundby up to 80%, volumetric growth must contribute to closure,relieving the high tension generated by contraction.

We further note that the circumferential stress at thewound edge can be computed from Eqs. (8), (9) and (21)revealing that it is highly compressive. This can be observedfor the planar case (λ = 1) in Fig. 6 where we plot theradial and circumferential stresses as functions of positionsfor contraction only.

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

µ (MPa)

Trr(A

0)

(MP

a)

range for µ

tearing threshold

Fig. 5 Radial stress on the inner boundary required for dermal closurewith no growth. Equation (21) is solved for varied values of µ and Nzwith reference variables: A= 4 mm, aR = 4.4 mm, B= 20 mm and finalwound radius ac = 1.8 mm. For a wide range of parameter values, theradial stress calculated is higher than the dermal tissue can physicallywithstand

Fig. 6 Radial and circumferential stress across the annulus for dermalclosure with no growth. Equations (8), (9) and (11) are solved withTrr(B) = Tres. We take µ = 0.2 and λ = 1 with reference variables:A = 4 mm, aR = 4.4 mm, B = 20 mm and final wound radius ac = 1.8mm. The radial, Trr , and circumferential, Tθθ , stress are plotted asfunctions of position.

2.7 Anisotropic homogeneous growth

Before proposing an evolution law for spatially dependentgrowth, it is instructive to analyse the effect of puregrowth in a simple setting. We neglect all body forces andboundary loads, and consider the effect of time independentanisotropic but homogeneous planar growth on an annulusof tissue. That is, we take γr and γθ as constants (with γz = 1and λ = 1), and consider the deformation and stress statedue to growth alone in the (γr,γθ ) plane.


0 1 2 3 4 5 6 7 8 9 100.0

0.5

1.0

1.5

2.0

2.5

3.0

γr (radial growth)

γ θ (

circ

um

fere

nti

al g

row

th)

a > A0

a < A0

1

2

34

5

6b < B

0

b > B0

Trr < 0T

rr > 0

a = A0

b = B0

γrγ

θ = 1

γr = γ

θ

Fig. 7 Analysis of compatible growth laws. Equation (23) is solvednumerically using the root-finding algorithm fsolve in MatLab tofind the curves in (γr,γθ )-space such that a = A0 (solid blue) andb = B0 (dashed red). The dashed black line represents no growth, i.e.γrγθ = 1. Regions 1 to 6 represent different grown configurations ofthe dermal cylinder. Reference variables are A0 = 4 mm and B0 = 5mm corresponding to a proliferative band of width 1 mm surroundingthe wound

The deformation and stress satisfy

∂ r∂R0

= γrγθ

R0

r, (22a)

∂Trr

∂R0=

µγrγθ R0

r2

(r2

γ2θ

R20−

γ2θ

R20

r2

), (22b)

Solving Eqs. (22) subject to stress-free boundary conditions,we obtain an implicit equation for the current wound radiusa in terms of γr,γθ and the reference variables:

0 =γr

2γθ

log(

B20

A20

)− γθ

2γrlog(

γrγθ (B20−A2

0)+a2

a2

)+

γθ

2γr

(a2− γrγθ A2

0)( 1

a2 −1

γrγθ (B20−A2

0)+a2

). (23)

Similarly, we can obtain an implicit equation for the outerradius of the cylinder, b, by using Eq. (22a) to substitute fora. Equation (23) allows us to explore the effect of differentialradial (γr) and circumferential (γθ ) growth. In Fig. 7 weshow how the (γr,γθ ) parameter space can be decomposedinto distinct regions, each associated with a different grownconfiguration.

For normal healing, the inner radius should decrease asthe wound repairs. At the same time, the outer radius shouldnot change markedly so that material is not compressed orpulled far from the wound. This behaviour with a < A0and b ≈ B0, corresponds to those parts of regions 5 or 6 inFig. 7 that are close to the curve b = B0. We observe that,close to the curve b = B0, as γr increases, γθ decreases,indicating growth must be anisotropic. This is consistentwith the modelling assumptions of Wu and Ben Amar(2015), in which the authors assign γθ < 1 in order to obtain

wound closure. If growth were isotropic, i.e. γr = γθ theconfiguration of the cylinder would lie in region 4 (along thedotted line) and growth would cause the wound to expand.We also observe that, in the regions of interest, growthcreates a compressive radial stress, Trr < 0 (see Fig. 8).The growth thus serves to counteract the excessive tensioncreated by contraction. On the other hand, both growthand contraction create a compressive hoop stress near thewound edge. A highly compressive hoop stress can resultin circumferential buckling (see Wu and Ben Amar (2015)for example), which could mark the onset of scarring. Theeffect of contraction and growth is counteracted by the factthat the hoop stress is tensile in the recoiled wound; in anycase, we leave a stability analysis within this framework forfuture work.

Fig. 8 Radial and circumferential stress in the case of homogeneousplanar growth. Equations (22) are solved with stress-free boundaryconditions and the radial, Trr , and circumferential, Tθθ , stresses areplotted as functions of position. We take µ = 0.2 and λ = 1 withreference variables: A = 4 mm, aR = 4.4 mm and B = 20 mm. Thegrowth components are γr = 1.6 and γθ = 0.95 and are chosen to beclose to the line b = B0 in Fig. 7

2.7.1 On geometry

The above analysis highlights the interplay of geometryand mechanics in the wound healing process. In essence,filling a wound in a circular geometry involves a packingproblem. In a circular geometry, in order to extend inwardradially the tissue must also vary in the orthogonal direction,undergoing circumferential resorption and/or being putin circumferential compression. In a one-dimensionalCartesian geometry – a ‘slash wound’ – no such geometricalissues exist. The tissue can grow in one Cartesian directionwith no change or stress induced in the orthogonaldirections.

Such issues, clearly visible at the tissue level, mustbe resolved through coordinated activities at the cell level.


From a tissue-level modelling viewpoint, such activities aremanifest in a feedback between stress and growth. For this,we next turn to heterogeneous growth laws.

2.8 Mechanosenstitive growth laws

In general, our growth laws comprise an evolution equationfor the growth tensor of the form G = H(G,T, . . .), andmay incorporate the effects of, for example, biochemistryand temperature. In this work, as we are interested inmechanically stimulated growth, we postulate the form G=

K(T−T∗)G, with K a diagonal tensor so that stress in agiven direction induces growth only in that direction. Incomponent form this is expressed as

∂γr

∂ t= k(Trr−Tres)γr, (24a)

∂γθ

∂ t= m(Tθθ −Tres)γθ , (24b)

∂γz

∂ t= nTzzγz. (24c)

In Eqs. (24) k, m and n are growth rate parameters and Tresis the resting residual stress. We note that contraction leadsto high radial tension ensuring that Trr > 0 and close tothe wound edge Tθθ < 0. This observation suggests that γrand γθ will naturally tend to the “wound filling” locationof the phase space in Fig. 7 if k,m, and n are non-negative.To begin with and for simplicity we will take k = m = n,that is, the sensitivity of the tissue to stress induced growthis equal in all directions. Initially, no growth occurs sothat G|t=0 = I. Equations (24), with k,m,n > 0, assumethat stress higher than the resting value Tres stimulatesgrowth, whereas stress lower than the residual stress causesresorption. This has been observed in wound healing wherefibroblast proliferation and collagen synthesis is enhanced inthe presence of tension (Kessler et al, 2001) and release oftension can induce apoptosis (Grinnell et al, 1999; Chipevand Simon, 2002). Similar assumptions and growth lawshave been used in other biological applications such asembryonic wound closure (Taber, 2009) and remodelling ofarteries (Alford et al, 2008).

2.9 Time scales and recoil dynamics

The recoiled state S2 is defined to be in mechanicalequilibrium. From wound healing data (Bowden et al, 2014)we observe that the recoil of the wound occurs over a periodof up to one day. As the dermis moves this causes frictionagainst the underlying subdermis. Assuming that there is africtional body force resisting radial motion of the dermis,Eq. (6) can be written as

ρ x−∇ ·T+qx = 0, (25)

where q is a friction coefficient. We now use dimensionalarguments to justify whether the time dependent termsin Eq. (25) are relevant during healing. Let χ denote acharacteristic length and te, td and tg the elastic, drag andgrowth time scales, respectively. From Eqs. (7) and (8),we deduce the Cauchy stress tensor scales with the shearelastic modulus, µ . The elastic time scale can be determinedby balancing the first two terms in Eq. (25), giving te =

(ρχ2/µ)1/2.The healing of the wound occurs on the growth time

scale, which is on the order of days (Ghosh et al, 2007).Typical values for the tissue density, 10−6 kg·mm−3,shear elastic modulus, 0.2 MPa (see Section 2.5), andcharacteristic length, 4 mm reveal that te ≈ O(10−4)

seconds. Unsurprisingly, te � tg and it is appropriate toneglect inertial terms in Eq. (25).

Balancing the second and third terms in Eq. (25) givesa drag time scale of td = qχ2/µ . The drag time scale setsthe time of wound recoil, which is on the order of hours.This gives q ≈ O(10−3) MPa·days·mm−2. Once the recoilis complete, the tissue velocity is much slower, and thedrag plays a negligible role, i.e the tissue is essentially inmechanical equilibrium during the growth and contractionphase.

2.10 Contraction

We model wound contraction by prescribing the radial stressat the inner boundary such that

Trr|r=a = fc. (26)

In practice, contraction is achieved by fibroblasts whichmigrate into the tissue and localise around the wound edge.For simplicity we represent the contractile activity of thefibroblasts as

fc(t) = f a(t)H (t; tc, tm), (27)

where f is the contraction coefficient, a is the wound radiusand H is a linear switch function, parametrised by tc andtm, given by

H (t) =

0 for t < tct−tc

tm−tcfor tc < t < tm

1 for t > tm.(28)

The switch function approximates the time taken forfibroblasts to migrate to the wound edge and begincontracting the tissue. Contraction begins approximately2 days post wounding and attains its maximal valueapproximately 5 − 14 days after injury (Monaco andLawrence, 2003). In Eq. (28), tc is the time at whichfibroblasts start to localise around the margin and tm isthe time at which the fibroblasts exert their maximumcontractile effect.


2.11 The full model

For completeness, we now state the full model, in terms ofthe independent variables R0 and t:

∂ r∂R0

= γrγθ γzR0

λ r, (29a)

∂Trr

∂R0=

γrγθ γzR0

λ r

[αWα

r+q

∂ r∂ t

], (29b)

Tθθ = Trr +αWα , (29c)

Tzz = Trr +λWλ , (29d)

∂γr

∂ t=

{0 for t ≤ tRk(Trr−Tres)γr for t > tR

, (29e)

∂γθ

∂ t=

{0 for t ≤ tRm(Tθθ −Tres)γθ for t > tR

, (29f)

∂γz

∂ t=

{0 for t ≤ tRnTzzγz for t > tR

, (29g)

In Eqs. (29) t = tR is the time at which the wound has fullyrecoiled. The recoil phase occurs on the order of hours,ending before growth and contraction begin (tR < tc). Theboundary conditions are given by

Trr(A0, t) = f a(t)H (t) and Trr(B0, t) = Tres, (29h)

r(A0, t) = a(t), (29i)

Nz(t) =2π

λ (t)

∫ B0

A0

γrγθ γzTzzR0 dR0 ≡ η(1−λ (t)), (29j)

where the two extra boundary conditions (Eqs. 29i and 29j)are used to determine the time-dependent unknowns a(t)and λ (t). Note that the axial boundary condition (Eq. 29j)is of the form of a spring axial load, which we use to modelthe resistance of the epidermal layer to axial deformation ofthe underlying dermal layer. The initial conditions for thecomponents of the growth tensor are given by

γr(R0,0) = 1, γθ (R0,0) = 1, and γz(R0,0) = 1. (29k)

Also, W is an auxiliary strain-energy function of only twovariables, using incompressibility:

W (α,λ ) =µ

2

(γ2

z

α2λ 2 +α2 +

λ 2

γ2z−3). (29l)

3 Results

3.1 Numerical implementation

Equations (29) were solved numerically in MatLab. Alldependent variables are stored as functions of R0 withspatial discretisation dx = 0.1. For each time increment(dt = 0.05 days), the growth in Eqs. (29e)-(29g) wereupdated via a forward Euler scheme (Press et al, 1994). The

integral boundary condition in Eq. (29j) was re-written as adifferential equation by letting

∂ I∂R0

= γrγθ γzTzzR0, (30a)

with

I(B0) =Nz

2πand I(A0) = 0. (30b)

The radial deformation and radial stress were thendetermined using the boundary value problem solver bvp4cfor a system of three differential equations given byEqs. (29a), (29b) and (30a) subject to boundary conditionsin Eqs. (29h), (29i) and (30b). The unknowns a and λ

were included in the boundary value problem solver as freeparameters which were determined, for each time increment,by prescribing the two extra boundary conditions.

3.2 Model behaviour

In this section we present typical model solutions, obtainedusing the parameter values stated in Table 1. These areeither taken from the biological literature or were estimatedby matching model behaviour to previous healing curves(Bowden et al, 2014).

The plot of the dermal wound radius, a(t), presentedin Fig. 9 reveals three distinct phases of healing. Duringthe first phase, which lasts for approximately two days, thewound retracts to a radius 110% of its initial size as a resultof the residual tension in the surrounding tissue and thewound radius plateaus before growth and contraction areactivated. During the second phase (2 < t < 14 days), thewound radius decreases rapidly due to contraction at thewound edge and proliferation of the surrounding tissue. Atlater times (t > 14 days), as the wound radius decreases,contraction slows down, since it is proportional to the woundradius. As a result of the mechanosensitive properties of thetissue, growth also slows down and the wound radius beginsto plateau. We note that the healing curve is qualitativelysimilar to experimental data (McGrath and Simon, 1983;Bowden et al, 2014).

In Fig. 10 we plot the corresponding growth and stresscomponents, associated with the simulation in Fig. 9, asfunctions of the undeformed radius R0 at days 2, 14 and25. As expected, since growth is mechanosensitive, with thegrowth rates in Eqs. (24) depending linearly on the stress,the growth and stress profiles are qualitatively similar withthe radial stress and growth increasing over time as thecontraction force increases. Contraction reaches a maximumat day 14 and we observe that the radial stress at day 25is less than that on day 14. As expected, radial growth andstress are larger closer to the wound edge. The dermis is incompression in the circumferential and axial directions and,


Table 1 Model parameters. Table summarising the parameters that appear in our mathematical model along with their physical interpretation,default values and supporting references. Values in brackets are adopted after Section 3.4

parameter physical interpretation units physical range reference default

k radial growth MPa−1·days−1 k > 0 0.2m circumferential growth MPa−1·days−1 m > 0 0.2 (0.1)n axial growth MPa−1·days−1 n > 0 0.2 (0.1)f contraction MPa·mm−1 0 < f < 1.325 (Uhal et al, 1998; Wrobel et al, 2002) 0.2tc contraction begins days 2−5 (Monaco and Lawrence, 2003) 2tm maximum contraction days 5−14 (Monaco and Lawrence, 2003) 14q friction MPa·days·mm−2 q = O(10−3) estimated (see Section 2.9) 0.002µ shear elastic modulus MPa 0.02 < µ < 0.52 estimated (see Section 2.5) 0.2Nz axial load Nη load coefficient N 10A initial wound radius mm A > 0 (McGrath and Simon, 1983) 4aR recoiled wound radius mm aR > A (McGrath and Simon, 1983) 4.4B domain size mm B > aR estimated (see Section 2.4) 20L tissue thickness mm L > 0 (Bowden et al, 2014) 0.1

Time (days)

0 5 10 15 20 25

a (m

m)

0

1

2

3

4

phase I phase II phase III

Fig. 9 Typical model solution with healing phases. Equations (29)are solved numerically and the wound radius a(t) is plotted as afunction of time. Reference variables and parameters are specifiedusing the default values in Table 1. The healing curve can be separatedinto three distinct phases representing the initial recoil, a proliferativeand contracting phase and finally the rate of closure slows down ascontraction and growth decrease

as a consequence, material is resorbed in these directions.With γθ < 1 < γr, we are in a regime associated with normalhealing (see Section 2.7), for which the wound radiusdecreases whilst the outer radius remains approximatelyconstant. We interpret the removal of tissue from thecircumferential direction (γθ < 1) as tissue remodelling – thenet change in volume is locally given by (detG−1), whichis positive.

3.3 Sensitivity of final wound radius to model parameters

Ultimately, we are interested in the time to closure and thequality and mechanical properties of the healed tissue. InFig. 11 we show how the wound radius at day 25 changesas we vary the default model parameters by ±10%, oneparameter at a time. We observe that increasing the radialgrowth sensitivity k causes the wound radius to decrease.

Increasing the rate of circumferential absorption (since thecircumferential stress is negative) accelerates wound closurein a manner similar to that seen by increasing the contractioncoefficient. Interestingly, circumferential resorption haspreviously been used as a contractile mechanism in a modelof epidermal wound closure (Wu and Ben Amar, 2015).The effect of axial growth on the wound radius is negligibleand as expected increasing tissue stiffness has a detrimentaleffect on wound closure, since a stiffer tissue will deformless when subject to a given force.

Fig. 11 Effect of model parameters on final wound radius. Equa-tions (29) are solved numerically. Each model parameter is varied by±10% of its default value and the wound radius at day 25 is recorded.The dashed line indicates the wound radius when all parameters arefixed at their default values. All reference variables and parameters arespecified using the default values in Table 1. Increasing the radial andcircumferential growth parameters, k and m, and the contraction co-efficient, f , decrease the wound radius at day 25. Increasing the axialgrowth parameter, n, has negligible effect on the wound radius at day25. Increasing the tissue stiffness, µ , decreases the wound radius at day25


R0 (mm)

4 6 8 10

Trr

(M

Pa)

0

0.2

0.4

t=2

t=14

t=25

(a) Radial stress (b) Circumferential stress

R0 (mm)

4 6 8 10

Tzz (

MP

a)

-0.04

-0.02

0

(c) Axial stress

(d) Radial growth (e) Circumferential growth (f) Axial growth

Fig. 10 Typical model solution for the stress and growth. Equations (29) are solved numerically. (a) The radial stress Trr , (b) the circumferentialstress Tθθ , (c) the axial stress Tzz, (d) the radial growth γr , (e) the circumferential growth γθ and (f) the axial growth γz are plotted as functions ofposition at 2 (−), 14 (- -) and 25 (· -) days. Reference variables and parameters are specified using the default values in Table 1. The radial stressis higher closer to the wound edge, resulting in higher radial growth there. The tissue is in compression circumferentially and axially, resulting inresorption of material in these directions

3.4 Anisotropic growth rate constants

We expect that, as the wound heals, the dermal tissueexperiences a net gain in material. In Figure 12 wedemonstrate the effect on the net change in volume onday 25 as we vary the growth rate parameters by ±50%.Interestingly, we find that the default set of parameters (withall growth rates equal) results in a net loss of material. Inorder for the wound to heal with a net gain in material, werequire a higher radial than circumferential (k > m) growthrate. This anisotropy in growth rate essentially requires thatthat a cell can be aware of the radial direction; in fact, thismay not be so surprising. Fibroblast cells migrate radially inresponse to chemical growth factor gradients (Pierce et al,1989; Chung et al, 2001) and hence a directional bias forradial growth is feasible. Hence, we take k > m in theremainder of this work so as to incorporate a radial bias(k = 0.2, m = 0.1, n = 0.1).

3.5 Remodelling after closure

We anticipate that, after the wound has closed, the dermaltissue remodels to its natural state of residual stress. Theformation of granulation tissue in the wound space (laterremodelled into scar tissue) prevents the dermis fromclosing completely. Therefore there is a point at which the

Fig. 12 Effect of model parameters on net change in volume.Equations (29) are solved numerically. Each parameter is varied by±50% of its default value and the net change in volume at day 25 isrecorded. The dashed line indicates the net change in volume whenall parameters are fixed at their default values. All reference variablesand parameters are specified using the default values in Table 1. Forthe given default parameters there is a net loss in material. A net gainin material is achieved by either increasing the radial growth rate k ordecreasing the circumferential growth rate m.

dermal edge halts. The mechanism by which this occurs isnot known but could involve a combination of contraction


ceasing (due to inactivation of fibroblasts), the radial stressreaching a threshold value and the granulation tissue in thewound space acting as a physical barrier, preventing furtherinward movement of the dermis. The questions we considerare whether the halted tissue will reach an equilibrium state,returning to the base residual stress and, if so, how long theremodelling will take.

We simulate the remodelling period of the dermal tissueby halting the wound edge at some point t = T . Wechoose T = 25 days as a representative time by which thegranulation tissue fills the wound space but note that thestimulus that halts movement of the dermal edge couldeasily be modified to account for other effects.

We solve Eqs. (29) for 0 ≤ t ≤ T . For t > T we switchthe boundary conditions from the fixed load specified inEqs. (29h)-(29j) to a fixed displacement so that

r(A0, t) = a(T ), r(B0, t) = b(T ), λ (t) = λ (T ). (31)

This fixes the deformation, preventing the wound fromclosing further. Due to incompressibility and a fixeddeformation, there can be no net change in material so that∂

∂ t (det(G)) = 0. We therefore have the following system fort > T (=25 days):

∂γr

∂ t= k(Trr−Tres)γr, (32a)

∂γθ

∂ t= m(Tθθ −Tres)γθ , (32b)

∂γz

∂ t= nTzzγz, (32c)

Tθθ = Trr +αWα , (32d)

Tzz = Trr +λWλ , (32e)

∂

∂ t(γrγθ γz) = 0. (32f)

Expanding Eq. (32f) and inserting Eqs. (32a)-(32c) yieldsa relationship for the stress in terms of the growth and thestrain

Trr =(k+m)Tres−mαWα −nλWλ

k+m+n, (33a)

Tθθ = Trr +αWα , (33b)

Tzz = Trr +λWλ , (33c)

where W is defined in Eq. (29l). The components of thegrowth tensor satisfy Eqs. (32a)-(32c) and the stress isupdated according to Eqs. (33).

In Figs. 13(a)-(c) we plot the components of the stresstensor at days 25, 50 and 100. At day 25, we observe alarge radial tension at the wound edge, with circumferentialand axial compression of similar magnitude. Over theremodelling period these profiles relax to a steady state withthe stress returning to the isotropic planar stress state Trr =

Tθθ = Tres, Tzz = 0. In Figs. 13(d) and (e) the components

of the stress and growth tensors are plotted as functions ofposition at day 200, after the system has reached steadystate. Figure 13(f) shows how the spatial average of thecomponents of the growth tensor evolve over time. We findthat growth has slowed down by day 50 and that by day 100the remodelling phase is essentially complete.

3.5.1 Sensitivity of remodelling time to model parameters

We now investigate how the duration of the remodellingphase changes as we vary parameters associated with thegrowth and tissue stiffness. The results are given as a bargraph of the time taken for the tissue to reach a steady statein Fig. 14. Steady state is defined as2

∂γr

∂ t=

∂γθ

∂ t=

∂γz

∂ t= 0. (34)

In order to make a reliable comparison, the modelparameters are not changed for the first 25 days of healing.After day 25, when the tissue begins to remodel, eachmodel parameter is varied by ±10% of its default value.We observe that the tissue stiffness has the greatest effecton the remodelling time, such that as µ is increased by10%, the time taken for the tissue to reach steady state isreduced by approximately 10 days. For the remaining modelparameters tested, the remodelling time is between 120and 150 days. Comparing the sensitivity of the remodellingtime to the model parameters with that of the final woundradius we observe some significant differences. In Fig. 11the radial and circumferential growth parameters had asimilar effect on the final wound radius, however in Fig. 14circumferential growth has a much greater effect thanradial growth. This may be due to the degree of radialversus circumferential stress the tissue needs to recover. Weobserve in Fig. 13(a) and (b) that at day 25 the radial andcircumferential stresses at the wound edge are 0.2451 and−0.2621 MPa respectively. With a positive resting stress(Tres = 0.0376 MPa) the growth rates in Eqs. (24) imply thatthe circumferential growth rate dominates the radial growthrate. Surprisingly, varying the axial growth rate, which hadnegligible effect on the healing curve, has a significant effecton the remodelling time. We postulate that this is due to thefixed axial deformation3.

2 Numerically we define steady state to be such that each ∂γi∂ t is

below a positive threshold value close to zero. For the simulations inFig. 14 the threshold was taken to be 10−5.

3 To test this we ran simulations where the axial deformation wasnot fixed according to λ (t) = λ (T ) but instead satisfied the springcondition in Eq. (29j). In this case, the sensitivity of the remodellingtime to the axial growth parameter was significantly lower than thatpresented in Fig. 14.


4 6 8 100

0.1

0.2

0.3

R0 (mm)

Trr (

MP

a)

t=25

t=50

t=100

(a) Radial stress

4 6 8 10

−0.2

−0.1

0

0.1

R0 (mm)

Tθ

θ (

MP

a)

(b) Circumferential stress

4 6 8 10

−0.1

−0.05

0

0.05

R0 (mm)

Tzz (

MP

a)

(c) Axial stress

0 5 10 15 200

2

4

6

R0 (mm)

Gro

wth

γr

γθ

γz

(d) Steady state growth

0 5 10 15 20

0

0.02

0.04

R0 (mm)

Str

ess

(M

Pa)

Trr

Tθθ

Tzz

(e) Steady state stress

0 50 100 150 200

0.9

1

1.1

1.2

Time (days)

Gro

wth

γr

γθ

γz

(f) Average growth

Fig. 13 Remodelling of the dermal tissue. Equations (29) are solved numerically for 0 < t < 25. At t = 25 the position of the wound edge is fixedby switching the boundary condition from a fixed load (Eqs. 29h-29j) to a fixed displacement specified by Eq. (31). (a) The radial stress Trr , (b)the circumferential stress Tθθ and (c) the axial stress Tzz are plotted as functions of position at 25, 50 and 100 days. (d) The radial, circumferentialand axial growth and (e) stress are plotted as functions of position at 200 days. (f) The average radial, circumferential and axial growth are plottedas functions of time. Reference variables and parameters are specified using the default values in Table 1. The growth reaches a steady state byapproximately day 100 and the stress returns to the isotropic planar stress T∗

k m n125

130

135

140

145

150

Model parameter

Tim

e (d

ays)

µ

−10%

+10%

Fig. 14 Effect of model parameters on remodelling time. Equa-tions (29) are solved numerically for 0 < t < 25. At t = 25 the po-sition of the wound edge is fixed by switching the boundary conditionfrom a fixed load (Eqs. 29h-29j) to a fixed displacement specified byEq. (31). During the remodelling phase, each model parameter is var-ied by ±10% of its default value and the time taken to reach steadystate is recorded. The dashed line indicates the remodelling time whenall parameters are fixed at their default values. Reference variables andparameters are specified using the default values in Table 1

3.5.2 Steady state solution

Figure 13 shows that, for the default parameter values inTable 1, the stress reaches a steady state for which T =

T∗. The growth laws in Eqs. (24) ensure that this is theonly steady state solution. A natural question is whetherthe growth profile (for example, that shown in Fig. 13(d))needed to reach this state is unique. At steady state we have

∂ r∂R0

=γθ γrγzR0

λ r(35a)

Tθθ = Trr +αWα (35b)

Tzz = Trr +λWλ . (35c)

For a fixed deformation with T = T∗, solving Eqs. (35) forthe components of the growth tensor gives

γθ =

√λ

γz

rR0

(36a)

γr =

√λ

γz

∂ r∂R0

(36b)(γz

λ

)3− Tres

µ

(γz

λ

)2−1 = 0. (36c)


Since λ ,µ > 0, Eq. (36c) has only one real root for Tres >

−( 27

4

) 13 µ given by

γz =

(C6µ

+2T 2

res

3µC+

Tres

3µ

)λ , (37)

where C =

[12µ

2

√12T 3

res +81µ3

µ+8T 3

res +108µ3

] 13

and three distinct real roots for Tres < −( 27

4

) 13 µ . That

is, if the residual stress is negative (i.e. the skin is incompression before wounding) and large enough, there arethree possible choices for γz at steady state. Therefore, theroots of Eq. (36c) depend on the residual stress where there

is always one positive root. For Tres < −( 27

4

) 13 µ there is a

fold bifurcation with two extra roots, however these are bothnegative. The qualitative behaviour of this bifurcation doesnot change for different values of µ and λ .

Since the growth profiles of γr and γθ depend on γzand we require γz > 0 in Eqs. (36) for physiologically realsolutions we conclude that the steady state profiles of thegrowth components in Fig. 13(d), associated with T = T∗

and Tres > 0, are unique.

3.6 Model experiments

The model we have developed provides a frameworkfor investigating wound healing from a mechanical pointof view. This can have particular value in the contextof experiments in which the mechanical environment isexplicitly altered. In this section we consider two suchscenarios, rewounding and the application of pressureduring healing.

3.6.1 Rewounding

The mechanical model gives access to the stress state in thetissue at any point or time. It is difficult to access the stressprofile experimentally, but the effect of stress can clearly beseen when it is relieved. Rewounding, i.e. recutting a woundin the same location, provides one means of accessing themechanical state. In the same manner as the degree of recoilupon initial wounding can be used to measure the residualtension in the healthy skin, measuring the degree of recoilupon rewounding gives a measure of the stress state in theskin during the healing process. Similar experiments havepreviously been used to determine whether the embryonic‘purse-string’ mechanism is in tension (Davidson et al,2002).

A hypothetical rewounding experiment also providesa means of calibrating our model. We have found thatdifferent balances of growth and contraction can generate

similar healing profiles. Additionally, wound closure canbe obtained with conceptually opposing growth laws andmechanical boundary conditions (Taber, 2009). Currently,most experimental data comprise time series of averagedwound areas. With such data it is not possible to distinguishsuch cases: additional information about the stress duringhealing is needed to calibrate the model. In theory,a rewounding experiment could be repeated at regularintervals during healing to give a time series of spatiallyaveraged stress measurements.

Here, we simulate rewounding, using the model topredict the retraction of a wound in response to a secondinjury. By varying the radial growth and contractioncoefficients, k and f , we compare two cases: (a) healingwith high growth and low contraction and (b) healing withlow growth and high contraction. While the evolution ofthe wound radius in each case is similar (see Fig. 15), thestress generated in the tissue differs markedly. When therate of contraction is high the stress at the wound edge ismuch greater than when the contraction rate is small (resultsnot shown). In the absence of measurements of the stressprofile, making a new wound and recording the retractedarea would enable us to estimate the tension in the skinand to determine whether contraction or growth dominatehealing. The retraction of the second injury in Fig. 15(b),when the dermal tissue is subject to low growth and highcontraction, is much greater than that in Fig. 15(a) wherethe dermal tissue is subject to higher growth and lowercontraction.

3.6.2 Pressure treatments

Pressure therapy involves the application of pressure tocontrol wound healing. The most common applicationof pressure therapy for the management of hypertrophicscars is via pressure bandages (Kischer, 1975). Despitetheir widespread use, the mechanism by which pressurebandages improve healing is not fully understood and theclinical effectiveness of this non-invasive therapy remainsto be scientifically proven (Anzarut et al, 2009). It hasbeen suggested that mechanical pressure induces a hypoxicatmosphere resulting in death of fibroblasts and thereforereduced tissue synthesis (Kischer, 1975). Alternativelymechanical pressure may inhibit the activity of fibroblastsby reducing the rate at which they secrete TGFβ , a growthfactor which stimulates fibroblasts to contract and producetissue (Martin, 1997).

As well as the application of pressure to retard theproliferative activity of fibroblasts during healing, negativepressure wound therapy (NPWT), or vacuum assistedclosure (VAC), is used to stimulate tissue regeneration innon-healing wounds. NPWT and VAC have been shownto increase blood flow and granulation tissue formation,


Time (days)

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m)

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Fig. 15 Predicted outcome of rewounding experiments. Equations (29) are solved numerically and at t = 25 a wound of radius 1 mm is cut. Thewound radius is plotted as a function of time for (a) high growth and low contraction (k = 0.8, f = 0.3), (b) low growth and high contraction(k = 0.1, f = 0.5). All other reference variables and parameters are specified using the default values in Table 1

to decrease bacteria, and to stimulate tissue synthesisthrough increased mechanical tension (Mendez-Eastman,2001; Huang et al, 2014).

The pressure needed for effective treatment is unknown,although values in the range 9-90 mmHg (0.001-0.01 MPa)have been reported (Giele et al, 1997). Alternatively NPWTtends to generate up to 125 mmHg of suction in the woundarea (approximately 0.017 MPa). In this section we useour mechanical model to investigate the effects of applyingpressure to the surrounding dermal tissue during healing.Using Eq. (29j) with a modelling domain of radius 20 mm,the pressures discussed here can be approximated by axialloads ranging from −12.5 to 21.5 N.

We simulate treatment for a period of 15 days. Thewound begins to heal load free (Nz = 0 N). On day 5 pressureis applied by assigning a non-zero axial load (Nz = 10 orNz =−10 N). Pressure is released on day 20 and the woundcontinues to heal load free until day 25.

In Fig. 16(a) we show how the wound radius respondsto different axial loads. We observe that the effect of axialload on the wound radius is negligible. However Fig. 16(b)reveals that the tissue volume is greatly affected by applyinga constant axial load. When the load is positive, for examplecorresponding to NPWT, the tissue mass added during thetreatment period is 3 times greater than in the control case.When the load is negative, for example in the applicationof a pressure bandage, there is a loss in total tissue volume.In Fig. 16(c) the tissue thickness is plotted as a function oftime. We observe that, although the changes in volume donot affect the wound radius, they contribute to the thicknessof the tissue. We also observe that when the load is releasedthe thickness of the dermal tissue returns to a value closerto the control case but there is still a noticeable difference,most likely due to the difference in tissue volume observedin Fig. 16(b). These results suggest that although applyingan axial load has negligible affect on the healing curve itcould be used to control the total amount of tissue producedin the surrounding dermis during healing.

4 Discussion

In this paper we have developed a mechanical modelof dermal wound healing in which closure is driven bycontraction of the wound edge and mechanically inducedvolumetric growth. Our first step was to establish the degreeof residual tension in the pre-wounded tissue by analysingthe recoil that is observed after circular punch biopsiesin mice. Volumetric tissue growth was motivated by theunrealistically high stresses that would be generated inthe tissue surrounding the wound area if healing wereto occur through contraction alone. A simple analysis ofhomogeneous growth revealed that normal healing shouldbe characterised by anisotropic growth, with the additionof material in the radial direction and removal of materialin the circumferential direction. In the full model, we havepostulated simple mechanosensitive growth laws that enableus to isolate the effects of such anisotropy during healingsimulations.

Typical solutions of the model revealed three distinctphases of healing. Initially the wound retracts and theradius plateaus due to no growth or contraction (up to 2days). The wound radius then decreases over approximately14 days as a result of contraction and mechanosensitivegrowth. Finally, contraction and mechanosensitive growthslow down. Over a much longer time period (approximately100 days) the dermal tissue remodels, returning to itshomeostatic resting stress. We show that there is only onesteady state for the stress, corresponding to T = T∗, withone associated non-negative steady state growth profile.The system takes approximately 120-150 days to remodel,depending on the choice of model parameters.

Fundamental difficulties in linking modelling and exper-imental efforts in wound healing relate to over-parameterisationof the mathematical models and the technical difficulty as-sociated with obtaining suitable experimental data for modelvalidation. While our modelling framework neglects manyfeatures of the wound healing process, such as an explicit


0 5 10 15 20 250

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Fig. 16 Effect of axial load on the healing dermis. Equations (29) are solved numerically. From day 5 to 20 and axial load is assigned for NPWT(Nz = 10 N, −), control (Nz = 0 N, - -) and pressure bandage (Nz =−10 N, · -). (a) The wound radius a(t), (b) the total tissue volume and (c) thetissue thickness l(t) are plotted as functions of time. Reference variables and parameters are specified using the default values in Table 1

description of fibroblast activity, it has the advantage of char-acterising wound healing of a nonlinear elastic tissue in asimple manner and with a small number of parameters. Wehave shown that the model captures well the qualitative fea-tures of healing curves obtained in mice. The problem is thatsuch features are not unique, even while varying a set of only4 free parameters. For instance, we find that if one is onlytracking the wound radius over time, two different param-eter regimes – one with high growth and low contraction,and one with low growth and high contraction – can bothproduce the same healing curve for the wound radius. How-ever, such cases can be distinguished if additional mechani-cal information is available. We have simulated rewoundingexperiments, in which the wound is recut before it is fullyhealed. The degree of recoil can be used to distinguish be-tween cases and calibrate the model. Such experiments canalso provide useful information about the mechanical stateof the healing tissue and the mechanosensitivity of the grow-ing tissue.

Our framework is also well suited to investigate pressuretreatments. By applying an axial load to the healing dermiswe observed that loads reported in the literature do notchange the behaviour of the healing curve but can have asignificant effect on the total volume of tissue producedduring healing. Therefore if the dermis is hyper-proliferativeduring healing, applying pressure to the area can normalisethe overall tissue volume. Similarly, if the dermis in under-proliferative, applying suction can normalise the volume oftissue produced.

We have considered circular wounds, as arising for ex-ample from a circular punch biopsy. We have, therefore,restricted our modelling to a cylindrical geometry. Assum-ing that the tissue is homogeneous, we showed that resid-ual stress in the unwounded skin is isotropic. Under this as-sumption a circular wound will remain circular when cutand, hence, we have restricted attention to symmetric defor-mations. A stability analysis would be straightforward andcould provide information on scarring; see for instance Wu

and Ben Amar (2015). We employed the simplest isotropicconstitutive equation for a hyperelastic material (neo-Hookean)allowing for analytical progress. Other constitutive equa-tions, such as Mooney-Rivlin or Fung, could easily be incor-porated. However, considering a more realistic anisotropicstrain-energy, such as that determined by Annaidh et al (2012),would be much more complicated and the simplicity and an-alytical power of the proposed model would be lost.

Due to the lack of experimental data on wound healingin humans we have presented results using parameter valuesand tissue dimensions obtained from experiments in mice.Given accurate experimental data, the parameters in themodel could be adapted to simulate healing in humans.

The mechanosensitive growth laws in our model aresimilar to those used in a model of embryonic wound healing(Taber, 2009). In embryonic wound healing contractionis generated by a ‘purse-string’ (Martin and Lewis,1992; Davidson et al, 2002), modelled by circumferentialresorption at the wound edge in both Taber (2009) andWu and Ben Amar (2015). By combining experimentaland computational approaches Wyczalkowski et al (2013)investigate in more detail possible mechanisms of the‘purse-string’. It has been shown that the contractionmechanism in dermal wound closure does not proceedvia a ‘purse-string’ (Gross et al, 1995) and was thereforenot considered in the present study. While the growthprofiles in Fig. 10 are qualitatively similar to those foundin Taber (2009), there are fundamental differences. Taber’sgrowth laws are formulated with zero net change involume, and feedback such that compression relative tothe target stress stimulates growth. In comparison, ourmodel, has the feedback in the opposite direction (withtension stimulating growth), and a net volumetric growththat serves to counteract the tension due to contraction.The key to such opposing forms is a difference in activityat the wound edge and how this translates to a boundarycondition for the mechanics. In the Taber model, the woundedge is stress free, hence below the resting stress, hence


a negative growth law induces the needed radial growth.Here, the contractile force puts the wound edge above theresting tension, hence a positive growth law is needed.Nevertheless, that models with contrasting feedback cangenerate similar healing profiles suggests a clear needfor further investigation both on the tissue and cellularscales. Different modelling approaches will generally havediffering mechanical signatures, and experiments such asthe rewounding discussed in the present work could providevaluable insight in discerning between them.

In order to focus on the mechanical environment inthe tissue, we have restricted to mechanically stimulatedgrowth, such that stress greater (less) than the residualstress results in addition (removal) of material. With thegrowth laws formulated in this manner, growth will notoccur unless contraction is active. It is quite likely that basalor chemically stimulated growth occurs in the absence ofcontraction and we intend to include this in future modellingefforts. Our sensitivity analysis in Section 3.4 showedthat, to achieve a net gain in material, mechanosensitivegrowth should have a radial bias. This could be possiblewith a growth factor gradient in the radial direction. Anatural next step is to couple a mathematical descriptionof fibroblast and growth factor activity to the contractionfunction and the evolution of growth laws providing a moremechanistic approach to the main processes driving dermalwound closure and the inclusion of chemically stimulatedgrowth. Validation of our model against experimentaldata is required, for example the rewounding experimentsdiscussed in this paper are currently underway.

Acknowledgements LGB gratefully acknowledges the UK’s Engi-neering and Physical Sciences Research Council for funding througha studentship at the Systems Biology programme of the University ofOxford’s Doctoral Training Centre. The authors thank Dr. PY Liu andDr. XT Wang (Department of Plastic Surgery, Alpert Medical Schoolof Brown University) for valuable discussions.

Conflict of interest The authors declare that they have no conflict ofinterest.

References

Alford P, Humphrey J, Taber L (2008) Growth andremodeling in a thick-walled artery model: effects ofspatial variations in wall constituents. Biomech ModelMechanobiol 7(4):245–62

Ambrosi D, Mollica F (2004) The role of stress in the growthof a multicell spheroid. J Math Biol 48(5):477–99

Annaidh A, Bruyere K, Destrade M, Gilchrist M, Maurini C,Ottenio M, Saccomandi G (2012) Automated estimationof collagen fibre dispersion in the dermis and itscontribution to the anisotropic behaviour of skin. AnnBiomed Eng 40(8):1666–78

Anzarut A, Olson J, Singh P, Rowe B, Tredget E (2009)The effectiveness of pressure garment therapy for theprevention of abnormal scarring after burn injury: a meta-analysis. J Plast Reconstr Aes 62(1):77–84

Bermudez D, Herdrich B, Xu J, Lind R, Beason D,Mitchell M, Soslowsky L, Liechty K (2011) Impairedbiomechanical properties of diabetic skin implicationsin pathogenesis of diabetic wound complications. Am JPathol 178(5):2215–23

Bowden L, Maini P, Moulton D, Tang J, Wang X, Liu P,Byrne H (2014) An ordinary differential equation modelfor full thickness wounds and the effects of diabetes. JTheor Biol 361:87–100

Chipev C, Simon M (2002) Phenotypic differences betweendermal fibroblasts from different body sites determinetheir responses to tension and tgfβ1. BMC Dermatol2(1):13

Chung CY, Funamoto S, Firtel R (2001) Signaling pathwayscontrolling cell polarity and chemotaxis. Trends BiochemSci 26(9):557–566

Ciarletta P, Ben Amar M (2012) Papillary networks inthe dermal–epidermal junction of skin: a biomechanicalmodel. Mech Res Commun 42:68–76

Ciarletta P, Foret L, Ben Amar M (2011) The radial growthphase of malignant melanoma: multi-phase modelling,numerical simulations and linear stability analysis. J RSoc Interface 8(56):345–368

Clark R (1988) The Molecular and Cellular Biology ofWound Repair. Plenum, New York

Cumming B, McElwain D, Upton Z (2010) A mathematicalmodel of wound healing and subsequent scarring. J R SocInterface 7(42):19–34

Dallon J (2000) Biological implications of a discrete math-ematical model for collagen deposition and alignment indermal wound repair. Math Med Biol 17(4):379–393

Dallon J, Sherratt J, Maini P (1999) Mathematical modellingof extracellular matrix dynamics using discrete cells:fiber orientation and tissue regeneration. J Theor Biol199(4):449–71

Davidson L, Ezin A, Keller R (2002) Embryonic woundhealing by apical contraction and ingression in xenopuslaevis. Cell Motil Cytoskel 53(3):163–176

Diridollou S, Patat F, Gens F, Vaillant L, Black D, Lagarde J,Gall Y, Berson M (2000) In vivo model of the mechanicalproperties of the human skin under suction. Skin ResTechnol 6(4):214–221

Ehrlich H (1988) Wound closure: evidence of cooperationbetween fibroblasts and collagen matrix. Eye 2(2):149–157

Ehrlich H, Rajaratnam J (1990) Cell locomotion forces ver-sus cell contraction forces for collagen lattice contrac-tion: an in vitro model of wound contraction. Tissue Cell22(4):407–417


Eringen A (1962) Nonlinear Theory of Continuous Media.McGraw-Hill, New York

Flynn C, Taberner A, Nielsen P (2011) Modeling themechanical response of in vivo human skin under a richset of deformations. Ann Biomed Eng 39(7):1935–46

Ghosh K, Pan Z, Guan E, Ge S, Liu Y, Nakamura T, RenX, Rafailovich M, Clark RAF (2007) Cell adaptationto a physiologically relevant ECM mimic with differentviscoelastic properties. Biomaterials 28(4):671–679

Giele H, Liddiard K, Currie K, Wood F (1997) Direct mea-surement of cutaneous pressures generated by pressuregarments. Burns 23(2):137–141

Goriely A, Vandiver R (2010) On the mechanical stability ofgrowing arteries. IMA J Appl Math 75(4):549–570

Goriely A, Moulton D, Vandiver R (2010) Elastic cavitation,tube hollowing, and differential growth in plants andbiological tissues. Europhys Lett 91(1):18,001

Grinnell F, Zhu M, Carlson MA, Abrams JM (1999)Release of mechanical tension triggers apoptosis ofhuman fibroblasts in a model of regressing granulationtissue. Exp Cell Res 248(2):608–619

Gross J, Farinelli W, Sadow P, Anderson R, Bruns R(1995) On the mechanism of skin wound “contraction”:a granulation tissue “knockout” with a normal phenotype.P Natl A Sci USA 92(13):5982–5986

Hex N, Bartlett C, Wright D, Taylor M, Varley D (2012)Estimating the current and future costs of type 1 andtype 2 diabetes in the UK, including direct health costsand indirect societal and productivity costs. Diabetic Med29(7):855–62

Huang C, Leavitt T, Bayer L, Orgill D (2014) Effect ofnegative pressure wound therapy on wound healing. CurrProb Surg 51(7):301–31

Jacquet E, Josse G, Khatyr F, Garcin C (2008) Anew experimental method for measuring skin’s naturaltension. Skin Res Technol 14(1):1–7

Jeffcoate W, Price P, Harding K (2004) Wound healing andtreatments for people with diabetic foot ulcers. Diabetes-metab Res 20(Suppl 1):S78–89

Kessler D, Dethlefsen S, Haase I, Plomann M, Hirche F,Krieg T, Eckes B (2001) Fibroblasts in mechanicallystressed collagen lattices assume a synthetic phenotype.J Biol Chem 276(39):36,575–36,585

Kischer C (1975) Alteration of Hypertrophic Scars Inducedby Mechanical Pressure. Arch Dermatol 111(1):60

Lin I, Taber L (1995) A model for stress-induced growth inthe developing heart. J Biomech Eng 117(3):343

Maggelakis S (2004) Modelling the role of angiogenesis inepidermal wound healing. Discrete Contin Syst (4):267–273

Martin P (1997) Wound healing: aiming for perfect skinregeneration. Science 276(5309):75–81

Martin P, Lewis J (1992) Actin cables and epider-mal movement in embryonic wound healing. Nature360(6400):179–83

McDougall S, Dallon J, Sherratt J, Maini P (2006) Fi-broblast migration and collagen deposition during dermalwound healing: mathematical modelling and clinical im-plications. Philos T Roy Soc A 364(1843):1385–405

McGrath M, Simon R (1983) Wound geometry and thekinetics of wound contraction. Plast Reconstr Surg72(1):66–72

Mendez-Eastman S (2001) Guidelines for Using NegativePressure Wound Therapy. Adv Skin Wound Care14(6):314–323

Monaco J, Lawrence W (2003) Acute wound healing anoverview. Clin Plast Surg 30(1):1–12

Moulton D, Goriely A (2011) Possible role of differentialgrowth in airway wall remodeling in asthma. J ApplPhysiol 110(4):1003–12

North J, Gibson F (1978) Volume compressibility of humanabdominal skin. J Biomech 11(4):203–207

Olsen L, Maini P, Sherratt J, Dallon J (1999) Mathematicalmodelling of anisotropy in fibrous connective tissue.Math Biosci 158(2):145–170

Pierce G, Mustoe T, Lingelbach J, Masakowski V, Griffin G,Senior R, Deuel T (1989) Platelet-derived growth factorand transforming growth factor-beta enhance tissue repairactivities by unique mechanisms. J Cell Biol 109(1):429–440

Press W, Vetterling W, Teukolsky S, Flannery B, Green-well Yanik E (1994) Numerical recipes in fortran–the artof scientific computing. SIAM Rev 36(1):149–149

Rachev A, Stergiopulos N, Meister J (1998) A modelfor geometric and mechanical adaptation of arteries tosustained hypertension. J Biomech Eng 120(1):9

Rivlin R (1948a) Large elastic deformations of isotropicmaterials I. Philos T Roy Soc A 240(822):459–490

Rivlin R (1948b) Large elastic deformations of isotropicmaterials III. Philos T Roy Soc A 240(823):509–525

Rodriguez E, Hoger A, McCulloch A (1994) Stress-dependent finite growth in soft elastic tissues. J Biomech27(4):455–467

Segal R, Diegelmann R, Ward K, Reynolds A (2012) Adifferential equation model of collagen accumulation ina healing wound. B Math Biol 74(9):2165–82

Sherratt J, Murray J (1990) Models of epidermal woundhealing. P Biol Sci 241(1300):29–36

Sherratt J, Murray J (1991) Mathematical analysis of abasic model for epidermal wound healing. J Math Biol(29):389–404

Sherratt J, Murray J (1992) Epidermal wound healing: theclinical implications of a simple mathematical model.Cell Transplant 1(5):365–371


Singer A, Clark R (1999) Cutaneous wound healing. N EngJ Med, 341:738–746

Taber L (1998) A model for aortic growth based on fluidshear and fiber stresses. J Biomech Eng 120(3):348

Taber L (2001) Biomechanics of cardiovascular develop-ment. Annu Rev Biomed Eng 3:1–25

Taber L (2009) Towards a unified theory for morphome-chanics. Philos T R Soc A 367(1902):3555–83

Taber L, Eggers D (1996) Theoretical study of stress-modulated growth in the aorta. J Theor Biol 180(4):343–57

Tracqui P, Woodward D, Cruywagen G, Cook J, Murray J(1995) A mechanical model for fibroblast-driven woundhealing. J Biol Sys 3(4):1075–1084

Tranquillo R, Murray J (1992) Continuum model offibroblast-driven wound contraction: Inflammation-mediation. J Theor Biol 158(2):135–172

Uhal B, Ramos C, Joshi I, Bifero A, Pardo A, Selman M(1998) Cell size, cell cycle, and alpha -smooth muscleactin expression by primary human lung fibroblasts. AmJ Physiol Lung Cell Mol Physiol 275(5):998–1005

Vermolen F, Javierre E (2012) A finite-element modelfor healing of cutaneous wounds combining contraction,angiogenesis and closure. J Math Biol 65(5):967–96

Wahl S, Wong H, McCartney-Francis N (1989) Role ofgrowth factors in inflammation and repair. J Cell Bioch40(2):193–9

Wrobel L, Fray T, Molloy J, Adams J, Armitage M,Sparrow J (2002) Contractility of single human dermalmyofibroblasts and fibroblasts. Cell Motil Cytoskel52(2):82–90

Wu M, Ben Amar M (2015) Growth and remodellingfor profound circular wounds in skin. Biomech ModelMechanobiol 14(2):357–370

Wyczalkowski M, Varner V, Taber L (2013) Computationaland experimental study of the mechanics of embryonicwound healing. J Mech Behav Biomed 28:125–146

Yang L, Witten T, Pidaparti R (2013) A biomechanicalmodel of wound contraction and scar formation. J TheorBiol 332(null):228–248

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