Finite Elements in Analysis and Design 45 (2009) 519 -- 529

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Finite Elements in Analysis andDesign

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Computational modeling of reconstructive surgery: The effects of the natural tensionon skinwrinkling

A. Cavicchi∗, L. Gambarotta, R. MassabòDepartment of Civil, Environmental and Architectural Engineering, University of Genova, Via Montallegro, 1, 16145 Genova, Italy

A R T I C L E I N F O A B S T R A C T

Article history:Received 2 April 2008Received in revised form 9 December 2008Accepted 10 February 2009Available online 10 April 2009

Keywords:Reconstructive surgeryComputational modelingSkin wrinklingFinite strainsMembrane hyperelasticityRelaxed energy densityConstitutive modeling

A computational model is presented for the simulation of procedures of reconstructive surgery charac-terized by the excision of a cutaneous defect and the closure and suture of the wound edges. The skin ismodeled as a plane membrane with zero flexural stiffness. The membrane undergoes large deformationsand is characterized by a Fung type constitutive response in biaxial tension. Skin wrinkling, which is atypical outcome of the surgery in the form of extrusion of the wound edges and dog-ears, is consideredthrough a modification of the elastic potential as originally proposed by Pipkin's Relaxed Energy Densitytheory [A.C. Pipkin, The relaxed energy density for isotropic elastic membranes, IMA J. Appl. Math. 36(1986) 85–99; A.C. Pipkin, Relaxed energy density for large deformations of membranes, IMA J. Appl.Math. 52 (1994) 197–308]. The post-buckling analysis of a stretched annular membrane performed byGeminard et al. [Wrinkle formations in axi-symmetrically stretched membranes, Eur. Phys. J. E 15 (2004)117–126] is used to validate the model under conditions similar to those of the surgery and to discussthe influence of a pre-existing tension in the membrane on the extension of the wrinkled regions. Themodel is applied to simulate different surgical procedures and investigate the effects of the natural stateof the skin and the shape and size of the excisions. The results explain and validate current practice.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

The numerical simulation of the mechanical response of skinwithin procedures of reconstructive surgery finds potential applica-tion in virtual reality systems for surgeon training and computer as-sisted surgery. These systems can also be used to optimize surgicalprocedures and minimize negative outcomes, which include perma-nent marks and high stresses that reduce blood perfusion and facil-itate the occurrence of hypertrophic scars, necrosis or dehiscence.This is a critical aspect of the actual practice, which is based solelyon preliminary geometrical and anatomical considerations and post-surgical corrections.

Dog-ears or extrusion of wound edges are common sequelaeof procedures of reconstructive surgery where portions of skin areexcised to remove cutaneous defects, such as cancerous and non-cancerous growths, burn wounds, lacerations or birth defects. Theycan be described as wrinkling phenomena and therefore stronglydepend on the mechanical behavior of the skin during surgery. Sim-ilarly, the mechanical behavior of the skin determines the level ofthe stresses in wound closing.

∗ Corresponding author. Tel.: +390103532516.E-mail address: cavicchi@dicat.unige.it (A. Cavicchi).

0168-874X/$ - see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.finel.2009.02.006

Early applications of linear elasticity to simulate reconstruc-tive surgery and calculate the stresses in the skin in wound clos-ing date back to the analytical works of Danielson [5], Danielsonand Natarajan [6] and Furnas and Fischer [10] in the 70s. Finiteelement applications first appeared in the 80s [7,16]. More re-cently, advances in biomechanics of the skin and computationalmethods have given a strong push forward in the mechanicalsimulation of procedures of reconstructive surgery [2,17] thatalso involved the medical community [3,38,39,23,27,28,24,14,15,18]).

Existing models for reconstructive surgery typically make use ofsimplified constitutive equations, often calibrated only on in vitroskin, and neglect the natural tension of skin whose importance hasbeen recently highlighted by Cerda [1]. In this paper, a numericalmodel to simulate the different steps of a common reconstructivesurgery is presented. The model takes account of the large strainsand displacements imposed by the surgery and of the natural stressof the skin. It uses an isotropic Fung type constitutive equation forbiaxial tension [37,9] that reproduces the highly non-linear responseof the skin and can be calibrated in vivo following the procedure setup in [11]. Wrinkling is simulated using the finite strain model re-cently formulated in [21], where wrinkling is considered through amodification of the elastic potential as originally proposed by Pipkin'sRelaxed Energy Density theory [25,26,34,35]. Alternative approaches

520 A. Cavicchi et al. / Finite Elements in Analysis and Design 45 (2009) 519 -- 529

that are not based on energy functions can be found in the literature(e.g. [22,8,30,33]).

Wrinkling is a post-critical phenomenon that skin shares withall natural and man-made soft thin structures and is a consequenceof their limited bending stiffness. The plane model used in thenumerical procedure here presented simplifies the analysis bydescribing the skin as a membrane, so that flexural stiffness isneglected, and assuming that the contractions in the plane of themembrane take place through a diffused infinitesimal folding in theabsence of any compressive forces. As a consequence, the modelis able to locate the regions of the domain where wrinkling takesplace but cannot distinguish between fine wrinkles and deep folds[33,4]. The effects of this approximation on the description of theonset and extension of wrinkling are discussed using the refer-ence problem of a stretched annular thin plate that was solved byGeminard et al. [12] by means of a post-buckling analysis. The nu-merical model is then applied to simulate simple procedures of re-constructive surgery and investigate the effects of the natural tensionof the skin and the size and shape of the excision on the wrinklingonset and extension and on the level of the stresses along the woundmargins.

2. Computational modeling of skin wrinkling

The work refers to procedures of reconstructive surgery that con-sist of incision and undermining of the skin, excision of a cutaneousdefect and suture of the wound edges. In such procedures relevantcontractions are typically imposed on the skin; these contractionslead to geometric instabilities, due to the limited flexural rigidity,that manifest themselves as dog-ears. In the simulations, the skin ismodeled as a plane homogeneous, isotropic and hyperelastic mem-brane subject to in-plane forces and displacements applied along theboundaries. The geometric instabilities, which are associated withnegative stress states in linear elastic models [2,17], are consideredusing the plane wrinklingmodel formulated in [21]. Themodel refersto the short-term response of the skin and does not take account oftissue growth and remodeling that may occur over time and reducethe effects of the contractions (see for example [31]).

The assumption of a plane homogeneous and isotropic mem-brane requires a clarification since surgery is typically performedon curved surfaces and the skin is heterogeneous and anisotropic.The assumption of a plane membrane allows highlighting relevantmechanisms related to the wrinkling processes while avoiding ad-ditional complications. Neglecting the layered structure of the skinand introducing effective homogeneous properties is commonly ac-cepted provided the scale of observation is large compared to thematerial microstructure as in the problems considered in this pa-per. Isotropy was assumed in [11] to simplify the calibration of theconstitutive model parameters from in vivo experimental tests andin [21] to derive the model of the wrinkling membrane in a closedform. The assumption is partially justified when dealing with skin atcertain anatomical locations, e.g. the scalp. As observed by Chaudhryet al. [2], the actual response of anisotropic skin will differ mainlyquantitatively and relevant conclusions on the wrinkling process areexpected to hold true.

2.1. Hyperelastic constitutive model for skin wrinkling

Skin is modeled as an elastic membrane whose reference config-uration is described by a region S0 with boundary �S0 in the plane(e1, e2); in this configuration the skin is fully detached from the sub-coutaneous attachments and unstressed. In the current configura-tion the material points occupy the surface S with boundary �S. Thespatial position vector x defines the position of the material pointp in the current configuration and is related to the displacement

vector u by u= x−p. Prescribed tractions t per unit initial length ofboundary act as dead loads in the (e1, e2) plane.

The membrane is assumed to be hyperelastic and the constitutiveequation for states of biaxial tension is defined by the strain energydensity per unit reference area W(E), where E = 1

2 (FTF − I) is the

Green–Lagrange strain tensor and F = RU is the deformation gradi-ent, R being the rotation tensor and U the stretch tensor. In accor-dance with this assumption, the second Piola–Kirchhoff membraneforce tensor is R = �W/�E and is related to the Cauchy membraneforce tensor T through R = JF−1TF−T , where J = det F. Among thecurrent configurations, the natural configuration Sn with boundary�Sn, which describes the skin in its natural or physiological state, hasspecial relevance. Under the previous hypotheses this configurationis homothetic to S0 and the strain and stress fields are homogeneousand isotropic, En = 1/2(�2

n − 1)I and Rn = �nI, with �n the naturalstretch and �n the corresponding natural tension; in this configura-tion Rn = Tn = �nI.

The equilibrium of the membrane is posed as an energy-minimization problem. Minimum energy states in membranes caninvolve continuous distributions of infinitesimal wrinkles (oscilla-tions with very high frequency characterized by magnitude andwavelength tending to zero and leading to a discontinuous defor-mation gradient) that cannot be described by ordinary membranetheory [20,19]. In Massabò and Gambarotta [21], the problem ofincluding wrinkling in the constitutive model for a isotropic ma-terial characterized by a Fung type behavior was solved followingthe approach of Pipkin [25,26]. The method is briefly recalled in thefollowing.

The formulation is in terms of the principal values �1 and �2 ofthe Green–Lagrange tensor E. It is assumed that wrinkling occurs inthe directions of the principal strains, which coincide with the direc-tions of the principal values of the second Piola–Kirchhoff membraneforce tensor R (PKII) since the material is isotropic. For vanishing ro-tations R=I, R and T are coaxial and the principal directions describethe actual distribution of the wrinkles in the current deformed con-figurations. The wrinkling criterion refers to the natural contraction,which is defined as the transverse contraction of a membrane sub-ject to simple extension. For a given �i >0, the natural contraction isthe strain �j at which the minimum of the elastic potential W(�1, �2)is attained, �j = w(�i) for i, j = 1, 2.

If both principal strains are larger than the corresponding natu-ral contractions, �j� w(�i) for i, j = 1, 2, the membrane is in a stateof biaxial tension (taut) and the analysis can be performed usingthe elastic potential W(E). If either �1>0 and −1/2��2<w(�1) or�2>0 and −1/2��1<w(�2) holds, then classical membrane theoryand the elastic potential W(E) would predict compressive membraneforces. This state is not admissible and it is assumed that the mem-brane stays in simple tension and the additional decrease in trans-verse width with respect to the natural contraction is accomplishedby wrinkling. The tension field is included in the theory introduc-ing a Relaxed Energy function Wr(�1, �2) that represents the averageenergy function per unit area in the wrinkled region and substitutefor W(E). Since the material is isotropic and surface forces are ig-nored, the principal directions of PKII and Cauchy membrane forcesare straight lines (tension rays). If both principal strains are negative,then the principal membrane forces would be negative according tothe classical membrane theory. Instead, the membrane is assumedto be in a state of biaxial wrinkling (slack) and inactive, �1 = �2 = 0.As for the previous case this can be included in the model througha Relaxed Energy function Wr(�1, �2) = 0.

In order to reproduce the in vivo behavior of human skin, theclassical phenomenological model proposed by Fung [9] for statesof biaxial tension has been used. The model reproduces the mainfeatures of the mechanical response of soft biological tissues subjectto short-term biaxial tension. In particular it reproduces the highly

A. Cavicchi et al. / Finite Elements in Analysis and Design 45 (2009) 519 -- 529 521

non-linear strain stiffening response of two-phase tissues, like theskin, that at low strains ismainly controlled by the elastin and at largestrain by the collagen fibers after they have become uncrimpled.

Under the assumption of isotropy, the elastic strain energy perunit surface area is

W(E) = c exp{B[�211 + �222 + 2��11�22 + 2(1 − �)(�12�21)]}, (1)

where c, B and � are material parameters to be determined exper-imentally. The parameter c has the dimensions of a force per unitlength and � and B are dimensionless. These parameters must satisfythe constitutive inequalities

c>0, B>0, −1<�<1 (2)

in order for the strain energy function to be convex and the responseof the skin to be strain stiffening. Inequalities (2) were derived in [11]imposing the convexity of W at incipient deformations (F = I), bothin the plane of the Green strains and in the plane of the principalstretches �i, for all states characterized by positive principal strains.The natural contractions of the membrane follow as w(�1) = −��1and w(�2) = −��2. At incipient deformation, � equals the general-ized Poisson ratio (��11/��22)(��11/��11)

−1 and coincides with thePoisson ratio of a linear elastic model.

The Relaxed Energy Density that describes states of uniaxial wrin-kling can be defined in the plane of the principal Green–Lagrangestrains by noting that if a principal contraction �j is increased abovethe natural contraction, the consequent folding process occurs withno expense of energy and the elastic potential W must retain theminimum value W(�i, �j = w(�i)), for i, j = 1, 2. The Relaxed EnergyDensity is then obtained substituting �j = w(�i)= −��i into the elas-tic potential of Eq. (1). When the criterion for biaxial wrinkling issatisfied, the membrane becomes slack and inactive and its energyfunction is set equal to zero. In case of anisotropy, which is notconsidered here, strain-based wrinkling criteria cannot be used andmore complex stress–strain mixed criteria must be considered (seefor instance [29,13,32]). The constitutive model is summarized in thetable below.

Criteria for wrinkling and Relaxed Energy Density

Taut if

{�1� w(�2) = −��2�2� w(�1) = −��1

⇒ Wr(�1, �2) = W(E) = c exp{B[�21 + �22 + 2��1�2]}

Wrinkled

if

{�1�0

−1/2��2� w(�1) = −��1⇒ Wr(�1) = c exp{B�21(1 − �2)}

or

if

{�2�0

−1/2��1� w(�2) = −��2⇒ Wr(�2) = c exp{B�22(1 − �2)}

(3)

Slack if

{�1�0

�2�0⇒ Wr(�1, �2) = 0

2.2. Finite element analysis of partially wrinkled membranes

The equilibrium configurations of partially wrinkled mem-branes were analyzed in the case of prescribed mixed in-planedisplacement–traction boundary conditions: in-plane displace-ments u(p) = u(p) were imposed on a part �S0u of the boundaryof S0; in-plane tractions S(p)n0(p) = t(p), with S and n0 the firstPiola–Kirchhoff membrane force tensor (S= FR) and the unit vectornormal to �S0t at a point p, were applied on the remainder �S0t , andassumed to be independent of displacements.

The highly non-linear equilibrium problem of a Fung hyperelas-tic membrane subject to wrinkling and large strains does not allow

analytical solution of cases of practical importance, so that a FiniteElement procedure was used. The potential energy functional basedon the Relaxed Energy Density (3) is defined in the Lagrangian for-mulation:

�(u) =∫S0Wr(E(u))da −

∫�S0t

t · uds, (4)

where da and ds are surface and linear elements, respectively, and(u,E) define kinematically admissible states. Assuming a finite el-ement representation of the displacement field u = Na, where acollects the nodal displacements andN the shape functions, the equi-librium of the membrane is obtained by applying the principle ofstationary potential energy that provides the non-linear equationW(a) = qi(a) − qe = 0, qi(a) and qe being the internal and externalforce vectors, respectively. The solution of the non-linear equationis obtained through a Newton-type iterative scheme K(ak)�ak+1 =qe − qi(ak), where K is the stiffness matrix. The internal force vec-tor qi(ak) is calculated by using the components of the membraneforce tensor R evaluated at each Gauss point. The stiffness matrixK(ak) is evaluated at each iteration as the sum of two standard con-tributions, the material and the geometric stiffness matrices. Thematerial stiffness matrix is calculated only at the first iteration andthen kept constant until convergence to the solution is obtained. Itcorresponds to the unstretched membrane in the absence of wrin-kling and its non-singular contribution prevents degenerate K due tonodes surrounded by inactive elements (wrinkled/slack states). Thematerial stiffness matrix is then multiplied by a coefficient largerthan unity in order to take into account, in an approximate way, thestiffening behavior of the constitutive model and improve the con-vergence rate. The geometric stiffness matrix is instead updated ateach iteration using the components of the membrane force tensorR provided by Eq. (3), which takes into account the wrinkling of themembrane using the strain tensor E compatible with the elementdisplacement vector a evaluated at the previous iteration. Once themembrane force tensor has been calculated, the geometric stiffnessmatrix is obtained in a standard way by the FE program used to im-plement the numerical procedure.

The proposed model was implemented in the finite element pro-gram FEAP [36]. Quasi-static simulations were performed using thestandard bilinear quadrilateral isoparametric membrane elementthat is available in FEAP [36]. The accuracy of the finite elementsolution and the non-linear solution algorithm, as well as the capa-bility of the proposed model to predict states of incipient, partialand full wrinkling, were checked referring to some simple problems(i.e., membranes in simple shear, uniaxial and biaxial contraction)and to the problem of a stretched annular membrane that will bepresented in Section 3.1.

The model showed a rather fast and stable convergence to thesolution and the solution procedure appeared not to be sensitive to

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incremental load histories when applied to the problems examinedin this work. It is important to note on this regard that in the prob-lems analyzed the actions applied are in most part imposed displace-ments; the only static boundary conditions are zero forces on someparts of the boundary (for instance to simulate the line of incisionin the surgical procedure), so that intermediate situations of appliedloads that cannot be equilibrated are not found.

3. Validation and applications

The influence of the natural tension of the skin on the onset andgrowth of wrinkling is described in two examples that follow. Thefirst example (Section 3.1) refers to an annular membrane and isproposed in order to highlight the fundamental aspects of the prob-lem, check the reliability and the range of applicability of the planewrinkling model with respect to post-buckling analyses and validatethe numerical procedure. In the second example (Section 3.2), a sim-ple procedure of reconstructive surgery is simulated and the effectsof the natural tension and the size and shape of the excision on theoutcome of the surgery are discussed.

3.1. Stretched annular membrane (contracting scar)

An annular membrane subjected to uniform radial displacementsalong its inner and outer boundaries is considered (inset of Fig. 1).The application of the radial displacement u0 to the external bound-ary generates a positive strain field that represents the natural stateof the skin; the internal radial displacement ui could represent theeffect of a contracting scar. This problem has analytical solution un-der the assumptions of isotropic linear elasticity, small strains andnon-compressive stress fields (Appendix A).

For a fixed external displacement u0 and when the internal dis-placement ui is above a critical value, the circumferential strains ��in an inner region of the membrane overcome the natural contrac-tion and wrinkling occurs. The natural contraction is w(��) = −��r ,where �r is the radial strain and � the Poisson ratio. The solutionof the problem is summarized by the relation between the imposedboundary displacements u0 and ui and the wrinkling extension, interms of its external boundary position rw, expressed in the form

ui = − 2w[� + Log(w)]

(1 − �) − 22w(1 + �)

u0, (5)

where =a/b, a and b are the inner and outer radii of the membrane,u0 =u0/b, ui =ui/a and w = rw/a (ui and rw have been normalized tothe inner radius, a, to have dimensionless parameters that directlyrelate to the initial dimension of the defect that contracts and leadsto wrinkling). The displacement u0 imposed at the external boundaryhas been normalized instead to the external radius, b, so that thenatural strain field is fully defined by u0 for a given geometry ratio and is independent of the actual dimensions of the membrane (seeAppendix A).

Wrinkling occurs at the critical displacement ratio ui/u0 =2�[(1−�) − 2(1 + �)]−1, which is obtained by substituting w = 1 in (5),extends throughout the membrane on increasing the ratio ui/u0and approaches the asymptotic value c

w = −1√(1 − �)/(1 + �)for ui/u0 → ∞. Wrinkling does not take place if the inequality>

√(1 − �)/(1 + �) holds. The dependence of the solution on the

Poisson ratio � is described in Fig. 1a, where the normalized wrin-kling extension w is shown as a function of the displacement ratioui/u0 for a fixed geometry ratio = 1

20 . The diagram highlights thescreening effect the Poisson ratio � has on the progress of wrinklingand on its asymptotic extension c

w. The effects of the geometry ra-tio =a/b are similar, as shown in Fig. 1b. The asymptotic wrinklingextension c

w increases on decreasing and tends to infinity as

→ 0; as a consequence, the smaller is , the larger is the wrinklingextension w for a given value of the ratio ui/u0. The equation

uiu0

= −2w(1 − �)[� + Log(w)] (6)

describes the response of a membrane whose inner radius, a, is neg-ligible with respect to the outer radius b ( → 0).

The analytical results were reproduced using the computationalmodel presented in the previous section. Because of the axisymmetryof the problem, the analysis was performed referring to an angularsector of the membrane with suitable boundary conditions. The do-main was discretized with the four-node elements described aboveand only one element was used in the circumferential direction.Quick convergence in terms of both membrane forces and wrinklingextension was obtained on refining the mesh in the radial direction,and accurate solutions were obtained using only a few elements. Inthe small strain limit and for � = �, the numerical results correctlyconverged to the analytical solution with negligible numerical errors.

Geminard et al. [12] solved the problem of an annular thin plateby performing a post-buckling analysis. They considered a geometrydefined by a/h=50 and b/h=1000 (so that = 1

20 ), with h the thicknessof the plate and assumed a Poisson ratio � = 0.5. The diagram inFig. 2 shows the normalized wrinkling extension as a function of theimposed normalized inner displacement ui/h, for different values ofthe normalized external displacement u0/h. The dashed lines in thediagram are the results of the post-buckling analysis; the solid linesare the analytical results obtained using the plane model, Eq. (5).

The analytical and numerical curves show a similar dependenceof the normalized wrinkling extension (rw − a)/h on the normalizedexternal displacement u0/h, which includes the asymptotic behavior.In addition, the wrinkling extensions predicted by the two modelsare in agreement for large values of the ratio u0/h (u0/h=12.5, 37.5 inthe diagram), which include the case of prestressed plates with verysmall thickness. In these cases, the compressive stresses in the post-critical configurations are much smaller than the tensile stresses,thereby validating the assumption of non-compressive stress fieldsof the plane model. In the procedure of reconstructive surgery thatwill be examined in the next section the closure of the wound afterthe excision imposes large displacements and stretches that put themembrane under similar conditions.

The differences between the solutions become larger for verysmall values of the ratio u0/h (u0/h = 0.05, 0.01, 0 in the diagram).Small values of the ratio u0/h correspond to either a plate of finitethickness or a small external displacement u0. In both cases, com-pressive stresses are present in a large portion of the plate beforebuckling takes place and they are no longer negligible in the globalequilibrium. As a consequence, the assumption of non-compressivestress fields leads to an evident overestimation of the wrinkling ex-tension but for large values of ui when wrinkling approaches theasymptotic extension.

3.2. Simulation of reconstructive surgery of the skin

The excision of a portion of skin and the closure and suture ofthe wound within a surgical operation generate a final configura-tion that differs from the original natural state and where the skinmay be subjected to higher tractions and folding (Fig. 3). In this sec-tion, simulations of surgical procedures characterized by three fun-damental steps are presented: the incision and undermining of aportion of skin (Fig. 3c); the excision of a portion of skin (Fig. 3d);the closure and suture of the wound (Fig. 3e). The natural stretch�n and the shape and size of the excision deeply influence the re-sponse at all stages. Different natural configurations and excisionswere examined using the FE model presented in Section 2 in order to

A. Cavicchi et al. / Finite Elements in Analysis and Design 45 (2009) 519 -- 529 523

Fig. 1. Diagrams of the size of the wrinkled region as a function of the normalized applied displacements in an annular membrane. (a) Influence of the Poisson ratio; (b)influence of the geometry.

Fig. 2. Diagram of the size of the wrinkled region as a function of the normalized internal displacement in an annular membrane. Comparison of the results of the planemodel, Eq. (5), and the post-buckling analysis of [12].

investigate their effects on the stress and strain states and the ex-tension of the wrinkled regions at different stages during surgery.

The geometry of the skin flap used in the simulations matches theprocedure of cosmetic surgery (hair replacement) on scalp flaps thatwas used in Gambarotta et al. [11] to calibrate the parameters of theconstitutive model (Fig. 3). The incision has length 180mm and thedomain is squared and symmetric about it. Three different shapes ofthe excision are considered: a spindle, with axes rmax =30mm alongthe incision and rmin = 20mm, and two circles with radii rmax andrmin (Fig. 4). All measures are in the natural configuration.

The parameters of the constitutivemodels used in the simulationsare those inferred in vivo by Gambarotta et al. [11]: c= 1.13kPa cm,B = 0.89 and � = 0.6. The effect of the natural stretch on the resultswas analyzed by performing a parametric analysis where the natu-ral stretch was varied around the experimental value �n = 1.1 [11].

The range considered is from �n=1.0 (no natural stretch) to �n=1.3,which is a reference value for which the skin, with the assumedconstitutive parameters, enters the stiffening regime due to the re-orientation of the collagen fibers in the direction of the load (seeFig. 7b, inset); this regime does not represent a physiological stateand is not considered.

The natural configuration of the skin (Fig. 3b), which is approxi-mated by an equibiaxial stretch �n, was recreated in the numericalmodel by imposing the displacements u = (�n − 1)p to the externaledges �S0 of the reference, fully detached, domain (Fig. 3a). The linesof incision–excision, �S02, �S03 and �S04, are subject to differentboundary conditions during the different steps of the operation. Thephase of the incision (Fig. 3c) was obtained by assuming a free edgealong �S02 and �S03; the phase of the excision (Fig. 3d) was obtainedby assuming a free edge along �S02 and �S04; the final configuration

524 A. Cavicchi et al. / Finite Elements in Analysis and Design 45 (2009) 519 -- 529

a

Fig. 3. (a) Skin flap in its reference configuration (fully detached); (b) natural configuration of the skin flap; (c) incision and undermining; (d) excision of a portion of theskin; (e) closure and suture of the wound.

Fig. 4. Finite element meshes of the reference configurations used to simulate (a) spindle, (b) large circular and (c) small circular excisions.

(Fig. 3e) was obtained by imposing suitable vertical displacementsand allowing a free adaptation along x1 of the boundaries �S02 and�S04.

Because of the symmetry of the problem, the finite element anal-yses refer to a quarter of the domain shown in Fig. 3. The meshesof Fig. 4 were used to simulate the natural state of the skin and theincision (Fig. 3b, c). The excision and closure (Fig. 3.d,e) were simu-lated using modified meshes where the elements of the subdomaincorresponding to the portion of the skin to be excised (thick solidlines in Fig. 4) were removed. The mesh refinement was chosen af-ter a convergence analysis where convergence on membrane forcesand wrinkling extension was checked. The convergence analysis alsoexcluded the presence of membrane force singularities in the finalconfiguration. The meshes were generated automatically and com-parisons between the results from different mesh geometries ob-tained by slightly changing the nodal positions showed negligibledifferences.

In Fig. 5 the final configurations corresponding to different shapesof the excision and �n = 1.1 are presented. Figs. 5a, c, e show thewrinkling extension and Fig. 5b, d, f themaximumCauchymembrane

forces. The large circle gives rise to the largest wrinkling extensionwhile the spindle leads to the smallest (Fig. 5c, e). The differences aremore evident in terms of membrane forces: the largest excision leadsto much higher tractions in the skin (Fig. 5d), with a maximum valueat the wound margin (tI = 7.98kPa cm) that is about two times thevalues obtained with the other excisions. The traction fields relatedto the spindle and small circular excisions are similar; the spindlegives rise to slightly higher tractions at the center of the wound(tI = 4.62kPa cm vs tI = 3.43kPa cm), but differences are limited to avery small region.

Fig. 6 shows in details the extension of wrinkling (Fig. 6a, c, e) andthe maximum Cauchy membrane forces (Fig. 6b, d, f) in the inter-mediate configurations of the surgical operation (after the incision,Fig. 6a, b, after the excision, Fig. 6c, d, and after the closure, Fig. 6e, f)with a spindle shape excision and �n = 1.1. The scale of values isthe same for the three configurations (apart from the maximum val-ues) in order to show the evolution of the membrane forces duringthe operation. The regions in orange and red correspond to valueshigher than the natural traction �n and highlight the progressive in-crease of the membrane forces in the domain during the operation.

A. Cavicchi et al. / Finite Elements in Analysis and Design 45 (2009) 519 -- 529 525

Fig. 5. (a, c, e) Wrinkling extension (in red uniaxial wrinkling, in blue biaxial wrinkling) and (b, d, f) maximum Cauchy membrane force fields after the suture (the regionin white represents zero membrane forces). Natural stretch: �n = 1.1. Units: kPa cm. (For interpretation of the references to color in this figure legend, the reader is referredto the web version of this article.)

The incision and the excision relax the natural membrane forces al-most everywhere, with the exception of small areas at the end ofthe incision and at the top of the excision, where stress concentra-tion takes place. The excision generates a large region of wrinklingwhich is reduced by the following closure. The diagram in Fig. 6fclearly highlights the changes of the final configuration with respectto the natural: the membrane forces are substantially larger than

�n = 0.348kPa cm everywhere in the domain with the exception ofa region around the wrinkled area.

The effects of the natural stretch on the outcome of the surgicalprocedure are summarized in the diagrams in Fig. 7, where areas ofslack skin (Fig. 7a) and maximum Cauchy membrane forces (Fig. 7b)in the final configuration (spindle case), normalized to the valuescorresponding to a fully relaxed natural configuration (�n =1.0), are

526 A. Cavicchi et al. / Finite Elements in Analysis and Design 45 (2009) 519 -- 529

Fig. 6. (a, c, e) Wrinkling extension and (b, d, f) maximum Cauchy membrane force fields corresponding to the different steps of the surgery. Natural stretch: �n =1.1. Units:kPa cm.

shown as functions of the natural stretch �n. The results correspond-ing to �n=1.0 are shown in Fig. 8 in details. The extension of the slackskin appears to be very sensitive to �n in the range 1<�n<1.05;on increasing �n there is a more gradual reduction of the slack areauntil it vanishes for �n�1.2. It is important to note that the resultsshown in this section depend on the constitutive parameters used,on the geometry of the domain and on the loading conditions.

4. Conclusions

A computational model has been presented to simulate proce-dures of reconstructive and cosmetic surgery characterized by theincision and undermining of the skin, the excision of a cutaneousdefect and the closure and suture of the wound edges. The modelhas been applied to investigate the effects of the natural tension of

A. Cavicchi et al. / Finite Elements in Analysis and Design 45 (2009) 519 -- 529 527

Fig. 7. (a) Normalized area of slack membrane and (b) normalized maximum tensile membrane forces as functions of the natural stretch �n in the spindle case. In the inset:natural stress as function of the natural stretch.

Fig. 8. (a) Wrinkling extension and (b) maximum Cauchy membrane force field after the suture of a spindle excision in an unstretched natural configuration. Natural stretch:�n = 1.0. Units: kPa cm.

the skin and the size and shape of the excision on the outcome ofthe surgery.

The model is formulated in the framework of finite strain hyper-elasticity and refers to plane and isotropic membranes characterizedby a Fung type constitutive response in biaxial tension. It assumesthat wrinkling occurs as a continuous distribution of infinitesimalwrinkles described by a Relaxed Energy Density function. The con-stitutive parameters can be identified in vivo by applying the nu-merical/experimental procedure formulated and applied to humanscalp skin in [11].

The plane wrinkling model and the numerical procedure havebeen validated referring to the problem of a stretched annular mem-brane. The membrane is pre-stressed by an imposed external dis-placement and subject to an inner displacement that could representthe effect of a contracting scar. The validation has been performedreferring to the analytical solution of the plane model obtained un-der the assumption of small strains and to the results of the numer-ical post-buckling analysis of Geminard et al. [12]. The comparisonsdemonstrate the accuracy of the numerical solution and the capa-bility of the plane wrinkling model to accurately predict the size of

the wrinkled regions under conditions similar to those of the sur-gical procedure, where large displacements are applied to close thewound. The proposed model overcomes the difficulties of a post-buckling analysis in a regime of finite strains and large displace-ments: the post-critical evolution of a very thin plate subjected tolarge displacements and strains such as those typical of proceduresof reconstructive surgery is characterized by complex boundary layerphenomena and bifurcations of the equilibrium path that representnumerical challenges and that, if not correctly considered, wouldlead to wrong solutions.

The problem of the annular membrane highlights fundamentalaspects and gives basic insight into the behavior of pre-stressedwrinkling membranes. Two important results have come out of theanalysis: the first is the strong screening effect that a pre-stress fieldhas on the wrinkling onset and extension, even in the case of smallcontracting defects in very large membranes; the second is the pres-ence of an asymptotic limit of the wrinkling extension that dependson the Poisson ratio.

The proposed model emerges as a suitable tool for surgical op-eration planning and computer-assisted surgery. The simulations of

528 A. Cavicchi et al. / Finite Elements in Analysis and Design 45 (2009) 519 -- 529

reconstructive surgery presented in the paper highlight the capabil-ity of the model to predict the presence of wrinkled regions thatresemble the dog-ears commonly observed after the suture of thewounds in actual surgery. The size of the wrinkled regions is shownto be strongly affected by the natural tension of the skin so confirm-ing the dependence observed in the analytical solution of the annularmembrane and the importance of this parameter in predicting wrin-kling extension. The results show the ability of the numerical modelto describe the main steps of the surgery and give quantitative in-formation on membrane forces and wrinkling extension on varyingthe natural stress of the skin. This information can help predictingthe outcome of the surgical procedure and improving its technique.Used in combination with the in vivo identification procedure de-veloped in Gambarotta et al. (2005), the model can be applied tooptimize size and shape of the excisions, while considering the in-dividual constitutive parameters and the natural state of the skin,in order to reduce the dog-ears and the stresses along the woundmargin.

The results presented in this paper have been obtained under theassumption of isotropy, previously used for the closed form deriva-tion of the constitutive model [21] and the calibration of its param-eters [11]. While the anisotropy of the skin could be numericallyintroduced in the constitutive model by posing a mixed wrinklingcriterion in terms of principal stresses and strains (see for instance[29,13,32]), the in vivo calibration of the constitutive parameterswould be more challenging and would require more complex ex-perimental setups and a larger number of experimental measure-ments (see comments in [11]). However, while it is expected thatthe quantitative outcome of the simulations would change if the ac-tual anisotropy of the skin is considered, the general conclusions ob-tained in this paper are expected to hold true. This is confirmed bythe analyses of Chaudhry et al. [2] and Lott-Crumpler and Chaudhry[17].

All simulations have been performed on skin flaps that have beenpreviously detached from the subcutaneous attachments. This is atypical procedure of hair replacement techniques where large por-tions of the skin are excised and large undermined areas help to re-duce stresses after suture. Undermining of the skin is often avoidedwhen dealing with reconstructive surgery and the excision of smallportions of skin to remove skin cancers or other defects. The simu-lations presented above describe these cases if the stresses exertedby the subcutaneous attachments are negligible. This assumptionseems to be satisfied ([1] and references therein) but needs furtherconfirmations.

Appendix A.

An elastic annular membrane subject to uniform radial displace-ments ui and u0 along its inner and outer boundaries is considered(inset of Fig. 1). The axisymmetric displacement field is describedby the radial component ur and by the circumferential componentu� =0. The solution of the linearized problem (small strain) with theassumption of isotropy is

ur()a

=

1 − 2

[(1 − 1

2

)u0 +

(2 − 1

2

)ui

], (7)

where = a/b, a and b being the inner and outer radii of the mem-brane, = r/a, r being the radial position, u0 =u0/b and ui =ui/a. Thestress field is given by the relations

�r()E

= 11 − �2

{[(1 + �) + (1 − �)

12

]u0

+[(1 + �)2 + (1 − �)

12

]ui

}1

1 − 2 , (8)

��()E

= 11 − �2

{[(1 + �) − (1 − �)

12

]u0

+[(1 + �)2 − (1 − �)

12

]ui

}1

1 − 2 , (9)

where �r and �� are the radial and the circumferential components,E is Young's modulus and � the Poisson ratio. If u0 and ui are non-negative, �r is tensile throughout the membrane; �� is compressivein an inner portion of the membrane if the inequality ui>uic =2u0�[(1 − �) − 2(1 + �)]−1 holds.

Under the assumption of non-compressive stress fields and forui>uic, the membrane is fully taut only in an external region definedby w��−1, where w is a priori unknown. The displacementand the stress fields in this region are still described by Eqs. (7)–(9)by substituting w, /w and −ur(w)/a for , and ui, whereur(w) is a priori unknown. The inner domain defined by 1�<wis wrinkled and the radial equilibrium yields �r()=�r(w)w/. Theradial displacement follows as

ur()a

= −ui + �r()E

Log. (10)

The outer boundary of the wrinkled region defined by w is ob-tained by imposing the condition ��(w) = 0 and the continuity ofthe radial stress and displacements at w. This condition is analogousto impose that the circumferential strain �� is equal to the naturalcontraction at w, namely ��(w)=−��r(w) with �r the radial strain.By imposing these conditions, w can be expressed as a function ofthe boundary displacements through the relation

ui = − 2w[� + Log(w)]

(1 − �) − 22w(1 + �)

u0. (11)

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