automated estimation of collagen fibre dispersion in the...

Automated Estimation of Collagen Fibre Dispersion in the Dermis

and its Contribution to the Anisotropic Behaviour of Skin

AISLING NI ANNAIDH,1,2,3 KARINE BRUYERE,4,5,6 MICHEL DESTRADE,1,7 MICHAEL D. GILCHRIST,1

CORRADO MAURINI,2,3 MELANIE OTTENIO,4,5,6 and GIUSEPPE SACCOMANDI8

1School of Mechanical and Materials Engineering, University College Dublin, Belfield, Dublin 4, Ireland; 2UPMC, Univ Paris 6,UMR 7190, Institut Jean Le Rond d’Alembert, Boıte courrier 161-2, 4 Place Jussieu, 75005 Paris, France; 3CNRS, UMR 7190,Institut Jean Le Rond d’Alembert, Boıte courrier 161-2, 4 Place Jussieu, 75005 Paris, France; 4Universite de Lyon, 69622 Lyon,

France; 5IFSTTAR, LBMC, UMR_T9406, 69675 Bron, France; 6Universite Lyon 1, Villeurbanne, France; 7School ofMathematics, Statistics and Applied Mathematics, National University of Ireland Galway, Galway, Ireland; and 8Dipartimento

di Ingegneria Industriale, Universitadegli Studi di Perugia, 06125 Perugia, Italy

(Received 19 December 2011; accepted 1 March 2012; published online 17 March 2012)

Associate Editor Jane Grande-Allen oversaw the review of this article.

Abstract—Collagen fibres play an important role in themechanical behaviour of many soft tissues. Modelling ofsuch tissues now often incorporates a collagen fibre distri-bution. However, the availability of accurate structural datahas so far lagged behind the progress of anisotropicconstitutive modelling. Here, an automated process is devel-oped to identify the orientation of collagen fibres usinginexpensive and relatively simple techniques. The methoduses established histological techniques and an algorithmimplemented in the MATLAB image processing toolbox. Ittakes an average of 15 s to evaluate one image, compared toseveral hours if assessed visually. The technique was appliedto histological sections of human skin with different Langerline orientations and a definite correlation between theorientation of Langer lines and the preferred orientation ofcollagen fibres in the dermis ðp<0:001;R2 ¼ 0:95Þ wasobserved. The structural parameters of the Gasser–Ogden–Holzapfel (GOH) model were all successfully evaluated. Themean dispersion factor for the dermis was j ¼ 0:1404�0:0028: The constitutive parameters l, k1 and k2 wereevaluated through physically-based, least squares curve-fitting of experimental test data. The values found for l, k1and k2 were 0.2014 MPa, 243.6 and 0.1327, respectively.Finally, the above model was implemented in ABAQUS/Standard and a finite element (FE) computation was per-formed of uniaxial extension tests on human skin. It isexpected that the results of this study will assist those wishingto model skin, and that the algorithm described will be ofbenefit to those who wish to evaluate the collagen dispersionof other soft tissues.

Keywords—Fibre orientation, Anisotropic, Skin, Collagen

Fibres.

INTRODUCTION

Collagen fibres govern many of the mechanicalproperties of soft tissues, in particular their anisotropicbehaviour. Due to the complex nature of fibrearrangements, such tissues are often represented aseither isotropic or transversely isotropic.1,2,5 Theavailability of quantitative structural data on the ori-entation and concentration of collagen fibres is crucialin order to describe the behaviour of these soft tissuesaccurately.

A number of studies have used statistical distribu-tions to describe the fibre arrangement in soft tissues.Lanir18 was the first to attempt to account for fibredispersion. Lanir’s method expresses the mechanicalresponse in terms of angular integrals. This techniqueaccounts for the contribution of infinitesimal fractionsof collagen fibres in a particular orientation. Thisapproach leads to accurate results but it is not apractical option for efficient numerical implementa-tion.3 The second approach uses generalised structuretensors (GST), which are assumed to represent thethree-dimensional distribution of the fibres. The strainenergy function is then calculated by using the averagestretch rather than by using a stretch for each indi-vidual fibre. GST is a simpler method than Lanir’s,and is easily implemented in FE algorithms.9

In this paper we implement the GST method pro-posed by Gasser et al.,9 described in detail in ‘‘Gasser–Ogden–Holzapfel Model’’ section. In that model,structural parameters such as the fibre dispersionparameter and mean orientation of fibres are required.In order to quantify the orientations of collagenfibres in the human dermis an automated process is

Address correspondence to Aisling Nı Annaidh, School of

Mechanical and Materials Engineering, University College Dublin,

Belfield, Dublin 4, Ireland. Electronic mail: [email protected]

Annals of Biomedical Engineering, Vol. 40, No. 8, August 2012 (� 2012) pp. 1666–1678

DOI: 10.1007/s10439-012-0542-3

0090-6964/12/0800-1666/0 � 2012 Biomedical Engineering Society

1666

developed for detecting collagen orientations in his-tology slides.

Histology is the microscopic study of cells and tis-sue. It is an important diagnostic tool in medicinewhich is utilised here for the purpose of analysingcollagen orientation. Traditionally, histology slides areexamined by expert observers who assess individualslides visually. This method is both time consumingand subjective. An automated process would enablelarge volumes of images to be analysed quickly andimprove the objectivity of the task. Much research hasbeen conducted recently on imaging techniques ofbiological soft tissue for the extraction of structuraldata; however these techniques require either expensiveequipment or manual post processing.6,27,28,29 Therehas been little research to date carried out on theautomated analysis of histology slides. Four notableexceptions are the work of Van Zuijlen et al.,26

Noorlander et al.,21 Elbischger et al.4 and Jor et al.15

Elbischger et al.4 successfully automated the analy-sis of collagen fibre orientation in the human adven-titia, (the outermost membrane of the artery). Theirtechnique was based on the automated analysis ofTransmitted Light Microscopy images which werestained with Elastica van Gieson. The algorithm uses aridge and valley analysis to detect the orientation ofcollagen fibres and segments regions of homogeneousfibre orientations. This is a somewhat complicatedtechnique which requires substantial coding to imple-ment. There are also assumptions made which cannotbe applied to the dermis, i.e. that the image contains atleast 50% collagen and that the fibres have a commonorientation. In contrast, Noorlander et al.21 used arelatively simple technique upon which our own algo-rithm is based. In their study, Picrosirius red stainingwas used together with epipolarised light to image thefibres. Individual collagen fibres were detected andellipses fit to the longest ten fibres. While their tech-nique is quantitative it is not an automated techniqueand requires the user to identify fibre orientationsvisually. The results refer to the ‘collagen alignmentindex’ which is the mean length of the best fit ellipseand has no connection with the angular orientation offibres, which itself is not measured. Van Zuijlen et al.26

used Fourier analysis to measure the level of anisot-ropy of collagen in the skin. Those authors concludedthat analysis of ‘orientation index’ by Fourier analysisis superior to conventional techniques. However, the‘orientation index’ is a measure of the anisotropy ofthe matrix but, just like the ‘collagen alignment index’of Noorlander et al.,21 it fails to provide informationon the mean orientation of the fibres. In a recentpublication Jor et al.15 modelled the orientation ofcollagen fibres in porcine skin. Their technique

involved the staining of porcine skin sections withPicrosirius red and image acquisition by confocal laserscanning microscopy. Their study provided quantita-tive results on the orientation of collagen fibres inporcine skin, however the plane of interest was normalto the epidermis and it is generally believed that thepreferred orientation of collagen fibres is parallel to theepidermis.11

The chief advantage of the technique proposed hereover other automated techniques is that it is an inex-pensive and relatively simple technique. It can be easilyimplemented in MATLAB using the Image ProcessingToolbox so that the algorithm can be easily amendedas required for the user’s specific application. It is afully automated process (aside from the image acqui-sition phase) and is capable of analysing a greaternumber of images, and faster, than a manual analysis.It also eliminates the subjectivity which is present usingtraditional methods. The technique provides quanti-tative data on both the collagen orientation and thelevel of anisotropy in the dermis, but it could be easilyamended to identify any tissues/objects within othersoft tissues.

Due to the anisotropic properties of collagen basedsoft tissues,20 modelling now often incorporates acollagen fibre distribution. The accuracy of thesestructural models rely heavily on knowledge of colla-gen fibre orientations. While this data has recentlybeen published for porcine skin by Jor et al.,15 to thebest of the authors’ knowledge, quantitative data onthe orientation of collagen fibres in the human dermishas not been previously published. With the quanti-tative data obtained here, structural parameters suchas c and j from the Gasser–Ogden–Holzapfel model9

can be evaluated independently. This structural data,coupled with the mechanical testing of the same skinsamples20 provides us with sufficient data to modelhuman skin using the GOH model.

MATERIALS AND METHODS

Tensile Tests and Histology

In vitro tensile tests of human skin were performed inIFSTTAR (Institut francais des sciences et technologiesdes transports, de l’amenagement et des reseaux),France. French law allows human corpses that havebeen donated to science to be used for research pur-poses. The ethics committee within IFSTTARapprovedthe use of human biological material.

The tensile tests were performed using a UniversalTensile Test machine at a strain rate of 0.012 s�1: Thetensile loadwasmeasuredwith a 1 kNpiezoelectric loadcell and the strain was measured via a displacement

Automated Estimation of Collagen Fibre Dispersion 1667

actuator. Skin biopsies were excised adjacent to thetensile test samples and stained with Van Gieson to dyecollagen fibres pink/red. Further details of the tensiletest experimental procedure and histology protocol areoutlined in Nı Annaidh et al.20

Automated Detection of Fibre Orientations

Images were taken of slides parallel to the epidermis(see ‘‘Fibre Dispersion’’ section) using an AperioScanScope XT scanner and ScanScope software. Thisslide scanner has the ability to scan up to 120 slidesautomatically. The images to be analysed were selectedfrom the reticular layer of the dermis which forms themain structural body of the skin. The optical magni-fication used was 59, which is quite low; however, thiswas the optimal magnification to obtain a large enoughfield of view to be representative while also maintain-ing enough detail to recognise the boundaries of fibrebundles. Multiple images were taken of each sample inorder to capture the entire area. The field of view foreach image taken at this magnification was 2.24 mm2:The orientation of collagen fibres were then calculatedin a fully-automated customised MATLAB routineusing the Image Processing Toolbox. The algorithm isdescribed below and is illustrated in Fig. 1.

� A global threshold level was computed usingthe graythresh function. This function chooses aglobal image threshold using Otsu’s methodwhich aims to minimise the intraclass varianceof black and white pixels.23 This automaticallydistinguishes collagen fibres from other areaswhich are not of interest in the analysis, such ascells or histological ground substance. Theimage was then binarised based on this thresh-old level producing an image containing blackpixels for collagen and white pixels for all otherareas;� Morphological operations were performed on

the binary image using the bwmorph function.An erosion step was performed to detach cross-linking fibres from each other. This stepremoved pixels from the boundaries using astructural element of size 3 pixels 9 3 pixels.One iteration only of this step was performedwhich was the optimal number of iterations todetach cross-linking fibres while also leavingsmaller fibres intact. The second morphologicaloperation was performed using the imfill func-tion. This function fills in the ‘holes’ within thebinary image, where a hole is defined as a set ofisolated pixels which cannot be reached byfilling in the background pixels from the edge.This step resulted in a ‘cleaner’ image toanalyse.

� Individual fibre bundles are identified usingbwlabel. This function identifies connectedcomponents (8-connected) in a 2D binary imageand labels each component individually. It usesthe general procedure outlined by Haralick andShapiro10;� The regionprops function outputs a set of

properties for each labelled component. Thefunction also fits an ellipse to each componentby matching the second order moments of thatcomponent to an equivalent ellipse followingthe procedure by Haralick and Shapiro10;� Only ellipses which were elongated and of a

certain area were selected1

� The orientation of the major axis of each ellipsewas calculated and taken as the approximateorientation of each component. The advantageof fitting an ellipse about each component isthat the orientation is measured in a systematicand repeatable manner which can account forthe non-uniformities of the shape of eachcomponent;� The orientations of each component was then

plotted on a histogram (see Fig. 2). Two dis-tinct peaks were evident from this figure. It isassumed that these two peaks correspond to thepreferred orientation of two crossing families offibres as shown in Fig. 3. A von Mises proba-bility density function was fit to the data andthe mean orientation and dispersion factor werecalculated as described in ‘‘Fibre Dispersion’’section.

Validation

As explained by Van Zuijlen et al.,26 there are dif-ficulties involved with validating automated techniquesbecause the true result of each histology slide is un-known. For the validation of their algorithm, com-puter generated images representing collagen fibreswith known orientations were used. However, in orderto create an accurate computer generated image, amodel which represents the structure of the collagen

1Out-of-plane fibres manifest themselves as circular areas. This

condition was introduced to exclude out-of-plane fibres from the

analysis. A sensitivity analysis was carried out to determine the

sensitivity of the algorithm to varying the area and eccentricity cri-

teria. It was found that a ±10% change in both the area and

eccentricity criteria led to a ±1% change in the mean orientation.

This indicates that the selected criteria do not significantly affect the

result of mean orientation. For this study it was specified that in-

plane fibres must have an eccentricity larger than 0.7 and an area

greater than 1000 pixels, which for our images, captured at 59,

corresponds to 5 lm2:

ANNAIDH et al.1668

matrix is required, but no such model exists.4 To val-idate our technique, a selection of slides in the planeperpendicular to the epidermis (see Fig. 4) were man-ually segmented and their mean orientation comparedto those calculated through the automated process.Although it is the plane parallel to the epidermis that isused for the collection of structural data, the perpen-dicular plane was chosen for validation purposes be-cause in general it displays a more orientated patternand is easier to segment manually. Manual segmenta-tion was performed by marking bundles of elongatedfibres with a line as shown in Fig. 5. The orientation ofeach line was then measured manually and the meancalculated.

The dermis possesses a complex interwoven collagenstructure, which is difficult to assess. It is quite possiblethat the collagen pattern may change over a smallvolume. For this reason six images were taken at eachdepth, whereby each image spanned 3.75 mm2 of the50 mm2 sample. The analysis was performed at sixlevels of increasing depth and the mean orientation wastaken as the average over these six levels.

Gasser–Ogden–Holzapfel Model

The Gasser–Ogden–Holzapfel (GOH) model appliesto incompressible solids with two preferred directionsaligned along the unit vectors a1 and a2 (say) in thereference configuration (see Fig. 3). Its strain energydensity W is of the form

W ¼ WðC;H1;H2Þ ð1Þ

where C is the right Cauchy-Green strain tensor, andthe structure tensors H1;H2 depend on a1 and a2 andon the dispersion factors j1; j2 (to be detailed later),respectively, as follows

Hi ¼ jiIþ ð1� 3jiÞa1�a2; ði ¼ 1; 2Þ: ð2Þ

Specifically, the GOH model assumes that W de-pends on I1 ¼ trðCÞ; trðH1CÞ and trðH2CÞ only, asfollows

W ¼ l2ðI1 � 3Þ þ l

X

i¼1;2

ki12ki2

eki2½trðHiCÞ�1�2 � 1n o

; ð3Þ

(a)

(d) (e) (f)

(b) (c)

FIGURE 1. Images output from automated algorithm. (a) Original histology slide. Scale bar is 1 mm. (b) Binarised image afterautomated thresholding. (c) Binarised image after erosion step. (d) All identified collagen bundles outlined in green. (e) Remainingbundleswhich meet area and eccentricity criteria. (f) Best fit ellipse about each fibre that meets the specified criteria.


where l; ki1; ki2 are positive material constants, andfrom Eq. (2),

trðHiCÞ ¼ jiIi þ ð1� 3jiÞI4i; ð4Þ

with I4i � ai�Cai; two anisotropic invariants. Note thatthe constitutive parameter l has the dimensions ofstress: it would be the shear modulus of the solid ifthere were no fibres (ki1 = 0); whilst the parameters ki1

and ki2 are dimensionless stiffness parameters: the ki1are related to the relative stiffness of the fibres in thesmall strain regime, and the ki2 are related to the largestrain stiffening behaviour of the fibers.

Now we focus on homogeneous uniaxial tensiletests. These can be achieved for anisotropic tissueswhen two families of fibres are mechanically equivalentk11 ¼ k21 � k1 (say) and k12 ¼ k22 � k2 (say), with thesame dispersion factor j1 ¼ j2 � j (say), and when thetension occurs along the bisector of a1 and a2 (seeFig. 3). Let us call c the angle between a1 and thetensile direction, so that now

a1 ¼ cos ciþ sin cj; a2 ¼ cos ci� sin cj: ð5Þ

Here, i is the unit vector in the direction of tension, andj is the unit vector in the lateral direction, in the plane ofthe sample. The stretch ratios along those unit vectorsare k1 and k2, respectively. Then I41 ¼ I42 ¼ k21 cos

2 cþk22 sin

2 c � I4 (say) and W reduces to

W ¼ l2ðI1 � 3Þ þ l

k1k2

ek2½jI1þð1�3jÞI4�1�2

� 1n o

; ð6Þ

giving the following expression for r; the Cauchy stresstensor

r ¼ �pIþ 2@W@I1

FFT þ @W@I4

Fa1�Fa1 þ Fa2�Fa2½ �; ð7Þ

wherep is aLagrangemultiplier introducedby the internalconstraint of incompressibility and F is the deformationgradient. Note that Fa1�Fa1 þ Fa2�Fa2 ¼ 2ðk1 cos cÞ2i� iþ 2ðk2 sin cÞ2j� j; showing that r is diagonal in thefi; j; kg basis. Its components are

r11 ¼ �pþ 2ðW1 þW4 cos2 cÞk21 6¼ 0;

r22 ¼ �pþ 2ðW1 þW4 sin2 cÞk22 ¼ 0;

r33 ¼ �pþ 2W1k�21 k�22 ¼ 0;

ð8Þ

where

2W1 ¼ lð1þ 4k1jaek2a2Þ;

2W4 ¼ 4lk1ð1� 3jÞaek2a2 ;a ¼ jðk21 þ k22 þ k�21 k�22 Þþ ð1� 3jÞðk21 cos2 cþ k22 sin

2 cÞ � 1:

ð9Þ

Now, eliminate p from the stress components to getthe two equations

r11 ¼ lðk21 � k�21 k�22 Þ þ 4lk1aek2a2 jðk21 � k�21 k�22 Þ�

þ ð1� 3jÞk21 cos2 c�; ð10Þ

0 ¼ k22 � k�21 k�22 þ 4k1aek2a2 jðk22 � k�21 k�22 Þ�

þ ð1� 3jÞk22 sin2 c�: ð11Þ

0 100 200 3000

0.01

0.02

0.03

0.04

Orientation (θ)

p, P

roba

bilit

y D

ensi

ty2γ

FIGURE 2. Histogram of collagen orientations. The two dis-tinct peaks correspond to the preferred orientation of the twofiber families. The angle, c, is half the distance between thetwo peaks i.e. c 5 41�.

FIGURE 3. Lattice structure of crossing collagen fibres withfibre dispersion taken into account.

ANNAIDH et al.1670

Equation (11) gives the relationship between thetensile stretch and the lateral stretch and allows,implicitly, k2 to be expressed in terms of k1. Substi-tuting then into (10) gives the r11–k1 stress-stretchrelationship. In the isotropic limit, j = 1/3 (see ‘‘FibreDispersion’’ section), and (11) yields the well-knownrelationship k2 = k1

21/2 for uniaxial tension in incom-pressible solids.

The two Eqs. (10) and (11) form the basis of anumerical determination of the constitutive parame-ters l, k1 and k2, assuming that the structural param-eters j and c are known. It should be noted here thatthe inclusion of l in the anisotropic term of Eq. (6) isnot standard for the GOH model and has been addedhere for ease of calculation. We now quantify furtherthose latter parameters, j and c.

Fibre Dispersion

The GOH model assumes that the mean orientationof collagen fibres has no out-of-plane component. Ourhistological examination of the skin indicates that themajority of collagen fibres in the dermis run parallel to

the epidermis. Slides parallel to the epidermis had, onaverage, three times less cross-sectioned fibres thanslides perpendicular to the epidermis. This is in agree-ment withHolzapfel et al.11 who state that the preferredorientation of the 3D collagen fiber network lies parallelto the surface, but to prevent out-of-plane shearing,some fiber orientations have components which are out-of-plane. Despite the assumption that the fibres have noout-of-plane component in the GOH model, the three-dimensional nature of the adopted distribution impliesthat although the preferred orientation of the fibers arein the plane parallel to the epidermis, some fibers ori-entations have an out-of-plane component.9

Here we assumed that each of the two families ofcollagen fibres is distributed according to a p-periodicVon Mises distribution, which is commonly assumedfor directional data. The standard p-periodic VonMises Distribution is normalized and the resultingdensity function, qðHÞ; reads as follows,

qðHÞ ¼ 4

ffiffiffiffiffiffib

2p

rexp½bðcosð2HÞ þ 1�

erfiðffiffiffiffiffi2bpÞ

; ð12Þ

where b is the concentration parameter associated withthe Von Mises distribution and H is the mean orien-tation of fibres (for a graph of the variation of thedispersion parameter j with the concentrationparameter b, see Gasser et al.9).

The parameters b and h were evaluated using themle function in MATLAB. Analogous to least squarescurve-fitting, the maximum likelihood estimates(MLE), is the preferred technique for parameter esti-mation in statistics. Then j is calculated by numericalintegration of the integral given by Gasser et al.,9

j ¼ 1

4

Zp

0

qðHÞ sin3 HdH: ð13Þ

The structural parameter j describes the material’sdegree of anisotropy. Itmust be in the range 0 £ j £ 1/3:

FIGURE 4. Biopsies of skin samples for purpose of histological staining. Note that the biopsies have been sliced in threeorthogonal planes.

FIGURE 5. Manual segmentation of collagen. Elongated fi-bres were marked by black lines, and their orientation waslater measured manually.


the lower limit, j = 0, relates to the ideal alignment ofcollagen fibres and the upper limit, j = 1/3, relates tothe isotropic distribution of collagen fibres. Figure 6 isa 3D graphical representation of the orientation ofcollagen fibres for different values of j.

The fibres were assumed to form an interweavinglattice structure as first postulated by Ridge andWright24 and shown in Fig. 3. These authors suggestedthat the mean angle of the two families of fibres indi-cates the direction of the Langer lines. More recentin vitro15 and in vivo25 studies have also supported thishypothesis. The lattice structure proposed by Ridgeand Wright24 is an idealised one, and the adoption hereof the dispersion factor creates a more realistic sce-nario. Finally, recall that the two families (directions)of fibres are assumed to have a common dispersionfactor.

Finite Element Representation

An FE computation was used to simulate uniaxialtensile tests of human skin which were described in aprevious publication.20 The simulation was carried outfor three samples: parallel, perpendicular, and non-symmetric about to the Langer lines. The test sampleswere of the dimensions shown in Fig. 7. The lengthof the unclamped specimen was 68 mm and thethicknesswas 2.25 mm. 1512 reduced integration hybrid

hexahedral (C3D8RH) elements were used for themesh. The numerical analyses were performed usingthe static analysis procedure in ABAQUS/Standard.Displacement was applied through a smooth ampli-tude boundary condition, and the top and bottom ofthe sample were encastred to represent the clamping ofsamples. The material model used was the anisotropicGOH model, which is an internal material model inABAQUS.

RESULTS

Structural Parameters

As expected, it was found from the histology thatthe collagen fibres were locally orientated. This meantthat each sample had a different mean orientation andfibre dispersion. Table 1 tabulates the results of 12different human skin samples, with different orienta-tions, which were procured from the backs of twodifferent subjects.2 See Fig. 8 for details of specimenlocation and orientation. The preferred orientation, bH;refers to the bisector of the two families of fibres. The95% Confidence Interval of the mean of these imagesranged from 1.16� to 2.77� thereby indicating that thepreferred orientation does not change significantlyover this small area. The results of our validation re-vealed that the automated process differed from themanual segmentation by an average of 5� ± 4� givingus further confidence in the validity of our technique.

A Pearson correlation test was carried out to test fora correlation between the measured preferred orienta-tion obtained through histology and the perceivedorientation of Langer lines (the natural lines of tensionin the skin). The orientation of Langer lines wasassessed using generic maps, described further in NıAnnaidh et al.20 The correlation was deemed to besignificant (p< 0.001) with an R2 value of 0.95. Thisshows that the Langer lines have an anatomical basis, a

FIGURE 6. Three dimensional representation of the orienta-tion of collagen fibres. (a) j 5 0.0085, (b) j 5 0.25, (c) j 50.33.

FIGURE 7. Dimensions of test specimen (mm).

2It should be noted here that these results have been obtained using

the mle method described in ‘‘Fibre Dispersion’’ section. An alter-

native, but less reliable method exists whereby data is clustered into

two intersecting fibre families using an agglomerative clustering

algorithm such as was performed in Nı Annaidh et al.20

ANNAIDH et al.1672

point which had previously been suggested but untilnow had not been quantitatively assessed.

Constitutive Parameters

It was assumed that the orientation of collagen fi-bres is symmetric about the axis of applied stress. Inreality, this is not always the case; however someassumptions must be made in order to ensure that the

constitutive relations remain practical for numericalimplementation. In particular, as explained in ‘‘Gasser-Ogden-Holzapfel Model’’ section, this assumptionleads to a homogeneous deformation of the sample,and in turn, to an explicit stress-strain solution. In thissection, three illustrative examples from Table 1 (ital-icized) have been chosen for further investigation. Twoof these examples have been chosen because they arethe samples that are closest to being symmetrical aboutthe axis of applied stress. The third sample has beenchosen to illustrate how its behaviour can be modelledusing FE analysis and is described further in ‘‘FiniteElement Simulations’’ section.

The constitutive parameters for the GOH model areobtained by using Eqs. (10) and (11), obtained in‘‘Gasser-Ogden-Holzapfel Model’’ section. When lin-earized in the neighbourhood of small strains,ki ’ 1þ ei; say, we find that they read as follows,

r11 ¼ 4l½1þ 2k1ð1� 3jÞ2 cos4 c�e1þ 2l½1þ 4k1ð1� 3jÞ2 sin2 c cos2 c�e2; ð14Þ

0 ¼ ½1þ 4k1ð1� 3jÞ2 cos2 c sin2 c�e1þ 2½1þ 2k1ð1� 3jÞ2 sin4 c�e2: ð15Þ

These expressions reveal that the constitutiveparameters l and k1 are related to the early stages ofthe tensile tests, whilst k2 is a stiffening parameter,related to the latter (nonlinear) stages of the tensiletests. By solving (15) for e2; and substitution into (14),we find the linear stress-strain relation r11 ¼ E1e1;where E1 is the infinitesimal Young modulus in the1-direction, found here as

E1 ¼3þ 8k1ð1� 3jÞ2ð1� 3 cos2 c sin2 cÞ

1þ 2k1ð1� 3jÞ2 sin4 cl; ð16Þ

FIGURE 8. Location of tensile test samples shown in Table 1(figure amended from Langer17).

TABLE 1. Local mean orientation of fibres and dispersion factor.

Age Gender Location Orientation of Langer lines� Preferred orientation bH� Dispersion factor j

81 Female 5 0/180 174 ± 3 0.1306 ± 0.0054

4 0/180 20 ± 3 0.1439 ± 0.0088

6 45 38 ± 7 0.1314 ± 0.0054

1 45 46 ± 8 0.1675 ± 0.0023

2 90 88 ± 8 0.1535 ± 0.0059

3 135 121 ± 5 0.1485 ± 0.0026

89 Male 5 0/180 178 ± 4 0.1462 ± 0.0053

4 0/180 0 ± 5 0.1456 ± 0.0055

6 45 61 ± 4 0.1289 ± 0.0046

1 45 13 ± 3 0.1276 ± 0.0054

2 90 89 ± 5 0.1009 ± 0.0085

3 135 118 ± 7 0.1602 ± 0.0095

Orientations are given with respect to the axis perpendicular to the axis of applied stretch. The three samples italicized are those taken as

illustrative examples for further analysis. (Note that the data given is axial data i.e. it represents undirected lines and does not distinguish

between h and p + h14 e.g. the orientation of 0� and 180� are equivalent).


(which is consistent with the formula E = 3l in linearisotropic (j = 1/3) incompressible elasticity). Hence, byplotting the values of r11 for the early part of the tests(first 1000 data say, corresponding to a tensile stretch ofless than 2%), we can determine E1 by linear regressionanalysis, see Fig. 9a. Here we have plotted the ‘parallel’sample italicized inTable 1, data forwhichwas collectedfrom tensile tests of human skin samples.20

Once E1 is determined, l can be expressed in termsof E1 and k1 using Eq. (16). Then the remainingmaterial parameters k1 and k2 are found through thenonlinear least squares fitting with experimental testdata of Eq. (10), subject to the definition of k2 in termsof k1 given by Eq. (11). The data fitting was performedusing the lsqnonlin MATLAB routine in the Optimi-sation Toolbox where the objective function, Err(k),was given as

ErrðkÞ ¼Xn

i¼1ðyexpi � y

modelðkÞi Þ2 ð17Þ

where n is the number of experimental data points, yiexp

is the experimental value and yimodel(k) is the value

predicted by the model using the current materialparameters, k.

Non-linear optimisation procedures are often sen-sitive to the initial starting point provided by theuser.22 In our case, the initial estimate for k1 was foundby calculating the slope of the non-linear part of thestress-stretch curve. Since we have shown that k1 isrelated to the stiffening stage of the tensile test, ourinitial estimate therefore has a physical meaning.Furthermore, the initial estimates of both k1 and k2were varied over a large range (0-1e6 for k1 and 0-1e3

for k2) and led to the same set of optimal parameterseach time, illustrating that the results of the optimi-sation procedure are not sensitive to this initialestimate.

The results of our optimisation procedure gave us avalue of 243.6 for k1 and 0.1327 for k2, with an R2 of99.5%. Figure 9b shows the GOH model fit to theexperimental data. It can be seen that these materialparameters provide an excellent fitting to the ‘parallel’sample, at least from a descriptive point of view.Turning now to the predictive capabilities of the GOHmodel, we examine a sample ‘perpendicular’ to theLanger lines. In Fig. 10 we use the material parametersk1 and k2 obtained through least squares fitting, lcalculated by linear regression and Eq. (16), coupledwith the unique structural data for the ‘perpendicular’sample in Table 2. We compare the model predictionfor a tensile test occurring perpendicular to the Langerlines to the experimental data. The fit remains goodwith R2 = 97.96%. This shows that the model iscapable of predicting the behaviour of skin once thestructural parameters have been evaluated.

1 1.005 1.01 1.015 1.02−0.01

0

0.01

0.02

0.03

0.04

λ1

σ 11 (

MP

a)Data Linear

(a)

1 1.1 1.2 1.3 1.40

10

20

30

40

λ1

σ 11 (

MP

a)

SlopeExp DataGOH Model

(b)

FIGURE 9. Nonlinear curve fitting to obtain the constitutiveparameters: l and k1 are related to the early (infinitesimal)stress-strain part of the graph, see (a); k2, to the rest of thecurve.

1 1.1 1.2 1.3 1.4 1.50

10

20

30

40

λ1

σ 11 (

MP

a)

FEA ParallelData ParallelData PerpendicularFEA Perpendicular

R2Par

= 99.54%

R2Perp

= 97.96%

FIGURE 10. Comparison of sample parallel to Langer linesand perpendicular to Langer lines.

ANNAIDH et al.1674

Finite Element Simulations

The conventions used by ABAQUS are related toours through C10 ¼ l=2; k01 ¼ k1l and k02 ¼ k2; sothat the ABAQUS parameters used were C10 =

0.1007 MPa, k01 ¼ 24:53 MPa and k02 ¼ 0:1327:The FE simulation results of the uniaxial tensile

tests for both the parallel and perpendicular sampleswere identical to the analytical solution. The resultswere independent of both mesh density and elementtype. Figure 11 shows the Cauchy stress distributionacross the parallel and perpendicular samples at theend of the test. The large difference in magnitudesbetween the two is due to the variation in the meanorientation of fibres. Note the uniform distribution ofstress in the middle section of the test samples, thanksto the dog-bone shape of the specimen, and the sym-metry of fibres about the axis of applied stress.

As discussed in ‘‘Constitutive Parameters’’ section,for ease of determining the material parameters, it wasassumed that the orientation of collagen fibres issymmetric about the axis of applied stress. This is foran idealised scenario only, where one knows the exactorientation of collagen fibres prior to testing, and cantherefore apply the stress in this orientation. However,with the model parameters that have now been deter-mined, a FE simulation can provide results for sampleswhere the collagen fibres are not symmetric about theaxis of applied stress. Examining the non-symmetricexample in Fig. 12a, we can see a non-uniform distri-bution of stress throughout the test specimen. A localmagnification of stress occurs near the neck regions ofthe test specimen. This is due to the non-symmetry of

(a) (b)

FIGURE 12. Cauchy stress in Pa of a non-symmetric sample strained by 30% (a) r11 (b) r12.

(a) (b)

FIGURE 11. Cauchy stress in Pa of sample strained by 50%(a) Parallel to the Langer lines (b) Perpendicular to the Langerlines.

TABLE 2. Values obtained through curve-fitting for theparameters l, k1 and k2.

Sample l (MPa) k1 k2 c bH� j R2 (%)

Parallel 0.2014 243.6 0.1327 41 88 0.1535 99.54

Perpendicular 0.2014 243.6 0.1327 41 0 0.1456 97.96

Non-symmetric 0.2014 243.6 0.1327 41 118 0.1602 94.40

Also displayed is R2, a measure of goodness of fit, and j and cobtained directly through histology.


collagen fibres about the axis of applied stress andmakes this problem a much more complicated one tosolve analytically. The presence of significant levels ofshear in Fig. 12b (which is absent from both the par-allel and perpendicular samples) indicates further theeffect of this non-symmetry on the sample response.

Because the stress distribution in this sample is non-uniform, to examine the predictive capabilities of theGOH model here, we must plot the experimental force-displacement data against the values predicted byABAQUS for a node at the top of the test sample, seeFig. 13. Again, we have found that the model predictsthe behaviour well with an R2 of 94.4%, showing thatthe GOH model is capable of predicting the aniso-tropic response of human skin.

DISCUSSION

For this paper, histology slides in three differentplanes were examined; however, after capturing theimages from all three planes it was observed that threetimes as many cross-sectioned fibres were present in theplane normal to the epidermis, therefore an assump-tion was made to ignore the fibres normal to the epi-dermis. Hence, the further analysis of the samples wasrestricted to the plane parallel to the epidermis alone,making this a 2D analysis. Physically however, thereare a percentage of fibres that run normal to the epi-dermis and this information has not been capturedhere. Furthermore, unloaded collagen fibres have acrimped nature: Their relative orientation may varywith respect to the epidermal plane. Here we have triedto overcome this limitation by taking an averagemeasure over six levels of depth spanning 30 lm,withthe view that the general orientation of the collagen

fibres are still captured. Ideally, a full 3D analysis ofthe dermal structure could consider the effects of col-lagen crimping. A 3D analysis was beyond the scope ofthis paper, but the current technique could be extendedby creating a montage of overlapping images throughthe thickness of the dermis, as described in Jor et al.,15

therefore turning the 2D analysis into a 3D analysis.A further limitation of this technique is that we have

assumed that the structure of the skin biopsy removedis representative of the entire tensile test sample. Inreality, the structure, and therefore the properties ofskin may vary considerably over a small area. Thevariation of the mean fibre orientation over the volumeof the biopsy was quantified, however, the size of thebiopsy is very small relative to the tensile test sampleand we cannot infer that the variation would be neg-ligible. At a minimum, future studies should excise anumber of biopsies along the length of the test speci-men to investigate the variation of the structure.

While this technique has been described as ‘auto-mated’, there are still a number of ‘manual’ steps thatmust first be performed. The first manual task is thehistological staining, combined with the mounting ofskin biopsies. The collagen detection process demandsa high quality of histological staining for the method tobe successful and therefore, care must be taken tofollow standard procedures carefully. The second‘manual’ task is the image acquisition phase. Modern‘slide scanners’ automatically scan multiple slides atonce meaning that tedious image acquisition tech-niques using a manual microscope are no longer nec-essary, however images must still be captured from the‘digital slide’.

While this technique has provided quantitativestructural data of human skin, it can, of course, onlybe applied in-vitro. Considering the effect that themean orientation of collagen fibres has on themechanical response of skin, the development of in vivomethods for establishing the orientation of fibres is ofthe utmost importance. Advanced imaging techniquessuch as ultrasonic surface wave propagation mayeventually provide real-time, in vivo structural data.

It should be noted that the nonlinear curve fittingtechnique rests only onmeasurements of k1 and r11, andthat k2, along with k1 and k2 are obtained during thesimultaneous optimisation of Eq. (10) and Eq. (11).Ideally, a more complete analysis would include, com-pare, and contrast experimental data for k2 and/or k3.An extensive experimental data set would include pla-nar biaxial tests with in-plane shear and separatethrough thickness shear tests,7,13,16 however in theabsence of these advanced testing protocols tensile testscoupled with a histological study of the collagen fibrealignment can be used for reasonable determination ofmaterial parameters.12 Of course non-uniqueness of

0 5 10 15 200

100

200

300

400

Displacement (mm)

For

ce (

N)

FEAData

R2 = 94.4%

FIGURE 13. Comparison between predicted model responseand experimental force-displacement data for a non-symmet-ric sample.

ANNAIDH et al.1676

optimal material parameters is an intrinsic problem innon-linear fitting. It is possible that the optimisationprocedure finds a local minimum and assumes that thisis the global minimum.22 To ensure that the optimisa-tion procedure is providing a unique set of materialparameters a number of checks are available: Theproperties of the Hessian matrix can be investigated atthe optimum,8,19 or alternatively, one can plot theobjective function as a function of the varying materialparameters. In this case the objective function wasinvestigated and the plot (not reproduced) shows thatour procedure calculates a global minimum and notmerely a local minimum.

In this study we have developed a simple automatedprocess which can detect the orientation of collagen fi-bres. This technique can be easily implemented inMATLAB and can be adapted to detect other biologicalfeatures, such as certain cells, leading to applications indiagnostics. We have applied this technique to skinbiopsies and provided new quantitative data on theorientation of collagen fibres in the human dermis. Sofar, the availability of accurate structural data has lag-ged behind the progress of anisotropic constitutivemodelling. Here we have provided the structural datarequired to accurately make use of advances in consti-tutive modelling, and help fill the void of experimentaldata. The model parameters of the GOH model havebeen evaluated for skin using experimental data fromthe same skin samples. These sets of parameters willprovide invaluable data for those wishing to model theanisotropic behaviour of skin. Finally, an FE simula-tion of a uniaxial tensile test on three separate humanskin samples was performed which predicted theresponse of these three samples well.We have illustratedthat the Gasser-Ogden-Holzapfel model can success-fully model the anisotropic behaviour of human skinand that it can be implemented in ABAQUS with ease.

ACKNOWLEDGMENTS

The authors acknowledge gratefully the advice andassistance of Mr. Ciaran Driver, Dr. Michael Curtisand Prof. Marie Cassidy, of the Office of the StatePathologist (Ireland), in the area of histology. Thisresearch was supported by a Marie Curie Intra Euro-pean Fellowship within the 7th European CommunityFramework Programme, awarded to MD; by the IrishResearch Council for Science, Engineering and Tech-nology; by the Office of the State Pathologist (IrishDepartment of Justice and Equality); and by the Ile-de-France region. G.S. is supported by the PRIN 2009project ‘‘Matematica e meccanica dei sistemi biologicie dei tessuti molli’’.

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