Journal of Biomechanics 44 (2011) 182–188

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Journal of Biomechanics

0021-92

doi:10.1

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E-m

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Growth and surface folding of esophageal mucosa: A biomechanical model

Bo Li, Yan-Ping Cao, Xi-Qiao Feng n

Institute of Biomechanics and Medical Engineering, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China

a r t i c l e i n f o

Article history:

Accepted 8 September 2010Mucosal folding in such biological vessels as esophagus and airway is essential to their physiological

functions and can be affected by some diseases, e.g., inflammation, edema, lymphoma, and asthma. A

Keywords:

Esophagus

Mucosal folding

Residual stress

Finite volumetric growth

90/$ - see front matter & 2010 Elsevier Ltd. A

016/j.jbiomech.2010.09.007

esponding author. Tel.: +86 10 62772934; fa

ail address: fengxq@tsinghua.edu.cn (X.-Q. Fe

a b s t r a c t

biomechanical model within the framework of finite deformation theory is proposed to address the

mucosal folding induced by the growth and residual stresses in the tissue. A hyperelastic constitutive

law is adopted for the mucosal layer, which grows in a cylindrical lumen. The fields of the engendered

displacements and residual stresses are solved analytically. Furthermore, the instability analysis

predicts the folding number, which agrees well with our experimental observations. This study not only

sheds light on the biomechanical mechanisms underlying mucosal folding but also provides a

promising approach for determining the residual stress level in living tissues under different

physiological or pathological conditions according to their folding features.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Esophagus exhibits mucosal folding at its luminal surfacealong the circumferential direction (Stiennon, 1995; Yang et al.,2007). Surface folds may appear due to the contraction of themuscle layer in an in vivo state and is also observable in theabsence of any active muscle contraction or external loading(Liao et al., 2003; Noble et al., 2010). Besides esophagus, similarmucosal folding phenomena have also been seen in many otherbiological vessels, ranging from pulmonary airways, arteries,blood vessels in the myocardium, eustachian tubes to thegastrointestinal tracts. On the one hand, it is believed thatsurface folding plays a significant physiological role in healthybiological tissues; on the other hand, the mucosal folding and itsphenotypic characteristics are also pathologically associated withsuch diseases as inflammation, edema, lymphoma and asthma.Abnormal growth in mucosa and the alteration in its foldingpatterns are important clinical signs and symptoms of thesediseases (Stiennon, 1995; Wiggs et al., 1997). In view of its clinicalrelevance and significance, much effort has been directed duringthe past decade towards developing appropriate mechanicalmodels to elucidate the mechanisms behind the mucosal folding(Carroll et al., 2000; Hrousis et al., 2002; Lambert et al., 1994;Seow et al., 2000; Wiggs et al., 1997; Yang et al., 2007). In theseprevious models, mucosal folding is usually treated as theconsequence of smooth muscle contraction or external mechan-ical loads. In most real situations, however, the formation and the

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x: +86 10 62781824.

ng).

alteration of folding patterns result from the constraint growth ofthe soft tissues.

The growth of living tissues and organs results in residualstresses. For instance, recent experiments indicate the existenceof residual stresses in a normal esophageal mucosa (Liao et al.,2003; Lu and Gregersen, 2001; Yang et al., 2007). It is wellrecognized that residual stresses play an important role in variousfunctions of living tissues (Dunlop et al., 2010; Fung, 1990;Holzapfel et al., 2000; Humphrey, 2003; Skalak et al., 1996).A sufficiently large residual stress can buckle the biological tissues(Ben Amar and Goriely, 2005; Ben Amar and Ciarletta, 2010; Liangand Mahadevan, 2009; Volokh, 2006). As aforementioned, suchesophageal diseases as inflammation and edema engenderabnormal growth in mucosae, which may alter the distributionsof residual stresses in the soft tissues and further affect theobserved folding patterns. In the previous models of mucosalwrinkling (Carroll et al., 2000; Hrousis et al., 2002; Lambert et al.,1994; Seow et al., 2000; Wiggs et al., 1997; Yang et al., 2007), thecorrelation between the growth and the folding patterns has notbeen explicitly addressed.

In the present paper, the physiological or pathological growthand folding of esophageal mucosa (Fig. 1) are investigated byusing the theory of finite volumetric growth (Rodriguez et al.,1994). A hyperelastic constitutive law is adopted to model themucosal layer, which grows in a cylindrical and stiff lumen. Thetotal deformation of the mucosa is decomposed into a growth partdescribing the addition of a material and an elastic deformationpart characterizing the reorganization of the mucosa needed toensure geometric compatibility and structural integrity. The fieldsof the engendered inhomogeneous displacements and residualstresses are solved analytically. Instability analysis is further

Fig. 1. Photograph of bovine esophagus.

B. Li et al. / Journal of Biomechanics 44 (2011) 182–188 183

performed and the folding number predicted by using the tissuegrowth model agrees with our experimental observations.

2. Biomechanical model of tissue growth

2.1. Growth and deformation

Fig. 1 shows a typical ring cut from a bovine esophagus, whichmainly consists of an inner mucosa–submucosa layer and anouter muscle layer. For simplicity, the mucosa and submucosa arehere treated as a ‘‘mucosal tube’’. Because the elastic modulus ofthe muscle layer is far higher than that of the inner mucosa (Yanget al., 2007), the former is assumed to be rigid in our analysis. Anincompressible hyperelastic constitutive law is adopted to modelthe mucosal layer, which grows in a cylindrical lumen. The initialand stress-free state of the mucosa before its growth and thedeformed state after growth are defined as the referenceconfiguration and the current configuration, respectively. Thecoordinates of a representative material point in the initial andcurrent configurations are traditionally described by X and x,respectively. Let F¼ @x=@X denote the total geometric deforma-tion gradient. Then, according to the theory of volumetric growth,the geometric deformation tensor can be decomposed into(Rodriguez et al., 1994)

F¼AUG ð1Þ

where A is an elastic deformation tensor resulting from stressesand G a growth tensor describing the addition of materials. Thesymbol dot between two tensors denotes their inner product. Theelastic incompressibility requires that detA¼ 1.

In terms of the hyperelastic assumption, the deformation ofthe mucosa can be described by a strain energy function WðAÞ andthe nominal stress S is related to the elastic deformation tensor Aby (Ben Amar and Goriely, 2005)

S¼ JGG�1U@W

@A�pA�1

� �ð2Þ

where JG ¼ detG stands for the volume change due to the growthand p is the hydrostatic pressure. For convenience, we alsointroduce the Cauchy stress r, which is related to the nominalstress S by

r� J�1FUS¼AU@W

@A�pI ð3Þ

where J¼det F and I is a second-order unit tensor.

In the absence of body forces, the mechanical equilibriumequation reads

DivðSÞ ¼ 0 or divr¼ 0 ð4Þ

where ‘‘Div’’ and ‘‘div’’ stand for the divergence operators in theinitial and current configurations, respectively. Besides theequilibrium Eq. (4), the stress field of the mucosa must satisfythe following boundary conditions:

STUN¼ 0 or rUn¼ 0 ð5Þ

at the free boundary, and

x�X¼ 0 ð6Þ

at the fixed boundary, where N and n are the unit outward normalvectors in the initial and current configurations, respectively.

2.2. Incremental form of governing equations

Instability is defined here as a bifurcation of deformation orthe existence of a new nontrivial solution to the incrementalboundary problem of tissue growth. The incremental equationsaccounting for tissue growth have been proposed by Ben Amarand Goriely (2005). Here we reformulate their equations in adifferent and simpler way.

With growth and deformation, a line element dX in the initialconfiguration transforms to dx in the current configuration; thenthis process can be described by the deformation gradient tensorF, that is, dx¼ FUdX. Hence, the incremental connection readsðdxÞd¼ _FUdX. When the current configuration is chosen as thereference configuration, the incremental connection becomesðdxÞd¼ _F0Udx, with _F0 ¼ grad _x being the gradient of _x withrespect to x. Here, the superposed dot over a quantity signifies itsincrement, and the subscript 0 indicates a quantity calculatedwith respect to the deformed configuration. From the abovedefinitions, we have the relation

_F ¼ _F0UF ð7Þ

Suppose that the growth of the tissue is controlled biologicallyor externally and not influenced by the state of its stresses orstrains (Ben Amar and Goriely, 2005; Ben Amar and Ciarletta,2010). Then, from Eq. (1), we obtain

_F ¼ _AUG ð8Þ

Substituting Eqs. (1) and (8) into (7) yields

_A ¼ _F0UA ð9Þ

The incompressibility of the material leads to ðdetAÞ d ¼ 0,which further gives

trð _F0Þ ¼ 0 ð10Þ

The incremental form of the nominal stress is given by

_S ¼ JGG�1UðCUU _A� _pA�1

þpA�1U_AUA�1

Þ ð11Þ

where the elasticity tensor C is defined as

C¼ @2W

@A@Að12Þ

By using Eq. (9), the incremental constitutive law (11) can bewritten in a push-forward form, namely, _S0 ¼ J�1F _S, whichyields

_S0 ¼LUU _F0þp _F0� _pI ð13Þ

B. Li et al. / Journal of Biomechanics 44 (2011) 182–188184

where L is the fourth-order tensor of instantaneous modulidefined in the current configuration. The relation between thetensors L and C is (Ogden, 1984)

LUU _F0 ¼AUCUUð _F0UAÞ ð14Þ

The push-forward form of the incremental equilibriumequation can be written as

div _S0 ¼ 0 ð15Þ

The boundary conditions in Eqs. (5) and (6) become

_ST0Un¼ 0, _x ¼ 0 ð16Þ

respectively. Surface folding will occur if the system of equationsgiven by Eqs. (15) and (16) has a nontrivial solution.

3. Residual stress analysis

The constrained growth of a mucosa creates an inhomogeneousstress distribution. Consider a cylindrical mucosal layer with theinitial inner radius A and outer radius B, as shown in Fig. 2. In thecylindrical coordinate system, a representative material point withthe coordinate X¼ ðR,Y,ZÞ in the stress-free initial configurationtransforms to a new position x¼ ðr, y, zÞ in the current configurationafter growth. We focus on the case of axisymmetric growth, i.e., r¼

rðRÞ. After growth, the inner and outer radii become a¼ rðAÞ and b¼

rðBÞ, respectively. Then, the geometric deformation gradient tensor is

F¼ diagðl1, l2, l3Þ ð17Þ

where li are the stretches and the index i¼1, 2, and 3 denotequantities in the radial, circumferential, and longitudinal directions,respectively. Here, l1 ¼ @r=@R, l2 ¼ r=R, and l3 will be determinedbelow. The growth tensor is assumed to be

G¼ diagðg1,g2,g3Þ ð18Þ

where gi denotes the growth factors. gi41 represents growth and0ogio1 indicates shrinking or resorption. For illustration, weconsider the space-independent growth by assuming that gi isuniform in the mucosal layer. In the present analysis, the mucosa issupposed to deform and grow under the plane-strain conditions,without deformation and growth in the longitudinal direction.Thereby, l3 ¼ 1 and g3 ¼ 1. In the special case of g1 ¼ g2, an isotropicgrowth or resorption is achieved. From Eq. (1), the elastic deformationtensor is derived as

A¼ diagða1, a2, a3Þ ð19Þ

where a1 ¼ g�11 @r=@R, a2 ¼ g�1

2 r=R, and a3 ¼ 1 stand for the elasticdeformations. Let a¼ a2, then detA¼ 1 leads to a1 ¼ a�1. Besides,the incompressibility condition requires that detF¼ detG, i.e.,R�1r@r=@R¼ g1g2. Integration of this equation gives

r2 ¼ B2�g1g2ðB2�R2Þ ð20Þ

Fig. 2. Growth of mucosa: (a) the initial state before growth, (b) the current state after

where the fixed boundary condition of rðBÞ ¼ B has been used.Eq. (20) describes the inhomogeneous deformation due to growth.The parameter a satisfies

a2 ¼B2

g22R2�

g1

g2

B2

R2�1

� �ð21Þ

Define ab ¼ aðBÞ ¼ g�12 and aa ¼ aðAÞ. Then one has

a2a ¼

B2

g22A2�

g1

g2

B2

A2�1

� �ð22Þ

The necessary condition of a2Z0 requires

g1g2rB2

B2�A2ð23Þ

This condition can also be understood as follows. Thevolume of the mucosa per unit length along the longitudinaldirection before growth is V0 ¼ pðB2�A2Þ, and its maximumvolume per length after growth is V ¼ pB2. Therefore, theincompressible assumption of elastic deformation leads toJGrV=V0, i.e., Eq. (23).

The stress field can be derived from Eqs. (2) and (3) togetherwith a specified strain energy function WðAÞ. Here we adopt theisotropic Neo-Hookean constitutive model to characterize thenonlinear hyperelasticity of mucosa

W ¼m2ða2

1þa22þa

23�3Þ ð24Þ

where m is the shear modulus at the ground state.According to Eqs. (3) and (19), the components of the Cauchy

stress tensor are given as

srr ¼ ma�2�p, syy ¼ ma2�p ð25Þ

The equilibrium Eq. (4) in the current configuration reducesto

@srr

@rþsrr�syy

r¼ 0 ð26Þ

Using Eq. (25) and replacing the variable r by a, Eq. (26)becomes

@srr

@a¼mða2�a�2Þ

a�ga3ð27Þ

where g¼ g2=g1. From the boundary condition in Eq. (5), theresidual stresses are derived analytically as

srr ¼m2g

gða�2�a�2a Þþðg

2�1Þlnga2�1

ga2a�1þg2ln

a2a

a2

� �ð28Þ

syy ¼ srrþmða2�a�2Þ ð29Þ

growth, and (c) the buckling state when the residual stress exceeds a critical value.

B. Li et al. / Journal of Biomechanics 44 (2011) 182–188 185

In the special case without growth in the circumferentialdirection (i.e., g2 ¼ 1), Eq. (29) indicates that the relationshipsrr r ¼ b ¼ syy r ¼ b

���� always holds since ab ¼ g�12 .

4. Instability analysis

The incremental displacement field is expressed by the vector

_x ¼ uðr,yÞerþvðr,yÞey ð30Þ

where er and ey are the unit base vectors in the radial andcircumferential directions, respectively, and the functions u and v

are the incremental displacements along the er and ey directions.Then the increment of the deformation tensor _F0 is calculated by

_F0 ¼@u

@rererþ

1

r

@u

@y�v

� �ereyþ

@v

@reyerþ

1

r

@v

@yþu

� �eyey ð31Þ

The incompressible condition in Eq. (10) becomes

@u

@rþ

1

r

@v

@yþu

� �¼ 0 ð32Þ

Substituting Eqs. (13) and (31) into (15) leads to theincremental equilibrium equations

L1111@2u

@r2þ

dL1111

drþL1111

rþ

dp

dr

� �@u

@r

þL21211

r2

@2u

@y2þ

1

rL1122þL1221ð Þ

@2v

@r@y

þ1

r2r

dL1122

dr�L2121�L2222

� �@v

@y�@ _p

@r¼ 0 ð33Þ

L1212@2v

@r2þL2222

1

r2

@2v

@y2þ

dL1212

drþL1212

r

� �@v

@r

�dL1221

drþL2121

rþ

dp

dr

� �v

rþ

1

rðL1122þL1221Þ

@2u

@r@y

þ1

r2r

dL1221

drþL2121þL2222þr

dp

dr

� �@u

@y�

1

r

@ _p

@y¼ 0 ð34Þ

Then incremental boundary conditions read

L1111�L1122þar@W

@ar

� �@u

@r� _p ¼ 0 ð35Þ

@v

@rþ

1

r

@u

@y�v

� �¼ 0 ð36Þ

for the traction-free condition at r¼a and

u¼ 0, v¼ 0 ð37Þ

for the fixed displacement condition at r¼b.The differential equation system given by Eqs. (32)–(37)

holds for any incompressible hyperelastic constitutive law. Thecritical condition of surface instability of the tissue is that theequations have a nontrivial solution different from that derived inSection 3.

To determine the critical condition of buckling, we use thefollowing functions to describe the initial folding pattern shownin Fig. 1:

u¼ u0ðrÞcosmy,

v¼ v0ðrÞsinmy,_p ¼ _p0ðrÞcosmy ð38Þ

where m is the circumferential mode-number, u0, v0, and _p0 arefunctions of the variable r. Adopting Eqs. (32) and (38) inconjunction with the specified strain energy function (24), the

equilibrium Eqs. (33) and (34) are recast, after the elimination ofv0 and _p0, as

u0ð4Þ þC1u0

ð3Þ þC2u000þC3uu0þC4u0 ¼ 0 ð39Þ

Correspondingly, the boundary conditions in Eqs. (35)–(37)become

½u0ð3Þ þC5u000þC6uu0þC7u0� r ¼ a ¼ 0,

��½ruu0þðm

2�1Þu0� r ¼ a ¼ 0j ð40Þ

at r¼a, and

u0 r ¼ b ¼ 0, uu0 r ¼ b ¼ 0���� ð41Þ

at r¼b, where a prime denotes the derivative with respect to r andthe coefficients CK (K ¼ 1�7) are given in the Appendix. Thedeterminantal method is applied to find the values of theparameters for which Eqs. (39)–(41) have a nontrivial solution.

5. Results

5.1. Deformation and residual stresses

The growth, either natural or abnormal, of a mucosa constrainedby a muscle layer would elicit inhomogeneous deformation andcreate residual stresses. For convenience, the Cauchy stresses arenormalized as srr ¼ srr=m and syy ¼ syy=m. The relative stretchesand the normalized stresses are shown in Fig. 3, where the inner andouter radii of the mucosa are taken as A¼ 0:6 and B¼ 1,respectively. As can be seen from Fig. 3(a), although the mucosagrows isotropically, the mucosa elongates in the radial direction(l141) but shrinks in the circumferential direction (l2o1). Both l1

and l2 decrease with increase in R. The circumferential stretchreduces to one at R¼ B due to the rigid constraint. Fig. 3(b) clearlydemonstrates that, as R increases, the compressive radial normalstress markedly increases, while the circumferential normal stressdecreases. srr vanishes at R¼ A since the inner boundary is assumedto be traction-free. In the case of isotropic growth, the bigger thegrowth factor, the larger the deformation and the higher the residualstresses. The circumferential normal stress has a larger value in thevicinity of the luminal surface, whereas the radial normal stress ishigher near the mucosa/muscle interface.

In the case of anisotropic growth with g1ag2, some distinctfeatures are observed, as shown in Fig. 4, where we still take A¼ 0:6and B¼ 1. Despite the relative growth volume ratio JG ¼ g1g2 istaken the same value, the circumferential growth ðg1 ¼ 1, g241Þwill engender a higher residual stress level than the radial growthðg141, g2 ¼ 1Þ. When g2 ¼ 1, the normal stresses in radial andcircumferential directions are equal adjacent to the mucosa/muscleinterface for the reason elucidated in Section 3.

5.2. Surface folding of mucosa

When the residual stresses engendered by tissue growth aresufficiently large, the mucosa may buckle into a shape withperiodic wrinkles on its inner surface, as shown in Fig. 2. Thetransformation of circumferentially uniform deformation to sur-face wrinkling will lead to partial release of the elastic strainenergy of the system. Among all possible folding patterns, thesystem will prefer the mode, referred to as the critical mode mc,that minimizes the elastic strain energy. For example, Fig. 5 showsthat in the case of isotropic growth (i.e., g1 ¼ g2 ¼ g) with theradius ratio A/B¼0.6, the critical mode mc of the mucosa is nine.The corresponding growth factor at the occurrence of wrinkling isdeemed as the critical growth factor, gc, analogous to the

Fig. 3. Distributions of (a) the stretches and (b) residual stresses in the mucosa

created by isotropic growth.

Fig. 4. Distribution of the residual stresses created by anisotropic growth.

Fig. 5. Variation of the growth factor g with the mode-number m.

Fig. 6. Dependence of the critical characteristics on the ratio A/B.

Table 1Comparison of the prominent characteristics of mucosal folding between isotropic

and anisotropic growth.

A/B g1 ¼ g2 ¼ g g1 ¼ 1, g2 ¼ g g1 ¼ g, g2 ¼ 1

gc JcG mc gc Jc

G mc gc JcG mc

0.20 1.01400 1.02820 6 1.02783 1.02783 6 1.02857 1.02857 6

0.25 1.02214 1.04477 6 1.04382 1.04382 6 1.04572 1.04572 7

0.30 1.03236 1.06577 6 1.06370 1.06370 6 1.06783 1.06783 7

0.35 1.04485 1.09171 6 1.08766 1.08766 6 1.09574 1.09574 7

0.40 1.05984 1.12326 7 1.11593 1.11593 6 1.13064 1.13064 7

B. Li et al. / Journal of Biomechanics 44 (2011) 182–188186

definition of the critical pressure for the instability of a shellunder an exterior pressure (Timoshenko and Gere, 1961).

For the case of isotropic growth, the critical growth factor andthe critical mode of the mucosa are plotted in Fig. 6 as functions of

the ratio A/B. It is found that the initial thickness of the mucosahas a pronounced influence on the folding pattern. The thinnerthe mucosa, the larger the critical mode mc. It is interesting tonote that the critical growth factor gc increases with the decreasein mucosal thickness. This conclusion is distinct from the case of acylindrical shell under an exterior pressure, in which the criticalload of buckling increases rapidly with the increase in shellthickness. The dependence of the surface folding pattern on thethickness of mucosa is attributed to the constraint effect the outermuscle layer.

The influence of the anisotropic growth of the mucosa on thesurface folding characteristic is clarified in Table 1, in comparisonwith the case of isotropic growth. For simplicity, we consider

B. Li et al. / Journal of Biomechanics 44 (2011) 182–188 187

two special modes of anisotropic growth: (i) g1 ¼ 1 and g2 ¼ g

(circumferential growth), and (ii) g1 ¼ g and g2 ¼ 1 (radialgrowth). Since the circumferential growth creates higher stressthan the radial growth (Fig. 4), the critical relative growth volumeratio Jc

G at the occurrence of folding in the former case is smallerthan that of the latter.

6. Discussion

The growth of a mucosa can not only create inhomogeneousdeformation and residual stresses but also affect its mechanicalproperty and biological functions (Hrousis et al., 2002; Wiggset al., 1997). Though much effort has been devoted to elucidatingthe mechanisms of mucosal wrinkling, the role of growth-inducedresidual stresses in the phenotypic characteristic of foldingpatterns remains elusive. In this paper, therefore, we make anattempt to establish a biomechanical model for investigating thesurface folding of mucosae.

Our analysis reveals that for both the cases of isotropic andanisotropic growth, the maximal circumferential stress occursnear the inner luminal surface, but the maximal radial stress isfound adjacent to the mucosa/muscle interface. It is demonstratedthat when the residual stresses induced by tissue growth aresufficiently high, the mucosa can spontaneously wrinkle. In otherwords, the contraction of the outer smooth muscle or externallyapplied load is not necessary for the occurrence of surfacewrinkling. The folding pattern of a mucosa closely hinges on itsinitial thickness. For a mucosa with a representative thicknessobserved in healthy esophagus, the fold number is approximately6. Our experimental observations show that the fold number ofbovine esophagus is in the range of 5–7, as shown in Fig. 7.Therefore, the theoretical prediction, which does not contain anyfitting parameter, has a good agreement with those in real bovineesophagus. A thinner mucosa is prone to develop more folds.Though surface wrinkling patterns with a larger folding numberrarely occurs in esophageal mucosa, we do observe such patternsin pulmonary airways, which have a much thinner mucosal layer.Fig. 8 gives a few pictures of porcine airways, which has anapproximate value of mc¼30.

Fig. 8. Mucosal folding in porcine airway.

6.1. Model limitations

A limitation of the proposed model is the plane-strainassumption in the deformation analysis. The axial growth mayalter the residual stresses in both the longitudinal and circumfer-ential directions and, thus, exert an influence on the mucosalfolding pattern. Theoretically, both longitudinal and circumfer-ential buckling may happen, leading to more complex three-dimensional surface patterns. However, our observations showthat the surface folds in the inner surfaces of such long organs as

Fig. 7. Mucosal folding i

esophagi and airways have long ridge shapes along their lengthdirection. This implies a relatively weaker influence of the axialgrowth. Therefore, the plane-strain assumption, which greatlysimplifies the mathematical manipulation, appears to be reason-able for most realistic situations.

In recognition of its superior property of elastic deformation,we have modeled the mucosa as a hyperelastic material. Thetheoretical framework in Section 2 works for any hyperelasticconstitutive relation. For illustration, we adopt the Neo-Hookeanconstitutive law in the analysis given in Sections 3–5. It is worthpointing out that this simple quasi-linear material approximationis reasonable for the problem under study yet its validationrequires testing.

Another assumption made in our model is that the mucosa andsubmucosa are regarded as a single ‘‘mucosal layer’’. The multi-layer or functionally graded model may provide a more exactprediction for the folding pattern, especially in the evaluation ofthe stiffening/weakening effects of some diseases. However, asthe first attempt to examine the mucosal folding by directlyincorporating the growth effects, we formulate a generaltheoretical framework in Section 2 that involves the fundamentalfeatures of esophagus. This study may help elucidate themechanism of folding induced by growth from the viewpoint ofbiomechanics. Refinement of the present analysis is of interestand deserves further effort.

6.2. Clinical relevance

Thickening of mucosa is an intrinsic pathogenesis of suchdiseases as inflammation, edema, lymphoma of esophagi, andasthmatic airways (Carroll et al., 2000; Hrousis et al., 2002; Seowet al., 2000; Stiennon, 1995; Wiggs et al., 1997). It has beenrecognized that the thickening of mucosa originates fromabnormal growth. Nevertheless, the direct mechanical effect ofabnormal growth on the folding patterns has not yet beendocumented. In this study, we have elucidated this phenomenonby using the method of surface instability. The abnormal growth

n bovine esophagus.

B. Li et al. / Journal of Biomechanics 44 (2011) 182–188188

of a mucosa alters the residual stress field in the tissue, which, inturn, affects its surface patterns. Recently, Noble et al. (2010)observed the appearance of mucosal folds in a relaxed airwayutilizing a new optical technique, anatomical optical coherencetomography. Their findings of surface folding can be wellexplained by the present biomechanical model.

The residual stresses in living tissues created by growth arephysiologically important in remodeling and maintaining thenormal biological functions of tissues. The presence of residualstresses in a tissue can often be revealed by making cuts andobserving its corresponding shape change (Fung, 1990). However,this method is not suitable for evaluating the residual stressesin vivo. Our theoretical study suggests that one can nondestruc-tively evaluate the residual stress level engendered by growthaccording to the observed folding patterns.

6.3. Significance

The mucosal folding of living tissues is vital to the main-tenance of their normal functions. Understanding the biomecha-nics of this phenomenon can find wide applications in mucosalphysiology and pathology. By considering the geometric andmechanical effects of growth, a general theoretical model isproposed. Despite its simplicity, this model provides substantialinsight into the residual stresses and the folding patternsassociated with the normal and abnormal growth of mucosae.Of importance is to notice that many other biological vessels,e.g., pulmonary airways, arteries, blood vessels in the myocar-dium, eustachian tubes, and gastrointestinal tracts, share similaranatomical structures to esophagi. Therefore, although ourpresent analysis focuses on esophageal mucosa, the biomechani-cal model described here is also applicable to many other softtissues.

Conflict of interest statement

The authors have no financial and personal relationships thatcould inappropriately influence or bias this work.

Acknowledgements

Supports from the National Natural Science Foundation ofChina (Grant nos. 10972121 and 10732050), Tsinghua University(2009THZ01001), and the 973 Program (2010CB631005) areacknowledged.

Appendix A. Supporting information

Supplementary data associated with this article can be foundin the online version at doi:10.1016/j.jbiomech.2010.09.007.

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