Pergamon .I. Biomchanics, Vol.21, No. 4, pp. 455-467,1994 Copyright01994 EllevierScienaLtd PrintedinGnatBritain.Allrightsrescmd 0021-929oD4s6.00+.00 STRESS-DEPENDENTFINITEGROWTHINSOFTELASTIC TISSUES EDWARDK.RODRIGUEZ,ANNE HOGER andANDREW D.MCCULLOCH Institute for Biomedical Engineering, University of California, San Diego,La Jolla, CA 92093-0412, U.S.A. Abe&&-Growthandremodeling in tissues maybe modulatedby mechanical factors such as stress. For example, in cardiac hypertrophy, alterations in wallstress arising from changes in mechanical loading lead tocardiac growthandremodeling. A general continuumformulation for finite volumetric growthinsotI elastic tissues is therefore proposed. The shape change of an unloadedtissue during growth is described by amappinganalogoustothedeformationgradienttensor.Thismappingisdecomposedinto atransformationofthelocalzero-stress reference stateandanaccompanyingelasticdeformationthat ensuresthecompatibilityofthetotalgrowthdeformation.Residualstressarisesfromthiselastic deformation. Hence, acompletekinematicformulation for growthingeneral requires a knowledgeof the constitutive law for stress in the tissue. Since growth may in turn be affected by stress in the tissue, a general formfor thestressdependentgrowthlawisproposedasarelationbetweenthesymmetric growth-rate tensor andthe stress tensor. Withathick-walledhollowcylinder of incompressible, isotropic hyperelastic material asanexample,themechanicsof left ventricular hypertrophy are investigated. Theresults show thattransmurallyuniformpurecircumferential growth,whichmaybesimilartoeccentricventricular hypertrophy, changes the state of residual stress in the heart wall. A model of axially loaded bone is used to test a simple stress-dependent growth law in which growth rate depends on the difference between the stress duetoloadingandapredetermined growthequilibrium stress. INTRODUCIION Growthandremodelingarefundamentalmechanical processesbothinthenormaldevelopmentoftissues andinvariouspathologicalconditions.Mechanical quantitiessuchasthestressandstraininthetissue canmodulateitsgrowth.Forexample,cardiacand vascularhypertrophyarethoughttobedue,atleastin part,toincreasedwallstress(Fung,1990, Grossman, 1980). Itis alsowellknownthatgrowthandremodel- inginlongboneareaffectedbythestateofstressin thelimb(CarterandHayes,1977; Cowin,1983,1986; GuoandCowin,1992). Indeed,considerableprogress hasbeenmadeindevelopingmodelsofgrowthand remodelinginboneandcartilage.Cowin(1985)de- velopedananalyticaldescriptionofthemicrostruc- turalevolutionof fibroustissuesandboneintermsof afabrictensor.Theymadetheoreticalpredictionsfor diaphysealsurfaceremodelingandinternalremodel- ingofbones(CowinandFiroozbakhsh,1981). More recently,FiroozbakhshandAleyaasin(1989)studied theeffectsofstressconcentrationsininternalbone remodeling,andGuoandCowin(1992)studiedsur- facegrowthundertorsionalloading.Hart(1990)de- velopedafiniteelementanalysisforadaptiveelastic materialsandstudiedthetimecourseofstrain-in- ducedsurfaceremodeling.HarriganandHamilton (1992)determinedtheconditionsrequiredforthe Received in Jinal form24 July1993. Addresscorrespondenceto:AndrewD.McCulloch, InstituteforBibmedicEngineering,Universityof California,SanDiego,9500GilmanDrive,LaJolla,CA 92093-0412, U.S.A.Tel. 619-534-2547. Fax.619-534-5722. stabilityof strainenergybasedlawsforboneremodel- ing.Incontrasttotheextensiveworkonbones,very littlehasbeendoneonmodelingtherelationship betweenstress,strainandgrowthinsofttissues,prob- ablybecausetheyusuallyexhibitlargeelasticdefor- mationsunderphysiologicalloadingandbecausesoft tissues,unlikebones,mayberesiduallystressed. Oneof thefirstapplicationsof continuummechan- icstothestudyofgrowthindeformabletissueswas a modelof homogeneousstress-dependentgrowthfor linearlyelasticmaterialsdevelopedbyHsu(1968). Later,acontinuumdescriptionoffinitegrowthkin- ematicswasformulatedbySkalak(1981)andSkalak etal.(1982).Theyexaminedvolumetricgrowth, growthbyaccretiononsurfacesandgrowthfields withdiscontinuities.Skalak(198 1) pointedoutthatif thegrowthstrainisincompatible,forexampleifthe growthof thedifferentcellscomprisingabodyoccurs insuchamannerthatthecontinuityofthebodyis compromised,continuitymaybemaintainedbyan elasticstress.Therefore,residualstressoccursasare- sultoftheelasticdeformationrequiredtokeepall cellscontinuouswitheachotherwithoutintroducing gapsorsuperpositions.Thefundamentalimportance of residualstressinintacttissueswasdemonstratedby Fung(1990) inbloodvessels.Thepresenceof aresid- ualstressfieldcanbeidentifiedif deformationoccurs whencutsarema&intheunloadedtissue.Usingthis experimentalapproach,Fungandcoworkershave showntheexistenceofresidualstressinarteries (ChoungandFung,1986; LiuandFung,1988), veins (Xie etal.,1991), ventricularmyocardium(Omensand Fung,1990) andtrachea(HanandFung,1991). The deformationthatreturnsthecutstateofthetissueto 455 456E. K. RODRIGUEZ et al. theresiduallystressed,intactstatedefinestheresidual strain.Fung(1990) suggeststhatchangesintheresid- ualstrainrevealtheeffects of nonuniformgrowthand remodeling.Hemeasuredchangesinresidualstrainin bloodvesselsbymeasuringtheopeninganglesthat resultedwhenringsdissectedfrombloodvesselswere cutacrossthewallrelievingcircumferentialresidual stress.Remodelingafteri&arena1aorticbandingin theratcorrelatedwithsignificantchangesinresidual strain. Stressinatissuemaynotonlybecausedoraltered bygrowth,butitmayalsoaffectthegrowthitself. Grossman(1980)proposedthatcardiachypertrophy andnormalcardiacgrowthdevelopinresponseto increasedhemodynamicloadingandalteredsys- tolicanddiastolicwallstresses.Hesuggestedthatthe resultingpatternsof hypertrophyreflectthenatureof thestresschanges.Avarietyoflawsforgrowthas afunctionofstress,strainorstrainenergyhavebeen proposedforbone(Cowin,1983).Forsofttissues, Fung(1990) proposedanequationforthemassrateof growthasafunctionofsomesuitablemeasureofthe stress.Inhisformulation,growthorresorptionoccur so thatthestressduetoloadinginthetissuereturnsto oneofthreeequilibriumstates.Aroundoneofthese equilibriumstates,ifloadingstressishigh,then growthoccurstoreducestress.Ifthestressislower thantheequilibriumstate,thenresorptiontakes place.However,if thestressis toohighortoolow,one of theothertwoequilibriumstatesgovernsgrowth.In thiscase,highstressesmayleadtoresorptionas mayoccur,for example,arounda stressconcentration inbone.Similarideashavealsobeenproposedby Pauwels(1980) forbone.However,growthisathree- dimensionalprocesswhoserateis onlyfullydescribed byatensormeasureandnotasinglescalargrowth rate.Sincethestressisalsoatensorinthethree- dimensionalbody,ageneralformulationforstress- dependentgrowthrequiresatensorialconstitutive relation.Theformofthisrelationandtheconditions restrictingithavenotbeenpreviouslydescribed. Ageneralcontinuumformulationforgrowthmust thereforeallowforthepossibilityofresidualstress arisingfromgrowth,andshouldnotrelyontheexist- enceof aglobalzero-stressstate.Moreover,sincethe kinematicsofgrowthcannotalwaysbeseparated fromthestressinthetissue,theconstitutivelawfor stressinthetissuewillgenerallyberequired.The purposeofthispaperistodevelopageneralthree- dimensionaltheoryforthecontinuummechanicsof finitevolumetricgrowthinsoftelastictissues.This theorymaybeusefulforstudyingtherelationship betweenstressandnormaltissuegrowthorpatho- logicgrowthsuchascardiachypertrophy. METHODS Inthissection,thekinematicsoffinitegrowthare introducedusingthenotationoffinitedeformations. Ingeneral,theshapechangethatoccursduringthe growthof anunloadedbodyisduetotwoprocesses: (1) materialmaybeaddedorremoved,changingthe localstress-freereferencestateofthetissue;(2)an elasticdeformationmayberequiredtoaccommodate thischangeintissueconfigurationandvolumein ordertomakethetotalgrowthdeformationcompat- ible(i.e. sothatthematerialcanundergothegrowth withoutintroducingdiscontinuitiesin- thebody).Re- sidualstressarisesfromtheelasticpartofthetotal deformation.Therefore,inourdescriptionofgrowth kinematics,thetotalshapechangeduringgrowthis decomposedintothesetwoparts:thechangeintissue stress-freereferencestateandtheelasticdeformation. Thedependenceofthegrowthratetensoronthe stresstensorisalsoexaminedinthissectionand a generalconstitutivelawforstress-dependentgrowth isformulated.Finally,wegivetwoexamplesof finite growthinthree-dimensionalelasticbodieswithpos- sibleapplicationstoventricularhypertrophyandcon- nectivetissue. Toanalyzethekinematicsof growth,we definethe volumetricgrowthof a soft elastictissueas thechange inthelocalzero-stressstateofthebodywithout requiringthatthereexistsacorrespondingglobal zero-stressstate.LetB(t,)denotea stress-freebodyat timeto. Wewillseelaterthattheanalysisgeneralizes directlyforthecasewhenB(t,)isnotstress-freewhen itis unloaded.Thebodymaychangeinshape,density andmaterialpropertieswithtimeasitgrowsand remodelsintheabsenceofanyexternalloads.Letit growintoanewstress-freestate,B(t,). Themassrate ofgrowthperunittissuevolumeVatapointis definedby Ifp,themassdensity,isconstantwithtimethen dV tflpdt. Conservationofmass(Spencer,1980) requiresthat ti=$+div(pv), (3) wherevisthegrowthvelocityvector(Skalaketal., 1982),whichrepresentstherateanddirectionof motionateachpointinthetissueduringgrowth. Skalaksuggestedthatvelocityfieldsareusefulfor describinggrowthbecausetheyembodythecontinu- ousgrowthrateratherthandiscretechanges.Cowin (1983), CowinandFiroozbakhsh(1981)andLuoetal. (1991)usedgrowthvelocitiestodescribesurfacere- modelinginboneasafunctionofstrain.Ifdensityis constantwithtimeandpositionthen ti=pdivv. (4) Densityisconsideredheretobeconstantintimeand positionbecauseweassumethatmaterialproperties Growthinsoft tissues457 donotchangewithgrowthalone.Thisassumption wasalsoadoptedbyHsu(1968)andothers(Cowin andFiroozbakhsh,1981; GuoandCowin,1992; Mat- theckandHuber-Betzer,1991).Weconsiderthe growthtoconsistoftheadditionorremovalofthe sametissuematerial.Hence,fromequations(1)and (4), therateofunittissuevolumechangeis dV dt=divv=trD,, whereD,is therateof growthtensor,whichis analog- oustotherateofdeformationtensorincontinuum mechanics(Spencer,1980); itisthesymmetricpartof thegradientofthegrowthvelocityfield.Thetraceof D,istherateofvolumetricgrowthordilation.The tensorD,hastheadvantagethatgrowthratefunc- tionscanbedescribedwithoutaninitialreference configuration,whichmaybedifficulttodefineexperi- mentally.Alternatively,thegrowthratemaybede- scribedbyanequivalentLagrangianmeasure,therate ofgrowthstretchtensorO,,whichisreferredto a definedreferencestate.ThegrowthstretchtensorU, canbeobtainedsimplybyintegrating0,intime.The relationbetweentheratetensorsD,and0,is givenin AppendixB. ThegrowthstretchtensorU,isrelatedtothe growthdeformationgradienttensorF,bytheright polardecomposition* Fs=RsUs,(6) whereR,istherotationpartofthegrowthdeforma- tion.ThegrowthdeformationgradientF,isanalog- oustothedeformationgradienttensorincontinuum mechanics(Spencer,1980). Itscomponentsaregiven inAppendixA.Toformulateaproblem,0,maybe specifiedandintegratedintimetoobtainUS. Itwillbe seenbelowthatR,maybechosentoequaltheident- itytensorsothatU,=F,withoutlossofgenerality. LettheCauchystresstensorTinthe_originalelastic bodyB(t,)be givenbythqfunctionT: T=?(C), (7) whereC =FT FistherightCauchyereendeforma- tiontensorreferredtomaterialcoordinatesinthe originalbody.Followingvolumetricgrowthwithno changeintheelasticresponseofthetissue,thestress inthegrownbodyisgivenintermsofdeformation componentsreferredtotheoriginalmaterialcoordin- atesby T=$(F;TCF;). (8) Theprecedingdescriptionestablishesthekin- ematicsof thelocalchangeinthezero-stressreference state.Wenowconsiderhowgrowthaffects thestateof stressinthebody.Evenintheabsenceofexternal loads,thestateofstressinagrownbodymaybe *Right as opposedtoleft because it leads naturally to the Lagrangian formulation. alteredbytheintroductionofresidualstressduring growth.Thisarisesbecausethereisnorequirement thatthegrowthdeformationgradientF,corresponds toacompatibledisplacementfield,i.e.thechangein localzero-stressstateduringgrowthneednotbe continuousfrompointtopoint.Forexample,acell maygrowindependentlyof itsneighbors.Laterwe see thattherearealsosmoothgrowthfieldsthatarenot compatible.Hence,ifwerequirethatthetissuere- mainsintactandcontinuousasitgrows,theobserved growthdeformationintheabsence.ofexternalloads maynotbethesameasF,definedabove.As shownin Fig.1,thedeformationdefinedbyF,describesthe growthfromtheoriginalzero-stressreferencestate B(t,)toanewlocallystress-freestateB(t,),which maydiffer fromtheobservedintactgrownstateB(t,). Thepartofthetotalgrowthdeformationthatmaps B(t,)totheunloadedstateB(t,)isanelasticdefor- mationF,thatensuresthecontinuityofthebody. Fromtheprecedingdiscussion,theoverallob- servedgrowthdeformationF,,thatmapsB(t,)to B(t,)isthengivenby Fes=F,F,. (9) Itis nowevidentwhyR,inequation(6) canbe chosen tobetheidentitywithoutlossof generality;anyrota- tionalpartofF,canbeincorporatedintotheelastic partofthetotalgrowthdeformation. SinceB(t,)isthenewzero-stressreferenceconfig- uration,thestateofstressinB(t,)mustbe T=?(F;F,). (10) If thebodyis inequilibriumintheabsenceof external loads,thenihestressgivenbyequation(10) is aresid- ualstressT.Therefore,incompatiblegrowthfields B(t,)BO,1 T=O 0 F, T-t \ -0 f WJ T=O Fig.1.Three states in finite growth of a stress-free tissue: (a) theoriginalzero-stress reference stateB(t,);(b) thegrown zero-stress B(t,k(c) the observedjntactand unloadedgrown stateB(t,)withresidual stress T.Thegrowthdeformation gradient F,mapsB(t,)intoB(tl).However, F,maynotbe compatible so B(t,) is shownas a collection of discontinuous andsuperimposedmaterial.Sothattheoverallgrowth deformationiscompatible,theelasticdeformationF.is required tomapB(t,)intotheointact grownstateB(t,).F. gives rise to the residual stress T in B(t,). The overall growth deformationisthenthecompositiongiven byequation(9). 458E. K.RODRIGUEZetal. leadtoresidualstressthatarisesasadirectresultof theelasticdeformationsrequiredtomaintaincontinu- ityofthebody.Clearly,theanalysisoffinitegrowth kinematicsinanelastictissuewill,ingeneral,require thattheconstitutivelawforstressinthetissuebe known.Whengrowthoccursunderconditionsofex- ternalloading,F,alsoincludestheelasticdeforma- tionduetotheload.Whentheoriginalundeformed referencestateisresiduallystressed,theaboveana- lysisgeneralizesdirectlyprovidedthatthemapping thatdescribesthelocalstress-freestateateachpoint inthereferencebodyisknown,eventhoughthat mappingwillnotbecompatible.Theproblemthen becomesthesameasthatofasubsequentgrowth deformationreferredtoapreviouslygrownstate.If thislocalstress-freeconfigurationofthereference residu;llystressedbodyis notknown,buttheresidual stressTis, thentheanalysiswillrequireaformforthe constitutivelawthatisvalidforaresidualstressfield. Forthecaseoffiniteelasticdeformations,avalid generalformisnotpresentlyknown.However,Hoger (1993)haspresentedaconstitutivelawforresidually stressedelasticbodiesthatundergosmallstrainsand arbitraryrotations. Ingeneral,growthmaydependonthestateof stress inthetissueateachpoint.Todescribestress-depend- entgrowth,werequireaconstitutivelawforthe mathematicalformof thegrowthtensorasafunction ofthestresstensor.SincethegrowthdeformationF, doesnotincludearotation,thegrowthlawmaybe formulatedbywritingthegrowthstretchU,asa func- tionoftheCauchystressT: U,=U&T),(11) where6,isatensor-valuedfunctionthatmapssym- metrictensorsintosymmetrictensors.Inmostcases detU,#1. Another,perhapsmorephysiological,approachto theformulationof stress-dependentgrowthfunctions istowritethegrowthrateasafunctionofthestress. Thiswouldadmitagrowthlawofthetypedescribed byFung(1990). Recallthattherateof volumechange atapointisgivenbythetraceofthegrowthrate tensorD,[equation(5)].However,acompletede- scriptionofthegrowthraterequiresallthecompo- nentsofD,.Thus,thegrowthfunctionmaytakethe form D,=&(T),(12a) whereD8 isasymmetricfunctionthatmapssymmet- rictensorsintosymmetrictensors.SinceU#andD, arerelatedbyequations(Bl)and(B2), equation(12a) canbewrittenas U,=&(T),(12b) whereU,isalsoasymmetricvaluedfunctionof sym- metrictensors. AformulationintermsofU,iseasiertointegrate thanoneintermsof D,sinceintegratingU,provides U,(=F,).ThegrowthstretchtensorU,isinvertible andpositive-definite. If, assuggestedbyFung,thereexistsa stressstate! correspondingtogrowthequilibrium,thenitiswell motivatedtodefinethegrowthratetensortobezero whenthestressisatthegrowthequilibriumstate,i.e. (13a) and U,=Q(T-R+R~). (I3b) Intheseequatio:s,therotationRisrequiredtoen- surethatTandTaremeasuredinthesameframeof reference,asrequiredbyobserverindependence(Gur- tin,1981). Twomodelswereusedtostudyindependentlyhow growthleadstoresidualstressandhowstressmay determinethegrowthpatternof atissue.Forthefirst case,agrowthdisplacementfieldisspecifiedinan unloadedcylindricaltubeandtheresidualstressfields thatresultfromdifferentgrowthfieldsareexamined. Inthesecondexample,aloadedspecimenisusedto studystress-dependentgrowthusingasimplelaw. Thebasicequationsofthisboundaryvalueproblem aregiveninAppendixC. Residualstressarisingfromgrowth Ahollowcylindricaltubemodelcomposedofan incompressibleisotropicelasticmaterialisusedto illustratehowcircumferentialgrowthgivesriseto a transmuraldistributionof residualstressthatwould causethecylindertochangeshapewhencut.Circum- ferentialgrowthservesasafirstapproximationfor eccentricventricularhypertrophy,aclinicaltermthat describesventricularenlargementinresponsetochro- nicvolumeoverload(elevatedfillingpressure).The residualstresspresentinthecylinderaftergrowthis calculatedassumingthatgrowthstrainsgenerate stressessimilartothoseof loadingsothataconstitut- iveequationforanisotropicmaterialcanbeused.To obtainthegrowthdeformationF,weprescribethe followinggrowthdisplacementfield: p=R,~P=&(R)O, [=Z. (14) ApointPintheoriginalstress-freeconfiguration B(t,)hasthecoordinates(R,0,Z).F,mapstheorig- inalstateB(t,-,) intothenewlocallystress-freestate B(t,),inwhichPhascoordinates(p,cp, 6).Theterm KB(R)isacircumferentialgrowthstretchratiothat dependsontheradius,butforsimplicitywe usea con- stantwhichissufficienttointroduceanonuniform residualstress.A valueK,>1.0 simulatesgrowth,and whenK,c1.0resorptionoccurs.Thisdisplacement fieldis incompatiblesincerp is notuniqueat0=0,2x. TheincompatibilitycanbeseeninFig.2,wherethe locallystress-freestateB(c,)isshownwhenKg=0.9 andKg=1.1. WhenKB=0.9thereisdiscontinuityin Growthinsofttissues459 A)KC) = 0.9 lacal ly dress-free state beforecircumferentialresorption Gr o wn unloadedstate with residualstress Loeallystress-freestate aftercircumferentialresorption Locallystress-freestate beforecircumferentialgrowth Grownunloadedstate with residualstress Local lystress-freestate aftercircumferentialgrowth Fig. 2.Cylindrical modelsof theleftventricleinthe zero-stressstateafteruniformcircumferentialgrowth: (A) resorption(K,=O.9k(B) growth($71.1).Theoriginalzero-stressis a hollowcylinder(B(t,)).Notethat thegrownstateB(t,)is stress-freeaaddrseontinuous inbothcases.Inthecaseof resorption,toachievean intactgrownstate,anelasticdeformationthattakesthegrownopenconfigurationintoa closedcylinderis required.Thiselasticdeformation,inturn,willgiverisetoresidualstress.Afteruniformcircumferential resorptiontheB(t,)statewillappearasshowninpanelB.Thisstateisstress-freebutitshows asuperpositionofmaterial.Toachieveanintactconfiguration,anelasticdeformationthatopensthe cylindertopreventanysuperpositionisrequired.Asbefore,thiselasticdeformationwillalsogiveriseto residualstressinthefinalgrownstate. thecylinderwhichisnotpermissibleifthetotal growthdeformationistobecompatible.When KI=1.1thereisasuperpositionofmaterialwhich shouldnotoccurinacompatibledeformation.The displacementfieldof equation(14) definesthefollow- inggrowthdeformationgradient: 10o- F,=0;Ko0 . (1% _o0l_ Inorderthattheoverallgrowthdeformationfieldis compatible,anadditionalelasticdeformationisre- quiredthatwillbeusedintheconstitutiveequation forthematerialtodeterminetheresidualstressthat mustsatisfyequilibriumandthezerotractionbound- arycondition.Adisplacementoftheform r=r(p),f9=t&)p,z=eZ,(16) whichmapsthestateB(t,)tothefinalgrownstateof theintactbodyB(t,)inwhichPhascoordinates (r,8, z),givesrisetotheelasticdeformation c9 W)oo dp F,=0 $II,0. L 0Oe 1 (17) 460 E. K.RODRIGUEZetal. Anappropriatechoice.ofq@(p) allowstheoverall growthdeformationF,F,tobecompatiblebyrestor- ingcontinuityofdisplacementsat0=0,2n.Forthe casewhenKB isaconstant,thesimplestchoiceis qe=l/KB.Theelasticdeformationalsoallowsfor someaxialchangeinlengthviatheparametera. Theincompressibilityconstraintisonlyappliedto F,sothatthethirdprincipalinvariantofCisunity, i.e. arr () 2. I,= ap-Ette=A(18) whichcanbeintegratedtoobtainanexpressionfor thenetgrownradius,r. Fromequation(7) withC=F&Fep,thestressbe- comesafunctionofF,.Lagrangianstraincompo- nentsreferredtothecoordinatesinstateB(t,)cal- culatedfromE =3(FfF,-I)aregivenby Epp=;[(y-l], E~q=&---l]? E,, = 4 (E-1). (19) Forthisexample,weadopttheformforthe stress-strainrelationshipusedbyGuccioneetal. (1991)forrestingpassiveventricularmyocardium. Theyusedthestrainenergyfunctionsuggestedby ChoungandFung(1986): W=i(eQ-1), (20) wherecisaconstantandQ isafunctionofthethree principalstraincomponentsthatdefinesthematerial symmetryofthetissueunderconsideration.Foran isotropicmaterial,Qis Q=2h(b+&p+~%(21) whereblisaconstant.Thestressisthenobtained from wherep is thehydrostaticpressurethatentersintothe constitutiveequationasaLagrangemultiplier. Theequilibriumequationsare dT,rr,-r,=, dr+ 1 r dT*2T, z+-=o, (24) r dT,zT,, x+7=0. However,sincethezerotractionboundarycondition attheinnerandouterwallswasused,onlyequation (23) hastobesolved.Integratingthisequationgives T,,= s T~e+Tzr -dr+T,,l,=,,,(26) 12 r whereT,atr=rzistheradialstressattheouter grownwallandwasspecifiedaszero.Theinternal pressureisequalto-T,,atr=rl.Themodelis solvedbyspecifyingthegrowninnerradiusr,and solvingforthefinalvalueofinnerradiusthatgives zerotransmuralpressure.Transmuralresidualstress distributionsarethencalculatedforthegrownstate. Stress-dependentgrowth Inthisexample,weanalyzestress-dependent growthintheabsenceof residualstressbyconsidering asimplehomogeneousstress-field.Sincereliabledata onstress-dependentsofttissuegrowtharenotavail- able,wesolvearectangularmodelofbonetissue subjectedtocompressivestressandsmallstrainsbut withoutassuminginfinitesimallinearelasticity.The blockgrowsinexternaldimensionsasalinearfunc- tionofthedifferencebetweentheloadingstressand a preestablishedno-growthequilibriumstateof stress. Weassumeelasticpropertiesfortissuefromthe diaphysialregionofhumanfemurwithaYoungs modulusalongthelongaxis(z-axis)estimatedtobe 18.4 GPa(Cowin,1983).Theblockiscompressed alongitslongaxiswiththetwootherdimensions correspondingtotheradial(x-axis)andcircumferen- tial(y-axis)directions(Fig.3).Sincetheproblemis homogeneous,thereis noresidualstress(Hoger,1986; Hsu,1968). TheoriginalspecimenB(t,)wassubjected tocompressiveloadingthatgaverisetothedeformed statecharacterizedbytheelasticdeformationgradi- entF..Ifthecorrespondingaxialstressisdifferent fromthegrowthequilibriumstress,thenthetissue growsorresorbsalongitsxandydimensionsuntil equilibriumisrestored. Theelasticdeformationgradientthatarisesfrom compressiveloadingisgivenby ;;I F,=0 J;i; (27) 001s whereAs isthestretchratio(511. Spencer,A.J.M.(1980)ContinuumMechanics.Longman, London. Stalsberg,H.(1970)Mechanismofdextralloopingofthe embryonicheart.Am. J.Cardiol.25,265-271. Taber,L.A.,Hu,N.,Pexieder,T.,Clark,E.B. andKeller, B.B.(1993) Residualstrainintheventricleofthestage 16-24chickembryo.Circ.Res.72,455-462. Takamizawa,K.andHayashi,K.(1987) Strainenergyden- sityfunctionanduniformstrainhypothesisforarterial mechanics.J.BiomechanicsM,7-17. Terracio,L.,Tingstorm,A.,Peters,W.H.andBorg,T.K. (1990) Apotentialroleformechanicalstimulationincar- diacdevelopment.Ann.N.Y.Acad.Sci.S&48-60. Xie, J.P.,Liu,S. Q.,Yang,R. F.andFung,Y. C. (1991) The zero-stressstateofratveinsandvenacava.ASMEJ. biomech.Engng113,36,41 APPENDIXB RELATIONSBETWEENTHERATEOFDEFORMATION TENSORSD,AND0, TheratetensorsD,and0,arerelatedby DI=f(ti,U;+U,-0,). TheinverseisgivenbyHogerandCarlson(1984): (Bl) . f_J,-.!..- 1112-13 {-U:D,U:+I~(U:D,U,+U,D,U:) +(I:+I,)U,D,U,+Z(U:D,+D,U:) -I,(D,U,-U,D,)+II~~DIJ,(B2) whereII,I,andI,aretheprincipalinvariantsofU,. APPENDIXA APPENDIXC ConsiderabodyB withthethreestatesof growthdefined FORMULATIONOFABOUNDARYVALUEPROBLEM byB(t,), B(t,) andB(t,).A pointPinB(t,) hascoordinates {Xx},inB(t,)ithascoordinates{x.}andinB(t,)ithas Tousethepresentgrowthformulationinaboundary coordinates{xi}.Wecannowexplicitlywritethegrowth valueproblem,weassumethatB(t,)isknownandthe deformationgradientsdefinedinthepresentformulationas materialpropertiescanbeexperimentallydetermined. -AformforF.canthenbenostulated.anditmavbestress- CF,l=$=F,,R, R axiCR1=~=Ksn. II axi CF,,l=,=Fe,,. AR (Al)dependents&estressesa;ebelievedtoaffect&modeling andgrowthinsofttissues.InthiscaseF,= F,(T)wherethe dependence.ont,, is implicitandthestressinthisexpression 642) isrelatedtotheloadingthatthebodyexperiences.The grownstateB(t,) anditsresidualstressesmaynotbe known WV butmaybe. obtainablebyasglutionofaboundaryvalue problemfortheresidualstressT. Thisboundaryvalueprob- lemtakestheform Usingindicialnotationwe canwritethefinalcompositionof F,andF,givenbyequation(9) as div+=O. F %R =F,I_F,*R. (A4) Thestresstensorofequation(10) canbewrittenas Sincefromequation(lo),i=T(FzF,), tion(9) , (A5) ;= *(F;T F& F,,F; ) since (C2) (C3) axR axiax, axs F,=F,,F;. WI HereT=&)is subjecttothetraction-freeboundarycondi- Thegrowthratesgivenbyequations(12a), (12b), (13a) andtion (13b) are,respectively,giveninindicialnotationby n (Cl) thenfromequa- (A7) ?n=O0ndB. (C4)