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Title Method for the unique identification of hyperelastic material properties using full-field measures. Application to the passive myocardium material response.

Permalink https://escholarship.org/uc/item/1vs1s3jk

Journal International journal for numerical methods in biomedical engineering, 33(11)

ISSN 2040-7939

Authors Perotti, Luigi E Ponnaluri, Aditya VS Krishnamoorthi, Shankarjee et al.

Publication Date 2017-11-01

DOI 10.1002/cnm.2866

Peer reviewed

eScholarship.org Powered by the California Digital Library University of California eevd... . . Received an with show stiffness and myocardial point identifying material to the identification. application at property its data unique experimental admit myocardium passive not using do reliability approach that the our evaluate laws of we material Lastly, analysis for the even is properties, approach material our identified of the outcome of optimal An to one data. provide strategy experimental and available numerical properties the material new the of a in description linear is are framework that laws our uniquely energy yielding of material convex, polyconvex component formulate is important function objective An error outline properties. force We material least-square catheterization. identified the through which e.g., under measurements, conditions pressure of set from a computed imaging be resonance magnetic may using forces clinically acquired the model. be a while can in data forces external displacement and the internal applications, balanced vivo to in lead the For of that interior parameters the stiffness within seek the also and and assume — boundaries we interest the identified of i.e., on body both uniquely — known data is to forces displacement applied leads estimating field the to full that to due with framework field work displacement literature a to the approach formulating to an by in design We poised work methods properties. this is material several extend currently tissues we are biological and There stiffness of material tool. stiffness) diagnostic (e.g., powerful properties a material become the of measurement Quantitative h etventricle. left the ug .Perotti E. Luigi ril sIt .Nmr eh imd nn. 086 o:10.1002/cnm.2866 doi: this e02866. cite Engng., Please Biomed. Record. Meth. of Numer. Version J. the Int. and which as version process, article this between proofreading differences and not to has pagination lead but typesetting, review may peer copyediting, full the undergone and through publication been for accepted been has article This 3 nttt fMcaisadSelSrcue,T rse,Gray n rse etro opttoa Material Computational of Center Dresden and Germany, Dresden, TU Structures, Shell and Mechanics of Institute 1 rprisuigfl edmaue.Apiaint h passive the to Application measures. field full using properties 1 eateto ailgclSine n eateto iegneig nvriyo aiona o Angeles Los California, of University Bioengineering, of Department and Sciences Radiological of Department orsodnet:Dprmn fRdooia cecs CA 04 eCneAeu,LsAgls A90095. CA Angeles, Los Avenue, Conte Le 10945 UCLA, Sciences, Radiological of Department to: Correspondence ehdfrteuiu dnicto fhprlsi material hyperelastic of identification unique the for Method 2 Accepted Article Angeles Los California, of University Engineering, Aerospace and Mechanical of Department 1 ∗ dtaV Ponnaluri V. Aditya , ycrimmtra response material myocardium ailB Ennis B. Daniel hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This SUMMARY 2 hnajeKrishnamoorthi Shankarjee , Science 1 ila .Klug S. William , nsilico in xeietmdln h asv ligof filling passive the modeling experiment 2 2 ailBalzani Daniel , 3 , hr h isepesr emcno emaue ietytruhiaigtcnqe [ techniques imaging through directly measured be biomechanics, cannot in implement term to pressure difficult is tissue it and the problems where ideal of set specific a formulation to original applicable This only discretization. is difference finite the of context the in approach this proposed methods on focus we Here [ literature. measures the full-field to in applied presented and developed been have methods iterative heart. functioning the the as study such organs and diseased models and biomechanical healthy the construct of define to input which critical properties, a material are identified stiffness, cogent the a material of addition, part an In is strategy. identification as its diagnostic used and emerging therapy, be to and response may and stiffness disease tissue of presence the the Therefore, of indicator counterparts. tissue healthy their than stiffness -al [email protected] E-mail: in mm [ 2x2 e.g., (see, (e.g., wall locations myocardial spaced the homogeneously inside resolution) of plane hundreds many filling at the For measured to interior. due be its field in can displacement also pressure heart’s and the interest (MRI), of imaging domain resonance the magnetic of using boundaries example, the or surface the on measured are progression. disease with associated [ are tumors tissues cancerous examples, biological numerous Among of stiffness the in changes Significant material convexity; myocardium; passive identification; unique problems; Inverse WORDS: KEY 2 emnRsac onain xelneInitiative Excellence Foundation, Research German HL78931 P01 number: contract/grant Health; of Institutes National 14POST19890027 number: contract/grant Association; Heart American et Sumi al., et Skovoroda to modulus Similar the boundaries. but the (inclusions), of stiffness parts low near accurate and not high was of reconstruction regions between distinguish They elasticity. to of able equations Navier’s were the in unknowns the [ from pressure al. undetermined et the removed Skovoroda problem, this circumvent to order [ Yagle and Raghavan for. solved and be properties to material unknowns the the as of treated terms are in which rewritten pressure, the are problem forward the of equations discretized ie h motneo hrceiigtemtra epneo ilgcltsu,mn ietand direct many tissue, biological of response material the characterizing of importance the Given nas-ald“iet praht aeilprmtrietfiainuigfl ldmaue,the measures, filed full using identification parameter material to approach “direct” so-called a In 0

Accepted Article modeling 5 ,weeete tan rdslcmnsdet nw ple loads applied known to due displacements or strains either where ], .INTRODUCTION 1. .E EOT TAL. ET PEROTTI E. L. hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This 1 , 2 n boi ycrim[ myocardium fibrotic and ] 11 6 , sue aeilicmrsiiiyand incompressibility material assumed ] 7 , 8 ]) 3 , 4 aeadifferent a have ] 9 aefirst have ] 10 .In ]. os n etre esrmns aejee l [ al. et Banerjee measurements. perturbed and noisy [ method (FEMU) iterative Updating used Model commonly Element Finite A the displacements. is method) method element finite the with equations equilibrium biomechanics. in problems boundaries general specific to of apply set easily idealized not an does to values and restricted — further absolute is not assumption — This relative moduli. to elastic leads the strain and for boundaries 1D) domain’s the not along (but constant be 2D [ to al. moduli using et modulus Barbone by shear indicated material as Nevertheless, the measurements. reconstruct successfully to able were eann ftemnsrp,w ee oorfaeokb h he ntaso t ai components: basic its of initials three the by EMS. framework our to refer we manuscript, the of remaining measures. experimental noisy approximately from properties that material demonstrated the have identify to and sufficient term are penalty iterations ten a on scheme continuation a through term the the modify [ to used al. typically et is Cottin matrix weighting Furthermore, a function. method, objective FEMU the (FEMU- In displacements (FEMU-F). computed forces numerically or U) and measured between difference squares least the the as computed usually function, objective non-linear a L of optimization the requires approach [ al. n,i u prah ed ouiu aeilpoete.I otatwt te ietmtos our methods, direct efficiently other solved with contrast be In can properties. equations material unique of to system leads approach, This our properties. in and, material unknown the of terms in framework a S propose 3) we and objectives, these E 1) achieve of: to consisting biaxial order simplified problems, or In kinematics kinematics scenarios. finite linearized to 3D loading/plane-strain confined general been to have studies applicable previous our method several Additionally, a whereas identified algorithm. formulate solution their the to that of is fine-tuning so objective or second properties guesses initial stiffness on material depend not mechanical do values uniquely identify to method a [ applied stresses between internal difference and the loads the forces, traction minimizes computed approach and (CEGM) applied Method between Gap Equations difference properties. Constitutive the material minimizes the of method terms FEMU-F in the system linear While a yields method FEMU-F the properties, material 2 eodapoc o ovn nes rbesi h trtv prah e .. [ e.g., see approach, iterative the is problems inverse solving for approach second A u prahsae h tegho h ietmtosadrslsi iersse fequations of system linear a in results and methods direct the of strength the shares approach Our construct to is objective first our above, presented approaches iterative and direct from Starting L ro ombteneprmnal esrdadnmrclycmue eg,sligthe solving (e.g., computed numerically and measured experimentally between norm error 12 2 nltclysbttt h neemndpesr emwt ennra tes They stress. normal mean a with term pressure undetermined the substitute analytically ] ipaeeterrnr saddt h betv ucint dniyeatcmdl with moduli elastic identify to function objective the to added is norm error displacement aiiyaduiuns nlsso h dnie aeilpoete.Frrfrnei the in reference For properties. material identified the of analysis uniqueness and tability Accepted Article MEASURES FIELD FULL USING PROPERTIES MATERIAL OF IDENTIFICATION UNIQUE ulbimbsdojciefnto;2 M 2) function; objective based quilibrium 19 .I oie eso fteCG ehd [ method, CEGM the of version modified a In ]. hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This 16 hwta ftemtra a slna nthe in linear is law material the if that show ] 21 oto h otiuinfo h additional the from contribution the control ] 13 tra nryfnto optimization; function energy aterial ,ti prahrqie h elastic the requires approach this ], 15 , 16 , 17 , 18 .FM minimizes FEMU ]. 20 14 , 21 .This ]. , 22 3 ] h S n sascae ihtesgsadsmtm fhatfiue(.. ype,ftge xrieitlrne even intolerance) exercise fatigue, dyspnea, (e.g., failure ( heart normal of is symptoms fraction and ejection signs when the with associated is and US, the (HFpEF) fraction as ejection such preserved conditions with pathological failure understanding heart to critical is which filling, during passive response the heart’s identify the to it (Section apply heart we the tool, of diagnostic properties a material as application potential its show and approach [ guaranteed are problems value and stability boundary material the that of so minimizers polyconvex of be to existence law the energy material the require we formulation this (Section forces internal and (external) applied equations. [ of al. system et linear Cottin a by yield observed the will also in if as linear case, is method this density In energy FEMU-F material properties. chosen the material the the and with identity recovered the to is equal is approach matrix weighting our FEMU-F the in particular, In formulate stresses. we or forces function computed objective and measured between difference the minimizing of hogottemnsrp,w osdrtegnrlstigo body a of setting general the Γ consider we manuscript, the Throughout have treatment [ and difficult diagnosis been whose HFpEF, far of so progression the monitor and onset the detect to serve (Section properties the material analyze identified to the us of allows uniqueness laws energy and material stability of class and function objective the of definition chosen (Section data experimental available the on based in laws consists energy approach material formulate our to of strategy component a second The properties. material identified the guarantee not of do uniqueness and displacements measured and computed between difference the minimize which require not does and elasticity assumptions. finite of settings general the in applicable is approach 4 easm httebd ssbetdt oyforces body to subjected Γ is body the that assume We h rmwr ffiieeatct n,i re opeev aeilfaeidfeec,tematerial the indifference, frame material preserve to order in and, elasticity finite of framework the N N e oorapoc stedfiiino h betv ucinbsdo h ifrnebetween difference the on based function objective the of definition the is approach our to Key objective the schemes iterative several with shares approach our non-iterative, although Further, n ipaeetbudr condition boundary displacement and , ∪ 2 FE sadblttn n rwn elhpolm[ problem health growing and debilitating a is HFpEF Γ D .FRUAINO NEULBIMBSDOJCIEFUNCTION OBJECTIVE BASED EQUILIBRIUM AN OF FORMULATION 2. , Γ N ∩ Γ

AcceptedD Article = 29 {} ]. where , > 50% Γ ). N steNuanbudr and boundary Neumann the is .E EOT TAL. ET PEROTTI E. L. 4 .Tepsiemoada aeilpoete characterize properties material myocardial passive The ). hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This u ¯ on 2 .Ti si otatwt te omnapproaches, common other with contrast in is This ). Γ 27 D , oevr eew omlt h rbe in problem the formulate we here Moreover, . 28 2 nqeyietfidmtra rprismay properties material identified Uniquely . fetn oeta 0 fhatfiueptet in patients failure heart of 50% than more affecting ] B in Ω rcinbudr condition boundary traction , Γ 3 16 .Fnly nodrt etour test to order in Finally, ). D ,teotmzto problem optimization the ], steDrcltboundary. Dirichlet the is Ω 23 ihboundary with , 24 dhoc ad , 25 , 26 physical .The ]. 3 T .In ). = Γ on iceie form discretized ( problem equilibrium of form weak the as setting continuum the in problem equilibrium hyperelastic the summarize We clarity. for needed field properties displacement the of density function energy a material The configuration. reference the from measured etruigadul usrp,eg,tecomponent the e.g., where subscript, double component the a indicate using also We vector indices. repeated over summation indicate node to with notation associated function shape element finite the is where discretize to order In chosen. are methods numerical ( eqn. other if obtained is derivation equivalent an gradient. deformation and displacement ( the of conditions variations boundary the account into taking as energy potential where mapping, body the of behavior qiiru rbe ( problem equilibrium tensor, deformation Green where eoecniun,w eettefiieeeetmto FM odsrtz h ekfr fthe of form weak the discretize to (FEM) method element finite the select we continuing, Before 2 D D ,w prxmt h otnu ipaeetfil as field displacement continuum the approximate we ), P NQEIETFCTO FMTRA RPRISUIGFL IL MEASURES FIELD FULL USING PROPERTIES MATERIAL OF IDENTIFICATION UNIQUE a ai stefis il-ichf testno and tensor stress Piola-Kirchhoff first the is α stedslcmn ttelocation the at displacement the is = ntefloig ewl xlctyidct h eednyon dependency the indicate explicitly will we following, the In . u i ∫ ( x Ω Accepted Article X a P ) iJ yisrigen ( eqn. inserting By . eoe h oyrfrnecngrto and configuration reference body the denotes Ω 2 δD ∫ .W ou nti ueia ehdi h eane fti ok but work, this of remainder the in method numerical this on focus We ). sdsrbdb neeg density energy an by described is Ω ai F P N = : a,J δ ∇ F · ∇ d X d V V φ P − stedfraingain and gradient deformation the is − + u P ∫ u ∫ N ∂ B P through ˆ 3 u ∂ ( 1 hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This Ω noen ( eqn. into ) x Ω a eotie ntrso h rtvraino the of variation first the of terms in obtained be can ) = = = = T = ) T i x u T 0 δD ¯ ∂W · a 1b D δ ffiieeeetnode element finite of ai u φ ∂ a n ( and ) ( N d N i.e., , α F S a a , N 2 ˆ d C ( − i x n sn niilntto,w banthe obtain we notation, indicial using and ) S ) steui omlto normal unit the is ) fdisplacement of ∫ 1c W − C a , Ω si h olwn,w sdEinstein used we following, the in As . on in on ( .Herein, ). ∫ B in ( φ α Ω ; Ω Γ = ) · Γ Ω , B δ N C D , u i ; δD ; ) C d . 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In and , eea tnadapoce eg,[ (e.g., approaches standard Several f t ( ai , int · 4 ) , f f n sn asinqartr ocmueteitgasnmrcly h oa force nodal the numerically, integrals the compute to quadrature Gaussian using and ) r on od ple ietyt node to directly applied loads point are 6a ai f enstedrvtv ihrsetto respect with derivative the defines ext ai int ext L )-( f = = and , 2 ai ext u 6b D norm n ∑ q ∂ , Acceptedexp Article D =1 QP ), exp f ∫ ai δ int Ω seautda iceelctosacrigt hsnojciefnto,e.g., function, objective chosen a to according locations discrete at evaluated is n stegoa etrcnann h oa ipaeet.I predetermined a If displacements. nodal the containing vector global the is D B srahd tec trto,svrlagrtm,sc sGntcAlgorithms, Genetic as such algorithms, several iteration, each At reached. is u are: ) f W QP i g ai exp int r h iceegoa etr otiig epciey h components the respectively, containing, vectors global discrete the are ∂u ( ⋆ X ( X ,etra ocs( forces external ), α = stenme fqartr points, quadrature of number the is ai q srcre.I hsstig h displacements the setting, this In recorded. is q , ) fqartr point quadrature of ∥ C N D ( a u − ( )) X D q = ) exp w δ ∫ D ∥ .E EOT TAL. ET PEROTTI E. L. q Ω 1 2 + 30 hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This P ) hrfr,b hnigtemtra rpris tis it properties, material the changing by Therefore, )). · = f ( iJ t , ai f ext ai D N int 31 ,advrain fdslcmns( displacements of variations and ), . ∆ T a,J q − ( )t dniyukonmtra rprisin properties material unknown identify to ]) D , · ) a w f d ncmatclm arxntto,e ( eq notation, matrix column compact In . ∆ ext q ndirection in = Ω with , stewih soitdwt udauepoint quadrature with associated weight the is ) = u n ∑ q exp =1 QP 0 D , and P ∆ D iJ F i := r eoptdutlasatisfactory a until recomputed are h components The . 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Figure ymnmzn h ifrnebtentecmue n h xeietlfre (Figure forces experimental properties the material and the computed the identify between to difference them used the from then minimizing by are and These inputs, forces. known internal as corresponding displacements the experimental compute the to apply similar we approach, our method, the in FEMU-F computes data, and the displacement forces experimental the known with the standard compared applies the be approach turn to we standard displacements guesses, the initial Whereas to head. respect its with on robust is approach that solution unique a with problem a the that implies This problem. equilibrium nonlinear a of solution function the objective the via and equilibrium dependence implicit Through this but only identified, problem. be is to equilibrium properties material forward the on depends nonlinear field displacement a computed solving by obtained field minimizing on based approaches esnw antgaateauiu ouint minimizing to solution unique a guarantee cannot we reason function objective the ee oaodti mlctdpnec nmtra rpris n ihtega fformulating of goal the with and properties, material on dependence implicit this avoid to Here, standard the that guarantee no however, is, there elasticity finite of setting general the In Accepted Article MEASURES FIELD FULL USING PROPERTIES MATERIAL OF IDENTIFICATION UNIQUE g ⋆ g h etro iceie ipaeetbudr conditions. boundary displacement discretized of vector the g tefi olna mlctfnto ftemtra rprisadfrthis for and properties material the of function implicit nonlinear a is itself ⋆ ⋆ h esrddslcmnsaecmae ihtecmue displacement computed the with compared are displacements measured the , ni notmlsto rprisi obtained. is properties of set optimal an until g ⋆ edt nqemtra rpris ned o aheauto of evaluation each for Indeed, properties. material unique to lead hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This g ⋆ . 1 D ¯ -right). denotes 7 B of only function a become f domain discretized the in body the of response material npeetn u M rmwr n betv ucin esatb enn rrqiie 1and P1 prerequisites defining by start we P2: function, objective and framework EMS our presenting In level model the at function Objective 2.1. 8 otcoeyblneteapidetra ocs ocmuetemtra rprista edt the to lead that properties between material equilibrium the closest compute To forces. external applied the to balance closely respect most with minimization least-squares by possible between as closely for as basis imposed the Given as be minimized. here considered be is to residual, function satisfied. material objective the approximately i.e., the suitable only equilibrium, is inserting perfect equilibrium from when deviation the the consequence, momentum, Thus, linear a of As balance the field loads. into displacement applied properties real the the using exactly computed material match real forces the cannot internal of the response therefore the and adopted describe the approximately always of will only imperfections model as material well the as Indeed, pressure, model. and material displacements measured the in errors to due ai ext P2 P1 new aecoe naporaemtra nryfunction energy material appropriate an chosen have we Once ngnrl h qiiru ewe xenl(xeietl n nenlfre ilb inexact be will forces internal and (experimental) external between equilibrium the general, In n tractions and eed ntecoe ueia ehd(.. q.( eqn. 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( u 6 R )) α )). ai a encoe,ie netefunctions the once i.e, chosen, been has W orsodt h nenlfre that forces internal the to correspond R ai a eslce rmteltrtr or literature the from selected be can W , u ( α in , W C f Ω ( ( ai int α u u oasto externally of set a to due u )) , ndirection in oc qiiru can equilibrium force , C n consequently and ( u 3.1 )) α edsrb the describe we odsrb the describe to α ftebody the of fteerror the of i tnode at f ai int and f (7) ai int Ω a dnie as identified H to corresponding properties material the Then where ecrano t oiiedfiieestruhu h dnicto rcs.Mroe,once number, Moreover, condition process. its compute identification can the we throughout known, definiteness positive its of certain be stiffness the characterize to available in contained are modes data deformation the enough all i.e., with posed, associated well choose is to properties possible material is its it field, displacement recorded the of the in but contained properties function. cases, material energy certain the chosen characterize in to in method used properties are proposed data material the insufficient the that of indicates all inadequacy rather identify an to show expect not not does do Hessian we semi-positive available, is ratio stretch fixed a in used in included modes deformation the all “excite” not H ihasrcl oiiedfiieHsin(.. 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Accepted to respect with density energy material the of convexity requires Polyconvexity . Article 2 α . . . , ( F N ) ] .FRUAINO AEILLAWS MATERIAL OF FORMULATION 3. oevr oyovxt ietygaate aeilsaiiyi the in stability material guarantees directly polyconvexity Moreover, . T 36 with α ,adtxiemmrnsi [ in membranes textile and ], i.e., , α W i α ocagsi h isesrcue o xml,acag nthe in change a example, For structure. tissue the in changes to ( i α > , C .E EOT TAL. ET PEROTTI E. L. 0 ( 2.1 hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This ∀ Φ u )= )) i i ( h aeileeg density energy material the , C { ∈ ( u dre l [ al. et oder ∑ ¨ i 1 =1 N )) W , 2 α ersnsdfeetiorpcadanisotropic and isotropic different represents ( N . . . , α i Φ , C i ( C ( } u 25 37 ( ; F )) u Φ .Ti seti togylne with linked strongly is aspect This ]. .Apicpefrtecntuto of construction the for principle A ]. t oatrcof cofactor its , )) ota h xeietlyobserved experimentally the that so i ( , C dradNf [ Neff and oder ¨ ( u )) r oyovxfunctions polyconvex are W ( α ( F , ( = ) C 26 35 ( I u .Teebases These ). .I addition, In ]. .Snethen, Since ]. 24 )) det .Hr we Here ]. hudbe should ( 34 F )) .The ]. F (19) − T tesfe eeec ofiuain h orhfnto dpsteie rsne nGse ta.[ al. et Gasser in presented idea the adapts function fourth The configuration. reference stress-free a function. this of advantages the of analysis term the including by materials biological of typical response incompressible almost the model We Table of column ‘ third exponent the the in of reported values different for (anisotropic) polyconvex remain fiber they terms and and the law (isotropic) power choose we matrix Moreover, the myocardium. model passive to of e.g., response tissues, biological in responses both functions polyconvex candidate of choice Our A h aeileeg a ob sdwt u prahcnb eetdfo h ieaueprovided literature the from selected be in can approach linear our is with it used be to law energy material The laws material viable identify to Strategy 3.2. constructing in used be to functions polyconvex of Examples I. 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Accepted Article a Anisotropic Isotropic type Response symbol ≥ I I 4 1 1 = = and tr tr ⟨·⟩ ( ( β C C ersnsteMcua brackets Macaulay the represents ) stepnlyprmtr e atanadNf [ Neff and Hartmann see parameter; penalty the is ( I , A ⊗ ⟨ W 2 b A I = ⟨ I h motneo hsapc shglgtdi Sections in highlighted is aspect this of importance The . vol 3 I )) 1 ( 1 / ( I 1 2 3 I , 1 = hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This I ( (3 + I ( 2 3 C I 4 tr b/ ⟨ I Φ β 3 b 3 I b/ I ( − 2 1 i b n h tutrltensor structural the and 5 ( 4 C b ( 3 Φ − C − I = − I ) − i 5 3 b 2 ( 3 nldsiorpcadaiorpctrst model to terms anisotropic and isotropic includes ) 3 1 tr u + 3 b − 3 b ⟩ b/ )) b ( − ) a ) tr I I C 2 1 a 3 4 b 2 ) ( 2 C b − a − ( ⟩ A 2 3 a 2 ) ⟩ ) ) ⊗ ⟨·⟩ a I , a A 0 = W ) a ) 3 l h em eotdhr ensure here reported terms the All . . a a a a ≥ . (( 5 Φ = ≥ ≥ ≥ ≥ i 1 det Notes A · ota hi xrsini a is expression their that so , + ) 1 1 1 1 b , , , , ⊗ ( b b b b ∈ C | A ( ≥ ≥ ≥ ≥ a [0 ) · ) ne h conditions the under ’ , : | , 1 1 1 1 ) 1] . 38 o detailed a for ] W ae on based 39 (20b) (20a) ;the ]; (21) 3.2 13 natsu ape ecl hsdt set D data test this stretch call biaxial We a sample. during tissue recorded a relation on strain-stress the e.g., stresses, and strains relating data S2. S1. properties material linear the identify to finally and non-linearity), material the (i.e., after only case, either In data. experimental available 14 sdsrbdi h olwn tp n umrzdi Algorithm in summarized and steps following the in described as uniquely. identified be can k norsrtg,w i oslc h terms the select to aim we strategy, our In experimental available of set a describe to law energy material a construct to want we general, In ( = opt nenlparameters internal Compute candidate a Choose ecntuttevector the construct we e fmtra properties material of set parameters internal of initialization the includes ifrn eomto states deformation different properties material the how describe pairs These vector the of component term every for Lastly, each for Furthermore, i with denote We arrange we where ∑ i.e., , F i N k =1 , P α k i ) Φ , i Accepted Articlek en ( (eqn. 1 = α , 2 i M . . . , 19 h etro h aeilproperties material the of vector the )uigaseicsto functions of set specific a using )) W i F eascaeeeycmoeto h vector the of component every associate we , . k Φ ednt with denote We . α osrc addt aeileeg est fteform the of density energy material candidate a Construct D ∈ i ik F n opt h pairs the compute and o ahsri-tesstate strain-stress each for k k γ ftecniee nrydniywsal opretycapture perfectly to able was density energy considered the If . eeaut h functions the evaluate we , i α Φ α .E EOT TAL. ET PEROTTI E. L. in D k i i α hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This Φ n aheeetdsrbn tansrs state strain-stress a describing element each and [ = [Φ = [ = k i . argmin = γ α Φ α sn h rcdr hw nSection in shown procedure the Using i 1 i i ( i [ = 1 1 k α M , α , ob nlddin included be to α , i Φ α , a h ieo h set the of size the i 2 i Φ 2 i 2 k b , W .,α ..., , ..., , i .,α ..., , α ) g ˜ i i . ] k scoe,telna aeilproperties material linear the chosen, is T soitdwt ahfunction each with associated Φ ( . α iM iM Nk α ) D Φ ] ik ] , ] T T T i k . optdfrdifferent for computed . ftefr itdi Table in listed form the of . Φ 1 neednl by independently : ik W n,smlrt h vector the to similar and, D ocmueterexponents their compute to , . α i otecorresponding the to α 2.2 eproceed We . Φ k dniya identify , i n fixed and D ayfor vary I k This . W i.e., , (22) α = α i , γ functions α eiaiefe ehd a eue swl n pcfi rbesmyrqiedifferent require may problems specific in and presented well examples as algorithms. used the be see may (e.g., methods studied derivative-free cases the all in Section well performed it because until modified is law material where behavior, material the ssoni Section in shown as ( of form problem to solution the that remark We implementation the [ use MATLAB We only. in evaluations provided function on based and derivative-free is which Since identify to used law Therefore, law. material once, at states stress/strain experimental the properties all material changing possible without as well as represents law material ( solving of aim the Indeed, considered. states average their to possible as close as i.e., possible, function objective The ( to according solved function the of evaluation each For function objective The i i NQEIETFCTO FMTRA RPRISUIGFL IL MEASURES FIELD FULL USING PROPERTIES MATERIAL OF IDENTIFICATION UNIQUE until o hnigdfrainsae ngnrlti snttecs n h olnaiyo the of nonlinearity the and case the not is this general In state. deformation changing a for h γ α si eea o ovxfnto,w ov tuigteagrtmpeetdi [ in presented algorithm the using it solve we function, convex a not general in is W 3.3 sdfie stecleto falsets all of collection the as defined is i Φ eoe scntn spsil over possible as constant as becomes

a encoe ae nteidentified the on based chosen been has Accepted Article i n Figure and ed ob dpe.Ti a eahee ymdfigteitra parameters internal the modifying by achieved be can This adapted. be to needs h ( γ , α 2.2 α 22 i = ) . α h nodrt obtain to order in ) . h i 41 γ sdfie as defined is 5 ol ecntn over constant be would ∑ smnmzdoc h aeilproperties material the once minimized is i n o t ipiiyades cesblt.Hwvr other However, accessibility. easy and simplicity its for and ) per mlctyi ( in implicitly appears hog h function the through ] =1 N α t u u v i sa ls spsil to possible as close as is ∑ k M =1 γ hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This γ h ( α argmin = [ = eiso nenlotmzto rbeshst be to has problems optimization internal of series a , i α e ahsae ychanging By state. each per ik γ 23 α ¯ α 1 − γ i , sgnrlyntuiu.Hwvr neafinal a once However, unique. not generally is ) k γ α 23 . ¯ γ 2 i i ..., , h st dniytenonlinearities the identify to is ) ) i.e, , α Φ ¯ 23 ( 2 i i γ Φ vrterange the over , fminsearch o hsproe ecompute we purpose, this For . γ γ hog h omlto ftematerial the of formulation the through ) , i N h nqeesof uniqueness the , α n hr ol en edt adjust to need no be would there and , with ] ) . , α ¯ i α ¯ o vr term every for i = ecoeti algorithm this chose We . ∑ { ,...,M , . . . i, γ k M h omlto fthe of formulation the , =1 M α i α α r scntn as constant as are ik ssilguaranteed still is { } ,...,N , . . . i, . fdeformation of γ ota the that so γ by } nthe in (24) (23) 40 15 ], a ( law tp 1truhS above. S3 through S1 steps Table in listed type the of ( eqn. of form particular a is which different (Figure configuration loading stretch equibiaxial an of combination other any and values a ecnie eleprmna aa hr ed o aeapir nweg fteterms the of knowledge priori a have in not included do we where Section data, in experimental Subsequently, real data. consider pseudo-experimental we the generate candidate to the used of ones functions the basis a with energy from coincide the generated choose data pseudo-experimental we use here Specifically, first law. we material approach, known our of part central a is which W, data. synthetic using Example 3.2.1. 1 Algorithm S3. 16 1 In Compute while end while Initialize candidate Choose 2 = dniymtra properties material Identify 25 tpS1 Step hog h rcdr rsne nSection in presented procedure the through data experimental ∀D Minimize Evaluate . ,w rce ognrt e fped-xeietldata pseudo-experimental of set a generate to proceed we ), 0 and , h α k ( compute γ and W γ α , dnicto fitra aaeesadmtra properties material and parameters internal of Identification i eslc addt nryfnto ftesm ompeetdi ( in presented form same the of function energy candidate a select we , a α sa xml,w osdrteeeg law energy the consider we example, an As . argmin = [ = 2 h γ ) h 3 = ( i.e., ,

Accepted Article minimized not is a γ ( γ i , b , α W . , α 0 α k i h eeaiyo h xml rsne eede o eedo h chosen the on depend not does here presented example the of generality The . = ) ] with W ) T = argmin = α D sn rdetfe algorithm free gradient using I ( ∑ g ˜ ∑ and , α k ( , i N α i N iutnosyfrall for simultaneously =1 α C =1 ¯ W ) i ( b α = u √ 1 α = i 19 )= )) do Φ = ∑ ∑ g ˜ α i α with ) k b k M k M 1 α nodrt lutaetefruaino h nryfunction energy the of formulation the illustrate to order In ( =1 2 . M =1 .E EOT TAL. ET PEROTTI E. L. α ( α 1 1 = I sn h values the Using α , 1 α ) hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This 1 ( ik ( I N 2 I α utemr,w chose we Furthermore, . 3 a , 1 -1/3 ik α 2 = I ¯ − 1 3 i -1/3 a , − α ¯ 2.2 i 2 nldn n stoi n n nstoi term anisotropic one and isotropic one including ) 3 − ) 2 2 seulysial.Uigtemtra energy material the Using suitable. equally is . k ) et eapytepoeuedsrbdin described procedure the apply we Next, a). + 3 eietf nqemtra properties material unique identify we , ) α a 1 2 ⟨ + I γ 4 α i − 2 dnie rvosyadalthe all and previously identified ⟨ 1 I ⟩ 4 D ◃ , ◃ − dniyitra parameters internal Identify fsize of α dniymtra properties material Identify 1 1 ⟩ a 5 = 2 , M . 0 kPa, 20 = α ◃ eutn from resulting 2 25 Initialization 10 = ,btwith but ), Φ . i 0 W obe to kPa, (25) (26) 3.3 γ to α α osdrtersligsxdfeetfrso nrydniiswe pligtesrtg described strategy the applying when densities Section energy in of forms different six resulting the consider [ purpose, in this reported For data tissue. experimental myocardial the of use response we constitutive passive the describes that law energy Table in listed form the that betv function objective of value the report we density term anisotropic one and choice isotropic our one limit we of simplicity, composed For laws present. material be to should components anisotropic and isotropic both Section in described strategy the of applicability the data demonstrate We experimental on based law material viable a of Identification 3.3. ( in shown form general the of S3. step in identified be will current pairs the the in data, as data experimental the exactly ( represent where example can function energy candidate the if case since step previous ( in terms pairs The eol piie h exponents the optimized only we Figure in shown response the given However, data. measured the represent best density energy stretch fiber (Figure data hr h nta us specifies guess initial the where ecnld yepaiigta h taeypeetdi plcbet n aeileeg law energy material any to applicable is presented strategy the that emphasizing by conclude We the of end the at identified already were properties material correct the case, specific this In In In α D tpS2 Step tpS3 Step i iutnosy ecmuetemtra properties material the compute we simultaneously, r o osatwt epc to respect with constant now are 3 NQEIETFCTO FMTRA RPRISUIGFL IL MEASURES FIELD FULL USING PROPERTIES MATERIAL OF IDENTIFICATION UNIQUE .Teeeprmnsaecridotfrtredfeetrto ffie stretch fiber of ratios different three for out carried are experiments These ). 25 ( 3.2 α .W ov rbe ( problem solve We ). λ i o every for , sn h values the using , , 2 n olc h eut nTable in results the collect and ( Φ .. : (equibiaxial, 1:1 i.e., , W α 25 i i

Accepted Article) , g sukonapir nti xml .. ed o nwwihterms which know not do we i.e., — example this in priori a unknown is ˜ n ( and ) for Φ ( ( α α i ) ) i i ilntb osata h n fse 2adtefia aeilproperties material final the and S2 step of end the at constant be not will , I k Φ en ( (eqn. 1 = . Φ 26 1 1 = k i ) oti h aetrs fti snttecs,a ihra experimental real with as case, the not is this If terms. same the contain ) , = 2 eeeatycntn n qa to equal and constant exactly were h 19 to 16 a r o osat(Figure constant not are ( ( a 1 I ,ee fhr ersrc usle opolyconvex to ourselves restrict we here if even ), γ 20 ) n h dnie aeilproperties material identified the and )), a 1 23 1 2 = , i I = eidentify we , α n efie h values the fixed we and n eoe h orc ausfor values correct the recover and ) 3 -1/3 λ ) a , 1 en ( (eqn. 2 a Φ − = 2 1 = i hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This 3 = 3 λ (Figure 42 ) 2

n einitialize we and 24 ,21( 2:1 ), II F ,wihwsotie nbailsrthexperiments stretch biaxial in obtained was which ], rvosycmue n h ulsto experimental of set full the and computed previously npriua,frec addt omo h energy the of form candidate each for particular, In . k ) h piie exponents optimized the )), α 1 k 2 , λ )adeult h ausue n( in used values the to equal and c) and 1 Φ 2 = 2 2 )since b) α k 2 λ α = k b 2 i 1 n compute and ,ad12( 1:2 and ), mn l obntoso isotropic of combinations all Among . ⟨ α 5 = I 1 4 Φ = α − and i 1 α 1 α , n( in 3.2 ⟩| 2 2 F α γ 1 = Φ oee,ti sol the only is this However, . λ 26 k i.e., , 2 ycntutn material a constructing by { i 2 α . 10 = r o h aea the as same the not are ) ae rmTable from taken . 2 = { 1 a α , 1 a a , λ 1 2 (Figure 1 } 2 = 2 .Tefr fthe of form The ). 3 nti example, this In . } W eepc that expect we , h minimized the , , ihtrsof terms with a 25 2 λ 2 3 = d). 1 ). ocross to Φ Note . I i We . will 17 α arta etdsrbstemtra eaira ifrn ttso eomto seSection (see deformation of states different at behavior material the describes best that pair of values minimized the 1) report: we exponents density optimized energy candidate each For II. Table 18 iue2 h rcdr ocmueteitra parameters internal the compute to procedure The 2. Figure for value minimum the to leads that one the choose we terms anisotropic and ( ( I 2 I 3 I 3 / 3 I 1 2 1 / 3 − Φ − 1 3 3 3 / ) 2 a ) 1 a

Accepted Article1 { a ⟨ ⟨ 1 4 4 1 1 xmlfidwt ytei xeietldt nsesa-d. steps in data experimental synthetic with exemplified a , I I ⟨ ⟨ 3 3 I I 1 1 2 ( ( 1 1 / / } I I 3 3 n )temnmzdvle of values minimized the 2) and ; 1 1 (3 + (3 + I I ⟨ ⟨ 4 4 I I 4 4 − − − − Φ − − I I 2 5 5 1 1 ) ) 4 4 3 3 ⟩ ⟩ ) ) − − a a I I 2 2 4 4 2 2 .E EOT TAL. ET PEROTTI E. L. properties − − ⟩ ⟩ a a hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This 2 2 3 3 ⟩ ⟩ a a 2 2 h { α ( 9.76 5.53 5.26 9.68 4.77 4.50 γ 1 α , , α 2 ) } . γ g ˜ ( { { { { { { α htdfietemtra o-iert is non-linearity material the define that .9 1.56 1.39, 2.22 2.28, 2.26 2.30, 1.89 1.58, 2.84 2.59, 2.82 2.60, { ) a n orsodn dnie material identified corresponding and 1 a , 2 } } } } } } } h 353.64 280.40 61.12 58.14 31.17 31.69 ( g ˜ γ ( α , α ) ) h n corresponding and ( { { γ { { { { 82,0.66 28.25, 7.25 30.48, .4 1.74 1.04, 0.63 1.28, 4.36 1.35, 2.66 5.82, , { α α ) 1 .. the i.e., , α , 3.2 2 } } } } } and } } eenealsa piie omlto fsri nrydniisi em fcmuigteinternal the computing of terms parameters in densities energy strain of formulation optimized an enables herein (Figure experiments stretch biaxial during al. et Yin properties α material the compute we simultaneously data experimental the all and nrylw erpr h omlzdpisbfr piiigteexponent the optimizing before pairs normalized the report We law. energy iue4 Normalized 4. Figure high of variability the how illustrates Φ h icsinbfr q.( and eqn. (left) before fiber discussion the cross the and in shown fiber are the data in strain-stress ratios stretch The respectively. different 1:2, for and tests 2:1, stretch 1:1, biaxial i.e, in directions, obtained fiber are data [ strain-stress in The reported data Experimental 3. Figure rs br(ih)drcinuigtedfraingain tensor gradient deformation the using direction (right) fiber cross 2 2 sn h taneeg function energy strain the Using 7 = = Φ ⟨ i I . / 25 4 max NQEIETFCTO FMTRA RPRISUIGFL IL MEASURES FIELD FULL USING PROPERTIES MATERIAL OF IDENTIFICATION UNIQUE − P.Teietfideeg ucinacrtl ecie h tansrs aaotie by obtained data strain-stress the describes accurately function energy identified The kPa. γ 1 (Φ ⟩ n nqeietfiaino h aeilproperties material the of identification unique a and a 2 i ) n identify and

Accepted. Article α i / q.( eqn. α ¯ versus W 23 23 ocluaeteoptimal the calculate to ) a ) codn oti rtro,w select we criterion, this to According )). = 1 Φ α 2 = i α / srdcdb minimizing by reduced is 1 max ( . 42 I 60 1 (Φ eciigteaiorpcrsos ftepsiemyocardium. passive the of response anisotropic the describing ] I and 3 hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This i -1/3 ) o h stoi a n nstoi b em ntematerial the in terms (b) anisotropic and (a) isotropic the for − a 2 3 2 = ) 2 . 5 60 . .I sepaie httepoeuepresented procedure the that emphasized is It ). 82 a + i iial oFigures to Similarly . aus(e triangles). (red values α 2 ⟨ I h F 4 ( − γ n h rtPoaKrhoftensor Kirchhoff Piola first the and , 1 α α ⟩ 2 ) . . a seilyfritreit to intermediate for especially , 82 i bu ice)adatrsolving after and circles) (blue Φ 1 = 2 ( α and b I 1 1 I 30 = 3 -1/3 2 . − ,Figure c, 48 3 P and kPa ) a 1 (27) and P 19 . 4 oa xeietldt,te9%cndneelpei within considering is even ellipse confidence that 95% observe the We data, data. experimental total experimental of set full the using computed add we Second, [ properties. from material data experimental identified the the to on noise points measurement of number the of impact markers nteietfidmtra rpris h xeietldt eotdi [ in reported data Section in experimental presented The example properties. material identified the on [ from data perform the we subsample properties, we material First, identified tests. the preliminary on two noise experimental of effect the address to order In noise experimental of Effect 3.4. stretch fiber between ( eqn. in reported law [ data Experimental 5. Figure 20 ( corresponding the plot we (Fig. properties ellipses material identified confidence the 95% set on a Based choosing time. randomly each by points times thousand material one of analyses [ the parameters repeat form we internal the ten) nonlinear not and twenty, properties, the (thirty, level material and linear law the energy identify material only the we of i.e., data, experimental of set full the ten and twenty, properties thirty, material and choose the identify randomly to we states analyses strain/stress experimental subsequent In points. data experimental aeilpoete r eaieyrbs ihrsett h ubro vial xeietldata experimental available of number the to respect with robust relatively are properties material α 1 ntefis etw dniytemtra rprisuigol usto h hryfieavailable thirty-five the of subset a only using properties material the identify we test first the In n brdrcin( direction fiber and ) 3 a h ecnaeerri optdwt epc otemtra rprisietfidi Section in identified properties material the to respect with computed is error percentage The 2 ntemtra asaekp osatadeult h auscmue nSection in computed values the to equal and constant kept are laws material the in # gemn bandfraltresrthrto yuigasnl e fmtra properties. material of set single a using by ratios stretch three all for obtained agreement ,21( 2:1 ), λ

Accepted Article1 2 = 27 λ 1 n h aeilpoete n aaeesrpre nTable in reported parameters and properties material the and ) λ n rs brstretch fiber cross and 2 α qaemarkers square - 2 42 aeilpoete,rsetvl.Ti eutsget htteidentified the that suggests result This respectively. properties, material ) mres essmdldrsos sldlns banduigtematerial the using obtained lines) (solid response modeled versus (markers) ] 3.3 6 o h aeilpoete omlzdwt epc otevalues the to respect with normalized properties material the for ) . 42 obte nesadteipc fmaueetuncertainty measurement of impact the understand better to ] .E EOT TAL. ET PEROTTI E. L. 2 hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This ,ad12( 1:2 and ), λ 2 r osdrd .. : (equibiaxial, 1:1 i.e., considered, are λ 2 2 = 42 ihu elcmn oudrtn the understand to replacement without ] ± λ 1 7% raglrmarkers triangular - and a α 1 ± 1 , 18% a and 2 .Frec subsampling each For ]. 42 α error II sas sdi the in used also is ] 2 he ifrn ratios different Three . h exponents The . 3.3 3 λ ▽ o h isotropic the for 1 .Nt h good the Note ). = ≈ λ 30% 2 3.3 circular - fthe of using a 1 tnaddeviation standard at is percentile 97.5% the that choose: we Specifically, directions. fiber cross and normalized of covariance slight and rprisietfiainoetosn ie n lttersligcndneelpe (Fig. ellipses confidence resulting the plot and times thousand one identification properties mean with distribution Gaussian a from chosen where are plots All points. experimental triangles) red - a c, (panel (panel the 30 10 with considering Together and obtained points. squares), properties experimental black material line) b, identified solid (panel the (blue 20 of 10 subset circles), and a blue line), with plot dashed obtained we data (black ellipses, experimental 20 of confidence subsets line), to dotted correspond experimental subfigure (red available each 30 the in replacement) ellipses (without confidence subsampling three by The data. obtained ellipses confidence 95% 6. Figure [ at centered are ellipses the (i.e., bias of absence the note also We points. oei et tde r arne.Mroe,teniecniee eewsadddrcl othe to much directly and added was preliminary here only considered noise are the here Moreover, at warranted. presented noise are tests and studies data depth the in experimental that more of remark amount We the level. to point respect location material with the the stable also but be minimum, to unique appears a minimum has its function of objective the Therefore, experiments. only stretch not biaxial framework in this noise and under data experimental of amount the to respect with within to are equal percentile ellipses 97.5% a with noise experimental an For ± aeilproperty material epciey si h usmln et eosreasih oainead oeimportantly, more and, covariance slight property a material observe anisotropic we the that test, subsampling the in As respectively. 10% nascn etw d os otecmoet ftedfraingain tensor gradient deformation the of components the to noise add we test second a In ohtetsssgetta h ehdpeetdhri edt dnie aeilpoete robust properties material identified to lead herein presented method the that suggest tests the Both omlzdwt epc otemtra properties material the to respect with normalized , i ± 1 = NQEIETFCTO FMTRA RPRISUIGFL IL MEASURES FIELD FULL USING PROPERTIES MATERIAL OF IDENTIFICATION UNIQUE 20% α 1 fie ieto)and direction) (fiber and , / α ˆ

Accepted Article1 α σ and 1 ± [+13% nti ae hr sas la istwr oe ausof values lower toward bias clear a also is there case, this In . orsodn oadfeetlvlo xeietlnie erpa h material the repeat we noise, experimental of level different a to corresponding 30% α 2 / α α ro ntefie n rs-brsrths(Figs. stretches cross-fiber and fiber the in error , ˆ 1 2 − F and axes). 11%] ii ± 1 = 0 α . i 1 2 and . 2 = , ( + 0 α ie,teae fteelpe r lotaindwt the with aligned almost are ellipses the of axes the (i.e., ± 2 0 [+5% smr estv oeprmna os hnteisotropic the than noise experimental to sensitive more is hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This cosfie ieto) rand direction), fiber (cross . F 2 and , ii − , − )( + (1 1) α µ ˆ 31%] ± 1 0 = , 0 α . ˆ 3 2 .. h 75 ecniecrepnsto corresponds percentile 97.5% the i.e., , dnie sn l 5eprmna points. experimental 35 all using identified ro o aeilproperties material for error n tnaddeviation standard and rand 30% ( ,σ µ, tec ro,te9%confidence 95% the error, stretch )) , (0 σ , α ) 1 / 7 sarno number random a is α and a ˆ 1 σ 1 = α eselect We . 2 . , 7 F α ) o each For b). α 2 ntefiber the in 1 / α and ˆ 2 1 = σ 7 α c). 21 so 2 ]) , and ...Aayi fmtra nrydniislna in linear densities energy material of Analysis 3.5.1. condition the on depends properties material identified number the energy of candidate robustness the the in particular, properties In material density. the identify to sufficient are data experimental available Section in discussed As densities energy material of Analysis 3.5. compute to required differentiation the by amplified significantly be may gradient deformation of side ( each noise on of levels regions three (shaded the reported to is corresponding 30% ellipses and confidence 20%, The 10%, markers). stretch experimental to fiber the corresponding between limit percentile ratios in 97.5% different the markers) Three (discrete directions. data (b) experimental stretch fiber recorded cross the and stretch to (a) added fiber noise the distributed Gaussian 7. Figure 22 C deformation the on only depend that will fact definiteness) The state. positive its (and number condition its therefore ( functions basis energy of combination linear ( g ˜ λ ( hssmlfisgetyteaayi of analysis the greatly simplifies This . 2 α npatc,i sotnpsil oexpress to possible often is it practice, In 2 = ) ± sidpneto h nta us o h aeilproperties. material the for guess initial the of independent is 30% λ λ 2 κ 1 r osdrd .. : ( 1:1 i.e., considered, are ( raglrmarkers triangular , r eotdi ae c.Tecndneelpe r omlzdwt epc otematerial the to respect with normalized are ellipses confidence The (c). panel in reported are ) H g ˜ ) Accepted ( eqns. see , Article H g ˜ F sidpneto h aeilpoete urnesta h iiiainof minimization the that guarantees properties material the of independent is nfl Dapiain h os fet ietytedslcmn edand field displacement the directly affects noise the applications 3D full In . 2 h Hessian the , 10 ▽ properties n ( and ) .O ahsd fteeprmnal esrdcre npnl a n (b), and (a) panels in curves measured experimentally the of side each On ). λ 1 = 11 λ ). H α ˆ 2 1 .E EOT TAL. ET PEROTTI E. L. iclrmarkers circular , g ˜ and hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This κ fteojciefnto ( function objective the of ( H H α ˆ g ˜ 19 g ˜ 2 nltclyi em fteaporaeivrat of invariants appropriate the of terms in analytically ) dnie ihu noise. without identified ), o n yeo eomto.Freape o the for example, For deformation. of type any for H g ˜ α sidpneto h aeilpoete and properties material the of independent is . # o l h aeillw omltda a as formulated laws material the all For ,21( 2:1 ), λ 1 2 = 16 losu odtriei the if determine to us allows ) λ 2 F qaemarkers square , . λ 1 n rs fiber cross and ± 10% 2 ,ad1:2 and ), , ± 20% , o h aeileeg est eotdi q.( eqn. in reported density energy material the for the identify to performed) been already have experiments the properties. if material sufficient (or suited best are data invariant anisotropic iue8 odto number Condition 8. Figure (Figure direction fiber the along stretch the of ratio the vary ( we example this In experiments. simulated ( in reported density energy material eairo asv ycrim mn hm h aeillwpooe yHmhe ta.[ al. et Humphrey by ( proposed law see material the them, Among myocardium. passive of behavior with to equal is direction fiber the in stretch maximum λ 1 o ntnew compute we instance For that observe We eea te aeileeg ashv enpooe nteltrtr odsrb h material the describe to literature the in proposed been have laws energy material other Several n rs brdrcin( direction fiber cross and ) 29 λ 1 ,rset h osritstfrhi q.( eqn. in forth set constraint the respects ), /λ 8 NQEIETFCTO FMTRA RPRISUIGFL IL MEASURES FIELD FULL USING PROPERTIES MATERIAL OF IDENTIFICATION UNIQUE ). 2 & 1 H H H Accepted lower with associated are Article g, g, g, ˜ ˜ ˜ H 11 22 12 H g ˜ I 4 8 = 8 = 8 = = g ˜ uhcoe-omepesosfrteHsinaeueu o sesn which assessing for useful are Hessian the for expressions closed-form Such . eed xlsvl ntetreiorpcivrat ( invariants isotropic three the on exclusively depends   a a a H κ sym. 1 2 2 2 1 ( ⟨ a

κ g, ˜ ( H I κ g 2 11 4 10 10 10 10 10 10 10 10 ( g ˜ I ( λ H − ) 3 I 1 0.5 2 0 1 2 3 4 5 6 7 1 / I safnto fsrthratio stretch of function a as g ˜ .Ec aasti opsdo e tec/tespit n the and points stretch/stress ten of composed is set data Each ).

3 1 3 I H H 1 ) 1 / ⟩ − 27 g, g, 2( ˜ ˜ 3 codn oen ( eqn. to according 22 12 a − 3 seas Section also (see ) 2 )   − H 3 2( 1) hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This ) g ˜ ; κ a I ( 27 a eepesdas expressed be can 1 ( 4 a − H 1 1 . n o h uprye l [ al. et Humphrey the for and ) − 1) g ˜ 1) ) I 1 3 19 λ − n hrfr edt oerbsl identified robustly more to lead therefore and I . 5 1 3 − 2 / eosreta,i hseape experiments example, this in that, observe We . n hrfr ti neapeo nexisting an of example an is it therefore and ) λ / 1 3 2 / 3.5.2 I 3 1 ⟨ 11 I 1.5 ( 4 o ifrn iuae ixa stretch biaxial simulated different for ) ). I − 9 W W 1 λ I I 1 3 1 candidate Humphrey 2 /λ ⟩ a − 2 2 − 1 uigbailsrthexperiments stretch biaxial during 1 ) ( 2 ,

I 4 − 43 aeileeg density energy material ] 3 1 I 1 I ) 1 , , I 2 , I 3 n the and ) (28) 43 23 α ], aclto edn oteeTyo xasosi ie nAppendix in given is expansions Taylor these to leading calculation ( since that Note algorithm. Appendix (see invariants deformation the of only function a as computed be alrsre ntepoiiyo e fmtra properties material of set a of proximity the in series Taylor ( In [ al. et Guccione by [ proposed al. et myocardium Holzapfel passive and for laws material the in consider we nonlinear example, terms including densities energy Section material of Analysis 3.5.2. ( eqn. in reported of density orders energy three the to for up than is larger model magnitude material Figure Humphrey instance the using for obtained number (see condition properties the example, material identified the of robustness the number condition the determine to used be Hessian the before, As adopted. be can foregoing the in reported analysis same the which for law energy material 24 o h oe aaeescnb bandb aiainwt xeiet.I oe rdcin are predictions model If experiments. with validation by obtained be estimates can reliable parameters that provided model case the the for be will This response. myocardium predictive passive for the foundation solid of a modeling provides which level, microscopic the at structure and physics tissue function as given have are vectors direction associated The respectively. directions, ako nqeesi h dnie aeilpoete (Figure properties material identified the in uniqueness of lack h aeillw fGcin ta.[ al. et Guccione of laws material The function objective the expand We 30a I 4 f n ( and ) 2 = g ˜ ti o osbet urne nqeesof uniqueness guarantee to possible not is it , sntcne o hs aeillw,epsn h xsec fmlil iiaadthe and minima multiple of existence the exposing laws, material these for convex not is f · Cf 30b W W

Accepted, Article Holzapfel Guccione I ), W 4 s 45 E Humphrey = ] = H s + = = g · ˜ 2 1 Cs ( soitdwt ( with associated α 2 2 C 19 α α α α 1 5 and , 1 6 − sntsatisfied, not is ) exp 2 = + { x [ exp I x [ exp ) [ ,s n ersn h br rs br n sheet-normal and fiber, cross fiber, the represent n and s, f, , α α I α g ˜ 8 1 4 α fs 2 + 2 ( ( α ( E √ I 2 16 .E EOT TAL. ET PEROTTI E. L. 6 = 1 ( ff ( 2 I I 44 α − banduigteeeg este ( densities energy the using obtained ) hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This I 4 1 f + 4 4 − s n ozpe ta.[ al. et Holzapfel and ] − · ( 3)( κ α E 29 − Cs ( 1) ) + 3)] 3 H fs √ 1) sidpneto h aeilpoete n can and properties material the of independent is ) ( E 2 . E g ˜ H I 2 + sf ) 4 ss 2 ] g ˜ ln ifrn eomto ah n evaluate and paths deformation different along + − − α + 2 α eed on depends α 2 E )+ 1) 1 3 E ( 4 } √ fn { nn 2 + E x [ exp I 2 + 4 α nf α α 2 − α 5 ) α 27 init ] ( 7 α fen ( eqn. if E 9 8 I 1) ; ). 4 1 { ). sn 2 sdt ntaieteidentification the initialize to used ( 3 x [ exp − α )+ I + 4 6.2 45 ealddsrpino the of description detailed A . 3) f α − r ae nkoldeof knowledge on based are ] α 2 entc htteobjective the that notice We . f 3 19 8 . 1) ( and I I sntstse.A an As satisfied. not is ) 1 8 2 2 fs ] − − ] s − n codnl we accordingly and , 8 3) 1 α .Nt ht nthis in that, Note ). } 1 30a . 6.1 } sdsusdin discussed As . n ( and ) ). H g ˜ 30b a then can (30b) (30a) na in ) (29) 44 ] est ed oacne betv ucinsince function objective convex a to leads density parameters internal the to similar way a available in is values computed their possibly about or knowledge properties preliminary — if material — The priori al. a set et be Holzapfel may by nonlinearly appear proposed that law material the in uniquely identified be may ( eqn. in entirety [ its in al. approach et our Guccione apply the can we with quantify Furthermore, to are. analysis stability properties a material as the serve degenerate can how eigenvalues its and function objective our of Hessian the atclryrlvn eas h ulfil esrsnee oapyormto a eacquired [ Hadjicharalambous be by al. investigated can et been method recently is has our application case apply similar This A to HFpEF. filling. with needed patients passive measures for field during full properties the scenario material because case myocardium relevant real the particularly a with measure consistent to setting it a apply in framework and EMS our of applicability the show We of of eigenvalues expansion smallest Taylor [ the al. of et term Quadratic 9. Figure fixing by Equivalently, solution. unique a to priori a setting by example For recovered. is properties identified the of uniqueness properties, ( eqns. in nonlinearly appearing a in that nonlinear models crucial using is method our it apply For then to properties. appropriate — material not of work is set it unique present reason a material this the to data a of experimental of of concern set use chief each then on associates the scheme is — modeling — focus models tool two the diagnostic these when a However, for as concern. case model much the of be not to is appears uniqueness it parameter as — variations parameter to insensitive α 3 eetees u ehdcnsilb fvlewt uhnniermdl.Frt vlainof evaluation First, models. nonlinear such with value of be still can method our Nevertheless, and , 44 46 α n c ozpe ta.[ al. et Holzapfel (c) and ] 19 NQEIETFCTO FMTRA RPRISUIGFL IL MEASURES FIELD FULL USING PROPERTIES MATERIAL OF IDENTIFICATION UNIQUE ,woas tde h rsigpolmo nqeietfiblt fmtra properties material of identifiability unique of problem pressing the studied also who ], 4 ,wt iercefiinsa h nyukon.B dniyn nytelna material linear the only identifying By unknowns. only the as coefficients linear with ), n( in

Accepted Article30a n dniyn only identifying and ) .APIAINT ETVNRCLRMODEL VENTRICULAR LEFT TO APPLICATION 4. H g ˜ SeAppendix (See . 44 n ozpe ta.[ al. et Holzapfel and ] 30a 45 n ( and ) aeilmodels. material ] W Humphrey α 6.2 30b hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This 1 hntemtra a rpsdb ucoee l leads al. et Guccione by proposed law material the then , o oedtis.Nt htol h uprye l energy al. et Humphrey the only that Note details). more for .I on o hs aeileeg astk h form the take laws energy material these so, doing In ). α sntpolyconvex.) not is 2 W , α g ˜ Humphrey 4 at , v α 45 1 α 6 and init aeilmdl fw xteparameters the fix we if models material ] and , slna ntemtra rpris(however properties material the in linear is o a uprye l [ al. et Humphrey (a) for v 2 α r h ievcosascae ihthe with associated eigenvectors the are 8 n( in γ 30b . ,termiigproperties remaining the ), α pcs(Figure spaces 43 ,()Guccione (b) ], α 9 25 2 ). , oncinwt h ti,adtelaigeetdb h ih ventricle. the right pericardium, the the by with exerted contact loading the the the and to that atria, instance, note the for with we due, connection assumption, forces realistic other to a subjected is is this ventricle Although left filling. passive during acting force main (Section function objective our to input as data synthetic (Section geometry prpit.Atog mlyn oeavne nt lmnsfricmrsil aeil may materials incompressible deemed for is elements formulation finite element advanced current more the employing and limit Although this appropriate. element reach not mixed do advanced we examples more following However, the locking. a in without that incompressibility myocardium note enforce We to required mm. be average would 1.25 formulation corresponding approximately The is nodes. contains length 30908 herein edge and generated element elements mesh tetrahedral coarsened (quadratic) The mesh ten-noded elements.) element 19834 hexahedral finite biventricular 800k original than (the more count preservecontains element and the cavity reducing inner smooth while a quality create element to muscles papillary the removed also We Hyperworks). (Figure model biventricular the from [ element in finite provided and are protocol the generation imaging the at mesh of Committee details UCLA The Use the Committee.) and by Research approved Animal Care Care was Chancellor’s #2008-161-12 the Animal protocol for we (Animal Institutional Angeles Guide data, the Los Health California, DTMRI and of of University and Animals Institutes National MRI Laboratory the of of by Use acquisition forth and the set guidelines During the rabbit. to Zealand adhered biventricular have New the on white based a mesh element of finite model and geometry (LV) ventricular left the generate We generation mesh ventricular Left 4.1. filling multiple or one from data (Section using steps and properties material regionally-defined or uniform with solving and (Section problem properties value tissue boundary reference the prescribing by obtained field, displacement corresponding (Section [ can DTMRI data and displacement MRI field through full acquired and be orientations, fiber geometry, Cardiac myocardium. passive for 26 aiaetedrvto formto n oso t oeta plcblt ihra lncldata. clinical real with applicability potential its show to and and method verify our to of intended may derivation are that the present error validate we tests and The noise properties. experimental material of of identification presence the the compromise them, Among not stage. challenges additional this several at be here will there considered scenario case real a in that remark We catheterization. nodrt etorapoc e )Gnrt nt lmn ehbsdo R acquired MRI on based mesh element finite a Generate 1) we: approach our test to order In 4.2 ;3 eeaesnhtceprmna aacnitn fitaetiua rsueand pressure intraventricular of consisting data experimental synthetic Generate 3) );

Accepted Article4.4 .I l h xmlspeetd eaueta nrvnrclrpesr sthe is pressure intraventricular that asume we presented, examples the all In ). 4.1 ;2 nld irsrcua brifraini u nt lmn mesh element finite our in information fiber microstructural Include 2) ); 48 .Frti td,w xrce h etvnrclrgeometry ventricular left the extracted we study, this For ]. 4.3 10 .E EOT TAL. ET PEROTTI E. L. 47 yecsn h ih etil sn yems (Altair Hypermesh using ventricle right the excising by ) ;4 dniytemoada aeilpoete sn the using properties material myocardial the Identify 4) ); hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This hl h nrvnrclrpesr smaue through measured is pressure intraventricular the while ] 4.4 .W eetteetssuigmodels using tests these repeat We ). ehwiei seulto equal is it while mesh tt eapoiaeypeevd h aiyt ycrimrtoi qa to equal is for ratio required myocardium to and cavity myocardium The LV preserved. and approximately cavity be LV to the it of volumes the between ratio the computed we emn mrcnHatAscainmdl[ seventeen- model the Association using Heart domain LV American the segment two subdivide construct we model, we non-uniform regions, first different the In in models. properties additional material different identify to ability method’s our exact enforcing on depend not does incompressibility. presented quasi method or the applications, clinical real in required be iuaetepeec fa nace rawt tfe stoi n brdrce aeilresponse material directed fiber and isotropic stiffer (Figure with area infarcted an of presence the simulate nodrt sesta h ehpoesn tp rsre h vrl Vsrcueadgeometry, and structure LV overall the preserved steps processing mesh the that assess to order In 11 NQEIETFCTO FMTRA RPRISUIGFL IL MEASURES FIELD FULL USING PROPERTIES MATERIAL OF IDENTIFICATION UNIQUE Acceptedb). Article 0 . 35 iue1.Lf etiua ehgeneration mesh ventricular Left 10. Figure ntecasndms sdi h olwn.Fnly nodrt test to order in Finally, following. the in used mesh coarsened the in hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This 49 (Figure ] 11 ) ntescn ouiommdl we model, nonuniform second the In a). 0 . 36 ntefieoriginal fine the in 27 o apesie hog h Vaesoni Figure in shown are LV the through slices sample for Accepted endocardium and epicardium the to midwall direction: the its the orientation from with myocyte Notice moving in associated directions. changes is transverse longitudinal the bar orthogonal more and two each each view) to the of top In in - circumferential color slices. (left red The from organization representative and fibers. myocyte green four the circumferential and midwall of on direction, clear direction mesh longitudinal the the element in in finite plotted blue LV is the bar a throughout element orientation Fiber 12. Figure [ method of interpolation barycenter neighbor the nearest to the grid DT-MRI using the elements element from tetrahedral per interpolated each orientation are fiber data one orientation consider fiber we the subsequently study therefore and this and In grid mesh. structured element a finite the on to acquired interpolated are data DT-MRI (DT-MRI). magnetic imaging tensor diffusion resonance using measured i.e., is orientation microstructure, LV. Fiber myocardial the the throughout of orientation fiber inclusion the the Article is model LV realistic a constructing in Essential interpolation infarcted fiber and versus microstructure Myocardial normal 4.2. and (a) segments AHA seventeen the into segmentation Domain 11. Figure 28 .E EOT TAL. ET PEROTTI E. L. rgt-ltrlview). lateral - (right ycrim(b). myocardium hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This 12 . 50 .Fbrorientations Fiber ]. oe mlyn h aepoeueadeprmna aadsrbdi Section in described data experimental and procedure same the Employing role. ntefloigeape ecmuetedslcmn eda ecie nSection in described as field displacement the compute we examples following the In filling passive during properties myocardial of Identification 4.4. where reads energy complete the that such codnl otecmue ipaeetfil,w oetefiieeeet oe n aclt the calculate and nodes elements finite the move we field, displacement computed the to Accordingly Figure ( in law reported material is data original slightly the only using differ obtained density properties energy the material from regularized the with associated properties material the exponents the acltosb digalna stoi emt h nrydensity: energy the to term isotropic numerical linear the a regularize adding we by method, calculations Newton-Raphson in the However, using approach. problem our forward with the used solve be to may order and viable not be was still would problem law equilibrium material forward a a If such necessary, solving problem. value and boundary MRI FEM through available the were solve data to displacement used be not could method Newton-Raphson the turn, at matrix stiffness β ( eqn. in described law Section in properties material reference input the recover and pressure identify corresponding to the aim and field we displacement load, computed the Given heart. the the of at base conditions the boundary at displacement nodes zero assign we pressure. problem, filling equilibrium diastolic forward human the healthy solving a In of representative is which kPa, 1.5 to equal pressure filling ahrgo ftemoadu.Sbeunl,w pl igeo utpeitaetiua pressure intraventricular multiple increments or single to a properties apply we material Subsequently, myocardium. reference the different of assign region through each first acquired We be problem. would equilibrium the applications solving clinical by in MRI, which field, displacement full the compute We data input displacement of Generation 4.3. 100 = naltefloigeape ecos omdltemoadu codn otematerial the to according myocardium the model to choose we examples following the all In θ . 0 = NQEIETFCTO FMTRA RPRISUIGFL IL MEASURES FIELD FULL USING PROPERTIES MATERIAL OF IDENTIFICATION UNIQUE 0 kPa, ∆ . 025 p { 0 = a b h parameter The . Accepted1 Article a , 2 = F . 05 W 2 } = . 0 P n opt h orsodn ipaeetfil.W ec maximum a reach We field. displacement corresponding the compute and kPa n h aeilproperties material the and = and , I 27 ic hntecefiinso h lsiiytensor elasticity the of coefficients the then since , 13 W ,tgte ihtefunction the with together ), vol n gi hw odareeto h oe ihteexperiments. the with model the of agreement good a shows again and a 1 = + θ α sitninlycoe ml eas thsol regularization a only has it because small chosen intentionally is . 1 0 [( oee,wtotaymdfiain ( modification, any without However, . Φ reg I 3 I hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This 1 1 / = 3 θ − ( 3 I ) 1 { I α a 3 1 -1/3 1 α , Φ + − 2 } W reg 3 srpre nTable in reported as ) vol ] 27 + pcfidi ( in specified .Tenwfi fteexperimental the of fit new The ). α 2 ⟨ I 4 − 1 ⟩ a 27 2 ∂ 21 , F ed oasingular a to leads ) III 3.3 P ,wt parameters with ), eosrethat observe We . erecomputed we , eoezr.In zero. become 4.4 . (32) (31) 4.3 29 . oei Sections in linear the done only identify we section, on this based in properties either Therefore, material chosen, data. been literature on has based law or material tests the sample of tissue form the that assume we examples following hs r h eesr aat ul n iiieteojciefunction objective properties. the material minimize reference and build the to using data obtained necessary the was are field These displacement input the pressure which the for to corresponding load forces external the compute we Finally, forces. internal corresponding [ data Experimental 13. Figure [ al. et Yin of data experimental biaxial the fit to computed exponents and properties Material III. Table 30 civn nacrc ls omciepeiini h rsne et ae ni iiodt san is data silico in on based tests presented the in precision machine to close ‘true’ However, accuracy and defined. an be identified achieving cannot between properties material error ‘true’ the since computed, applications be vivo cannot in properties material In with precision. field displacement machine input to the close generate error to an used properties material material correct the the recovers identifies it always i.e, distributed method properties, our uniformly properties, multiple material the Given for approach. guesses EMS initial random our using properties material reference the load and pressure a to due i.e., step, filling single ( properties material properties material nafis xml,w sueta h ycrimi ooeeu n hrfr nytwo only therefore and homogeneous is myocardium the that assume we example, first a In α 2 7 = sn q.( eqn. using . 06 P.Te,b pligtecmue ipaeetfil siptdt,w re-identify we data, input as field displacement computed the applying by Then, kPa.

Accepted Articleθ θ law Material 3.3 0 = 0 = 32 α α α with ) ewe elh ujcsadptet n ouete sdansi markers. diagnostic as them use to and patients and subjects healthy between n o h aeillwsterms law’s material the not and and . [ = 025 α 4.3 θ euaie aeillwdsrbdb q.( eqn. by described law material regularized 1 α , 0 = iigamtra a sncsayt opr ietytelinear the directly compare to necessary is law material a Fixing . 42 2 h ] mres essmdldrsos cniuu ie)uigthe using lines) (continuous response modeled versus (markers) ] r dnie.W opt h nu ipaeetfil u oa to due field displacement input the compute We identified. are ) oiia aeillw or law) material (original ( 4.62 4.50 γ , α ) .E EOT TAL. ET PEROTTI E. L. hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This { { .7,2.82 , 2.87 2.82 , 2.60 ∆ { a p 1 0 = a , 2 } . 05 Φ } } P,adcrepnigto corresponding and kPa, i θ n hi non-linearities their and 0 = 34.86 31.69 g ( α . 025 ) rglrzdmtra law.) material (regularized 32 { { 51 7.06 , 35.19 7.25 , 30.48 ). { α 1 g α , (Section 2 } } } γ α i 1 2.1 spreviously as 35 = .I l the all In ). . 19 kPa 42 ] h trtv iiiaino h ifrnebtencmue n xeietldslcmns(see displacements experimental Section and computed between difference the of minimization iterative the to equal pressure applied an to corresponds point rprisis properties to point time third increment pressure a to corresponding fteiiilges(e Table (see (Figure guess field initial displacement the input of the generate to used properties material correct where expression the case, this In function filling. objective passive the during of points time multiple at available clinically be will the constructing in once function at objective considered pressure are single states pressure a multiple with when obtained also achieved one was the increment to equivalent Accuracy method. our of validation important ecmuete2nr odto ubro h Hessian from the ranges of number condition 2-norm the compute we is time computation total example this for generated field (Figure displacement pseudo-experimental the with material associated the identify properties we expected, As myocardium. homogeneous otherwise an in properties material and iteration AHA one in the properties material using Table problem (see properties region the material each anisotropic setup and to isotropic We different assign model. and LV regions seventeen in inhomogeneous segmentation an in properties material our methods. iterative by to offered compared speedup when the magnitude of employed, orders method several alternative of be the could on approach depends this and Although speedup. estimate significant an a only provide can is method our hundred, one iterations, reach of easily number the could on which Depending herein. used resources computational and model same the for nodrt eti h ierssessle nteietfiainpoesaewl conditioned, well are process identification the in solved systems linear the if test to order In different by characterized infarct an containing model LV an consider we example last the In the identify to proceeded we points, time multiple at data displacement and pressure Using t 14 2 ersnstetm onsa hc aaaeaalbe loi hscs,w dniythe identify we case, this in Also available. are data which at points time the represents NQEIETFCTO FMTRA RPRISUIGFL IL MEASURES FIELD FULL USING PROPERTIES MATERIAL OF IDENTIFICATION UNIQUE ilrqiesvrlsltoso h owr rbe,ec trto requiring iteration each problem, forward the of solutions several require will ) neednl fteiiilges(e Table (see guess initial the of independently ) ≈ ≈ 7 nasadr igecr ne P.I otat lsi nes ehdbsdon based method inverse classic a contrast, In CPU. Intel core single standard a on 17s 7

Acceptedto Article 0 g . ≈ hsrpeet oeraitcseai ic rsueaddslcmn data displacement and pressure since scenario realistic more a represents This . 35 470 P,ads n eosreta h oa ouintm oietf h material the identify to time solution total the that observe We on. so and kPa, g ≈ ntedfeeteape niaigwl odtoe iersystems. linear conditioned well indicating examples different the in ssml oie as modified simply is V 23s. .As nti ae u prahwsal oietf h different the identify to able was approach our case, this in Also ). IV g .I hsadtefloigeape,w s e iepit,each points, time ten use we examples, following the and this In ). ( α , u = ) ∆ ≈ p 3 seFigure (see 23s hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This N ∑ 0 = t =0 steps . R 15 t ( P trigfrom starting kPa 0 α . 05 , u ) P,tescn iepitto point time second the kPa, VI T R 14 κ .Smlryt h rvoseape,the examples, previous the to Similarly ). t ( ). ( H α g , ) u srpre nFigure in reported As . ) , p 0 = . 05 P,ie,tefis time first the i.e., kPa, 14 independently ) 0 . 20 ≈ 14 0hours 10 P,the kPa, , κ ( H (33) 31 g ) nadto,w opt xlctyteHessian the measurements. explicitly displacement compute field we full material on addition, uniquely based In identifying laws of material goal optimal the formulating with and EMS properties termed framework new a developed have We the report model we stiffer a each case containing each per model the In cases and infarct. regions, representative AHA an five seventeen in representing for segmented region model reported the results model, uniform identification the using properties Material 14. Figure 32 napiain hr aeilpoete edt eietfidquickly. identified be to need properties material benefit where key a applications is in speed method’s our needed, simulated is study and to comparative measured thorough respect a between Although differences with displacements. the speedup of significant sought minimization the a the on to based identify approaches lead to standard could sufficient method are our data Furthermore, experimental properties. available material the if assess number, condition at available are information microstructure the the comparing by computed error properties) material the identifying after (i.e., final and guess) random Accepted C in-house The Article ++ oeue oietf h aeilpoete,teL nt lmn eh and mesh, element finite LV the properties, material the identify to used code xc n urn aeilpoete nalregions. all in properties material current and exact .CONCLUSIONS 5. .E EOT TAL. ET PEROTTI E. L. hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This https://github.com/luigiemp/voom2 H g forojciefnto n,bsdo its on based and, function objective our of L 2 omo h nta ie,bsdo h initial the on based (i.e., initial the of norm . ewe elh ujcsadptet.Fxn aeillwhwvri eesr ocompare to necessary example, is patients. for however HF change, and law subjects to material healthy law between a properties material material Fixing the directly patients. allow the and not and subjects do law healthy we material time, between the same identifying not the By at law. properties material invariants material deformation the the in on dependence also non-linear but properties, their material material and The linear uniquely. the identified on are only properties not material depends linear response the chosen, been has law material a h functions the be to properties material the in linear are they as far as identified. literature the from selected identify to laws approach material proposed the but laws, material optimal formulating exponents the identify to together used be would γ locations deformation different of and states time the in case, instants this different In at filling. passive ventricle left the describing data experimental (i.e., invariants deformation parameters of internal functions (i.e., the non-linearities choosing by achieved is latter The data. densities energy material These function. objective our with together xmlspeetdhri,w on that found we herein, presented examples and Section in described strategy the that remark resulting The aata r curda ato otn lncleas h dnicto ftemtra rprisat properties material the of identification The exams. clinical routine displacement of part as and patient-specific acquired of are availability geometry that the data on which based designed from is framework exams, Our catheterization extracted. be MRI ventricular can information left and both pressure undergo intraventricular may measure Patients HFpEF to progression. advanced and to in onset increase moderate disease an from with HFpEF, suffering of associated study is the stiffness in material example, myocardium For passive response. and health therapeutic between monitor distinguish to to and measures disease case, diagnostic latter become this may In properties applications. material biomechanical identified to attention particular developed with have approach we available, our are measures) pressure intraventricular LV (e.g., conditions boundary described, [ as in problem example, minimization for constrained inequality an solving by imposed be may constraint i norcretfaeokw eaaetefruaino h aeillw(.. h dnicto of identification the (i.e., law material the of formulation the separate we framework current our In used be to laws energy material formulate to strategy new a present we approach, our of part As lhuhorfaeokcnb ple nayisac hr ulfil esrsadtraction and measures field full where instance any in applied be can framework our Although properties material identified the that recall We n hrceietennierrsos ftemtra.Ti taeymypoieisgt in insights provide may strategy This material. the of response nonlinear the characterize and 3.3 oietf h exponents the identify to ) NQEIETFCTO FMTRA RPRISUIGFL IL MEASURES FIELD FULL USING PROPERTIES MATERIAL OF IDENTIFICATION UNIQUE Φ W i n non-linearities and 32 Accepted Article experimental available the of description optimal one represents and polyconvex is ]. γ γ i i rmteietfiaino h iermtra rpris Once properties. material linear the of identification the from ) ol eaatdadue tte3 oe ee,eg,with e.g., level, model 3D the at used and adapted be could γ α hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This i codn oteshm rpsdi Section in proposed scheme the to according ) eeawy oiie fntatmtclystse,this satisfied, automatically not If positive. always were 3.2 n sda igemtra on (Sections point material single a at used and α utb rae hnzr seen ( eqn. (see zero than greater be must W r ierin linear are α sas plcbewith applicable also is Φ α i 19 n their and ) (eqn. ) nthe In )). uniquely 3.2 3.2.1 We . 19 33 ). nftr okw ilaayeteefc ftsu r-tesi oedti n osdradditional consider and detail more in pre-stress tissue in of terms effect the analyze will we themselves. properties work material future the In in pre- effect our pre-stress In the the measures. absorb of vivo to in chosen alteration have on we based (e.g., approach approaches current organ other several the by shared from is extracted advantage This is state). stress sample tissue a environment the when and tissue occurring vivo ex changes in occurring changes material avoiding vivo, in tissues biological from independently properties material values. myocardium initial their passive the identify this overcome uniquely identified can and the approach our of challenge how uniqueness shown nonlinear, have is we Here problem challenging. equilibrium becomes the properties heart) material the for true is it data. displacement (as MRI When and computed possible as well as matching by properties myocardial passive yESbt sdsrbdi Section in described as required but, data diastolic EMS same by estimate the use to approaches These methods protocols. susceptible other research as less approach, emerging are loop be stiffness ventricular PV may standard measures the based Beside PV-loop noise. estimate, experimental global to a being [ that, increases note stiffness to tissue myocardial important local while decrease apparent can of estimates stiffness PV-loop global geometry), ventricle or geometric thickness to wall due myocardial fact, in In stiffness. (changes material effects the in changes characterize the to of unable stiffness are global and “apparent” ventricle left the measure only based estimates are These stiffness loops. ventricular (PV) diastolic pressure-volume of on estimates common most the setting, clinical myocardium a passive In measure stiffness. to approaches previous over advantages several carries level tissue the 34 a eahee ycosn mn rhgnlivrat ofr h aeileeg est to density energy material the form to invariants orthogonal among choosing by achieved modeling be material accurate may describe more and properties noise material to sensitivity identified Reduced behavior. and myocardium model passive the material chosen the closely their on how based test and, The displacements agreement, measured loading. the pressure with compared measured be and then problem can properties value material displacements material computed boundary real identified forward the the the is of using solving filling it idealization in passive as an during consists — only test exist are concluding not models important does material An solution since response. mis-calibration. exact applications to an real subject when in higher case is be the pressure may intraventricular affected noise of be to may measure sensitivity data Moreover, the identifying fiber while of and noise aim displacements imaging the MRI with by standard properties, applications of material cardiac presence myocardial in passive in instance, the robustness For its error. investigate and to noise experimental is general, in applications biomechanical ial,w ihih httepooe M rmwr esrstemtra rprisof properties material the measures framework EMS proposed the that highlight we Finally, n formjrojcie nfrhrdvlpn u M rmwr o lncl and clinical, for framework EMS our developing further in objectives major our of One W Accepted phenomenon. this for account explicitly to Article 2 mlyteetra ocsa nw nusadietf the identify and inputs known as forces external the employ , .E EOT TAL. ET PEROTTI E. L. hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This 51 .Hwvr tis it However, ]. eouino h xeietldt,ado hi quality. their on and temporal data, and experimental spatial the the of on resolution depend likely will properties avoid material identifiable 1) of number to: optimal determined The be to needs properties material singular large identifiable a of to lead number in may However, optimal properties mesh. the material element work, unknown finite of the number high of very regions a different practice, to assigned different be principle, may In constant. properties properties material material of location and number the held have we models) to required are measures. vivo analyses in Further with pericardium. combined the when simplifications and from these ventricle evaluate excluded right the have both we model For involved, heart assessed. complications first carefully this and be uncertainties to additional needs the properties given material example, identified the on simplifications model of them. on material based formulating densities by energy and invariants deformation orthogonal of functions polyconvex form to how [ attributes strain invariants deformation or orthogonal using properties material identifying of advantage the shown have eue nES eea tde eg,[ (e.g., studies Several EMS. in used be ial,i h xmlsivligihmgnosL oes(H emnainadinfarct and segmentation (AHA models LV inhomogeneous involving examples the in Finally, influence the noise, and error experimental of effect the mitigate and evaluate to addition, In κ ( H NQEIETFCTO FMTRA RPRISUIGFL IL MEASURES FIELD FULL USING PROPERTIES MATERIAL OF IDENTIFICATION UNIQUE g ) Accepted properties. material varying of terms in resolution spatial highest the achieve 2) and ; Article 54 .W lnt noprt hs datgsi u prahb investigating by approach our in advantages these incorporate to plan We ]. 52 , hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This 53 )hv aesgicn dacsi hsdrcinand direction this in advances significant made have ]) H g nfuture In . 35 ..Aayia omof form Analytical 6.1. 36 H H H H H H H H H H H H H AcceptedH H Article g, g, g, g, g, g, g, g, g, g, g, g, g, g, g, ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ H 11 44 34 35 33 25 24 23 22 15 14 13 12 55 45 g ˜ 8( = 8 = 4 = 16 = 8 = 24 = 6 = 12 = 18( = 16 = 4 = 8 = 12( = = 32 = 8 = H             g ˜ I √ I √ √ √ ( H o h aeilmdlb uprye l[ al et Humphrey by model material the for √ 1 1 I √ √ √ I 2 ( √ √ 1 1 g, I I I I ˜ , √ I ( ( I 4 4 4 4 I I I 11 1 I I 4 ( ( I I 4 4 4 ( ( ( 1 1 I I √ 4 4 ( ( ( − √ 4 1 I √ I 2( 2( − − − − 1 1 I − − H H 1) I sym. √ √ 4 I 3) 3) − − 4 1) 1) 4 g, g, ˜ ˜ 1) 8 + 3) 2 − − I I 2 3)( 3)( 4 3 , − 22 12 , 4 4 2 .E EOT TAL. ET PEROTTI E. L. . 1) , , )+ 1) − − 1) 8 + √ √ .APPENDIX 6. hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This , 2 H H H )+ 1) + 1) I I I , 1 √ 4 4 g, g, g, ˜ ˜ ˜ √ ( √ 33 23 13 − − I I 4 I I 4 I 1) 1) ( 1 1 √ √ ( 4 I I H H H H − − 1 2 I I , − 1 4 4 , − g, g, g, g, ˜ ˜ ˜ ˜ 3 3 − 1) 44 34 24 14 ) ) 3)( 3) , ( ( ) √ √ √ H H H H H ( I I I I g,, ˜ g, g, g, g, ˜ ˜ ˜ ˜ 4 4 4 1 45 55 35 15 25 − − − −             1) 1) )+2( + 1) 3) ; 2 , , , 43 ] I 1 − 3) 2 , (34) o lblycne o qs ( eqns. for convex globally not minimum its to close i.e., oteaslt iiu eaeal ofidteeatmtra rpris tews,since Otherwise, properties. material exact the find to able are we minimum absolute the to ycrimbhvo ( behavior properties myocardium material of sets Both densities). energy material and al. et Holzapfel and al. et Guccione al V eeec aeilpoete n aeilpoete nta ausue nfierpeettv test representative five in used values initial properties material and properties material Reference IV. Table material the for guess tests. subsequent initial initialize the to and used properties properties) material (reference procedure Section optimization in the presented in results the all of reproducibility for allow Section to in and shown cases completeness test For the in used properties material guess initial and Reference 6.3. g 18 where fiber the along stretch combines to that according deformation shear and incompressible direction an consider we example an As of expansion Taylor 6.2. eoti Tables in report found. be can minimum global the to corresponding choose We identified. and be to properties material the for guesses initial compute We ersnsara iuto hn nodrt dniyukonmtra rpris ewudbase literature. would the we in properties, reported material already values unknown on identify guess to initial our order in when, situation real a represents ˜ (Figure entc ht if that, notice We . 472 α α H,init H , 16 α aebe rvosyietfidi h ieaueadaerpeettv fraitcpassive realistic of representative are and literature the in identified previously been have NQEIETFCTO FMTRA RPRISUIGFL IL MEASURES FIELD FULL USING PROPERTIES MATERIAL OF IDENTIFICATION UNIQUE 9 . 026 = rudaohrsto aeilproperties material of set another around ) aisfo o11and 1.1 to 1 from varies { , 2 2 P IV . Property 280 . iJ 481 Accepted Articleext - VI α α α , 2 1 , 9 using G,init 11 h aeilpoete sdt eeaetedslcmn edapida input as applied field displacement the generate to used properties material the . 726 α . G g 120 ˜ g ˜ = , o ucoee l [ al. et Guccione for sfo al n[ in 2 Table from is slclycne.I te od,ol fteiiilgesi ls enough close is guess initial the if only words, other In convex. locally is 1 Reference , α . α 685 0 35.19 7.06 G G . 216 30a = and , ae ihteuiomL model. LV uniform the with cases 15 , { n ( and ) 11 0 . F α 779 . 831 . H,init 436 = et1 Test , 2.69 7.49 hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This ,      0 30b } 14 . 0 = 0 00 00 α . , for . 0 γ 3 30 ,teei ogaateta h aeilproperties material the that guarantee no is there ), 0 α 44 , nta us e etnumber test per guess Initial . H 0 4 and ] W aisfo o02i v increments. five in 0.2 to 0 from varies 0 , √ hnteTyo xaso of expansion Taylor the then , n ozpe ta.[ al. et Holzapfel and ] . et2 Test 19.60 49 1 0 3.11 . . α 0 0 H . 0 , , hr G n H ee,rsetvl,t the to respectively, refer, ‘H’ and ‘G’ (here 0 α 0 α . √ 762 . γ γ H 1 0 G,init α } sfo al n[ in 1 Table from is      et3 Test } 2.67 8.36 rmtesm alsctdaoe This above. cited tables same the from , and for α W et4 Test 31.52 H,init 3.73 α G G,init hc ersn potential represent which , 45 and aeillaws material ] = et5 Test 1.32 2.15 45 { α 1 g ˜ . ete expand then We ].) H . 2 slclyconvex, locally is = , 26 { 0 . 7 . 059 , 2 . 0 4.4 4.4 , , 8 14 . we , 023 (35) g ˜ α . 37 7 is G } , etcsswt h Vmdlsgetdacrigt h eete H ein.Teisotropic The regions. representative AHA five seventeen in the used to values according initial segmented model properties LV material the and with properties cases test material Reference V. Table 38 4OT9907t ug .Prti h ainlIsiue fHat rn 0 L83,adthe and HL78931, P01 grant Health of insight Institutes whose National the Klug, The Perotti, fellowship S. biomechanics. postdoctoral E. William computational Association Luigi Heart in colleague American to contributions the and 14POST19890027 from other friend support many the our acknowledge and gratefully to work authors work this inspired this knowledge dedicate and to like would We Region 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Accepted Article anisotropic Property α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 α 2 aeilpoete r sindsprtl nec region. each in separately assigned are properties material Reference 28.59 26.39 24.19 21.99 19.79 17.59 10.59 52.78 50.59 48.39 46.19 43.99 41.79 39.59 37.39 35.19 32.99 30.79 9.74 8.94 8.19 7.50 6.87 6.29 5.76 5.29 4.88 4.52 4.22 3.97 3.78 3.64 3.56 3.53 ACKNOWLEDGEMENTS .E EOT TAL. ET PEROTTI E. L. hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This et1 Test 51.46 11.32 24.89 19.55 19.27 30.16 19.16 26.13 10.33 19.67 23.49 21.86 23.71 1.99 0.54 0.16 0.49 2.29 9.44 0.55 7.05 7.42 8.91 7.32 1.64 4.44 3.12 7.43 1.80 2.38 2.27 2.52 4.38 3.59 nta us e etnumber test per guess Initial et2 Test 11.03 72.00 13.56 19.94 58.80 10.18 73.01 11.61 38.96 56.33 22.46 18.28 12.83 37.55 14.64 11.88 12.82 0.83 1.25 2.38 3.08 1.21 3.13 9.13 8.03 6.14 6.65 5.77 1.19 7.53 0.73 1.66 1.12 5.06 101.05 et3 Test 89.66 12.84 66.74 36.26 25.55 37.12 19.30 47.14 49.30 41.12 37.87 40.13 10.83 12.82 11.57 12.44 12.19 26.68 16.02 28.77 15.06 9.88 2.84 8.65 3.16 5.52 1.97 2.24 2.90 4.78 0.63 2.39 0.43 et4 Test 76.38 10.12 57.86 56.70 35.50 42.93 15.72 46.59 34.06 14.77 50.69 32.57 13.79 20.38 72.81 16.44 76.62 11.51 25.67 2.27 4.55 3.94 7.29 0.25 6.01 7.44 5.75 2.85 4.89 7.05 0.26 0.32 2.24 4.74 et5 Test 22.72 31.48 23.44 10.23 18.25 53.24 17.79 19.65 23.29 41.76 22.57 33.62 47.02 11.13 16.93 59.61 15.54 3.69 6.44 6.53 6.77 7.63 4.22 9.07 5.69 5.25 0.42 3.57 2.72 2.48 2.56 3.52 2.17 3.83 α 1 and ntttoa taey“h yegtcUiest”a ato h xelneIiitv yteGerman the by Initiative Excellence the of Balzani. part Daniel to as (DFG) Foundation University” Research Synergetic “The Strategy Institutional subsequent initialize to region AHA each in Different assigned 16). (region are infarct properties an material representing test the region representative stiffer five for significant in values a used initial containing values random model initial LV properties the material with and cases properties material Reference VI. Table NQEIETFCTO FMTRA RPRISUIGFL IL MEASURES FIELD FULL USING PROPERTIES MATERIAL OF IDENTIFICATION UNIQUE Region Infarct 17 16 15 14 13 12 11 10 Accepted9 8 Article7 6 5 4 3 2 1 Property α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α α 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 Reference 35.19 35.19 35.19 35.19 35.19 35.19 35.19 14.12 70.38 35.19 35.19 35.19 35.19 35.19 35.19 35.19 35.19 35.19 7.06 7.06 7.06 7.06 7.06 7.06 7.06 7.06 7.06 7.06 7.06 7.06 7.06 7.06 7.06 7.06 hsatcei rtce ycprgt l ihsreserved. rights All copyright. by protected is article This REFERENCES et1 Test 15.72 12.61 20.79 18.15 12.78 10.71 31.10 23.40 31.70 23.38 34.05 28.11 30.27 3.47 4.45 1.74 6.31 4.57 9.41 2.40 2.71 8.20 0.76 1.15 3.16 6.54 3.83 4.86 0.80 4.70 0.40 0.40 4.02 0.44 tests. nta us e etnumber test per guess Initial et2 Test 14.21 10.94 35.16 24.77 13.04 19.52 33.60 21.60 23.87 12.34 11.47 12.78 1.19 5.78 6.77 0.46 6.73 4.43 2.50 4.88 5.48 6.41 8.94 2.07 3.88 9.11 2.01 5.93 6.37 1.74 5.97 9.38 1.08 5.66 et3 Test 11.78 58.33 52.24 10.90 36.11 12.62 47.52 12.01 15.11 9.290 31.28 14.10 17.46 63.42 11.47 48.93 34.92 61.02 46.87 14.07 17.13 18.17 12.31 66.20 12.90 5.80 4.38 0.32 2.05 0.88 1.59 9.53 1.99 0.88 132.76 et4 Test 40.51 64.56 57.84 59.21 10.24 63.37 34.40 10.81 49.96 48.06 12.95 11.11 49.37 12.39 32.88 28.58 64.65 8.45 5.41 9.23 9.86 7.95 8.10 9.64 1.55 4.16 4.48 9.26 4.93 1.02 1.41 9.85 0.25 et5 Test 20.46 39.80 46.62 41.29 12.17 62.22 60.33 69.56 25.94 10.80 60.84 13.17 49.93 10.07 60.02 89.09 12.48 9.93 2.20 3.77 3.89 1.59 3.97 7.60 2.15 8.52 8.27 1.89 0.73 5.63 4.83 5.21 1.98 7.35 39 40 20. 19. 18. 17. 16. 15. 14. 13. 12. 11. 10. 9. 8. 7. 6. 5. 4. 3. 2. 1. upyM,Jh utnIIKG adc ,MyrF,LnioG orsJ,FlleJ,EmnRL. 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Hunter isotropic in AS, terms response Douglas stress orthogonal JD, Humphrey a in JC, failure Criscione to transition the in properties 291 diastolic dilatation. of cardiac Role the of JH. model Omens mouse AD, McCulloch LR, Frank PN, Costandi classification. pattern neighbor 13 Nearest PE. Hart TM, Cover T, Ryan JA, Rumberger DJ, Pennell WK, Laskey S, Kaul AK, MS, Verani Jacobs V, Dilsizian NJ, Weissman MD, Cerqueira Ennis WS, Klug Z, 2014; Qu JN, Weiss AV, Ponnaluri A, Frid DB, OA, Ajijola NP, Borgstrom LE, and Perotti structure S, of Krishnamoorthi MRI using modelling mechanics MRI. ventricular Passive tagged M. 3D function. Nash A, N, Young using D, Smith estimation Ennis R, H, parameter Lam Razavi V, Wang for J, laws Lee mechanobiology constitutive G, in modeling Carr-White cardiac and J, passive Biomechanics Wong of E, Analysis Sammut D. 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