Myocardial material parameter estimation

Biomechan Model Mechanobiol (2008) 7:161–173DOI 10.1007/s10237-007-0083-0

ORIGINAL PAPER

Myocardial material parameter estimationA non–homogeneous finite element study from simple shear tests

H. Schmid · P. O’Callaghan · M. P. Nash · W. Lin ·I. J. LeGrice · B. H. Smaill · A. A. Young ·P. J. Hunter

Received: 7 November 2006 / Accepted: 7 March 2007 / Published online: 9 May 2007© Springer-Verlag 2007

Abstract The passive material properties of myocardiumplay a major role in diastolic performance of the heart. In par-ticular, the shear behaviour is thought to play an importantmechanical role due to the laminar architecture of myocar-dium. We have previously compared a number of myocardialconstitutive relations with the aim to extract their suitabilityfor inverse material parameter estimation. The previous studyassumed a homogeneous deformation. In the present studywe relaxed the homogeneous assumption by implementingthese laws into a finite element environment in order to obtainmore realistic measures for the suitability of these laws inboth their ability to fit a given set of experimental data, aswell as their stability in the finite element environment. Inparticular, we examined five constitutive laws and comparethem on the basis of (i) “goodness of fit”: how well they fita set of six shear deformation tests, (ii) “determinability”:how well determined the objective function is at the optimalparameter fit, and (iii) “variability”: how well determined thematerial parameters are over the range of experiments. Fur-thermore, we compared the FE results with those from theprevious study.

It was found that the same material law as in the previ-ous study, the orthotropic Fung-type “Costa-Law”, was themost suitable for inverse material parameter estimation formyocardium in simple shear.

H. Schmid (B) · M. P. Nash · W. Lin · I. J. LeGrice · B. H. Smaill ·A. A. Young · P. J. HunterBioengineering Institute, University of Auckland,Private Bag 92019, Auckland, New Zealande-mail: [email protected]

P. O’CallaghanAgResearch, Hamilton, New Zealand

1 Introduction

Cardiovascular disease is a frequent cause of death andmorbidity world-wide (Reddy and Yusuf 1998). The mechan-ical properties of ventricular muscle substantially influencethe pumping function of the heart. The mechanical propertiesof passive myocardium play a major role during the diastolicfilling phases of the heart cycle. Stiffening of the myocardiumleads to impaired filling, which can lead to increased fillingpressure, increased cardiac work, and ultimately decreasedpump function via the Frank-Starling mechanism. Such dia-stolic dysfunction is often associated with heart failure andmay be observable before appreciable evidence of systolicdysfunction (Mandilov et al. 2000). Thus, an understandingof the passive mechanical properties of the myocardium iscentral to the understanding of these disease processes.

Early mathematical models of the whole heart focussedon the relationship between blood pressure and cavity vol-ume, which has been used by clinicians for many years (Sugaet al. 1973; Janicki and Weber 1977). In recent decades,it has become apparent that an improved understanding ofregional variation of myocardial tissue properties is impor-tant to understand the fundamental mechanisms underlyingventricular mechanics, such as wall thickening and shear-ing deformations. The apparent heterogeneous, anisotropicmechanical properties have since been represented using avariety of material laws (ML) based on a different theoreticalframeworks, ranging from elastic to viscoelastic, and fromphenomenological to microstructurally based approaches. Ithas been demonstrated that myocardial tissue exhibits a non-elastic response (Dokos et al. 2002). Several studies havefitted rheological constitutive relations to viscoelastic data(Bischoff et al. 2004; Miller and Wong 2000), however, atpresent there are insufficient data on the viscoelastic proper-ties of passive ventricular myocardium at the physiological

123

162 H. Schmid et al.

strain rates occurring in vivo, particularly in shear modes ofdeformation (relative to the laminar architecture). Many stud-ies have focussed the analysis to the hyperelastic properties,with particular attention paid to anisotropy.

Guccione et al. (1991) modelled the equatorial region ofthe canine left ventricle as a thick–walled cylinder consist-ing of an incompressible transversely isotropic exponentialFung-type hyperelastic material (Fung 1965, 1993). Subse-quently, LeGrice et al. (1995a) showed that the microstruc-ture is a composite of discrete layers of myocardial fibres ofusually four to six cells thick, which suggested an orthotropicmechanical response. To this end, the transversely isotropicFung-type relation was extended to account for orthotropy byCosta et al. (2001). Another approach using an orthotropic“pole-zero” formulation was proposed by Nash and Hunter(2000).

Various studies have shown the significance of shear defor-mation in cardiac mechanics, (Arts et al. 2001; LeGrice et al.1995b) and the shear properties of passive ventricular myo-cardium remain poorly characterized. We recently compareda number of myocardial constitutive relations with the aimto extract their suitability for inverse material parameter esti-mation (Schmid et al. 2006). In the context of three dimen-sional simple shear experiments from Dokos et al. (2002),and assumed a homogeneous deformation. In this study, werelax this assumption, by using finite element analysis meth-ods to account for the non-homogeneous deformations thatoccur during simple shear due to the Poynting-effect (Poynt-ing 1909). This allowed us to compare the finite elementresults with our previous homogeneous results (Schmid et al.2006). This comparison is very useful for experimentalistswho are generally concerned with the assumption of homo-geneity in simple shear and biaxial tests, e.g. Gardiner andWeiss (2001).

In this study, we compared five constitutive laws:

1. Costa law (CL)2. Separate Fung-type law (SFL)3. Pole-zero law (PZL)4. Tangent law (TL)5. Langevin Eight-chain law (LECL).

Further details of these material laws are presented in the nextsection. SFL and TL were designed to have similar featuresto CL and PZL, respectively, and were investigated to deter-mine whether any improvement could be obtained over thelatter relations. These constitutive relations provide charac-terizations of experimental data and material properties thatare useful in practice. Note that theoretically desirable prop-erties such as polyconvexity and fibre dispersion, etc. (Itskovand Aksel 2004; Gasser et al. 2006; Leonov 2000; Lainé et al.1999) were not accounted for in the design of these materiallaws and will be the subject of future studies.

To compare parameter estimation results for the differentlaws, we examined three major criteria. “Goodness of fit” isthe ability of a material law to minimize a given objectivefunction. This is quantified by two measures being the rela-tive error of the estimate, and the Akaike Information Crite-rion (AIC) (Burnham and Anderson 2002), which penalisesthe number of material parameters in each model. “Deter-minability” quantifies how well the material parameters aredetermined at the optimum by means of optimality criteria,see Lanir et al. (1996). Parameter “Variability” over the rangeof experiments for a given material law was also examined.

We start this report by presenting the theoretical back-ground for the material parameter estimation process (i.e.the details of five material laws, and their similarities), afinite element implementation, a model convergence anal-ysis, and the objective function we used. This is followedby a descriptive summary of the three comparison measures,the results of the numerical studies, and the subsequent sta-tistical analysis. We also compare these finite element studyresults with the results from our previous homogeneous simu-lations (Schmid et al. 2006). Finally, we end with a discussionof the advantages and limitations of the various constitutiverelations.

2 Methods

We first briefly summarize the fundamentals of continuummechanics. The deformation is described by the deformationgradient tensor F. The strain is quantified using the Greenstrain tensor E = 1

2 (FTF− I). The balance of linear momen-

tum is expressed using Eq. (1), where S is the second Piola–Kirchhoff stress tensor, (Holzapfel 2000).

Div(F S) = 0 (1)

Assuming hyperelasticity the remaining relationship betweenthe stress S and the strain E is then specified by a function,the strain energy density Ψ = Ψ (E).

2.1 Tissue experiments

We base our modelling investigations on experimental datataken from Dokos et al. (2002). Passive shear properties ofsix pig hearts were examined. Samples (∼3 × 3 × 3 mm)were cut from adjacent regions of the lateral left ventricularmidwall, with sides aligned with the microstructural mate-rial axes ( f, n, s; f iber, normal, sheet). Sinusoidal cycles ofsimple shear (shear displacement range [−50%, 50%]) wereapplied separately to each specimen in two orthogonal direc-tions. Three specimens from each heart were tested in twodirections, giving all six possible modes of shear with respectto the microstructural axes. Data for the fitting of material

123

Myocardial material parameter estimation 163

properties were taken from cycles after strain softening haddiminished. The forces on the top face of the cubes weremeasured and taken as the data for the material parameteroptimization, which is described below.

2.2 Constitutive laws

Here we list the strain energy density functions for all con-stitutive relations. For further details see Schmid et al. (2006)or the references therein.

The CL (Costa et al. 2001), has the following strain energydensity function with seven material parameters (a, bαβ ):

ΨCL(Ef f , Efn, Efs, Enf , Enn, Ens, Es f , Esn, Ess)

= 1

2a(eQ − 1)

where

Q = bf f E2f f + 2bfn

(1

2(Efn + Enf )

)2

+2bfs

(1

2(E fs + Esf )

)2

+ bnn E2nn

+2bns

(1

2(Ens + Esn)

)2

+ bss E2ss (2)

The (SFL) was motivated by the desire to decouple the mate-rial parameters from the single exponential in the CL and has12 material parameters (aαβ, bαβ ):

ΨSFL(Ef f , Efn, Efs, Enf , Enn, Ens, Esf , Esn, Ess)

= 1

2af f (e

bf f E2f f − 1) + 1

2afn(e

bfn( 12 (Efn+Enf ))

2 − 1)

+ 1

2afs(e

bfs (12 (Efs+Esf ))

2 − 1)

+ 1

2ann(ebnn E2

nn − 1) + 1

2ans(e

bns (12 (Ens+Esn))2 − 1)

+ 1

2ass(e

bss E2ss − 1) (3)

The PZL has the following strain energy density (Nashand Hunter 2000) with 12 material parameters (kαβ, aαβ ):

ΨPZL(Ef f , Efn, Efs, Enf , Enn, Ens, Esf , Esn, Ess)

= kf f E2f f

|af f − |Ef f ||2 + kfn(12 (Efn + Enf ))

2

|afn − | 12 (Efn + Enf )||2

+ kfs(12 (Efs + Esf ))

2

|afs − | 12 (Efs + Esf )||2

+ knn E2nn

|ann − |Enn||2

+ kns(12 (Ens + Esn))2

|ans − | 12 (Ens + Esn)||2 + kss E2

ss

|ass − |Ess ||2 . (4)

The TL was adapted to relax the infinite slope of the PZLat the poles and has 12 material parameters (aαβ, bαβ ):

ΨTL(Ef f , Efn, Efs, Enf , Enn, Ens, Esf , Esn, Ess)

= 1

2af f IntTan

(bf f E2

f f

)

+1

2afnIntTan

(bfn

(1

2

(Efn + Enf

))2)

+1

2afsIntTan

(bfs

(1

2

(Efs + Esf

))2)

+1

2annIntTan

(bnn E2

nn

)

+1

2ansIntTan

(bns

(1

2(Ens + Esn)

)2)

+1

2assIntTan

(bss E2

ss

), (5)

where IntTan(x) is the indefinite integral of Tan(x), a trun-cated Taylor series expansion of the tangent function to thefifth order.

Bischoff et al. (2002) presented the Langevin eight chainmodel (LECL), which differs from the above relations in thatit is based on micro–structural modelling of macromolecules.The deviatoric part of the strain energy density function has4 material parameters (a, b, c, n):

ΨLECL(Ef f , Efn, Efs, Enf , Enn, Ens, Esf , Esn, Ess)

= Ψ0 + nkθ

4

(N

4∑i=1

[ρ(i)

Nβ(i)

ρ + lnβ

(i)ρ

sinh β(i)ρ

]

− βP√N

ln[λa2

a λb2

b λc2

c ])

. (6)

The inverse Langevin function is used during the computa-tion of the stress strain relationship. Since no closed formof the inverse function exists we utilize the so–called Padéapproximant function (Cohen 1991):

L−1(x) = x3 − x2

1 − x2 + O(x6) . (7)

2.3 Finite element implementation

We describe the implementation of this model into the finiteelement environment CMISS (http://www.cmiss.org). Afinite element model was created for each of the three sepa-rate tissue blocks in each of the six sets of experiments. Eachblock was given a cuboid geometry with the recorded dimen-sions. Two shear modes were applied in each block in orderto cover the 6 different shear modes for each experiment.

The forward solution of the finite elasticity equations overthe mathematical representations of the ventricular myocar-dium were solved using the Galerkin finite element methodincorporating tri-linear 8-node elements. Figure 1 shows atypical finite element model as used in this study. The simple

123


Fig. 1 This graph shows the undeformed and deformed finite elementmesh. The mesh has five elements in each direction. The boundary con-ditions are imposed on the bottom and top surface. The bottom surfaceis fixed and the top surface is displace into the positive x-direction byhalf the height of the cube. Note the bulging as typical for the non-homogeneous simple shear deformation

shear deformation was modelled as a xz-shear, i.e. the topface with its normal in the z-direction was displaced in the x-direction. Each element incorporated the fibre and the sheetorientation of the tissue as appropriate for each of the exper-imental tests. A variety of mesh resolutions were tested andthe results of the convergence analysis is presented in thenext section.

The constitutive relations were implemented using Cell-ML (http://www.cellml.org), an XML based markup lan-guage which is compatible with the finite element environ-ment. All laws were validated against the same functionalform of the stress–strain relationship implemented using Mat-lab (http://www.mathworks.com).

The incompressibility constraint was enforced through aLagrange multiplier as presented by Nash and Hunter (2000).

2.4 Objective function

The top face force from the finite element model was uti-lized in the objective function to build a modified least squareerror Ω between the experimental (texp) and the model results(tmod) for a given set of material parameters ϑ . See Appendix1 for the derivation of this objective function.

Ω(ϑ) = 1

2

∑modes

∑x,z-force

G∑j=1

ω j(tmod(ϑ, x j ) − texp(x j )

)2

(8)

where G is the number of Gaussian quadrature points foreach of the 12 displacement–force curves and ω j and x j arethe associated weights and Gauss points, respectively.

The use of 12 Gauss points reduced the fit error to less than0.01% (Schmid et al. 2007). This also reduced computationaltime by 98% due to the fact that just 144 data points wererequired as opposed to the approximately 3,000 provided inthe full data set.

By setting t jmod(ϑ, x) = 0 in Eq. 8, we obtain a physi-

cally meaningful measure of error for each experiment as itprovides an estimate of the “energy content” ΩT. We use thisto scale the magnitude of the objective function to provide arelative error ΩRel = Ω(ϑ0)/Ω

T.

2.5 Optimization kernel

A sequential quadratic programming (SQP) algorithm wasused to optimize the material parameters for each constitutivelaw. At each step in the optimization process SQP involvesthe solution of a quadratic problem with linear constraints.The Hessian is approximated using the local gradient, as iscommon for sums of squares problems. The derivatives of theobjective function with respect to the optimization variables(the material parameters) was performed using one-sideddifferences. Initial estimates of the material parameters weretaken from the homogeneous studies Schmid et al. (2006).Constraints in terms of interval bounds on the material param-eters were imposed to ensure a valid forward solution. Foreach optimization iteration, a series of finite elasticity prob-lems must be solved. One for the current solution, and onefor each finite difference derivative approximation. In eachof these finite element solutions, the solver is started fromthe previous deformation solution.

2.6 Comparison amongst material laws and models

This sections presents the measures to compare the featuresof the different material laws and models:

Goodness of fit The first measure is the objective functionvalue at the optimum Ω(ϑ0). We utilize the energy contentto normalize this value to form a “relative objective functionvalue” ΩRel = Ω(ϑ0)/Ω

T.

Mean, standard deviation and coefficient of variation Stan-dard measures to assess the behaviour of a given quantity arethe mean µ and standard deviation σ of e.g. ΩRel over allexperiments for each material law. A standard measure toassess the relative variation of a given quantity is the coeffi-cient of variation CoV= σ

µ. These measures have also been

used for the other criteria introduced below.

Akaike Information Criterion (AIC) In order to comparematerial laws effectively one must also take into account (and

123


penalise) the number of material paramaters in each law. Thisis achieved using the (AIC):

AIC = N ln

(1

NΩ (ϑ)

)+ 2K , (9)

where K denotes the number of material parameters and Nthe number of data points. The best model is defined as themodel with the lowest AIC.

Determinability Here we investigate the shape of the multi-dimensional space in the neighbourhood of the optimal pointusing three measures.

1. det(H0) at the optimal point (where H0 is the Hessianof the objective function at the optimal point) representsthe volume of the so-called indifference region (Laniret al. 1996) and is also referred to as the D-optimalitycriterion. The higher the value of this number, the lowerthe variance the material parameters.

2. The condition number of the Hessian at the optimum,cond(H0), describes the ratio between the highest andthe lowest eigenvalues of H0 which indicates the eccen-tricity of the hyperellipsoid.

3. The so-called M-optimality criterion relates to the inter-actions between material parameters and it is defined as:

det(H0) where Hi j = Hi j

Hii Hj j(no sum) . (10)

Equation 10 describes the alignment of the hyperellip-soid with the axes of the material parameters. (Hi j = δi j

for perfect alignment, which corresponds to zero corre-lation between the material parameters.)

Material parameter variability amongst models In the com-parison of the values between the homogeneous simulationsand the FEM simulations, or between several refinementsof the finite element study for the convergence analysis, itis important to account for the varying magnitudes of theindividual parameters. We therefore utilized the following

measure ∆mα\mβκ in order to compare a specific quantity κ

between two different models α and β:

∆α\βκ = |κα − κβ |

κβ

. (11)

For example ∆homo\FEMa would denote the comparison of the

material parameter a of the homogeneous model with the oneobtained from the finite element solution, whereas ∆

222\333Ω

denotes the difference in the objective function between twodifferent mesh resolutions, i.e. a model with 8 = 2×2×2 ele-ments (two elements in each direction) versus a model with27 = 3 × 3 × 3 elements (three elements in each direction).

Furthermore it is helpful to employ an overall measurethat compares all material parameters (MP):

∆α\βM P =

K∑i=1

∆α\βγi

, (12)

where K denotes the number of material parameters for thegiven constitutive relation and γi a material parameter.

3 Results

This section firstly presents the convergence analysis of theFE mesh, secondly the results of the FE simulations for allexperiments and lastly a comparison between the homoge-neous and the FE model.

3.1 Convergence analysis

We used tri-linear models with an equal number of elementsin each direction. We started with an eight element cube (222cube) (two elements in each direction), and refined this upto a 888 cube (512 elements). We checked for convergencewith respect to two criteria: the objective function value and

the ∆mα\mβ

M P criterion.The sequence was repeated with differing starting values

and limits, until a minimum least squares error between thepredicted and observed reaction forces was obtained. Theinitial estimates from the homogeneous study proved to bevery close to the optimum for almost all cases.

Table 1 shows the numerical results of the convergenceanalysis for the CL. The analysis indicated that a 555 cubewas sufficiently converged for the study since the 666 cube

improved ∆mα\mβ

M P by just 0.91% and Ω by just 0.50%.Furthermore the convergence analysis for the SFL, PZL

and TL showed very similar results, which again confirmedour choice of the 555 cube. However, the LECL did notconverge for any of the experiments when starting from thehomogeneous values. We performed a considerable numberof tests from varying initial parameters, but this did not resultin a successful optimisation.

3.2 Results for FEM simulations

The detailed numerical results for all material laws are givenin Tables 2 and 3 (Tables 4, 5 can be found in Appendix 2). Welist all material parameter values and for each of these entrieswe present the mean µ, standard deviation σ and coefficientof variation (CoV= σ

µ) across the experiments. We also list

the total pseudo–energy content (ΩT) for each experiment inthe last column of Table 2.

123


Tabl

e1

Con

verg

ence

anal

ysis

for

the

CL

CL

Ω∆

α\β

Ωm

ax(∆

α\β

γi

)∆

α\β MP

a∆

α\β

ab

ff∆

α\β

bff

bfn

∆α\β

bfn

bfs

∆α\β

bfs

b nn

∆α\β

b nn

b ns

∆α\β

b ns

b ss

∆α\β

b ss

(%)

(%)

(%)

(%)

(%)

(%)

(%)

(%)

(%)

(%)

111

127.

20.

171

34.0

11.1

12.6

19.3

9.0

13.1

222

121.

44.

8225

.710

.60.

189

9.3

36.7

7.3

10.4

6.6

12.1

3.9

22.8

15.5

8.5

5.7

17.6

25.7

333

121.

30.

0622

.412

.40.

184

2.6

32.2

13.7

11.6

10.1

13.4

9.5

19.3

18.3

9.5

10.3

14.4

22.4

444

122.

61.

055.

02.

90.

185

0.38

31.2

3.5

11.9

2.7

13.7

2.6

18.4

5.0

9.7

2.5

13.8

3.9

555

123.

91.

073.

21.

90.

187

0.94

30.5

2.3

12.1

1.7

13.9

1.7

17.8

3.2

9.9

1.3

13.6

2.1

666

124.

60.

501.

40.

910.

187

0.51

30.2

0.97

12.2

0.90

14.1

0.94

17.6

1.4

9.9

0.74

13.4

0.89

777

124.

80.

210.

990.

670.

188

0.36

30.0

0.65

12.3

0.68

14.2

0.72

17.4

0.99

10.0

0.57

13.3

0.70

888

124.

90.

090.

580.

420.

189

0.20

29.9

0.35

12.3

0.48

14.3

0.50

17.3

0.58

10.0

0.43

13.3

0.38

Itca

nbe

seen

that

for

the

555

cube

max

(∆α\β

γi

)is

3.2%

and

∆α\β

Ω1.

07%

.Itw

asth

eref

ore

conc

lude

dth

atth

e55

5cu

bew

assu

ffici

ently

conv

erge

dto

mod

elth

esi

mpl

esh

ear

defo

rmat

ion.

See

text

for

expl

anat

ion

and

defin

ition

ofsy

mbo

ls

Tabl

e2

Com

pari

son

ofm

ater

ialp

aram

eter

estim

ates

acro

ssC

Lfo

ral

lexp

erim

ents

.See

text

for

expl

anat

ion

and

defin

ition

ofsy

mbo

ls

CL

ΩΩ

Rel(%

)A

ICR

ank

det(

H)

cond

(H)

det(

H)

ab

ffb

fnb

fsb n

nb n

sb s

sΩ

T

Exp

198

2.4

2.5

134.

13

2.2E

+22

4.1E

+08

1.4E

-14

0.41

36.3

10.7

12.7

12.3

7.96

11.4

39,7

48

Exp

217

986.

917

1.9

3−2

.4E

+18

5.4E

+08

−1.6

E-0

90.

3320

.714

.411

.30.

0017

.033

.826

,217

Exp

312

3.9

1.5

4.6

42.

6E+

202.

1E+

094.

0E-1

10.

1930

.512

.113

.917

.89.

8613

.68,

021

Exp

422

2.8

2.6

41.3

36.

6E+

183.

4E+

082.

3E-1

10.

2338

.011

.311

.54.

4311

.29.

608,

457

Exp

543

2.4

1.5

82.8

4−1

.3E

+22

1.1E

+10

−2.0

E-1

20.

2035

.711

.410

.513

.08.

3518

.928

,332

Exp

617

5.2

1.3

26.3

1−1

.4E

+21

1.2E

+09

−2.4

E-1

20.

1862

.312

.012

.37.

0610

.926

.313

,613

µ62

2.4

2.7

76.8

1.2E

+21

2.6E

+09

−2.5

E-1

00.

2637

.212

.012

.09.

1110

.918

.920

,731

σ65

6.5

2.1

65.4

1.1E

+22

4.3E

+09

6.4E

-10

0.09

113

.81.

301.

216.

493.

289.

4612

,746

CoV

105.

5%77

.285

.1%

904.

8%16

1.1%

−256

.6%

35.8

%37

.0%

10.9

%10

.1%

71.2

%30

.1%

50.0

%61

.5%

123


Tabl

e3

Com

pari

son

ofm

ater

ialp

aram

eter

estim

ates

for

SFL

acro

ssal

lexp

erim

ents

.See

text

for

expl

anat

ion

and

defin

ition

ofsy

mbo

ls

SFL

ΩΩ

Rel(%

)A

ICR

ank

det(

H)

cond

(H)

det(

H)

aff

bff

afn

bfn

afs

bfs

a nn

b nn

a ns

b ns

a ss

b ss

Exp

174

2.4

1.9

126.

61

4.2E

+61

4.0E

+07

2.8E

-35

0.85

42.2

0.02

458

.40.

014

75.2

0.00

8715

7.9

0.04

035

.81.

0211

.7

Exp

214

995.

717

0.5

22.

1E+

561.

2E+

114.

3E-2

00.

1266

.60.

012

75.7

0.05

945

.42.

950.

380.

082

52.0

0.78

34.2

Exp

394

.21.

2−2

.51

5.9E

+54

2.4E

+12

4.7E

-10

0.28

50.3

0.01

751

.60.

022

55.0

0.18

36.4

0.01

546

.10.

045

56.8

Exp

418

8.7

2.2

40.9

21.

5E+

473.

3E+

136.

0E+

010.

2273

.70.

051

37.9

0.02

548

.80.

0010

0.0

0.05

833

.00.

011

89.1

Exp

530

1.2

1.1

70.2

29.

7E+

619.

0E+

126.

6E-1

50.

2757

.60.

016

54.1

0.01

847

.50.

1039

.60.

0066

57.0

0.01

712

3.2

Exp

617

5.7

1.3

36.4

25.

9E+

577.

2E+

102.

5E-2

20.

2598

.40.

0061

70.3

0.02

151

.31.

492.

610.

0090

59.3

0.15

53.8

µ50

0.2

2.2

73.7

2.3E

+61

7.4E

+12

9.9E

+00

0.33

64.8

0.02

158

.00.

026

53.9

0.79

56.1

0.03

547

.20.

3461

.5

σ54

0.9

1.8

64.0

4.0E

+61

1.3E

+13

2.4E

+01

0.26

19.9

0.01

613

.60.

016

11.0

1.20

61.5

0.03

010

.90.

4439

.7

CoV

108.

1%79

.586

.8%

171.

8%17

4.6%

244.

9%78

.0%

30.7

%76

.5%

23.5

%62

.0%

20.4

%15

2.6%

109.

6%87

.2%

23.2

%13

1.9%

64.6

% Before making any comparison it is important to note thatexperiments 2 and 4 yielded comparably poor results for allmaterial laws. Leaving out these experiments would there-fore yield a much closer material parameter set for all materiallaws. These poor results are most likely due to the fact thatthe non–homogeneous aspect of micro-structural fiber orien-tation was not measured. Inverse finite element studies thatinclude this aspect may shed some more light on the possiblereasons.

Comparing the mean of the relative goodness of fit of thefinite element study amongst all four material laws, the SFLobtained the best relative goodness of fit (2.2%), whereas thecoefficient of variation of the objective function was lowestfor the TL (66.7%). The AIC confirms the result for the SFL.

Comparing the CoV of material parameters for all lawswe find that the CL has the lowest (71.2%) for the parameterbnn whereas PZL has the highest (210%) for ann .

The CL did converge without any complications and tookthe shortest computational time (∼8 h) on an IBM 1.9 GHzPower 5 Processor when starting from the homogeneousparameter estimation values. Varying the initial starting pointof ϑ had no effect on the final outcome. We therefore con-cluded that the CL was very stable for the estimation process.

The TL also converged when starting from the homoge-neous values. However, the step size in the parameter spaceneeded to be decreased for a stable optimization. Thisincreased the optimization time to between 4 and8 days.

The other laws were rather unstable and required moresophisticated estimation strategies which we outline now.Since the estimation process did not work initially the searchspace was restricted. The approach itself can be subdividedinto two parts:

1. fixing a given set of material parameters while estimatingfor the rest(a) fix axial parameters, estimate shear parameters(b) fix shear parameters, estimate axial parameters(c) estimate all parameters

2. refinement of mesh.

For the SFL and PZL the above approach was used. We hadto start at the 222 cube and we needed to repeat this for allintermediate meshes to finally obtain a fully converged 555cube.

The first part consisted of three substeps which can beexplained as follows. The mode of deformation is simpleshear, so the shear parameters account for the majority ofthe energy content of all modes. Fixing the axial ones andestimating shear parameters therefore ensures that the shearparameters are allowed to optimise the objective function first(fixing the shear parameters and estimating the axial param-eters as a first substep lead to an immense overestimationof the axial terms, since they would attempt to minimise the

123


Fig. 2 Experimental (dotted)and fitted force–displacementcurves (solid) of the SFL for allsix modes for experiment 3.Groups of two pictures show thex- and z-force, respectively. Theoverall error is 1.2%. Note thedifferent scales on each graph.The abscissa shows thedisplacement in mm, whereasthe ordinate shows the top faceforce in m N , where e.g. NSxindicates the x-force for theNS-mode

−1 0 1

−40

−20

0

20

40

NSx

−1 0 1

0

10

20

NSz

−1 0 1

−50

0

50

NFx

−1 0 1

0

10

20

30

NFz

−1 0 1

−10

0

10

SNx

−1 0 1

0

5

10

SNz

−1 0 1

−40

−20

0

20

40

SFx

−1 0 1

0

5

10

15

SFz

−1 0 1

−50

0

50

FSx

−1 0 1

0

20

40

60

FSz

−1 0 1

−50

0

50

FNx

−1 0 1

0

20

40

FNz

error of a shear deformation through overly stiff axial behav-iour). Naturally after the first two substeps all parameterswere being estimated for a given refinement.

Guccione et al. (1991) published a transversely isotropicmaterial which we also fitted to all six experiments. It exhib-ited very poor behaviour since it was only able to fit threeout of the six modes, those with the highest partial energycontent. It can therefore be concluded that a transversely iso-tropic material is not suitable to model the passive myocardialbehaviour in simple shear.

The D-optimality for all laws reflect the stability of theoptimization process. The higher the numbers, the worse theconvergence. The condition numbers for all material lawsshow that the SFL had the highest eccentricity with 7.4×1012

whereas CL was lowest 2.6 × 109. The M-optimality againshows that the CL has the lowest material parameter corre-lation whereas the PZL has the highest.

One of the advantages of the homogeneous model werethought to be that it gives good first estimates for more real-istic finite element studies. It is therefore also useful to com-

ment on the performance of the material laws with respectto the convergence behaviour and the computational timeinvolved when compared to the homogeneous model.

3.3 Comparison of FEM and homogeneous models

When we compared the homogeneous study with the FEMstudy we found that the difference in information by addingthe finite element study in terms of the mean of the goodnessof fit criterion ∆

homo\FEMΩ was lowest for the CL (8.19%)

and highest for the SFL (21.7%). The same was true for themean increase in the ∆

homo\FEMAIC , where CL has the lowest

value (0.05%) and the SFL had the highest value (24.1%).These numbers were obtained by comparison of the tableslisting the results of the homogeneous model in Schmid et al.(2006).

When comparing the individual ∆homo\FEMγi then SFL,

PZL and TL have outliers in the order of 105 and higherfor ann, knn, ann for the fourth experiment, respectively. The

CL again had the highest ∆homo\FEMγi for bnn for the fourth

123


Fig. 3 Experimental (dotted)and fitted force–displacementcurves (solid) of the CL for allsix modes for experiment 3.Groups of two pictures show thex- and z-force, respectively. Theoverall error is 1.5%. Note thedifferent scales on each graph.The abscissa shows thedisplacement in mm, whereasthe ordinate shows the top faceforce in m N , where e.g. NSxindicates the x-force for theNS-mode

−1 0 1−40

−20

0

20

40NSx

−1 0 1

0

10

20

NSz

−1 0 1

−50

0

50

NFx

−1 0 1

0

10

20

30

NFz

−1 0 1

−10

0

10

SNx

−1 0 1

0

5

10

SNz

−1 0 1−40

−20

0

20

40SFx

−1 0 1

0

10

20SFz

−1 0 1

−50

0

50

FSx

−1 0 10

20

40

60

FSz

−1 0 1

−50

0

50

FNx

−1 0 1

0

20

40

60FNz

experiment (104.1%). On one hand this is pointing towardsthe poorer material parameter estimation capability of thehomogeneous simulations, since it usually reached large neg-ative values for this experiment, as well as towards the factthat the optimization package of the finite element envi-ronment reached the lower bound imposed on the materialparameter. Furthermore this points towards the fact that theparameters of the normal direction are those being most diffi-cult to estimate due to the lowest partial energy content ofthe NF-mode (4.0%) versus (44.9%) in the FN-mode.

The comparison of ∆homo\FEMΩRel

and ∆homo\FEMAIC may be

interpreted the following way. Firstly it means that the SFLis ideally used to minimise the relative error in the finite ele-ment environment where it performs best. The CL, however,seems to perform almost identically in the homogeneoussimulations and in the finite element simulations, while per-forming almost as well as the SFL in terms of the goodnessof fit criteria, see also Figs. 2, 3 and 4.

When comparing the material parameter consistency by

looking at the mean of all ∆homo\FEMγi then CL performs best

with 15.3%, see also Fig. 5. If one disregards the second andfourth experiment, then ∆

homo\FEMγi for the TL (27.1%) and

therefore also performs well. The PZL and SFL, however,have values of 67.7 and 138.0%, respectively.

The LECL exhibited comparably poor behaviour in thehomogeneous study, i.e. it usually was not capable of fittingthe weaker modes. This result was confirmed by forwardsimulations in this study and is most likely the reason why itwas not possible to use it in the inverse material estimationprocess. Please note that the underlying assumption of themicrostructure of the LECL does not resemble the laminarstructure of the myocardium. Using macromolecular basedconstitutive laws might therefore not be suitable for the myo-cardial sheet structure.

4 Discussion

In this paper, we have examined five alternative forms ofconstitutive laws for representing the stress-strain behav-iour of passive myocardial tissue. In order to examine the

123


CL SFL PZL TL

50

100

150

200

250

300

max CoV

CL SFL PZL TL

10

20

30

40

50

60

70

CoV

CL SFL PZL TL

0.5

1

1.5

2

2.5

Rel

Fig. 4 Top Comparison of ΩRel of all laws between the homogeneousresults (grey) and the FE results (black). The ordinate shows percent-ages.; all laws perform comparably similar for both models. Middlecomparison of µCoV for all laws; bottom comparison of maxCoV for alllaws. CL stands out for both variability measures

effectiveness of these laws, we examined their applicationto experimental shear tests. Three measures were used toassess the five constitutive laws. The first (goodness-of-fit)was a measure of how well each optimized constitutive lawfitted the experimental data from the six tests, and thesecond (determinability) measured how sensitive theoptimal fit was to small errors in the data and the third(variability) measured the variance of the material param-eters over the range of experiments. Furthermore the

Homo FE

8

12

16

bns

Homo FE

10

20

30

bss

Homo FE

10

12

14bfs

Homo FE

0

5

10

15

bnn

Homo FE

30

50

70

bff

Homo FE

10

12

14

bfn

Homo FE

0.2

0.3

0.4

a

Fig. 5 Comparison of variability of CL parameters across experiments.Modified box whisker plots of all material parameters compare thehomogeneous (Homo) data set (left, dark grey) and the FE (FE) data set(right, light grey). The dashed line indicates the mean of the materialparameter. The box encapsulates all values between the lower and upperquartile and the “whiskers” indicate the lowest and highest value. Thegraphs indicate the good agreement between homogeneous and finiteelement values

∆α\βκ -criterion was utilised to quantify the difference in

material parameters between the homogeneous case and thefinite element simulations as well as for the convergenceanalysis.

Our results show that the CL performed best for bothhomogeneous simulations and inverse finite element mate-rial parameter estimations. This is clear from the fact thatalthough the goodness of fit and AIC of the SFL is slightlybetter than that for the CL in the FEM study, the CL hasby far the highest material parameter consistency and the

123


lowest computational time involved when compared to theother laws.

There are some issues regarding CL that require furtherattention. It exhibits a theoretical cross-coupling of strainterms for each stress component, whereas this is not thecase for the other three laws. In the homogeneous simulation(which has a sparsely populated strain tensor), this cross-coupling did not occur in the analytic expression of the topface force. We were therefore cautious that this might differ-entiate the CL from the other three laws when using FEMinverse parameter estimations, especially with more com-plex deformation modes. The results of this study indicatethat the cross-coupling does not play a major role for thefinite element simulations.

It is worth pointing out that in our experience the CLalso performed the most stable in forward solutions. Theother laws, however, are certainly still suitable for forwardsolutions. In further studies, we will extend the experimentalprotocol from merely simple shear modes by adding uniax-ial extension modes. Furthermore, this study provides a solidbackground for identifying a constitutive relation for the sys-tem identification process of multi–scale constitutive models(Schmid et al. 2005).

The shear modes are assumed to play a critical role in myo-cardial deformation (Arts et al. 2001; LeGrice et al. 1995b).Smaill and Hunter (1991) found that there was little mechan-ical coupling between the fiber and sheet direction in mid-myocardial specimen in biaxial tests. It therefore remains anopen question whether results for the constitutive relationsof inverse material parameter estimation procedures woulddiffer in biaxial extension tests.

Acknowledgments Holger Schmid was funded by the InternationalDoctoral Scholarship of the University of Auckland. The authors wouldlike to thank Socrates Dokos from the University of New South Walesfor making the data available.

Appendix 1

Numerical computations become expensive when perform-ing inverse finite element parameter estimations. The tra-ditional method of using a least squares objective functioncan be modified to avoid such expensive computations. Themodified objective function used in this study is described asfollows.

The conventional least squares objective function involvesthe summation over all six modes, all three directions of thetop face force and all data points of each mode and forcedirection, resulting in approximately 6 × 2 × 250 = 3, 000data points:

Ω(ϑ) = 1

2

∑modes

∑x,z-force

∑data

points

(tana(ϑ) − texp

)2, (13)

where ϑ is the vector of all material parameters. By addinga “weight” to each addend, namely the width ∆x of eachinterval of two successive data points, the objective functionapproximates the following integral, assuming that the datapoints imply a piecewise linear function.

Ω(ϑ) = 1

2

∑modes

∑x,z-force

∑data

points

(tana(ϑ) − texp

)2∆x

≈ 1

2

∑modes

∑x,z-force

12 γi∫

− 12 γi

(tana(ϑ, x) − texp(x)

)2 dx

(14)

By choosing this weighting the integral forms a L2-normin the functional space of squared integrable functions, andcan therefore serve as a measure of the length of the error.This also holds for the piecewise linear approximating func-tions. This measure can be interpreted as a “pseudo-energycontent” (pseudo, because the dimensions of the integrals areJ 2/m) and serves as a reference for the minimized objectivefunction to obtain a relative error.

The above formulation suggests that it would be numeri-cally more efficient to approximate the integral via a Gaussianquadrature integration method, see for example Press et al.(1989). This would then read:

12 γi∫

− 12 γi

(tana(ϑ, x) − texp(x)

)2 dx

≈G∑

j=1

ω j(

tana(ϑ, x j ) − texp(x j ))2

, (15)

where G is the number of Gauss quadrature points for eachof the twelve displacement–force curves. The objective func-tion then reads:

Ω(ϑ) = 1

2

∑modes

∑x,z-force

G∑j=1

ω j(

tana(ϑ, x j ) − texp(x j ))2

(16)

The convergence analysis of this modified objective functionis detailed in Schmid et al. (2007).

Appendix 2

This appendix presents the detailed tables for the PZL andthe TL (Tables 4, 5).

123


Tabl

e4

Com

pari

son

ofm

ater

ialp

aram

eter

estim

ates

for

PZL

acro

ssal

lexp

erim

ents

.See

text

for

expl

anat

ion

and

defin

ition

ofsy

mbo

ls

PZL

ΩΩ

Rel(%

)A

ICR

ank

det(

H)

cond

(H)

det(

H)

kff

aff

kfn

afn

kfs

afs

k nn

a nn

k ns

a ns

k ss

a ss

Exp

180

4.4

2.0

131.

62

−3.E

+10

79.

5E+

10−5

.E-6

12.

460.

430.

049

0.34

0.03

80.

310.

050

0.24

0.03

20.

3710

.91.

37

Exp

216

986.

517

8.3

46.

E+

101

5.2E

+06

1.E

-86

0.09

80.

240.

040

0.32

0.18

90.

430.

000

0.58

0.51

0.49

0.80

0.34

Exp

396

.81.

2−0

.82

2.E

+11

39.

5E+

046.

E-9

70.

790.

390.

037

0.36

0.03

90.

340.

350.

390.

031

0.38

0.16

0.36

Exp

418

3.2

2.2

39.0

11.

E+

117

1.2E

+07

3.E

-107

0.59

0.32

0.08

00.

400.

040

0.35

0.00

00.

240.

106

0.44

0.03

10.

28

Exp

528

3.3

1.0

66.3

17.

E+

123

2.3E

+06

3.E

-109

0.83

0.38

0.02

90.

340.

031

0.36

0.30

0.46

0.01

10.

330.

037

0.23

Exp

617

8.9

1.3

37.6

37.

E+

115

1.4E

+07

2.E

-91

0.55

0.26

0.01

50.

310.

037

0.35

0.34

0.60

0.02

10.

340.

430.

38

µ54

0.8

2.4

75.3

1.E

+12

31.

6E+

10−9

.E-6

20.

890.

340.

040.

350.

060.

360.

170.

420.

120.

392.

050.

49

σ62

1.4

2.1

66.9

3.E

+12

33.

9E+

102.

E-6

10.

810.

080.

020.

030.

060.

040.

170.

160.

200.

064.

320.

43

CoV

114.

9%87

.488

.8%

244.

9%24

4.8%

−244

.9%

91.8

%22

.7%

52.7

%8.

7%10

0.1%

10.7

%10

0.0%

38.1

%16

4.5%

15.9

%21

0.7%

87.8

%

Tabl

e5

Com

pari

son

ofm

ater

ialp

aram

eter

estim

ates

for

TL

acro

ssal

lexp

erim

ents

.See

text

for

expl

anat

ion

and

defin

ition

ofsy

mbo

ls

TL

ΩΩ

Rel

(%)

AIC

Ran

kde

t(H

)co

nd(H

)de

t(H

)a

ffb

ffa

fnb

fna

fsb

fsa n

nb n

na n

sb n

sa s

sb s

s

Exp

110

342.

614

7.3

41.

4E+

621.

2E+

082.

4E-3

63.

3214

.60.

1613

.70.

1114

.80.

3118

.70.

092

12.9

1.08

11.2

Exp

214

705.

616

9.3

1−8

.4E

+68

1.3E

+11

−1.3

E-4

00.

5318

.10.

043

17.5

0.21

14.1

0.00

18.1

0.49

13.4

3.62

9.91

Exp

310

2.6

1.3

2.8

31.

9E+

651.

1E+

092.

4E-3

41.

2514

.60.

1113

.10.

1513

.50.

7112

.30.

078

12.7

0.23

14.8

Exp

419

3.8

2.3

42.6

4−2

.8E

+65

2.1E

+09

−1.0

E-3

01.

2617

.40.

2611

.40.

1512

.80.

0087

19.2

0.23

11.2

0.11

16.9

Exp

532

3.9

1.1

74.7

35.

6E+

691.

1E+

098.

6E-3

81.

3015

.70.

1013

.50.

1112

.70.

3613

.40.

049

13.6

0.16

20.7

Exp

625

0.2

1.8

58.6

48.

9E+

601.

1E+

083.

1E-3

41.

8419

.70.

039

14.6

0.17

12.0

0.69

6.44

0.08

912

.70.

6217

.4

µ56

2.5

2.5

82.5

8.0E

+68

2.2E

+10

−1.7

E-3

11.

5816

.70.

1214

.00.

1513

.30.

3514

.70.

1712

.70.

9715

.2

σ55

6.5

1.6

63.7

2.4E

+69

5.1E

+10

4.3E

-31

0.95

2.06

0.08

42.

000.

038

1.03

0.31

4.96

0.17

0.84

1.35

4.05

CoV

98.9

%66

.777

.2%

299.

2%23

4.5%

−245

.1%

59.9

%12

.4%

70.5

%14

.3%

25.2

%7.

7%90

.0%

33.8

%98

.5%

6.6%

138.

9%26

.7%

123


References

Arts T, Costa K, Covell J, McCulloch A (2001) Relating myocardiallaminar architecture to shear strain and muscle fiber orientation.Am J Physiol 280:H2222–H2229

Bischoff J, Arruda E, Grosh K (2002) A microstructurally basedorthotropic hyperelastic constitutive law. J Biomech Eng 69:570–579

Bischoff J, Arruda E, Grosh K (2004) A rheological network modelfor the continuum anisotropic and viscoelastic behaviour of softtissue. Biomech Model Mechanobiol 3(1):56–65

Burnham K, Anderson D (2002) Model selection and multi-modelinference: a practical information-theoretic approach, 2nd edn.Springer, New York

Cohen A (1991) A padé approximant to the inverse langevin function.Rheol Acta 30:270–273

Costa K, Holmes J, McCulloch A (2001) Modelling cardiacmechanical properties in three dimensions. Philos Trans R Soc359(1783):1233–1250

Dokos S, Smaill B, Young A, LeGrice I (2002) Shear properties ofpassive ventricular myocardium. Am J Physiol Heart Circ Physiol283:H2650–H2659

Fung Y (1965) Foundations of solid mechanics. Prentice-Hall, Inc.,Englewood Cliffs

Fung Y (1993) Biomechanics: mechanical properties of living tissues,2nd edn. Springer, New York

Gardiner J, Weiss J (2001) Simple shear testing of parallel-fibered pla-nar soft tissues. J Biomech Eng 123(2):170–175

Gasser T, Ogden R, Holzapfel G (2006) Hyperelastic modelling of arte-rial layers with distributed collagen fibre orientations. J R SocInterface 3:15–35

Guccione J, McCulloch A, Waldmann L (1991) Passive material proper-ties of intact ventricular myocardium determined from a cylindricalmodel. J Biomech Eng 113:42–55

Holzapfel G (2000) Nonlinear solid mechanics. Wiley, ChichesterItskov M, Aksel N (2004) A class of orthotropic and transversely isotro-

pic hyperelastic constitutive models based on a polyconvex strainenergy function. Int J Solids Struct 41:3833–3848

Janicki J, Weber K (1977) Ejection pressure and the diastolic left ven-tricular pressure–volume relation. Am J Physiol 232(6):H545–H552

Lainé E, Vallée C, Fortuné D (1999) Nonlinear isotropic constitutivelaws: choice of the three invariants, complex potentials and con-stitutive inequalities. Int J Eng Sci 37:1927–1941

Lanir Y, Lichtenstein O, Imanuel O (1996) Optimal design of biaxialtests for structural material characterization of flat tissues. J Bio-mech. Eng 118:41–47

LeGrice I, Smaill B, Chai L, Edgar S, Gavin J, Hunter P (1995a) Lam-inar structure of the heart: ventricular myocyte arrangement andconnective tissue architecture in the dog. Am J Physiol Heart CircPhysiol 38(269):H571–H582

LeGrice I, Takayama Y, Covell J (1995b) Transverse shear along myo-cardial cleavage planes provides a mechanism for normal systolicwall thickening. Circ Res 77:182–193

Leonov AI (2000) On the conditions of potentiality in finite elasticityand hypo-elasticity. Int J Solids Struct 37:2565–2576

Mandilov L, Eberli F, Seiler C, Hess O (2000) Diastolic heart failure.Cardiovasc Res 45:813–825

Miller C, Wong C (2000) Trbaeculated embryonic myocardium showsrapid stress relaxation and non-quasi-linear viscoelastic behaviour.J Biomech 33:615–622

Nash M, Hunter P (2000) Computational mechanics of the heart. J Elast61:113–141

Poynting J (1909) On pressure perpendicular to the shear planes in finitepure shears, and on the lengthening of loaded wires when twisted.Proc R Soc Lond A82:546–549

Press W, Flannery B, Teukolsky S, Vetterling W (1989) Numerical rec-ipes. Cambridge University Press, Cambridge

Reddy K, Yusuf S (1998) Emerging epidemic cardiovascular disease indeveloping countries. Circulation 97:596–601

Schmid H, Nash M, Walker C, Sands G, Pope A, LeGrice I, YoungA, Nielsen P, Hunter P (2005) A framework for multi-scale mod-eling of the heart. IFMBE Proceedings, IFMBE, Prague: ISSN1727-1983. In: Hozman J, Kneppo P (eds) (Proceedings of the 3rdEuropean medical and biological engineering conference—EM-BEC 05. Prague, Czech Republic, 20-25.11.2005), Id. 2535 11,pp 4201–4205

Schmid H, Nash M, Young A, Hunter P (2006) Myocardial materialparameter estimation—a comparative study for simple shear. J Bio-mech Eng 128(5):742–750

Schmid H, Nash M, Young A, Röhrle O, Hunter P (2007) A com-putationally efficient optimization kernel for material parameterestimation procedures. J Biomech Eng 129(2):279–283

Smaill B, Hunter P (1991) Theory of heart, chap 1, Structure and func-tion of the diastolic heart: material properties of passive myocar-dium. Springer, Heidelberg, pp 1–29

Suga H, Sagawa K, Shoukas A (1973) Load independence of the instan-taneous pressure-volume ratio of the canine left ventricle andeffects of epinephrine and heart rate on the ratio. Circ Res 32:314–322. www.cellml.org, 2006. www.cmiss.org, 2006. www.math-works.com, 2006

123

Documents

Myocardial material parameter estimation