Ansys Myocardium Constitutive

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Biomechanics and Modeling in Mechanobiology (2019) 18:1213–1232

Original Paper

On the AIC‑based model reduction for the general Holzapfel–Ogden

Myocardial Constitutive Law

Debao Guan1 · Faizan Ahmad2 · Peter Theobald2 · Shwe Soe2 · Xiaoyu Luo1   · Hao Gao1 Received: 29 November 2018 / Accepted: 19 March 2019 / Published online: 3 April 2019

Abstract

Constitutive laws that describe the mechanical responses of cardiac tissue under loading hold the key to accurately model the biomechanical behaviour of the heart. There have been ample choices of phenomenological constitutive laws derived from experiments, some of which are quite sophisticated and include effects of microscopic fibre structures of the myocardium. A typical example is the strain-invariant-based Holzapfel–Ogden 2009 model that is excellently fitted to simple shear tests. It has been widely used and regarded as the state-of-the-art constitutive law for myocardium. However, there has been no analysis to show if it has both adequate descriptive and predictive capabilities for other tissue tests of myocardium. Indeed, such an analysis is important for any constitutive laws for clinically useful computational simulations. In this work, we perform such an analysis using combinations of tissue tests, uniaxial tension, biaxial tension and simple shear from three different sets of myocardial tissue studies. Starting from the general 14-parameter myocardial constitutive law developed by Holzapfel and Ogden, denoted as the general HO model, we show that this model has good descriptive and predictive capabilities for all the experimental tests. However, to reliably determine all 14 parameters of the model from experiments remains a great chal- lenge. Our aim is to reduce the constitutive law using Akaike information criterion, to maintain its mechanical integrity whilst achieving minimal computational cost. A competent constitutive law should have descriptive and predictive capabilities for different tissue tests. By competent, we mean the model has least terms but is still able to describe and predict experimental data. We also investigate the optimal combinations of tissue tests for a given constitutive model. For example, our results show that using one of the reduced HO models, one may need just one shear response (along normal-fibre direction) and one biaxial stretch (ratio of 1 mean fibre : 1 cross-fibre) to satisfactorily describe Sommer et al. human myocardial mechanical properties. Our study suggests that single-state tests (i.e. simple shear or stretching only) are insufficient to determine the myocardium responses. We also found it is important to consider transmural fibre rotations within each myocardial sample of tests during the fitting process. This is done by excluding un-stretched fibres using an “effective fibre ratio”, which depends on the sample size, shape, local myofibre architecture and loading conditions. We conclude that a competent myocardium material model can be obtained from the general HO model using AIC analysis and a suitable combination of tissue tests.

Keywords  Akaike information criterion (AIC) · Holzapfel–Ogden (HO) constitutive law · Reduced HO models · Simple shear tests · Uniaxial tests · Biaxial tests · Myocardial mechanical tests

1  Introduction

Cardiac diseases remain a major public healthy burden, especially the adverse remodelling of cardiac function after acute myocardial infarction. Studies have demonstrated that stress/strain in myocardium can have great effects on pathological processes such as hypertrophy and myocardial infarction (Zile et al. 2004; Costa et al. 2001; Mangion et al.

2017). Accurate prediction of myocardial stress relies on the choice of constitutive laws. Determining the constitutive laws and their parameters from limited experimental data,

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however, remains a great challenge for the cardiac model- ling community. In general, biological tissue, including myocardium, mainly consists of proteins such as collagen, elastin and ground substance. Published in vitro/ex vivo experimental tests of the mechanical behaviour of human myocardium (Pinto and Fung 1973) have shown strong anisotropy and transmural variations. Similar conclusions have also been reported by other studies, with Langdon et al. (1999) inves- tigating the effect of biaxial constraint caused by glutaralde- hyde cross-linking on the equal-biaxial mechanical proper- ties of bovine pericardium. Dokos et al. (2002) examined the shear properties of passive ventricular myocardium through six modes of simple shear tests on samples from porcine hearts, reporting that simple shear responses are highly non- linear along the micro-structural axes of the tissue. Later, Sommer et al. (2015b) determined biaxial extension and tri- axial shear properties, characterizing the underlying micro- structure of the passive human ventricular myocardium.

Results showed it is a nonlinear, anisotropic (orthotropic), viscoelastic and history-dependent soft biological material that undergoes large deformations. Very recently, Ahmad et al. (2018) studied biomechanical properties of neonatal porcine cardiac tissue by using uniaxial tensile, biaxial ten- sile and simple shear loading modes with samples from the anterior and posterior walls of the right and left ventricles.

The compressibility of myocardial tissue is quantified by McEvoy et al. (2018) using a joint experimental-computa- tional approach, investigating volumetric changes in excised porcine myocardium tissue under both tensile and confined compression loading conditions.

Over the years, a number of models have been developed to describe myocardial mechanical properties, ranging from linear elastic to hyperelastic, from isotropic to anisotropic, and from phenomenological to microstructurally based con- stitutive laws (Holzapfel and Ogden 2009). Nowadays, it is a common practice to characterize myocardium as an ani- sotropic, hyperelastic material. One approach employs the angular integration of each collagen fibre’s contribution fol- lowing a distribution map. Lanir (1983) developed a general multi-axial theory for the constitutive relations in fibrous connective tissues on the basis of micro-structural and ther- modynamic considerations. Sacks et al. (2016) developed a rigorous full structural model (i.e. explicitly incorporating various features of the collagen fibre architecture) for exog- enously cross-linked soft tissues, which made an extension to the collagenous structural constitutive model, meaning the uncross-linked collagen fibre responses could be mapped to the cross-linked configuration. Based on Sack’s study, Avaz- mohammadi et al. (2017b) proposed a fibre-level constitutive model for the passive mechanical behaviour of the right ven- tricular free wall, which explicitly separated the mechanical contributions of myocytes and collagen fibre ensembles, whilst accounting for their mechanical interactions.

Another widely used approach employs strain compo- nents directly or strain invariants when developing such constitutive laws. For instance, Guccione et al. (1991) used a transverse isotropic exponential Fung-type hyperelastic material model to characterize the equatorial region of the canine left ventricle, in which the strain energy function con- sists of six strain components. Soon afterwards, LeGrice et al. (1995) found that the micro-structure of myocardium was a composite of discrete fibre layers, which suggested an orthotropic mechanical response according to a local ortho- tropic material axes: the fibre direction 퐟 , the sheet direc- tion 퐬 and the sheet–normal 퐧 . The transversely isotropic Fung-type relation was then extended to account for the orthotropy described by Costa et al. (2001). There are many constitutive laws that use strain-invariant-based orthotropic or transversely isotropic constitutive laws to characterize passive myocardial tissue, which were recently reviewed in Holzapfel and Ogden (2009). Based on the simple shear data from Dokos et al. (2002), Holzapfel and Ogden proposed a simplified formulation (HO2009) derived from a more gen- eral strain-invariant-based material model (the general HO model) (Holzapfel and Ogden 2009). The HO2009 model has one term related to the matrix responses, two terms related to the stress responses along 퐟 and 퐬 , and a final term for interaction between 퐟 and 퐬.

The HO2009 model and its variation have been widely used in the cardiac modelling community such as the Living- Heart Project (Baillargeon et al. 2014). Göktepe et al. (2011) developed a general constitutive and algorithmic approach to the computational modelling of passive myocardium using the HO2009 model, which is embedded in a nonlinear finite element method. Wang et al. (2013) studied the fibre ori- entation on left ventricular diastolic mechanics using the HO2009 model and further extended it to include residual stresses (Wang et al. 2014). Gao et al. (2017) implemented the HO2009 model into an immersed boundary framework combined with finite element to study left ventricle (LV) biomechanics both in diastole and systole. Simplified forms of the HO2009 model were also used by Asner et al. (2016) with personalized ventricular dynamics derived from in vivo data. General structural tensors accounting for collagen fibre dispersion were introduced by Eriksson et al. (2013), fol- lowed by the recent extension of Melnik et al. (2018) to account for fibre dispersion in the coupling term between the fibre and sheet directions. Inverse estimation of unknown parameters in the HO2009 model from in vivo data was first investigated by Gao et al. (2015), and later by Nikou et al.

(2015), and by Palit et al. (2018). The HO2009 model has also been applied to simulate various heart diseases such as myocardial infarction (Gao et al. 2017; Baillargeon et al.

2014).

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No study has previously investigated the descriptive and predictive capability of HO-type strain energy functions. A competent constitutive law should be able to describe as many deformation modes (uniaxial, biaxial, simple shear, etc.) as possible in qualitative point and then from quantita- tive point, with acceptable errors of simulation with respect to the experimental data (Destrade et al. 2017), and have the fewest terms. Mechanical properties of myocardium are traditionally measured by a single series of either uniaxial (Pinto and Fung 1973), biaxial tests (Demer and Yin 1983) or simple shear deformations (Dokos et al. 2002), despite it being demonstrated that combined biaxial data (with differ- ent loading protocols) and simple shear data (with various loading directions) are required to adequately capture the tis- sue’s direction-dependent nonlinear response (Holzapfel and Ogden 2009). For example, Holzapfel and Ogden (2009), and Schmid et al. (2009) both only used simple shear data of Dokos et al. (2002) to demonstrate the good descriptive capability of selected constitutive laws. Only recently Som- mer et al. (2015b) have performed both biaxial and shear tests on similar human myocardial samples, whilst Ahmad et al. (2018) reported their experiments on neonatal por- cine myocardium samples with uniaxial, biaxial and shear tests. An unanswered question is whether a selected material model, such as the HO2009 model, can adequately fit to dif- ferent types of mechanical tests.

A competent constitutive law should also be able to pre- dict stress responses from different deformation modes. A constitutive law with parameters derived from simple shear test data can, for example, be used to accurately predict the biaxial test data. This predictive capability is critical for achieving accurate cardiac modelling, where the deformation states can differ significantly from the original experimen- tal data. Some studies describe the predictive capability of constitutive laws for arterial tissues, but rarely consider myo- cardium. For example, Hollander et al. (2011) compared the descriptive and predictive powers of a Fung-type exponen- tial phenomenological model, a strain-invariant-based par- tial structure model and a structural model based on angular integration, by characterizing coronary arterial media. They found that different test protocols (extension, inflation, and twist) are necessary to reliably predict mechanical response.

Polzer et al. (2015) studied the ability of a material model to predict the biaxial response of porcine aortic tissue with a predefined collagen structure. Schroeder et al. (2018), recently, showed that the Holzapfel–Gasser–Ogden model with generalized structure tensors (Gasser et al. 2006) can- not predict the biaxial arterial wall behaviour when deter- mined from only uniaxial tests, whilst the four-fibre-family constitutive law is the most robust when predicting uniaxial or biaxial behaviour of porcine aortic tissue.

This study first considers the descriptive capabilities of the general and specific HO models proposed in Holzapfel and Ogden (2009), using Dokos et al. simple shear data of porcine myocardium (Dokos et al. 2002), Sommer et al.

biaxial and simple shear data of human myocardium (Som- mer et al. 2015b), and Ahmad et al. uniaxial, biaxial and simple shear data of neonatal porcine myocardium (Ahmad et al. 2018). Secondly, the Akaike information criterion (AIC) (Schmid et al. 2006; Ten Eyck and Cavanaugh 2018; Avazmohammadi et al. 2017a) is used to analyse the good- ness of fit of the general HO model to the experimental data, with AIC values determined when excluding different strain invariants. Based on the AIC analysis, reduced HO mod- els for different experimental studies are then proposed by excluding those strain invariants with little contribution to the overall goodness of fit. Finally, we use predictive capa- bility of the reduced HO models to find the optimal combi- nation of experiments for each species of tissues that uses minimal mechanical tests.

2.1  Selected Myocardial Experiments

In this study, the experimental data are taken from three ex vivo myocardial biomechanical studies: Dokos et al. (2002) investigating porcine myocardium; Sommer et al.

(2015b) investigating human myocardium; and Ahmad et al. (2018) investigating neonatal porcine myocardium. These are briefly summarized below. For details, please refer to the original papers.

• Dokos et al. (2002) published shear data of passive

Myocardium From Porcine Hearts With Six Different

shear modes, shown in Fig. 1a where (ij) is used to refer to shear in the j direction within the ij plane, where i ≠j ∈{f, s, n} . Myocardial samples were cut from adja- cent regions in the left lateral ventricular mid-wall with a size of ∼3 × 3 × 3 mm.

• Sommer et al. (2015b) performed similar six shear-mode experiments, with samples from human hearts (size: ∼4 × 4 × 4 mm). They also performed biaxial testing with different stretch ratios (1:1, 1:0.75, 1:0.5, 0.75:1, 0.5:1) along the mean fibre direction (MFD) and the cross-fibre direction (CFD) (Fig 1b). MFD is the aver- age angle of the dominant orientation of collagen fibres on the upper and lower surfaces of each sample (Sommer et al. 2015a), with CFD perpendicular to MFD. Square

Specimens With Dimensions ∼25 × 25 × 2.3 Mm Were

used in biaxial tests, with tension applied along the MFD and CFD. They recorded the collagen fibre rotation within samples, which was   14.8 ± 6.9◦ per mm depth in the transmural direction.

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• Ahmad et al. (2018) performed uniaxial (Fig. 1c), biaxial and simple shear experiments on myocardial samples from neonatal porcine left and right ventricular free

Walls. Sample Dimensions Were ∼15 × 5 × 3 Mm For

uniaxial tests, ∼15 × 15 × 3 mm for biaxial tests and ∼3 × 3 × 3 mm for simple shear tests. Shearing was only performed in the sheet–fibre and sheet–normal planes, whilst the MFD was determined based on the external surface texture and not the average angle of the dominant orientation of collagen fibres as in Sommer et al. (2015a).

is related to the applied force components fij in the tests and

Cauchy Stress Tensor 흈 As

where 퐅 is deformation gradient tensor.

• Uniaxial Tests

For uniaxial stretch experiments along MFD, we have in which 휆1 is the stretch ratio, f1 is the applied force along MFD direction, and in this case f1 = f11 , 휎11 is the Cauchy stress component, and A1 is the reference cross- sectional area perpendicular to MFD. Similarly, for uni-

Axial Stretch Along Cfd

where the applied force f2 = f22.

• Biaxial Tests

For the shear-free biaxial test along MFD and CFD, since

A1 = A2 = A , Then

Again, in this case, we have f1 = f11 , f2 = f22. If shear exists in the biaxial test as in Fig.  2,

훾12 ≠0 And 훾21 ≠0 , Then

Because the measured force is the sum of forces along the directions 1 and 2, there is no force applied in the third direction, f1 and f2 satisfy the following equations

⎥⎦

and 휎11 = 휆1 P11 + 훾12 P12, 휎22 = 휆2 P22 + 훾21 P21.

(6)

f1 = f11 + f12, and f2 = f21 + f22. Fig. 1   a A sketch of all six possible shear modes, 퐟 , 퐬 , and 퐧 denote the fibre, sheet, and normal direction, respectively. (ij) refers to shear in the j direction within the ij plane, where i ≠j ∈{f, s, n} . b A sample with fibres (red dash lines), which is stretched along the two orthogonal directions (MFD and CFD) in fibre-normal plane during a biaxial test; c uniaxial tension tests along the MFD and CFD; f1 and f2 are the loading force along the MFD and CFD. L is the initial length of specimen, and 휆1 and 휆2 are stretch ratios In the following, we refer these three sets of experiments as Dokos’s data, Sommers’s data, and Ahmad’s data. Let 1, 2, 3 represent the components in MFD, CFD and sheet (transmural) directions and assume that the test sample is incompressible. To make use of the experiments, it is con- venient to use the first Piola–Kirchhoff (P–K) stress 퐏 , which

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Finally, the relationship between the first P–K stress

Components And The Applied Forces Is

Note when shear is present, 휎11 ≠휆f1∕A and 휎22 ≠휆f2∕A . Therefore, we need to determine the P–K stress compo- nents from experiments using (7), and then recover the Cauchy stress components from (1).

We further assume the shear increases linearly with

Stretch, That Is

where k1 and k2 are the maximum values of 훾12 and 훾21.

• Simple Shear Tests

For the simple shear tests, shown in Fig. 1a, we have and in this case, the stress components are determined

A ,

i ≠j ∈{1, 2, 3}.

2.2  The General Ho Model

To characterize the mechanical behaviours of myocardium, the general HO strain energy function proposed by Holzap- fel and Ogden (Holzapfel and Ogden 2009) is employed,

Which Is

where a, b, ai, bi, aij, bij are the 14 material constants, I1 = trace(퐅T퐅) , representing the overall squared stretch, I4f , I4s and I4n are squared stretches along each direction, in which 퐟0, 퐬0, 퐧0 are the initial fibre, sheet and normal direc- tions. The max() in (11) will ensure the collagen fibres can only bear the load when stretched but not in compression.

I4N = 퐧0 ⋅(퐅T퐅퐧0),

Fig. 2   The recorded image in a biaxial tensile specimen in Ahmad et al. (2018) (a). The four white markers in the centre of the experi- mental sample in a are also shown in b, in which the solid rectan- gle represents the initial shape; and the deformed shape is shown in dashed lines. f1 and f2 are the loading forces in MFD and CFD I8fs , I8fn and I8sn are invariants representing the coupling

I8Fn = 퐟0 ⋅(퐅T퐅퐧0),

I8sn = 퐬0 ⋅(퐅T퐅퐧0).

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2.3  Effective Fibre Contribution

The rotation of collagen fibres from epicardium to endocar- dium plays a significant role in the myocardial mechanical response. Thus, it is necessary to consider fibre rotation in tested samples when fitting constitutive laws to experimental data. We further assume collagen fibres (along with myo- cytes) only lie in the 퐟−퐧 plane. Considering a myocardial sample with linearly rotated fibres from 휃1 to 휃2 as shown in Fig. 3a, the local fibre angle related to the MFD with a

Depth Of H Is

where H0 is the total thickness of the sample, and the local

And

퐬= (0, 0, 1). Fig. 3   a The fibre direction varies along the thickness of myocar- dium. b The effective area (blue) when the fibre direction is 휃 under uniaxial loading in the MFD test. The collagen fibres (red dot line) within the region enclosed by the two blue dashed lines are defined as effective fibres that are stretched both sides. The effective fibre ratio is defined by rectangle area dividing blue effective area Because collagen fibres can only bear the load when stretched, factors depending on fibre angle ( 휃 ) are introduced in the general HO law for I4f , I4s and I4n which measure the squared stretches of different fibre families, but not for I8fs , I8fn and I8sn which are dependent on the angels between dif-

+ 훼4S(휃)휓4S + 훼4N(휃)휓4N + 휓8Fs + 휓8Fn + 휓8Sn,

in which 휓i is the strain energy term associated with the invariant of Ii . 훼4f(휃) , 훼4s(휃) and 훼4n(휃) values will depend on the experimental loading conditions and the fibre structure of tested samples. For example, in an uniaxial test along the MFD as shown in (Fig. 3b), only when fibres are attached to both ends (the most left and right sides), or in the shaded area in Fig. 3b, can they be stretched along the MFD and contribute to the stress response. If only one end of the fibres is stretched (e.g. the unshaded area in Fig. 3b), they will not contribute to the stress response (i.e. the other end is unconstrained). 훼4f(휃) is defined as the ratio between the shaded blue area and the total area of the sample as shown in Fig. 3b, denoting the effective fibre ratio 휀휃 . Similarly, collagen aligned in the CFD may contribute to the stress response depending on the size of the sample, the fibre angle and the experimental set-up. Collagen in the sheet direction is not stretched, which means they do not contribute to the uniaxial test; therefore, for the uniaxial test in Fig. 3b, the

Effective Fibre Ratios Are

where W0  and L0  are the width and length of the tested

L0  . Effective Ratios

for biaxial and simple shear tests can be found in the appendix. The stress tensor in a myocardium layer ( det (퐅) = 1 ) with

Others ,

𝛼4s(𝜃) = 0.

휕퐅

퐅−T −p퐅−T.

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Because the local fibres in a test sample rotate from 휃1 to 휃2 transmurally (as shown in Fig. 3), the total Cauchy stress

2.4  Parameter Estimation

For Dokos et al. study, all six shear experiments are used for formulating (17). For Sommer et al. study, we fit the strain energy functions using both the biaxial and simple shear tests.

All three modes of experimental data from Ahmad et al. study are combined together. Material parameters are estimated using a nonlinear least square minimization function (fmin- con from MatLab, MathWorks 2017), with the loss function where 휷 denotes the set of unknown parameters, N is the

N Are The Experimen-

tal values. The relative and absolute differences of the area-under-the-curve between the experimental and fitted stress–strain curves ( errRelative , errAbsolute ) are introduced to

Quantitatively Describe The Goodness Of Fit,

in which 휆min and 휆max are the minimum and maximum stretch or shear, respectively. A value of 0 indicates a per- fect fitting.

2.5  Reduced Ho Models

Some of the invariants may be excluded in the general

Ho Model When Applied To Human Myocardium, Whilst

still achieving a good agreement with experimental data. For example, Holzapfel and Ogden (2009) reported that after dropping I4n, I8fn and I8sn from the general HO model, they could still fit the six shear tests of Dokos et al. (2002) very well; hence, they proposed an 8-parameter HO model

휃1

퐏휃d휃.

[Exp(Bfsi2

8fs) −1]. The reason for excluding I8fn and I8sn is because the two shear responses marked as (nf) and (ns) were not distin- guishable based on Dokos et al. data. There lacks, however, a study investigating whether the general HO and HO2009 can fit all other myocardial experiments well, such as human myocardium in Sommer et al. (2015b).

Reducing the general HO model (11) is advantageous, as so many strain invariants and material parameters prevent efficient personalized cardiac simulations. Furthermore, multiple sets of optimal material parameters from limited experimental data can lead to different simulation results for a given boundary-value problem (Ogden et al. 2004). To derive a simplified but competent strain energy function, the AIC analysis (Burnham and Anderson 2003) is employed in this study to reduce the general HO model, which is defined

As

where K is the number of model parameters. AIC is typi- cally used for model selection by considering both the model complexity and the loss function. The best model is the one with the lowest AIC value. This approach has previously been used by Schmid et al. (2006) to compare five different myocardial strain energy functions. Note the AIC in (20) is negative when the fitting is good. Therefore, for any two dif- ferent models and the same experimental data, the one with the more negative AIC value suggests a better fitting. Similar AIC values represent comparable models. In this study, we consider various reduced forms of (11). This allows us to drop the terms in (11) that make little change in the AIC value. This way we can select the simplest strain energy function that fits to the test data. To this end, we introduce

The Aic Ratio:

where 휂 represents the ratio of AIC values of a reduced and the general HO model for the same experimental data. We aim to simplify the general HO model with a subset of strain invariants ( {I4f , I4s , I4n , I8fs , I8fn , I8sn} ), for effectively characterizing different experimental studies. The steps for

Reducing The General Ho Model Are

1. Compute the AIC value for the general HO model and

휂= 1;

2. Compute 휂 values for reduced models whilst removing

One Strain Invariant At A Time;

3. The invariant associated with the least changed 휂 value

May Be Dropped, Leading To A Reduced Ho Model,

4. Repeat 2–3 for the remaining set of the strain invariants,

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5. If the 휂 value is reduced by a predetermined threshold 휖 , stop; otherwise, go to 2. In this study, we chose 휖 to be 0.05.

We Further Compare The Modelling Accuracy Between

the general and various reduced HO models using a three- dimensional (3D) finite element (FE) bi-ventricular model, which is reconstructed from 3D computed tomography (CT) data. Details of the data acquisition can be found in Ahmad et al. (2018). The 3D CT data are first segmented using Seg3D;1 then, the boundary contours are exported into SolidWorks (Dassault Systemes, MA USA) for 3D geometry reconstruction, and then meshed with ICEM (ANSYS, Inc.

PA USA). Finally, explicit Abaqus (Dassault Systemes, MA USA) is used for the FE simulation. User subroutines are implemented for different strain energy functions. Diastolic filling in the left ventricle (Fig. 4a) is simulated with layered myofibre rotating from the epicardial to endocardial surface (Fig. 4b), with rotation angles measured from experimental studies using a rule-based approach (Wang et al. 2013).

Through Predictive Analysis

Likewise, we can use AIC method to determine the opti- mal combination of experiments using minimum tests. For a given strain energy function, we firstly fit it to a subset of experimental data with Ns data points and then use it to predict the remaining points ( N −Ns ). We introduce a

All Are Computed Using Parameters

optimized from a subset or all combinations of experimental

All

. data, respectively. We do not consider cases when 훿 becomes negative. Hence, 훿 denotes the AIC change using different combinations of experiments for the same model. We chose the criterion for the best combination to be the minimum group of tests which satisfies 훿≥0.8 . This corresponds to about 5% change of the relative error in (18). The pseudo- code for this analysis is listed in Algorithm 1.

Algorithm 1 The predictive analysis for determining the

3.1  The General Ho Strain Energy Function

Figure 5a shows the results by fitting the general HO model to the Dokos et al. shear tests. Improved agreement can be found when including the effective fibre ratio (AIC: −589.3 ) compared to without (AIC: −464.7 ), whilst the mean rela- tive error also decreases from 15.9% to 9.3%. When fitting

Sure Linearly Increases From 0

to 4 mmHg in a period of 0.5s.

Dium ( 60O To −60O)

1  http://www.sci.utah.edu/cibc-softw​are/seg3d​.html

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all test data from Ahmad et al. study with the effective fibre ratio, the AIC value is reduced significantly from −338.5 to −1170.3 (shown in Figs. 5b–d), whilst the relative errors for the uniaxial test along the MFD decrease from 36.06% to 4.25%, and from 26.76% to 6.97%, for the biaxial test along the CFD.

In Ahmad et al. data, we estimate k1 = 0.18 and k2 = 0.05 using markers for the sample angle changes (Fig. 2). In Som- mer et al. (2015b), no information on the shear measure- ments is available. However, we assume the maximum shear angles in both MFD and CFD in Sommer et al. biaxial tests are around 6◦ , i.e. k1 = k2 ≈0.1 (Sommer et al. 2015a; Bil- liar and Sacks 2000), which is necessary for a good fit to their experiments. As for Ahmad et al. and Sommer et al.

biaxial tests, the difference with and without shear for the same model in Fig. 6 indicates including the shear compo- nent is critical when fitting biaxial experimental tests, since for fibre-reinforced material, it is almost impossible to con- duct biaxial tests without inducing shear (Freed et al. 2010).

Figure 7a demonstrates that both the general HO (AIC: −589.3 ) and HO2009 (AIC: −559.3 ) models can fit Dokos et al. shear test data very well, whilst noticeable differ- ences can be found when fitting the two material models to Sommer et al. data (Fig. 7b, c, where only plotting one set (MFD:CFD=1:1) experimental data, whilst the remain- ing  four sets have similar results and are included in the appendix (Fig. 12). Better agreement is achieved for the general HO model (AIC: −1102.6 ) than the HO2009 model

(Blue) Using Same Strain Energy

function. a is for Sommer et al.

Biaxial Test And A Minimum

shear angle of 6o is introduced.

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(AIC: −849.5 ). Figure 7d–f shows the results when fitting the two models to Ahmad’s data. Again, much better agree- ment can be found when using the general HO model (AIC: −1170.3 ) compared to the HO2009 model (AIC: −423.1 ); in particular, the HO2009 model fails to fit the shear test in (Fig. 7f).

3.2  Reduced strain energy functions based on AIC

Analysis

Although the general HO model can fit the three selected experimental studies very well as shown in Fig. 7, it includes seven invariants with fourteen unknown parameters, which can be extremely challenging to obtain an unique solution when fitting to limited experimental data. A reduced form, such as the HO2009 model, is desirable for constructing personalized models (Gao et al. 2017; Palit et al. 2018; Nikou et al. 2015).

However, since HO2009 is derived from fitting the Dokos et al. data only, if such a model fails to describe other experimen- tal data, we need to have strategies in place to derive a better reduced model with test data available for tissues of interests.

In this work, alternative reduced strain energy functions are identified from the general HO model for selected exper- imental studies based on the AIC analysis. Figure 8a reports 휂 values when individually excluding each invariant from the general HO model (11) and fitting to three experimental studies. For Dokos et al. data, I4n , I8fn and I8sn have much less contribution to the agreement compared to I4f , I4s and I8fs , because 휂 remains more than 0.95 when dropping these terms. This means I4n , I8fn and I8sn can be dropped from (11), and it can now be denoted as HO-D, which actually equates to the HO2009 model (19). In other words, 휂 is reduced by 3% when using the HO-D model (HO2009) to replace the general HO model. Similarly, invariants I4s , I8fn and I8sn may be excluded from the general HO model when fitting to Sommer et al. biaxial and simple shear data. This gives us

Ho And The Ho2009 Models For

the three experimental studies.

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