On the AICbased model reduction for the general Holzapfel–Ogden myocardial constitutive law
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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 straininvariantbased Holzapfel–Ogden 2009 model that is excellently fitted to simple shear tests. It has been widely used and regarded as the stateoftheart 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 14parameter 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 challenge. 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 normalfibre direction) and one biaxial stretch (ratio of 1 mean fibre : 1 crossfibre) to satisfactorily describe Sommer et al. human myocardial mechanical properties. Our study suggests that singlestate 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 unstretched 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 tests1 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, however, remains a great challenge for the cardiac modelling 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) investigating the effect of biaxial constraint caused by glutaraldehyde crosslinking on the equalbiaxial mechanical properties 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 nonlinear along the microstructural axes of the tissue. Later, Sommer et al. (2015b) determined biaxial extension and triaxial shear properties, characterizing the underlying microstructure of the passive human ventricular myocardium. Results showed it is a nonlinear, anisotropic (orthotropic), viscoelastic and historydependent 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 tensile 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 experimentalcomputational 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 constitutive laws (Holzapfel and Ogden 2009). Nowadays, it is a common practice to characterize myocardium as an anisotropic, hyperelastic material. One approach employs the angular integration of each collagen fibre’s contribution following a distribution map. Lanir (1983) developed a general multiaxial theory for the constitutive relations in fibrous connective tissues on the basis of microstructural and thermodynamic considerations. Sacks et al. (2016) developed a rigorous full structural model (i.e. explicitly incorporating various features of the collagen fibre architecture) for exogenously crosslinked soft tissues, which made an extension to the collagenous structural constitutive model, meaning the uncrosslinked collagen fibre responses could be mapped to the crosslinked configuration. Based on Sack’s study, Avazmohammadi et al. (2017b) proposed a fibrelevel constitutive model for the passive mechanical behaviour of the right ventricular 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 components directly or strain invariants when developing such constitutive laws. For instance, Guccione et al. (1991) used a transverse isotropic exponential Fungtype hyperelastic material model to characterize the equatorial region of the canine left ventricle, in which the strain energy function consists of six strain components. Soon afterwards, LeGrice et al. (1995) found that the microstructure of myocardium was a composite of discrete fibre layers, which suggested an orthotropic mechanical response according to a local orthotropic material axes: the fibre direction \({\mathbf {f}}\), the sheet direction \({\mathbf {s}}\) and the sheet–normal \({\mathbf {n}}\). The transversely isotropic Fungtype relation was then extended to account for the orthotropy described by Costa et al. (2001). There are many constitutive laws that use straininvariantbased 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 general straininvariantbased 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 \({\mathbf{f}}\) and \({\mathbf{s}}\), and a final term for interaction between \({\mathbf{f}}\) and \({\mathbf{s}}\).
The HO2009 model and its variation have been widely used in the cardiac modelling community such as the LivingHeart 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 orientation 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), followed 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).
No study has previously investigated the descriptive and predictive capability of HOtype 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 quantitative 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 different loading protocols) and simple shear data (with various loading directions) are required to adequately capture the tissue’s directiondependent 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 Sommer 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 porcine 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 different types of mechanical tests.
A competent constitutive law should also be able to predict 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 experimental data. Some studies describe the predictive capability of constitutive laws for arterial tissues, but rarely consider myocardium. For example, Hollander et al. (2011) compared the descriptive and predictive powers of a Fungtype exponential phenomenological model, a straininvariantbased partial 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) cannot predict the biaxial arterial wall behaviour when determined from only uniaxial tests, whilst the fourfibrefamily 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 (Sommer 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 goodness 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 models 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 capability of the reduced HO models to find the optimal combination of experiments for each species of tissues that uses minimal mechanical tests.
2 Method
2.1 Selected myocardial experiments

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 \(i\ne j\in {\{\text {f, s, n}\}}\). Myocardial samples were cut from adjacent regions in the left lateral ventricular midwall with a size of \(\sim \,3\times 3\times 3\) mm.

Sommer et al. (2015b) performed similar six shearmode experiments, with samples from human hearts (size: \(\sim \,4\times 4\times 4\) 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 crossfibre direction (CFD) (Fig 1b). MFD is the average 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 \(\sim \,25\times 25\times 2.3\) mm were used in biaxial tests, with tension applied along the MFD and CFD. They recorded the collagen fibre rotation within samples, which was \(14.8\pm \,6.9^\circ\) per mm depth in the transmural direction.

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 \(\sim \,15\times 5\times 3\) mm for uniaxial tests, \(\sim \,15\times 15\times 3\) mm for biaxial tests and \(\sim \,3\times 3\times 3\) 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).

Uniaxial tests
For uniaxial stretch experiments along MFD, we havein which \(\lambda _1\) is the stretch ratio, \(f_1\) is the applied force along MFD direction, and in this case \(f_1=f_{11}\), \({\sigma }_{11}\) is the Cauchy stress component, and \(A_1\) is the reference crosssectional area perpendicular to MFD. Similarly, for uniaxial stretch along CFD$$\begin{aligned} {\mathbf{F}}=\left[ \begin{matrix}\lambda _1 &\quad 0 &\quad 0 \\ 0 &\quad \frac{1}{\sqrt{\lambda _1}} &\quad 0 \\ 0 &\quad 0 &\quad \frac{1}{\sqrt{\lambda _1}} \end{matrix}\right] \,\,\,\text {and}\,\quad {\sigma }_{11}=\lambda _1\frac{f_1}{A_1}= \lambda _1 P_{11}, \end{aligned}$$(2)where the applied force \(f_2=f_{22}\).$$\begin{aligned} {\mathbf{F}}=\left[ \begin{array}{ccc}\frac{1}{\sqrt{\lambda _2}} &\quad 0 &\quad 0 \\ 0 &\quad \lambda _2 &\quad 0 \\ 0 &\quad 0 &\quad \frac{1}{\sqrt{\lambda _2}} \end{array}\right] \,\,\,\text {and}\,\quad {\sigma }_{22}=\lambda _2\frac{f_2}{A_2}=\lambda _2 P_{22}, \end{aligned}$$(3) 
Biaxial tests
For the shearfree biaxial test along MFD and CFD, since \(A_1 = A_2 = A\), thenAgain, in this case, we have \(f_1=f_{11}\), \(f_2=f_{22}\).$$\begin{aligned} {\mathbf{F}}=\left[ \begin{matrix} \lambda _1 &\quad 0 &\quad 0 \\ 0 &\quad \lambda _2 &\quad 0 \\ 0 &\quad 0 &\quad \frac{1}{\lambda _1\,\lambda _2} \end{matrix}\right] \, \text {and} \, {\sigma }_{11}=\lambda _1\frac{f_1}{A}=\lambda _1 P_{11}, \, {\sigma }_{22}=\lambda _2\frac{f_2}{A}=\lambda _2 P_{22}. \end{aligned}$$(4)If shear exists in the biaxial test as in Fig. 2, \(\gamma _{12}\ne 0 \,\text {and}\,\gamma _{21}\ne 0\), then
$$\begin{aligned} {\mathbf{F}}=\left[ \begin{matrix} \lambda _1 &\quad \gamma _{12} &\quad 0 \\ \gamma _{21} &\quad \lambda _2 &\quad 0 \\ 0 &\quad 0 &\quad \frac{1}{\lambda _1\,\lambda _2\gamma _{12}\,\gamma _{21}} \end{matrix}\right] \, \text {and} \, {\sigma }_{11}=\lambda _1\,P_{11}+\gamma _{12}\,P_{12},\, {\sigma }_{22}=\lambda _2\,P_{22}+\gamma _{21}\,P_{21}. \end{aligned}$$(5)Because the measured force is the sum of forces along the directions 1 and 2, there is no force applied in the third direction, \(f_1\) and \(f_2\) satisfy the following equations (Sommer et al. 2015a)Finally, the relationship between the first P–K stress components and the applied forces is$$\begin{aligned} f_1=f_{11}+f_{12},\,\,\text {and}\,\,f_2=f_{21}+f_{22}. \end{aligned}$$(6)Note when shear is present, \(\sigma _{11}\ne \lambda f_1/A\) and \(\sigma _{22}\ne \lambda f_2/A\). Therefore, we need to determine the P–K stress components from experiments using (7), and then recover the Cauchy stress components from (1).$$\begin{aligned} \begin{aligned}&P_{11}+P_{12}=({{\varvec{\sigma }}}{\mathbf{F}}^{\text {T}})_{11}+({{\varvec{\sigma }}} {\mathbf{F}}^{\text {T}})_{12} = \frac{f_1}{A} ,\\&P_{21}+P_{22}=({{\varvec{\sigma }}}{\mathbf{F}}^{\text {T}})_{21}+({{\varvec{\sigma }}} {\mathbf{F}}^{\text {T}})_{22} = \frac{f_2}{A}. \end{aligned} \end{aligned}$$(7)We further assume the shear increases linearly with stretch, that iswhere \(k_1\) and \(k_2\) are the maximum values of \(\gamma _{12}\) and \(\gamma _{21}\).$$\begin{aligned} \gamma _{12}=k_1\frac{\lambda _11}{\lambda _1^{\max }1},\quad \text {and} \quad \gamma _{21}=k_2\frac{\lambda _21}{\lambda _2^{\max }1}, \end{aligned}$$(8) 
Simple shear tests
For the simple shear tests, shown in Fig. 1a, we haveand in this case, the stress components are determined from$$\begin{aligned} \text {(ns):} \quad {\mathbf{F}}=\left[ \begin{matrix} 1 &\quad 0&\quad 0 \\ 0 &\quad 1 &\quad 0 \\ 0 &\quad \gamma _{32} &\quad 1 \end{matrix}\right] \quad \text {(fn):} \quad {\mathbf{F}}=\left[ \begin{matrix} 1 &\quad 0 &\quad 0 \\ \gamma _{21} &\quad 1 &\quad 0 \\ 0 &\quad 0 &\quad 1 \end{matrix}\right] \quad \text {(sf):} \quad {\mathbf{F}}=\left[ \begin{matrix} 1 &\quad 0 &\quad \gamma _{13} \\ 0 &\quad 1 &\quad 0 \\ 0 &\quad 0 &\quad 1 \end{matrix}\right] \nonumber \\ \text {(nf):} \quad {\mathbf{F}}=\left[ \begin{matrix} 1 &\quad \gamma _{12} &\quad 0 \\ 0 &\quad 1 &\quad 0 \\ 0 &\quad 0 &\quad 1 \end{matrix}\right] \quad \text {(fs):} \quad {\mathbf{F}}=\left[ \begin{matrix} 1 &\quad 0 &\quad 0 \\ 0 &\quad 1 &\quad 0 \\ \gamma _{31} &\quad 0 &\quad 1 \end{matrix}\right] \quad \text {(sn):} \quad {\mathbf{F}}=\left[ \begin{matrix} 1 &\quad 0 &\quad 0 \\ 0 &\quad 1 &\quad \gamma _{23} \\ 0 &\quad 0 &\quad 1 \end{matrix}\right] \end{aligned}$$(9)$$\begin{aligned} \sigma _{ij}={P}_{ij} =\frac{f_{ij}}{A},\,\,\,\,i\ne \,j\in \{1,2,3\}. \end{aligned}$$(10)
2.2 The general HO model
2.3 Effective fibre contribution
2.4 Parameter estimation
2.5 Reduced HO models
 1.
Compute the AIC value for the general HO model and \(\eta =1\);
 2.
Compute \(\eta\) values for reduced models whilst removing one strain invariant at a time;
 3.
The invariant associated with the least changed \(\eta\) value may be dropped, leading to a reduced HO model,
 4.
Repeat 2–3 for the remaining set of the strain invariants,
 5.
If the \(\eta\) value is reduced by a predetermined threshold \(\epsilon\), stop; otherwise, go to 2.
2.6 Optimal combination of experiments through predictive analysis
Algorithm 1 The predictive analysis for determining the optimal combination with minimal tests
3 Results
3.1 The general HO strain energy function
3.2 Reduced strain energy functions based on AIC analysis
The estimated parameters for the reduced HO models fitting to corresponding experimental studies
a (kPa)  b  \(a_\text {f}\) (kPa)  \(b_\text {f}\)  \(a_\text {s}\) (kPa)  \(b_\text {s}\)  \(a_\text {n}\) (kPa)  \(b_\text {n}\)  \(a_\text {fs}\) (kPa)  \(b_\text {fs}\)  \(a_\text {fn}\) (kPa)  \(b_\text {fn}\)  \(a_\text {sn}\) (kPa)  \(b_\text {sn}\)  

Dokos et al  HOD  0.073  15.529  25.992  9.348  4.822  0.001  –  –  0.178  16.740  –  –  –  
Sommer et al  HOS  0.809  7.474  1.911  22.063  –  –  0.227  34.802  0.547  5.691  –  –  –  – 
Ahmad et al  HOA  0.075  18.143  7.067  1.339  –  –  2.745  4.497  1.859  4.066  3.541  8.222  –  – 
Relative and absolute errors for the reduced HO models when fitting to corresponding experimental studies
Experiment  Model  Relative Error (\(\%\)) and Absolute Error (kPa)  Mean  

Dokos et al  (fs)  (fn)  (sf)  (sn)  (nf)  (ns)  
HOD  %:  2.96  4.43  12.55  12.10  13.37  16.87  10.38  
kPa:  0.12  0.14  0.14  0.11  0.08  0.10  0.12  
Sommer et al  1:1  1:0.75  0.75:1  1:0.5  0.5:1  (fs)  (fn)  (sf)  (sn)  (nf)  (ns)  
MFD  CFD  MFD  CFD  MFD  CFD  MFD  CFD  MFD  CFD  
HOS  %:  10.12  12.99  13.19  15.83  16.76  9.57  15.29  16.23  31.24  18.22  6.39  15.10  12.34  21.92  5.31  8.54  14.32  
kPa:  0.18  0.13  0.16  0.10  0.15  0.06  0.16  0.07  0.14  0.08  0.11  0.23  0.13  0.22  0.05  0.08  0.13  
Ahmad et al  Uniaxial  Biaxial  Simple shear  
MFD  CFD  MFD  CFD  (sf)  (sn)  
HOA  %:  4.72  4.08  5.95  6.96  7.15  14.61  7.24  
kPa:  0.02  0.01  0.04  0.04  0.001  0.005  0.02 
3.3 Optimal combination of experimental tests
To find the optimal combination of tissue tests, we use reduced HO models and a random initialization strategy to get the average value of \(\delta\), \(avg(\delta )\), and its corresponding standard deviation, std.
Combinations from Dokos et al. data As shown in Fig. 10a, in additional to all tests, case 25 (\(\text {(fs)}+\text {(fn)}+\text {(ns)}\)), case 42 (\(\text {(fs)}+\text {(fn)}+\text {(sf)}+\text {(sn)}\)), case 57 (\(\text {(fs)}+\text {(fn)}+\text {(sf)}+\text {(sn)}+\text {(nf)}\)) meet the criterion of \(avg(\delta ) \ge 0.8\). Clearly, case 25 is the optimal combination.
Combinations from Sommer et al. data Figure 10c displays partial \(avg(\delta )\) values of Sommer et al. AIC analysis when combining different biaxial and simple shear test data using the HOS model; for clarity, only group 1, 2, 3, 5 and 6 are shown. The best combination is case 20 ((1:1)+(nf)). In particular, case 562 ((1:1)+(1:0.75)+(0.75:1)+(1:0.5)+(0.5:1)) is the combination of all biaxial data and has negative \(\delta\) value, suggesting using biaxial data only cannot predict the simple shear responses. Likewise, using simple shear tests only, case 1484 (\(\text {(fs)}+\text {(fn)}+\text {(sf)}+\text {(sn)}+\text {(nf)}+\text {(ns)}\)) is unable to predict biaxial data. Therefore, both biaxial and simple shear test data are needed when characterizing myocardial properties. This agrees with the observation by Holzapfel and Ogden (2009). Figure 10d, e shows the stress of biaxial tests and simple shear tests with parameters determined from stress responses in (1:1)+(nf).
Combinations from Ahmad et al. data In Fig. 10f, apart from all tests, none of the other combinations meet \(\delta \ge 0.8\) in Ahmad et al. study. The fitting curves using all the tests are already shown in Fig. 8e, g, f.
4 Discussion
This study focuses on a rational reduction of the general HO model for the myocardial tissue responses. Three different myocardial experiments are selected, including Dokos et al. study on porcine myocardium over a decade ago (Dokos et al. 2002), Sommer et al. study on human myocardium published several years ago (Sommer et al. 2015b), and the very recent experimental data from Ahmad et al. (2018) on neonatal porcine myocardium (Ahmad et al. 2018). To our best knowledge, these are the most comprehensive myocardial mechanical experiments. Dokos et al. (2002) is the first presenting simple shear tests to characterize the directiondependent myocardial mechanical property, which has driven new developments in strain energy function and led to the extensive use of the HO2009 model (Holzapfel and Ogden 2009). Sommer et al. (2015b) included biaxial and simple shear tests, with both needed for characterizing an orthogonal hyperelastic material (Holzapfel and Ogden 2009). We show, for the first time, that the general HO model is very good as describing stress responses from different deformation types as shown in Fig. 7.
A number of studies have used the HObased strain energy functions (mostly HO2009 model) to construct personalized biomechanical models (Gao et al. 2017; Asner et al. 2016; Baillargeon et al. 2014). The widely successful application of the HOtype models suggests it is good for characterizing myocardial mechanical properties and provides the natural starting point to optimize the general HO model for specific tissue types, aiming to achieve the least terms and yet retaining sufficient descriptive and predictive capability. However, it has been recognized that the HO2009 model has its limitations (Fig. 7). This is because their model reduction is based on Dokos et al. simple shear data only, which did not include all responses of the myocardial tissues.
In the past several decades, efforts have been made to develop a strain energy function with fewest terms, whilst accurately describing the test data and predicting the dynamics (Zhang et al. 2019). A simplified but competent material model not only reduces computational cost, but is also easy to implement and personalize from limited test data. In this study, the AIC analysis is employed to systematically reduce the general HO model, whilst maintaining good descriptive and predictive capabilities. An invariant is excluded from the general HO model if it causes only a small change in the resultant AIC value. For instance, Fig. 8a suggests that \(I_{4\text {n}}\), \(I_{8\text {fn}}\) and \(I_{8\text {sn}}\) could be excluded when fitting to the Dokos et al. data, which is the same formulation as the HO2009 model. Other approaches can also be used for model reduction and selection such as parameter sensitivity analysis, by setting those insensitive parameters to constant values or zero (Snowden et al. 2017).
Interestingly, the reduced HO models are different for the selected experimental studies. Presumably, this is because these tests were for different species and ages; Dokos et al. (2002) used adult porcine myocardium, Ahmad et al. (2018) used the neonatal porcine myocardium, and Sommer et al. (2015b) worked on human myocardium. When fitting to the biaxial tests only from Sommer et al. data, the general HO model can be simplified to a reduced form consisting of only \(I_1\) and \(I_{4\text {f}}\), similar to the findings reported in Holzapfel and Ogden (2009). This is because in the biaxial tests, collagen fibres are only stretched in fibrenormal plane, but not in the sheet direction, thus \(\max ({I_{4\text {s}}}, 1) = 1\) and \(I_{8\text {fs}} = 0\). When fitting to the biaxial and simple shear tests together, the term with \(I_{4\text {n}}\) needs to be included, which is different from the reduced formulation when fitting only to Dokos et al. data. One reason is that the shear responses along (fs) and (fn) are closer to each other in Sommer et al. human myocardium, than in Dokos et al. porcine myocardium. This is similar to shear responses along (sf) and (sn), and along (nf) and (ns), which suggests there may be a difference in passive myocardial properties between human and porcine myocardium. The reduced HO model from Ahmad et al. data needs to incorporate \(I_{8\text {fn}}\), which might be explained by: (1) the asymmetric fibre structure in relation to the stretching axis; and (2) limited test data with only 2 shear responses, 2 biaxial tests and 2 uniaxial tests. There is, however, no conclusion as to the number of tests required with different deformation types to fully characterize myocardium.
The AIC analysis can also be used to choose the best combination of experiments. As shown in Fig. 10, different combinations of test data affect the prediction accuracy. Specifically, within the shear responses (Fig. 10a), the groups containing (fs) and (fn) always have better predictive capability than other groups. One reason is that the shear responses along (fs) and (fn) are much stiffer than other directions in both Dokos et al. and Sommer et al. data. For the biaxial test, most combinations have good predictive capability, which suggests that not all the biaxial tests in Sommer et al. data are needed to fit the general HO model or the HOS model. For instance, one stretching ratio with 1(MFD):0.75(CFD) from Sommer et al. biaxial tests has good predictions for other stretching ratios. But if the stretch ratio is largely nonequal, such as 1(MFD):0.5(CFD) or 0.5(MFD):1(CFD), the prediction is poor (see Fig. 13 in Appendix), partially because the material response with lower stretch ratios is still within the toe regime with nonstretched collagen fibres (Cheng et al. 2018; Lanir 1979). Prediction between different deformation types is poor, as shown in Fig. 10b, using biaxial tests only (case 563) and simple shear only (case 1484). This might be because one experiment type is inadequate to capture the nonlinearity and anisotropy of myocardium. Ahmad et al. (2018) included simple shear, biaxial and uniaxial tests, which allows investigation of uniaxial data in characterizing myocardium property. However, even with Ahmad et al. data, the predictions of uniaxial tests using the two biaxial and simple shear tests (case 25) are poor. As discussed in Holzapfel and Ogden (2009), biaxial tests are insufficient for characterizing a hyperelastic anisotropic material. When using stress responses from both the simple shear and biaxial tests, the least test data for the HOS model with good prediction are one shear test along (nf), together with a biaxial test 1(MFD):1(CFD). Our results presented here suggest uniaxial tests are still needed for an experiment like Ahmad et al. study, whilst further studies may be needed for experiments like Sommer et al. study using uniaxial tests.
In general, the stiffness aligned to the collagen fibre direction is much greater than the extracellular matrix, which is considered homogeneous and isotropic. Many studies have demonstrated the importance of excluding compressed fibres which cannot bear load (Zhuan et al. 2018; Holzapfel and Ogden 2017). Here we use a simpler approach, effective fibre ratios, to consider this effect. Because of the gradual fibre rotation transmurally, we assume the collagen fibres will experience the same deformation as the extracellular matrix only when both ends are stretched. A simplified FEM model based on Fig. 3b is simulated under uniaxial stretch along the MFD (Fig. 11), showing that the stress is much higher in the effective fibre area. The inclusion of the effective fibre ratio is also supported by Fig. 5, where the goodness of fit for the general HO model is much better than without it. The effective fibre ratio is a geometrical effect and depends on the sample size, loading direction and the local collagen fibre structures. It does not affect the fit to biaxial tests since the inplane collagen fibres will be physically stretched at both ends, but will affect the fit to the uniaxial and simple shear tests.
This study also demonstrates that biaxial stretch of myocardium cannot be free of shear. The shearfree scenario is only possible if fibres are strictly aligned in both stretching directions and without crossfibre effects. Both are not true in mycardium tissue tests. The assumption of no shear in the model leads to the poor outcome of predicting biaxial test data from simple shear tests, even if the general HO model is used. Indeed, we show that assuming shearfree behaviour in Sommer et al. biaxial testing produced relatively poor goodness of fit for both the general HO and HOS models; however, this is significantly improved when including a small shear component as per biaxial tests of fibrereinforced anisotropic material (Sommer et al. 2015a; Billiar and Sacks 2000)(Figs. 6a). As the shear components in the biaxial tests are not reported by Sommer et al. (2015b), the maximum shear angles are assumed to be the same along the CFD and MFD, respectively, at around \(6^o\). In Ahmad et al. data, the shear components in the biaxial tests are estimated, with the results presented here (Fig. 6b) suggesting that measuring of shear components in biaxial testing is necessary for myocardium and potentially, other anisotropic materials.
The average value avg and standard deviation std of optimized parameters from 100 random generated initial starts in interval (0.001, 50)
Experiment  Model  Parameters  

a (kPa)  b  \(a_\text {f}\) (kPa)  \(b_\text {f}\)  \(a_\text {s}\) (kPa)  \(b_\text {s}\)  \(a_\text {n}\) (kPa)  \(b_\text {n}\)  \(a_\text {fs}\) (kPa)  \(b_\text {fs}\)  \(a_\text {fn}\) (kPa)  \(b_\text {fn}\)  \(a_\text {sn}\) (kPa)  \(b_\text {sn}\)  
Dokos et al  HOD  avg  0.073  15.517  26.040  9.333  4.869  0.001  –  –  0.170  16.955  –  –  –  – 
std  4.0E–3  2.3E–1  1.1E–1  4.3E–2  4.7E–2  5.6E–5  –  –  5.0E–3  1.2E–1  –  –  –  –  
General HO  avg  0.019  8.576  25.790  9.668  4.281  0.010  0.001  0.868  0.250  16.037  0.025  13.826  0.252  8.773  
std  1.6E–2  7.7E+0  3.4E–2  1.5E–2  3.0E–2  3.8E–2  8.3E–4  6.5E–1  1.8E–2  1.8E–1  2.0E–2  5.7E+0  1.0E–1  4.7E+0  
Sommer et al  HOS  avg  0.809  7.474  1.911  22.063  –  –  0.227  34.802  0.547  5.691  –  –  –  – 
std  9.5E–4  4.9E–3  1.2E–3  4.8E–3  –  –  1.3E–3  2.2E–2  8.8E–4  1.8E–2  –  –  –  –  
General HO  avg  0.180  9.762  2.204  21.597  0.098  49.878  0.508  27.719  1.291  5.295  1.345  2.017  0.947  4.514  
std  4.4E–3  7.8E–3  5.1E–3  1.3E–2  1.8E–2  2.6E–1  3.3E–3  1.7E–2  4.9E–3  2.2E–2  1.6E–2  7.2E–1  2.2E–3  5.2E–1  
Ahmad et al  HOA  avg  0.075  18.143  7.067  1.339  –  –  2.745  4.497  1.859  4.066  3.541  8.222  –  – 
std  2.0E–4  1.6E–2  6.4E–4  9.1E–4  –  –  2.7E–3  6.2E–3  1.1E–3  3.4E–3  2.0E–3  7.2E–3  –  –  
General HO  avg  0.005  0.484  7.212  1.25  2.244  13.414  3.223  3.747  1.069  8.961  3.344  11.016  0.421  5.773  
std  3.5E–18  6.5E–19  9.0E–4  1.3E–3  3.0E–2  8.4E–1  4.8E–3  9.8E–3  7.7E–4  4.0E–2  2.6E–3  9.5E–3  5.5E–4  1.1E–1 
Many other constitutive models exist such as the “polezero” model (Nash and Hunter 2000), various Fungtype models (Costa et al. 2001; Guccione et al. 1991) and the constitutive framework with minimized crossterm covariance proposed by Criscione et al. (2002). The AIC analysis can be readily applied to select different types of material models. For instance, we can compare the HOD model and the Fengtype Guccione’s model (Guccione et al. 1991) with Dokos et al. shear data. We find that better fitting results can be achieved using the HOD model, which has a much lower AIC value (\(559.3\)) than the value from the Guccione’s model (\(65.8\)). This is because the Guccione’s model is a transversely isotropic material model, but myocardium is known to be orthotropic.
5 Conclusion
Footnotes
Notes
Acknowledgements
We are grateful for the funding provided by the UK EPSRC (EP/N014642/1). D. Guan also acknowledges funding from the Chinese Scholarship Council and the fee waiver from the University of Glasgow. F. Ahmad is grateful to the Ser Cymru NRN in Advanced Engineering & Materials for funding his PhD scholarship.
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