An investigation of uniaxial mechanical properties of excised sheep heart muscle fibre – fitting of different hyperelastic constitutive models

: This paper presents the investigation of biomechanical behaviour of sheep heart fibre using uniaxial tests in various samples. Non-linear Finite Element models (FEA) that are utilised in understanding mechanisms of different diseases may not be developed without the accurate material properties. This paper presents uniaxial mechanical testing data of the sheep heart fibre. The mechanical uniaxial data of the heart fibre was then used in fitting four constitutive models including the Fung model, Polynomial (Anisotropic), Holzapfel (2005) model, Holzapfel (2000) model and the Four-fibre Family model. Even though the constitutive models for soft tissues including heart myocardium have been presented over several decades, there is still a need for accurate material parameters from reliable hyperelastic constitutive models. Therefore, the aim of this research paper is to select five hyperelastic constitutive models and fit experimental data in the uniaxial experimental data of the sheep heart fibre. A fitting algorithm was made used to optimally fitting and determination of the material parameters based on selected hyperelastic constitutive model. In this study, the evaluation index (EI) was used to measure the performance and capability of each selected anisotropic hyperelatic model. It was observed that the best predictive capability of the mechanical behaviour of sheep heart fibre the Polynomial (anisotropic) model has the EI of 100 and this means that it is the best performance when compared to all the other models.


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function that depends on exponential function. Recently, a comparative study was presented for unfilled and highly filled rubbers using hyperelastic constitutive models [19].
The aim of this study is to use uniaxial test data of the sheep muscle fibre to generate material parameters for application to five different hyperelastic constitutive models. The Hyperfit® was utilised in determining the material parameters and errors associated with fitting selected constitutive models. These material parameters could be utilised in the future for the development of computational models in understanding how myocardial infarction affects the global functioning of the heart.

Tissue acquisition and preparation
The fresh sheep hearts were collected from the local abattoir (See Figure 1(a)). The hearts were then placed in the cooler box before transportation to the Biomechanics Lab for preparation and testing. The muscle fibre was then isolated and taken out from the left ventricle (LV) as shown in Figure 1  ventricle (LV). The collected fresh sheep was then placed in an iced cooler box for 1 hour 20 minutes before testing.

Uniaxial mechanical testing
Ten (N = 10) specimen of the left ventricle muscle was cut out from the sheep hearts from an unknown age, weight, and pre-heart disease. The sheep hearts were delivered from the local abattoir three hours after the slaughter. The delivery was done in a temperature-controlled bag at between 0 0 C and 3 0 C. On arrival the temperature on the bag was measured to confirm and to ensure that mechanical properties were not To magnify the difference between coefficient of determination (R 2 ) values, the evaluation index (EI) that can assess which models are better than the others was defined [19].
Where r is the quantity that is based on coefficient of determination (R 2 ) and is expressed in Equation (5). The rminimum and rmaximum values are minimum r value for worst hyperelastic model and maximum r value for best hyperelastic model, respectively. When EI is 100 then it means that the hyperelastic model is the best. In addition, the greater the coefficient of determination (R 2 ) the greater the EI value.

Selected hyperelastic anisotropic constitutive models
It has been proven that like most soft biological tissues, when subjected to the mechanical force, the heart tissue exhibits strong nonlinearity, large strains and anisotropic elastic behaviour due to the presence of preferred directions in its microstructure and reorientation of the fibre directions with deformation. To develop computational models for simulation, the material parameters from hyperplastic constitutive models are required. These parameters are normally utilised to predict the mechanical response of the myocardium of the sheep. In this study, Fung, Polynomial (Anisotropic), Holzapfel (2000), Holzapfel (2005) and Four-fibre Family models were selected. The selected hyperelastic constitutive models were fitted using Hyperfit® [20].

Fung model
The Fung model is a hyperelastic anisotropic material model proposed by Fung et al [21] for the stress-strain description of the arterial wall. The model is fully phenomenological and formulated through the components of Green-Langrange strain tensor in cylindrical polar coordinates of the artery (R-radial, -circumferential, Z-axial).
Exponential stress-strain behaviour is a characteristic feature of this model: Where: for the specific coordinate system (corresponding to the supported data protocols) and for the symmetric configuration of fibre families: Where: Ψ -is strain energy density properties -is material parameter defining the orientation angle of fibres (measured from axis "1") in the undeformed configuration. Where: Ψ -is strain energy density Incompressible formulation of this model is used in Hyperfit implementation. one axial, one circumferential and two symmetrical in diagonal directions. Its SEF adopts the following express: Where:

Numerical analysis
This algorithm is the COBYLA method implemented as third-party library. SciPy library is adopted for this implementation. COBYLA method performs constrained optimisation by the linear approximation method. Coefficient of determination (R 2 ) (also known as Nash-Sutcliffe coefficient) Range: For perfect fit: Where: ye is experimental (observed) value of the fitted function ym is the model (theoretical) value of the fitted function i is the data point index n is the number of data-points ye is the mean value from the experimental values calculated as follows: Correlation coefficient (r) is defined as follows: Where: is the mean value from the models values (calculated below): Normalized RMS error (NRMSE) is defined as follows: Normalised error (NE) is defined as follows:

Results
In these experiments, ten (N = 10) specimens were extracted from ten sheep hearts and there was no visible tissue damage to all the specimens. The average measurements of the specimens were 7.45 ± 0.43 mm and 15.27 ± 0.62 mm for both thickness and width, respectively (See Table 1). The stress vales were calculated from the average crosssectional Area (ti x wi) from the average ti and wi shown in Table 1.

Uniaxial mechanical response
All ten (10) specimens were subjected to uniaxial forces at the same strain rate. The average stress at 0.5 strain for all ten specimens were found to be 31.44 ± 12.32 kPa. The maximum stress at 0.5 strain was found to be 48.49 kPa. Ten (N=10) specimens were subjected to uniaxial testing at the same strain rate. All the thickness and width of the sample were measured before testing to ensure that the cross-sectional area of the specimen can be calculated. The average thickness and width of all specimens were calculated to be 7.76 mm ± 0.43 mm and 15.27 mm ± 0.62 mm, respectively (See Table 1). Figure 3 shows the stress vs strain of the ten specimens of the sheep fibre of the LV. The mechanical response of the sheep fibre of the LV exhibits the hyperelatic and non-linear material behaviour (See Figure 3). However, it was observed that specimen 3 exhibited the lowest engineering stress when compared to the rest of the specimens that were tested.
In addition, the stress at 50% strain was captured and the average stress at 50% was calculated to be 33.4 kPa (see Figure 4). The maximum and minimum stress at 50% strain was captured to be 48.49 kPa and 4.98 kPa, respectively as shown in Figure 4. The standard deviation (STDEV) was calculated to be 12.32 kPa. Figure 5 shows the typical fitting of experimental data using the hyperelastic Fung model.         Comparison of how experimental data fits into the selected hyperelastic models is shown in Figures 6, 7, 8 and 9. Figure 4 shows that the best hyperelatsic model fit on the sheep heart fibre is the polylomial (anisotropric). However, the worse hyperelastic model identified was Holzapfel (2000). The Four-fibre Family hyperelastic model was seen to be the second best when fitted in the sheep fibre mechanical uniaxial test data. In addition, Figure 6 shows that the hyerelastic Fung model is third when compared to other models that were considered in this study. As expected, the least performing model, Holzapfel (2000) was recorded to have the highest normalised RMS error in most of the ten samples (See Figure 7). In comparing the hyperelastic Holzapfel (2005) and Polynomial (Anisotropic) models, the polynomial (Anisotropic) model was seen to have the lowest normalised RMS error in most of the ten samples (See Figure 7). This trend is like what has been observed in looking at the normalised error of all hyperelastic models (See Figure   8). Figure 9 shows that the hyperelastic Polynomial (Anisotropic) model has the highest correlation error when compared to others. The box plot shown in Figure 10 shows that the hyperelastic Holzapfel (2000) has the largest range of the coefficient of determination (R 2 ) when compared to other four hyperelastic models. However, it is to be noted that the lowest range of the coefficient of determination (R2) is seen on the hyperelastic Fung model followed by the Polynomial (Anisotropic) model (See Figure 9). In Figure 11, the hyperelastic Holzapfel (2005) has the widest range of the normalised error (NE) when compared to other models whereas the hyperelastic Polynomial (Anisotropic) model seems to have the lowest range of normalised error as shown in Figure 11. It was observed that the widest correlation coefficient range (r) is under the hyperelastic Holzapfel (2000) model while the Fung model has the closet range of the correlation coefficient (r).       It was found that the Polynomial (Anisotropic) model is better than all the other constitutive models based on the average high correlation coefficient (R 2 ) of 0.99. In this paper, the average correlation coefficient was utilised as a deciding factor, however, it is reported that averaging the stress-strain curve is the best option [28]. It has been observed that the average correlation coefficient (R 2 ) of experimental fitted data may not reproduce average hyperelastic constitutive model. Also, the individual correlation coefficient (R 2 ) generated from each sample (from N = 10) fitted in five hyperelastic models was compared.
This approach may assist in coming to the right conclusion since the plotted trend of selected constitutive models for each R 2 .
To fully characterise the mechanical response of the cardiac tissue biaxial testing is normally utilised [3,7]. However, in order to capture anisotropic mechanical response non-linear the shear tests are also required [16,29]. In this paper only uniaxial experimental data was utilised to fit the strain energy of various constitutive models. Even though this model may not fully characterise the mechanical response of the cardiac tissue of the LV of the sheep, enough evidence may be obtained on how best the constitutive model fitted the selected constitutive models. This is mainly because the material parameters obtained from fitting the strain energy of hyperelastic constitutive models may be directly utilised in developing computational models [8][9][10][11][12][13][14].
The mechanical uniaxial data of the heart fibre was fitted into four constitutive models to better characterise the underlying mechanics of the tissue. Soft tissue like the myocardium shows distinctive mechanical properties such as elasticity, viscoelasticity, The typical stress-strain curves (N =10) were fitted with all hyperelastic constitutive models shown in Equations (6), (8), (12), (15) and (20).
Lastly, it is observed that the full characterisation of the mechanical behaviour of the heart myocardium requires both biaxial and shear test. Anisotropic behaviour may also be fully understood by utilising inverse finite element modelling [8,9,30]. In conclusion,

Study limitations and future extensions
Determination of the mechanical properties of the sheep heart fibre on the LV of the myocardium tissue is based not only on the distribution and orientation of fibre tissue elements but also on collagen orientation, which was not investigated here. Hence, continued research is required to identify the related mechanics of tissue collagen orientation. In addition, a relatively small number (N=10) of heart tissue samples were investigated, so a meaningful correlation between biomechanical properties of the different heart fibre tissue could not be quantified. Uniaxial and biaxial mechanical tests are the most common methods to investigate mechanical properties of sheep heart fibre.
While the uniaxial mechanical test offers a quicker and easier examination of the material mechanical property, the biaxial mechanical test better mimics the loading conditions.
Hence, additional tests should be conducted with the biaxial testing system to determine the biaxial properties to expand the present data. Composition of the sheep heart fibre specimens may also vary throughout their dimension due physiological differences such as the myocardial structure, morphology, and underlying heart conditions and this was not investigated during the study. Future studies should characterise the important aspect of viscoelastic properties LV under different physiological conditions.

Conclusion
A critical issue in cardiovascular solid mechanics is the determination of the mechanical properties of various tissues under various loading conditions. Depending on the desired mechanical response to be measured, the tests can be conducted as uniaxial or

Conflicts of interest
The authors of this paper have no financial or personal relationships with other people or organisations that could inappropriately influence (bias) our work.