Biomechanical behaviour and hyperelastic model parameters identication of sheep omasum

The function of the omasum is incompletely understood; however, the omasum plays an important role in the transport of appropriately sized feed particles from the reticulorumen to the abomasum, oesophageal groove closure, fermentation of ingesta, and absorption of water, volatile fatty acids, and minerals. The aim of this study is to evaluate the suitable hyperelastic anisotropic model based on biomechanical properties of sheep omasum. The results show that all five (5) hyperelastic models may be suitable for the evaluation of sheep omasum. The average coefficient of determination (R 2 ) of Fung, Polynomial (Anisotropic), Holzapfel (2000), Holzapfel (2005) and Four-Fiber-Family hyperelastic models were found to be 0.79 ± 0.19, 0.95 ± 0.05, 0.92 ± 0.07, 0.93 ± 0.05 and 0.94 ± 0.03, respectively. Also, it was found that the best hyperelastic model for fitting uniaxial data of the sheep omasum was Polynomial (Anisotropic) with EI of 100.0 followed by the Four-Fiber-Family model with EI of 96.18. of the tissue. The established anisotropic materials properties could be utilised in the understanding of tumours of the omasum. The identified material parameters will provide new platform for FE investigations of mechanical aspects of various therapies when using sheep omasum models.


Introduction
Ruminant digestive physiology is largely dependent on pregastric fermentation in the forestomach. Therefore, in contrast to monogastric species the ruminant stomach system consists of a non-secretory forestomach functioning as a specialized mixing and fermentation vat and an acid and pepsinogen-secreting compartment functioning as stomach in the classical sense, the abomasum. The forestomach system anatomically consists of three primary structures: rumen, reticulum, and omasum [1]. Tumours of the omasum, reticulum, and rumen occur most often in cattle but are uncommon in most parts of the world. Papilloma, fibro papilloma, and fibroma are the most common tumours in the forestomach of cattle [2] .
There are many challenges that the field of tissue engineering must overcome to realise its clinical promise. These include concerns related to infection, adverse immune response, biochemical elements, and mechanical performance [3]. A biomechanical approach can be useful to investigate the stomach mechanical functionality [4]. The development of a computational model of the stomach can allow to examine in depth its functioning, related with complex neuro-gastrointestinal phenomena [5]. Models definition requires experimentations to characterise the organ geometrical conformation and the tissues mechanical behavior as such it is paramount to investigate the mechanical properties of omasum [6].
Computational modelling remains useful modality to validate experimental data and the mechanical behaviour of soft tissues [7][8][9][10][11]. However, to study the behaviour of these models it is essential to develop constitute model [12][13][14][15][16]. The Fung model is a widely utilised hyperelastic model which was originally developed for artery tissues. This hyperelastic model uses the strain energy function that depends on exponential function. Recently, a comparative study was presented for unfilled and highly filled rubbers using hyperelastic constitutive models [17]. In previous work, determination of layer-specific mechanical properties of human coronary arteries using Holzapfel constitute model was carried out [18][19][20]. The to develop these models the use of curve fitting is paramount [12]. Uniaxial testing remains the most common modality of mechanical analysis for biological and other soft materials [19]. It has been observed that Specimen mounting presents one of the most challenging tasks during soft tissue testing due to the fact that there is no widely accepted universal standard for clamp design for soft tissues. As a result there are many techniques that have been used to ensure that appropriate boundary conditions are imposed during soft tissue testing [21].
Study of mechanics of soft tissue is still relevant especially for the possible development of accurate computational models [7,8,11,12,16,[22][23][24][25][26] to study how diseases develop and advances. Sheep have been previous utilised to study mechanical properties of various important organs [12]. Currently there is no mechanical data on the sheep omasum. Therefore, this study aims to understand the mechanical properties of sheep omasum under uniaxial loading. Also, various hyperelastic models like Fung, Polynomial (Anisotropic), Holzapfel (2000), Holzapfel (2005) and Four-Fiber-Family were fitted in the biomechanical data to evaluate the best model that may be utilised for the development of computational models.

Experimental methods 2.1. Tissue acquisition
The fresh sheep stomach was collected from the local abattoir. The stomach were then placed in the cooler box before transportation to the Biomechanics Lab for preparation and testing. The omasum was then isolated and taken out from the stomach.

Uniaxial mechanical testing
Twelve (N = 12) specimen of the omasum was cut out from the sheep stomach from an unknown age, weight, and pre-stomach disease. The sheep omasum were delivered from the local abattoir four 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 compromised. The uniaxial testing was performed on the sheep strip of tissue (See Figure 1). Immediately after the arrival of the specimens, the sheep stomach were dissected to extract the selected omasum and thereafter stored in a 0.9 % NaCl physiological saline solution (PSS) for 20 minutes before mechanical testing. A total of 12 uniaxial tests were done on the omasum. During testing there were minimal challenges like tearing of tissue, fixation movement and tissue rapturing prematurely.
The dissected omasum was then mechanically tested using CellScale UStretch (Watreloo, Canada) with a load cell of 44 N and force accuracy of 0.2 % of the load capacity. The specimens were deposited in a bath filled with 0.9 % NaCl physiological saline solution. Each sample was immersed for four minutes before actual testing happened. To minimise the effect of strain measurements, the needles of the BioRake were inserted at a minimum of 0.2 mm from the edge.

Tissue stress-stretch analysis
The engineering stress and strain is determined from the force and displacement obtained during elongation of the muscle fibre of the sheep during uniaxial testing. Where: Ai -is the cross sectional area of the sheep muscle fibre Δ -is the change in length 2 -Original length " -Engineering stress " -Engineering strain?
-average thickness of the muscle fibre / -Breadth of the muscle fibre

Evaluation index (EI) of the capability of anisotropic hyperelastic models
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 [13].
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
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 omasum 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® [27].

Fung model
The Fung model is a hyperelastic anisotropic material model proposed by Fung et al [28] for the stress-strain description of the arterial wall. The model is fully phenomenological and formulated through the components of Green-Lagrange 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: -is the strain energy density "e -are components of Green-Lagrange strain tensor E referred to cylindrical polar coordinates (R, Ꝋ, Z) in the reference configuration C -is the stress-like material parameter b1, b2…b3 are dimensions material parameters

Polynomial (Aniso) Model
Polynomial (Aniso) model is hyperelastic anisotropic material model defined through polynomial series of strain invariants (Ii). The model definition is adopted from the ANSYS finite element package and simplified to symmetrical configuration of fibre families and to only I1, I2, I4 (I6) dependence. Incompressible formulation of this model is used in Hyperfit implementation. ( 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 P , U , b , d -are (pseudo-) invariants of C (and A0, B0) C -is right Cauchy-Green def. tensor Q , Q -are structural tensors referenced to individual family of fibres Q , Q -are the direction vectors defining the orientation of fibre families " , e -are stress-like material parameters referenced to isotropic (matrix) properties p = (= p ) -are stress-like material parameters referenced to anisotropic (fiber) properties -is material parameter defining the orientation angle of fibres (measured from axis "1") in the undeformed configuration.

Holzapfel (2000) Model
The Holzapfel (2000) model is a hyperelastic anisotropic material model proposed by Gerhard Holzapfel et al [29] for stress-strain description of arterial layers. The model is characterised by an isotropic matrix of neo Hookean form and anisotropic contribution of two symmetrical (and mechanically equivalent) families of (collagen) fibres reinforcing a vessel wall layer. Incompressible and Nearly-Incompressible (NI) formulations of this model are available in Hyperfit.
Where: Ψ -is strain energy density is stress-like material parameter referenced to matrix properties k1 -is stress-like material parameter referenced to fibre properties k2 -is dimensionless material parameter referenced to fibre properties -is material parameter defining the orientation angle of fibres (measured from axis "1") in the undeformed configuration d -is material parameter related to volumetric compressibility

Holzapfel (2005) Model
The Holzapfel (2005) model is a hyperelastic anisotropic material model proposed by Gerhard Holzapfel et al [18] for stress-strain description of arterial layers. The model represents a composite material reinforced by two families of (collagen) fibres that are mechanically equivalent and arranged in symmetrical spirals (in arterial layer).
Incompressible formulation of this model is used in Hyperfit implementation.

Four-fibre Family model (FFF)
Four-fiber Family model is a hyperelastic anisotropic material model adopted from Ferruzzi et al [21] for the stress-strain description of aortas and aneurysms. It is a variation of the HGO model, in which two more fibre families are added in longitudinal and circumferential orientations, respectively. This model therefore presents a total of four families of fibres; one axial, one circumferential and two symmetrical in diagonal directions. Its SEF adopts the following express: Where: Ψ is strain energy density

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: Where: q '''' 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: Figure 2 shows the stress-strain curves for all the samples (Test 1-12) tested on the sheep Omasum. The stresses are plotted up to 50 % strain, although half of the samples were tested up to 35% strain. Some of the stress-strain curves exhibit long toe regions that extend up to 25% strain.

Figure 2. Stress vs Strain of the sheep Omasum for all the tested samples
The elastic modulus was calculated two strain levels (15 and 20%). Figure 3 shows the elastic modulus of sheep omasum under axial loading for all the samples. Sample 5 and 6 shows the highest elastic modulus at both the 15% and 20% strain. To examine the exact trend of the stress-strain curves of all the samples relative to each other, the stress was calculated at 30% strain as it covered all the samples and plotted in Figure 4.

Figure 4. Stress value of all the samples at 30% of strain
The study investigated material parameters differences across the sheep Omasum samples using five hyperelastic different models and their prediction capabilities. The models were the Fung, the Polynomial model, the two Holzapfel (2000 and2005) and four-fiber-family models. The model material properties are presented for the five models in Table 1 to Table 5.    model material parameters (a1, a2,  a3, b1, b2, b3 , c2, c3, c5, c6, and R 2      To examine the exact differences between the five models, Normalised Error (NE) and Normalised RMS Error (NRMSE) were calculated and plotted in Figure 6. The NE and NRMSE results show that the errors from Fung model has the biggest error as compared to the other models.

Discussion
This is the first study to investigate the mechanical materials properties of sheep Omasum using different hyperelastic models. The sheep omasum samples showed large variability in stress strain response during uniaxial testing ( Figure 2). The large variability could be a result from variations in the regions of the Omasum where the samples were cut.
The results of the uniaxial tension tests of sheep Omasum stress-strain response were utilised to fit in the strain energy functions of Fung, Polynomial (Anisotropic), Holzapfel (2005), Holzapfel (2000) and Four-fiber family model. A direct comparison of hyperelastic constitutive models was made based on correlation coefficient (R 2 ) and evaluation index (EI). The Polynomial (anisotropic) hyperelastic constitutive model was found to have the EI of 100. This means that it is the best performance constitutive model when compared to other four hyperelastic model considered in this study. In addition, Four-Fibre family hyperelastic constitutive model was found to have the EI of 96 and therefore regarded as second best in terms of performances when fitted on the uniaxial experimental data of the sheep Omasum. This was followed by both the Holzapfel models (2005Holzapfel models ( , 2000 with EI of 83 and 63, respectively. Finally, the worst performer with EI of 0 was found to be Fung model.

Limitations
The measurement of sample thickness which was made difficult by the variable thickness of the Omasum. Errors in estimating the true sample thickness would affect the cross-sectional area, and hence the calculated stress.

Conclusion
As a first attempt to describe the mechanical material properties of the sheep omasum, five different hyperelatic models were used. This is the first step in understanding the mechanical material properties of Omasum and full characterisation of the tissue. The established anisotropic materials properties could be utilised in the understanding of tumours of the omasum. The identified material parameters will provide new platform for FE investigations of mechanical aspects of various therapies when using sheep omasum models.