## 3 Assumptions for Numerical Modelling  
It is very common for studies to simplify by using a linear elastic model to describe the properties of soft tissues. Together with the assumption of incompressibility and isotropy of these tissues, this greatly simplifies calculations. The linear elastic model also makes it possible to describe material subjected to different types of loading. This idea is too general and already has an error in its assumption due to the nature of the linearity of the rheological response of the material [13; 14]. Another approach involves approaching tissue structure by describing it with hyperelastic models [10]. Both of these classes of models can provide useful predictions of the mechanical behaviour of different elastomers/soft tissues. Table 1 summarises the strain energy density functions of commonly used hyperelastic models. \begin{table}
\begin{tabular}{c c} \hline \hline
**Model** & **Equation** \\ \hline Neo-Hooke & \(W=C_{1}(l_{1}-3)\) \\ [PERSON] & \(W=\Sigma C_{ij}(l_{1}-3)^{i}(l_{2}-3)^{j}\) \\ [PERSON] & \(W=\Sigma(\mu_{k}/\alpha_{k})(\lambda_{1}{}^{ak}+\lambda_{2}{}^{ak}+\lambda_{3}{ }^{ak}-3)\) \\ [PERSON] & \(W=\Sigma C_{i}(l_{1}-3)^{i}+\Sigma\frac{1}{D_{i}^{i}}\left(\int^{l}\int^{l} \int^{l}\int^{l}\int^  
Due to the complexity of the model itself and the number of component inputs in the further stage of the study, it was decided to consider two hyperelastic rheological models, the Ogden model and the [PERSON] model. The aforementioned models do not require complex calculations and are successfully applied in the literature to fit the mechanical response of the liver. After preliminary simulations of the excitation of mechanical wave propagation in the liver by simulating the loading of the surrogate model with spherical, flat, and rounded tip indenters, the tuning of the ultrasound wave in biological media proceeded. Tuning was carried out with forcing by an ultrasonic signal with a frequency of \(f=3.0\) MHz. [PERSON]'s hyperelastic material model is a phenomenological model of deformation of nearly incompressible, nonlinear elastic materials such as elastomers. The model is based on [PERSON]'s observation that the elastic properties of rubber can be described by a strain energy density function, which is a power series in strain invariants. Since the polynomial form of the strain energy density function is used, but not all three invariants of the left Cauchy-Green strain tensor, [PERSON]'s model is also called the reduced polynomial model. [PERSON]'s mathematical model is shown below:  
\[U=C_{10}\big{(}\overline{I}_{1}-3\big{)}+C_{20}\big{(}\overline{I}_{1}-3\big{)} ^{2}+C_{30}\big{(}\overline{I}_{1}-3\big{)}^{3}+\frac{1}{D_{1}}\bigg{(}J^{el}- 1\big{)}^{2}+\frac{1}{D_{2}}\bigg{(}J^{el}-1\bigg{)}^{4}+\frac{1}{D_{3}}\bigg{(} J^{el}-1\bigg{)}^{6} \tag{1}\]  
\[\overline{I}_{1}=\overline{\lambda}_{1}^{2}+\overline{\lambda}_{2}^{2}+ \overline{\lambda}_{3}^{2},\ \mu_{0}=2C_{10},\ K_{0}=\frac{2}{D_{1}} \tag{2}\]  
where \(C_{i0}\), \(D_{i}\) are the material constants; \(I_{1}\) represents invariants of the Cauchy-Green deformation tensor; \(J^{el}\) is the volume ratio after and before deformation; \(\lambda_{i}\) represents principal stretches; \(\mu_{0}\) is the shear modulus; and \(K_{0}\) is the bulk modulus. The values of the material constants used for the Yeoh model were obtained through in-house experiments. The results are shown below in Table 2.  
The [PERSON] material model is a hyperelastic material model used to describe the nonlinear stress-strain behavior of complex materials such as rubbers, polymers, and biological tissue. The model developed by [PERSON], like other hyperelastic material models, assumes that their behavior can be described by a strain energy density function from which the stress-strain relationship can be derived. In [PERSON]'s material model, the strain energy density is expressed in terms of principal stretches \(\lambda_{j},j=1,2,3\).