## 2 Description of the Problem  
Based on an analysis of the literature on the subject and our own research [8], it was found that it is very important to determine the parameters necessary for the construction of numerical models, including rheological models, and to choose such a model that will provide an optimal solution and approximation of the physical model. One of the important elements is the determination of the value of the propagation velocity of the ultrasonic longitudinal wave, c, ranging from about 1410 m/s in fat to about 1630 m/s in muscle. Other key parameters are [PERSON]'s modulus and Poisson's ratio for the elastic range of the test medium. On the other hand, the measuring head generates two types of waves: a low-frequency (50 Hz) transverse wave of high amplitude and a high-frequency (ultrasound 2-3 MHz) longitudinal wave. Based on the analyses, it is concluded that the speed and length of the generated transverse wave with a frequency of 50 Hz [8] do not significantly affect the transmission of the mechanical transverse wave to the liver parenchyma. The reason is that the thicknesses of all the layers on the path from the surface of the measuring head to the liver parenchyma (skin, adipose tissue, muscle tissue, pleural cavity with serous fluid, diaphragm, peritoneal fluid between the peritoneal membranes, liver capsule membrane) are smaller than the length of the applied wave.  
The liver is the largest gland in the human body. It weighs an average of 1.5 kg and contains more than 450 mL of blood. Averaged dimensions for the population are 280 \(\times\) 160 \(\times\) 80 mm. Figure 1 shows an image of the liver viewed from the diaphragmatic and visceral surfaces [9].  
The FibroScan(r) device is mainly used for transcutaneous examinations, in which wave propagation is carried out sequentially through soft tissue and the liver. The examination takes place on the surface between the ribs, and their close presence can affect wave propagation. A typical in vivo study is shown in Figure 2a. The authors of the study paid special attention to the use of the measuring head of the FibroScan(r) device on the so-called living liver, rather than its percutaneous use, as in the classical application. In view of the above, the head should be reconstructed by developing an additional overlay to replicate the surface between the skin and the liver. In the process of creation, special attention should be paid to matching the appropriate properties to replace the soft structures, i.e., the fraction from the skin to the liver (Figure 2b). This operation will allow the soft structures present to be taken into account during the test, so that the waveform parameters for the test on the living liver are the same as for the transcutaneous test.  
Figure 1: Liver: (**a**) anterior view, diaphragmatic surface, (**b**) posterior view, visceral surface [9].  
Modelling the soft tissue system and liver from the point of view of numerical analysis poses significant challenges. Few studies have been devoted to these issues in the literature [10]. Numerical simulation of ultrasound waves in the range of 2.0-3.0 MHz requires an appropriate methodological approach. The first stage of the study consisted of modelling the system (the liver, skeletal system, muscular system, and skin) and an indenter imitating a scanning head corresponding to the system of human anatomy. The thicknesses of the various layers were developed based on human cross sections taken in vivo at the level of the vertebral bodies of the tenth thoracic and first lumbar vertebrae [11]. Based on the literature review, two mathematical models were selected in describing the rheology of liver tissue structure, namely, the so-called parenchymal structure models of Yeoha and Ogden [12].