# Modeling the Stress-Strain State of a Filled Human Bladder
## 2 Materials and Methods
### Finite Elements Model  
Finite element modeling was performed using a powerful engineering simulation software, Ansys 2022 R2 Workbench. A numerical and finite element model of the bladder and surrounding areas was developed. The geometry of the model was built using Ansys 2022 R2 DesignModeler tools. The bladder is in the pelvis (Figure 7). In the unfilled state, the bladder is completely protected by the pelvic girdle. However, when the bladder is filled, it can extend beyond the pelvic girdle. Figure 7: The location of the bladder inside the pelvic girdle (**a**) and the digital model of the bladder (**b**): 1—filled bladder, 2—fiber (images created by the authors). The research conducted in [40], in which the urodynamic study of the human bladder using ultrasonic vibrometry was carried out, shows that the increase in the stiffness of the bladder wall begins when its volume reaches 300 mL. As can be seen from Figure 6, the onset of the stiffening of biological tissue corresponds to a 20% strain. According to this, we will consider the volume of 300 mL as the initial undeformed state of the bladder. The stresses accumulated in zone 1 will be neglected because they are small compared to the stresses that occur during further deformation (Figure 6). The stress-strain state of the bladder wall will be described by the curve from zone 2 (2'). This is due to the fact that the beginning of this zone represents the beginning of an increase in the rigidity of the wall and the volume of the bladder by 300 mL. It was decided not to take zone 3 into account in modeling, since it corresponds to an area in which the biological tissue is damaged. As a first attempt to solve the problem, we will assume that the material of the bladder wall is a linear elastic isotropic material. The average value of the heel modulus (E = 0.83 Mpa) was taken as the value of the elastic modulus. To simplify, suppose that with an increase in volume, the bladder retains a spherical shape and will be in a homogeneous stressed state. Then, for the initial state, we can take any point in zone 2 with the corresponding initial volume of the bladder. In this calculation, the value of 400 mL is used as the initial volume of the bladder, since this volume was used to validate the model. In this case, the rupture of the bladder will appear at a lower strain, since the point on the curve in zone 2 will be moved to the right. The fiber surrounding the bladder was defined as a spherical shell with a variable thickness. The thickness of the fiber at the top of the bladder was 2 mm, and at the base it was 5 mm [41]. The fiber material was given by an incompressible linear elastic material with an elastic modulus of 3.5 kPa [42]. The surrounding organs were defined by a homogeneous incompressible linear elastic material with an elastic modulus of 5 kPa. This value was obtained by averaging the values of the elastic modulus of the abdominal cavity organs taken from the literature [42]. The surrounding organs were included in the model to obtain a believable distribution of pressure on the bladder. It was assumed that the fiber and surrounding organs do not have a prestressed state. The geometry of the surrounding organs (Figure 7) was built based on the distance (25 cm) between the distantia spinarum and the direct size of the conjugata externa (20 cm) taken from the literature [43,44]. With the help of boundary conditions in the form of a fixed support, the influence of the pelvic and spinal bones was considered. The case of thin subcutaneous fat was considered, so the distance from the bladder wall to the outer border of the body was set to 20 mm. The impact of blunt abdominal trauma was modeled by pressing an ultrasound sensor into the pelvic area. For simplification, a rigid sphere with a radius of 75 mm, which corresponds to the radius of the curvature of the sensor, was considered as an ultrasound sensor. The pressure was considered as crossing the sphere by 15 mm along the \(X\)-axis.