# Modeling the Stress-Strain State of a Filled Human Bladder
## 2 Materials and Methods
### Biomechanical Properties  
There are not enough data in the literature about the mechanical characteristics of the human bladder. For this reason, uniaxial tensile testing experiments were conducted using a testing machine (Figure 3a). To conduct this standardized test, samples are cut from the material. Next, the samples are placed between the grippers and the material is stretched until it breaks. Measuring samples were cut in a certain way from the human bladder. The flowchart of the experiments is shown in Figure 3b. Figure 1: Structure of the bladder (the image created by the authors). Figure 2: Ultrasound of a full bladder: (**a**) 510 mL; (**b**) 420 mL (photographs get by the authors). A sample section pattern is shown in Figure 4. This pattern is shown since the material of the bladder walls is an anisotropic material. Samples 2, 4, 6, 8, 10, and 12 were cut in the direction from the base to the apex and samples 1, 3, 5, 7, 9, and 11 were cut in the perpendicular direction. The sample dimensions are 40 mm \(\times\) 20 mm. The thickness of the samples was about 2.5 mm. The samples were taken from five human bladders. They were not subjected to rotting or freezing processes. Next, the samples were treated with a mixture with formaldehyde. Then, the samples were wrapped in foil and placed on an ice substrate. Within 12 h, the samples were transported for the experiments to determine the mechanical properties of the material. Figure 5 shows the process of preparing human bladder samples and the image of the prepared samples. Figure 4: Pattern of cutting samples from the human bladder. Figure 5: Sample preparation process: (**a**) measurement and excision of bladder samples; (**b**) prepared samples (photographs got by the authors). Figure 3: Experimental setup: (**a**) uniaxial testing machine for determining the mechanical characteristics of a material; (**b**) the flowchart of the experiments. A uniaxial tensile test was performed to determine the mechanical characteristics of the human bladder wall. The distance between the grips was calibrated to 20 mm. Preconditioning was performed by repeatedly loading and unloading (cyclic) the specimens at a strain rate of 50%/min. In each test, load and displacement data were obtained and converted into stress (\(\sigma\)) and strain (\(\varepsilon\)) rates using the cross-section area (true stress). The preconditioning of biological soft tissue is considered as an essential step towards establishing a repeatable set of experiments [32; 33]. A preconditioning procedure was carried out with a deformation amplitude of 10%. This amplitude was chosen to establish tissue resistance to subsequent loading. The hysteresis curve showed repeatability starting from the 7\({}^{\text{th}}\) series. To keep the loading history constant throughout the study, all samples were preconditioned at a fixed number of series. Deformation \(\varepsilon\) is the reaction of a material to an applied force and is defined as follows:  
\[\varepsilon=\frac{\Delta L}{L_{0}}\]  
where \(\Delta L\) is the changing in the length of the sample and the tensile length, \(L_{0}\), is the original length of the sample. The true stress \(\sigma\) is the force per unit area that is created in the material's cross-section due to deformation. Given the incompressibility of the material, the true stress can be defined as follows:  
\[\sigma=\frac{F(1+\varepsilon)}{bh}\]  
where \(b\) is the thickness of the sample, \(h\) is the width of the sample, and \(F\) is the applied force. The rupture of the samples was observed in the center. Samples cut from the base to the apex and perpendicular directions demonstrate different behavior. In the case of samples cut from the base to the apex, there is greater rigidity compared to the other case. The mechanical properties of soft biological tissues are determined by three main components. In this regard, soft tissue is characterized by nonlinear elastic behavior [34].