Modeling the Stress–Strain State of a Filled Human Bladder

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The results of this work can be used for solving medical problems in emergency and reconstructive surgery, as well as to build intelligent surgical decision-making systems.

Abstract

In this paper, the problems of modeling the human bladder and its stress–strain state under an external static influence are considered. A method for the identification of the anisotropic biomechanical characteristics of the bladder tissue is proposed. An FEM model was created, which takes into account the fact that the bladder is surrounded by fiber, affected by surrounding organs, and partially protected by pelvic bones. The model considers the presence of constant hydrostatic pressure on the walls of the bladder when it is full. It has been shown that the isotropic mechanical characteristics of biological tissue can be used for studying the deformed state of a filled bladder if a filled bladder of 300 mL is considered as the initial non-deformed stage. This was shown by the modeling and verification of the effect of the external static force on the bladder. Numerical experiments were conducted based on the constructed model. To validate the results obtained, a series of natural experiments on the effect of external pressure on the bladder under ultrasound control were conducted. In the future, there are plans to use the constructed model to study rupture deformations of the bladder under the influence of static and dynamic loads.

1. Introduction

Trauma to the bladder leads to violations in the integrity of the organ wall. Usually, such injuries are caused by mechanical damage. At the same time, the mechanism of damage depends on different factors [1]. For example, with a blunt blow to the hypo-gastrium, intravesical pressure increases sharply and the load on the bladder wall grows. Also, the incident of a hydrodynamic effect contributes to the intraperitoneal rupture of the organ around the least developed muscles (near the tip on the back wall of the bladder) [2,3,4]. With a lower mechanical effect force, the impact causes closed damage (bruises, hemorrhages). Also, this pathogenesis is characteristic in the presence of urological diseases with a decrease in urine passage. A significant displacement of the bladder in the case of mechanical injuries leads to a sharp tension of the supporting lateral and vesico-prostatic ligaments with an extraperitoneal rupture of the soft-elastic wall of the organ [5]. A strong blow can cause the rupturing of ligaments, as well as urinary blood vessels.

Impacts to the bladder, for example, falling on a solid object, in some cases can lead to blunt trauma to the bladder, which is associated with a risk of serious complications. In some situations, such exposure leads to damage or rupturing of the back wall of the bladder. At the same time, this can be quite difficult to diagnose. This is because the clinical picture may be implicit, and an ultrasound analysis may not show damage to the tissues of the bladder. As a result, this can lead to peritonitis and death. Therefore, for the correct planning of the operation at the stage of the admission of the patient, it would be prudent to know how and how badly the bladder could have been damaged because of a fall or impact.

The criteria for the systematization of traumatic injuries are severity, connection with the environment, location of the rupture relative to the peritoneum, and the combined damage to other organs. In this regard, it is important to know how the human bladder is deformed under the effect of a blunt blow to the pelvic region. Knowledge of deformations of the bladder walls will allow us to predict the course of pathological processes and complications. In this regard, it will be possible to choose the optimal patient management tactics [6,7].

The problems involved in the mathematical and numerical modeling of the stress–strain state of internal organs are of interest to researchers. So, in [8], a finite element model of the human thorax and abdomen was developed based on computed tomography images from the Visible Human Project [9] for research on blunt trauma. In [10], the effect of blunt abdominal trauma on liver ruptures was studied using a finite element model. For the model, the authors used a detailed geometry of the liver obtained by tomography. Also, a simplified model of the torso was used. The verification of the model was carried out by comparing the curves of the liver displacement at a certain load. The authors in [11] showed the mechanism of damage to the spleen because of blunt force trauma based on finite element analysis. The stress distribution and damage to the spleen were considered by simulating the effect of a knuckle with a certain velocity in the spleen area on a THUMS human body model [12]. The results obtained using the model were compared with clinical cases of spleen injury. The authors concluded that the damage to the spleen was caused by a direct kick to the protuberance, and the compression of neighborhood tissues and organs. In [13], a finite element model of the eye was constructed for the dynamic modeling of blunt force trauma. After constructing the model, an analysis of the potential causes of damage to the optic nerve was carried out by evaluating the stress–strain state. In [14], a 3D model of the pelvic floor was created based on high-resolution images of a thin section to study pelvic organ prolapse in women. The Valsalva maneuver was simulated by applying pressure to the uterus and bladder from 0 to 10 kPa. The findings of the study suggest that an imbalance in abdominal pressure during obstetric surgery and on pelvic floor support structures can lead to pelvic floor prolapse in women.

Currently, there are not enough works devoted to the study of the deformation of the bladder under any influence. At the same time, the creation of new valid operational techniques is impossible without the use of methods for the preoperative prediction of the results of operations. It is necessary to use mathematical models to improve the viability of medical rehabilitation after operative interventions on the bladder. Mathematical models should consider the geometric parameters and mechanical properties of structures. It is also important to consider the methods of studying and evaluating the condition of the bladder and the preoperative prediction of the present state of the bladder as a result of operation.

Most of the works related to the modeling of the bladder are most often associated with the study of fluid dynamics and the behavior of the walls of the bladder during its filling and emptying [15]. For example, in [16], the relationship between changes in the basic size of the bladder and an increase in the volume of the bladder during filling was investigated. The results of finite element modeling showed that the size of the bladder increases linearly with its volume during filling. In another study [17], a three-dimensional finite element model was used to study the interactions between the bladder and the pelvic organs caused only by changes in the volume of the bladder.

There are also several studies related to the development of bladder substitutes. These include the creation of bioengineered tissues and implants, as well as other innovative methods to restore bladder function. For example, [18] presents urinary bladder phantom mimicking mechanical properties and pressure during filling. The authors note that soft hollow organ phantoms can be used to complement animal experimentations for developing and validating medical devices. In [19], models of the lower urinary tract based on silicone with a fibrous structure and based on PVA were presented. Modeling using Ansys shows that the deformation of the created silicone model during urination corresponds to the normal bowl-shaped deformation of the real analog. The issue under discussion is important and has attracted the attention of scientists from all over the globe. In [20], researchers conducted experiments on simulating the filling and emptying of the bladder, considering its mechanical properties. The aim of this research was to investigate the behavior of the bladder and identify a material that could be used to replace it in cases of certain diseases.

Most of the works related to the study of the bladder are devoted to the study of mechanical properties using various methods. Similar studies began in the 70s and one of the first works [21] confirmed that the bladder has viscoelastic properties. Thus, in [22], using step-by-step cytometry, the viscoelastic properties of the bladder wall tissue were analyzed. During the study, the authors obtained the modulus of elasticity and viscosity. Because a pig’s bladder is similar in characteristics to a human’s bladder, most studies conduct experiments on it. In [23], the viscoelastic properties of pig bladder tissue were studied using experiments on uniaxial stretching, ball bursting, and DMA. The researchers received differing information and concluded that using the results of all three methods in combination could be useful for other studies. The authors of [24] investigated the quasi-static uniaxial mechanical properties of the bladder walls. The study revealed a significant directional anisotropy of the mechanical properties of the bladder. The authors showed that the data obtained can be used to develop the most realistic computer modeling for predicting deformations in various conditions. In [25], the mechanical properties of a pig’s bladder were investigated by conducting stress relaxation tests and analyzing the results using a modified Maxwell–Wiechert model. The findings suggest that the tissue near the neck of the bladder exhibits different viscoelastic properties compared to the rest of the organ. There are also studies that have investigated the mechanical conduct of the human bladder. For example, [26] investigated the biomechanical properties of women’s bladders using uniaxial tensile tests. Tissue samples were obtained from cadavers without impaired pelvic floor functions. The obtained characteristics show that age can influence the mechanical conduct of the bladder. In addition, in [27], the biomechanical properties of the bladder of women with and without urinary incontinence were evaluated using reverse finite element analysis. The researchers used a method based on magnetic resonance imaging (MRI) techniques. The findings showed that the bladder tissue of women with urinary incontinence was more rigid than that of those who did not experience the condition.

Thus, there are not enough publications dedicated to modeling the stress–strain state of the human bladder, considering the surrounding tissues. Therefore, the objective of this work is to create a model of the human bladder and study the stress–strain state under the influence of a blunt blow to the suprapubic region. When creating the model, the fact that the bladder is surrounded by fiber was considered. The fact that the bladder is exposed to the surrounding organs and is partially protected by the pelvic bones was also considered. Of course, it was considered that the bladder was initially in a tense state due to the fact that it was full of fluid.

The remainder of this paper is organized as follows. The assumptions and limitations of current research, biomechanical properties, and finite element model of the bladder are described in Section 2. The research results are stated in Section 3. The validation of the suggested approach is presented in Section 4. Finally, Section 5 and Section 6 discusses the study’s findings, results, and the main conclusion of the work.

2. Materials and Methods

2.1. Assumptions and Limitations

The structure of the bladder consists of three regions (Figure 1): the mucosa, detrusor muscle, and adventitia [28,29]. The mechanical properties of each region are different and cannot be determined. For this reason, we will consider the bladder wall as a homogeneous spherical shell, the mechanical properties of which represent the properties of each region.

The bladder can expand 15 times compared to its unfilled state [30]. The volume of a full human bladder ranges from 330 to 440 mL [31]. In some cases, the bladder can enlarge to more than 600 mL. Existing research shows that a person’s bladder can be considered as a sphere if its volume is at least 100 mL [15]. Also, the ultrasound of a full bladder shows that its form remains spherical or elliptical (Figure 2). Therefore, to simplify the modeling of the stress–strain state of a filled bladder, we accept the assumption that its form is a sphere.

2.2. 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.

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 × 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.

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 (σ) and strain (ε) 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^th series. To keep the loading history constant throughout the study, all samples were preconditioned at a fixed number of series.

Deformation ε is the reaction of a material to an applied force and is defined as follows:

$$\epsilon ={\displaystyle \frac{∆L}{L_0}}$$

where $∆L$ 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:

$$\sigma ={\displaystyle \frac{F\left(1+\epsilon \right)}{bh}}$$

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]. Nonlinear behavior can be observed on the averaged graphs of the stress–strain curves of the bladder tissue obtained from the experiments (Figure 6). The arc-length corridor method was used to average the experimental curves.

Three characteristic regions can be seen in the stress–strain curves. The first area (Figure 6, zone 1) is a plateau, where there is a linear interrelation between stresses and deformations. In this area, the deformation curves for the two orientations have no statistically significant differences. The toe modulus is 0.11 ± 0.02 Mpa. Next, a nonlinear transition is observed in the region of 20% deformations.

The transition is followed by a linear area (Figure 6, zone 2). In this area, tissue hardening is observed due to the stretching of collagen fibers [35]. The heel modulus in the direction from the base to the apex of the human bladder is 0.42 ± 0.03 Mpa. In the perpendicular direction, the heel modulus is 1.24 ± 0.17 Mpa. The average value of the heel modulus is 0.83 Mpa.

In the last area (Figure 6, zone 3), due to damage to the sample, the resistance decreases. This is conducive to a breaking of the sample. The samples cut in the direction from the base to the apex show a high degree of hardening in zone 2. Also, these samples have a greater tension drop in zone 3.

Zones 1 and 2 (2′) in Figure 6 represent the physiological state of the bladder tissue. To simulate the stress–strain state of the bladder during a blunt impact, we will consider the mechanical characteristics relevant to zone 2. In zone 3, the bladder tissue is damaged, and its behavior is not physiological. For this reason, in modeling, we ignore the characteristics corresponding to zone 3. In this case, we will associate the transition points between zones 2 and 3 (points B and B′) with the limit state. The reason for this is that the destruction of the sample begins at these points [36,37,38].

The data obtained during the experiments correlate with the nature of the curves obtained in [24]. The study presented the results of a uniaxial tensile test on pig bladder samples in the axial and transverse directions. It has been shown that samples stretched in the transverse direction have a lower maximum stress than samples stretched in the axial direction. At the same time, when stretched in the axial direction, the maximum stress corresponds to a smaller deformation than in the transverse direction.

The ultimate tensile strain is 75 ± 3% for the direction from the base to the apex and 60 ± 6% in the perpendicular direction. These values turned out to be similar to the results from [39], where the ultimate tensile strain was 69 ± 17%.

2.3. 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.

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. A frictionless contact area was defined between the sphere and the front surface of the body model. A bonded contact was established between the bladder and the fiber (the geometry of the surrounding organs, fiber, and bladder was set so that all these components at the border had common nodes).

In this study, we will solve the problem of static deformation. In this regard, we assume that the volume of fluid inside the bladder is constant and there is no displacement of fluid inside the bladder. This assumption made it possible to use the hydrostatic three-dimensional fluid element HSFLD242 [45] (Figure 8) for modeling.

The HSFLD242 element is designed to study the effect of the volume and pressure of a fluid in hydrostatic problems related to the interaction of a fluid and a solid. The HSFLD242 element is defined by a set of nodes that are common to the nodes of the inner surface of a solid. The main nodes have three degrees of freedom. Also, there is a pressure node, which is in the volume of the fluid.

A script has been developed that permits the use of the HSFLD242 element in modeling applications. The script was developed because it is not possible to use this element through the mechanical Ansys Workbench interface.

Second-order tetrahedral SOLID187 elements were used to create a finite element mesh of the bladder, fiber, and surrounding organs. The calculation was carried out for 42,218 nodes and 26,014 elements.

The mathematical model of deformation can be written, taking into account the assumptions that were formulated earlier: the bladder is in the form of a sphere, the material is incompressible, and constant internal pressure persists. If we introduce a spherical coordinate system ($r,\theta , \phi$) with the origin at the center of the sphere, then stresses and strains will be related by the following relations:

$${\sigma }_{rr}-\sigma ={\displaystyle \frac{2{\sigma }_i}{3{\epsilon }_i}}{\epsilon }_{rr}, {\sigma }_{\theta \theta }-\sigma ={\displaystyle \frac{2{\sigma }_i}{3{\epsilon }_i}}{\epsilon }_{\theta \theta }, {\sigma }_{\phi \phi }-\sigma ={\displaystyle \frac{2{\sigma }_i}{3{\epsilon }_i}}{\epsilon }_{\phi \phi },$$

$$\sigma ={\sigma }_{rr}+{\sigma }_{\theta \theta }+{\sigma }_{\phi \phi }, \theta ={\epsilon }_{rr}+{\epsilon }_{\theta \theta }+{\epsilon }_{\phi \phi }$$

$${\epsilon }_i={\displaystyle \frac{\sqrt{2}}{3}}\sqrt{{\left({\epsilon }_{\theta \theta }-{\epsilon }_{\phi \phi }\right)}^2+{\left({\epsilon }_{\theta \theta }-{\epsilon }_{rr}\right)}^2+{\left({\epsilon }_{\phi \phi }-{\epsilon }_{rr}\right)}^2}$$

$${\sigma }_i={\displaystyle \frac{\sqrt{2}}{3}}\sqrt{{\left({\sigma }_{\theta \theta }-{\sigma }_{\phi \phi }\right)}^2+{\left({\sigma }_{\theta \theta }-{\sigma }_{rr}\right)}^2+{\left({\sigma }_{\phi \phi }-{\sigma }_{rr}\right)}^2}$$

${\sigma }_i=g\left({\epsilon }_i\right)=f\left(r,\theta , \phi \right)$—nonlinear law of elastic deformation that needs to be determined.

Due to the nonlinearity of the boundary and the geometry, a nonlinear static structural analysis was performed. In this analysis, the displacement vector of the nodes U is iteratively calculated using the finite element static equilibrium equations.

$$L=K U$$

where K = K(U) is the stiffness matrix and L = L(U) is the external force vector.

3. Results

For the numerical simulation, according to the experiments conducted by the authors and described in Section 2.2, the following mechanical parameters of the bladder were taken: a Young’s modulus of 0.83 MPa and Poisson’s ratio of 0.5. The characteristics of the fiber around the bladder are a Young’s modulus of 3.5 kPa and Poisson’s ratio of 0.5 [42]. The effective characteristics of the surrounding organs are a Young’s modulus of 5 kPa, and Poisson’s ratio of 0.5 [42].

As a load, pressing on the abdominal area of a blunt object in the form of a rigid sphere was simulated (Young’s modulus of 100 GPa, Poisson’s ratio of 0.3) at a 15 mm depth into the fiber. The values of the Young’s modulus and Poisson’s ratio are chosen to be large enough to represent an absolutely solid body in relation to the fiber and tissues. The place of the application of the load was slightly higher than the bladder, since most of the bladder is normally protected by the pelvic girdle. The general pattern of deformation in the cross-section of the model is shown in Figure 9; Figure 10 shows the deformation of the bladder in different projections. The maximum displacement in the bladder was 12 mm.

4. Validation

The ultrasound of a bladder filled to 400 mL was performed to validate the constructed FEM model. Two volunteers with a small amount of adipose tissue in the pelvic girdle area drank water. Then, when filling the bladder, an ultrasound was performed in two ways: without pressing the ultrasound sensor on the area with the bladder and by pressing on the volunteer’s abdomen in the projection of the Lieto triangle. It should be noted that the ultrasound was conducted immediately after the volunteer assumed a horizontal position. It is important to emphasize this, since when in a horizontal position, the organs are additionally deformed. Figure 11a shows the volunteer’s bladder immediately after it has assumed a horizontal position, and Figure 11b shows the state of the bladder after 15 min. This effect is probably due to the influence of gravity, the surrounding organs, and the heterogeneity of the surrounding fiber.

Since a significant biological deformation of the bladder is observed during a long stay in a horizontal position, this must be considered when constructing models of organ deformation in bedridden patients. The model proposed in this article can only be used for patients who were in an upright position before the application of exposure or have just assumed a horizontal position.

Figure 12a shows an undeformed bladder. Figure 12b shows the bladder when pressing on the patient’s abdomen by 1.5 cm. The deflection of the bladder was approximately 42 px, which is 11.07 mm.

Figure 13 shows the results of a similar experiment on a second volunteer and on another ultrasound device. The deflection of the bladder was also about 11 mm.

The shape of the deformed bladder obtained during the experiments and the values of the maximum deflections correspond to the digital FEM model described in this paper.

5. Discussion

In the experiments, the values of the anisotropic mechanical characteristics of the human bladder were obtained. The results of stretching in the axial and transverse directions are qualitatively similar to the results of the study of the pig’s bladder. Also, the average marginal deformation relevant to points B and B′ falls within the confidence interval of the human bladder values obtained in [39].

The stress–strain state of the human bladder under the influence of a blunt impact was studied using mathematical modeling. The FEM model of the bladder was constructed considering the influence of fiber and abdominal organs. The FEM model of the bladder was validated through the experiments by volunteers. The maximum displacement of the bladder wall in the FEM model turned out to be close to the same value in the experiment. The difference was less than 10% (

Table 1. The values of the maximum displacements (mm) in the FEM model and in the experiment.

	The Maximum Displacement (mm)
FEM model	12
Experiment 1	11.07
Experiment 2	11

).

It has been shown that the isotropic mechanical characteristics of the human bladder wall can be used to determine the stress–strain state of a bladder with a volume of 300 mL or more.

This was shown by modeling the force effect on the bladder. Also, this is correct when the volume of the bladder increases.

In this paper, this was demonstrated by modeling the force effect on the bladder with a volume of 400 mL. A similar simulation can be given for another bladder volume above 300 mL. In further studies, there are plans to use this approach to assess the impact force at which a rupture of the human bladder occurs.

6. Conclusions

Currently, there are not enough materials in the scientific literature on biomechanical studies of the state of the bladder during normal, pathological, and postoperative changes. The unavailability of necessary knowledge about the emerging stress–strain state of the bladder in pathological or postoperative changes explains the fact that, due to the constraints of necessary information support, there has not been a general opinion on the technology of operations to date. Also, the conditions for the minimal traumatization of biological structures have not been studied; as a result, operative interventions are often performed “blindly”, and the surgeon is guided only by previous experience, relying on intuition. The development of a mathematical model of the stress–strain state of a filled bladder allows us to consider the particularity of the geometric dimensions of the structures of the bladder, as well as to predict possible pathological changes in a particular patient after falling on a solid object. Modeling also makes it possible to determine the tactics of surgical intervention necessary to restore and preserve functional properties, and thereby carry out the preoperative prediction of results.

In this paper, models of the static deformation of the spherical bladder were considered. The limitation of the constructed model is that it is applicable only to patients who are in an upright position during impact or who have fallen on a solid body from an upright position. This is due to the fact that when staying in a horizontal position for a long time, the internal organs are additionally deformed by gravity. The shape of the bladder after such a deformation is difficult to predict, since most likely it will be due to the heterogeneity of the surrounding fiber, the presence and volume of fat deposits, and the location and interaction of internal organs. In other words, the deformation of the internal organs of a patient who has been in a horizontal position for a long time will be specific to each patient.

For further research, it will be necessary to consider the effect of the elliptical shape on the deformation of the bladder. There are also plans to study the dynamic stress–strain state of the bladder, considering its fullness with fluid. Since in some cases, the fall or impact of a solid body on the bladder leads to the rupturing of the back wall of the bladder, at the next stages of the study, it is necessary to study the mechanism of the transmission of the impact from the front to the back wall and identify the conditions that lead to damage to the tissues of the bladder.