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. Author manuscript; available in PMC: 2025 Sep 23.
Published in final edited form as: Biomech Model Mechanobiol. 2024 Dec 31;24(1):347–359. doi: 10.1007/s10237-024-01914-7

Modeling bladder mechanics with 4D reconstruction of murine ex vivo bladder filling

Eli Broemer 1, Pragya Saxena 2, Sarah Bartolone 3, Grant Hennig 4, Gerald M Herrera 4, Bernadette Zwaans 3,5, Nathan R Tykocki 2, Sara Roccabianca 1,6
PMCID: PMC12452039  NIHMSID: NIHMS2080379  PMID: 39741201

Abstract

This study presents a novel methodology for high-resolution 3D bladder modeling during filling, developed by leveraging improved imaging and computational techniques. Using murine bladder filling data, the methodology generates accurate 3D geometries across time, enabling in-depth mechanical analysis. Comparison with a traditional spherical model revealed similar stress trends, but the 3D model permitted nuanced quantifications, such as localized surface curvature and stress analysis. This advanced 3D model captures complex tissue behavior crucially influenced by tissue-specific microstructural characteristics. This methodology can also be extended to other tissues such as lungs, uterus, and gastrointestinal tract tissues. Applying this analysis to different tissues can uncover mechanisms driven by localized mechanics, such as the sensation of fullness in the bladder due to microcontractions, uterine contractions during labor, and peristaltic contractions in the gastrointestinal tract. This broader applicability underscores our approach’s potential to advance the understanding of tissue-specific mechanical behaviors across various biological systems.

Keywords: Bladder, Murine, Imaging, Reconstruction, Stress

1. Introduction

The urinary bladder exhibits intricate mechanical characteristics, undergoing significant stretching during filling while maintaining low pressures. As it nears its elastic limit, pressure rises, signaling the brain’s need for voiding (Griffiths 2007; Nenadic et al. 2016). However, the precise biomechanical mechanisms underlying the sensation of fullness remain elusive because bladder mechanics are more sophisticated than they seem. The bladder wall tissue is described as viscoelastic, anisotropic, and inhomogeneous (Nagatomi et al. 2004; Morales-Orcajo et al. 2018), properties stemming from a complex microstructure that is finely tuned to facilitate extensive deformations within a duty-cycle lasting hours while still being able to generate significant force for voiding contractions (Smith et al. 2012). While many characteristics have been described individually (Nagatomi et al. 2004; Morales-Orcajo et al. 2018), there is a need for comprehensive mechanical descriptions capable of fully capturing the bladder’s sophisticated mechanics. Moreover, it is essential that enhanced mechanical testing and analysis are used to first characterize the healthy bladder, thereby providing a foundational understanding to tackle a spectrum of bladder dysfunctions (Wyndaele et al. 2011). Furthermore, the sensation of fullness in the bladder is influenced by microcontractions (Heppner et al. 2016) which are highly localized, and therefore crucially influenced by local tissue properties and mechanical loading.

The task of capturing the physiological behavior of the bladder in terms of its mechanics is indeed challenging. Common ex vivo methodologies to assess its mechanical properties, such as uniaxial or biaxial testing, often require tissue sectioning prior to testing (Zwaans et al. 2022; Chen et al. 2013; Hornsby et al. 2017). While these methods aid in material characterization, they do not fully reflect the physiological loading conditions experienced by the organ as a whole in vivo. On the other hand, in vivo bladder pressurization via catheterization and cystometry provides valuable insights into organ function but collect incomplete information with regard to mechanical properties (Smith et al. 2012; Mingin et al. 2015; 2014; Chang et al. 2009; Cornelissen et al. 2008; Yu et al. 2014). Quantifying the thickness and microstructure of the bladder wall as they relate to physiological functioning is necessary for characterizing the mechanical nature of the bladder wall. However, quantification of thickness and composition is challenging with in vivo methodologies due to the invasiveness of obtaining accurate measures. While some innovative methods using human, pig, and rat models have quantified the mechanics of pressurized bladders, research on whole-organ bladder mechanical modeling remains limited (Trostorf et al. 2022; Makki et al. 2022; Cheng et al. 2022). Thus, it is apparent that further advancement in whole-organ level mechanical modeling of the bladder is necessary to derive comprehensive mechanical measures that correlate with organ function. This is where the method presented here delivers value: We show that our ex vivo whole bladder filling method allows for both bladder pressurization and material mechanical modeling, so that the relationship between bladder function and mechanics can be explored.

Digital image correlation (DIC) is an established method for tracking deformations of a material under load. It has been used to study stresses in the bladder wall in both biaxial (Morales-Orcajo et al. 2018) and whole-organ pressurization (Trostorf et al. 2022). Typically, markers or a spackle pattern are applied to the material surface for DIC methods, although deformations and stresses without marking the surface have been modeled in other pressurized organs (Genovese et al. 2014). In the current study, bladder mechanics were obtained without the use of a surface pattern due to challenges with surface adhesion, and the complexity of accurate 3D point tracking. Using techniques inspired by Genovese et al., we were able to successfully model deformation and stresses of the bladder wall without typical DIC surface treatments.

The conventional approach for modeling bladder mechanics at the organ level employs a thin-walled spherical pressure vessel model (Nagle et al. 2017; le Feber et al. 2004; Watanabe et al. 1981; Regnier et al. 1983; Korkmaz and Rogg 2007; Korossis et al. 2009; Damaser 1999; van Mastrigt et al. 1978; van Beek 1997; Schüle et al. 2021). This method utilizes the thin-walled assumption in conjunction with Laplace’s law (Basford 2002), which relates wall tension to internal pressure, vessel radius, and wall thickness. Although this model is straightforward and produces results comparable to more complex models (Monteiro 2013), it oversimplifies the underlying mechanics and overlooks critical factors (such as anisotropy and inhomogeneity) that are essential for a comprehensive understanding of bladder behavior. Developing non-spherical models, however, requires substantial data and expertise, presenting significant challenges. As a result, efforts to model bladder mechanics in three dimensions over time often face limitations in spatial or temporal resolution, demand extensive labor, or are prohibitively expensive (Chai et al. 2012; Costa et al. 2007).

This study introduces a method for mechanical modeling of whole-organ biomechanics under physiological loading conditions. This work aims at generating datasets for spatially accurate modeling of the unique tissue mechanics of the urinary bladder with high temporal resolution. The methodology combines high-frequency imaging of the bladder during filling with computer vision algorithms to generate detailed analyses of geometry and wall stresses from empty to full capacity. We then compared our 3D model to the traditional spherical model, highlighting both differences and similarities. Additionally, the data processing and modeling steps are compiled into a highly automated pipeline utilizing data from a low-cost experimental setup (Hennig et al. 2022). As such, this work provides a necessary tool to expand our knowledge about the healthy bladder in mice by employing a novel methodology capable of producing detailed mechanical models in an automated and unbiased manner.

2. Methods

2.1. Animal experiments

Data were collected from 8 male C57BL/6 mice ages 10–12 weeks old. The complete experimental protocol has been previously described (Hennig et al. 2022). Briefly, the bladder was extracted and tested on a custom Pentaplanar Reflective Imaging Microscopy (PRIM) device illustrated in Fig. 1a as well as in Fig. 1 of (Hennig et al. 2022). Each bladder was cannulated in the device and pressurized with three loading–unloading cycles to allow for tissue acclimation. Bladders were filled to a maximum luminal pressure of 25.5 mmHg at a constant rate of approximately 30 μL/min. Internal bladder pressure was measured via pressure transducer and synchronized with video imaging, each at a rate of 10 Hz. The data from the last filling cycle for each bladder were analyzed. Data were processed with Python 3.7.9 and custom code. Data collected and analyzed included synchronous recording of luminal bladder pressure measurement and video imaging. From each test, data were subsampled at 0.5 mmHg pressure steps from 0 to 25.5 mmHg (52 data points for each bladder). For each pressure step desired, the pressure value nearest to the step in the data was used.

Fig. 1.

Fig. 1

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3D reconstruction pipeline. a Data were collected from 8 male C57BL/6 mice (10–12 weeks of age) using the Pentaplanar Reflected Image Macroscopy (PRIM) system as previously described (Hennig et al. 2022). Image of the full field of view (b) and the result of binary segmentation (c) of a representative bladder. Silhouettes were projected on their respective axes and a solid primitive is initialized (d). Both sides and top silhouettes were used to carve a voxelized bladder reconstruction (e). For comparison, wireframe mesh of a reconstructed bladder at 25 mmHg pressure was overlaid with the fitted sphere for its same geometry (f). g–h, bladders became more spherical with increasing pressure. Fitted sphere error decreased with pressure as measured by effective sphere root-mean square deviation (RMSD) (g). Sphericity was lowest at low pressures (< 5 mmHg) across all samples, and approached 1 (perfect sphere) as pressure increased (h). 2D graphs were created with Matplotlib (Hunter 2007), and 3D renders were created with Mayavi (Ramachandran and Varoquaux 2011)

2.2. 3D bladder reconstruction

The shape-from-silhouette method (Baker 1977; Chien and Aggarwal 1986; Martin and Aggarwal 1983) was used to generate bladder 3D reconstructions at all time points in the subsampled data. Initially, the image data showed 5 views of the bladder (Hennig et al. 2022). However, only three orthogonal silhouettes were essential for this reconstruction method, and the two remaining silhouettes were redundant (Chien and Aggarwal 1986). First, silhouettes were obtained by masking three orthogonal bladder views from the background in the image data (Fig. 1b, c). A U-Net machine learning model (Ronneberger et al. 2015) was employed for this binary segmentation task. The U-Net architecture was sourced from an open-source implementation (Alexandre 2017) and trained on our dataset. For training the neural network, 1500 individual bladder images were collected from the complete video data, and corresponding segmentation masks were created manually. The training dataset did not intersect with the subsampled data. Data augmentation allowed the dataset to be expanded to 12,000 image-mask pairs via rotations and cropping. The augmented dataset was split 80% for training and 20% for validation with random selection. To optimize the model’s performance for this specific application, various hyper-parameter combinations of learning rate and weight decay were explored. The best model was selected based on lowest training and validation errors, convergence without over-fitting or under-fitting, as well as qualitative assessment of model segmentation results. Final segmentations were reviewed and manually repaired, when necessary, before the next steps of the analysis.

Next, the 3D bladder shapes were obtained from the volume intersection of the three orthogonal silhouettes at each time point (Fig. 1d, e). For each instance of reconstruction, a 3D array with dimensions corresponding to the silhouettes was initiated and populated with ones as V^(x,y,z)=1 (Fig. 1d). Let Sx(y,z), Sy(x,z) , and Sz(x,y) be the binary silhouettes projected on the planes normal to the x-, y-, and z-axes, respectively, oriented as specified in Fig. 1d. The silhouettes allocate a value of 1 to the portion of the image occupied by bladder tissue, and a 0 to the background. Each axis of the V^ array was then multiplied by its respective silhouette. Specifically, Sx and Sy were first multiplied element-wise on the array by V^(x,y,z)⋅Sx(y,z)⋅Sy(x,z) (Fig. S1a–c). However, to overcome the visual hull effect (Laurentini 1994), Sz was dynamically modified for each z-axis layer of the array. For each layer k along the z-axis, the bladder cross section created by Sx and Sy was identified, and Sz had its silhouette region rescaled to correspond to the cross-section dimensions. The rescaling step effectively eliminated sharp corners in the final shape which are inherent to shape-from-silhouette volume carving from three perspectives (Fig. S1d). Obtaining the final bladder volume shape can be expressed by

V(x,y,z)=V^(x,y,z)⋅Sx(y,z)⋅Sy(x,z)⋅Szk(x,y), (1)

where Szk be the rescaled Sz for layer k and V(x,y,z) is a 3D array containing ones where there exists bladder volume, and zeros where there exists empty space. This process gave a voxel representation of the bladders at each subsampled time point (Fig. 1e). Measure of the volume occupied by the bladder in voxels was obtained by multiplying the total sum of the array by the cubed pixel-to-mm scale factor, which is 2.62·10−4 for this specific system.

Finally, the voxelized bladder reconstructions were transformed into triangular surface meshes using a marching cubes algorithm (Lewiner et al. 2003; Virtanen et al. 2020). This allowed the tissue surface to be represented with 3D point coordinates in space, rather than an array. To improve the quality of the mesh generated by the marching cubes algorithm, the outer surface of the bladder in the voxel array was labeled first. In essence, for each reconstruction, each voxel which did not have an empty voxel in its 26 neighbors was deemed a non-surface voxel and had its value incremented. More specifically, a new 3D array was initialized as Vs(x,y,z)=V(x,y,z). Then, for each i-th solid voxel, where Vxi,yi,zi=1, the sum of values in its 3 × 3 × 3 neighborhood, was computed as

Σxi,yi,zi=∑a=−11∑b=−11∑c=−11Vxi+a,yi+b,zi+c. (2)

The values in the array Vs were then updated such that Vsxi,yi,zi=2 if Σxi,yi,zi=27. This resulted in Vs being zero where there is empty space, one at outer surface of the bladder, and two inside the bladder. The marching cubes algorithm was then used to generate surface meshes for each voxel array at the boundary, i.e., where Vs(x,y,z)=1. Constructing Vs allowed the algorithm to generate meshes with nearly half the number of vertices and faces and minimal topological differences compared to running the marching cubes algorithm on the binary V array. Note that the V arrays were padded with zero on all sides to ensure Eq. (2) is always valid, and to provide the marching cubes algorithm a clear transition boundary to develop the surface from. As a last step, the meshes, which were originally quite faceted, were smoothed with Laplacian smoothing (Vollmer et al. 1999; Dawson-Haggerty 2013) using a lambda of 0.9 and 50 iterations.

2.3. Definition of metrics for assessing the shape of the bladder

After creating surface meshes for the data of interest (i.e., all bladders at subsampled time points), several metrics for assessing the shape of the bladders were obtained. These include: fitting an effective sphere, sphericity (Wadell 1935), and principal curvature.

The radius of the effective sphere was obtained by minimizing the difference between the surface mesh vertex points of the reconstructed geometry and a sphere. The general equation to describe the surface of a sphere centered at the origin is x2+y2+z2=a2 where a is the sphere radius. The radius of the best fitting sphere, a¯, can be calculated analytically as the average distance of all vertices on the mesh surface from the centroid. To simplify this process, the meshes were translated such that their centroid was aligned with the coordinate system origin. Thus, for the i-th vertex on a mesh the distance from the origin was found as di=xi2+yi2+zi21/2, with i=1,N. It follows that the effective radius was

a¯=∑i=0NdiN (3)

To calculate the goodness-of-fit for the sphere, we quantified a surface deviation parameter as the root-mean square deviation (RMSD) of di and a¯ of the fitted sphere as

RMSD=∑i=1Ndi−a¯2N (4)

Sphericity (Wadell 1935), a measure of how spherical a surface is, is defined as a ratio of volume V and surface area A as

Ψ=π1/3(6V)2/3A (5)

Both Eqs. (4) and (5) can be interpreted as quantifying how spherical the bladder surfaces were. The RMSD of the best fitting sphere provided a more localized assessment since it is based on specific surface irregularities at each point in a mesh. Sphericity gave a more global assessment of the surface since it derives from the total volume and surface area.

2.4. Stress calculation

Principal curvatures of each surface mesh (i.e., κ1 and κ2 ) were quantified using an established algorithm (Panozzo et al. 2010; Jacobson and Panozzo 2018). In short, at each vertex in a mesh, a local quadric surface is fit to nearby vertices and the principal curvatures of the quadric were found using differential geometry principles. Essentially, the eigenvalues and eigenvectors of the negative differential of the fitted quadric surface’s normal vector field give the principal curvatures and principal directions at the vertex (Porteous 2001). Quantifying curvature of the bladder surface provides localized geometrical data, as well as informing a non-spherical mechanical model of pressurization.

Mechanical analyses were formulated for the full 3D meshes and the spherical approximation. Generally, at each point, the Cauchy stress tensor is defined as

σ=σr000σ1000σ2 (6)

where σr is the principal stress in the radial direction, and σ1 and σ2 represent the principal stresses along the in-plane principal directions defined at each point. For a thin-walled, axisymmetric membrane with negligible bending stiffness, we can define the principal stresses following the generalized Laplace’s law (Humphrey and Kyriacou 1996) as

σr=−P2,σ1=P2hκ2,σ2=Phκ21−κ12κ2 (7)

where κ1 and κ2 are the in-plane principal curvatures, P is the luminal pressure, and h is the wall thickness in the current configuration.

The effective sphere was also used to model the mechanical stresses under the spherical approximation with Laplace’s law. In this case, the principal stresses become

σr=−P2,σ1=σ2=Prm2h (8)

where rm and h are the mid-wall radius and wall thickness calculated for all subsampled time points (i.e., current configuration). Thickness is assumed to be uniform over the entire surface. The method for quantifying wall thickness is described in the Supplementary Material section.

Finally, to aid in location-specific stress analysis the von

Mises stress σv was also calculated as

σv=12σr−σ12+σr−σ22+σ1−σ22 (9)

3. Results

3.1. This reconstruction method accurately describes bladder geometry during inflation

The 3D mesh geometry overlaid on the effective sphere for a representative bladder at luminal pressure 25 mmHg is shown in Fig. 1f. This figure highlights how the shape of the bladder can deviate quite significantly from the effective sphere, even if the difference between the two volumes is small. The average difference between the 3D bladder mesh volume and effective sphere volume across samples was 4.93 mm3 for all pressures and 6.05 mm3 at 25 mmHg (2.6% and 2.2% average volume percent error, respectively). Figure 1g shows the RMSD of mesh geometry from the effective sphere for all samples as a function of luminal pressure. For most samples, the RMSD was largest at low pressures, with the spherical approximation becoming more accurate as the pressure increased. Analyzing sphericity also revealed that at low volumes, the bladder is poorly approximated by a sphere (Fig. 1h). Figure S2 illustrates bladder shape changes from 1 to 20 mmHg, for a representative bladder (same as what included in Fig. 2). The 3D geometry at pressures of 1, 2, 3, 4, and 20 mmHg is overlapped to highlight the shape change.

Fig. 2.

Fig. 2

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Von Mises stress distribution for a bladder at 5, 10, 15, and 20 mmHg. a–d, 3D mapping of the stress distribution on the bladder surface. e–h, Equivalent stress distribution displayed in 2D. The 3D vertices were fattened such that the center of the plots corresponds to the dome of the bladder, and the outer circumference corresponds to the bladder neck. Axes are mm. Scale bar is in kPa. Same stress scale for all plots

3.2. Novel location‑specific mapping of bladder wall stress using 3D reconstruction

The analysis performed here yielded localized, point-wise values of the principal Cauchy stresses and the von Mises stress, which are essential for determining the local mechanical effects of bladder pressurization. Figure 2a–d shows a 3-dimensional view of the resulting point-wise values of the von Mises stress. To allow for visualization of the whole stress field, the 3D bladder surface was flattened to 2D, as shown in Fig. 2e–h. To following approach was employed to obtain this visualization. Briefly, the highest z-coordinate in the mesh (i.e., the dome) was set as the origin, and the distances to all other vertices on the mesh were calculated using Dijkstra’s algorithm (Dijkstra 1959). The mesh vertices were then remapped to the 2D representation at their respective distance and angle relative to the dome. The stress distribution is non-uniform over the surface, but the non-uniformity is preserved with increasing pressure. The principal in-plane Cauchy stresses (σ1 and σ2), the von Mises stress, and the ratio of σ1 to σ2 were then mapped for a representative bladder at luminal pressure of 25 mmHg (Fig. 3). This representation allows appreciation of how both the in-plane principal stresses and the von Mises vary in a localized manner. In contrast, a spherical model would show a uniform stress across the entire surface. From these stress maps, it was found that stresses were highest near the bladder neck (outer circumference of the plots in Fig. 2e–h and Fig. 3), but the stress distribution was non-uniform across the rest of the surface. However, it was also found that when the von Mises stress over the entire surface was averaged (e.g., 60.35 kPa), the result was of similar magnitude as the uniform spherical stress at the same pressure (e.g., 57.29 kPa for the sample in Fig. 3). The ratio of σ1 to σ2 provides more insight into the distribution of stresses in the bladder wall (Fig. 3d). It was found that the distribution of stresses visualized this way was not uniform, with potential stress concentrations around the bladder neck and around the dome.

Fig. 3.

Fig. 3

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In-plane principal stresses (a, b), von Mises stress (c), and in-plane principal stresses ratio (d) distribution on a representative bladder surface at 25 mmHg. Axes are mm, scale bars are kPa for (a–c). This is a 2D remapping of the 3D bladder surface/stresses. The 3D mesh vertices were remapped to the 2D representation at their respective distance and angle relative to the dome. The origin of the plot corresponds to the bladder dome, and the outer circumference corresponds to the bladder neck

3.3. Bladder volume quantification using novel 3D reconstruction

Bladder volumes were validated by comparison to experimental parameters and a manual volume estimation. Specifically, the volume of the 3D bladder meshes was linearly dependent upon time, which is in agreement with the constant infusion rate employed during the experiments. Figure 4a shows the reconstructed volume for each bladder as a function of time during filling. The best fit line was found to have a slope of 25.13 μL/min while the experimental slope (i.e., infusion rate) was approximately 30 μL/min. Figure 4b shows the comparison between the volumes obtained from this reconstruction method to the volume obtained with our previous manual method (Hennig et al. 2022).

Fig. 4.

Fig. 4

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Validation of 3D bladder reconstruction. a Reconstructed bladder volume over time from 0 to 25 mmHg internal bladder pressure. Experimental fill rate was approximately 30 μL/min (gray line), and the average fill rate observed from volume reconstruction was 25.13 μL/min (black line). The best fit line had an R2 of 0.96, and RMSE of 20.74 μL. b Pressure–volume plot for the volume obtained from the reconstruction method, and a manual method by which final volume was estimated by measuring ellipsoidal axis lengths in the images, and all other volumes were based on the constant infusion rate (Hennig et al. 2022). Standard errors shown

4. Discussion

Although this work does not include a model of bladder dysfunction, the methodologies demonstrated here lay the groundwork for assessing dysfunction in terms of the mechanical behavior of the healthy bladder. The literature is lacking in consensus data on measures of bladder health, and deeper understanding of the bladder mechanics may be the bridge to this gap. Quantifying bladder shape is relevant as a non-spherical bladder may indicate functional deficiencies or irregularities in the bladder wall microstructure. Tissue stiffening, as seen in bladder fibrosis, can also alter the elasticity of the tissue and drive non-spherical deformation. A non-spherical bladder may also present altered emptying characteristics compared to a more spherical bladder (Hirahara et al. 2006; Gray et al. 2019; Bih et al. 1998). Furthermore, diseased conditions such as diabetes have shown increased bladder capacity, and can present altered bladder shapes compared to healthy subjects (Pitre et al. 2002; Fagerberg et al. 1967).

Mechanical stress in the bladder wall was modeled via Laplace’s law as a function of the local curvature of the reconstructed volume, and also as a thin-walled spherical pressure vessel (Fig. 2). The results of both stress models were of the same magnitude. The agreement between the two models indicates that the spherical approximation is comparable to the average wall stress based on surface curvature. However, the spherical model can only provide an average uniform stress across the entire surface. Modeling the stress using local curvature allows the wall stresses to be discretely mapped over the entire surface which can be of value to study the anisotropic behavior observed in bladders.

The method of reconstruction proved to be accurate based on the linear volume results shown by Fig. 4a. Volume reconstruction allows the actual shape and deformations of the bladder wall to be analyzed. Shape analysis presented here showed that the spherical model is more accurate at higher pressures, but less accurate at lower pressures. When the bladder is at a low volume/pressure, the organ can appear more collapsed which leads to a more non-spherical shape and may even be non-convex. Furthermore, Fig. 1g shows that the bladder is more poorly approximated by a sphere at low pressures but approaches a more spherical shape as pressure/volume increase. Comparing the pressure–volume results shown by Fig. 4b, the methods of estimating bladder volume via a manual method and the reconstruction method showed substantial overlap. This validates the reconstruction method with a method that is commonly used to estimate bladder volume in the clinical setting (Bih et al. 1998). There was some difference at low volumes, which is likely due to the manual methodology by which only the final volume was estimated, and all preceding volumes were inferred from the infusion rate. This discrepancy may be due to the infusion rate being affected by the bladder reaching max capacity. Near capacity, there may be more system losses or leakage which would reduce the filling rate. This can be seen in Fig. 4a, where the volume with respect to time is highly linear for the majority of the filling, but at higher volumes, the trend became nonlinear as shown by the plotted data falling away from the linear trend.

Bladder capacity in healthy C57BL6 mice has been shown to be between 150 and 250 μL (Herrera et al. 2003, Dörr 1992, Kim and Hill 2017, Durnin et al. 2016). Threshold pressure (i.e., pressure which initiates voiding) has been found between 7 and 15 mmHg (Herrera et al. 2003, Kim and Hill 2017, Smith et al. 2010 ), and maximum bladder pressure may be between 18 and 35 mmHg (Herrera et al. 2003, Dörr 1992, Kim and Hill 2017, Smith et al. 2010, Durnin et al. 2016). As such, we chose 25 mmHg as an upper limit for pressurization. Average bladder volumes (as shown by Fig. 4b) agreed with the literature, accounting for the fact that total bladder volume included the tissue volume as well.

The reconstruction method involved generating voxelized representations of the bladders as the geometries were sourced from the pixelated image data. The conversion from voxel to mesh representation has many benefits. Meshes are a more standard 3D data representation than voxels since they are much more efficient in memory usage and graphical rendering (Frey et al. 1994; Huang et al. 1998). In this work, converting the voxel data to mesh data reduced the memory footprint by an order of magnitude while retaining data resolution, and rendering time was reduced by 72%. Creating meshed geometries also lend the data to future analysis by integration with finite element methods. A limitation that must be overcome however is that the meshes are all created independently from each other. In other words, the faces and vertices for a mesh at a given time point in the experiment have no connection to the faces and vertices of the mesh at the previous or following time point. Furthermore, each mesh has a different number of vertices and faces since they are independent of meshes of the same bladder at different time points. This limitation may be overcome by development of a method in which the vertices of the final mesh are projected onto the preceding mesh, and repeating this projection for all meshes back to the first bladder mesh surface.

To address the limitation of the shape-from-silhouette method, which does not completely capture the whole geometry of the bladder surface due to the visual hull effect (Laurentini 1994), we implemented a scaling technique for the silhouette on the z-axis for each z layer of the reconstruction. This approach effectively resolved the visual hull issue by assuming that the geometry across the entire z-axis can be characterized by a single silhouette from a top-down perspective. Although this assumption introduces some constraints, it provides a practical solution within the limitations of our experimental setup.

This 3D reconstruction method may be comparable to DIC methods, another frequently used approach is to track surface deformations (Morales-Orcajo 2018, Trostorf 2022). DIC allows surface tracking and stress/strain quantification via individual markers, or a speckle pattern applied to the material surface. The outermost mesothelial layer of the bladder wall is resistant to marker/ink adhesion which makes DIC methods challenging for the bladder (Murakamo et al. 1995). Furthermore, due to the layered structure of the bladder wall, a DIC mapping only models deformation of the outermost layers (i.e., mesothelial and adventitial layers) which do not significantly contribute to the bladders’ stiffness, especially under physiologically relevant loads. The method shown here allows for a simpler experimental setup and analysis. The setup is simpler because no surface preparation is necessary. The analysis is simpler in that only silhouettes are required for the reconstruction, and silhouettes provide a high contrast boundary which is clearly identifiable. DIC registration would be more complex and error-prone due to the bladder being a 3D surface, which makes accurate point tracking more challenging. Additionally, a speckle pattern applied to the surface would obscure the translucency of the bladder wall which is relied upon to measure its thickness. While both methods, DIC or 3D reconstruction, have their merits and drawbacks, the 3D reconstruction method was chosen as it was best suited for the goals of this study.

More accurate shape-from-silhouette reconstruction typically requires additional imaging perspectives, achievable with more cameras or by rotating the object. However, these approaches were not feasible for our experimental setup. Future work could explore other reconstruction methods, such as texture mapping, which includes lighting conditions, or advanced machine learning techniques. These methods hold potential for improved bladder reconstruction but require further development and expertise.

An innovation of the work is the amount of data generated from reconstruction and mechanical modeling, and visualizing this substantial amount of 4D data. Significant time was dedicated to development of the reconstruction method. Methods for visualization were developed; however, more work can be done when it comes to the ability of comparison between samples. Improved visualization is a subject of future work, and integrating the mesh and mechanical data into existing 3D visualization software may be a potential route to explore in the future. Nonetheless, the geometry and mechanical modeling of the bladders throughout pressurization were still able to be created and useful for the analyses here.

The mechanical model presented from the 3D reconstruction is useful since, due to the mechanics and material properties of the bladder wall, stresses are not expected to be uniform across the surface, and with this method, the distribution of stresses can be quantified and studied.

This is an indication that the material properties at these regions may be different than the rest of the bladder wall. Analyzing the stress distributions in this way with dysfunctional bladders may provide even more insight into how the mechanics may be altered as a consequence of disease.

The modeling method presented is also valuable for understanding healthy function of the bladder or other pressurized organs. It has been hypothesized that local bladder contractions are a mechanism by which the nervous system senses bladder fullness (Drake et al. 2017). Further study on these contractions, which are observable from geometry alone, may provide insights into bladder health and dysfunction. Alterations in bladder contraction location, magnitude, or frequency coinciding with treatment or dysfunction would improve our understanding of bladder sensory mechanisms. It has also been shown that the bladder has location-specific material properties (Korossis et al. 2009, Tuttle et al. 2023). Further investigation of the 3D stress mappings developed in the current work may inform localized active (in the case of contraction) or passive mechanical behavior of the bladder.

While the system does not image microstructural components per se (e.g., collagen fibers), it does provide sufficient spatial and temporal resolution to identify areas where microstructural changes are likely taking place. Due to the high resolution (10 Hz) achievable with this system, active localized contractions can be investigated as they manifest in the material geometry. For example, local bladder contractions appear as transient deviations in the surface geometry, which can be identified and measured using this approach, resulting in the quantification of surface area size and the depth of surface deviation. The size and location of active contractions in different organs can better inform our understanding of active tissue behavior and dysfunction. This system can also provide microstructural insights from tracking the geometry over a large range of pressures. For example, the change in the overall shape of the bladder can indicate whether the tissue behaves isotropically or anisotropically.

This novel methodology is highly versatile, allowing it to be extended to the study of the mechanics of various other hollow organ tissues beyond the bladder, including the lungs, uterus, and gastrointestinal tract tissues. Application of this advanced 3D analysis to different types of tissues permits the testing of novel hypotheses for mechanics-based mechanisms driven by localized forces or tissue characteristics. For instance, localized uterine contractions play a crucial role during labor (Young et al. 2023), and peristaltic contractions are essential for the movement of food through the gastrointestinal tract (Bortoff and Ghalib 1972). Each of these processes is driven by localized mechanical interactions specific to the tissue’s microstructural characteristics and mechanical loading. The ability to model and analyze these localized mechanics in high resolution provides a deeper understanding of the underlying biological functions and behaviors. This broader applicability of our methodology highlights its potential of significantly advancing the comprehension of tissue-specific mechanical behaviors across a wide range of biological systems. By extending this approach to different tissues, we can gain valuable insights into the diverse mechanical properties and interactions that occur within various organs, ultimately contributing to the development of more effective treatments and interventions for a multitude of physiological conditions.

Future work could include using these methods to compare the effects of sex on the mechanics and shape of the murine bladder. More complex shape analysis, including spheroidal approximation/fitting, could also be studied in future research. While the bladders in this study were analyzed in their true 3D shapes and as simplified spheres, other researchers have shown that a spheroidal shape may better approximate bladder geometry (Damaser and Lehman 1995 Jun 1). Indeed, as we have shown that bladder shape is poorly approximated by a sphere, especially at low pressure, we intend to include a spheroidal model of the bladder as a next step in this research. While our results show that the wall stresses obtained from the full 3D model were of the same magnitude as those from the simplified spherical model, a spheroidal model, being non-isotropic, would better reflect the deformation the bladder undergoes during filling. In conclusion, our work shows that it is possible to create 3D models of ex vivo bladder filling accurately with low-cost laboratory equipment and open-source code. These methods can be used to inform better models of the mechanics of bladder filling. In this work, a detailed 3D model of the bladder was compared to a spherical approximation of the bladder. Results for average stress on the wall were similar. However, the full 3D model allowed for more in-depth analysis which is relevant to study the non-isotropic nature of the bladder as it deforms.

Supplementary Material

Supplementary Material

Supplementary Information The online version contains supplementary material available at https://doi.org/10.1007/s10237-024-01914-7.

Acknowledgements

The Pentaplanar Reflective Imaging Macroscopy (PRIM) device is protected under US patent 10,585,270 (Inventors: Nathan R. Tykocki and Grant Hennig).

Funding

This study was funded by grants from the National Institute of Diabetes and Digestive and Kidney Diseases: P20-DK127554, R01DK135696, R01DK119615.

Footnotes

Code availability The code developed for the data analysis has been published on an online repository: DOI: https://doi.org/10.5281/zenodo.6649159.

Declarations

Conflict of interest The authors declare no competing interests.

Ethical Standards Animal care approval: Animal Care Committee (ACC) protocol AL-2020–04 approved via Beaumont Hospital in Royal Oak, MI, USA, an Office for Laboratory Animal Welfare (OLAW) approved institution (D16–00255 (A3408–01)).

Data availability

Data are available upon request. Contact corresponding author.

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Supplementary Materials

Supplementary Material
NIHMS2080379-supplement-Supplementary_Material.pdf (284.7KB, pdf)

Data Availability Statement

Data are available upon request. Contact corresponding author.

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