A large-scale balloon model of confluent cells validates stress inference from geometry

Abstract

The mechanical environment surrounding cells influences their behavior. Multiple methods have been developed to measure local mechanics with mechanical perturbation, which is invasive. For sheets of confluent cells, the geometry of the boundaries of the cells can be used to estimate the local membrane tension and internal cell pressure through stress inference (SI), which is image-based and therefore non-invasive. SI is sensitive to noise and has not been validated in a physical system using a direct pressure measurement. We developed a large-scale model of confluent cells using densely-packed balloons with interior pressure sensors to allow for simultaneous measurement of pressure and geometry. We also developed a new SI method which resulted in a correlation coefficient >0.8 with the balloon model, less noise sensitivity in computational models, and measured pressure change in red onion epithelial cells. These results demonstrate the validity and utility of SI in multiscale physical and computational applications.

Introduction

Cells exist within and are affected by a micromechanical environment that plays a pivotal role in their behavior from development to organ function. For example, the mechanical properties of the microenvironment have a profound effect on large-scale systems, such as during embryogenesis¹ stem cell differentiation² in wound healing³ and in tumor growth^4,5. Cells perceive their mechanical environment through various mechanisms including stretch-activated ion channels and other mechanosensitive components⁶. Current experimental methods used to explore mechanics at the cellular scale include micro-pipetting⁷, laser ablation⁸, polyacrylamide beads⁹, and atomic force microscopy¹⁰. However, these methods are invasive and thus can change the behavior of the cells due to mechanical perturbations⁷ They also have limited resolution, can take a significant amount of time to apply, may destroy the tissue sample, and require expensive training and equipment^11,8,7).

To address the need for a faster non-invasive method of studying the local mechanics of cells during development or within tissue systems, considerable research has gone into Stress Inference (SI), which estimates local boundary tension and cell pressure by analyzing the geometry of cell boundaries captured through standard imaging techniques (^8,12. Tension in this context refers to the force along a measurable boundary, such as those of cell membranes in mammalian cells or cell walls in plant cells. This tension can change due to active biological phenomena such as cell contraction driven by actin-myosin activation, passive phenomena such as osmosis, or environmental perturbations such as a cell being stretched or compressed. Pressure refers to hydrostatic pressure within the cell (also known as turgor pressure in plant cells). As described by Roffay et al.⁸, stress inference (SI) offers several advantages over traditional methods - it is non-destructive, allows rapid measurement, and enables longitudinal tracking of cellular responses.

SI relies on the assumption that the mechanical equilibrium of confluent cells can be adequately described by a 2D projection of the 3D cell boundaries. SI further assumes that the effects of out-of-plane forces, shear forces, membrane bending stiffness, and the mechanics of the cytoskeleton can be neglected, and therefore the curvatures of the cell boundaries reflect the pressure differences across the boundaries and the tangential tensions within the boundaries themselves. SI methods have been used in various applications⁸. For example CellFIT¹³ and Foambryo¹⁴ have been used to study tissues in mice, ascidians, worms¹⁴, quail and fruit flies¹⁵, as well as plant models including arabidopsis¹⁰.

Validating SI typically is based on the results of either 2D computational model simulations or indirect measurements of intracellular tension and pressure¹⁵. This is problematic because computational models may not recapitulate all critical aspects of real physical systems¹³, while indirect measurements of pressure are conditional upon the validity of the assumptions on which they are based¹⁰. SI has yet to be validated in a physical system in which pressures are measured directly. Accordingly, we developed a large-scale physical model consisting of densely-packed balloons, each containing an internal air pressure sensor. Using these pressure measurements together with the geometry of the balloon walls, we tested the accuracy of balloon pressure estimation obtained using a novel iterative SI algorithm and compared these estimates to those obtained with two existing SI methods. In various systems, we found a high correlation between known and estimated pressures demonstrating the validity and utility of SI in multiscale physical and computational applications.

Results

Direct SI validation with balloon frame

Images from 20 unique configurations were collected from the balloon frame with the balloon pressures and geometry recorded. The processed images of the model were then analyzed using the IM, MM, and UM. The median correlation coefficients for the IM and UM were 0.83 and 0.81 respectively, while the MM was 0.65. The coefficients for the IM and UM were significantly greater than the MM, but were not significantly different from each other (Fig. 1A). The RMSE was greatest for the MM but was not different between the IM and UM (Fig. 1B). The MM had the greatest CV (Fig. 1C). The IM and UM had the lowest CVs and were not significantly different from each other. These results are summarized in Fig. 1.

Fig. 1. Comparison of measured and estimated pressures.

A−C Box and whisker plots of correlation coefficients, root mean square errors (RMSE), and coefficient of variation (CV), respectively, for the three estimation methods. * indicates p < 0.05 and ** indicates p < 0.01 using sign-rank test. D Heat maps of one balloon image. i shows the actual measured gauge pressure, where ii-iv show the estimated pressures with the three methods, where each value is the median from 10 networks.

Figure 1 Di shows the measured pressure map in one of the balloon images, and panels 1Dii−iv show the pressure maps estimated by the different SI methods. The IM and UM produced similar maps, but the MM map has the greatest pressure heterogeneity indicated by one region of high pressure surrounded by low pressure regions.

Tension estimation in computational networks

Using the 2D computational models, 43 small networks with an average of 11 cells, 104 medium networks with an average of 39 cells, and 20 large networks with an average of 192 cells were created and analyzed.

In networks where the MM was applied, tension varies on a length scale comparable to the length of the whole system, resulting in regions of high- and low-tension estimates (Fig. 2A). This behavior was more visible in large networks. These local extrema were not consistent between runs in the Monte-Carlo approach on the same image (Fig. 2A), indicating run-to-run sensitivity to noise. This run-to-run tension variation is measured with the CV and is shown to be significantly higher in the MM compared to the IM (Figs. 2D, S1–6). In the networks where the IM was applied, local maxima were not seen (Fig. 2B). Increased run-to-run heterogeneity in the MM compared to the IM was evident using both imaging noise and Gaussian noise at all noise levels tested (Figs. S1–6). Repeated iterations of the IM in cases with no noise resulted in high CC in 2D computational models (Fig. S7). When noise was present, however, repeated iterations resulted in increased CV and decreased CC (Fig. S8). Accordingly, in all tests using the IM, only one iteration was used. The IM does not converge to zero after the first step (see results).

Fig. 2. Heterogeneity in tension estimates between network iterations.

A Tension in three unique networks using the Mean method. B Tension in three unique networks using the Iterative method. C Left, median tensions from 10 networks using the Mean method; Center, actual tension in solved network; right, median tensions from 10 networks using the Iterative method. D Correlation coefficient and coefficient of variation for Mean and Iterative methods for tension. The correlation coefficient is between the median network and the known network, where the coefficient of variation is a measure of heterogeneity between the 10 networks made for that image. The dotted line represents an alpha level of 0.05, and the dotted-dashed line indicates the p value of the difference between the Mean and Iterative methods using sign-rank test, therefore anywhere the dotted-dashed line is below the dotted line indicates significant difference.

The tension RMSE values obtained with simulated network images to which noise had been added were not consistently significantly different between the MM and IM for any network size (Fig. S1–3). For Gaussian noise in medium and large networks, the RMSE for the IM was significantly larger than for the MM for low noise levels but then was not significantly different for larger noise levels (Figs. S5, 6).

The CC for tension in the imaging noise showed consistent behaviors across all size groups, where the CC for the IM was greater than that of the MM (Figs. S1–3). For the Gaussian noise, the results were similarly consistent, where the CC was greater for the MM at low noise, but the CC for the IM was significantly greater at high levels of noise (Figs. S4–6). These results are consistent with the correlation coefficient values published by Chiou et al.¹², who used a method similar to the MM.

Pressure estimation in computational networks

For computational networks where imaging noise was applied, the CV was significantly greater in the MM estimates than the IM estimates for all network sizes except low noise levels in the large networks (Figs. S1–3), which is consistent with the results from the balloon frame. In the Gaussian noise networks, the CV was not consistently significantly different between the IM and the MM (Figs. S4–6). The pressure RMSE was not consistently significantly different between the MM and the IM in the imaging noise or Gaussian noise networks, except for the medium-sized imaging noise networks (Fig. S2).

The pressure CC was significantly higher for the IM in the small and medium networks from the imaging noise (Figs. S1, 2) but was not significantly different from the MM in the large networks at high noise levels (Fig. S3). The CC in the Gaussian noise networks was not significantly different between the MM and the IM except for the small networks at low and high noise levels, and medium networks at high noise levels, where the IM had a higher CC (Figs. S4 and 5). These results are consistent with correlation coefficient values published by Chiou et al.¹², which used a method similar to the MM.

Pressure estimate in onion cells

The sheet of living red onion cells provided clear boundaries between cells in living tissue, and changes in pressure behaviors were evident at the two tonicity levels (Fig. 3). Under hypotonic conditions, the protoplasts filled the space of the cell walls (Fig. 3A). However, under hypertonic conditions, the plant cells underwent plasmolysis and the protoplasts shrank away from the cell walls (Fig. 3B). Furthermore, a pressure gradient that was evident in the hypotonic conditions (Fig. 3C) was eliminated in the hypertonic conditions (Fig. 3D). This change in tonicity was enough to cause a change in the pressure distributions from right skewed to left skewed (Fig. 3E) that were significantly different (p < 0.05). This change in tonicity also lead to a significant change in the cell areas (p < 10^-5). Furthermore, there was a negative correlation between the estimated pressures and the cell areas (Fig. 3G, H). The correlation for the hypotonic cells was −0.35, and for the hypertonic solution was −0.73, with R² values of 0.12 and 0.54, respectively.

Fig. 3. Pressure maps in red onion cells.

A, B Cells in hypotonic and hypertonic solutions, respectively. C, D Estimated pressures in hypotonic and hypertonic solutions, respectively. E Normalized pressure distributions in pressure maps; the distributions were significantly different using a sign-rank test, p < 0.05. F Area distributions in cell images; the distributions were significantly different using a sign-rank test, p < 10^-6_. G, H Pressure vs area in hypotonic and hypertonic solutions, with R² values of 0.12 and 0.54, respectively.

Discussion

Cellular mechanics play an essential role in cell behavior, so there is a need to quantify this behavior in a reliable and non-invasive way. Previous studies have proposed SI as a means of estimating local mechanics from cell geometry. In this work, we validated the ability of SI to estimate tension and pressure in cells based on geometry with a physical balloon frame. Furthermore, we evaluated a novel SI method using the balloon frame and computational models and applied these methods to map pressure differences in red onion epithelial cells. We found that SI is able to predict relative pressures in a physical system, and that the novel IM shows greater robustness against noise than the MM.

Using SI as a means of estimating the local micromechanics of tissue is not a new concept, and the strengths and limitations of this method have been explored using many indirect methods. To our knowledge, the present study is the first to validate SI using a physical model where pressures and geometry were measured directly and simultaneously without mechanically influencing the system. For the three tension estimation methods used, the highest CC was 0.83 from the IM (Fig. 1A). This high CC indicates that SI, and in particular the IM implementation, can be used to estimate local pressures in physical systems.

The balloon frame was first constructed using opaque balloons, but we found that when a high-pressure balloon pressed onto an adjacent low-pressure balloon, the low-pressure balloon extruded out of the plane of the image so it is overlapped with the contour of the high-pressure balloon, which made it impossible to see the relevant wall curvatures. It was thus necessary to use transparent balloons and to measure the internal geometry at the middle of the vertical dimension of the balloons, where the curvature between adjacent balloons is representative of the pressure difference between them. This midline-curvature is therefore a more accurate representation of the 2D projection of a 3D system along the vertical dimension than the surface of the balloons. For biological applications, this midline-curvature can be determined from compressed z-stacks or brightfield images, as seen in the onion cells in Fig. 3A, B where the cell walls appear darker because they are perpendicular to the plane of the image.

We found in the balloon frame that the CC for the uniform tensions were both similar to those from the IM, whereas the MM resulted in lower CC. This indicates that the pressure estimates are relatively insensitive to wall tension. Thus, the converse problem, estimating wall tension from pressures, is impractical. These observations highlight the value of the IM because it provides both tension and pressure estimates. Our computational analyses show that the tension estimates are more robust against noise than the MM.

Despite the use of lubrication on the balloons, friction was still observed with balloons repeatedly sticking and slipping along the acrylic while being compressed. While this conflicts with our assumptions about the geometry representing a system in equilibrium without shear forces, we were still able to achieve highly correlated estimates using SI.

It has been previously established^8,13 that SI is highly sensitive to noise. To evaluate this, we introduced two types of noise into the computational models, namely Gaussian noise and imaging-based noise. We tested networks similar in size to the balloons (approximately 10 cells), to see which type of noise is most similar to that from the physical system. Although neither seems to fully capture the behavior shown in the balloon frame, the CCs in the imaging noise in the small networks appear closer than the CCs in the Gaussian noise, so we posit that the noise introduced by imaging is a more accurate representation of realistic noise.

One of the clear behaviors noted from the MM is the presence of local tension extrema that vary between runs (Fig. 2A). These local extrema are caused by high sensitivity to noise. Previous studies have also encountered this sensitivity in large networks, referring it to as error propagation¹⁵. While at low levels of noise, this sensitivity is a benefit (Figs. S4–6), in systems with moderate noise levels, it is a detriment. This is in contrast to the IM, which does not show these characteristic network-scale tension variations and extrema (Fig. 2B). The MM requires that the average of all estimated tensions be 1, but this allows for wildly fluctuating values. The IM provides an initial guess of 1 for all tensions, but then allows those tensions to move from that initial estimate towards the solution from the geometric element (G in Eq. 1). Repeated iterations move away from the initial estimate towards the geometric solution (Fig. S8), indicating convergence of the IM to the correct solution. In the case of no noise, we also found that the average magnitude of the solutions decreased with each iteration (not shown). The solution is, however, relative, rather than absolute; while the magnitudes of the solution approach zero, the relative magnitudes of the tensions become more correlated to the actual values. Furthermore, because calibration is necessary to convert the estimated values to those corresponding to the scale of the tissue, the magnitude of the solution does not matter. However, we also found that the repeated iterations are sensitive to noise (Fig. S7). Using a single iteration of the IM strikes a balance between stability and sensitivity. The IM therefore acts like low pass filter, preventing local extrema from occurring. This is also likely why the IM behaves similarly to the UM (Fig. 1A–C). These results indicate an increased stability of the IM in response to noise compared to the MM, and this is further enhanced by the Monte-Carlo approach to estimating the pressure and tension with multiple iterations of the same image.

Interestingly, not including the boundaries on the edges of the networks was not a detriment to the IM. In cases where the entire cell sheet was considered (such as the balloons and computational models) and in cases where the selected cells comprised only a portion of the whole sheet (such as the onion cells), the IM was still able to produce meaningful estimates. Other studies have included exterior pressure in cells as a boundary condition¹⁴; however, for our tests, we found that removing the exterior boundaries allowed for measurements to be made in all tested conditions.

While the use of balloons and computational models is important to validate the efficacy of SI, it is also important to show results using living cells. One application of SI is the estimation of local turgor pressure in plant cells, so we tested the effects of osmotic pressure on epithelial cells from a red onion. Images of red onion epithelial cells in hypotonic and hypertonic conditions were collected and showed significant changes in pressure distributions in response to different tonicities. It is important to note that the pressures shown in Fig. 3C–E are relative pressures, so the actual pressures at different tonicities are not measurable without further calibrations. However, the relative pressures and the change in the shape of the distributions can still provide key insights into the behavior of living tissue, such as gradients in pressure, and the loss thereof as demonstrated in Fig. 3.

Interestingly, there was a negative correlation between cell area and pressure (Fig. 3G, H), which is consistent with the results presented in Long et al.¹⁰. This negative relationship becomes more pronounced in the hypertonic solution (Fig. 3H) than the hypotonic solution (Fig. 3G). A possible explanation is that the more pronounced negative correlation is due to the higher osmotic pressures having a greater effect on the wall curvature and outweighing mechanical influences, such as bending stiffness in the cell walls or linkages to the underlying cytoskeleton.

It is worth noting that the gradient shown in Fig. 3C is not an artifact caused by noise, as is present with the MM (Fig. 2A) because the gradient was consistent across repeated tests and was present in a second hand-tracing of the cell boundaries. The gradient largely disappeared in the hypertonic conditions, likely caused by the greater osmotic pressure dominating the mechanical effects. This further demonstrates the robustness of the IM method in a biological context.

As with the other methods of estimating the local mechanics within tissue, there are important limitations to address with this study. First, as discussed in Roffay et al.⁸, SI tends to be sensitive to noise, and we have observed that behavior in this study. Regardless of implementation, the SI approach also requires that the system be at equilibrium which prevents the ability to study transients and dynamics. However, we have also introduced the IM which shows resilience against moderate noise levels, and we have shown in the balloon frame that the pressure estimates are robust against noise in the tension estimates.

The number of balloons within the frame was small compared to typical micrographs of plant or animal cells. These steps were taken to maintain simplicity, and therefore allow for generalization of the models’ applications. The balloon frame does not measure the tensions in the balloon surfaces. Also, while the balloon frame approximates a 2D cell network, it is actually 3D which presents issues related to out-of-plane forces and displacements. Despite these limitations, however, we still observed behaviors in the balloon frame that were consistent with those seen in biological cell networks.

Onion cells have very clearly defined cell boundaries that do not require staining to observe, which while helpful in this test is not true of all cells. Also, onion cell walls have a high bending rigidity, which is contrary to the assumptions typically associated with SI. This bending rigidity led to cell walls that were not perfectly circular, as can be seen in Fig. 3. However, we were able to see a change in the wall curvatures in the hypertonic vs hypotonic solutions, indicating a noticeable change in structure in response to pressure changes.

In practice, many closely packed cells do not have clear curvature that is representative of the underlying mechanics. In this case, SI would struggle to return meaningful results. However, this is a conceptual limitation of SI, rather than the IM. Nevertheless, despite this limitation, SI has already been shown to be useful in biological applications such as in plant cells and drosophila wings^10,15. Additionally, SI can be computationally taxing for large systems. Sparse matrices were used during computation to limit the computational load; however, in future studies where large 3D systems are analyzed, this may become more pronounced of a limitation. An alternative is to use stress tensors, as described by Kong et al.¹⁵, which is requires less computation. However, this approach is fundamentally different from the methods presented in this research, and cannot estimate pressure gradients, whereas the IM and MM can.

In this study, we validated SI directly with a physical balloon model using multiple SI implementations. A new IM was also introduced and found to outperform previous methods. When applied to red onion cells, the IM provided biologically meaningful insight.

Methods

Tension and pressure estimation

The theory whereby pressure and tension are estimated based on geometry is well established⁸. Consider a sheet of confluent cells of varying internal pressures and surface tensions; the curvature of the boundaries that separate the cells reflects the equilibrium of forces within and between the cells. These boundaries represent the structures through which the cells interact with each other, such as the cell membrane or the cell wall. SI posits that the tension and pressure can be estimated from a 2D projection of the boundaries between the cells.

At an intersection of three cells in the 2D network, a node is placed with three unit-vectors $\hat{a}$, $\hat{b}$, $\hat{c}$ multiplied by magnitudes $A$, $B$, and C. These three vectors represent the forces applied to the node by the tensions in the intersecting cell boundaries and sum to give a unit-vector $\hat{m}$ with magnitude M as shown in Fig. 5 and Eq. 1.

$$\left[\begin{array}{ccc}{x}_a& x_b& x_c\\ y_a& y_b& y_c\end{array}\right]\left[\begin{array}{c}A\\ B\\ C\end{array}\right]=\left[\begin{array}{c}M{x}_m\\ M{y}_m\end{array}\right]$$ 1

Fig. 5. Forces at equilibrium. Each region represents a cell, with the gray lines marking the cell boundaries.

At the junction of the three cells, shown on the left, a node is placed, and the tensions from each cell boundary apply a force, represented by vectors $A\hat{a}$, $B\hat{b}$, $C\hat{c}$, with the resulting vector $\mathrm{M}\hat{m}$. For a system at equilibrium, $M\hat{m}$= 0. Each unit vector is split into x and y components, shown applied to vector $\hat{b}$. The angles of the vectors are calculated as the tangent of the best-fit curve (red dashed line) of radius ρ, shown on the right, at the boundary.

Equation 1 is of the form $\mathit{GT}=M$, where G contains the geometries of the nodes, x_i and y_i represent the x and y components of the i^th force vector, T contains the magnitudes of all the forces, and M contains all the resulting net forces on the nodes. This is illustrated in Fig. 5 for a single node at which three boundaries intersect; however, Eq. 1 incorporates all nodes and boundaries within a system. Each node contributes two rows to G corresponding to the sum of the x and y components of each boundary attached to it, and each boundary contributes a column to G. In general, cell boundaries are curved, even where they intersect at nodes, so the directions of the boundary force vectors were determined from the tangents to circular arcs fit to each wall segment as illustrated in Fig. 5¹³.

Since G is known from geometry and M is known to be zero at equilibrium, T can be calculated by solving Eq. 1. However, this is typically an overdetermined inversion problem¹³, so Eq. 1 is solved for T using the Matlab function lsqminnorm, which solves for the least-squares solution to a set of linear equations. The G matrix also tends to be sparse, which can be handled efficiently by lsqminnorm. For the cells at the perimeter of the system, only the boundaries that are shared with other cells are included in Eq. 1; i.e., the boundaries that are on the perimeter are ignored.

At equilibrium, $M=0$, so Eq. 1 has the trivial solution $T=0$. To avoid this trivial solution, Chiou et al.¹² and Brodland et al.¹³ imposed the condition that the mean value of the elements in T must equal 1, hereafter referred to as the Mean Method (MM). This condition is implemented by extending Eq. 1 to become

$$\left[\begin{array}{ccc}{x}_a& x_b& x_c\\ y_a& y_b& y_c\\ 1& 1& 1\end{array}\right]\left[\begin{array}{c}A\\ B\\ C\end{array}\right]=\left[\begin{array}{c}M{x}_m\\ M{y}_m\\ n\end{array}\right]$$ 2

where n is the number of tensions, equal to the number of boundaries, within the network.

We have developed an alternate solution strategy in which an identity matrix is incorporated into G, and an initial set of values for T are incorporated into M, shown in Eq. 3. These elements act as an initial estimate for T, and are all initially set to 1. The new expression is then solved for T, and the solution is reincorporated into M in place of the initial values of 1. Solving the expression again produces a new estimate of T, and so on. This procedure can be repeated iteratively and is hereafter referred to as the Iterative Method (IM).

$$\left[\begin{array}{ccc}{x}_a& x_b& x_c\\ y_a& y_b& y_c\\ 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\begin{array}{c}\left[\begin{array}{c}A\\ B\\ C\end{array}\right]\\ \hookrightarrow \end{array}=\left[\begin{array}{c}M{x}_m\\ M{y}_m\\ 1\\ 1\\ 1\end{array}\right]\to \left[\begin{array}{ccc}{x}_a& x_b& x_c\\ y_a& y_b& y_c\\ 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\begin{array}{c}\left[\begin{array}{c}{A}_1\\ B_1\\ C_1\end{array}\right]\\ \hookrightarrow \end{array}=\left[\begin{array}{c}M{x}_m\\ M{y}_m\\ A\\ B\\ C\end{array}\right]\to \left[\begin{array}{ccc}{x}_a& x_b& x_c\\ y_a& y_b& y_c\\ 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\begin{array}{c}\left[\begin{array}{c}{A}_2\\ B_2\\ C_2\end{array}\right]\\ \hookrightarrow \end{array}=\left[\begin{array}{c}M{x}_m\\ M{y}_m\\ A_1\\ B_1\\ C_1\end{array}\right]=\left[\begin{array}{ccc}{x}_a& x_b& x_c\\ y_a& y_b& y_c\\ 1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{A}_n\\ B_n\\ C_n\end{array}\right]=\left[\begin{array}{c}M{x}_m\\ M{y}_m\\ A_{n-1}\\ B_{n-1}\\ C_{n-1}\end{array}\right]$$ 3

Where n is the number of iterations. Once the tensions in all the intersecting cell boundaries are estimated, Laplace’s law is used to estimate the pressures within the cells using the curvatures of the individual cell boundaries as follows.

$$\left[\begin{array}{ccc}{c}_{1,2}& c_{2,1}& 0\\ c_{1,3}& 0& c_{3,1}\\ 0& c_{2,3}& c_{3,2}\end{array}\right]\left[\begin{array}{c}{P}_1\\ P_2\\ P_3\end{array}\right]=\left[\begin{array}{c}{A}_n/{\rho }_1\\ B_n/{\rho }_2\\ C_n/{\rho }_3\end{array}\right]$$ 4

which is of the form $\mathit{CP}=D$, where C includes the concavity of adjacent pairs of cells, $c_{i,j}$ represents the concavity of the boundary between cell i and cell j, and is 1 when i is convex relative to j, and -1 when i is concave. Since cellular concavity is determined from the image, C is known. The vector P contains the intracellular pressures, which are unknown. The D vector contains the estimated pressure differences between adjacent cells based on the estimated tensions ($A_n,B_n,C_n$) and the radii of curvature (${\rho }_1,{\rho }_2,{\rho }_3$) all of which are known. Therefore, P can also be solved by inverting Eq. 4 using lsqminnorm. Equation 4 above is written for three cells but in general, C has rows equal to the number of boundaries, and columns equal to the number of cells within the network. For cells at the perimeter, the pressure is calculated using the boundaries they share with other cells, and the exterior boundaries are not considered.

To evaluate the sensitivity of Eq. 4 for estimating intracellular pressures on the accuracy of the boundary tensions, we compared estimates of P_i obtained using the IM and MM as well as when the values of $A_n$, $B_n$, and $C_n$ in Eq. 4 were all set to 1, referred to as the uniform method, UM. Previous studies have compared different SI methods to a uniform tension model as a means of evaluating the efficacy of more complex SI methods¹⁶.

The tension and pressure estimates provided by these methods are relative rather than absolute. To obtain absolute values, the relative values must be calibrated to one or more known pressure and tension values in the system. In the present study, we calibrated the relative values of pressure to two measured values.

Balloon frame

We developed a macroscopic physical model of confluent cells similar to that of Mead et al.¹⁷. Cells were represented by transparent semi-inflated vinyl balloons (Aredpoook, 20 inches, Bobo) containing barometric pressure sensors (GY-BMP280 Atmospheric Pressure Sensor Module). The balloons were inflated to random volumes to introduce heterogeneity and then lubricated (J-Lube) to reduce shear forces between adjacent walls as they were compressed vertically between two acrylic sheets within a steel frame. For each image, the positions of the balloons in the frame were randomized. The balloons were then compressed laterally with a foam bar to force them to become space-filling. In this configuration, the balloons approximated confluent epithelial cells at equilibrium. Intra-balloon pressures were recorded via an Arduino board (LAFVIN Nano v3.0). The edges of the balloons were traced by hand on the front acrylic sheet with a dry-erase marker, and photographed (Fig. 6A). The edges were digitized by hand from the photograph and converted into a binary image for segmentation (Fig. 6C).

Fig. 6. Balloon frame model.

A Full balloon frame, where balloons are packed between two acrylic sheets, and further compressed with a yellow foam bar (i). Each balloon has a labeled pressure sensor (ii) that leaves the enclosure to connect to the Arduino board. The heterogeneity in the system also allows for gaps between balloons (iii). The edges of the balloons were then traced by hand using a dry-erase marker, represented here as dashed lines. B These lines were then digitized into a binary image. Networks were then made by stochastically placing nodes along the borders of the cells, allowing for similar but not identical networks for Monte-Carlo analyses of networks. Two such networks are shown in red and blue in C.

While the balloon model allows the balloon pressures to be measured, the wall tensions cannot be measured. Therefore, the primary purpose of the balloon model is to allow evaluation of the pressure estimates created by the different SI methods by comparing the measured pressures to the estimated pressures. The correlation coefficient (CC) and root mean square error (RMSE) between the measured and estimated pressures were used as SI performance metrics. The coefficient of variation (CV) of estimated pressure from ten different realizations of each balloon network was used to assess the robustness of the methods. These measures were statistically evaluated using sign-rank tests offset by the Bonferroni correction, and we considered an alpha value of 0.05 to be significant.

Creating a network from images

Images of confluent cells were converted into binary images. A watershed algorithm (Matlab, 2024a) was then used to convert the cell-cell boundaries into contiguous paths having widths of 1 pixel (Fig. 6B). Two types of nodes were then identified: intersection nodes, and wall nodes. Intersection nodes are those where 3 or more cells meet, while wall nodes are placed along the boundary where two cells meet. To identify the nodes, a $3\times 3$ pixel window was placed on every pixel in the image in random order. If the window contained pixels from 3 or more different cells, the center pixel was identified as an intersection node. If a window contained pixels from only 2 different cells, then it was identified as a wall node. Including all the wall nodes is computationally expensive, therefore a wall node was only accepted if it was a predetermined distance from all other nodes and satisfied an additional probability constraint. This additional probability allowed for modification of the density of points within the network, i.e., if the probability was low, there would be fewer points accepted along the cell boundary, and vice versa. Through trial and error, we found that a minimum distance of 20% of the mean diameter of all the cells in the network with a 50% probability constraint worked well. That is to say, if a wall node was identified, it would only be accepted if it was greater than 20% the mean diameter of the cells in the network away from any other node and then had a 50% chance of not being accepted by chance. This effectively allowed for random sampling of points along the walls with a distance constraint. It was also found that if a rastered order was used (i.e., each pixel was tested in order from top to bottom) a directional bias was introduced that interfered with the pressure estimates. This stochastic approach to defining the nodes along the boundaries allows for a Monte-Carlo approach to estimating tension and pressure values because different network realizations, while not identical, are statistically similar as shown in Fig. 6C. The median tension and pressure values from the 10 runs were used as the final values for comparison.

2D computational model and sensitivity analysis

To explore the behaviors of SI using the IM in systems with more cells than the balloon frame, a computational model of a pre-stressed pressurized network¹³ was also utilized. In the computational models, both the pressures and tensions throughout the network are known and can therefore be compared to the estimated values. To create these computational networks, a Voronoi-based diagram was created based on a Poisson disk sampling¹⁸, then each cell within the network was assigned a random pressure, and each edge of the network was segmented to allow for curved walls. Segments were then converted into pre-stressed springs. Simulated annealing was used to find the equilibrium configuration of the network¹⁹, accounting for both the tensile forces from the springs and the pressures acting on each node. For these tests, pressure was kept constant within the cells, and therefore was not changed as the cells changed sizes. A range of network sizes were tested to evaluate the robustness of pressure and tension estimates relative to network size. These simulations provided known geometries with known tensions and pressure values that could then be estimated from the geometry using SI.

The sensitivities of the IM and MM were compared by adding noise to the computational models in two different ways. First, 2D computational networks (Fig. 4A) were discretized into binary images at various resolutions and then re-converted back into a network using the approach described previously (Fig. 6C), a process that we refer to as imaging noise. Resolutions were standardized based on the average diameter of the cells within the image, such that there were between 30 and 100 pixels per cell diameter (Fig. 4B), and 1 pixel width per boundary. This discretization introduces noise as the fine details of the network are lost in coarse discretization. This represents the process of imaging cells at various magnifications.

Fig. 4. Noise in the computational networks.

A Example of a small, solved network. The colors of the walls indicate tension, where blue is low and red is high. The numbers in the cells represent the cell’s internal pressure (AU). B Discretized region of the network at various resolutions: i, 30 pixels/diameter, ii, 65 pixels/diameter, iii 100 pixels/diameter. Red arrows indicate the same wall. Note that the clear curvature at high resolution on panel iii is lost at low resolution on panel i, indicating introduced noise. C, D Effects of Gaussian noise on the network, where black indicates the original network boundary, and red represents the network with added noise: c represents the network with 1% noise added, and d represents the network with 5% noise added.

Second, random noise following a Gaussian distribution was added to the positions of the nodes in the 2D networks, which we refer to as Gaussian noise. The standard deviation of the Gaussian was set as a fraction of the average cell diameter in order to standardize noise across all networks. This fraction was scaled between 10^-4 and 10^-1 to provide a wide range of noise amplitudes (Fig. 4C, D).

All computational networks were tested with the MM and IM, and the CC, RMSE, and CV between estimated and known pressure and tension values were compared at their respective noise levels. Significance was evaluated using the sign-rank test. We considered an alpha value of 0.05 to be significant.

Onion cells analysis

The IM approaches were also applied to living cells. Red onion epithelial cells were harvested and placed in a drop of deionized water to induce hypotonicity and then imaged. A 2 M NaCl solution was then added to induce hypertonicity, causing a change in cell pressure, and the same regions of cells were imaged again. The cell walls, rather than the cell membranes, were considered as the boundaries of the cells, consistent with Long et al.¹⁰. The cell walls were hand-traced and subjected to the IM. Note that although the plasma membranes of the cells withdraw inward from the cell walls, the walls between adjacent cells still stay in contact. The estimated pressures and cell areas were compared in both tonicities using a sign-rank test, with an alpha value of 0.05 considered to be significant.

Author contributions

J.K.H. carried out balloon experiments and the modeling; J.K.H. and Y.D. carried out the onion experiments; K.K. helped with imaging; J.K.H., X.V., J.H.T.B., K.R.L., and B.S. conceptualized the studies. All authors reviewed the manuscript.

Data availability

All data generated or analysed during this study are included in this published article and its supplementary information files.The underlying code for this study is not publicly available but may be made available to qualified researchers on reasonable request from the corresponding author.

Code availability

The underlying code for this study is not publicly available but may be made available to qualified researchers on reasonable request from the corresponding author.

Competing interests

The Authors declare the following Competing Financial Interests: BS is a co-founder of Mechanobiologix, LLC; JHTB is a consultant for and shareholder in Oscillavent, Inc. and Respiratory Sciences, Inc., and a consultant for Chiesi USA, Inc. HTN is an Editorial Board Member for the journal, but was not involved in the editorial review of, or the decision to publish this article. All other authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s44341-025-00031-5.