Computational Modeling of Developing Cartilage Using Experimentally Derived Geometries and Compressive Moduli

Abstract

During chondrogenesis, tissue organization changes dramatically. We previously showed that the compressive moduli of chondrocytes increase concomitantly with extracellular matrix (ECM) stiffness, suggesting cells were remodeling to adapt to the surrounding environment. Due to the difficulty in analyzing the mechanical response of cells in situ, we sought to create an in silico model that would enable us to investigate why cell and ECM stiffness increased in tandem. The goal of this study was to establish a methodology to segment, quantify, and generate mechanical models of developing cartilage to explore how variations in geometry and material properties affect strain distributions. Multicellular geometries from embryonic day E16.5 and postnatal day P3 murine cartilage were imaged in three-dimensional (3D) using confocal microscopy. Image stacks were processed using matlab to create geometries for finite element analysis using ANSYS. The geometries based on confocal images and isolated, single cell models were compressed 5% and the equivalent von Mises strain of cells and ECM were compared. Our simulations indicated that cells had similar strains at both time points, suggesting that the stiffness and organization of cartilage changes during development to maintain a constant strain profile within cells. In contrast, the ECM at P3 took on more strain than at E16.5. The isolated, single-cell geometries underestimated both cell and ECM strain and were not able to capture the similarity in cell strain at both time points. We expect this experimental and computational pipeline will facilitate studies investigating other model systems to implement physiologically derived geometries.

Introduction

The mechanical loading environment regulates the metabolism and behavior of chondrocytes, the cells responsible for building and maintaining cartilage [1]. Perturbations in the way in which loads are transmitted to chondrocytes, due to injury or defects in development, can lead to pathological phenotypes such as osteoarthritis or chondrodysplasias [1,2]. Alterations in cell metabolism can induce remodeling of the surrounding extracellular matrix (ECM), which will then lead to changes in matrix composition and stiffness, further affecting how cells are deformed in response to external loads. Subtle variations in the amount of strain cells experience can directly impact cell metabolism [3,4].

To understand how chondrocyte mechanics correlates with the surrounding environment, it is critical that cells and ECM are investigated within the same native environment [5]. Previous studies on the micro- and nanomechanics of cartilage were unable to resolve ECM and cell stiffness within the same tissue due to limitations in methodology [6–10]. To address these limitations, we developed a novel atomic force microscopy (AFM) method wherein the mechanics of cells and ECM are resolved in situ using a vibratome to create homogenous sections of viable tissue [5]. Using a murine model of Schwartz-Jampel syndrome (SJS), in which the pericellular matrix (PCM) molecule perlecan is knocked down, we showed how our method could measure significant differences in cell and matrix compressive moduli as a function of development and perlecan content [11]. Focusing on the developing humerus of embryonic and neonatal mice, we found that overall ECM stiffness increased concomitant with increasing age and, at each time point, perlecan knockdown significantly decreased ECM stiffness. While the stiffness of the resident chondrocytes was an order of magnitude lower, the values followed the same trends with respect to age and perlecan knockdown. During this time, immunohistochemical visualization of the cartilage indicated the organization of the cells and ECM substantially changed as well, leading us to question if the chondrocytes and ECM were remodeling to maintain specific stress or strain profiles in response to both growth and changes in ECM composition due to SJS.

Experimentally measuring how the stress and strain of chondrocytes change in vivo in response to loading is technically challenging. To investigate how mechanical forces are distributed at the cellular level, researchers rely on in silico models that enable the investigation of a wide array of simulated loading regimes that would otherwise be infeasible experimentally [12,13]. Most models use single-cell geometries, and the material parameters employed have been approximated or taken from the literature and in many cases, based off separate model systems. The density of native chondrocytes and the composition of the surrounding ECM significantly change as a function of tissue location, development, and disease [11,14,15], which is not captured in silico when considering an isolated cell.

Establishing a physiologically derived geometry for finite element analysis (FEA) of cartilage has been hindered by the difficulty in imaging deep enough into the ECM-rich cartilage. Cao et al. demonstrated the feasibility of generating three-dimensional (3D) geometries based off confocal z-stacks for FEA; however, the imaging was confined to the immediate surface of the tissue and subsequent analyses only considered discrete clusters containing 1–4 cells [16,17]. The limitation in imaging is largely due to light scattering from lipids and mismatches in refractive index. Recently, various clearing methods have been developed that greatly enhance the 3D visualization of tissues [18], which we previously demonstrated can resolve the 3D organization of the ECM deep within developing and adult cartilage [19–21].

Here, we adapt these imaging techniques to provide us with physiologically derived geometries of developing cartilage to combine with our experimentally determined compressive moduli for FEA. The primary goal of this study was to establish a methodology that can segment, quantify, and generate mechanical models of developing cartilage to explore how variations in geometry and material properties affect strain distributions within the tissue. Specifically, we wanted to investigate how the variations in chondrocyte and ECM stiffness experimentally observed in the SJS model could affect the distributions of strains within developing tissue. We used decellularization and confocal imaging to build multi-cellular geometries from embryonic day (E)16.5 and postnatal day (P)3 developing cartilage. These geometries were used for FEA in which the compressive moduli for the cells and ECM were obtained from our previous study [11]. The finite element models based on the true geometry and standard isolated, single-cell geometry were subjected to unconfined compression. While the magnitude of force that the articulating surfaces of developing and neonatal cartilage experience are unknown, it is likely that the developing skeleton is subjected to considerable shear and compressive loading due to the contractions of embryonic muscles and maternal movement [22,23]. Direct loading of articular cartilage using physiologically relevant loads (i.e., ½ body weight) has been reported to generate strains up to 10% in humans [24,25]. Furthermore, chondrocyte metabolism is affected by deformations as low as 1% [3], leading us to use a compression of 5% in ANSYS for this initial study.

Simulations indicated that chondrocytes at E16.5 and P3 had similar strains in response to 5% compression, even though the compressive moduli were significantly different as a function of both component (cell and ECM) and developmental time point. In contrast, the ECM at P3 took on significantly more strain. The isolated geometries underestimated both cell and ECM strain and were not able to capture the similarity in cell strain between time points. These data suggest that the stiffness and organization of the cartilage change during development to maintain a constant average strain within cells. With this initial framework confirming the feasibility and repeatability of using physiologically derived geometries, future studies will focus on computationally investigating a wider range stress and strain profiles. In addition, experimentally determining the time-dependent and multiphasic response of this model system will enable us to build more comprehensive constitutive models to govern the FEA.

Materials and Methods

Animal Model.

To examine the organization of developing chondrocytes and cartilage at different time points, wild-type dilute brown non-agouti (DBA) mice were time mated to produce embryonic day (E)16.5 embryos and postnatal day (P)3 pups. Experiments were approved by the Purdue Animal Care and Use Committee (PACUC; protocol 1310000973). PACUC ensures that all animal programs, procedures, and facilities at Purdue University adhere to the policies, recommendations, guidelines, and regulations of the USDA and the United States Public Health Service in accordance with the Animal Welfare Act and Purdue's Animal Welfare Assurance. Dams were euthanized via CO₂ inhalation, confirmed by cervical dislocation. E16.5 embryos were removed from the uterine horns and P3 pups were euthanized via decapitation. Femurs were isolated from the surrounding soft tissues and placed in ice-cold phosphate buffered saline (PBS) until processed for imaging.

Tissue Decellularization, Fixation, and Staining.

To prepare the tissue for imaging, samples were decellularized in 0.05% sodium dodecyl sulfate (SDS) in PBS at 4 °C with gentle rocking for 48–72 h. The SDS solution was replaced after 24 h. Following decellularization, samples were washed 3 × 30 min in PBS at room temperature then fixed in 4% paraformaldehyde for 24 h at 4 °C with gentle rocking. Samples were rinsed in PBS overnight at 4 °C on a rocker and transferred to fresh PBS for storage at 4 °C until staining. Before staining, samples were incubated in blocking buffer (10% donkey serum (Lampire, Pipersville, PA) in PBS with 0.1% Triton X-100 (PBST)) for 24 h at 4 °C with gentle rocking. Samples were then incubated in 20 μg/ml wheat germ agglutinin (WGA) AlexaFluor488™ conjugate (Invitrogen, Carlsbad, CA), diluted in blocking buffer, for 48 h at 4 °C on a rocker. WGA is a lectin that binds to sialic acid and N-acetylglucosamine in proteoglycans enabling the global visualization of ECM architecture. After staining, samples were washed 2 × 1 h with PBST at room temperature, then overnight at 4 °C. Finally, samples were rinsed in PBS at 4 °C for at least 1 h then transferred to fresh PBS for storage at 4 °C until imaging.

Image Acquisition and Region of Interest Substack Generation.

Image acquisition of the tissue samples was performed within three days of staining using a 63× oil-immersion Plan-Apochromat (NA = 1.4) on a Zeiss LSM880 confocal microscope (Carl Zeiss, Oberkochen, Germany). Image stacks were acquired at 1024 × 1024 pixels, 2× line averaged, with 0.22 μm/pixel xy-resolution and 0.39 μm z-axis intervals. The same image acquisition parameters were used for both E16.5 and P3 developmental time points. Widefield images were acquired using a Leica M80 stereo microscope (Leica Microsystems, Wetzlar, Germany).

To improve image quality and prepare scans for quantification, imaged Z-stacks were first processed using Zen Blue (Carl Zeiss) to perform nearest-neighbor deconvolution. Then, region of interest substacks (ROIs) were isolated using ImageJ (NIH) for further segmentation analysis. The top layer of cells from the articular surface was excluded so that ROI substacks included only subsurface cellular geometries to avoid surface irregularities and provide a consistent volume for comparisons. For each developmental time point, n = 3 ROIs from each of n = 3 biological replicates (i.e., three individual animals) were generated (see Fig. S1 available in the Supplemental Materials on the ASME Digital Collection). All ROIs were the same dimensions (28.55 × 28.55 × 28.80 μm, x × y × z), except for P3 3-1, P3 3-2 and P3 3-3, which were (28.55 × 28.55 × 23.67 μm, x × y × z).

Image Processing and Analysis.

To quantify the developing tissue and construct a geometric model for simulations, a custom, automated ROI processing and segmentation matlab algorithm was developed and implemented. In brief, the algorithm processed image stacks are enhanced using median filtering, histogram correction to improve cell to ECM contrast, followed by adaptive thresholding (Otsu's method), morphological filtering, Chan-Vese active contours, and watershed distance transform to segment cells from the rest of the volume before proceeding to structural mesh generation.

Preprocessing and Filtering.

First, isotropic voxel dimensions were enforced by interpolating the image stack in the z-direction to compensate for differences between the x × y and z resolutions. Stacks were then median filtered; twice through a 5 × 5 pixel two-dimensional filter, followed by a 50% weighted 11 × 11 × 3 pixel 3D filter. Filter parameters were optimized to minimize subcellular geometric components and prevent distortion of cellular boundaries. Intensity correction was performed on a slice-by-slice basis using the built-in matlab imadjust function. Stacks were padded in three dimensions using replicate values to mitigate edge effects on contour fitting.

Binary Mask Generation and Refinement.

After image stacks were intensity corrected and contrast enhanced, they were binarized into masks using adaptthresh and imbinarize, following Otsu's method [26,27] to segment chondrocytes from the surrounding ECM. Masks were processed to remove holes using regionprops and imfill [28]. Neighboring cells were separated by image opening with imopen, then small artifacts were removed by bwareaopen. Masks were refined using Chan-Vese active contour method [29] and constrained against unchecked growth by a dilated version of the input mask. Morphological filtering using imopen and bwareaopen was applied again to eliminate fused, neighboring regions and isolated, small artifacts. Finally, to ensure separation of neighboring cells, watershed distance transform was implemented using imextendedmin and imimposemin to force unique, morphologically central, regional minima. Masks were trimmed of padding values before mesh generation.

Structural Mesh Generation and Import to ANSYS.

To allow finite element analysis of scanned regions, structural meshes were generated from the 3D binary masks using the ISO2MESH toolbox [30]. Mesh node/face matrices were written to .stl files using stlwrite. .stl files were then converted to .sat files using CONVERT_stl_to_sat to enable importation into ANSYS for subsequent finite element modeling.

Three-Dimensional Finite Element Analysis.

To analyze the impact of tissue mechanical properties and geometry, ROIs were simulated using ANSYS version 18.0 (ANSYS, Inc., Canonsburg, PA). Simulations utilized the geometry extracted from the original image sets prepared as described above, defined as “true geometry,” or geometry simulated with volume fraction-matched ellipsoids, defined as “isolated geometry.” The volume fraction of an imaged tissue volume or simulated volume is defined as the percentage of the total volume filled by cells relative to the overall volume of the ROI. Simplified single cell analyses were simulated using an ellipsoid, which was 23 μm long along the main axis and 17 μm long along the minor axes contained within a bounding box which varied in size from 15,069 μm³ to 249,307 μm³ yielding volume fractions varying from 1.5%, 14%, and 25% to simulate observed cellular volume fractions in scanned tissue samples. In all simulations, cells were bonded to the surrounding ECM. Simulations examining ROIs built off true geometries had between 103,581 and –377,884 nodes and 60,260–220,102 elements with an average 254,114 nodes and 149,193 elements per simulation. Single cell simulations had between 45,989–82,180 nodes and 28,505–49,486 elements, with an average of 66,087 nodes and 40,310 elements per simulation.

The compressive moduli for the cells and the ECM used for the mechanical properties in simulations were obtained from in situ AFM measurements of viable, developing cartilage previously reported in Ref. [11] (Table 1). These compressive moduli directly correlate with the species of mouse and developmental time points used to generate the true geometries. The loading rate and trigger force (11.2 nN) were small enough such that the compressive modulus reported in Ref. [11] can be assumed to be a Hookean initial modulus, as demonstrated by the minimal hysteresis observed. Poisson's ratios of 0.4 and 0.45 were used for cells and the ECM, respectively [31,32]. All simulations of true and isolated geometries in this study were performed by compressing the total sample volume by 5%. Volumetric von Mises stress and strain were collected for all simulations and segmented by tissue type (i.e., cell, ECM).

Table 1.

Compressive moduli of cells and ECM from Ref. [11]

	WT (kPa)	Neo/Neo (kPa)
E16.5 cell	2.06	1.22
E16.5 ECM	24.34	15.28
P3 cell	4.10	2.83
P3 ECM	54.64	27.59

To avoid noise from edge artifacts in the simulation and investigate the effect of varied ROI cellular geometry, cells were analyzed in two phases. The first analysis examined all cells in the ROI while the second analysis only looked at core cells, which were defined as any cell not within 1 μm of the ROI boundary to avoid unrealistic stresses and strains caused by direct contact with the compressing edge or by cells that were only partially imaged in the ROI. While examining the core cells, all other cells were still present in the geometry of the simulation; however, the stress and strain measures from noncore cells were discarded.

To examine the effect neighboring cells have on average cell strain, two sets of simulations were performed. The first simulation, defined as “side-by-side,” examined two simplified single cells, side by side along the y-axis while being the entire volume was compressed in the −z direction by 5%. The second simulation, defined as “in-line,” examined two cells in line with the direction of compression (−z). In both simulations, the distance between neighboring cells was varied between 1 and 160 μm before loading. Simulations had between 27,290–63,524 nodes and 16,299–38,790 elements, with an average of 49,221 nodes and 29,818 elements per simulation.

Statistical Analyses.

Prism 7.0c (GraphPad Software, La Jolla, CA) was used for statistical analyses. Two-way analysis of variance (ANOVA), followed by Tukey's post hoc analysis (α = 0.05), was used to analyze differences in strain as a function of geometry and cell/ECM compressive moduli combinations. Error bars represent standard deviation. A 95% confidence interval was accepted and labeled with (*) for 0.01 < p ≤ 0.05, (**) for 0.001 < p ≤ 0.01, (***) for 0.0001 < p ≤ 0.001 and (****) for p ≤ 0.0001.

Results

To accurately analyze how changes in cell and ECM stiffness affect the distribution of strain within developing cartilage, geometries based off the native physiology were generated. Femurs from E16.5 and P3 mice were decellularized and stained with AF488-conjugated WGA, which stains a subset of proteoglycans and enables the visualization of the ECM (Fig. 1). Confocal image stacks were obtained at 3 different positions for 3 biological replicates at each time point and processed in matlab to generate regions of interest (ROIs) of the true geometry for analysis in ANSYS (Figs. 1 and S1 available in the Supplemental Materials on the ASME Digital Collection). ROIs that only included a single, isolated cell were also generated using various cell volume fractions (ϕ) to mimic either the ROI typically used to model chondrocytes (ϕ_adult cell = 0.015) [3], or the average volume fraction of the E16.5 (ϕ_E16.5 cell = 0.25 ± 0.04) and P3 ROIs (ϕ_P3 cell = 0.14 ± 0.02; see Figs. S1 and S2(a) available in the Supplemental Materials on the ASME Digital Collection). Preliminary simulations indicated that volume fraction greatly affected average von Mises strain within cells after 5% compression (see Fig. S2(b) available in the Supplemental Materials on the ASME Digital Collection). Therefore, volume fractions of the isolated geometries were matched with those of the true geometry being compared.

Fig. 1.

Model system and experimental workflow: (a) relative scale of E16.5 embryo and P3 pup. Bars = 5 mm, (b) immunohistochemical staining for perlecan (red) and type VI collagen (green) reveal a dramatic change in ECM organization from E16.5 to P3. Confocal image stacks to build geometries were acquired starting at the articular surface (*). Nuclei stained with 4′,6-diamidino-2-phenylindole (blue). Bars = 25 μm, and (c) workflow showing the processing of a P3 femur from harvest to analysis in ANSYS. (See color figure online.)

The response of cells in the isolated geometry (ϕ_E16.5 cell = 0.25) was compared with the true geometry of E16.5 cartilage to determine how chondrocytes were influenced by neighboring cells. Different cell and ECM combinations, representing the compressive moduli from wild type (WT) and homozygous perlecan knockdown (Neo) tissues, were analyzed (Table 1), where WT/WT indicates wild-type cell and wild-type ECM material properties were used. The geometries were compressed by 5% in ANSYS and the equivalent, von Mises strains were compared. The cell in the isolated geometry had approximately the same strain as cells in the E16.5 geometries; however, simulations indicated that cells on the boundary experienced lower strains than the core (arrowheads, Fig. 2). The strain of the core cells (i.e., any cell not within 1 μm of the ROI boundary) was then analyzed and found to be significantly higher (p < 0.0001) than when strain was averaged over all cells (Fig. 2). This was not unexpected due to boundary effects when loading under unconfined compression. To test if the presence of the boundary cells affected the mechanical response of the core cells, the geometries were modified so that only core cells were present during the simulation. The strain of the core cells without boundary cells was significantly higher than when the boundary cells were included (p < 0.001) indicating that neighboring cells have a strain-shielding effect (Fig. 2). In contrast, when true P3 geometries were compared with the isolated geometry (ϕ_P3 cell = 0.14), the isolated geometry generated approximately 10% more cell strain when compared with strain averaged over all cells in the true geometry (see Fig. S2(c) available in the Supplemental Materials on the ASME Digital Collection). These results highlight that both volume fraction and distribution of that volume within a geometry significantly affect the mechanical response.

Fig. 2.

Single-cell simulations underestimate cell strain. Geometries based off E16.5 cartilage were compressed 5% in the –z direction and modeled using compressive moduli from Ref. [11]. Materials were mixed to determine the effect of different cell/ECM stiffness ratios on calculated strain. Strain in isolated cells (ϕ_{E16.5 cell} = 0.25) was similar to strain in all cells. However, analysis of all cells versus core cells (i.e., any cells at least 1 μm away from a boundary) indicated that inclusion of strain from boundary cells significantly reduced the strain for all cell/ECM combinations (***p < 0.001, ****p <0.0001; Tukey's post hoc test). When boundary cells were removed from the simulation, strain was also significantly greater than all cells (p < 0.0001) and core cells (p < 0.001). Two-way ANOVA revealed the effect of cell/ECM combination (p < 0.001) and geometry (p < 0.0001) on strain were significant. Error bars = standard deviation. N = 3 biological replicates, where each biological replicate is an average of 3 ROIs.

To test how cell orientation and location of neighboring cells affect strain magnitude, two cells either in line or side by side were compressed in the –z direction and spacing between cells was varied. When the two cells were in-line with the direction of loading, a smaller amount of strain was generated intracellularly than if the cells were side by side (Fig. 3). Furthermore, only the in-line cells showed a decrease in strain as the cells became closer together. There was no difference in average ECM strain between the two orientations (Fig. 3).

Fig. 3.

Effect of spacing on cell strain. The in-line conformation had a larger influence on strain than side-by-side and generated values that more closely approximated the core cell strain determined using the true geometries. Variation in cell strain increased as spacing decreased. The amount of ECM between outer cell edges and the boundary of the simulation was keptconstant as distance between cells varied, and the volume fraction of the cells was constant for the two orientations for agiven cell spacing. Simulations used the E16.5 WT/WT materialcombination. Geometries were compressed 5% in the –z direction (arrow) and the cell or ECM strain was averaged across intracellular nodes of both cells. Error bars = standard deviation. See color figures online.

To assess how developmental changes in compressive moduli affect the response of cells and ECM to 5% compression, 3 sets of geometries from E16.5 and P3 (see Fig. S1 available in the Supplemental Materials on the ASME Digital Collection) were compared with simulations of isolated cells with the same volume fractions (see Fig. S2(a) available in the Supplemental Materials on the ASME Digital Collection). Each set of geometries was modeled using compressive moduli that matched the time point, except for the E16.5Neo/P3WT configuration. Simulations of isolated cells indicated that strain within P3 cells was approximately 8% higher than E16.5 (Fig. 4(a)). In contrast, core cell strains in the E16.5 and P3 true geometry were much closer, with P3 trending slightly lower than E16.5 (p < 0.05) and two-way ANOVA indicating that cell/ECM combination had a significant effect (p < 0.01). Deviations in strain occurred when cell/ECM genotypes did not match; strain in the WT/Neo combination was lower and Neo/WT and E16.5Neo/P3WT combinations were higher than WT/WT and Neo/Neo. ECM in the isolated and true geometries followed the same trend, where P3 strain was higher than E16.5. Similar to cell strain, the isolated geometries underestimated ECM strain (Fig. 4(b)). Two-way ANOVA indicated that ECM strain was significantly affected by cell/ECM combination (p < 0.01) and age (p < 0.0001).

Fig. 4.

Strain in core cells from E16.5 and P3 geometries showed similar trends when material properties were altered. (a) Cell strain in the isolated geometries indicated that strain increased approximately 8% in cells from E16.5 to P3; however, the strain remained relatively constant when the core cells of the true geometry were analyzed. Two-way ANOVA revealed the effect of cell/ECM combination (p < 0.01) and age (p < 0.05) were significant on strain. (b) Both isolated and true geometries showed an increase in ECM strain between E16.5 and P3. There was little variation of ECM strain due to age or material in the isolated geometries. ECM strain at P3 was significantly higher than at E16.5 for all cell/ECM combinations (****p < 0.0001; Tukey's post hoc test). Two-way ANOVA indicated the effect of cell/ECM combination (p < 0.01) and age (p < 0.0001) was significant on strain. Geometries were compressed 5% and the cell or ECM strain was averaged. Error bars = standard deviation. N = 3 biological replicates, where each biological replicate is an average of 3 ROIs.

To determine how age-specific geometry affects cell strain, E16.5 geometries were simulated with P3 material properties and vice versa. The effect of geometry had a larger influence on cell strain than the material properties (Fig. 5). Different age/geometry combinations did not significantly affect cell response, indicating that cells are adapting their stiffness to maintain a constant strain during these stages of development. In contrast, the age/geometry combinations significantly affected ECM strain (p < 0.0001, Fig. 5). Taken together, these data indicate that the presence of neighboring cells significantly affects cell strain response and that cell organization influences strain separate from the influence of volume fraction.

Fig. 5.

Age-specific geometry affects strain more than cell/ECM material properties. There was no significant difference between strains when E16.5 geometries were simulated with E16.5 or P3 material properties, and vice versa. Two-way ANOVA indicated that age/geometry combination had no effect on cell strain, whereas cell/ECM combination did (p < 0.01). For ECM, both age/geometry (p < 0.0001) and cell/ECM combination (p < 0.01) had an effect. For each cell/ECM combination, simulations that used the P3 geometry had significantly higher strains than those that used E16.5 geometries (bars, p < 0.0001; Tukey's post hoc test); however, there was no difference when the same geometry was used and materials were varied. Geometries were compressed 5% and the core cell or ECM strain was averaged. Error bars = standard deviation. N = 3 biological replicates, where each biological replicate is an average of 3 ROIs.

Discussion

Our study demonstrates that physiologically derived geometries can generate repeatable computational analyses of cartilage mechanics that enable robust comparisons between biological variables. With the advent of new imaging technologies and increased computational power, the use of tissue-specific geometries based off 3D images is feasible. Here, using mild decellularization conditions, we demonstrate that the 3D organization of cells in developing murine cartilage be visualized and that there are significant differences in geometry between time points. Furthermore, cell volume fraction and organization significantly influenced the outcome of FEA indicating the need for these accurate representations of geometry. This methodology is not constrained to only developing tissues as we previously showed how the organization of cells within adult bovine cartilage can be visualized up to 500 μm deep using a fructose-based clearing method [20]. We expect this experimental and computational pipeline will enable future studies to implement true geometries with more advanced material descriptions.

Notably, our simulations using the true geometries indicated that cell strain remained constant even though the organization and volume fraction varied between E16.5 and P3 (Fig. 3(a)). In addition, strain within the cells was amplified by ∼1.8-fold, similar to other reports [16,17,33–35]. In contrast, simulations that used the isolated geometries predicted ∼10% increase in strain between E16.5 and P3 even though the volume fractions were matched (Fig. 3(a)). If the native volume fraction was not taken into account and cells were instead modeled using a smaller volume fraction closer to what is typically used in other studies (ϕ_adult cell = 0.015), the simulations would overestimate strain by 50% (from 0.08 to 0.12; see Fig. S2(b) available in the Supplemental Materials on the ASME Digital Collection). These discrepancies can have a profound effect on future studies that aim to use these computational models as guidelines for experimental parameters as subtle changes in loading conditions have been shown to differentially affect cellular behavior in vitro [3,4]. For example, Han et al. showed that calcium signaling in meniscus fibrocartilage cells is altered in response to a uniaxial tensile strain of 0.015 [4]. Overall, these results suggest that cells mediate their stiffness and organization to maintain a constant strain, and potentially metabolism, across the developmental time window investigated here.

While the density of chondrocytes in murine cartilage is higher than larger mammals, others have demonstrated that cell spacing has an effect even in tissues with a low volume fraction of cells. Sibole et al. generated a random distribution of 11 cells within a representative volume element having a volume fraction of 5.8% [34], reported to be the density of chondrocytes in the middle layer of human articular cartilage [14]. These simulations found that strain was significantly more variable than the isolated cell model [34]. In addition, the location of cells within the geometry also influences the mechanical response [17,33,34,36], providing further evidence that the traditional single, centralized cell does not accurately capture chondrocyte mechanics.

The primary limitation of our study is that the simulations only captured the initial response to loading and assumed the cells and ECM were made up of a single phase. The compressive moduli used in this model were taken from our previous study that used AFM to measure the mechanics of E16.5 and P3 cartilage. Freshly harvested tissue was vibratomed and cell and ECM stiffness across a 15 × 15 μm² area were measured using a 5 μm diameter tip at a rate of 10 μm/s [11]. There was very little hysteresis using these testing parameters, allowing us to approximate the behavior to be linearly elastic [11]. However, it is well established that cartilage is biphasic and shows a time-dependent response to loading. Both the solid and fluid phases are thought to contribute to the time-dependent response, which has been described as viscoelastic and/or poroelastic [12,37,38]. The organization and composition of the solid phase (e.g., cytoskeleton in the cell and fibrillar matrices in the ECM) will greatly affect compressive and tensile moduli as well as permeability, indicating that separate constitutive relationships will need to be developed to describe material behavior at different time points.

There are substantial changes in ECM organization during chondrogenesis (Fig. 1(b)). The ECM in cartilage can be characterized as having two different compartments, a PCM that tightly surrounds cells and the bulk, interstitial matrix. At E16.5 there is no defined PCM (Fig. 1(b)). Indeed, the bulk of the ECM appears to be comprised of “traditional” PCM components, perlecan and type VI collagen, which assemble into a mesh-like network with nidogens and laminins [39]. In contrast, the architecture at P3 more closely resembles the adult phenotype where perlecan and type VI collagen are restricted to the PCM and the bulk matrix is dominated by fibrillar type II collagen and a hydrated gel made up of hyaluronic acid and aggrecan [40]. Due to the contrasting materials present at these two time points, the permeability and resistance to axial and shear forces will substantially vary. In particular, the addition of a PCM at P3 will also influence the mechanical response as it is thought to contribute to the biomechanical, biophysical, and biochemical interactions between chondrocytes and the bulk ECM [41]. Computational models have indicated that the PCM influences fluid flow and stress distribution around the chondrocytes [33,42]. The PCM was not included in our geometries as our initial AFM study did not have the resolution to identify the mechanics of the PCM separate from the bulk ECM. Nevertheless, we expect to refine the constitutive equations that drive our FEA as we continue to experimentally quantify how the material composition of the cartilage changes in our model system.

Another limitation of this model is that it was simulated under unconfined compression. The primary goal of this study was to investigate the role of geometry on the mechanical response of cells and not to model the multiscale mechanics of developing cartilage. To incorporate these ROIs into future multiscale models, it will be necessary to impose periodic boundary conditions. However, with the heterogeneous geometry of the ROIs and differences in compressive moduli of cells and ECM (Table 1), this will generate discontinuities at the boundaries. Since we demonstrated that the cells at the boundaries influence core cell mechanics (Fig. 2), the ROIs will have to be refined so that the cells at the edges form continuous cells of the same volumes.

The simulations performed in this study highlight the importance of accurate geometry when analyzing biological tissues and provide insights on cellular strain in response to compression. Our future studies aim to develop constitutive equations that better capture the time-dependent and biphasic nature of cartilage by first experimentally characterizing how the material properties of cartilage change as a function of development and disease. Proteomic analyses will identify the components of the solid phase (e.g., cytoskeleton and ECM) [15], 3D imaging will reveal the dimensions and organization of ECM networks [19–21] and time-dependent micro- and nano-indentation tests will measure how the mechanical response changes in concert with variations in solid phase composition and network organization [10,37,38]. Implementation of constitutive equations based on these experimentally derived parameters with physiologically derived geometries will generate comprehensive models that can be used to better understand how cells respond to the mechanical environment.

Funding Data

• National Institutes of Health (R01 AR071359 and DP2 AT009833; Funder ID: 10.13039/100000002).

Nomenclature

AFM =

atomic force microscopy

ECM =

extracellular matrix

FEA =

finite element analysis

Neo =

homozygous for perlecan knockdown

PCM =

pericellular matrix

ROI =

region of interest

WT =

wild type

ϕ =

volume fraction