Superficial zone chondrocytes can get compacted under physiological loading: A multiscale finite element analysis

Abstract

Recent in vivo and in vitro studies have demonstrated that superficial zone (SZ) chondrocytes within articular layers of diarthrodial joints die under normal physiologic loading conditions. In order to further explore the implications of this observation in future investigations, we first needed to understand the mechanical environment of SZ chondrocytes that might cause them to die under physiological sliding contact conditions. In this study we performed a multiscale finite element analysis of articular contact to track the temporal evolution of a SZ chondrocyte’s interstitial fluid pressure, hydraulic permeability, and volume under physiologic loading conditions. The effect of the pericellular matrix modulus and permeability was parametrically investigated. Results showed that SZ chondrocytes can lose ninety percent of their intracellular fluid after several hours of intermittent or continuous contact loading, resulting in a reduction of intracellular hydraulic permeability by more than three orders of magnitude. These findings are consistent with loss of cell viability due to the impediment of cellular metabolic pathways induced by the loss of fluid. They suggest that there is a simple mechanical explanation for the vulnerability of SZ chondrocytes to sustained physiological loading conditions. Future studies will focus on validating these specific findings experimentally.

1. Introduction

Articular cartilage is a specialized tissue lining diarthrodial joints in the body, where it serves to reduce friction while sustaining contact forces for over 108 million cycles of loading in a 60-year lifespan. As with any mechanical system with moving parts, cartilage must experience some normal ‘wear and tear’ over normal activities of daily living. In some cases, this ‘wear and tear’ may progress to further degradation of the tissue, resulting in osteoarthritis (OA). Yet primary (idiopathic) OA is rare in individuals younger than 45. This counter-intuitive observation suggests that there must exist an intrinsic repair mechanism to compensate for this daily wear and tear in diarthrodial joints, and to maintain normal tissue function over many decades of life.

This routine damage initiates in the superficial zone (SZ) of cartilage tissue [1–3] where its resident chondrocytes have been shown to die even under normal physiologic loading conditions [4–6]. As such, it is probable that this expected repair mechanism is a SZ cell replenishment mechanism, most likely from the synovium lining [7–9]. However, the mechanism by which these SZ cells preferentially die has not yet been explained. The objective of the current study is to better understand mechanical factors that may lead to SZ chondrocyte death, as part of a broader effort to understand the mode of repair of normal ‘wear and tear’.

The SZ is known to exhibit lower compressive [10] and shear moduli [11] compared to the middle and deep zones (MZ, DZ) of the articular layer, likely resulting in excessive SZ compaction under physiological loading. Such compaction may wring out the interstitial fluid of SZ chondrocytes, significantly altering their fluid transport properties and possibly causing their death. Chahine et al. [12] showed that SZ chondrocytes died after sustained 12 h compaction of live cartilage explants by 50–80% strain. Using suitable control experiments they reported that this death was not a result of higher local compressive strains in the soft SZ, but instead the higher vulnerability of SZ chondrocytes to loading than their MZ and DZ counterparts.

Albro et al. [13] showed that chondrocytes can survive a volume loss of 50%, induced by hyperosmotic loading, for a short period of time (up to 1 h, at room temperature). However, lasting changes in cell volume can affect the ability of the cell to regulate its volume via ion channel transport, significantly affecting its homeostasis and viability [14]. A theoretical analysis by Ateshian et al. [15] demonstrated that chondrocytes under compression in situ experience fluid pressure differentials across the semi permeable cell membrane on the order of tens of kilopascals, causing water outflow and resulting in cell volume reduction greater than that of the extracellular matrix (ECM). Tissue compaction, especially of the pericellular matrix (PCM), and reduced cytoplasm permeability as fluid volume is lost, may also decrease diffusivity of glucose, insulin, other nutrients, and waste products between the cell and the synovial fluid, harming cell metabolism and potentially resulting in cell death. Further, SZ chondrocytes have been shown to be metabolically different and less phenotypically mature [16–18] than MZ and DZ chondrocytes, and may therefore be more sensitive to this loss of nutrients.

The evidence summarized above, though suggestive, is limited by experimental constraints that cannot examine field variables throughout the entire cellular environment. Computational models of the type proposed here provide access to data experiments cannot obtain. A few multiscale models of chondrocytes embedded in tissue have begun to investigate the mechanical role of the ECM and PCM on chondrocyte deformation under compressive loading [19–25]. As such, in this study we performed a multiscale finite element analysis of articular contact to understand the fate of a SZ chondrocyte, by tracking the temporal evolution of its interstitial fluid pressure, hydraulic permeability, and volume change under physiologic loading conditions. This deeper understanding of the mechanical loading environment of SZ chondrocytes, and the response of the chondrocytes to that load, will provide insight into potential repair mechanisms that maintain tissue viability despite regular loading, and may be used to inform and guide future experiments.

2. Materials and methods

This section presents the computational approach used to develop a multiscale biphasic contact analysis of a chondrocyte in cartilage tissue undergoing loading. The scales of analysis range from tissue (macroscopic) down to cellular (microscopic). Preprocessing, analysis, and postprocessing were performed using the open-source finite-element software FEBio Studio (www.febio.org) [26].

2.1. Creation of finite element model and meshing

A previously-reported model by Zimmerman et al. [27] of biphasic frictional sliding contact between a cartilage plug and strip (publicly available on the FEBio Studio repository), which accounts for symmetry about the mid-plane, was modified by replacing one finite element in the cartilage strip SZ with a finite element domain containing an embedded 15 × 15 × 5 μm [28] ellipsoidal chondrocyte (Fig. 1A).

Fig. 1.

(A) Overview of multiscale model of cartilage plug in contact with a cartilage strip, magnifying the interface where one element has been replaced with a finite element domain containing a chondrocyte in the strip SZ, called the ‘container’, in purple. (B) A closer view of the fine meshes of the container, PCM, membrane, and cytoplasm.

The dimensions of the cartilage strip were 20 × 5.5 × 2.5 mm, and the cartilage plug was modeled as a 16 mm diameter core of a hollow sphere with an outer radius of 18 mm, and a thickness of 2 mm. For both the plug and strip, the SZ zone was ~67 μm thick [27], consisting of the first layer of elements. The dimensions of the chondrocyte ‘container’ domain, which replaced a single element in the strip SZ, were 375 × 380 × 66.5 μm, and the dimensions of the PCM surrounding the chondrocyte were 30 × 30 × 10 μm, resulting in a PCM thickness of 7.5 × 7.5 × 2.5 μm.

All meshes employed 8-node hexahedral elements, with the exception of the cell membrane which used 4-node porous-permeable shells [29]. The strip and plug respectively included 7743 and 3780 elements, biased towards the articular surfaces. Increasingly fine meshes were used to model the chondrocyte ‘container’ domain, which replaced a single element in the strip SZ, and the PCM, with 3840 and 2688 elements respectively (Fig. 1). The cytoplasm consisted of 3200 8-node hexahedral elements, biased towards the membrane, which consisted of 384 4-node porous shell elements [29]. A mesh convergence analysis was performed, showing that all variables of interest (including cell volume, hydraulic permeability, fluid pressure, contact pressure, and friction coefficients) changed by less than 5% compared with a coarser mesh.

2.2. Constitutive models and material properties

All materials in this model used the biphasic theory to model deformable porous-permeable media [30] under finite deformation. The cartilage matrix was modeled as two layers: a SZ consisting of a single layer of elements, and a MZ/DZ comprising the rest of the tissue. The matrix materials were as previously described in Zimmerman et al. [27], as a biphasic material with a referential solid volume fraction of 0.2, constant isotropic permeability of $k=0.0011{\text{mm}}^4\cdot {\text{N}}^{-1}\cdot {\text{s}}^{-1}$, and a compressible neo-Hookean solid matrix [31] with Young’s modulus of $E=306\text{kPa}$ and (drained) Poisson’s ratio $v=0$. In the SZ, two orthogonal fibril families were implemented parallel to the articular surface, using a power-law strain energy density,

$${\Psi }_r=\frac{\xi }{\beta }{\left({\lambda }_n^2-1\right)}^{\beta }$$ (1)

where ${\lambda }_{\text{n}}$ is the fiber stretch ratio and $\xi$ is a measure of the fiber modulus, with $\xi =17.5\text{MPa}$ and $\beta =2$. In the DZ, three orthogonal fibril families were implemented parallel and perpendicular to the articular surface, with $\xi =17.5\text{MPa}$ and $\beta =2.5$. These fibril models can only sustain tension (Fig. 2A, B).

Fig. 2.

(A) Schematic of fibril orientation in the cartilage Plug and Strip (B) Individual fibril orientations shown in red, green, and blue. The first two red and green families run parallel to the articular surface and orthogonal to each other. They are found in the cartilage Plug SZ and DZ, and Strip SZ and DZ. The third blue family is oriented perpendicular to the articular surface, orthogonal to red and green, and is found only in the DZ of the Plug and Strip. (C) Schematic of fibril orientation in the PCM. Two orthogonal fibril families in red and green run circumferentially around the chondrocyte, representing the collagen fibril architecture seen in the PCM of chondrocytes.

The PCM was modeled similarly to the SZ, but with the two orthogonal fibril families oriented circumferentially around the chondrocyte (Fig. 2C). Young’s modulus of the PCM has been found to be inhomogeneous around the cell [32–35]. To account for all possibilities, the modulus of the solid matrix was parametrically varied between 25 and 306 kPa, the latter equivalent to that of the ECM. The permeability was selected as either the same properties as the SZ or $k=4.71\times {10}^{-5}{\text{mm}}^4\cdot {\text{N}}^{-1}\cdot {\text{s}}^{-1}$ [36]. The container employed the same properties as the SZ. The cell cytoplasm was modeled as a biphasic material with a referential solid volume fraction of ${\phi }_r^S=0.4$ [13,37–39], a porous neo-Hookean solid matrix (not to be confused with the previously described compressible neo-Hookean solid matrix; see below) with $E=0.78\text{kPa}$ [29], and a Holmes-Mow volumetric strain-dependent isotropic hydraulic permeability ($k_0=0.002{\text{mm}}^4\cdot {\text{N}}^{-1}\cdot {\text{s}}^{-1}$ [29], $M=2.2$ and $\alpha =2$ [40]) to account for decreasing permeability as the cytoplasm gets compacted, thus providing a measure of intracellular transport hindrance due to cell compaction.

$$k=k_0{\left(\frac{J-{\phi }_r^s}{1-{\phi }_r^s}\right)}^{\alpha }{e}^{\frac{1}{2}M\left(J^2-1\right)}$$ (2)

Here, $J$ is the relative volume (current volume divided by referential volume) of the porous solid matrix. In theory, when a biphasic material loses its entire interstitial fluid content due to complete compaction, its porous solid matrix reduces to an incompressible elastic solid, unable to undergo further volumetric loss. The volumetric strain-dependent hydraulic permeability also reduces exactly to zero in this theoretical limit. This occurs when the relative volume $J$ of the solid matrix reduces to the referential solid volume fraction ${\phi }_r^S$. Computationally, some investigators have proposed to model this material model transition – from compressible to incompressible – explicitly in a biphasic model [41]. In this study, we relied instead on the theoretical principle that complete loss of interstitial fluid content only occurs after an infinite amount of time. To minimize the chance of complete compaction due to numerical round-offs in a discretized time domain analysis, we employed a porous neo-Hookean model whose constitutive model uses the strain energy density,

$$W=\frac{\mu }{2}\left({\overline{I}}_1-3\right)-\mu \ln \overline{J},\overline{J}=\frac{J-{\phi }_r^s}{1-{\phi }_r^s}$$ (3)

where $\mu$ is the shear modulus of the porous solid matrix and ${\overline{I}}_1=\mathrm{tr}\overline{\mathit{C}}$, with $\overline{\mathit{C}}={\overline{\mathit{F}}}^T\cdot \overline{\mathit{F}}$ and $\overline{\mathit{F}}={(\overline{J}/J)}^{1/3}\mathit{F}$. This constitutive model lets the strain energy density $W$ tend to infinity as $J$ reduces to its minimum value ${\phi }_r^s$ (i.e., as $\overline{J}$ goes to zero), thus combining with the strain-dependent hydraulic permeability to produce numerical constraints that minimize the chance of achieving $J<{\phi }_r^s$ due to numerical round-offs [42].

The cell membrane was assigned a thickness of 10 nm [43]. Its elasticity was modeled with a Mooney-Rivlin solid with a shear modulus of 0.5 kPa (comparable to the cytoplasm modulus) and bulk modulus of 2.5 MPa, the latter to ensure the membrane would not be crushed by the transmembrane fluid pressure gradient. Its hydraulic permeability was set to 7 × 10⁻¹⁰ mm⁴ · N⁻¹ · s⁻¹ based on experimental measurements [13,29].

2.3. Boundary conditions and contacts

Two modeling conditions were created in this study, one which employed reciprocal sliding contact to reproduce physiological loading conditions realistically. The second model employed static contact conditions (without sliding). The justification for each of these models is provided in the Discussion below.

The substrates (the boundary surfaces that would normally interface with subchondral bone) of the cartilage strip and plug were modeled as rigid bodies. The degrees of freedom of the strip rigid substrate were all fixed. When modeling sliding contact, the plug rigid substrate had a prescribed reciprocal motion in the direction tangential to the strip articular surface, with a sawtooth profile producing 10 mm unidirectional travel in 100 s, with each reciprocal cycle requiring 200 s. A compressive load of 3.15 N was prescribed on the plug substrate, in the direction normal to the articular surface, to be consistent with the model of Zimmerman et al. [27] and the experimental study of Caligaris et al. [44]. This load was ramped up in 10 s then maintained constant for the duration of the finite element analysis. Symmetry conditions were enforced on the symmetry plane of the model (zero displacement and fluid flux normal to the symmetry plane). The remaining lateral boundaries of the biphasic strip and plug were set to free-draining (zero fluid pressure) and traction-free conditions. A sliding frictional biphasic contact interface [27] was employed between the articular surfaces of the plug and strip, with the equilibrium (drained) friction coefficient set to 0.201 and the solid-on-solid contact area fraction set to 0.0144, as explained in an earlier study [27]. These values were consistent with previously-reported experimental measurements of frictional contact between immature bovine articular layers [44]. To engage the finite element contact interface at the very first iteration of the first time step in this load-control configuration, the articular surfaces of the plug and strip were overlapped slightly, by ~0.08 mm. This was a necessary requirement to satisfy force balance from the start of the analysis, though the overlap resolved itself completely at the end of the first time step, due to the proper enforcement of the contact interface.

The container mesh domain was ‘attached’ to the rest of the cartilage strip using tied biphasic contact interfaces, which enforce continuity of solid matrix displacements and mixture tractions, and fluid pressure and normal flux across dissimilar meshes, to within a specified tolerance. The description of tied biphasic contact interfaces was provided in the recently-reported formulation of sliding frictional biphasic contact [27], where it was used to model the ‘stick’ phase of ‘stick-slip’ frictional contact. In this study, the tied biphasic interfaces used a scaled penalty parameter set to 1000 for the static contact models and 10 for the migrating contact (implying that the contact penalty scales the penalty values automatically calculated from the material properties of finite elements across the tied interface [27] ), with Lagrange augmentation used to enforce tied conditions within a gap (overlap) tolerance of 1 μm and a differential pressure tolerance of 1 kPa.

2.4. Finite element analyses

Finite element models were executed on a single node of a shared high-performance computing cluster, model Dell 6420 with dual Intel Xeon Gold 6126 processors (2.6 GHz, 12 cores per CPU, total of 24 cores), with 192 GB of dedicated memory. The model was subjected to the following two loading conditions: Migrating Contact: The reciprocal sliding analysis was prescribed to run for 1 h of loading. The computational analysis completed 1792 timesteps before terminating prematurely due to exceeding the requested maximum number of reformations, completing a simulation of 1582 s (24 min). It ran for 64 h and 42 min (wall clock). Static Contact: This analysis was set to run for 8 h of static compressive loading. The computational analysis completed 244 time steps before terminating prematurely after 10,846 s (181 min or ~3 h) due to element inversion produced by excessive localized compaction. It ran for 33 h and 37 min (wall clock).

3. Results

The deformed mesh of the migrating contact area model is presented in (Fig. 3A) after 2.5 min of loading. The corresponding contact pressure distribution is shown on the side of the strip, in (Fig. 3B). Plots of the peak and mean contact pressure magnitudes as a function of time are presented in (Fig. 3C) for both migrating and static contact conditions. The corresponding creep deformations (the vertical displacement of the plug as a function of time) are shown in (Fig. 3D). As interstitial fluid flowed out of the tissue, the fluid pressure in the cytoplasm decreased and the stresses were transferred to the solid matrix, exhibiting significant creep deformation (Fig. 3D). As the tissue deformed and contact area between the plug and strip increased, the contact pressure on the strip SZ decreased accordingly (Fig. 3C).

Fig. 3.

(A) Mesh deformation of migrating contact area model, at 0 and 2.5 min. (B) Corresponding contact pressure at 2.5 min; top: our model containing the SZ chondrocyte and surrounding PCM and container; bottom: model without chondrocyte. The discontinuity in contact pressure between the container and surrounding mesh in our model is due to the transition from the more compliant refined container mesh to the coarser mesh. The contact pressure on the cell container surface location is otherwise consistent between models. (C) Maximum and average contact pressure on the strip SZ. As the tissue deformed, the contact area between the cartilage plug and strip increased, decreasing the contact pressure. (D) Creep deformation of the cartilage tissue, or vertical displacement of the plug over time. As interstitial fluid flowed out of the tissue and stresses were transferred to the solid matrix, the pressure decreased and the tissue exhibited significant creep deformation.

The cell volume relative to its reference state ($t=0$) was recorded as a function of time (Fig. 4A). In both loading configurations, the cell volume decreased steadily throughout loading, following the same trend. After 24 min of migrating load, cell volume was reduced by 10% of its initial value (Fig. 4A); in contrast, the strip and plug respectively lost less than 0.7% and 1.9% of their initial volume. In the static contact model, after 30 min of loading, a cell volume loss greater than 20% and a PCM volume loss of ~10% were observed (Fig. 4B). After 3 h of static contact loading, the cell volume had reduced to ~46% of its initial value (~54% volume loss), indicating that almost all the cytoplasmic fluid had left the cell (the complete loss of cytoplasmic fluid would lead to a cell volume reduction to 40% of its initial value, based on ${\phi }_r^S=0.4$). The PCM volume decreased by ~21%, but notably remained higher in regions directly adjacent to the cell membrane (Fig. 4B). The strip and plug respectively lost 4.3% and 9.8% of their initial volume.

Fig. 4.

(A) Volume of the cytoplasm relative to its reference state ($t=0$) decreased over time. (B) Color-map visualization of the cell cytoplasm, membrane, and PCM compaction in the static model at time 0, 30 min, and 3 h. At 3 h of loading, the cell compacted to 46% of its initial volume, indicating almost all cytoplasmic fluid has left the cell. The PCM volume decreased by ~20%, but remained higher in regions directly adjacent to the cell membrane.

The fluid pressure in the cytoplasm decreased over time as fluid flowed out of the cell and the tissue (Fig. 5A). For the migrating contact model, the fluid pressure oscillated as it declined due to the intermittent sliding of the plug over the cell domain of the strip, reaching a maximum when the plug was directly over the cell, and dropping to zero when the plug was furthest from the cell. A comparison of the peak cytoplasmic fluid pressure (Fig. 5A) and peak contact pressure (Fig. 3C) shows that these values closely tracked each other for the first five minutes, then the contact pressure headed toward a steady-state non-zero value whereas the fluid pressure headed toward a steady-state zero value.

Fig. 5.

(A) Fluid pressure in the cytoplasm decreased over time. In the migrating contact model, the fluid pressure oscillated, reaching a maximum when the plug passes over the chondrocyte, and dropping to zero when the plug was furthest from the cell. (B) Fluid pressure differential across the cell membrane, calculated as the difference between the average fluid pressure in the cytoplasm and the PCM. In the static model, the difference reached a maximum value of 18.5 kPa at ~16 s after initiation of loading, then decreased progressively toward zero.

The fluid pressure drop across the membrane was calculated as the difference between the average fluid pressure in the cytoplasm and the PCM (Fig. 5B). In the static model, the difference reached a maximum value of 18.5 kPa at ~16 s after initiation of loading, then decreased progressively toward zero. At this time point the volumetric strain-dependent hydraulic permeability of the cytoplasm was still near its peak value (Fig. 6A), as was the transmembrane fluid flux (Fig. 6B). The pressure difference then rapidly dropped by five-fold over the next 500 s, approaching 0 kPa, consistent with declining fluid flux out of the chondrocyte.

Fig. 6.

(A) Log-log plot of hydraulic permeability of the cytoplasm over time, permeability remained at the zero-dilatation value for the first ~10 s of loading, then decreased as the cytoplasm was compacted. (B) Fluid flux out of the chondrocyte cytoplasm over time in the static contact model. The size and orientation of the white arrows correspond to the volume of fluid flowing and the direction it is flowing. The fluid flowed out rapidly, and reached a maximum at ~10 s; fluid flow then slowed down as the cell was compacted and the permeability dropped.

As the cytoplasm progressively became compacted, its hydraulic permeability decreased considerably (Fig. 6A). In both models, the initial (zero-volumetric-strain) permeability of $k_0=2\times {10}^{-3}{\text{mm}}^4\cdot {\text{N}}^{-1}\cdot {\text{s}}^{-1}$ was maintained for the first ~10 s of loading, at which point the migrating contact model had passed over the cell once. In the static contact model, permeability decreased by four orders of magnitude over the loading duration due to compaction, dropping below 6 × 10⁻⁶ mm⁴ · N⁻¹ · s⁻¹ as nearly all fluid exuded from the cell, leaving only the solid fraction of the cytoplasm.

Parametrically varying the PCM modulus from the ECM modulus ($E_{Y-PCM}$) value of 306 kPa, down to 100, 50, and 25 kPa revealed no major changes in the trends or results discussed above (Fig. 1A–C). However, for all PCM moduli lower than the ECM modulus (25–100 kPa), fluid did exude from the chondrocyte more quickly, reaching volume equilibrium in less time (Fig. 7A), after about 2 h for all three cases. Cases with faster fluid volume loss also displayed decreased hydraulic permeability (Fig. 7B). The fluid pressure differential for $E_{Y-PCM}=306\text{kPa}$ also had the highest maximum pressure differential of 18 kPa, followed by the fastest decay to zero. Meanwhile, with $E_{Y-PCM}=25\text{kPa}$, the pressure differential reached a maximum of only 6 kPa, though it was maintained for the longest time (Fig. 7C).

Fig. 7.

(A) Volume of the cytoplasm relative to its reference state ($t=0$) decreased over time until mostly the solid volume fraction remains. The time to reach equilibrium was faster for the three cases with ${\text{E}}_{\text{Y-PCM}}<{\text{E}}_{\text{Y-ECM}}$. (B) Log-log plot of the hydraulic permeability of the chondrocyte over time. Permeability decreased faster for lower values of ${\text{E}}_{\text{Y-PCM}}$, as the cytoplasm was compacted. (C) The maximum fluid pressure differential across the cell membrane correlated positively with ${\text{E}}_{\text{Y-PCM}}$, while the decay of this differential correlated negatively with ${\text{E}}_{\text{Y-PCM}}$. (D) Cytoplasm fluid volume loss was not affected by lowering $k_{PCM}$. High $E_Y=306\text{kPa}$, Low $E_Y=100\text{kPa}$, High $k=4.71\times {10}^{-5}{\text{mm}}^4\cdot {\text{N}}^{-1}\cdot {\text{s}}^{-1}$, Low $k=4.71\times {10}^{-5}{\text{mm}}^4\cdot {\text{N}}^{-1}\cdot {\text{s}}^{-1}$. (E) Hydraulic permeability of the cytoplasm was not affected by lowering $k_{PCM}$. (F) Fluid pressure differential across the cell membrane was not affected by lowering $k_{PCM}$. Values in blue are the same model dataset between columns, for comparison.

Changing the PCM permeability $k_{PCM}$ between 4.71 × 10⁻⁵ mm⁴ · N⁻¹ · s⁻¹ (Low $k_{PCM}$) and 0.0011 mm⁴ · N⁻¹ · s⁻¹ (High $k_{PCM}$) did not significantly change the results found at $E_{Y-PCM}=100\text{kPa}$ (Low $E_Y$) and 306 kPa (High $E_Y$) (Fig. 7D–F).

In the migrating contact configuration, the effective friction coefficient ${\mu }_{eff}$, which was calculated as the average over each 100 s of the absolute value of the ratio of the tangential and normal forces on the plug, exhibited the previously-reported time-dependent increase due to progressive loss of interstitial fluid pressurization [45,46], as shown in (Fig. 8).

Fig. 8.

The effective friction coefficient of the migrating contact model exhibited the previously-reported time-dependent increase due to progressive loss of interstitial fluid pressurization.

4. Discussion

The objective of this study was to understand the mechanical environment of SZ chondrocytes in articular cartilage under load conditions that replicate moderate activities of daily living. The longer-term objective of this investigation was to provide a cogent explanation for experimental observations of loss of viability of SZ chondrocytes under normal loading conditions [4–6]. Here, we explored the hypothesis that SZ chondrocytes may get significantly compacted under sustained contact loading of articular layers, leading to significant transport hindrance of intracellular fluid. We adopted a finite element modeling approach to facilitate this investigation, since experimental approaches for obtaining these types of results in situ would be exceedingly complex, if not impossible, given the large deformations involved under physiologic loads.

Results of this study showed that loading articular cartilage under physiological conditions and durations, representative of common activities of daily living such standing and walking, resulted in significant reduction in the volume of SZ chondrocytes (Fig. 4), producing cell compaction and considerable reduction in the fluid transport properties (Fig. 6). Within 3 h of sustained loading under low contact pressures, the SZ chondrocyte reduced to 46% of its initial volume and the cytoplasmic hydraulic permeability decreased by more than three orders of magnitude. Since the non-fluid content of chondrocytes was taken to be ${\phi }_r^S=40\%$ of the cell’s referential volume based on our earlier experimental studies [13], a reduction of the cell’s volume to 46% of its initial value implies that the interstitial fluid content reduced from 60% to 6% (a 90% reduction). Though experimental results of the in situ compaction of superficial zone chondrocytes are not currently available, several experimental studies have examined fluid loss in chondrocytes using osmotic loading [13,37–39], demonstrating that chondrocytes can indeed lose a large fraction of their interstitial water. For example, Xu et al. used NaCl at 1290 mOsm to sustainably reduce the chondrocyte volume to 52% of its initial (isotonic) volume. These authors also reported a solid volume fraction of 41% of the initial cell volume. Therefore, the cell fluid volume fraction reduced from 100% − 41% = 59% to 52% – 41% = 11%, implying an 81% reduction in fluid volume. Based on these experimental findings, it is reasonable to expect that mechanical loading of the chondrocyte can produce large interstitial fluid losses, as observed in our computational study.

These results strongly suggest that cell viability would be lost under these circumstances, since normal cellular metabolic activity would be considerably hindered by this excessive and sustained loss of intracellular fluid. Indeed, in our earlier investigation of cell viability under compressive loading of immature bovine cartilage [12], loss of SZ chondrocyte viability was much more pronounced after 12 h than after 6 h following load application. Another study by Rana et. al has shown that hyperosmotic-induced skrinkage can cause macromolecular crowding, possibly leading to apoptosis under sustained periods of time [47]. Thus, results from this study, when also interpreted in the context of prior findings, suggest that SZ chondrocytes may lose viability when subjected to extended durations of loading within a day’s activities.

In this study, migrating contact with friction and static loading of cartilage-on-cartilage were chosen as circumstances representative of physiologic joint articulation. Static loading is representative of activities such as standing or sitting, while migrating contact is representative of activities during which a joint is loaded intermittently, allowing some recovery time between loading and unloading phases of a sliding cycle. This could include shifting one’s weight back and forth while standing, slow walking, finger typing, or other daily manual activities for upper extremity joints. The peak contact pressure observed in these finite element analyses was 0.55 MPa (Fig. 3), considered to be in the low range of contact pressures for activities of daily living, which normally range from 0.5 to 5 MPa [48]. Therefore, the finite element simulations of this study may be viewed as representative of activities such as loading of facet joints of the spine during sitting, loading of finger joints during typing, or loading of lower extremity joints for extended durations of standing or slow ambulation.

A discontinuity in the contact pressure can be seen along the interface between the container and the strip SZ (Fig. 3B). However, the chondrocyte has a characteristic size of 15 μm whereas the ‘cell container’ has a characteristic size of 380 μm (or cell height of 5 μm versus ‘cell container’ height of 60 μm). This greater-than-tenfold difference in size means that the chondrocyte cannot possibly perceive the conditions outside of the ‘cell container’ domain based on St-Venant’s principle. Furthermore, as shown in Fig. 3B, the contact pressure in our multiscale model is consistent with what would be found had we not replaced an element with our container element (Fig. 3B).

Several factors can influence the compaction time of SZ chondrocytes due to loss of interstitial fluid. In addition to the tissue material and transport properties, the most significant factor is the path length for the pressurized fluid to escape the loaded regions of the articular layers. In a static contact configuration this means that the size (radius) of the contact area influences the compaction time, with larger contact areas requiring longer times. In this study the contact radius increased from 2.7 to 5.0 mm over 3 h of loading. The characteristic time constant for the creep response of cartilage under static loading, extracted from a simple exponential fit, was on the order of ~91 min, implying that it would take more than 4 ½ h to reach 95% of creep equilibrium conditions. For comparison purposes, in the experimental study of Guilak et al. [28], loaded cartilage explants with a radius of 1.5 mm required only 20 min to produce steady-state values for their chondrocyte volume. These outcomes are self-consistent in the context of porous media mechanics and interstitial fluid transport. In other words, realistic physiologic loading conditions in articular joints require several hours of loading to produce significant cartilage deformation. Thus, the deleterious effects of cell compaction observed in these finite element analyses may not occur when joints are subjected to shorter durations of loading. Nevertheless, in normal activities of daily living it may not be uncommon to subject joints to moderate loading for several hours, as may be encountered in professional settings that require sustained manual labor, or extended times of sitting (as would occur for truck drivers or office workers).

From our prior studies, the orientation of collagen fibrils tangential to the articular surface in the SZ enhances fluid pressurization in this region under loading, and reduces the friction coefficient [45,49]. The results of this study reproduced this expected functional response, with high fluid pressurization producing a low effective friction coefficient (Fig. 8). An inverse relationship has been shown between the tensile stiffness of the cartilage matrix and the characteristic time constant for fluid to escape the tissue [49]. Since the tensile modulus is highest in the SZ of articular cartilage [50,51], the SZ may undergo a relatively more rapid loss of extracellular and intracellular fluid than the MZ and DZ. Thus, the structural attribute of the ECM which confers its functionally low frictional response may also be responsible for increasing the vulnerability of SZ chondrocytes to loss of their intracellular fluid. These observations are consistent with experimental observations that SZ chondrocytes die under normal physiologic daily loading, and must be replenished, likely by a supply of cells from the synovium [7–9].

In a previous related study [15], it was shown analytically that mechanical loading of a chondrocyte in situ can lead to much faster loss of cell volume and equilibration than loading of isolated chondrocytes (~1 h versus several days), when incorporating a semipermeable cell membrane with realistic hydraulic permeability assessed from osmotic loading experiments [13]. This differential temporal response was attributed to the greater pressure differential across the cell membrane, which is largely determined by the PCM stiffness. The computational results of the current study similarly confirmed that a larger pressure difference between the PCM and cell cytoplasm resulted in greater fluid flux magnitude out of the cell (Figs. 5B, 6B).

The results of the parametric study indicated that the lower modulus of the PCM allowed fluid to exude from the cytoplasm faster (Fig. 7A–C). Decreasing the permeability of the PCM by a factor of 100 did not have a notable effect on the equilibration time. Future work could further explore the role of the PCM properties by varying additional material properties and geometries. The PCM consists mainly of collagen type VI and a high concentration of proteoglycans [52], which contributes to an elevated charge density [53]. Incorporating these parameters into our analysis with a multiphasic model could elucidate the effect of this charge on intracellar fluid transport. Additional insights may be gained from the recent study of Sibole et al. who investigated the effect of PCM shape on strains experienced by SZ chondrocytes, and found that PCM asymmetry may have protective properties [24]. Other chondrocyte multiscale computational studies by Kazemi and Williams implemented a rounded, thinner PCM, between 0.5–2 μm around the chondrocyte [21,25]. Implementing a similar, more physiologically accurate PCM geometry could complement these other future efforts to investigate the role of additional PCM properties on intracellular fluid loss and cell death. However, from our parametric PCM study, it appears our results are not overly sensitive to PCM geometry, and so we would not expect our present conclusions to change.

A potential limitation of this computational analysis was the inability to compute the finite element model response to migrating contact for more than 24 min of loading. Though migrating contact is more representative of physiological loading than static contact, the computational challenge of solving this configuration was apparent from the disparity in the wall clock solution times between the two models. However, since both contact configurations led to consistent changes in cell volume, contact pressure, creep deformation, fluid pressure, and permeability (Figs. 3–5), it is reasonable to assume that the conditions achieved under static contact loading after 3 h would be reproduced under migrating contact at later times, while remaining consistent with loading durations for activities of daily living (~8 h). This is due to the ability of the cell in the migrating contact model to briefly recover its volume when not directly under the load. Further, with significant fluid volume loss, some finite elements become thin enough that nodal load increments during time stepping may cause element inversion, causing a model to terminate prematurely. In principle this could be overcome by employing even shorter time increments; however, this would require even longer wall clock solution times.

Another potential limitation of the current study is that we employed exclusively a biphasic representation of the chondrocyte, PCM and the ECM of cartilage. A multiphasic analysis that can account for the charged nature of proteoglycans and ions of the interstitial fluid would require a considerably more elaborate set up. Fortunately, we have illustrated this type of analysis in a recent study [29], where the chondrocyte membrane was only permeable to the chloride ion, whereas the PCM and ECM were ascribed distinct material properties and fixed charge densities. That analysis allowed us to explore the transient evolution of the cell and tissue volume, osmolarities, and electric potential resulting from the compression of the cartilage matrix. Those results demonstrated that intracellular and extracellular steady-state responses could be achieved after approximately one hour of sustained compressive loading of the ECM. Therefore, we expect that a multiphasic analysis would not substantially change the conclusions of the current study.

An additional limitation is that we did not account for the ability of chondrocytes to actively regulate their volume in response to the contact loading conditions considered here. However, in an earlier study [54] we showed that chondrocytes subjected to hyperosmotic loading at 580 mM at body temperature initially lose 20% of their initial volume to osmotically driven fluid exudation. After 24 h of sustained osmotic loading, they only recover 5% of their volume via regulatory volume increase. Therefore, regulatory volume changes may not be very significant in chondrocytes, justifying our assumption to not account for this effect in our simulations.

In this study, we did not choose to examine the magnitude of stresses in the chondrocyte or its PCM to determine its viability, because there is currently no reliable literature data on the influence of those stresses on cell survival. Furthermore, since we focused on mechanisms associated with cell apoptosis rather than cell necrosis, there was no effort to assess whether the cell membrane could be ruptured mechanically as a result of articular contact loading. Indeed, it has been shown that the chondrocyte membrane has ruffles that can unfurl to expand the cell surface area during volume expansion, by a considerable amount without stretching it [37,55]. These findings suggest that membrane rupture would require a much larger areal strain expansion than observed in the current study. Therefore, our evaluation of finite element results focused primarily on interstitial fluid volume loss and transport inhibition.

The drastic decrease in hydraulic permeability of the cell cytoplasm and the creep deformation of the soft SZ also support the hypothesis that compressive loading may reduce the diffusivity of nutrients from the synovial fluid to the cell [56]. Durney et al. [57] demonstrated that culturing explants with physiologic levels of the metabolic mediators found in synovial fluid, including glucose, cortisol, insulin, and ascorbic acid, is critical in maintaining tissue homeostasis. Furthermore, Cigan et al. [58] showed that insulin and ascorbic acid are essential for chondrocyte viability in engineered construct culture, and that glucose has a dose-dependent effect on cell metabolism and proliferation. Constructs cultured in low or no glucose media exhibited lower DNA content, indicating some cell death or lack of cell proliferation, but were not separated by zonal subpopulation. Together these results support our premise that access to nutrients at the proper dosage is critical to maintaining cell function. The orders-of-magnitude decrease in hydraulic permeability observed in the static loading model after 3 h could very reasonably prevent transport of nutrients from synovial fluid or culture media into a cell, or hinder intracellular signaling processes triggered by ligand-membrane binding events, which could potentially cause SZ cell death. DZ and MZ chondrocytes may be less sensitive to changes in nutrient transport, as they are accustomed to low nutrient access and may be more phenotypically mature. Thus, the results of this study provide a cogent hypothesis for the normal occurrence of SZ chondrocyte death reported in prior in vitro and in vivo studies [4–6].

Based on these findings, we may design future experimental studies that investigate the specific hypothesis that SZ chondrocytes are vulnerable to loss of intracellular fluid under sustained durations, perhaps comparing their response to that of MZ and DZ chondrocytes. If needed, these experimental studies may be complemented by additional finite element analyses, which may include multiphasic transport analyses of various electrically neutral or charged solutes and their chemical interactions with the cell membrane and intracellular environment, as reported in our recent study [29].

Statement of significance.

As with any mechanical system, normal ‘wear and tear’ of cartilage tissue lining joints is expected. Yet incidences of osteoarthritis are uncommon in individuals younger than 45. This counter-intuitive observation suggests there must be an intrinsic repair mechanism compensating for this wear and tear over many decades of life. Recent experimental studies have shown superficial zone chondrocytes die under physiologic loading conditions, suggesting that this repair mechanism may involve cell replenishment. To better understand the mechanical environment of these cells, we performed a multiscale computational analysis of articular contact under loading. Results indicated that normal activities like walking or standing can induce significant loss of intracellular fluid volume, potentially hindering metabolic activity and fluid transport properties, and causing cell death.

Abbreviations:

OA

osteoarthritis

SZ

superficial zone

MZ

middle zone

DZ

deep zone

ECM

extracellular matrix

PCM

pericellular matrix