Figure 1.
The autonomous approach to couple joint, tissue, and cell scales is shown above.
Left: The method began with the FEA of a macro-scale model for a kinetic/kinematic scenario. Deformation gradients at the centroids of finite elements in tissue regions of interest were calculated from element nodal positions. Right: Displacement boundary conditions were prescribed to surface nodes of independent micro-scale models corresponding to the deformation gradient of each finite element experiencing deformation in tissue regions considered. Results of micro-scale FEA solutions were post-processed to calculate deformation metrics of interest.
Figure 2.
Macro and micro-scale models used in study.
A. A model of the tibiofemoral joint was employed at the macro-scale. Boundary conditions were prescribed for the bones, modeled as rigid bodies, to approximate compressive loading of the joint. The tibia was fixed in space. Femoral compression was prescribed at fixed flexion with other degrees of freedom unconstrained. B. The single (top) and 11 cell (bottom) models. The single cell model was the popular scenario considered in previous studies [39], while the 11 cell model better represented the cell densities observed in situ for the middle layer of articular cartilage in the knee [40].
Figure 3.
A finite element model of the tibiofemoral joint with representation of ACL, PCL, MCL, LCL, tibial and femoral cartilage, lateral and medial menisci, femur, and tibia.
Soft tissue structures were discretized with 56433 linear hexahedral finite elements with an average edge length of approximately . The bones were discretized with 25220 quadrilateral shell elements with an assigned thickness of
. An enlarged model region is shown on the right to illustrate the mesh resolution. The femoral and tibial cartilage each had 3 hexahedral elements through their thickness.
Table 1.
Material parameters for articular cartilage [30], ligaments [31]–[33], and meniscus [34]–[46].
Table 2.
Material parameters for micro-scale model [41].
Figure 4.
A cross-section of the 25889 finite element mesh of the single cell micro-scale case, which was determined to be mesh-converged.
The extracellular matrix (ECM), pericellular matrix (PCM), and chondrocyte (Cell) were discretized with linear hexahedral elements. The chondrocyte had 9 elements across its diameter and the pericellular matrix had 2 elements through its thickness. For the eleven cell case, the chondrocyte and pericellular matrix had the same mesh densities.
Figure 5.
Fringe plots showing macro-scale and micro-scale deformation metric regional distributions in the middle layer of cartilage of the femur (left) and tibia (right).
A. The effective strain which occurred at the macro-scale. B. The change in cellular aspect ratio (major-minor) calculated for the single cell model. C. The maximum change in cellular aspect ratio (major-minor) experienced by a cell in the 11 cell model.
Figure 6.
The change in cellular aspect ratio measured in each micro-scale model for the single cell (left) and 11 cell (right) cases was plotted as a function of macro-scale effective strain occurring at the corresponding finite element.
The solid lines were linear least-squares regressions performed with the requirement that the lines passed through the origin. The slopes of each of these lines were provided in the legend. In the 11 cell case, the line labeled “Max” considered only data from the cell which experienced the greatest change in aspect ratio. Likewise, the line labeled “Min” was the cell which experienced the least change in aspect ratio, and the “Avg” was the average of the linear regressions for all 11 cells. These slopes indicated the distribution of the 11 cell change in aspect ratios was skewed left.
Table 3.
Linear regression slopes and sum of squared residuals of single and 11 cell cases for change in cellular aspect ratio vs macro-scale effective strain.
Figure 7.
An example of the internal effective strain distributions observed in a single (top right) and 11 cell (bottom right) micro-scale solution for the same deformation gradient passed from the macro-scale (left).
A macro-scale finite element experienced a combined loading deformation gradient which resulted in a shape change from the gray cube to the blue hexahedron shown on the left. The single and 11 cell models were deformed in the same way with the resulting effective strain distributions in a cross-section of the model shown on the right. While the cells experienced qualitatively similar strains in both cases, the strain in the ECM around the cells in the 11 cell model differed from the single cell in regions of close cellular proximity. See Table 4 for cell-specific quantitative results.
Table 4.
Chondrocyte deformation metric set for a single and eleven cell model with boundary conditions prescribed via a specific deformation gradient passed from the macro-scale.
Figure 8.
The quartile analyses of the ratios of average cellular effective strain to extracellular matrix (ECM) effective strain for the four cases considered in the material sensitivity study of the 11 cell model are shown above.
In agreement with [41], scaling of material properties while maintaining ratio (Study and Study ×10) does not strongly impact chondrocyte deformation, but modifying the stiffness ratio between micro-scale components, i.e. by decreasing pericellular matrix stiffness by tenfold (Study PCM/10), does.
Figure 9.
The volume-averaged maximum shear strain (left) and volumetric strain (right) for the 11 cell model plotted against the macro-scale effective strain.
Deformation metrics such as those shown here, may have additional implications to understand cellular damage mechanisms as well as the onset of mechano-biological function, which change in aspect ratio may not provide. For example, pure cell dilation will result in a volumetric strain, but not a change in aspect ratio. The large bandwidth of the volumetric strain also suggests that a direct linear relationship between the macro- and micro-scales may not be acceptable for all characterizations of deformation.
Figure 10.
To investigate the potential error due to sampling of the deformation gradient at element centroids, 11 cell micro-scale models were solved for the eight Gaussian integration points of the macro-scale element determined to have the highest variation in deformation gradient.
A. Deformation gradients were calculated at the eight integration points of macro-scale element ID 20585. B. The micro-scale model boundary conditions, assigned for each integration point and identified in the image sequence, resulted in a range of deformations. C. The variation in deformation gradient associated with integration point sampling resulted in a variation in cellular deformation metrics. The quartile analysis of the resulting volume-averaged cellular effective strain and change in cellular aspect ratio is shown above with the value determined from the analysis performed at the macro-scale element centroid overlaid.