Fig 1.
Overview of the multiscale modeling approach.
(A) Two idealized models were constructed consisting of epiphyseal bone, growth plate (GP) cartilage and metaphyseal bone (about 7 x 7 x 7 mm) with variations of mammillary processes: flat and ‘m’ shaped. (B) Set-up of the multiscale modeling approach. At the macroscale level, quarter models were used for analysis. About 0.69 mm thick growth plate cartilage was partitioned into two sections to represent the reserve zone (RZ) and the proliferative/hypertrophic (P/H) zone. Calcified cartilage (CC) was also included in the macroscale model. At the mesoscale, three individual layers were generated in the P/H zone to represent the gradient change of elastic modulus through the thickness of the growth plate. The microscale model of the chondron consisted of interterritorial matrix (ITM), territorial matrix (TM) and 46 chondrocytes with gradually changing cellular shape along with the same number of ITM sections. The elastic modulus of ITM increased from the RZ to the metaphyseal side to represent the gradual change of its material properties.
Table 1.
Young’s modulus (E) and Poisson’s ratio (ν) of the growth plate components used in the multi-scale models.
Fig 2.
Determination of the elastic moduli of the growth plate utilized in the models.
(A) Stained histological slice (haematoxylin and eosin) of a 4-month old bovine growth plate; chondrocytes are dispersed in the reserve zone (RZ) and are stacked in tubes (chondrons) extending from the zone of proliferation (P) through the zones of maturation and hypertrophy (H). The walls of the chondron tubes (interterritorial matrix) become increasingly calcified toward the metaphyseal side. Following chondrocyte death, the walls of tubes or bars of calcified cartilage (CC) matrix form the scaffolding upon which bone is laid down (primary spongiosa); scale bar = 100 μm. (B) Distribution of the elastic modulus of the P/H zone (3-layer mesoscale and 10-layer mesocale models) and interterritorial matrix (ITM) (microscale model).
Fig 3.
Different levels of axial strain were experienced by different zones (layers) of the sample.
Although a compressive displacement equal to 20% of the growth plate thickness was applied to the top surface of the whole sample, the combined reserve zone and proliferative/hypertrophic (P/H) layers were subjected to engineering strains of only about 15%. Strains in the calcified cartilage (CC) zone reached about 3% strain.
Fig 4.
Distribution of the axial logarithmic strain and deformation of the chondrons.
(A) A non-uniform distribution of axial logarithmic strain, in the y-direction, exists within the ‘m’ shape growth plate in the mesoscopic model. Chondrocytes were not included in this model. The distribution of strain was significantly different at these locations from the central or internal regions of the growth plate and reached values approaching double the nominal engineering strain of 20%; (B) Deformed chondrons obtained from the microscale model with cellular detail is overlaid on a deformed macroscale model displaying only the growth plate. Chondrons buckled within 300 μm of free surfaces, but not within the growth plate interior when 20% overall compressive strain was applied across the macroscale model. Deformed models are not scaled up. The ‘m’ shape growth plate exhibited transverse outward buckling deformation of 125 μm at the edge. The contour plot is obtained by extrapolating the integration point values to the nodes and then averaging.
Fig 5.
Logarithmic strain vector plots at the center and at a plane 50 μm from one of the two free surfaces of the ‘m’ shape growth plate undergoing axial (Y-) compression of 20%.
The red and blue arrows represent strains in X and Z directions, respectively, at the integration points. The results were obtained from the mesoscale model where detailed chondrons are not included. Regions close to the outer edge experienced a significant level of transverse (X- and Z-) strains. These strains suggest that observations made near the surface would lead to different assessment of transverse outward strain distribution compared to the interior of the growth plate. The centerline represents the center of the full model, as a quarter of the actual growth plate layer is shown here (XY and YZ planes are symmetry planes).
Fig 6.
Principal logarithmic strain vector plots at center and at 50 μm from one of the free surfaces of the ‘m’ shape growth plate.
The arrows represent the maximum and minimum principal strains. The results were obtained from the mesoscale model without the details of chondrons. Regions close to the outer edges experienced a significant level of ‘out-of-plane’ strain. The free surfaces caused significant changes in the principal strains near the epiphyseal bone border. Tensile strains were greatest at this border and in the hypertropic zone at the calcified cartilage border and smallest at the base of the proliferative zone where cell division takes place. The centerline represents the center of the full model, as a quarter of the actual growth plate layer is shown here (XY and YZ planes are symmetry planes).
Fig 7.
Distribution of minimum principal logarithmic strains within the ‘m’ shape growth plate obtained from the mesoscale model without cellular detail.
Peak compressive strains occurred at the base of the proliferative zone where cell mitosis occurs. This contrasts with regions close to the free surface, where compressive strains were at a minimum around the base of the proliferative zone and maximum in the reserve and hypertrophic zones.
Fig 8.
Chondrocyte deformation measured as change in cell height at two locations in the chondron for an overall axial compressive strain of 20%.
(A) Cellular strains vary differently within each chondron depending on the location of the chondron. Within interior chondrons compressive axial cell strains peaked at the transition from reserve to proliferative zone where cell division occurs. Chondrons located close to the free surface (300 μm from the surface) of the growth plate experienced a reverse strain pattern with peak compressive strains occurring in the hypertrophic zone where no cell division occurs, but where cell size increases 10-fold. (B) Contour plot of the logarithmic strain (LE) at the chondron scale in the y-direction at two locations, at the center (P4) and at 300 μm from the edge (P3). The contour plot is obtained by extrapolating the integration point values to the nodes and then averaging.
Fig 9.
Chondrocyte aspect ratio change with deformation at the center and 300 μm away from the edge.
Aspect ratio is defined as the ratio of the height over the width of the cells.
Table 2.
Comparison of mean computed strains to reported mean experimental strains measured with confocal microscopy on compressed growth plate explants.
Fig 10.
Results for two cases with the cellular Young’s modulus increased 20-fold over the instantaneous short time scale value, for cellular Poisson’s ratio = 0.50 and 0.49 in an effort to approximate long-time scale compression with a purely elastic approach to compare with the experimental values in the literature.
Results are very sensitive to assumed values of cellular Poisson’s ratio and to the Young’s modulus ratio between the cell and the tightly bound territorial matrix and we do not believe this is the correct approach for modeling long-time scale events since the real value of cellular Poisson’s ratio cannot be accurately determined.
Fig 11.
Chondrons appear to be oriented along the minimum principal strain directions, becoming roughly perpendicular to the epiphyseal bone plate, whereas the hypertrophic portion along with the calcified cartilage bars and primary spongiosa align more with the primary compressive load direction along the tibial long axis (Y).
(A) Histology slice of a 4-month old bovine proximal tibial growth plate stained with H&E, scale bar = 200 μm. (B) Minimum principal strain vector plot at the central region of the ‘m’ shape growth plate. The results were obtained from the mesoscale model without the cellular detail.