An axisymmetric boundary element model for determination of articular cartilage pericellular matrix properties in situ via inverse analysis of chondron deformation

Abstract

The pericellular matrix (PCM) is the narrow tissue region surrounding all chondrocytes in articular cartilage and, together, the chondrocyte(s) and surrounding PCM have been termed the chondron. Previous theoretical and experimental studies suggest that the structure and properties of the PCM significantly influence the biomechanical environment at the microscopic scale of the chondrocytes within cartilage. In the present study, an axisymmetric boundary element method (BEM) was developed for linear elastic domains with internal interfaces. The new BEM was employed in a multiscale continuum model to determine linear elastic properties of the PCM in situ, via inverse analysis of previously reported experimental data for the three-dimensional morphological changes of chondrons within a cartilage explant in equilibrium unconfined compression (Choi, et al., 2007, "Zonal Changes in the Three-Dimensional Morphology of the Chondron Under Compression: The Relationship Among Cellular, Pericellular, and Extracellular Deformation in Articular Cartilage," J. Biomech., 40, pp. 2596-2603). The microscale geometry of the chondron (cell and PCM) within the cartilage extracellular matrix (ECM) was represented as a three-zone equilibrated biphasic region comprised of an ellipsoidal chondrocyte with encapsulating PCM that was embedded within a spherical ECM subjected to boundary conditions for unconfined compression at its outer boundary. Accuracy of the three-zone BEM model was evaluated and compared with analytical finite element solutions. The model was then integrated with a nonlinear optimization technique (Nelder-Mead) to determine PCM elastic properties within the cartilage explant by solving an inverse problem associated with the in situ experimental data for chondron deformation. Depending on the assumed material properties of the ECM and the choice of cost function in the optimization, estimates of the PCM Young's modulus ranged from approximately 24 kPa to 59 kPa, consistent with previous measurements of PCM properties on extracted chondrons using micropipette aspiration. Taken together with previous experimental and theoretical studies of cell-matrix interactions in cartilage, these findings suggest an important role for the PCM in modulating the mechanical environment of the chondrocyte.

Figure 1

Confocal microscopy images of in situ chondron deformation within the ECM of a cylindrical porcine articular cartilage explant under static unconfined compression, adapted from Choi et al. (2007) [with permission]. (left) cartilage explants were fixed at 0% and 10% compression and immunolabeled for type VI collagen to quantify the boundaries of the chondrocyte and PCM, (right) 3D reconstructions were made of chondrons in control and compressed tissue explants.

Figure 2

Illustration of two-scale axisymmetric continuum model of in situ deformation of chondrons in the middle zone of a cylindrical cartilage explant subjected to static unconfined compression. (a) A single chondron is located at 50% depth (middle zone), along the symmetry axis of a cylindrical explant. (b) The unconfined compression analytical solution for a homogeneous cylinder of ECM is used to formulate boundary conditions for a three-zone (cell-PCM-ECM) interface problem at the microscopic scale of the chondrocyte.

Figure 3

The local environment of a single chondron is represented as a three-zone domain in an axisymmetric BEM model of coupled cell-PCM-ECM deformation in articular cartilage.

Figure 4

Verification of the three-zone BEM model in the case ν_ECM = ν_PCM = ν_Cell = 0.2 and E_ECM = E_PCM = E_Cell = 2.0kPa by comparison to known analytical solutions from linear elasticity theory. Deformed displacement profiles for the BEM model (circles) are compared to each analytical solution (solid lines) on the cell-PCM and PCM-ECM interfaces for: (a) plane stress, (b) plane strain, (c) confined compression, and (d) unconfined compression.

Figure 5

Illustration of convergence for the BEM model when an identical number of elements are used on I_CP, I_PE and Γ_E. Errors were calculated as a least squares sum comparing radial (a) and axial (b) deformed coordinates of the BEM model at all nodes of I_CP to corresponding values for the unconfined compression analytical solution.

Figure 6

Verification of the three-zone BEM model against an independent finite element solution. Deformed profiles for the BEM simulations (solid circles) and finite element simulations (open circles) are compared along the cell-PCM and PCM-ECM interfaces for the cases of macroscopic: (a) plane stress, (b) plane strain, (c) confined compression, and (d) unconfined compression. Note that the undeformed chondron was centered at z/h= 0.5 and its deformed shape is a combination of an axial translation and deformation.

Figure 7

BEM simulations of deformed chondron shape for estimated parameter values in the case of one parameter optimization using the cost function of Eq. 11. Computational predictions (dots) are compared to deformed (black line) and undeformed (grey line) shapes of the chondron based on Table 1 for the cases of Table 2: (a) Case I, (b) Case II, and (c) Case III.

Figure 8

BEM simulations of deformed chondron shape for estimated parameter values in the case of two parameter optimization using the cost function of Eq. 11. Computational predictions (dots) are compared to deformed (black line) and undeformed (grey line) shapes of the chondron based on Table 1 for the cases of Table 3: (a) Case IV, and (b) Case V.

Figure 9

BEM simulations of deformed chondron shape for estimated parameter values in the case of two parameter optimization using the cost function of Eq. 12. Computational predictions (dots) are compared to deformed (black line) and undeformed (grey line) shapes of the chondron based on Table 1 for the cases of Table 4: (a) Case VI, and (b) Case VII.