Pericellular Matrix Mechanics in the Anulus Fibrosus Predicted by a Three-Dimensional Finite Element Model and In Situ Morphology

Abstract

Anulus fibrosus (AF) cells have been demonstrated to exhibit dramatic differences in morphology and biologic responses to different types of mechanical stimuli. AF cells may reside as single cell, paired or multiple cells in a contiguous pericellular matrix (PCM), whose structure and properties are expected to have a significant influence on the mechanical stimuli that these cells may experience during physiologic loading of the spine, as well as in tissue degeneration and regeneration. In this study, a computational model was developed to predict the micromechanical stimuli, such as stress and strain, fluid pressure and flow, of cells and their surrounding PCM in the AF tissue using three-dimensional (3D) finite element models based on in situ morphology. 3D solid geometries of cell-PCM regions were registered from serial confocal images obtained from mature rat AF tissues by custom codes. Distinct cell-matrix units were modeled with a custom 3D biphasic finite element code (COMSOL Multiphysics), and simulated to experience uni-axial tensile strain along the local collagen fiber direction. AF cells were predicted to experience higher volumetric strain with a strain amplification ratio (relative to that in the extracellular matrix) of ~ 3.1 - 3.8 at equilibrium, as compared to the PCM domains (1.3 - 1.9). The strain concentrations were generally found at the cell/PCM interface and stress concentration at the PCM/ECM interface. Increased numbers of cells within a contiguous PCM was associated with an apparent increase of strain levels and decreased rate of fluid pressurization in the cell, with magnitudes dependent on the cell size, shape and relative position inside the PCM. These studies provide spatio-temporal information on micromechanics of AF cells in understanding the mechanotransduction in the intervertebral disc.

Figure 1

Geometry registration and meshing of the cell-matrix interaction model for a representative cell-matrix unit (CMU). (a) 3D reconstructed CMU; (b) 3D registered geometry of the CMU in tetrahedral element meshes, viewed in YZ plane; (c) Elliptical surface contours from multiple slices were stacked into 3D solid objects for the pericellular matrix and cell, respectively; (d) The CMU in tetrahedral element meshes under 3D iso-view; (e) The 3D geometry model for finite element modeling including three sub-domains – cell, pericellular and extracellular matrix; (f) The 3D geometry model in tetrahedral element meshes under 3D iso-view. For clarity, only meshes on the outer surfaces of the sub-domains are shown here.

Figure 2

Registered 3D solid geometries of the pericellular matrix and cell(s) in the annulus fibrosus (AF) in tetrahedral element meshes. Examples shown are from the inner AF (1, 3, 5) and outer AF (2, 4, 6), including 1 cell (1&2), 2 cells (3&4), and 3+ cells cell-matrix unit subgroups (5&6). For clarity, only meshes on the pericellular matrix and cell surfaces are shown here (not shown on equivalent scales).

Figure 3

Equilibrium volumetric strains at multiple length scales shown for loading conditions of Case 1 and Case 2. Strain distribution was generally uniform in the extracellular matrix (ECM) and cells, but non-uniform in the pericellular matrix (PCM). Strain concentrations were observed on the cell/PCM interface. Individual cells residing in the same PCM may exhibit different magnitudes of strain amplification over values for the far-field ECM depending on cell size, shape and position within the PCM. Note that the color bars for Case 1 and Case 2 are in different ranges.

Figure 4

Temporal response of average volumetric strain at multiple length scales in Case 1. Comparisons between the pericellular matrix (PCM, a&b) and cell (c&d) domains in the 1 cell, 2 or 3+ cell cell-matrix unit subgroups showed that the average volumetric strain in the extracellular matrix (ECM, ~ 0.05) was amplified in the PCM domain (0.08 – 0.09), and furthermore in the cell domain (0.15 – 0.19). Values for average volumetric strain generally increased with the number of cells enclosed in one PCM in both inner (a&c) and outer (b&d) annulus fibrosus (AF) regions. Open and closed signs represent two different models in the same cell-matrix unit subgroup.

Figure 5

Temporal responses of deviatoric strain at multiple length scales in the outer annulus fibrosus (AF) for loading described in Case 1. A gradient of deviatoric strain was shown at early times with the highest value observed in the cell domain. Strain concentrations were seen mainly at the cell/pericellular matrix interface (up to ~ 0.10), but not at the pericellular/extracellular matrix interface.

Figure 6

Temporal responses of the effective von Mises stress in the pericellular matrix of the inner annulus fibrosus for loading described in Case 1. The stress was highly heterogeneous in the pericellular matrix and relaxed over the time. Stress concentrations were seen mainly at the pericellular/extracellular matrix interfaces (shown here).

Figure 7

Temporal responses of average fluid pressure in the pericellular matrix (PCM) and cell for loading conditions in Case 1. The average fluid pressures in the PCM (a&b) and cell (c&d) domains in the 1 cell, 2 or 3+ cell cell-matrix unit subgroups showed similar trends with the lowest fluid pressurization rate in the cell domain or in 3+ cell cell-matrix unit subgroups (See insets). The decay of fluid pressure with the onset of equilibrium occurred more rapidly for all cell-matrix unit models in the outer (b&d) annulus fibrosus (AF) than those in the inner AF (a&c) and exhibited a trend towards decreased pressure decay with increasing numbers of cells within one cell-matrix unit. Open and closed signs represent two different models in the same cell-matrix unit subgroup.

Figure 8

Temporal responses of the pressure gradient predicted for the outer annulus fibrosus for loading conditions of Case 1. A pressure gradient, indicating an overall inward fluid flow after tensile deformation existed in all sub-domains at early times. The pressure gradient was largely reduced in the extracellular matrix over time but was maintained for longer times in a narrow spatial region of the pericellular matrix (PCM), and primarily directed along the transverse directions. The spatial distribution of the pressure gradient varied within one PCM or cell, depending on the local PCM geometry and cell position. Arrows indicate the local direction of fluid flux.

Figure 9

Average fluid pressure in the pericellular matrix (PCM) and cell for the loading conditions of Case 2. A complex pressurization pattern was observed at the pericellular (a&b) and cellular (c&d) scales for both inner (a&c) and outer (b&d) annulus fibrosus (AF) model predictions. The time to peak fluid pressure was significantly greater in Case 2 loading as compared to Case 1, particularly noted for the inner AF. Similarly, the pressurization decay occurred more slowly as compared Case 1 models; predictions exhibited a similar trend for sustained pressures in the cell and PCM domains with increasing numbers of cells within one PCM. Open and closed signs represent two different models in the same cell-matrix unit subgroup. Note that the y-axis for the inner and outer AF was on different scales.