Three-dimensional finite element modeling of pericellular matrix and cell mechanics in the nucleus pulposus of the intervertebral disk based on in situ morphology

Abstract

Nucleus pulposus (NP) cells of the intervertebral disk (IVD) have unique morphological characteristics and biologic responses to mechanical stimuli that may regulate maintenance and health of the IVD. NP cells reside as single cell, paired or multiple cells in a contiguous pericellular matrix (PCM), whose structure and properties may significantly influence cell and extracellular matrix mechanics. In this study, a computational model was developed to predict the stress-strain, fluid pressure and flow fields for cells and their surrounding PCM in the NP using three-dimensional (3D) finite element models based on the in situ morphology of cell-PCM regions of the mature rat NP, measured using confocal microscopy. Three-dimensional geometries of the extracellular matrix and representative cell-matrix units were used to construct 3D finite element models of the structures as isotropic and biphasic materials. In response to compressive strain of the extracellular matrix, NP cells and PCM regions were predicted to experience volumetric strains that were 1.9-3.7 and 1.4-2.1 times greater than the extracellular matrix, respectively. Volumetric and deviatoric strain concentrations were generally found at the cell/PCM interface, while von Mises stress concentrations were associated with the PCM/extracellular matrix interface. Cell-matrix units containing greater cell numbers were associated with higher peak cell strains and lower rates of fluid pressurization upon loading. These studies provide new model predictions for micromechanics of NP cells that can contribute to an understanding of mechanotransduction in the IVD and its changes with aging and degeneration.

Fig. 1

Registered 3D solid geometries in tetrahedron meshes in the nucleus pulposus. Models include different cell–matrix unit (CMU) subgroups: 1 cell CMUs 1 and 2; 2 cell CMUs 3 and 4; and 3 or 4 cell CMUs 5 and 6. For clarity, only meshes on the surfaces of the pericellular matrix and cells are shown above (not shown on equivalent scales)

Fig. 2

Average fluid pressure and volumetric strain at multiple material domains in the nucleus pulposus. The average fluid pressures in the PCM domain a in the 1 cell, 2 cell, and 3 or 4 cell cell–matrix unit subgroups showed similar temporal responses with negative pressures throughout time, while the overall fluid pressures in the cell domain b showed a period of positive fluid pressure characteristic of flow reversal after some time of loading. The pressurization rate was lower in the cell domain and decreased with increasing cell numbers in one PCM. Comparisons of average volumetric strain among the PCM c and cell d domains in the 1 cell, 2 cell, and 3 or 4 cell cell–matrix unit subgroups showed that the average volumetric strain in the extracellular matrix (−0.006) was amplified in the PCM domain (∼ −0.01) and furthermore in the cell domain (−0.007–0.021) with a value depending on the cell number and relative position within the PCM. Open and closed signs represent two different models in the same cell–matrix unit subgroup

Fig. 3

Temporal responses of volumetric strain at multiple material domains in the nucleus pulposus. High strain concentration (generally with positive values) was seen near the cell/PCM interface and decreased over time. Similar trends were seen among 1 cell a, 2 cell b, and 3 or 4 cell c cell–matrix unit subgroups

Fig. 4

Deviatoric strain at multiple material domains in the nucleus pulposus at equilibrium. The deviatoric strain in the cell domain was 25% lower than that of the extracellular matrix. The strain concentrations were seen at the PCM/extracellular matrix interface, mainly along the long axis of the PCM

Fig. 5

Effective von Mises stress at multiple material domains in the nucleus pulposus. The stress was highly heterogeneous in the PCM, when compared to the extracellular matrix and cells (viewed in YZ plane at x = 0). The stress concentration (up to ∼2,000 Pa) was mainly at the cell/PCM interface and significantly higher than the average value in the PCM (∼800 Pa)

Fig. 6

Average volumetric strain of individual cells enclosed in one PCM containing multiple cells in the nucleus pulposus. Similar responses were seen for all cells, but the magnitude of strain can be different for individual cells in both the 2 cell a and b and 3 or 4 cell c and d cell–matrix unit subgroups