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TECHNICAL PAPERS: Bone/Orthopedic

The Micromechanical Environment of Intervertebral Disc Cells Determined by a Finite Deformation, Anisotropic, and Biphasic Finite Element Model

[+] Author and Article Information
Anthony E. Baer, Lori A. Setton

Department of Biomedical Engineering, Duke University, Durham, NC 27708

Tod A. Laursen

Department of Civil and Environmental Engineering, Duke University, Durham, NC 27708

Farshid Guilak

Division of Orthopaedic Surgery, Department of Surgery, Duke University Medical Center, Durham, NC 27710

J Biomech Eng 125(1), 1-11 (Feb 14, 2003) (11 pages) doi:10.1115/1.1532790 History: Received April 01, 2002; Revised September 01, 2002; Online February 14, 2003
Copyright © 2003 by ASME
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Figures

Grahic Jump Location
Example confocal images from each anatomic zone. Cells from the anulus fibrosus and transition zone were elongated in oriented clusters, with a small number of cells exhibiting short cytoplasmic projections. Cells from the nucleus pulposus were larger, highly vacuolated, and found in closely packed clusters. Scale bar: 10 μm.
Grahic Jump Location
Example of a three-dimensional reconstruction showing the best-fit ellipsoid. Data are shown in the three principal planes of the best-fit ellipsoid and in perspective. Dots indicate the coordinates of the surface polygon vertices from the three-dimensional reconstruction of an anulus fibrosus cell; bold lines indicate the projections of the best-fit ellipsoid onto the principal planes. The radii of the best-fit ellipsoid (R1,R2,R3) were used to characterize the geometry of the cells. For the example shown, R1=12.1 μm,R2=6.7 μm,R3=3.8 μm. The total sum of squared error for this fit was 2015.4 μm2 , with an average of 0.216 μm2 per data point.
Grahic Jump Location
Equilibrium stress-stretch curves for the extracellular matrix. Curves show the axial component of the total stress tensor (σzz) vs. the applied axial stretch (λz) at equilibrium. Material properties were selected from previous measurements (Table 2). The tension-compression nonlinearity of the anulus fibrosus and transition zone was simulated by including contributions from the fiber term (F2) in the strain energy function (ψs) only when λz>0; this lead to dramatically higher stresses in tension compared to compression.
Grahic Jump Location
The multi-scale, axisymmetric finite element mesh. A macro-scale mesh simulated the stress-relaxation behavior of a cylindrical tissue sample in tension. The fiber direction was defined by the vector a and coincided with the z-axis. The micro-scale mesh simulated a cell and its surrounding extracellular matrix within a small region at the center of the tissue sample. The reference cell geometry was varied by adjusting the aspect ratio of the cells (r1/r2). To couple the two meshes, the time-history of the displacements and pressures from the macro-scale mesh were applied as boundary conditions for the micro-scale mesh.
Grahic Jump Location
Mean values of the cell aspect ratios. Results show the aspect ratios of the best-fit ellipsoid. Mean±SD;n=45–49.* Significantly different from anulus fibrosus values (P<0.05, ANOVA).
Grahic Jump Location
Volume changes in the vicinity of cells in the three anatomic zones. Cell aspect ratios (r1/r2) were set to baseline line values (Table 1). Color maps show the deformed volume normalized to the reference volume, computed by the determinant of the deformation gradient tensor (J=det F=dV/dV0). Results are shown for a ramp-and-hold displacement with a ramp time of 100 s and a final axial stretch of λz=1.05.  
Grahic Jump Location
Stress and strains in the vicinity of the cell at equilibrium. Cell aspect ratios (r1/r2) were set to baseline line values (Table 1). Color maps show the values of (A) the axial component of the total stress tensor (σzz); (B) the mean isotropic total stress (σm=[σzzrrθθ]/3); (C) the axial component of the Eulerian finite strain tensor (ezz); and (D) the volume change (J). Values for stress (A and B) in the nucleus pulposus were scaled by a factor of 105; stresses in this region were actually on the order of 10 Pa.
Grahic Jump Location
Effect of cell geometry on equilibrium strain values. Color maps show equilibrium values of the axial component of the Eulerian finite strain tensor (ezz) for cells in the anulus fibrosus and transition zone. Cell aspect ratio (r1/r2) was set to (A) spherical values; (B) baseline values; (C) elongated values (Table 1).
Grahic Jump Location
Strains at equilibrium within cells of varying geometries. Plots show both maximum and minimum values within the cell for (A) the axial component of the Eulerian finite strain tensor (ezz) and (B) the volume change (J). As a reference, plots also show tissue values from the macro-scale mesh. For the anulus fibrosus and transition zone, three different cell geometries were used based on experimental measurements (Table 1).
Grahic Jump Location
Transient volume changes in cells of various geometries. Plots show both maximum and minimum values of the volume change (J). Results are shown for a ramp-and-hold displacement with a ramp time of 100 s and a final axial stretch of λz=1.05. As a reference, plots also show tissue values from the macro-scale mesh. For the anulus fibrosus and transition zone, three different cell geometries were used based on experimental measurements (Table 1).
Grahic Jump Location
Maximum strain concentration within the cell. Plot shows model predictions for the maximum value of ezz within a cell normalized to the far field value within the tissue (ezz0). Results are shown at equilibrium (λz=1.05) for different values of r1/r2. The open symbols represent the baseline values of r1/r2 (Table 1).

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