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TECHNICAL PAPERS: Soft Tissue

# Dependence of Mechanical Behavior of the Murine Tail Disc on Regional Material Properties: A Parametric Finite Element Study

[+] Author and Article Information

Orthopaedic Mechanobiology Laboratory, Department of Mechanical Engineering, Graduate Program in Bioengineering, University of Maryland, College Parkhsieh@umd.edu

Diane R. Wagner, Louis Y. Cheng, Jeffrey C. Lotz

Orthopaedic Bioengineering Laboratory, Department of Orthopaedic Surgery, University of California, San Francisco

J Biomech Eng 127(7), 1158-1167 (Jun 08, 2005) (10 pages) doi:10.1115/1.2073467 History: Received March 15, 2004; Revised June 08, 2005

## Abstract

In vivo rodent tail models are becoming more widely used for exploring the role of mechanical loading on the initiation and progression of intervertebral disc degeneration. Historically, finite element models (FEMs) have been useful for predicting disc mechanics in humans. However, differences in geometry and tissue properties may limit the predictive utility of these models for rodent discs. Clearly, models that are specific for rodent tail discs and accurately simulate the disc’s transient mechanical behavior would serve as important tools for clarifying disc mechanics in these animal models. An FEM was developed based on the structure, geometry, and scale of the mouse tail disc. Importantly, two sources of time-dependent mechanical behavior were incorporated: viscoelasticity of the matrix, and fluid permeation. In addition, a novel strain-dependent swelling pressure was implemented through the introduction of a dilatational stress in nuclear elements. The model was then validated against data from quasi-static tension-compression and compressive creep experiments performed previously using mouse tail discs. Finally, sensitivity analyses were performed in which material parameters of each disc subregion were individually varied. During disc compression, matrix consolidation was observed to occur preferentially at the periphery of the nucleus pulposus. Sensitivity analyses revealed that disc mechanics was greatly influenced by changes in nucleus pulposus material properties, but rather insensitive to variations in any of the endplate properties. Moreover, three key features of the model—nuclear swelling pressure, lamellar collagen viscoelasticity, and interstitial fluid permeation—were found to be critical for accurate simulation of disc mechanics. In particular, collagen viscoelasticity dominated the transient behavior of the disc during the initial $2200s$ of creep loading, while fluid permeation governed disc deformation thereafter. The FEM developed in this study exhibited excellent agreement with transient creep behavior of intact mouse tail motion segments. Notably, the model was able to produce spatial variations in nucleus pulposus matrix consolidation that are consistent with previous observations in nuclear cell morphology made in mouse discs using confocal microscopy. Results of this study emphasize the need for including nucleus swelling pressure, collagen viscoelasticity, and fluid permeation when simulating transient changes in matrix and fluid stress/strain. Sensitivity analyses suggest that further characterization of nucleus pulposus material properties should be pursued, due to its significance in steady-state and transient disc mechanical response.

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## Figures

Figure 4

Preferential compaction of the solid matrix at the periphery of the nucleus pulposus (along both endplate and inner annulus borders) as evidenced by (a) spatial gradients in porosity at the 2h time point during 0.8MPa axial compression. Time-course plots of porosity at various points in the nucleus along the (b) axial and (c) radial directions illustrate the rates at which matrix is compacted. Letters in the plot legends refer to the points along the disc axis and midheight illustrated in (a).

Figure 1

A finite element mesh was generated based on the morphology of the mouse tail disc. (a) Histologic HBQ-stained section near the midsagittal plane of the mouse tail disc. Images of the finite element mesh (b) overlaid on the histologic section to illustrate correspondence with disc morphology and (c) schematically diagrammed for clarity of the various tissues simulated.

Figure 3

Comparison between finite element predictions and experimental values of absolute and normalized compressive displacements for mouse discs subjected to 20min creep loading. (a) Absolute displacements were in excellent agreement for mouse discs loaded at 0.8MPa. (b) Normalized displacements, which accentuated the transient response of discs during creep, likewise exhibited excellent agreement for 0.8MPa stress. Labels tFEM and texp refer to the time taken to reach 75% of endpoint displacement (after 20min) for the model and experiments, respectively. Shaded areas represent mean experimental values ±1 S.D.

Figure 5

Sensitivity of mechanical response to varying stiffness of (a) the nucleus and (b) the annulus. The four outcome measures to assess mechanical response were most sensitive to annulus stiffness changes. In the nucleus, only when stiffness exceeded that of the annulus was the model’s response affected.

Figure 6

Sensitivity of time to reach 0.75 of total displacement, t0.75 to soft tissue permeability. The other three outcome measures (compressive displacement, annular dilatational stress, octahedral shear strain) were not at all affected by variations in permeability. However, because permeability of the endplate was high relative to that of the nucleus and annulus, endplate changes had no effect on t0.75 within this range.

Figure 7

Finite element analysis demonstrated that omission of swelling pressure from the nucleus results in (a) prolonged creep behavior and higher levels of compressive strain, together with (b) decreased bulging of the disc. These observations are due to the decreased tendency for nucleus elements to expand, a phenomenon that functions to pressurize the disc and support axial loads.

Figure 2

Stress-strain curves for a denucleated disc in tension and compression illustrating agreement of FEM predictions within one standard deviation of in vitro mechanical testing. Both experimental and finite element simulation were performed at slow strain rates of 0.0003mm∕min.

Figure 8

To assess the contributions of collagen viscoelasticity and poroelasticity to the transient response of the disc to creep compression, separate FE simulations of discs with linearly elastic collagen and with no-flow boundary conditions were compared with the validated baseline model. While viscoelasticity clearly inhibited strain rate in the initial portions of creep, the role of fluid permeation eventually exceeded that of viscoelasticity in prolonging creep. This point was determined to occur approximately 2200s after load application.

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