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Research Papers

The Importance of Intrinsic Damage Properties to Bone Fragility: A Finite Element Study

[+] Author and Article Information
M. R. Hardisty

Lawrence J. Ellison Musculoskeletal
Research Center,
Department of Orthopaedic Surgery,
University of California,
Davis, Sacramento, CA 95817;
Biomedical Engineering,
College of Engineering,
University of California,
Davis, Davis, CA 95616

R. Zauel

Bone and Joint Center,
Department of Orthopaedic Surgery,
Henry Ford Hospital,
Detroit, MI 48202

S. M. Stover

J. D. Wheat Veterinary Orthopedic
Research Laboratory,
School of Veterinary Medicine,
University of California,
Davis, Davis, CA 95616

D. P. Fyhrie

Lawrence J. Ellison Musculoskeletal
Research Center,
Department of Orthopaedic Surgery,
University of California,
Davis, Sacramento, CA 95817;
Biomedical Engineering,
College of Engineering,
University of California,
Davis, Davis, CA 95616
e-mail: dpfyhrie@ucdavis.edu

Contributed by the Bioengineering Division of ASME for publication in the Journal of Biomechanical Engineering. Manuscript received January 18, 2012; final manuscript received October 5, 2012; accepted manuscript posted November 28, 2012; published online December 27, 2012. Assoc. Editor: Sean S. Kohles.

J Biomech Eng 135(1), 011004 (Dec 27, 2012) (9 pages) Paper No: BIO-12-1019; doi: 10.1115/1.4023090 History: Received January 18, 2012; Revised October 05, 2012; Accepted November 28, 2012

As the average age of the population has increased, the incidence of age-related bone fracture has also increased. While some of the increase of fracture incidence with age is related to loss of bone mass, a significant part of the risk is unexplained and may be caused by changes in intrinsic material properties of the hard tissue. This investigation focused on understanding how changes to the intrinsic damage properties affect bone fragility. We hypothesized that the intrinsic (μm) damage properties of bone tissue strongly and nonlinearly affect mechanical behavior at the apparent (whole tissue, cm) level. The importance of intrinsic properties on the apparent level behavior of trabecular bone tissue was investigated using voxel based finite element analysis. Trabecular bone cores from human T12 vertebrae were scanned using microcomputed tomography (μCT) and the images used to build nonlinear finite element models. Isotropic and initially homogenous material properties were used for all elements. The elastic modulus (Ei) of individual elements was reduced with a secant damage rule relating only principal tensile tissue strain to modulus damage. Apparent level resistance to fracture as a function of changes in the intrinsic damage properties was measured using the mechanical energy to failure per unit volume (apparent toughness modulus, Wa) and the apparent yield strength (σay, calculated using the 0.2% offset). Intrinsic damage properties had a profound nonlinear effect on the apparent tissue level mechanical response. Intrinsic level failure occurs prior to apparent yield strength (σay). Apparent yield strength (σay) and toughness vary strongly (1200% and 400%, respectively) with relatively small changes in the intrinsic damage behavior. The range of apparent maximum stresses predicted by the models was consistent with those measured experimentally for these trabecular bone cores from the experimental axial compressive loading (experimental: σmax = 3.0–4.3 MPa; modeling: σmax = 2–16 MPa). This finding differs significantly from previous studies based on nondamaging intrinsic material models. Further observations were that this intrinsic damage model reproduced important experimental apparent level behaviors including softening after peak load, microdamage accumulation before apparent yield (0.2% offset), unload softening, and sensitivity of the apparent level mechanical properties to variability of the intrinsic properties.

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Figures

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Fig. 1

FEM Method: Trabecular bone cores were obtained from male human cadaveric T12 vertebrae. Bone voxels were directly converted from μCT to eight noded hexahedra. Isotropic and homogenous nonlinear damaging material properties (Fig. 2) were applied to each element. Axial loading was applied and a strain-based damageable elastic material property (Fig. 2) was applied iteratively. The strains within the trabecular bone cores were determined and used to apply the damaging criteria to five elements at a time; once the damaged elements were determined, their modulus was reduced. The model was rerun to determine the changes to the strain field and the next elements that would be damaged.

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Fig. 2

Intrinsic Material Properties of the FEM: (a) Smooth dashed curve shows stress versus strain of a body with infinite sites to break, behaving initially linearly, followed by a smooth softening region and then failure. Jagged curves show the mechanical response of a body with 10, 5, 2 and 1 finite site(s) to break. (b) Idealization of a body with one site to break that breaks in two stages. A two stage intrinsic material model was used in this study. Damage was introduced in two stages (primary and secondary events) by reducing the intrinsic Young's Modulus (Ei1) of elements exceeding chosen principal tensile strain values. Primary failure occurred at an intrinsic damage strain (εi1), which reduced the modulus to the Intrinsic damaged material modulus (Ei2) until loaded past the intrinsic rupture strain (εi2), where a secondary failure occurs further reducing the modulus to Ei35 = 100 MPa. Wi,total the intrinsic toughness modulus was estimated as the area under the damage curve.

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Fig. 3

Trabecular Bone Core Damage Site Prediction: 3D rendering of finite element model geometry obtained from Micro-CT imaging with damage sites resulting from axial compression with: Ei2 = 500 MPa, εi1 = 0.005, εi2 = 0.05. Left: Damage sites and deformation of the trabecular bone volume just prior to yield (0.002 offset method), εa = 0.018. Center: Damage sites and deformation of the trabecular bone volume after apparent yield (0.002 offset method). Right: Apparent axial stress-strain curve of the whole bone core. Red circle denotes the yield (0.002 offset method) point. The pre-yield (0.002 offset method) geometry (Left) clearly shows damage predicted during a nearly linear region of apparent deformation.

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Fig. 4

Mechanical Input Contour Plot: Color maps depict (a) the normalized apparent toughness modulus (Wa/Ea) and (b) normalized apparent yield strength (σay/Ea) resulting from FEA with εi1 = 0.005, εi2 (x-axis), and Ei2 (y-axis). The color map illustrates the nonlinear relationship between the intrinsic damage material level properties and the apparent mechanical properties. The similarity of the plots for toughness and strength is consistent with the correlation found between the apparent yield strength (σay) and apparent toughness modulus both experimentally and in the present computer modeling study. Additionally, the plots demonstrate that many combinations of intrinsic material level properties lead to effectively equivalent apparent mechanical properties.

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Fig. 5

Stress-strain behavior of the FEM of trabecular bone cores at the apparent level with changes in the intrinsic material properties as a function of intrinsic damaged material modulus, Ei2 (Left) and intrinsic rupture strain, εi2 (Right). All models had the same geometry, intrinsic modulus, Ei1 = 10 GPa, intrinsic damage strain, εi1 = 0.005. The mechanical response of the trabecular bone core to increasing either εi2 or Ei2 was nonlinear as increasing damage properties corresponded with increased apparent level behavior until εi2 = 0.05 or Ei2 = 1100.

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Fig. 6

Comparison of damage quantity and distribution due to changes in intrinsic rupture strain (εi2): four whole trabecular bone core models with Ei1 = 10,000 MPa, εi2 = 0.005,Ei2 = 1100 MPa, and εi2 as indicated. The damage depicted shows the first 10,000 damage events predicted within each model. The left most top panel is the only model shown with the bone. The sites of damage in the trabecular bone core model are shown in the other panels without the bone volume rendering, enabling all damage to be visualized. Both spatial clustering and temporal clustering (color) can be observed. The lower the failure strain the more clustering appears to occur, both spatially and temporally. The blue damage occurred first with red occurring at the end. The clustering of damage spatially and temporally does not entirely explain the differences in apparent toughness moduli. The εi2 = 0.05 and εi2 = 0.1 appear to have nearly identical damage distributions; however, the apparent toughness moduli (Wa) differed by 15%. The number of secondary failures was different across the four models. The number of primary and secondary damage events were both moderately negatively correlated with normalized apparent toughness modulus (Wa/Ea,init; R2 = 0.1 for primary and R2 = 0.15 for secondary and R2 = 0.19 cumulatively). Apparently, the small number of secondary failures 0.015% in the εi2 = 0.05 model compared to the εi2 = 0.1 model's 0% secondary failures has a large effect on the apparent level mechanical properties.

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Fig. 7

Damage volume versus normalized apparent toughness modulus (Wa/Ea,init): The normalized apparent toughness modulus (Wa/Ea) and the amount of primary and secondary damage are only loosely related. The top of a line segment is the total damage that occurred in a model, and the bottom of the line segment is the amount of primary damage that occurred in a model. Therefore, the length of the line is the amount of secondary damage that occurred. A general observation is that the larger values of apparent toughness associate with short line segments (i.e., small amounts of secondary damage). In statistical analysis, however, apparent toughness (Wa/Ea) was weakly negatively related with primary (p < 0.001) and secondary damage volume in a linear multiple regression (R2model = 0.4 (p < 0.001), R2Primary|Secondary = 0.19 (p < 0.001), R2Secondary|Primary = 0.37 (p < 0.001)). Certain combinations of intrinsic material properties led to large amounts of damage without deteriorating apparent properties.

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