Identification of material parameters based on Mohr–Coulomb failure criterion for bisphosphonate treated canine vertebral cancellous bone

doi:10.1016/j.bone.2008.05.023

Bone

Volume 43, Issue 4, October 2008, Pages 775-780

https://doi.org/10.1016/j.bone.2008.05.023 Get rights and content

Abstract

Nanoindentation has been widely used to study bone tissue mechanical properties. The common method and equations for analyzing nanoindentation, developed by Oliver and Pharr, are based on the assumption that the material is linearly elastic. In the present study, we adjusted the constraint of linearly elastic behavior and use nonlinear finite element analysis to determine the change in cancellous bone material properties caused by bisphosphonate treatment, based on an isotropic form of the Mohr–Coulomb failure model. Thirty-three canine lumbar vertebrae were used in this study. The dogs were treated daily for 1 year with oral doses of alendronate, risedronate, or saline vehicle at doses consistent, on a mg/kg basis, to those used clinically for the treatment of post-menopausal osteoporosis. Two sets of elastic modulus and hardness values were calculated for each specimen using the Continuous Stiffness Measurement (CSM) method (E_CSM and H_CSM) from the loading segment and the Oliver–Pharr method (E_O–P and H_O–P) from the unloading segment, respectively. Young's modulus (E_FE), cohesion (c), and friction angle (ϕ) were identified using a finite element model for each nanoindentation. The bone material properties were compared among groups and between methods for property identification. Bisphosphonate treatment had a significant effect on several of the material parameters. In particular, Oliver–Pharr hardness was larger for both the risedronate- and alendronate-treated groups compared to vehicle and the Mohr–Coulomb cohesion was larger for the risedronate-treated compared to vehicle. This result suggests that bisphosphonate treatment increases the hardness and shear strength of bone tissue. Shear strength was linearly predicted by modulus and hardness measured by the Oliver–Pharr method (r² = 0.99). These results show that bisphosphonate-induced changes in Mohr–Coulomb material properties, including tissue shear cohesive strength, can be accurately calculated from Oliver–Pharr measurements of Young's modulus and hardness.

Introduction

Nanoindentation has been widely used to study tissue mechanical properties (elastic modulus and hardness) of both cortical [1], [2], [3] and cancellous bone [4], [5], [6], [7], [8]. The method and equations used to calculate bone tissue modulus and hardness in these studies were based on the assumption that the bone was linearly elastic and, therefore, the material properties remained constant independent of the indentation depth [9]. In the present study, we altered the constraint of linearly elastic behavior and use nonlinear finite element analysis to determine the change in cancellous bone material properties caused by drug treatment.

Bone tissue ultrastructure and the accumulation of damage at the ultrastructural level are not completely understood [3]. In particular, the failure criterion for bone tissue at the nanoscopic level was not known until recently. Tai et al. [3] determined that a cohesive-frictional model, specifically a Mohr–Coulomb pressure dependent failure criterion (in short, Mohr–Coulomb criterion), could accurately reproduce experimental force-displacement data measured using an atomic force microscope.

In Tai et al.'s study [3], they performed a series of indentation tests on bovine cortical bone tissue using an atomic force microscope. To identify the failure properties of the tissue, they used a finite element model of the indentation process. In their models they assumed a Young's modulus of 18 GPa and a Poisson's ratio of 0.3. They showed that the models matched the experimental data when the friction angle (ϕ) and the cohesion (c) were set to 15° and 100 MPa (Fig. 1), respectively. This was an important study, because it demonstrates that the Mohr–Coulomb criterion appears to be a good material model for compressive loading of bone tissue. A limitation of the study is that their approach required assuming that the Young's modulus was fixed. In the current study, we have developed a method to determine all of the material parameters (modulus, cohesion, friction angle) except the Poisson's ratio, which we left as unchanged.

Bisphosphonates (BPs) increase average mineralization of trabecular bone tissue [10], [11], [12]. Studies in beagle dogs, using doses at and above those used for treatment of osteoporosis, have shown that BP treatment also results in microdamage accumulation and a reduction in bone toughness in vertebrae [13], [14], [15], [16]. With an increase in both bone mineralization and microdamage, it is not clear how BPs might affect bone tissue nano-level mechanical properties (elastic modulus and hardness), since mineralization and microdamage have opposite effects. The results of the current study are intended to partially clarify this mixed effect of increasing both bone tissue mineralization and microdamage because nanoindentation can directly measure the bone tissue mechanical properties.

In this study, we (1) develop a new method for identifying the material parameters of the Mohr–Coulomb criterion for bone tissue using nanoindentation; (2) determine whether the material properties identified by either of the methods (Oliver–Pharr (O–P), Continuous Stiffness Measurement (CSM), and the Mohr–Coulomb FE model) differ with bisphosphonate treatment; (3) compare the material property results among the O–P, CSM and FE method, and (4) demonstrate that the material properties from the finite element results can be predicted using the Oliver–Pharr experimental results.

Section snippets

Bone samples

Thirty-three canine second lumbar vertebrae were used in this study. The specimens were collected during a previously completed BP treatment study [14]. Briefly, the beagle dogs were treated daily for 1 year with oral doses of alendronate sodium (ALN, 0.20 mg/kg/day, n = 12), risedronate sodium (RIS, 0.10 mg/kg/day, n = 10), or saline vehicle (VEH, n = 11). These bisphosphonate doses approximate, on a mg/kg basis, those used for the treatment of post-menopausal osteoporosis.

The vertebrae were

Results

E_CSM was higher in the RIS group, but not in the ALN group, compared to the VEH group (p = 0.025). No difference was observed in E_O–P among the three groups (p = 0.11, Fig. 5a). H_CSM and H_O–P were significantly higher in both the ALN- and RIS-treated groups compared to VEH (p = 0.002 and 0.0028, Fig. 5b), but there was no significant difference between the two BP groups.

E_CSM was linearly correlated with H_CSM in the ALN, RIS and VEH groups (r² = 0.47, 0.74 and 0.84, p < 0.018), respectively. The slopes

Discussion

At doses consistent with those used to treat post-menopausal osteoporosis, both alendronate and risedronate significantly increased tissue hardness of dog vertebral cancellous bone at the nano-level. Hardness is a useful tool for estimating bone strength [18], but it is not a mechanical property in the same sense as Young's modulus, cohesion or friction angle. As a result, the goal of this study was to develop a finite element method to analyze bone tissue nanoindentation load-displacement data

Acknowledgments

This work was supported by NIH Grants AR40776 (DPF), R01 AR047838 (DBB), and T32 AR007581 (DBB), and a research grant from The Alliance for Better Bone Health (Procter and Gamble Pharmaceuticals and Sanofi-Aventis). Merck and Co. kindly provided the alendronate. This investigation utilized an animal facility constructed with support from Research Facilities Improvement Program Grant Number C06RR10601 from the NIH National Center for Research Resources. This indenter used in this study was from

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Identification of material parameters based on Mohr–Coulomb failure criterion for bisphosphonate treated canine vertebral cancellous bone

Abstract

Introduction

Section snippets

Bone samples

Results

Discussion

Acknowledgments

Biomaterials

Bone

Bone

Bone

Bone

Mater Charact

Bone

J Biomech

An application of nanoindentation technique to measure bone tissue lamellae properties

J Biomech Eng-Trans ASME

Age, gender, and bone lamellae elastic moduli

Bone

Nanogranular origins of the strength of bone

Nano Lett