Thread 1. Computational Methods in Biomechanics and Mechanobiology

directly measurable tissue-independent elastic properties), which are here quantified through a multiscale homogenization scheme delivering effective elastic properties of bone materials [2]: At a scale of 10nm, long cylindrical collagen molecules, attached to each other at their ends by - l . 5nm long crosslinks and hosting intermolecular water inbetween, form a contiguous matrix called wet collagen. At a scale of several hundred nanometers, wet collagen and mineral crystal agglomerations interpenetrate each other, forming the mineralized fibril. At a scale of 5 microns, the solid bone matrix is repre- sented as collagen fibril inclusions embedded in a foam of largely disordered (extrafibrillar) mineral crystals [3]. Remarkably, needle and sphere type repre- sentations of disordered minerals deliver quasi-identical mechanical behavior of such extrafibrillar porous polycrystalsa [4]. At a scale above the ultrastructure lacunae are embedded in extracellular bone matrix, forming the extravascular bone material. Model estimates predicted from tissue-specific composition data agree remarkably well with corresponding stiffness experiments across cortical and trabecular materials, which opens new possibilities in the exploitation of computer tomographic data for nano-to-macro mechanics of bone organs [5], especially in combination with currently investigated extensions towards dam- age and failure.

References [1] S.J. Gould, R.C. Lewontin. Proceedings of the Royal Society London 1979;

205B(1161): 581-598. [2] Ch. Hellmich, J.-F. Barthelemy, L. Dormieux. European Journal of Mechanics

A/Solids 2005; 23: 783-810. [3] Ch. Hellmich, F.-J. UIm. Journal of Biomechanics 2005; 35: 1199-1212. [4] A. Fritsch, L. Dormieux, Ch. Hellmich: Comptes Rendus Mecanique 2006; in

press. [5] S. Torabia, Ch. Hellmich. International Bone Densitometry Workshop, Annecy,

France, June 2004.

5777 Fr, 11:00-11:15 (P52) Indirect determination of trabecular bone effective tissue properties using micro-finite element simulations

E. Verhulp 1, B. van Rietbergen 1, R. M~ller 2, R. Huiskes 1. 1Dept. of Biomedical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands, 2Institute for Biomedical Engineering, University and ETH Zuerich, Switzerland

Trabecular bone can resist considerable forces even when it is loaded in the post-yield range. A better understanding of this behavior is of particular importance for improved predictions of the mechanical integrity of collapsed vertebrae, a condition commonly found in osteoporosis. Understanding this post-yield behavior requires information about bone tissue failure parameters. For trabecular bone, however, these parameters are not well known since these are difficult to obtain from mechanical tests on individual trabeculae. The objective of this study was to determine trabecular bone tissue failure parameters in a novel indirect way by comparing results of nonlinear micro- finite element qtFE) analyses that account for the full trabecular architecture with those of real mechanical tests of the same bone specimens. Seven bovine trabecular bone specimens of 5 mm in size were scanned using a ~tCT scanner device. The specimens were subsequently compressed to failure and the compression experiments were simulated using ~tFE models created from the CT scans. After testing, another ~tCT scan was made of the deformed sample. A plasticity model based on Hill's yield function was used to describe to behavior of the tissue. Initial material properties were based on cortical bone and iteratively adjusted in order to fit the simulated and measured load-displacement curves. The geometrically and materially nonlinear models consisted of 80,000-160,000 elements and were solved using MSC.Marc. The results showed that the ~tFE models were capable of accurately simulating trabecular bone failure after fitting of the tissue properties. The local defor- mations observed in the CT scans also compared well with those obtained from the ~tFE simulations. The tissue material properties that resulted from the fit procedure were different from those of cortical bone since significant compression softening had to be included. The results also suggest that there is considerable variation in tissue properties between specimens.

6171 Fr, 11:15-11:30 (P52) Variations in hard-tissue properties do not significantly affect trabecular strength G. Bevill, T.M. Keaveny. Department of Mechanical Engineering, University of California at Berkeley, Berkeley, CA, USA

High-resolution finite element (FE) models obtained from micro-CT scans are widely used to gain insight into both the elastic and failure behavior of trabecular bone [1,2]. One outstanding issue regarding these types of models is whether accurate prospective predictions of strength can be made using averaged hard-tissue material properties, or whether knowledge of specimen- specific properties is required.

T1.10 Computational Biomechanics of Bone Damage and Failure $417

To address this issue, high-resolution FE models having a spatial resolution of 40-60 microns were created from micro-CT images of n = 72 cylindrical cores of normal human trabecular bone (age = 70±11; n =34 male, n =38 female). Fifty-four cores, taken from four different anatomic sites, were machined along the direction of principal trabecular orientation (on-axis); 18 cores from the femoral neck were taken at an oblique angle (n=9 at 15 °, n=9 at 45 °) to the principal orientation (off-axis). The specimens were tested to failure in on- axis tension (n =32), on-axis compression (n =22), or off-axis compression (n = 18). All FE models used the same hard-tissue properties in the nonlinear material model, which included tension-compression asymmetry, and allowed for geometrically large deformations. The FE yield stress predictions were then compared against the corresponding experimental results. The agreement between experimental measurement and the FE strength predictions was outstanding. Linear regression analysis between FE prediction and experiment revealed that R 2 ranged from 0.86 to 0.98 within any single anatomic site, and increased to at least 0.96 when the data were pooled across sites. On a paired basis, FE predictions of yield stress were statistically indistinguishable from experiment in all three loading cases (p=0.29-0.58). These results establish that high-resolution FE models can be used to prospec- tively predict trabecular strength for on- and off-axis loading for normal bone from a range of anatomic sites without a priori knowledge of specimen-specific hard-tissue properties. This suggests that, at least in healthy bone, natural variations in hard-tissue material properties do not appreciably contribute to variations in apparent strength.

References [1] Van Rietbergen B., et al. J Biomech 1995. [2] Bayraktar H.H., Keaveny T.M. J Biomech 2004.

7076 Fr, 11:30-11:45 (P52) Damage and failure of trabecular bone with non-linear geometry and inhomogeneous material properties N. Harrison, D. O'Mahoney, E McDonnell, E McHugh. National Centre for Biomedical Engineering Science and Department of Mechanical and Biomedical Engineering, National University of Ireland, Galway, Ireland

High resolution finite element modelling (HRFEM) based on the voxel method has been widely used for simulating trabecular bone (TB) mechanical be- haviour. In this experimental-computational work, standard HRFEM procedures combined with a novel inhomogeneous material properties assignment proce- dure and non-linear geometry analysis is used to investigate TB damage and failure. Experimentally determined apparent modulus of 10 ovine lumbar vertebral TB cored samples was determined by mechanical compression to failure at room temperature using protocols that minimized end artefacts. Before testing, each cored sample was pOT scanned and a small TB sample was cut from the extracted core and embedded in Epoxy. A range of Young's modulus values, determined by nanoindentation of embedded TB samples, was mapped directly to the range of pOT grayvalues. This resulted in inhomogeneous HRFEM models with 10 grades of bone, each at 36 pm resolution (1.5 million brick elements). Within each element, material properties are isotropic. Using a parallel element by element solver, a low apparent level strain was first applied to each model to determine a computational apparent modulus. Very good agreement was found with the experimentally determined apparent modulus for each specimen. However, even at these low apparent strains, the range of tissue level strains over which linear kinematics is valid, was exceeded. Previous authors have modelled failure, using larger apparent level strains, effective isotropic tissue modulus and a bi-linear material model without the inclusion of non-linear geometry [1]. For the second part of this work, our inhomogeneous models were loaded past the experimentally measured apparent failure strain. Damage and failure of trabecular bone was modelled by the reduction in modulus of each element, regardless of its initial modulus, that exceeded tissue yield strain, with the inclusion of non-linear geometry. Excellent correlation was found between computational and experimental results. This suggests that using inhomogeneous material properties and non- linear geometry provides an improved prediction of TB failure sites.

References [1] Bayraktar H.H., et al. J Biomech 2004; 37(1): 27-35.

6112 Fr, 11:45-12:00 (P52) Damage modeling of human vertebral trabecular bone specimens under experimental loading

T.L. Bredbenner 1 , D.E Nicolella 1 , D.T. Davy 2. 1Materials Engineering Dept., Southwest Research Institute, San Antonio, TX, USA, 2Mechanical and Aerospace Engineering Dept., Case Western Reserve University, Cleveland, OH, USA

There are obvious advantages to investigating the mechanical behavior of tra- becular bone using microstructural models incorporating trabecular structure;

$418 Journal of Biomechanics 2006, Vol. 39 (Suppl 1) Oral Presentations

however, they quickly become very complex and computationally intensive, even with simple models for tissue behavior. Alternatively, continuum damage mechanics (CDM) models offer the potential for characterizing the damage accumulation process and the risk of fracture in real skeletal structures. We implemented a phenomenological constitutive model based on CDM, along with a computational solution scheme, to describe the elastic and inelastic response of human vertebral trabecular bone to low-cycle transient applied loading. The evolution of stiffness loss was described by a continuum damage mechanics model, residual stress and strain were described by an independent plasticity model, and overall time-dependence was described by a viscoelas- ticity model. The evolution of stiffness loss is described by a combination of viscoelastic behavior and damage accumulation and residual stress and strain are described by an independent model of material plasticity. Simulations using finite element analysis methods demonstrate that it is possible to determine corresponding model parameters from experimental data and to characterize the experimental response. The model successfully predicted damage mea- sures, such as evolving and accumulated modulus degradation, despite the complicated nature of the material response and the limited amount of data for multiaxial material characterization. The ability to model the basic features of the response and reasonably predict the highly nonlinear behavior of low- density heterogeneous vertebral trabecular bone allows us to move closer to the goal of predicting the in vive response or the risk of fracture in a clinical setting.

In order to better understand the processes of damage accumulation in cortical bone involved with ESWT, finite element and fatigue behaviour analysis were performed. Methods: Previously developed FE models were expanded with material properties obtained from experimental work and calculated using LS-DYNA (LSTC, Livermore, CA). Parametric studies were conducted to compare the implications of different values on the resulting peak stresses and strains. The focusing effect of the parabolic reflector, and the consequences of impedance mismatch at boundaries were also implemented in the analysis. Fatigue loading was investigated in terms of both classical and fracture-mechanics-based approaches. Parametric studies were carried out on tensile, compressive and torsional loading conditions, and on stress intensity factors resulting from the fracture mode associated with the microscopical features of cortical bone. Results and Discussion: Peak pressure is influenced and altered by the chosen focal depth. Acoustic impedance mismatch between bone and water considerably modified pressure and energy bolus transmitted to the tissue. Estimation of fatigue life were calculated for specific configurations of mean stress, stress amplitude, intensity factors, and failure modes. Damage anal- ysis in terms of potential for spalling events and interlamellar debonding corroborated the microdamage induced experimentally. This study shows that disruptions and visible microcracks may accumulate and propagate as a result of the energy dissipated in the tissue even without exceeding material failure properties in each single shock wave.

6184 Fr, 12:00-12:15 (P52) A relationship between damage accumulation and fabric in trabecular bone based on computational simulations G.L. Niebur. Tissue Mechanics Laboratory, University of Notre Dame, Indiana, USA

Trabecular bone is damaged by fatigue and overloading, which results in reduced stiffness and affects whole bone mechanics. Most experimental in- vestigations of trabecular bone damage have focused on loading along the principal trabecular orientation. However, recent experiments have shown that both modulus decreases and microdamage depend on the orientation of the loads with respect to the trabecular orientation. [1,2] The ability to relate the damage to the trabecular architecture would provide improved insight into the mechanisms of damage. The goal of this study was to apply computational models to investigate relationships between trabecular architecture and micro- damage. Nonlinear, high-resolution finite element models were used to simulate ap- parent level monotonic overloads to 1.2% compressive strain in the on-axis and transverse directions and 1.75% shear strain in the axial-transverse plane of seven bovine tibial trabecular bone samples. Asymmetric tissue level yield properties were calibrated from previous experimental data. The total volume of tissue exceeding the yield strain at the end of the simulation was calculated as a measure of the damaged tissue. An apparent level damage parameter was defined as the scalar magnitude of the product of the fabric tensor and the applied strain, and calculated for each specimen. The volume of damaged tissue was linearly related to the product of fabric and applied strain (R 2 = 0.92, p <~: 0.01). Computational models provided a powerful mechanism for investigating dam- age accumulation in trabecular bone. Although the models have not been validated as a predictor of microdamage, previous linear elastic models have shown that the locations of microdamage and high stress magnitudes are correlated in high-resolution models. 3 The results suggest a plausible model of microdamage accumulation that can be further validated by experimental testing.

References [1] Wang X., et al. European Journal of Morphology 2005; 42: 61-70. [2] Wang X., et al. Journal of Biomechanics, 2006; in press. [3] Nagaraja S., et al. Journal of Biomechanics 2005; 38: 707-716.

7473 Fr, 12:15-12:30 (P52) Acoustic energy: Numerical analysis on damage accumulation and shock wave focussing in bone

A.E. Tami 1 , U. Knothe 2, D.T. Davy 1 . 1Dept. efMechanical & Aerospace Engineering, Case Western Reserve University, Cleveland, OH, USA, 2Orthopaedic Research Center, The Cleveland Clinic Foundation, Cleveland, OH, USA

Introduction: Naturally occurring microdamage appears to trigger mecha- nisms of repair and regeneration as a response to trauma to the cells, disruption of fluid flow and the consequent impaired transport of nutrients and signaling molecules. Thus, we hypothesized that extracorporeal shock wave therapy (ESWT) can be exploited to induce remodeling processes and maintain bone health. In vitro and in vivo experiments have shown the feasibility of producing physiological, low-level microdamage using this novel approach.

T1.11 Computational Biomechanics of the Spine 4222 Tu, 08:15-08:30 (P17) Influence of curved lines of action for trunk extensor muscles on load redistribution and stability of the spine during lifting N. Arjmand, A. Shirazi-Adl, B. Bazrgari. Department ef Mechanical Engineering, Ecole Polytechnique, Montr6al, Qu6bec, Canada

A realistic estimation of trunk stability margin and load redistribution among passive and active systems of the spine under various activities leads to a better assessment of risk of occupational injuries. For an improved estimation of spinal loads in lifting, biomechanical models require precise data on the lever arm and line of action of trunk extensor muscles. The straight line of action of thoracic erector muscles are often assumed despite the apparent curved paths of these muscles in forward flexion postures. In this study, curved paths for thoracic extensor muscles are formulated in our Kinematics-driven finite element model and subsequently used to investigate the spinal response with either straight or curved muscles. Lever arms of global muscles at different levels are restrained either to remain the same or decrease by up to 10% relative to upright postures. In cases with wrapping muscles, reaction forces are developed at wrapping points between the muscles and the spine. Isometric lifting tasks at two trunk flexions of 400 and 650 and three lumbar postures of free style, lordotic and kyphotic with 180N in hands are analyzed. Comparison of trends in predicted activity in global muscles with their mea- sured EMG as posture changed indicates a better qualitative agreement had the reduction of 10% in lever arms been considered only for kyphotic and free style postures and not for the lordotic ones; this suggests that the reduction in lever arms of wrapping muscles may be posture dependent. The assumption of straight and not curved global muscles results in much greater spinal compression and muscle forces whereas smaller shear forces at lower levels. Despite much smaller muscle activities, wrapping of global muscles slightly improves spinal stability. While providing similar stability margin, active muscle forces as well as maximum compression and shear forces decrease in kyphotic postures; these results advocate postures with moderate flexion in lifting. The extent of alterations in lever arm of global muscles during flexion tasks influences results as such reductions substantially increase muscle and spinal compression forces.

5051 Tu, 08:30-08:45 (P17) A multiphasic FE analysis of an L4-L5 motion segment with focus on the inhomogeneous intervertebral disc W. Ehlers, B. Markert, N. Karajan. Institute efApplied Mechanics (CE), Chair II (Continuum Mechanics), University of Stuttgart, Stuttgart, Germany

Motivated by a remarkable number of patients suffering from low back pain, modern numerical simulation techniques concerning the modeling of the human intervertebral disc (IVD) are of growing interest. As one of the most vulnerable parts of the human spine, the IVD belongs to the class of soft avascular cartilaginous tissues consisting of mostly ionized water (interstitial fluid) and collagen of type I and II embedded in an extracellular meshwork of charged protein compounds. Moreover, the organized type-I collagen is the dominant structural part of the anulus fibrosus (AF) enclosing the electro- chemomechanically active nucleus pulposus (NP). Bearing this in mind, the established Theory of Porous Media (TPM) as a continuum-mechanical frame- work is perfectly suitable for the description of such multi-component continua