The Accuracy of Digital Image- Based Finite Element ...

The Accuracy of Digital ImageBased Finite Element Models R. E. Guldberg School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332

S. J. Hollister Orthopaedic Research Laboratories, The University of Michigan, Ann Arbor, Ml 48109

G. T. Charras School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332

Digital image-based finite element meshing is an alternative approach to time-consuming conventional meshing techniques for generating realistic three-dimensional (3D) models of complex structures. Although not limited to biological applications, digital image-based modeling has been used to generate structure-specific (i.e., nongeneric) models of whole bones and trabecular bone microstructures. However, questions remain regarding the solution accuracy provided by the digital meshing approach, particularly at model or material boundaries. The purpose of this study was to compare the accuracy of digital and conventional smooth boundary models based on theoretical solutions for a two-dimensional (2D) compression plate and a 3D circular cantilever beam. For both the plate and beam analyses, the predicted solution at digital model boundaries was characterized by local oscillations, which produced potentially high errors within individual boundary elements. Significantly, however, the digital model boundary solution oscillated approximately about the theoretical solution. A marked improvement in solution accuracy was therefore achieved by considering average results within a region composed of several elements. Absolute errors for Von Mises stress averaged over the beam cross section, for example, converged to less than 4 percent, and the predicted free-end displacement of the cantilever beam was within 1 percent of the theoretical solution. Analyses at several beam orientations and mesh resolutions suggested a minimum discretization of three to four digital finite elements through the beam cross section to avoid high numerical stiffening errors under bending.

Finite element analysis (FEA) has become an essential part of biomechanical investigations focused on better understanding structure-function relationships associated with tissue homeostasis, adaptation, and failure. Through advances in computer memory capacity and FEA application software, the technology has evolved over the last decade from analyzing simple twodimensional (2D) and idealized models to realistic three-dimensional (3D) models (Huiskes and Hollister, 1993). Unfortunately, the time required to generate accurate 3D finite element models of complex biological structures is considerable and often necessitates the use of simplified or representative geometries. The first step in the modeling process traditionally involves estimation of structural topography using idealized geometries or surface-fitting methods followed by finite element mesh generation within the defined surfaces. For this study, models generated using the traditional surface-based approach will be referred to as smooth boundary finite element models. Significant effort has been directed toward developing modeling techniques that produce valid finite element discretizations without user intervention (Frey et al., 1994; Lo, 1991; Shephard and Georges, 1991; Zienkiewicz, 1991). Automated meshing algorithms of varying robustness have been developed, which for some structures can generate 2D or 3D meshes within previously defined surfaces. The preliminary requirement to define the structure's topography, however, will continue to be a timeintensive step for biomechanical investigators. Digital imagebased finite element modeling was developed as an alternative modeling approach, which automates both topography estimation and mesh generation (Fyhrie et al, 1992; Hollister and Kikuchi, 1992; Keyak et al., 1990). Any digital imaging method that produces a gray-scale pixel or voxel map of finite resolution can be used as a basis for

finite element mesh generation. In particular, several investigators have used reconstructed 3D computed tomography (CT) images or stacked 2D images from serial sections to model various levels of bone structure (Fig. 1) (Hollister et al., 1994; Keyak et a l , 1990; van Rietbergen et al, 1995). After thresholding a gray-scale image into different material domains, the individual voxels are directly converted into hexahedral finite elements by assigning nodal connectivity and material properties. Using this approach, it is possible to efficiently generate nonidealized 3D models of arbitrarily complex structures. Although digital image-based modeling offers many practical advantages, the solution accuracy provided by digital meshes has been questioned due to an inexact representation of curved boundaries (Hollister and Riemer, 1993; Jacobs et al, 1993; Keyak et al, 1993). In a 2D proximal femur model, Jacobs et al (1993) reported higher surface strain errors using a digital mesh relative to a triangular mesh fitted within smooth model boundaries. However, Keyak et al. (1993) found a significant relationship between strain gage measurements from a femur mechanically tested in compression and surface strains predicted using a CT-based 3D digital model of the tested femur. Hollister and Riemer (1993) concluded that 2D digital and smooth models provided similar strain energy density distributions around circular plate inclusions but also noted oscillations in the digital model solution at material boundaries. Previous validation studies have used solutions from smooth boundary models as a gold standard to evaluate the accuracy provided by digital models. However, conventional smooth meshes also represent numerical approximations in which solution accuracy depends on element type, refinement, and distortion. The purpose of the current study was to quantify errors associated with digital image-based modeling using the theoretical elasticity solutions for simple 2D and 3D problems and thereby establish practical guidelines for modeUng complex structures such as the microstructure of trabecular bone.

Contributed by the Bioengineering Division for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Division January 21, 1996; revised manuscript received August 15, 1997. Associate Technical Editor: R. T. Hart.

Methods An infinite 2D compression plate with a central circular defect was modeled specifically to evaluate error patterns along

Introduction

Journal of Biomechanical Engineering

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APRIL 1998, Vol. 120 / 289

Subcase 1 STRAIN ENERGY DENSITY

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