The effect of the density–modulus relationship selected to apply material properties in a finite element model of long bone
Introduction
In many areas of orthopaedic biomechanics, such as implant design, properly developed finite element (FE) models can be a great companion to in vitro studies, as they may allow a wider range of experimental variables to be explored in a cost-effective and timely manner. One challenge in developing these models is the assignment of accurate material properties to bone. Through the use of computed tomography (CT), many recent studies have developed subject-specific FE models, where the geometry and material properties of bone are assigned based on information derived from the scans. Inhomogeneous material properties can be applied by extracting density information from each CT voxel and relating the density to elastic modulus. This involves the use of an equation to relate density and elastic modulus, and there are many such relationships from which to choose in the literature, as reviewed by Helgason et al. (2008) and Wirtz et al. (2000). Most FE studies tend to use one of these multiple equations without justification or investigation into its appropriateness for the model.
The selection of the density–modulus relationship can have a great effect on how well the FE results compare to experimental results, as shown by Schileo et al. (2007). This recent study investigated the effect of three different density–modulus relationships on FE results and compared the results to experimental strains in a femur model. They concluded that of the three equations tested, applying a femur-specific equation developed by Morgan et al. (2003) to their model resulted in values that most closely matched their experimental strains, highlighting the need to investigate site-specific equations for applications in other bones.
No known studies have directly compared the reported density–modulus relationships for material property assignment in the distal ulna. Identifying the most accurate density–modulus relationship for the ulna will aid in developing accurate FE models of the ulna for future investigations into various implant designs. There are no relationships in the literature specifically for the ulna; however several studies provide pooled data from various long bone sites. The purpose of this study was to apply six different density–modulus equations from the literature (Carter and Hayes, 1977; Keller, 1994; Lotz et al., 1990, Lotz et al., 1991; Morgan et al., 2003; Snyder and Schneider, 1991; Wirtz et al., 2000) to subject-specific FE models of the distal ulna, and observe which produce bone strains closest to those previously found in experimental testing with strain gauges.
Section snippets
Experimental model
The experimental methodology used has been previously published in detail (Austman et al., 2007) and is therefore only described briefly here. Eight fresh-frozen isolated ulnae (right specimens, mean age=68±8 years) were thawed and stripped of all soft tissues. The proximal portion of each ulna was cemented in a custom-designed jig that allowed a 20 N medially directed force to be applied to the distal articular surface of the ulna via a materials testing machine (Instron 8872, Canton, MA, USA) (
Results
The strain output found experimentally and the FE strain results found using each of the density–modulus relationships are shown for a representative specimen in Fig. 5. For this specimen, the experimental values always fell somewhere between those predicted using Carter and Hayes (1977) and Morgan et al. (2003). Table 2 summarizes the percent error found for each specimen and each of the six density–modulus relationships, as well as an average percent error over all eight specimens. The lowest
Discussion
Proper development of a subject-specific FE model requires care during multiple model steps including geometry extraction, meshing, material property assignment, and applying boundary conditions. The selection of an appropriate density–elastic modulus relationship to assign inhomogeneous properties to bone is one area that has received little attention in the literature, and this study was undertaken to address this void. The goal of this work was to evaluate the effect of the density–elastic
Conflict of interest statement
None declared.
Acknowledgments
Funding for this study was provided by Natural Sciences and Engineering Research Council (NSERC), the Canadian Foundation for Innovation (CFI), the Ontario Innovation Trust (OIT), the Early Researcher Award (ERA) program, and the Canadian Institutes for Health Research (CIHR 67018). The authors wish to thank Todor Ivanov for his assistance preparing the CT files for analysis and Ruochu Gao for her assistance with the statistical analysis.
References (21)
- et al.
The effect of distal ulnar implant stem material and length on bone strains
Journal of Hand Surgery American
(2007) - et al.
Comparison of an inhomogeneous orthotropic material models used for FE analyses
Medical Engineering & Physics
(2008) - et al.
Validation of a finite element model of the human metacarpal
Medical Engineering & Physics
(2005) - et al.
Design and implementation of an instrumented ulnar head prosthesis to measure loads in vitro
Journal of Biomechanics
(2006) - et al.
Mathematical relationships between bone density and mechanical properties: a literature review
Clinical Biomechanics
(2008) Predicting the compressive mechanical behavior of bone
Journal of Biomechanics
(1994)- et al.
Mechanical properties of metaphyseal bone in the proximal femur
Journal of Biomechanics
(1991) - et al.
Trabecular bone modulus–density relationships depend on anatomic site
Journal of Biomechanics
(2003) - et al.
Comparison of isotropic and orthotropic material property assignment on femoral finite element models under two loading conditions
Medical Engineering & Physics
(2006) - et al.
Tetrahedral versus hexahedral finite elements in numerical modelling of the proximal femur
Medical Engineering & Physics
(2006)