Finite element analysis of a femur to deconstruct the paradox of bone curvature Sameer Jade a , Kelli H. Tamvada b , David S. Strait b , Ian R. Grosse a,n a Department of Mechanical and Industrial Engineering, 160 Governor's Drive, University of Massachusetts, Amherst, MA 01003, USA b Department of Anthropology, Arts and Sciences 237,1400 Washington Ave., University of Albany, NY 12222, USA HIGHLIGHTS  FEA was used to study structural strength and bending predictability in long bones.  Load carrying capacity can be compromised by bone curvature.  Load carrying capacity can be also be increased by bone curvature.  Curvature does increase bending predictability.  Probability density functions can be generated for bending predictability. article info Article history: Received 22 October 2012 Received in revised form 6 September 2013 Accepted 11 September 2013 Available online 4 October 2013 Keywords: FEA Bone curvature Bending predictability abstract Most long limb bones in terrestrial mammals exhibit a longitudinal curvature and have been found to be loaded in bending. Bone curvature poses a paradox in terms of the mechanical function of limb bones, for many believe the curvature in these bones increases bending stress, potentially reducing the bone's load carrying capacity (i.e., its mechanical strength). The aim of this study is to investigate the role of longitudinal bone curvature in the design of limb bones. In particular, it has been hypothesized that bone curvature results in a trade-off between the bone's mechanical strength and its bending predictability. We employed finite element analysis (FEA) of abstract and realistic human femora to address this issue. Geometrically simplified human femur models with different curvatures were developed and analyzed with a commercial FEA tool to examine how curvature affects the bone's bending predictability and load carrying capacity. Results were post-processed to yield probability density functions (PDFs) describing the circumferential location of maximum equivalent stress for various curvatures in order to assess bending predictability. To validate our findings, a finite element model was built from a CT scan of a real human femur and compared to the simplified femur model. We found general agreement in trends but some quantitative differences most likely due to the geometric differences between the digitally reconstructed and the simplified finite element models. As hypothesized by others, our results support the hypothesis that bone curvature can increase bending predictability, but at the expense of bone strength. & 2013 Elsevier Ltd. All rights reserved. 1. Introduction In a classic study, John Bertram and Andrew Biewener examined the mechanics of bone curvature in the design of the long (i.e., limb) bones of mammals based on elementary analytical expressions (Bertram and Biewener 1988). They noted that although a distinct longitudinal curvature was ubiquitous in mammals, a straight bone should be a more efficient design because the maximum mechanical strength of a straight bone should be higher. Bertram and Biewener hypothesized that the benefit of bone curvature is that it makes more predictable the manner in which the bone bends. Bending predictability was defined by Bertram and Biewener as the prob- ability of the bone to bend in a certain direction, thereby yielding consistent stress patterns within the bone. In other words, curvature restricts the range of bending directions when a bone is subjected to loads in variable directions. In contrast, a straight bone subjected to the same loads will have no restriction on the range of its bending direction and hence will have no bending predictability (Bertram and Biewener, 1988). Although other explanations for bone curva- ture have been proposed (Bertram and Biewener, 1988; Yamanaka et al., 2005; Taylor et al., 1996; Rubin and Lanyon, 1982; Biewener, 1983; Biewener, 1986; Frost, 1979; Lanyon, 1980), this study focuses Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/yjtbi Journal of Theoretical Biology 0022-5193/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jtbi.2013.09.012 n Corresponding author. Tel.: þ1 413 545 1350; fax: þ1 413 545 1027. E-mail address: grosse@ecs.umass.edu (I.R. Grosse). Journal of Theoretical Biology 341 (2014) 53–63