title: To what extent can cortical bone millimeter-scale elasticity be predicted by a two-phase composite model with variable porosity?
abstract: An evidence gap exists in fully understanding and reliably modeling the variations in elastic anisotropy that are observed at the millimeter scale in human cortical bone. The porosity (pore volume fraction) is known to account for a large part, but not all, of the elasticity variations. This effect may be modeled by a two-phase micromechanical model consisting of a homogeneous matrix pervaded by cylindrical pores. Although this model has been widely used, it lacks experimental validation. The aim of the present work is to revisit experimental data (elastic coefficients, porosity) previously obtained from 21 cortical bone specimens from the femoral mid-diaphysis of 10 donors and test the validity of the model by proposing a detailed discussion of its hypotheses. This includes investigating to what extent the experimental uncertainties, pore network modeling, and matrix elastic properties influence the model's predictions. The results support the validity of the two-phase model of cortical bone which assumes that the essential source of variations of elastic properties at the millimeter-scale is the volume fraction of vascular porosity. We propose that the bulk of the remaining discrepancies between predicted stiffness coefficients and experimental data (RMSE between 6% and 9%) is in part due to experimental errors and part due to small variations of the extravascular matrix properties. More significantly, although most of the models that have been proposed for cortical bone were based on several homogenization steps and a large number of variable parameters, we show that a model with a single parameter, namely the volume fraction of vascular porosity, is a suitable representation for cortical bone. The results could provide a guide to build specimen-specific cortical bone models. This will be of interest to analyze the structure-function relationship in bone and to design bone-mimicking materials.
authors.affiliation: Sorbonne Universités, UPMC Université Paris 06, UMR 7371, UMR_S 1146, Laboratoire d'Imagerie Biomédicale, F-75005, Paris, France; CNRS, UMR 7371, Laboratoire d'Imagerie Biomédicale, F-75005, Paris, France; INSERM, UMR_S 1146, Laboratoire d'Imagerie Biomédicale, F-75005, Paris, France. Electronic address: mathilde.granke@gmail.com.
authors.forename: Mathilde
authors.initials: M
authors.lastname: Granke
authors.affiliation: Sorbonne Universités, UPMC Université Paris 06, UMR 7371, UMR_S 1146, Laboratoire d'Imagerie Biomédicale, F-75005, Paris, France; CNRS, UMR 7371, Laboratoire d'Imagerie Biomédicale, F-75005, Paris, France; INSERM, UMR_S 1146, Laboratoire d'Imagerie Biomédicale, F-75005, Paris, France.
authors.forename: Quentin
authors.initials: Q
authors.lastname: Grimal
authors.affiliation: School of Mathematics, Alan Turing Building, University of Manchester, Manchester M139PL, UK.
authors.forename: William J
authors.initials: WJ
authors.lastname: Parnell
authors.affiliation: Julius Wolff Institute & Berlin-Brandenburg School for Regenerative Therapies, Charité-Universitätsmedizin Berlin, 13353 Berlin, Germany.
authors.forename: Kay
authors.initials: K
authors.lastname: Raum
authors.affiliation: Fachbereich Mathematik, Technische Universität Darmstadt, 64289 Darmstadt, Germany.
authors.forename: Alf
authors.initials: A
authors.lastname: Gerisch
authors.affiliation: ESRF, 6 rue Jules Horowitz, 38043 Grenoble, France; CREATIS INSERM U1044, CNRS 5220, INSA Lyon, Université de Lyon, 69621 Villeurbanne, France.
authors.forename: Françoise
authors.initials: F
authors.lastname: Peyrin
authors.affiliation: Sorbonne Universités, UPMC Université Paris 06, UMR 7371, UMR_S 1146, Laboratoire d'Imagerie Biomédicale, F-75005, Paris, France; CNRS, UMR 7371, Laboratoire d'Imagerie Biomédicale, F-75005, Paris, France; INSERM, UMR_S 1146, Laboratoire d'Imagerie Biomédicale, F-75005, Paris, France.
authors.forename: Amena
authors.initials: A
authors.lastname: Saïed
authors.affiliation: Sorbonne Universités, UPMC Université Paris 06, UMR 7371, UMR_S 1146, Laboratoire d'Imagerie Biomédicale, F-75005, Paris, France; CNRS, UMR 7371, Laboratoire d'Imagerie Biomédicale, F-75005, Paris, France; INSERM, UMR_S 1146, Laboratoire d'Imagerie Biomédicale, F-75005, Paris, France.
authors.forename: Pascal
authors.initials: P
authors.lastname: Laugier
mesh_terms: D001842:Bone and Bones / Q000502:physiology*; D004548:Elasticity; D008962:Models, Theoretical; D016062:Porosity
keywords: Anisotropic elasticity; Cortical bone; Effective properties; Mechanical model; Porosity