Mimetization of the elastic properties of cancellous bone via a parameterized cellular material

📝 Abstract

Bone tissue mechanical properties and trabecular microarchitecture are the main factors that determine the biomechanical properties of cancellous bone. Artificial cancellous microstructures, typically described by a reduced number of geometrical parameters, can be designed to obtain a mechanical behavior mimicking that of natural bone. In this work, we assess the ability of the parameterized microstructure introduced by Kowalczyk (2006) to mimic the elastic response of cancellous bone. Artificial microstructures are compared with actual bone samples in terms of elasticity matrices and their symmetry classes. The capability of the parameterized microstructure to combine the dominant isotropic, hexagonal, tetragonal and orthorhombic symmetry classes in the proportions present in the cancellous bone is shown. Based on this finding, two optimization approaches are devised to find the geometrical parameters of the artificial microstructure that better mimics the elastic response of a target natural bone specimen: a Sequential Quadratic Programming algorithm that minimizes the norm of the difference between the elasticity matrices, and a Pattern Search algorithm that minimizes the difference between the symmetry class decompositions. The pattern search approach is found to produce the best results. The performance of the method is demonstrated via analyses for 146 bone samples.

💡 Analysis

Bone tissue mechanical properties and trabecular microarchitecture are the main factors that determine the biomechanical properties of cancellous bone. Artificial cancellous microstructures, typically described by a reduced number of geometrical parameters, can be designed to obtain a mechanical behavior mimicking that of natural bone. In this work, we assess the ability of the parameterized microstructure introduced by Kowalczyk (2006) to mimic the elastic response of cancellous bone. Artificial microstructures are compared with actual bone samples in terms of elasticity matrices and their symmetry classes. The capability of the parameterized microstructure to combine the dominant isotropic, hexagonal, tetragonal and orthorhombic symmetry classes in the proportions present in the cancellous bone is shown. Based on this finding, two optimization approaches are devised to find the geometrical parameters of the artificial microstructure that better mimics the elastic response of a target natural bone specimen: a Sequential Quadratic Programming algorithm that minimizes the norm of the difference between the elasticity matrices, and a Pattern Search algorithm that minimizes the difference between the symmetry class decompositions. The pattern search approach is found to produce the best results. The performance of the method is demonstrated via analyses for 146 bone samples.

📄 Content

MIMETIZATION OF THE ELASTIC PROPERTIES OF CANCELLOUS BONE VIA A PARAMETERIZED CELLULAR MATERIAL

Lucas Colabellaa, Adrián P. Cisilinoa, Guillaume Häiatb and Piotr Kowalczykc
a INTEMA- Faculty of Engineering, National University of Mar del Plata – CONICET Av. Juan B. Justo 4302, Mar del Plata B7608FDQ, Argentina
bCNRS – Laboratoire Modélisation et Simulation Multiéchelle, UMRS CNRS 8208, 61 avenue du gal de Gaulle, 94010 Creteil, France c Institute of Fundamental Technological Research, Polish Academy of Sciences, Pawinskiego 5B, 02-106 Warsaw, Poland ABSTRACT Bone tissue mechanical properties and trabecular microarchitecture are the main factors that determine the biomechanical properties of cancellous bone. Artificial cancellous microstructures, typically described by a reduced number of geometrical parameters, can be designed to obtain a mechanical behavior mimicking that of natural bone. In this work, we assess the ability of the parameterized microstructure introduced by Kowalczyk (2006) to mimic the elastic response of cancellous bone. Artificial microstructures are compared with actual bone samples in terms of elasticity matrices and their symmetry classes. The capability of the parameterized microstructure to combine the dominant isotropic, hexagonal, tetragonal and orthorhombic symmetry classes in the proportions present in the cancellous bone is shown. Based on this finding, two optimization approaches are devised to find the geometrical parameters of the artificial microstructure that better mimics the elastic response of a target natural bone specimen: a Sequential Quadratic Programming algorithm that minimizes the norm of the difference between the elasticity matrices, and a Pattern Search algorithm that minimizes the difference between the symmetry class decompositions. The pattern search approach is found to produce the best results. The performance of the method is demonstrated via analyses for 146 bone samples. 1 INTRODUCTION Bones are hierarchical bio-composite materials with a complex multiscale structural geometry (Carretta et al. 2013). Bone tissue is arranged either in a compact pattern (cortical bone) or a spongy pattern (cancellous bone). Cancellous bone can be found in vertebral bodies and at the epiphyses of long bones. In the vertebral body, it is the main load bearing structure, where as in the appendicular skeleton, it transfers mechanical loads from the articular surface to cortical bone. Furthermore, trabecular bone quality is an important determinant of the overall bone strength and affects fracture risk. To better understand the mechanics of cancellous bone is of interest for the diagnosis of bone diseases (osteoporosis), the evaluation of the risk of fracture, and for the design of artificial bone (Cowin 2001). Cancellous bone can be assimilated to a composite material with hierarchical structure. In a bottom-up description, the structure starts in the nanoscale (mineralized collagen fibril) and moves up to the sub-microscale (single lamella), the microscale (single trabecula), and mesoscale (trabecular bone) levels. Trabeculae are organized into a three- dimensional lattice oriented mainly along the lines of stress, which forms a stiff and ductile structure that provides the framework for the soft bone marrow filling the intertrabecular spaces. Trabeculae consist of a nanometric extracellular matrix that incorporates hydroxyapatite, the bone mineral that provides bones rigidity (Sansalone et al. 2010, 2012), and collagen, an elastic protein which improves fracture resistance
(Keaveny et al. 2001). Bone tissue mechanical properties and trabecular architecture are the main factors determining the mechanical properties of cancellous bone, which show a high dependency on species, anatomic site, age and size of the sample (Fritsch and Hellmich 2007; Parkinson and Fazzalari 2013). The small dimensions of the trabeculae (of the order from tens to a cent of microns) hinder their mechanical characterization at tissue- level. In recent years, nano-indentation has provided the means for the direct measurement of the elastic properties of trabecular bone tissue (a complete review of the available techniques, many of them indirect, can be found in the recent paper by Oftadeh et al. (2015)). By means of high resolution nano-indentation, Brennan et al. (2009) studied the tissue property variations within a single trabecula; they found that Young´s modulus and hardness increase towards the core of the trabeculae. Despite this findings, it is the
common assumption that the mechanical inhomogeneity and anisotropy of bone tissue has a minor impact on the apparent properties of cancellous bone and, consequently, it can be approximated by an isotropic tissue modulus (Kabel et al. 1999a, b). Different experiments have shown that linear elasticity can predict the behavior of cancellous bone (Keaveny et al. 1994). The trabecular a

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