Examination of continuum and micro-structural properties of human vertebral cancellous bone using combined cellular solid models

Abstract

Two- and three-dimensional structural models of the vertebral body have been used to estimate the mechanical importance of parameters that are difficult to quantify experimentally such as lattice disorder, trabecular thickness, trabecular spacing, connectivity, and fabric. Many of the models that investigate structure-function relationships of the vertebral body focus only on the trabecular architecture and neglect solid-fluid interactions. We developed a cellular solid model composed of two idealized unit cell geometries to investigate the continuum and micro-structural properties of human vertebral cancellous bone in a mathematically tractable model. Using existing histomorphological data we developed structure-function relationships for the mechanical properties of the solid phase, estimated the micro-structural strains, and predicted the fluid flow characteristics. We found that the micro-structural strains may be 1.7 to 2.2 times higher than the continuum level strains between the ages of 40 and 80. In addition, the predicted permeability agrees well with the experimental data.

Modeling the mechanical consequences of vibratory loading in the vertebral body: microscale effects

“…It should also be noted that Sander et al (2003) assumed a rule of mixtures combination of the two different unit cells. There were two important reasons for this assumption.…”

“…In the vertebral body, cancellous bone appears to contain a somewhat regular array of trabecular struts (Fig. 1), and thus has been idealized as a cellular solid consisting of two repeating unit cells: a rectangular unit cell and an offset unit cell (Sander et al 2003). Each unit cell consists of a single vertical trabecula, modeled as a circular beam of length L V and diameter D V , and two horizontal trabeculae, also idealized as a circular beam with length L H and diameter D H (Fig.…”

“…In their study, Sander et al examined the mechanical and fluid flow characteristics of vertebral cancellous bone and studied the effect of aging on these properties. Using the cellular solid model in conjunction with relevant phenomenological relationships, equations of initial solid volume fraction, offset unit cell content fraction, and Poisson's ratio were formulated (Sander et al 2003). The model developed by Sander et al was based on architectural measures obtained by Mosekilde (1988) and calibrated to the experimental results obtained by Kopperdahl and Keaveny (1998).…”

“…In order to describe the mechanical consequences of vibratory loading, a mixture theory approach is adopted and supplemented with the cellular solid model presented in Sander et al (2003). The vertebral body consists of a cancellous bone core and several fluid components intermixed within the intertrabecular space.…”

Permeability calculations in three-dimensional isotropic and oriented fiber networks

“…[6][7][8][9][10] The most common measure used to characterize such flows is the hydraulic permeability k, which for an isotropic medium is defined by Darcy's law as 11…”