RVE Procedure for Estimating the Elastic Properties of Inhomogeneous Microstructures Such as Bone Tissue

word count: 249 Total word count: 3393Attachments: 5 figures and 2 tables + 1 figure and 4 tables in the appendices 2 Disclosure pageThe authors have no conflict of interest to report. AcknowledgmentsThe script manager Medtool (www.dr-pahr.at) was used to generate the µFE models, start the computations and analyse the results. The authors would like to thank Prof. Bert van Rietbergen for sharing the µCT data. Karol Daszkiewicz is supported by grants from the are also gratefully acknowledged. AbstractBoundary conditions (BCs) and sample size affect the measured elastic properties of cancellous bone. Samples too small to be representative appear stiffer under kinematic uniform BCs (KUBCs) than under periodicity-compatible mixed uniform BCs (PMUBCs). To avoid those effects, we propose to determine the effective properties of trabecular bone using an embedded configuration. Cubic samples of various sizes (2.63, 5.29, 7.96, 10.58 and 15.87 mm) were cropped from µCT scans of femoral heads and vertebral bodies. They were converted into µFE models and their stiffness tensor was established via six uniaxial and shear load cases. PMUBCs-and KUBCs-based tensors were determined for each sample. "In situ" stiffness tensors were also evaluated for the embedded configuration, i.e. when the loads were transmitted to the samples via a layer of trabecular bone. The Zysset-Curnier model accounting for bone volume fraction and fabric anisotropy was fitted to those stiffness tensors and model parameters ν0 (Poisson's ratio), E0 and µ0 (elastic and shear moduli) were compared between sizes. BCs and sample size had little impact on ν0. However, KUBCs-and PMUBCs-based E0 and µ0 respectively decreased and increased with growing size, though convergence was not reached even for our largest samples. Both BCs produced upper and lower bounds for the in situ values that were almost constant across samples dimensions, thus appearing as an approximation of the effective properties. PMUBCs seem also appropriate for mimicking the trabecular core, but they still underestimate its elastic properties (especially in shear) even for nearly orthotropic samples.

Section: Discussionsupporting

confidence: 72%

Section: Discussionmentioning

confidence: 54%

“…Other authors Zysset 2008, Blöß andWelsch 2015) inferred that PMUBCs were in fact the best boundary conditions for approximating the effective properties of cancellous bone regions.…”

Section: Discussionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The effective elastic properties of human trabecular bone may be approximated using micro-finite element analyses of embedded volume elements

Daszkiewicz

Maquer

Zysset

2016

“…This explains why just few rod-like trabeculae surrounding sample 1 better transmit the deformations from the outer layer when KUBCs are applied. Nevertheless, exactly because PMUBCs are more compliant, we can reasonably assume that PMBCs-based embedded configuration lead to the effective yield properties as demonstrated for elasticity (Jiang et al 2001;Blöß and Welsch 2015;Daszkiewicz et al 2017).…”

Section: Discussionmentioning

confidence: 89%

Estimation of the effective yield properties of human trabecular bone using nonlinear micro-finite element analyses

Wili

Maquer

Panyasantisuk

et al. 2017

Micro-finite element ([Formula: see text]FE) analyses are often used to determine the apparent mechanical properties of trabecular bone volumes. Yet, these apparent properties depend strongly on the applied boundary conditions (BCs) for the limited size of volumes that can be obtained from human bones. To attenuate the influence of the BCs, we computed the yield properties of samples loaded via a surrounding layer of trabecular bone ("embedded configuration"). Thirteen cubic volumes (10.6 mm side length) were collected from [Formula: see text]CT reconstructions of human vertebrae and femora and converted into [Formula: see text]FE models. An isotropic elasto-plastic material model was chosen for bone tissue, and nonlinear [Formula: see text]FE analyses of six uniaxial, shear, and multi-axial load cases were simulated to determine the yield properties of a subregion (5.3 mm side length) of each volume. Three BCs were tested. Kinematic uniform BCs (KUBCs: each boundary node is constrained with uniform displacements) and periodicity-compatible mixed uniform BCs (PMUBCs: each boundary node is constrained with a uniform combination of displacements and tractions mimicking the periodic BCs for an orthotropic material) were directly applied to the subregions, while the embedded configuration was achieved by applying PMUBCs on the larger volumes instead. Yield stresses and strains, and element damage at yield were finally compared across BCs. Our findings indicate that yield strains do not depend on the BCs. However, KUBCs significantly overestimate yield stresses obtained in the embedded configuration (+43.1 ± 27.9%). PMUBCs underestimate (-10.0 ± 11.2%), but not significantly, yield stresses in the embedded situation. Similarly, KUBCs lead to higher damage levels than PMUBCs (+51.0 ± 16.9%) and embedded configurations (+48.4 ± 15.0%). PMUBCs are better suited for reproducing the loading conditions in subregions of the trabecular bone and deliver a fair estimation of their effective (asymptotic) yield properties.

Bone turnover and mineralisation kinetics control trabecular BMDD and apparent bone density: insights from a discrete statistical bone remodelling model

Castoldi,

Pickering,

Sansalone

et al. 2024

The mechanical quality of trabecular bone is influenced by its mineral content and spatial distribution, which is controlled by bone remodelling and mineralisation. Mineralisation kinetics occur in two phases: a fast primary mineralisation and a secondary mineralisation that can last from several months to years. Variations in bone turnover and mineralisation kinetics can be observed in the bone mineral density distribution (BMDD). Here, we propose a statistical spatio-temporal bone remodelling model to study the effects of bone turnover (associated with the activation frequency $$\mathrm {Ac.f}$$ Ac . f ) and mineralisation kinetics (associated with secondary mineralisation $$T_\textrm{sec}$$ T sec ) on BMDD. In this model, individual basic multicellular units (BMUs) are activated discretely on trabecular surfaces that undergo typical bone remodelling periods. Our results highlight that trabecular BMDD is strongly regulated by $$\mathrm {Ac.f}$$ Ac . f and $$T_\textrm{sec}$$ T sec in a coupled way. Ca wt% increases with lower $$\mathrm {Ac.f}$$ Ac . f and short $$T_\textrm{sec}$$ T sec . For example, a $$\mathrm {Ac.f}=$$ Ac . f = 4 BMU/year/mm$$^3$$ 3 and $$T_\textrm{sec}$$ T sec = 8 years result in a mean Ca wt% of 25, which is in accordance with Ca wt% values reported in quantitative backscattered electron imaging (qBEI) experiments. However, for lower $$\mathrm {Ac.f}$$ Ac . f and shorter $$T_\textrm{sec}$$ T sec (from 0.5 to 4 years) one obtains a high Ca wt% and a very narrow skew BMDD to the right. This close link between $$\mathrm {Ac.f}$$ Ac . f and $$T_\textrm{sec}$$ T sec highlights the importance of considering both characteristics to draw meaningful conclusion about bone quality. Overall, this model represents a new approach to modelling healthy and diseased bone and can aid in developing deeper insights into disease states like osteoporosis.