Quantification of stiffness measurement errors in resonant ultrasound spectroscopy of human cortical bone

Abstract

Resonant ultrasound spectroscopy (RUS) is the state-of-the-art method used to investigate the elastic properties of anisotropic solids. Recently, RUS was applied to measure human cortical bone, an anisotropic material with low Q-factor (20), which is challenging due to the difficulty in retrieving resonant frequencies. Determining the precision of the estimated stiffness constants is not straightforward because RUS is an indirect method involving minimizing the distance between measured and calculated resonant frequencies using a model. This work was motivated by the need to quantify the errors on stiffness constants due to different error sources in RUS, including uncertainties on the resonant frequencies and specimen dimensions and imperfect rectangular parallelepiped (RP) specimen geometry. The errors were first investigated using Monte Carlo simulations with typical uncertainty values of experimentally measured resonant frequencies and dimensions assuming a perfect RP geometry. Second, the exact specimen geometry of a set of bone specimens were recorded by synchrotron radiation micro-computed tomography. Then, a "virtual" RUS experiment is proposed to quantify the errors induced by imperfect geometry. Results show that for a bone specimen of ∼1° perpendicularity and parallelism errors, an accuracy of a few percent ( <6.2%) for all the stiffness constants and engineering moduli is achievable.

Homogenization of cortical bone reveals that the organization and shape of pores marginally affect elasticity

“…The orientation of the three-dimensional volume of each specimen was corrected by slightly rotating the ensemble of voxel using Fiji [61] so that the image reference frame coincides with the orientation of the specimen's faces. As the shape quality of our specimens was quite good ( perpendicularity and parallelism errors were about 18, see [56]), the potential misalignments due to orientation correction should be small. In each specimen, a rectangular parallelepiped volume of interest (VOI, sized 2.8 Â 3.9 Â 4.8 mm 3 ) was selected manually on the principle of retaining the maximum volume for morphometric analysis, as well as for the evaluation of the effective stiffness using FFT homogenization (see §2.6).…”

“…The experimental errors, e.g. irregularity of specimen geometry and uncertainties of the extracted resonant frequencies, following this protocol, typically cause an error of approximately 1:7% for the shear stiffness constants and approximately 3:1% for the longitudinal and off-diagonal stiffness constants [56]. Bone stiffness constants of the specimens measured by RUS are denoted as C EXP ij in the following text.…”

Assessment of elastic coefficients of child cortical bone using resonant ultrasound spectroscopy

“…The minimum number of frequencies identified for a specimen was 21. However, following the work of Bernard et al (2016Bernard et al ( , 2015, we used the Bayesian approach to circumvent this problem and to solve the inverse problem of calculating stiffness coefficients from a number of resonant frequencies lower than 5 times the number of coefficients to calculate and with a minimal error (Cai et al, 2017).…”

“…A smart protocol using resonant ultrasound spectroscopy (RUS) has recently been developed to assess the whole stiffness tensor of cortical bone at millimeter scale. The relevant protocol and method are well presented and discussed in the articles by Bernard et al (Bernard et al, 2016, and recently supplemented by works of Cai et al (Cai et al, 2017). This work, as most of the work conducted on bone, has been carried out on adult bone and more precisely on ageing human bone, which is nowadays a well-documented material.…”

Elastic properties measurement of human enamel based on resonant ultrasound spectroscopy

“…3. These shape imperfections would result in errors in the forward model for calculating the theoretical value of the resonant frequencies (Cai et al, 2017).…”