Analyzing the anisotropic Hooke׳s law for children׳s cortical bone

Abstract

Child cortical bone tissue is rarely studied because of the difficulty of obtaining samples. Yet the preparation and ultrasonic characterization of the small samples available, while challenging, is one of the most promising ways of obtaining information on the mechanical behavior of non-pathological children׳s bone. We investigated children׳s cortical bone obtained from chirurgical waste. 22 fibula or femur samples from 21 children (1-18 years old, mean age: 9.7±5.8 years old) were compared to 16 fibula samples from 16 elderly patients (50-95 years old, mean age: 76.2±13.5 years old). Stiffness coefficients were evaluated via an ultrasonic method and anisotropy ratios were calculated as the ratio of C33/C11, C33/C22 and C11/C22. Stiffness coefficients were highly correlated with age in children (R>0.56, p<0.01). No significant difference was found between C11 and C22 for either adult or child bone (p>0.5), nor between C44 and C55 (p>0.5). We observe a transverse isotropy with C33>C22=C11>C44C55>C66. For both groups, we found no correlation between age and anisotropy ratios. This study offers the first complete analysis of stiffness coefficients in the three orthogonal bone axes in children, giving some indication of how bone anisotropy is related to age. Future perspectives include studying the effect of the structure and composition of bone on its mechanical behavior.

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

“…Results for bones are presented in Table 1 and Table 2. Table 2 presents the stiffness coefficients of adult and child cortical bones obtained by Bernard et al (2016), Lefèvre et al (2015), and in this case study. Lefèvre et al (2015) evaluated only the diagonal stiffness coefficients for the NPCCB while Bernard et al (2016) evaluated the whole stiffness matrix for adult cortical tibia specimens with millimetric dimensions comparable to the dimensions of the present NPCCB specimens (from 5 to 37 mm 3 ).…”

“…During the last years, pioneer studies have been published on non-pathological child cortical bone (NPCCB) (Bala et al, 2016;Berteau et al, 2012aBerteau et al, , 2012bLefèvre et al, 2015). Lefèvre et al (2015) measured the diagonal stiffness coefficients of NPCCB (fibula) specimens, which were collected under the same conditions as in the present study (same anatomical site). They provided the values of the six diagonal stiffness coefficients, and their statistical sample shows a trend toward transverse isotropy for this kind of material.…”

“…As a result, the frequency range to be considered was defined so as to enclose 30 resonant frequencies for each specimen. The initial values of the coefficients of the presumed elasticity tensor were the six diagonal coefficients values published in (Lefèvre et al, 2015) ( Table 1). The three off-diagonal coefficients were supposed to be equal and deduced from 12 = 13 = 23 = 11 − 2 66 = 10.24 .…”

A critical assessment of the in-vitro measurement of cortical bone stiffness with ultrasound

“…Most of the knowledge on the stiffness of cortical bone, which is an orthotropic or transverse isotropic material (Ashman et al, 1984;Rho, 1996;Orías et al, 2009;Rudy et al, 2011) has been obtained through the measurement of ultrasonic bulk wave velocity (BWV) ex vivo along different anatomical directions using cuboid specimens. Since the method was introduced to measure bone in the 1960's (Lang, 1969), and until recently, this technique has been applied to assess bone elasticity and relate it to age (Lefèvre et al, 2015), anatomical location (Orías et al, 2009;Schwartz-Dabney and Dechow, 2002) or others bone properties such as microstructure (Granke et al, 2011) and extravascular bone tissue properties (Baumann et al, 2012). …”

“…It appears that there is no generally accepted protocol to measure BWV in cortical bone in terms of US frequency (frequencies in the range 1 to 20 MHz have been used), signal processing technique and specimen size (specimens of characteristic size between 0.5 and 10 mm have been used) (Lang, 1969;Kohles et al, 1997;Ashman et al, 1984;Lefèvre et al, 2015). Furthermore, the signal processing method used to estimate the velocity is rarely specified in the studies.…”