Exploration of trabecular bone nonlinear elasticity using time-of-flight modulation

Abstract

Bone tissue contains microcracks that may affect its mechanical properties as well as the whole trabecular structure. The relationship between crack density and bone strength is nevertheless poorly understood. Linear ultrasound techniques being almost insensitive to the level of damage, we propose a method to measure acoustic non- linearity in trabecular bone using time-of-flight modulation (TOFM) measurements. Ultrasonic short bursts times-of- flight (TOF) are modulated as a result of nonlinear interaction with a low-frequency (LF) wave in the medium. TOF variations are directly related to elastic modulus variations. Classical and nonclassical nonlinear parameters beta, delta, and alpha can be derived from these measurements. The method was validated in materials with classical, quadratic, nonlinear elasticity. In dense trabecular bone region, TOFM related to classical, quadratic, nonlinear elasticity as a function of the LF pressure exhibits tension-compression asymmetry. The TOFM amplitude measured in dense areas of trabecular bone is almost one order of magnitude higher than in a low-density area, but the linear parameters show much smaller variations: 5% for ultrasound propagation velocity and 100% for broadband ultrasonic attenuation (BUA). In high-density trabecular bone regions, beta depends on the LF pressure amplitude and can reach 400 at 50 kPa.

Relationships of linear and nonlinear ultrasound parameters with porosity and trabecular spacing in trabecular-bone-mimicking phantoms

“…The measurements of the B/A in the phantoms may be compared with previously reported measurements of the B/A in human trabecular bone. 19,20 For instance, Renaud et al measured a value of 22 for the B/A in a low-density trabecular bone region and 142 in a high-density region of a human calcaneal trabecular bone sample by using a time-of-flight modulation method. 20 Hence, the phantom data obtained in the present study fall within the range of the human calcaneus data.…”

Characterizing the nonlinear interaction of S- and P-waves in a rock sample

“…First developed in medical characterization of bone and other materials, [20][21][22] it was later applied to rocks. [23][24][25][26] The method relies on monitoring the material several times during a resonant period.…”

Evaluation of a wave vector frequency domain method for nonlinear wave propagation.

“…27 The nonlinearity coefficient b was 72. 42 The forward projected sound field was obtained from a concave circular transducer with an aperture radius of 10 mm and a focus distance of 60 mm. The excitation signal was the same in Fig.…”