Time-domain model of the ultrasonic wave propagation in an inhomogeneous anisotropic viscoelastic fluid/solid multilayer medium: application to cortical bone

Abstract

Cortical bone quality is assessed in clinical practice using axial transmission (AT) devices, which allow the measurement of quantitative ultrasonic parameters such as the first arriving signal (FAS) velocity. However, the physical interaction between an ultrasonic wave and cortical bone remains poorly understood due to the complex nature of the bone structure. Cortical bone and the surrounding soft tissues are attenuating media, which might affect the analysis of the results obtained with AT devices. Moreover, cortical bone is highly heterogeneous and a gradient of material properties from the outer to inner part of the cortical shell has been reported. The aim of this work is to evaluate the effect of anisotropic heterogeneous dissipative phenomena occurring in bone and in soft tissues on the ultrasonic response of the bone structure. A two-dimensional finite element time-domain method is derived to model transient wave propagation in a three-layer medium composed of an inhomogeneous transverse isotropic viscoelastic solid layer sandwiched between two dissipative acoustic fluid layers. The model couples acoustic propagation in both dissipative acoustic fluid media with the response of the solid whose constitutive equation is based on the linear theory of viscoelasticity without memory. Bone viscoelasticity is assumed to be heterogeneous and a constant spatial gradient of viscoelastic properties is considered for a value of bone thickness corresponding to a relatively thick bone. Realistic variations of the viscosity of the soft tissues within the physiological range do not affect the FAS velocity in any configuration. However, when the viscoelastic bone parameters vary within their physiological range, changes of the value of the FAS velocity (up to 301 m/s) are comparable to what has been obtained for variation of the elastic parameters. The components of the viscoelastic tensor affecting the results are the same as those of the stiffness tensor found. Our results highlight the importance of accounting for absorption phenomena occurring in cortical bone for the analysis of ultrasonic measurements with AT device.