Acoustic radiation force in tissue-like solids due to modulated sound field

The focus of this study is the sustained body force (the so-called acoustic radiation force) in homogeneous tissue-like solids generated by an elevated-intensity, focused ultrasound field (Mach number=O(10 ^-3)) in situations when the latter is modulated by a low-frequency signal. This intermediate-asymptotics problem, which bears relevance to a number of emerging biomedical applications, is characterized by a number of small (but non-vanishing) parameters including the Mach number, the ratio between the modulation and ultrasound frequency, the ratio of the shear to bulk modulus, and the dimensionless attenuation coefficient. On approximating the response of soft tissues as that of a nonlinear viscoelastic solid with heat conduction, the featured second-order problem is tackled via a scaling paradigm wherein the transverse coordinates are scaled by the width of the focal region, while the axial and temporal coordinate are each split into a fast and slow component with the twin aim of: (i) canceling the linear terms from the field equations governing the propagation of elevated-intensity ultrasound, and (ii) accounting for the effect of ultrasound modulation. In the context of the focused ultrasound analyses, the key feature of the proposed study revolves around the dual-time-scale treatment of the temporal variable, which allows one to parse out the contribution of ultrasound and its modulation in the nonlinear solution. In this way the acoustic radiation force (ARF), giving rise to the mean tissue motion, is exacted by computing the fast time average of the germane field equations. A comparison with the existing theory reveals a number of key features that are brought to light by the new formulation, including the contributions to the ARF of ultrasound modulation and thermal expansion, as well as the precise role of constitutive nonlinearities in generating the sustained body force in tissue-like solids by a focused ultrasound beam.