Elastography is a non-invasive imaging technique that can assess in vivo tissue stiffness. In shear wave elastography imaging, the acoustic radiation force (ARF) produced by focused ultrasound generates a local force that produces shear waves. The authors compare three existing formulations for the ARF: its full expression in the second-order approximation and two simplified formulations using a quasi-plane wave and an attenuated plane wave approximation. Analytical expressions for the ARF are derived for the special cases of a concave spherical source and a quasi-Gaussian beam. They provide expressions for the resulting ARF and show discrepancies between the different formulations. For strongly divergent or highly focused beams the ARF expressed by the second-order approximation significantly differs from both simplified formulations. However, despite those differences the second-order and quasi-plane wave approximations create identical shear displacements in tissue. To compute the ARF and the displacements produced by a conventional ultrasound probe, the three formulations were incorporated into the k-Wave simulation package. The second-order and quasi-plane wave approximations give different forces but nearly identical displacements while the plane wave approximation significantly differs. It is concluded that to properly take into account the ultrasound field structure, the second-order or quasi-plane wave approximations should be preferably used.
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