A theoretical model was previously developed to evaluate the relationship between the dynamics of ultrasonic speckle and its underlying tissue. The model is divided into an instrumental part represented by the point spread function (in the far field) of the ultrasonic apparatus and a moving tissue component described by a collection of scatterers. By computing the convolution of these terms and then the envelope, one obtains a simulated ultrasonic speckle pattern sequence which shows speckle motions closely linked to the tissue dynamics when small motion amplitudes are involved. Here, a theoretical study of the correlation between various linear transformations of the tissue and the corresponding ultrasonic speckle motions is performed, based on a 2D extension of the envelope cross-correlation analysis of a narrow-band Gaussian noise. In the linear scan case, obviously, tissue translation generates an identical speckle translation. However, tissue/speckle motion correlation decreases with increasing rotation and/or biaxial deformation, lateral deformation (perpendicular to the beam propagation axis) being much less sensitive. With respect to the transducer frequency, the rotation and the axial deformation of the tissue show a better relationship with their respective speckle motion at lower frequencies while lateral deformation correlation is independent of the pulse frequency. With respect to beam (pulse) size parameters, tissue/speckle correlation decreases with rotation when a wide ultrasonic beam is used while the axial deformation correlation decreases with the axial duration of the pulse. This study sets the ground for the development of an ultrasonic strain gauge particularly useful for the assessment of biomechanical soft tissue and fluid flow properties based on speckle tracking.
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