Errors in Estimates of Motion and Strain-Tensor in Ultrasound Elastography

Abstract

Error analysis was performed for quantifying noise in ultrasound elastography images of biologically soft tissue. The interaction of ultrasonic bean with tissue was modeled in 3D. Static tissue deformation applied by an external mechanical source was represented by a second order strain tensor. Complex motion induced in response to deformation was tracked using standard cross correlation based estimator. Pre- and postcompression echo signals were windowed by the same kernel and cross correlated. The amount of shift where the cross correlation function peaked was considered as the estimate of the local tissue motion. Covariance matrix of the errors made in estimating motions within two windows with certain amount of overlap was derived analytically. The components of the covariance matrix were related to the variances of the displacement errors and the errors made in estimating the elements of the strain tensor. The results were combined to investigate the dependencies of these errors on the experimental and signal-processing parameters as well as to determine the effects of one strain component on the estimation of the other. The expressions were evaluated for special cases of axial strain estimation in the presence of axial, axial-shear and lateral-shear type deformations. The signals were shown to decorrelate with any of these deformations; with strengths depending on the reorganization and interaction of the tissue scatterers with the ultrasonic point spread function following the deformation. The loss of signal coherence resulted in more degradation in the estimation performance. The precision of the estimates was sensitive to the direction of the motion. Motion parallel to the transducer array axis produced more error than the motion perpendicular to this axis, and axial shear type deformation introduced more error than that of the lateral shear depending on the features of ultrasonic point spread function as it was defined by its width, length and wavelength. Conditions that favor the improvements in the motion estimation performance were discussed, and advantages gained by signal companding and pulse compression were also illustrated.