Regularization for ultrasonic measurements of tissue displacement vector and strain tensor

Abstract

In this report, we present the regularization results of displacement vector measurement by the multidimensional cross-spectrum phase gradient method (MCSPGM), multidimensional autocorrelation method (MAM), and multidimensional Doppler method (MDM) without lateral modulation. In addition to the results of spatially variant regularization in displacement vector measurement using displacement variances, the results of displacement component-dependent regularization are presented. That is, according to the measurement accuracy of displacement components, the regularization is properly applied to the respective displacement components. For instance, only the regularization on the lateral displacement is effective. That is, for the lateral nonmodulation case, the accuracies and stabilities of lateral/elevational displacement measurements significantly increase. In conjunction, the convergence speed of phase matching also increases. The demonstrated measurements of the displacement vector distributions in experiments using inhomogeneous shear modulus agar phantoms confirm that displacement-componentdependent regularization enables better strain tensor measurement and shear modulus reconstruction from the viewpoints of accuracy and stability. The contrast-to-noise ratio (CNR) is useful to set regularization parameters properly for the displacement regularization, whereas the CNR is useless for shear modulus regularization in the sense that the accuracy of reconstruction value cannot be evaluated.

Ultrasound-Based Guidance for Partial Breast Irradiation Therapy

“…The computation time of these methods is reported to be few minutes and therefore are not immediately suitable for real time elastography. In [37] and [38], few phase-based methods are regularized and strain and elasticity modulus images are obtained. The regularization term is the Laplacian (second derivative) of the motion field and is spatially variant based on the peak-value of the correlation coefficient.…”

Ultrasound-Based Guidance for Partial Breast Irradiation Therapy

“…The computation time of these methods is reported to be few minutes and therefore are not immediately suitable for real time elastography. In [37], [38], few phase-based methods are regularized and strain and elasticity modulus images are obtained. The regularization term is the Laplacian (second derivative) of the motion field and is spatially variant based on the peak-value of the correlation coefficient.…”

A limitation in the use of spatially stationary strain variance estimation in the regularization of shear modulus reconstruction

“…We previously reported the effectiveness of regularization for stabilizing strain (displacement) measurement methods [26][27][28] and shear modulus reconstruction methods [20,23,27,[29][30][31] for decreasing the instabilities of the measurement and reconstruction due to the contamination of echo data. Such regularization enables the incorporation of the a priori knowledge that the targets are spatially smooth.…”

“…Regarding the regularizations of strain measurement and shear modulus reconstruction, we further proposed to set suitable regularization parameters at each position (i.e., spatially variant regularization parameters) [20,23,27,28,31], because the measurement accuracies of displacements and strains spatially vary. That is, although the cross-validation method (i.e., using the independence of the raw vectors of the matrix) is used in such a modulus reconstruction for a region of interest (ROI) (e.g., [5]), our measurement and reconstruction should be properly regularized at each position according to the measurement accuracies of displacements and strains, respectively.…”

“…When using displacement vector measurement methods (i.e., MCSPGM, MAM and MDM) without lateral modulation methods (LMMs) [18,20,[33][34][35][36] or a multidirectional synthetic aperture method (MDSAM) [18,20], because the stability and accuracy of lateral/elevational displacement measurements are much lower than those of axial displacement measurement [15][16][17][18][19], the regularization of only lateral/elevational displacements is also effective (i.e., displacement-component-dependent regularization) [28]. Spatially variant regularizations using an echo correlation coefficient [27] and displacement variances [28] have also been reported. Because both MDSAM and LMMs require a special data acquisition system, the regularization is effective.…”