Limitations of Curl and Directional Filters in Elastography

Abstract

In the approaches to elastography, two mathematical operations have been frequently applied to improve the final estimate of shear wave speed and shear modulus of tissues. The vector curl operator can separate out the transverse component of a complicated displacement field, and directional filters can separate distinct orientations of wave propagation. However, there are practical limitations that can prevent the intended improvement in elastography estimates. Some simple configurations of wavefields relevant to elastography are examined against theoretical models within the semi-infinite elastic medium and guided waves in a bounded medium. The Miller–Pursey solutions in simplified form are examined for the semi-infinite medium and the Lamb wave symmetric form is considered for the guided wave structure. In both cases, we examine simple but practical wave combinations that can prevent the curl and directional filter operations from directly providing an improved measure of shear wave speed and shear modulus. Additional factors including signal-to-noise and the support of filters also restrict the applicability of these strategies for improving elastographic measures. Thus, some implementations of shear wave excitations applied to the body and to bounded structures within the body are shown to involve waves that are not easily resolved by the vector curl operator and directional filters. These limitations may be overcome by more advanced strategies or simple improvements in baseline parameters including the size of the region of interest and the number of shear waves propagated within.