An error analysis of Helmholtz inversion for incompressible shear, vibration elastography with application to filter-design for tissue characterization

For over fifteen years there has been significant effort in elastography, which describes the general area of imaging material mechanical properties. Shear vibration elastography uses dynamic tissue displacements to infer material properties from the physics of motion. The method can be used with both magnetic resonance and ultrasound data, which can both be modeled with the time-harmonic, Helmholtz equation if the material is linear, isotropic, incompressible, and piecewise-homogeneous. In this work, we develop a unified perspective on direct Helmholtz inversion. Using the fundamental theorem of statistics and a Gaussian noise model, we present a closed form for the joint conditional probability distribution of the real and imaginary parts of the squared wavenumber given the data and an arbitrary set of weights. An approximate distribution can be used in the case of high SNR which allows a figure-of-merit to be established to objectively compare inversion approaches. Adaptively choosing the inversion weights for each subregion as the smoothed and windowed conjugate of the data results in a narrow conditional probability distribution function and, consequently, high-quality estimates of complex shear modulus. To test the results, we used experimental ultrasound data-collected using a focused 5 MHz transducer with a pulse-repetition frequency of 4 kHz in a block of 15% bovine gel. The gel was harmonically compressed using a signal containing equal amplitudes at frequencies of 200, 300, 400 and 500 Hz. Noise on the measured displacement was estimated from the magnitude of the complex (baseband) correlation function and used with the conditional probability distribution function to report error bars on single-region estimates of complex shear modulus, wave-speed and attenuation.