Direct Methods for Dynamic Elastography Reconstruction: Optimal Inversion of the Interior Helmholtz Problem
Travis E. Oliphant, Ph.D. May 2001
Elastography is a broad area of inquiry encompassing several measurement modalities and employing a plethora of post-processing techniques to obtain data that allows interpretation of the mechanical properties of tissues. Dynamic elastography uses a time-series of material displacement (typically obtained with phase-based ultrasound or magnetic resonance techniques) to infer local tissue mechanical properties. Magnetic resonance measurement results in a time-filtered displacement where the applied motion-sensitizing gradients are the impulse response of the linear filter. Ultrasound measurement results in instantaneous measurement of the low-frequency displacement. Both techniques can be used to obtain a harmonic displacement field in 1, 2, and/or three dimensions. If the material of interest can be well modeled as piecewise-constant-so that an assumption of local homogeneity is appropriate-then the material properties for an arbitrary linear material can be determined from an algebraic relation involving derivatives of the data. The data are not as sensitive to some of the material constants as they are to others. Therefore, in the presence of random noise, the error in some of the reconstructed estimates may render those estimates useless. In these case, it is appropriate to recast the material model in terms of only the important material properties. For example, many tissues of interest for elastography can be considered as isotropic and incompressible so that the only parameter of interest is the shear modulus. Ignoring the longitudinal vibration allows writing the equations of motion as a Helmholtz equation, so that finding shear modulus from displacement measurements becomes an inverse Helmholtz problem where the data are known inside the region of interest (the interior Helmholtz inverse problem). Solutions to the interior Helmholtz inverse problem are the major focus for much of the presented work.
Algebraically inverting the Helmholtz problem under conditions of local homogeneity results in an estimate of squared wavenumber, which is proportional to the inverse of shear modulus that is in turn the ratio of weighted integrals (sums) of the local displacement data. One weighting function must be the negative Laplacian of another weighting function that can be selected to reduce the sensitivity to noise of the resulting estimate. Comparing several weighting functions shows that many Gaussian-like weighting functions provide comparable results as long as the spatial-support of the weighting functions are less than one wavelength in the underlying displacement field. If the support of the weighting function becomes larger than one-wavelength, then the weighting function must be modulated in order to continue to benefit from the increased smoothing of the larger window. It is shown that the error in the reconstructed squared wave-number using Gaussian-like weighting functions is a strong power-law function of window-size-to-wavelength, so that regions where the wavelength is high require larger window sizes in order to keep the same level of reconstruction error. A local frequency estimation technique is shown to be a solution to the lossless interior inverse Helmholtz problem that provides a weighted average of reconstructed images at several resolutions such that regions with a large wavelength use mostly the low-resolution images while regions with small wavelengths are weighed more heavily to the high-resolution images.
An optimal weighting scheme is derived for the direct-method of (discrete) interior Helmholtz inversion. It is shown that the set of weights that results in the minimum-variance estimate of modulus satisfies a nonlinear fixed-point equation dependent on the local displacement. The gain curve in one-dimension for the ideal set of weights is determined from simulations. The ideal weights can be approximated using the local data resulting in a biased estimator where the bias is dependent on the input noise-level. Simulations in one- and two-dimension reveal that the data-dependent optimal approach gives minimum error but that the fixed weighting strategies perform very close to the optimal case for small window sizes (less than a wavelength). It is also noted that for one dimensional data the interior inverse Helmholtz problem is equivalent to frequency estimation, and the one-dimensional version of the optimal algorithm significantly outperforms a popular frequency estimation algorithm.
The conditional probability density function (CPDF) is described for the case of fixed-weight reconstruction methods and used to develop an uncertainty metric for estimated modulus that can be reported along with the estimate of modulus. The CPDF for the case of non-fixed weights is also described and approximated.
The fixed-window strategies and the approximately-optimal adaptive weighting strategies are applied to magnetic resonance phase data to compare their performance and to demonstrate the uncertainty values and their use in magnetic resonance elastography. Fixed-weight strategies are the best choice for for high-resolution magnetic resonance reconstruction.
The data-dependent optimal weighting strategy is applied to ultrasound data collected to demonstrate a prototype of an ultrasound "biopsy" application, where local dynamic displacement information is gathered and used to estimate complex-valued shear modulus (with uncertainty) for a region of interest.
For both magnetic resonance and ultrasound displacement measurements, the error predicted using the random-noise on the data and the CPDF of the modulus under-predicts the true error as evidenced by averaging over a presumably constant modulus area. This disagreement depends on the level of averaging used and it is conjectured that other correlated (system) noise components are present in both the magnetic resonance and ultrasound displacement data.