Analysis of Dynamic Magnetic Resonance Elastography Data - Advance Image Processing in Magnetic Resonance Imaging - page 453

This necessarily implies inaccurate results at or near the boundaries between regions
and a limit of resolution on the order of the local window size. However, the local
window can be as small as desired, subject only to increased noise in derivative
estimates, and in practice, inversions “fail gracefully” (with a gradual transition) across
regional boundaries. The resolution of MRE inversion is limited by the accuracy of
the spatial derivative estimation and thus ultimately by the SNR (with noisy data,
averaging over larger spatial windows may be required). Stiffer objects are more
difficult to estimate accurately because their spatial derivatives of displacement change
more slowly.
A different possibility is to not make this assumption and solve Equation 14.4
allowing the mechanical properties to vary in the physical model. This method
has two confounding effects: (1) the equation remains a differential equation and
not an algebraic one, and (2) the assumption of incompressibility does not decou-
ple the equations of motion for shear modulus, and so all components of motion
are once again necessary. This approach is computationally more challenging but
in principle models more accurately the physics of motion for arbitrary materials.

14.5.9 F INITE E LEMENT A NALYSIS

Van Houten et al. [50,51] have described a finite-element-based subzone tech-
nique for solving Equation 14.4. In their approach, a solution is iteratively refined
on small overlapping subzones of the overall domain by updating the solution
based on differences between forward calculations of the displacement from the
current solution and measured values. After an update is performed on one
subzone, the subzone with the greatest residual error is determined and updated.
Local homogeneity is not assumed. The approach is elegant, and good results
have been demonstrated on synthetic and actual data sets. It is computationally
very intensive, and it seems to be very sensitive to the data being acquired in a
true steady state, i.e., with enough motion cycles before acquisition so that the
wave field reaches equilibrium. Although this is technically a requirement on all
harmonic inversion techniques, analysis and practice have shown that violation
of these conditions — i.e., excitation with only a few cycles before acquisition
such that the wave field has not reached equilibrium — has only minor effects
on the other inversions [43]. No direct comparisons have been made between
finite element inversions and the other methods mentioned earlier, but the results
appear to be comparable in quality.
Finite element methods are also widely used for forward simulations of MRE
experiments. Another approach termed coupled harmonic oscillator simulation
has also been used to simulate and, to a limited extent, interpret and analyze
MRE data [52,53].

14.5.10 A NISOTROPIC I NVERSIONS

Certain tissues are far from being isotropic; for example, muscle tissue is highly
anisotropic, and it is known that shear waves propagate preferentially along