Multi-frequency inversion in Rayleigh damped Magnetic Resonance Elastography
Introduction
In time-harmonic Magnetic Resonance Elastography (MRE), a single input frequency is a common experimental approach for mechanical property reconstruction. It is less time consuming and only requires a single patient examination. A number of studies investigated various models for both static [1], [2] and dynamic [3], [4], [5], [6], [7], [8], [9], [10], [11] MRE methods using single frequency data. These models include linearly elastic [12], [13], [14], viscoelastic (VE) [15], [16], [17], [18], [19], [20] and even poroelastic [21], [22], [23] models.
Two main limitations prevent single frequency approaches from providing accurate approximations of the tissue response. The first is associated with the dispersive nature of the complex shear modulus at different frequencies and thus quantitative estimates acquired at a particular excitation frequency do not represent the true static (at 0 Hz) values of material constants. The second arises from complex boundary conditions and undesirable wave interaction between propagating and reflected waves inside the tissue. This effect can cause amplitude nulls where no elasticity information is available. The first issue can be mitigated by simultaneous inversion of multiple wave data sets acquired at different excitation frequencies and finding an appropriate model that can accurately account for dispersion characteristics and associated frequency dependent behaviour [24]. However, the second issue presents challenging limitations to current inversion methods.
Thus, a multi-frequency (MF) approach offers a number of benefits over a single frequency approach. First, it allows dispersion measurement of the rheological tissue parameters by analysing multiple propagating wave velocities corresponding to different driving frequencies. This can enable more accurate measurement of the viscosity which might be potentially useful in tissue characterisation. Second, it can avoid wave amplitude nulls accruing in single-frequency wave patterns. Previous MF elastography studies found a correlation between biological tissue response to various excitation frequencies in a form of power-law (PL). A PL implies frequency dependant behaviour of the VE parameters.
More complex models, such as the Rayleigh or proportional damping (RD) model [25], [26], capture this frequency dependant behaviour. However, they are not identifiable with single frequency data [27]. Hence, multiple actuation frequencies are required for more complex models, as well. Previous studies of MF applications have mainly utilized two parameter (Voigt model, Maxwell model) and three-parameter (Zener model, Jeffreys model) VE models [28], [29], [30], [31], while the RD model has been previously discussed in the context of simulation studies for a homogeneous material [32], [33]. Preliminary single-frequency RD MRE experiments compared reconstruction results for the RD and linear VE models with mixed results [34]. However, Petrov et al. [27] concluded that the RD model was not uniquely identifiable for single frequency data. Without a guarantee of a priori identifiability, the parameter estimates obtained might be unreliable or random. Two alternative approaches are suggested to overcome non-identifiability of the RD model: 1. simultaneous MF inversion and 2. parametric inversion, when only single frequency data is available.
Consequently, a MF approach might contribute towards theoretical identifiability of the RD model [27] and is expected to produce more robust, data driven results than possible with a single frequency data. This study evaluates MF inversion in RD MRE in comparison to a parametric approach to accurately delineate the RD parameters and assess the efficacy of MF for RD reconstructions. Two alternative material models are also exploited, a zero-order model and a PL model.
Section snippets
RD model in time-harmonic MRE
The RD model is implemented through Finite Element (FE) based solution of a nearly-incompressible linear isotropic Navier's equation, defined:
where u is the displacement within the medium; λ is the first Lamé's parameter (λ = 1/3 for a nearly-incompressible case), μ is the second Lamé's parameter, also known as the shear stiffness; ρ is the density of the material, ∇P is a pressure term, related to volumetric changes through the bulk modulus, K, via the
Results
Fig. 2 shows the T2*-weighted multi slice MR images for the P1 and P2 phantoms, where the top slices from left to right in the figure correspond to the bottom slices located near the actuation plate and upwards. Fig. 3 shows three-parameter RD reconstruction results based on the zero-order model computed by simultaneous inversion over 4 available frequencies for both phantom configurations, while Fig. 4 shows median and IQR of the quantified parameters at each slice throughout the phantoms.
The
MF three-parameter RD reconstruction
Simultaneous MF based inversion over a range of relatively higher frequencies did not lead to a better quality reconstruction results, compared with previously reported independent SF reconstructions. The differences between quality reconstruction for the different parameters were evident, with the real shear modulus, μR, having the highest quality image and the images of the RD parameters, μI and ρI, being significantly poorer. These trends verify results from the structural model analysis,
Conclusion
This research compares full simultaneous MF inversion with parametric approach to investigate accuracy of elastographic based reconstruction of RD properties in tissue simulating damping phantoms. Although overall reconstruction of real shear modulus, μR, and damping ratio, ξd, was successful, accurate identification of RD parameters failed. The results confirm that close range of frequencies that was practicable for the available and typically used piezoelectric actuation did not resolve
Acknowledgments
The authors would like to acknowledge Matthew McGarry from Dartmouth College of Engineering (NH, USA) for help in data collection, and Dr Elijah Van Houten from the University of Sherbrooke (Quebec, Canada) for providing valuable expertise in MRE.
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