The purpose of the study was to investigate the identifiability of homogeneous, transversely isotropic, linear elastic material properties from a displacement field obtained using magnetic resonance elastography (MRE). In addition, the study investigated using finite element modelling (FEM) as an inversion method which takes into account a subject-specific geometry and realistic boundary conditions in order to solve for homogeneous and anisotropic material properties.

FEM Simulations: Harmonic displacement was simulated in two models, a cylinder and LV geometry with a histological muscle fibre field (Figure 1a and Figure 2b). Abaqus 6.13 (Dassault Systèmes Simulia Corp., Providence, USA) was used to simulate the steady-state harmonic motion; Gaussian noise was added to the displacement field; and a least squares objective function was used to determine the optimal transversely isotropic material properties by minimizing the difference between modelled and reference displacement fields. A Monte-Carlo simulation (n=30) was run, randomly generating the Gaussian noise distribution for each simulation. Four parameters were optimised: structural damping coefficient (s), transverse Young’s modulus (E₁), fibre Young’s modulus (E₃) and Poisson’s ratio (v). In the reference model, these parameters were set to: s = 0.25, E₁ = 18.0 kPa, E₃ = 60.0 kPa and v = 0.495. A parameter sweep was run and the determinability of the four parameters was further evaluated based on three identifiability criteria obtained from the Hessian at the minimum of the objective function. The three criteria were: 1) the D-optimiality criterion related to the volume of the indifference region, 2) the eccentricity criterion related to the ratio of the most identifiable to least identifiable parameter, and 3) the M-optimality criterion related to the parameter interaction.

MRE Experiment: A cyclindrical phantom (r = 76.2 mm and h = 127 mm) was vibrated at 60 Hz using a pneumatic driver and imaged using a GRE MRE sequence to collect wave images at four phase offsets relative to the harmonic displacement in a 3T MRI scanner (Tim Trio, Siemens Healthcare, Germany). In the FE model, displacements from the MRE images were applied on all sides of the phantom. A transversely isotropic material law was used and the fibre direction was arbitrarily set to run longitudinally through the phantom. A least squares objective function was used to minimize the difference between modelled and MRE displacements, at the internal nodes, in order to determine the optimal material properties.

Results from the two Monte-Carlo simulation experiments and the phantom material parameter optimisation are listed in Table 1. The D-optimality criterion for each experiment was large which showed that the volume of the indifference region, or minimum, was small, and hence identifiable. The parameters were largely independent of one another, based on the M-optimality criterion values of 0.84 and 0.94, for the cylinder and LV experiments, respectively. When parameters are completely independent, the M-optimality value is one. The eccentricity criteria were greater than one for each case indicating that the parameters are not equally identifiable.

The average shear stiffness of the phantom has previously been measured to be 5.6 kPa using local frequency estimation (LFE), a common inversion algorithm used to calculate isotropic shear stiffness from MRE displacement maps1. Since the material is isotropic, the shear modulus can be equated to a Young’s modulus of 16.8 kPa. The Young’s modulus determined from the FE method differed from the LFE method by 5%.

The Monte-Carlo simulations with added Gaussian noise resulted in values which were extremely close to the true parameters for both the cylinder and LV geometries. The parameter sweeps also revealed one clear global minimum within the parameter space (Figure 1b and Figure 2b). Since the parameters are largely independent of one another, each eigenvector is primarily associated with one parameter direction. The size of each eigenvalue gives relative information on the determinability of the corresponding parameter. In both the cylindrical and LV simulations, the Poisson’s ratio was the least identifiable parameter when compared with the damping coefficient and Young’s moduli. In the phantom experiment, between v = 0.4995 and v = 0.4998 (the optimal value), the objective function only changed by %0.3. Above 0.4995, Poisson’s ratio had little effect on the objective function. In subsequent experiments, the Poisson’s ratio will be fixed to save valuable computational time without sacrificing accuracy of the other parameters.Overall, this initial work shows that using displacement data from MRE to identify anisotropic material properties is a well-posed problem even in the presence of noise.