We describe an efficient gradient computation for solving inverse problems arising in magnetic resonance elastography (MRE). The algorithm can be considered as a generalized ‘adjoint method’ based on a Lagrangian formulation. One requirement for the classic adjoint method is assurance of the self-adjoint property of the stiffness matrix in the elasticity problem. In this paper, we show this property is no longer a necessary condition in our algorithm, but the computational performance can be as efficient as the classic method, which involves only two forward solutions and is independent of the number of parameters to be estimated. The algorithm is developed and implemented in material property reconstructions using poroelastic and viscoelastic modeling. Various gradient- and Hessian-based optimization techniques have been tested on simulation, phantom and in vivo brain data. The numerical results show the feasibility and the efficiency of the proposed scheme for gradient calculation.

Intrinsic actuation MR elastography (IA-MRE) exploits natural pulsations of the brain as a motion source to estimate mechanical property maps. The low frequency motion of IA-MRE introduces new considerations for inversion algorithms relative to traditional external actuation MRE. Specifically, inertial forces become very small, which leaves low frequency viscoelastic inversions with a non-unique scalar multiplier. Biphasic poroelastic inversions include additional fluid–solid interaction forces to balance the elastic forces, which avoids the non-uniqueness. Analyzing the convergence behavior from different starting values using 1 Hz simulated data, IA-MRE data from a gelatin phantom and in vivo brain IA-MRE data reveal that higher frequency (50 Hz) viscoelastic inversion reaches the correct, unique solution regardless of initial property estimate; whereas, low frequency viscoelastic inversion recovers relative values of shear modulus. In the presence of measurement noise, the non-unique scalar multiplier is determined by the softest material reaching the prescribed lower bound on shear modulus. Poroelastic inversion produces a unique solution at both 50 Hz and 1 Hz; however, hydraulic conductivity must be known or accurately estimated in order to recover quantitatively accurate shear modulus maps at low frequency.

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This study evaluated non-linear inversion MRE (NLI-MRE) based on viscoelastic governing equations to determine its sensitivity to small, low contrast inclusions and interface changes in shear storage modulus and damping ratio. Reconstruction parameters identical to those used in recent in vivo MRE studies of mechanical property variations in small brain structures were applied. NLI-MRE was evaluated on four phantoms with contrast in stiffness and damping ratio. Image contrast to noise ratio was assessed as a function of inclusion diameter and property contrast, and edge and line spread functions were calculated as measures of imaging resolution. Phantoms were constructed from silicone, agar, and tofu materials. Reconstructed property estimates were compared with independent mechanical testing using dynamic mechanical analysis (DMA). The NLI-MRE technique detected inclusions as small as 8mm with a stiffness contrast as low as 14%. Storage modulus images also showed an interface edge response distance of 11mm. Damping ratio images distinguished inclusions with a diameter as small as 8mm, and yielded an interface edge response distance of 10mm. Property differences relative to DMA tests were in the 15%-20% range in most cases. In this study, NLI-MRE storage modulus estimates resolved the smallest inclusion with the lowest stiffness contrast, and spatial resolution of attenuation parameter images was quantified for the first time. These experiments and image quality metrics establish quantitative guidelines for the accuracy expected in vivo for MRE images of small brain structures, and provide a baseline for evaluating future improvements to the NLI-MRE pipeline.

A numerical framework for interstitial fluid pressure imaging (IFPI) in biphasic materials is investigated based on three-dimensional nonlinear finite element poroelastic inversion. The objective is to reconstruct the time-harmonic pore-pressure field from tissue excitation in addition to the elastic parameters commonly associated with magnetic resonance elastography (MRE). The unknown pressure boundary conditions (PBCs) are estimated using the available full-volume displacement data from MRE. A subzone-based nonlinear inversion (NLI) technique is then used to update mechanical and hydrodynamical properties, given the appropriate subzone PBCs, by solving a pressure forward problem (PFP). The algorithm was evaluated on a single-inclusion phantom in which the elastic property and hydraulic conductivity images were recovered. Pressure field and material property estimates had spatial distributions reflecting their true counterparts in the phantom geometry with RMS errors around 20% for cases with 5% noise, but degraded significantly in both spatial distribution and property values for noise levels > 10%. When both shear moduli and hydraulic conductivity were estimated along with the pressure field, property value error rates were as high as 58%, 85% and 32% for the three quantities, respectively, and their spatial distributions were more distorted. Opportunities for improving the algorithm are discussed.