Gradient-Based Optimization for Poroelastic and Viscoelastic MR Elastography

Abstract

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.

Stiffness reconstruction methods for MR elastography

“…170,171,[178][179][180][181] Associated studies include investigations into the validity of anisotropic material assumptions, 182 results of applying isotropic reconstructions to anisotropic data, 60,150 experimental requirements, 183 and hardware to produce adequate wave data. 184 Poroelastic material assumptions have also been studied in MRE 46,56,57 as well as US elastography. 185 Nonlinear, usually neo-Hookean, constitutive laws have been investigated in the context of static elastography.…”

“…The mixed displacement‐pressure formulation of the static deformation equations are solved via FEM in 2D and the minimization is regularized via a Tikhonov approach. Tan et al introduced a generalized adjoint method applicable to poroelasticity and also provided comparisons between conjugate‐gradient, quasi‐Newton, and Gauss–Newton methods by computational complexity and by results across various data sets, including phantom and brain (Figure ).…”

Analytical solution for converging elliptic shear wave in a bounded transverse isotropic viscoelastic material with nonhomogeneous outer boundary

“…Numerical finite element (FE) techniques have been useful in evaluating and comparing inversion algorithms, and in some cases have themselves been integrated into iterative inversion algorithms. 4,19,[36][37][38] Typically, the FE simulation provides a forward model to calculate displacements based on assumed material parameters. These are then compared to the measured displacements, and the material parameters are iteratively updated based on differences between simulation and experiment until suitable convergence is reached.…”

Stabilized finite elements for time‐harmonic waves in incompressible and nearly incompressible elastic solids

“…Elastomeric materials, for example, are often accurately modeled as incompressible, and modeling their dynamical response is of current interest in understanding elastomeric actuators . Biological tissues are also often modeled as incompressible elastic or viscoelastic materials, and propagation of shear elastic waves in such media is of interest in a new medical imaging field called elastography . Therefore, it is of interest to have accurate, efficient, and stable methods to compute elastic wave fields in incompressible and nearly incompressible materials.…”

“…[1][2][3] Biological tissues are also often modeled as incompressible elastic or viscoelastic materials, and propagation of shear elastic waves in such media is of interest in a new medical imaging field called elastography. [4][5][6][7][8][9][10][11][12][13][14][15] Therefore, it is of interest to have accurate, efficient, and stable methods to compute elastic wave fields in incompressible and nearly incompressible materials. Unfortunately, classical Galerkin discretization struggles with both incompressibility and wave dispersion.…”