Modeling of Soft Poroelastic Tissue in Time-Harmonic MR Elastography

Abstract

Elastography is an emerging imaging technique that focuses on assessing the resistance to deformation of soft biological tissues in vivo. Magnetic resonance elastography (MRE) uses measured displacement fields resulting from low-amplitude, low-frequency (10 Hz-1 kHz) timeharmonic vibration to recover images of the elastic property distribution of tissues including breast, liver, muscle, prostate, and brain. While many soft tissues display complex time-dependent behavior not described by linear elasticity, the models most commonly employed in MRE parameter reconstructions are based on elastic assumptions. Further, elasticity models fail to include the interstitial fluid phase present in vivo. Alternative continuum models, such as consolidation theory, are able to represent tissue and other materials comprising two distinct phases, generally consisting of a porous elastic solid and penetrating fluid. MRE reconstructions of simulated elastic and poroelastic phantoms were performed to investigate the limitations of current-elasticity-based methods in producing accurate elastic parameter estimates in poroelastic media. The results indicate that linearly elastic reconstructions of fluid-saturated porous media at amplitudes and frequencies relevant to steady-state MRE can yield misleading effective property distributions resulting from the complex interaction between their solid and fluid phases.

Gradient-Based Optimization for Poroelastic and Viscoelastic MR Elastography

“…As in the viscoelastic case, a linear system is given as $$false[\mathrm{bold-italicK}false]false\{Ufalse\}=false\{ffalse\},$$ results in elements of the stiffness matrix, K , and the forcing vector, f , consistent with [44], [45]. Diagonal terms in the stiffness matrix are given by $$K_{4false(j-1false)+1,4false(k-1false)+1}=true\langle false(2\mu +\lambda false)\frac{\partial {\varphi }_k}{\partial x}\frac{\partial {\varphi }_j}{\partial x}+\mu true(\frac{\partial {\varphi }_k}{\partial y}\frac{\partial {\varphi }_j}{\partial y}+\frac{\partial {\varphi }_k}{\partial z}\frac{\partial {\varphi }_j}{\partial z}true)-{\omega }^{2}false(\rho -\beta {\rho }_{f}false){\varphi }_k{\varphi }_{j}true\rangle ,$$ $$K_{4false(j-1false)+2,4false(k-1false)+2}=true\langle false(2\mu +\lambda false)\frac{\partial {\varphi }_k}{\partial y}\frac{\partial {\varphi }_j}{\partial y}+\mu true(\frac{\partial {\varphi }_k}{\partial x}\frac{\partial {\varphi }_j}{\partial x}+\frac{\partial {\varphi }_k}{\partial z}\frac{\partial {\varphi }_j}{\partial z}true)-{\omega }^{2}false(\rho -\beta {\rho }_{f}false){\varphi }_k{\varphi }_{j}true\rangle ,$$ $$K_{4false(j-1false)+3,4false(k-1false)+3}=true\langle false(2\mu +\lambda false)\frac{\partial {\varphi }_k}{\partial z}\frac{\partial {\varphi }_j}{\partial z}+\mu true(\frac{\partial {\varphi }_k}{\partial x}\frac{\partial {\varphi }_j}{\partial x}+...$$…”

“…In this paper, we develop and implement a generalized adjoint method based on a Lagrangian that no longer requires the stiffness matrix to be self-adjoint, but reverts back to the classic adjoint scheme when the stiffness matrix is self-adjoint. We present the details of implementation via the finite element method in a common mathematical framework for three widely used mechanical models in MRE: compressible viscoelasticity [42], [34], incompressible viscoelasticity [32], [43] and poroelasticity [44], [45]. Numerical experiments with data from simulations, tissue mimicking phantoms and in vivo brain data show that the generalized adjoint algorithm preserves efficiency of the gradient calculation.…”

Suitability of poroelastic and viscoelastic mechanical models for high and low frequency MR elastography

“…17,18 A PE material is modeled as a biphasic continuum, comprising a porous elastic matrix with an infiltrating pore fluid. Volumetric deformation of the solid matrix leads to fluid flow in the material.…”

Scattering and Diffraction of Elastodynamic Waves in a Concentric Cylindrical Phantom for MR Elastography

“…Perriñez et al . used a bean curd (tofu) to mimic soft poroelastic tissue in MR elastography [8], estimating shear modulus from simulated data with a finite-element-based nonlinear inversion scheme. Othman et al .…”