Real-time solution of the finite element inverse problem of viscoelasticity

Abstract

The linear dynamic finite element model can be formulated such that the elasticity and viscosity of the elements appear as the parameters in a linear system of equations. The resulting system of equations can be solved directly using singular value decomposition or a similar technique or through defining a quadratic functional. A priori knowledge and regularity measures can be added as equality or inequality constraints. The sensitivity of the inverse problem solution to the displacement noise and model imperfections are tested in simulations, where the parameters were successfully reconstructed with a displacement signal-to-noise ratio as low as 20 dB. Also, the viscoelastic parameters have been successfully estimated for a breast phantom with an embedded hard inclusion. The study of the computation speed demonstrates the potential of the new method for real-time implementations.

Simulation-Based Joint Estimation of Body Deformation and Elasticity Parameters for Medical Image Analysis

“…Alternatively, in dynamic elastography (for example, magnetic resonance elastography (MRE) and vibro-elastography), an MRI or ultrasound machine in tune with an applied mechanical vibration (shear wave) or focused ultrasound beams is used to find the displacement field [4], [9], [10]. With known external forces and displacement field, the elasticity can be computed by solving a least-squares problem [11], [12], [13], if the algebraic equations resulting from the physical model is linear. A real-time performance has been reported using this direct method with a simplified 2D domain that assumes homogeneous material within a region [13].…”

“…With known external forces and displacement field, the elasticity can be computed by solving a least-squares problem [11], [12], [13], if the algebraic equations resulting from the physical model is linear. A real-time performance has been reported using this direct method with a simplified 2D domain that assumes homogeneous material within a region [13]. Another type of method uses iterative optimization to minimize the error in the displacement field generated by the simulator [14], [15], [16].…”

Constraint-Based Soft Tissue Simulation for Virtual Surgical Training

“…It is still a challenging problem to obtain material properties as Young's modulus and Poission ratio. In this paper the elasticity parameter estimations is carried out as an inverse problem by minimizing the quadratic norm of the difference between the measured displacements obtained with the data acquisition system and the displacements resulted from the simulation model [Eskandari, et al, 2011]. The recovered Young's modulus and Poisson ratio through an off-line optimization program are then used in the multi-resolution strain limiting simulations.…”

“…In recent years efforts have been made to model the parametrization of complex biomechanical models as an inverse problem that are solved by optimization. For examples, the estimation of elastic properties of soft tissue is solved as a least squares problem in [Eskandari, et al, 2011], and an evolutionary algorithm is used to estimate the parameters of a complex organ behavior model taking into account of various real patient data sets in [Vidal et al, 2012]. Previous methods for accelerating nonlinear finite element computations include pre-computing various quantities of a constitutive model [Müller et al, 2001], modal analysis to compute deformations in a reduced subspace [James and Pai, 2002], as well as GPU executions [Taylor et al, 2008] and data-driven approaches [Bickel et al, 2009].…”

Identifying malignant and benign breast lesions using vibroelastography

“…Given the spatial displacements at each frequency, equation (2) can be solved for the value of Young's modulus and viscosity. Different methods for solving the Helmholtz equation have been proposed in the literature including direct inversion of the wave equation, the finite element method and local frequency estimation [14], [15]. In this paper we use the local frequency method which measures the local wavelength of the displacement phasor image to obtain the shear wave velocity and therefore the shear modulus.…”