Shear modulus reconstruction in dynamic elastography: time harmonic case

Abstract

Abstract. This paper presents a direct inversion approach for reconstructing the elastic shear modulus in soft tissue from dynamic measurements of the interior displacement field during time harmonic excitation. The tissue is assumed to obey the equations of nearly incompressible, linear, isotropic elasto-dynamics in harmonic motion. A finite element discretization of the governing equations is used as a basis, and a procedure is outlined to eliminate the need for boundary conditions in the inverse problem. The hydrostatic stress (pressure) is also reconstructed in the process, and the effect of neglecting this term in the governing equations, which is common practice, is considered. The approach does not require iterations and can be performed on subregions of the domain resulting in a computationally efficient method. A sensitivity study is performed to investigate the detectability of abnormal regions of different size and shear modulus contrast from the background. The algorithm is tested on simulated data on a 2D domain, where the data is generated on a very fine mesh to get a near exact solution, then down-sampled to a coarser mesh that is similar to the spatial discretization of actual data, and noise is added. Results showing the effect of the hydrostatic stress term and noise are presented. A reconstruction using MR measured experimental data involving a tissue-mimicking phantom is also shown to demonstrate the algorithm.

3D Heterogeneous Stiffness Reconstruction Using MRI and the Virtual Fields Method

“…-static experiments: the tissue is compressed quasi-statically [Barbone and Gokhale, 2004,Skovoroda et al, 1995, Skovoroda et al, 1999, Fehrenbach et al, 2006, -dynamic experiments: a time harmonic excitation made on the boundary creates a time harmonic shear wave in the tissue [McLaughlin and Yoon, 2004, Weaver et al, 2005, Manduca et al, 2001, Romano et al, 2001, Liew and Pinsky, 2005, Park and Maniatty, 2006.…”

“…The advantage over quasi-static experiments is that a direct relationship between the measured strains and the unknown moduli can be derived, provided that only shear waves are present in the tissues [McLaughlin andYoon, 2004,Park andManiatty, 2006]. Such a direct relationship is usually not available in static experiments.…”

A generalized computationally efficient inverse characterization approach combining direct inversion solution initialization with gradient-based optimization

“…2). Alternatively, the elastic modulus of the matrix (i.e., healthy) material was assumed to be fixed at 15 kP a, which was based on normal glandular breast tissue [6,5]. The two values of the amplitude parameter used were 1 and 3 (i.e., modulus at the hard inclusion center of approximately 30 kP a and 60 kP a), and the two values of the breadth parameter used were 5 mm and 15 mm.…”

“…A common direct approach is to relate the measured response to the unknown material property distribution parameters based on the manipulation of the forward boundary value problem and a least-squares criteria, which creates a solution process similar to that of solving the forward problem itself (e.g., similar in process to a finite element analysis to predict the deformation response of a solid given geometry, material properties, and boundary conditions) [5,6,7]. Thus, the solution estimate can be obtained at a cost on the order of a single numerical analysis of the forward problem, even with a relatively generalized parameterization of the spatial distribution of the unknown property (e.g., finite element-type parameterization).…”

Identification of Materials Mechanical Properties from Full-Field Measurements: Latest Advances in the Virtual Fields Method

“…Determination of the mechanical behaviour of biological tissues is a key issue in the early detection of cancerous tumours (stiff tumours embedded in softer tissues) or in the production of data to establish models of the human body for crash simulation, for instance. The determination of stiffness from such measurements is known as "elastography" in the medical community [23,24]. These tissues also exhibit highly non linear behaviour (hyperviscoelastic for muscle tissues, for instance) and there is still much work to do both on the measurements themselves (moving from images to high quality deformation measurements) and on the identification procedures.…”