Paul E. Barbone

Quantitative elasticity imaging: what can and cannot be inferred from strain images.

We examine the inverse problem associated with quantitative elastic modulus imaging: given the equilibrium strain field in a 2D incompressible elastic material, determine the elastic stiffness (shear modulus). We show analytically that a direct formulation of the inverse problem has no unique solution unless stiffness information is known a priori on a sufficient portion of the boundary. This implies that relative stiffness images constructed on the assumption of constant boundary stiffness are in error, unless the stiffness is truly constant on the boundary. We show further that using displacement boundary conditions in the forward incompressible elasticity problem leads to a nonunique inverse problem. Indeed, we give examples in which exactly the same strain field results from different elastic modulus distributions under displacement boundary conditions. We also show that knowing the stress on the boundary can, in certain configurations, lead to a well-posed inverse problem for the elastic stiffness. These results indicate what data must be taken if the elastic modulus is to be reconstructed reliably and quantitatively from a strain image.

Linear and nonlinear elasticity imaging of soft tissue in vivo: demonstration of feasibility.

We establish the feasibility of imaging the linear and nonlinear elastic properties of soft tissue using ultrasound. We report results for breast tissue where it is conjectured that these properties may be used to discern malignant tumors from benign tumors. We consider and compare three different quantities that describe nonlinear behavior, including the variation of strain distribution with overall strain, the variation of the secant modulus with overall applied strain and finally the distribution of the nonlinear parameter in a fully nonlinear hyperelastic model of the breast tissue.

Elastic modulus imaging: on the uniqueness and nonuniqueness of the elastography inverse problem in two dimensions

We examine the uniqueness of an N-field generalization of a 2D inverse problem associated with elastic modulus imaging: given?N?linearly independent displacement fields in an incompressible elastic material, determine the shear modulus. We show that for the standard case, N=1, the general solution contains two arbitrary functions which must be prescribed to make the solution unique. In practice, the data required to evaluate the necessary functions are impossible to obtain. For N=2, on the other hand, the general solution contains at most four arbitrary constants, and so very few data are required to find the unique solution. For N=4, the general solution contains only one arbitrary constant. Our results apply to both quasistatic and dynamic deformations.

BOUNDARY INFINITE ELEMENTS FOR THE HELMHOLTZ EQUATION IN EXTERIOR DOMAINS

A novel approach to the development of infinite element formulations for exterior problems of time-harmonic acoustics is presented. This approach is based on a functional which provides a general framework for domain-based computation of exterior problems. Special cases include non-reflecting boundary conditions (such as the DtN method). A prominent feature of this formulation is the lack of integration over the unbounded domain, simplifying the task of discretization. The original formulation is generalized to account for derivative discontinuities across infinite element boundaries, typical of standard infinite element approximations. Continuity between finite elements and infinite elements is enforced weakly, precluding compatibility requirements. Various infinite element approximations for two-dimensional configurations with circular interfaces are presented. Implementation requirements are relatively simple. Numerical results demonstrate the good performance of this scheme.

Solution of the nonlinear elasticity imaging inverse problem: the compressible case

We have recently developed and tested an efficient algorithm for solving the nonlinear inverse elasticity problem for a compressible hyperelastic material. The data for this problem are the quasi-static deformation fields within the solid measured at two distinct overall strain levels. The main ingredients of our algorithm are a gradient based quasi-Newton minimization strategy, the use of adjoint equations and a novel strategy for continuation in the material parameters. In this paper we present several extensions to this algorithm. First, we extend it to incompressible media thereby extending its applicability to tissues which are nearly incompressible under slow deformation. We achieve this by solving the forward problem using a residual-based, stabilized, mixed finite element formulation which circumvents the Ladyzenskaya-Babuska-Brezzi condition. Second, we demonstrate how the recovery of the spatial distribution of the nonlinear parameter can be improved either by preconditioning the system of equations for the material parameters, or by splitting the problem into two distinct steps. Finally, we present a new strain energy density function with an exponential stress-strain behavior that yields a deviatoric stress tensor, thereby simplifying the interpretation of pressure when compared with other exponential functions. We test the overall approach by solving for the spatial distribution of material parameters from noisy, synthetic deformation fields.

Linear and Nonlinear Elastic Modulus Imaging: An Application to Breast Cancer Diagnosis

We reconstruct the in vivo spatial distribution of linear and nonlinear elastic parameters in ten patients with benign (five) and malignant (five) tumors. The mechanical behavior of breast tissue is represented by a modified Veronda-Westmann model with one linear and one nonlinear elastic parameter. The spatial distribution of these elastic parameters is determined by solving an inverse problem within the region of interest (ROI). This inverse problem solution requires the knowledge of the displacement fields at small and large strains. The displacement fields are measured using a free-hand ultrasound strain imaging technique wherein, a linear array ultrasound transducer is positioned on the breast and radio frequency echo signals are recorded within the ROI while the tissue is slowly deformed with the transducer. Incremental displacement fields are determined from successive radio-frequency frames by employing cross-correlation techniques. The rectangular regions of interest were subjectively selected to obtain low noise displacement estimates and therefore were variables that ranged from 346 to 849.6 mm . It is observed that malignant tumors stiffen at a faster rate than benign tumors and based on this criterion nine out of ten tumors were correctly classified as being either benign or malignant.

Wave Propagation in Piezoelectric Layered Media with Some Applications

In this article we introduce a systematic methodology to investigate wave propagation in piezoelectric layered media. It is based on a matrix formalism and the con cept of the surface impedance tensor which relates the components of particle displace ment and the normal component of the electric displacement along a surface to the electric potential and the components of traction acting along the same surface. Once the surface impedance tensor for a single layer is calculated, a simple recursive algorithm allows the evaluation of the surface impedance tensor for any number of layers. As an example, the surface impedance tensor is used to find the dispersion curves for a bilaminated piezoelec tric plate. Also the dispersion curves for the subsonic interfacial waves when the plate is in contact with a nonconducting acoustic fluid is investigated. Floquet theory is also ap plied to study wave propagation when the layered medium is periodic.

Elastic modulus imaging: some exact solutions of the compressible elastography inverse problem

We consider several inverse problems motivated by elastography. Given the (possibly transient) displacement field measured everywhere in an isotropic, compressible, linear elastic solid, and given density rho, determine the Lamé parameters lambda and mu. We consider several special cases of this problem: (a) for mu known a priori, lambda is determined by a single deformation field up to a constant. (b) Conversely, for lambda known a priori, mu is determined by a single deformation field up to a constant. This includes as a special case that for which the term [see text]. (c) Finally, if neither lambda nor mu is known a priori, but Poissons ratio nu is known, then mu and lambda are determined by a single deformation field up to a constant. This includes as a special case plane stress deformations of an incompressible material. Exact analytical solutions valid for 2D, 3D and transient deformations are given for all cases in terms of quadratures. These are used to show that the inverse problem for mu based on the compressible elasticity equations is unstable in the limit lambda --> infinity. Finally, we use the exact solutions as a basis to compute non-trivial modulus distributions in a simulated example.

Quantitative three-dimensional elasticity imaging from quasi-static deformation: a phantom study

We present a methodology to image and quantify the shear elastic modulus of three-dimensional (3D) breast tissue volumes held in compression under conditions similar to those of a clinical mammography system. Tissue phantoms are made to mimic the ultrasonic and mechanical properties of breast tissue. Stiff lesions are created in these phantoms with size and modulus contrast values, relative to the background, that are within the range of values of clinical interest. A two-dimensional ultrasound system, scanned elevationally, is used to acquire 3D images of these phantoms as they are held in compression. From two 3D ultrasound images, acquired at different compressed states, a three-dimensional displacement vector field is measured. The measured displacement field is then used to solve an inverse problem, assuming the phantom material to be an incompressible, linear elastic solid, to recover the shear modulus distribution within the imaged volume. The reconstructed values are then compared to values measured independently by direct mechanical testing.

Nearly H1-optimal finite element methods

Abstract We examine the problem of finding the H 1 projection onto a finite element space of an unknown field satisfying a specified boundary value problem. Solving the projection problem typically requires knowing the exact solution. We circumvent this issue and obtain a Petrov–Galerkin formulation which achieves H 1 optimality. Requiring weighting functions to be defined locally on the element level permits only approximate H 1 optimality in multi-dimensional configurations. We investigate the relation between our formulation and other stabilized FEM formulations. We show, in particular, that our formulation leads to a derivation of the SUPG method. In special cases, the present formulation reduces to that of residual-free bubbles. Finally, we present guidelines for obtaining the Petrov weight functions, and include a numerical example for the Helmholtz equation.