We consider the problem of determining the shear modulus of a linearelastic, incompressible medium given boundary data and one component of the displacement field in the entire domain. The problem is derived from applications in quantitative elasticity imaging. We pose the problem as one of minimizing a functional and consider the use of gradient-based algorithms to solve it. In order to calculate the gradient efficiently we develop an algorithm based on the adjoint elasticity operator. The main cost associated with this algorithm is equivalent to solving two forward problems, independent of the number of optimization variables. We present numerical examples that demonstrate the effectiveness of the proposed approach.

Recently a new adjoint equation based iterative method was proposed for evaluating the spatial distribution of the elastic modulus of tissue based on the knowledge of its displacement field under a deformation. In this method the original problem was reformulated as a minimization problem, and a gradient-based optimization algorithm was used to solve it. Significant computational savings were realized by utilizing the solution of the adjoint elasticity equations in calculating the gradient. In this paper, we examine the performance of this method with regard to measures which we believe will impact its eventual clinical use. In particular, we evaluate its abilities to (1) resolve geometrically the complex regions of elevated stiffness; (2) to handle noise levels inherent in typical instrumentation; and (3) to generate three-dimensional elasticity images. For our tests we utilize both synthetic and experimental displacement data, and consider both qualitative and quantitative measures of performance. We conclude that the method is robust and accurate, and a good candidate for clinical application because of its computational speed and efficiency.

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.

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.

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.

The acoustic cloaking theory of Norris [Proc. R. Soc. London, Ser. A 464, 2411-2434 (2008)] permits considerable freedom in choosing the transformation f from physical to virtual space. The standard process for defining cloak materials is to first define f and then evaluate whether the materials are practically realizable. In this paper, this process is inverted by defining desirable material properties and then deriving the appropriate transformations which guarantee the cloaking effect. Transformations are derived which result in acoustic cloaks with special properties such as (1) constant density, (2) constant radial stiffness, (3) constant tangential stiffness, (4) power-law density, (5) power-law radial stiffness, (6) power-law tangential stiffness, and (7) minimal elastic anisotropy.

Previous work has demonstrated improved diagnostic performance of highly trained breast radiologists when provided with B-mode plus elastography images over B-mode images alone. In those studies we have observed that elasticity imaging can be difficult to perform if there is substantial motion of tissue out of the image plane. So we are extending our methods to 3D/4D elasticity imaging with 2D arrays. Further, we have also documented the fact that some breast tumors change contrast with increasing deformation and those observations are consistent with in vitro tissue measurements. Hence, we are investigating imaging tissue stress-strain nonlinearity. These studies will require relatively large tissue deformations (e.g., > 20%) which will induce out of plane motion further justifying 3D/4D motion tracking. To further enhance our efforts, we have begun testing the ability to perform modulus reconstructions (absolute elastic parameter) imaging of in vivo breast tissues. The reconstructions are based on high quality 2D displacement estimates from strain imaging. Piecewise linear (secant) modulus reconstructions demonstrate the changes in elasticity image contrast seen in strain images but, unlike the strain images, the contrast in the modulus images approximates the absolute modulus contrast. Nonlinear reconstructions assume a reasonable approximation to the underlying constitutive relations for the tissue and provide images of the (near) zero-strain shear modulus and a nonlinearity parameter that describes the rate of tissue stiffening with increased deformation. Limited data from clinical trials are consistent with in vitro measurements of elastic properties of tissue samples and suggest that the nonlinearity of invasive ductal carcinoma exceeds that of fibroadenoma and might be useful for improving diagnostic specificity. This work is being extended to 3D.

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